Matsushima & Matsumura 2003 - Mixed Oligopoly and Spatial Agglomeration

8/10/2019 Matsushima & Matsumura 2003 - Mixed Oligopoly and Spatial Agglomeration

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Mixed Oligopoly and Spatial AgglomerationAuthor(s): Noriaki Matsushima and Toshihiro MatsumuraSource: The Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 36, No. 1(Feb., 2003), pp. 62-87Published by: Wiley on behalf of the Canadian Economics Association

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Mixed oligopoly and spatial

agglomerationNoriaki Matsushima Faculty of Economics, Shinshu UniversityToshihiro Matsumura Institute of Social Science, University of

Tokyo

Abstract. We investigate a mixed market where a state-owned welfare-maximizingpublic firm competes against profit-maximizing private firms. We use a circular citymodel with quantity-setting competition. In contrast to a pure market case discussed byPal (1998a), spatial agglomeration of private firms always appears in equilibrium. Allprivate firms locate at the same point, and the public firm locates at the opposite side.We also find that this equilibrium pattern of the location is second best provided thatoutput of each private firm cannot be controlled by the social planner. JEL Classifica-tion: H42, L13

Oligopole mixte et agglomeration spatiale. Les auteurs examinent un marche mixte ouiune entreprise publique possedee par l'Etat et cherchant 'a maximiser le niveau de bien-etre est en concurrence avec des entreprises privees qui cherchent 'a maximiser leursprofits. On utilise un modele de cite circulaire oiu la concurrence se fait en choisissant laquantite produite. En contraste avec le cas du marche parfait discute par Pal (1998a),l'agglomeration spatiale des entreprises privees parait etre en equilibre. Toutes lesentreprises privees se localisent au meme point, et l'entreprise publique se localise duc6te oppose. I1 appert que ce pattern d'equilibre de localisation est un equilibre desecond ordre compte tenu du fait que la production de chaque entreprise privee ne peutetre contr6lee par le planificateur social.

We are grateful to Koichi Futagami, Shingo Ishiguro, Yasushi Iwamoto, Murdoch MacPhee,Yoshiyasu Ono, Akihisa Shibata, Hiroki Yoshino, and participants of the seminars at ShinshuUniversity and University of Tokyo, and Macroeconomics workshop for their helpful commentsand suggestions. We are also indebted to two anonymous referees for their precious andconstructive comments and suggestions. Needless to say, we are responsible for any remainingerrors. The first author and the second author gratefully acknowledge financial supports from the

Zengin Foundation for Studies on Economics and Finance and Grant-in-Aid from the JapaneseMinistry of Education, Science and Culture, respectively. This paper does not reflect any views ofthese foundations. Email: [email protected]

Canadian Journal of Economics / Revue canadienne d'Economique, Vol. 36, No. IFebruary / fevrier 2003. Printed in Canada / Imprim& u Canada

0008-4085 / 03 / 62-87 / ' Canadian Economics Association

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Mixed oligopoly and spatial agglomeration 63

1. Introduction

Studiesof mixed markets, n which state-owned welfare-maximizing ublic irmscompete against profit-maximizing rivate firms, have become increasinglypopular n recent years.1 Mixed oligopolies are common in developed, devel-oping, and former communist ransitional conomies.2 n Japan, n particular,competition between private and public firms exists in many oligopolisticmarkets such as those for banking services, housing loans, life insurance,broadcasting ervices, nd overnight eliveries see,e.g., Ide and Hayashi 1992).

In many mixed markets, herd behaviour' by private irms s often observed.It is often the case in mixed markets that private firms adopt very similarstrategies, which are

completelydifferent rom

thoseof

public irms. Notoriousherd behaviour by Japanese city banks is a typical example. Japanese privatebanks compete in the domestic market against strong public banks, such asPostal Bank and Public House Loan Corporation. Many arge Japanese privatebanks rushed into the international inancial markets to avoid competitionagainst strong public banks.

There are many other examples f herd behaviour y private irms n Japan.Postal savingsand postal ife insurance,which are organized y Postal Bank, dealmainlywith small ransactions. n the other hand, arge city banks and major ife

insurance ompanies deal primarily with large transactions. ostal Bank stillprovides avingsaccounts or elderly and handicapped eople (welfaredeposit),while most large city banks simultaneously ave ceased providing hese services.PublicHouse Loan Corporation as intensively rovided ong-term ousing oanswith fixed interest rates since 1950, which private firms were not doing untilrecently. Large Japanese ity banks locate many branches n large cities. Forconvenience,most of their branches are close to railway and subway stations.Hence, in many localities, here s an agglomeration f private bank branches.Postal Bank has branches ll over Japan and in large cities,but most are incon-

veniently ocated are not near stations). Hence, Postal Bank branches re rarelyamong his agglomeration f private banks. Television rograms uppliedby theJapan Broadcasting orporation NHK) are quite different rom those of privatebroadcasting ompanies,which are quite similar o each other.

