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Aspects of Mathematical Economics, Social Choice and Game Theory

A Thesis Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Applied Mathematics

Haradhan Kumar Mohajan Registration No. 32 Session: 2009–2010 Jamal Nazrul Islam Research Centre for Mathematical and Physical Sciences, University of Chittagong, Chittagong, Bangladesh. March 2018 1

Declaration

I hereby declare that the research work presented in this thesis, entitled “Aspects of Mathematical Economics, Social Choice and Game Theory” being submitted to the Jamal Nazrul Islam Research Centre for Mathematical and Physical Sciences (JNIRCMPS), University of Chittagong, Chittagong, Bangladesh for the award of the degree of Doctor of Philosophy (PhD), was carried out entirely by me that it has not, either wholly or in part, been submitted for any other degree at any university. Any error in the thesis is entirely my responsibility.

______________________ Haradhan Kumar Mohajan Registration No. 32 Session: 2009–2010 Jamal Nazrul Islam Research Centre for Mathematical and Physical Sciences (JNIRCMPS), University of Chittagong, Chittagong, Bangladesh. March 2018

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Certification

This is to certify that the work entitled “Aspects of Mathematical Economics, Social Choice and Game Theory”, to the best of my knowledge, is original and was carried out under my supervision by Haradhan Kumar Mohajan in partial fulfillment of the requirements for the degree of Doctor of Philosophy (PhD) in Mathematical Economics and Social Science at the Jamal Nazrul Islam Research Centre for Mathematical and Physical Sciences (JNIRCMPS), University of Chittagong, Chittagong, Bangladesh.

Supervisor

____________________ Professor Dr. Mohammed Abul Hossain Department of Economics University of Chittagong, Chittagong, Bangladesh. March 2018

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Dedicated to my beloved mother Chabi Mohajan & Reverend father Late Niranjan Kumar Mohajan

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Acknowledgement

In the way of my research, it became quite clear to me that a researcher can hardly complete a PhD thesis alone. Many people and institutions deserve thanks and appreciations for their valuable contributions. As the list of individuals and institutions I wish to thank cannot be accommodated in this limited space. I, therefore, would like to thank some specific ones for their dedicated supports. I feel immense pleasure to acknowledge my debt of gratitude to my reverend supervisor Professor Dr. Mohammed Abul Hossain, Department of Economics, University of Chittagong, Chittagong, Bangladesh for his constant, unfailing and everready guidance and assistance at every stage of this work. He has been extremely generous with his time and knowledge, and I am deeply appreciative of his help and support. He has always encouraged me to think about my research in different ways, and offered helpful suggestions about alternative approaches to design studies, test multilevel models, and frame and present my research. He has also supported me, even in the most difficult times, and for this I will always be grateful to him. I am extremely lucky to have him during these research periods, and I wish all PhD students could have such a great supervisor. I acknowledge my eternal debt of gratitude to my reverend first PhD supervisor Late Emeritus Professor Dr. Jamal Nazrul Islam, Research Centre for Mathematical and Physical Sciences (RCMPS), University of Chittagong, Chittagong, Bangladesh, for his guidance and assistance at every stage of my research work. Under his supervision I have submitted my PhD thesis. I firmly believe that no one has ever had a more supportive, generous, international famous, excellent talent, and encouraging supervisor. He has suggested me such an interesting topic of modern mathematics with applications to the problems in economics, political science, and social science. Unfortunately, he has passed away before the defense of my PhD research. I have also successfully awarded my M. Phil. Degree in 2007, under his direct supervision on “Singularity Theorems in General Relativity” from RCMPS University of Chittagong, Chittagong, Bangladesh.

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I am greatly indebted to my examiner and convener of my PhD research Professor Dr. Abdul Wadud, Department of economics, University of Rajshahi, Bangladesh for providing the suggestions during the 1st and 2nd defenses, and for the revision of my thesis before the 3rd time submission. I am ever grateful to my respected teacher UGC Professor Dr. M. A. Mansur Chawdhury for the efficient guidance to prepare this research. He is a generous, supportive, and cooperative person. I owe a heavy debt of gratitude for my respected teachers Professor Dr. Sujit K. Sen and Professor Dr. Quamrul Islam, JNIRCMPS, University of Chittagong, Chittagong, Bangladesh for splendid cooperation, constant advice accorded to me by them. I am also grateful to Professor Dr. Anjan Kumar Chawdhury, Director, JNIRCMPS, University of Chittagong, Chittagong, Bangladesh for the superb support of my research work. My thanks go to Professor Dr. Anupam Sen, Vice Chancellor, Premier University, Chittagong, Bangladesh, who has encouraged me during this research. Additionally, my thanks go to all the teachers and all the staff of the JNIRCMPS; and all my colleagues and all the staff of the Premier University. I am indebted for their inspiration. Special thanks to all the PhD students from the JNIRCMPS with whom I shared these years of hard work. Congratulations to those who have finished before me, and good luck to all the others! I am grateful to the authors of the books, journals, and publications which I have consulted unsparingly. I gratefully acknowledge my mother, my wife Sucharita, my two sons Devajit and Indrajit, and my only daughter Subasri for their love and patience that allowed me with enough time during the course of research. Without their help and stimulation this work could not have been completed. I am owed greatly to my respected father who unfortunately passed away from us in 1994, during the study of my M. Sc. in the Department of Mathematics, University of Chittagong. Last but not least; I am grateful to my parents-in-laws Late Sushil Ranjan Bhowmik and Minati Prova Bhowmik for their support to the procedure of this research.

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All the glory goes to Almighty God, who gives me the inner strength, courage, patient, good health, wisdom, and long life for all the achievements.

___________________ Haradhan Kumar Mohajan JNIRCMPS, University of Chittagong, Bangladesh Cell: +8801716397232 Email: [email protected] Email: [email protected] March 2018

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Acronyms AV:

Approval voting

BCBI:

Bangladesh Country Background Information

BEA:

The Bureau of Economic Analysis

BMT:

Billions metric tons

BTB:

Backwards Tie-Breaking

CAFT:

Clean Air Task Force

CCC:

Chittagong City Corporation

CCR:

Collective choice rule

CEH:

Centre for Ecology & Hydrology

CES:

Constant Elasticity of Substitution

CH4:

Methane

CID:

Cubic inches of displacement

CO2:

Carbon dioxide

CO2e:

Carbon dioxide equivalents

CRS:

Constant returns to scale

DWL:

Deadweight loss

EEE:

External Economic Environment

EIA:

Energy Information Administration

EPA:

Environmental Protection Agency

EPM:

Emissions per mile

EU:

European Union

FTB:

Forwards Tie-Breaking

GDP:

Gross domestic product

GE:

The green economy

GFC:

Global financial crisis

GHG:

Greenhouse gas

GNI:

Gross National Income

GNP:

Gross National Product

GT:

Giga tons 8

GWP:

Global warming potential

HDI:

Human Development Index

HFC:

Hydrofluourocarbon

IEA:

International Energy Agency

IIA:

Independent of irrelevant alternatives

ILO:

International Labour Organization

IMF:

International monetary fund

IPCC:

Intergovernmental Panel on Climate Change

LPN:

Liga para a Protecção da Natureza

MCPF:

Marginal cost of public funds

MED:

Marginal environmental damages

MMT:

Million metric tons

MNA:

Middle East and North Africa

MP:

Members of Parliament

mpg:

Miles per gallon

MSW:

Municipal solid waste

MVT:

Median voter theorem

NAS:

National Academy of Sciences

NASA:

National Aeronautics and Space Administration

NBR:

National Board of Revenue

NDP:

Net domestic product

NDRC:

National Development and Reform Commission

NNP:

Net national product

N2O:

Nitrous dioxide

NO2:

Nitrogen dioxide

PCE:

Pollution control equipment

PFC:

Perfluourocarbon

ppb:

Parts per billion

ppm:

Parts per million

PPP:

Purchasing Power Parity

R&D:

Research and development

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RRADs:

Respiratory-related restricted active days

SC:

Single-crossing

SCC:

Social cost of carbon

SD:

Sustainable development

SF6:

Sulphurhexafluouride

SPM:

Summary for Policymakers

SPSS:

Statistical Package for Social Sciences

SROs:

Statutory Regulatory Orders

STV:

Single transferable vote

SWF:

Social welfare function

TWh:

Terawatt-hours

UN:

United Nations

UNDP:

The United Nations Development Programme

UNEP:

The United Nation Environment Programme

UNFCCC:

The United Nations Framework Convention on Climate Change

USA:

The United States of America

USAID:

The United States Agency for International Development

VAT:

Value addition tax

WCED:

World Commission on Environment and Development

WHO:

World Health Organization

WTP:

Willing to pay

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Abstract

The goal of this PhD thesis is to contribute to the analysis on mathematical terms of economics and social choice for the sustainable development and social welfare of the global humanity. Sustainable economic development is considered to be an essential factor in the development of countries. Economic growth increases social welfare through the optimization in production and consumption, building healthy human capital, electing patriotic leaders, creating green environment policy, reducing global climate change that increases sever natural calamities, and by the sustainable use of the natural resources. At the end of the 20th century and at the beginning of the 21st century there is an enormous change in global economy. In March 2018 the population of the world became about 7.5 billion. For the daily needs of the increased population, industrial production increases proportionally. Day by day the life expectancy increases, and economic competitions of the nations have changed due to open economy. Again the natural calamities, such as, cyclone, flood, earthquake, and tsunami comparatively increase recently, which decrease sustainable economic development. This thesis stresses on these items, and tries to develop models with the mathematical analysis. Each chapter of the thesis is a distinct contribution, and provides a different perspective on the formation of optimization, voting methods, social welfare, sustainability, and beneficial environment tax. These different approaches relate to the fields of political economy, mathematical economics, social choice, and social welfare. Although the chapters have different contributions, but aim of all chapters is the sustainable welfare of the global humanity. It is empirically applied research, and the results of these separate enquiries are provided by the mathematical techniques. This thesis focuses on optimization in economics, Arrow’s impossibility theorem, political economy, method of voting system, the manipulation and tie breaking of voting, green accounting, environmental pollution and healthcare, social welfare and sustainability, sustainable development, open and closed economy, open economy of Bangladesh, greenhouse gas emissions and global warming, and environment tax to reduce environment pollution with mathematical investigation. The thesis analyzes 11

limited economical and social choice problems with some detail mathematical calculations and theoretical presentations. It also displays diagrams, and adds examples and definitions where necessary. It also discusses some propositions to make it understandable to the readers. Here examples, diagrams, tables, theorems, and propositions are from author’s own concepts and related with respective topics. A section 8.3.1 ‘Theoretical Calculations of the Marginal Physical Product’ is established by the author. The thesis also includes a survey data on the garments workers of Bangladesh about willing to pay (WTP). The thesis shows that the use of mathematics in detail makes the economical and social choice concepts easier and interesting to the readers. The purpose of this thesis is to explore mathematical techniques in easier and in detailed manner. The thesis examines that the economic, environmental and social objectives are needed for sustainable development globally. It also examines the development of mathematical models and practical tools for understanding and making decisions for economic development and social welfare in a modern globalized world. The thesis includes both primary and secondary data to conduct the research. The results of the data are expressed in terms of statistical analysis, mathematical calculations, propositions, theorems, and illustrative examples. Generally, this thesis provides some new insights to modern social and economical viewpoint by exploring the influential role of mathematical exploration. It is concluded that this thesis will encourage the researchers of economics and social science to develop their knowledge both in theoretical perspectives and mathematical calculations. Finally, a number of illustrative policy frameworks have been suggested to the future researchers for working efficiently to expand this thesis for sustainable economic development and social welfare.

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Contents Page Declaration

2

Certification

3

Acknowledgement

5

Acronyms

8

Abstract

11

Contents

13

List of Figures

18

List of Tables

19

Chapter–I: General Introduction

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1.1 Introduction

21

1.2 Background of the Study

22

1.3 Research Questions

25

1.4 Aims and Objectives of the Study

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1.5 Scope of the Study

28

1.6 Contribution to the Study

29

1.7 Significance of the Study

30

1.8 Outline of the Thesis

30

1.9 Conclusion

34

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Chapter–II: A Review of the Literature

35

2.1 Introduction

35

2.2 Purpose of Literature Review

36

2.3 Importance of Literature Review

36

2.4 Literature Review in Section-wise

36

2.5 Conclusion

51

Chapter–III: Research Methodology

52

3.1 Introduction

52

3.2 Research Approaches

53

3.3 Focus of Research

54

3.4 Data Collection

55

3.5 Research Design

57

3.6 Two Criteria for Good Measurement

63

3.7 Ethical Reflections

66

3.8 Conclusion

67

Chapter–IV: Elementary Discussions

68

4.1 Introduction

68

4.2 Some Related Definitions

68

4.3 Optimization, Social Choice and Pareto Optimality

70

4.4 Game Theory

73

4.5 Environmental Economics, Social Welfare and Sustainability

75

4.6 Global Warming and Environment Taxes

77

4.7 Conclusion

78

Chapter–V: Optimization in Economics with Lagrange Multiplier

79

5.1 Introduction

79

5.2 Three Examples on Optimization

79

5.3 Mathematical Discussion of the Models

83

5.4 Sufficient Conditions for Implicit Functions

87

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5.5 Cost Minimization of a Competitive Firm

90

5.6 Output Maximization of an Agency

94

5.7 Utility Maximization Subject to Multiple Constraints

96

5.8 Conclusion

99

Chapter–VI: Social Choice, Arrow’s Theorem and Game Theory

101

6.1 Introduction

101

6.2 Social Choice and Political Relation in the Light of Game Theory

101

6.3 Unitary and Federal Democracy of two States

106

6.4 International Relation between two Adversary Countries

109

6.5 Arrow’s Impossibility Theorem

112

6.6 Single-Profile Arrow’s Impossibility Theorem

117

6.7 Conclusion

122

Chapter–VII: Methods of Voting System

123

7.1 Introduction

123

7.2 Condorcet Method

124

7.3 Borda Count

124

7.4 Single Transferable Voting System

125

7.5 Approval Voting

127

7.6 Median Voter Model

133

7.7 Majority Judgment Voting

138

7.8 Conclusion

147

Chapter–VIII: Environmental Pollution and Healthcare

149

8.1

Introduction

149

8.2

Environmental Accounting and Roles of Economics

149

8.3

Environment Pollution Decreases Economic Development

154

8.4

Valuing Health Impacts from Air Pollution in Bangladesh

160

8.5 Conclusion

169

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Chapter–IX: The NNP, Sustainability and Social Welfare Economy

170

9.1 Introduction

170

9.2 The Real Net National Product to Measure the Sustainability

171

9.3 Green Net National Product for the Sustainability and Social Welfare

179

9.4 Global Sustainable Development

185

9.5 Open Economy of Bangladesh

190

9.6 Conclusion

192

Chapter–X: Greenhouse Gas Emissions and Global Warming

193

10.1 Introduction

193

10.2 Greenhouse Gas Emissions

194

10.3 Greenhouse Gas Emissions of the USA and Mitigation Policies

198

10.4 Greenhouse Gas Emissions of China and Mitigation Policies

201

10.5 Comparison of GHG Emissions between China and the USA

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10.6 Effect of Methane Gas in Atmosphere

209

10.7 Recent Natural Calamities

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10.8 Conclusion

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Chapter–XI: Green Taxes on Environment Pollution

212

11.1 Introduction

212

11.2 Cap and Trade System or Carbon Tax System

213

11.3 Taxes on Car and Gasoline

213

11.4 Optimal Environmental Taxes Due to Health Effect

225

11.5 Conclusion

231

Chapter–XII: General Conclusion of the Thesis

233

12.1 Introduction

233

12.2 Brief Summary of the Thesis

234

12.3 Major Findings of the Thesis

236

12.4 The Value of this Approach

239

12.5 Strengths, Weaknesses, and Limitations of the Research

239

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12.6 Personal Reflection

241

12.7 Directions for Future Researches

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12.8 Conclusion and Recommendations

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Appendix–I

245

Appendix–II

249

References

253

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List of Figures Figure

Page

4.1: Three-dimensional Euclidean space

68

4.2: The rectangular hyperbolae lying in the positive quadrant

69

4.3: The plot of y = x3 with a saddle point at 0

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6.1: There are 216 points in the lattice cube

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7.1: Single-peaked preferences

134

7.2: Preference profile for Example 7.5

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7.3: Preference profile for Example 7.6

137

7.4: Preference profile for Example 7.7

138

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List of Tables

Table

Page

4.1: Battle of Sexes game

74

4.2: The payoff matrix for suspects

74

4.3: The payoff matrix for pricing

75

6.1: Strategic form game

102

6.2: A simple Battle of Sexes game

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6.3: Prisoners’ Dilemma game

110

6.4: The four strategies of A’s action when A can observe B’s prior action

110

6.5: A game where player A moves after observing B’s action

110

6.6: Player A loses reputation R if A is aggressive

111

7.1: ERS97 rules where 60 voters are electing 2 candidates from 6

126

7.2a: Tie-breaking by the majority judgment

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7.2b: First step of tie-breaking by the majority judgment

141

7.3: Second step of tie-breaking by the majority judgment

141

7.4: Tie Breaking in M

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7.5: Majority judgment for example 7.11

143

7.6: Majority judgment for example 7.12

143

7.7a: Majority judgment for city I

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7.7b: Majority judgment for city II

144

7.7c: Majority judgment of combination of city I and city II

144

7.8a: Majority judgment by Bishop

145

7.8b: Majority judgment if two opposite voters would vote

145

7.8c: Majority judgment if two voters would vote in same grade

145

8.1: WTP per person in Bangladesh currency of sub-sample A

165

8.2: WTP per person in Bangladesh currency of sub-sample B

166

8.3: Sub-sample B: sub-sample A, for 7 different symptoms

167

8.4: WTP to avoid coughing days among one of their children

168

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10.1: The global warming potential of six GHGs

194

10.2: Selected statistics for China and the USA in 2005

208

11.1: Tax on private cars/jeeps

224

11.2: Tax on transports used for hire

225

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Chapter–I

General Introduction

1.1 Introduction This chapter provides a brief description and the issues to be dealt with in the thesis. It explains the structure of the thesis by introducing the main area of interest and its problems. This thesis is a mathematical work on economics and social choice. It provides an empirically applied research of welfare economics. In our research to achieved the results we have proved propositions and theorems. We have also used examples to make the concepts easier to the readers. Further, we have added diagrams and tables to provide the desired information in the study. Modern economy is both mathematical and theoretical basis, and this thesis tries to develop models with detail mathematical calculations and theoretical explanations. If the economic development is in unsustainable way, it will not provide true social welfare to the society. This thesis is interested to determine the total sustainable social welfare in terms of mathematical models. This thesis stresses on optimum economic structures, behavior of political institutions and social choice theory, various voting methods, environmental pollution and healthcare, sustainable use of manmade and natural capitals, effects of greenhouse gas (GHG) emissions and mitigation policies, and environment taxes to reduce GHG emissions. This chapter is a theoretical study, but it gives the clear idea to the readers about the works of the thesis. This chapter presents background of the study, research questions, aims and objectives of the study, scope of the study, contribution to the study, and significance of the study. It explains scientific contribution of the thesis, importance and applicability of research. Outline of the thesis is presented at the end of this chapter.

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1.2 Background of the Study In the 21st century sustainable development is an essential issue for the humankind. At the end of the 20th century and in the beginning of the 21st century there is an enormous increase in global economy. At present the global economy is more than 5 times the size it was half a century ago. The scale of the present world economy is estimated to produce $90 trillion of output per year. Poverty has declined rapidly in many countries of the world, especially in Asia. At the same time, general living standards of people have increased resulting in greater demands, and participation in decision making at various levels of economic structures and social standards. In many cases the economic developments are not in sustainable ways. So that it will be difficult to survive of the future generations. Therefore, sustainable development is essential for the true economic development and social welfare. Due to global economy, the gap of rich and poor has increased that creates discrimination of modern economy. On the other hand, due to scientific development, the global mortality rate decreases and the population of the world increase rapidly. At the start of the industrial revolution in 1750, the estimated global population was 800 million. At present the populations of the world are increasing about 75–80 million per year (Mohajan, 2015a). Most of the high-income world and much of the middle-income world has already reached a low fertility rate, but fertility rates in Sub-Saharan Africa, North Africa, and parts of the Middle-East, remain very high (Population Division, 2011). It is estimated that in March 2018, populations of the world become about 7.5 billion, which is more than 9 times of the people of 1750 (Industrial Revolution). It is projected to increase by almost one billion within the next thirteen years and estimate of reaching as many as 8.6 billion by 2030, 9.8 billion by 2050, and 11.2 billion by 2100. Almost all of the additional 2.3 billion people will enlarge the population of developing countries whereas, in contrast, the population of more developed regions will experience a minimal change by 2050 (Population Division, 2018). For the daily needs of the increased population, and due to economic development, industrial production increases to mitigate the demand of the people. As a consequence, the anticipated population increase will have profound effects on the future political stability, food security, and energy consumption of the world as a whole. Day by day the life expectancy increases,

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and economic competitions of the nations have changed due to open economy. It has also delivered unprecedented environmental damages, for example, an estimated 60% of the world’s ecosystems have been degraded in the last 5 decades. Again the natural calamities such as floods, cyclone, earthquake, and tsunami comparatively increase, which decrease economic development. It is estimated that the main causes of these irregular natural disasters are due to global climate change, and it is conjecture that main cause of the climate change is GHG emissions. So that, in the 21st century economy faces with difficult challenges. Human has become a serious threat to its own future wellbeing, and perhaps even survival in the 21st century. Present world is divided into three categories (Sachs, 2005): i) 55 high-income (above $20,000 in per capita income) countries, such as, the USA, Canada, Western Europe, Japan, Australia, and New Zealand (about 1.3 billion people), ii) 103 middleincome (between $4,000 and $20,000 in per capita income) countries, most of them are in Eastern Asia (Korea, Singapore, etc.), Central Europe, the former Soviet Union, and Latin America (about 4.9 billion people), and iii) 36 low-income ($4,000 and below in per capita income) countries, in South America, Southern Asia, part of Eastern Asia, and Sub-Saharan Africa (0.8 billion people) (Mohajan, 2014c). Recently the illegal migrations have increased globally. In Bangladesh more than one million Rohingya refugees have migrated illegally from Myanmar since March 2018, which creates a great problem for the Government of Bangladesh and local citizens (Mohajan, 2018c). The three richest people in the world have assets that exceed the combined gross domestic product (GDP) of the 48 least developed countries. Recent Oxfam (Oxfam, 2014) analysis has found that the richest 85 people in the world have more money than the poorest 3.5 billion people (about 50% of the world population). In the 21st century about 1.2 billion people are enjoying longevity and good health, and 83% of total global income, and most of them are in developed countries. Almost 90% of the wealthiest and healthiest adults are in North America, Europe, and Japan. At least 1.2 billion people live in absolute poverty, whose income is $1.25/day, equivalent to 22% of the world population (in the USA cost of a half-dozen eggs is $1.25), that they struggle for mere survival every day (Mohajan, 2013g). About 2.4 billion people, 35% of the world population, live on less than $2.00 per day. These

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poorest of the poor people face the daily life-and-death challenges of insufficient nutrition, lack of health care, unsafe shelters, and the lack of safe drinking water and sanitation (Shepherd et al., 2014; Mohajan, 2014b, 2018b). Sustainable development tries to make sense of the interactions of three complex systems: i) the world economy, ii) the global society, and iii) the earth’s physical environment. The development is sustainable if economic progress is widespread, extreme poverty is eliminated, social trust is encouraged through policies that strengthen the community, and the environment is protected from human-induced degradation (Welfare for the Future, 2002). With competitive economy the industrialized countries emit GHGs. These GHG emissions are due to human activities for the global economic competition. Scientifically it is estimated that the GHG emissions are the main causes of global climate change. We have suggested a green environment tax to reduce GHG emissions (Mohajan, 2015a). Bangladesh is a densely populated developing country in Southern Asia and its area is 147,872 km2. In 2013, its population became more than 160 million. The population density of Bangladesh is about 1,082/km2, which is the highest in Southern Asia. About 77% of the population lives in the rural areas. Agriculture is the backbone of the economy. Thus, about 80% of the population is agrarian (Mohajan, 2013e). In 2015, Bangladesh becomes a lower-middle income country in the Southern Asia. In 2018, its populations become more than 161 million. So, to become middle income country by 2020 we have to use all our natural resources and manmade capitals efficiently (Mohajan, 2018b). Side by side we have to elect patriotic national leaders in democratic ways. On the other hand, we have to build a healthy and efficient human capital for the social welfare of the humanity. If global warming cannot be reduced, it is estimated that onefifth of coastal land of Bangladesh will submerge in the middle of the 21st century. Few or more, every country of the world will be victim of global climate change. I have prepared the thesis thinking for the sustainable economic development in future. I hope, my thesis will contribute to develop every country in a sustainable way. The choice of the research topic is purely based on the researcher’s interest. I have chosen the field “Aspects of Mathematical Economics, Social Choice and Game Theory” for my PhD thesis thinking for the sustainable development of economics and social

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science in the modern globalized world. Usually the research topic is selected on a single field, but my research topic is not chosen such a way. As I was new in this field, my first supervisor advised me to continue work on economics and social choice which are related to mathematics. But, my research target is a single topic, which is sustainable social and economic welfare. Present economy is interrelated to social science, political science, and environment science. In this thesis I have discussed some important items of these subjects that are closely related to each other. The thesis focuses on optimization, healthcare, social choice, social welfare, sustainable development, and reduction of environment pollution. I hope, the global humanity will be benefited from this thesis. This thesis is a partial work on mathematical economics and social choice, but I have used detail mathematical calculations to make it interesting to the readers.

1.3 Research Questions Well-defined research questions will enable researchers to specify their research objectives, find out proper information needs, decide what is to be investigated, and determine the suitable research designs (Hair et al., 2006). The research question forms the basis of the research design, data collection, and eventually of the data analysis and interpretation. It is very useful to express the purposes of this study in terms of a set of questions. This study focuses on the partial social and economic development of the global humanity. The following research questions are identified based on the research agenda to discuss the scenario, and better analyze the situation: i.

Do the economic models of optimization have enough mathematical calculations and theoretical analysis to adjust them in any changing situation?

ii.

How can we identify the main opportunities as well as challenges and factors those are affecting the productivity and environment?

iii.

Do the discussions of various voting methods are fruitful to build a sustainable society?

iv.

Does the real and green net national product (NNP) provide net social profit, welfare equivalent and sustainable income to improve the human life, and establish sustainable economic development in the society?

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v.

What are the effects of GHG emissions that cause global warming, and do they really liable to global climate change that decreases economic development, and reduce social welfare?

vi.

Is the green environment tax is enough to reduce GHG emissions globally? The above research questions will facilitate the readers to understand the study. This

study is intimately linked to sustainable economic development thoughts and practices, and it also takes social, economic and environmental objectives into consideration.

1.4 Aims and Objectives of the Study The purpose of any research is to obtain solutions of problems through the application of scientific procedures. In the research background we have highlighted that use of mathematics is essential in modern economics and social choice theory. The major goal of this thesis is to find out the truth which is not known, and which is yet to be revealed. The theme of this thesis is to create and explore mathematical frameworks for different aspects of the economic development and social welfare. At present there is close relationships among politics, economics, social choice, climate change, and sustainable development. This thesis thoroughly examines by the mathematical techniques of the economic optimization, political relations, social choice, health valuation, voting system, sustainable development and social welfare, climate change, and environment taxes to reduce natural calamities. My aim in this research is to help to create a global sustainable economic development, and create a healthy human capital for the optimum social welfare. The aim of this study is to develop a mathematical representation of economics and social choice to estimate the economic consequences of improving the social welfare for sustainability. The work deals with optimality, voting system, social welfare, sustainability and environment pollution. The aim of the research is also to explain economics problems with detailed mathematical techniques by using definitions, tables, examples, and diagrams where appropriate. Another aim is to make economic materials interesting to the readers, and to show the importance of use of mathematics in economics and social science.

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The main objectives of this study are the construction of mathematical models that are accurate, valuable, and globally acceptable, which are capable of effective use for optimization for the economic sustainability and social welfare. Particular objectives of this thesis are as follows: •

To review the literature on “Aspects of Mathematical Economics, Social Choice and Game Theory”, that will serve as a guideline to develop a mathematical framework, and will facilitate sustainable economic development and social welfare.



To highlights of the study area that we use throughout the thesis, especially the mathematical frameworks of the research, and to provide the guidelines to the future researchers.



To develop economic models with detail mathematical calculations and theoretical analysis, also displaying diagrams and citing examples to make the thesis easier to the readers, and to show that there is a close relationship between mathematics and economics for an efficient modernized economic environment.



To identify the main opportunities as well as challenges and factors those are affecting the productivity and environment in the way of sustainable economic development.



To explain the consequences of optimization by comparative statics with the help of method of Lagrange multipliers introducing explicit examples of mathematical models for the implicit functions.



To discuss the Arrow’s impossibility theorem that makes social choice theory more challenging; advantages and disadvantages of unitary and federal democracy, and international relation between two adversary countries in the light of game theory.



To discuss the various voting methods, manipulation of voting, and importance of voting system to create a developed society.



To provide mathematical presentations of NNP in detail, and discuss models that will improve the human life, and establish sustainable economic development in the society.

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To analyze the sources and effects of GHG emissions that cause global warming and climate change.



To impose green environment tax in the society to reduce global GHG emissions, and help those nations who are affected seriously due to climate change severely.



To identify the scope of further works to future researchers in the mathematical economics and social choice. To achieve the above mentioned objectives I have discussed every chapter in detail

theoretical analysis, and some chapters with sufficient mathematical formulations. Although every chapter has discrete result, but aim of every chapter is to present an environment friendly sustainable healthy society for the welfare of the global humanity.

1.5 Scope of the Study The scope of the study refers to the parameters under which the study will be operating, what the study covers, what is in the study domain and what is not, which factors are within the accepted range of the study, and which is closely connected to the framing of the problem. All researches are bounded in a number of ways. Actually a research which has an unlimited scope and few boundaries may be incapable of leading to conclusions. I have discussed some such scopes in the background of the study, and some will be discussed in the methodology chapter. The scope also includes collection of the data for the study, and period taken to complete the study.

1.5.1 Data for the Study There are various procedures of collecting data, such as, tests, questionnaires, interviews, classroom observations, diaries, journals, etc. The data collection for this study is primary sources of data collection, and secondary sources of date collection. Primary collection sources of data are performed by the interview in some 500 female garments workers of Chittagong City Corporation (CCC), Bangladesh. The secondary data are collected from published and unpublished papers, theses, websites, various research reports, case studies, and books of various authors.

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1.5.2 Period for the Study I have taken period under this PhD research since February 2010 to March 2018. During this period I have learnt a lot of in my research area. My two supervisors have helped me cordially, and have guided me efficiently for the preparation of the thesis.

1.6 Contribution to the Study By incorporating works of mathematical economics and social choice theory, the thesis measures of social welfare for all nations. This study makes a contribution to empirical knowledge of planning for sustainable economic development and social welfare. The original contributions of this thesis are the application of mathematical techniques in the economics and social choice. It makes a relationship between economic growth and social welfare. To conduct this research, several contributions to both theoretical and mathematical analysis are considered.

1.6.1 Theoretical Analysis Chapters I to III and XII are prepared on the basis of theoretical contributions. In these chapters I have tried to convince the readers to show the overall structure of the thesis. Chapter I displays the introductory map of the research. Chapter II is about the literature review, chapter III provides the methodology of the literature, and final chapter XII is about the conclusion of the thesis. Also other chapters have some theoretical analysis with the mathematical calculations.

1.6.2 Mathematical Analysis Mathematical modeling is widely used in economics and social science to improve understanding of the behavior of economic analysis and social choice, sustainable economic development and social welfare, explore new theoretical concepts and mathematical theories, and predict performance of future economic development. Chapters IV to XI of this study are prepared on the basis of detailed mathematical analysis with theoretical presentations. In these chapters main research works are given with mathematical techniques. We hope mathematical analysis will help the readers to understand the thesis easily. We have also used diagrams, tables, examples, propositions,

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and theorems to clarify the mathematical representations to the readers. Results of the research are provided through the mathematical proofs of propositions and theorems. Some cases examples and diagrams facilitate to clarify the results.

1.6.3 Statistical Analysis In chapter VIII we have discussed statistical analysis of the survey data collected from the 500 female garment workers of Chittagong City Corporation (CCC) of Bangladesh. We have used SPSS 17.0 software to calculate the results of the survey.

1.7 Significance of the Study This study spreads out a set of crucial sustainable issues of mathematical economics and social choice. It tries to find out the underlying socioeconomic causes of modern globalized world. This study can be helpful as a reference for future investigations and monitoring in the mathematical economics and social sciences.

1.8 Outline of the Thesis According to Vickers (2003), economics is considered as “the science of incentives”. In this doctoral thesis we have tried to incentive the economical and social science materials in terms of mathematical models and theoretical analysis. The thesis is structured with a conventional sequence of chapters on literature review, research methodology, obtained results, discussion of results, and general conclusion. This thesis provides distinctive contributions in fulfilling the obligation of the research objectives. The thesis is divided into two parts, and organized in twelve chapters. The first part includes the first three chapters, and is devoted for providing the background preparation and tools necessary for problem definition, modeling, and implementation. The second part includes chapters IV to XI, and is dedicated to problem statement, formulation, solution, and discussion. The second part is prepared on the basis of detail theoretical analysis and sufficient mathematical representations. This thesis is on the mathematical economics and social choice to develop the sustainable economic structure for the global social welfare. The dissertation contains twelve chapters, and this section provides a brief snapshot of the contents of each chapter.

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Chapter I is a general introduction. It sets the scene for the main body of the thesis. It reflects the contributions of the entire thesis. It represents background of the study, research questions, aims and objectives of the study, scope of the study, contribution to the study, significance of the study, and outline of the thesis. Chapter II is devoted to presenting a review of the literature related to conceptual issues addressed in this thesis. The purposes of the literature review are to discuss some major theoretical frameworks and concepts of relevance to the topic being researched, establish a conceptual background against which to further explore the research questions and to identify key concepts and relationships between ideas and practice in the field of mathematical economic development and sustainable social welfare. These include; optimization in economics with necessary and sufficient conditions, social choice and democratic leadership, healthcare strategy for pollutions, real NNP and sustainable social welfare, GHG emissions and climate change, and environment tax for the reduction of pollution. The literature review also covers the green environment policy throughout the thesis. It indicates the working procedures of the previous scholars that influence on our research activities. Chapter III covers the research design and the research methodology used in this research to collect data for the research. This is followed by research approaches on mathematical economics and social choice to complete a successful research. The construction of the research instruments, the sources of data for the research, and a detailed account of how the mathematical analysis is conducted are also indicated here. Chapter IV only includes some definitions which are collected from the different references given in the thesis. This chapter will be helpful to understand the thesis perfectly. Although the chapter discusses definitions, but it has mathematical analysis, and diagrams to make the thesis easier to the readers. In the economic theory of the firm, activities are usually considered that maximize profits and minimize costs. Chapter V presents optimization in economics with Lagrange multiplier. The method of Lagrange multipliers is a very useful and powerful technique in multivariable calculus. For optimal values the necessary condition is provided by Lagrange multipliers, and sufficient condition is provided by considering the determinant of Jacobian matrix negative and the determinant of the Hessian matrix positive.

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Comparative statics provide conditions under which an optimization problem’s endogenous variables increase or decrease when exogenous variables change. It mathematically shows the behavior of the firm, and recommends that if the cost of a particular input increases, the firm needs to consider decreasing the level of that particular input; at the same time, and there is no effect on the level of other inputs. This part also discusses maximization of output subject to both linear and nonlinear budget constraints, maximization of utility function subject to multiple constraints, and minimization of cost subject to output as a constraint. Chapter VI provides Arrow’s impossibility theorem, and single-profile Arrow’s impossibility theorem. These are elucidated with examples, propositions and theorems to make the social choice interesting to the readers. Game theoretical models are introduced to describe how the political institutions are formed, how these are developed to create efficient political leaders, and to form stable democracy in the states. A brief description is given here in the light of prisoner dilemma game to show the nature of the behavior between the two adversary countries. Chapter VII is on methods of voting system. Here different types of voting system and manipulation of them are given with sufficient theoretical properties and mathematical calculations. It discusses Condorcet method, Borda count, single transferable voting, approval voting, median voter model, and majority judgment voting. This chapter is a part of political economy. Every section of this chapter has both mathematical and theoretical analysis. In addition, it has sufficient definitions, examples, theorems, and propositions with proof to make it understandable to the common readers. Chapter VIII deals with environmental pollution and healthcare, and contains beneficial economic models. The aim of this chapter is to serve as an indicator of wealth changes, performance of environment policy, and sustainable use of natural and manmade capitals. It also provides a new mathematical structure, and a survey result on the garments workers of Bangladesh. In addition, it highlights future labor sector of Bangladesh, and stresses on the development of this sector for the sustainable economic development in Bangladesh. Chapter IX provides the NNP, sustainability and social welfare economy. The NNP is an important item for a country and can serve as an indicator of sustainability. It

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emphasizes on optimal growth and growth without optimality, and examines sustainability in these two cases. It highlights on discrimination in modern economy, present unsustainable development practices, and sustainable development goals. It also contains the recent world economy and the open economy of Bangladesh. Chapter X describes global warming which is due to GHG emissions that cause global climate change. As climate change is a global problem, every nation in the world is affected less or more by climate change. Due to global climate change the natural calamities are increased which cause huge destruction of both humanity, and manmade and natural assets. As a result economic development becomes slow. The chapter discusses GHG emissions of the USA and China, and mitigation policies in some details. These two countries are responsible about 40% global GHG emissions. This chapter briefly discusses effect of methane gas in atmosphere, and benefits from the reduction of methane gas emissions. Finally, it tries to highlight GHG emissions due to global transportation, and recent natural calamities. Chapter XI includes some policies that would influence people to drive fewer miles and to buy smaller cars, use better pollution control equipment, and cleaner fuel. Here Lagrange multiplier techniques are applied to calculate environmental taxes. It advocates on cap and trade system or carbon tax system to reduce GHG emissions. It also stresses on optimal environmental taxes to reduce global environment pollution. Pigovian tax and deadweight loss are introduced by calculations to make the policy acceptable to the readers. The final chapter XII, ‘General Conclusion’ of the thesis discusses brief summary of the thesis that is guided the entire study. This chapter also draws conclusions on the major findings of the study, contribution of the thesis, strength, weakness and limitations of the thesis, and outlines areas for further research to extend the findings of our research. This discussion is followed by highlighting the contribution of the research to the state of the art knowledge. In the thesis we have included two appendices. The first one is about the welfare effect of the optimum environmental tax which is equal to the Pigovian rate. Here we have derived equation (11.54) of chapter XI. The second one is the revised form of equation (11.54). Here we consider the neutral assumption that two goods are equal

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substitutes for leisure. For this we have derived equation (11.55) of chapter XI. Both of the appendices are prepared in detail mathematical calculations.

1.9 Conclusion This chapter is introductory and discusses only the summary work procedures. It discusses partially on background of the research, research questions, aims and objectives of the study, scope of the study, contribution to the study, significance of the study, and outline of the thesis. We hope, from this chapter the readers will find a clear map of our works in the thesis.

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Chapter–II

A Review of the Literature

2.1 Introduction This chapter reviews the literature of past and present works on the mathematical economics and social choice that are useful for the establishment of the strong sustainable society. The purpose of this literature review is to establish the current state of knowledge on the social economy, and to identify key trends and gaps in this knowledge. The chapter also reviews at various ways how this thesis can be analyzed through the theoretical frameworks and mathematical representations. Literature review is an integral part of entire research process, and makes valuable contribution to every operational step as the researcher moves on. It can be timeconsuming, daunting and frustrating, but it is also rewarding. It brings clarity and focuses to the research problem, helps to understand the subject area better, broaden knowledge, and contextualize the findings clearly and precisely. It tells us about the others have used procedures and methods similar to the ones that we are proposing, which procedures and methods have worked well for them, and what problems they have faced with them (Legesse, 2014). The literature review can play an extremely important role in shaping research problem because, the process of reviewing the literature helps to understand the subject area better. Therefore, it helps to conceptualize the research problem clearly and precisely, and makes it more relevant and pertinent to the field of enquiry. It also helps to understand how the findings of the research fit into the existing body of knowledge (Martin, 1985). This chapter provides detailed explanations of how to undertake a literature review, and why it is a pivotal element of research in mathematical economics and social choice.

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Here we have considered the purpose of literature review, the importance of literature review, literature review in section-wise, and a brief conclusion of this chapter.

2.2 Purpose of Literature Review The purpose of literature review in any academic writing is which to be equated to it essence in any published work. It allows for more lights to be thrown on the background to the study. It also helps to further define the problem a researcher is hoping to resolve. The purposes of this literature review are as follows: •

To discuss some major theoretical and mathematical frameworks and concepts of relevance to the topic being researched.



To establish a conceptual background against which to further explore the research questions for identifying key concepts that will help to analyze specific issues.



To identify relationships between ideas and practice in the field of mathematical economic development and sustainable social welfare.



To represent the effects of GHG emissions and reduction policies by mathematical structures.

2.3 Importance of Literature Review In any study it is vital to understand what has already been done in the selected field, and what has been done in the wider subject area of that topic. The significances of inclusion of it in the research are as follows: •

To improve the methodology that is capable of providing valid answer to the research questions.



To understand the subject area study clearly and precisely.



To study widely around the subject area that intends to conduct research.



To develop existing knowledge and findings something different from those of others.

