Bill and Herb have provided an illuminating and interesting presentation of Irving Fisher's Ph.D. dissertation Mathematical Investigations in the Theory of Value and Prices. They correctly emphasize that Fisher's fundamental contribution to the early theory of general equilibrium was the construction of a machine to compute the equilibrium quantities in a Walrasian model of competitive markets. As Fisher notes, Walras deserves priority for deriving a system of equations that characterize equilibrium in competitive markets. Even in this area, however, Fisher makes a subtle and interesting contribution in his system of equilibrium equations, discussed later.
The first modern treatment of computing equilibrium prices in Walrasian economies is due to Herb, as everyone in this audience knows. His seminal paper, “On the Computation of Equilibrium Prices,” appears, most appropriately, in Ten Essays in Honor of Irving Fisher. The modern treatment of the existence question in the general equilibrium model, due to Arrow and Debreu, converts the equilibrium conditions into a fixed point of a continuous map, say from the price simplex into itself. It follows from Brouwer's fixed point theorem that this map will have a fixed point. The Scarf algorithm computes approximate fixed points of any continuous map of the simplex into itself. Hence the Scarf algorithm can be used to compute equilibrium quantities.
Subsequent to Scarf's research, a more direct method of solving nonlinear systems of equilibrium equations was suggested by Eaves. The so-called homotopy method deforms a set of equations whose solution we know into the equilibrium equations, tracing out a path of solutions terminating in a solution for the equilibrium equations. Unfortunately, there is no price-adjustment interpretation of the disequilibrium prices along the homotopy path. In this way they are similar to the prices generated by the Fisher machine out of equilibrium.
Below we give a system of equations characterizing equilibrium in an exchange economy with two agents and three goods. Agents are assumed to be endowed with money income and additive separable utility functions, which are monotone, strictly concave, and smooth. The unknowns in our equations are the state variables of Fisher's machine, in other words, prices, individual consumptions, expenditures, marginal utilities of income, and marginal utilities of the consumptions implied by expenditures and prices. Equilibrium values are computed using the homotopy method.