In this paper, we attempt o explain herd behaviour y private irms n mixedmarkets. We show that such behaviour nevitably rises n equilibrium ithoutassuming, s do standard modelsof herd behaviour, nyinformational xternalityor network xternality see,e.g., Banerjee 992;Scharfstein nd Stein 1990).

1 For pioneering work on mixed oligopolies, see Merrill and Schneider (1966) and Harris and Wiens(1980). See also B6s (1986, 1991), Vickers and Yarrow (1988), and De Fraja and Delbono (1990)for excellent surveys.

2 The interest in mixed oligopolies is due to their importance to the economies of Europe, Canada,and Japan more than to that of the United States. However, there are examples of mixedoligopolies in the US such as the packaging and overnight-delivery ndustries.



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64 N. Matsushima and T. Matsumura

In most existing work on mixed oligopoly, quantity-setting models withsingle homogeneous goods are investigated, without considering locationmodels. An exception is Cremer, Marchand, and Thisse (1991), who examinea mixed market by using a traditional Hotelling-type location-price model andfind that neither maximum nor minimum differentiation arises in equilibrium.In their model, herd behaviour by private firms never arises in equilibrium.

In this paper, we also use a location model in order to investigate multi-dimensional choice in mixed markets. We formulate a mixed oligopoly modelin a circular city with quantity-setting and locational choice (a spatial pricediscrimination model with quantity competition). We show that each privatefirm has a strong incentive to locate far away from the public firm, whichresults in an agglomeration of private firms. We also show that such anequilibrium location is second best, provided that the output of each privatefirm cannot be controlled by a social planner.

Since the seminal work of Hotelling (1929), a rich and diverse literature onspatial competition has emerged. Location models fall into two categories:those in which firms bear the transportation costs are shipping or spatialprice discrimination models; those in which consumers pay for transport areshopping models. In each type, one can have either Bertrand-type price settingor Cournot-type quantity-setting. Our model is a spatial price discrimination

model with Cournot competition. Whether a price-setting or quantity-settingmodel is appropriate depends on the nature of the market being analysed. Ifthe quantity of each firm is more (less) flexible than the price, a price-setting(quantity-setting) model is more appropriate. For example, mail-order retai-lers that send catalogues to consumers incur substantial costs when theychange price. In such a market, price-setting competition may be more likely.On the other hand, in markets in which output capacity decisions must bemade before pricing decisions, which Kreps and Scheinkman (1983) discuss,quantity-setting competition is more likely.3

Most papers on location theory use shopping models with Bertrand com-petition. D'Aspremont, Gabszewicz, and Thisse (1979), and Neven (1985) pointout that firms have a tendency to differentiate themselves in order to relaxprice competition, which does not produce an agglomeration of firms.4Although Cournot-type and Bertrand-type non-spatial models are equallypopular, the body of literature on spatial competition that uses Cournot-typemodels is relatively small. Economists have recently considered spatial pricediscrimination models with Cournot competition. Hamilton, Thisse, and

3 See also Eaton and Lipsey (1989) and Friedman (1983, 1988). For empirical work on thisproblem, see, among others, McBride (1983), Gisser (1986), and Brander and Zhang (1990).

4 We must emphasize that important studies show that the minimum Hotelling-type differentiationmay arise in equilibrium. De Palma et al. (1985) and Rhee et al. (1992) show this by introducingunobservable attributes in brand choice. Jehiel (1992) and Friedman and Thisse (1993) considerprice collusion after firms have made location choices. Mai and Peng (1999) discuss cooperationbetween firms in the form of information exchange through communication.



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2. The model

We formulate an oligopoly model in a mixed market, in which a welfare-maximizing public firm competes with profit-maximizing rivate firms. Firm0 is the public firm. Other firms (firm 1, firm 2,..., firm n) are private.8 LetN {_ 1, 2,..., n} denote the set of all private firms and N+ {O,1, 2,..., n}denote the set of all firms.

We now present a two-stage ocation-quantity ame. The basic structure fthe model is from Pal (1998a), although we consider a more general trans-portation cost function. Let xi (i E N+) be the location of firm i. Variable xi isthe point on the circle ocated at a distance rom 0 (measured lockwise).