2.4 Literature Review in Section-wise Here we have discussed the literature review of our study in section-wise as follows:

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2.4.1 Optimization Models in Economics A classical optimization method is an analytical method that can be solved by differentiable functions (Yogesh, 1998). Optimization could be generally defined as the process of finding the best solution to a problem. Therefore, optimization is the minimization or maximization of a mathematical function to find the best solution that satisfies a number of equality and/or inequality constraints. It is an important tool for solving various applied economics analytical and practical problems (AlHajri & ElHawary, 2007). According to E. L. Lawler, in optimization problems, the goal is to find an optimal arrangement, grouping, ordering, or selection of discrete objects usually finite in number (Lawler, 1976). Optimization theory is a part of the applied mathematics including the use of mathematical models and methods to find the optimal alternative of actions in different decision situations (Yogesh, 1998). Prajit K. Dutta and Roy Radner have examined that competitive firms must behave as if they were maximizing profits; otherwise they would go bankrupt, or even fail to be financed in a competitive capital market (Dutta & Radner, 1999). Allen M. Featherstone, Ghassan A. Moghnieh, and Barry K. Goodwin investigate non-parametrically the optimizing behavior of a sample of 289 Kansas farms under profit-maximization and cost-minimization hypotheses. They have shown both deterministic and stochastic nonparametric tests (Featherstone et al., 1995). The efficiency argument for profit maximization indicates that the organizations should maximize profits because of the course of action that will lead to an ‘economically efficient’ or ‘welfare maximizing’ outcome (Jensen, 2002). Ajoy Kumar Dey has identified ten different approaches for maximization of profit as: innovation, brand image, customization–mass customization, customer collaboration, long tail effect, operational excellence, outsourcing, value engineering, moving away from unprofitable customers, and reducing quality. Out of these approaches a manager should select the one that maximizing profit (Dey, 2009). The method of Lagrange multipliers (named after Joseph Louis Lagrange) is a very useful and powerful technique in multivariable calculus, and has been used to facilitate

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the determination of necessary conditions. These are mathematical tools for constrained optimization of differentiable functions (Mohajan, 2017a). Leonid Hurwicz and Marcel K. Richter have extended the applicability of Lagrange multipliers to a wider class of problems, by reducing smoothness hypotheses for classical Lagrange inequality constraints as well modern inequality constraints, and by reducing constraint qualifications to minimal levels (Hurwicz & Richter, 1995). John V. Baxley and John C. Moorhouse (1984) considered implicit functions with assumed characteristic qualitative features generating meaningful economic behavior. They expressed that there is a ‘widespread economic folklore’ which assumes the determinant of Jacobian matrix is negative, and the determinant of the Hessian matrix is positive (Mohajan et al., 2013). Comparative statics are formalized by John R. Hicks (1946) and Paul A. Samuelson (1947) to predict the economic optimization outcomes. To stay competitive by creating higher value for consumers firms are in constant search for strategies and tactics that will maximize profit. We have used twelve comparative statics for the prediction of the optimization problems efficiently. Mathematical modeling and optimization techniques are utilized in the study to derive the optimal policy that minimizes the total costs, and maximizes the utility function and also maximizes the output. Here implicit function theorem has been applied to show the sufficient condition more effectively (Mohajan, 2017a). Vedran Kojić and Zrinka Lukač have proposed a new original method to solve the production cost minimization problem with Cobb-Douglas production function by using the weighted arithmetic-geometric-mean inequality. They have not used the techniques derivatives or the Lagrange multiplier method that we have used. They have derived the minimum costs and global minimizers in the case of the Cobb-Douglas production function in the direct way (Kojić & Lukač, 2014). Vedran Kojić presents a new, noncalculus approach to solve the utility maximization problem with Constant Elasticity of Substitution (CES) utility function, as well as with Cobb-Douglas utility function in case of many commodities. He has not use Lagrange multiplier method or some other method based on differential calculus, and has solved by using Jensen’s inequality and weighted arithmetic-geometric mean inequality (Kojić, 2017). Adriana Agapie and Tony Lima have assumed a two variables Cobb-Douglas production function for the minimization of

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the cost of a firm. They have emphasized that constant factor prices and having a production function that meets the necessary conditions to allow the use of the Lagrange multiplier method for the constrained optimization problem of the minimizing cost assuming technical efficiency (Agapie & Lima, 2013). The modern ‘utility thought’ originated from Bentham’s nominal measures of pleasure, happiness, etc. His outstanding contribution to the utility theory was to make the utility concept be viewed as a numerical magnitude (Bentham, 1789). Thomas Juster provides a brief history of the development of utility theory, and suggests a reconceptualization of the basic sources of utility for the best output of different goods (Juster, 1990). The utility maximization problems are complex, because of the relations among the objective function, constraints, and other inputs (e.g., capital, labor, land, raw materials, etc.) (Fischione et al., 2009). Ying Hu, Gechun Liang, and Shanjian Tang have studied the maximization of the expected exponential utility of terminal wealth for an investor with an unbounded random endowment by the use of differential equations (Hu et al., 2017). Lingqi Gu, Yiqing Lin, and Junjian Yang have discussed the numerairebased utility maximization problem in markets with proportional transaction costs, in which the investor aims to solve the maximization problem on the expected utility over his/her terminal wealth (Gu et al., 2017).

2.4.2 Social Choice and Social Welfare Social choice theory has a long history. The independent works of Bergson (1948) and Samuelson (1947) on social welfare functions was remarkable. Perhaps the most significant work in social choice theory is that of Kenneth J. Arrow, and he has been credited for creating the modern field of social choice theory (Arrow, 1963). Social choice theory can be used to determine the weights assigned to different components of social welfare (Sen, 1999). It allows the normative significance of economic and noneconomic events to be evaluated in a formal framework. It influences on society’s choices, preferences, and value judgments on issues of economic equity and efficiency, intergenerational equity, aggregation, value judgments, justice, poverty measurement, and market perspectives versus social perspectives are considered (Boadway & Bruce, 1984). Social choices can be made by the individual within a new framework that

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considers alternatives from a social perspective that includes considerations other than their own wellbeing (Sen, 1995). Arrow has established Arrow’s impossibility theorem which indicates that it is impossible for a social welfare function to satisfy five conditions namely: i) completeness and transitivity, ii) universality, iii) Pareto consistency, iv) independence of irrelevant alternatives (IIA), and v) non-dictatorship simultaneously (Arrow,1963). Allan M. Feldman and Roberto Serrano have revised the works of Arrow in some detail, and they have discussed the single-profile Arrow impossibility theorem with some examples, propositions, and theorems (Feldman & Serrano, 2008). In single-profile models of Arrow’s theorem, neutrality is the natural assumption to substitute for Arrow’s IIA assumption. In both multi-profile and single-profile cases of course there is a dictator. But, dictators in single-profile models are sometimes innocuous than that are in multiprofile case (Mohajan, 2012b). Drew Fudengerg and Jean Tirole have discussed the game theory in some details (Fudenberg & Tirole, 1991). John Jr. Nash has provided Nash equilibrium which is an essential part in game theory (Nash, 1951). Roger B. Myerson has used the “Battle of Sexes” game to show how an efficient political institution is formed, and has established the political relation between two adversary countries by using “Prisoners’ Dilemma” game (Myerson, 1996, 2006). Myerson has also discussed the unitary and federal democracy of two states (Myerson, 2006).

2.4.3 Voting Procedures French political philosophers Jean-Charles Borda (1781) and Marquis de Condorcet (1785) introduced modern voting system, but they have not mentioned about manipulation of voting. Condorcet, Borda, and even many modern politicians believe that elections are logically imperfect (Mohajan, 2013b). Duncan Black first successfully introduced the manipulation of voting in 1958 in his book ‘Theory of Committee and Elections’ (Black, 1958). The single transferable voting (STV) was first proposed in the mid-19th century, quite independently, by Thomas Hare, an English Lawyer, and Carl George Andrae, a Danish Mathematician and Politician (Hart, 1992). The STV in multi-member

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constituencies is little used in public elections. Currently, just two relatively small European countries, Ireland and Malta, organize it to elect Parliament (Reynolds et al., 2005). The STV is a system of preferential voting designed to minimize wasted votes. In STV, a constituency elects two or more representatives per electorate (Mohajan, 2012e). Nicolaus Tideman has given comparison of STV with other voting procedures. He has also provided history, choice of a quota, transfer of votes, limitations, and a further refinement of this voting system (Tideman, 1995). STV is used to elect the federal Senate in Australia, and the City Council and School Committee in Cambridge, Massachusetts, and to elect Community School Boards in New York City in the USA (Farrell, 2001). It is also used in dozens of unions and religious, charitable, and professional organizations in many countries (Tideman, 1995). Christopher Banks has indicated that STV is the most suitable electoral system to guarantee both an increased degree of proportionality and maintain direct accountability of all Members of Parliament (MP) to the electorate for the UK (Banks, 2017). Henry R. Droop has provided the STV quota. Droop quota is one of the best methods in STV counting (Droop, 1881; Mohajan, 2012e). Robert A. Newland and Frank S. Britton have discussed tie-breaking system in STV. The counting rule of STV commonly used is ERS97 rule, has been given by Newland and Britton (1997). Toby Walsh has studied empirically whether computational complexity is a barrier to the manipulation for the STV rule (Walsh, 2009). Yvo Desmedt and Kaoru Kurosawa have presented an electronic voting scheme that allows pointing the winner, without revealing the final tally, that is, no information is disclosed except for the name of the winner(s). Joanna Boron and Marek Klonowski have suggested an extension of the Desmedt-Kurosawa scheme for STV procedures (Boron & Klonowski, 2008). Approval voting (AV) is a single winner voting system has used for multicandidate elections (Mohajan, 2011e). Steven J. Brams and Peter C. Fishburn have introduced approval voting and strategies of it (Brams & Fishburn, 2005). Steven J. Brams and M. Remzi Sanver have developed the notion of a ‘critical strategy profile’ under AV, which facilitates the identification of all possible outcomes that can occur under AV (Brams & Sanver, 2006). Marc Vorsatz has shown that AV is the only social choice function that satisfies anonymity, neutrality, strategy-proofness and strict

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monotonicity of social welfare function (Vorsatz, 2007). Jean-François Laslier has highlighted on the strategic AV for a large population of voters (Laslier, 2004). Steven J. Brams, D. Marc Kilgour, and Richard F. Potthoff have proposed a modification of AV to elect multiple winners, who may be either individuals or members of a political party (Brams et al., 2017). Krzysztof Przybyszewski and Honorata Sosnowska have discussed the applications of AV in Polish 2015 presidential and parliamentary elections. They have proposed a possibility to predict a winner of the second round of presidential elections (Przybyszewski & Sosnowska, 2016). Ulle Endriss has stressed on sincerity and manipulation under AV (Endriss, 2013). Jordi Massó and Marc Vorsatz have introduced the weighted AV when the weights are identical and strictly positive for all alternatives that are a natural extension by relaxing the assumption of neutrality (Massó & Vorsatz, 2008). AV is comparatively better than other voting systems, and encourages sincere voting, and all votes are of equal weight (Mohajan, 2013b). The median voter theorem is among the most prominent results of formal political theory. It has a long theoretical and empirical history within public economics. It has been widely used to study the interactions between economic and political behavior (Cukierman & Spiegel, 2003). It states that given single-peaked preferences and majority voting, the median demand for the public good is what is going to be supplied (Gupta, 2004). Black’s median voter theorem is “If all voters’ preferences are single peaked on a single dimension then the bliss point of the median voter is a Condorcet winner” (Black, 1958). Miguel Angel Balleste and Guillaume Haeringer have identified two properties that characterize the domain of single-peaked preferences on the line. The first property states that for any subset of alternatives the set of alternatives considered as the worst by all agents cannot contain more than two elements. The second property states that two agents cannot disagree on the relative ranking of two alternatives with respect to a third alternative, but agree on the relative ranking of a fourth one (Balleste & Haeringer, 2011). Swati Dhingra has investigated the empirical validity of the median voter theory of trade policy (Dhingra, 2014). Joshua S. Gans and Michael Smart (1996) analyzed single-crossing property, Salvador Barbera (2010) discussed single-dipped and single-peakedness, Barbera and Bernardo Moreno (2011) elucidated top-monotonicity and single-plateaued.

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In Arrow’s impossibility theorem preference relation xPy for individual 1 and individual 2 express same preferences where x and y are two candidates (Arrow, 1963). But, individual 1 can express x is Very Good and y is Rejected. On the other hand, individual 2 can express x is Excellent and y is Very Good. Michel Balinski and Rida Laraki first have introduced this type of judgment which is the median values of the grades given to a candidate is taken as the final grade of that alternative. It is quite similar to range voting, but using the median instead of the average and with an interesting tie braking rule. It allows the voters to convey more information in the input of the choice and ranking functions (Balinski & Laraki, 2006, 2007, 2010, 2014). Francis Galton has shown the logic that if the number of voters may be odd, then there is one middlemost value; if the number of voters be even, there are two middlemost values, the mean of which must be taken in any operation (Galton, 1907). Balinski and Laraki have revealed that when the number of voters is odd, the majority grade is the median, but in the case of an even number of voters and a candidate’s two middle grades are different the lower of the two middle grades must be the majority grade (Balinski & Laraki 2006; Mohajan, 2012a). Manzoor Ahmad Zahid and Harrie de Swart show that the resulting Borda Majority Count avoids the counter-intuitive results and has a number of other nice properties as well. They have also presented interesting tie braking rules in majority judgment voting (Zahid & de Swart, 2015).

2.4.4 Sustainable Development for Social Welfare Modern economic development started by Adam Smith’s inquiry in The Wealth of Nations (Smith, 1937). Economic development incorporates the social factors of education and health improvements, and environmental protection; with the economic benefits of efficient allocation of resources, and sustainable growth (Todaro, 1989). In the modern world the state of the economy of a country is determined by its GDP (Mohajan, 2013a). Economic growth is the leading indicator for this task, as it can be measured through GNP or GDP, and these are generally used as a proxy for overall economic development (Sen, 1988). National income can be considered a better measure of society’s welfare than either GDP or GNP (Weitzman, 1976). Robert Costanza, Maureen Hart, Stephen Posner, and John Talberth have described inappropriate use of Gross

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Domestic Product (GDP) as a measure of national wellbeing. They have analyzed various alternatives and complements to GDP in terms of their motives, objectives, and limitations (Costanza et al., 2009). Welfare economics is concerned with the principles of maximizing social welfare. It defines social welfare and its criterion, identifies those factors that prohibit achieving optimal levels of social welfare, and sets out policies to maximize social welfare (Oser & Brue, 1988). G. Heal and B. Kriström have highlighted the present and future production, use of ecosystem services, and sustainable use of natural resources (Heal & Kriström, 2002). Natural capital consists of a variety of ecosystems, such as, wetlands, lakes, forests, agricultural landscape, and coastal water (Mohajan et al., 2012). Sometimes the emission of pollutants amounts directly to a degradation of ecosystems (Dasgupta & Mäler, 2000, 2001). The term ‘ecosystem services’ was first coined by Ehrlich and Ehrlich (1981), but it was not until the late 1990s that the concept received wide public attention, following publication of the work of Costanza et al. (1997) and Daily (1997). Energy production and consumption activities have been linked to local health impacts, global climate change, air and water pollution, soil contamination, biodiversity loss, resource depletion, security implications and land-use conflicts (Bagstad & Shammin, 2012). Meadows et al. (2005) stressed that the people of the world have to act soon to establish a sustainable world; otherwise, the global population will face enormous challenges to provide sufficient goods, energy, and food to a growing population. They also predicted that future generations may experience recession, hunger, conflicts, and reduced living standards. Therefore, we have to develop sustainable energy production and improved energy efficiency for the welfare of the future generations. Karl-Göran Mäler, Sara Aniyar, and Åsa Jansson have provided a brief and consistent basis for accounting for sustainable development focusing on ecosystem services (Mäler et al., 2008). Medical expenditures to cure diseases due to air pollution should not be deducted from NNP, but hamper of production due to pollution related illness should be subtracted from NNP (Dasgupta & Mäler, 2000). Air pollution causes disutility, and the disutility of pollutants, in the form of “pain and suffering” can be alleviated with inputs for healthcare and mitigation (Huhtala & Samakovlis, 2003).

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The concept of sustainable development, which is proposed to reconcile economic, social, and ecological dynamics, was initiated during 1980s. Recently the economic development through sustainable concept has become an important issue to ensure the effectiveness of long term perspective. Sustainable development is a complex concept describing developments at different time-scales, geographical scales, and across domains (Munasinghe, 2001). Baker (2006) distinguishes the following normative principles of sustainable development: i) common but differentiated responsibilities, ii) inter-generational equity, iii) intra-generational equity, iv) justice, v) participation, and vi) gender equality. R. E. Munn argues that development, to be sustainable, requires profound changes in political, social, economic, institutional, and technological order, including redefinition of relations between developing and developed countries, and a succession of technological break-through (Munn, 1989). Within the concept of sustainable development, there is a central tension between advancement in human wellbeing and environmental conservation for future generations (Qizilbash, 2001). W. D. Nordhaus (2002) has shown that healthy human capital gives optimal product in the society and decreases all kinds of medical expenditures related with pollutions. As a result society gains rich economy. In the study human capital in macroeconomics demonstrated that not only education but also good health has a significant positive effect on aggregate output. Partha Dasgupta and Karl-Göran Mäler (2000) have stated that medical expenditures to cure diseases due to air pollution should not be deducted from net national product (NNP), but hamper of production due to pollution related illness should be subtracted from NNP. Arrow et al. (2010) and Dasgupta (2008) have discussed the dynamics of the resource for the sustainable development of wealth. Martin L. Weitzman (1976), Robert M. Solow (1986), John Hartwick (1990), and Mäler (1991) lay the foundation for a concept of NNP which is adjusted for the depletion of natural and environmental resources. Solow sees this primarily as the level of consumption of future generations not being less than the consumption levels of current generations. This can be assured if the total stock of natural and man-made capital. Thus, true NNP measures the maximum current level of consumer satisfaction that can be sustained forever; it is therefore a measure of sustainable income (Solow, 1986).

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In Dasgupta-Heal-Solow model of capital accumulation and resource depletion, eventually the welfare of the society is optimally decreases along the discounted utilitarian path (Dasgupta & Heal, 1974, 1979; Solow, 1974). Martin Weitzman published his seminal paper on the significance for dynamic welfare of comprehensive national accounting aggregates, where he had included important theoretical contributions on welfare and sustainable accounting (Weitzman, 1976). In the aftermath of the World Commission on Environment and Development (WCED, 1987), it became important to investigate whether the concept of NNP can serve as an indicator of sustainability. Solow goes on to argue that traditional measures of Gross Domestic Product (GDP) and Gross National Product (GNP) are not bad for studying fluctuations in employment or analyzing the demand for goods and services. However, these measures pay little attention to capital depreciation; do not provide an accurate picture of welfare. Hence, national statisticians also compile the NNP which accounts for depreciation in fixed capital (Mohajan, 2011f). Weitzman’s fundamental results for closed economy and Hartwick’s open economy facilitate to understand the NNP and open economy. Many authors seem to indicate that Hartwick rule is relevant to an open economy whose reproducible capital is defined to include foreign assets. Hartwick (1977) refers to it as a ‘Saudi Arabian’ rule, but Hoel (1998) expresses it for a single resource-exporter. Geir B. Asheim (2010) has highlighted on global welfare comparisons for the welfare of growing economies. Great Philosopher and the father of modern economics Adam Smith’s famous law of the invisible hand stated two fundamental theorems of welfare economics on the basis of competitive markets tend toward an efficient allocation of resources, in 1776, in his book ‘The Wealth of Nations’ (Smith, 1937). According to Justine Ram, the idea of sustainability has conceived from the notion of having growth without compromising or impoverishing future generations. Thus, many economists have attempted to answer the question of how growth can be accommodated without leaving a depleted or degraded stock of natural resources (Ram, 2012). Social welfare is maximized when producers and consumers maximized their utility in a healthy way. Our social environment is polluted in different ways mainly by

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air and water which creates different diseases, and we need to invest an extra amount in health sector due to this pollution (Mohajan, 2012f). M. L. Smith and his coauthors have proposed that nine domains be included in the index for the USA as: health, social cohesion, spiritual and cultural fulfillment, education, safety and security, living standards, life satisfaction and happiness, leisure time, and connection to nature. Performance in these nine domains of wellbeing is used to evaluate three ‘wellbeing elements’, which they have identified as environmental wellbeing, societal wellbeing and economic wellbeing, all of which are combined to describe human wellbeing (Smith et al., 2013). For the unsustainable economic development the economic growth can be antieconomic because, the marginal physical throughput may cause environmental costs to increase faster than production benefits, thereby making society poorer, not richer (Daly, 1996).

2.4.5 GHG Emissions and Global Warming The GHGs were first discovered in 1896 by a Swedish chemist, Svante Arrhenius, who estimated that doubling the concentration of CO2 would increase the global average temperature by 5°C–6°C (Houghton, 2004). The atmospheric CO2 concentration is increasing at the rate 2 ppm per year which has become global environment problem over the last decades (Budzianowski, 2013). Global warming is now +0.60C in the past three decades and +0.80C in the past century, and continued warming in the first half of the 21st century is consistent with the recent rate of +0.20C per decade (Intergovernmental Panel on Climate Change, IPCC, 2007). It is predicted that that the profitable fossil-fuel resources are probably sufficient to increase the CO2 concentrations well beyond 750 ppm, with the risk of dangerous climate change impacts (Stern, 2007). Current predictions indicate that the future increase of the mean annual temperature of 5°C for the Arctic by the end of the 21st century (Stendel et al., 2006). The living organisms are in dangerous position and some species have already extinct and some more will extinct in future if global warming cannot be controlled (Mohajan, 2013d). The National Academy of Sciences (NAS, 2010) expressed its expert

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opinion that concentrations of CO2 in the atmosphere have increased and continue to increase more rapidly due to human activities. Both NAS (2010) and IPCC (2007) expressed that humans, largely through the ever increasing burning of fossils are changing the earth’s climate. Jane A. Leggett (2011) and the Organization for Economic Co-operation and Development (OECD, 2012) have discussed GHG emissions mitigation policies. Many national and international organizations have been taken various mitigation policies of recent GHG emissions (Carvalho, 2012). Methane is dangerous greenhouse gas, since it is 21 times more global warming potential than CO2. All the nations talk about the reduction of only CO2 and no nation stress on methane reduction. Methane mitigation provides opportunity to improve air quality globally, which can be a cost-effective component of international ozone management, bringing multiple benefits for air quality, climate, agriculture and human health (Mohajan, 2012d). Mohajan (2013f) discusses mathematical calculations of GHG emissions from small industries in those use biomass and fossil fuels to run the mills which cause global warming in the atmosphere. Simple calculations are presented to estimate three GHGs, CO2, methane, and nitrous oxide emissions from small industry. Here the techniques of the efficiency methods of World Resources Institute, World Business Council for Sustainable Development, National Council for Air and Stream Improvement, Inc. and Intergovernmental Panel on Climate Change are followed to show the total GHG emissions from a small mill. The three simple rules can help the sustainable limits to material and energy throughput (Meadows et al., 2005): •

For a renewable resource, the sustainable rate of use can be no greater than the rate of regeneration of its source.



For a non-renewable resource, such as fossil fuels, the sustainable rate of use should be less than the rate at which a renewable resource can substitute it.



For a pollutant, the sustainable rate of emission can be no greater than the rate at which that pollutant can be recycled, absorbed, or rendered harmless.

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Nordhaus and Boyer (2000) found that a warming of 2.5°C would cause damage of about 2% of GDP while a warming of 6°C would cause damage costs of 10% of GDP. They estimated the cost of future climate damages to around $4 trillion in present value. Andre Barbe has expressed that in 2016; about 30% of US electricity generation came from coal. He stressed that the reduction of coal consumption is a goal of many countries’ energy and environmental policies. He has used a modified version of the GTAP-E model to quantify the effects for a restriction on US coal consumption (Barbe, 2017).

2.4.6 Environment Tax At present there are two classic alternatives for regulating GHG emissions, which are a cap and trade policy, and a carbon tax policy. Cap and trade is a quantity control policy, and carbon tax is a price control policy. The cap and trade system provides a price which is a secondary result of regulating the quantity of GHG emissions. On the other hand, the carbon tax effectively reduces the quantity of GHG emissions which is a secondary result of setting a price (Mohajan, 2012i). Finland first enacted carbon tax in 1990 on fuels, and then Norway, Sweden, and Denmark implemented carbon taxes in 1991 and 1992 (Anderson et al., 2000). Germany implemented an ecological tax on heating fuel, gasoline, natural gas, and electricity in 1999 (IEA, 2007a). Japan enacted a tax on heavy polluting vehicles in 2001, but reduced the tax on low-pollution vehicles to encourage the development and purchase of greener vehicles (IEA, 2007c). In 2001, the UK implemented a climate change levy which adds about 15% to the cost of electricity (IEA, 2007d). Hungary introduced a New Environmental Burden Tariff in 2004 which taxes pollution of the soil, air, and water (IEA, 2007e). Recently the USA has established social cost of carbon (SCC) for analysis of federal regulations (Interagency Working Group on Social Cost of Carbon, IWGSCC, 2010). At present the world carbon trade includes fulfillment markets in the EU, the USA, and New Zealand, representing over $140 billion in traded value and as much as 5 gigatons of emissions per year (Linacre et al., 2011). To enlarge the world carbon trade with proposed markets in Australia and Japan, the international market is projected to reach magnitudes of $2-3 trillion by 2020 (Lazarowicz, 2009; Calel, 2011).

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Gordon Tullock (1967) and David Terkla (1984) were the first who suggested that revenues from environmental taxation could be used to finance reductions in pre-existing taxes. The optimal environmental tax should be less than the marginal environmental damages; since the presence of pre-existing distortionary taxes, increasing the welfare costs is associated with the overall tax code (Mohajan, 2011a). Don Fullerton and Sarah E. West (2002) have expressed that emissions from vehicles pollute air that worsened human health, diminishing visibility and caused global warming. They have shown that if government is applied a tax for engine size and gasoline, and the motorists obey the rules accordingly then environment pollution must be decreased. Arthur Cecil Pigou (1932) has suggested that the environmental tax levels must be at the marginal cost of environmental damage. Fullerton and West (2010), and Islam et al. (2011d) have discussed the taxes on cars and gasoline. They have shown that if government is applied a tax for engine size and gasoline, and the motorists obey the rules accordingly then environment pollution must be decreased. Blundell and Shephard (2011) recently have analyzed in detail about optimal taxation of low income families which will be beneficial to the family and to the government. Bossi et al. (2013) have studied the implementation of the social optimum in a model of habit formation. They have considered taxes that address inefficiencies due to negative consumption externalities, imperfect competition, and self-control problems. David and Sinclair-Desgagn´e (2010), and Hattori (2011) have characterized optimal environmental policy in a case where innovation in clean production technologies is developed and provided by a monopoly. Fullerton and West (2010) have used the example of a proposed carbon permit system to illustrate and has discussed six different types of distributional effects. Himmes and Weber (2011) have analyzed the optimal policy design in a context with more than one externality while taking explicitly into account uncertainty surrounding future emission damage costs. Kaplow (2011) has given the technique to illuminate and has extended Tax by Design’s analysis regarding the value addition tax (VAT), environmental taxation, wealth transfer taxation, and income transfers. Mohajan (2012f) shows that due to environment pollution workers suffer from illness; as a result economic development decreases. Baumol and Oates (1988) have

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proposed that pricing GHG emissions is a fundamental lesson from environmental economics and the theory of externalities.

2.5 Conclusion In this chapter we have aimed to examine the existing literature review on optimization in mathematical economics, social choice theory, NNP and sustainable development, voting system, GHG emissions and climate change, and beneficial environmental taxes. This chapter stresses on social welfare, poverty reduction, and sustained economic growth for the welfare of the global humanity.

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Chapter–III

Research Methodology

3.1 Introduction This chapter presents all the tools and systems used for this dissertation. It discusses the methodological epistemologies (what are known to be true) and approaches that support mathematical economics and social choice research. The research is based on: mathematical modeling of economic analysis in optimizations, social choice and game theory, voting system, environmental pollution, healthcare, and sustainable economy, NNP, social welfare and sustainability, GHG emissions, global warming and climate change that effects on modern economy, and finally green taxes on environment pollution to reduce GHG emissions. This chapter introduces the research strategy and the empirical methods for the general approach, and specific techniques to address the objectives for the research. It also presents the research design and the methods used in the selection of the research participants, and for data collection. Research methodology indicates the logic of development of the process used to generate theory that is procedural framework within which the research is conducted (Remenyi et al., 1998). It provides the principles for organizing, planning, designing, and conducting research. Methodological decisions are determined by the research paradigm that a researcher is following. The research paradigm not only guides the selection of data gathering and analysis methods but also the choice of competing methods of theorizing (Sayer, 1992). This study is based on both primary and secondary data that are collected from various sources. Primary data have been collected from the 500 female garment workers of the slum areas of Chittagong City Corporation (CCC) of Bangladesh by random sampling technique through questionnaire. The secondary data are collected from the

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websites, books and e-books, previous published articles, theses, conference papers, case studies, magazines, and various research reports. Here we have tried to discuss in brief, and clarify how evidence in this study was collected and analyzed, as well as to introduce the research strategy and the empirical techniques applied in this research. The research strategy adopted was face-to-face interview of the garments workers. The mathematical and theoretical data are collected and developed to make this empirical research fruitful. The purpose of this chapter is to introduce the research instruments that we have developed and utilized in the pursuit of our research goals. This chapter presents an overview of issues related to the procedures involved in the study which include research approaches, focus of research, data collection, research design, two criteria for good measurement, ethical reflections, and conclusion.

3.2 Research Approaches A research approach is a plan of action that gives direction to conduct research systematically and efficiently. There are three main research approaches as (Creswell, 2009): i) quantitative (structured) approach, ii) qualitative (unstructured) approach, and iii) mixed methods research. All researches must involve an explicit, disciplined, and systematic approach to find out most appropriate results. Our research falls in the third category (Mohajan, 2018a). Researchers typically select the quantitative approach to respond to research questions requiring numerical data, the qualitative approach for research questions need textural data (Mohajan, 2018b), and the mixed methods approach for research questions want both numerical and textural data (Williams, 2007). The quantitative method supports the positivist paradigm, whereas the qualitative method also very closely supports to the naturalistic paradigm. In the past, quantitative research has been considered the more rigorous of the two, but recently qualitative has gained more credibility in the modern classic research. Both methods are appropriate for conducting research, and each method can contribute greatly to the scientific research. Also both have their strengths and weaknesses, and advantages and disadvantages; so that, ‘neither one is markedly superior to the other in all respects’ (Ackroyd & Hughes, 1992).

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Most of the research involves 4 p’s as: people, problems, programs, and phenomena. In our research we have tried to include all of these to enrich our work. Our study is consists of both positivistic and naturalistic paradigms. We have proved propositions and theorems, and provide mathematical examples to augment the research. In most of the cases of our research we have used mathematics as a tool of our research approach. Accordingly we have obtained our results in term of mathematics. We have also analyzed survey data on about 500 female garments workers of the CCC to develop WTP in the industrial sector of Bangladesh.

3.3 Focus of Research My research has focused on problems in optimization, social choice, voting system, NNP and social welfare, reduction of environment pollutions, sustainable development, GHG emissions, and environment tax. The research strategy has been employed to identify important problems in the research area, and then find accurate solutions of them. The research is conducted both in the field to collect primary data to calculate WTP in the industry to provide medical facilities to the workers to increase productivity, and to collect secondary data to develop models and obtain mathematical solutions. The research works of the thesis focused on the followings: ▪

To provide a mathematical basis for economics and social welfare, including the theoretical arguments for their application in this empirical research.



To find the linkages between mathematics and economics based on sustainable development and social welfare.



To set up a methodological framework to analyze the works for the sustainable development of the global economy.



To present the empirical research results of the green accounting, environment pollutions and behavioral effects of environmental taxes.



To explore how the environment pollution can be reduced for the welfare of humanity.



To find out the ways how the GHG emissions can be reduced.

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3.4 Data Collection Data refer to raw facts without any processing, organizing or analysis, and hence they have little meaning, and few benefits to the managers and decision-makers. They are un-interpreted materials on which a decision is to be based, and depend on facts which may include anything known to be true or exist. They are bits of content in either text or numerical format (sequences of numbers, letters, pictures, etc.). They are meaningless in themselves. They are indicated by a set of ‘discrete intention details about events’. They are normally structured, but do not bear any information to use them in a particular context (Mohajan, 2016). Researchers can identify and use relevant data at the following stages of the data “life cycle” as (Osorio, 2014): i) study concept, indicating key elements, definitions and concepts, ii) data collection, including questionnaires and coding instruments, iii) data processing, containing the data and specifying the content of the information, iv) data archiving, indicating procedures to guarantee the preservation of data and confidentiality, v) data distribution, indicating the terms of use and citation, vi) data analysis, providing replication codes and publications, and vii) data repurposing, indicating the procedures for post hoc harmonization and data transformation. In the thesis we have used both the primary and secondary data to perform the job properly. This study focuses the collection and analysis of only two primary data, and some secondary data that are available from the qualitative and quantitative data collection processes. This thesis analyzes all the chapters with sufficient theoretical analysis and mathematical calculations to make it interesting to the readers. The models contain detail mathematical calculations to reflect the core concepts of the models. The results of the research are presented in some chapters in types of examples, propositions, and theorems with proof. So that future researchers can also develop them by identifying if there are lacks of skill in the thesis.

3.4.1 Primary Data Collection Primary data are collected from the garments workers of the slum areas of the Chittagong City Corporation (CCC) of Bangladesh. Sample size of the study was 500. Furthermore, the surveys data are given in the tables reflect the opinions of the garment

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workers for willing to pay (WTP) system in the industries of Bangladesh. In this study, interviews have been conducted on a one-to-one a single participant, and scientific honesty is maintained for the validity of the data, that is, manipulation of design and methods were not applied in the research. To calculate the results we have used Statistical Package for Social Sciences (SPSS) 17.0. The SPSS is a commercial computer software package that has been used in research since the early 1960s. For the data analysis of the primary data various statistical techniques such as, mean, median, standard deviation, etc., have been used depending upon the requirements. In this survey anonymity and confidentiality were maintained strictly. The anonymity was censured by not disclosing the participants’ name on the questionnaire and research reports. Confidentiality means that the information they provide will not be publicly reported in a way which identifies them. Finally, we can demand that ethical standards, reliability and validity of the data collection and statistical analysis were followed for the better result. We have divided the sample into two sub-samples A and B. Sub-sample A, is asked about their WTP to avoid one or more additional days of symptoms for the next 12 months in the light of experience obtained in the last 12 months. Sub-sample B, is asked about their WTP to avoid 14 additional days of symptoms for the next 12 months.

3.4.2 Secondary Data Collection At present, a lot of secondary data are being collected and archived by researchers all over the world for research that are becoming more widespread (Andrews et al., 2012). Secondary data are collected by someone else for his primary research purposes which provide basic research principles. The researchers who have limited time and resources, they can use the secondary data for their researches. For the collection of secondary data we have used both published and unpublished data sources. The published data are collected from: i) various publications of the federal or local governments (e.g., census reports, annual reports and financial statements of companies, statistical statement, reports of government departments), ii) various publications of foreign governments or of international bodies and their subsidiary organizations (e.g., UNO, IMF, World Bank, ILO, WHO, etc.), iii) various research reports are prepared by research scholars, universities, economists, etc., in different

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fields, iv) books of various authors, magazines, and newspapers, v) various sources from university libraries, vi) technical and trade journals, vii) websites, and viii) public records and statistics, historical documents, and other sources of published information. The unpublished data are collected from many sources. They are found in diaries, letters, unpublished biographies and autobiographies, and also from scholars and research workers, trade associations, labor bureaus, and other public/private individuals and organizations. Secondary data are classified as ‘internal or external’ in terms of its source. Internal secondary data are information acquired within the organization where research is being carried out. On the other hand, external secondary data are obtained from outside sources. The two major advantages of using secondary data in the research are time and cost savings. We have studied research works of various scholars in details during our research works. We have tried our best to present every chapter in some detailed mathematical techniques with some new concepts. The analysis was undertaken to develop the skill of the techniques of mathematics in economics and social choice. In addition, this thesis includes two appendices to provide full satisfaction when one goes through the related topics. The major advantages of analysis of secondary data are the cost effectiveness and convenience it provides. When good secondary data are available, researchers can utilize them for high quality empirical researches. These provide researchers with opportunities to work effectively to test new ideas, theories, frameworks, and models of research design (Smith, 2008).

3.5 Research Design The research design is the conceptual structure within which the research is conducted; it constitutes the blueprint for the collection, measurement and analysis of data. So the research design can be defined as a master plan for the determined methods, structure, and strategy of a research to find out alternative tools to solve the problems, and to minimize the variances (Kothari, 2004). Research design ‘deals with a logical problem and not a logistical problem’ (Yin, 2003).

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It optimizes the validity of data for a given research problem. Research design is the overall configuration of a piece of research to gather good results from the collected data. We have used SPSS 17.0 to find our results of the survey data. In our study we have introduced some propositions, and prove them with mathematical procedures. We also provide sufficient examples to make the study easier to the readers. We have also displayed diagrams to describe the theoretical analysis and mathematical procedures efficiently. This mathematical economic research provides various aspects of economic theories, methods, and analysis to present in a coherent, logical, reliable, and useful manner. We have divided our research into seven sections as follows:

3.5.1 Mathematical Techniques in Optimization Optimization is defined as an act, process, or methodology of finding fully perfect, functional, or effective as possible the best solution that can be applied to all the quantifiable problems. The general optimization problem consists of finding minimum cost or maximum profit of a quantified parameter, objective function, by varying design variables under given design constraints. Selection of an optimization method for a given problem depends on the following considerations (Wetter, 2000): ▪

structure of the objective function (linear, non-linear, convex, continuous, number of local minima or maxima, etc.),



availability of analytic first and second order derivatives,



number of design variables, and



design constraints. In our optimization problem we have used the method of Lagrange multipliers in

multivariable calculus, and have been used to facilitate the determination of necessary conditions which is considered as a device for transferring a constrained problem to a higher dimensional unconstrained problem. We have used sufficient conditions for implicit functions by considering the determinant of Jacobian matrix is negative, and the determinant of the Hessian matrix is positive. We have also used the above techniques to explain cost minimization of a competitive firm, output maximization of an agency subject to both linear and nonlinear constraints, and utility maximization subject to multiple constraints, such as, Cobb-Douglas production function. We have used twelve

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comparative statics to predict the situation of production and consumption. We have introduced three explicit models which satisfy the optimization conditions. In our study of optimization we have used detail mathematical analysis which satisfies necessary and sufficient conditions mentioned above.

3.5.2 Social Choice and Game Theory Arrow’s theorem shows that it is impossible for a social welfare function to satisfy five conditions namely: i) Completeness and Transitivity, ii) Universality, iii) Pareto Consistency, iv) Independence of Irrelevant Alternatives (IIA), and v) Non-dictatorship simultaneously (Arrow, 1963). By the Arrow’s theorem we have shown in the study that there is a flaw in democratic vote to reflect the preferences of all the individuals in the society. In this study we have shown both combinatorial approach and geometrical approach to Arrow’s Theorem with mathematical analysis, and displaying a geometrical diagram. We have also discussed two simple versions of Arrow’s Theorem for singleprofile case only. To analyze these we have proved some propositions and theorems. We have used the ‘Battle of Sexes’ game theoretic model to show how political institutions can be formed to elect democratic leaders, and how a patriotic leader can serve the country efficiently. We have tried to show the different aspects of unitary and federal democracies with their benefits and drawbacks by the use of mathematical techniques. In a present dangerous perturbed world the main victim is innocent common people. We use Prisoners’ Dilemma game to describe two rival countries’ strategy for establishing a peaceful society in the world. We have shown that the war gives only violence and destruction. For example, Psychopathic militarists like Hitler become a threat to our civilization only when ordinary rational people become motivated to support them as leaders.

3.5.3 Research on Voting Procedures In voting system every voter’s preference ordering, taken collectively, form the input, the output is usually a single certain winner or a set of winners. Here we have discussed voting system, and manipulation of voting. We have discussed Condorcet

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method and Borda voting in brief. The single transferable vote (STV) is a system of preferential voting designed to minimize wasted votes. We have analyzed Droop Quota of STV, and tie-breaking in STV in mathematical model. We have discussed strategies under approval voting (AV) by a series of propositions with proof and examples. In median voter model we have analyzed along with single-peakedness and single-crossing properties with the illustrative examples, and displaying diagrams. Majority judgment voting and the majority count of Borda and their tie-breaking are discussed with providing examples and propositions.

3.5.4 Environmental Pollution and Healthcare In our environmental economic model we consider the production of goods and services where we require labor, manufactured capital, and natural resources; and have stressed on the performance of environment policy, and sustainable use of natural capital (wetlands, lakes, forests, agricultural landscape, and coastal water). We have used mathematical techniques to obtain our results of the study. We have tried to highlight on environmental accounting and roles of economics by the sustainable development of wealth; and then have calculated green national accounts. We have calculated that environment pollution decreases economic development. We have developed the three terms of equation (8.25) in our own procedure. Healthy persons are the human asset of a country. We have discussed a mathematical model on the health impacts from air pollution in Bangladesh. We have also surveyed on about 500 female garments workers of Bangladesh to establish WTP for the improvement of the labor sector of the country. Finally, we have suggested the ways to develop the future labor sector of Bangladesh.

3.5.5 Research on Sustainable Development Sustainability is a pioneer policy for the governments, international organizations, and corporations for the concerns over the climate change, environment degradation, and economic instability. The NNP is an important item for a country, and it represents the maximized value of flow of goods and services that are produced by the productive assets of the society. It is important to investigate whether the concept of NNP can serve as an indicator of sustainability (Weitzman, 1976). We have used the mathematical tools to

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furnish this portion. Here we have mathematically shown that the real NNP in sustainable development by the optimal growth in the society, and growth without optimality. We have tried to show the green NNP for the sustainability and social welfare by using the mathematical relations on welfare equivalence income, sustainable income, net social profit, and wealth equivalence income. We have established following relations with sufficient mathematical calculations: i) between green NNP and wealth equivalent income, ii) between green NNP and net social profit, iii) between green NNP and welfare equivalent income, and iv) between green NNP and sustainable income. Then we have discussed global sustainable development in terms of global green economy, sustainable economy, showing discrimination in modern economy. Later, we have tried to find present unsustainable development practices and the recent economy of the world. Finally, we have tried to discuss in brief the open economy of Bangladesh. We have provided propositions to understand the effects of NNP for the sustainable social welfare clearly.

3.5.6 GHG Emissions and Reduction Procedures A major part of the current world energy supply comes from fossil fuels: oil, coal, and natural gas. As a result, these non-renewable resources are being rapidly depleted. Due to their increased consumption; driven by economic and population growth in the dominating energy markets, it is estimated that the only fossil fuel remaining after 2042 will be coal (Raj & Singh, 2012). On the other hand, it is also widely accepted that the burning of fossil fuels releases GHGs which contribute to the global warming, and overall have a negative impact on the environment (IPCC, 2007). The current concentrations of GHG in space have increased since 1750 (Industrial Revolution) due to human activities from a CO2 equivalent of 280 ppm (parts per million) to 450 ppm, but proposed boundary is 350 ppm (Stern, 2007; Mohajan, 2015b). In this portion we have used theoretical analysis, and the statistical results of secondary data that are collected from the various survey data of other scholars. Actually we have not done any experiment to show the effect of GHG emissions. In our study we have stressed on the GHG emissions of the USA and China. Because, these two countries are in the leading position in GHG emissions, together emit about 40% of global CO2 emissions, and about

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35% of total GHGs. We have tried to show the effects of GHG emissions, and the ultimate fate of the living organisms. We have also shown the reduction policies of global GHG emissions. We have demanded that recent increase of natural calamities and climate change is due to global warming that is due to GHG emissions. These over natural calamities already diminished the global economic development and social welfare, and will also sloth the sustainable economic improvement. The production of renewable fuel should be increased to reduce GHG emissions.

3.5.7 Green Taxes on Environment Pollution In the last part of the 20th century and in the beginning of the 21st century the area of cities of the world has expanded, and new cities and towns have grown rapidly. As a result in vehicle-miles traveled increases. Emissions from vehicles pollute air that worsened human health, diminishing visibility, and caused global warming (Fullerton & West, 2002). Actual vehicle emissions depend not only on vehicle size and age, but also on qualities of the fuel, maintenance of the car’s pollution control equipment (PCE), frequency of cold start-ups, temperature of the air, speed of the vehicle, and aggressive driving (Fullerton & West, 2002). For regulating GHG emissions, there are two policies; a cap and trade policy, and a carbon tax policy. Cap and trade is a quantity control policy, and carbon tax is a price control policy. Price increases through energy or carbon tax would be necessary to limit energy demand, and increase of GHG emissions (Stern, 2007). We have given a brief discussion on these two policies. In the two mathematical models we have calculated taxes on car and gasoline; one for homogenous consumers, and the other for heterogeneous consumers. We also show a detail mathematical analysis on second-best taxes on gasoline and size. Yet Bangladesh has not imposed emission taxes on vehicles properly. We have tried to encourage Government of Bangladesh to develop vehicle tax rates of Bangladesh to reduce environment pollution in the cities of this country. We have shown optimal environmental taxes due to health effect, and beneficial sides of environmental tax; and finally, optimal environmental tax to reduce GHG emissions, and to create a healthy environ in the society.

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3.6 Two Criteria for Good Measurement Measurement is the assigning of numbers to observations in order to quantify phenomena. There are two criteria for evaluating measurements in research: i) validity, and ii) reliability. These two criteria are the most important and fundamental characteristics of all researches. A research becomes good or poor depending, respectively, on the strength or weakness of these two aspects (Brink, 1993). Before and after collecting the data, the researchers need to consider the validity and reliability of their data to do a good research. In our study we have tried to create reliable and valid tests and questionnaires in order to enhance the accuracy of their assessment and evaluations. When we compare reliability and validity, latter is more essential. If a research is not valid, it is of course not reliable. But, to enrich our research we will try to show that our study will be both reliable and valid.

