Spatial monopoly with product differentiation.

• Author: Peng, Shin-kun

1. Introduction

   In the literature on spatial monopoly, it is common to assume that the firm produced (or sold) only one good. This assumption, however, departs from the real world. It is restrictive and simplifies the consumer preference and demand characteristics (Greenhut and Ohta 1972; Holahan 1975; Beckmann 1976; Hsu 1983; Claycombe 1991, 1996; Hwang and Mai 1990; Peng 1992; Chu and Lu 1998). In reality, a trip to the supermarket or the grocery store shows that many similar products are sold to the various tastes and requirements of different consumers, as with a restaurant offering a variety of meals. Therefore, a firm may be regarded as producing (or selling) multiple products or similar types of products with a differentiated variety. Moreover, a consumer may consume many products or prefer to consume one good with a differentiated variety (Dixit and Stiglitz 1977; Ottaviano, Tabuchi, and Thisse 2002). In this paper, I develop a model to incorporate both demand- and supply-side considerations of product variety in a spatial monopoly.

   From the viewpoint of consumer preferences, it is well recognized that a consumer exhibits a preference for variety in consumption. Hanly and Cheung (1998) point out that demand complementarity is one source of the advantages that accrue to a multiple-product firm. In practice, even consumers tend to purchase only one, or at most a few, of the varieties offered. A representative consumer approach can be used to generate a utility function to incorporate the aggregate preference for a differentiated product. The representative consumer is a simplifying construct that is frequently used in the theoretical analysis of differentiated product markets. (1) In addition, in numerous fields, including industrial organization (Dixit 1979; Vives 1990), international trade (Anderson, Schmitt, and Thisse 1995), and demand analysis (Philips 1983), the utility function is specified in order take into account the consumption of differentiated products. However, they all assume that a firm produces only one product. Thus, the results are based on a framework of monopolistic competition. Here, I consider a model with a quadratic utility function and highlight the economic effect with the choice of product differentiation and the consumption of product variety.

   Classical (nonspatial) economics describes various interesting issues related to a multiple-product monopoly (Klinger-Monteiro and Page 1998; Armstrong 1999). Thisse and Vives (1988) analyze the simultaneous choice of policy and price in a product differentiation context, and their examination is closely related to a firm's variety offer. In a product differentiation context, it is also interesting to examine the economic effect of the choice of variety and pricing in the spatial monopoly. It may establish some interesting results that differ from some conventional results in the literature that are based on the assumption of a single-product firm.

   This paper analyzes the optimal decision of the firm when consumers have preferences for product variety in a spatial monopoly. In particular, I employ a quadratic utility function and assume that the consumer consumes all the goods produced by the firm. Thus, the demand function for each variety is linear. After comparing the results with those in the literature where demand is linear, it is also important to examine the economic effect with these spatial pricing policies based on a multiproduct monopoly and make comparisons with existing literature on single-product firms.

   Based on a survey of firms in the United States, West Germany, and Japan, Greenhut (1981) finds that firms in the United States tend to practice price discrimination. Of 174 sampled firms, less than one-third adopted mill pricing (f.o.b.), and only one-fifth used uniform pricing. The remaining 46% resorted to discriminatory pricing. The tendency for price discrimination was even greater in West Germany and Japan, where the percentages of firms engaged in price discrimination were approximately 47% and 55%, respectively. This evidence clearly shows that discriminatory pricing is not only possible in countries where the practice is illegal, such as the United States, but is the most common pricing method.

   The main findings of this paper are as follows. First, despite fixed or variable market size, the quantity produced of each product variety is not identical under the three spatial pricing policies, and the spatial monopoly produces more product varieties under discriminatory pricing than under both mill and uniform pricing. The discriminatory pricing also yields a larger total output for all product varieties than do mill and uniform pricing. These findings stand in sharp contrast to conventional analysis in a single-product monopoly with the same assumption of a linear demand function. Second, the comparison of consumer surpluses among the three alternative spatial pricing policies under the given market size is dependent on the characteristics of both the demand and the supply side (i.e., market size, consumer preferences, and the fixed costs associated with each variety), and the outcome of this comparison differs from that in the case of a single good. More interestingly, the results of my welfare comparison also differ with those in the literature as shown in Beckmann (1976), Hsu (1983), and Peng (1992), which also specially depend on consumer preferences for variety. Finally, with the assumption of variable market size, I find that spatial price discrimination provides more varieties and a great level of consumers' surplus than mill pricing. This result confirms de Palma and Liu (1993) by the framework of a random utility model, but it contrasts Holahan's (1975) result, where mill pricing is always preferred by customers and depends on the single-product monopoly. Therefore, this may explain why a government could allow discriminatory pricing adopted by a spatial monopoly. Particularly, it is based on the scheme of a multiproduct firm.