Inthe

firststage, each firm (i

eN+) simultaneously hooses ts location xi.Let qi(x) denote firm 's output offered at each point x E [0, 1];x is the point on

the circle ocated at a distance rom 0. In the second stage, each firm (i E N+)observes ts competitors' ocations and simultaneously hooses qi(x) E [0, oo) forx E [0, 1].Let p(x) denote the price of the product at x and

n

q(x) E qi(x)i=O

denote the total quantity upplied at x. We assume hat the demand unction at

each point x is linear and is given byp(x) = a - bq(x),

where a and b are positive constants.9 Let d(x, xi) denote the distance betweenx and xi. In our model, d(x, xi) is given by

d(xxiI-x xi if xi+ 1/2 ?x?xi

XI + if + 1/22 xi >x

I -xi+ x if xi > x + 1/2.To ship a unit of the product rom where the firm is located to a consumer atpoint x, each firm i (i E N+) pays a transportation cost t(d(x, xi)): [0, 1/2] * 91+.We assume hat t is a strictly ncreasing unction. Firms are able to discriminate

8 In this paper we do not allow the government to nationalize more than one firm. As Merrilland Schneider (1966) point out, the most efficient outcome is achieved by the nationalizationof all firms, provided that nationalization does not change the costs of fins (i.e., there isno X-inefficiency in the public firm). An analysis of mixed oligopoly is

necessary, because it isimpossible or undesirable, for political and/or economic reasons, to nationalize an entire sector.For example, in the absence of competition, public firms may have no incentive to reduce costs, inwhich case there is a loss of social welfare. Thus, we rule out the possibility of nationalizing allfirns.

9 As do many existing studies that use location-quantity models, we assume that the demandfunction is linear. Our main results (proposition 1 and 2) hold if p' < 0 and p" < 0. The conditionthat p' < 0 is a sufficient but not a necessary condition for our results.



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among consumers, since they control transportation. Consumer arbitrage isassumed to be prohibitively costly.10 Each of the (n + 1) firms has identicaltechnology and constant marginal costs of production, which is normalized tozero.11 These are standard assumptions that are made in many other location-quantity models.

3. Equilibrium

In this section, we discuss the equilibrium of the model formulated above. Weuse subgame perfection as the equilibrium concept. The game is solved bybackward induction. First, we discuss the equilibrium outcomes in the second-

stage subgames given the location of each firm.

3.1. Quantity choiceWe follow the Cournot assumption that firms compete in quantities at eachpoint in the market. Since marginal production costs are constant, quantitiesset at different points by the same firm are strategically independent. Cournotequilibrium in this game can then be characterized as a set of independentCournot equilibria, one for each point x. Let wi(x) denote firm i's (i c N) profitat x, given the locations of all firms:

7i(x) = (a - bq(x) - t(d(x, xi)))qi(x). (1)

Let w(x) denote the social surplus (consumer surplus plus the profits of allfirms) at x:

q(x) ii

w(x) ] (a - bm)dm - 3t(d(x, xi))qi(x). (2)i=o

The first-order conditions for firm 0 and firm i (i c N), are, respectively:

a - bq(x) - t(d(x, xo)) = 0 (3)a - b(qi(x) + q(x)) - t(d(x, xi)) = 0. (4)

10 This assumption is not essential. Unless transportation costs for consumers are strictly lowerthan for firms, consumer arbitrage plays no role in our model. For a discussion of this issue,see Hamilton, Thisse, and Weskamp (1989).

11 We assume that the public firm is as efficient as each private firm. This assumption is notessential. Suppose that the marginal production cost of firm 0 is co and that of each private firmis zero. If a is sufficiently large (a > (n + 1) (co + t(1/2)), our main results (proposition 1and 2)hold for any co> -t(1/2). See Yarrow (1986) and Vickers and Yarrow (1988) for excellentsurveys of the empirical literature on the relative efficiency of public and private firms.



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We assume that the entire market will always be served by firm 0. Thisassumption is satisfied if a > t(I /2) (n + 1).12

LEMMA 1. In equilibrium, p(x) = t(d(x, xo)).

Proof. From (3), we have bq(x) = a - t(d(x, xo)). Substituting this into thedemand function (p = a - bq(x)) yields lemma 1. QED

Lemma 1 implies that the price at each local market is equal to the unittransportation cost of the public firm. As De Fraja and Delbono (1989) havepointed out, a welfare-maximizing public firm chooses its output to equalizeprice and its marginal cost. Since the marginal cost of firm 0 is its unit

transportation cost, lemma 1 is established.Let cs denote consumer surplus at x. Because the circular city is sym-

metrical, we can assume xo < 1/2 in equilibrium, without loss of generality.Substituting p(x) = t(d(x, x0)) into the demand function, we obtain

q(x) = a - t(d(x, xo)) (5)b

Simple calculations yield

cs= (a - t(d(x, xo)))2 (6)

2b(6Let cs denote the total consumer surplus, which is given by

cs= 2bj (a - td(x, xo))2dx

1 Frxo Xo + 2b i:/(a - t(xo - x))2dx + 2 (a - t(x - xo))2dx

+ | (a -t(I _ x + Xo))2dx . (7)[JO J~~~~~~x0

LEMMA 2. cs does not depend on xi (i E N+).

