Chapter 21 The computation of equilibrium prices: An exposition

A demonstration of the existence of equilibrium prices for a general Walrasian model of competitive behavior necessarily makes use of some variant of Brouwer's fixed point theorem as an essential step. The strategy for calculating equilibrium prices is to render that step constructive by a numerical approximation of the fixed point implied by Brouwer's theorem or one of its alternatives. This chapter provides a brief review of the competitive model and the role of fixed point theorems. For the purpose of computing equilibrium prices, the demand functions are frequently more natural to work with than the underlying utility function or preference relationship. A general equilibrium model is fully specified to obtain a numerical solution. On the consumer side, this is typically done by providing a numerical or algebraic description of the functions. A few examples of some of the more familiar utility functions and their associated demand functions are appropriate at this point. A model of equilibrium involves a number of consuming units, individuals, aggregations of individuals, or countries involved in foreign tradeeach of whom has a stock of assets, prior to production and trade, and each of whose demands are functions of the prevailing prices. The market demand functions specify the total consumer demand, for the goods and services in the economy, as a function of all prices. The chapter also explains that market demand functions cannot be arbitrarily specified if they are derived from individual demand functions by the process of aggregation.