A modified diversity index and its application to crop ...

A modified diversity index and its application to crop diversity in Assam, India Premananda Bharati, Utpal Kumar De, and Manoranjan Pal Citation: AIP Conference Proceedings 1643, 19 (2015); doi: 10.1063/1.4907421 View online: http://dx.doi.org/10.1063/1.4907421 View Table of Contents: http://scitation.aip.org/content/aip/proceeding/aipcp/1643?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Gender diversity in physics in India: Interventions so far and recommendations for the future AIP Conf. Proc. 1517, 106 (2013); 10.1063/1.4794242 Use of modified smooth exterior scaling method as an absorbing potential and its application J. Chem. Phys. 134, 094301 (2011); 10.1063/1.3558737 Fundamentals of the Mechanical Index and caveats in its application J. Acoust. Soc. Am. 105, 1324 (1999); 10.1121/1.426204 PHOTOINDUCED REFRACTIVE INDEX INCREASE IN POLY(METHYLMETHACRYLATE) AND ITS APPLICATIONS Appl. Phys. Lett. 16, 486 (1970); 10.1063/1.1653076 A Modified Atomic Orbital Method with its Application to Ethylene and Oxygen J. Chem. Phys. 22, 1267 (1954); 10.1063/1.1740363

This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 14.139.207.126 On: Thu, 05 Feb 2015 05:15:27

A Modified Diversity Index and its Application to Crop Diversity in Assam, India Premananda Bharatia, Utpal Kumar Deb and Manoranjan Palc a

Biological Anthropology Unit, Indian Statistical Institute, 203 BT Road, Kolkata 700108, India Department of Economics,North-Eastern Hill University, Shillong, 793022, Meghalaya, India c Economic Research Unit, Indian Statistical Institute, 203 BT Road, Kolkata 700108, India

b

Abstract. A new measure of diversification index is proposed taking the correlation structures of the shares into consideration. This index is a generalization of the Herfindahl-Hirschman Index (HHI) proposed in the context of concentration in industries. When this index is applied in the crop production along with other related data in Assam, India, it is seen that many of the interrelations are changed from that of found by HHI. The analysis helps us to plan for improvement of crop patterns in a region prudently. Keywords: Crop Diversity, Herfindahl-Hirschman Index, Hall and Tideman Index, Cropping Pattern of Assam. PACS: *91.62.Gk, 92.05.Df, 02.50.Sk

INTRODUCTION Diversity and concentration are two sides of a coin. While concentration index is used in the firm size distribution, the diversity index is used in the analysis of political parties, export of items or agricultural production. Measurement of concentration is needed to reveal the extent to which a few giant firms have control over the market. In other words it gives the likelihood of monopolistic practices. In the context of market or socio-political scenario, too much of concentration may lead to unethical practices. On the other hand, diversity is believed to be linked with performances. More is the diversity, better is the performance in terms of profit, rate of return etc. In the context of agricultural, production, can be increased either through the expansion of area under cultivation or through the rising yields. In any situation however, total returns from agriculture can further be increased with the diversification of area under cultivation towards high remunerative crops from the low yielding crops ([7] and [8]). Crop diversification is an important instrument for economic growth. Food and nutrition security, growth of income and employment, poverty alleviation, judicious use of land, water and other resources, sustainable agricultural progress as well as for sustainable environmental management. It is inevitable for ensuring productivity, profitability and sustainability ([7] and [48]). In order to know the benefit accrued to diversity in the agricultural production, one needs to compute a diversity index, which is opposite to that of concentration index. There are some measures for concentration that can be modified to get corresponding diversity indices. There are various existing methods of examining diversity of crop. The very crude measure of diversity is the number of crops cultivated and proportion of area under various crops. The simple measure of number of crops however does not speak about the evenness of the distribution of the area under cultivation. Thus the other popular measures were developed and the widely used index was the Hirschman Herfindahl Index (HHI), which was first used to examine the regional industrial concentration ([27] and [51]). It is defined as the sum of squares of proportion of each crop to gross area under cultivation. In symbol ; where N is the total numbers of crops Pi be the proportion of acreage under ith crop to total cropped area. The value of HHI ranges from 0 (for perfect diversification) to 1 (for perfect specialization). This index however cannot assume theoretical minimum value especially in case of smaller number of crops. The Simpson Index, defined as D 