Proof. See the appendix.

The intuition behind lemma 2 follows. Lemma 1 states that, at any localmarket x, the price does not depend on the location of each private finn, xi (i E N).Thus, obviously, total consumer surplus cs does not depend on these locations.

12 A similar assumption (of a sufficiently large a) is often made in mixed oligopoly and quantity-setting spatial models. See, among others, Anderson and Neven (1991), and Pal (1998a).



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The price at each local market depends on the location of the public firm, x0.However, the distribution pattern of the prices (the greater the distancebetween x and x0, the higher the price at x) does not depend on x0. This iswhy the total consumer surplus does not depend on xo.

Since the equilibrium price must equal the marginal cost (the unit transport-ation cost) of the public firm, each private firm does not produce at a market inwhich its unit transportation cost is higher than that of the public firm. Hence,

LEMMA 3. Firm i (i E N) produces positive output at x e [0, 1] if and onlyif d(x, xo) > d(x, xi).

From (4), we have

bqi(x) = a - bq(x) - t(d(x, xi)) = p - t(d(x, xi)) = t(d(x, x0)) - t(d(x, xi)),

(8)

where the second equality is derived from the demand function and the lastequality is derived from lemma 1. This implies lemma 4.

LEMMA 4. If firm i (i e N) produces positive output at x, then qi{x) =

{t(d(x, xo)) - t(d(x, xi))}/b.

This result is important. Lemma 4 indicates that the output of each privatefirm depends on its own location and that of firm 0, but does not depend onthe locations of the other firms. We now explain the intuition behind lemma 4.Suppose that n = 2, x0 = 0, xl = 1/4, and x2 = 1/2. Now, suppose that firm 2gets close to firm 1. Since the marginal cost of firm 2 decreases at markets near1/4, firm 2 increases its output at that location. In a pure market in which thereis no public firm, the output of firm 1 decreases at markets near 1/4 under theassumption of strategic substitutes. However, in a mixed market, an increase infirm 2's output reduces firm 0's output. Given marginal cost pricing by firm 0,the decrease in firm 0's output offsets the increase in firm 2's output. Thus, thecombined output of firm 0 and firm 2 remains constant. This is why firm 1does not change its output when the location of firm 2 changes.

Firm 0's profit is zero due to marginal cost pricing.13 If firm i (i e N) doesnot produce at x, its profit at x is zero. Otherwise, its profit at x is

Xi(x) = (t(d(x, xO)) - t(d(x, xi))) t(d(x, x b))-t(d(x, xi))

(t(d(x, xO)) - t(d(x, Xi)))2b ~~~~~~~~~~~

13 In this paper, our use of the term 'marginal cost pricing' might be misleading. In our quantity-setting model, firms do not choose prices. By marginal cost pricing, we mean that the public firmchooses its output to equate its marginal cost to the market price.



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From (9), 7ri oes not depend on xj (i,j E N, i $1). Let 1i(xo,xi) (i e N) denotethe total profit of firm i, which is given by fI 7ri(x)dx. ince 7ri epends on x0and

xibut does not depend on

xj, Hlidoes not

dependon

xj (i, jE

N,i

$1).Hence, the following emma.

LEMMA 5. The profit of each private firm does not depend on the locations ofother private firms.

3.2. Location equilibriumIn this subsection, we discuss he equilibrium ocation of each firm. We restrictour attention o pure strategy equilibria.

In the firststage,

eachprivate firm chooses location to maximize profit.Proposition 1states that all private irms choose the same ocation, which s on

the side of the circle opposite the location of the public firm.

PROPOSITION 1. In equilibrium, X1 =X2 = , = Xn and d(xo, xl) = 1/2.


We now explain why each private irm prefers o locate as far away from thepublic firm as possible. Suppose that x0 =0. Owing to welfare-maximizingbehaviour by the public firm, the price in each market equals marginal cost,which in our model is the unit transportation ost of firm 0. Therefore, heequilibrium rice is highest at x = 1/2 and lowest at x = 0. In other words, foreach private firm, markets near the location of the public firm are thin andthose far away from that location are thick. To minimize ransportation ostsat the largest market or the private irms, each firm prefers he location that isfurthest rom the public firm. Total consumer urplus cs) does not depend onthe location pattern (lemma 1). Since marginal cost is constant, the profit ofthe public firm s zero given marginal ost pricing. Thus, total social surplus smaximized when profits of private firms are maximized. Since each privatefirm wants to be a long distance from the public firm, the public firm alsowants to be a long distance rom the private irms.

