interregional mixed duopoly

Regional Science and Urban Economics 39 (2009) 233–242

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Regional Science and Urban Economics

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Interregional mixed duopoly

Tomohiro Inoue ⁎, Yoshio Kamijo, Yoshihiro TomaruFaculty of Political Science and Economics, Waseda University, 1-6-1, Nishi-waseda, Shinjuku-ku, Tokyo 169-8050, Japan

⁎ Corresponding author. Tel.: +81 3 3203 7391.E-mail address: [email protected] (T. Inoue)

0166-0462/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.regsciurbeco.2008.10.001

a b s t r a c t
a r t i c l e i n f o
Article history:
We investigate an interregi Received 17 December 2007Received in revised form 8 October 2008Accepted 10 October 2008Available online 25 October 2008
JEL classification:H42L13R32

Keywords:Interregional mixed duopolyLocal public firmSpatial model

onal mixed duopoly wherein a local public firm competes against a private firm.We employ a spatial model with price competition. The public firm is owned by the local government of theleft half of the linear city called Region 1, and maximizes its welfare. We demonstrate that our two-stagegame comprising location choice and price competition has two types of equilibria. In one equilibrium (E1),the local public firm locates in Region 1, and the private firm locates outside the region. In the otherequilibrium (E2), both firms are located in Region 1. We find that although the two firms are closely located inE2, E2 payoff-dominates E1. Moreover, E2 is robust in the sense that the sequential choice of location adoptsthis equilibrium, regardless of whether the public firm is a leader or a follower.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

During the recent wave of privatization, not only state-ownedfirms but also local public firms have been privatized. Nevertheless,local public firms still exist in many developing countries as well as indeveloped countries. This is because they usually provide essentialservices such as natural gas, electricity, water, medical facilities, andeducation. In most cases, such goods and services are also provided byprivate firms. The purpose of this paper is to investigate mixedmarkets wherein private and local public firms compete.

Competition among public and private firms has been studied inliterature on mixed oligopolies (e.g., De Fraja and Delbono, 1989). Itusually assumes one country or one market in which one public firmand several private firms compete, and it examines the effect of theprivatization of the public firm on social welfare. Thus, the literaturehas not established an appropriate model reflecting the behaviors oflocal public firms in a country comprising a number of regions orprovinces. Certainly, a few previous studies such as Fjell and Pal(1996), Pal andWhite (1998), andMatsushima andMatsumura (2006)have investigated the effect of imports from foreign firms on thedomestic mixed market. If we regard the domestic and foreigncountries as provinces or counties, it appears that some worksanalyzed mixed markets, which include a local public firm. However,in the real world, local public firms in one region often supply goodsand services to consumers in other regions. In fact, state (or city)

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universities, local airports, and city hospitals supply services toresidents of other regions. For example, in Japan, Yokohama CityUniversity, which is owned and managed by Yokohama City, admitsnot only students who live in Yokohama but also those hailing fromthe other regions. Another example is Kobe Airport owned by KobeCity, which is a representative airport in the Kansai area of Japan. Inthis paper, we establish amodel wherein a local public firm in a regioncompetes against a private firm and supplies goods and/or services toconsumers who live outside the region.

For this purpose, we employ a Hotelling-type spatial model(Hotelling, 1929) in which the population is dispersed and eachconsumer has a specific personal address on the line with a length ofunity (hence, the so-called linear city). In this model, a firm locates at apoint on the line, and the purchase of goods from one of them involvestransportation costs that vary according to the consumer's location.Since consumers have to incur the transportation costs of goods, theyselect a firm to purchase goods from, taking into account thetransportation costs in addition to prices. Studies onmixed oligopoliesusing a spatial model have been conducted earlier (see, e.g., Cremeret al., 1991; Matsumura and Matsushima, 2003, 2004; Matsushimaand Matsumura, 2003, 2006). Cremer et al. (1991) conducted apioneering work on spatial mixed oligopoly, in which they assumedthat the state-owned and private firms exist in a linear city and decidetheir own locations and prices. We extend their model by dividing thecity into two symmetric districts, Regions 1 and 2, each of which is runby a local government, and thus, a firm owned by the government isregarded as a local public firm. We assume that the local governmentof Region 1 owns the public firm, and the owners of the private firmreside in Region 2. In addition, we assume that the local public firm

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http://dx.doi.org/10.1016/j.regsciurbeco.2008.10.001

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1 Ohsawa (1999) also considers the regional division of the linear city in the contextof tax competition.

2 We implicitly assume that each consumer derives a surplus from consumptionequal to s, which is so large that every consumer consumes one unit of the product.However, the value of s is irrelevant to the result. Thus, we omit the surplus as withCremer et al. (1991).

234 T. Inoue et al. / Regional Science and Urban Economics 39 (2009) 233–242

aims at maximizing local welfare in Region 1 and that the local welfaredoes not include the profit of the private firm.

In the above setting, we find that our model of location choice andprice competition has multiple equilibria. In one equilibrium(henceforth, we refer to this equilibrium as E1), the local public firmlocates in the region run by the government, and the private firmlocates outside the region. This equilibrium well fits hospitals, whichexist ubiquitously regardless of whether they are owned by the publicor private sector. In this equilibrium, the local public firm suppliesgoods and services to all the residents in Region 1. Moreover, this firmalso provides goods to some of the residents in Region 2. In the otherequilibrium (E2), both firms locate in Region 1. In Japan, variousuniversities including private and local public universities agglomer-ate in large cities such as Tokyo, Osaka, and Kobe. Such universitiespresent an example of equilibrium E2. Furthermore, in contrast to E1,goods are supplied to a large number of the consumers in Region 1 bythe private firm, and the local public firm monopolizes the demand ofthe residents in Region 2.

The results of our paper are very peculiar compared to those of theexisting works. d'Aspremont et al. (1979) show that in privateduopoly, one firm is located at one endpoint of the linear city, andthe other firm is located at the other endpoint. Cremer et al. (1991)investigated the mixed duopoly model wherein a private firmcompetes against a state-owned firm that maximizes the socialwelfare of the entire linear city. They show that one firm is located atpoint 1/4 and the other is located at point 3/4 of the city, whichindicates that competition between the state-owned and private firmsyields the first-best locational configuration. The difference betweenthe result of our paper and those of the existing works arises fromthe fact that the local public firm in our model takes into accountonly the benefits of residents in one region (Region 1). This impliesthat the local public firm has two incentives; one is to decrease thetransportation costs of the residents in Region 1 and the other is toincrease its profits from Region 2. Due to these two incentives, whichdo not appear in the existingworks, our result differs from those of theexisting works.

