Pricing Policy in Spatial Competition É Takatoshi Tabuchi y January 1, 1999 Abstract In this paper, we analyze a spatial oligopoly ía la Hotelling assuming Norman and Thisse's (1996) spatial non-contestability. Each årm selects a pricing strategy in the årst 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 årms is too large in comparison to the social optimum one, whereas the equilibrium number of discriminatory pricing årms is too small. Finally, we observe a hysteresis in the spatial arrangements of pricing strategies. JEL Classiåcation: L13 Keywords: pricing policy, Hotelling, spatial competition, excess theorem, hysteresis É forthcoming in Regional Science and Urban Economics y Faculty of Economics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan. Email: ttabuchi@e.u-tokyo.ac.jp 1
1 Introduction The major focus of this paper is the pricing policy used by retail årms 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 årms apply exclusively the mill pricing strategy, while Hoover (1937) supposed spatially discriminatory pricing, in which the pricing policy is usually predetermined. However, it is the proåt-maximizing årms 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 årms (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 årms. Similar examples are found in liquor and food delivery årms. Discriminatory pricing årms 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 årms can choose mill or discriminatory pricing strategies. Spatial competition in geographic distance is often interpreted as competition in product diãerentiation in characteristics space. Some årms 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 årms to choose their pricing strategies, we study impacts of various cost diãerences on the structure of industries. In reality, mill pricing predominates in some industries while discriminatory pricing prevails in other industries. We will argue 2
that this is ascribed to the diãerences in exogenous parameters such as transportation costs, marginal costs, and åxed costs. Furthermore, we identify several reasons for the diãerence in the number of årms 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 årms (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 årms 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. 2 The Model There are n årms producing an identical good. For a moment, n is assumed to be åxed, 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 2 [0; 1]. Each consumer purchases one unit of the good from the årm oãering the lowest full price. The full price is deåned by the mill price plus the transportation costs in the case of mill pricing, and deåned by the delivered price in the case of discriminatory pricing. The transportation cost is assumed to be a quadratic function of distance with the coeécient of c. It is incurred by consumers in the case of mill pricing and by årms 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 åxed costs of entry are F. F implies the existence of scale economies in production 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. 3
and retailing. Throughout the paper, we assume that the cost parameters c and F are the same whereas f is diãerent between the mill pricing and the discriminatory pricing årms. This simpliåcation is used to indicate the situation that the unit retailing cost would be very diãerent in each case. Thus, we denote the marginal costs of discriminatory pricing årms by f d and those of mill pricing årms by f m. Given a symmetric location on the circumference of a circle, each årm simultaneously chooses selects a pricing strategy in the årst stage, and simultaneously chooses a single mill price or a delivered price schedule in the second stage. However, if neighboring årms take diãerent strategies in the årst stage, we alter the second stage game as follows. Instead of giving a simultaneous choice, we assume that the mill pricing årm becomes a price leader and the discriminatory årm is a follower reacting optimally the mill price (Thisse and Vives, 1988). Such a leader-follower relationship may be justiåed by the çexibility of the price schedule used by the discriminatory pricing årms since they could easily cut the price at each location in secret if it were proåtable. We seek subgame perfect Nash equilibrium. Due to mathematical tractability, we focus only on symmetric equilibrium although asymmetric equilibrium analysis is also important, as exempliåed by Tabuchi and Thisse (1995). In sections 2 and 3, we consider the case of aåxed number of årms. It will be endogenized in sections 4 and 5. Let us denote the proåt ofthei-th årm by ô i (S i ; S 1 ; :::; S iä1 ;S i+1 ; :::; S n ), where strategy S is either D (discriminatory pricing) or M (mill pricing). These two price variables will be deåned exactly in the next two subsections. 2.1 Discriminatory pricing If each årm chooses the discriminatory pricing strategy, the full price is equivalent to the delivered price in each location. The equilibrium price schedule of årm i is set equal to the second lowest level of the delivery (transportation) costs plus marginal costs among all årms: p d i (x) = c(x Ä x iä1) 2 + f d for x 2 [(x iä1 + x i )=2;x i ]; = c(x Ä x i+1 ) 2 + f d for x 2 [x i ; (x i + x i+1 )=2]: Note that given the symmetry of location and cost structures, the market boundaries are the midpoints between two neighboring årms. 4