1



N

W i2 i 1

, and E 

1  N

1



N

, can be traced to a paper by Simpson Wi N

i 1

[46], titled “Measurement of Diversity” published in Nature. It is almost identical to the Hirschman-Herfindahl Index of industry concentration used in economics and developed independently by both Herfindahl [24] and Hirschman [25]. Here, D represents the number of times one would have to take pairs of individuals at random from a population in order to select a pair of the same species. On the other hand, E refers to the evenness of distribution The 2nd ISM International Statistical Conference 2014 (ISM-II) AIP Conf. Proc. 1643, 19-29 (2015); doi: 10.1063/1.4907421 © 2015 AIP Publishing LLC 978-0-7354-1281-1/$30.00

19 to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject 14.139.207.126 On: Thu, 05 Feb 2015 05:15:27

in the population, sample, or portfolio. E can have a maximum value of one in a situation where species, or industries, are equally weighted. Values decreasing below one indicate increasing dominance by relatively few species, or industries. N is the number of species, or industries, in the population, while w is the population weight of each species, firm, industry, or other unit of measurement. The Simpson Index has also been referred to as the Yule Index after the similar measure devised to characterise the vocabulary used by different authors [56]. Yule [56] is cited by Simpson [46] as a key reference for his index, which is a combination of the ideas of Yule [56] with those of Fisher et al [13] and Williams [55]. The Industrial diversity was also examined by using Ogive Index (OI), which was computed by taking into account the deviation from the ideal or equal distribution of acreage [52]. It is defined as ; where N is the total number of crops cultivated in the region. Like HHI, it is also subtracted from unity to convert it into an index of diversification and make it comparable with other indices. Also, Entropy Index (EI) and Modified Entropy Index (MEI) have been widely used in agricultural diversification literature ([44], [17], [47] and [45]). EI is defined as . Its value varies from 0 (perfect concentration) to Log N (perfect diversity). The upper value of EI can exceed one or be less than one when N is greater or less than the base of logarithm. Thus it does not correspond to any standard scale of measuring degree of diversification. In MEI number of crops, N is considered as the base of logarithm. Symbolically, The lower and upper value of MEI is zero (when only one crop is cultivated in the whole area) and one (perfect diversification). Though it is better than those other measures (due to its uniform scale) it does not consider the change in value due to changes in number of crop activities [45]. Hannah and Kay [19] stated that most of the indices are special cases of the general index !

where Si is the share of ith item and ϕ is a parameter, such that ϕ ≥ 0 and ϕ ≠ 1. For ϕ = 2 the index becomes " $ # , which is the inverse of the Herfindahl Index of measuring disparity. In extreme case of ϕ = 1 the index becomes the Entropy Index. This general index measures both the number of items and the evenness of item shares, with the parameterϕ determines the weighting of emphasis on number of items versus evenness. When, ϕ is zero it simply counts the number of items [39]. Using this general index, Tauer and Seleka [50] did not find significant relationship between the cash receipt variability with the reduction in diversification of agricultural production across 38 US states during 1960 to 1989. Since they used cash receipts to measure diversity across commodities, it does not capture diversity by production practice. Hoffmann [26] however tried to provide a distribution based measure of diversity which is more flexible and represents a useful complement to models of generalized feature based diversity, such as Nehring and Puppe’s [38] “Theory of Diversity”. There are two fundamentally different generalised models of distribution based diversity indices. The so-called ‘Rényi diversity’ is an additive and non-concave one parameter generalisation of the Shannon Entropy [41]. It is used to measure biodiversity [42], statistical sampling [35], economic diversity modelling [4] etc. The other generalisation of Shannon entropy, which is concave but not additive was derived by information theorists Havrda and Charvát [23], statisticians Patil and Taillie [39], physicist Tsallis [53] and also used for biodiversity measurement [30]. Also, researchers used a number of inequality indices to examine concentration of the whole in a few items as opposed to diversification ([5], [15], [43] and [49]). Various concentration and diversification indices are used by Meilak [37] to examine the export concentration and its link with the size of the economy. Most of these concentration indices exhibit the general form: CI = ∑ SiWi ; i = 1, 2, ..., n, where Wi = the weight attached to the export share of a particular export category, Si = the share of export category i and n = the number of export categories. Most of these indices are based on the shares of each category and used arbitrary weights to the respective items. These measures ignore relative size variations in commodity groups and these can equally describe a country exporting one product and a country exporting product groups with similar shares. Though, these measures satisfy the general characteristics of a concentration or diversification measure, it is necessary to improve upon such indices in order to incorporate the interdependencies of the variables. In the paper we have thus tried to develop an alternative measure of diversification by taking into account the bi-variate correlation coefficients.