Other interesting features of the multiple equilibria are thatequilibrium E2 payoff-dominates equilibrium E1, and that the socialwelfare of the entire city is larger in E1 than in E2. The reasons forthese occurrences are as follows. In equilibrium E2, the local publicfirm sets a higher price to earn higher profits from Region 2 since itmonopolizes the demand of Region 2. As a result, not only the localpublic firm but also the private firm enjoys higher profits due tostrategic complementarity in the price-setting stage. Since the profitof the public firm in E2 is so large that it should increase the localwelfare to a level higher than that in E1, E2 is payoff-dominant to E1.Moreover, the residents in Region 2 incur higher transportation costsin E2 due to the one-sided location of both firms, which results inlower social welfare in E2 than in E1.

As shown in Matsumura et al. (2005), in the context of the spatialcompetition, in the sequential-move model, the efficient equilibriumis chosen from among the multiple equilibria in the simultaneous-move model. We investigate the sequential location choice game inour setting. Similar to Matsumura et al. (2005), we find that E2 ischosen in the sequential-move game, although it is not efficient fromthe social welfare viewpoint. Further, this is robust in the sense that E2is chosen regardless of whether the public firm is a leader or afollower.

Our results related to the order of moves have the followingsignificances in the literature on mixed oligopoly as well as that onpure oligopoly. First, whether a public firm should become a leader ora follower has been discussed by several researchers in the context ofmixed oligopoly such as Matsumura and Matsushima (2003) andOgawa and Sanjo (2007). They show that the equilibrium locationpattern is different between the public leadership and the privateleadership in the location choice. In contrast to these studies, we show

that the same equilibrium location pattern arises, regardless ofwhether the public firm is a leader or a follower. Second, themultiplicity of equilibria in the simultaneous location choice case isresolved in the case of sequential location choice in our model. That is,the sequential location choice serves as an equilibrium selectionbetween E1 and E2. Similar results are observed in Matsumura et al.(2005) who consider a shipping model with quantity-setting incircular markets. Finally, we also consider the issue of the endogenousorder of moves in mixed oligopoly by using the observable delay gameof Hamilton and Slutsky (1990), for example, Pal (1998b) for Cournotcompetition and Bárcena-Ruiz (2007) for Bertrand competition. Weshow that in equilibrium, two types of Stackelberg competition(public leadership and public followership) arise.

There exist some studies that also consider public firms that supplygoods to consumers outside their region without using a spatialmodel. Bárcena-Ruiz and Garzón (2005b) analyze the model wheretwo regions (or countries) trade with one another and theirgovernments strategically decide whether to privatize their publicfirms. They show that the decision of privatization and the tradepatterns are determined by the difference in the marginal costsbetween the local public and private firms. Using a similar model,Bárcena-Ruiz and Garzón (2005a) analyze whether national govern-ments should decidewhether to privatize public firms or whether thisdecision should be delegated to a supra national authority. Since thesetwo studies use non spatial models, they do not discuss firms' locationpatterns, which can be abundantly analyzed in our spatial model.Meanwhile, some works investigate a spatial model with pluralregions. Tharakan and Thisse (2002) analyze the model in which tworegions are divided by a boundary point on the linear city, and eachregion has a private firm. However, they assume that each private firmlocates at the center of its region, although in our model, firms'locations are determined endogenously, and they can locate in eithercountry.1

The remainder of this paper proceeds as follows. In Section 2, weexplain the basic framework of the spatial model. In Section 3, we firstexplore the subgame perfect equilibrium for the two-stage game: Inthe first stage, a local public firm and a private firm choose theirlocation, and in the second stage, they compete in price. We thendiscuss the properties of the two types of equilibria. In Section 4, weextend the basic model to a sequential-move game. In Section 5, weoffer some concluding remarks and discuss possibilities for futureresearch.

2. Model

A linear city represented by the interval [0, 1] exists, andconsumers are uniformly distributed with a unit density in the city.We assume two regions that divide this city into two symmetric areas.These areas [0, 1/2) and [1/2, 1] are referred to as Regions 1 and 2,respectively.

There are two firms—A and B—that produce a homogeneous goodat the same constantmarginal production cost. Here, we introduce theassumption of zero marginal production cost to simplify the analysisbecause our results do not depend on it. Each consumer purchases oneunit of the good from the firm offering the lowest full price, defined asthe mill price charged by the firm plus the transportation costbetween the firm and the consumer.2 Thus, the demand is perfectlyinelastic. Let a∈ [0,1] and b∈ [0,1] denote the locations of Firms A andB, respectively. The mill price of Firm i is Pi∈ [0, ∞) (i=A, B), and the

S

Table 1Classification of W1

∏A C1 T1

Case 1 abb x≥1/2PAx

PA/2 ∫1=20 t a−zð Þ2 dzCase 2 abb xb1/2 PAx+PB (1/2−x) ∫x0 t a−zð Þ2 dz + ∫1=2x t b−zð Þ2 dzCase 3 a>b x≥1/2

PA (1−x)PB/2 ∫1=20 t b−zð Þ2 dz

Case 4 a>b xb1/2 PBx+PA (1/2−x) ∫x0 t b−zð Þ2 dz + ∫1=2x t a−zð Þ2 dz

Case 5 a=bPAbPB PA PA/2

∫1=20 t a−zð Þ2 dz = ∫1=20 t b−zð Þ2 dz� �

PA=PB PA/2 PA/2 (=PB/2)PA>PB 0 PB/2

235T. Inoue et al. / Regional Science and Urban Economics 39 (2009) 233–242

transportation cost is quadratic in distance. Then, for example, the fullprice the consumer residing at point y bears equals the mill price PAplus the transportation cost t (y−a)2 when he purchases the goodfrom Firm A. The value of t (>0) does not affect the results obtainedfrom our analysis.

We assume that each region is ruled by a local government, andFirm A is owned by the local government of Region 1, that is, Firm A isthe local public firm of Region 1. The other firm, Firm B, is a privatefirm owned by private shareholders in Region 2.3 Since our mixedduopoly represents the competition between a local public firm and aprivate firm from outside the region, we describe it as an interregionalmixed duopoly.

