Dynamic implications of capital market perfection.

• Author: Sinha, Deepak K.

1. Introduction

   Capital market perfection is a key assumption often made by scholars in the area of finance. Similarly, the assumption of a rational, utility maximizing consumer giving rise to demand functions which are homogeneous of degree zero in prices and income is central to a large body of research in economics. Interestingly, with firms facing menu costs, the two assumptions together impose certain dynamic restrictions on all prices and quantities in an economy. These restrictions are independent of the nature of technology, market structure, and the like. Nor do they depend upon any particular functional forms for the demand and production relationships. For these reasons they provide valuable insight into the nature and sources of economic growth. If the capital market is perfect, consumers are rational, and firms face menu costs then the time paths of prices and outputs from any model of economic growth must conform to these restrictions. Hence their importance.

   The paper is divided into five parts. In the second part I demonstrate that in the general case, when firms face menu costs and preferences are not homothetic, the dynamic restrictions imposed by capital market perfection and the homogeneity of demand require that the prices of all firms in an industry move together. In the third and fourth parts of the paper I consider the particular case of homothetic preferences when products are weak gross substitutes. I provide an alternative proof of co-movement of prices in an industry and show that if there are no changes in consumer's tastes then the elasticities of demand for firms are constant. If technology exhibits constant returns then it can be shown that, in the Nash non-cooperative equilibrium, firms will incur menu cost in the form of investment in productivity to reduce real cost and price. As a result the output of the economy will go up. In the fifth part I discuss some implications of the model for consumer's surplus and welfare.

2. Capital Market Perfection and Economic Dynamics

   Capital market perfection implies that, in equilibrium, the expected rates of return on any two assets with identical risk characteristics are equal [4]. For simplicity, in this paper I limit my analysis to the case when all firms have identical risk characteristics, in which case capital market perfection implies that all firms in the economy have the same expected real rate of return equal to r. I do not assume r to be constant. Let [Mathematical Expression Omitted]; [Rho] itself need not be constant. Then capital market perfection can be shown to impose a restriction on the movements of real profit of all firms in the economy. To see this consider a firm u which enters an industry at t = T by investing [V.sub.uT]. Its real profit during a period dt at t is [[Pi].sub.u]dt. Then capital market perfection implies the following dynamic restriction on profits:

   [[Pi].sub.u]dt = r[V.sub.uT] (1a)

   [Mathematical Expression Omitted].

   The above restriction, in turn, results in some dynamic constraints on prices, quantities, and returns on sales of firms. To show this I consider an economy with n firms (u = 1,..., n), each producing a single good, Firm u's output, return on sales, and nominal price during a period dt at T are [q.sub.uT]dt, [s.sub.uT], and [P.sub.uT] (= [p.sub.uT][e.sup.[Mu](t-T)]; [p.sub.ut] is the real price; [Mathematical Expression Omitted] is the rate of growth of consumer's expenditure; m is consumer's expenditure in current dollars), so that its nominal profit during the period is [s.sub.uT][P.sub.uT][q.sub.uT]dt. It will be convenient to write [Mathematical Expression Omitted], [Mathematical Expression Omitted], and [Mathematical Expression Omitted] and -[[Phi].sub.u] are the rates of change in firm u's output, return on sales, and real price during the time period dt at T. [[Omega].sub.u], [[Omega].sub.u], [[Phi].sub.u], and [Mu] may not be constant over time. From equation (1b):

   -[[Phi].sub.u] + [[Omega].sub.u] + [[Omega].sub.u] = -[[Phi].sub.v] + [[Omega].sub.v] + [[Omega].sub.v] = [Rho]. (2)

   For simplicity, in this paper I do not allow the formation of new industries, although I do allow entry by new firms in established industries. I define industries by cross price elasticities. Goods in an industry are relatively good substitutes for other goods within the industry but are poor substitutes for all other goods in the economy. All cross price elasticities across industries are therefore zero. In the absence of formation of new industries, budget shares of all industries are constant. The model, however, can easily be modified to allow for the formation of new industries, in which case the budget shares of industries will fall over time.

   The budget share of the industry in which firm u operates is [b.sub.u], which I assume is constant. Although there are several industries in the economy, it will not be necessary to use a suffix for them. Different industries have different budget shares, so that, if for two firms u and v, [b.sub.u] = [b.sub.v] then the two firms are in the same industry, [q.sub.u] (= [q.sub.u]([P.sub.1],..., [P.sub.u],..., [P.sub.v],..., [P.sub.n], [a.sub.u], [b.sub.u], m)) is a function of all nominal prices, m, [b.sub.u], and the non-price influences on firm u's share of industry revenue, [a.sub.u], which declines over time due to entry; [Mathematical Expression Omitted] may not be constant. Since [b.sub.u] and [a.sub.u] are independent of prices, I can write [q.sub.u] as [b.sub.u][a.sub.u][f.sub.u] where [f.sub.u] = [f.sub.u]([P.sub.1],..., [P.sub.u],..., [P.sub.v],..., [P.sub.n], m) is a function of nominal prices and m alone. Therefore, [[Delta].sub.[q.sub.u]]/[[Delta].sub.[a.sub.u]] = [q.sub.u]/[a.sub.u].

   In the general case when preferences are not homothetic, the dynamic restriction on the real profit of firms in equation (1b), together with menu cost and the assumption of consumer rationality which results in demand functions which are homogeneous of degree zero in prices and income, imposes a constraint on the movements of all prices in...