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Pricing policy in spatial competition Takatoshi Tabuchi* Faculty of Economics, University of Tokyo, Hongo 7 -3 -1, Bunkyo-ku, Tokyo 113 -0033, Japan Received 26 June 1996; received in revised form 20 October 1998; accepted 7 January 1999

Abstract In this paper, we analyze a spatial oligopoly a´ la Hotelling assuming Norman and Thisse (1996, Product variety and welfare under discriminatory and mill pricing policies. Economic Journal 106 (1996) 76–91) spatial non-contestability. Each firm selects a pricing strategy in the first stage and chooses a price (schedule) in the second. Seeking subgame perfect Nash equilibrium, we obtain the following. First, mill pricing strategy may become prevalent due to improvements in transportation technology, whereas the discriminatory pricing strategy would be dominant when economies of scale become large. Second, for any pricing strategy, the equilibrium number of mill pricing firms is too large in comparison to the social optimum one, whereas the equilibrium number of discriminatory pricing firms is too small. Finally, we observe a hysteresis in the spatial arrangements of pricing strategies. 1999 Elsevier Science B.V. All rights reserved. Keywords: Pricing policy; Hotelling; Spatial competition; Excess theorem; Hysteresis JEL classification: L13

1. Introduction The major focus of this paper is the pricing policy used by retail firms in the framework of spatial competition. Most of the literature on spatial competition does not consider the choice of pricing strategies. Hotelling (1929) assumed that firms apply exclusively the mill pricing strategy, while Hoover (1937) supposed

*Corresponding author. Tel.: 181-3-3812-2111 ext. 5603; e-mail: [email protected] 0166-0462 / 99 / $ – see front matter 1999 Elsevier Science B.V. All rights reserved. PII: S0166-0462( 99 )00006-X

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spatially discriminatory pricing, in which the pricing policy is usually predetermined. However, it is the profit-maximizing firms that determine the pricing strategy, i.e., the pricing strategy is not exogenously given but should be endogenously determined within a model. Although mill pricing is ubiquitous in many retail industries, discriminatory pricing can sometimes be observed as well. For example, books are usually sold by mill prices at retail stores. Recently, however, mail-order firms (which have no stores) have begun to take orders by fax, mail or internet and deliver books. The advances in high technology have resulted in the increase of such retail firms. Similar examples are found in liquor and food delivery firms. Discriminatory pricing firms enjoy some advantages. First, they do not have operating costs of retail stores. In addition, they only need to have small inventories. Second, the (opportunity) costs of visiting stores are much higher than the costs of delivering goods in some industries. That is why there are importers, door-to-door salespeople and peddlers. Therefore, it may also be natural to consider that firms can choose mill or discriminatory pricing strategies. Spatial competition in geographic distance is often interpreted as competition in product differentiation in characteristics space. Some firms sell ready-made products by mill prices as well as custom-made products by discriminatory prices. Examples are automobiles, houses, clothes, and so forth. Ready-made products in characteristics competition correspond to mill-priced products in spatial competition while custom-made products in characteristics competition correspond to discriminatory-priced products in spatial competition since the transportation costs in geographical space are translated into the distaste costs in characteristics space. In this paper, by allowing firms to choose their pricing strategies, we study impacts of various cost differences on the structure of industries. In reality, mill pricing predominates in some industries while discriminatory pricing prevails in other industries. We will argue that this is ascribed to the differences in exogenous parameters such as transportation costs, marginal costs, and fixed costs. Furthermore, we identify several reasons for the difference in the number of firms between industries and discuss prospective changes in the industrial structure. We also conduct a welfare analysis. By calculating the social total costs, we obtain the optimum number of firms (the optimum number of varieties in characteristics space), and compare it with the equilibrium ones. The organization of this paper is as follows. In Section 2, we describe the model, presenting two kinds of spatial arrangements of pricing strategies. In Section 3, examining the sustainability of each equilibrium strategy, we compute dominant strategies for all the ranges of parameter values. In Section 4, regarding the number of firms as an endogenous variable, we investigate long run equilibrium with free entry and compare it with the social optimum. In Section 5, we analyze the transition in the spatial arrangement of pricing strategies by changing a parameter value. Section 6 concludes the paper.

