Multi-product Supply Function Equilibria

ABSTRACT: We solve for Nash equilibria in a procurement auction with two heterogeneous divisible goods and a uniform price for each good. There are (dis)economies of scope in production and goods could be substitutes or complements for the procurer. Before demand is realized, each firm offers a vector of supply functions where supply of a good depends on the prices of both goods, as in the product-mix auction and some electricity markets. For quadratic costs and linear demand, we identify examples with loss leader pricing and we prove that it is welfare enhancing to separate arbitrage technologies, such as storage and transportation, from oligopoly producers. We consider linear pricing, without price discrimination, and show that outcomes are not influenced by bundling of the goods. This can be used to simplify the analysis and the operation of a multi-product auction.