3.6.1 Validity in Research Validity in research is alarmed with the accuracy and truthfulness of scientific findings (Le Comple & Goetz, 1982). It is a very important feature in a measuring instrument. It refers to the methodological soundness or the appropriateness of the instruments used (Hashim et al., 2007). It is the ability of a measure to measure what is supposed to measure (Robson, 2011). It indicates that how well the data collection, and data analysis of the research captures the reality being studied (Mohajan, 2017c). An important aspect of any research should always be the degree of validity present in the procedures and conclusions (Graziano & Raulin, 2006). According to Burns (1999) “Validity is an essential criterion for evaluating the quality and acceptability of research.” According to Bond (2003) research validity is “Foremost on the mind of those developing measures and that genuine scientific measurement is foremost in the minds of those who seek valid outcomes from assessment.” Campbell and Stanley (1963) have defined two major forms of validity that encompass the many types: i) internal validity, and ii) external validity. The both validities are important to the overall validity of the study. Internal validity refers to whether or not the manipulation of the independent variable really makes a true reflection

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or representation of reality on the dependent variable. In brief, it refers to whether a study can be replicated (Willis, 2007). It is the extent to which factors influencing a true reflection of reality rather than the result of the effects of extraneous or chance variables, not necessarily related to factors influencing contraceptive non-utilization. It is mainly concerned with the congruence of the research findings with the reality. It also deals with the degree to which the researcher observes and measures what is supposed to be measured. External validity refers to the degree or extent to which representations or reflections of reality are legitimately applicable across groups. It is concerned with the applicability of the findings in other settings or with other subjects outside of the sample (Graziano & Raulin, 2006). We not only want to the findings to be due to the intervention, but we would also like to generalize those findings to a larger population (Wilson, 2010). In our research primary data are collected in random questionnaires basis, and anonymity and confidentiality are strictly followed. We have tried to represent the true information for the data collection. Directly after the interviews we have compiled the data from interviews, and transformed them into precious information. The validity of the result has been discussed with my supervisors with valuable feedback. The secondary data are verified with mathematical calculations, illustrative examples, and propositions with proof. Most of the secondary data are collected from scholarly sources or reliable news sources, improving the accuracy and trustworthiness of this study. Therefore, we hope that in our study the results seem to us have of high validity.

3.6.2 Reliability in Research One of the main requirements of any research process is the reliability of the data and findings. Reliability deals with the consistency, dependability, and replicability of the results of any research. It is an important concept in research because; it can be used to reduce errors during the analysis of responses to questionnaires (Neuman, 2012). It indicates that the scores of an instrument are stable and consistent (Creswell, 2009). Reliability coefficients range from 0 to 1, with higher coefficients indicates higher levels of reliability (Traub & Rowley, 1991). Reliable data are dependable, genuine, trustworthy, sure, unfailing, authentic, and reputable (Mohajan, 2017c).

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Reliability is the strength of the quantitative research. It is the degree to which measures are free from error, and therefore, yield consistently the same results over repeated testing periods. It means that the operation of a study, such as, the data collection procedures, can be repeated with the same results every time. Therefore, it is concerned with the consistency, stability and repeatability of the informant’s accounts as well as the researchers’ ability to collect and record information accurately (Seltiz et al., 1976). Hence, the researchers can develop scoring results by reliability to reduce measurement errors. For example, a tailor measuring fabric with a tape measure obtains a true value of the fabric’s length. If he takes repeated measures of the fabric, and each time comes up with the same length, it is assumed that the tape measure is reliable. Therefore, reliability is the steadiness, constancy, and sureness of a measurement tool. We can assure that reliability is maintained throughout our research. Any researcher can depend on our works for the future research. Many qualitative researchers avoid the terms validity and reliability, and use terms such as credibility, trustworthiness, truth, value, applicability, consistency and conformability for evaluating the scientific merit of qualitative research (Leininger, 1991). Neutrality and trustworthiness in research increase the reliability and validity (Golafshani, 2003). Reliability does not imply validity. For example, while there are many reliable tests of specific abilities, not all of them would be valid for predicting. Reliability is a necessary, but not sufficient item of a research. For a test to be reliable, it also needs to be valid. For example, if a scale is off by 10 kg, it reads the weight every day with an excess of 10 kg. The scale is reliable because it consistently reports the same weight every day, but it is not valid because it adds 10 kg of true weight. It is not a valid measure of the weight. Reliability may be improved by clarity of expression, lengthening the measure, and other informal means. Psychometric analysis is considered the most effective way to increase reliability (Cortina, 1993). It is possible to have reliability without validity, but it is logically impossible to demonstrate that an unreliable test is valid.

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However, tests that are reliable are not always valid. For example, let a thermometer was a degree off. It would be reliable by giving the same results each time, but not valid because, the thermometer was not recording the correct temperature. Because of the lack of time, and difficulties of collecting primary data, we have not had a possibility to conduct the survey study more than once. Therefore, it is difficult to draw any conclusions on the reliability of our survey results on the garments workers of CCC. But, in the mathematical calculations to establish propositions, prove theorems, provide examples, display diagrams, and prepare tables we have studied them several times. We hope that these studies are more reliable. Throughout this thesis a consistent and conscious effort was made to ensure that a high level of reliability was accomplished.

3.6.3 Threats to Validity and Reliability The multiple factors can create risks to the validity and reliability of the findings of a researcher. Error is one of them. Researchers thus must be careful of the sources of errors in plans and implementation of their studies. The major sources of research errors can be obtained from the careless of researcher, the subjects participating in the study, the social context, and the methods of data collection and analysis. Errors of measurement that affect reliability are random errors, and errors of measurement that affect validity are systematic or constant errors. Threats to research reliability and validity can never be totally eliminated, so a researcher needs to try his best to minimize the threats as much as possible (Mohajan, 2017c).

3.7 Ethical Reflections According to W. L. Neuman (2012) “Ethics begins and ends with you, the researcher”. Ethic is an important characteristic in any research. Some general ethical issues in research that result in some prohibitions are (Neuman, 2012): •

never cause unnecessary or irreversible harm to participants,



secure prior voluntary consent when possible and never unnecessarily humiliate,



degrade, or release harmful information about specific individuals that was collected for research purposes.

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In the study ethics are maintained by keeping the answers acquired strictly confidential. We have taken prior permission from the respondents before conducting the research, and no false information was given in the research. In the theoretical analysis we have given proper references in the research. We have maintained the ethical formalities throughout our research.

3.8 Conclusion This chapter has provided an outline and description of the research methodology undertaken in the thesis, and how that is used to draw up the specific research plan for this study. In this chapter, we have presented the methodology that is used throughout the thesis. In this chapter we have mentioned the methodology section-wise that is used in our study. We have also discussed the choices of research methods, and validity and reliability of the thesis.

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Chapter–IV

Elementary Discussions

4.1 Introduction In this chapter we have discussed some definitions (with diagrams where necessary). The definitions are very elementary, and will be used later throughout the thesis. The definitions are collected from the various references given both in the text and reference lists.

4.2 Some Related Definitions

4.2.1 Preference Relations We consider the 3-dimensional Euclidean space in which a typical point P is denoted by x = P(x1 , x2 , x3 ) , where x1 , x 2 and x3 are real numbers. We can define a scalar

product

between

x.y = x1 y1 + x2 y2 + x3 y3

and

two

vectors

the

x = (x1 , x2 , x3 ) and

3-dimensional

plane

is

y = ( y1 , y 2 , y3 )

as

represented

by

p1x1 + p2 x2 + p3 x3 = k , where the vector p = ( p1, p2 , p3 ) is perpendicular to the plane

• x = P(x1 , x2 , x3 )

x3

O

x2 x1 Figure 4.1: Three-dimensional Euclidean space.

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(Figure 4.1). We can write, xPy if x is strictly prefer to y, xIy if x is indifferent to y and xRy to mean that either x is preferred to y or x is indifferent to y, so that y is not preferred to x (Mohajan, 2017b).

4.2.2 Convex and Non-Convex Sets A convex set is defined as a set C which is such that, if x and x are in C, so are all the vectors of the form t x + (1 – t) x with 0  t  1 ; otherwise it is called a non-convex set. A non-convex set cannot be a preferred set (Mohajan, 2017b).

4.2.3 Utility Function If u i (x ) and ui ( y ) be the numerical values of utility of an individual i to the alternatives x, y  Y then ui (x )  ui ( y ) implies xRi y . In preference relation we can write u(x) > u (y)  xPy (Cassels, 1981).

4.2.4 Indifference Curves and Hypersurfaces The curves in the x1 x 2 -plane are given by; u(x1 , x2 ) = c , u(x1 , x2 ) = c1 ,

u(x1 , x2 ) = c2

where 0  c  c1  c2 (say). Here, x1 x2 = c , x1 x2 = c1 , x1 x2 = c2 are

rectangular hyperbolae (Figure 4.2) (Mohajan, 2017b). x1 x2 = c2

x2 x1 x2 = c

x1 x2 = c1 x1

O

Figure 4.2: The rectangular hyperbolae lying in the positive quadrant with

0  c  c1  c2 .

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In 3-dimensional case x1 x 2 x3 = c , x1 x 2 x3 = c1 ,

x1 x2 x3 = c2 are called rectangular

hyperboloid. The individual is indifferent to the bundles represented by points on the same curve. These types of curves are called indifference curves which do not intersect each others. By a hypersurface we mean the set of points in the n-dimensional Euclidean space R n for which

f(x) = constant. For different values of the constant, we find

corresponding different hypersurfaces.

4.2.5 Price Vector and Budget Constraint Let p1 be the cost of 1 kg of rice, and p2 be the cost of 1 kg of wheat in dollar. We call p = ( p1, p2 ) the price vector of possible bundles of rice and wheat. The total cost of the bundle x1 , x 2 is, p1 x1 + p2 x2 = p.x . For bundle x with a price vector p let us consider one has maximum c amount of dollars to spend, then we can write, p. x  c ; ( p. x is the price of the bundle x) which is referred to as budget constraint (Mohajan,

2017b).

4.3 Optimization, Social Choice and Pareto Optimality

4.3.1 Optimization Let us consider a function f(x), where x = (x1, x2 ,..., xn ) . For a function f(x) to be 2

optimum (maximum or minimum)

d f df  0 at x = x0 the function is = f ( x ) = 0 . If dx dx 2

d2 f  0 at x = x0 the function is minimum at a maximum at a point x = x0 , and if dx 2 point x = x0 (chapters V, VIII, and XI). If f (x, y ) be a function of two variables x and y then

for

optimum

f (i.e. f x ) = 0 = f (i.e., f y ) x y

,

and

f xx f yy − f xy2  0.

If

f xx  0 (and f yy  0 ) , then the function has a minimum point, if f xx  0 (and f yy  0 ) then the function has a maximum point. For f xx f yy − f xy2  0 , there is neither a maximum nor a minimum, but a saddle point. In all cases, the tangent plane at the extremum

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(maximum or minimum) or a saddle point to the surface z = f (x, y ) , is parallel to the zplane (Islam et al., 2010, 2011c; Moolio et al., 2009) .

4.3.2 Saddle Point A saddle point for a smooth function is a stationary point such that the curve in the neighborhood of that point is not entirely on any side of the tangent space at that point (Figure 4.3). In one dimension, a saddle point is a point which is both a stationary point

Figure 4.3: The plot of y = x3 with a saddle point at 0.

and a point of inflection. Since it is a point of inflection, it is not a local extremum. For example, for the function y = x3 there is a point of inflection at x = 0 and it is a saddle point and it is not a local extremum.

4.3.3 Necessary and Sufficient Conditions If A→B and B→A, then we often say “A if and only if B”, or “A is necessary for B and B is sufficient for A”. Let us consider an example (Johnson, 2012); •

To be a lawyer, it is necessary to have attended law school.



To be a lawyer, it is sufficient to have passed the bar examination.

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So a law degree is a necessary condition to be a lawyer, but not sufficient. On the other hand, if someone has passed the bar examination, he is allowed to represent clients, so it is a sufficient condition to be a lawyer.

4.3.4 The Implicit Function Theorem Suppose f (x0 , c0 ) = 0 and

f (x0 , c0 )  0 (Johnson, 2012). Then there exists a x

continuous implicit solution x(c), where c is some parameter, with derivative

x(c ) f ( x(c ), c ) =− c for c close to c0. x f x ( x(c ), c )

4.3.5 Comparative Static Analysis In the society the behavior of the buyers and sellers often changes, which causes the shift of demand and supply curves to itself over time. In economics it is important to analyze how these shifts affect equilibrium. This analysis is called comparative static analysis. For example, if P be the price of a commodity X, then

X  0 indicates that if P

the price of commodity X increases, the level of consumption of X will decrease. In chapter V we have used twelve comparative statics to predict economic situations (Islam et al., 2011c).

4.3.6 Shadow Price The shadow price of a commodity is defined as its social opportunity cost, i.e., the net loss (gain) associated with having 1 unit less (more) of it. In chapter V we have shown that Lagrange multiplier is a shadow price. For example, if

C =  , then if the Q

firm wants to increase (decrease) 1 unit of its production, it would cause total cost to increase (decrease) by approximately  units.

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4.3.7 The Pareto Criterion An alternative x belongs to the set X will be described as Pareto-optimal if there is no other alternative in the set which Pareto-wise better than x. Most of the modern welfare economics has been based on Pareto optimality (Islam et al., 2009b). Weak Pareto Principle: For all x, y  X if xPi y for all i, then xPy. Strong Pareto Principle: For all x, y  X if xRi y for all i, and xPi y for some i, then xPy.

4.3.8 Social Welfare Function (SWF) The definition of SWF by Arrow is a collective choice rule (CCR) that specifies orderings for the society is called a social welfare function F. An SWF is a rule such that each social CCR such that each social preference that is determined is an ordering. In SWF there must be no dictator.

4.3.9 Social Choice Function and Monotonic Function Let, N = 1,2..., n be the set of individual voters, and let Y = x, y, z,... be the complete and transitive finite set of alternatives. Let, L(Y ) denotes the set of strict transitive ordering of the alternatives in Y, and L(Y ) denotes the set of profiles of such N

preference orderings, one for each individual voter. A function f : L(Y ) → Y will be N

called a social choice function. A social choice function f is monotonic if whenever

f (L1 ,..., LN ) = x for any alternative x and for every individual i, and every alternative y the ranking Li ranks x above y if Li does, then f (L1,..., LN ) = x (Islam et al., 2009a).

4.4 Game Theory

4.4.1 Battle of Sexes Game In game theory, ‘Battle of the Sexes’ is a 2-player coordination game. The husband would most of all like to go to the football game. The wife would like to go to the opera. Both would prefer to go to the same place rather than different ones (Fudenberg & Tirole,

73

1991). Since the couple wants to spend time together, if they go separate ways, they will receive no utility (0, 0). If they go either the opera or a football match, both will receive

Table 4.1: Battle of Sexes game. Husband

Opera 5, 3

Opera 5, 3

Football 0, 0

Wife

0, 0

0, 0

3, 5

some utility form the fact that they are together, but one of them will actually enjoy the activity. The description of this game in strategic form is given as Table 4.1.

4.4.2 Prisoner Dilemma Game Two members A and B of a criminal-gang are arrested and imprisoned. Each suspect is interrogated separately, and no communication is allowed between the two suspects. Each prisoner is given the opportunity either to betray or to cooperate with the other by remaining silent. The offer is as follows (Fudenberg & Tirole, 1991): •

If A and B each confess the other, each of them serves 2 years in prison.



If A confesses B but B remains silent, A will be set free and B will serve 5 years in prison and vice-versa.



If both A and B remain silent, both of them will only serve 1 year in prison (on the lesser charge).

The payoff matrix is given in Table 4.2. The concept of prisoners’ dilemma can be used to analyze price and non-price competition in oligopolistic markets, as well as the

Table 4.2: The payoff matrix for suspects.

Suspect A

Suspect B Confess Silence Confess 2, 2 0, 5 Silence 5, 0 1, 1

incentive to cheat in a cartel. Oligopolistic price competition in the presence of the prisoners’ dilemma can be examined with the payoff matrix in Table 4.3. The payoff

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matrix of Table 4.3 shows that if both firm A and B charge a low price, each of them would earn a profit of 3. If A charge high price and B charge low price, A will earn 1 and B will earn 6 and vice-versa. If both A and B charge high price, both of them will earn 4.

Table 4.3: The payoff matrix for pricing.

Firm A

Firm B Low price 3, 3 1, 6

Low price High price

High price 6, 1 4, 4

4.4.3 Maximin and Minimax In decision theory, statistics, philosophy and game theory maximin is the best decision in which one whose worst outcome is at least as good as the worst outcome of any other decisions. Maximin is a term commonly used for non-zero-sum games to describe the strategy which maximizes one’s own minimum payoff. Minimax is used in zero-sum games to denote minimizing the opponent’s maximum payoff. The minimax values are also very important in the theory of repeated games (Wikipedia, the Free Encyclopedia).

4.5 Environmental Economics, Social Welfare and Sustainability The welfare is increasing instantaneously over time if and only if real net national product (NNP) is increasing instantaneously over time. World Commission (1987) defined sustainable development as follows: “Sustainable development is an economic program in which, lightly speaking, the wellbeing of future generations is not jeopardized.” Again we can define sustainability as follows (Arrow et al., 2012; Dasgupta, 2007, 2010; Mohajan et al., 2012): “Sustainable development is an economic program along which average wellbeing of present and future generations, taken together, does not decline over time.” An economic development is sustainable if

dU  0 , where U is the utility is obtained dt

from consumption of good (Dasgupta & Mäler, 2000, 2001).

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4.5.1 Externalities In economics, in the course of producing and consuming, harmful or beneficial side effects that are borne by firms and people not directly involved in the production and consumption. These side effects are called externalities. Externalities are called external costs when they are harmful and external benefits when they are beneficial. Air pollution from motor vehicles is an example of a negative externality (Buchanan & Stubblebine, 1962).

4.5.2 Nominal Price and Real Price Estimated price of an item that may bear little or no relation to its market price, and which is quoted to initiate a negotiation or transaction. Nominal price is used where both the recent market price has not been established or where demand and supply situation makes the market-price uncertain (Sen, 1979). Nominal price is adjusted for inflation. Real price is obtained by removing the effect of price level changes from the nominal price of time-series data, so as to obtain a truer picture of economic trends.

4.5.3 Gross Domestic Product Gross Domestic Product (GDP) is the best way to measure a country’s economy. It includes everything produced by all the people and companies that are in the country (Coyle, 2014).

4.5.4 Nominal and Real GDP In 2013, the US GDP was $17.09 trillion. This is known as nominal GDP, which is the raw measurement that leaves price increases in the estimate. To compare GDP from one year to another, it is important to take out the effects of inflation. To do this, the Bureau of Economic Analysis (BEA) calculates real GDP (Wikipedia, the Free Encyclopedia): Income from the US companies and people from outside the country are not included, so the impact of exchange rate and trade policies do not muddy up the number. The effects of inflation are taken out.

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Only the final product is counted, so that if someone in the US makes shoelaces, and it is used to make shoes in the US, only the value of the shoe gets counted. Real GDP is, therefore, lower than nominal (Coyle, 2014).

4.5.5 Gross National Product Gross National Product (GNP) is the total value of all final goods and services produced annually by a nation during a given period (usually 1 year) plus income earned by its citizens (including income of those located abroad), minus income of non-residents located in that country (Coyle, 2014). GNP is calculated as follows (Wikipedia, the Free Encyclopedia): GNP = GDP + NR (Net income inflow from assets abroad or Net Income Receipts) – NP (Net payment outflow to foreign assets).

4.5.6 Net National Product Net National Product (NNP) is calculated as follows: NNP = GNP–Depreciation.

4.6 Global Warming and Environment Taxes The world has realized that global warming is continually increasing due to greenhouse gas (GHG) emissions. The living organisms are in dangerous position and some species has already extinct and will extinct in future if global warming cannot be controlled. It is clear to environment experts of all nations that emissions of carbon dioxide (CO2) and other GHGs are liable to global warming (Mohajan, 2011c).

4.6.1 Deadweight Loss In economics, a deadweight loss (DWL) is a loss of economic efficiency that can occur when equilibrium for a good or service is not Pareto optimal. DWL occurs when supply and demand are not in equilibrium and total society welfare is not maximized. Causes of DWL can include monopoly pricing, externalities, taxes or subsidies, and binding price ceiling or floors (Case & Fair, 1999).

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4.6.2 Pigovian Tax Pigovian tax is a special tax which is often levied on companies that pollute the environment or create excess social costs, called negative externalities, through business practices. Pigovian tax is applicable only because market economies often fail to provide a proper incentive to reduce negative externalities (Pigou, 1932).

4.7 Conclusion In this chapter we have given some definitions which are very easy and will be used in the subsequent chapters of this thesis. The readers who are new in this field will be benefited more from this chapter. Here we have not done any new work and definitions are collected from the references given both in the text and reference list.

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Chapter–V

Optimization in Economics with Lagrange Multiplier 5.1 Introduction The method of Lagrange multipliers is a very useful and powerful technique in multivariable calculus, and has been used to facilitate the determination of necessary conditions; normally, this method was considered as device for transferring a constrained problem to a higher dimensional unconstrained problem (Baxley & Moorhouse, 1984; Islam et al., 2010, 2011c). We examine a set of related examples to highlight the following features (Baxley & Moorhouse, 1984): To begin with, functions are not explicitly given but they have some assumed characteristic features, which are meaningful for and give insight into economic behavior. Later, explicit functions are considered to clarify the characteristics. Assuming, for example, that a firm wishes to minimize the cost of producing a given output, one may want to know how changes in the input prices will affect the situation. So the problem is not: “find the minimum”, but, “assuming the minimum is obtained, what consequences can be deduced”. The Lagrange multipliers  or i , (i =1, …, m ) for some m >1, as indicated, have usually been used as a device. In economic problems, as we shall see, the Lagrange multipliers can be interpreted as rates of change of optimal values relative to some parameters. In these considerations and discussions the Implicit Function Theorem is important for solving a system of non-linear equations for the endogenous variables and calculating partial derivatives of these variables with respect to the exogenous variables.

5.2 Three Examples on Optimization Assume that an individual consumes two commodities x and y; the amounts he purchases in the market place are X and Y kg respectively. He keeps a certain quantity L

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of his leisure time (Baxley & Moorhouse, 1984). Let P1 and P2 be the prices of per unit of x and y respectively, and let T be the total time period available. Let his wage per unit time be w, so that his total income is (T − L) w . His budget constraint is;

(T − L) w = XP1 + YP2 .

(5.1)

The utility U of the individual is given by a utility function u unique to him, as a function of X, Y, and L;

U = u ( X ,Y , L ) .

(5.2)

We now impose certain general and reasonable conditions on the function u as follows (where a subscript denotes partial derivative with respect to the subscript) (Mohajan et al., 2013):

uX  0,

uY  0,

uL  0 ,

(5.3a)

uXX  0,

uYY  0,

uLL  0 ,

(5.3b)

uYL  0 ,

(5.3c)

uXL  0, either

uXY  0

or

uXY  0

uXY = 0 .

or

(5.3d)

We now formulate the maximization problem for the utility function u given by (5.2) in terms of a single Lagrange multiplier  , by defining the Lagrange function as follows:

v ( X , Y , L,  ) = u ( X , Y , L ) +  ((T − L) w − XP1 − YP2 ).

(5.4)

Maximization of utility occurs for values X *, Y *, L*, * of X, Y, L,  that must satisfy the following equations:

v = (T − L ) w − XP1 − YP2 = 0 ,

(5.5a)

vX = uX −  P1 = 0 ,

(5.5b)

vY = uY −  P2 = 0 ,

(5.5c)

vL = uL −  w = 0 .

(5.5d)

Mathematically we can write twelve partial derivatives as follows:

 X P1 YP1 LP1   X P2 YP2 LP2 X Y L w  w w

P  1



P  . 2

w 

80

(5.6)

These twelve partial derivatives are called the comparative statics of the problem. Now we introduce three explicit models (A), (B), and (C) as follows (Mohajan, 2017a): 5.2.1 Model (A): u XY  0 Consider the function u is given by; u ( X,Y,L ) = u0 X aY b Lc ,

(5.7)

where u0 , a, b, c are constants. Taking partial derivatives we get, u X = u0 aX a −1Y b Lc , uY = u0bX aY b −1Lc , u L = u0cX aY b Lc −1

(5.8a)

u XX = u0 a(a − 1)X a − 2Y b Lc , uYY = u0b(b − 1)X aY b − 2 Lc ,

u LL = u0c (c − 1)X aY b Lc − 2

(5.8b)

u XL = u0 acX a −1Y b Lc −1 , uYL = u0bcX aY b −1Lc −1 , u XY = u0 abX a −1Y b −1Lc .

(5.8c)

If we now assume the constants a, b, and c to satisfy the following inequalities: 0 < a < 1,

0 < b < 1,

0 < c< 1,

(5.9)

and assume X, Y, L to be positive, as is required by the nature of the problem, we readily see that the conditions (5.3 a, b, c) are satisfied, and also the first condition in (5.3d) (Mohajan, 2017a). 5.2.2 Model (B): u XY  0 Consider the function u is given by,

((

)

)

u ( X,Y,L ) = u0 A 1 − e − aX − bY + CXYe − f ( L ) ,

(5.10)

where u0 , a, b, A, and C are positive constants, and f (L ) is a function of L, and is given by;

f (L) = c(L0 + L) , −1

(5.10a)

with c, L0 positive constants which are distinct to those of Model (A). Taking partial derivatives of (5.10) we get (Mohajan et al., 2013);

( = u (Abe

) ( ) ),

u X = u0 Aae− aX − bY + CYe − f ( L ) , uY

0

− aX − bY

+ CXe− f

L

uL = u0Cc(L0 + L) XYe− f (L ) , −2

81

(5.11a) (5.11b) (5.11c)

with f (L) = df dL = −c(L0 + L) . −2

(5.11d)

Taking the second partial derivatives of (5.11) we get, u XX = −u0 a 2 Ae− aX − bY , uYY = −u0b 2 Ae− aX − bY ,

(

)

uLL = −u0CcXY f (L) − ( f (L)) e− f (L ) 2

(5.12a)

u XL = −u0Cf (L )Ye − f ( L ) , uYL = −u0Cf (L )Xe− f ( L ) ,

(

u XY = −u0 − abAe− aX − bY + Ce− f ( L )

)

(5.12b)

with f (L) = 2c (L0 + L) , so that; −3

( f (L) − ( f (L)) )= 2c (L + L)

−3

2

0

− c2 (L0 + L) . −4

(5.13)

Since u XX and uYY given in (5.12a), are clearly negative, as required by (5.3b). Again

uLL , given in (5.12a), to be negative, the quantity on the right hand side of (5.13) must be positive, so that;

2(L0 + L )  c , which is satisfied by all positive L if we choose L0 , c so that 2 L0  c ; this we assume to be the case. Hence, all the conditions of (5.3 a, b) are satisfied, as can be verified from (5.11 a–d) and (5.12a). Consider now u XY is given by the last relation in (5.12b). We shall see that the constants or parameters A, C, a, b, and c can be chosen so that u XY < 0 is satisfied (Mohajan, 2017a). 5.2.3 Model (C): u XY = 0 It is similar to Model (A), but u consists of two parts as follows:

u ( X , Y , L ) = u1 X a Lc + u2Y b Lc ,

(5.14)

where u1, u2 ; a, b, c are new constants. Taking partial derivatives of (5.14) we get;

(

)

u X = u1aX a −1Lc , uY = u2bY b −1Lc , uL = c u1 X a + u2Y b Lc −1 ,

(5.15a)

(

)

u XX = u1a (a − 1)X a − 2 Lc , uYY = u2b (b − 1)Y b − 2 Lc , uLL = c (c − 1) u1 X a + u2Y b Lc − 2 , (5.15b) u XL = u1acX a −1Lc −1 , uYL = u2bcY b −1Lc −1 , u XY = 0 .

82

(5.15c)

Hence, (5.3 a, b, c) and the last relation in (5.3d) are satisfied, if we choose u1, u2 to be positive and a, b, c to satisfy (5.9) of Model (A) (Mohajan, 2017a).

5.3 Mathematical Discussion of the Models Consider the four equations (5.5a–d) in seven variables X, Y, L, , P1, P2 , w . We solve for X, Y, L,  in terms of P1, P2 , w , and denote the solution as follows (Moolio & Islam, 2008; Moolio et al., 2009; Islam et al., 2010):

(

)

U = u X *, Y *, L* = u~ (P1 , P2 , w) .

(5.16)

If the left hand sides of (5.5a–d) are assumed to be continuously differentiable, then by the implicit function (will be discussed later) X *, Y *, L*, * will all continuously differentiable functions of P1, P2 , w provided the following Jacobian matrix is non-

(

)

singular at X *, Y *, L* :

 o − P1 − P2 − w − P u u XY u XL  XX . H = 1 − P2 uYX uYY uYL    − w uLX uLY uLL 

(5.17)

Using chain rule in (5.16) we get,

U * X Y L Y L   X = uX + uY + uL =   P1 + P2 + w . w w w w w w   w

(5.18)

From (5.5a) we get,

wT = XP1 + YP2 + wL ,

(5.19)

so that taking partial derivative we get:

T = P1

X Y L + P2 +w + L. w w w

(5.20)

1 U * . (T − L ) w

(5.21)

Using (5.20) in (5.18) we get,

=

Let w (T − L) , the money earned by the individual, be denoted by B:

83

B = f (P1, P2 , w)  Tw − L (P1, P2 , w) w .

(5.22)

Solution of (5.22) is as follows:

w = g (P1, P2 , B) , say,

(5.23)

and we express U * in terms of P1, P2 , B ; ~ U * = u~ (P1 , P2 , B ) = u~ (P1, P2 , g (P1, P2 , B)) = u~ (P1 , P2 , B ) .

(5.24)

Taking partial derivative with respect to B we obtain;

 ~ w U *  ~ U * w u (P1 , P2 , B ) . = u~ (P1 , P2 , B ) = = B B B w B B

(5.25)

We define a function as follows (Mohajan, 2017a):

h (P1, P2 , w, B) = f (P1, P2 , w) − B , dh =

h h h h dP1 + dP2 + dw + dB . P1 P2 w B

(5.26) (5.27)

We set dP1 = 0 = dP2 , dh = 0 in (5.27), that is, we hold P1 , P2 constant, and confine

P1, P2 , w, B to the ‘surface’ (5.22), i.e., consider B to be given by (5.22), (or (5.23)). Now (5.27) becomes; dw h B h =− =− B . dB h w hw

(5.28)

Since P1 , P2 are constants we can write, dw dB = w B , also from (5.26), hw = f w ,

hB = −1 , so that , w h (P , P , w, B ) 1 =− B 1 2 = , B hw (P1 , P2 , w, B ) f w (P1 , P2 , w)

(5.29)

where f w (P1 , P2 , w)  0 . Taking the partial derivative of (5.22) with respect to w we get; f w (P1 , P2 , w) = T − L + w

where  S =

 (T − L ) = (T − L ) 1 + w  (T − L ) = (T − L )(1 +  S ) w  T − L w 

(5.30)

w  (T − L ) is the individual’s elasticity of labor supply. The quantity T − L w

 S can be interpreted as the ratio of a fractional change in work time to that in wage rate. Using (5.25), (5.29), and (5.30) we get;

84

U * 1 U * = , (T − L )(1 +  S ) w B

(5.31)

where  S  −1 and so that the expression for λ is given by (5.21) now becomes,

 = (1 +  S )

U * . B

(5.32)

From (5.32) we see that the Lagrange multiplier λ is proportional to the marginal utility of income, the proportionality being the elasticity of labor supply plus unity; λ equals the marginal utility of income if  S = 0 , i.e., if there is no supply response to change in wage rate. Baxley and Moorhouse (1984) state the two conditions as follows: •

the determinant of H is negative,



the determinant of the Hessian matrix

 o − P1 − P2 − P u u XY XX  1 − P2 uYX uYY

 ,  

(5.33)

is positive. In model (A), with the use of (5.8 b, c), the determinant of the Jacobian matrix (5.17) can be written as follows:

o − P1

− P1

− P2

u0 a (a − 1)X a − 2Y b Lc

−w

u0 abX a −1Y b −1Lc

u0 acX a −1Y b Lc −1 .

a −1

b −1 c

− P2

u0 abX

−w

u0 acX a −1Y b Lc −1

Y

L

u0b (b − 1)X Y a

b−2 c

u0bcX aY b −1Lc −1

L

a

u0bcX Y

(3.34)

b −1 c −1

L

u0c (c − 1)X aY b Lc − 2

After expanding and simplifying we get,

bc (b + c − 1)X 2 aY 2b − 2 L2c − 2 P12 + ac (a + c − 1)X 2 a − 2Y 2b L2c − 2 P22  Hˆ = H = u02 + ab (a + b − 1)X 2 a − 2Y 2b − 2 L2c w2 − 2abc X 2 a − 2Y 2b −1L2c −1P w + X 2 a −1Y 2b − 2 L2c −1P w + X 2 a −1Y 2b −1L2c − 2 P P 2 1 1 2 

(

We consider a = b = c; o < a < 1, then we can write (5.35) as follows:

85

   . (5.35)  

)



(

)



Hˆ = u02a2 X 2a − 2Y 2a − 2 L2a − 2 (2a − 1) P12 X 2 + P22Y 2 + w2 L2 − 2a (P1P2 XY + P1wXL + P2 wYL ) . (5.36) Now for this model the determinant of the Hessian matrix (5.33) can be written as follows: − P1

− P2

u0 a (a − 1)X a − 2Y b Lc

u0 abX a −1Y b −1Lc

o H  = − P1 − P2

,

(5.37)

u0b (b − 1)X aY b − 2 Lc

u0 abX a −1Y b −1Lc

for general values of a, b and c. After expanding and simplifying we get,





H  = u0 X a − 2Y b − 2 Lc b (1 − b )P12 X 2 + a (1 − a )P22Y 2 + 2abP1P2 XY .

(5.38)

Now we choose in the range of (5.9) a = b = c, to obtain the following expression for H  :



(

)



H  = u0 aX a − 2Y a − 2 La (1 − a ) P12 X 2 + P22Y 2 + 2aP1P2 XY .

(5.39)

In Model (C) the determinant of the Jacobian matrix (5.17) can be written as follows:

o

− P1

− P1

u1a(a − 1)X a − 2 Lc

− P2

−w u1acX a −1Lc −1

0

.

u2b (b − 1)Y

− P2

0

−w

u1acX a −1Lc −1

b−2 c

L

u2bcY

(5.40)

b −1 c −1

L

(

)

c (c − 1) u1 X a + u2Y b Lc − 2

u2bcY b −1Lc −1

Similarly, as before after expanding and simplifying we get,





u2bc (b + c − 1) u2Y 2b − 2 + (b + c − bc − 1) u1 X aY b − 2 L2c − 2 P12    2a − 2 a − 2 b 2c − 2 2 ( ) ( ) + u ac a + c − 1 u X + a + c − ac − 1 u X Y L P   1 1 2 2 Hˆ =   . (5.41) a − 2 b − 2 2c 2 ( )( ) + u u ab 1 − a b − 1 X Y L w  1 2  − 2u u abc 2 X a −1Y b −1L2c − 2 P P + a (1 − a )X a − 2Y b −1L2c −1P w + b (1 − b )X a −1Y b − 2 L2c −1P w  1 2 2 1   1 2





We set, a = b = c =





1 , so that for Model (C) inequalities in (5.9) are satisfied. Hence, 2

(5.41) can be written as: 3 1 − 2 Hˆ = − u1u2 ( XYL ) 2 L−1 (P1 X + P2Y + wL ) , 16

86

(5.42)

which is negative definite, as required. The determinant for the Hessian matrix (5.33) for Model (C) can be written as; − P1

o H = − P1

− P2

u1a (a − 1)X a − 2 Lc

− P2

0

.

(5.43)

u2b (b − 1)Y b − 2 Lc

0

Similarly, as before after expanding and simplifying we get,





H  = u2b (1 − b)Y b − 2 P12 + u1a (1 − a )X a − 2 P22 Lc

(5.44)

which is positive definite for allowed values of a, b, and c. Hence, for suitable values of the parameters, Models (A, B, and C) satisfy sufficient conditions (i) and (ii) of (5.33) (Mohajan, 2017a).

5.4 Sufficient Conditions for Implicit Functions Since u is a function of the endogenous variables X, Y, L, the functions u X , uY , u L also depend on the same variables;

u X = u X ( X , Y , L) ,

uY = uY ( X ,Y , L) ,

uL = uL ( X ,Y , L) .

(5.45)

We denote the left hand sides of (5.5a–d) by the four components of a vector F, which all depend on , X ,Y , L, P1, P2 , w, which may be regarded as points in a seven-dimensional Euclidean space, R7 . Hence,

F = (F1 , F2 , F3 , F4 ) , Fi = Fi ( , X ,Y , L, P1, P2 , w) = 0 ; i = 1, 2, 3, 4,

(5.46)

the latter representing the four equations (5.5a–d). Hence, F is a four-vector valued function taking values in R 4 and defined for points in R7 . The solution of (5.46) be,

  X    = G (P1 , P2 , w) , Y    L 

(5.47)

where G = (G1 , G2 , G3 , G4 ) , being a four vector valued function of P1, P2 , w . The Jacobian matrix for G, J G is given by;

87

   P  1  X JG =   P1  Y   P1

 P2 X P2 Y P2

  w  X  . w  Y   w 

(5.48)

Assuming the solution λ, X, Y, L to be given as functions of P1, P2 , w as in (5.47), we write (5.5a–d) explicitly (with the use of (5.46)) as follows (Mohajan, 2017a):

w T − L(P1, P2 , w)− X (P1, P2 , w)P1 − Y (P1, P2 , w)P2 = 0 ,

(5.49a)

uX X (P1, P2 , w),Y (P1, P2 , w), L(P1, P2 , w)−  (P1, P2 , w)P1 = 0 ,

(5.49b)

uY X (P1, P2 , w),Y (P1, P2 , w), L(P1, P2 , w)−  (P1, P2 , w)P2 = 0 ,

(5.49c)

uL X (P1, P2 , w),Y (P1, P2 , w), L(P1, P2 , w)−  (P1, P2 , w)w = 0 .

(5.49d)

Appling the first partial derivatives with respect to P1, P2 , w respectively, of (5.49a), to get the following three equations: L X Y + P1 + P2 = − X , P1 P1 P1

(5.50a)

L X Y + P1 + P2 = −Y , P2 P2 P2

(5.50b)

w

w

w

L X Y + P1 + P2 = T − L . w w w

(5.50c)

Similarly, taking the second order partial derivatives of (5.49b–d) we get as follows: u XX

X Y L  + u XY + u XL − P1 =  , P1 P1 P1 P1

(5.51a)

u XX

X Y L  + u XY + u XL − P1 = 0 , P2 P2 P2 P2

(5.51b)

u XX

X Y L  + u XY + u XL − P1 = 0 . w w w w

(5.51c)

uYX

X Y L  + uYY + uYL − P2 = 0 , P1 P1 P1 P1

88

(5.52a)

uYX

X Y L  + uYY + uYL − P2 =  , P2 P2 P2 P2

(5.52b)

uYX

X Y L  + uYY + uYL − P2 = 0 . w w w w

(5.52c)

u LX

X Y L  + u LY + u LL − P1 = 0 , P1 P1 P1 P1

(5.53a)

u LX

X Y L  + u LY + u LL − w = 0, P2 P2 P2 P2

(5.53b)

X Y L  + u LY + uLL − w=. w w w w

(5.53c)

u LX

Assuming that the Jacobian matrix H is given by (5.17), and the Jacobian matrix for G,

J G , given by (5.48), four sets of equations (5.50) to (5.53) can be written as follows (Mohajan, 2017a):

− X − Y T − L   0 0  −1  −  JG = −H .  0 − 0     0 0 − 

(5.54)

According to the rules of matrices, we get;

H −1 =

1 C T , where C = (Cij ) , the matrix of cofactors of H, and T for transpose. From det H

(5.17) and (5.54) we get, X 1 (− XC12 − C22 ) , =− P1 det H

(5.55)

where C12 , C22 are given by;





C12 = P1 uYY uLL − (uLY ) 2 − P2  u XY uLL − uLY u XL  + w  u XY uLL − uYY u XL , (5.56a) C22 = − P12uLL − w2uYY + 2wP2uLY .

(5.56b)

We confine ourselves to Models (A) and (C), so that u XY  0 , or u XY = 0 . With u XY  0 , and conditions (5.3b, c), the second and third terms in C12 are positive. Let us assume that;

89

uLY  u 2 det  LL = uLLuYY − (uLY )  0 ,  uYL uYY 

(5.57)

“as economist generally do”, say Baxley and Moorhouse (1984). Then C12 > 0; from (5.3b,c), C22  0 . Thus

X  0 , that is, if the price of x increases, then the amount of x, P1

given by X, decreases, which is reasonable. For Model (A) we have, from (5.8b,c), after some calculations,

uLLuYY − (uLY ) = u02bc (1 − b − c)X 2aY 2b − 2 L2c − 2 , 2

(5.58)

which is positive if b + c < 1, which is valid for suitable choice of b, c, consistent with (5.9). Now we will use mathematical techniques to describe some such problems in details.

5.5 Cost Minimization of a Competitive Firm We consider that, a competitive production firm produces Q units of a commodity during a specified time, with the use of K quantity of capital, L quantity of labor, and R quantity of other inputs (e.g., land and other raw materials) into its production process (Humphery, 1997). If the firm seeks to maximize its profit while meeting the terms of the contract, its production policy can be characterized as a constrained cost minimization problem in which the firm chooses the least cost combination of three factors: K, L, and R to produce quantity Q units of the product (Baxley & Moorhouse, 1984). The firm minimizes the cost function (Moolio & Islam, 2008):

C (K , L, R) = rK + wL + R ,

(5.59)

subject to the constraint of production function:

Q = f (K , L, R) ,

(5.60)

where r is the rate of interest or services per unit of capital K, w is the wage rate per unit of labor L, and  is the cost per unit of other inputs R, while f is a suitable production function. In terms of single Lagrange multiplier  , the Lagrange function Z as follows:

Z (K , L, R,  ) = C (K , L, R) +  (Q − f (K , L, R)) .

(5.61)

For optimization we can write from (5.61) as;

Z  = Q − f (K , L, R ) = 0 ,

90

(5.62a)

Z K = CK −  f K = 0 ,

(5.62b)

Z L = CL −  f L = 0 ,

(5.62c)

Z R = CR −  f R = 0 .

(5.62d)

From (5.64b–d), the Lagrange multiplier is obtained as;

=

CK CL CR . = = fK fL fR

(5.63)

Considering the infinitesimal changes dK, dL, dR in K, L, R respectively, and the corresponding changes dQ and dC, we get:

dC = CK dK + CL dL + CR dR ,

(5.64)

dQ = f K dK + f L dL + f R dR .

(5.65)

With the use of (5.62b–d), or (5.64) and (5.65), we obtain;

dC C K dK + C L dL + C R dR = =. dQ f K dK + f L dL + f R dR

(5.66)

Thus, the Lagrange multiplier may be interpreted as the marginal cost of production; that is, it represents the increase in total costs incurred from the production of an additional unit Q. If, for example, one of the inputs, say K, is held constant, then (5.66) represents the partial derivative:  C Q  , (with dK = 0), and so on.  K

5.5.1 Interpretation of Lagrange Multiplier Before we discuss sufficient conditions, we provide an interpretation of Lagrange multiplier, with the aid of chain rule, from (5.66) we get:

C * K L R . = CK + CL + CR Q Q Q Q

(5.67)

From (5.59), we get: C K = r , C L = w , C R =  , and from (5.62b–d), we get: r =  AK  −1L R , w =  AK  L −1R ,  =  AK  L R −1 .

Therefore, we write (5.67) as follows:

 C * K L R  = *  AK  −1L R +  AK  L −1R +  AK  L R −1 . Q Q Q Q   From (5.62a), we have:

91

(5.68a)

Q = AK  L R  .

(5.68b)

Applying partial differentiating we get; 1 =  AK  −1L R

K L R , +  AK  L −1R −1 +  AK  L R −1 Q Q Q

(5.68c)

which allows us to rewrite (5.68a) as:

C * = * . Q

(5.69)

Therefore, (5.69) verifies (5.66). Thus, in this particular illustration, if the firm wants to increase (decrease) 1 unit of its production, it would cause total cost to increase (decrease) by approximately * units, Lagrange multiplier is a shadow price.