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2. The model There are n firms producing an identical good. For a moment, n is assumed to be fixed, but in the later sections, it will be endogenously determined by a free entry condition. It is assumed that n is large enough to preclude fraction problems. Firms choose either mill (fob) pricing or spatially discriminatory pricing in pure strategies 1 . Consumers are uniformly distributed over the unit-circumference of a circle with the density normalized to one, and their location is denoted by x [ [0,1]. Each consumer purchases one unit of the good from the firm offering the lowest full price. The full price is defined by the mill price plus the transportation costs in the case of mill pricing, and defined by the delivered price in the case of discriminatory pricing. The transportation cost is assumed to be a quadratic function of distance with the coefficient of c. It is incurred by consumers in the case of mill pricing and by firms in the case of discriminatory pricing. The transaction costs for the resale of goods between consumers are prohibitively high. The marginal costs of production and retailing are f and the fixed costs of entry are F. F implies the existence of scale economies in production and retailing. Throughout the paper, we assume that the cost parameters c and F are the same, whereas f is different between the mill pricing and the discriminatory pricing firms. This simplification is used to indicate the situation that the unit retailing cost would be very different in each case. Thus, we denote the marginal costs of discriminatory pricing firms by f d and those of mill pricing firms by f m . Given a symmetric location on the circumference of a circle, each firm simultaneously chooses a pricing strategy in the first stage, and simultaneously chooses a single mill price or a delivered price schedule in the second stage. However, if neighboring firms take different strategies in the first stage, we alter the second stage game as follows. Instead of giving a simultaneous choice, we assume that the mill pricing firm becomes a price leader and the discriminatory firm is a follower reacting optimally the mill price (Thisse and Vives, 1988). Such a leader–follower relationship may be justified by the flexibility of the price schedule used by the discriminatory pricing firms since they could easily cut the price at each location in secret if it were profitable. We seek subgame perfect Nash equilibrium. Due to mathematical tractability, we focus only on symmetric equilibrium although asymmetric equilibrium analysis is also important, as exemplified by Tabuchi and Thisse (1995). In Section 2 and 3, we consider the case of a fixed number of firms. It will be endogenized in Section 4 and 5. 1

In reality, a uniform delivered pricing strategy is also common. However, we do not include this strategy simply because in theory this is a special case of the spatially discriminatory pricing strategy.

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Let us denote the profit of the ith firm by pi (Si ; S1 , . . . ,Si21 , Si 11 , . . . ,Sn ), where strategy S is either D (discriminatory pricing) or M (mill pricing). These two price variables will be defined exactly in the next two subsections.

2.1. Discriminatory pricing If each firm chooses the discriminatory pricing strategy, the full price is equivalent to the delivered price in each location. The equilibrium price schedule of firm i is set equal to the second lowest level of the delivery (transportation) costs plus marginal costs among all firms: p di (x) 5 c(x 2 x i 21 )2 1 f d for x [ [(x i 21 1 x i ) / 2, x i ], 2

5 c(x 2 x i 11 ) 1 f

d

for x [ [x i , (x i 1 x i 11 ) / 2].

Note that given the symmetry of location and cost structures, the market boundaries are the midpoints between two neighboring firms. For each location x, firm i has to pay the delivery costs and the marginal costs: c(x2x i )2 1f d for the range of x[[(x i 21 1x i ) / 2, (x i 1x i 11 ) / 2]. Hence, the profit of firm i is given by

p *i (Di ; $2i ) 5

E

(x i 11 2x i 21 ) / 2

cx 2 dx 1

(x i 2x i 21 ) / 2

E 2E

(x i 2x i 21 ) / 2

2

E

(x i 11 2x i 21 ) / 2

cx 2 dx

(x i 11 2x i ) / 2

cx 2 dx

0

(x i 11 2x i ) / 2

cx 2 dx 2 F

0

c 5 ] (x x11 2 x i 21 )(x i 11 2 x i )(x i 2 x i 21 ) 2 F 4 where $ 2i j;(D1 , . . . ,Di21 , Di 11 , . . . ,Dn ). Due to the assumption of symmetric location, x i 11 2x i 51 /n, ; i51, 2, . . . ,n21. Therefore, (1) is reduced to c p *i (Di ; $21 ) 5 ]3 2 F, 2n

(2)

which is illustrated by the shaded area in Fig. 1.

