Euler equations ∗

An Euler equation is a difference or differential equation that is an intertemporal first-order condition for a dynamic choice problem. It describes the evolution of economic variables along an optimal path. It is a necessary but not sufficient condition for a candidate optimal path, and so is useful for partially characterizing the theoretical implications of a range of models for dynamic behavior. In models with uncertainty, expectational Euler equations are conditions on moments, and thus directly provide a basis for testing models and estimating model parameters using observed dynamic behavior. An Euler equation is an intertemporal version of a first-order condition characterizing an optimal choice as equating (expected) marginal costs and marginal benefits. Many economic problems are dynamic optimization problems in which choices are linked over time, as for example a firm choosing investment over time subject to a convex cost of adjusting its capital stock, or a government deciding tax rates over time subject to an intertemporal budget constraint. Whatever solution approach one employs — the calculus of variations, optimal control theory or dynamic programming — part of the solution is typically an Euler equation stating that the optimal plan has the property that any marginal, temporary and feasible change in behavior has marginal benefits equal to marginal costs in the present and future. Assuming the original problem satisfies certain regularity conditions, the Euler equation is a necessary but not sufficient condition for an optimum. This differential or difference equation is a law of motion for the economic variables of the model, and as such is useful for (partially) characterizing the theoretical implications of the model for optimal dynamic behavior. Further, in a model with ∗Prepared for the New Palgrave Dictionary of Economics. For helpful comments, I thank Esteban Rossi-Hansberg, Per Krusell, and Chris Sims. First draft June 2006. †Department of Economics, Bendheim Center for Finance, and Woodrow Wilson School, Princeton University, Princeton, NJ 08544-1013, e-mail: [email protected], http://www.princeton.edu/∼jparker