When a≠b, for a consumer residing at

x =a + b2

+PA−PB2 a−bð Þt ; ð1Þ

the full price of purchasing from either of the two firms is the same.Thus, this point denotes the boundary of the demand for each firm. IfFirm A is located at the left of Firm B, that is, abb, the consumers wholive on the left-hand side of x purchase from Firm A, whereas thoseliving on the right-hand side of x purchase from Firm B, and vice versa.Accordingly, Firms A and B face the demands given by

DA PA; PB; a; bð Þ =f x if a b b;1−x if a > b;0 if a = b and PA > PB;12 if a = b and PA = PB;1 if a = b and PA b PB;

DB PA; PB; a; bð Þ = 1−DA PA; PB; a; bð Þ;where, in the case of both firms locating at the same point (a=b), allthe consumers purchase from the firm offering a lower price becausethe transportation costs are the same for both firms. When both firmsset the same price, we assume that the total demand is equally dividedbetween them.

Social welfare is defined by

W =∏A +∏B−PADA PA; PB; a; bð Þ−PBDB PA; PB; a; bð Þ−T

=−∫x0t a−zð Þ2dz−∫1x t b−zð Þ2dz if a V b;−∫x0t b−zð Þ2dz−∫1x t a−zð Þ2dz otherwise;

(

where PiDi denotes the sum of the burden of the mill price from Firmi; T, the total transportation cost; and∏i, the profit of each firm, whichis given by

∏i = PiDi PA; PB; a; bð Þ i = A;B:

Since individual demands are perfectly inelastic, positive prices(i.e., the prices above marginal costs) do not distort the allocation ofresources. Thus, themaximization of social welfare is equivalent to theminimization of the total transportation cost.

We assume that the local public firm maximizes the local welfareof its own region, whereas the private firm maximizes its own profit.Thus, Firm A maximizes the following local welfare of Region 1.

W1 =∏A−C1−T1;

where C1 denotes the sum of their price burden, and T1, the sum of thetransportation costs borne by the residents of Region 1. Note that ∏A,C1, and T1 vary with the locations of the two firms and thecorresponding boundary x. In addition, when both firms locate atthe same point (a=b), the local welfare of Region 1 also depends on

3 As described later, the local welfare of Region 1 does not include the profit of FirmB due to this assumption.

the prices set by the firms. Thus, we describe these relations in Table 1.Henceforth, W1 in Case j is denoted by Fj (j=1, 2, 3, 4, 5).

Since social welfare represents the sum of the local welfare of thetwo regions, the local welfare of Region 2 is given by

W2 =W−W1:

We consider the following two-stage game: In the first stage, eachfirm chooses its location simultaneously, and in the second stage, thefirms choose their prices simultaneously, having observed theirlocations chosen in the first stage. We assume that each firm canlocate at any point in the interval [0, 1] without any restriction.4 Weuse a subgame perfect equilibrium as our solution concept, and thus,the game is solved backwards.

3. Results

3.1. Price-setting

In the second stage, Firms A and B compete with respect to price inthe given locations. Since their objectives vary with their locations, toanalyze the equilibrium, we should separate them into three cases:(I) a>b, (II) bba, and (III) a=b. Then, we obtain the equilibrium pricesas shown in the following lemma.

Lemma 1. The equilibrium prices in the second stage are as follows:

(I) a b b : P1A = −

a−bð Þ a + bð Þt3 ; P1

B = −a−bð Þ 3−a−bð Þt

3 :

(II) a > b : f P3A =

a−bð Þ 4−a−bð Þt3

; P3B =

a−bð Þ 2 + a + bð Þt3

if a + b > 1;

P4A = a−bð Þt; P4

B =a−bð Þ 1 + a + bð Þt

2otherwise:

(III) a = b : P5A = 0; P5

B = 0

uperscript j of Pij corresponds to Case j in Table 1 (j=1, 3, 4, 5).

Proof. See Appendix. □Here, we present some remarks on Lemma 1. First, when Firm A

locates in Region 1 and to the left of Firm B (abb), Firm A becomes atough player in the price-setting game because most of itscustomers are Region 1 residents, and it has no incentive to increaseits profit fromRegion 1,which is offset by the decrease of the consumersurplus of Region 1. Thus, it does not hesitate to charge a low price. Infact, PA1−PB1=(a−b)(3−2a−2b) t/3b0 when a≤1/2. On the other side,however, Firm A may charge a higher price than Firm B when a>b. Infact, PA4−PB4=(a−b)(1−a−b) t/2>0. This is because, in this case, Firm Aattempts to earn a large amount of profit from Region 2.

Second, except for (III), Firm A consistently has customers inRegion 2. It is notable that, in (I), the boundary x is greater than 1/2(Case 2 does not realize in equilibrium). The reason is that since, in

4 This assumption allows the local public firm to locate outside its home region.However, Firm A does not locate in the outside region in equilibrium. This is becausethe firm has an incentive to reduce the transportation costs of the residents in Region1, as will be described in detail later.

Fig. 1. The ranges of cases in the second-stage equilibrium.


contrast to the profit earned from Region 1, the increase of its profitfrom Region 2 improves the local welfare of Region 1, Firm A sets a lowprice to capture the demand from Region 2.

Finally, similar to other spatial models such as those presented byd'Aspremont et al. (1979) and Cremer et al. (1991), the equilibriumprices represents the increasing functions of the distance betweenthe two firms. For example, PA1 and PB

1 are decreasing functions of a,whereas they are increasing functions of b.

In the next subsection, we consider the location problem in thefirst stage.

3.2. Location choice

Now, we consider the location choices of the two firms in the firststage. In this stage, the objective functions of both firms changeaccording to their locations. Thus, we represent Fig. 1 based onLemma 1 (Case 5 is on the line a=b). This figure shows the locationpair that establishes each case in the second-stage equilibrium.