2.2. Mill pricing Consider instead that every firm chooses the mill pricing strategy. The full price is the mill price plus the transportation costs, pi m 1c(x i 2x)2 . Let b i be the market boundary between firm i and firm i 11. Since the marginal consumer located at b i is indifferent as to whether to buy a certain good at i or i 11, the neighboring full prices should be equated as:

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Fig. 1. Discriminatory pricing.

2 m 2 pm i 1 c(b i 2 x i ) 5 p i 11 1 c(x i 11 2 b i ) .

Fig. 2 illustrates the situation. The profit is then given by

pi (Mi ; }2i ) 5 ( p im 2 f m)(bi 2 bi 21 ) 2 F,

(3)

where } 2i ;(M1 , . . . ,Mi 21 , Mi 11 , . . . ,Mn ). Differentiating (3) with respect to pi m and employing Theorem 6 in Economides (1989), we have a symmetric equilibrium mill price pi m * 5c /n 2 1f m and the profit c p *i (Mi ; }2i ) 5 ]3 2 F, n

Fig. 2. Mill pricing.

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both of which are the same for all firms. The shaded area in Fig. 2 is equal to (4).

3. Short run equilibrium

3.1. Discriminatory pricing Let us investigate the sustainability of equilibrium. Suppose that every firm takes discriminatory pricing but firm i changes its strategy to mill pricing. As assumed at the beginning of the preceding section, firm i is a mill price leader while the other firms are followers using discriminatory price schedules. Thus, given pi m , the neighboring firms would undercut prices wherever possible. Anticipating this reaction, firm i, the deviating leader, maximizes its profit with respect to its mill price in the following manner: max pi (Mi ; $2i ) 5 ( p im 2 f m)(bi 2 b i 21 ) 2 F, m pi

where 2 2 f d 2 pm p im 2 f d 1 c(x i2 2 x i221 ) i 1 c(x i 11 2 x i ) b i 5 ]]]]]]] and b i 21 5 ]]]]]]] 2c(x i 11 2 x i ) 2c(x i 2 x i21 ) ]] for ( f d 2 f m)n 2 1 4Œ2cFnn # 5c.

Note that the last inequality ensures that neighboring firms i61 have a nonnegative profit, which limits the profit of deviating firm i. Computing the first-order condition for maximum with x i 11 2x i 51 /n ; i51, . . . ,n21, we obtain the mill price of firm i as pi m (Mi ;$ 2i )5[c1( f d 1f m)n 2 ] / 2n 2 , and the profit of firm i as [c 1 ( f d 2 f m)n 2 ] 2 p i* (Mi ; $2i ) 5 ]]]]]] 2 F. 4cn 3

(5)

Firm i will not choose mill pricing unless the profit is larger than it was before. This allows us to establish a sustainability condition:

p *i (Mi ; $2i ) # p *i (Di ; $2i ).

(6)

The LHS is given by (5) while the RHS is (2). As long as (6) is satisfied, the discriminatory pricing strategy should prevail everywhere.

3.2. Mill pricing Next, consider the situation that every firm takes mill pricing but one firm i changes its strategy to discriminatory pricing. This time, unlike the previous subsection, we should take a chain effect into account when computing the profit

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of the deviating firm as in Eaton and Wooders (1985). This is because anticipating the price schedule of one deviating firm, each of the other firms would change its optimal mill prices, which vary according to the distance from the deviating firm. Let i51 be the deviating firm without losing generality. For n$3, the first-order conditions for profit maximization are expressed as: 2c m f d 1 pm 1 ]2 5 4p m 3 1 2f 2, n 2c d m pm 1 ]2 5 4p nm , n21 1 f 1 2f n In addition, there are the following first-order conditions for n$4: 2c m m pm 1 ]2 5 4p m j 1 p j 12 1 2f j 11 for j 5 2, . . . ,n 2 2. n Solving the set of difference equations, we obtain 1 ] 4

pm j 5

5

H

c 3c ] ] [(2 1Œ3)A 1 (n) 1 (2 2Œ3)A 2 (n)] f d 2 f m 2 ]2 1 f d 1 3f m 1 ]2 n n

s

d

J

c c ] ] [(2 1Œ3) j22 A 1 (n) 1 (2 2Œ3) j22 A 2 (n)] f d 2 f m 2 ]2 1 f m 1 ]2 n n

s

d

for j 5 2, n,

for j 5 3, . . . ,n 2 1,

where the second line of pj m is for n$4, and ] 1 2 (2 2Œ3)n 22 A 1 (n) 5 ]]]]]]]]]]]]]]]] ] ] ] ] (2 2Œ3)[1 2 (2 2Œ3)n22 ] 1 (2 1Œ3)[(2 1Œ3)n22 2 1] ] (2 1Œ3)n22 2 1 A 2 (n) 5 ]]]]]]]]]]]]]]]] . ] ] ] ] (2 2Œ3)[1 2 (2 2Œ3)n22 ] 1 (2 1Œ3)[(2 1Œ3)n22 2 1] Notice that the mill prices are different from one another. The profit of the deviating firm is then computed as