THE DIVERSITY MEASURES As mentioned earlier, all the measures proposed so far are based on shares. Suppose pi is the share of ith item. This represents the proportion of activities controlled by ith item. We may without loss of generality arrange these shares in decreasing order denoting p1 with the largest share and pn with the smallest share, n being the number of items in the group of items concerned. Naturally, ∑pi = 1. Suppose P is the measure of concentration. From this we can have a corresponding measure of diversity by taking either reciprocal or complementary of P, i.e., 1/P or 1 – P. If P lies in between 0 and 1 then 1/P will lie in between 1 and infinity, whereas 1 – P will still lie in between 0 and 1. Because of this property we prefer to take 1 – P as a measure of diversification. There are many measures of concentrations. Among these we shall discuss a few of the indices. Absolute Concentration Ratio is the oldest and a very popular measure of concentration. It is defined as ACR 

K

i1 Si ,

(1)

where, K is the top K firms in the industry. This is also known as K -firm concentration ratio. Within an industry, K is often taken as a number between 3 and 5, because it is often the case that 3-5 largest firms have about 70%80% share of the industry. K is thus known as the focal point of concentration. However, K should be fixed if we want to compare among different industries. If firms of all the industries are taken then the value of K may be very large. The corresponding diversity index is Absolute Diversification Index &

ADI = 1 –

%.

(2)

The most popular index is the so-called Herfindahl-Hirschman Index (HHI), defined as the sum of the squared values of firms’ shares. HHI =

$

% .

(3)

The corresponding diversity index is Herfindahl-Hirschman Diversification Index HHDI = 1 –

* ) ( + '( .

(4)

We can define a more general form of HHI by taking the following weighted sum of si: GHHI =

$

,%,

(5)

where wi is the weight attached to si, so that $ , = 1. HHI is then viewed as the weighted sum of si, where the weight of si is nothing but si itself. Thus smaller firms contribute less to the value of the index. Hall and Tideman [18] proposed some desirable properties which should be satisfied by a good measure of concentration. HHI satisfies all the axioms proposed by Hall and Tideman. Another set of axioms was proposed by Hannah and Kay [19]. These axioms, apart from some of the Hall and Tideman axioms, are based on entry of new firms, merger of firms etc. There are many other concentration indices proposed in the literature. One can also define the corresponding diversification index to each of the concentration indices. Since we shall mainly concentrate on HHDI, we refrain ourselves of narrating all these indices. These indices are given in the Appendix I. HHDI was proposed in the literature to understand the concept of fractionalization in party system. Douglas Rae ([9] and [10]) defined an index of fractionalization in the party system as -./

"

$

0 ,

(6)

where DDR = the index of fractionalization in the party system and Ti is the proportion of party i of votes given (See also [11] and [54]). Greenberg's measure of linguistic diversity [16] is also same as the above index.


$

$!

2 2 $ A similar index of fragmentation, defined by Rae and Taylor [11], is /1 " , which is the $ $ probability that a randomly selected pair of individuals in a society will belong to different groups. ni is the number of members of the ith group.

