Technical Note–Dynamic Duopolistic Competition with Sticky Prices

A paradoxical conclusion arises in a series of game-theoretic models: the limit equilibria retain frictional qualities even as frictions seemingly vanish. This originates in textbook models, such as the differential game by Fershtman and Kamien [Fershtman C, Kamien MI (1987) Dynamic duopolistic competition with sticky prices. Econometrica 55(5):1151–1164] on duopolistic competition with sticky prices. We show that this paradox is an artifact of the type of limit restricted by continuous-time modeling. Fershtman and Kamien find that the closed-loop equilibrium remains surprisingly distinct from the static Cournot equilibrium as price adjustment becomes infinitely fast. We formulate and solve a discrete-time analog that nests their continuous-time model. Contrary to their conclusion, we show that the frictionless closed-loop equilibrium converges to the static Cournot equilibrium. Price stickiness persists instantaneously in the continuous-time setting because this approach cannot control the extent of price adjustment per period. Because of this subtle limitation, limit results differ between continuous- and discrete-time formulations of the model.