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: jparker@princeton.edu, http://www.princeton.edu/∼jparker 1 A version of this paper has been pulished as,“Euler Equations.” Parker, Jonathan A. In New Palgrave Dictionary of Economics, edited by Steven N. Durlauf and Lawrence E. Blume, 1851-1854. London, England: Palgrave MacMillan uncertainty, the expectational Euler equation directly provides moment conditions that can be used both to test these theoretical implications using observed dynamic behavior and to estimate the parameters of the model by choosing them so that these implications quantitatively match observed behavior as closely as possible. The term ‘Euler equation’ first appears in text-searchable JSTOR in Tintner (1937), but the equation to which the term refers is used earlier in economics, as for example (not by name) in the famous Ramsey (1928). The mathematics was developed by Bernoulli, Euler, Lagrange and others centuries ago jointly with the study of classical dynamics of physical objects; Euler wrote in the 1700’s ‘nothing at all takes place in the universe in which some rule of the maximum . . . does not appear’ (Weitzman (2003), p. 18). The application of this mathematics in dynamic economics, with its central focus on optimization and equilibrium, is almost as universal. As in physics, Euler equations in economics are derived from optimization and describe dynamics, but in economics, variables of interest are controlled by forward-looking agents, so that future contingencies typically have a central role in the equations and thus in the dynamics of these variables. For general, formal derivations of Euler equations, see texts or entries on the calculus of variations, optimal control theory or dynamic programming. This entry illustrates by means of example the derivation of a discrete-time Euler equation and its interpretation. The entry proceeds to discuss issues of existence, necessity, sufficiency, dynamics systems, binding constraints, and continuous-time. Finally, the entry discusses uncertainty and the natural estimation framework provided by the expectational Euler equation. The Euler equation: Consider an infinitely-lived agent choosing a control variable (c) in each period (t) to maximize an intertemporal objective: P∞ t=1 β t−1u (ct) where u (ct) represents the flow payoff in t, u0 > 0, u00 < 0, and β is the discount factor, 0 < β < 1. The agent faces a present-value budget constraint: