Convergence of Myopic Firms to Long-Run Equilibrium Via the Method of Characteristics

Abstract

Sonnenschein (1981, 1982) models the evolution of firms from short run to long run equilibrium. The short run equilibrium is based on Rosen’s (1974) model of an economy with a continuum of differentiated products and a continuum of consumers. The economy contains a standard commodity, the numeraire, amounts of which are denoted by y, and a one-dimensional continuum of differentiated products, which are parametrized by a type index x that lies on the circle. Each consumer demands one unit of one type of the differentiated commodity, and each firm produces exactly one type of commodity at each point in time. Given an initial distribution of firms, the initial equilibrium price function for commodities is defined by the condition that supply equals demand for all commodities. As time evolves, firms adjust the commodity they produce. They perform this adjustment in a myopic way by maximizing the rate of change in profit subject to a quadratic cost of adjustment. This has them move in the direction of higher profit at a velocity which is proportional to the rate of change of profit and defines a vector field. The specification of the differentiated product and the availiability of a continuum of firms provides a natural setting in which one can make precise the idea that firms are myopic. As firms move, the density of firms changes and prices are assumed to adjust continually so that at each instant the economy is in short run equilibrium. As prices change, the gradient of profit changes, which leads to further changes in the distribution of firms, and so on.