# Capacity Games with Supply Function Competition

## TL;DR

This paper analyzes capacity games with supply function competition, where suppliers with complex costs compete for a buyer with uncertain demand, revealing conditions for optimal reservation and profit distribution in equilibrium.

## Contribution

It introduces a supply function competition framework for capacity games with non-constant marginal costs and characterizes equilibrium properties under submodularity conditions.

## Key findings

- Buyer’s reservation maximizes overall supply chain profit in equilibrium.
- Suppliers earn profits equal to their marginal contribution to the supply chain.
- Submodularity holds for two suppliers and certain cost structures with more than two suppliers.

## Abstract

This paper studies a setting in which multiple suppliers compete for a buyer's procurement business. The buyer faces uncertain demand and there is a requirement to reserve capacity in advance of knowing the demand. Each supplier has costs that are two dimensional, with some costs incurred before demand is realized in order to reserve capacity and some costs incurred after demand is realized at the time of delivery. A distinctive feature of our model is that the marginal costs may not be constants, and this naturally leads us to a supply function competition framework in which each supplier offers a schedule of prices and quantities. We treat this problem as an example of a general class of capacity games and show that, when the optimal supply chain profit is submodular, in equilibrium the buyer makes a reservation choice that maximizes the overall supply chain profit, each supplier makes a profit equal to their marginal contribution to the supply chain, and the buyer takes the remaining profit. We further prove that this submodularity property holds under two commonly studied settings: (1) there are only two suppliers; and (2) in the case of more than two suppliers, the marginal two-dimension costs of each supplier are non-decreasing and constant, respectively.