For each location x, årm i has to pay the delivery costs and the marginal costs: c(x Ä x i ) 2 + f d for the range of x 2 [(x iä1 + x i )=2; (x i + x i+1 )=2]. Hence, the proåt ofårm i is given by ô É i (D i ; D Äi ) = Z (xi+1 Äx iä 1 )=2 (x i Äx iä 1 )=2 Z (xi+1 Äx i )=2 Ä 0 cx 2 dx + cx 2 dx Ä F Z (xi+1 Äx iä 1 )=2 (x i+1 Äx i )=2 Z (xi cx 2 Äx iä 1 )=2 dx Ä cx 2 dx 0 = c 4 (x i+1 Ä x iä1 )(x i+1 Ä x i )(x i Ä x iä1 ) Ä F (1) where D Äi ë (D 1 ; :::; D iä1 ;D i+1 ; :::; D n ). Due to the assumption of symmetric location, x i+1 Ä x i =1=n; 8i =1; 2; :::; n Ä 1. Therefore, 1 is reduced to ô É i (D i ; D Äi )= which is illustrated by the shaded area in Figure 1. c Ä F; (2) 2n3 2.2 Mill pricing Consider instead that every årm chooses the mill pricing strategy. The full price is the mill price plus the transportation costs, p m i + c(x i Ä x) 2. Let b i be the market boundary between årm i and årm i + 1. Since the marginal consumer located at b i is indiãerent as 5
to whether to buy a certain good at i or i +1, the neighboring full prices should be equated as: p m i + c(b i Ä x i ) 2 = p m i+1 + c(x i+1 Ä b i ) 2 : Figure 2 illustrates the situation. The proåt isthengivenby ô i (M i ; M Äi )=(p m i Ä f m )(b i Ä b iä1 ) Ä F; (3) where M Äi ë (M 1 ;:::;M iä1 ;M i+1 ; :::; M n ). Diãerentiating (3) with respect to p m i and employing Theorem 6 in Economides (1989), we have a symmetric equilibrium mill price p mé i = c=n 2 + f m and the proåt ôi É (M i ; M Äi )= c Ä F; (4) n3 both of which are the same for all årms. The shaded area in Figure 2 is equal to (4). 6
3 Short run equilibrium 3.1 Discriminatory pricing Let us investigate the sustainability of equilibrium. Suppose that every årm takes discriminatory pricing but årm i changes its strategy to mill pricing. As assumed at the beginning of the preceding section, årm i is a mill price leader while the other årms are followers using discriminatory price schedules. Thus, given p m i, the neighboring årms would undercut prices wherever possible. Anticipating this reaction, årm i, the deviating leader, maximizes its proåt withrespect to its mill price in the following manner: max ô i (M i ; D Äi )=(p m p m i Ä f m )(b i Ä b iä1 ) Ä F; i where b i = f d Ä p m i + c(x 2 i+1 Ä x2 i ) 2c(x i+1 Ä x i ) and b iä1 = pm i Ä f d + c(x 2 i Ä x2 iä1 ) 2c(x i Ä x iä1 ) for (f d Ä f m )n 2 +4 p 2cF nn î 5c: Note that the last inequality ensures that neighboring årms i Ü 1 have a non-negative proåt, which limits the proåt of deviating årm i. Computing the årst-order condition for maximum with x i+1 Ä x i =1=n 8i =1; :::; n Ä 1; we obtain the mill price of årm i as p m i (M i ; D Äi )=[c +(f d + f m )n 2 ]=2n 2, and the proåt ofårmi as ô É i (M i ; D Äi )= [c +(f d Ä f m )n 2 ] 2 4cn 3 Ä F: (5) Firm i will not choose mill pricing unless the proåt is larger than it was before. This allows us to establish a sustainability condition: ô É i (M i ; D Äi ) î ô É i (D i ; D Äi ): (6) The LHS is given by (5) while the RHS is (2). As long as (6) is satisåed, the discriminatory pricing strategy should prevail everywhere. 3.2 Mill pricing Next, consider the situation that every årm takes mill pricing but one årm i changes its strategy to discriminatory pricing. This time, unlike the previous subsection, we should 7
take a chain eãect into account when computing the proåt of the deviating årm as in Eaton and Wooders (1985). This is because anticipating the price schedule of one deviating årm, each of the other årms would change its optimal mill prices, which vary according to the distance from the deviating årm. Let i = 1 be the deviating årm without losing generality. For n ï 3, the årst-order conditions for proåt maximization are expressed as: f d + p m 3 +2f m + 2c n 2 = 4p m 2 ; p m nä1 + f d +2f m + 2c n 2 = 4p m n : In addition, there are the following årst-order conditions for n ï 4: >< p m j = >: p m j + p m j+2 +2f m + 2c n 2 =4pm j+1 for j =2; :::; n Ä 2: Solving the set of diãerence equations, we obtain 8 ö 1 [(2 + p 3)A 1 (n)+(2ä p 3)A 2 (n)] íf d Ä f m Ä c ì 4 n 2 + f d +3f m + 3c õ n 2 for j =2;n; [(2 + p 3) jä2 A 1 (n)+(2ä p í 3) jä2 A 2 (n)] f d Ä f m Ä c ì n 2 + f m + c n 2 for j =3; :::; n Ä 1; wherethesecondlineofp m j is for n ï 4, and A 1 (n) = A 2 (n) = 1 Ä (2 Ä p 3) nä2 (2 Ä p 3)[1 Ä (2 Ä p 3) nä2 ]+(2+ p 3)[(2 + p 3) nä2 Ä 1] (2 + p 3) nä2 Ä 1 (2 Ä p 3)[1 Ä (2 Ä p 3) nä2 ]+(2+ p 3)[(2 + p 3) nä2 Ä 1] : Notice that the mill prices are diãerent from one another. The proåt of the deviating årm is then computed as í ôi É (D i ; M Äi ) = n 2c = n 32c p m 2 Ä f d + c ì 2 Ä F n ö 2 [(2 + p 3)A 1 (n)+(2ä p 3)A 2 (n)] íf d Ä f m Ä c n 2 ì Ä3f d +3f m + 7c õ 2 Ä F: (7) n 2 Since the share of neighboring årms i Ü 1 is non-negative, the value of braces does not exceed 12c=n 2. The sustainability condition of mill pricing equilibrium is then given by ô É i (D i; M Äi ) î ô É i (M i; M Äi ); (8) 8