5.5.2 Comparative Static Analysis

(

)

As like (5.47) the solution of equation F * , K * , L* , R * , r , w,  , Q = 0 , may be solved in the form of;

 *   *  K  = G(r , w,  , Q ) .  L*   *  R 

(5.70)

Moreover, the Jacobian matrix (5.48) for G is given by;

 *   r *  K  r  *  L  r  R *   r

* w K * w L* w R * w

*  K *  L*  R * 

*   Q  C 21 K *   1 C 22 Q  = −  L*  J C 23   Q C 24 *  R  Q 

C31 C 32 C33 C 34

C 41 C 42 C 43 C 44

C11  C12  . C13   C14 

(5.71)

Now, we study the effects of changes in r , w,  , and Q on K, L and R. Firstly, we find out the effect on capital K when its interest rate r increases. From (5.71), we get: K * 1 1 = − C22  = − − QLQL Z RR + 2 QLQR Z LR − QRQR Z LL . r J J

92

K * 1 = ( +  )A 3 K 3 L3 − 2 R 3 − 2 . r J

   +         1   ( K  +  )      w      Q     =− , where J = H . r    2−         1             A  r *

Since  ,  ,  , A  0, and r , w ,   0 , and Q is the output of the firm that can never be negative, therefore, K *  0, r

(5.72)

which indicates that if the interest rate or services of the capital K increases, the firm may consider decreasing the level of input K. Secondly, we examine the effects on labor L when the interest rate of capital K increases. Similarly, from (5.71), we get;

   +     +       1   L 1 1             Q     = − C23  =    +     +       1   . r J                 r   w    A     *

Again, since  ,  ,  , A  0, and r , w ,   0 , and Q is the output of the firm that can never be negative, therefore, L*  0, r

(5.73)

which indicates that when the interest rate or services of capital K increases the firm can increase the level of labor L, because both inputs are unrelated to each other, as C KL = 0 . Again from (5.71), we get;

   +          1−   K 1 1     w      Q     = − C12  =    +          1   . Q J                          A  r  *

Again, since  ,  ,  , A  0, and r , w ,   0 , and Q is output of the firm that can never be negative, therefore,

K *  0, Q

(5.74)

93

which verifies our assumption and common sense that when the demand of the product increases the firm may consider increasing its level of inputs: capital, labor, and other inputs.

5.6 Output Maximization of an Agency For the fixed annual budget, same as section 5.5 a government agency maximizes the output function (Moolio et al., 2009):

Q = g (K , L, R) ,

(5.75)

subject to the budget constraint; B = rK + wL + R ,

(5.76)

where symbols are as section 5.5. Same as (5.62) we can write,

Z  = B − rK − wL − R = 0 ,

(5.77a)

ZK = QK −  r = 0 ,

(5.77b)

ZL = QL −  w = 0 ,

(5.77c)

Z R = QR −   = 0 .

(5.77d)

Similarly, after tedious calculations as (5.68) the Lagrange multiplier is obtained as follows:

=

QK QL QR = = . r w 

(5.78)

Let the function g is given by;

Q = g (K , L, R ) = AK  L R  . The Lagrange function is defined as follows:

Z (K , L, R,  ) = AK  L R +  (B − rK − wL − R ) . Therefore, (5.77a–d) become:

Z  = B − rK − wL − R = 0 ,

(5.79a)

Z K =  AK  −1L R − r = 0 ,

(5.79b)

Z L =  AK  L −1R − w = 0 ,

(5.79c)

Z R =  AK  L R −1 −   = 0 .

(5.79d)

94

Using the method of successful elimination and substitution, we solve above set of equations, and obtain the optimum values of K, L, R, and  : K = K* =

B , r

(5.80a)

L = L* =

B , w

(5.80b)

R = R* =

B , 

(5.80c)

         r w 

 B( −1)     .   

(

)

 = * = A

(5.80d)

Thus, the stationary point is K * , L* , R * . After calculating same process as (5.68) the production can be written as follows:

       B Q* = A     r w  

  . 

(5.81)

Similarly, as (5.69) the Lagrange multiplier is obtained as follows: Q * = * . B

(5.82)

Therefore, (5.82) verifies (5.66). Hence, if the agency wants to increase (decrease) 1 unit of its output, it would cause the total budget to increase (decrease) by approximately * units. Similarly, as (5.70) after straightforward but tedious calculation, we get;

  2  2   2 B 2 H = − A  2 2  2 2 r w   2

 r 2 w2  2 5    . 4  B 

(5.83)

Since A  0,   0,   0,   0, and r , w,  are the costs of inputs and hence, are positive, while B is budget that will never be negative, therefore, from (5.83), H  0 .Thus, the value of the output function obtained in (5.81) is indeed a relative maximum value. Similar way after tedious calculations as like as in section 5.5 we get; K *  0. B

95

(5.84)

This verifies our assumption and common sense that when the budget size increases, the agency may consider increasing its level of inputs: capital, labor, and other inputs, in order to increase the output services. Our results and discussion are true for

L*  0, B

R *  0 as well. B

5.6.1 Output Maximization Subject to a Nonlinear Constraint We consider explicitly a simple algebraic form of the output function in three variables:

Q = g (K , L, R) = KLR subject to particular nonlinear budget constraint: B = rK + wL +  ( R) R , where

 (R ) = 0 R − 0 , with  0 being the discounted price of the inputs R. Therefore, the budget constraint takes the form: B = rK + wL +  0 R 2 −  0 R . Following straightforward L* 1  B4  steps of calculation as like section 5.6, we get; = −  . Again, since for 0 J  16w0 2 

this specific case, we have taken the value of B = 2 0 . Therefore, re-substituting it into the above equation and after simplifying, we get:

L* 1 . Since w  0 , therefore, = 0 7 w

L*  0 , which indicates that even if the discounted cost of the other inputs R increases,  0 the agency still may like to consider increasing the level of labor L. Following Islam et al. K * 3 L* R* (2011c) we can write,  0 , which indicate =  0 . And accordingly, 0, B B 14 B

that when budget size increases the level of input of capital K, labor L, and other inputs R also increase, in order to increase the output services provided by the agency.

5.7 Utility Maximization Subject to Multiple Constraints Let us confine ourselves to two-commodity world, assuming an individual consumer obtains his/her utility from the consumption of two goods x and y which are

96

purchased in the marketplace in quantities X and Y, respectively (Islam et al., 2010). Then, individual consumer’s utility function U (X, Y ) must be maximized subject to the budget constraint,

B = P1 X + P2Y

(5.85)

R = r1 X + r2Y

(5.86)

and the coupon constraint,

where P1, P2 are the prices, and r1 , r2 are ration coupons required per unit of commodity x, y respectively. We introduce two Lagrange multipliers 1 , 2 to define the Lagrange function L as follows:

L( X ,Y , 1, 2 ) = U ( X ,Y ) + 1 (B − P1 X − P2Y ) + 2 (R − r1 X − r2Y ) . (5.87) Setting up partial derivatives of (5.87) equal to zero, we get the following necessary conditions for maximization; L1 = B − P1 X − P2Y = 0 ,

(5.88a)

L2 = R − r1 X − r2Y = 0 ,

(5.88b)

LX = U X − 1P1 − 2r1 = 0 ,

(5.88c)

LY = UY − 1P1 − 2r2 = 0 .

(5.88d)

In principle, (5.88a–d) lead to the optimal solutions X * , Y * , 1* , *2 , each quantity being a function of the parameters P1, P2 , r1, r2 . If, for instance, we consider the money constraint to remain constant (not to change), that is, if dB = 0, then we get; dU U X dX + U Y dY = = 2 , dR rX dX + rY dY

(5.89)

where (5.88a–d) have been used with 1 = 0 . The Lagrange multiplier  2 can be

 U   U  interpreted as   . Similarly, 1 is equated to   . Therefore, the Lagrange  R  R  B  R multipliers, in this specific illustration, give the changes in the utility consequent to one of the constraints being operative, but not the other. We now consider an explicit form of the utility function U in two commodities, and provide a detailed discussion;

U = U ( X , Y ) = XY .

97

(5.90)

Solution of the set of four simultaneous equations (5.88a–d) produced by the first order conditions for the optimum values of 1 , 2 , X, and Y gives the following optimal values: X* =

r2 B − P2 R , P1r2 − P2 r1

(5.91a)

Y* =

P1R − r1B , P1r2 − P2 r1

(5.91b)

1* =

(P1r2 + P2r1 ) R − 2r1r2 B , (P1r2 − P2 r1 )2

(5.91c)

*2 =

(P1r2 + P2r1 ) B − 2 P1P2 R , (P1r2 − P2r1 )2

(5.91d)

(

)

with P1r2 − P2r1  0 . As before, the stationary point is X * , Y * and the utility of an individual consumer is;

U* =

(P1r2 + P2r1 ) BR − P1P2 R 2 − r1r2 B 2 . (P1r2 − P2r1 )2

(5.92)

Assuming the money constraint does not change; that is, if dB = 0 , then 1 = 0 ; and as before (5.92) can be written as follows:

 U *    = *2 .  R  R

(5.93)

Equation (5.93) verifies (5.66). Thus, the Lagrange multiplier *2 obtained in (5.93) may be interpreted as the marginal utility; that is, the change in total utility incurred from an additional unit of coupon R. In other words, if an individual wants to increase (decrease) a 1 unit of his utility, it would cause the total coupon quantity to increase (decrease) by approximately *2 units; here we assume that budget constraint remains unchanged. Similarly, as before for 2 = 0 we get;

 U *    = 1* .  B  B

(5.94)

Equation (5.94) also verifies (5.66). Thus, the Lagrange multiplier 1* obtained in (5.97) may be interpreted as the marginal utility. Proceeding as like the calculations as in section 5.4 we get,

98

X * r (r B − P2 R ) . =− 2 2 P1 (P1r2 − P2r1 )2

(5.95)

Since P1, P2 , r1, r2 are prices and ration coupons for goods x and y, and so are greater than zero, as well as B, R are budget and ration coupons, so are positive. Therefore, the sign of

X * depends on the term (r2 B − P2 R) , assuming that P1r2 − P2r  0 . Then, there seems to P1 be three situations: •

X * If r2 B  P2 R , then  0 , which indicates that if the price of commodity x P1 increases, the level of consumption of x will decrease. This situation seems reasonable result in the sense that commodity x has many substitutes; and hence, consumers switch to substitutes when price of commodity x goes up.



If r2 B  P2 R , then

X *  0 , which indicates that even if the price of commodity x P1

increases, the level of consumption of x will also increase. It seems that commodity x is a superior good in this situation and it has no other substitutes. •

If r2 B = P2 R , then

X * = 0 , which indicates that if the price of commodity x P1

increases, there seems no effect on the level of consumption of goods x. It looks like commodity x is a necessity and it has neither complementary nor supplementary goods.

5.8 Conclusion In this chapter we have used Lagrange multiplier method to obtain a firm’s cost minimization problem subject to Cobb-Douglas production function as an output constraint, an agency’s output maximization problem subject to both linear and nonlinear budget constraint, maximize utility function subject to two constraints: i) budget constraint, and ii) coupon constraint. It is demonstrated that the value of the Lagrange multiplier is positive, and sometimes it indicates shadow price. We have used necessary and sufficient conditions to obtain optimal value in each case. With the help of comparative static analysis and application of Implicit Function Theorem, we 99

mathematically showed the behavior of the firm (including explicit examples), and recommend that if the cost of a particular input increases, the firm needs to consider decreasing the level of that particular input; at the same time, and there is no effect on the level of other inputs. In this chapter we have done mathematical calculations in some details.

100

Chapter–VI

Social Choice, Arrow’s Theorem and Game Theory

6.1 Introduction Mathematical economics is closely related with social choice theory. In this chapter, an attempt has been made to show this relation by Arrow’s impossibility theorem with easier mathematical calculations. A social welfare function is a procedure for aggregating properties of individual preferences into social orderings. Arrow’s theorem shows that it is impossible for a social welfare function to satisfy five conditions namely: i) Completeness and Transitivity, ii) Universality, iii) Pareto Consistency, iv) Independence of Irrelevant Alternatives (IIA), and v) Non-dictatorship simultaneously. Arrow’s theorem makes social choice theory more challenging and interesting. The chapter includes aspects of both federal and unitary democracies where we have shown that both democracies have some difficulties, but comparatively federal democracy is better. A brief description is given on the behavior between the two adversary countries.

6.2 Social Choice and Political Relation in the Light of Game Theory Let us consider an island and in that island, there is a forest that has only two kinds of animals, deer and hare. Suppose there are two hunters in that island. If both of them hunt a deer, they will share equally. If both hunt for hare they each will catch one hare. If one hunts for hare while the other for deer then the former will catch and the latter will catch nothing. Let the cost of a deer is $100, and that of a hare is $10. So, each of the hunters prefers half a deer to one hare (Fudenberg & Tirole, 1991). It is a simple example of a game theory. The hunters in the game are the players. A strategy for a player is a complete plan for hunting a deer or a hare which are the players’ choice. The payoff to their choice is preying what they want to maximize. If each player

101

believes the other will hunt a hare, each is better off hunting himself which is a noncooperative outcome is also Nash equilibrium (Nash, 1951). The game theorist’s assumption is that all the individuals are perfectly rational and intelligent. A game in strategic form is a special case of the multistage form in which there is only one stage and each player has only one possible information state (Myerson, 1986). That is, it has three elements; a set of players N = 1,2,..., n, for each player i there is the pure-strategy space S i and payoff function u i that gives player i’s von Neumann-Morgenstern utility u i (s ) for each profile s = (s1 ,..., s n ) of strategies. For some given player i we will use all players other than i as player i’s opponent by “–i”. First we consider the finite games where S = i S i is finite.

Table 6.1: Strategic form game. L

M

R

U

5, 4

6, 2

8, 3

M

3, 2

10, 5

4, 8

D

4, 0

11, 8

3, 10

Strategic form s for finite two-player games are often depicted as payoff matrices as in Table 6.1, where players 1 and 2 have three pure strategies each: U, M, D (up, middle and down), and L, M, R (left, middle and right), respectively, i.e., s1 = U , M , D ,

s2 = L, M , R. The first entry in each box is player 1’s payoff for the corresponding strategy profile; the second is player 2’s. A mixed strategy  i is a probability distribution over pure strategies. The space of player i’s mixed strategies is denoted by



where

i

 i (si ) is the probability that  i assigns to s i . The space of mixed-strategy profiles is denoted by

=   i

, with element  . The players are assumed to randomize

i

independently in a game without communication, so that player i’s expected payoff to profile  would be,

102

 n  u i ( ) =     j (s j ) u i (s ) j = 1  sS 

when the players used the randomized strategies ( 1 ,...,  n ) . A combination of randomized strategies ( 1 ,...,  n ) is a Nash equilibrium if, for every player i and every randomized strategy  i ,

ui ( 1 ,...,  n )  ui ( 1 ,...,  i −1 ,  i,  i +1 ,...,  n ) . That is, each player i cannot increase his expected payoff by using any other randomized strategy  i instead of  i , when every other player j is using  j . A mixed strategy profile  * is a Nash equilibrium if, for all players i,

u i   i* ,  −*i   u i  si ,  −*i  si  S i .     Nash equilibrium is strict if each player has a unique best response to his rivals’ strategies (Harsanyi, 1973). That is, s* is a strict equilibrium if and only if it is a Nash equilibrium and for all i and all si  si* ,

ui  si*, s*− i   ui  si , s*− i  .     By definition, a strict equilibrium is necessarily pure strategy equilibrium. Here, we observe that in Table 6.1 no matter how player 1 plays, R gives player 2 a strictly higher payoff than M does which is called strategy M is strictly dominated. So that, a rational player 2 should not play M. Again, if player 1 knows that player 2 will not play M, then U is a better choice than M or D. If player 2 knows that player 1 knows that player 2 will not play M, then player 2 should play L. This process is called iterated strict dominance (Islam et al., 2009b). Now, we discuss varying the strategies of a single player i while holding the strategies of his opponents fixed. Let, s −i  S −i denotes a strategy selection for all players but i, and we can write,

(si, s−i ) = (s1 ,..., si−1 , si, si+1 ,..., sn ) . Similarly, for mixed strategies we can write,

( i, −i ) = ( 1 ,..., i −1 , i, i+1 ,..., n ) . 103

Definition: Pure strategy s i is strictly dominated for player i if there exists  i   such i

that,

ui ( i, s−i )  ui (si , s−i ),

s−i  S−i .

The strategy s i is weakly dominated if,

ui ( i, s−i )  ui (si , s−i ),

s−i  S−i . 2

A two-player zero-sum game is a game such that,

 u (s ) = 0 for all s ; where one player i =1

i

wins and the other loses. This type of game studied in game theory, but most of the games used in social sciences are non-zero-sum. Now, we introduce a simple game theoretic model that tells how political institutions may be founded. Let us consider a simple “Battle of Sexes” game shown in Table 6.2 (Myerson, 1996, 2004). The two players in this game are called player A and player B, must independently choose one of the two possible strategies: to grab or to defer.

Table 6.2: A simple Battle of Sexes game.

Player B

Player A

Grabs

Defers

Grabs

0, 0

8, 4

Defers

4, 8

0, 0

If the players both grab or both defer then neither player gets anything; if exactly one player grabs then he gets payoff 8 while deferential player gets payoff 4. This simple game has three equilibria. There is an equilibrium in which player A grabs while player B defers, giving payoffs (8, 4). There is another equilibrium in which player A defers while player B grabs, giving payoffs (4, 8). There is a third equilibrium in which both players independently apply the same randomize strategy, where the 2  2 expected payoff are  2 , 2  which is worse for both players than either of the non3  3

symmetric equilibria (Islam et al., 2009b).

104

Now, consider the island mentioned above with a large population of individuals. Every day they randomly matched into pairs, and play the simple Battle of Sexes once. Each player’s objective is to maximize a long-run discount average of his sequence of payoffs from these daily Battle of Sexes matches. In this situation the islanders need a leader to provide focal arbitration. The leader must be any eligible person from the islanders, and can be elected by a public election. The islanders might obey his instruction as long as everyone else is expected to obey him. The leader will instruct them everyday who should grab and who should defer, which is of course a self enforcing equilibrium. The islanders have ways to remove a leader in the next election when they observe that the leader is a corrupt person. Any deferring player will obtain payoff 4 when justice demand it, by the simply grabbing player will obtain payoff 8. So, the higher class societies always want to grab, and lower class societies always compel to defer if laws in the society is deteriorated. So, the establishment of justice in our island is an essential issue, and the political institutions must be impartial, justful and non-selfish to build up a standard society in this island. The impartial leader will educate every people to become rational and intelligent so that a civilized society will be found in the whole island. But, in the real society leaders or political institutions are not so. Everyone including leader is selfish materialist, that is, everybody in the society wants his/her own maximum total welfare (Islam et al., 2009b). Suppose that, the player A’s home is in the east of the field, player B’s home is in the west of the field, and there is an old fence that crosses the field from north to south. So the player A only expected to grab in the east of the fence, and player B is expected to grab the west of the fence. Now suppose that, one bush has grown right through the fence, so that, there is confusion about which side it is on. Then both of the players will want to grab it simultaneously. If any player ever left the boundary without grabbing then the other player will grab it confidently always, and he also will grab fruits of both sides of the fence successfully in all future matches. Hence, saving boundary is an important issue and patriotic symbols for the islanders of both states. The islanders will sacrifice their best efforts to save the boundary of their state; otherwise they cannot save their state (Islam et al., 2009b).

105

6.3 Unitary and Federal Democracy of Two States Let us consider further that, at present the east state is ruled by an elected leader with unitary democracy, and the west state is ruled by an elected leader with federal democracy. First, we analyze the unitary democracy of the east state. The leader is elected by public election for a fixed period, and then run for re-election again in each period until rejected by the voters. In each period he may rule responsibly or corruptly (Myerson, 2006). Let, r denotes the leader’s payoff each period if he rules responsibly, and r +c be the payoff if he rules corruptly. So that, c is the ruler’s additional benefit in each period if he rules corruptly instead of being responsible. In each period, each voter gets welfare w from the government if the leader is responsible and each voter gets 0 if the leader is corrupt. Let, voter transition cost is x when they elected a new leader to oust the previous leader. So that, in the case of changing a leader each voter’s payoff is

(w − x) if the new leader is responsible, and (0 − x) if the new leader is corrupt. Each politician wants to maximize the expected total discount value of payoffs, where payoffs in future periods are discounted by some discount factor  per period. Voters also discount their future payoffs by the same factor  per period. Here, the parameters are r , c, w, x,   0 with   1 . In equilibrium of this game, we say that democracy succeeds

when the voters expect that their leaders will always serves responsibly, with probability 1, and the voters in this case get maximum payoffs. So, the voters always elect a leader who is responsible always, and oust a leader who is ever deviated to corruption. When the leader is expected to always serve responsibly, the voters’ expected discounted value of future payoff is,

(

)

w 1 +  +  2 + ... =

w , 1− 

and the leader’s expected discounted value of future payoff is

r . In this case, for 1− 

success of democracy to be an equilibrium, w r  x , and  r+c. 1−  1− 

Hence, the voters want to replace corrupt leaders, and leaders prefer a long responsible career over a short corrupt career.

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If the voters always re-elected a leader with probability 1, regardless of whether acting responsibly or corruptly then of course the democracy will be frustrated, and the equilibrium is maximum for the leader. Again, democracy will fail if the voters expect that their leader will always act corruptly, and they do not want to oust the leader since they have to expend a transition cost x if they oust the corrupt leader. So that, the corrupt leader always take positive benefit c if the democracy is frustrated, and fails which we call bad equilibrium. In this situation let there is a small probability p  0 that a politician may be intrinsically virtuous, and serve responsibly. A politician who is not intrinsically virtuous may be called normal, so that any politician has probability (1 − p) of being normal. For a bad equilibrium we get, p

x(1 −  ) pw , i.e., x  . w 1− 

So that, transition cost x is greater than the expected gain

pw from better government 1− 

in the unlikely p-probability event of getting a new leader who is virtuous. Hence, possibility of democratic failure still exists if the probability of intrinsic virtue is small enough. The ultimate fate of this state is that democracy will be abolished, and the current leader was guaranteed to rule forever. Then, the voters would get payoffs w forever if the current leader is virtuous, and they get 0 forever if that leader is normal. Now, we describe the federal democracy of the west state where each of N provinces has an elected leader called a governor, and there is also an elected leader of the nation called the president (or prime minister). In federal democracy at the beginning of each period, voters in the nation first choose a president, and then voters in each province choose a governor of their province. Then, elected leaders both national and provincial will serve responsibly or corruptly. A virtuous leader of course serves responsibly. As above, a president gets payoff r1 or r1 + c1 each period, depending on whether responsible or corrupt. Voters get a payoff w1 from the national government when the president serves responsibly, but they get 0 from corruption, and they pay transition cost x1 whether they elected a new president. Similarly, a governor gets payoff

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r0 or r0 + c0 each period, depending on whether responsible or corrupt. Provided voters get a payoff w0 from their provincial government when the governor serves responsibly, but get 0 from corruption, they pay a transition cost x 0 whenever they change a new governor. As before, each politician has probability p of being virtuous type, and otherwise normal. Voters and politicians discount their future payoffs by the discount factor  in each period. We assume that, national election is not influenced by the provincial governors, and vice-versa. In federal equilibrium, the democracy in either level succeeds if the elected leader always serves responsibly, with probability 1. The democracy is frustrated if they (leaders) expect to be re-elected always with probability 1, even though they serve corruptly. The democracy fails if the voters expect that the leaders that they elect will always serve corruptly if they are normal. As before, we can write for provincial democracy; r x0 (1 −  )  1 and r0 + c0  0 , 1−  w0

p and similarly in national democracy; p

r x1 (1 −  )  1 and r1 + c1  1 . w1 1− 

We assume that, a politician would always prefer being president over being governor, so that,

r1  r0 + c0 . We observe that, there are multiple equilibria in this case. First, we consider an equilibrium where provincial democracy is frustrated and fails. In this case, corrupt governors would not be re-elected, so that all the governors would serve responsibly. The national voters always expect that they will not re-elect a corrupt leader (Islam et al., 2009b). Again, we consider an equilibrium where provincial democracy is frustrated and fails, but national democracy succeeds. In this case, a corrupt president would not be reelected so that any president would serve responsibly. All the governors expect to be reelected again, since they know that voters do not elect new leaders. If a governor serves responsibly he has a little chance of winning for higher national offices, so that the

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governors always trend to be corrupted. In both equilibria, there is an inconsistency between national and provincial politics. There is a third equilibrium where democracy always succeeds at both the provincial and national levels. In this equilibrium, the governors and the president always serve responsibly because; they know that otherwise they would not be re-elected. In a real democratic state, the democracy cannot consistently be frustrated at both levels in a federal system which is the feature of a true democratic state. In unitary democracy, there is a good chance of the leader being corrupt, but in federal democracy it is not possible being so easily. If the citizens are rational and intelligent then there is a little chance of the leaders being corrupt in both cases, since then the citizens will not re-elected the corrupt leaders. Hence, we can suggest that the unitary democracy is comparatively fragile, but the federal democracy is stronger and lives long (Islam et al., 2009b).

6.4 International Relation between Two Adversary Countries In a dangerous perturbed world, we need to create peaceful atmosphere for the innocent common people. Expansion of nuclear weapons threatens to the innocent people. In 1945, two towns of Japan, Hiroshima and Nagasaki, were destroyed by atom bombs killing millions of innocent citizens. Here, we use Prisoners’ Dilemma game to describe two rival countries’ behavior as shown in Table 6.3 (Schelling, 1960; Myerson, 2006). Two countries’ problems have to be solved by the successful deterrent strategy which is basis on balance between resolve and restraint provided that both countries have sufficient knowledge about these. As before, this game has two players, player A and player B. Here, asterisks (*) indicate each player’s best payoff. The cell that has two asterisks is a Nash equilibrium of the game which is (–5*, –5*). Of course, for both players’ cooperation would be better off, but mutual cooperation is not an equilibrium, as each player will always be tempted to aggression. Consider that player A will move according to the motion of B. When player A gets to move second after observing what B does, player A has four

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Table 6.3: Prisoners’ Dilemma game.

Player B Cooperative Aggressive Player A

Cooperative

0, 0

–10, 2*

Aggressive

2*, –10

–5*, –5*

possible strategies which are listed in Table 6.4 (Myerson, 2006). From Tables 6.3 and 6.4 we can form Table 6.5. In each row of Table 6.5, an asterisk in the second number indicates the best payoff of B. Observe that, there is only one such asterisk where A does Table 6.4: The four strategies of A’s action when A can observe B’s prior action. A’s strategy

B cooperative

B aggressive

A is cooperative always

A cooperative

A cooperative

A does the same as B

A cooperative

A aggressive

A does the opposite of B

A aggressive

A cooperative

A is aggressive always

A aggressive

A aggressive

the same as B. So, the player A has one deterrent strategy that motivates B to act cooperatively. But, in this case A would get highest payoff 2 when A would act aggressively. So, A does not want to actually follow his deterrent strategy when B Table 6.5: A game where player A moves after observing B’s action.

B cooperative

B aggressive

A is cooperative always

0, 0

–10, 2*

A does the same as B

0, 0*

–5*, –5

A does the opposite of B

2*, –10

–10, 2*

A is aggressive always

2*, –10

–5*, –5*

110

cooperates, and B should not believe that A would use this deterrent strategy, unless A can somehow constrain himself to follow this strategy. Without such restraint, this game still has only one equilibrium, where both players are aggressive and both get payoff –5. For being B cooperative A must make a credible commitment, and to be sure this there need some outside force such as UN to restrain player A from acting aggressively when B has cooperated. Suppose that, player A has a reputation that A is always cooperative. So, if player A ever lost that reputation, by acting aggressively against a cooperative player B, then the world will believe that A will betray always. Let, the value of reputation is R then if A is aggressive then we will subtract reputation value R. So that, by the above scenario we can form Table 6.6, where as long as the reputation value R is greater than 2, there is a good equilibrium in which B is cooperative and A does the same as B. As before, in this game has a bad equilibrium where both are aggressive always both will get the bad payoff ‘–5’. So, remembering the bad equilibrium, and lost of reputation both players will focus on the better equilibrium according to Schelling’s focal-point effect. The world wants this type of equilibrium, since it tends to a focal point say peace. So that, we can say that if America is cooperative with Iran always, then Iran, and also the other countries will be cooperative, and the innocent people will suffer no more. If America acts aggressively with Iran and with other countries of the middle-east, then, the whole Arab countries may be unified. In that case America may have to be paid a greater value for this aggression.

Table 6.6: Player A loses reputation R if A is aggressive.

B cooperative

B aggressive

A is cooperative always

0*, 0

–10, 2*

A does the same as B

0*, 0*

–5*, –5

A does the opposite of B

2–R, 10

–10, 2*

A is aggressive always

2–R, –10

–5*, –5*

We have mentioned that, game-theoretic analysis is based on an assumption that people are rational. Of course, nobody is perfectly rational. In the history, we have seen

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that there are people in the world who irrationally drawn to violence and destruction. For example, Psychopathic militarists like Hitler become a threat to our civilization only when ordinary rational people become motivated to support them as leaders. Now, we can say that, American people are rational, and can elect a president who will be cooperative to the leaders of the other countries. As, America is the most powerful country at this moment, the citizens of America should act rationally, otherwise they lose the reputation forever. 6.5 Arrow’s Impossibility Theorem Arrow’s impossibility theorem is very subtle but delicate (Arrow, 1963). Arrow showed that the preferences of many individuals be aggregated into social preference, but there is a flaw in this aggregation. Because, a social welfare function cannot be derived by democratic vote to reflect the preferences of all the individuals in the society (Islam et al., 2009a, b; Mohajan, 2012b). Arrow’s theorem shows that it is impossible for a social welfare function to satisfy five conditions namely: i) Completeness and Transitivity, ii) Universality, iii) Pareto Consistency, iv) Independence of Irrelevant Alternatives (IIA), and v) Non-dictatorship simultaneously.

For voter paradox, suppose the preference

relation for A, B, and C are as follows:

xA PyA PzA

(6.1a)

yB PzB PxB

(6.1b)

zC PxC PyC .

(6.1c)

Now we impose two conditions on the group preference of x, y, z as follows: i) it must be transitive, ii) it should satisfy the majority rule, that is, if out of three people two prefer x to y, then the group prefers x to y. From (6.1) we see that x is preferred to y by A and C, so that, by the majority rule, x is preferred to y by the group. Again, we see that y is preferred to z by A and B, again by the majority rule y is preferred to z by the group. Since we claim that the group choice be transitive, so that x will be preferred to z by the group. If we now require that the group choice be transitive, we deduce that x is preferred to z by the group. However, from (6.1 b, c) we see that in fact z is preferred to x by B and C, so that by the majority rule z should be preferred to x. Thus, we see that in the situation that the 112

individual choice is given by (6.1a-c) it is not possible to impose the requirements of transitivity and majority rule simultaneously, although these conditions are fairly reasonable. Now we describe combinatorial approach and geometrical approach to Arrow’s Theorem as follows (Islam et al., 2009a): 6.5.1 A Combinatorial Approach to Arrow’s Theorem Let us consider the sets U and S to have three elements each. We denote by x, y, z the group choices, and by xA , y A , z A , etc., the individual choices. Now there are six possibilities for the group preference ordering, as follows (Islam et al., 2009a):

xPyPz(W1 )

(6.2a)

xPzPy(W2 )

(6.2b)

yPzPx(W3 )

(6.2c)

yPxPz(W4 )

(6.2d)

zPxPy(W5 )

(6.2e)

zPyPx(W6 ) .

(6.2f)

Corresponding to (6.2a-f), we have the individual preferences, six of each individual which we denote by w A1 , w A2 , w A3 , w A4 , w A5 , w A6 , wB1 , ,... , etc. The possibilities for the arguments of the function W are as follows:

(

)

W wAi , wB j , wCk ; i, j, k = 1,2,3,...,6 .

(6.3)

Thus, there are 63 = 216 possibilities for the arguments of W, and there are six possible values (6.2a-f); so, the function W represents a map from a set consisting of 216 elements to a set consisting of six elements. Arrow’s theorem guarantees that one of the following three possibilities must necessarily hold;

( W (w W (w

) )= W )= W .

W wAi , wB j , wC k = Wi Ai

, wB j , wC k

Ai

, wB j , wC k

j

(6.3a)

k

That is, the group preference coincides with one of the individual preferences, so that there has to be a ‘dictator’ if conditions (i) to (v) are to be satisfied. Now we state briefly

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how Arrow’s theorem is to be considered in the combinational approach. In this case (6.3a) can be introduced as follows:

(

)

W wAi , wB j , wCk = Wa

(6.4)

where a  1,2,3,...,6, and i, j, k runs independently the values over the same set. The six values of ‘a’ give six possibilities (6.2a-f) for the group preference. Hence, we can write; a = a(i, j, k) = a(i j k).

(6.5)

Arrow’s theorem implies that if conditions (i) to (iv) are satisfied, this map must reduce to one of the following three: a (i j k) = i ; a (i j k) = j ; a (i j k) = k.

(6.6)

First we consider {x, y};

xA PyA , xB PyB , xC PyC  xPy .

(6.7)

We see from (6.7) that xPy obtains for W1 ,W2 ,W5 . If we denote the set of integers {1, 2, 5}, then i, j, k  1,2,5  a(ijk ) 1,2,5 . Let, (i, j, k ), (i, j, k ) be two possible set of values of the indices i, j, k, and let T={x, y}. If these two sets of values corresponds to the same ordering for x, y; then a(i, j, k ) and a(i, j, k ) must induce the same ordering on x, y. So that; i, i  {1, 2, 5} or {3, 4, 6}

(6.8a)

j, j  {1, 2, 5} or {3, 4, 6}

(6.8b)

k , k   1, 2, 5 or {3, 4, 6}

(6.8c)

then, a(i, j, k ) and a(i, j, k ) are both from the set {1, 2, 5} or both from {3, 4, 6}.

6.5.2 A Geometrical Approach to the Combinational Formalism Here we introduce equations (6.2a-f) in the new notation: 0 : xPyPz

(6.9a)

1 : xPzPy

(6.9b)

2 : yPzPx

(6.9c)

3 : yPxPz

(6.9d)

4 : zPxPy

(6.9e)

5 : zPyPx.

(6.9f)

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Thus (000), for example, gives the group decision or preference (6.9a) denoted by the integer 0. There are 63 = 216 such possibilities, which can be grouped into 6 groups (Islam et al., 2009a). The above six groups (6.9a–f) corresponds to A’s choice. In the first group A’s choice is uniformly ‘0’ in the second group A’s choice is ‘1’, and so on. A more symmetric way of representing these 216 values of the function a(i j k) in which choices of A, B, C are represented symmetrically, is through a cubic lattice in a threedimensional Euclidean space containing 6×6×6 = 216 points (Figure 6.1). By Arrow’s theorem if A’s choice prevails then all the points on any one lattice plane parallel to the

(000)

(001)

(002)

(003)

(004)

(005)

(010)

(011)

(012)

(013)

(014)

(015)

(020)

(021)

(022)

(023)

(024)

(025)

(030)

(031)

(032)

(033)

(034)

(035)

(040)

(041)

(042)

(043)

(044)

(045)

(050)

(051)

(052)

(053)

(054)

(055)

(100)

(101)

(102)

(103)

(104)

(105)

(110)

(111)

(112)

(113)

(114)

(115)

(120)

(121)

(122)

(123)

(124)

(125)

(130)

(131)

(132)

(133)

(134)

(135)

(140)

(141)

(142)

(143)

(144)

(145)

(150)

(151)

(152)

(153)

(154)

(155)

(200)

(201)

(202)

(203)

(204)

(205)

(210)

(211)

(212)

(213)

(214)

(215)

(220)

(221)

(222)

(223)

(224)

(225)

(230)

(231)

(232)

(233)

(234)

(235)

(240)

(241)

(242)

(243)

(244)

(245)

(250)

(251)

(252)

(253)

(254)

(255)

(300)

(301)

(302)

(303)

(304)

(305)

(310)

(311)

(312)

(313)

(314)

(315)

(320)

(321)

(322)

(323)

(324)

(325)

(330)

(331)

(332)

(333)

(334)

(335)

115

(340)

(341)

(342)

(343)

(344)

(345)

(350)

(351)

(352)

(353)

(354)

(355)

(400)

(401)

(402)

(403)

(404)

(405)

(410)

(411)

(412)

(413)

(414)

(415)

(420)

(421)

(422)

(423)

(424)

(425)

(430)

(431)

(432)

(433)

(434)

(435)

(440)

(441)

(442)

(443)

(444)

(445)

(450)

(451)

(452)

(453)

(454)

(455)

(500)

(501)

(502)

(503)

(504)

(505)

(510)

(511)

(512)

(513)

(514)

(515)

(520)

(521)

(522)

(523)

(524)

(525)

(530)

(531)

(532)

(533)

(534)

(535)

(540)

(541)

(542)

(543)

(544)

(545)

(550)

(551)

(552)

(553)

(554)

(555).

(6.10)

(j k) plane must have the same value, the value given by the i entry in (i j k), for B all the points in any lattice plane parallel to the (i k) plane has the same value, corresponding to the entry j in (i j k); similar condition holds for C. We now explain how one can use the above formalism to give a ‘combinatorial’ proof of Arrow’s theorem for the particular case of three individuals and three choices. Let us consider the basic assumption; (0 1 2) = 0.

(6.11)

Here we are simply fixing on A as the dictator. If instead of (6.11) we had chosen (0 1 2) =1, 2; we would have chosen B, C respectively as the possible dictator. The points are grouped into six lattice planes, each containing 36 points. Again, we consider (0 1 0). From (6.9a-f) we see that in this case all three individuals prefer x to y, and prefer x to z. The value of (0 1 0), that is, the group preference must also reflect this. So that it is clear from (6.9a-f) that,

(010) 0, 1.

(6.12)

Let us consider the choices (010), (012), and the subset {y, z}; then (010) and (012) are both in the set {0,2,3} or both in the set, {1,4,5}.

(6.13)

116

From (6.11) it follows that (012) is in the set {0,2,3}, and so (010) must also be in this set. But, from (6.12), (010) is also in the set {0,1}. The only common value between the sets {0,1} and {0, 2, 3} is 0, and so we must have (0 1 0) = 0.

Figure 6.1: There are 216 points in the lattice cube where some of the points are displayed. 6.6 Single-Profile Arrow’s Impossibility Theorem In section 6.5 we have discussed multi-profile version of Arrow’s theorem. In this section we will discuss two simple versions of the same theorem, but for single-profile case only (Feldman & Serrano, 2008; Mohajan, 2012b). Now we use some related definitions that are described in section 4.3 of chapter IV.

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6.6.1 Illustrative Examples in a single-profile and two Individuals Model Let us consider there are two individuals and three alternatives, and also assume no individual indifferences between any pair of alternatives (Feldman & Serrano, 2006). Observe that if one of the five examples discards, the remaining four may be mutually consistent. In all the examples 6.1 to 6.4 we use the same profile, and all satisfy simple diversity. In example 6.5 we will modify the individual preferences.

Example 6.1:

Individual 1

Individual 2

x

z

y

x

z

y

Society (majority rule)

xPy, xIz & zIy

Decision: Transitivity assumption fails, since xIz and zIy should imply xIy, but society has xPy.

Example 6.2:

Individual 1

Individual 2

x

z

y

x

z

y

Society

xIyIz

Decision: Pareto (weak or strong) fails, since xP1 y and xP2 y should imply xPy, but society has xIy.

Example 6.3:

Individual 1

Individual 2

Society

x

z

x

y

x

z

z

y

y

Decision: Neutrality fails, consider preference of x vs. z, two individuals are spilt, i.e., individual 1 prefers x to z, and individual 2 prefers z to x, so individual 1 gets her way. Again in y vs. z the two individuals are split, and individual 2 gets her way.

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Example 6.4:

Individual 1

Individual 2

Society

x

z

x

y

x

y

z

y

z

Decision: Individual 1 is a dictator, so that no dictatorship assumption fails.

We have observed that though in examples 6.1 to 6.4 we use same profile, but social preferences are different. Also all satisfy simple diversity assumption. Following example 6.5 modifies the individual preferences.

Example 6.5:

Individual 1

Individual 2

Society

x

z

x

z

x

z

y

y

y

Decision: Simple diversity fails, since options are no longer split over two pairs of alternatives. 6.6.2 Some Elementary Arrow Paradoxes and Arrow’s Theorem in a Single-profile Version when n = 2 Although independent of irrelevant alternatives (IIA) is an essential part in multiprofile Arrow’s impossibility theorem but it does not play any role in single-profile case. We use neutrality assumption in single-profile instead of IIA. First, we introduce Samuelson’s reduction ad absurdum (Samuelson, 1977; Feldman & Serrano, 2008) as follows: Example 6.6 (Samuelson’s Chocolates): There are two individuals. There is a box of 100 indivisible chocolates to be distributed between them. Both of the individuals are fond of chocolates, and each is hungry enough to eat all of them. The alternatives are x0 = (100, 0); x1 = (99,1) ; x2 = (98, 2),…, where the first number in each pair is the number of chocolates going to individual 1, and the second is the number going to individual 2.

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Here any rational individual would say that x1 is better than x0 , i.e., x1Px0 . So that it would be legal to take a chocolate from individual 1, when he has 100 of them, and gives it to individual 2. Let, n  100 . The individual preferences are xn P1 x100 and x100P2 xn . By neutrality, x100P2 xn ! That is, society should give all the chocolates to individual 2! Samuelson’s chocolates example is a serious problem on neutrality. Neutrality would have implied that all the x’s are socially indifferent. So that it is illogical if x1Px0. In fact, any social decision procedure that simply counts instances of xPi y , yPi x , and

xIi y , but does not weigh strength of feelings, satisfies neutrality.

Proposition 6.1: Assume n = 2, and the strong Pareto principle and neutrality hold. Suppose there are two alternatives x and y, and for the two individuals i and j, xPi y and yPj x . Suppose that social preference is xPy. Then, individual i is a dictator.

Proof: Let, p and q be two alternatives in the society, and let individual i’s preference is

pPi q. We will show that social preference is pPq. Let, contrary, so that pPj q . Strong Pareto implies, pR j q for all j, and pPj q for some j, and then social preference is pPq. So that, person j is a dictator, but we have person i is a dictator, a contradiction. Again, neutrality implies for four alternatives p, q, w, and z, wPi z implies pPi q . Hence, wPz implies pPq, so that individual i is a dictator. Q.E.D.

6.6.3 Single-profile Arrow Impossibility Theorem 6.1 Let us consider there are two individuals say i =1, 2, and four alternatives x, y, z, and w in the society. Here we only discuss single-profile version of Arrow’s impossibility theorem (Feldman & Serrano, 2008). We will show how five conditions above are mutually inconsistent. By strong and weak Pareto principle we can say for i = 1, 2; xPi y implies xPy, and both individuals are dictators. Again by simple diversity xPi y for i = 1, 2, but opinions are split on x vs. z, and y vs. z. Since opinions are split on x vs. z, one individual prefers x to z, while the other prefers z to x. If xPz, then by Proposition 6.1, the individual who prefers x to z is a dictator. Similarly, if zPx, then by Proposition 6.1, the

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individual who prefers z to x is a dictator. Again if zIx, then with xPy we yield transitivity result zPy. But, opinions are split on y vs. z. Then as before one person prefers y to z, and another person prefers z to y. By Proposition 6.1, the individual who prefers y to z is a dictator, and similarly who prefers z to y is another dictator. Neutrality implies for any four alternatives x, y, z, and w for i = 1, 2, xPi y if and only if wPi z , and zPi w if and only if yPi x . Then wRz if and only if xRy, and zRw if and only if yRx. In both cases, as before, we find a dictator. Now, we are in a position to introduce Arrow’s Impossibility theorem for a singleprofile and two-individuals case as follows:

Theorem 6.1 (Feldman & Serrano, 2008): Assume n = 2. The assumptions of complete and transitive social preferences, strong Pareto, neutrality, simple diversity and no dictator are mutually inconsistent. Proof: Assume there are two individuals i = 1 and 2 in the society. For three alternatives x, y , z  X we can write xPi y . By simple diversity preference we can split xPi y as

follows: Some individuals prefer x to z, some individuals prefer z to x. By proposition 6.1, if xPz then individual 1 is a dictator; and if zPx then individual 2 is a dictator. Also some individuals prefer y to z, and some individuals prefer z to y, so that by proposition 6.1, if yPz then individual 1 is a dictator, and if zPy then individual 2 is a dictator. Again, completeness implies if xIy then yIx. By the transitivity if xIy and yPz implies xPz. Now we split this relation between x and y. In this situation if individual 1 prefers x to y, then individual 2 prefers y to x. By proposition 6.1, if xPy then individual 1 is a dictator; and if yPx then individual 2 is a dictator. Again, if both individual i’s preference relation is xPi y then by weak or strong Pareto optimality we have xPy. Now we can split this preference relation as follows: If we consider the split between x vs. z we can write; if individual 1 prefers x to z, then individual 2 prefers z to x. So that by proposition 6.1, if xPz then individual 1 is a dictator; and if zPx then individual 2 is a dictator.