If Firm B is located at [0, 1/2), as Firm A moves toward the right(that is, a increases), the price of Firm A changes from PA

1 to PA4 at a=b

and from PA4 to PA

3 at a=1−b. If Firm B is located at [1/2, 1], it changesfrom PA

1 to PA3 at a=b as a increases.5 Accordingly, in order to obtain

the reaction function of Firm A, we need to distinguish between thetwo cases. When b∈ [0, 1/2), the objective of Firm A is given by

W1 =

F1 = P1A x1− 1

2

� �−∫1=20 t a−zð Þ2dz if a V b;

F4 =P4A2 −P4

Bx4−∫x40 t b−zð Þ2dz−∫1=2x4 t a−zð Þ2dz if b b a V 1−b;

F3 = P3A 1−x3� �

− P3B2 −∫1=20 t b−zð Þ2dz otherwise;

8>><>>: ð2Þ

where xj denotes the boundary in Case j (j=1, 3, 4).6 When b∈ [1/2, 1],it is given by

W1 =F1 = P1

A x1− 12

� �−∫1=20 t a−zð Þ2dz if a V b;

F3 = P3A 1−x3� �

− P3B2 −∫1=20 t b−zð Þ2dz otherwise:

(ð3Þ

5 Henceforth, we consider Pi5 as Pi

1 when a=b (i=A, B). This does not affect ourresult.

6 The boundaries in the second-stage equilibrium are as follows:

x1 =3 + a + b

6; x3 =

2 + a + b6

; x4 =1 + a + b

4:

From Eqs. (2) and (3), we can obtain the reaction function of Firm Aas follows:

RA bð Þ =10−b−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi73−20b + 4b2

p

3if b b b;

−18−2b +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi378 + 72b + 16b2

p

6otherwise;

8>><>>: ð4Þ

where b≈0.366. Following the same procedure, we can also obtain thereaction function of Firm B as follows:

RB að Þ = 1 if a b a;0 otherwise;

�ð5Þ

where ā≈0.380. See Appendix for the derivation of reaction functions(4) and (5).

Fig. 2 describes the reaction functions RA(b) and RB(a). RA(b) isjumped at b=b and RB(a) is jumped at a=ā. As shown in this figure,our model has two subgame perfect equilibria, E1 and E2. Let (ai⁎, bi⁎)denote the pair of equilibrium location points in Ei (i=1, 2). We havethe following proposition.

Proposition 1. There are two subgame perfect equilibria, E1 and E2, inthe two-stage game. The location points, prices, and the boundaries inequilibrium are as follows:

E1fa⁎1 =−20 +

ffiffiffiffiffiffiffiffiffi466

p

6≈ 0:265

b⁎1 = 1

PA a⁎1; b⁎1

� �=

−14 +ffiffiffiffiffiffiffiffiffi466

p� �26−


p� �t

108≈ 0:310t

PB a⁎1; b⁎1

� �=

32−ffiffiffiffiffiffiffiffiffi466

p� �26−


p� �t

108≈ 0:426t

x⁎1 =4 +


p

36≈ 0:711

E2fa⁎2 =10−

ffiffiffiffiffiffi73

p

3≈ 0:485

b⁎2 = 0

PA a⁎2; b⁎2

� �=

10−ffiffiffiffiffiffi73

p� �t

3≈ 0:485t

PB a⁎2; b⁎2

� �=

13−ffiffiffiffiffiffi73

p� �10−


p� �t

18≈ 0:360t

x⁎2 =13−


p

12≈ 0:371

where xk⁎ denotes the boundary in equilibrium Ek (k=1,2).

Fig. 2. The reaction curves in the first-stage.

7 If the shareholders of Firm B reside in Region 1, the local welfare of Region 1includes its profit. Then, the market share effect and the payment effect vanish, andthus, the price-raising effect relatively increases. Therefore, the distance between thetwo firms increases, and the prices are raised. For details on the equilibrium, see Inoueet al. (2008).

Table 2Equilibrium comparison

∏A2 ∏B1 T1 T2 W1

(=∏A2−∏B1−T1)∏B W

(=−T1−T2)W2

E1 0.065t 0 0.011t 0.033t 0.055t 0.123t −0.044t −0.099tE2 0.243t 0.134t 0.018t 0.045t 0.091t 0.134t −0.063t −0.154t


Proposition 1 shows that our two-stage game has two types ofequilibria. Each firm is located in its home region in E1, whereas in E2,both firms are located in Region 1 whose government owns Firm A.

As shown by d'Aspremont et al. (1979), if both firms are private,they locate at both edges of the linear city. However, Proposition 1shows that this is not the case when one of the firms is owned by alocal government. Our local public firm, Firm A, hasW1 as its objectivefunction. Let∏ij denote the profit of Firm i earned from the consumersin Region j (i=A, B; j=1, 2), and Dij denote the demand for Firm i fromthe residents of Region j (in other words, Dij denotes the market shareof Firm i in Region j). Then, W1 can be rewritten as

W1 =∏A2−∏B1−T1 = PADA2−PBDB1−T1: ð6Þ

Totally differentiating this function, we obtain

dW1 =DA2dPA + PAdDA2−d PBDB1ð Þ−dT1: ð7Þ

This equation states that the local welfare of Region 1 improves ifthe price PA and market share DA2 increase or if the transportationcost T1 or the payment to Firm B by residents of Region 1, PBDB1,decreases. Henceforth, we refer to the terms DA2dPA, PAdDA2, and dT1as the price-raising effect, market share effect, and transportation costeffect, respectively. We also term d (PBDB1) as the payment effect.

We now explain why the locations in Proposition 1 represent theequilibrium locations. First, we consider equilibrium E1. To examinethis equilibrium, we analyze what happens if Firms A and B locate atpoints 0 and 1, respectively, in the first stage. In this case, by Eq. (1) andLemma 1, the demand for Firm A is given by DA=x1=2/3. Hence, theresidents of Region 1 purchase the goods from only Firm A, andwe canreduce Eq. (7) as follows:

dW1 =DA2dPA + PAdDA2−dT1:

First, the transportation cost effect provides Firm A with anincentive to move to point 1/4 where the transportation costs inRegion 1 are minimized. Further, at point 1/4, Firm A moves itslocation toward the right because this move leads to an expansion ofits market share in Region 2 through the market share effect. It iscertain that Firm A has another incentive to move toward the left toavoid the severe price competition (the price-raising effect). However,this price-raising effect is small since local welfare W1 is free of theinfluence of Firm A's profit from Region 1, which weakens the price-raising effect. In fact, when evaluated at a=1/4 and b=1 (thetransportation cost effect vanishes), from Lemma 1,

AW1

Aa=DA2

APAAa

+ PAADA2

Aa=

5t288

> 0:

When a=a1⁎, we find that ∂W1/∂a=0, and thus, Firm A is located ata=a1⁎. In addition, Firm B has an incentive to remain at b=1. If Firm Bmoves toward the left from b=1, it faces tough competition, whichreduces its profits. Hence, (a, b)=(a1⁎, 1) is an equilibrium outcome.

