The goal of this appendix is to explore the simplest possible differentiated products environment where (pure) Nash-Bertrand equilibrium prices do not exist due to price cycles a la Edgeworth and to suggest an alternative equilibrium concept as better suited to analyzing such environments.

We develop and characterize a concept called an Undercut-Proof equilibrium. In an Undercut-Proof equilibrium, each firm chooses its price so as to maximize profit while ensuring that its price is sufficiently low that any rival firm would not find it profitable to set a lower price in order to grab all of the first firm's customers. Thus, unlike the Nash-Bertrand behavior, where each firm assumes that the rival firm does not alter its price, in an Undercut-Proof equilibrium environment, firms assume that rival firms are more sophisticated in that they are “ready” to reduce their prices whenever undercutting and grabbing their rivals’ customers is profitable. These beliefs are pervasive amongst firms competing in differentiated products using pricing strategies. Finally, the Undercut- Proof equilibrium can be calculated easily for any number of firms in the industry.

The Simplest Product Differentiation Model

Consider the following example (see Shilony 1977, Eaton and Engers 1990, and Shy 1996, Ch. 7), of a market with two stores called A and B which sell differentiated brands. Assume that production costs are zero.