Optimal Procurement Auctions of Divisible Goods with Capacitated Suppliers

The literature on procurement auctions (reverse auctions) typically assumes that the suppliers are uncapacitated (see, e.g. Dasgupta and Spulber, 1990; Ankolekar et al., 2005; Chen, 2004; Che, 1993). Consequently, these auction mechanisms award the contract to a single supplier. We consider a model where suppliers have limited production capacity, and both marginal costs and the production capacities are private information. We construct the optimal direct mechanism that maximizes the retailer’s expected profit. We provide a closed-form solution when the distribution of the cost and production capacities satisfies a modified regularity condition (Myerson, 1981). We also present a sealed low bid implementation of the optimal direct mechanism for the special case of identical suppliers, i.e. symmetric environment. This implementation requires each supplier to submit a bid consisting of the desired marginal payment and total available production capacity. These bids serve as the input to a simple optimization problem that computes the quantity allocation for each firm. We extend the model to multi-product procurement with complementarities. The results in this paper are applicable to a number of principle-agent mechanism design problems where the agents have privately known upper bound on the allocations. Examples of such problems include monopoly pricing with adverse selection and forward auctions.