Equilibrium Existence and Approximation for Incomplete Market Models with Substantial Heterogeneity∗

This paper contains an analysis of incomplete market models with finitely but arbitrarily many heterogeneous agents. We discuss the mathematical foundation for equilibrium conditions which leads to two findings. First, we establish existence of equilibria for small and large risks. Second, we develop a simple but general solution technique which handles many state and choice variables for each agent and thus an extremely high-dimensional state space. The method is based on perturbations around a point at which the solution is known. The novel idea is to exploit the symmetry of the problem to overcome the curse of dimensionality. We use the analysis to demonstrate the impact of heterogeneity on macroeconomic quantities and the pricing of risk. Furthermore, we set our technique apart from the standard method used in the literature. ∗We thank Marios Angeletos, Dave Backus, Fernando Duarte, Darrell Duffie, Xavier Gabaix, John Geanakoplos, Sydney Ludvigson, Hanno Lustig, Semyon Malamud, Marti Subrahmanyam, Stijn van Nieuwerburgh, Kjetil Storesletten, and the audiences at the New York Fed, NYU Stern, Wharton, Duke, 2011 SCE conference, SITE 2011, the Chicago Booth-Deutsche Bank symposium, the Cowles GE conference, the SED, and the EFA for helpful comments. An earlier version of the paper was circulated under the title ”Solving General Incomplete Market Models with Substantial Heterogeneity” †New York University, Stern School of Business; Postal Address: 44 W Fourth Street, Suite 9-73, New York, NY 10012, USA; E-mail: [email protected]. ‡Hoover Institution, Stanford University; Postal Address: 434 Galvez Mall, Stanford, CA 94305, USA; E-mail: [email protected]