This paper develops a stochastic general equilibrium model for an oligopoly, in which all inventory constraint parameters are endogenously determined. We propose several systems of demand processes whose distributions are dunctions of all retailers' prices and all retailers' service levels. We proceed with the investigation of the equilibrium behavior of infinite-horizon models for industries facing this type of generalized competition, under demand uncertainty.
We systematically consider the following three competition scenarios. (1) Price competition only: Here, we assume that the firms' service levels are exogenously chosen, but characterize how the price and inventory strategy equilibrium vary with the chosen service levels. (2) Simultaneous price and service-level competition: Here, each of the firms simultaneously chooses a service level and a combiend price and inventory strategy. (3) Two-stage competition: The firms make their competitive choices sequentially. In a first stage, all firms simultaneously choose a service level; in a second stage, the firms simultaneously choose a combined pricing and inventory strategy with full knowledge of the service levels selected by all competitors. We show that in all of the above settings a Nash equilibrium of infinite-horizon stationary strategies exists and that it is of a simple structure, provided a Nash equilibribum exists in a so-called reduced game.
We pay particular attention to the question of whether a firm can choose its service level on the basis of its own (input) characteristics (i.e., the cost parameters and demand function) only. We also investigate under which of the demand models a firm, under simultaneous competition, resonds to a change in the exogenously specified characteristics of the various competitors by either: (i) adjusting its service level and price in the same direction, thereby compensating for price increases (decreases) by offering improved (inferior) service, or (ii) adjusting them in opposite directions, thereby simultaneously offering better or worse prices and service.
Bernstein, Fernando, and Awi Federgruen. "A general equilibrium model for industries with price and service competition." Operations Research 52, no. 6 (2004): 868-886.