تبعیض قیمت چند بعدی

We examine the profitability and welfare implications of price discrimination in a multi-dimensional model. First, when firms price discriminate on one and the same dimension, uniform price lies in between discriminatory prices and price discrimination raises profits relative to uniform pricing. This is in contrast to common findings in existing one-dimensional models featuring best-response asymmetry, suggesting that price discrimination can have qualitatively different implications in one- and multi-dimensional models. Second, price discrimination on one and the same dimension is the likely outcome when price discrimination decisions are endogenized using a two-stage discrimination-then-pricing game. Correspondingly, an observation of one-dimensional price discrimination in practice does not necessarily indicate that the underlying model should be one-dimensional.

Consumer characteristics usually are multi-dimensional (e.g., location, gender, age, income and student status). Correspondingly, if endowed with information on such consumer characteristics, a firm may be able to price discriminate on multi-dimensions.1 In this paper, we examine the welfare implications of third-degree price discrimination in a multi-dimensional model. We ask two questions: First, how should price discrimination be conducted? Second, what are its welfare implications? A relatively large literature has answered these questions in one-dimensional settings where consumer heterogeneity occurs on a single dimension along which firms can price discriminate (see Armstrong, 2006 and Stole, 2007 for surveys of this literature). One strand of this literature assumes best-response symmetry (e.g., Holmes, 1989) and common findings are that uniform price lies in between discriminatory prices and price discrimination may raise or lower profits.2 Another strand assumes best-response asymmetry and usually finds that price discrimination intensifies competition, benefiting consumers at the cost of firms (e.g., Bester and Petrakis, 1996, Chen, 1997, Shaffer and Zhang, 1995 and Thisse and Vives, 1988).3 This paper extends the existing analysis to a multi-dimensional setting and several new questions emerge. Would firms have an incentive to price discriminate on some dimensions but not others? And if they do, would they price discriminate on the same or different dimensions? These questions do not fit existing studies, because their underlying one-dimensional models do not give firms the option of price discriminating on some dimensions but not others.4 Moreover, as we will show later, even when product differentiation occurs on multi-dimensions, firms may still choose to price discriminate on only one dimension. Correspondingly, an observation of one-dimensional price discrimination is not necessarily a confirmation that the underlying model is one-dimensional. To obtain concrete results, we employ a model commonly used in the existing literature featuring best-response asymmetry (e.g., Liu and Serfes, 2004, Shaffer and Zhang, 1995 and Thisse and Vives, 1988). In particular, we modify the standard one-dimensional Hotelling model to two-dimensions in our main analysis and to general n-dimensions in the extensions. Similar to much of the literature, we assume that price discrimination along a given dimension takes the form of two-group price discrimination. 5 Our results are qualitatively different from those in one-dimensional models. For example, we find that when firms price discriminate on one and the same dimension, profits go up and uniform price lies in between the discriminatory prices. This is in sharp contrast to the common findings in one-dimensional models featuring best-response asymmetry. On the other hand, price discrimination on one but different dimensions and price discrimination on both dimensions intensify competition and reduce profits, similar to the results in one-dimensional models. While these exact results are specific to our model, their general message is clear: the welfare implications of price discrimination can be qualitatively different in one- and multi-dimensional models. Correspondingly, results based on a one-dimensional model can be inaccurate when the underlying model is multi-dimensional. Our study takes a step in examining this issue and calls for more research. We identify two effects of price discrimination in our multi-dimensional model. First, the well-understood intensified competition effect exists in both one- and multi-dimensional models. That is, the ability to price discriminate enables a firm to be more aggressive in its weak market. However, when both firms do so, they force their rivals to be more aggressive in their strong markets as well, leading to all-out competition (all discriminatory prices are below the uniform price). Second, price discrimination has a reduced demand elasticity effect which exists in multi-dimensional models but not in one-dimensional models. This effect reduces competition and raises prices. Our results suggest that the reduced demand elasticity effect dominates the intensified competition effect when firms price discriminate on one and the same dimension, but the results are reversed when firms price discriminate on one but different dimensions or when they price discriminate on both dimensions. We then endogenize firms' decisions on price discrimination using the framework that firms acquire information about consumers which enables them to price discriminate. Our results show that price discrimination on one and the same dimension can emerge as a SPNE if consumer information is not too costly to acquire. Price discrimination on both dimensions can also be supported as an SPNE. However, it is dominated by price discrimination on one and the same dimension from firms' perspective. Taken together, they suggest that if firms price discriminate in equilibrium, the likely outcome is price discrimination on one and the same dimension which lowers consumer surplus. This is contrary to the results in existing studies where price discrimination is characterized as a prisoners' dilemma game. Moreover, it suggests that an observation of one-dimensional price discrimination in practice does not necessarily imply that the underlying model is one-dimensional.

We examine the issue of price discrimination in a multi-dimensional model. Firms have the option of price discriminating on some dimensions but not others. We find that price discrimination on one and the same dimension raises prices in firms' strong markets but lowers prices in their weak markets, leading to higher overall profits relative to uniform pricing. These results are contrary to predictions from one-dimensional models. On the other hand, price discrimination on one but different dimensions and price discrimination on both dimensions lead to lower prices on average and lower profits, similar to the results in existing literature. We then endogenize price discrimination decisions and find that the likely outcome is price discrimination on one and the same dimension. Correspondingly, observed one-dimensional price discrimination in practice does not necessarily mean that the underlying model should be one-dimensional. Our results have clear managerial implications regarding pricing strategies in multi-dimensional settings. Relative to one-dimensional settings, firms may also have more incentives to acquire consumer information which facilitates price discrimination. Our analysis can be extended in several directions. We have assumed that consumer distributions on different dimensions are independent. What if they are dependent? The framework in Chen and Riordan, 2010 and Chen and Riordan, 2013 allows for such an extension. Another direction is to consider general n-dimensions but allow unit transport cost ti and consumer information costs to vary across dimensions. This creates a hierarchy of questions. If firms price discriminate on one and the same dimension, which dimension should it be, after weighing the benefit and cost? Would firms have an incentive to price discriminate on more than one dimensions? A third direction is to consider multi-dimensional price discrimination in settings other than best-response asymmetry (e.g., Holmes, 1989 and Schmalensee, 1981) and compare the results in one- vs. multi-dimensional models.