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Spatial monopoly with product differentiation.

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 relative one when choosing prices. This specification for the firm's structure (i.e., allowing to produce multiple varieties) is completely different than that in the case of a single good in the conventional monopolistic competition model. (4) That is, the monopolist is regarded as a multiple-product firm that produces the same type of good, but with different varieties, or else produces many products. For simplicity, I also assume that the consumer will consume every good or variety in a given period of time, (5) but my model differs from conventional analysis in that the monopolist decides not only the price it charges for each variety but also the measure of product varieties it wishes to produce. Thus, its profit objective function can be specified as

(5) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

By considering Equation 4 and assuming symmetry among varieties within the firm, we obtain

(6) [pi] = n[p([alpha] - p)/[beta] + n[gamma]] - nf.

The monopolist has a single plant that is assumed to be located at the origin, and it produces and sells its product varieties to consumers. The transportation cost is borne by the firm at a constant transportation cost rate t per unit of distance per unit of product for each variety, and consumers pay the full price for the delivered differentiated product, with the full price determined by the pricing policy adopted by the monopolist. Furthermore, consider the scenario where the consumers are homogeneously distributed over the linear space. Based on the assumption of symmetry in space, it suffices to examine the spatial monopoly decision on the right half of the linear space where x [greater than or equal to] 0.

3. Spatial Monopolist Decision

When the spatial monopolist transports the differentiated products from the plant (or store) to consumers, the consumers have to pay the cost of delivery. I define full price p(x) as the price the consumer residing in location x should pay for each variety. I consider three spatial pricing policies in this examination: (i) mill pricing; that is, the firm charges the same price, [p.sup.M], at the firm's door for each variety regardless of the destination of its products; (ii) uniform pricing; that is, the firm selects the same delivered price, [p.sup.U], for each variety regardless of the consumers' location; and (iii) spatial discriminatory pricing; that is, the firm chooses different delivered prices, [p.sup.D](x), for the consumer who resides in location x.

Fixed Market Size

Given the market size R, assume that the firm offers each of the differentiated products to the consumers at every location x within this fixed market boundary. I will relax this assumption later and examine the monopolist's decisions among the three alternative pricing strategies with variable market size.

Consider the spatial monopoly and consumer's demand for each product variety at location x. Profit [pi] for the monopolist is specified as follows:

(7) [pi] = [[integral].sup.[bar]R.sub.0] n {[p(x) - tx][[alpha] - p(x)]/[beta] + n[gamma]} dx - nf,

where p(x) is the full price at x for each variety, which is [p.sup.M] + tx, [p.sup.U], and [p.sup.D](x) under mill pricing, uniform pricing, and discriminatory pricing, respectively. Thus, the monopolist's profit under mill pricing, uniform pricing, and discriminatory pricing can be formulated, respectively, as follows:

(8.1) [[pi].sup.M] = [[integral].sup.[bar]R.sub.0] [n.sup.M] {[p.sup.M]([alpha] - [p.sup.M] - tx)/[beta] + [gamma][n.sup.M]} dx - [n.sup.M]f

(8.2) [[pi].sup.U] = [[integral].sup.[bar]R.sub.0] [n.sup.U] {[p.sup.U] - tx)([alpha] - [p.sup.U])/[beta] + [gamma][n.sup.U]} dx - [n.sup.U]f

(8.3) [[pi].sup.D] = [[integral].sup.[bar]R.sub.0] [n.sup.D] {[p.sup.D](x) - tx)([alpha] - [p.sup.D](x))/[beta] + [gamma][n.sup.D]} dx - [n.sup.D]f.

The profit function of a spatial monopoly here is very different from what is used in conventional analysis with a one-product firm. Within this framework, the firm can choose both price and the measure of product varieties to maximize its profit. Accordingly, I set the first-order derivatives of Equation 8 with respect to p and n, respectively (see Appendix B for the sufficient condition).

We now derive both the optimal price and measure of product varieties under the three alternative spatial pricing policies: (6)

(9.1) [p.sup.M] = 1/2 ([alpha] - 1/2 t[bar]R)

(9.2) [p.sup.U] = 1/2 ([alpha] + 1/2 t[bar]R)

(9.3) [p.sup.D](x) = 1/2 ([alpha] + tx)

and

(10.1) [n.sup.M] = [n.sup.U] = 1/2[gamma] [square root of ([beta][bar]R/f)]([alpha] - 1/2 t[bar]R) - [beta]/[gamma]

(10.2) [n.sup.D] = 1/2[gamma] [square root of ([beta][bar]R/f)][square root of ([[alpha].sup.2] - [alpha]t[bar]R + 1/3 [(t[bar]R).sup.2]])] - [beta]/[gamma].