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 årms choose mill pricing while even-numbered ones choose discriminatory pricing. While we can prove this very structure in which årms alternate in their pricing strategies is not sustainable, 3 we can conjecture that the two diãerent pricing strategies never coexist in general. 3.3 An example When the number of årms n is three, the sustainability conditions (6) and (8) in subsections 3.1 and 3.2 can be analytically obtained as follows. For discriminatory pricing, (6) is equivalent to f d Ä f m c î p 2 Ä 1 9 ' 0:0460: For mill pricing, (8) is equivalent to f d Ä f m c ï 5 Ä 3p 2 18 ' 0:0421: Putting these two together, we can classify the three cases in accordance with the value of the parameters and establish the following. Remark 1 Whenthenumberofårmsisthree, (a) if (f d Ä f m )=c < 0:0421, each årm takes the discriminatory pricing strategy; (b) if 0:0421 î (f d Ä f m )=c î 0:0460, each årm takes the discriminatory pricing strategy or each årm takes the mill pricing strategy; and (c) if (f d Ä f m )=c > 0:0460, each årm 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 årms 2 It should be noted that the åxed cost F vanishes in conditions (6) and (8), but that F will aãect the entry decision shown below and in Section 4. 3 Its proof is straightforward. It is available upon request to the author. 9
(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 diãerence between the two marginal costs is small and/or the transportation costs c are large. That is, årms tend to adopt the discriminatory pricing strategy so long as the marginal costs of discriminatory pricing årms are not much higher than those of mill pricing årms. Second, comparing (4) with (2), we can observe that the proåt of mill pricing is always greater than that of discriminatory pricing. However, when (f d Ä f m )=c is small, each årm takes discriminatory pricing, leading to a smaller proåt than in the case of mill pricing. As a result, each årm falls into Pareto inferior state in equilibrium like the prisoners' dilemma. This is a similar ånding by Thisse and Vives (1988). 4 Long run equilibrium and social optimum 4.1 Equilibrium number of årms So far, the number of årms n is exogenously åxed. In reality, however, n should be endogenously determined by the free entry of årms. Free entry means that an entrant årm cannot cover its åxed 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 årms, 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 proåts. 5 In the case of discriminatory pricing, the long run equilibrium number of årms is computed as í ì c 1=3 n d = : (9) 16F This is simply obtained by setting the proåt (1) of a discriminatory pricing entrant equal 4 If relocation does not incur any costs, incumbent årms would also earn zero proåt 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. 10
to zero with the three parentheses of the RHS being 1=n Ç 1=2n Ç 1=2n. In the case of mill pricing, the long run equilibrium number of årms should be computed in a way similar to Eaton and Wooders (1985). Solving the n+1 årst-order conditions for maximum of (3) and setting the proåt of a mill pricing entrant to zero, 6 we get the equilibrium number of årms: n m = í c 32F ì 1=3 B(n m ) 2=3 ; (10) where B(n) = (5 + 2p 3)(2 + p 3) nä2 +(5Ä2 p 3)(2 Ä p 3) nä2 Ä 32 (1 + p 3)(2 + p 3) nä2 +(1Ä p 3)(2 Ä p 3) nä2 Ä 10 : It should be noted that n m is not explicitly expressed since both sides of (10) contain n m. Since B(2) = 11=4 andb 0 (n) > 0; 8 n ï 2, we can compare (9) with (10) and show that the number of årms is always larger in the case of mill pricing. 4.2 Social optimum number of årms Since the demand is inelastic, the social optimum is obtained by just minimizing the sum of the total transportation costs, the total åxed costs, and the total marginal costs. It is formulated as Z 1=2n min n TC(n) =2n cx 2 dx + nf +min(f d ;f m ): (11) 0 Solving this yields í ì c 1=3 n o = : (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) ï 11=4, we conårm 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 årms with mill pricing but too few årms with discriminatory pricing as compared to the social optimum. 6 We do not have to consider a discriminatory pricing entrant since its proåt is always lower than that of the deviating årm 7. 11