   This paper proceeds as follows. In section 2, I develop a simple model with a quadratic utility function to consider the consumption of differentiated products. In section 3, I examine the alternative spatial monopolist's decisions by considering fixed and variable market size, respectively. In section 4, I compare the economic effects and discuss the economic implications of the three spatial pricing policies based on a fixed and variable market size. In section 5, I provide concluding remarks and suggest avenues for future research.

2. The Model

   The Consumer

   In a spatial context, a representative consumer is an agent whose utility embodies aggregate preference for diversity in every given location. I assume that there are two goods in the economy. The first good is homogeneous and is chosen as the numeraire. The second good is a horizontal-differentiated product following Salop (1979) and Wolinsky (1984). Preferences are identical across consumers and described by the following quasi-linear utility function, which is symmetric in all varieties (2) (see Appendix A for more details). Since this is a restrictive assumption, we can have natural asymmetry because of graduated physical differences in varieties, with a pair close together being better mutual substitutes than a pair farther apart.

   The utility function is given as follows:

   (1) u[[q.sub.0], q(i)] = [alpha] [[integral].sup.n.sub.0]q(i)di - 1/2 [beta] [[integral].sup.n.sub.0][[q(i)].sup.2]di - [gamma] [[integral].sup.n.sub.0][[integral].sup.n.sub.0] q(i)q(j)di dj + [q.sub.0] for j [not equal to] i [member of] [0, n],

   where q(i) is the quantity of variety i [member of] [0, n], n is the measure of variety, and [q.sub.0] is the quantity of the numeraire. Assume that the parameters [alpha] > 0 and [beta] > 2[gamma] > 0 both hold on the whole context. In this utility function, [alpha] is a measure of the consumer's maximum willingness to pay since it expresses the intensity of preferences for the differentiated product, and [beta] > 2[gamma] implies that the representative consumer has a taste for variety, and a large value for [beta] means that the representative consumer is biased toward a dispersed consumption of more differentiated products. The parameter [gamma] > 0 indicates that all differentiated goods are assumed to be substitutes for each other. For a given value of [beta], a higher value of [gamma] implies that the varieties of substitutes will be closer to each other. (3)

   The representative consumer's budget constraint can then be written as follows:

   (2) [[integral].sup.n.sub.0] p(i)q(i)di + [q.sub.0] = Y,

   where Y is the consumer's income, which is assumed to be given; p(i) is the price of product variety i; and the price of the homogeneous good is normalized to one since I treat it as a numeraire. The consumer chooses [q.sub.0], q(i) to maximize utility subject to the budget constraint, which yields

   (3) p(i) = [alpha] - [beta]q(i) - [gamma][[integral].sup.n.sub.0] q(j)dj, i [member of][0, n].

   Therefore, the demand for variety i is given by

   (4) q(i) = [alpha] - p(i)/[beta] + n[gamma] + [gamma]/[beta]([beta] + n[gamma]) [[integra].sup.n.sub.0][p(j) - p(i)]dj.

   Hence, the demand function for variety i has the desirable properties that the demand is decreasing in [beta], [gamma] and the measure of product variety n and is increasing in both [alpha] and the price of other varieties.

   The Firm

   The horizontal differentiated products are assumed to be a continuum. In choosing n, the monopolist picks the range of variety produced, [0, n], and each variety has a fixed cost f. All varieties are produced in the same place, and the marginal cost of production of a variety is set equal to zero. This simplifying assumption, which is standard in many models of industrial organization, makes sense here, unlike in Dixit and Stiglitz (1977), because our preferences imply that firms use an absolute markup instead of a...