From Equations 9 and 10, we find that the spatial price depends on both the market size and the transportation cost and is independent of consumer preference [beta] and the degree of substitutability [gamma]. (7) However, the optimal measure of varieties will be determined by these demand and supply characteristics in addition to the market size and transportation cost. Furthermore, from Equations 4, 9, and 10, the spatial demand of each variety for a consumer at location x is given by

(11.1) [q.sub.M](x) = ([alpha] + 1/2 t[bar]R) - 2tx/([alpha] - 1/2 t[bar]R) [square root of (f/[beta][bar]R)]

(11.2) [q.sub.U](x) = [square root of (f/[beta][bar]R)]

(11.3) [q.sub.D](x) = [alpha] - tx/[square root of [[[alpha].sub.2] - [alpha]t[bar]R + 1/3 [(t[bar]R).sup.2])] [square root of (f/[beta][bar]R)]

If we denote Q = [[integral].sup.[bar]R.sub.0] q(x) dx and [OMEGA] [equivalent to] nQ as the quantity being produced for each variety and the total output for all differentiated varieties, respectively, then we have

(12.1) [Q.sup.M] = [Q.sup.U] = [square root of (f[bar]R/[beta])]

(12.2) [Q.sup.D] = [alpha - 1/2 t[bar]R/[square root of ([[[alpha].sup.2] - [alpha]t[bar]R + 1/3 [(t[bar]R).sup.2)] [square root of (f[bar]R/[beta])]

and

(13.1) [[OMEGA].sup.M] = [[OMEGA].sup.U] = 1/[gamma] [1/2[bar]R ([alpha] - 1/2 t[bar]R) - [square root of ([beta)f[bar]R])]

(13.2) [[OMEGA].sup.D] = 1/[gamma] [1/2 [bar]R ([alpha] - 1/2t[bar]R) - [alpha] - 1/2t[bar]R/[square root of ([[[alpha].sub.2] - [alpha]t[bar]R + 1/3 [(t[bar]R).sup.2)]] [square root of ([beta]f[bar]R)]]

Thus, with the assumption of a fixed market size, I make comparisons on the output of each variety, the measure of variety, and the total output of the firm as shown in Table 1.

Table 1 shows that the results differ sharply from those obtained in the conventional analysis based on the assumption of the single-good firm. When both the market size and the firm's location are given, and based on a linear demand function, Beckmann (1976) obtains the result that total output is identical under all three spatial pricing policies. Peng (1992) also demonstrates that, regardless of the shape of consumer density, monopoly output is the same under all three spatial price schemes. In addition, Hwang and Mai (1990) assume that the firm's location is endogenously determined, and they treat linear markets as two submarkets in which the two linear demand curves have the same quantity intercept. They found that total output under discriminatory pricing is definitely smaller than under mill pricing.

It is also interesting to find that the spatial monopolist will offer more varieties but less quantity for each variety under discriminatory pricing than those decisions for mill and uniform pricing. In addition, both the output for each variety and the total output for all varieties are decreasing in [beta]. Such a finding does make sense because more profit can be extracted, and this means that more marginal profit arises as a result of offering another product variety.

Variable Market Size

I next examine the case of a variable market size where market size R is endogenously determined. In other words, the fringe of the linear market, R, that is served by the firm is either limited to the price charged or, in the case of uniform pricing, by the monopolist who restricts the service area. Thus, the following relationships can be employed to determine the extent of the market fringe under alternative spatial pricing policies, respectively (see Beckmann 1976; Hsu 1983; Pang 1992):

(14) [q.sup.M] (R.sup.M) = 0

(15) [p.sup.U] - t[R.sub.U] = 0

(16) [q.sup.D] (R.sup.D) = 0

These relationships are based on the assumption that a consumer will refuse to buy any quantity of each variety if the selling price is more than the maximum he or she is willing to pay under both mill pricing and discriminatory pricing. Alternatively, the consideration is that the firm will refuse to sell beyond the market boundary at which the amount it receives for each unit of variety equals the cost of transporting the products. By substituting Equations 9 and 11 into Equations 14-16, the optimal market size under different spatial pricing policies is given as follows:

(17.1) [R.sup.M] = [R.sub.U] = 2/3 [alpha]/t

(17.2) [R.sub.D] = 1/t [alpha].