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Now, it is clear that in every case we have a dictator. So that if all the assumptions except no dictatorship condition are satisfied, then we have a dictator always. Hence, the conditions are mutually inconsistent. Q.E.D.

6.6.4 Innocuous Dictators in Simple-profile Arrow Theorem We have seen that in multi-profile case a dictator may be dangerous. But, in singleprofile case, a dictator may be innocuous sometimes, but not always. Now we will give examples of such dictators. •

If an individual i is indifferent between all pairs of alternatives by definition he is a dictator.



If weak Pareto is satisfied by the society then everyone is a dictator.



By rule if in a society of 10 individuals, 8 have identical preferences, and then these 8 individuals are dictators.

6.7 Conclusion In this chapter we have indicated how a political institution is formed, and how this serves the society by creating efficient and democratic leaders. The Nash equilibrium has been clarified by game theory, which describes and explains an essential part of the society. Game theory plays an important role to explain more clearly the problems of economics and political science. Here we have discussed briefly unitary and federal democracy to show their importance in the society. We have given a brief description on the behavior between the two adversary countries using Prisoner Dilemma. It is hoped that easier discussion of Arrow’s theorem will give a better idea to the readers about dictatorship. We hope that this chapter will help the readers to find partial concept of social choice and political economy.

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Chapter–VII

Methods of Voting System

7.1 Introduction A voting system is manipulable whenever some individual misrepresents her preferences in order to secure an outcome preferred to the outcome when she is honest; otherwise, it is strategy-proof. Duncan Black (1958) first introduced the concept of manipulation of voting. In voting system every voter’s preference ordering, taken collectively, form the input, the output is usually a single certain winner or a set of winners. This chapter is an exposition of voting system, and of the manipulation of voting. French political philosophers Borda (1781) and Condorcet (1785) introduced modern voting system, but they had not mentioned about manipulation of voting. Condorcet, Borda, and even many modern politicians believe that elections are logically imperfect. The single transferable vote (STV) is a system of preferential voting designed to minimize wasted votes. Droop quota is one of the best methods in STV counting. Approval voting (AV) is a single winner voting system used for multi-candidate elections. In this method each voter may vote for as many of the candidates as she wishes. In AV no ranking is involved, so all the votes have equal weight (Brams & Fishburn, 1978). In majority judgment voting scope is given to the voters to evaluate the candidates in some common grading system. Borda voting is one kind of majority judgment voting where voters can express their opinion by ranking the candidates (Balinski & Laraki, 2007).

7.2 Condorcet Method A Condorcet method is any single-winner election method which always selects the Condorcet winner (i.e., an alternative that beats every other alternative in sequence of

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pair-wise majority contests); the candidate who would beat each of the other candidates in a run-off election if such a candidate exists. Now we discuss the Condorcet voting paradox in which there is no Condorcet winner (Condorcet, 1785; Risse, 2005). Let us assume that there are 17 voters of three types, and three alternatives x, y, z. Let preference relations being as follows: Type 1: xPyPz

by 8 voters,

Type 2: yPzPx

by 5 voters,

Type 3: zPxPy

by 4 voters.

In an election a vote between x and y; x collects 8+4 = 12 votes, and y collects 5 votes, so that x wins. Again a vote between y and z; y collects 8+5 = 13 votes, and z collects 4 votes, so that y wins. Again a vote between x and z; x collects 8 votes, and z collects 4+5 = 9 votes, so that z wins. We observe that there is a cycle in the voting results where x is defeated by y, y is defeated by z, and also z is defeated by x which is a voter paradox. Condorcet’s ad hoc judgment is that x is the Condorcet winner, since x wins by 7 votes and defeats by 1 vote, y wins by 9 votes and defeats by 7 votes, z wins by 1 vote and defeats by 9 votes. But, this is not a satisfactory and acceptable decision (Islam et al., 2011a).

7.3 Borda Count Jean-Charles Borda (1733–1799) developed another voting method named “method of marks” in 1770 (Borda, 1781). Each elector ranks the alternatives according to her order of preference (ties disallowed). Once all votes have been counted, and the candidate with the most points is the winner. In this method if there are m alternatives, an elector’s first choice is assigned (m − 1) points, his second (m − 1) points and so on down to her last choice, which is assigned 0 point. Borda votes in the above preference relations be as follows (Islam et al., 2012): For x : 8×2+5×0+4×1 = 20 marks, For y : 8×1+5×2+4×0 = 18 marks, For z : 8×0+5×1+4×2 = 13 marks. Here x gets highest marks 20, so x wins. We observed that Borda method has no voter paradox, and it is a sincere voting.

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7.4 Single Transferable Voting System The single transferable vote (STV) is a system of preferential voting designed to minimize wasted votes. STV initially allocates an elector’s vote for her most preferred candidate and then, after candidate have been either elected or eliminated, transfers surplus or unused votes according to the voter’s stated preferences (ties disallowed) (Mohajan, 2012e).

7.4.1 Droop Quota In an STV election, a candidate requires a certain minimum number of votes ‘the quota’ to be elected. A number of different quotas can be used; the most common is the Droop quota. Henry Droop defined his quota as (Droop, 1881);  V    +1  S +1

where, V = the total number of valid votes cast, and S = the number of seats to be filled. The Droop quota is the smallest quota such that no more candidates can be elected than there are seats to be filled. STV is a step procedure, in each step voters cast votes for their most preferred candidate. It proceeds according to the following steps: Any candidate who touched or exceeded the required quota is declared elected. If not enough candidates have been elected, the count continues. If a candidate casts more vote than the quota, then her surplus is transferred to other candidates according to the next preference on each voter’s ballot. If none meets the quota, the candidate with the fewest votes is eliminated, and her votes are transferred. This process continues until the last candidate survives, which is the last winner in the election. Again in quota system, voting procedure is stopped when the numbers of remaining candidates instead of counting votes until all candidates have reached a quota. In STV, candidates who receive excess votes, and candidates who are excluded have their votes transferred to other candidates. That is why, it is said to be minimizing wasted votes (Mohajan, 2012e).

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7.4.2 Tie-Breaking in STV Although we have mentioned above that in STV ties are disallowed, sometimes ties may occur for several different reasons, and the ties need to be broken (Newland & Britton, 1997; O’ Neill, 2004). The ties can be broken by the following three rules: Forwards Tie-Breaking (FTB): Choose the candidate who has the most (least) votes at the first stage where they had unequal votes. Backwards Tie-Breaking (BTB): Choose the candidate who has the most (least) votes at the previous stage or at the latest point in the count where they had unequal votes. Borda Tie-Breaking: Choose the candidate with the highest (least) Borda score. A weak tie occurs when candidates have the same number of votes at a given stage. A strong tie occurs when candidates are still tied after applying a tie-breaking rule. A strong tie would be broken by lottery. Here we will use ERS97 rules of tie-breaking (Newland & Britton, 1997; O’ Neill, 2004). The difference between FTB and BTB is given in Table 7.1 which is from Newland and Britton (1997) without any change. Here we have to eliminate one candidate at stage-4, and there is a tie between candidates z and u. From Table 7.1 we see that u has one more vote than z at stage-1. So that candidate z is eliminated. If z and u had been tied at stage-1, then we would have to be looked to subsequent stages. If z and u would have been tied in all stages, then we would have been a strong tie which would have been broken by lottery. But, in BTB we have to look at the previous stage to break ties, and if necessary to the preceding stages. In Table 7.1 we see Table 7.1: Example tally with ERS97 rules where 60 voters are electing 2 candidates from 6.

Stage x y z u v w Non-Transferable

1 23 13 6 7 7 4 0

Surplus of x Eliminate w 2 3 20.00 20.00 13.00 13.00 6.50 10.00 7.50 9.50 7.50 7.50 5.50 – 0.00 0.00

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Eliminate v 4 20.00 15.00 12.00 12.00 – – 1.00

Eliminate z 5 20.00 15.00 2.00 18.00 – – 5.00

in preceding stage-3 that z is ahead to u, so that u would be eliminated. One problem arises with FTB where the elimination order is: 4, 1, 2, 3; which is not sequential and is undesirable. If we make a meaningful sequence starting from 4 then the order is: 4, 3, 2, 1; which is BTB. Again FTB does not use the most relevant information than BTB to break the tie. Hence, BTB is better than FTB in tie-breaking system (Mohajan, 2011e).

7.5 Approval Voting At first blush it seems that AV is less appealing than other voting systems. But, if we compare other voting systems with AV, we will find AV is comparatively better than other voting systems. Assume that there are k candidates, numbered 1, 2,…, k. A scoring rule for an election is a collection of vote-sets, where each vote-set consists of k numbers. The candidate assigned the greatest total across all voters wins the election (Mohajan, 2011e).

7.5.1 Strategies under AV An AV strategy S is a subset of candidates. Choosing a strategy under AV means that voting for all candidates in the subset, and no candidates outside it. The list of strategies of all voters is called a strategy profile S. The number of votes that candidate i receives at S is the number of voters who include i in the strategy S that they select. Admissible strategies under AV involve always voting for a most preferred candidate, and never voting for a least preferred candidate (Brams & Fishburn, 1978, 1983). An AV strategy is sincere if, given the lowest-ranked candidate that a voter approves of, she also approves of all candidates ranked higher. Thus, if S is sincere, everyone ranked above the lowest ranked candidate that a voter approves of is also approved, and everyone ranked below is not approved. A strategy profile S is said to be sincere if and only if the strategy S that every voter chooses is sincere based on each voter’s preference P (Brams & Fishburn, 1978). Let us consider there are four candidates x, y, z, and u; and a voter’s preference profile being as; xPyPzPu. We can write her possible sincere approval votes as follows (Mohajan, 2011e): i) vote for x, y, z, and u, ii) vote for x, y, and z, iii) vote for x and y, iv) vote for x, and v) vote for no candidates. If a voter be indifferent between y and z, but still

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x is her most preferred candidate, then also (i) to (v) conditions are sincere. Now we can also include a new combination as a sincere vote which is; vote for x and z. Let us consider an example where there are 11 voters, and a set of four candidates {x, y, z, u}. Example 7.1: Voters are grouped into three different types as follows: Type 1: xPyPzPu by 5 voters, Type 2: yPzPxPu by 3 voters, Type 3: uPyPzPx by 3 voters. Voters of Type-1 have three sincere strategies as; i) vote for x, ii) vote for x and y, and iii) vote for x, y, and z. The sincere strategies of other two types of voters are analogous. For simplicity, among the voters of each type choose the same strategy S. For example, a typical strategy profile of the 11 voters is as; S = (x, x, x, x, x, yz, yz, yz, uyz, uyz, uyz). The number of votes of each candidate at S is 6 votes for y and z, 5 votes for x, and 3 votes for u. We see that AV selects candidates {y, z} as tied winners at S. We can define an AV critical strategy profile for candidate i at preference profile P as follows: Every voter who ranks i as her worst candidate votes only for the candidate that she ranks top. The remaining voters vote for i, and all candidates they prefer to i. Let, Ci (P ) be the AV critical strategy profile of candidate i at P. In Example 7.1, the critical strategy profile for candidate x is, Cx (P ) = (x, x, x, x, x, yzx, yzx, yzx, u, u, u), collecting 8 votes for x compared to 3 votes each for y, z, and u. The critical strategy profile for candidate y is, C y (P ) = (xy, xy, xy, xy, xy, y, y, y, uy, uy, uy), collecting 11 votes for y compared to 5 votes for x and 3 votes for u. The critical strategy profile for candidate z is, Cz (P) = (xyz, xyz, xyz, xyz, xyz, yz, yz, yz, uyz, uyz, uyz), collecting 11 votes for each of y and z compared to 5 votes each for x, and 3 votes for u. The critical strategy profile for candidate u is, Cu (P ) = (x, x, x, x, x, y, y, y, u, u, u), collecting 5 votes for x and each of 3 votes for y and u. Here we observe that x with 5 votes elects compared to u, and y collecting 3 votes each. Hence, in Example 7.1, the set of AV outcome that are possible is {x, y, {y, z}} (Mohajan, 2011e).

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Proposition 7.1: At critical strategy profile of any candidate i, Ci (P ) , candidate i gets maximum votes compare with any other strategy profile. Proof: From the definition of critical strategy profile S, and Example 7.1 we see that at

Cx (P ) , x gets more votes than other candidates, similar case for candidates y, and z. On the other hand, at Cu (P ) , u loses, but gets more votes than any other strategy profile. Clearly, Ci (P ) is the best strategy profile for any candidate i. Q.E.D. A candidate is a Pareto candidate if there is no other candidate that all voters rank higher. In Example 7.1 we see that x and y are Pareto candidates and AV outcomes; z is not a Pareto candidate, but is a component of AV outcome as z ties with y in Cz (P) ; u is a Pareto candidate, but is not an AV outcome. The following Proposition 7.2 will give clear idea about Pareto candidate and AV outcomes.

Proposition 7.2: At every preference profile P a Pareto candidate i may be an AV outcome or a tied AV outcome or not an AV outcome at her critical strategy profile

Ci (P ) . Proof: At preference profile P let i be a Pareto candidate. If every voter votes only for her top choice, then if i is chosen at top position by most of the voters, at critical strategy profile Ci (P ) , i will be an AV outcome. Consider at the critical strategy profile Ci (P ) , i is a Pareto candidate, and also consider another candidate j who is not a Pareto candidate; then both cast equal but more votes than other candidates. At this situation i tied with j, and both of them are AV outcomes. If more voters place i at the last position, and few voters place i at first position then i is a Pareto candidate; and at critical strategy profile

Ci (P ) , i will not be an AV outcome. Q.E.D.

Proposition 7.3: Condorcet winners must be an AV outcome, but Condorcet loser sometimes be an AV outcome. Proof: Assume i is a Condorcet winner. In this case more voters place i as their best choice, and less voters place her as their worst choice. Again those voters who place i

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among their best and worst choices, then more voters prefer i to any other candidate j. Hence, total votes of i will be more than j. As a result i will be always an AV outcome. Again now assume i is a Condorcet loser. In this case two situations arise. First, at

Ci (P ) sometimes candidate i collects more votes than any other candidate j. Since at Ci (P ) , those voters who place i at the last place their first choices will be render in different candidates j, k, etc. (say), and those voters who will place i as their best choice will collect votes only for i. Finally, i will collect more votes than any other candidate j. Consequently, i will be an AV outcome. Second, at Ci (P ) , if i collects less votes than any other candidate j, obviously i will not be an AV outcome. Q.E.D. In voting system if voters vote for a predetermined number of candidates then it is called fixed rule. No fixed rule may elect a unique Condorcet winner, but flexibility of AV is needed to elect a unique Condorcet winner (Brams & Sanver, 2005). The following two Examples 7.2 and 7.3 illustrate the above concept clearly. The first example is due to Moulin (1988).

Example 7.2: Let us consider 17 voters and 3 candidates election being as follows: Type 1: xPyPz by 6 voters, Type 2: yPxPz by 4 voters, Type 3: yPzPx by 4 voters, Type 4: zPxPy by 3 voters. Vote for one candidate elects candidate y, and vote for two candidates also elect candidate y defeating candidate x. Candidate x is a Condorcet winner, but by fixed rule x is loser. Therefore, no fixed rule elects the unique Condorcet winner.

Example 7.3: Let us consider 6 voters and 3 candidates election is as follows: Type 1: xPyPz by 3 voters, Type 2: yPxPz by 2 voters, Type 3: zPxPy by 1 voter.

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Clearly, x is a Condorcet winner. Here different number of voters vote for different number of candidates elect x. Thus, flexibility of AV may be needed to elect a unique Condorcet winner.

Proposition 7.4: No fixed rule may elect a unique Condorcet winner, but flexibility of AV may be needed to elect a unique Condorcet winner. Proof: Assume i is a unique Condorcet winner in a preference profile P. According to fixed rule voters may vote for one candidate or they may vote for two candidates, and so on. Consider another candidate j which is not a unique Condorcet winner. If more voters’ first choice is j, then j will be an AV outcome, but not i, the unique Condorcet winner. Similar result arises if voters vote for two candidates, and so on. Hence, no fixed rule may elect a unique Condorcet winner. If the voters do not follow fixed rule and different voters vote for different number of candidates may elect i, i.e., the flexibility of AV may be needed to elect a unique Condorcet winner. Q.E.D.

Proposition 7.5: At a preference profile P, a candidate elected by STV is an AV outcome, but the converse is not true. Proof: Consider two candidates i and j. Let, candidate i always collects more votes than candidate j. According to STV, j will be eliminated and i is an STV winner. Obviously, candidate i is an AV outcome. Therefore, an STV outcome is also an AV outcome. Again suppose candidate i is an AV outcome. In critical strategy profile C j (P ) , j may be AV outcome, but in STV it is impossible. Q.E.D.

There are two kinds of stability as follows: •

Given a preference profile P, a non-tied AV outcome is stable if there exists a strategy profile S such that no voters of a single type have an incentive to switch their strategy to another sincere strategy in order to induce a preferred outcome.



Given a preference profile P, an outcome is strongly stable if there exists a strategy profile S such that no types of voters, coordinating their actions, can form

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a coalition χ, all of whose members would have an incentive to switch their AV strategies to other sincere strategies in order to induce a preferred outcome.

We will cite the following example (Brams & Sanver, 2005) to clarify these two definitions as follows:

Example 7.4: Let us consider 9 voters and 3 candidates election is as follows: Type 1: xPzPy by 4 voters, Type 2: yPzPx by 2 voters, Type 3: zPyPx by 3 voters. In Example 7.4 neither candidate x nor candidate y is stable AV outcome. The critical strategy profile, Cx (P ) = (x, x, x, x, y, y, z, z, z) renders candidate x an AV outcome if 2 voters of Type-2 switch to strategy yz; candidate z, whom the Type-2 voters prefer to candidate x, wins. In C y (P ) = (x, x, x, x, y, y, zy, zy, zy) that renders candidate y an AV outcome; the Type-1 voters have an incentive to switch to strategy xz to induce the selection of candidate z, when they prefer to candidate y. In Cz (P) = (xz, xz, xz, xz, yz, yz, z, z, z); AV outcome z is obviously stable, since candidate z is the unanimous choice of all voters at the critical strategy profile of z. Here it is not possible to switch on the part of the 4 voters of Type-1 to x, or 2 voters of Type-2 to y, or 3 voters of Type-3 to zy. Here we assume that the coordinating players in χ are allowed to communicate to try to find a set of strategies to induce a preferred outcome for all of them. In Example 7.1 at Cx (P ) , 3 voters of Type-2 cannot upset the outcome by switching from yzx to yz or y, nor can the 3 voters of Type-2 upset the outcome by switching from u to uy or uyz. But, if there two types of voters cooperate, and form a coalition χ, with 3 voters of Type-2 choosing strategy y, and the 3 voters of Type-3 choosing strategy uy, they can induce the selection of Condorcet winner y, whom both types prefer to candidate x. Hence, at Cx (P ) , AV outcome x is stable, but not strongly stable at C y (P ) . If an AV outcome is neither strongly stable nor stable, it is called unstable (Mohajan, 2011e).

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Proposition 7.6: At critical strategy profile P of candidate i is Ci (P ) , then a non-tied AV outcome i is stable. Proof: If a candidate i is tied with any other candidate j at Ci (P ) then i is not stable, since i’s votes renders and j switch to win in the election. If at Ci (P ) , i is non-tied with any other candidate j then according to the definition of Ci (P ) , i is non-renders by any other candidate j. Q.E.D.

7.6 Median Voter Model The weak form of the Median voter theorem (MVT) says that the median voter casts her vote in favor of the outcome that wins in the election. The strong form of the MVT says that the median voter always gets her most preferred policy (Islam et al., 2011b).

7.6.1 Single-Peakedness and Single-Crossing We have two basic versions of the MVT: (i) Single-peaked preference (Black, 1958) and (ii) Single-crossing property (Gans & Smart, 1996). These two versions are as follows:

Single-peakedness: Single-peaked preferences have played an important role in the literature ever since they were used by Black (1948) to formulate a domain restriction that is sufficient for the exclusion of cycles according to the majority rule. A set of preference relations is single-peaked if there is linear order of the alternatives such that every preference relation has a unique most preferred alternative or ideal point, over this ordering, and the preference for any other alternatives

monotonically decreases by

moving away from the ideal point. Now consider three voters V1 , V2 , and V3 (Figure 7.1). The utility of each voter depends from their ideal point which gives a single peak for each voter. Therefore, they have a single- peaked of the preferences. Single-peakedness is the oldest and probably the best known restriction on individuals’ preferences guaranteeing the existence of voting equilibria (Black, 1958).

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Utility

V1 V2 V3



L

x

y





z

R

Figure 7.1: Single-peaked preferences. Single-Crossing Property: For x, y  Y , we may write x  y to mean that x is left to y in the spatial voting model. Let the voters’ preferences are transitive ordered in some political spectrum say from leftist to rightist (Myerson, 1996). We mean i  j that voter i is to the left of voter j in this political spectrum. For any two voters i and j such that i  j , for any two policy alternatives x and y such that x  y , if ui (x )  ui ( y ) then u j ( x )  u j ( y ) but if u j (x )  u j ( y ) then ui (x )  ui ( y ) . This assumption is called the single-crossing (SC) property. It does not exclude individuals’ preferences, but which do not monotonically decrease on both sides of the ideal point as single-peakedness does. We can also define an easier way SC as follows (Saporiti, 2009): Let, > is linear order of Y, and  is a linear order of SC, and SC  P(Y ) . For all x, y  Y and for all P, P  SC the single-crossing property N

indicates,

 y  x,

P  P & yPx  yPx &

 y  x,

P  P & xPy  xPy .

If the number of voters is odd and their order is complete and transitive, then there is some median voter M such that;

#  i  N : i  M  =#  j  N : M  j . For any pair of alternatives x, y  Y such that x  y , if the median voter M prefers x then all voters to the left of the median voter agree with him, but if the median voter prefers y

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then all the voters to the right of the median voter agree with him. In both cases majority grows where median voter supports. Analysis of Top-Monotonicity (Barbera & Moreno, 2011): For all i  N for any A  Y we denote by t i ( A) the set of maximal elements of Ri on A. So that

t i ( A) = x  A : xRi y, y  A . We call t i ( A) the top of i in A. When t i ( A) is a singleton, it will be called individual i’s peak on A. Now if we will weaken the notion of single-peakedness about indifferences by considering that there are more individuals which are indifference then majority rule may destroy. But, good news is that in this case does not create any cycle, and yet Condorcet winner exists. Before define top-monotonicity first we define these types of preferences, such as, single-plateaued and order-restricted preferences as follows (Islam et al., 2011b): Single-plateaued: A preference profile P is single-plateaued iff there exists a linear order > of the set of alternatives such that;

a) the set of alternatives in the top of each of the voters is an interval

t i ( A)= [ pi− ( A) , pi+ ( A) ] relative to >, called the plateau of i, and b) for all i  N , for all t i ( A) , and for all y , z  A ; [ z  y  pi− ( A) or z  y  pi+ ( A) ]

 yPi z . In this situation we yet can say that Condorcet winners exist under singlepeakedness preferences, and that they coincide with the median(s) of the distribution of the voters’ peaks. We see that single-peakedness and order-restriction are equivalent, and have been proven by Gans and Smart (1996). Now we are in a position to define topmonotonicity condition.

Top-monotonicity: A preference profile R is top-monotonic iff there exists a linear order > of the set of the alternatives, such that for all A  Y for all i, j  N , all x  t i ( A) , all y  t j ( A) , and any z  A ,

z  y  x

or z  y  x  yRi z if z  t i ( A)  t j ( A) and yPi z if z  t i ( A)  t j ( A) .

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We observe that when we compare top-monotonicity with single-peakedness and single- plateauedness, it represents a significant weakening of these conditions. Finally, we can say that top-monotonicity satisfies MVT. In the following two examples we will see that single-peakedness and SC do not satisfy simultaneously, but both satisfy top-monotonicity. As SC is equivalent to orderrestriction, then SC can be changed to order-restriction (Barbera & Moreno, 2011). Utility

P1

P2

• z

• y

P3

• x

• u

Figure 7.2: Preference profile (Example 7.5). Example 7.5: In this example we will see that the given preference profile satisfies single-peakedness, but not SC. Let, A = x, y, z, u and N = 1, 2, 3 . The preference relations being as follows: zPyPxPu for individual 1, yPxPuPz for individual 2, xPyPzPu for individual 3. This preference profile can be expressed as in Figure 7.2. Individual 1’s peak is z, individual 2’s peak is y, individual 3’s peak is x relative to z  y  x  u . But, the profile violates SC relative to z  y  x  u , for any order P of the individuals. If P2  P3 then we see that xP3 y , and yP2 x which violates SC. Again if P3  P2 then we can write

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uP2 z , and zP3u which is also the violation SC. By the definition, this example satisfies top-monotonicity relative to z  y  x  u .

Example 7.6: In this example we will see that the given preference profile satisfies SC, but not single-peakedness. Let, A =  x, y, z  and N = 1, 2, 3 . The preference relations being as follows: xPyPz for individual 1, xIyPz

for individual 2,

zPxIy for individual 3. This preference profile can be expressed as in Figure 7.3. For P2  P1 we can write yP2 z , and yP1 z , also for P3  P2 we can write xI3 y , and xI2 y . So that preference profiles satisfy SC on A, relative to x  y  z . From Figure 7.4 we see that individuals 2 and 3 have no single-peak or single-plateau. But, according to the definition the preference profile this example satisfies top-monotonicity relative to x  y  z . Utility

P1

P2

• x

P3





y z Figure 7.3: Preference profile (Example 7.6).

Example 7.7: Let, A =  x, y, z, u  and N = 1, 2, 3 . The preference relations being as follows: This preference profile can be expressed as in Figure 7.5. Here individual 3 has no single-peak. If P3  P2  P1 then xP1 y , and yP2 x , but xP3 y . If P2  P3  P1 then

zP1u , and uP3 z , but zP2 u . At last if P3  P1  P2 then uP2 y , and yP1u , but uP3 y .

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Hence, preference profile is not order-restricted, and not SC. But, by the definition the preference profile is top-monotonic relative to x  y  z  u .

Utility

P1 P2 P3

• x







y z u Figure 7.4: Preference profile (Example 7.7).

7.7 Majority Judgment Voting In Arrow’s impossibility theorem preference relation xPy for individual 1 and individual 2 express same preferences where x and y are two candidates (Arrow, 1963; Islam et al., 2011a). But, in majority voting xPy may give different results. Suppose in majority judgment voting both of the individuals’ preference is xPy. Individual 1 expresses x is Good, and y is Rejected. On the other hand, individual 2 expresses x is Excellent, and y is Very Good. We see that this gives more accurate information of the voters’ opinion. Balinski and Laraki (2006, 2007, 2010, 2014) first have introduced this type of judgment which is the median values of the grades given to a candidate is taken as the final grade of that alternative. They argue that an individual should choice the middlemost aggregation functions, and call the resulting system majority judgment. Let, a finite set of m candidates is defined by C =  c1, c2 ,..., cm , and a finite set of n voters is defined by V =  v1, v2 ,..., vn . Let, G =  g1 , g 2 ,..., g k  be the set of grades where

g1  g 2  ...  g k . Here gi  g j means that g i is the higher grade than g j or gi = g j . An maj

input profile is m×n matrix (g ij ) of grades where each row i contains the grades provide by the voters to candidate c i , and each column j contains the grades vector v j assign to

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the different candidates. A social grading function F is defined by F : G m n → G m such that (Zahid & Swart, 2015);  g11 g12 ... g1n     g 21 g 22 ... g 2 n   . .. ... ... ...    → ( f (g11, g12 ,..., g1n ) ,..., f (g m1 , g m 2 ,..., g mn )) .  . .. ... ... ...     . .. ... ... ...  g   m1 g m 2 ... g mn 

(7.1)

That is, F (a ) = ( f maj (c1 ),..., f maj (cm )) , where f maj (ci ) is the majority grade which is the median value of the candidate ci . The k th order function f k indicates an n-tuple of grades, and supplies an output the k th highest grade (Balinski & Laraki, 2007). When the number of voters is odd, the majority grade is the median. If the number of voters is even, then the lower of the two middle grades must be the majority grade. Let, a be the number of grades is given to a candidate above its majority grade  , and b be the number of grades is given to a candidate below its majority grade  . Hence, we can write the majority grade  * as follows (Mohajan, 2012a):  + if a  b ;   * =  0 if a = b ;  − if a  b. 

7.7.1 The Majority Count of Borda The Borda rule (Borda, 1781) belongs to the class of point ranking rules where points are given to each candidate or alternative according to her rank in the preference of the voters. The majority count of Borda is a function M (a ) : G mn → N m which assigns to any profile the output (M (c1 ),..., M (cm )) , where M (ci ) is the majority count of Borda of candidate ci . Let ‘c’ be a candidate, and let G =  g1 , g 2 ,..., g k  be the set of grades where

139

g1  g 2  ...  g k . Let vi be the number of voters, then the Borda majority count (M) can be defined by (Mohajan, 2012a);

M (c ) = v1  0 + v2  1 + ... + vk  (k − 1) k

=  vi  (i − 1) .

(7.2)

i =1

We assign 0 point to the R grade, 1 point to the P grade, 2 points to the A grade, 3 points to the G grade, 4 points to the VG grade, and 5 points to the E grade. Hence, this system is the Borda count.

7.7.2 Tie-Breaking in Majority Judgment The general majority ranking  between two competitors ci and c j is determined maj

as follows:

( )



If f maj (ci )  f maj c j , then ci  c j .



If f maj (ci ) = f maj c j , then we drop one majority grade from the grades of each

maj

( )

competitor.

If the tie is not broken then the procedure is repeated step by step, by dropping grades from lower to higher until we receive a winner between ci and c j . Now we set an example related to this type of tie-breaking as follows:

Example 7.8: Let there are two candidates x and y. They make tie in an election and the tie-breaking procedures are given as follows: Table 7.2a: Tie-breaking by the majority judgment.

x y

a 37 36

E 19 17

VG 18 19

G 24 25

A 2 20

P 17 15

R 20 4

b 39 39

Total 100 100

In the Example 7.8, both x and y have the majority grades at G. The type of tie is G − .

(

)

According to simple tie-breaking case the majority value of x is 37, G − ,39 , and for y is

140

(36, G ,39). Since 37>36, obviously x is the majority judgment winner in the election. −

Now we apply the general tie-breaking rule for the Example 7.8. First, we drop the E grade and then we obtain the Table 7.2b from Table 7.2a as follows:

Table 7.2b: First step of tie-breaking by the majority judgment.

x y

a 18 19

VG 18 19

G 24 25

A 2 20

P 17 15

R 20 4

b 39 39

Total 81 83

(

)

Hence, from Table 7.2b we see that the majority value of x is 18, G − ,39 , and for y is

(19, G ,39). Since 19>18, obviously y is the majority winner in the general tie-breaking −

rule. In the case of large elections Balinski and Laraki (2007) introduce another type of tie breaking rule. Candidate’s three majority values are sufficient to determine the candidate’s position in the majority ranking as follows:

a = the number of grades above the majority grade, (a, , b) where  = majority grade, and b = the number of grades below the majority grade .  Now we set an example related to this type of tie-breaking for a large election as follows: Example 7.9: Let us consider an election where there are two candidates x and y, and 1259617 voters. The results are given in Table 7.3, and we see that there is a tie in G+.

(

)

(

)

The majority value of x is 361572, G + ,199285 , and for y is 361572, G + ,353725 . Table 7.3: Second step of tie-breaking by the majority judgment, Example 7.9. a E VG G A P R b total x 361572 158976 202596 698760 100000 91485 7800 199285 1259617 y 361572 162848 198724 445320 240034 104742 8949 353725 1259617 Since 353725>199285, according to Balinski and Laraki (2007) the winner in the election is y, but according to the general tie-breaking rule the winner is x (Balinski & Laraki, 2006). From the above examples we observe that in majority tie-breaking winner in the election depends on which method is followed in the tie-breaking process. 141

7.7.3 Tie-Breaking in Borda Majority Count Sometimes there is a tie in M, and then the tie can be broken by dropping the R grade and re-calculating for the M. Again if the tie arises in this case the P grade must be dropped, and re-calculate for the M. This process is continued by dropping grades step by step from lower to higher until the tie is broken. The candidate, who acquires the greatest M, is the Borda majority winner. The following example shows the procedures of tie breaking in M (Zahid & Swart, 2015): Example 7.10: Let us consider 100 judges who give their judgments for three candidates

c1 ,c2 , and c3 as in the Table 7.4. In Table 7.4, we see that all the three candidates with Borda score 310 tied in M-1. Hence, we drop the R grades of all the candidates and then re-calculate for M, but same condition arises in M-2. At this situation we drop the P Table 7.4: Tie Breaking in M.

c1 c2 c3

E 11 13 13

VG 33 34 29

G 21 18 22

A 29 24 31

P 2 7 1

R 4 4 4

M-1 310 310 310

M-2 214 214 214

M-3 120 125 119

grades of all the candidates, and after re-calculating for M we observe that now tie breaks, and candidate c2 with highest score 125 wins in the election.

7.7.4 Drawbacks in Majority Judgment Voting No voting method is stainless, so that majority judgment voting also has some counter-intuitive results. Some drawbacks of majority judgment voting are discussed with some examples as follows:

Example 7.11: In majority judgment sometimes the winner may lose. Consider there are 1,000 voters, and two candidates x and y. Voters rank the two candidates as in the Table

(

)

7.5. Here the majority grade of x and y is G + . The majority value of x is 500, G + ,0 , and

(

)

the majority value of y is 510, G + ,290 . According to both tie-breaking rules y is the

142

winner. Here no voter gives x lower grade than G, but 290 voters do the job for y. Here y is the winner because 10 extra voters vote VG to y, and not taking into account the 290 voters evaluate y as lower than G. Here x’s performances are G or better than G; and y’s performances are not so, but according to majority voting y wins. If we consider nonmajority judgment voting system then obviously x would win. Hence, in majority judgment voting winner may lose. Table 7.5: Majority judgment for Example 7.11.

x y

a E VG G A 500 200 300 500 0 510 200 310 200 160

P 0 70

R 0 60

b total 0 1000 290 1000

Example 7.12: In majority judgment sometimes looser may be the majority judgment winner. Now we consider a 100 round competitive contest where four players x, y, z, and w are competitors. They play 100 rounds, and one judge gives them grades as Table 7.6 (Mohajan, 2012a): Table 7.6: Majority judgment for Example 7.12.

x y z w

a 50 50 50 0

E 20 0 0 0

VG 10 30 10 0

G 20 10 10 0

A 0 10 20 0

P 0 0 10 70

R 50 50 50 30

b 0 0 0 30

At the first sight we can say that w is the looser, and x is the winner because x performs five times G or better than G, and w performs 0 in A or better than A. But, according to Balinski and Laraki (2006, 2007) the lower majority middle grade be the majority grade. The majority grade of x, y, and z is R, but the majority grade of w is P, so that w is the majority judgment winner. Hence, in majority judgment voting sometimes looser may be the majority judgment winner.

Example 7.13: In two cities electoral cases majority judgment violates both winner consistency and rank consistency. First, we consider the following example with candidates x and y, and the set of grades be {0, 1, 2, 3, 4, 5, 6}. 143

For city I,

Table 7.7a: Majority judgment for city I. Candidates Scores x 6 4 y 6 6

3 2

1 2

0 0

Majority grades 3 2

In city I, x  y , and x is ranked above y, so that x wins in the election (Table 7.7a). maj

For city II, Table 7.7b: Majority judgment for city II. Candidates Scores x 6 6 y 6 5

5 4

1 4

0 1

Majority grades 5 4

In city II, x  y , and x is ranked above y, so that x wins in the election (Table 7.7b). In maj

the combination of two cities elections, y  x , and y is ranked above x, so that y wins in maj

the two cities combination elections (Table 7.7c).

Table 7.7c: Majority judgment of combination of city I and city II. Candidates x y

Scores 6 6 6 6

6 6

5 5

4 4

3 4

1 2

1 2

0 1

0 0

Majority grades 3 4

Example 7.14: In an election two friends A and B have different opinions. A supports candidate x and B supports candidate y. The set of grades be {0, 1, 2, 3, 4, 5, 6}. Voter A decided to give the highest grade 6 to x, and second highest grade 5 to y. Voter B decided to give the highest grade 6 to y, and second highest grade 5 to x. Both of them think their votes will give no fruitful result, so that they decided not to cast their votes. First, we consider the example due to Bishop (2010). In the election there are two candidates x and y, and their scores are as follows:

144

Table 7.8a: Majority judgment by Bishop. Scores Candidates x 1 y 2

Majority grades

2 3

4 3

4 6

6 6

4 3

Here x  y , so that x wins in the election. Now we apply this result for the case of voters maj

A and B. If they would have vote then adding their votes to the Table 7.8a of Bishop we get as follows:

Table 7.8b: Majority judgment if two opposite voters would vote.

Candidates Scores x 1 2 y 2 3

4 3

4 5

5 6

6 6

Majority grades 4 5

6 6

From the Table 7.8b above we see that y  x , and y wins in the election. In Bishop’s maj

example if both A and B votes 6 grade for x, and 5 grade for y; the voting situation

Table 7.8c: Majority judgment if two voters would vote in same grade. Candidates Scores x 1 2 y 2 3

4 3

4 5

6 5

6 6

Majority grades 6 4 6 5

becomes as Table 7.8c. Here we observe that y  x , and y is winner in the election. This maj

result is the same in the following two cases: •

If both A and B be with opposite grades supporters, but absent in the election.



If both supporters A and B give the highest rank to x.

This is a drawback of majority judgment voting.

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7.7.5 Theoretical Properties of the Majority Judgment and Borda Count The majority grade of a candidate is her median grade. It is simultaneously the highest grade approved by a majority, and the lowest grade approved by a majority. The winner consistency of an electorate is defined as follows: If there are two separate districts of an electorate, and a candidate wins in both electorates; then he must wins in the combinations of the two districts (Mohajan, 2012a).

Proposition 7.7: Majority judgment voting is not winner consistent. Proof: Let, there are two candidates x and y. The elections are held in two cities. Let, in city-I majority judgment gives x  y , so that x wins in the election. Again let in city-II maj

majority judgment gives the same result, i.e., x is also winner here. If we combine the two cities according to Balinski and Laraki (2010) we observe that y  x always. So that y maj

wins in the combination of two cities election. In Example 7.14 we have obtained the same result. Hence, majority judgment voting is not winner consistent. Q.E.D.

The majority ranking orders the candidates according to their majority grades. The rank consistency of an electorate is defined as follows: If there are two separates cities of an electorate, and the ranking of two candidates x, y in two cities of a consistency are x  y , then in the whole electorate the ranking of the candidates will be the same, that is, x  y , and hence x is the winner.

Proposition 7.8: Majority judgment voting is not ranking consistent. Proof: Let, there is an election in two separate districts A and B, and there are two candidates x and y. Assume in district A majority judgment gives x  y , i.e., x is ranked above y in the election, consequently x wins in the election. Again let in district B majority judgment gives the same result, i.e., x is ranked above y in the election here. If we combine the two districts’ outcomes according to Balinski and Laraki (2010) we observe that y  x always, i.e., y is ranked above x in the election, consequently y wins in the election. Hence, ranks of x higher in both separate districts, but y is ranked above x

146

in the combination of two districts election. Hence, majority judgment voting is not ranking consistent. Q.E.D.

The grade consistency in majority judgment is defined as follows: If there are two separate towns of an electorate, and the majority grade of a candidate in each town is  ; then the majority grade of the whole electorate will be  always.

Proposition 7.9: Grade consistency is satisfied by the majority judgment. Proof: Let, there is an election in two separate towns P and Q, and there are two candidates x and y. According to Balinski and Laraki (2010), let the majority grade of x in town P be  . In another town Q the majority grade of x also be  . Then the majority grade of x in the combination of P and Q must be  . Q.E.D.

The Borda majority count has no so many drawbacks as the majority judgment has. It also contradicts the Propositions 7.7 and 7.8. So that it is both winner and rank consistent. Hence, we see that even in the 21st century politicians cannot present a voting system which is better than the 18th centurion politicians Condorcet and Borda provided (Mohajan, 2012a).

7.8 Conclusion In this chapter we have described various types of voting system, and analyze various types of manipulable and non-manipulable voting systems using some easier methods. A Condorcet method is a single-winner election method which always selects the Condorcet winner. We have discussed Borda voting with some detail calculations. The STV minimizes the wasted votes than any other voting method. The main drawback of STV is in vote counting system. Again the STV is not manipulation free, which is also a drawback of the system. Another drawback of STV is tie-breaking, and we have shown different methods of tie-breaking. Here we have discussed AV with strategies and stability of it. We have discussed aspect of MVT using some propositions, examples, and diagrams. We investigate the majority judgment by mathematical calculations displaying a series of tables, examples and propositions with proof. We have shown that majority

147

judgment voting is not winner and rank consistent. Voting system is a very complicated field, but we have tried our best to make it easier. In this chapter we have included examples, tables, and propositions with proof very clearly with detailed mathematical calculations where appropriate. We have shown that some methods have Condorcet winner, where there is no voting manipulation, and the individuals sincerely declare their preferences.

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Chapter–VIII

Environmental Pollution and Healthcare

8.1 Introduction This chapter is about environmental pollution, healthcare, and sustainable economy. It contains a beneficial economic model where government pursues optimal economic policies. The aim of this chapter is to serve as an indicator of wealth changes, performance of environment policy, and sustainable use of natural capital. The chapter also indicates the present and future production, use of ecosystem services, and sustainable use of natural resources. Here we have related wealth to society’s capital asset (Heal & Kriström, 2002). Due to air pollution human capital cannot be utilized properly, and net national product (NNP) of a country decreases. Healthy human capital gives optimal product in the society, and decreases all kinds of medical expenditures related with pollutions. In the study about human capital in macroeconomics shows that not only education but also good health has a significant positive effect on aggregate output (Bloom et al., 2001; Nordhaus, 2002). Medical expenditures to cure diseases due to air pollution should not be deducted from NNP, but hamper of production due to pollution related illness should be subtracted from NNP (Dasgupta & Mäler, 2000). Willing to pay (WTP) to avoid 1 and 14 additional days is important both for workers and factory authorities. It helps to mitigate respiratory related diseases which are mainly caused by air pollution. Since no WTP is yet established in the labor sector of Bangladesh, so that this chapter will show new ways to the people of this country (Mohajan, 2012c).

