  This article presents a proof of the existence of Bertrand-Nash equilibrium
prices with multi-product firms and under the Logit model of demand that does
not rely on restrictive assumptions on product characteristics, firm
homogeneity or symmetry, product costs, or linearity of the utility function.
The proof is based on conditions for the indirect utility function, fixed-point
equations derived from the first-order conditions, and a direct analysis of the
second-order conditions resulting in the uniqueness of profit-maximizing
prices. Several subsequent results also demonstrate that price equilibrium
under the Logit model of demand cannot adequately describe multi-product
pricing.