The result of equal market size under mill and uniform pricing extends those results obtained in the existing literature involving single-product firms such as Holahan (1975), Beckmann (1976), Hsu (1983), and Peng (1992). Furthermore, by substituting Equation 17.1 into Equations 8.1 and 8.2 and Equation 17.2 into Equation 8.3, respectively, we can now derive both the measure of varieties and price of each product variety based on the assumption of a variable market fringe for the three spatial pricing policies, respectively, as follows: (8)

(18.1) [n.sup.M.sub.v] = [n.sup.U.sub.v] = 1/3 [alpha]/[gamma] [square root of (2/3 [alpha][beta]/tf)] - [beta]/[gamma]]

(18.2) [n.sup.D.sub.v] = 1/2 [alpha]/[gamma] [square root of (1/3 [alpha][beta]/tf)] - [beta]/gamma]]

and

(19.1) [p.sup.M.sub.v] = 1/3 [alpha]

(19.2) [p.sup.U.sub.v] = 2/3 [alpha]

(19.3) [p.sup.D.sub.v] = [p.sup.D (x) = 1/2 ([alpha] + tx).

We in turn obtain the quantity for each variety and total output for all differentiated varieties as follows:

(20.1) [Q.sup.M.sub.v] = [Q.sup.U.sub.v] = [square root of (2/3 [alpha]f/t[beta])]

(20.2) [Q.sup.D.sub.v] = [square root of (3/4 [alpha]f/t[beta])]

and

(21.1) [[OMEGA].sup.M.sub.v] = [[OMEGA].sup.U.sub.v] = 2/9 [[alpha].sub.2]/t[gamma] - [beta]/[gamma] [square root of (2/3 [alpha f/t[beta])]

(21.2) [[OMEGA].sup.D.sub.v] = 1/4 [[alpha.sub.2]/t[gamma] - [beta] /[gamma] [square root of (3/4 [alpha]f/t[beta])].

Thus, if the spatial monopolist can determine the market size, aside from both the measure of varieties and the price of each variety, then we have the comparison results as revealed in Table 2.

As depicted by Table 2, under the assumption of a variable market size, with respect to the comparison of the optimal measure of varieties and total output, we obtain the same result as that on the assumption of a fixed market size. However, with regard to the comparison of the quantity produced of each variety, we show a different result from those based on a fixed market size. The different result stems from the larger market size chosen by the monopolist under discriminatory pricing relative to that under mill and uniform pricing. By using the same assumption of a variable market size in the literature relating to a single-good spatial monopoly, Greenhut and Ohta (1972) claim that total output is greater under discrimination than under simple f.o.b. mill pricing regardless of the shape of the gross demand curve. Holahan (1975) examines this issue with variable market size and finds that spatial price discrimination results in a firm producing larger outputs than under a mill pricing policy. Therefore, even when we consider the situation where the multiple-product spatial monopolist can choose the measure of varieties, the total output for all differentiated varieties in our model supports that of Greenhut and Ohta (1972) and Holahan (1975). With regard to the total output for each variety, our result is still consistent with theirs.

4. Economic Effects of Spatial Pricing Policies

In this section, I examine the relative effects of alternative spatial pricing policies on economic benefits in the case of a multiple-product monopoly under a given market size and a variable market size and make a comparison of these results with those in conventional analysis that are based on the framework of a single-product spatial monopoly. Because both the firm and the consumers are involved in the model, a comparison is made of the effects of these policies on consumer surplus, monopolist's profits, and social surplus.

Fixed Market Size

Under the assumption of a fixed market size, the spatial monopolist serves all the consumers within the market size [bar]R. First, I examine the consumer surplus with product differentiation under the three spatial pricing policies. Since the demand function for each variety is linear as shown in Equation 5, the consumer surplus can be specified as

(22) CS = 1/2 [[integral].sup.[bar]R.sub.0] [[integral].sup.n.sub.0] [[alpha] - [p.sub.i] (x)] [q.sub.i](x)di dx.

