Stock Market Education of Participation Technology ...

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Stock Market Education of Participation Technology. Soumitra K. Mallick,. Indian Institute of Social Welfare & Business Management,. Management House,.
Stock Market Education of Participation Technology

Soumitra K. Mallick, Indian Institute of Social Welfare & Business Management, Management House, College Square West, Calcutta 700 073 India; Email : [email protected]; June 2011

Abstract: This paper considers the problem of developing an optimal stock market education system of participation technology by considering growth in a mixed (stock market) economy in the steady state with a representative agent, where stock market education and therefore participation in the stock market process is dependent endogenously on a participation technology. With such a specification the differential equation characterizing the choice of stock market education and participation technology for the intertemporal ”golden rule” consumption problem using general concave utility specification and Riemannian aggregation procedures for multi-dimensional representative felicity is setup and analyzed. This equation is partly characterized for sequential stock market education and therefore participation dynamics. Key words : education technology of market participation, optimal growth, choice equation, stock market participation . JEL Classification # I20, G10, C54. The author acknowledges helpful discussions with Jess Benhabib,Yaw Nyarko,Jota Ishikawa,Fumio Hayashi, Ed Prescot on related work & Sandra Dawson for some free time to complete this work. This paper is an abridged version of a paper presented at the Far Eastern meetings of the Econometric Society 2001, Kobe & a continuing education school at the National Stock Exchange of India Ltd., Mumbai, 2001. The author thanks Kobe University & the Indian Institute of Social Welfare & Business Management for financial support. The usual disclaimer applies.

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Introduction :

One of the common problems facing stock markets in India is the problem of participation. Since a majority of people live in villages, many of whom are remotely located from the financial centers of metropolitan cities the only source of access to capital markets are the few local branches of nationalized banks or the post office. It is well known that this gives rise to ”black markets” in finance like the often usurious village moneylenders. In a country where official lending rates still hover around 12 percent perannum and the growth rate of the economy is around 6 percent so that competitive market theory dictates that inflation be around 6 percent to justify the interest rates at equilibrium real incomes are bound to suffer, thus further reducing the incentives to invest in stock markets. This further encourages highly riskprone quickgain activities. The picture in the cities is not very bright either. People often cite ignorance about stock markets as reasons for not investing in them. In fact there are no facilities like network of broker terminals, like bank branches, where small savers can invest in shares. Coincidentally, the optimal growth model is a representative agent planning model, for a context with private property and decentralized decisions. However the exact decentralized properties of such a model in mixed economies with stock markets is yet relatively unknown (Mallick (1993)). The purpose of this paper is to analyze paths of market participation in such a model in the presence of an exogenous stock market education participation technology. Recently, there has been a renewal of interest in the economic implications of learning about future wealth. On this see for e.g. Buera, Monge-Naranjo & Primiceri (2011), Banbura, Giannone & Reichlin (2010), Conley & Udry (2010), El-Gamal & Sundaram (1993), Sokoloff & Engerman (2000). These papers discuss the implications of learning about the budget constraint entries and the expected utility simultaneously and their result on the equilibrium which includes the learning costs like fixed effect of political cost in a model where income depends on political uncertainty. This paper takes the similar modelling approach which is simultneous impact of learning about wealth on the budget constraint as well as on the time horizon in the objective function with no explicit role of uncertainty except possibly in the motivation to such knd of learning and therefore the learning

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or education for participation in the economc decision process after learning. The resultant representative agent optimal growth model is constructed under certainty. It is shown that exogenously chosen education systems of participation technologies may not always ensure a dynamically optimal solution (”golden rule”) by characterizing a dynamic functional education system of participation equation. Mazumdar and Mitra (1994) introduced wealth effects in the optimal growth model through utility and discover chaotic paths. Nyarko & Ohlson (1994) studied the role of uncertainty in an optimal growth problem where utility depends on consumption and resource stock and find that the first best growth path is attainable under otherwise standard assumptions on the process of growth. Khan & Mitra (1986) in contrast to Radner’s (1961) turnpike model proved the existence of a stationary path of capital market participation (which is optimal), where there are multiple sectors in the economy, without any exogenous imposition of path of expansion. Ohlson & Roy (1996) in contrast to the decentralized and single budget constraint framework of this paper,have analyzed the growth properties of an optimal growth model with utility depending on both consumption and resource stock and derive conditions under which the dynamically optimal resource utilization does not lead to extinction of the resource in the limit. In all of the above papers, which are variants of the optimal growth set up, aggregate wealth is seen to affect the characteristics of the optimal solution. However, it is not clear that the interiority assumptions made on the optimal path of stock market education of participation would necessarily remain valid if the capital stock affected utility at some fundamental stage of the model. Section 2 discusses the model, section 3 discusses dynamic stock market education of participation equation, section 4 concludes.