What if Firm A passes through point a1⁎ and arrives at point ā? Inthis case, the competition becomes more severe if Firm B remains atb=1. Suppose that Firm B moves to b=0 in order to avoid such severecompetition. Then, Firm A sets a higher price because only Firm Asupplies to the residents of Region 2 and because it does not take intoaccount their benefits. This enables Firm B to set a higher price due tothe strategic complement in the price-setting stage. As a result, Firm Bcan earn higher profits. Accordingly, it has an incentive to move topoint 0. In addition, under (a, b)=(ā, 0), we find that PBDB1>0 becauseFirm B is located at the left of Firm A, and that the market share effectPAdDA2 is zero because only Firm A supplies to Region 2. Therefore, weobtain

AW1

Aa=DA2

APAAa

−A

AaPBDB1ð Þ−AT1

Aa≈ 0:117t > 0;

that is, the price-raising effect dominates the payment and transporta-tion cost effects, which indicates that Firm A moves toward the right.Firm A is located at a=a2⁎ at which the three effects are balanced.Furthermore, Firm Bwants to remain at b=0, because the competitiongetsmitigated as Firm Amoves away from Firm B. Thus, the pair (a, b)=(a2⁎, 0) represents the other equilibrium outcome.7

3.3. Equilibrium comparison

In this subsection, we compare two equilibria, E1 and E2. Table 2describes the values of equilibrium payoffs of Firms A and B,equilibrium social welfare, equilibrium local welfare of Region 2,and other relevant variables, respectively.

From this table, we observe that E2 is preferable to E1 for bothfirms. The reason for this is as follows. Although in E1, the two firmsstay away from each other (the distance between the location of FirmsA and B is |a1⁎−b1⁎|≈0.735), Firm B faces severe competition from FirmA in the price-setting stage because, as well-described in Eq. (6),Firm A has a strong incentive to explore the demand from Region 2.On the other hand, despite the close distance between the twofirms (|a2⁎−b2⁎|≈0.485) in E2, the competition between the twofirms is milder in E2 than in E1. This is because, since Firm Amonopolizes the market in Region 2 in E2, it maintains a relatively highlevel of price, and allows a large share of Firm B in Region 1 in order toearn higher profits from Region 2. As a result, the share of Firm B isgreater in E2 than in E1 (i.e., x2⁎−0≈0.371>0.289≈1−x1⁎), and the profitof Firm B is also larger in E2 than in E1.

From the above discussion, it is easily understood that the payoff ofFirm A or the local welfare of Region 1 is larger in E2 than in E1. On theone hand, Firm A's profit from Region 2 is much greater in E2 than inE1 because it monopolizes the market of Region 2 in E2. On the otherhand, residents in Region 1 incur more transportation costs in E2 thanin E1, and a large number of residents in Region 1 incur high expensesfor purchasing goods from Firm B. However, the former positive effecton the local welfare is so large that it should overcome the com-bination of the latter two negative effects on the local welfare, andthus, the local welfare of Region 1 is larger in E2 than in E1.

A particular feature of E2 is that in this equilibrium, both firmsobtain benefits at the expense of consumer surplus in Region 2. On theone hand, Firm A receives benefits from Region 2 by selling theproducts at a high price as shown by Proposition 1. On the other hand,Firm B, which locates at the leftmost point in Region 1, receivesbenefits because it can sell a larger volume in E2 than in E1 due to thehigh price set by Firm A. Consequently, the residents of Region 2 facedual hardships. First, they have to bear with the high prices set by FirmA, and second, the transportation costs incurred by them are very highsince both firms are located in Region 1. Thus, the local welfare ofRegion 2 in E2 is lower than that in E1. In fact, these hardships forRegion 2 are so severe that in E2, not only the local welfare of Region 2but also social welfare decreases as compared to E1, as shown inTable 2. These results are summarized in the following proposition.


Proposition 2. Equilibrium E2 is preferable to equilibrium E1 for the twofirms in terms of their payoffs. However, social welfare in E2 is lower thanthat in E1.

In themixed duopolywith a state-owned firm (Cremer et al., 1991),the social welfare of the entire city is maximized. On the other hand, inthe duopoly with private firms (d'Aspremont et al., 1979), socialwelfare decreases because of a larger distance between the two firms.In contrast, in our interregional mixed duopoly with a local publicfirm, social welfare is intermediate between their models, regardlessof E1 or E2.

4. Sequential location choice

In the preceding section, we analyzed a simultaneous-move gamein which a local public firm competes with a private firm. In thissection, we investigate whether the timing of entry of a firm into themarket is of any consequence.

In our model, whether Firm B is a private firm of Region 2 or aforeign private firm does not affect the behavior of the local publicfirm because the local welfare does not include the profit of the privatefirm. Thus, we can consider our interregional mixed duopoly as aninternational mixed duopoly.8 One scenario is that the public firm hasan advantage over the foreign firm in entering the market. Thus, weconsider the timing as follows: First, the public firm (Firm A) choosesits location. Second, having observed the location of the public firm,the private firm (Firm B) chooses its location. After the sequentialchoice of location, the two firms simultaneously set their prices.

In this sequential location choice game, Firm A chooses its locationby considering the best response behavior of Firm B against itslocation. Thus, the equilibrium exists on Firm B's best response curve,which comprises two vertical lines, as shown by Fig. 2. Solving thissequential game, we obtain the following proposition.

Proposition 3. In the subgame perfect equilibrium of the sequential-move game of the public leader, first, the local public firm is located at a2⁎,and then, the private firm is located at b2⁎.

Proof. See Appendix. □Thus, the location pair of E2 is realized through the sequential-

move game of the public leader.On the other hand, another scenario that the foreignprivate firm is an

incumbent and the public firm is a new entrant also makes sense,especially indeveloping countrieswhere foreign companies are attractedin an early stage of the industry. In this case, the timing is as follows: First,the private firm (Firm B) chooses its location. Second, having observedthe location of the private firm, the public firm (Firm A) chooses itslocation. In this sequential-move game of the public follower, we have,interestingly, a result similar to that of Proposition 3.

Proposition 4. In the subgame perfect equilibrium of the sequential-move game of the public follower, first, the private firm is located at b2⁎,and then, the local public firm is located at a2⁎.

