The Generalized Euler Equation and the Unilateral Default Problem

We show how to characterize Markov equilibria in a class of models of unilateral debt default by means of analytical functional equations. By treating equilibrium conditions as constraints in the decision problem of the agent,the equilibrium turns into a game between the borrower and its future selves. Given the time-inconsistent nature of the decision problem, the Generalized Euler Equation includes derivatives of decision rules as its arguments. The functional equations give insights into and explicitly identify the various economic relevant margins. The computational method relies on Hermitian splines to provide controlled accuracy. A comparison with the functional equations for the problem under commitment provides additional insights into the environments without commitment. We apply this approach to several problems from the recent literature on sovereign debt with incomplete markets.