S H

D

n c 2 p *i (Di ; }2i ) 5 ] p m2 2 f d 1 ]2 2 F 2c n n c ] ] 5 ] [(2 1Œ3)A 1 (n) 1 (2 2Œ3)A 2 (n)] f d 2 f m 2 ]2 32c n 7c 2 2 3f d 1 3f m 1 ]2 2 F. n

S

J

D (7)

Since the share of neighboring firms i61 is non-negative, the value of braces does not exceed 12c /n 2 . The sustainability condition of mill pricing equilibrium is then given by

p i* (Di ; }2i ) # p *i (Mi ; }2i ),

(8)

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where the LHS is (7), while the RHS is (4)2 . Throughout this paper, we limit our analysis to mill pricing equilibrium and discriminatory mill pricing equilibrium. In principle, there are other spatial constellations such that odd-numbered firms choose mill pricing while evennumbered ones choose discriminatory pricing. While we can prove this very structure in which firms alternate in their pricing strategies is not sustainable 3 , we can conjecture that the two different pricing strategies never coexist in general.

3.3. An example When the number of firms n is three, the sustainability conditions (6) and (8) in Section 3.1 and (3.2) can be analytically obtained as follows. For discriminatory pricing, (6) is equivalent to ] f d 2 f m Œ2 2 1 ]]] # ]] . 0.0460 c 9 For mill pricing, (8) is equivalent to ] f d 2 f m 5 2 3Œ2 ]]] $ ]]] . 0.0421. c 18 Putting these two together, we can classify the three cases in accordance with the value of the parameters and establish the following. Remark 1. When the number of firms is three, (a) if ( f d 2f m) /c ,0.0421, each firm takes the discriminatory pricing strategy; ( b) if 0.0421#( f d 2f m) /c #0.0460, each firm takes the discriminatory pricing strategy or each firm takes the mill pricing strategy; and (c) if ( f d 2f m) /c .0.0460, each firm takes the mill pricing strategy. Two implications are drawn from Remark 1. First, the discriminatory pricing is likely to take place. It prevails when the marginal costs of the discriminatory pricing firms ( f d ) are lower than those of the mill pricing ones ( f m). Moreover, it prevails even when the former marginal costs are higher insofar as the difference between the two marginal costs is small and / or the transportation costs c are large. That is, firms tend to adopt the discriminatory pricing strategy so long as the marginal costs of discriminatory pricing firms are not much higher than those of mill pricing firms. Second, comparing (4) with (2), we can observe that the profit of mill pricing is 2 It should be noted that the fixed cost F vanishes in conditions (6) and (8), but that F will affect the entry decision shown below and in Section 4. 3 Its proof is straightforward. It is available upon request to the author.

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always greater than that of discriminatory pricing. However, when ( f d 2f m) /c is small, each firm takes discriminatory pricing, leading to a smaller profit than in the case of mill pricing. As a result, each firm falls into Pareto inferior state in equilibrium like the prisoners’ dilemma. This is a similar finding by Thisse and Vives (1988).

4. Long run equilibrium and social optimum

4.1. Equilibrium number of firms So far, the number of firms n is exogenously fixed. In reality, however, n should be endogenously determined by the free entry of firms. Free entry means that an entrant firm cannot cover its fixed costs. We assume prohibitively high relocation costs 4 , whose state is called ‘spatially non-contestable’ by Norman and Thisse (1996). That is, location is assumed to be once-for-all. Given the symmetric location of incumbent firms, an entrant would locate at a midpoint between two neighboring incumbents while incumbents do not relocate. So, the free entry equilibrium is such that an entrant just fails to break even but incumbents earn positive profits 5 . In the case of discriminatory pricing, the long run equilibrium number of firms is computed as

S D

c n d 5 ]] 16F

1/3

.