Proof. See Appendix. □The two propositions are intuitively explained as follows. First,

consider the case that the local public firm is a leader. After thelocation choice by the public firm, as illustrated in Fig. 2, a private firmlocates at either b1⁎=1 or b2⁎=0. Since the best responses of the publicfirm to b1⁎=1 and b2⁎=0 are a2⁎ and a1⁎, respectively, the public firmfaces a binary choice problem between E1 and E2, and thus, it choosesequilibrium E2 on the basis of Proposition 2.

Next, consider the case that the private firm is a leader. When itslocation choice b is smaller than b, the public firm chooses, as a best

8 In this case, the shareholders of the private firm are assumed to reside outside thelinear city.

response for Firm B's location choice, a location between 0.485 and0.500, where as b increases, the best response of Firm A also increases.However, Firm A does not move toward the right as much as Firm Bdoes, and thus, Firm B chooses b=0 to maintain a distance from thelocation of the public firm (when bbb). On the other hand, considerb>b. Then, the best responses of Firm A for the location choice ofFirm B lie in the range from 0.238 to 0.265. Thus, similar to the formercase, Firm A does not move toward the right as much as Firm B does,resulting in b=1, the location where Firm B is farthest fromFirm A, being the location choice of Firm B. Comparing thelocation pair (a1⁎, b1⁎) with (a2⁎, b2⁎), we know from Proposition 2 thatFirm B prefers (a2⁎, b2⁎) to (a1⁎, b1⁎).

Our results obtained in this section have the following signifi-cances in the literature on mixed oligopoly as well as that on pureoligopoly. First, whether a public firm should become a leader or afollower has been discussed by several researchers in the context ofmixed oligopoly. In the quantity-setting duopoly model with homo-geneous goods, it is known that the public firm should be the follower.In the case of differentiated goods, Matsumura and Matsushima(2003) consider a spatial model with price competition between apublic firm and a (domestic) private firm and show that if there is noprice regulation, the public firm should be a leader. Recently, Ogawaand Sanjo (2007) extended Matsumura and Matsushima's model suchthat a public firm competes against a private firm that is partiallyowned by foreign capital. They find that the equilibrium locationpattern in the case of a public leader is the same as that in asimultaneous location choice, but it is different from the equilibriumlocation pattern in the case of a private leader. In their model, thepublic leader is preferable to the private leader in terms of socialwelfare.9 In contrast to these studies, we show that the sameequilibrium location pattern arises, regardless of whether the publicfirm is a leader or a follower.

Second, as Propositions 3 and 4 show, the multiplicity of equilibriain the simultaneous location choice case is resolved in the case ofsequential location choice in our model. That is, the sequentiallocation choice serves as an equilibrium selection between E1 and E2.Similar results are observed inMatsumura et al. (2005) who consider ashipping model with quantity-setting in circular markets. They showthat when firms choose their locations simultaneously, the results ofPal (1998a) (dispersion) and Matsushima (2001) (partial agglomera-tion) emerge in equilibrium. However, they also show that when firmssequentially choose their location, the Pal-type equilibrium alwaysexists but the Matsushima-type equilibrium fails to exist if thetransportation cost is significantly convex or concave. Moreover, theyshow that the profits of firms in the Pal-type equilibrium are neversmaller than those in theMatsushima-type equilibrium, and this pointis also similar to our model (Propositions 2, 3, and 4). However, interms of the equilibrium location pattern obtained from the sequentiallocation choice, our results contrast with those of Matsumura et al.(2005). As observed in Proposition 1, in our model, both firmsagglomerate in Region 1 in E2.

Finally, in contrast to the above literature where the sequentialorder of moves is exogenously given, there are some studies thataddress the issue of the endogenous order of moves in mixedoligopoly by using the observable delay game of Hamilton and Slutsky(1990), for example, Pal (1998b) for Cournot competition and Bárcena-Ruiz (2007) for Bertrand competition. In the (simple) observable delaygame, firms simultaneously choose the timing (either t=1 or t=2) ofthe decisions, and then, the original game is played according to thetiming chosen by them. Thus, when they choose the same period, theyplay the simultaneous-move game, and when they choose differentperiods, they play the sequential-move game with the orders

9 Ogawa and Sanjo (2007) do not confirm this point in their paper because theirinterest lies in the equilibrium location pattern rather than in social welfare. Weconfirm this point by using MATHEMATICA6.


determined by their choice of periods. Applying the observable delaygame to our model in order to address the issue of the endogenousorder of location choice, we easily find that on the assumption that inthe case of the simultaneous-move game, equilibrium E1 occurs, theNash equilibrium of the observable delay game is either (tA, tB)=(1, 2)or (tA, tB)=(2, 1), where tA and tB denote the decisions of Firms A andB, respectively, in the observable delay game.

5. Conclusion

This paper investigates a mixed duopoly in which a private firmand a local public firm compete, by using a spatial model. To introducea local public firm, we divide a linear city into two symmetric regions.Similar to other literature on the spatial model, we construct a two-stage gamewhere, in the first stage, firms choose their location, and inthe second stage, they compete in price. We show that the game hastwo subgame perfect equilibria (E1 and E2). In E1, both local public andprivate firms are located in different regions, whereas in E2, both firmsare located in the same region—Region 1. Moreover, we also considerthe sequential location choice and demonstrate that E2 is realizedregardless of whether the local public firm is a leader or a follower.

On the basis of our analysis, we conclude that the local public firmsupplies its goodsor services outside its home region similar to that in thereal world, wherein we often observe similar phenomena in publicairports andpublic universities. This is in contrast tomostof the literatureon mixed oligopolies, which assume that public firms supply goods andservices only to their own regions. In addition, we find some policyimplication that the government might attract foreign private firms intoits country by locating a public firm near the boundary (E2). This can beinterpreted as a foreign firm's direct investment, and the welfare of thecountry improves as a result of the entry of the foreign private firm.However, thismight be detrimental to the neighboring country, that is tosay, it might result in the beggar-thy-neighbor behavior.

In ourmodel, we ignore the aspect of a spatial model that is viewedas a model of product differentiation. However, it might be possible toapply our model for product differentiation such as garnering supportfor a political party.10 Therefore, our model does not completelyeliminate the aspect of product differentiation.