8.2 Environmental Accounting and Roles of Economics In our environmental economic model we consider the production of goods and services where we require labor, manufactured capital, and natural resources. Natural

149

capital consists of a variety of ecosystems, such as, wetlands, lakes, forests, agricultural landscape, and coastal water (Mohajan et al., 2012). Here we consider the time t  0 is continuous, where t = 0 denotes the present time, and t  0 denotes the future time. For simplicity, all marketed goods and services are supposed in the compounded good G which uses natural capital N  0 (at a finite time t) as a production factor. Environmental services are represented by the single compounded resource R. The marketed good and environmental services also need man-made capital K, and P is emit pollutants, as by products, which are treated as inputs into production of all marketed goods. Let, K 0 and N 0 be the initial stocks. The all purpose goods can be produced with its man-made capital K, and environment resource R. Hence the production function can be written as; G = G(K , R) . Again environmental resource R, can be expressed as a function of natural capital N, and environment pollution P, i.e.,

R = R(N , P) . Now the production function can be rewritten as; G = G(K , N , P) . We assume that G is an increasing, non-concave, and continuously differentiable function each of its variables. Natural capital can be changed by emitted pollution to produce goods, and concentration by environment. The change in natural capital is determined by its own growth g, ecosystem management and pollutant deposition, i.e., g = g (N , P) , where it is assumed that g N  0 and g P  0 (Mohajan, 2013c). Ecosystem management is made at the expenditure X which depends on N, and assumes to be increasing and convex in N. Let, C  0 denotes aggregate consumption at time t, then the net accumulation of physical capital satisfies the condition;

dK = K = G (K , N , P ) − C − K − X ( N ) , dt

(8.1)

dN = N = g (N , P ) , dt

(8.2)

where δ is the capital depreciation rate. In some cases the emission of pollutants amounts directly to a degradation of ecosystems (Dasgupta & Mäler, 2000, 2001). Hence, resources are good and pollution which is degrader of resources is bad. Let, the natural rate of the resource base be D(R) which is continuously differentiable function. We can augment it by the expenditure X. The expenditure X

150

consists of the costs in the case of minerals and fossil fuels, clean up costs in the case of polluted water, etc. Now we define, t

Z=

 Edt

(8.3)

−

where Z would be the measure of stock at time t. In differentiate form we can write (8.3) as; dZ / dt = E . Let us consider Q (X, Z, N) the rate at which the augmentation occurs. Here Q is continuous and differentiable where QX  0 , and QZ  0 . The dynamics of the resource base can be expressed as (Arrow et al., 2002, 2010; Dasgupta, 2008, 2010; Dasgupta & Mäler, 2000, 2001);

dN = D( N ) − R + Q( X , Z , N ) . dt

(8.4)

The utility in the society is determined by consumption of both marketed goods and services of resource, i.e., U = U (G, R). Again pollutants (P) affect utility directly through its impact on health; therefore, the utility can be written as; U = U (G, R, P) which is assumed to be non-decreasing in all its arguments except P. Marketed and non-marketed goods G is the consumption of goods C, so that we can express the utility as;

U = U (C, R, P).

8.2.1 Sustainable Development of Wealth The social welfare for t  0 is determined by current and discounted future streams of utility as follows: 

W =  U (C , R, P ) e −rt dt ,

(8.5)

t =0

where r is the utility discount rate. Now we consider a time autonomous problem, then (8.5) can be written in terms of initial stock parameters as;

W = W ( K , N , ) where N  R n , (n-dimensional Euclidean space), θ is an optimal resource allocation mechanism which describes the institutional set up for allocating resources among goods and services. The value of change in wealth for t > 0 can be defined as;

151

dW dW dK dW dN , = . + . dt dK dt dN dt

dW = K + N , dt

(8.6)

where λ is the shadow price of capital, and ν is the accounting price of the natural asset which can be derived by the maximization of the Hamiltonian (will be discussed below). World Commission (1987) defined sustainable development as; “Sustainable development is an economic program in which, lightly speaking, the wellbeing of future generations is not jeopardized.” For sustainability we can write (8.6) as; dW = K + N  0 . dt

Again we can define sustainability as (Arrow et al., 2010; Dasgupta, 2007, 2010): “Sustainable development is an economic program along which average wellbeing of present and future generations, taken together, does not decline over time.” An economic development is sustainable if (Dasgupta & Mäler, 2000, 2001),

dU 0 dt

(8.7)

which offers greater flexibility in ethical reasoning. It permits initial sacrifices in the current standard of living, but requires that no future generation should have to experience a decline in their standard of living. If we consider the utility be a function of consumption, C and labor, L then we can write (8.7) as;

dU dC dL = UC +UL  0. dt dt dt

(8.8)

Hence in this situation we can write the sustainability as (Dasgupta & Mäler, 2000, 2001): “If sustainable development is taken to mean that, starting from now, utility must never decline, then an economic program corresponds to sustainable development if and only if, the value of changes in the flow of consumption services is always non-negative.”

8.2.2 Green National Accounts We can write the Hamiltonian as (Dasgupta, 2008; Gren, 2003):

152

H =U +

dW , dt

H = U (C, R, P) +  (G − C −  K − X (N )) + g (N , P) .

(8.9)

Hence we see that all changes in market goods and services are captured by net domestic product (NDP), and can be represented in utility terms actually which is the Hamiltonian (8.9). Taking partial differentiation of (8.9) for maximization we get (Gren, 2003; Mohajan et al., 2012);

UC −  = 0 ,

(8.10)

U R RN + U R + GR + g N = 0 ,

(8.11)

 =  (r − GK +  ) ,

(8.12)

 =  (r − g N ) − U R RN −  (GN − X N ) .

(8.13)

From (8.10) we can write,  = U C , which indicates the marginal utility of consumption equals the shadow price of capital. From (8.11) we see that optimal use of pollutants is determined where marginal benefit from production of marketed and non-marketed goods and services equals marginal cost. Integrating (8.13) we get the accounting price of the natural asset ν (t) for t  0 ; 

 (t ) =  U R RN + U C (GN − X N ) e(r − g

N

)( − t )

d .

(8.14)

0

From (8.14) we see that the accounting price of the natural asset in time t is thus the discounted streams of current and future net utility from marketed and non-marketed goods and services of a marginal change in N(t). In practical life we see that increased stock enhance growth at relatively low stock level, but at larger levels a further increase in the stock may imply a reduction in growth. The current net domestic product, NDP C , can be expressed as (Gren, 2003):    NDP C (t ) = NDP (t ) +  U (P, R ) +  U RU S S e(r − g N )( − t )d  . 0  

(8.15)

where  = 1/ U C . Equation (8.15) indicates that current utility from pollutants and ecosystem services, and change in future utility from ecosystem services caused by the period’s change in the stock of natural capital.

153

8.3 Environment Pollution Decreases Economic Development Social welfare is maximized when producers and consumers maximized their utility in a healthy way. Our social environment is polluted in different ways mainly by air and water which creates different diseases, and we need to invest an extra amount in health sector due to this pollution (Mohajan, 2012f). Assume utility U(C) is obtained from consumption of good C, inputs (labor) in health sector and mitigation is L2 , environment pollution is P, and disutility is D(P, L2 ) such that DP  0 and DL2  0 . Let, the weight is i(q), where i is a function of personal characteristic q. The additional demand for services of healthcare sector h( L2 ) due to environment pollution is modeled by j(P), where hl  0 and jP  0 , so that j(P)i(q)h( L2 ) constitutes the unnecessary consumption of healthcare services due to pollution (Huhtala & Samakovlis, 2003). The net utility in the presence of pollution is, NU = U (C ) − D(P, L2 ) . Therefore, the aggregated net utility, discounted by a constant interest rate r, is maximized (Huhtala & Samakovlis, 2003) 



0

0

max  NUe− rt dt = max  U (C ) − D(P, L2 ) e− rt dt subject to

K = f (K , L1, P) − C − j (P) i(q) h(L2 ) −  K

(8.16)

where K (0) = K0 is given initial level of capital,

n(P)L = L1 + L2 . Here, K = stock of capital, δ

= depreciation rate of capital stock,

L  = total labor available in the economy,

L1 = labor input used in producing the consumption commodity C, L2 = labor input used in healthcare sector, f

= production function for the composite commodity, where f K , f L1 , f P  0 , and

n(P) = the effect of air pollutants on the productivity of labor, where nP  0 .

154

Without environment pollution there is no additional demand for healthcare, i.e., j(0) =1, and if environment pollution exists then, j(P)>1. Similarly without pollution, there is no productivity adjustment that is, n(P) = n(0) = 1, and if environment pollution found then 0 0. For maximization we get;

 * u (C* (0)) + W (K 0 ) I* (0) =  * (u(C* (0)) + ψ(0) I* (0)) , where  *  0 . (9.5a)

172

The term u C* (0) + Ψ(0) I* (0) is the net national product in terms of utility (utility NNP). For both cases discounted utilitarianism and maximin, utility NNP represents dynamic welfare globally. For optimal growth equation (9.1) maximizes, and maximized welfare at time t can be written as; 

(

)

W (t ) =  u C* ( ) e−  ( −t ) d . *

(9.6)

t

Proposition 9.1: Utility NNP represents dynamic welfare globally. Proof: From the definitions of discounted utilitarianism and maximin it is realized that the term u C* (0) + Ψ(0) I* (0) is the NNP in terms of utility (utility NNP). The discounted utilitarianism in (9.4) indicates that utility NNP is maximized, and hence dynamic welfare is maximized. Again the maximin in (9.5a) expresses (C*(0), I*(0)) maximizes

 * u (C) + W (K 0 ) I subject to (C, I)  S(K0), for ρ* > 0. As a result utility NNP maximizes, which is the dynamic global welfare. Q.E.D. Let Ψ(t ) represent the trajectory of the dual vector of shadow investment prices, relative to utility being the numeraire. The current-value Hamiltonian is given by (Asheim & Weitzman, 2001):

H (C(t ) , I(t ) ; Ψ(t )) = u C(t ) + Ψ(t ) I(t ) for all t.

(9.7)

Now (C*(t),I*(t)) maximizes H(C(t),I(t) ; Ψ(t ) ) subject to (C(t), I(t))  S(K*(t)) as follows:

H * (t ) = H (K*(t), Ψ(t ) ) =

max

(C(t ) ,I (t ))S (K* (t ))

u C(t ) + Ψ(t ) I(t ) = u C* (t ) + Ψ(t ) I* (t ) .

(9.8)

Since Ψ(t ) I* (t ) is the value of net investments so that the co-state differential equation can be written as (Asheim & Weitzman, 2001);

 (t ) .  HK (K*(t), Ψ(t ) ) =  Ψ(t ) − Ψ Since  = −

(9.9)

 (t ) then (9.9) becomes,  (t )

 HK (K*(t, Ψ(t ) ) = −

 (t )  (t ) . Ψ(t ) − Ψ  (t )

(9.9a)

Differentiating (9.8) with respect to t, and using (9.9) we get;

(

)

 (t ) I* (t ) +  H Ψ  (t ) =  Ψ(t ) − Ψ  (t ) . H * (t ) =  HK I * (t ) +  H Ψ Ψ Ψ

173

(9.10)

Differentiating right side of (9.8) with respect to t we get;

 * (t ) + d (Ψ(t ) I* (t )) . H * (t ) = u (C* (t )) C dt

(9.11)

From (9.10) and (9.11) it follows that:

 * (t ) + u (C* (t )) C

d ( Ψ(t ) I* (t )) =  Ψ(t ) I* (t ) . dt

(9.12)

Differentiating (9.6) and using (9.12) we get; 

(

)

d W * (t ) = − Ψ( ) I* ( ) e−  ( −t ) d = Ψ(t ) I* (t ) . d  t

(9.13)

Proposition 9.2: Ψ(t ) be the trajectory of the dual vector of shadow investment prices, relative to utility being the numeraire and I * (t ) is the maximum investment then maximized welfare increases if and only if Ψ(t ) I* (t )  0 . Proof: Suppose welfare increases, that is, W * (t )  0 . From the definition of shadow prices we can write, Ψ(t )  0 for each t. If the total invest I(t ) is maximized then maximized investment becomes I* (t )  0 for each t. Accordingly, Ψ(t ) I* (t )  0 . Conversely let, Ψ(t ) I* (t )  0 then the social development will increase, and the society will gain maximum welfare, and maximum welfare will increase continuously. Consequently, W * (t )  0 . Q.E.D. Proposition 9.2 indicates that maximized welfare is increasing if and only if Ψ(t ) I* (t )  0 . Moreover it indicates that welfare is increasing if and only if measurable NNP exceeds the value of consumption, this is a different kind of welfare significance than which was shown by Weitzman (1976), where higher welfare is indicated by higher NNP (Mohajan, 2011f). If C( ) =t is the implemented path given the initial stock K(t) = K, then the 

dynamic welfare of this path, 

V (K ) =  u (C( )) e−  ( −t ) d , t

174

(9.14)

is a function solely of K. In particular,

d (V (K (t ))) = V (K (t ) I(t ))  0 dt

(9.15)

means that dynamic welfare is increasing at time t. Again the partial derivatives of V in accounting prices can be written as (Asheim, 2003); q(t ) =

V (K (t )) Ψ(t ) , =  (t )  (t )

(9.16)

where  (t ) >0 is the not-directly-observable marginal utility of current expenditures, which may depend on the ‘quantity of money’ at time t. From (9.16) we can write; q(t ) I(t ) =

1 d V (K (t )) I(t ) (V (K (t )))  0 . =  (t )  (t ) dt

(9.17)

Here q(t)I(t) represents the value of net investments, and it is often referred to as the genuine savings indicator (Hamilton, 1994). Hence q(t)I(t) indicates that dynamic welfare is increasing. Now consider that resource allocation mechanism is Markovian, then by differentiating (9.14) with respect to time t we get;

u(C(t )) + V (K(t ) I(t )) = V (K(t )) .

(9.18)

Now again differentiating (9.18) with respect to t we get,

 (t ) + (u (C(t ))) C Using  = −

d V (K (t ) I(t )) =  V (K (t )) I(t ). dt

(9.19)

 (t ) and V (K(t )) = Ψ(t ) we can write (9.19) as (Dixit et al., 1980);  (t )  (t ) + d (Ψ(t ) I(t )) = −  (t ) Ψ(t ) I(t ) . (u (C(t ))) C dt  (t )

(9.19a)

Equation (9.19a) indicates that the change in utility NNP equals the supporting utility discount rate times the value of net investments. The consumption prices in terms of the numeraire can be written as (Asheim, 2003);

p(t ) =

(

)

u C* (t ) .  (t )

(9.20)

Using (9.20) in (9.19) we get;

 (t ) + d  (q(t ) I(t )) = r (t ) (q(t ) I(t )) , p(t ) C dt

175

(9.21)

r=−

where

(t )  (t )

(9.22)

is the nominal interest rate at time t. Using (9.22) we get;

 HK =  (t ) r(t ) q(t ) − (t ) q (t ) .

(9.23)

Combining (9.12) with (9.23) we can write (9.21) for maximization as follows:

 * (t ) + d  (q(t ) I* (t )) = r (t ) (q(t ) I* (t )). p(t ) C dt

(9.24)

The comprehensive NNP in nominal prices, y(t), is the sum of nominal value of consumption and the nominal value of net investment as follows (Asheim & Weitzman, 2001):

y(t ) = p(t ) C* (t ) + q(t ) I* (t ) .

(9.25)

Since the level of NNP in nominal prices at t depends on arbitrary  (t ) , so that (t )  0 cannot signify welfare improvement. From (9.13) we get maximized welfare is increasing if and only if NNP exceeds the value of consumption, i.e.,

W * (t )  0  y(t ) − p(t ) C* (t ) = q(t ) I* (t )  0 . Hence for a change in NNP to indicate a change in welfare, NNP must be measured in real prices. By the application of price index  (t ) nominal prices  p(t ) , q(t ) turns into real prices P(t ) , Q(t ) as follows (Asheim & Weitzman, 2001): P(t ) =

p(t ) q(t ) , and Q(t ) = .  (t )  (t )

Then the real interest rate, R(t ) at time t in terms of nominal interest rate, r(t) of (9.22) is given by; R(t ) = r (t ) −

 (t ) .  (t )

(9.26)

A Divisia price index satisfies P (t ) C* (t ) = 0 . Hence we can write, i.e.,

 (t ) p (t ) C* (t ) = .  (t ) p(t ) C* (t )

(9.27)

Now we define comprehensive NNP in real Divisia prices, Y(t), as the sum of the real value of net investments from (9.25) as follows:

Y (t ) = P(t ) C* (t ) + Q(t ) I* (t ) . 176

(9.28)

Differentiating (9.28) and then using (9.24) we get;

 * (t ) + d (Q(t ) I(t )) = R(t ) Q(t ) I(t ) . Y (t ) = P(t ) C dt

(9.29)

Since Q(t) is proportional to V (K(t )) , then (9.29) implies that Y (t )  0 , and hence it indicates the welfare is improving in the society (Mohajan, 2011f).

Proposition 9.3: For the increase of welfare NNP must be measured in real prices. Proof: If NNP is measured by nominal prices then (9.25) indicates that comprehensive NNP, y(t) is the sum of nominal value of consumption and the nominal value of investment. But, NNP in nominal prices at t depends on arbitrary  (t ) , that is why

y (t )  0 cannot signify welfare improvement. Again (9.29) indicates that real Divisia prices Y (t )  0 is measured in terms of real prices P(t) and Q(t), where P(t) > 0, Q(t) > 0, I(t) > 0, and real interest rate R(t )  0 , hence Y (t )  0 . Consequently, W (t )  0 . Q.E.D.

9.2.2 Growth without Optimality Let us consider the global welfare comparisons (Asheim, 2003), either in one society over time, where K = K(t) is the vector of capital stocks at time t  and K = K(t) is the vector of capital stocks at time t  , or across different societies, where K  is the vector of capital stocks in the one society and K  is the vector of capital stocks in the other society. Under stationary technology and discount utilitarianism we get from (9.14); K 

V (K ) − V (K ) =  V (K ) dK ,

(9.30)

K

is a measure of welfare differences which is independent of the path between K  and

K. Consider that the factor of proportionality equal to one, so that P = u(C ) and

Q = V (K ) , and implying that, u(C) = u(C) C = PC .

(9.31)

Using (9.31) we can write (9.18) as follows:

Y = PC + QI = V (K )

177

(9.32)

Now equation (9.30) becomes; K 

V (K ) − V (K ) =  Q dK .

(9.33)

K

K 

Here

 Q dK

is not a difference in wealth, but rather a wealth-like magnitude, to use

K

Samuelson’s (1947, 1961) term. Equation (9.33) expresses that a positive welfare

 K  difference can be indicated by a positive real value of stock differences   Q dK  0  or  K  by a positive difference in real NNP (Y  − Y   0), (Asheim, 2003).

Proposition 9.4: Without optimality under stationary technology and discounted utilitarianism it is possible to obtain a positive welfare difference by real NNP, but it is not possible by nominal NNP. Proof: In nominal NNP, y(t) is a function of arbitrary  (t ) , so that a positive difference cannot be possible always, and hence ( y − y) cannot be always positive. Equation K 

(9.33) indicates that

 Q dK

is not a difference in wealth, but rather a wealth-like

K

magnitude, to use Samuelson’s (1947, 1961) term. But, a positive welfare difference can

 K  be indicated by a positive real value of stock differences   Q dK  0  . Hence a positive  K  welfare difference can be indicated by a positive difference in real NNP (Y  − Y   0) . Q.E.D.

Proposition 9.4 indicates that if stationary technology and discounted utilitarianism holds, but optimality does not hold, then the value of net investments and real NNP growth are quite unreliable indicators of sustainability (Mohajan, 2011f).

178

9.3 Green Net National Product for the Sustainability and Social Welfare Let, p(t) denotes the present value price of consumption (i.e., the consumption discount factor) at time t, and let q(t) denotes the vector of present value prices of the capital stocks at time t. Assume that (Mohajan, 2011b);

p(t ) = p(0) e− r t

(9.34)

is an exponentially decreasing function, where r is constant interest rate. Differentiating (9.34) with respect to t we get;

p (t ) = −r p(0) e− r t = −r p(t )

r=−

p (t ) . p(t )

(9.35)

The instantaneous interest rate is; r0 (t ) = −

p (t ) , p(t )

(9.36)

and the infinitely long-term interest rate is; r (t ) =

p(t ) 

 p( ) d

.

(9.37)

t

The term;

p(t ) C + q(t ) I + q (t )K ,

(9.38)

indicates instantaneous profit. We can write, Q(t ) =

q(t ) p (t )

(9.39)

for the capital prices in terms of current consumption. Differentiating (9.39) with respect to t we get; q (t ) = r0 (t ) Q(t ) − Q (t ) . p(t )

(9.40)

9.3.1 Welfare Equivalence Income Let a generation t has inherited the capital stocks K. The generation wants to 

maximize a social welfare functional,   (t ) u(C (t )) dt , over all feasible paths. Weitzman t

(1970) considers the level of utility v(t) at maximizing path (C * ( ), K * ( ), I * ( )) = t , so that;

179



  ( ) u (C ( )) d *

v(t ) =

t

.



  ( ) d

(9.41)

t

The consumption index of welfare w(t) is defined by (Asheim, 2000); 

 ( ) u (C ( )) d 1 *

w(t ) =

1 v(t ) = u u

t



  ( ) d

.

(9.42)

t

The notation w refers to Weitzman (1970), who first suggested stationary welfare equivalence. We assume that w(t) is continuous and differentiable everywhere. A country’s dynamic welfare is increased by moving K1 at time t1 to K 2 at time t 2 if and only if w(K1 )  w(K2 ) .

9.3.2 Sustainable Income A sustainable income s(t) at time t is the maximum consumption that can be sustained from time t on, given the capital stocks K that generation t has inherited;

(

)

s (t ) = sup inf (C ( )) .  t

Sustainable income is the best process in welfare economics, since the future generations will not suffer for national stocks. Consider a case where  ( ) =  (0) e− ,

 ( )  0 for all   t (Mohajan, 2011d). then the welfare sustainability is given by w

9.3.3 Green Net National Product Green NNP is the sum of consumption and the value of net investments;

g (t ) = C * (t ) + Q(t )I(t )

(9.43)

where the vector of capital goods, K, comprises all kinds of man-made capital (including stocks of accumulated knowledge), and all kinds of natural capital (including stocks of environmental resources). We see that green NNP includes current consumption and the value of net investments, but are not included capital gains Q (t )K (t ) .

180

Weitzman (1976) stated that if the own interest rate of consumption good is constant, the present value of future consumption equals the present value of consuming

C * ( ) + Q( )I( ) for all   t . Weitzman claimed that it is feasible to sustain a consumption equal to C * (t ) + Q(t )I(t ) which is correct in fact. But, in a closed economy with a constant population and no exogenous technological progress, NNP defined as

C(t ) + Q(t )I(t ) is not in general an exact indicator of sustainability, except in the uninteresting case with only one capital good (Asheim, 1994).

9.3.4 Net Social Profit A social cost-benefit is an index having the property that the acceptance of a small policy change increases the index if and only if the policy change leads to a welfare improvement (Dasgupta et al., 1972; Dasgupta et al., 1995, 1997). Hence, by a small policy change the net social profit S ( ) can be written as (Asheim, 2000); S ( ) = C * ( ) +

q( ) * q ( ) * I ( ) + K ( ) . p( ) p( )

9.3.5 Wealth Equivalence Income 

Let wealth at time t is

p( )

 p(t ) C ( ) d . Then the wealth equivalent income h(t) at *

t

time t is the consumption that if held constant will yield the same wealth as the wealth maximizing path (C * ( ), K * ( ), I * ( )) = t , i.e., 

 p( ) C ( ) d *

h(t ) =

t



 p( ) d

.

(9.44)

t

9.3.6 Relation between Green NNP and Wealth Equivalent Income By the relations (9.36) and (9.43) we can write; 

g (t ) =  r0 ( ) t

p( ) * C ( ) d . p(t )

181

(9.45)

Again by the relations (9.37) and (9.44) we can write; 

h(t ) = r (t ) t

p( ) * C ( ) d . p(t )

(9.46)

Suppose there is no exogenous technological progress. For constant interest rate, i.e.,

r0 ( ) = r (t ) = r for all τ, we get: 

g (t ) = h(t ) = r  t

p( ) * C ( ) d p(t )

(9.47)

which is Weitzman’s (1976) fundamental result on green national accounting. Again for constant consumption, i.e., for C * = C and for all τ, we get;

g (t ) = h(t ) = C * .

(9.48)

We have,   p( ) * 1  * d  p( ) * C ( ) d  . C ( ) d = C ( ) +   r0 (t )  dt  t p(t ) p(t ) 



 t

(9.49)

Using (9.49), we can write (9.46) as follows: h(t ) =

  r (t )  * d  p( ) *   . ( ) ( ) C  + C  d     r0 (t )  dt  t p(t ) 

(9.50)

The constant return to scale (CRS) implies that wealth is equal to the value of current capital stocks; 

 t

p( ) * K * (t ) C ( ) d = q(t ) = Q(t )K * (t ) . p(t ) p(t )

(9.51)

Using (9.51), the relation (9.50) becomes (Asheim, 2000);

h(t ) =

r (t )  * d  r (t ) C (t ) + Q(t )K* (t )  =  g (t ) + Q (t )K * (t ) .  r0 (t )  dt  r0 (t )

(



)



(9.52)

Relation (9.52) implies that to find wealth equivalent income h(t) we must add capital gains Q (t )K * (t ) to the green NNP, g(t), and the sum g (t ) + Q (t )K * (t ) must be adjusted for interest rate effects if there is not a constant interest rate, in which case r∞(t) / r0(t) need not equal 1 (Mohajan, 2011d).

182

Proposition 9.5: If there is no exogenous technological progress, then wealth equivalent income is related to green NNP according to resource allocation, consumption of resources and interest. Proof: If exogenous technological progress exists then we may face open economy condition and a regular path (C ( ), K ( ), I ( )) *

*

*

  =t

is not followed that

 p( ) C ( ) d

is

t

maximized over all feasible paths, given the capital stocks K that generation t has inherited. So that both conditions R–1 and R–2 will not satisfy. As a result equation (9.48) does not satisfy, i.e., h(t )  g (t ) for constant consumption, since some part of the instantaneous return on a country’s capital stock must be used to augment the country’s national wealth (Asheim, 1996). Again h(t )  g (t ) exists whenever both consumption and interest rate tend to decrease which violates Dasgupta-Heal-Solow model. Moreover CRS rule will not also satisfy, because wealth will not be equal to the value of current capital stocks, and obviously (9.52) will not satisfy. Therefore, in every case h(t )  g (t ) for all

  t . Q.E.D.

9.3.7 Relation between Green NNP and Net Social Profit The net social profit S ( ) can be written as (Asheim, 2000); S ( ) = C * ( ) +

q( ) * q ( ) * I ( ) + K ( ) p( ) p( )

(9.53)

where S ( ) is discounted by p( ) . Again we have, q ( )  = Q( ) − r0 ( ) Q( ) . p( )

(9.54)

Using (9.54), the equation (9.53) becomes:





S ( ) = C * ( ) + Q( )I * ( ) − r0 ( ) Q( ) − Q ( ) K * ( )





S (t ) = g (t ) − r0 (t ) Q(t ) − Q (t ) K * (t ) .

(7.55)

So that net social profit is obtained by subtracting the cost of holding capital

 r (t ) Q(t ) − Q (t )K (t ) from the green NNP. *

0

183

Proposition 9.6: Net social profit is always smaller than green NNP. Proof: Suppose, S ( )  g ( ) then equation (9.55) violates; since g ( ) is gross social profit, and S ( ) is net social profit for all τ. For a small policy change g ( ) is not a costbenefit index, as S ( ) is general, so that always S ( )  g ( ) is impossible. We know that opulence is not the same as well-being, so that always S ( )  g ( ). Q.E.D.

9.3.8 Relation between Green NNP and Welfare Equivalent Income If there is a constant utility discount rate under no exogenous technological progress, then it follows from a generalization of Weitzman’s (1976) result that; 



*   ( )  u(C (t )) + t



 q(t ) *  I (t ) d =   ( ) u C * (t ) d .  (t )  t

(

)

Which indicates that green NNP in terms of utility u (C * (t )) +

(9.56)

q(t ) * I (t ) , is equal to the  (t )

utility derived from welfare equivalent income, u(w(t)), i.e., 

(

) q((tt )) I (t ) =

u C * (t ) +

*

  ( )u (C ( )) d *

t



  ( ) d

= u(w(t)).

t

If the value of net investment Q(t )I(t ) = 0 , then we can write also g (t ) = C * (t ) = w(t ) . Again if there is no exogenous technological progress, and the utility discount factor

 ( ) is an exponentially decreasing function, then w(t )  g (t ) (Asheim, 2000). Proposition 9.7: Green NNP never exceeds welfare equivalent income in closed economy. Proof: For closed economy there is no exogenous technological progress and utility discount rate is constant. Hence for closed economy, maximum welfare equivalent income of a country never affected by the other countries. As a result g (t ) = w(t ) if u is linear or if Q(t )I(t ) = 0 then g (t ) = C * (t ) = w(t ) , i.e., both g(t) and w(t) are equal to

184

maximum level. Again if utility discount factor  ( ) is an exponentially decreasing function, then w(t )  g (t ) . Therefore, g(t) never exceeds w(t) in closed economy. Q.E.D.

9.3.9 Relation between Green NNP and Sustainable Income In the Ramsey model, one capital good technology is described by C + I  f (K ) , with f being an increasing and strictly concave function, hence green NNP measures sustainable income. If there is no exogenous technological progress then for constant consumption C * ( ) = C * for all τ, Q(t )I* (t ) = 0 . So that g (t ) = C * = s(t ) for a constant consumption path in a stationary technology. Asheim (1994) showed by Dasgupta-HealSolow model that, g (t )  C * (t )  s(t ) .

Proposition 9.8: If no exogenous technological progress then for constant consumption or for constant interest rate green NNP is greater than or equal to sustainable income. Proof: For no exogenous technological progress and constant consumption we have,

g (t ) = s(t ) . Again for constant interest rate and no exogenous technological progress,

g (t ) = w(t ) ; but we have w(t )  s(t ) which implies that g (t )  s(t ) . Hence for constant consumption or for constant interest rate and no technological progress, g (t )  s(t ) . Q.E.D.

9.4 Global Sustainable Development Sustainable development (SD) first became prominent in international discussion in the 1980s, and was the central theme of the United Nations (UN) Conference on Environment and Development in Rio de Janeiro in 1992. In the 21st century sustainable development is an essential issue for the humankind. In 2017, population of the world become 7.5 billion and it is estimated that this figure will be 9 billion by the early 2040s. At present at least 1.2 billion people live in absolute poverty, whose income is $1.25/day. They live in for mere survival every day, and most of them are in Sub-Saharan Africa, and in some countries of Asia. They face the daily life-and-death challenges of insufficient nutrition, lack of healthcare, unsafe shelters, and the lack of safe drinking water, and without sanitation facilities (Mohajan, 2015a). In order to improve human 185

wellbeing and social equity, we need SD by reducing environmental risks and ecological scarcities (UNEP, 2011a).

9.4.1 Global Green Economy The green economy (GE) proposal was introduced by Ms. Laura Altinger in October 2008, in a program launched by the United Nation Environment Program (UNEP) thinking the vulnerability of human welfare due to the unsustainability of economic development. A GE is defined as, growth in income, and employment is driven by public and private investments that reduce carbon emissions and pollution, enhance energy and resource efficiency, and prevent the loss of biodiversity and ecosystem services. GE improves human wellbeing and social equity significantly reducing environmental risks and ecological scarcities (UNEP, 2010). At present the global economy is more than 5 times the size it was half a century ago. Such rapid economic growth has given financial benefits, but it creates the gap between the rich and the poor. It has also delivered unprecedented environmental damages, for example, an estimated 60% of the world’s ecosystems have been degraded in the last 5 decades. Hence, current economy has not grown in sustainable ways. So, we have to implement a new kind of economy, which is elastic, sustainable, operates within the limits of our world’s resources, and creates a fairer society; and we can call it ‘Green Economy’ (Global Green Economy, 2011).

9.4.2 Sustainable Economy In a society if many people are very poor and some limited people are very rich, then we cannot say that the economy of this society a sustainable economy. A socially and environmentally sustainable economic system operating with the purpose of facilitating a good life with dignity for all while respecting nature as an integral part of life. To achieve this, a fundamental shift in economic rationality is required (Schildberg, 2014). A good society must give priority to save the natural environment. If a society breaks the physical systems of water and biodiversity, destroys the oceans and the great rain forests, then that society will not survive profitably and comfortably, and must face

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sever danger situation. Of course, a good society must think about environmental sustainability for the wellbeing of its children, and its future generations (Costanza, 2009). Investments in a sustainable world would be mainly for the replacement of capital and qualitative improvement rather than for quantitative expansion (Daly, 2005). Corruptions, lawlessness, untrustworthy politicians, unfair government services, discriminations of gender and minorities, etc., create a lot of unhappiness in the society. Citizens feel happier and better when they can trust their government, but unfortunately in many countries their governments not so to trust them (Ventura-Dias, 2013).

9.4.3 Discrimination in Modern Economy In 1776, Adam Smith observed that the inland parts of Africa and Asia were the least economically developed areas of the world (Smith, 1937). Nine of the twelve countries with the lowest Human Development Index (HDI) scores are landlocked (UNDP, 2002). Some countries situated in tropical region or landlocked, for example, Ethiopia, Niger, Mali, Kenya, Bolivia, Chad, Mali, Burkina Faso, Uganda, Rwanda, Zimbabwe, Zambia, Lesotho, and Laos still have not achieved the modern economic growth that some other countries of the world have experienced two centuries ago (Sachs, 2001). Landlocked countries like Bolivia, Ethiopia, Chad, Niger, Kyrgyzstan, Bhutan, and Nepal still face the disadvantages of high transport costs (Faye et al., 2004). These landlocked countries not only face the challenge of distance, but also the challenges that result from a dependence on passage through a sovereign transit country to access international shipping markets (Sachs, 2001). At least 1.2 billion people live in absolute poverty, whose income is $1.25/day, equivalent to 22% of the world population (in the USA cost of a half-dozen eggs is $1.25), that they struggle for mere survival every day (Raskin et al., 2010; Shepherd et al., 2014). About 2.4 billion people, 35% of the world population, live on less than $2.00 per day. These poorest of the poor people face the daily life-and-death challenges of insufficient nutrition, lack of healthcare, unsafe shelters, and the lack of safe drinking water and sanitation (Ortiz & Cummins, 2011; United States Agency for International Development, USAID, 2013; Shepherd et al., 2014). The inequitable distributions of wealth within countries have created social instability and public health concerns, and

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multiple forms of social deprivations (Sen, 1999; Pickett & Wilkinson, 2009). Although in 1807 the British Empire abolished the slave trade, and in 1833 abolished slavery in the British holdings entirely; but yet in the 21st century approximately 21 million people are working as modern day slaves, falling victim to trafficking, forced labor and sexual exploitation (Mohajan, 2012g; UN, 2013). Also those are above the extremely poor level are looking for improved prosperity, and a brighter future for their children. On the other hand, rich people are hoping that technological advances will offer them and their families even higher levels of wellbeing (Sachs, 2005). The economic development is not equally distributed throughout the world, rather it becomes in different rates in different parts of the world. At present the per capita income of the USA is more than $50,000 per year, and that of Niger is under $500 per year. This large gap between the rich and the poor, and also this close interconnection did not exist around 1750. Hence, present huge gap is due to modern economic growth since the start of the Industrial Revolution (Sachs, 2005). At present most parts of the world, for example, mass population of Sub-Saharan Africa and some countries of Southern Asia (Bangladesh, India, Pakistan, Nepal, Afghanistan, etc.) remain in the extreme poverty. Most people of these regions live in rural areas or in slums. They live without modern sewerage or household sanitation, often having to defecate in empty fields. They walk through unpaved muddy road that is not really passable by vehicles. They are deprived from emergency healthcare, electricity, adequate nutrition, clean cook stoves, safe water, and sanitation. They may earn just enough to buy a minimum of food, water, clothing, and shelter. Some people have no house and live in the streets or open fields. Some live in the refugee camps in an unhygienic ways (Shepherd et al., 2014).

9.4.4 Present Unsustainable Development Practices For the competitive global economic development every nation practices more or less unsustainable development policies. In the developing countries the wetland has been reduced in size due to unsustainable abstraction of water for use in agriculture (Dick, 2006). Water table has decreased for the unsustainable irrigation, and over ground water use (LPN, 2004). Support for the irrigation of dryland areas has been criticized by

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environment experts for being unsustainable, and has also had negative impacts on biodiversity (Baldock, 2000). China’s ecosystems are moving towards unsustainable level due to unsustainable economic development. Over the past few decades China faces difficulties for unsustainable production, distribution, and consumption practices (Economy, 2008). Biodiversity loss, climate change, food insecurity, chemical pollutions, and excess use of fossil fuels are for the unsustainable economic development. Tipping point and peak oil are obviously unsustainable (Rockström et al., 2009). Fishing industries in some countries have become unsustainable due to overexploitation of fish stocks (CEC, 2005).

9.4.5 The Recent Economy of the World In 2009, the world has suffered from the global financial crisis (GFC). This GFC is recovered in the next year and international monetary fund (IMF) has estimated that the global output has grown by 5%. It is estimated that in this recovery the contribution of China and India is about 7.3%, compares to 3% growth in advanced economies (External Economic Environment, EEE, 2010). Recent economic growth of China is roughly 10% per year in GDP growth. By the rule of 70, a growth of 10% means that China has been doubling its GDP roughly every 7 years. Recently China has become the largest trading country of the world. It is estimated that more than 200 million people of China have gathered from the countryside to the cities in search of new jobs in industry and services (Sachs, 2005). In the beginning of 2015, the population of China becomes about 1.398 billion, which is more than one-fifth of global population. The economy of China has proceeded from rural to urban, from agricultural to industrial and service-oriented. In 2014, China becomes the world’s firstbiggest economy in the world (Country Profile; China, 2015). The IMF forecasts that the Chinese economy will grow by 9.6% in 2011. But, the growth in Europe remained weak reflecting the ongoing impact of the GFC, and high government debt levels in several countries, namely Greece, Ireland, Italy, Portugal, and Spain. The IMF provided significant loans to alleviate the financial crisis. In 2011, Greece faces serious financial crisis. This financial crisis makes most of the people of Greece unemployed. France and Germany pledge to pay loan to recover this unexpected

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financial crisis. The IMF estimates that Europe grew by 2.1% in 2010, with individual growth rates ranging from 3.5% in Germany, and 5.5% in Sweden, to decline of 4.5% in Greece, and 1% in Ireland, but IMF’s forecast is partially fulfilled. Japan in 2010 recovers strongly with GDP growth of 3.9%. But, in March 2011, Japan experienced a large earth quake and tsunami that resulted in substantial loss of life and damage to infrastructure. The change in world economy influenced by the change of US improved in the second half of 2010, but unemployment increased about 9.6%. The real estate prices have been slowed, but the US government fails to recover it. The US passed a new fiscal stimulus package in late 2010 which includes further quantitative easing of up to $600 billion, and the IMF forecasts that the US economy will grow by 2.8% in 2011 (Mohajan, 2011b). The world economy is expected to continue to recover in 2012, with the IMF forecasting global growth of 4.5% compared to 4.4% in 2011. Growth in the emerging economies is expected to be led by Asian economies, including China, India, and Indonesia. Despite the positive outlook in developing Asia, significant inflation risks remain, particularly in China. For example, food prices and prices of other essential articles, in many developing economies increased significantly in 2010, placing upward pressure on wage growth (EEE, 2010). The IMF forecasts that US economic growth will strengthen slightly to 2.9% in 2012. But, this will be challenged due to unemployment, reverse weakness in housing sector, and to face Middle-East crisis. The IMF outlook for Europe is mixed. The forecast expresses that the growth in advanced European economies is expected to continue at below trend rates in 2012. The outlook of IMF may vary due to unrest civil war in the Middle-East and North Africa in early 2011. Since disrupted world oil supply will substantially increase oil prices which will affect the world economy. The IMF estimates that a 10% price increase in the crude oil reduces global GDP by between 0.2 to 0.3% (EEE, 2010).

9.5 Open Economy of Bangladesh Bangladesh is a densely populated (population in 2017 becomes more than 160 million) developing country of South Asia with the total area of 147,570 km2. About 90%

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of its populations are poor, and about 60% citizens are illiterate. Again natural disasters, such as, flood, cyclones, droughts, etc., constantly pursue its lot every year; which break the backbone of the economy, and frustrate future planning of economic development. Economy is sick due to high inflation rate, poor energy supply in industries and unemployment. As Bangladesh is a very small country, it depends on foreign aids. Its economy is open after the independent from Pakistan in 1971, but it failed to develop its economy due to political instability and failure to create a strong human capital. It depends on World Bank, Asian Development Bank, IMF, USAID, etc., who provide loan to Bangladesh with strong conditions. So that Bangladesh cannot use the loan properly to develop its economy (Mohajan, 2011b). The years after independence, the size of Real GDP, Per Capita GDP, and their growth rates were very small, and the condition slightly improved from 1990. Yet the growth trend and the structural changes of GDP in Bangladesh are not satisfactory. Many problems are responsible for this unsatisfactory GDP. These are the shortage of domestic food production, narrow structure of exports, increasing growth rate of imports, failure in the invocation of much Foreign Direct Investment, a defective banking system with cumulative interest of loans, continuous loss in the public enterprises, poor infrastructure, inefficient taxation, high inflation rate, social and political corruptions, political instability, and the serious deterioration of law and order situation (Bangladesh Country Background Information, BCBI, 2011). Major exports of Bangladesh are readymade garments, frozen fish and seafood, tea, chemical products, raw jute and jute goods, leather, manpower, etc. Major imports of Bangladesh are rice, wheat, sugar, edible oils, oil products, gasoline, fertilizer, scrap vessels, machinery and equipment, chemicals, steel, etc. To follow open economy Bangladesh cannot export competitive materials in the world markets, but much fortune losses to import necessary materials. As a result it failed to gain high benefits from open economy. Moreover, it has a permanent energy crisis due to shortage of electricity and natural gases. So that it cannot provide continuous energy supply in industries, and NNP is not satisfactory, which is a permanent problem of Bangladesh (BCBI, 2011).

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9.6 Conclusion In this chapter we have shown the growth with optimality and the growth without optimality, and have examined the sustainability in each case. The real NNP represents the maximized value of flow of goods and services that are produced by the productive assets of the society. We have discussed the relation between the Divisia index of real consumption prices and dynamic welfare evaluation. We also show the relations of green NNP with some other incomes. We have used mechanisms of Dasgupta-Heal-Solow model of capital accumulation and resource depletion. If green NNP approaches zero then unconstrained development is no longer sustainable. We have also introduced Weitzman’s fundamental results of closed economy, and Hartwick’s rule in open economy to realize NNP and open economy properly. Here we have shown all the mathematical calculations and theoretical concepts in some detail. We have added some propositions with proof to make the chapter easier to understand, and hope that readers take these as genially.

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Chapter–X

Greenhouse Gas Emissions and Global Warming

10.1 Introduction The world has realized that global warming is continually increasing due to greenhouse gas (GHG) emissions. The living organisms are in dangerous position, and some species have already extinct, and some more will extinct in future if global warming cannot be controlled. It is clear to environment experts of all nations that emissions of carbon dioxide (CO2) and other GHGs are liable to global warming. The current concentrations of GHG in space have increased since 1750 (Industrial Revolution) from a CO2 equivalent of 280 ppm (parts per million) to 450 ppm, but proposed boundary is 350 ppm (Stern, 2007; Mohajan, 2015b). The National Academy of Sciences (NAS) has expressed its expert opinion that concentrations of CO2 in the atmosphere have increased, and continue to increase more rapidly due to human activities (NAS, 2001, 2010). The intergovernmental Panel on Climate Change (IPCC) has expressed its expert opinion that the observed increase in global average temperatures since the mid-20th century is very likely due to the observed increase in anthropogenic GHG concentrations, and the temperature has been rising most rapidly since 1970 (IPCC, 2007; UN Foundation, 2007). After the industrial revolution the global average temperature increases about 0.760C. The global surface temperature has increased ≈ 0.20C per decade in the past 30 years. Global warming is now +0.60C in the past three decades, and +0.80C in the past century, and continued warming in the first half of the 21st century is consistent with the recent rate of +0.20C per decade (Mohajan, 2011c). The current average temperature of the world is 17oC. It is estimated that, in future the global average temperature will increase for the GHG emissions (Mohajan, 2013d).