(9)

This is simply obtained by setting the profit (1) of a discriminatory pricing entrant equal to zero with the three parentheses of the RHS being 1 /n31 / 2n31 / 2n. In the case of mill pricing, the long run equilibrium number of firms should be computed in a way similar to Eaton and Wooders (1985). Solving the n11 first-order conditions for maximum of (3) and setting the profit of a mill pricing entrant to zero 6 , we get the equilibrium number of firms:

S D

c n m 5 ]] 32F

1/3

B(n m)2 / 3 ,

(10)

where 4

If relocation does not incur any costs, incumbent firms would also earn zero profit with free entry. However, free relocation implies that incumbents may change their locations after the price competition, i.e., the stages of the game would be altered. 5 These characteristics were pointed out by a referee. 6 We do not have to consider a discriminatory pricing entrant since its profit is always lower than that of the deviating firm 7.

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] ] ] ] (5 1 2Œ3)(2 1Œ3)n 22 1 (5 2 2Œ3)(2 2Œ3)n22 2 32 B(n) 5 ]]]]]]]]]]]]]]] . ] ] ] ] (1 1Œ3)(2 1Œ3)n 22 1 (1 2Œ3)(2 2Œ3)n22 2 10 It should be noted that n m is not explicitly expressed since both sides of (10) contain n m . Since B(2)511 / 4 and B9(n).0, ; n$2, we can compare (9) with (10) and show that the number of firms is always larger in the case of mill pricing.

4.2. Social optimum number of firms Since the demand is inelastic, the social optimum is obtained by just minimizing the sum of the total transportation costs, the total fixed costs, and the total marginal costs. It is formulated as min TC(n) 5 2n n

E

1 / 2n 2

d

m

cx dx 1 nF 1 min( f , f ).

(11)

0

Solving this yields

S D

c 1/3 no 5 ] . (12) 6F In addition, a pricing strategy with lower marginal costs of production and retailing should be chosen.

4.3. Welfare comparisons From (9), (10) and (12) with B(n)$11 / 4, we confirm that n d ,n o ,n m . Hence, as in Norman and Thisse (1996), we can state the following: Proposition 1. There are too many firms with mill pricing but too few firms with discriminatory pricing as compared to the social optimum. Proposition 1 is against the ‘excess theorem’ by Salop (1979); Economides (1989). This is due to the assumption of the spatial non-contestability, which tends to blockade the entry of firms 7 . That is, if firms are able to price discriminatorily according to customers’ location and if relocation costs are prohibitively high, then more firms are needed. Next, let us compare the social total cost TC(n) of each pricing case by substituting (9), (10) and (12) into (11). Straightforward calculations yield 7

If relocation costs are zero (i.e., spatially contestable), then the equilibrium number of firms is 82% larger in mill pricing and 44% larger in discriminatory pricing. On the other hand, when relocation costs are prohibitively high (i.e., in our situation), it is only 12–22% larger in mill pricing and 28% smaller in discriminatory pricing.

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7 ? 22 / 3 TC(n d ) 5 ]] c 1 / 3 F 2 / 3 1 f d , 12 32 2 / 3 [8 1 3B(n m)2 ] 1 / 3 2 / 3 TC(n m) 5 ]]]]]] c F 1 f m, m 4/3 96B(n ) 62 / 3 TC(n o) 5 ]] c 1 / 3 F 2 / 3 1 ( f d , f m). 4 Directly comparing these social total costs, we obtain TC(n o),TC(n m),TC(n d ), i.e., Remark 2. Mill pricing is more efficient than discriminatory pricing. Suppose that the marginal costs are relatively small and negligible as compared to the fixed costs and the transportation costs. Then, we can approximately say from the above TCs that the social total cost is 12% higher in the discriminatory pricing equilibrium than that in the optimum. However, the social total cost is only 1.3–3.6% higher in mill pricing equilibrium, so it is very close to the optimum. Hence, we may state that when discriminatory pricing is prevalent, the government should encourage the entry of firms.