3.3. Welfare implicationsWe now briefly discuss welfare implications. First, consider the first bestsituation. Suppose that a social planner can control both the location andoutput of each firm. Without oss of generality, we suppose that xo= 0. Then,

the first-best outcome is given by x0= 0, xi = 1/(n + 1), x2= 2/(n +),...,Xn= n/(n + 1) with marginal ost pricing by all firms. This outcome s differentfrom the equilibrium outcome stated in proposition 1. In this sense, theequilibrium utcome in the mixed market s inefficient.

Next, consider the following second-best olution. Suppose that the socialplanner cannot control the output of each private firm but can control firm



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location. Proposition 2, which follows, states that the equilibrium outcome issecond best.14

PROPOSITION 2. Suppose hat the socialplanner annot control he output of eachprivate firm but can controlfirm location. In that case, the social planner choosesthe locations xl =X2= ... =xn and d(xo, xl)= 1/2.

Proof. Without loss of generality, we set x0 0. The total social surplus isn

W = cs+ 2,HO(, Xi). (10)

cs is constant (lemma 2). H1i(i N) is maximized when xi= 1/2. Thus, W ismaximized when xl = X2 = .... =Xn= /2. QED

This result is closely related to that of Lahiri and Ono (1988). They show thatwelfare is often decreasing in the output of an inferior firm whose cost is muchhigher than that of its rival. Suppose that x0 = 0. In our model, the public firm isinferior at the market near 1/2, and superior at the market near 0. Thus,additional production by a private firm greatly improves social welfare at themarket near 1/2 but not at the market near 0. If each private firm locates at 1/2,the additional output is supplied most intensively at the market where additionalsupply has the most value. This is why the equilibrium location is efficient.

Finally, we make a brief remark on the effect of privatization of the public firm.Henceforth, we assume linear transportation costs for simplicity. We assume that

t(d(x, xi)) = rd(x, xi), (T is a positive constant). (11)

PROPOSITION 3. Privatization ffirm 0 does not improvewelfare.


In this setting, in contrast to the results of De Fraja and Delbono (1989), theprivatization of firm 0 does not improve welfare. This result should not beemphasized too much. The result depends on the assumptions of constantmarginal costs, and on the assumption that equal public and private firmshave the same transport technology. If the public firms were less efficient thanprivate firms, full privatization could improve welfare.15

14 Cremer, Marchand, and Thisse (1991) discuss a location-price model and find that the locationpattern is efficient (first best) in a mixed duopoly. Our results are different from theirs in tworespects. First, in our model, the equilibrium outcome is the second best. Second, our results holdregardless of the number of the private firms, while their results do not hold in a triopoly.

15 De Fraja and Delbono (1989) show the possibility that full privatization improves welfare. Theyderive this result having assumed increasing marginal costs but not the relative nefficiency of thepublic firm. In our setting, it is very difficult to use increasing marginal costs. Were we to do so.the Cournot equilibrium at each market would not be independent, which would substantiallycomplicate the analysis.



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4. Extensions

In the preceding sections, we discussed a circular-city,quantity-setting,simultaneous-move mixed oligopoly model with a single public firm. In this

section, we briefly discuss how our agglomeration result depends on these features.

4.1. Linear cityWe consider a model with a linear-city of length 1. We show that the agglom-eration of private firms arises in equilibrium.

PROPOSITION 4. (i) There is an equilibrium n which all private irms agglomerate

regardless of the number of private firms. (ii) There is another equilibrium inwhich irms agglomerate at two points if and only if the number of the privatefirms is even.


If the number of private firms is odd, the equilibrium becomes unique andall private firms agglomerate at one point. If the number of the firms is even,two types of equilibria exist: one is as stated above; the other is the equilibriumin which private firms agglomerate at two points.

We can show that lemma 5 holds in this setting, too (i.e., the profits of eachprivate firm do not depend on the locations of the other private firms). Thus,the best location is the same for all private firms, which results in the agglom-eration of private firms. If the public firm locates on the left-hand side of thelinear city, all private firms prefer the right-hand side location and vice versa.When the public firm locates at the central point of the linear-city (i.e., at 1/2),the best locations for each private firm are 1/10 and 9/10. Locating centrally isbest for the public firm if and only if half of the private firms locate at 1/10 andthe others locate at 9/10. This is why there are multiple equilibria if and only ifthe number of private firms is even.

4.2. Leadership by the public firmWe now investigate whether the timing of entry into the market matters. Weconsider timing as follows. First, the public firm locates. Second, havingobserved the location of the public firm, the private firms simultaneouslylocate. In a circular city, the location pattern is not affected by the timing.

In the circular-city model, we can assume that the location of the public firm

is xo = 0 without loss of generality because of the symmetry of the circular city.Hence, exactly the same results are derived in this model as those discussed insection 3, too. This is not true in the case of a linear city.

