Optimal Control and Time Consistent Equilibria in Dominant Firm/ Competitive Fringe Dynamic Games

Abstract

We analyze an exhaustible resource market dynamic game between a dominant firm and a price-taking competitive fringe. Using discrete and continuous time frameworks, we examine two time consistent equilibria: the feedback Stackelberg equilibrium (FSE), which assumes the dominant firm commits its plan for one period, and the closed- loop Nash equilibrium (CLNE), which assumes no commitment for either agent. We show that: 1) the continuous time version of the CLNE and that of the FSE are not equivalent, 2) if the fringe reserves are inexhaustible, the FSE coincides with the open-loop Stackelberg equilibrium (OLSE), and 3) if the cartel reserves are inexhaustible the CLNE coincides with the open-loop Nash equilibrium (OLNE). We use optimal control theory to find explicitly the CLNE and the FSE in two continuous time examples.