Pricing with markups in industries with increasing marginal costs

We study a game that models a market in which heterogeneous producers of perfect substitutes make pricing decisions in a first stage, followed by consumers that select a producer that sells at lowest price. As opposed to Cournot or Bertrand competition, producers select prices using a supply function that maps prices to production levels. Solutions of this type of models are normally referred to as supply function equilibria. We consider a market where producers’ convex costs functions are proportional to each other, depending on the efficiency of each particular producer. We provide necessary and sufficient conditions for the existence of an equilibrium that uses simple supply functions that replicate the cost structure. We then specialize the model to monomial cost functions with exponent q > 0, which allows us to reinterpret the simple supply functions as a markup applied to the production cost. We prove that an equilibrium for the markups exists if and only if the number of producers in the market is strictly larger than 1+ q, and if an equilibrium exists, it is unique. The main result for monomials is that the equilibrium nearly minimizes the total production cost when the market is competitive. The result holds because when there is enough competition, markups are bounded, thus preventing prices to be significantly distorted from costs. Focusing on the case of linear unit-cost functions on the production quantities, we characterize the equilibrium accurately and refine the previous result to establish an almost tight bound on the worst-case inefficiency of equilibria. Finally, we derive explicitly the producers’ best response for series-parallel networks with linear unit-cost functions, extending our previous result to more general topologies. We prove that a unique equilibrium exists if and only if the network that captures the market structure is 3edge-connected. For non-series-parallel markets, we provide an example that does not admit an equilibrium on markups.