The consumer surpluses under mill pricing, uniform pricing, and discriminatory pricing are, respectively, given by

(23.1) C[S.sup.M] = [square root of ([bar]R)] / 4[gamma] [[alpha].sup.2] - [alpha]t[bar]R + 7/12 [(t[bar]R).sup.2] /([alpha] - 1/2 t[bar]R)[[square root of ([bar]R)] / 2] ([alpha] - 1/2t[bar]R) - [square root of (f[beta])]]

(23.2) C[S.sup.U] = [square root of ([bar]R)] / 4[gamma] ([alpha] - 1/2 t[bar]R) [[square root of ([bar]R/2)] ([alpha] - 1/2 t[bar]R) - [square root of (f[beta])]]

(23.3) C[S.sup.D] = [square root of ([bar]R)] / 4[gamma] [square root of ([[alpha].sup.2]) - [alpha]t[bar]R + 1/3 [(t[bar]R).sup.2])] [[square root of [([bar]R)]/2][[square root of ([[alpha].sub.2] - [alpha]t[bar]R + 1/3 [(t[bar]R).sup.2] - [square root of (f[beta])]].

In turn, we have

PROPOSITION 1.

(a) [CS.sup.D] > [CS.sup.M] > [CS.sup.U] if [square root of (f[beta])] > G ([alpha], t, [bar]R)

(b) [CS.sup.M] > [CS.sup.D] > [CS.sup.U] if [square root of (f[beta])] = G ([alpha], t, [bar]R)

(c) [CS.sup.M] > [CS.sup.D] > [CS.sup.U] if [square root of (f[beta])] < G ([alpha], t, [bar]R)

where

G ([alpha], t, [bar]R) [equivalent to] [t.sup.2] [([bar]R).sup.5/2]/8 [[[alpha].sub.2] - [alpha]t[bar]R + 7/12 [(t[bar]R).sup.2]/([alpha] - 1/2 t[bar]R) - [square root of ([[[alpha].sup.2] - [alpha]t[bar]R + 1/3 [(t[bar]R).sup.2]].sup.-1]]] >0.

As shown in Proposition 1, we know that the consumer surplus under uniform pricing is the smallest among these alternative spatial pricing policies independent of consumer preferences, production characteristics, and transportation cost. However, the comparison of the consumer surpluses under mill pricing and discriminatory pricing highlights that the ranking of consumer surpluses depends on the parameters a, [alpha], [beta], f, t, and [bar]R. (9) If all these parameters except [beta] are given, then it implies that when the representative consumer prefers to consume more variety, the consumer surplus under discriminatory pricing will be greater than that under mill pricing. It reveals that higher utility can be obtained as more varieties are consumed, while the quantity of each differentiated variety diminishes. This finding is similar to the work of de Palma and Liu (1993), who show, based on a binary logit utility model, that spatial price discrimination could provide the highest consumer surplus when consumers tastes are heterogeneous enough. This outcome differs sharply from that obtained in the linear demand model for the single-product case, for example, Beckmann (1976), Hsu (1983), Hwang and Mai (1990), and Peng (1992), who derive only result (c) of Proposition 1.

I next make a comparison of the firm's profit among these spatial pricing policies. Consider the profit function as shown in Equation 8. By substituting Equations 9 and 10 into Equation 8, the monopolist's profit under mill pricing, uniform pricing, and discriminatory pricing are, respectively, given by

(24.1) [[pi].sup.M] = [[pi].sup.U] = 1/[gamma] [[square root of ([bar]R)]/2 ([alpha] - 1/2 t[bar]R) - [square root of ([beta]f)].sup.2]

(24.2) [[pi].sup.D] = 1/[gamma] [[square root of ([bar]R)]/2 [square root of ([[alpha].sup.2] [alpha]t[bar]R + 1/3 [(t[bar]R).sup.2] - [square root of ([beta]f)].sup.2]]]

Therefore, we have the result that the firm's profit under discriminatory pricing is consistently the largest among the three alternative spatial pricing policies even when the monopolist can choose the measure of varieties, while the firm's profit under mill pricing and uniform pricing are identical as shown in Table 1. This result confirms what has been derived in the existing literature based on the single-product monopoly. When the firm can determine both the price of a good and the measure of product varieties, the multiple-product monopoly prefers discriminatory pricing to the other two spatial pricing policies since it has more decision variables and thus can obtain a greater profit than under mill pricing and uniform pricing.

By considering the comparison of consumer's surplus and firm's profit, we derive the interesting result that both consumers and the firm will prefer discriminatory pricing if the consumers prefer to consume more differentiated varieties. In other words, it claims that the spatial multiproduct firm will provide more product varieties with less quantity of each variety under discriminatory pricing than it does under both mill and uniform pricing. Therefore, when the consumer favors the consumption of more varieties, both the firm and the consumers benefit more from the discriminatory pricing than the other pricing policies. Interestingly, this finding contradicts the result obtained in conventional analysis that discriminatory pricing always only benefits the firm but harms the consumer under the framework of a single-product monopoly and a linear demand function.

