THE ENVELOPE THEOREM IN DYNAMIC OPTIMIZATION

The dynamic envelope theorem is presented for optimal control problems with nondifferential constraints. Some of these constraints may switch from binding to nonbinding, or vice versa, along the optimal path. Twice continuous differentiability of the optimal performance function and intertemporal symmetry and reciprocity conditions are shown to follow from the envelope theorem and twice continuous differentiability of the integrand, state equations, and constraints. Conditions implying convexity or concavity of the optimal performance function in the parameters are derived. Dynamic versions of Hotelling's Lemma, Roy's Identity, and the Slutsky equation are presented for an intertemporal consumption problem. © 1991.