How to do comparative dynamics on the back of an envelope for open‐loop Nash equilibria in differential game theory

Abstract

SummaryThe primal‐dual comparative statics method of Samuelson (1965) and Silberberg (1974) is extended to cover the class of non‐autonomous, finite horizon differential games in which a locally differentiable open‐loop Nash equilibrium exists. In doing so, not only is a one‐line proof of an envelope theorem provided but also the heretofore unknown intrinrsic comparative dynamics of open‐loop Nash equilibria are uncovered. The intrinsic comparative dynamics are shown to be contained in a symmetric and negative semidefinite matrix that is subject to constraint. The results are applied to a canonical differential game in capital theory, and the resulting comparative dynamics are given an economic interpretation. Copyright © 2016 John Wiley & Sons, Ltd.