Price-increasing competition.

In a discrete choice model of product differentiation, the symmetric duopoly price may be lower than, equal to, or higher than the single-product monopoly price. Whereas the market share effect encourages a duopolist to charge less than the monopoly price because a duopolist serves fewer consumers, the price sensitivity effect motivates a higher price when more consumer choice steepens the firm's demand curve. The joint distribution of consumer values for the two conceivable products determines the relative strength of these effects. The analysis provides precise conditions for price-increasing competition and reveals that it is unexceptional from a theoretical perspective.

1. Introduction

* Consider a market where initially a monopolist produces a single product. Suppose that a competitor enters the market with a differentiated product. Will competition from this second firm lead to lower or higher price(s) in this market? Although the standard insight of economics is that competition will lower prices, some recent empirical studies suggest otherwise. For example, Perloff, Suslow, and Seguin (2005) conclude that new entry raised prices in the anti-ulcer drug market J; Ward et al. (2002) provide evidence that new entry of private labels raised prices of national brands in the food industry; and Thomadsen (2007), simulating with estimated parameters from the fast food industry, finds that prices may be higher under duopoly competition than under monopoly. In this article, we shall explain why, from a theoretical perspective, there are plausible reasons that market price becomes higher in the presence of an alternative brand from a competitor; and, we shall provide precise conditions under which competition from a symmetric differentiated rival increases or reduces price.

We study a discrete choice model of product differentiation in which consumers' values for two substitute products have an arbitrary symmetric joint distribution. Each firm produces a single product, and the market structure is either monopoly or duopoly. We characterize under fairly weak assumptions a necessary and sufficient condition for the symmetric duopoly price to be higher than, equal to, or lower than the monopoly price. This condition balances two economic effects. At the monopoly price, a duopoly firm sells to fewer consumers than the monopolist. The larger this difference, the greater the incentive of a duopolist to reduce price below the monopoly level. We call this the market share effect. On the other hand, under product differentiation, a duopoly firm's demand curve may be steeper than the monopolist's, because consumers have a choice of products in the duopoly case and, therefore, are less keen to purchase the duopolist's product in response to a price cut. The steeper the duopolist's demand curve, relative to the monopolist's, the greater the incentive of the duopolist to raise price above the monopoly level. We call this the price sensitivity effect. When the second effect dominates, as, for example, if consumer values for the two products are drawn from a (Gumbel) bivariate exponential distribution, duopoly competition increases price compared to monopoly.

From the general necessary and sufficient condition, we derive some particular conditions under which price is higher under symmetric single-product duopoly than under single-product monopoly. If consumer values for the two products are independent, then a decreasing hazard rate is sufficient for a higher duopoly price. (2) In the more general case, a sufficient condition for price-increasing competition is that the conditional hazard rate is decreasing over a relevant range. The necessary and sufficient condition, however, also implies that price-increasing competition can occur when the hazard rate is nonmonotonic.

We also consider the implications of the dependence properties of consumer preferences when the marginal distribution of consumer values for a product is exponential. In the independent exponential case the monopoly price is the same as the symmetric duopoly price. Against this benchmark, and using copulas to characterize symmetric bivariate distribution functions, we show that the duopoly price is higher (lower) than the monopoly price if consumers' value for one product is stochastically decreasing (increasing) in their value for the other product. (3) The utility of the copula approach for studying product differentiation is that it allows us to vary the dependence relationship of two random variables while holding the marginal distributions constant. (4)

We further analyze a class of special cases in which the calculation of monopoly and duopoly prices is straightforward. In these special cases, consumer preferences for two products have a joint uniform distribution on a varying support allowing different degrees of negative or positive correlation. This analysis includes a treatment of two familiar models in oligopoly analysis. The Hotelling duopoly model (Hotelling, 1929) is a limiting case in which the preferences are perfectly negatively correlated, and the Bertrand duopoly model is a limiting case when the preferences are perfectly positively correlated. Duopoly competition raises price if consumers' preferences for the two products are sufficiently diverse and negatively correlated, as for instance in the Hotelling model when the market is fully served under duopoly but not under monopoly. Furthermore, the adverse price effect can be strong enough that aggregate consumer welfare goes down even though consumers are better served and the market expands under duopoly.

More competition can mean different things. Our main results establish conditions under which the addition of a second single-product firm raises equilibrium price. It is also true, however, that the symmetric multi-product monopoly price exceeds the duopoly price. The consolidation of two single-product firms into a multi-product monopoly raises price for the usual reason that the monopolist internalizes the profit externality. In this sense, less competition necessarily results in higher prices. Together, these results demonstrate that more product variety, whether from a multi-product monopolist or from a new entrant, can result in higher prices.

The standard view of the relationship between market structure and price has been challenged previously by several other theoretical studies. For instance, when consumers must search to find firms' prices, the presence of more firms makes it more difficult to find the lowest price in the market, reducing consumers' incentives to search. This can cause the equilibrium market price to increase with the number of firms (Satterthwaite, 1979; Stiglitz, 1987; Schulz and Stahl, 1996; Janssen and Moraga-Gonzalez, 2004). (5) An alternative approach assumes that each seller faces two groups of consumers, a captured loyal group and a switching group. With more sellers, each seller's share of the switching group is reduced, increasing its incentive to exploit the captured consumers through a higher price; but equilibrium prices under competition are in mixed strategies (Rosenthal, 1980). In contrast, in our analysis here, consumers have perfect information, and firms' prices are in pure strategies. Whereas Chen and Riordan (2007) and Perloff, Suslow, and Seguin (2005) have also shown that competition can increase price under perfect information and pure strategies, these papers rely on specific spatial models for which consumer valuations for two products are perfectly negatively correlated. (6) Our present analysis goes further by developing conditions for the symmetric duopoly price to exceed the monopoly price from a much broader perspective.

In the context of a symmetric oligopoly model with independently and identically distributed consumer preferences for alternative goods, Perloff and Salop (1985) recognize that the effect of more competitors on the symmetric equilibrium price is ambiguous, and Gabaix, Laibson, and Li (2005) study this relationship for particular distributions. Our analysis differs from Gabaix, Laibson, and Li (2005) in other ways beside relaxing the independence assumption. First, their main result approximates the equilibrium markup for a general class of preference distributions as a function of the number of firms. The approximation is valid if the number of firms is sufficiently large. Our analysis of monopoly and duopoly focuses on more concentrated markets. Second, following Perloff and Salop (1985), Gabaix, Laibson, and Li (2005) simplify by assuming consumers always purchase one of the available products; thus, firms compete only to steal market share from each other. In contrast, our model allows consumers the option to purchase none of the products. The no-purchase option clearly is necessary for a sensible study of monopoly, but also provides a second demand margin for duopoly and thus plays a key role in our comparison of the two market structures. Finally, we develop for the independence case a new necessary and sufficient condition for price to be higher under duopoly than monopoly based on the hazard rate of the preference distribution and, furthermore, characterize new conditions for price-increasing competition when preferences are dependent.

The rest of the article is organized as follows. Section 2 formulates and analyzes the general model, and compares monopoly and duopoly prices under different assumptions about the dependence properties of consumer preferences. Section 3 analyzes uniform distribution of preferences on a varying support, thus generalizing limiting cases of Hotelling duopoly and Bertrand duopoly. Section 4 shows how competition affects consumer welfare through a price effect and a variety effect, and how the balance can be either positive or negative. Section 5 compares single-product duopoly competition with multi-product monopoly. Section 6 draws conclusions.

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