Finally, since ourmodel is simple, it is expected to extend in variousdirections. For example, we can conceive of three extensions. First, inthis paper, we consider only a wholly-owned local public firm, butactually, there exist some quasi-public companies in rural industriessuch as local railways and local airports. Our model can deal with thecompanies by considering partial privatization of the local public firm,as in Matsumura (1998). Second, in spatial models of mixed oligopoly,there exist not only shopping models, as in our model, but alsoshippingmodels. Since shippingmodels can dealwith quantity-settingcompetition, it may provide us with some other insights with respectto the behavior of the local public firm. Third, the recent work ofMatsumura and Matsushima (forthcoming) solves the mixed strategyequilibria of the spatial model. Their methods may be applied to ourmodel. These three extensionswill be examined in our future research.

Acknowledgements

We wish to thank Toshihiro Matsumura, Dan Sasaki, NoriakiMatsushima, Kazuharu Kiyono, Yukihiko Funaki, Yoshihisa Baba, andthe participants at the ISS Industrial Organization Workshop at theUniversity of Tokyo, the 2007 Spring Meeting of JEA at Osaka GakuinUniversity, and the 2008 GLOPE-TCER Joint Junior Workshop onPolitical Economy at Waseda University for their helpful comments.

10 We might be able to suggest the public enterprises held by the Kuomintang Partyin Taiwan as an example because they supply their products even to people who do notsupport the party.

We would also like to thank two anonymous referees for theirvaluable comments and suggestions. We appreciate the financialsupport from the Japanese Ministry of Education, Culture, Sports,Science and Technology under theWaseda University 21st-COE GLOPEproject. Needless to say, we are responsible for any remaining errors.

Appendix A

Proof of Lemma 1.(I) abb. This case corresponds to Cases 1 and 2 in Table 1. Since

abb, we obtain

F1−F2 = −PA−PB + αð Þ24 b−að Þt = −

PA− PA

� �2

4 b−að Þt V 0;

where Fi denotes the local welfare of Region 1 in Case i, α≡ (b−a)(1−a−b) t, and PA≡PB−α. Thus, F1bF2 holds unless PA= PA. It is easilyverified that F1 and F2 represent the concave functions in PA. Moreover,

AF1APA jPA = PA

=AF2APA jPA = PA

=PB−α

2 a−bð Þt = −PA

2 b−að Þt :

Thus, the signs of the slopes of F1 and F2 at PA are changedaccording to the sign of PA. As a result, we have the relationship of F1and F2, as depicted in Fig. 3. The thin and thick curves denote F1 andF2, respectively.

By rewriting Eq. (1), we obtain

PA = a−bð Þ 2x−1ð Þt + PB + a−bð Þ 1−a−bð Þt = a−bð Þ 2x−1ð Þt + PA:

Thus, under abb,

PA V PA f xz12

Y Case 1 W1 = F1ð Þ;

PA > PA f x b12


8><>:and PA= PA ⇔ x=1/2. This indicates that the curves with the shadedportion in Fig. 3 represent W1. Thus, the maximum value of W1 isattained by the maximization of F2 when PAb0 and by the maxi-mization of F1 when PA≥0. By the first-order conditions for themaximization of F1 and F2, we obtain

rIA PBð Þ = f PB−α2 if PA z 0 PB z αð Þ;0 otherwise:

ð8Þ

In contrast with Firm A, the objective of Firm B is ∏B=PB (1−x),irrespective of Case 1 or 2. Thus, we have, by the first-order conditionof maximizing ∏B,

rIB PAð Þ = PA− a−bð Þ 2−a−bð Þt2

: ð9Þ

The reaction functions (8) and (9) yield the following equilibriumprices:

P1A a; bð Þ = − a−bð Þ a + bð Þt

3; P1

B a; bð Þ = − a−bð Þ 3−a−bð Þt3

:

Superscript 1 indicates that the equilibrium holds for the range ofCase 1 (x≥1/2).

(II) a>b. This corresponds to Cases 3 and 4 in Table 1. Consideringa>b, we obtain

F3−F4 = −PA−PB + αð Þ24 a−bð Þt V 0:

Fig. 3. The local welfare of Region 1 in (I) abb.


Thus, F3bF4 holds except for the case that PA= PA. Further, both F3 andF4 are concave in PA and

AF3APA jPA = PA

=AF4APA jPA = PA

= −PB−β

2 a−bð Þt ;

where β≡ (a−b) (a+b) t>0. We have the relationship of F3 and F4, asillustrated in Fig. 4. In this figure, the thin and thick curves representF3 and F4, respectively. Note that the signs of the slopes of F3 and F4 atPA= PA vary according to the sign of PB−β.

Eq. (1) implies that under a>b,

PA z PA f xz12


PA b PA f x b12


8><>:and PA= PA ⇔ x=1/2. The curves with the shaded portion representW1. Thus, we have, by the maximization of F3 and F4,

rIIA PBð Þ =PB + a−bð Þ 2−a−bð Þt

2if PB b β;

a−bð Þt otherwise:

8<:

The objective of firm B is ∏B=PBx, irrespective of Case 3 or 4. Thus,we have, by the first-order condition of maximizing ∏B,

rIIB ðPAÞ =PA + ða−bÞða + bÞt

2;

and the equilibrium prices are as follows.

P3A a; bð Þ = a−bð Þ 4−a−bð Þt

3; P3

B a; bð Þ = a−bð Þ 2 + a + bð Þt3

if a + b > 1;

P4A a; bð Þ = a−bð Þt; P4

B a; bð Þ = a−bð Þ 1 + a + bð Þt2

otherwise:

8><>:Superscripts 3 and 4 indicate that the equilibria hold for the ranges ofCase 3 (x≥1/2) and Case 4 (xb1/2), respectively.

Fig. 4. The local welfare o

(III) a=b. Finally, we consider Case 5. In this case, the firm that sets alower price captures all the demand in the market. Thus, Firm B(profit-maximizer) always prefers to set a price that is slightly lowerthan PAwhenever PA is positive. Moreover, for any price of Firm B thatsatisfies 0bPBbPA, there is another price PB′ such that PBbPB′bPA holds.Given such a price PB, the firm has an incentive to change the pricefrom PB to PB′ because its profit increases by doing so. Thus, Firm Bdoes not have an optimal action in the price-setting stage, providedPA>0. Consequently, PA

5(a, b)=PB5(a, b) =0 represents a uniqueequilibrium.