Proposition 1 is against the \excess theorem" by Salop (1979) and Economides (1989). This is due to the assumption of the spatial non-contestability, which tends to blockade the entry of årms. 7 That is, if årms are able to price discriminatorily according to customers' location and if relocation costs are prohibitively high, then more årms 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 TC(n d ) = 7 Å22=3 c 1=3 F 2=3 + f d ; 12 TC(n m ) = 322=3 [8 + 3B(n m ) 2 ] c 1=3 F 2=3 + f m ; 96B(n m ) 4=3 TC(n o ) = 62=3 4 c1=3 F 2=3 +min(f d ;f m ): 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 eécient than discriminatory pricing. Suppose that the marginal costs are relatively small and negligible as compared to the åxed costs and the transportation costs. Then, we can approximately say from the above TC's 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 to 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 årms. 5 Transition of long run equilibrium We obtained the equilibrium number of årms in section 4 and the sustainability conditions in section 3. Combining the two, we obtain long run equilibrium conditions by substituting the number of årms. 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 û2 (Ä1;û 2 ]; 7 If relocation costs are zero (i.e., spatially contestable), then the equilibrium number of årms 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 to 22% larger in mill pricing and 28% smaller in discriminatory pricing. 12
(2) mill pricing is stable if û2 [û 1 ; +1); where û ë f d Ä f m c 1=3 F 2=3 ; p p 32 2=3 7 Ä 4 2 Ä (2 + 3)A1 (n) Ä (2 Ä p 3)A 2 (n) û 1 = B(n) 4=3 3 Ä (2 + p 3)A 1 (n) Ä (2 Ä p 3)A 2 (n) û 2 = 16 2=3 ( p 2 Ä 1) ' 2:63: 2 (0:882; 0:969); Note that û 1 ' 0:882 for n = 3 and û 1! 0:969 for n!1. Since the RHS of û 1 equation contains n, the exact value of û 1 cannot be computed. Nonetheless, we conårm that û 1 <û 2 and establish the following proposition. Proposition 2 (a) If Ä1 <û<û 1, each årm takes the discriminatory pricing strategy. (b) If û 1 î û î û 2, each årm takes the discriminatory or each årm takes mill pricing strategy. (c) If û 2 <û<+1, each årm takes the mill pricing strategy. This is illustrated in Figure 3. From Proposition 2 and Remark 2, we observe that for û< û 1,eachårm 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 13
explained as follows. Since the marginal costs between the two pricings do not diãer much within that parameter range, the relative disadvantage of discriminatory pricing is not so major. However, discriminatory pricing is more çexible 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 Ä f m ) on the pricing strategies from Proposition 2. The impact of the diãerence f d Ä f 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 årms coexisted, then their market boundaries would be dependent on the marginal costs of discriminatory pricing årms ( f d ) and the mill price (p m i ). 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 of the discriminatory pricing årms remains unchanged. Consequently, the mill pricing årms 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 årms 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 årms such as retail stores and restaurants would be competing against on another. However, when these costs went up, mail-order årms would emerge and the market structure would become oligopolistic. In other words, low-cost goods would be sold by many mill pricing årms, while high-cost goods would be delivered by few discriminatory pricing årms. The former examples would be goods sold at convenience stores and the latter goods sold by mail-order årms. The number of convenience stores is much larger than that of mail-order årms. In this way, the cost conditions would determine the market structure. In the real world, mill pricing årms seem to be overwhelmingly prevalent in most retail industries. The value of û must therefore be large, which corresponds to small values of 14
c and F and a large value of f d Ä f m. Among these parameters, we infer that the large f d Ä f 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 årms. In general, the opportunity costs of transportation for workers are higher than those for consumers. Otherwise, årms would deliver goods rather than waiting for consumers to visit their stores. 8 This is another explanation for the above diãerence between convenience stores and mail-order årms. Concerning the future change in parameter values and their inçuence on the choice of strategies, û canbedecomposedinto1=c 1=3 and (f d Ä f m )=F 2=3 by deånition. 