A homotopy approach to the computation of economic equilibria on the unit simplex

In the model of a pure exchange economy with n + 1 commodities, the excess demand is a continuous function from the n-dimensional unit simplex Sⁿ to the (n + 1)-dimensional Euclidean space Rⁿ⁺¹. A zero point of this function is a price vector at which the demand is equal to the supply in the economy. Such a price vector yields an economic equilibrium. In this paper we present a simplicial homotopy method on the unit simplex to compute such an economic equilibrium, and show how it follows a unique piecewise linear path. This method has a clear economic interpretation. Along the path of generated prices the excess demand of each commodity is a multiple of the difference between the current and initial prices of that commodity.