Product Differentiation in a Hotelling City with Elastic Demand* - PDF document

Product Differentiation in a Hotelling City with Elastic Demand* Matt Birch Graduate Student Washington State University Robert Rosenman Professor Washington State University Abstract In the many variations and expansions on Hotelling’s original work on product differentiation, the most common assumption is that demand is perfectly inelastic. A common finding for goods with multiple characteristics is that duopolies will maximally differentiate along one dimension and minimally differentiate along all others. We analyze a “Hotelling square” (2-dimensional product characteristics) with elastic demand. We show that with elastic demand there will never be maximal or minimal differentiation. Partial differentiation will exist in at least one dimension, and possibly in both dimensions. We show the conditions under which different degrees and dimensionalities of differentiation occur. * This paper is very preliminary and incomplete. It is not for attribution or quotation. Most of the proofs are only sketched or outlined. JEL codes: L1, M3, R3 1

Birch and Rosenman – preliminary, not for attribution 1. Introduction For the better part of a century, spatial arguments have been utilized to explain product differentiation as an equilibrium behavior. Depending on the assumptions made, results have differed widely, including everything from minimal differentiation in all dimensions to maximal differentiation, partial differentiations and every other combination. To become acquainted with the literature stemming from Hotelling’s 1929 paper, “Stability in Competition,” is to read through a labyrinth of arguments concerning the structure of product differentiation. Hotelling concluded that firms competing in prices and products will offer the roughly the same product by locating as close to each other as possible (Hotelling, 1929), which is generally referred to as minimal differentiation. In 1979, Hotelling’s analysis was show n to be unsound because no equilibrium actually existed when firms were too close together (d'Aspremont, Gabszewiez, & Thisse, 1979). When the transportation costs are quadratic, rather than linear, maximal differentiation is the equilibrium outcome (d'Aspremont, Gabszewiez, & Thisse, 1979). 1 Subsequent papers has explored product differentiation along multiple dimensions. In a three- dimension Hotelling “cube,” for example, 2 firms competed with max -min-min differentiation in which firms differentiated only in the most salient dimension (Ansari, Economides, & Steckel, 1998). Irmen and Thisse extended this analysis to an n-dimensional hypercube. Their robust, but equivalent result is that firms engaged in max-min- … -min differentiation (Irmen & Thisse, 1998), from which they claimed that Hotelling was “almost” right. The hypercube concept was later expanded to fit into an evolutionary framework in which both evolutionary stability and stochastic stability suggested that differentiation was minimal on all dimensions (Hehenkamp & Wambach, 2010). 1 Economides (1986) analyzed the convexity of the transaction costs and found that while minimum differentiation never happened, maximum differentiation could occur if costs were convex enough, but otherwise would not. 2

Birch and Rosenman – preliminary, not for attribution Each of these studies relies on the assumptions that there are two firms, consumers have perfectly inelastic demand uniformly distributed across the market space, and consumers benefit from buying a good at any location differs only by the individual distance from the location. Different studies have relaxed these assumptions and gotten starkly different results. If consumers prefer quality as one of the product characteristics in a two-dimensional market, but quality is costly, there is max-min differentiation when quality costs are low or max-max differentiation when quality costs are high (Lauga & Ofek, 2011). If consumers are distributed non-uniformly, there can be partial differentiation in one dimension instead of max-min-..-min (Liu & Shuai, 2012). When there are more than two firms in the market, the result of maximal differentiation in one dimension and minimal in all others breaks down (Tabuchi, 2012). In a three firm, cube market, max-min-min does not hold. Instead there is partial differentiation along two dimensions and minimal along the third (Feldin, 2012). Our work is most closely related to Economides (1984) in which the assumption of perfectly inelastic demand was r emoved. His model has two firms on a “line,” and when demand is low enough, local monopolies form and Nash equilibrium exists for a larger set of locations than when demand is perfectly inelastic. Our analysis expands on the existing literature by incorporating both elastic demand and multiple dimensions for product charateristics. In our model two firms compete in a Hotelling “square” with elastic demand. When demand is elastic, we surmise that there will never be maximal or minimal differentiation in either dimension and that there is partial differentiation in at least one and possibly both dimensions. We analyze the conditions on demand that lead to these different outcomes. When S is sufficiently low, differentiation can occur on either or both dimensions. When S is higher, there will be differentiation in the dominant dimension and may be differentiation on the other dimension. When S is sufficiently high, there will be differentiation on both dimensions, with a higher degree of differentiation on the dominant dimension. 3