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 åxed 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 årms 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 årms. If, on the other hand, the progress in production and retailing technology is very fast, then they will be occupied by few discriminatory pricing årms. 9 Finally, let us examine Figure 3 in detail. It is observed that there exist multiple equilibria when û2 [û 1 ;û 2 ]. In such cases, initial conditions dictate which pricing equilibrium is realized. To see this, consider the two representative situations below. Suppose årst that ûis initially small but increasing over time. Discriminatory pricing prevails until it reaches û 2. When û exceeds û 2,årms alter their strategy to mill pricing, leading to mill pricing equilibrium. On the other hand, suppose û is initially large and 8 The reverse may be true for some industries competing in characteristic space, where årms produce a variety of diãerentiated products. In this case, the value of f d is considered to be relatively small. 9 In addition, there are other factors inçuencing the change in û. 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. The 15
is decreasing over time. Each årm selects mill pricing until û goes down to û 1. When û becomes smaller than û 1,eachårm changes from mill pricing to discriminatory strategy all at once. Thus, we observe a so-called hysteresis in that the spatial arrangements of pricings diãer between the two cases when û2 (û 1 ;û 2 ). Only within this range, government intervention is necessary. 6 Concluding remarks We have analyzed a spatial oligopoly in which årms compete in pricing strategy and in price (schedule). Consumers are uniformly distributed over the unit-circumference of a circle. Firms locating symmetrically on the circumference of a circle select their price strategies in the årst 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, årms tend to adopt discriminatory pricing as a dominant strategy inasmuch as the marginal costs between the mill and discriminatory pricing strategies do not diãer much. Thisisbecausediscriminatorypricingårm have a double advantage: they can price discriminatorily and they are second-movers. Such a situation is not only Pareto inferior for each årm, but also a market failure from a social welfare point of view. Second, the equilibrium number of mill pricing årms is too large as compared to the social optimum one whereas the equilibrium number of discriminatory pricing årms 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 årm is located at equal distance exogenously. What would happen if location were a strategic variable determined prior to the other stages, in other words, if we considered the game of location in the årst stage, price strategy in the second stage, and mill (or discriminatory) pricing in the third stage when n =3? As long as each årm selected the same pricing strategy, there would be little doubt as to the symmetric location in equilibrium. However, the mill pricing equilibrium in subsection 3.2 16
would become diãerent. In this case, due to the chain eãect caused by the defecting årm, mill prices would diãer from on another, which suggests that the location of årms would not be symmetric. In fact, we can conårm that location becomes asymmetric when there are three årms, two of which choose mill prices and one of which selects discriminatory pricing. Speciåcally, mill pricing årms tend to locate closer to the discriminatory årm when (f d Äf m )=c > 1=27 while they locate farther from the discriminatory årm when (f d Ä f m )=c < 1=27. In other words, relative strength in terms of the marginal costs determines the equilibrium distance between the årms. The location of the årms is symmetric if and only if (f d Äf m )=c =1=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. Acknowledgments 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 Okuno-Fujiwara, Noboru Sakashita, Konrad Stahl and the late Yuji Kubo. References [1] Eaton, B.C. and M.H. Wooders (1985) \Sophisticated entry in a model of spatial competition," Rand Journal of Economics 16, 282-297. [2] Economides, N. (1989) \Symmetric equilibrium existence and optimality in diãerentiated product markets," Journal of Economic Theory 47, 178-194. [3] Hoover, E.M. (1937) \Spatial price discrimination," Review of Economic Studies 4, 182-191. [4] Hotelling, H. (1929) \Stability in competition," Economic Journal 39, 41-57. 17
[5] Norman, G. and J.-F. Thisse (1996) \Product variety and welfare under discriminatory and mill pricing policies," Economic Journal 106, 76-91. [6] Salop, S. (1979) \Monopolistic competition with outside goods," Bell Journal of Economics 10, 141-156. [7] Tabuchi, T. and J.-F. Thisse (1995) \Asymmetric Equilibria in Spatial Competition," International Journal of Industrial Organization 13, 213-227. [8] Thisse, J.-F. and X. Vives (1988) \On the strategic choice of spatial price policy," American Economic Review 78, 122-137. 18