On the insufficiency of some conditions for minimal product differentiation
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On the insufficiency of some conditions for minimal product differentiation Kali P. Rath1 · Gongyun Zhao2 Received: 2 August 2019 / Accepted: 31 July 2020 © Society for the Advancement of Economic Theory 2020
Abstract The basic framework is Hotelling’s model of product choice with quadratic transportation cost. Duopolists choose locations in the initial period and compete in prices in subsequent infinite periods. The firms share profits on the profit possibility frontier. Friedman and Thisse (Rand J Econ 24:631–645, 1993) provides a set of sufficient conditions for a unique equilibrium and minimal product differentiation in this setting. This paper reexamines those conditions. In the presence of some mild continuity requirements, there is exactly one profit sharing rule which satisfies those conditions. Furthermore, given any discount factor(s), the corresponding profits cannot be the outcome of a subgame perfect Nash equilibrium at every pair of locations. This brings out an inconsistency in the conditions. A slight weakening of the conditions, to allow for a wider class of profit sharing rules, can result in multiple equilibria and minimal product differentiation need not obtain. Two examples demonstrate this. Thus, neither those conditions nor their weaker variants can be used to characterize a unique equilibrium. Keywords Minimal product differentiation · Profit possibility frontier · Profit sharing rule · Subgame perfect Nash equilibrium · Unique equilibrium JEL Classification C72 · C73
1 Introduction The basic framework is Hotelling’s (1929) model of spatial competition with quadratic transportation cost, as discussed in d’Aspremont et al. (1979), and extensively studied
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Kali P. Rath [email protected] Gongyun Zhao [email protected]
1
Department of Economics, University of Notre Dame, Notre Dame, IN 46556, USA
2
Department of Mathematics, National University of Singapore, Singapore 117543, Singapore
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K. P. Rath, G. Zhao
ever since. The consumers are uniformly distributed over the unit interval [0, 1]. Two producers are located in this interval at x1 and x2 respectively, x1 ≤ x2 . The production costs are zero. Each consumer buys a unit of the product from the producer with the lowest delivered price (price plus the transportation cost). In a one-shot, two stage game, where the duopolists choose locations first and prices next, the equilibrium locations are at the extreme endpoints of the market segment. Crawford (2012) contains a survey of the endogenous product choice literature. Now suppose that the time periods are given by 0, 1, 2, . . . The firms choose locations in the initial period and prices in subsequent infinite periods. Future payoffs are discounted. Given a pair of locations, the profit possibility frontier (PPF) is the set of Pareto optimal profit allocations between the two firms. Friedman and Thisse (1993) examines the choice of locations in this setting under “partial collusion,” the locations being chosen noncollusively and prices are such that the firms remain on the PPF. If the
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