THE PROPOSED MEASURE While measuring the oligopolistic power in the market one should consider the existing cooperation between firms. When there is more cooperation between the firms, there is more power of the firms in the market. Correlations between agricultural variables should reduce the diversity index. Almost all the productions of the commodities move in the same direction as time. Time here acts as an intervening variable. So we should first remove the effect of time from each variable before finding the correlation structure. Let us assume that the correlation between ith and jth production over time is ρij after eliminating the effect of time. This is in fact the corresponding partial correlation coefficient. The proposed measure is Modified Diversity Index (MDI), defined as 3 " 4 5 6 46 " 4784, where R is the matrix of ρij values and s is the 6 column vector of shares of production for the current period. Observe that 4784 can be taken as a measure of concentration index. There are some salient features of this index. (a) 0 ≤ MDI ≤ ADI, (assuming that any two variables are non-negatively correlated). It is 0 when 5 6 = 1 for all i and j, i.e., when any two variable move in the same direction without any error. In this case 4784 becomes (Σsi)2 = 1. It is HHI when the variables are completely independent of each other, 5 6 = 0 for all i and j. (b) 0 ≤ MDI ≤ 1, assuming that the variables can move freely, i.e., when 5 6 can take any value in the range – 1 to 1. [4784 9 47"4 = 4 =1, where " is the matrix consisting of 1’s only. Since R is a symmetric nonnegative definite matrix, we have 4784 ≥ 0.]

DESCRIPTION OF DATA AND CALCULATIONS The analysis is based on the secondary data on area under various crops since 1950 to 2011 collected from various issues of Statistical Hand Book of Assam, Economic Survey of Assam and Reports from Directorate of Economics and Statistics and Directorate of Agriculture, Government of Assam. Though several studies used earnings from the production of crops in order to compute the diversity index, here we use area under crops for the purpose. This is because the farmers’ maximise the returns from their limited land keeping in view the sustainable income possibilities. The land size allocation to different crops reflects the intention of the farmers which may not be realized in the same proportions through production. Moreover, the area of crop cultivation is more robust than the actual production, which is subject to technology available at the time of production and to the climatic behaviour of nature. We have divided the areas under different crops into six major groups, namely Cereals, Pulses, Oil Seeds excluding Cocoanuts, Fibres, Plantation Crops ad Misc. Crops. Areas under the six major crops are found by aggregating the areas of the individual crops in the respective groups under consideration. This is done for each year starting from 1950-51 to 2010-11. Thus we have data for 60 years. The following line diagrams give us a clear idea about the trend of area for each of the major groups.


FIGURE 1. Trends in the Areas under Major Crops.

It is clear from Figure-1 that all but one major group have increasing trend. The one which has decreasing trend is ‘Fibres’. Fibres do not come under food crops. The patterns of growth of the groups other than Cereals are not easily discernible because of Cereals itself, which captures more than 70 percent of the share. If we want to see the trend of these five groups more clearly, we can draw diagrams for these five groups only, as is done below.

FIGURE 2. Trends in the Areas under Major Crops except Cereals

If we look at the pattern of trends, we can see that there is an increasing movement in all but Fibre up to the year 1987-88. Also the increasing trend is somewhat slow up to 1975-76. After1987-88, it either decreases or increases at


a slower rate than the past. The similarity of the trends would imply high positive correlations among the groups except possibly with the areas under Fibres. Since from the Figures it is evident that the trends are similar for each group, we have computed the bivariate correlation coefficients among these groups to see whether it is really so. The correlation matrix is as follows. TABLE 1. Bivariate Correlations of the Areas under Major Crops

Cereals 1

Pulses .867** 1

Oil Seeds .905** .914** 1

Fibres -.661** -.477** -.613** 1

Cereals Pulses Oil Seeds Fibres Plantation Crops Misc. Crops ** Correlation is significant at the 0.01 level (2-tailed).

