Non-Linear Pricing Competition with Private Capacity Information

We study a common agency model with informed principals consisting of two capacity-constrained suppliers and a single retailer. The capacity of each supplier is her private information. Conditioned on their capacities, the suppliers simultaneously and non-cooperatively offer quantity-price schedules to the retailer. Then, the retailer decides on the quantities to purchase from each supplier in order to maximize his own utility. We prove the existence of a (pure strategy) Nash equilibrium for this game. We show that at the equilibrium each (infinitesimal) unit of the supply is assigned a marginal price which is independent of the capacities and depends only on the valuation function of the retailer and the distribution of the capacities. In addition, the supplier with the larger capacity sells all her supply.