Online - Only Appendix for Daniel Quint , “ Imperfect Com - petition with Complements and Substitutes ”

Theorem 2 establishes that if the marginal cost of one of the components of product 2 increases, the equilibrium prices of every product rise. It is tempting to assume that this must be good news for the makers of components of the other products. For products with just one component, this is indeed the case. For example, if n1 = 1, the lone firm in T1 sees the price of every competing product go up; and if it chooses in equilibrium to raise its own price P1, this can only be to further increase profits. However, when n1 > 1, it may not follow that an increase in a competing product’s cost is good news. The increases in P2, . . . , PK certainly increase the profits of each firm in T1; but at least in principle, it is possible that these increases also change the shape of the residual demand curve Q1 in such a way that the double-marginalization problem among the firms in T1 gets more severe, leading to P1 increasing “too much” in equilibrium and actually lowering the profits of these firms. To see an example of how this could happen (though not within the model of this paper), consider the following. Suppose there are two products, zero marginal costs, and n1 = 9 and n2 = 1. There are two populations of consumers: a measure 1 of “type-1” consumers, with v1 ∼ U [0, 1] and v2 = 0; and a measure 1 10 of “type-2” consumers, with v1 ∼ U [1, 3] and v2 = 10. Suppose the price of product 2 increases from 5 to 10, and the firms in T1 jointly best-respond. I will show that this change leads to a decrease in profits for the firms in T1. When P2 = 5, all type-2 consumers buy product 2 regardless of P1; so the residual demand for product 1 is the demand from the type-1 consumers, which is