On the Existence of Bertrand-Nash Equilibrium Prices Under Logit Demand

This article proves the existence of equilibrium prices in Bertrand competition with multi-product firms using the Logit model of demand. The most general proof, an application of the PoincareHopf Theorem, does not rely on restrictive assumptions such as singleproduct firms, firm homogeneity or symmetry, homogeneous product costs, or even concavity of the utility function with respect to prices. This proof relies on new conditions for the indirect utility function, along with fixed-point equations derived from the first-order conditions and a direct analysis of the second-order conditions that proves the uniqueness of profit-maximizing prices. The degree to which our conditions are as weak as possible is discussed. Models with finite purchasing power and convex total costs are also addressed. Analysis of equilibrium prices for multi-product firms with constant unit costs suggests that BertrandNash equilibrium cannot adequately describe multi-product pricing in differentiated product markets.