Let us finally turn to the comparison of welfare effects (consumers' surplus plus the firm's profit) under the three spatial pricing policies. Apparently, it is important to examine whether the result of conventional analysis can be supported in the case of a multiple-product monopoly. Using the results of Equations 23 and 24, we can derive the welfare under each spatial pricing policy as follows:

(25.1) W[S.sup.M] = 1/[gamma] {3/8[bar]R [[[alpha].sup.2] - [alpha]t[bar]R + 13/36[(t[bar]R).sup.2]] - [square root of ([bar]R/4)] 5[[alpha].sup.2] - 5[alpha]t[bar]R + 19/12 [(t[bar]R).sup.2]] / [alpha] - 1/2t[bar]R [square root of (f[beta])] + f[beta]}

(25.2) W[S.sup.U] = 1/[gamma] {3/8[bar]R [([alpha] - 1/2t[bar]R).sup.2] - 5/4 [square root of ([bar]R)] ([alpha] - 1/2t[bar]R) [square root of (f[beta])] + f[beta]}

(25.3) W[S.sup.D] = 1/[gamma] {3/8[bar]R [[[alpha].sup.2] - [alpha]t[bar]R + 1/3[(t[bar]R).sup.2]] - 5/4 [square root of ([bar]R)] [square root of ([[[alpha].sup.2] - [alpha]t[bar]R + 1/3[(t[bar]R).sup.2]]) [square root of (f[beta])] + f[beta]}.

These findings then lead to the following proposition:

PROPOSITION 2.

(a) W[S.sup.D] > W[S.sup.M] > W[S.sup.U] if [square root of (f[beta])] > G' ([alpha], t, [bar]R)

(b) W[S.sup.D] = W[S.sup.M] > W[S.sup.U] if [square root of (f[beta])] > G' ([alpha], t, [bar]R)

(c) W[S.sup.M] > W[S.sup.D] > W[S.sup.U] if [square root of (f[beta])] > G' ([alpha], t, [bar]R)

where

G' ([alpha], t, [bar]R) = 1/24 [t.sup.2] [([bar]R).sup.5/2] [5[[alpha].sup.2] - 5[alpha]t[bar]R + 19/12 [(t[bar]R).sup.2]/ [alpha] - 1/2t[bar]R - 5 [square root of ([[alpha].sup.2]) - [alpha]t[bar]R + 1/3[(t[bar]R).sup.2].sup.-1]]].

The comparison of welfare is different from that in existing literature with a single-product firm and linear demand function. For example, Beckmann (1976), Hsu (1983), and Peng (1992) claim that mill pricing is always preferred from the viewpoint of the welfare effect. However, based on Proposition 2, when we consider that the monopolist is a multiple-product firm and that the consumer consumes all product varieties, we show that this standard result is satisfied only in (c). In other words, the comparison of welfare between mill pricing and discriminatory pricing depends on the parameter of maximum willingness to pay [alpha] and the parameter of consumption preference for differentiated products [beta] as well as the transportation cost t.

From Proposition 2, we establish that the welfare under discriminatory pricing is the greatest among the alternative spatial pricing policies when [square root of (f[beta])] [greater than or equal to] G'([alpha], t, [bar]R). This implies that if the consumer favors the consumption of more varieties (i.e., 13 is large enough) and/or the monopolist has a higher fixed production cost for each variety, then the spatial monopolist will offer more varieties and the consumer will prefer to enjoy more varieties under the discriminatory pricing policy than under alternative pricing policies. In turn, the welfare arising from discriminatory pricing will be the largest under such a scenario. However, the parameter of substitutability between varieties [gamma] is irrelevant on the ranking of welfare but has a negative effect on the welfare for all spatial pricing policies. In a binary logit model, de Palma and Liu (1993) also show that discriminatory pricing could yield the highest welfare when the consumers' tastes are heterogeneous enough. The welfare under mill pricing may be larger than that under discriminatory pricing if [square root of (f[beta])] < G'([alpha], t, [bar]R). Furthermore, W[S.sup.M] is always larger than W[S.sup.U], as is the case in the existing literature.

