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Infinite Horizon Incomplete Markets

Michael Magill
Martine Quinzii

Abstract

The model of general equilibrium with incomplete markets is a generalization of the Arrow-Debreu model which provides a rich framework for studying problems of macroeconomics. This paper shows how the model, which has so far been restricted to economies with a finite horizon, can be extended to the more natural setting of an open-ended future, thereby providing an extension of the finite horizon representative agent models of modern macroeconomics to economies with heterogeneous agents and incomplete markets. There are two natural concepts of equilibrium over an infinite horizon which prevent agents from entering into Ponzi schemes, that is, from indefinitely postponing the repayment of their debts. The first is based on debt constraints which place bounds on debt at each date-event; the second is based on transversality conditions which limit the asymptotic rate of growth of debt. The concept of an equilibrium with debt constraint is a natural concept of equilibrium for macroeconomic analysis; however the concept of an equilibrium with transversality condition is more amenable to theoretical analysis since it permits the powerful techniques of Arrow-Debreu theory to be carried over to the setting of incomplete markets. In an economy in which agents are impatient (expressed by the Mackey continuity of their preference orderings) and have a degree of impatience at each date-event which is bounded below (a concept defined in the paper), we show that the equilibria of an economy with transversality condition coincide with the equilibria with debt constraints. An equilibrium with transversality condition is shown to exist: it follows that for each economy there is an explicit bound $M$ such that an equilibrium with explicit debt constraint $M$ exists, in which the constraint is never binding--this latter property ensuring that the debt constraint, whose objective is to prevent Ponzi schemes, does not in itself introduce a new imperfection into the model over and above the incompleteness of the markets.