We consider a linear-city model. We show that the agglomeration of privatefirms arises in the unique equilibrium. Without loss of generality we assumethat xo > 1/2.



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As is shown in the appendix (see proof of proposition 4), the best responseof each private firm is either xi = xo/5 or xi = (4 + xo)/5. The profits of the firmsthat locate at xi = xo/5 and xi = (4 + xo)/5 are, respectively:

16'r2x3 16'r2(1 -XO)

75b ' 75b

The former is larger than the latter when xo > 1/2, and the former is equal tothe latter when xo = 1/2. The agglomeration of private firms in one market isthe unique equilibrium outcome unless xo= 1/2. We then show that firm 0never chooses xo = 1/2. Suppose that firm 0 did choose xo = 1/2. Since each

private firm is indifferent between xi= xo/5 and xi = (4 + xo)/5, there are multipleequilibria in the subgame. However, as is discussed in the proof of proposition4, cs does not depend on the locations of the private firms. The profits ofprivate firms do not depend on the number of firms that choose xi=xo/5instead of xi= (4+xo)/5. Hence, welfare does not depend on which equilib-rium emerges in this subgame. Thus, welfare is given by

3a2 - 3(1 - 2xo + 2x2)aT + (1 - 3Xo + 3x2)T2 16T2x36b 75b

where the first term is cs. The first-order condition is

T(25(2a - T) - 50(2a - T)XO + 32Tnxo) 1250b =12.

The optimal location is

25(2a - T) - 5(2a - T)(50a - (32n + 25)T)xA0 32nT

Note that xo is larger than 1/2, sinceT > 0.16 The location pattern is similar tothe simultaneous location choice mentioned in section 4.1. That is, the secondtype of equilibrium location pattern stated in proposition 4 (ii) is ruled out.

4.3. Multiple public firmsIn this subsection, we introduce another public firm. Suppose there are twopublic firms in a circular city. We obtain the following result (see the appendixfor proof ).

16 Using L'Hopital's rule, we find that limT,O xo= 1/2. We also find that OXO/OT > 0.



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PROPOSITION 5. If the number of private firms, n, is odd, the following locationpattern is a unique subgame perfect Nash equilibrium: all private firms agglom-erate at Xa and each public firm locates at 0 and