Plantation Crops .929** .819** .887** -.834** 1

Misc. Crops .902** .772** .880** -.860** .989** 1

Area under Fibre Crop has negative correlations with all other area groups. This was expected from the trend diagram. But, the degrees of correlations, regardless whether negative or positive are too high in absolute value for the areas under crop to be of further effective use without trend corrections. Time seems to be an intervening variable. Thus we first correct all the variables for time and then compute diversity index for each year using the proposed formula. To compare these indices we also compute the diversity indices using existing method. This leads to the followingfour types of indices: 1. Herfindahl-Hirschman Diversification Index without correcting the variables for time. HHDI = 1 – 2.

4 .

(7)

Herfindahl-Hirschman Diversification Index after correcting the variables for time (Corrected HHDI). CHHDI = 1 –

4: ,

(8)

where% : is the share after correcting the variables for time. 3.

Modified Diversity Index (MDI). 3

"

6

4 W 6 46

"

4X84,

(9)

where ρij is the correlation between ith and jth areas, s is the vector of shares and R is the correlation matrix of areas. 4.

Modified Diversity Index after correcting the variables for time (Corrected MDI). ; 3

"

6

4: W:6 46:

"

4 : X8:4: ,

(10)

where % : is the share of area of ith crop group after correction of the variables for time, s* is the corresponding vector and R* is the correlation matrix of areas after correcting the variables for time. Here it should be noted that correction for time should be mean preserving. Otherwise the indices are meaningless. More precisely, we first regress each variable on time, find the residuals and add each residual with the mean value of the variable.


FINDINGS AND CONCLUSION From the data, we first compute the bivariate correlations ρij, and the shares si. We then compute HHDI and MDI. The variables are then corrected for time and translated so that they become mean preserving. The correlations : : < and the shares % are computed from corrected mean preserving areas. These may be termed as corrected correlations and corrected shares respectively. Below, we present the four diversity indices. TABLE 2. Year wise Diversity Indices of the Areas under Major Crops

HHDI Year 1951-52 0.431 1952-53 0.424 1953-54 0.418 1954-55 0.423 1955-56 0.428 1956-57 0.427 1957-58 0.432 1958-59 0.425 1959-60 0.424 1960-61 0.415 1961-62 0.419 1962-63 0.417 1963-64 0.426 1964-65 0.425 1965-66 0.425 1966-67 0.418 1967-68 0.418 1968-69 0.401 1969-70 0.406 1970-71 0.404 1971-72 0.419 1972-73 0.405 1973-74 0.411 1974-75 0.396 1975-76 0.377 1976-77 0.412 1977-78 0.418 1978-79 0.423 1979-80 0.439 1980-81 0.440

CHHDI .472 .467 .462 .464 .466 .464 .467 .459 .458 .450 .451 .448 .453 .451 .449 .442 .440 .424 .427 .423 .434 .420 .424 .408 .389 .419 .424 .426 .441 .441

MDI CMDI .217 .251 .217 .249 .203 .246 .205 .247 .226 .250 .207 .246 .220 .249 .219 .245 .222 .245 .215 .241 .225 .242 .218 .239 .222 .241 .223 .240 .218 .239 .213 .235 .215 .234 .176 .223 .190 .225 .195 .223 .207 .229 .192 .221 .205 .222 .173 .212 .149 .201 .159 .225 .154 .226 .156 .228 .159 .233 .158 .235

HHDI CHHDI MDI Year .157 1981-82 0.452 .451 .162 1982-83 0.470 .468 .157 1983-84 0.475 .473 .152 1984-85 0.483 .481 .166 1985-86 0.475 .472 .143 1986-87 0.489 .485 .142 1987-88 0.488 .484 .139 1988-89 0.485 .480 .133 1989-90 0.465 .457 .131 1990-91 0.461 .452 .135 1991-92 0.467 .458 .127 1992-93 0.463 .453 .113 1993-94 0.455 .442 .127 1994-95 0.469 .457 .123 1995-96 0.466 .452 .128 1996-97 0.479 .467 .129 1997-98 0.478 .465 .117 1998-99 0.483 .470 .108 1999-00 0.461 .443 .105 2000-01 0.459 .440 .107 2001-02 0.471 .453 .106 2002-03 0.465 .443 .103 2003-04 0.468 .446 .101 2004-05 0.470 .446 .099 2005-06 0.458 .428 .106 2006-07 0.492 .471 .105 2007-08 0.481 .456 .105 2008-09 0.470 .441 .104 2009-10 0.465 .433 .105 2010-11 0.471 .440