Variable Market Size

When substituting Equations 17-19 and demand function (Eqn. 4) into Equation 22, we achieve the consumer's surplus for the variable market size under mill pricing, uniform pricing, and discriminatory pricing, respectively, as

(26.1) C[S.sup.M.sub.v] = 1/3 [alpha]/[gamma] [4/27 [[alpha].sup.2]/t - 2/3 [square root of (2[alpha][beta]f/3t])]

(26.2) C[S.sup.U.sub.v] = 1/3 [alpha]/[gamma] [1/9 [[alpha].sup.2]/t - 1/2 [square root of (2[alpha][beta]f/3t])]

(26.3) C[S.sup.D.sub.v] = 1/3 [alpha]/[gamma] [1/8 [[alpha].sup.2]/t - 3/4 [square root of (2)] [square root of (2[alpha][beta]f/3t])].

Comparing Equations 26.1-26.3, we obtain

PROPOSITION 3.

(a) C[S.sup.U.sub.v] [greater than or equal to] C[S.sup.D.sub.v] > C[S.sup.M.sub.v] if [square root of (f[beta])] [greater than or equal] [C.sub.3]

(b) C[S.sup.D.sub.v] > C[S.sup.U.sub.v] [greater than or equal to] C[S.sup.M.sub.v] if [C.sub.2] [Less than or equal to] [square root of (f[beta])] [C.sub.3]

(c) C[S.sup.D.sub.v] [greater than or equal to] C[S.sup.M.sub.v] > C[S.sup.U.sub.v] if [C.sub.1] [Less than or equal to] [square root of (f[beta])] [C.sub.2]

(d) C[S.sup.M.sub.v] > C[S.sup.D.sub.v] > C[S.sup.U.sub.v] if [square root of (f[beta])] [C.sub.1],

where

[C.sub.1] [equivalent to] 5/48 [square root of (6)] - 54 [square root of (3)] [alpha] [square root of ([alpha]/t])], [C.sub.2] [equivalent to] 1/3 [square root of (2/3)] [alpha] [square root of ([alpha]/t)], and [C.sub.3] [equivalent to] 1/18 [square root of (3)] - 12 [square root of (6)] [alpha] [square root of ([alpha]/t)].

Considering the variable market size, Proposition 3 shows that spatial discriminatory pricing provides a greater level of consumers' surplus than that of mill pricing when the consumers prefer to consume more varieties (i.e., [beta] is large enough). This finding is completely different with Holahan (1975), who obtains that the consumers' surplus for mill pricing is always higher than that of discriminatory pricing when the spatial monopoly provides only a single product and can determine its market area. Notice that the higher surplus derived by distant customers and that from enjoying more varieties for all consumers under spatial discriminatory pricing compared to mill pricing outweigh the loss of nearby customers.

Substituting Equations 17-19 into Equation 8, the profits of spatial monopoly under alternative pricing policies are next written, respectively, as

(27.1) [[pi].sup.M.sub.v] = [[pi].sup.U.sub.v] = 1/3 [alpha]/[gamma] [2/9 [[alpha].sup.2]/t - 2 [square root of (2 [alpha][beta]f/3t])] + [beta]f/[gamma]

(27.2) [[pi].sup.D.sub.v] = 1/3 [alpha]/[gamma] [1/4 [[alpha].sup.2]/t - 3/[square root of (2)] [square root of (2 [alpha][beta]f/3t])] + [beta]f/[gamma].

Therefore, with the assumption of variable size, we also obtain the result as collected in Table 1, whereby the profit under discriminatory pricing is greater than mill and uniform pricing. This is the same with the result of a fixed market fringe. It is also consistent with that of Holahan (1975) by the assumption of a single-product spatial monopoly and variable market size.

For the sum of a firm's profit and consumers' surplus, we obtain the welfare measure for variable market size under mill, uniform, and discriminatory pricing as

(28.1) W[S.sup.M.sub.v] = 1/3 [alpha]/[gamma] [10/27 [[alpha].sup.2]/t - 8/3 [square root of (2[alpha][beta]f/3t])] + [beta]f/[gamma]

(28.2) W[S.sup.U.sub.v] = 1/3 [alpha]/[gamma] [1/3 [[alpha].sup.2]/t - 5/2 [square root of (2[alpha][beta]f/3t])] + [beta]f/[gamma]

(28.3) W[S.sup.D.sub.v] = 1/3 [alpha]/[gamma] [3/8 [[alpha].sup.2]/t - 15/4 [square root of (2)] [square root of (2[alpha][beta]f/3t])] + [beta]f/[gamma]

Thus, the welfare measure under the three spatial pricing policies can be compared by

PROPOSITION 4.

