Comparative Dynamics via Envelope Methods in Variational Calculus

Abstract

The primal-dual methodology of Samuelson (1965) is extended and applied to a nonautonomous variational calculus problem with a fixed vector of initial stocks, fixed initial and terminal time values, a free vector of terminal stocks, and a time-independent vector of parameters. It is shown that if the solution of the variational problem is smooth enough, the qualitative effects of parameter perturbations on the entire optimal arcs can be represented by a generalized Slutsky-type matrix, which holds in integral form and is symmetric negative semidefinite. Sufficient conditions for the optimal value function to be convex in the parameters are also given.