Comparative Dynamics and Envelope Theorems of Open-Loop Stackelberg Equilibria in Differential Games

Abstract

A direct proof of the envelope theorems and intrinsic comparative dynamics of locally differentiable open-loop Stackelberg equilibria (OLSE) is given using an extended primal-dual method. It is shown that the follower’s envelope and comparative dynamics results agree in form with those of any player in an open-loop Nash equilibrium, while those of the leader differ. This difference allows, in principle, an empirical test of the leader–follower role in a differential game. Separability conditions are identified on the instantaneous payoff and transition functions under which, for certain parameters, the intrinsic comparative dynamics of the leader’s time-inconsistent OSLE and those in the corresponding optimal control problem are qualitatively identical. However, similar conditions do not exist for time-consistent OSLE.