ECO 504 Spring 2006 Chris Sims

Lagrange multiplier methods are standard fare in elementary calculus courses, and they play a central role in economic applications of calculus because they often turn out to have interpretations as prices or shadow prices. You have seen them generalized to cover dynamic, non-stochastic models as Hamiltonian methods, or as byproducts of using Pontryagin’s maximum principle. In static models Lagrangian methods reduce a constrained maximization problem to an equation-solving problem. In dynamic models they result in an ordinary differential equation problem. In the stochastic models we are about to consider they result in, for discrete time, an integral equation problem or, in continuous time, a partial differential equation problem. Integral equations and partial differential equations are harder to solve than ordinary equations or differential equations — they are both less likely to have an analytical solution and more difficult to handle numerically. The application of Lagrangian methods to stochastic dynamic models therefore appears to be of less help in solving the optimization problem than is their application to non-stochastic problems. Consequently many references on dynamic stochastic optimization give little attention to Lagrange multipliers, instead emphasizing more direct methods for obtaining solutions. The economic literature has to some extent been guided by this pattern of emphasis. This is unfortunate, because Lagrangian methods are as helpful in economic interpretation of models in stochastic as in non-stochastic models. Also, in general equilibrium models, use of Lagrangian methods turns out sometimes to simplify the computational problem, in comparison to approaches that try to solve by more direct methods all the separate optimizations embedded in the general equilibrium.