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The Model :

There is one representative participant in the economy, who receives an exogenous wealth - wt in each period t.

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The participant makes three decisions - the current consumption (ct ), the planning horizon (θt ) - the period over which to roll over the present wealth in consumption and the time stationary consumption in each period in the future (ct+1 ) ∀t + n, n ∈ I+ . The capital stock which results at the end of each period is assumed to be invested entirely in the stock market to finance consumption in the future. The felicity function is the Riemann summation over time of a stationary, concave per period component given by u : R+ − > R+ , u(0) ≤ 0, u0 (.) > 0, u00 (.) < 0 The Riemann summation is specified for identifying the welfare aggregation procedure in this representative agent framework, as discussions on optimality would depend on this specification crucially. The representative planning problem at any t is : max

θ

θ +2−t

t ,θ )∈R t (ct ,(ct+1 )t+1 t +

u(ct ) + θt u(ct+1 )

s.t.ct + T (θt )ct+1 = wt

(1)

T (θt ) is the multiplicative education of participation investment function with the only assumption T (θt ) > 0 for θt > 0. Its some other properties which are endogenous to the planner’s problem in this paper will be derived later.

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Dynamic Stock Market education of participation equation:

Stock Market education of participation is given by : kt+1 = θt∗ c∗t+1 + wt+1

(2)

at the end of t where θt∗ , c∗t+1 solves (1). Thus, kt+1 measures the amount invested in the stock market at the end of t to finance consumption in the future. The amount invested serves as a instrument for the extent of education of participation. A direct measure of such extent of education of 3

participation in actual stock markets would be the ”trading volume” or the incremental ”market capitalization” realized in each period (the two are not the same it is acknowledged). However, both these measures have fundamental similarities and dissimilarities with ”sales volume” in goods market. These comparisons are beyond the scope of this paper. The budget balance constraint in (1) is different from an additive formulation as is usually done. This is done for ease of getting explicit solutions. However, consider the constraint : ct + θt ct+1 + γ(θt ) = wt

(3)

The following theorem establishes the qualitative equivalence. Theorem 1: If T (θt ) is differentiable, then for every T (θt ) ∃ a γ(θt ) such that the solution space of (1) is identical with the additive balance constraint replaced in (1). Proof : t) Consider the decomposition ct + T (θt )ct+1 = ct + ( γ(θ + θt )ct+1 and The c∗t+1 Kuhn-Tucker Saddle function of (1) is taken with the alternative resource constraints obtained by the above decomposition. By the concavity properties of u(.) and the linearity of the constraints an interior solution exists to the choice of ct , ct+1 in either case for a given T (θt ) or γ(θt ), holding c∗t+1 fixed in the second case at the latter solution (on the Kuhn-Tucker theorem see for e.g. Rockafeller (1970) theorem 28.4).

Since u is differentiable and positive the solutions are positive and locally unique by the same theorem. Hence, substitute the values of c∗t & c∗t+1 in the two cases and from the fact that differentiability of T (θt ) ensures invertibility one can solve the above decomposition as an equation to obtain the solution to γ(.). Since the objective function is linear in θt and T (θt ) is differentiable a solution to the optimum θt will also exist by the Kuhn-Tucker theorem for some T (.). The invertibility argument works using the Inverse function theorem 4

(Kolmogorov & Fomin (1970)) as before. Hence the theorem. This result helps in decomposing the impacts of stock market education of participation and the consumption streams which result over time, on the welfare levels of participants as well as optimal growth of the economy. Definition : Stock Market Education of Participation is ”golden rule” iff u0 (ct ) dkt+1 = 0 dkt u (ct+1 )

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(4)

Stock Market education of participation technology

From definition equation (2) kt+1 = θt∗ c∗t+1 + wt+1 ⇒ c∗t+1 =

kt+1 − wt+1 θt∗

Plugging this into (1) : kt+1 − wt+1 u θt∗

!

= u0 (ct )T (θt )[(wt − ct )

T 0 (θt ) ] T 2 (θt )

(5)

T 0 (θt ) 1 u0 (ct+1 ) dkt+1 = u0 (ct )T (θt ) 2 dwt θt T (θt ) ⇒

Now, if

T 0 (θt )θt T (θt )

dkt+1 u0 (ct ) T 0 (θt ) θt = 0 dwt u (ct+1 ) T (θt )

= 1, then

dkt+1 dkt

=

dkt+1 dwt

=

u0 (ct ) u0 (ct+1 )

in (6).

which becomes the same as the ”golden rule” (see definition). 5

(6)

Solving the differential equation we arrive at the particular solution

T (θt ) = Aθt This leads to the following theorem: Theorem 2 : Even when θ < ∞(∞ = lifetime of the agent) ∃T (θ) s.t. the capital market education of participation rule is ”golden rule”.

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Conclusion :

The above theorems characterize some features of the education of participation technology under certainty and participative optimal growth which results after taking into consideratio the system of education of prtiaicpation with respect to stock trading, the network of terminals, the distribution of trained agents of stock brokers and so on technology. Each of these types of ”education of participation technology” will have features of its own, which will be different from the more developed banking and non-banking-financial institutions like postoffice savings bank and insurance company networks, requiring further indepth modeling. However, that the nature of such ”stock market infrastructure” will determine the nature of stock market education of participation and hence the realized welfare and hence growth of the economy has been highlighted in this paper. References: 1. Buera, F.J., A. Monge-Naranjo & G.E.Primiceri : Learning the Wealth of Nations, Econometrica, 79, 2011, 1-45. 2. Banbura, M., D. Giannone & L. Reichlin : Large Bayesian VARs, Journal of Applied Econometrics, 25, 2010, 71-92. 3. Besley, T.J. & A. Case : Diffusion as a learning process : evidence from HYV Cotton, Research Program in Development Studies Discussion Paper 174, Woodrow Wilson School, 1994. 6

4. Conley,T. & C.Udry : Learning about a new technology : pineapple in Ghana, American Economic Review , 100, 2010, 35-69. 5. El-Gamal, M.A. & R.K.Sundaram : Bayesian Economist... Bayesian Agents I : An Alternative Approach to Optimal Learning, Journal of Economic Dynamics & Control , 17, 1993, 355-383. 6. Khan, M.A & T. Mitra : On the existence of a stationary optimal stock for a multisector economy : A primal approach,Journal of Economic Theory, 40, 1986, 319-328. 7. Mallick, S.K. : Bounded Rationality & Arrow-Debreu economies, unpublished Ph.D. dissertation, Dept. of Economies, New York University, 1993. 8. Mazumdar, M.K. and T. Mitra : Periodic and chaotic programs of optimal intertemporal allocation in an aggregative model with wealth effects,Economic Theory, 4, 1994, 649-676. 9. Nyarko, Y. and L. Ohlson : Stochastic growth when utility depends on both consumption and the stock level, Economic Theory, 4, 1994, 791-797. 10. Ohlson, L.J. & S. Roy : On conservation of renewable resources with stock-dependant return and non-concave production, Journal of Economic Theory, 70, 1996, 133-157. 11. Radner,R. : Paths of Economic growth that are optimal with regard only to final states, Review of Economic Studies, 28, 98-104, 1961. 12. Rockafeller, R.T. : Convex Analysis, Princeton University Press, Princeton, first edition, 1970. 13. Sokoloff,K.L. & S.L.Engerman, History Lessons : Institutions, Factor Endowments, and paths of development in the new world, Journal of Economic Perspectives, 17, 2000, 217-232. 14. Kolmogorov, A.N. & S.V Fomin : Introductory Real Analysis, Dover, New York, first edition, 1970.

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