Derivation of reaction functions (4) and (5)

(4): Reaction function of Firm A. As described in the main text,we need to distinguish between the cases of bb1/2 and b≥1/2. Whenbb1/2,W1 ismaximizedwhen a∈[0,1−b] because F3 is amonotonicallydecreasing function of a in a>1−b (as illustrated at the left-hand side ofFig. 5). Thus, we consider whether the maximum ofW1 exists in a∈ [0,b] (W1=F1) or in a∈(b, 1−b] (W1=F4). To derive the condition, let a1(b)=arg maxa∈[0, b] F1(a, b) and a4(b)=arg maxa∈ (b, 1−b] F4(a, b), and wecalculate the following equation:

F1 a1 bð Þ; b� �

−F4 a4 bð Þ; b� �

= −9 215 + 73γ−42δð Þ + 18b 213−10γ−4δð Þ + 4b2 27−34b + 9γ−4δð Þ�

t1944

;

γuffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi73−4b 5−bð Þ

q; δu

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi378 + 8b 9 + 2bð Þ;

q

This equation is a monotonically increasing function of b, and whenb=b≈0.366, the equation equals to zero. Thus, when b∈ [0, b], a4(b)maximizes W1, otherwise a1(b) maximizes W1.

In addition, when b≥1/2, W1 is maximized in a∈ [0, b] (W1=F1)because F3 is a monotonically decreasing function of a in a>b (as

f Region 1 in (II) a>b.

Fig. 5. Derivation of Eq. (4).


illustrated at the right-hand side of Fig. 5). Hence, in the first stage, thereaction function of Firm A is expressed by

RA bð Þ =

10−b−ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi73−20b + 4b2

p

3if b b b;

−18−2b +ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi378 + 72b + 16b2

p

6otherwise:

8>>>><>>>>:

(5): Reaction function of Firm B. As with Eqs. (2) and (3), theobjective function of Firm B is classified into two cases. When ab1/2,it is given by

∏B =G4 = P4

Bx4 if b b a;

G1 = P1B 1−x1� �

otherwise;

8<:

where Gj (j=1, 4) denotes the profit of Firm B corresponding toeach equilibrium price. In this case, ∏B is maximized when b=0(∏B=G4) or b=1 (∏B=G1), as illustrated at the left-hand side of Fig. 6.Whether the maximum value of ∏B exists at b=0 or b=1 dependson the value of a. To derive the condition, we calculate the followingequation:

G1 a;1ð Þ−G4 a;0ð Þ = 16−41a + 2a2−13a3� �

t72

;

This equation is a monotonically decreasing function of a, and whena=ā≈0.380, this equation equals zero.

In addition, when a≥1/2, the objective function is as follows.

∏B =

G4 = P4Bx

4 if b V 1− a;

G3 = P3Bx

3 if 1 − a b b V a;

G1 = P1Bx

1 otherwise:

8>><>>:

In this equation, ∏B is maximized when b=0 (∏B=G4) because G3 is adecreasing function of b in b ∈ (1−a, a] and the maximum value of G1,

Fig. 6. Derivatio

which denotes the value of G1 at b=1, is lower than the value of G4 atb=0 (as illustrated at the right-hand side of Fig. 6). Thus, on the basisof these relations, we obtain the reaction function of Firm B.

RB að Þ = 1 if a b a;0 otherwise:

�

Proof of Proposition 3

The fact that the pair of locations (a1⁎, b1⁎)=(a1⁎, 1) represents theequilibrium point of the simultaneous location choice game implies

W1 a⁎1;1� �

> W1 a;1ð Þ for any aa 0;1½ �; a≠a⁎1: ð10Þ

Applying a similar reasoning to (a2⁎, b2⁎)=(a2⁎, 0), we obtain

W1 a⁎2;0� �

> W1 a;0ð Þ for any aa 0;1½ �; a≠a⁎2: ð11Þ

Since W1(a2⁎, 0)>W1(a1⁎, 1) by Proposition 2, with Eq. (10), we obtain

W1 a⁎2;0� �

> W1 a;1ð Þ for any aa 0;1½ �: ð12Þ

By Eqs. (11) and (12), in the equilibrium of the sequential-movegame, the public firm chooses location a=a2⁎ in the first stage, andthen the private firm locates at b=0.

Proof of Proposition 4

Since Firm B is the first mover in the sequential game, consideringthe best response of Firm A against Firm B's location, Firm B faces thefollowing maximization problem:

maxba 0;1½ �

∏B RA bð Þ; bð Þ

Here, note that RA(b) has different expressions depending on whetherbbb or not. Therefore, we consider the two cases separately.

When bbb, RA bð Þ = R4A bð Þ = 10−b−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi73−20b + 4b2

p3 . Then, RA4(b) belongs to

the range from 0.485 to 0.5 if b∈ [0, b]. This implies that a=RA(b)>

n of Eq. (5).


b>b, i.e., Firm A locates at the right of Firm B. Moreover, RA4 (b)+bb1.Thus, by Eq. (1) and Lemma 1, ∏B can be rewritten as

∏B R4A bð Þ; b� �

= P4Bx

4 = P4B

R4A bð Þ + b

2+

P4A−P

4B

2 R4A bð Þ−b� �

t

" #:

Through some calculations, we have

∏B =1

216−13−2b +


p� �210−4b−


p� �t;

and it is easily verified that the above equation is maximized at b=0 inthe interval [0, b].

Whenb≥ b,RA bð Þ = R1A bð Þ = −18−2b +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi378 + 72b + 16b2

p6 . Then,RA1 (b) belongs

to the range from 0.238 to 0.265 if b∈ [b, 1]. This implies that a=RA1(b)bbbb, i.e., FirmA locates at the left of FirmB. Thus, by Eq. (1) and Lemma1,∏B can be rewritten as

∏B R1A bð Þ; b� �

= P1B 1−x1� �

= P1B 1−

R1A bð Þ + b

2−

P1A−P

1B

2 R1A bð Þ−b� �

t

" #:

Through some calculations, we have

∏B =1

388818 + 8b−

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi378 + 72b + 16b2

p� �−36 + 4b +

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi378 + 72b + 16b2

p� �2t;

and it is easily verified that the above equation is maximized at b=1 inthe interval [b, 1].

Finally, we compare the maximized profits of Firm B between bbb

and b≥b. However, we have already shown that the profit earnedin b=0 is greater than the profit earned in b=1 because E2 payoff-dominates E1.

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interregional mixed duopoly

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