5. Transition of long run equilibrium We obtained the equilibrium number of firms in Section 4 and the sustainability conditions in Section 3. Combining the two, we obtain long run equilibrium conditions by substituting the number of firms. In other words, we substitute (9) into (6) for the discriminatory pricing, and substitute (10) into (8) for mill pricing. Namely,

1. discriminatory pricing is stable if f [(2`, f2 ]; 2. mill pricing is stable if f [[f1 , 1`); where f d2f m f ; ]]] , c 1 / 3F 2 / 3 ] ] ] 32 2 / 3 7 2 4Œ2 2 (2 1Œ3)A 1 (n) 2 (2 2Œ3)A 2 (n) ]]]]]]]]]]]] f1 5 ]] [ (0.882, 0.969), ] ] B(n)4 / 3 3 2 (2 1Œ3)A 1 (n) 2 (2 2Œ3)A 2 (n) ] f2 5 16 2 / 3 (Œ2 2 1) . 2.63.

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Note that f1 .0.882 for n53 and f1 →0.969 for n→`. Since the RHS of f1 equation contains n, the exact value of f1 cannot be computed. Nonetheless, we confirm that f1 , f2 and establish the following proposition. Proposition 2 (a) If 2`,f,f1 , each firm takes the discriminatory pricing strategy. ( b) If f1 #f#f2 , each firm takes the discriminatory or each firm takes mill pricing strategy. (c) If f2 ,f, 1`, each firm takes the mill pricing strategy. This is illustrated in Fig. 3. From Proposition 2 and Remark 2 efficiency, we observe that for f , f1 , each firm chooses discriminatory pricing in equilibrium although mill pricing is more desirable for society as a whole. This is a so-called market failure. It is intuitively explained as follows. Since the marginal costs between the two pricings do not differ much within that parameter range, the relative disadvantage of discriminatory pricing is not so major. However, discriminatory pricing is more flexible in that it can vary the delivered price for each location. Hence, discriminatory pricing becomes the dominant strategy in spite of its minor handicap in the marginal costs. We can read the impacts of parameters (c, F and f d 2f m) on the pricing strategies from Proposition 2. The impact of the difference f d 2f m on the pricing strategy is straightforward, but the impacts of c and F are not. Therefore, let us investigate further what the economic implications of the change in the two parameters are. Supposing that the mill and discriminatory pricing firms coexisted, then their market boundaries would be dependent on the marginal costs of discriminatory pricing firms ( f d ) and the mill price ( pi m). Parameter c is not related to the former value but positively related to the latter. This means that as the transportation cost rate c gets smaller, the mill price decreases. On the other hand, the cost structure

Fig. 3. Parameter change and spatial arrangements of pricing strategies.

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of the discriminatory pricing firms remains unchanged. Consequently, the mill pricing firms would expand the market shares, which would lead to their dominance as c goes down. Next, when the entry cost F became small, entry would easily take place. This would increase the number of firms n, which would indirectly contribute to a decrease in the mill price since it is negatively related to n. Therefore, a decrease in F as well as a decrease in c leads to the prevalence of the mill pricing strategy. We may say that in the case of these two cost parameters going down, the market structure would approach monopolistic competition, where many small firms such as retail stores and restaurants would be competing against one another. However, when these costs went up, mail-order firms would emerge and the market structure would become oligopolistic. In other words, low-cost goods would be sold by many mill pricing firms, while high-cost goods would be delivered by few discriminatory pricing firms. The former examples would be goods sold at convenience stores and the latter goods sold by mail-order firms. The number of convenience stores is much larger than that of mail-order firms. In this way, the cost conditions would determine the market structure. In the real world, mill pricing firms seem to be overwhelmingly prevalent in most retail industries. The value of f must therefore be large, which corresponds to small values of c and F and a large value of f d 2f m . Among these parameters, we infer that the large f d 2f m is a crucial factor for the prevalence of mill pricing strategies. Especially, the marginal costs of retailing are considered to be relatively high for discriminatory pricing firms. In general, the opportunity costs of transportation for workers are higher than those for consumers. Otherwise, firms would deliver goods rather than waiting for consumers to visit their stores 8 . This is another explanation for the above difference between convenience stores and mail-order firms. Concerning the future change in parameter values and their influence on the choice of strategies, f can be decomposed into 1 /c 1 / 3 and ( f d 2f m) /F 2 / 3 by definition. The former term is simply interpreted as the transportation technology. The latter term may be considered an inverse measure of scale economies since it is approximately a ratio of the marginal (variable) costs to the fixed costs. The question is whether these two components increase or decrease over time. The technical progress in transportation reduces the transportation cost rate c, which contributes to the shift from discriminatory to mill pricing strategies. On the other hand, the technical progress in production and in retailing enables firms to enjoy further economies of scale, which leads to the reverse shift of the strategies. That is, there are two opposing factors. If the progress in transportation technology is very rapid, then the markets will be occupied by many mill pricing firms. If, on

8

The reverse may be true for some industries competing in characteristic space, where firms produce a variety of differentiated products. In this case, the value of f d is considered to be relatively small.