CMDI .237 .242 .243 .246 .244 .247 .244 .246 .237 .235 .239 .238 .232 .242 .240 .249 .248 .248 .234 .234 .241 .237 .238 .239 .234 .255 .247 .239 .234 .238

From Figure 3 and also from Table 3 it is clear that MDI is always decreasing whereas CMDI is almost stagnant. HHDI and CHHDI are very close to each other and have the same pattern of movements. CMDI though taking much different values from those of HHDI and CHHDI also show similar pattern. Thus it is expected that these three indices will have high positive values each other. The proposed Modified Diversity Index (MDI) and its corrected version are thus much different from the other two indices. The matrix of bivariate correlations is presented below which further confirms the trend behavior as discussed above. It is also seen from the diagram that there is a decreasing trend of all the indices up to 1975-76, after these more or less an increasing trend is observed up to 1987-88 and then it is almost stagnant or decreasing. Thus the indices should have been computed separately for data up to 1975-76 and for data after 1987-88 neglecting the observations for the middle 12 years. In conclusion, we can say that after Independence the diversity of crop areas more or less has a decreasing trend except for the period of 1976-77 to 1987-88. The reason for this is not clear and should be traced in migration patterns, floods or other economic and natural phenomena.


FIGURE 3. Trends in the Diversity Indices of the Areas under Major Crops TABLE 3. Bivariate Correlations of the Diversity Indices of the Areas under Major Crops

HHDI

CHHDI **

MDI

CMDI **

HHDI 1.000 0.639 -0.723 0.592** CHHDI 1.000 0.023 0.924** MDI 1.000 0.018 CMDI 1.000 **. Correlation is significant at the 0.01 level (2-tailed). It was already noted that the area under crops for each of the major groups were found by aggregating the areas under individual crops as follows: Rice (Paddy), Maize, Wheat and Other Cereals Cereals : Gram, Arhar and Other Rabi Pulses, Pulses : Rape and Mustard, Sesamum, Linseed and Castor Oil Seeds : Jute, Mesta and Cotton Fibres : Plantation Crops : Tea, Sugar Cane, Areca Nut and Tobacco Potato, Sweet Potato, Chilli, Turmeric, Tapioca, Banana, Onion and Pine Apple Misc. Crops : A closer look in the data reveals that data for some of the individual crops were not available up to 1975-76 and thus the aggregated values were found neglecting the unavailable observations. This further confirms our proposition that the indices should be calculated separately for all the sub-periods. To see how the correlations changes if we do so we present below the bivariate correlations separately for the three periods.


TABLE 4. Bivariate Correlations of the Areas under Major Crops Separately for the Three Periods Cereals

Pulses

OilSeeds Period 1 0.966** 0.917** 1.000

Cereals Pulses OilSeeds Fibres PlantationCrops Misc. Crops

1.000

0.888** 1.000


1.000

0.891** 1.000

Period 2 0.847** 0.952** 1.000


1.000

0.175 1.000

Period 3 0.459* 0.359 1.000

Fibres

PlantationCrops

MiscCrops

0.220 0.257 0.292 1.000

0.980** 0.879** 0.957** 0.177 1.000

0.939** 0.863** 0.920** 0.442* 0.911** 1.000

0.497 0.173 0.055 1.000

0.882** 0.982** 0.965** 0.165 1.000

0.864** 0.971** 0.963** 0.062 0.973** 1.000

0.388 0.249 0.859** 1.000

-0.142 0.025 -0.747** -0.739** 1.000

-0.044 0.085 -0.690** -0.751** 0.920** 1.000

All Periods Cereals 1 .867** .905** -.661** .929** ** ** Pulses 1 .914 -.477 .819** ** Oil Seeds 1 -.613 .887** Fibres 1 -.834** Plantation Crops 1 Misc. Crops **. Correlation is significant at 0.01 % level. *. Correlation is significant at 0.05 % level.

