The Generalized Euler Equation and the Bankruptcy-Sovereign Default Problem

We characterize the equilibrium of the standard sovereign default model with long-term, non-contingent debt. We show existence of the Markov equilibrium and uniqueness of equilibria that are the limit of finite economies. In general, the price and policy functions exhibit jumps and kinks; a suitable choice of arbitrarily small noise yields price and policy functions that are differentiable everywhere, which allows us to characterize the equilibrium using only the agents’ decision rules by means of a set of functional equations. We further describe the equilibrium objects via an Euler equation with derivatives on future actions—a Generalized Euler Equation (GEE) that disentangles the effects of default from those of dilution. The GEE yields computational strategies that search for continuous policy functions. A sufficient scale of the noise ensures concavity and a unique solution of the GEE. Applied to a calibrated model following Chatterjee and Eyigungor (2012), the GEE combined with the endogenous grid method delivers residuals orders of magnitude smaller than standard value function iteration, at roughly an order of magnitude lower computational cost.