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the other hand, the progress in production and retailing technology is very fast, then they will be occupied by few discriminatory pricing firms 9 . Finally, let us examine Fig. 3 in detail. It is observed that there exist multiple equilibria when f [[f1 , f2 ]. In such cases, initial conditions dictate which pricing equilibrium is realized. To see this, consider the two representative situations below. Suppose first that f is initially small but increasing over time. Discriminatory pricing prevails until it reaches f2 . When f exceeds f2 , firms alter their strategy to mill pricing, leading to mill pricing equilibrium. On the other hand, suppose f is initially large and is decreasing over time. Each firm selects mill pricing until f goes down to f1 . When f becomes smaller than f1 , each firm changes from mill pricing to discriminatory strategy all at once. Thus, we observe a so-called hysteresis in that the spatial arrangements of pricings differ between the two cases when f [(f1 , f2 ). Only within this range, government intervention is necessary.

6. Concluding remarks We have analyzed a spatial oligopoly in which firms compete in pricing strategy and in price (schedule). Consumers are uniformly distributed over the unitcircumference of a circle. Firms locating symmetrically on the circumference of a circle select their price strategies in the first stage, and choose mill prices or price schedules in the second stage. Seeking subgame perfect Nash equilibrium and later assuming spatial non-contestability, we obtained the following results. First, firms tend to adopt discriminatory pricing as a dominant strategy inasmuch as the marginal costs between the mill and discriminatory pricing strategies do not differ much. This is because discriminatory pricing firms have a double advantage: they can price discriminatorily and they are second-movers. Such a situation is not only Pareto inferior for each firm, but also a market failure from a social welfare point of view. Second, the equilibrium number of mill pricing firms is too large as compared to the social optimum one, whereas the equilibrium number of discriminatory pricing firms is too small. Third, the mill pricing strategy may become prevalent in the future if improvements in transportation technology are rapid enough. However, if the degree of scale economies in production and retailing becomes large, the discriminatory pricing strategy will become dominant. Finally, there exist multiple equilibria and a hysteresis in the spatial arrangements of pricing strategies. In this paper, we assumed that each firm is located at equal distance exogenously. What would happen if location were a strategic variable determined 9 In addition, there are other factors influencing the change in f. For example, F may decrease due to the recent tendency to apply various deregulations on entry restrictions; or the ratio of f /F may decrease due to increasing sunk costs of R&D investments.

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prior to the other stages, in other words, if we considered the game of location in the first stage, price strategy in the second stage, and mill (or discriminatory) pricing in the third stage when n53? As long as each firm selected the same pricing strategy, there would be little doubt as to the symmetric location in equilibrium. However, the mill pricing equilibrium in Section 3.2 would become different. In this case, due to the chain effect caused by the defecting firm, mill prices would differ from one another, which suggests that the location of firms would not be symmetric. In fact, we can confirm that location becomes asymmetric when there are three firms, two of which choose mill prices and one of which selects discriminatory pricing. Specifically, mill pricing firms tend to locate closer to the discriminatory firm when ( f d 2f m) /c.1 / 27 while they locate farther from the discriminatory firm when ( f d 2f m) /c,1 / 27. In other words, relative strength in terms of the marginal costs determines the equilibrium distance between the firms. The location of the firms is symmetric if and only if ( f d 2f m) /c51 / 27, whose measure is virtually zero. Hence, the endogenous location game hardly leads to symmetric locations under the coexistence of mill and discriminatory pricing strategies.

Acknowledgements I am grateful to participants of the Annual Meeting of the Japan Association of Economics and Econometrics in 1994, Workshops at the University of Tokyo and the University of Tsukuba, and the Seventh World Congress of Econometric Society in 1995. In particular, I wish to thank two anonymous referees, Michihiro Kandori, Yoshitsugu Kanemoto, Amoz Kats, Hiroshi Ohta, Masahiro OkunoFujiwara, Noboru Sakashita, Konrad Stahl and the late Yuji Kubo.

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