.902** .772** .880** -.860** .989** 1

The overall period correlations indicate inverse relation of fibres with cereals, pulses, oilseeds, plantations and other crops, which is a normal behavior where cultivation of natural fibre crops have declined over the years while others increased. However, significant positive associations among cereals, pulses, oilseeds and plantation crops are the indications of simultaneous growth without any substitution of land use. Only in the recent past decades minor substations have taken place and thus correlations have been weakened. Though HHDI is more or less stagnant over time, declining MDI indicates lowering diversity pattern over time. However, the CMDI also reflects more or less stagnant diversity of crop in Assam.

REFERENCES 1. M.A. Adelman, Review of Economics and Statistics, 51, 99-101, 1969. 2. Al-Samarrie Ahmad & Herman P. Miller, American Economic Review, 57, 59-72, 1967. 3. Nejat Anbarci & Brett Katzman, “Socially Beneficial Mergers A New Class of Concentration Indices”, GA, 30144-5 591, March 28, 2005. 4. H. Beran, “A generalisation and optimization of a measure of diversity”, in Modelling and decisions in economics, Physica, Heidelberg, edited by U. Leopold-Wildburger, G. Feichtinger and K. P. Kistner,pp. 119137, 2005. 5. Roger Clarke, Industrial Economics, Basil Blackwell, Oxford, 1985. 6. S.W. Davies, Economica, New Series, Vol. 46, No. 181, pp. 67-75, 1979. 7. U. K. De, Economics of Crop Diversification, New Dehli Akansha Publishing House, 2003. 8. U. K. De, and M. Chattopadhyay, Journal of Development and Agricultural Economics, 2 9, 340-350, 2010. 9. Rae Douglas, Political Consequences of Electoral Laws, Yale University Press, New Haven, pp. 53-58, 1967. 10. Rae Douglas, Comparative Political Studies, 1, 3, 414-416, 1968.