(a) W[S.sup.U.sub.v] > W[S.sup.D.sub.v] > W[S.sup.M.sub.v] if [square root of (f[beta])] > [W.sub.2]

(b) W[S.sup.D.sub.v] [greater than or equal to] W[S.sup.U.sub.v] [greater than or equal to] W[S.sup.M.sub.v] if [W.sup.1] [less than or equal to][square root of (f[beta])] [less than or equal to] [W.sub.2]

(c) W[S.sup.D.sub.v] > W[S.sup.M.sub.v] > W[S.sup.U.sub.v] if [square root of (f[beta])] < [W.sub.1],

where

[W.sub.1] [equivalent to] [square root of (6)]/9 [alpha] [square root of ([alpha]/t)], [W.sub.2] [equivalent to] [square root of (3)]/90 - 60 [square root of (2)] [alpha] [square root of ([alpha]/t)].

We also find that the comparison among the three pricing policies depends on the relative preference of consumers for variety with a given [alpha], t, and f. It is shown that the result under the variable market size is slightly different from that under a fixed market size. From Proposition 2, considering the fixed market fringe, we know that the welfare measure is smallest under uniform pricing regardless of the parameters' values. However, with the assumption of the variable market size, the welfare under uniform pricing can be the highest among three pricing policies when [beta] is large enough. The higher net benefit is derived by the larger consumers' surplus under the variable market size compared to that of a fixed market size. Proposition 4 also confirms Holahan (1975), where it appears that spatial price discrimination provides a greater level of welfare measure compared to mill pricing.

5. Concluding Remarks

I have examined the behavior of a spatial monopoly in the case of a multiple-product firm and with an assumption of a quadratic utility function with consumers assumed to be consuming all the product varieties. This paper analyzes the economic effects of a multiple-product monopolist's decision on the output of each product variety, the measure of product varieties being produced, the firm's total output, and consumers' surplus, profit, and welfare. The results are compared with those where a firm is assumed to produce a single good, which is commonly done in the existing literature. My examination has also helped clarify some issues related to the pricing policy of the spatial monopoly.

The main conclusions are as follows. First, the quantity of each variety produced is dependent on the consumer's preference for variety, and it is not identical under the three spatial pricing policies.

Next, the spatial monopolist produces more product varieties under discriminatory pricing than under mill pricing and uniform pricing despite a fixed or variable market size. Discriminatory pricing also yields a larger total output with all varieties than do mill and uniform pricing under both assumptions of a fixed and variable market size. This result differs with the result obtained in a single-product case under the same assumption of a linear demand function and homogeneous consumer distribution. Furthermore, with a given market size, the comparison of consumers' surplus among the three alternative spatial pricing policies is dependent upon the characteristic of the consumer's preferences in regard to product differentiation. This outcome is also inconsistent with that of a single-good case. In addition, the more interesting finding shows that the welfare comparison also departs from those commonly shown in the literature. Finally, with respect to the variable market size, the surplus comparison between pricing policies is highlighted, exhibiting that spatial discriminatory pricing yields a greater level of consumers' surplus than that of a mill pricing policy when the consumers prefer consumption of more varieties. This finding is also sharply different with the result in the existing literature for the single-product spatial monopolist, and it may support a government policy to adopt spatial discriminatory pricing policies.

Along these lines of thought, there are some possible avenues for future research. First, one could apply this framework to a few submarkets in a spatial economy (i.e., relax the assumption of continuous distribution on consumers). Second, one can explore the issue along a theoretical front, examining the monopolist's decision based on the assumption that one variety is being produced by one plant and how the spatial monopoly determines the location of each plant rather than measuring the product varieties. Third, one may study whether an endogenous-determined consumer distribution changes the firm's optimal decision. Fourth, it is possible to extend this consideration to an oligopolistic competition model. Finally, it would also be interesting to extend this examination to heterogeneous consumers with different consumption preferences.

Appendix A

The symmetric quadratic utility is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

When [gamma] [left arrow] 1/2 [beta], it expresses perfect substitution between varieties, and then this utility function reduces to a standard quadratic utility for a homogenous good. If we set [delta] [equivalent to] 2 [gamma] and [beta] > [delta] > 0, then the model would be equivalent to the specification by Ottaviano, Tabuchi, and Thisse (2002).

In the case of two varieties, the symmetric quadratic utility function is obviously reduced by

U([q.sub.0], [q.sub.1], [q.sub.2]) = [alpha] ([q.sub.1] + [q.sub.2]) - 1/2 [beta] ([q.sup.2.sub.1] + [q.sup.2.sub.2] - 2[gamma][q.sub.1][q.sub.2] + [q.sub.0].