xp:Xa = -(4a - (2n + 1)T) + 4(4a T)(4a (n + 1)T))Xa =

~~~~~2nTr

-(4a - (n + 1)T)+ (4a-)(4a-(n+ 1)T) (15)nT

If the number of private firms, n, is even, the following location pattern is alsoa subgame perfect Nash equilibrium: n/2 private firms locate at 1/4, while the rest

locate at 3/4, and each public firm locates at 0 and 1/2.

This result indicates that agglomeration of private firms emerges in the case ofmultiple public firms.

4.4. Bertrand competitionWe now consider a price-setting model.17 In the quantity-setting model, giventhe locations of the firms, the equilibrium is unique in the second-stagesubgame. In the price-setting model, the equilibrium is not unique in the

second-stage subgame. Hence, the equilibrium is never unique in the fullgame. We now discuss this multiplicity.Given any location pattern, the following pair of pricing strategies defines

a Nash equilibrium: the public and private firms set their prices at po(x) andpi(X), respectively:

po(x) = t(d(x, xo)), (16)pi(x) = max {t(d(x, xi)), min {t(d(x, xj))}}, (i e N, j e N+). (17)fri

Given these pricing strategies, the following location pattern arises in equilibrium.

PROPOSITION 6. The ollowing ocationpattern s a subgame erfect Nash equilib-rium: irms locate equidistantly rom each other in a circular city.


However, this is not a unique equilibrium. Suppose that the number ofprivate firms is two and that xo = 0, xi = 1/3, and x2 = 2/3. Then, for example

at market 1/3, any pricing po(1/3) e (0,I

-6ar - -r2) s the equilibrium strategy.3As long as po is greater than the private firm's cost (which is zero at 1/3), thebest reply for each private firm is to set the price just below po and hence obtain

17 For the discussions of price-setting shipping models of pure markets, see Hamilton, Thisse, andWeskamp (1989) and Hwang and Mai (1990).



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the entire market demand. At the market in which the private firm's cost isbelow that of the public firm, supply by the private firm at a price above thepublic firm's marginal cost is better than marginal cost pricing by the publicfirm, because supply by the private firm reduces transportation costs. At themarket 1/3, supply by the private firm at the price 6aT - T2 and supply bythe public firm at its marginal cost yields the same welfare. Thus, at this marketsetting any price between private firm's cost and v6aT - T2 defines theequilibrium strategy of the public firm. Since we assume that a is sufficientlylarge and r is positive, 0 < r/3 < 3 06ar - r2. Thus, in markets supplied byprivate firms, in equilibrium, both strategies that are tougher than, and strat-egies that are not as tough as, marginal cost pricing by the public firm can beequilibrium strategies. If the former (latter) strategy is expected by the privatefirms, each private firm has more (less) of an incentive to locate for away fromthe public firm than was the case in proposition 6. Therefore, the equilibriumlocation is not unique.

5. Concluding remarks

We developed a location-quantity model of mixed markets. We found that thepresence of a state-owned public firm induces an agglomeration of private

firms. We also found that the equilibriumlocation is second best if a social

planner cannot control the output of private firms.We admit that our results depend crucially on model specification. First,

results depend on the assumption a > t(1/2)(n + 1). Although existing studiesmake a similar assumption, the assumption may be inappropriate when n islarge. Hence, our results may not apply to mixed markets with a large numberof private firms. Second, we assume that the public firm maximizes welfare.While this is a common assumption in models of mixed markets,18 other modelformulations are possible. In our model, the assumption of a welfare-maximiz-

ing public firm induces the public firm to adopt marginal cost pricing. Somemodels assume average cost pricing rather than marginal cost pricing in thecontext of mixed oligopoly (see, e.g., Estrin and de Meza 1995; Fujita 1997;Futagami 1999). In our model, since we assume that marginal cost is constant,there is no difference between marginal and average costs. Our results mightchange if we introduce set-up costs. Third, we implicitly assume that privatefirms are domestic. If the private firms are foreign owned, the public firmmight maximize consumer surplus plus the profits of domestic firms ratherthan the profits of all firms. As Fjell and Pal (1996) have pointed out, the

presence of foreign-owned private firms affects the optimal behaviour ofthe

public firm. This problem remains a subject for future research.

18 See, among others, Anderson, de Palma and Thisse (1997), Cremer, Marchand and Thisse(1991), De Fraja and Delbono (1989), Fjell and Pal (1996), Matsumura (1998), and Pal (1998b).



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Appendix

Proof of lemma 2. Lemma 1 states that the price at each market x does notdepend on xi e N. Thus, cs does not depend on xi E N.

We now show that cs does not depend on xo. Because of the symmetry of thecircular city, we can assume xo < 1/2, without loss of generality. When xo =0, cs is

cs(0)f (a - t(d(x, O)))d =2 (a - t(d(x, O))) dx(18)s(O) Jo

~2b dx22b (8

When xo = y, cs is