11. Rae Douglas and Michael Taylor, The Analysis of Political Cleavages, Yale University Press, New Haven, 1970, pp. 24-33. 12. Güzinand Erlat and Oguzhan Akyuz, “Country concentration of Turkish exports and imports over time”, Paper presented at the 26th Annual Meeting of the Middle East Economic Association held in conjunction with the Allied Social Science Associations, New Orleans, January 5-7. http://www.luc.edu/orgs/meea/volume3/ gerlat/gerlat.html, Economics web institute, 2001. 13. R.A. Fisher, S.A. Corbet & C.B. Williams, Journal of Animal Ecology, 12, 1, 42-58, 1943. 14. Fred E Foldvary, Indian Economic Journal, 54, 3, September-December, 2006. 15. C. Gini, “Variabilita' e Mutabilata”, Reprinted in Memorie di Metodologia Statistica Rome: Libreria Eredi Virgilio Veschi, 1955 edited by A.V. Pizetti and T. Salvemini; Cited by E. W. Weisstein, Mathworld.com 19992004, 1912. 16. Joseph H. Greenberg, Language, 32, 1, March, 1956. 17. M.M. Hackbart and D.A. Anderson, Land Economics, 51, 4, 374-378, 1975. 18. M. Hall, and L. Tideman, Journal of the American Statistical Association, 62, 162-68, 1967. 19. L. Hannah, and J.A. Kay, Concentration in Modern Industry: Theory, Measurement and the U.K. Experience, London MacMillan Press, 1977. 20. P. E. Hart, Journal of the Royal Statistical Society, Series A General, 133,73-85, 1971. 21. P. E. Hart, Journal of the Royal Statistical Society, Series A General, 138, 423-434, 1975. 22. J.C. Hause, Annals of Economic and Social Measurement, 6, 73-107, 1977. 23. J. Havrda and F. Charvát, kybernetika, 3, 30-35, 1967. 24. O.C. Herfindahl, Concentration in the US Steel Industry, Doctoral Dissertation Unpublished, Columbia University, 1950. 25. A.O. Hirschman, The American Economic Review, 54, 761-762, 1964. 26. S. Hoffmann, “Concavity and additivity in diversity measurement: re-discovery of an unknown concept”, www.fww.ovgu.de/fww_media/femm/femm_2006/2006_28.pdf, 2007. 27. K. Hou and D. T. Robinson, ,Journal of Finance, 61, 4, August, 1927 -1956, 2006. 28. J. Hovarth, Southern Economic Journal, 36, 446-452, 1970. 29. P.E. Hart and R. Clarke, Concentration in British Industry 1935-75, Cambridge: Cambridge University Press, 1980. 30. C. Keylock, Oikos, 109, 203-206, 2005. 31. J. Kwoka, Antitrust Bulletin, 30, 915-947, 1985. 32. R. Linda, Economie Appliquee, 2-3, 325-472, 1972. 33. R. Linda, Méthodologie de l’analyse de la concentration appliquée àl’étude des secteurs et des marchés, European Commission, 1976. 34. Edward D. Mansfield, International Organization, 46, 3, 731-764, 1992. 35. M. Mayoral, Journal of Information Sciences, 105, 101-114, 1998. 36. R.A. Miller, Southern Economic Journal, 34 October, 259-267, 1967. 37. C. Meilak, Bank of Valletta Review, 37, 35-48, 2008. 38. K. Nehring and C. Puppe’s, Econometrica, 70, 1155-1198, 2002. 39. G. P. Patil and C. Taillie, Journal of the American Statistical Association, 77 379, 548-561, 1982. 40. J.L. Ray & J.D. Singer, Sociological Methods and Research, 1, 4, 403-437, 1973. 41. A. Rényi, On measures of entropy and information. In: Proceedings of the fourth Berkeley Symposium on Mathematical Statistics and Probability, Vol. 1, University of California Press, Berkeley, 547-561, 1961. 42. C. Ricotta, Journal Of Theoretical Biology, 222, 189-197, 2003. 43. R. Roll, Journal of Finance, 47, 1, 3-41, 1992. 44. C.E. Shannon, Bell System Technical Journal, 27, 3, 379-423, 1948. 45. R. L. Shiyani and H. R. Pandya, Indian Journal of Agricultural Economics, 53,4, 627-639, 1998. 46. E.H. Simpson, Nature, 163, 13, 688, 1949. 47. A. J. Singh; K. K. Jain and I. Sain, Indian Journal of Agricultural Economics, (403): 298-303, 1985. 48. R.B. Singh, Welcome Address, presented at the Expert Consultation on Crop Diversification in the Asia Pacific Region held at FAO Regional Office for Asia and the Pacific in Bangkok, Thailand, 4-6 July, 2000. 49. I.T. Tabner, Multinational Finance Journal, 2009, vol. 13, no. 3/4, pp. 209-228, 2009. 50. L.W. Tauer and T.B. Seleka, “Agricultural Diversity and Cash Receipt Variability for Individual States”, Cornell Agricultural Economics Staff Paper No 94-1, January, Cornell University, New York, 1994. 51. H. Theil, Economics and Information Theory, North-Holland Publishing Co., Amsterdam, 1967. 52. R. C. Tress, “Unemployment and Diversification of Industry”, The Manchester School, 9, 140-152, 1938.


53. C. Tsallis, Journal of Statistical Physics, 52, 1/2, 479-487, 1988. 54. Raimo Vayrynen, Scandinavian Political Studies, Bind 7 1972, 1970. 55. C.B. Williams, Nature, 157, 13, 482, 1946. 56. G. U. Yule, Statistical Study of Literary Vocabulary, Cambridge: Cambridge University Press, 1944.