Appendix B

The sufficient condition (e.g., in mill pricing) is given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]
Table 1. The Comparison of the Decision with the Fixed Market Size

 Comparisons among Alternative
 Spatial Pricing Policies

1. Output of each variety [Q.sup.D] < [Q.sup.U] = [Q.sup.M]
2. Measure of variety [n.sup.D] > [n.sup.U] = [n.sup.M]
3. Total output [[OMEGA].sup.D] > [[OMEGA].sup.U] =
 [[OMEGA].sup.M]
4. Firm's profit [[pi].sup.D] > [[pi].sup.U] = [[pi].sup.M]

Table 2. The Comparison of the Decision with the Variable Market Size

 Comparisons among Alternative
 Spatial Pricing Policies

1. Output of each variety [Q.sup.D.sub.v] < [Q.sup.U.sub.v.] =
 [Q.sup.M.sub.v.]
2. Measure of variety [n.sup.D.sub.v.] > [n.sup.U.sub.v.] =
 [n.sup.M.sub.v.]
3. Total output [[OMEGA].sup.D.sub.v.] > [[OMEGA]
 .sup.U.sub.v.] = [[OMEGA].sup.M.sub.v.]
4. Firm's profit [[pi].sup.D.sub.v.] > [[pi].sup.U.sub.v.]
 = [[pi].sup.M.sub.v.]
5. Market fringe [R.sup.D] > [R.sup.U] = [R.sup.M]


I would like to thank two anonymous referees for their valuable comments as well as the insightful comments and helpful suggestions from Simon P. Anderson, Jonathan Hamilton, Chao-cheng Mai, and J.-F. Thisse and participants at the International Conference on Industrial Economics in Taipei, Taiwan. The author is grateful to financial support from the National Science Council (NSC 90-2451-H-001-008) on this research.

(1) The other two major approaches to product differentiation that have been developed in the literature are the random utility models and address models. For a detailed comparison of these three approaches, see Anderson, de Palma. and Thisse (1992).

(2) Ottaviano, Tabuchi, and Thisse (2002) employed the same kind of quasi-linear utility function, u([q.sub.0], q(i)) = [alpha][[integral].sup.n.sub.0]q(i)di - ([beta] - [gamma])/2) [[integral].sup.n.sub.0][[q(i)].sup.2]di - ([gamma]/2)[[[[integral].sup.n.sub.0]q(i)di].sup.2] + [q.sub.0], to examine monopolistic competition and the agglomeration of firms. Dixit and Stiglitz (1977) proposed the other prime example using the CES utility function to specify the preferences of a representative consumer, which are also assumed to be symmetric in all commodities.

(3) When [beta] = [2.sub.[gamma]], it expresses perfect substitutability between varieties. Ottaviano, Tabuchi, and Thisse (2002) identify a simple condition in relation to the representative consumer to have a taste for variety, [beta] > [gamma]. In other words, the quadratic utility function exhibits a preference for variety when the product is differentiated.

(4) For example, Krugman (1991); Anderson, de Palma, and Thisse (1992); Gehrig (1998); Ottaviano, Tabuchi, and Thisse (2002).

(5) Armstrong (1999) assumes that the multiple-product firm faces consumers with unobservable tastes. Here, the relevance parameter of consumer preference [beta] is exogenously given for the firm.

(6) In order to ensure the number of varieties is positive, we assume 1/4 [bar]R[([alpha] - 1/2 t[bar]R).sup.2] > f[beta].

(7) In order to ensure that the mill price is positive, I assume that the condition [alpha] > (1/2) t[bar]R is satisfied in the context.

(8) In order to ensure that the number of variety is positive, it is necessary to assume that (2/27) [[alpha].sub.3]/t > f [beta].

(9) However, the parameter [gamma] has no effect on the ranking of consumer surplus, but the larger value of [gamma] (i.e., the substitutability between varieties is higher) will yield a smaller consumer surplus regardless of the pricing policy.

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Received April 2002; accepted April 2003.

Shin-kun Peng, Institute of Economics, Academia Sinica, and Graduate Institute of Building and Planning, National Taiwan University, Nankang, Taipei, Taiwan; E mail speng@econ.sinica.edu.tw. Present address: 128 Yen-Chiu Road, Sec. 2, Taipei 115, Taiwan.
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