Price Setting With Strategic Complementarities as a Mean Field Game

We study the propagation of monetary shocks in a sticky-price general equilibrium economy where the firms' pricing strategy features a complementarity with the decisions of other firms. In a dynamic equilibrium, the firm's price-setting decisions depend on aggregates, which in turn depend on the firms' decisions. We cast this fixed-point problem as a Mean Field Game and prove several analytic results. We establish existence and uniqueness of the equilibrium and characterize the impulse response function (IRF) of output following an aggregate shock. We prove that strategic complementarities make the IRF larger at each horizon. We establish that complementarities may give rise to an IRF with a hump-shaped profile. As the complementarity becomes large enough, the IRF diverges, and at a critical point there is no equilibrium. Finally, we show that the amplification effect of the strategic interactions is similar across models: the Calvo model and the Golosov-Lucas model display a comparable amplification, in spite of the fact that the non-neutrality in Calvo is much larger.