How to compute equilibrium prices in 1891

American Journal of Economics and Sociology, The,  Jan, 2005  by William C. Brainard,  Herbert E. Scarf

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I

Introduction

IRVING FISHER'S Ph.D. thesis, submitted to Yale University in 1891, is remarkable in at least two distinct ways. The thesis contains a fully articulated general equilibrium model presented with the broad scope and formal mathematical clarity that we associate with Walras and his successors. But what is even more astonishing is the presentation, in the thesis, of Fisher's hydraulic apparatus for calculating equilibrium prices and the resulting distribution of society's endowments among the agents in the economy.

Fisher's development of the general equilibrium model was done without any knowledge of the simultaneous achievement of Walras. In the introduction to his thesis, Fisher states that

   [t]he equations in Chapter IV, Sec. 10, were found by me two years
   ago, when I had read no mathematical economist except Jevons. They
   were an appropriate extension of Jevons' determination of the
   exchange of two commodities between two trading bodies to the
   exchange of any number of commodities between any number of
   traders.... These equations are essentially those of Walras.

Even though Fisher's construction of what we now call the Walrasian model of equilibrium was a fully original achievement, he did have contemporaries: the central ideas of equilibrium theory were independently discovered at several locations in the final decade of the last century. But the second theme of Fisher's thesis was entirely novel in conception and execution. No other economist of his time suggested the possibility of exploring the implications of equilibrium analysis by constructing specific numerical models, with a moderately large number of commodities and consumers, and finding those prices that would simultaneously equate supply and demand in all markets. The profession would have to wait until rudimentary computers were available in the 1930s before Leontief turned his hand to a simplified version of this computation.

Fisher's hydraulic machine is complex and ingenious. It correctly solves for equilibrium prices in a model of exchange in which each consumer has additive, monotonic, and concave utility functions, and a specified money income; the market supplies of each good are exogenously given. Both additivity and fixed money incomes make the model of exchange to which his mechanism is applied less than completely general. But we know of no argument for the existence of equilibrium prices in this restricted model that does not require the full use of Brouwer's fixed point theorem. Of course fixed point theorems were not available to Fisher and in that section of his thesis in which first-order conditions are presented for a general model of exchange, Fisher argues for consistency by counting equations and unknowns.

It is hard to discover the source of Fisher's interest in computation. He was a student of J. Willard Gibbs, and perhaps the theme of concrete models in mechanics was carried over to economics. But it is also possible that his hydraulic apparatus is simply an instance of an American passion for complex machinery that gets things done. Fisher, himself, had a passion for innovation. In the course of a long career, he invented an elaborate tent for the treatment of tuberculosis (described in the Journal of the American Medical Association, 1903), developed a mechanical diet indicator that permited easy calculation of the daily consumption of fats, carbohydrates, and proteins, copywrited (1943) an icosahedral world globe with triangular facets that when unfolded was allegedly an improvement on the Mercator projection, and patented an "index visible filing system" (1913) sold to Kardex Rand, later Remmington Rand, in 1925 for $660,000.

II

Fisher's Cisterns

LET US BEGIN by describing the special model of exchange that is solved by the Fisher machine. There are, say, m consumers, indexed by i = 1, ..., m and n goods, indexed by j = 1, ... n. Consumer i has the utility function

[u.sub.i]([x.sub.1], ..., [x.sub.n] = [summation over j=1, n][u.sub.ij]([x.sub.j]),

with each [u.sub.ij] increasing and concave. Society's endowment of good j is [E.sub.j], and consumer i's income is [Y.sub.i].

At prices p = ([p.sub.1], ... [p.sub.n]) [greater than or equal to] 0, consumer i is assumed to maximize utility subject to his budget constraint

max [u.sub.i]([x.sub.1, ..., [x.sub.n) subject to [summation over j=1,n][p.sub.j][x.sub.j] [less than or equal to] [Y.sub.i].

If the marginal utility of income for consumer i is [[lambda].sub.i] the demands, [x.sub.ij], will satisfy the first order conditions:

[u'.sub.ij]([x.sub.ij]) [less than or equal to] [[lambda].sub.i][p.sub.j] (= if [x.sub.ij] > 0).

A competitive equilibrium is given by a price vector p so that the market demands obtained by the summation of individual demands are equal, commodity by commodity, to the market supply. In other words, we are asked to find p = ([p.sub.1], ..., [p.sub.n]) [greater than or equal to] 0, [lambda] = ([[lambda].sub.1], ..., [[lambda].sub.m]) [greater than or equal to] 0 and a matrix of commodities [[x.sub.ij]], such that