Comparative Statics and Perfect Foresight in Infinite Horizon Economies

Abstract

Does a pure exchange economy with an infinite time horizon have determinate perfect foresight equilibria? When there is a finite number of infinitely lived agents equilibria are generically determinate. This is not true with overlapping generations of finitely lived agents. We ask whether the initial conditions together with the requirement of convergence to a steady state locally determine an equilibrium price path. In this framework there are many economies with isolated equilibria, many with continua of equilibria, and many with no equilibria at all. With two or more goods in every period not only can the price level be indeterminate but relative prices as well. Furthermore, such indeterminacy can occur whether or not there is fiat money and whether or not the equilibria are Pareto efficient. THIS PAPER CONSIDERS whether infinite horizon economies have determinate perfect foresight equilibria. This question is of crucial importance. If instead equilibria are locally indeterminate, not only are we unable to make comparative static predictions, but the agents in the model are unable to determine the consequences of unanticipated shocks. The idea underlying perfect foresight is that agents' expectations should be the actual future sequence predicted by the model; if the model does not make determinate predictions, the concept of perfect foresight is meaningless. We consider two extreme cases: the first with a finite number of infinitely lived consumers and the second with an infinite number of finitely lived consumers, an overlapping generations model. Both are models of stationary pure exchange economies. No production, including the storage of goods between periods, can occur. These models are unrealistic but are the easiest to study. Extensions of the results of this paper to models with production, infinitely lived assets, and mixtures of the two types of consumers are presented by Muller and Woodford [29]. When there is a finite number of infinitely lived consumers, we argue that equilibria are generically determinate. This is because the effective number of equations determining equilibria is not infinite, but equal to the number of agents minus one and must determine the marginal utility of income for all but one agent. Generically, near an equilibrium, these equations are independent and exactly determine the unknowns. When there are infinitely many overlapping generations, this reasoning breaks down: An infinite number of equations is not necessarily sufficient to determine