Birch and Rosenman – preliminary, not for attribution The rest of this paper is as follows. Section 2 introduces the geometry and the market framework. Section 3 sets up the firm maximization problem. Section 4 discusses equilibrium conditions. Section 5 concludes the paper. 2. The Model There are two profit-maximizing firms, referred to as A and B, competing on a Hotelling square. We assume that each firm faces a constant marginal cost, which we normalize to zero. 2 Firms simultaneously choose location and prices to maximize profits. Our equilibrium concept is Nash equilibrium. Consumers are distributed 𝑕(𝑦) , which we assume to be uniformly distributed over the interval [0,1] × [0,1] , and the population is normalized to 1. Consumer x located at (𝑦 1 , 𝑦 2 ) who buys from firm A located at (𝑏 1 , 𝑏 2 ) at a price 𝑞 𝐵 receives utility according to equation 1. 𝐵 = 𝑇 − 𝑢 1 (𝑦 1 − 𝑏 1 ) 2 − 𝑢 2 (𝑦 2 − 𝑏 2 ) 2 − 𝑞 𝐵 𝑉 (1) An equivalent statement holds for buying from firm B at (𝑐 1 , 𝑐 2 ) . The terms 𝑢 1 and 𝑢 2 are salience coefficients, which permit heterogeneity in mismatch costs between the two attributes. Intuitively, this allows for the different attributes to be weighted differently by the consumer. For simplicity and without loss of generality we assume that 𝑢 2 ≥ 𝑢 1 . Demand at firm A, which we denote as 𝐸 𝐵 is given by equation 2. 𝐸 𝐵 = ∫ 𝑕(𝑦)𝑒𝑦 (2) 𝑉 𝐵 ≥𝑉 𝐶 ,0 The marginal consumer who buys from firms A must have a reservation utility of zero. Setting equation 1 equal to zero gives 𝑢 1 (𝑦 1 − 𝑏 1 ) 2 + 𝑢 2 (𝑦 2 − 𝑏 2 ) 2 = 𝑇 − 𝑞 𝐵 , which defines an ellipse 3 2 Assuming zero cost helps us to focus on the competition strategies pertaining to location and price when demand is elastic, which has not been done in a multi-dimensional setting. It is a usual and convenient assumption in models of this sort. With this assumption, the firm ’ s profit is simply its demand. 3 If 𝑢 1 = 𝑢 2 then demand forms a circle rather than an ellipse. 4

Birch and Rosenman – preliminary, not for attribution centered at (𝑏 1 , 𝑏 2 ) . This is the set of consumers who will buy from firm A, assuming that it is sufficiently far from firm B and from the boundaries of the square, exists within the ellipse, as shown in Figure 1. At a given price, consumers within the ellipse will purchase the good and outside consumers will not. Figure 1: Depiction of an unconstrained demand ellipse and a constrained demand ellipse. Consumers on the boundary have zero surplus and are indifferent between buying and not buying. These marginal consumers are responsive to price changes, so the size of the ellipse is a function of price. The two most salient radii of the ellipse are the semi-major axis and the semi-minor axis, which are depicted in Figure 2 as 𝑠 1 and 𝑠 2 , respectively. These axes point us to the consumers who have the largest mismatch in either dimension. 5