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According to International Energy Agency (IEA) data (IEA, 2007a), the USA and China are approximately tied and leading global emitters of GHG emissions. Together they emit approximately 40% of global CO2 emissions, and about 35% of total GHGs. If GHG emissions cannot be controlled then the people of most of the countries will suffer for drinking water, shortage of foods, and various heat related diseases. Scientists declared that some plants and animals will extinct in the 21st century due to increase global warming (Mohajan, 2015b). Climate change is usually considered as a reduction in productivity (Nordhaus & Boyer, 2000).

10.2 Greenhouse Gas Emissions The six gases; Carbon dioxide (CO2), methane (CH4), nitrous oxide (N2O), sulphurhexafluouride (SF6), hydrofluourocarbon (HFC) and perfluourocarbon (PFC), together constitutes six GHG emissions. These six gases briefly called carbon dioxide equivalents (CO2e). CO2e gases covered in the Kyoto Protocol 1997, which is an international agreement linked to the United Nations Framework Convention on Climate Change (UNFCCC). The current concentrations of GHG in space have increased since 1750 from a CO2e of 280 ppm to 450 ppm (Stern, 2007; Mohajan, 2015b). Each GHG traps different amounts of heat, and stays in atmosphere for different lengths of time. So that it is necessary to measures of global warming potential to compare between gases. The Table 8.1 gives six GHGs global warming potential and atmospheric life in years (Sharma, 2007). The combined radiative forcing of CO2, CH4 and N2O is +2.30 Wm–2

Table 10.1: The global warming potential of six GHGs, (IPCC, 2001).

Gas

Global Warming Potential

Atmospheric Life (years)

CO2

1

5 to 200

CH4

21

12

N2O

310

114

HFC

140 to 11,700

1.4 to 260

PFC

6,500 to 9,200

10,000 to 50,000+

SF6

23,900

3,200

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compared to the radiative forcing of solar irradiance of +0.12 Wm–2. Oceans have warmed from surface of the sea to up to a depth of at least 3 km. It is estimated that absorbed 80% of the additional heat added to the climate. Warmer water taking more spaces of the sea than the colder water, as a result sea level is rising (Sharma, 2007). NAS has expressed its expert opinion that concentrations of CO2 in the atmosphere have increased and continue to increase more rapidly due to human activity (NAS, 2001, 2010). The NAS cites that the burning of fossil fuels is the primary source of anthropogenic CO2 emissions. IPCC (2007) has expressed its expert opinion that the observed increase in global average temperatures since the mid-20th century is very likely due to the observed increase in anthropogenic GHG concentrations and the temperature has been rising most rapidly since 1970 (UN Foundation, 2007). After the industrial revolution the global average temperature increases about 0.760C. The global surface temperature has increased ≈ 0.20C per decade in the past 30 years. Warming is larger in the Western Equatorial Pacific than in the Eastern Equatorial Pacific over the past century. The latest report (in 2007) shows that atmospheric concentrations of CO2 grew 80% from 1970 to 2004, and recently exceeds by far the natural range over the last 650,000 years (IPCC, 2007). Global warming is now +0.60C in the past three decades, and +0.80C in the past century, and continued warming in the first half of the 21st century is consistent with the recent rate of +0.20C per decade. Warming occurs over ocean areas, far from direct human effects, with warming over ocean less than over land, an expected result for a forced climate change because of the ocean’s great thermal inertia (Hansen et al., 2006). Both NAS (2010) and IPCC (2007) expressed that humans, largely through the ever-increasing burning of fossils are changing the earth’s climate. Africa is one of the most vulnerable continents due to global warming. It is estimated that water stress will affect between 75 and 250 million people of Africa by 2020. The cultivable land will decrease, the rain-fed agriculture could be cut in half and fisheries must be declined. As a result almost all African countries will seriously affect food security and malnutrition (IPCC, 2007). Forests will affect by pests, diseases, and fire. The citizens of most of the cities will suffer from heat waves, earthquake, tsunami, shortage of water supply, and energy supply by 2020. All the countries of the world those

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depend on rain for cultivation, their production of crops will decrease seriously due to droughts (IPCC, 2007). Coral reefs are very important because, they act as hatcheries and nurseries for open ocean fish. They protect coastal areas from storms, and provide fish, recreation, and tourism money. It is estimated that in Asia coral reef fisheries feed one billion people. The total economic value of coral is estimated about $30 billion. Rising carbon emissions might kill off the ocean’s coral reefs by 2050. The marine scientists said that global warming is seriously threatening the crucial component of the ocean biodiversity. If CO2 emissions keep stabilize at today’s levels of 380 ppm, coral reefs survive mostly intact. Sea water is acidifying as CO2 from power plants, cars, trucks, and other vehicles, and factories mix into the ocean. Acidified ocean water must be fatal to some fish eggs and larvae. IPCC (2007) expressed that 450 ppm is regarded by many climate scientists as the “tipping point” to contain rises in average temperatures to around 20C. That is, still enough to wipe out 20% to 30% of the earth’s animal and plant species, and for the world’s coral to be bleached, crop product will fall; and millions of people and other creatures suffer from water and food shortages. To decline in global emissions by 2020, it is particularly focused on the energy industry, where $30 trillion of new energy investment is required over the next decade. Methane is 21 times more powerful than CO2 to trapping heat. A vast expanse of permafrost in Siberia and Alaska has started to melt for the first time since it formed 11,000 years ago. It is caused by the recent 3°C rise in local temperature over the past 40 years which is more than four times the global average. Peat bogs cover an area of a million square miles (or almost a quarter of the earth’s land surface) to a depth of 25 m. This has the capacity to release billions of tons of methane trapped by ice below the surface. The whole world peat bogs store at least two trillion tons of CO2 which is equivalent to a century of emissions from fossil fuels. It is estimated that the west Siberian bog alone contains about 70 billion tons of CH4, a quarter of all the CH4 stored on the land surface of the world. This is equivalent to emitting 1.7 trillion tons of CO2, which is more GHG than has been emitted by humans in the past 200 years. Vast areas of wet peat land forests are being drained and logged in Indonesia and Malaysia. Along with the ensuing peat fires this contributes 2 billion tons of CO2, making South-East Asia the

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third largest polluter in the world behind the US and China. We can easily reduce our CO2 emissions from fossil fuels if we try, but we could not reduce methane emissions once if they started to emit (NAS, 2010). Experience of hotter summer days which could increase heat related mortality, ground-level ozone concentrations, storm water runoff, and negative impacts from erosion and invasive species. Rising temperatures may increase air pollution levels, with their attendant increases in respiratory illness and death. GHG emissions and climate change pose a serious threat to the economic well-being, public health, natural resources, and environment of the earth. The potential rise in sea levels threatens coastal communities, and has increased vector-borne diseases (EPA, 2010). IPCC and NASA advised that US should target to reduce GHG emissions 20% to 30% below 1990 levels by 2020 to avoid the risks of dangerous impacts of global warming. The four warmest years were, in ascending order are, 2002, 2003, 2005, and 1998. The last decade was the warmest on record, followed by the 1990s and then the 1980s; so the world is definitely warming up (Betts, 2010). The estimated uncertainty of global mean temperature implies that year 2005 was probably the warmest year which is based on the positive polar anomalies, especially the unusual Arctic warmth (Hansen et al., 2006). In a related development, scientists at the World Glacier Monitoring Service, based at the University of Zurich in Switzerland, reported that some 30 major glaciers around the world are shrinking fast, which threatening to increase floods in some regions, and to decrease precious water supplies in others. They reported that “Data from 30 glaciers in nine mountain ranges from Alaska, the Andes, Antarctica, the Alps and the Himalayas showed that between 2004 and 2006 the average rate of melting and thinning more than doubled.” In the survey they observed that the glaciers were melting at an average rate of about a foot a year between 1980 and 1999. The rapid melting of glaciers in every mountain region indicates the serious dangers, from drinking water shortages to flash floods to decreases in available water for irrigation (NAS, 2010). The IPCC’s (2007) Synthesis Report “Summary for Policymakers” (Table SPM-6) finds that using the best estimate sensitivity stabilizing at a warming of 20C to 2.40C requires stabilizing CO2 emissions in the range of 350–400 ppm CO2 or 445–495 ppm

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CO2e (IPCC, 2007). For 450 ppm CO2e target developed countries need to reduce their emissions 40% to below 1990 levels by 2020, and reduce the emissions 95% to still lower levels by 2050, even if developing countries make substantial reductions. At the international climate talks in Poland both the Chinese and Indian delegations told that the goal of merely returning to 1990 levels in 2020 is inadequate to fight global warming. In 2007, the EU agreed to “Slash GHG emissions by 20% within 13 years unilaterally and pledged to push for an agreement with the US and other industrialized countries to cut by 30% by the same deadline.”

10.3 Greenhouse Gas Emissions of the USA and Mitigation Policies CO2 emissions from energy use including transportation calculated for 83% of US GHG emissions in 2005 (EIA, 2006). US GHG emissions in 2007 were 16% higher than 1990 levels, so that US has to loss much of its credibility in the international community by failing to act already. The USA emits a number of different GHGs through a wide variety of activities in households and businesses. The EPA estimates that, in 2006 (EPA, 2008), US emissions of GHGs amounted about 7.1 BMTCO2e (billions metric tons CO2e) which is 85% in the form of CO2, 8% in the form of CH4, 5% in the form of N2O, and 2% in the form of other three GHGs. About 86% of those emissions were directly related to the generation and consumption of energy, but the remaining 14% came from industrial and agricultural processes, as diverse as, the production of cement and the management of landfills, wastewater and agricultural soils. About 94% of the CO2 was emitted directly through the combustion of fossil fuels, 40% from petroleum products, 35% from coal, and 19% from natural gas.

10.3.1 GHG Emissions in the US Transportation Sector The US transportation sector is the largest GHG emitter among the world’s transportation sectors. It was accountable for 31% of global transportation energy use and GHG emissions in 2006. In 2030, the US transportation sector is expected to use onefourth of global transportation energy. It is also estimated that CO2 emissions of the USA grow by about 10% by 2035 (EIA, 2010). In 2006, Americans traveled 5.2 trillion

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person-miles in vehicles, and moved 4.6 trillion ton-miles of freight (BTS, 1996), which consumed 28.6 quads of energy (EIA, 2009b). The US transportation sector faces following three major challenges to take any attempt to reduce higher GHG emissions (Mohajan, 2012h): The vehicle manufacturers want to make larger and more powerful vehicles which will be fuel economy. Any attempt to shift from petroleum fuels to lower-carbon alternatives such as hydrogen or electricity is failed, because the motorists want to use high-carbon fuels which give them excellent characteristics for transportation. The US population and economy are expected to continue to grow, increasing both freight and personal travel. The real GDP to be doubled for growing populations additional 85 million by 2035 compared to 2008 (EIA, 2010). The dependence on petroleum the US transportation system makes the US economy vulnerable to significant excess economic costs on the order of hundreds of billions of dollars per year (Greene, 2010). To buy gasoline US losses hundreds of billions of dollars each year that effect in economic development. In only 2008 the estimated economic cost of oil dependence was half a trillion dollars ($350 billion in wealth transfer, $150 billion in lost GDP) (Greene & Hopson, 2009). The US consumes more than 10 million barrels of oil per day only moving people and goods on roads and rails throughout the country which generates more than 23% of US anthropogenic GHG emissions. In 2010, Americans drove about 3 trillion miles (Burbank & Nigro, 2011). In January 2011, the Pew Center on Global Climate Change issued a report on all of the actions that can be taken by the US government across the transportation sector to save oil, and reduce GHG emissions (Greene & Plotkin, 2011). There are many ways to save oil, and to reduce GHG emissions from transportation as follows (Burbank & Nigro, 2011; Mohajan, 2011c): The new Annual Energy Outlook of EIA (EIA, 2010) forecasted that the transportation sector’s energy consumption will grow by about 21% from 2008 to 2035, compared to growth of about 33% for the commercial sector. At the same time the EIA (2010) forecasts that growth in CO2 emissions to be about 10% over the same period

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compared to 24% for the commercial sector. The forecast also anticipates that US oil imports will shrink dramatically, from 60% of total consumption in 2006 to 45% in 2035.

10.3.2 The US GHG Mitigation Policies In Kyoto Protocol the US government agreed that between 2008 and 2012 it would limit average annual emissions of GHGs to 7% below 1990 levels. But, the US government has not expressed by which technology will apply to implement Kyoto Protocol. S. 2191, the Lieberman-Warner bill, provides a useful illustration of the mechanics of a cap-and-trade system, which would have required the EPA to establish two cap-and-trade programs aimed at reducing the emission of GHGs in the US over the 2010–2050 periods. Under S. 2191, consumers of gasoline would not have needed to submit allowances for the CO2 emitted by their cars and trucks, but importers and refiners could not produce and sell the gasoline to consumers without submitting allowances, effectively bringing the consumers, the ultimate emitters increase the scarcity of gasoline, as a result raises its price. In the case of S. 2191, the number of allowances allocated under the main program would have declined from 5,775 MMTCO2e in 2012 to 1,732 MMTCO2e in 2050, at which point the number of allowances would be equal to about 28% of 2005 emissions in sectors covered by the program. The Low Carbon Economy Act of 2007, S. 1766, would have established a technology accelerator payment starting at $12 per metric ton of CO2e in 2012, and rising by 5% annually thereafter (McCarl & Schneider, 1999). In the USA, N2O emission reductions could be performing assuming relevant strategies are as follows: •

reduced nitrogen fertilizer applications,



use of nitrification inhibitors,



improved nitrogen nutrient management, and



reduced nitrogen content of animal feeds. Scientists estimated that about 0.13 MMTs of N2O emissions need to be reduced in

US in order to meet the Kyoto requirements (McCarl & Schneider, 1999). A strong research, development, demonstration, and deployment program will be crucial to the reduction of GHGs in the US transportation sector. There are enough

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alternative pathways that one can be reasonably assured of success if the US government commits to a strong effort to reduce GHG emissions (Mohajan, 2011c). Some experts suggest that market-based policies alone cannot achieve significant reductions in transportation emissions. In an EIA study of an economy-wide carbon capand-trade system a carbon price that rises from $20 per ton of CO2 in 2012 to $65 per ton in 2030 reduces emissions from the electric utility sector by 60%, but transportation emissions fall by only 5% (EIA, 2009c). On the other hand, bottom-up analyses of transportation options have claimed that emission reductions of 12% to 50% compared to projected levels in 2030, at costs of less than $50 per ton of CO2 (Greene & Schafer, 2003; Creyts et al., 2007). Mitigating transportation’s GHG emissions can save about 70% US petroleum use (EIA, 2009a).

10.4 Greenhouse Gas Emissions of China and Mitigation Policies People’s Republic of China is situated in the Eastern Asia on the western shores of the Pacific Ocean, Beijing is its capital city and Shanghai is its largest city. Its area is 9,640,821 km2 and it is considered as the 3rd largest country (after Russia and Canada) in the world. In 2010, its population becomes about 1.339 billion which is in the 1st position in the world (20% of the world’s total), and density of population is 138.96/km2, which is the 53rd in the world (Mohajan, 2014a). At present China faces four environmental problems such as: air pollution, water pollution, the emission of CO2 in the atmosphere which causes global warming, and shortage of future energy supply that relies on exhaustible resources. Environmental pollution mainly from coal combustion is damaging human health, air and water quality, agriculture, and ultimately the economy (Chow, 2008). According to International Energy Agency (IEA) data, the USA and China are approximately tied and leading global emitters of GHG emissions. Together they emit approximately 40% of global CO2 emissions (21% China and 19% the USA), and about 35% of total GHGs. India is the 3rd largest CO2 emitter in the world pushing Russia into 4th place (IEA, 2007b).

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10.4.1 Recent Climate Situation of China The climate of China is extremely varied, with tropical areas in the south to subarctic areas in the north. The northern zone has summer daytime temperatures of more than 300C, and winters of arctic severity, with the lowest temperature of –300C in northernmost Heilongjiang province. The central zone has a temperate continental climate, with very hot summer and cold winter. The unevenly seasonal and spatial distribution of rainfall in China may cause floods in South China, and droughts in North China. It experiences typhoons, monsoons, tsunamis, etc., which are also unusual worldwide due to mainly global warming (Mohajan, 2014a). At present China is facing severe environmental problems from its rapid economic growth. The latest information is released by the China Meteorological Administration shows that the average temperature of the earth’s surface in China has raised by 1.1 0C over the past century, from 1908 to 2007, and that China has experienced 21 warm winters from 1986 to 2007. Extreme climate phenomena, such as high temperatures, number of heat waves in summer, heavy rainfall and severe droughts, have increased in frequency and intensity. Heavy rainfall, rainstorms and floods (including the lowtemperature freezing rain and snow) have increased in southern China, spring and summer droughts in the middle and lower have affected of the Yangtze River, droughts have grown worse in northern China, and the occurrence of snow disasters and autumn rains have risen in western China and serious water logging in Beijing. In China’s coastal zones, the sea surface temperature and sea level have risen by 0.90C and 90 mm, (1 m =1,000 mm) respectively, over the past 30 years. Scientific research predicts that the above adverse effects will increase in future. Only in 2011, natural disasters have affected 430 million people, and caused direct economic losses of 309.6 billion yuan (Zhao et al., 2010). Environment current issues of China are (Country Profile; China, 2013): ▪

air pollution (GHGs, SO2) from reliance on coal produces acid rain,



water shortages, particularly in the northern side of the country,



water pollution from untreated wastes,



deforestation,

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estimated loss of one-fifth of agricultural land since 1949 to soil erosion and economic development, desertification, and trade in endangered species. Desertification is a great problem in China, due to large part to overgrazing, drought and environmental deterioration and leads to the loss of about 5,800 mile2 of grasslands every year. In addition, 31% of national land area experiences soil erosion, and 85% of the total grassland area is degraded (IMF, 2013).

10.4.2 GHG Emissions in China Environmental data from China are vague. The total GHG emission of China in 2004 was about 6,100 million metric tons (MMT) CO2e, of which 5,050 MMT was CO2, 720 MMT was CH4, and 330 MMT was N2O. Estimated GHG emissions in China in 2005 were around 7–7.5 BMT of CO2e with CO2 constituting 78–84% of the total, CH4 emissions were around 11–13%, N2O about 1%, and the synthetic gases (SF6, PFC, and HFC) together less than 1%. From 1994 to 2004, the average annual growth rate of GHG emissions is about 4% per annum. During 2001–2011 periods Chinese GHG emissions increased even more rapidly by 166% (Mohajan, 2014a). The emissions of CO2 of China are very high due to the large population, inefficient strong capital investment, heavy reliance on coal, and inefficient planed urbanization. The per capita income of the USA is very high, but that of China is very low. The GHG emissions of China are higher than the USA (Leggett et al., 2008). China produces about 80% of its electricity by the fossil fuel-fired technologies, and it emits one-fifth of world’s GHG emissions from power generation. In 2006, it has becoming the world’s largest GHGs emitter. The potential investors of China are confronted with uncertainty in the design of China’s future climate policy. IEA (2007b) expects that power generation in China will grow with an average 4.9% p.a. It is estimate that the installed capacity will reach 1,775 GW by 2030, which is nearly as high as the current installed capacity of the US and the EU combined (Schenker, 2011). Within the period of 1979 to 2007, the Chinese economy grew at an average 9.8% p. a. China acquired $1.5 trillion in foreign exchange resources by the end of 2007, and $3.2 trillion foreign reserve at the end of 2011; which is the world’s largest foreign

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reserve (The Prothom Alo, 2011). So that citizens have improved their standards of living. In 2011, some economic experts claim that China is now a developed country, but yet China is a developing country. World Bank in 2005 estimated that up to 200 million people in China lived on less than $1.25 per day. Hallding et al. (2009) indicate that although China has large foreign reserve and rich economic development, but about half of the populations live on less than $2 per day. So that China has not eradicated poverty, and cannot create field to increase per capita income which is a drawback to overcome poverty of the citizens of China. In most parts of China environ pollution has become so worst that social and political stability are at risk. In 2007, World Bank and the government of China estimated that the cost of outdoor air and water pollution to China’s economy totaled around $100 billion per annum which is 5.8% of China’s GDP (World Bank, 2007). Due to GHG emission China realized the effect of warming of Climate. Recently China has observed that impacts of storm intensity, rising sea levels, decrease in agricultural productivity, shifting water availability already affecting the people of China. Coal is the relatively cheap natural fossil energy source for China. In China coalfired power plants produce more than 2,500 TWh (terawatt-hours) electricity per year. Because of heavy reliance on coal, the electricity and heat sector is responsible for about 50% of China’s CO2 emissions from fuel combustion (IEA, 2010b). In 2007, it is estimated that China coal contribution is about 70%, in petroleum is 20%, in gas is 3%, and hydroelectric and nuclear contribute 7% for its total energy needs. While the USA used petroleum about 40%, coal and natural gas provide about 25%, and nuclear and hydroelectric contributing 10% for its total energy needs. Hence, China in 2007 consumed about twice as much coal each year as the USA. On the other hand, China is the world’s largest producer of hydroelectricity generating over 397 TWh per year, which is 16% to the total annual electricity production. China estimated in 2004 that its total GHG emissions in 2007 would be about 6,100 MMTCO2e which is a growth of 50% in one decade. This estimation is 83% from CO2, 12% from CH4, 5% from N2O, and 1% from SF6, HFC, and PFC.

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10.4.3 CH4 Emissions in China China is the single largest emitter of CH4 in the world (UNEP, 2011b). Global CH4 emissions in 2010 totaled 7.2 GT CO2e, of which China produced about 925 MMT CO2e, surpassing both India and the USA. CH4 emissions of China are due to its large population and economic activities, such as, energy use and production, waste disposal and agricultural processes (UNEP, 2011b). CH4 emissions from organic waste account for about 20% of China’s total CH4 emissions, which includes CH4 produced from the degradation of the organic fractions of municipal solid waste (MSW) in landfills, agricultural manure management, and wastewater treatment and discharge (UNEP, 2011b). CH4 leaked from fossil fuel production accounts for about 33% of China’s total CH4 emissions (Brink et al., 2013). China is rich in coal-related CH4 resources which are buried to a depth of 2 km are over 34 trillion m3, 12.5% of the world’s total, ranking the 3rd in the world (China University of Petroleum, 2008). In 2009, about 96 MMT CO2e was captured from Chinese coalmine, and 25 MMT CO2e was utilized (CAFT, 2012).

10.4.4 GHG Mitigation Policies of China In 1992, the United Nations Framework Convention on Climate Change (UNFCCC) supported 192 countries including China and the USA to stabilize “Greenhouse gas concentrations in the atmosphere at a level that would prevent dangerous anthropogenic interference with the climate system.” In the UNFCCC all the countries unanimously agrees to GHG concentrations (Leggett et al., 2008). One of the aims of these countries is to reduce GHG emissions by 55% of the 1990 levels by 2012. With the support from the UN and the USA, China hopes to board on a multi-million dollar renewable energy strategy to combat environment pollution. Due to GHG emissions China has realized the effect of warming of climate. Recently China has observed that impacts of storm intensity, rising sea levels, decrease in agricultural productivity, shifting water availability already affecting the people of the country. In the 11th Five-Year Plan China has taken attempts for the creation of clean and renewable energy as an important national policy. So that it is developing hydropower, solar power, wind power, natural gas, and biomass fuel technologies. It forecasts that the

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nuclear power use in electricity generation is expected to increase to 4% by 2025 from 1% current production which will contribute in the reduction of GHG emissions (Zhang, 2011b). In 2011, the Chinese Government issued the Work Plan for Controlling GHG emissions during the 12th Five-Year Plan Period, which assigns specific carbon intensity reduction targets to all provinces, autonomous regions, and municipalities directly under the central government. It targets to reduce energy consumption per unit of GDP by 16%, cut CO2 emissions per unit of GDP by 17%, and to raise the proportion of non-fossil fuels in the overall primary energy mix to 11.4% (China’s Policies and Actions, 2012). To reduce pollution and increase the shares of non-fossil fuels in the energy sector, China has set goals to improve its CO2 intensity of 40–45% by 2020, with an interim target in the 12th Five-Year Plan of 17% by 2015 (Leggett, 2011). China initiated circular economy during the 12th Five-Year Plan Period; some of them are recycling projects in 22 industrial parks, recovering mineral resources from city waste in 7 industrial parks, reuse of kitchen waste in 16 cities, re-use of industrial solid waste in 12 regions. The Ministry of Finance and the Ministry of Transport have allocated special funds for energy conservation and emission reduction to subsidize 402 projects in 2011 and 2012 that achieved a reduction of 1.837 MT of CO2 emissions (China’s Policies and Actions, 2012). China takes an attempt to produce 16% of all energy from renewable resources by 2020. It expects that wind, solar, geothermal and tidal energy will reduce 60 MMT CO2, biomass will reduce 30 MMT CO2, and hydroelectricity will reduce 30 MMT CO2 (Global Wind Energy Council, 2008; NDRC, 2008). Recently China has started to build energy saving buildings and announced that new buildings constructed from 2006 to 2010, the buildings should be designed in standard to energy conservation by 50%. The Government of China estimated that the standards and levels for refrigerators, air conditioners, washing machines, and color televisions will save 33.5 TWh; and reduce GHG emissions by 11.3 MMT CO2 by 2020 (Zhou, 2008). China, in 2004, set passenger vehicle fuel economy standards in step by step whose average speed will be 36 miles (1 mile = 1.61 km) per gallon (mpg) in 2008. It also emphasis same conditions on trucks and agricultural vehicles. After implementation of

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these standards China could reduce 488 MMT CO2 by 2030. The Chinese Ministry of finance adopted taxes on vehicles which is affected September1, 2008. This law doubled taxes on large vehicles and reduced taxes on small vehicles. Purchasers of cars with engines above 4 liters (1 gallon = 3.79 liters) capacity will pay a rise tax of 40%, the vehicles with engine capacity between 3 and 4 liters will rise 15% to 25%. On the other hand, engines with one liter capacity will reduce from 3% to 1% (Leggett et al., 2008).

10.5 Comparison of GHG Emissions between China and the USA The population of China is about 4.5 times larger than that of the USA, its economy, as measured using nominal exchange rates, was only about one-sixth as large. At present per capita carbon emissions in the USA are about 5 times than that of China, which implies that if China’s per capita GHG emissions rose to the US levels, then global carbon emissions would increase by more than 50%. About 40% of the US CO2 emissions are related to residential and personal transportation, but CO2 emissions are very few in these sectors (Table 10.2) in China. China requires 50% more energy to produce one billion dollars of GDP compared with the USA. When the UNFCCC was opened for signature in 1992, the already industrialized countries emitted almost 80% of the global CO2 from energy and industry. At that time the global CO2 emissions of the USA, the EU, and China were about 23%, 20%, and 11% (unfortunately in 2013, global CO2 emissions of China becomes 21%) respectively. At the same period all the developing countries contributed about one-third of the global CO2 emissions (Leggett, 2011). Coal is the relatively cheap natural fossil energy source for China. In China coalfired power plants produce more than 2,500 TWh electricity per year. Because of heavy reliance on coal, the electricity and heat sector is responsible for about 50% of China’s CO2 emissions from fuel combustion (IEA, 2010b). In 2007, it is estimated that China’s coal contribution is about 70%, in petroleum is 20%, in gas is 3%, and hydroelectric and nuclear contribute 7% for its total energy

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Table 10.2: Selected statistics for China and the USA in 2005, Source: (Leggett, 2011).

Country

China The USA

Population (millions)

1,339 307

Population growth (annual %)

0.5

GDP (billions $)

2, 244 12,398

GNI using PPP ($)

9,091 14,119

GNI per capita ($)

6,828 45,989

GDP growth (%)

14.2

0.9

1.9

Energy consumption per capita (kg oil equivalent per 1,316 7,893 capita) Electricity consumption per capita (kWh per capita)

2,791 13,506

CO2 emissions (MMT CO2) in 2010

8,333 6,145

GHG emissions (MMT CO2e) in 2005

7,527 7,282

GHG emissions per capita (metric tons per capita) in 2005 6

25

GHG emissions per GNI (tons per 1000 $ GNI, using 1.4

0.6

PPP) in 2005

needs. While the USA used petroleum about 40%, coal and natural gas provide about 25%, and nuclear and hydroelectric contributing 10% for its total energy needs. Hence, China in 2007 consumed about twice as much coal each year as the USA. On the other hand, China is the world’s largest producer of hydroelectricity generating over 397 TWh per year, which is 16% to the total annual electricity production. In 2004, China estimated that its total GHG emissions in 2007 would be about 6,100 MMTCO2e which is a growth of 50% in one decade. Of the estimated GHG emissions of China in 2004 were, about 83% CO2, 12% CH4, and 5% N2O, with less than 1% of SF6, HFC, and PFC.

10.6 Effect of Methane Gas in Atmosphere All the nations emphasized to the reduction of CO2 emissions, but no nation took CH4 emissions seriously. CH4 is present in the atmosphere low compared to CO2 (14%), but CH4 is 21 times more potent than CO2, so that all nations must take steps to reduce 208

fugitive emissions of CH4. At present CH4 emissions contribute more than one-third of anthropogenic warming (EPA, 2006). In the pre-industrial period CH4 was 715 ppb (parts per billion), but in 2005 it increased 148% to reach 1,774 ppb (IPCC, 2007). About half of this increase is due to decomposition of wastes in landfills, natural gas systems, and enteric fermentation (EPA, 2006; Mohajan, 2011c). Since the atmospheric lifetime of CH4 is short compared to that of the CO2, the Global warming potential (GWP) of CH4 varies considerably depending on the period of time chosen.

10.6.1 Benefits from the Reduction of Methane Gas Emissions Global emission of methane in 2000 is 352 million tons. This calculation would accurately be applied for a fifteen year period (1995–2010). Hence, in 15 years, total emissions of methane = 352 million tons ×15 = 5.3 billion tons. The total cost of 15 years global methane emissions is $600 billion. So that, the mean benefit = $600 billion ÷ 5.3 billion = $113 per ton of methane reduction. We can compare this with cutbacks of CO2 which gives benefits of between $10 and $50 per ton of CO2, with a mean value of $20. All values are calculated in 1990 dollars (Plambeck & Hope, 1996; Mohajan, 2012d). Only 5% of the benefits are in the EU, and 8% in the USA; the rest are benefited the developing countries. These benefits will continue to the 21st century (Hope, 2001). Reductions in CH4 emissions can slow the rate of near-term global warming, and reduce global air pollution in ozone sphere, as a result improve human health and reduce cropyield losses globally (USEPA, 2011).

10.7 Recent Natural Calamities In 2010, the Haitian earthquake killed hundreds of thousands. In terms of the overall number of disasters, 2011 was a terrible year (302 disasters): floods in Australia, a devastating earthquake in Christchurch, and a horrific tsunami and nuclear accident (estimated loss $210 billion) in Japan and more than 813 people had died. The USA was particularly hard hit as Mississippi River floods were followed by a string of deadly tornadoes, terrible wildfires, and then Hurricane Irene which closed down much of the country’s east coast for several days. In October 2011, Bangkok experienced serious

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floods resulting in $ 46.5 billion worth of economic losses, inundating 9.1% of the total land area (69 provinces) of the country affecting more than 13 million people, with 728 deaths (Chongvilaivan, 2012; Ferris & Petz, 2012; Poaponsakorn & Meethom, 2013). Over the past 30 years, disasters have affected more than 40 million people in the Middle East and North Africa (MNA) region, and have cost their economies about $20 billion (World Bank, 2014). Over the last thirty years, Bangladesh has experienced nearly 200 of these climate-related disasters, which have killed thousands of people, destroyed homes and livelihoods, and cost the nation around $16 billion in damage and economic losses (Give 2 Asia, 2014). In Bangladesh, the predicted rise in sea level will cover 17% of the country by 2050, displacing 18 million people (Harris, 2014). In 2012–2014, Beijing experienced massive floods following especially heavy rains. Indonesia experienced heavy flooding in early 2014 (East Asia Seasonal Analysis, 2014). In 2013–14, the UK has been affected by an exceptional run of severe winter storms, culminating in the coastal damage, and widespread flooding (Centre for Ecology & Hydrology, CEH, 2014). Deloitte Access Economics estimated that the annual economic cost of natural disasters would rise from $6 billion in 2012 to $12 billion by 2030, and $23 billion by 2050. All of these events were huge setbacks for the economy, with losses of life, massive losses of property, billions or even tens of billions of dollars of damage, and disruptions to the global economy (Deloitte Access Economics, 2013).

10.8 Conclusion In this chapter we have shown that GHG emissions increase global warming gradually which results global climate change. To keep the earth living place for all creatures we have to take immediate steps to reduce GHG emissions efficiently. Here we have discussed GHG emissions of the USA and China. In 2006, China became the largest GHG emitter in the world. China heavily depends on coal to produce electricity. Coal emits more CO2 in the atmosphere than other fossil fuels. For the survival of the humanity and sustainable economic development, both the USA and China can take necessary steps with other nations to reduce GHG emissions. Recently they have taken some steps to reduce GHG emissions. We have stressed that all the nations must take

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necessary steps to save the world from destruction. Industrial countries are emitting more GHGs than the developing countries. So that industrial countries will try to reduce GHG emissions efficiently.

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Chapter–XI

Green Taxes on Environment Pollution

11.1 Introduction In the last part of the 20th century and in the beginning of the 21st century the area of cities of the world has expanded, and new cities and towns have grown rapidly. As a result in vehicle-miles traveled increases. Again, most of the luxurious people like large vehicles which are increasing externalities from vehicle emissions. Emissions from vehicles pollute air that worsened human health, diminishing visibility, and caused global warming (Fullerton & West, 2002). Actual vehicle emissions depend not only on vehicle size and age, but also on qualities of the fuel, maintenance of the car’s pollution control equipment (PCE), frequency of cold start-ups, temperature of the air, speed of the vehicle, and aggressive driving (Fullerton & West, 2002). As like Fullerton and West (2002) we investigate some policies that would influence people to drive fewer miles and to buy smaller cars, better pollution control equipment, and cleaner fuel (Islam et al., 2011d). Tullock (1967) and Terkla (1984) were the first who suggested that revenues from environmental taxation could be used to finance reductions in pre-existing taxes. We see that this fund recycling process would significantly reduce the welfare costs associated with the overall tax code. But, it did not account for the effects of health damages from pollution. We show as like Williams (2003) and Caffet (2005) that this is not the case. Pollution taxes drive up the price of consumption goods, so that lower the real wage, causing households to work less, and consume more leisure which raise the environmental taxes (Mohajan, 2011a).

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11.2 Cap and Trade System or Carbon Tax System At present there are two classic alternatives for regulating GHG emissions, a cap and trade policy, and a carbon tax policy. Cap and trade is a quantity control policy, and carbon tax is a price control policy. The cap and trade system provides a price which is a secondary result of regulating the quantity of GHG emissions. On the other hand, the carbon tax effectively reduces the quantity of GHG emissions which is a secondary result of setting a price. To reduce GHG emissions both methods contribute, and in idealized circumstances they seem equivalent (Mohajan, 2012i). If we think the policies in the economic view we observe that quantity control (in cap and trade policy) is preferable when the marginal benefits from price control (in carbon tax policy) are sharply sloping as compared against the marginal costs, but price control is preferable when marginal benefits are relatively flat, and marginal costs are sharply sloping (Newell & Pizer, 2006). Hence, we observe that the marginal benefit function is flat, but the marginal cost function slopes sharply (Nordhaus, 1994). At present the world carbon trade includes fulfillment markets in the EU, the USA, and New Zealand, representing over 140 billion dollars in traded value, and as much as 5 GTs of emissions per year (Linacre et al., 2011). To enlarge the world carbon trade with proposed markets in Australia and Japan, the international market is projected to reach magnitudes of $2–3 trillion by 2020 (Lazarowicz, 2009; Calel, 2011). In post-Kyoto international framework the international carbon markets remain a key component of many countries’ carbon policy (Mohajan, 2012i).

11.3 Taxes on Car and Gasoline In our model we first consider homogenous consumers, and then consider for heterogeneous consumers that differ by income and two taste parameters, one for miles and one for vehicle size. The motorist can reduce their fees by repairing their vehicles, but not by driving less. Sevigny (1998) incorporates the choice of miles with a secondbest emissions tax, but this tax requires knowledge of each vehicle’s average emission per mile, and the accurate measurement of miles traveled. Emissions per mile (EPM) cannot be measured perfectly, because it depends on how the car is driven. Miles cannot be measured perfectly, because drivers can roll back the odometer.

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The efficiency of the emissions tax can be achieved from the homogeneous agents by a set of uniform tax or subsidy rates on choices such as fuel use, type of fuel, engine size, vehicle age, and pollution control equipment (PCE). Heterogeneous agents maximize different utility functions, so that they have different choices about miles driven, engine size, and vehicle age-three different important determinations of emissions (Fullerton & West, 2000, 2010). Let, a car drives m miles, so that total emission is m.EPM, and we treat a tax on those emissions as the ideal Pigovian tax. The motorist still has a variety of taxes or subsidies on observable choices such as gasoline, engine size, and vehicle age to induce individuals to drive fewer miles, to buy smaller cars, or to scrap older cars (Fullerton & West, 2010).

11.3.1 Model for Homogeneous Consumers We assume perfect information, perfect competition, and no market failure other than a negative externality from emissions for homogeneous consumers (Fullerton and West 2002). Let us consider a simple economy consists of n identical individuals each of which owns one vehicle. Each vehicle is composed of some attributes that affect emissions (such as engine size, fuel efficiency, and PCE) and other attributes that do not affect emissions (such as leather seats or a sunroof). They gain utility from driving miles m, the size of the vehicle s, and other goods and services, x. The size s of engine is measured as cubic inches of displacement (CID). The consumers may gain or lose utility from pollution-control equipment c, and per gallon fuel cleanliness f. Fuel cleanliness is an attribute of gasoline such as volatility or oxygenation. Again, household utility is affected by aggregate auto emissions, E. Thus, the household’s utility function is, u = u (m, s, c, f, x, E )

(11.1)

where u is continuous, differentiable, and strictly quasi-concave in its first five arguments. EPM = X discharge by a car depends positively on the size, and negative on PCE; and the clean-fuel characteristic, i.e., X = X (s, c, f ). Each of the households drives m miles, then aggregate emission E can be written as; E = n m X.

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(11.2)

The fuel efficiency is measured in MPG = Y, and depends on engine size and the quantity of the clean-car good on the vehicle, i.e., Y = Y (s, c). Cars with larger engines get lower gas mileage, so that, Ys =

Y  0 . Consumers do not purchase m directly, but through the s

combination they choose gasoline (g), size (s), and the clean car good (c), so that; g=

m . Y (s, c )

(11.3)

Consumers use (11.3) when they decide vehicle size, vintage, and how much gasoline will maximize utility (11.1). The individual is taxed or subsidized on consumption of m, s, c, f, and x. Let, p g = price per gallon of gasoline without any clean characteristic, p f = price per unit of the clean-fuel characteristic per gallon. The total price of a gallon of gasoline is = ( p g + p f f ), and the private cost of driving one mile is, ( p g + p f f )/Y (s, c). Again ps = price of s, which represents the price of adding a CID to an engine, pc = the price per unit of the clean-air good. For convenience we normalize the price of x equals one. The individual problem is to maximize (11.1) subject to budget constraint;

 p + pf f y =  g  Y (s, c )

  m + ps s + pc c + x . 

(11.4)

Hence the social planner Lagrangean is;

  p + pf f L= u (m, s, c, f, x, E= nmX ) +   y −  g   Y (s, c )

   m − ps s − pcc − x  

(11.5)

where δ is the marginal social value of income. The first-order conditions with homogeneous consumers for maximization are as follows:

 p + pf f um + uE n X =   g  Y (s, c )

  , 

(11.6a)

where bracketed term in (11.6a) is the total implicit price of a mile.

( p + p f f ) mYs  ,  us + uE n m X s =   ps − g  Y2  

215

(11.6b)

where the bracketed term in (11.6b) is the overall cost per unit of size, including the extra amount that must be paid for miles due to the lower MPG caused by the incremental unit of s.

( p + p f f ) mYc  ,  uc + uE n m X c =   pc − g  Y2  

(11.6c)

where the bracketed term in (11.6c) is the overall cost of PCE, including the extra amount that must be paid for miles due to the lower MPG.

 m pf u f + uE n m X f =    Y

  , 

(11.6d)

where the bracketed term in (11.6d) is the overall cost per unit of the clean-fuel characteristic.

ux =  .

(11.6e)

The term u E on the left-hand sides of (11.6a) to (11.6d) reflects the effect on utility of the increment to aggregate emissions from driving an additional mile, increasing vehicle size, adding PCE, or cleaner gas. An individual usually does not know that his own choices affect aggregate emission, but he may face taxes or subsides on its consumption of s, c, f, x, and g. The household’s budget constraint becomes;

 p +  + (p f +  f ) f y =  g g Y (s, c ) 

  m + ( ps +  s ) s + ( pc +  c ) c + (1 +  x ) x + m e X (s, c, f ) 

(11.7)

where  g is the tax per gallon of gas,  f is the tax per unit of clean-fuel characteristic,  s is the tax per unit size,  c is the tax per unit PCE, and  e is the tax per unit of emissions. Hence the household’s Lagrangean is;

L = u (m, s, c, f , x, E ) +   p +  + (p f +  f ) f   y −  g g Y (s, c )  

   m − ( ps +  s ) s − ( pc +  c ) c − (1 +  x ) x − m e X (s, c, f ) . (11.8)  

The first-order conditions for maximization are as follows:

 p +  g + (p f +  f ) f  um =   g +  e X (s, c, f ) , Y (s, c )  

216

(11.9a)

   − ( pg +  g + ( p f +  f ) f )Ys   + m e X s  , us =   ps +  s + m  2 Y    

(11.9b)

   − ( pg +  g + ( p f +  f ) f )Ys   + m e X c  , uc =   pc +  c + m  2 Y    

(11.9c)

 (p +  f )m  uf =   f + m e X f  , and Y  

(11.9d)

u x =  (1 +  x ).

(11.9e)

Emissions would be calculated to enter the consumer problem implicitly through the pollution tax  e . The price per unit mile, and similar emissions tax calculations would be for s, c, and f (Islam et al., 2011d).

11.3.2 Analytical Calculations for Taxes and Subsides The tax on emission  e , provides the basic efficient policy against which alternatives can be compared. Let, all other tax rates set to be equals to zero, i.e.,

 g =  f =  s =  c =  x = 0 . Then (11.6e) and (11.9e) imply u x =  =  . Now using  =  in (11.6a) we get;

 p + pf f   − uE n X . um =   g Y  

(11.10)

Using the value of um in (11.9a) we get;

e = −

uE n



.

(11.11)

Now we can define (11.11) as the marginal environmental damages (MED) per unit of emissions; which is the usual Pigovian tax, and it is greater than zero so long as uE  0 . Now we will calculate gas tax  g . For the impossible measurement of gas emission,

 e = 0 , and suppose all other tax rates be zero, i.e.,  f =  s =  c =  x = 0 , then from (11.6e) and (11.9e) we get,  =  , and (11.9a) now becomes;

 p + pf f g  um =   g + . Y Y  From (11.10) and (11.12) we get;

217

(11.12)

g = −

uE n



X (s, c, f ) Y (s, c ) ,

(11.13)

which represents the additional damage caused by an increase of one gallon of gas. From (11.13) we see that gas tax depends on fuel characteristic f, and on the characteristic of the vehicle at the pump (s and c). To calculate vehicle tax  v , suppose  f =  g =  s =  c =  x = 0 , and the Lagrangean of (11.8) is modified by subtracting a tax  v per vehicle. The vehicle tax would be as follows:

v = −

uE n



m X (s , c , f ) .

