# Dynamic Economics: Optimization by the Lagrange Method

By Gregory C. Chow | Go to book overview

CHAPTER FOUR Theories of Market Equilibrium

4.1 Asset Prices of an Exchange Economy

A simple and basic model of equilibrium in economics consists of a demand function and a supply function. These two functions jointly determine the price and output of a commodity. If the quantity of supply is fixed and given, the demand function evaluated at the fixed supply determines price, as in the following model of an exchange economy of Lucas ( 1978).

In this exchange economy, there exists only one consumer good (e.g., apples). The good is produced by k firms (e.g., farms), each using one type of capital good or asset (e.g., apple tree). Each firm is owned by the consumers. The ith firm has xit units of capital assets. The units of the assets of each firm are so defined that there is one asset per consumer. Each unit of the assets of the ith firm yields a random zit units of the consumer good in period t. In other words, each share of the stock of the ith firm yields a random zit units of dividends in period t. Let the vectors xt and zt denote (x1t, . . ., xktand (z1t, . . . , zkt)ʹ, respectively. Let p(zt) denote the vector (p1(zt), . . . , pk(zt)) of the prices of the k assets (shares of the k firms). The prices are functions of the dividends of all firms. The problem is to find the price functions p(zt). To solve this problem, first find the demand functions for the k assets, and then equate the demands with the fixed supplies of these assets.

The representative consumer in this economy is assumed to maximize expected discounted utility U(ct) of consumption subject to the budget constraint for period t

ct + pʹ(zt)xt+1 = [zt + p(zt)]ʹxt. (4.1)

The right-hand side of (4.1) gives the total revenue available to the consumer who owns the vector xt of assets of the k firms, as these assets yield dividends zt and can be sold at post-dividend prices p(zt). The left-hand side of (4.1) is the total expenditures for consumption and for keeping xt+1 units of assets at the beginning of period t + 1.

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