(11.14)

From (11.6b) and (11.9b) we get,

s = −

XY   m  Xs − s  .  Y  

uE n

(11.15)

Here the first term gives the direct damage caused by an increase of one unit of size, which is positive as long as emissions affect utility, uE  0 , and size affects emissions,

X s  0 . The second term is an indirect effect from an additional unit of size through its effect on fuel efficiency. Since uE  0 , so that  s  0 if;

X s − Ys  . X Y

(11.16)

We now solve PCE tax rates. From (11.6c) and (11.9c) we get for  =  ;

c = −

n uE m X c





n u E mYc X Y

(11.17)

which is analogous to the  s . The first term of (11.17) is negative to reflect the effect on damages of an added unit of PCE, and the second term is a rebate due to the effect that PCE has no fuel efficiency, and hence it is negative. So that  c is always negative, that is, it is necessarily a subsidy. Since  e = 0 the subsidy to PCE (either in  g or  e ) can only induce consumers to buy any such equipment if it is equal to the entire private cost of PCE, including both the direct cost pc , and the extra fuel cost incurred due to the negative effect that indeterminate (Islam et al., 2011).

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11.3.3 Model for Heterogeneous Consumers Let us assume parameter  to represent the household’s preference for miles, and β to represent the preference for size of the car. Together with income, these parameters are jointly distributed according to the distribution function h( ,  , y ) with positive support



  

on  ,   ,   y, y . The people who live fur from their work place have a high demand for miles (  ), but they may prefer either a large car for comfort and safety; or a small car for better gas mileage. For heterogeneity we ignore the clean-car, and clean-fuel characteristics. Hence, fuel efficiency and emissions per mile depend only on size, and each household generates m X (s) units of emissions. The total pollution is;

 mX (s ) h( ,  , y ) dddy .

E=

(11.18)

 y

A household’s utility function is;

U = u (m, s, x; ,  ) − E

(11.19)

where μ is the household’s change in welfare from additional pollution ( U / E ). The social planner must maximize a measure of social welfare such as a weighted sum of utilities of n households. We divide each household’s utility by its own marginal utility of income (  ). If  e is available, we want the maximization of our social welfare function to yield the solution of Pigou (1932). To evaluate * we use the prices at the Pigovian equilibrium, and weights are calculated as (1/ * ). The social welfare function is;

W=

 u (m, s, x )  −  E  h( ,  , y ) dddy . * 

    y

(11.20)

Again the social planner’s budget constraints; y=

pg

Y (s )

m − ps − x .

(11.21)

The social planner’s problem is to maximize this welfare function subject to a resource constraint, so that the Lagrangean is; L=



 y

 u (m, s, x )  −   m X (s ) h( ,  , y ) dddy  h( ,  , y ) dddy   *   y

219

  p   +     y − g m − ps − x  h( ,  , y ) dddy  Y (s )    y  

(11.22)

with respect to each consumer’s m, s, and x. Income plus tax rebates is y, and the marginal social value of income is δ. The first-order conditions for household i is as follows:

1 ui − n X (si ) =  * mi

 pg   ( ) ,  Y si 

(11.23a)

where the second term represents the external cost of an additional mile driven by individual i.   − pgYsi 1 ui − n mi X si =   ps + mi  2  Y * si 

  , 

(11.23b)

where the second term represents the external cost of an additional unit of size purchased by individual i.

1 ui = , * xi

(11.23c)

where each equation represents n first-order conditions, one for each individual i. A household does not identify that his own emissions add to aggregate emissions. The household’s budge constraint is;

yi =

pg +  g mi + ( ps +  s ) si + (1 +  x ) xi +  e X (si ) mi . Y (si )

(11.24)

Therefore, household problem is to maximize the Lagrangean;

L = ui (mi , si , xi ) −  E + 

 pg +  g  Y (si )

i  yi −  

   mi − ( ps +  s ) si − (1 +  x ) xi − mi e X (si ) .  

(11.25)

with respect to mi , si , and xi . The first-order conditions for maximization are as follows:

 p + g  ui = *  g +  e X (si ) , mi  Y (si ) 

220

(11.26a)

  ( pg +  g )Ysi ui = *  ps +  s − mi  si Y2  

   +  e mi X (si ) , and  

ui = * (1 +  x ) . xi

(11.26b)

(11.26c)

11.3.4 Analytical Calculations for Taxes To calculate Pigovian tax, we set all taxes except  e equals to zero, i.e.,

 s =  g =  x = 0 . Again we use δ =1 in (11.23a) and (11.26a) to equal each other. Hence, n

e =



= MED.

(11.27)

This is the first-best uniform Pigovian tax, and can be used to identify other first-best gas tax. In the heterogeneous-consumer model, this tax is as follows:

 g = n X (si ) Y (si ) .

(11.28)

i

Authorities might be able to impose a tax on each vehicle that depends on a direct measure of X (si ) , and multiply by a measure of mileage; then the vehicle tax be as follows:

 v = n X (si ) mi

(11.29)

i

which is similar to the consumer model. It indicates first-best, but the tax amount would differ among heterogeneous households. From (11.23b) and (11.26b) we get;

 s = n mi X s + n mi i

i

X (si )Ysi Y (si )

.

(11.30)

Suppose that the first three policies (11.28) to (11.30) are calculated above are not feasible, and policy is limited to a single uniform rate of tax on gasoline, and single uniform rate of tax on engine size or other vehicle characteristic. This policy achieves first-best in the homogeneous-consumer model, but not in the heterogeneous-consumer model. Moreover, a greater degree of heterogeneity means greater divergence from firstbest. For these reasons, we now consider how to set the second-best uniform tax rates on gasoline and engine size (Islam et al., 2011d).

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11.3.5 Second-best Taxes on Gasoline and Size We assume that producers’ prices are fixed which is equivalent to maximizing this weighted sum of indirect utilities;



 y

V ( s , g , x ; y, ,  )  −  E  h( ,  , y ) dddy ,  *  

(11.31)

with respect to  s and  g . First-order conditions of (11.31) are;

 

 1 V  −   ( A( s )) h( ,  , y ) dddy  h( ,  , y ) dddy = 0 , and (11.32a)   *  s   y

 

 1 V  −   (A( g )) h( ,  , y ) dddy  h( ,  , y ) dddy = 0 ,   *  g  y

y

y

(11.32b)

where A( i ) = g Y (s ) X s

s s g + g X (s ) Ys + X (s ) Y (s )  i  i  i

(11.33)

for i = s, g. Using Roy’s identity

V = −s (11.32a) becomes;  s

 − s  −   ( A( s )) h( ,  , y ) dddy  h( ,  , y ) dddy = 0 , *  y  

    y

(11.34a)

 − s   represents the change in welfare from a where the first term in the integral    *  change in the size tax, holding aggregate emissions constant, and the second term is the change in utility due to the effect that a size tax has on aggregate emissions. Again using Roy’s identity



 y

V = −g (11.32b) becomes;  g

 − g  −   (A( g )) h( ,  , y ) dddy  h( ,  , y ) dddy = 0  *  y   

(11.34b)

where the first term is the change in welfare from a change in the gas tax, holding aggregate emissions constant, and the second term the change in welfare from the effect that a gas tax has on aggregate emissions.

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To calculate second-best taxes we take average of all different gas tax rates in (11.28) as follows:

n X (s )Y (s ) h( ,  , y ) dddy   i

g =

y

i

 h( ,  , y ) dddy

 y

=

  X (s ) Y (s ) h( ,  , y ) dddy . i

i

(11.35)

y

Again if we take gas tax rate for the person with average choices then (11.28) becomes;

 g (s ) = n X (s ) Y (s )

(11.36)

Convexity of X(s) would mean that increases in size increase emissions per mile at an increasing rate which would raise the weighted average using X(si) in (11.35) relative to the tax rate using average size in (11.36). Similar result is obtained for Y(si). From (11.30) the average of the size tax rate becomes;

s =

 

y

 X (si )Ys

 X s mi h( ,  , y ) dddy + 

Y (si )

i

 y

i

h( ,  , y ) dddy .

(11.37)

Again, the size tax for the per person with average choices (11.30) becomes;

 s (s , m ) = n m X s + n m

X (s )Ys . Y (s )

(11.38)

Since both s and m have in both equations, the difference between the average size tax rate in (11.37), and the size tax rate using average miles and size in (11.38) depends both on whether preferences are correlated and on whether X(s) or Y(s) is non-linear. For linearity Xs and Ys be constants, then the first terms do not affect in either (11.37) or (11.38), but the second term of (11.38) must affects, since Ys 1 (Mohajan, 2011a). Williams (2003) defines marginal damages from pollution  P as the sum of two terms, the respective values of the direct utility loss from reduced health and the time lost to illness where the gross wage has been normalized to equal one, as follows:

P =

1 U H S H . −  H Q H Q

Since the representative agent still earns his wage during sick-days, so that Caffet (2005) corrected the second term of it. Caffet’s logic is that households only bear the distortionary cost introduced by the government transfers, which is the marginal deadweight loss of the labor tax relative to the wage keeping device. In this way, the Pigovian rate is given by;

P =

1 U H S H . − ( − 1)  H Q H Q

(11.53)

With the basis of (11.53) the welfare effect of the environmental tax being as (11.54) (for derivation see Appendix–I). The 1st term of (11.54) is the primary or Pigovian effect, dW P which is the effect of the tax on the pollution externality. In a first-best world

without pre-existing distortions, the optimum is reached for  X =  P , which means that the optimal environmental tax must equal to the Pigovian rate.

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  l  1 dU dX dX  S H  dM   −   L = ( X −  P ) + ( − 1)  X +  X +  M +   d X d X d X   X  H M  d X          dW P

dW R

dW I

 S H l  dQ . − +  L H Q Q  d X   

(11.54)

dW IB

The 2nd term is the gain from the marginal revenue-recycling effect, dW R , which is the welfare gain from using the pollution tax revenues to reduce the labor tax. This is happen when they are returned lump-sum, and have no efficiency consequences at all. So that dW R is the product of the efficiency value per dollar of revenue, and the incremental pollution tax revenue, where the marginal excess burden of taxation. The 3rd term dW I , and 4th term dW IB are respectively what Williams (2003) called cost-side, and benefit-side tax interaction effects. They activate when changes in households’ labor supply decisions interact with the labor market distortion. For the 3rd term, the pollution tax drives up the price of consumer polluting goods, which lower the real wage, and as a result it affects discouraging labor supply. We have seen that this fall in labor supply must exacerbate the private social cost of the labor tax by  L

l (Caffet, 2005). The beneficial side is that this also reduces labor tax revenues  X

equal to  L

l , and thus requiring a compensating increase in the labor tax rate, and  X

creating an efficiency loss which equals to the amount that the government has to refund multiplied by the efficiency cost per dollar of labor tax revenue, that is, the marginal excess burden of labor taxation. The tax interaction effect identified by prior literature on the double dividend hypothesis is exclusively the welfare loss from these two impacts, which equals to  L

l . In the analysis of Caffet (2005), there is however a second  X

term in dW I . In this term, first part indicates medical expenditures generate an efficiency gain, ( − 1) M

X which is related to the contraction of subsidies’ amount. Second part  X

indicates subventions cause this consumption to be under priced relative to its social cost.

229

As a result, any decrease in medical expenditures will lead to a general-equilibrium welfare gain. This gain equals the wedge between supply and demand prices multiplied by the reduction of subsidized consumption, which is equals to  M

X .  X

The 4th term dW IB is the benefit-side tax interaction effects, which expresses the impact of improved environmental quality on labor supply decisions. Caffet’s logic is that a rise in sick-days does not imply a decrease in government revenues. Because  L is levied on L +S instead of effective labor income L. So that welfare impacts of sick-days reduction can be resumed by the product of that decrease, and the sum of the efficiency gain and the private social benefit relative to that decrease (Mohajan, 2011a).

11.4.2 Optimal Environmental Tax Following Williams (2003), we now stress on the two opposite tax interaction effects, dW I and dW IB . Following Caffet (2005) we also consider the neutral assumption that goods X and Y are equal substitutes for leisure, allowing equation (11.54) to be re-written as follows (for derivation see Appendix–II);

1 dU dX S H  dM  S H  = ( X −  P ) −   M + + −   d X d X H M  d X  H Q 

 − ( − 1) LI L

M  M   dQ  1 − (1 −  M ) I  (1 −  M ) Q    d X −1

(11.55)

where L is the uncompensated labor supply elasticity and LI is the income elasticity of labor supply. The first term on the right hand side combines the primary welfare effect, the first component of cost-side tax interaction effect and the revenue-recycling effect. Without the other terms on the right hand side, it indicates that the optimal environmental tax is equal to the Pigovian rate divided by the MCPF. The second term symbolizes the interactions have already identified between subsidies to medical expenditures and environmental taxation. Finally, the last term shows that benefit-side tax interaction

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effects must improved health conditions, may significantly reduce the gross cost of an environmental tax reform. Health effects might lead to a double dividend if an environmental tax on both reduces pollution and raise welfare, which is exclusive of the environmental benefits (Goulder, 1995). If the second dividend is defined as a welfare gain exclusive of both direct and indirect benefits of reduction pollution, then excluding both the primary effect, and the benefit-side tax-interaction effect, the introducing health effects will not produce a double dividend. Even if double dividend only deals with an additional hypothesis, we have tried to show that it is a crucial point in studies modeling health effects in the agent’s utility function (Caffet, 2005). By introducing labor market imperfections in models which deal with the possibility of finding labor market imperfections in models which deal with the possibility of obtaining a second dividend in the form of a reduced unemployment level. This assumes the existence of a social security system seems that it is necessary to correctly estimate double dividend prospects in explicitly modeling health studies (Bovenberg & de Mooij, 1994; Caffet, 2005).

11.5 Conclusion By two models, we have tried to form a tax on emissions. In the first model we have considered homogeneous consumers where we have investigated the combination of a tax on gasoline that depends only on the cleanliness of the fuel, a flat rate of tax on engine size, and flat rate of subsidy to PCE, and this combination of course first-best. In the second model heterogeneous consumers differ by income, tastes for miles, and tastes for engine size. We show that if the engine size and driving miles are negatively correlated, and both X (s) and Y (s ) are linear then we would achieve second-best. If the taste for miles is negatively correlated with the taste for engine size, then the second-best uniform size tax would exceed the rate using means (size and miles). Yet Bangladesh has not imposed emission taxes on vehicles properly. We have tried to give a guideline to apply the taxes on vehicles, and the chapter will be helpful to the government and environment analysts of Bangladesh. We have discussed beneficial sides of optimal environmental taxes. We hope that this will help individuals to improve health damages due to air pollution. Derivation of equations (11.54) and (11.55) are given in Appendices with

231

detail calculations thinking for the readers who are new in this field. We hope that the readers will feel no bother when study this chapter, and realize the importance of the optimal environmental taxes due to health effects. Throughout the chapter we have tried to give mathematical calculations and physical interpretations in some details.

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Chapter–XII

General Conclusion of the Thesis

12.1 Introduction The aim of this doctoral thesis has been to describe, investigate and explain the problems of economics and social choice in terms of mathematical and theoretical analysis. The theoretical and mathematical perspectives that could be applied in economic development have been explained here in more detailed manner. We think that mathematical models with theoretical analysis make our study fruitful. Most of the chapters of this thesis are prepared giving the priority of detail mathematical calculations introducing definitions, examples, tables, theorems, and propositions where necessary. We have prepared Chapters I to IV and XII on the basis of theoretical analysis. The main theoretical and mathematical results of this thesis are provided in Chapters V to XI. The thesis has examined the ways to improve development policy of formulation and practice of economics and social choice in terms of mathematical analysis. The goal of the study is to contribute the methods of economic and social choice in terms of mathematical models. In this thesis we have investigated the optimization, healthcare and economic development, NNP and sustainable development policy, social welfare, GHG emissions, and environment tax. We have tried to explore the general framework of the study, and the practical and theoretical contributions of the research. This final chapter presents brief summary of the thesis, major research findings, the value of this approach, and personal reflection. The chapter highlights on strengths, weakness, and research limitations of this study. We have also suggested some interested valuable directions for the future researchers. Finally, we have provided conclusion and recommendations of the study.

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12.2 Brief Summary of the Thesis In this doctoral dissertation we have briefly discussed background of the study, research questions, aims and objectives of the study, scope of the study, contribution to the study, and significance of the study. We have also discussed literature review, and methodology of the study. The main contribution of this research has been given in chapters IV to XI. This thesis stresses on optimum economic structures, such as, to apply cost minimization techniques in production, maximization of output, maximization of utility in any changing situation of production and consumption (Moolio & Islam, 2008; Moolio et al., 2009; Islam et al.; 2010, 2011c). We have used the technique of the method of Lagrange multipliers which is a very useful and powerful technique in multivariable calculus. Here we have discussed necessary condition by using Lagrange multipliers, and sufficient condition by considering the determinant of Jacobian matrix is negative and the determinant of the Hessian matrix is positive for optimal values (Mohajan, 2017a). We have analyzed comparative statics mathematically to show the behavior of the firm (including explicit examples), and recommend that if the cost of a particular input increases, the firm needs to consider decreasing the level of that particular input; at the same time, and there is no effect on the level of other inputs (Mohajan, 2017a). Political institutions can provide a true patriotic leader to create a peaceful society. The thesis gives a direction how a democratic state can be formed. Game theory provides mathematical models of conflict and cooperation between intelligent rational decision makers. A society can take better decision to improve its payoffs by focusing on better equilibria. This thesis tries to give a game theoretical representation of international relation between two adversary countries to form a peaceful society for the welfare of all nations (Myerson, 2006; Islam et al., 2009b). We have discussed both federal and unitary democracies, and we show that both democracies have some difficulties, but comparatively federal democracy is more viable in some situations. Arrow’s impossibility theorem showed that the preferences of many individuals be aggregated into social preference, but there is a flaw in this aggregation. The theorem shows that it is impossible for a social welfare function to satisfy five conditions namely: i) completeness and transitivity, ii) universality, iii) Pareto consistency, iv) independence of irrelevant

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alternatives (IIA), and v) non-dictatorship simultaneously (Arrow, 1963). A combinatorial approach and a geometrical representation to Arrow’s theorem, and also single-profile of the theorem are given with illustrative examples, theorems, and propositions (Feldman & Serrano, 2008; Mohajan, 2012b). An attempt has been taken here to describe various types of voting system, and manipulation of them. The voting methods are discussed here in very simple, but in a detailed manner. This thesis discusses various voting methods, such as, Condorcet method (Condorcet, 1785), Borda count (Borda, 1781; Islam et al., 2011a), single transferable vote (Droop, 1881; Newland & Britton, 1997; Mohajan, 2012e), median voter model (Black, 1958; Gans & Smart, 1996; Islam et al., 2011b), approval voting (Mohajan, 2011e; Brams & Fishburn, 1983), and majority judgment voting (Balinski & Laraki, 2007; Mohajan, 2012a), with tie breaking and manipulations in some detail mathematical analysis with examples, propositions, theorems, and displaying diagrams. In the thesis we emphasize on environmental pollution and healthcare which contains a beneficial economic model. The study indicates the effects of wealth changes, performance of environment policy, and sustainable use of manmade and natural capital. It also indicates the present and future production, and use of ecosystem services to indicate the sustainability of natural resources. Medical expenditures to cure diseases due to air pollution should not be deducted from NNP, but hamper of production due to pollution related illness should be subtracted from NNP (Dasgupta & Mäler, 2000). In the thesis we emphasize on valuing health impacts from air pollution in Bangladesh. A survey on about 500 female garments workers in slum areas of Chittagong City Corporation (CCC), Bangladesh, conducted to estimate their willing to pay (WTP) to avoid additional days of seven light health symptoms, and of one major symptom, asthma. As WTP is not established in Bangladesh, no satisfactory outcome has found from this survey. In the thesis we also provide a guideline to develop the future labor sector of Bangladesh for the sustainable economic development and social welfare of the country (Mohajan, 2012c). In the thesis we have included the NNP, sustainability, and social welfare economy. The NNP is an important item for a country (Weitzman, 1976). In the aftermath of the World Commission on Environment and Development (WCED), it became important to

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investigate whether the concept of NNP can serve as an indicator of sustainability (WCED, 1987). We have also emphasized on social welfare comparisons based on national accounting aggregates. We also show the relation between the Divisia Index of real consumption prices and dynamic welfare evaluation. We emphasize on optimal growth and growth without optimality, and examine sustainability in these two cases. We also highlight the recent world economy and the open economy of Bangladesh (Mohajan, 2011b). We have discussed GHG emissions which cause global climate change (Mohajan, 2011c). The world has realized that global warming is continually increasing due to GHG emissions. If GHG emissions cannot be controlled then the people of most of the countries will suffer from drinking water, shortage of foods and various heat related diseases. As a result the nations will not achieve expected goal of economic development (IPCC, 2007). Due to global warming the living organisms are in dangerous position, and some species have already extinct, and some more will extinct in future if global warming cannot be controlled (Stern, 2007). We have investigated some policies that would influence people to drive fewer miles, and to buy smaller cars; use better pollution control equipment, and cleaner fuel (Fullerton & West, 2010). An attempt has been made to quote the vehicle tax rates of Bangladesh (Islam et al., 2011d). We have discussed the beneficial sides of environmental tax, and also have discussed optimal environmental tax with easier mathematical calculations (Williams, 2003; Caffet, 2005; Mohajan, 2011a). Throughout the thesis we have tried to flourish the research work by detail mathematical calculations, introducing definitions, examples, propositions and theorems with proof, and displaying diagrams where necessary. Finally, we have tried to give a precise conclusion of the thesis with recommendations for the future researchers.

12.3 Major Findings of the Thesis This study has investigated various concepts in the development of economics and social science with mathematical analysis. In the study we have a set of important findings which are commensurate to the objectives of the research. Major findings of the thesis are given section-wise as follows:

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12.3.1 Optimizations This part of the research has focused on maximum output with minimum cost to create maximum utility in the society. Here we have included examples, and have verified them with the method of Lagrange multiplier, and sufficient conditions for the Implicit functions. Here we have predicted by comparative statics that if there may be changed any input, then the change of other inputs, in order to increase the output. In this section we have obtained the techniques how the profit and utility will be maximized, and cost will be minimized in any changing situation. 12.3.2 Arrow’s Impossibility Theorem and Political Relations In this part we have explained how a political institution can be formed to elect patriotic leader. Here we have used game theory to explain benefits of unitary and federal democracy of two states. We have also obtained international relation between two adversary countries. Here we stress that only peace is the focal point in this perturbed world. We have introduced a combinatorial approach, and a geometrical approach to Arrow’s theorem. We have shown that in simple-profile Arrow theorem dictators are comparatively innocuous than multi-profile Arrow theorem.

12.3.3 Voting methods In this part we have shown the voting procedure, tie breaking, and manipulation of voting system. Here we have obtained the results with the help of propositions, diagrams, tables, and examples. To obtain the results we have used mathematical techniques with theoretical analysis.

12.3.4 Environment Pollutions In chapter VIII we have calculated different parts of equation (8.25) in our own procedure. Here we have tried to represent a survey data on the female garments workers of Bangladesh to estimate their WTP to avoid additional days of seven light health symptoms, and of one major symptom, asthma. We have also tried to give a guideline to make an efficient future labor sector of Bangladesh.

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12.3.5 The NNP and Sustainable Development In this portion we have tried to establish the relation that real NNP over time are an accurate indicator of true welfare improvements, and for the sustainability and social welfare. We have established the following relations with detail mathematical calculations: i) between green NNP and wealth equivalent income, ii) between green NNP and net social profit, iii) between green NNP and welfare equivalent income, and iv) between green NNP and sustainable income. Here we have provided some propositions with proof.

12.3.6 Greenhouse Gas Emissions In this portion we have shown that global warming is continually increasing due to GHG emissions. We have shown that due to global warming the living organisms are in dangerous position, and some species have already extinct, and some more will extinct in future. We have realized that if GHG emissions cannot be controlled then the people of most of the countries will suffer for drinking water, shortage of foods, and various heat related diseases. Also we have found that due to global warming most of the people in future will suffer from heat waves, earthquake, tsunami, shortage of water supply, and energy supply. As a result economic development will sloth if global warming cannot be reduced.

12.3.7 Environment Tax Here we have proposed a cap and trade policy, and a carbon tax policy to reduce GHG emissions. We have also shown that vehicle emissions depend not only on vehicle size and age, but also on qualities of the fuel, maintenance of the car’s pollution control equipment, frequency of cold start-ups, temperature of the air, speed of the vehicle, and aggressive driving. We have encouraged the drivers to drive fewer miles, and use cleaner fuel to reduce carbon emissions. We have explained beneficial sides of environmental tax and optimal environmental tax with mathematical calculations. We have observed that in Bangladesh the vehicle taxes are not adjusted properly. Hence the vehicle taxes in Bangladesh should be fixed depending on car size, engine size, and the type of gasoline are used by the motorists.

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12.4 The Value of This Approach This dissertation concludes that it creates value to enhance the economic and social welfare. We have highlighted the social welfare as: •

optimization in economic structures and social welfare, and



benefits of economic growth.

This thesis has focused on the social welfare implications of economic development. Therefore, the global humanity will be benefited from this work.

12.5 Strengths, Weaknesses, and Limitations of the Research 12.5.1 Strengths of the Study Excellent details of mathematical calculations and theoretical explanations are given on implementation of this research procedure. Every chapter of the thesis provides sufficient materials for a good research. Detail mathematical calculations, propositions, theorems, examples, tables, and figures provide validity and reliability of research. Although the thesis title is on mathematics basis, the thesis contains a survey report in the support of statistical analysis which enriches the study. I hope that the readers who are interested on mathematical model research in economics and social science will accept my work cordially. I think research objectives and research questions are stated clearly. Introduction, methodology, literature review, and concluding chapters are given in some details in third submission of this research. Possible explanations were provided for the outcomes measured. I think the references both in the text and reference list are given properly. Further, every chapter has introduction and conclusion.

12.5.2 Weaknesses of the Study Primary data collection for my research was limited on female garment workers of CCC of Bangladesh. If data are collected from the whole country, then I believe that my research would give more reliability than the present study. In the study I have included only female workers. I think inclusion of both gender would provide better result. Another weakness is that full thesis is not done on the basis of survey data. In my thesis I have used primary data only in chapter VIII, and most of the works on secondary data 239

which weaken the research. The thesis has more mathematical works, but recent researchers stress on statistical analysis. In my research I use mathematical framework, because my first supervisor advised me to do research on mathematical analysis of economics and social choice theory. Also my research is not confined on a single topic as most of the researchers do. I have submitted the thesis for three times and have faced three defenses due to my weakness of research procedure.

12.5.3 Limitations of the Study Limitations are the constraints that are largely beyond control of the researcher, but could affect the study outcome. These are considered as the potential weaknesses in the study that are out of our control. Actually everybody faces limitations when conduct his/her research, and this study is no different. For example, the use of only qualitative research, although considered appropriate, but could limit the validity and reliability of the result, in spite of the effects to reduce them. This is what Gray (2004) argues in any issue that weakens the reliability of data due to bias on the part of the responses provided. Therefore, some limitations were used in order to make thesis flourished. Lack of funding for the study is also a limiting factor. I have done my research work without any scholarship, and only depending on self funding. The continuity of the research has affected because of my financial constraints. I have submitted and resubmitted the thesis three times, and have faced three defenses due to my weakness of research. So I have spent a lot of money and time to do these activities, which have delayed the study. Another issue is apathy to face interview of some of the female garment workers in CCC for the collection of primary data. When I have gone to take interview I observe that most of the respondents are illiterate, and also WTP is new in Bangladesh. Some of the garment workers have refused to response, which is affected the continuous and speedy flow of the data collection. So that I have collected less date as I have expected to collect more. If I have collected more data my research work would be more reliable. This thesis concentrates on micro- and macro-economics, voting system, global warming and sustainability in partially. It does not cover all the concepts in the mentioned topics. For example, it does not reflect state level macroeconomic effects,

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effects on state budgets, the administration burden, deep area of game theory, complete solution of environment pollution and healthcare, full effectiveness of sustainable atmosphere, and efficient technology of the reduction of anthropogenic GHGs. In the study we have stressed on air pollution, but not on water and other pollutions. Hence, that portion of our research is not on full environment pollution, but most cases we have mentioned the work as environment pollution. Although all the chapters are relevant to the thesis title, but the thesis discusses chapters discretely. The thesis analyzes limited problems on economics, social choice, and environment science. Although we have tried our best to write the thesis in correct English, we do not demand that the thesis is free from grammatical mistakes and printing errors. Some cases mathematical calculations in the thesis are given in brief, thinking that the readers have sufficient knowledge to realize them (if we write them in brief).

12.6 Personal Reflection This thesis is the outcome of an extensive research effort designed to explore some interesting and important issues of mathematical economics and social choice. Although I have not succeeded two defenses, I think, during the research period, I have learned a lot. The research methodology chapter is new to me, and this is not done by me before this research. I have found a new experience when I have collected the survey data from the garment workers of the CCC. I have wondered how they can remain alive such a miserable environment. I have also learned the effects of GHG emissions that are causing sever natural calamities recently, which diminishing economic development. I have learned that there is a relation with health and economics. In the thesis I have stressed on green NNP for the sustainable development. I have written the thesis for three times, and have to face three defenses, and have found proper guidelines from the examination committee during the previous two defenses. This research have enriched my academic skills, and as well as capabilities of doing research with patient. I hope, in future I can do more research with patient to reach my goal. Although I have tried my best even in the third and final submission of this research, I believe, I have some weakness in the study. This is because of the lack of my knowledge in research methodology, literature review, and quantitative research on

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survey data. I have prepared my thesis giving the priority on mathematics, as my thesis title is mathematics related work on economics, social choice and game theory.

12.7 Directions for the Future Researches As I have already mentioned above that my thesis is limited work of the mathematical works on economic, social choice and game theory; there are more scopes for future researchers to do research in this field. To enhance the activities of the mathematical economics and social choice some of the instructions for the future researchers are as follows: •

As my thesis has included different topics, the future researchers can try to do PhD research individually on each chapter of my thesis. For example, a researcher can try to do PhD on optimizations, or on Arrow’s theorem or on GHG emissions, or on sustainability, or on NNP, or on voting system, or on environment tax, etc.



As my research is only on mathematics basis, the future researchers can try to do research by surveying questionnaire data on the works that I have done mathematically.



In my thesis, methodology of the research is not given richly. Future researchers no need to follow my research methodology chapter. They can enrich their methodology chapter according to their supervisors’ advice before submission of their thesis.



Optimization in economics of chapter V can be developed by using more examples with detail mathematical calculations. The researcher can do more works on both linear and non-linear cases.



I have a little contribution on game theory. A researcher can work on various sides of game theory. For example, he/she can perform research on Nash equilibrium, repeated games, zero-sum game, dynamic games, Bayesian games, etc.



In my thesis unitary and federal democracy are given briefly. These are related to economics, social science, and political science. We hope, a researcher has huge scopes to do more research on these fields.

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Different researchers can work on voting system. They can try on separate section of this chapter. For example, some can try on Condorcet method, some on Borda count, some on single transferable voting, etc.



I have discussed only one portion of environment pollution; namely, air pollution in Bangladesh. A researcher can try to work on water pollution, and reduction of water pollution in Bangladesh. He/she can also work on water borne diseases, and protection and cure technologies from them. Arsenic contamination is a great problem in the Northern areas of Bangladesh. Researchers can work in reduction of arsenic contamination to present arsenic free environment in Bangladesh.



A researcher can work on local and global open and closed economy with their advantages and disadvantages. They can collect data from home and abroad to enrich their research.



I have collected data only from the female garment workers of CCC. I recommend future researchers for collecting date from Dhaka and other districts to find more accurate result and try to establish WTP system in all labor sectors.



I have done limited works on GHG emissions. Future researchers have more scopes to work on this field. For example, they can try to work the effects of GHG emissions in developing countries, techniques of reduction of GHG emissions, etc.



I have not done any work on renewable energy, such as, on biofuels, electricity production from nuclear energy, windmills, water, etc., which reduce GHG emissions. A researcher can works on these areas.



I have done very little work on environment tax. Vehicle tax of Bangladesh is not adjusted properly. As a result air pollution is severing in this country. Future researchers can work on the environment tax on survey data of Bangladesh.

12.8 Conclusion and Recommendations Throughout this thesis we have tried to improve the social welfare and sustainable economic development. We have tried to introduce all the chapters with clear theoretical concepts and easier mathematical calculations. We have added some examples, theorems, and propositions with proofs in some chapters to make the thesis easier to the readers.

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Although we were very careful for the preparation of the thesis, there may be some remaining printing mistakes throughout the thesis; we hope the readers will be indulgent. In the modern globalized world optimization in production is essential for the sustainable economical development. The nations should use efficient strategies to the production and consumption of the welfare of the humanities. At present social instability, corruptions, and political unrest are major problems in the smooth economic development. All the nations will see the mutual welfare and try to reduce these problems. Voting is a better process to elect political leaders. No doubt a good leader will develop the nation. The weaknesses in voting system need to be identified and try to develop them. GDP, GNP, and NNP are indicators of sustainable development of a country. For social welfare these are needed to improve. All the developed and industrialized countries should try to reduce GHG emissions, and must be conscious in harmful chemical pollutions. They will also try to remove the absolute poverty from the society to make the world poverty free. The gap between the rich and the poor must be reduced in near future for the sustainable development of the modern economy. As the natural calamities have increased recently due to global warming, all the nations must be unifying to reduce the GHG emissions to save the world for future generation.

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Appendix–I

Derivation of equation (11.54): Taking the total derivative of utility (11.50) with respect to the corrective tax  X , substituting the consumer first-order conditions (11.51), and for constant public spending ( dG = 0 ), and finally dividing both sides by λ we obtain as follows: 1 dU dY dX dl dM 1 U H dQ = + (1 +  X ) + (1 −  L ) + (1 −  M ) + .  d X d X d X d X d X  H Q d X

(AI-1)

Taking a total derivative of the production function (11.46) with respect to  X gives; dL dY dX dG dM = + + + . d X d X d X d X d X

(AI-2)

Again taking the total derivative of the household time constraint (11.44) we obtain;

dL dl dT S  H dM H dQ   . + = − + d X d X d X H  M d X Q d X 

(AI-3)

Further consider the household’s time endowment T and the public good G be constant, so that using dG = 0 and dT = 0 , then from (AI-2) and (AI-3) we get: dL dY dX dM = + + , d X d X d X d X

(AI-4)

dL dl S  H dM H dQ   . + =− + d X d X H  M d X Q d X 

(AI-5)

From (AI-4) and (AI-5) we can write;

dl S  H dM H dQ  dY dX dM  . + + =− + + d X d X d X d X H  M d X Q d X 

(AI-6)

Using (AI-6) in (AI-1) can be written as;

1 dU dX dl dM 1 U H dQ S  H dM H dQ    . (AI-7) =X − L − M + − +  d X d X d X d X  H Q d X H  M d X Q d X  From (11.53) we obtain;

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 P + ( − 1)

S H 1 U H . = H Q  H Q

(AI-8)

Using (AI-8) in (AI-7) can be written as follows:

1 dU dX dQ S H dQ dl dM S  H dM H dQ  . =X +P + ( − 1) − L − M −  +  d X d X d X H Q d X d X d X H  M d X Q d X 

(AI-9)

Using dQ = − dX only in second term of right hand side of (AI-9) we get; 1 dU dX S H dQ dl  S H  dM . = ( X −  P ) + ( − 2) − L −  M +   d X d X H Q d X d X  H M  d X

We now take steps to make the term

(AI-10)

dl more explicit. This term symbolizes d X

interactions between the environmental tax and household labor supply decisions. According to the uncompensated demand equations and since the government adjusts its budget constraint with  L (i.e.,

d L = 0 ), we can write it as follows: d X

dl l l d L l dQ = + + . d X  X  L d X Q d X

(AI-11)

Taking the total derivative of (11.47) we get;

dL dl S  H dM H dQ   . =− − + d X d X H  M d X Q d X 

(AI-12)

Taking the total derivative of (11.49) we obtain;  dL S  H dM H dQ  dX dM   +  X +L  + + + X =M d X d X d X  d X H  M d X Q d X 

(L + S ) d L +

S  H dM H dQ   , + H  M d X Q d X 

− (L + S )

d L dL S  H dM H dQ =L − (1 −  L )  + d X d X H  M d X Q d X

 dX dM  +  X + X − M . d X d X 

Using (AI-12) in (AI-13) yield: d L 1  dX dM dl S  H dM H dQ    , =− − M − L − + X + X d X L+S  d X d X d X H  M d X Q d X 

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(AI-13)

d L 1 dl 1  dX dM S  H dM H dQ    . (AI-14) =L − −M − + X + X d X L + S d X L + S  d X d X H  M d X Q d X 

This expression gives the reduction in labor tax that can be financed by a marginal increase in the environmental tax, while maintaining budget balance. Using (AI-14) in (AI-11) we find: dl l 1 l dl 1 l  dX dM S  H dM H dQ    = +L − −M − + X + X d X  X L + S  L d X L + S  L  d X d X H  M d X Q d X  l dQ + , Q d X

(L + S )

dl l l dl l  dX dM S  H dM H dQ    = (L + S ) + L − − M − +  X + X d X  X  L d X  L  d X d X H  M d X Q d X  l dQ + (L + S ) , Q d X

(L + S )

dl l dl l l − L = (L + S ) − d X  L d X  X  L

 dX dM S  H dM H dQ    − M − +  X + X d  d   H  M d   Q d  X X X X    l dQ + (L + S ) , Q d X

 dl 1 l l  dX dM S  H dM H dQ    ( = L + S) − X + X − M − +   d X (L + S ) −  l   X  L  d X d X H  M d X Q d X  L  L

+ (L + S )

l dQ  . Q d X 

(AI-15)

From (11.52) we can write;

l l  L L + S (H ) −  L =  L  −1

L

then (AI-15) becomes;

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and L + S (H ) =

  −1

L

l ,  L

dl = d X

  l l l  − 1 l  − 1 L  L  X  L    − 1 L  L 1

+



L

 −1

 dX dM S  H dM H dQ    − M − +  X + X d  d   H  M d   Q d  X X X X   

l l dQ  .  L Q d X 

(AI-16)

Using (AI-16) in (AI-10) the equation (AI-16) becomes as follows:

 1 dU dX dX  S H  dM l   − ( − 1)  M + = ( X −  P ) + ( − 1)  X +  X −  L   d X d X d X  H M  d X  X  

− ( − 1)

S H dQ  S H  dM S H dQ , −  M + − ( − 2)  H Q d X  H M  d X H Q d X

  l  1 dU dX dX  S H  dM   −   L = ( X −  P ) + ( − 1)  X +  X +  M +   d X d X d X  H M  d X     X   S H l  dQ . − +  L Q  d X  H Q This is the required equation (11.54).

248

(AI-17)

Appendix–II

Derivation of equation (11.55): Assume that utility, the levels of public spending and environmental quality be constant. Now taking the total derivative of utility (11.50) with respect to  L we get;

l c l c =−  L (1 −  L )

U X c + = X U (1 −  L ) l =

U c Y Y + U  (1 −  L ) l

U c H M U  (1 −  L ) l

1 +  X X c 1 Y c 1 −  M M c + + , 1 −  L (1 −  L ) 1 −  L (1 −  L ) 1 −  L (1 −  L )

(AII-1)

where ‘c’ denotes a compensated derivative. The assumption that the cross elasticity of X and leisure is equal to the average (weighted by assumption shares) over all goods is given by: X , I = S X X , I + SY Y , I + S M M , I ,

where i , I =

i C 1 −  L and si = (1 −  L ) i

(AII-2)

pii respectively represent the compensated ( ) p i  i

elasticity of demand for good i with respect to the price of leisure and share of that good i in total consumption. Here i = X, Y, M. Now (AII-2) can be written as follows: X C 1 −  L p X X C 1 −  L pY Y C 1 −  L p M M C 1 −  L = X + Y + M . (AII-3) (1 −  L ) X  pX X (1 −  L ) X  pYY (1 −  L ) Y  pM M (1 −  L ) M

Substituting (AII-3) and the household budget constraint (11.47) in (AII-1) after some manipulation gives;

l c X c L + S = .  L (1 −  L ) X

(AII-4)

Since the change in the price of good X is equal to the change in the tax rate, the Slutsky equations give:

249

l l c l = − X,  X  X I

(AII-5)

l l c l = − (L + S ) .  L  L I

(AII-6)

Using (AII-4) in (AII-6) can be written as;

l X c 1 l 1 . = − I (1 −  L ) X  L (L + S )

(AII-6a)

The expression for the (cost-side) tax interaction effect from equation (11.54) becomes;

 l  S H  dM  . dW I = −  L +  M +  H M  d X    X 

(AII-7)

Using (AII-5) in (AII-7) we get;   l c l   S H dW I = −  L  − X  +  M + H M    X I  

 dM   d X

 . 

(AII-8)

Again the Slutsky symmetry property is as follows:

l c X c =− .  X  (1 −  L )

(AII-9)

Using (AII-9) and (AII-6a) in (AII-8) can be expressed as;   l c l   S H dW I = −  L  − X  +  M + H M    X I  

= − L

 dM     d X 

X l S H  dM  −   M +  L + S  L H M  d X 

S H  dM  . = (1 −  )X −   M +  H M  d X 

(AII-10)

From (11.54) the benefit-side tax interaction effects dW IB is:

 S H l  dQ  dW IB = − +  L . Q  d X  H Q

(AII-11)

By the partial derivative of the household budget constraint (11.47), the change in spending on X, Y and l for a change in Q becomes as follows:

(1 −  L ) l + (1 +  X ) X Q

Q

+

Y M . = −(1 −  M ) Q Q

250

(AII-12)

Similarly as like (AII-12) for the change in I, (11.47) becomes:

(1 −  L ) l + (1 +  X ) X I

I

+

Y M = 1 − (1 −  M ) . I I

(AII-13)

Weak separability of the health in the utility function implies that leisure demand is determined only by the relative prices of l, X and Y, and by total spending on those goods. Those relative prices are not affected by changes in Q or changes in I. As a result, the derivative of l with respect to Q will equal the derivative of l with respect to I times the ratio of the derivative of spending on those goods with respect to Q (AII-12) to the derivative of spending on those goods with respect to I (AII-13). Hence (AII-13) becomes; −1

l l  M  (1 −  M ) M . = − 1 − (1 −  M )  Q I  I  Q

(AII-14)

Again the uncompensated labor supply elasticity is; L =

 (L + S ) 1 −  L l 1 −  L  − 1 (1 −  L ) , = =  (1 −  L ) L + S  L L + S  L

(AII-15)

and the income elasticity of labor supply is;

LI =

(L + S ) (1 −  L )(L + S ) l = − (1 −  L ) . I L+S I

(AII-16)

Dividing (AII-16) by (AII-15) we get;

 L

l  = −( − 1) LI . I L

(AII-17)

Multiplying (AII-14) by  L and using (AII-17) we get; l l  L = − L Q I

−1

M  M  1 − (1 −  M ) I  (1 −  M ) Q

 = −( − 1) LI L

−1

M  M  1 − (1 −  M ) I  (1 −  M ) Q .

(AII-18)

Now using (AII-18) in (AII-11) we find that;    S H dW IB = − − ( − 1) LI L   H Q

M  M   dQ  ( ) ( ) 1 − 1 −  1 −  .  M M  I  Q  d  X  −1

(AIII-19)

Finally using the value of dW I and dW IB from (AII-10) and (AII-18) in (11.54) we get;

251

 1 dU dX dX  S H  dM   − (1 −  )X −   M + = ( X −  P ) + ( − 1)  X +  X   d X d X d X  H M  d X      S H + − − ( − 1) LI L   H Q

−1 M  M   dQ  , ( ) ( ) 1 − 1 −  1 −   M M   I  Q   d X

1 dU dX S H  = ( X −  P ) −   M +  d X d X H M   − ( − 1) LI L

 dM  S H + −   d X  H Q

−1 M  M   dQ  . ( ) ( ) 1 − 1 −  1 −   M M   I  Q   d X

This is the required equation (11.55).

252

(AII-20)

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