f1(a t(d(x, y)) Y (a - t(d(x,y)))2dxcs(y) J 2b dx=2 dx. (19)

Since d(x, 0) = d(x + y, y) we obtain

y0 2 Y (a --t(d(x, y)))2s(O) = 2 t((x)b dx = cs(y).

Proof of proposition 1. First, we assume that xo = 0 and show that each private

firm i chooses xi = 1/2. Since Hi does not depend on xj, firm Irs optimal locationdoes not depend on xj (i, j e N, i $j). Without loss of generality, we assumethat xi < 1/2. Firm i's total profit Hi(O, xi) is

Hi=/ r1r(x)dx = 0x2 2 (t(d(x, 0)) -t(d(x, x)))li iri()dx 0 dx b dx+ +i0d- JXi (t(x) - t(x, _ x))2 d + [(t(x) - t(x - xi))2 d

+Xi t(1- x)-t(x- X))2+ ~~~b dx. (20)

We now show that OHi/Oxi s increasing for xi < 1/2. From (20), we have that:

___I_ 2 F fidt(x1 - x)a _i = I[-|(t(x) - t(xi - x)) dx + (t(x)Oxi b dd

'+x.

- t(x-xi)) dt( dd i) dx + ] (t(1 - x) - t(x - Xi)) dd dx]. (21)

We now consider two cases: (i) xi < 1/3; and (ii) 1/3 < xi < 1/2.



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19/27


When xi= 1/2, the derivative in (21) is

OH,i I[-l(t(x) - t(1/2 - x))dt(1/2x)dxOx, b Ldd

+ (t(1 -x)-t(x- 1/2)) dt(x /2)dxXj- ''~ dd

- 2 [ (t(1/2 - y) - t(y)) dt(y) (-dy)b [d

+ j (t(1/2 - y) - t(y)) dd dy] =

Therefore, each private firm i chooses xi = 1/2.Second, we assume that xI = X2 = . .. = X,= 1/2 and consider the optimal

location of firm 0. Without loss of generality, we assume that xo < 1/2. LetW denote the total surplus (cs plus the profits of all private firms). Note thatthe profit of firm 0 is zero. Given xI =x2= ... = xn =1/2, the total social

surplus isn

W = csZ+E i. (26)i=l1

We have shown that the profit of each private firm i is maximized when thedistance between firm 0 and firm i is maximized. Thus, the profit of eachprivate firm is maximized when xo = 0. Since cs does not depend on xo, W(xo)

is maximized when xo = 0. QED

Proof of proposition 3. We compare the social surplus in the mixed market andthat in the pure market, in which the public firm is privatized. In the mixedmarket, the social surplus (now defined as swl) is

2w1 (a - rm)2 dm :(fr(m -0) -r(1/2 - m d

4f(T(1 m) -T(m - 1/2)) din2 b

dm

12a2-6a-i+?(n?+ l)224b



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In the pure market, the social surplus (now defined as Sw2) s

n(48(n + 2)a2 - 24(n + 2)aT + (2n2+ 7n + 8)T2)

(n is even)96(n + 1)2b

l n(48(n+ 2)a2 - 24(n + 2)aT+ (2n2 + 7n + 8)T2) K (n is odd),96(n + 1)2b

where K ((2n + 3)T2)/(96n2(n 1)2b) (see Matsushima 2001b for derivation).The difference between them is

48a2 _ 24aT + (2n3 + 5n2 + 4n + 4)T2

96(n + 1)2bSW] -w =W 2< 2T48a2 24aT+ (2n3+ 5n2+ 4n + 4)T2 + K (n is odd).

96(n + 1)2b(27)

To ensure that the public firm is able to supply all the points, we have assumed

that r < 2a/(n + 1). The difference is positive. Therefore, full privatization doesnot improve welfare. QED

Proof of proposition 4. Obviously, lemmas 1 and 2 hold in this setting.Consumer surplus is given by

cs = fu (a - T(xo _ M))2 dm + (( T( X))2 dmCS Jo~~2b

drn 2b

3a2 - 3(1 - 2xo + 2x2)aT+ (1 - 3xo + 3x2)T26b

By using the procedures applied in section 3, we can derive lemma 5 for thissetting, which states that the profit of each private firm i depends only on xoand xi. Let L denote the set of private firms that locate at xi C [0, x0] inequilibrium, let R denote the set of private firms that locate at xi e (xo, 1] inequilibrium; and let / denote the number of firms that belong to L. The profitof each private firm i e L is

xi ('r(xo - m) - T(Xi -rM))2 + (T(X -r)- Nm - xi))2 d

(xo - xi)2(xo + 5xi)92 (28)

6b



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This is maximized at

xi= xo (29)

The profit of each private firm i e R is

fi (-r(m xi) - T(x -iM)) d + f (-r(m xo) - r(m -x))2_R_=bdm ] dm

(6 - xo - 5xi)(xi - xo)2T26b

This is maximized at4 + xo

xi-= 5 (31)

If choosing xi=xO/5 yields strictly higher or lower profits than choosingxi= (4 + xo)/5, every private firm must choose the same location in equilibrium(i.e., either L or R must be empty). Otherwise, it is possible that some privatefirms choose x0/5 and other private firms choose (4 + xo)/5.

Given the locations of the private firms, welfare is given by

csZfiLZ fiR.S + E + E RiEL iER

The first-order condition is

T(2a - T)(I - 2xo) TE2(Xo - xi)(xo + 3xi)2b iEL 2b

E T2(Xo- xi)(4 - x0 - 3xi) ? . (32)

iE 2bSubstituting xi = x0/5 for i E L and xi= (4 + x0)/5 for i E R into (32) yields, if/ $ n-I,

(50a - (32(n - 1) + 25)-r)- (50a-(321 + 25)T)(50a-(32(n-1) + 25)7)32(21 n)-r

(33)

If1=

n-I, we obtain

x= 2 (34)

Suppose that I = n - / = n/2. Note that this is possible only if n is even.Substituting xo, given by (34), and xi, given by (29) or (31), into (28) and



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(30), respectively, we find that xi for i e L (R) is 1/10 (9/10). Given symmetry,HL = HIRI.Hence = n/2 can be an equilibrium.

Suppose that 1/ n/2. Substituting xo, given by (33), and xi, given by (29) or(31), into (28) and (30), respectively, we obtain

/ ~~~~~~~~~~~~3- (/(50a-(32(n -lI)?+25)T)) 3T

153600T (35)3

uR =15(3210025)

(36)

where

21 - n (50a-(32+25)T))0

If I> n/2 (res. I< n/2), then EL > 'R (res. EL n/2 (res. I< n/2),I must be n (res. 0). Given /= n, the equilibrium location pattern is

25(2a -T) - /25(2a - T)(50a - (32n + 25)T)

160n-r

xO= 25(2a - r) - /25(2a T)(50a (32n+ 25)r) QEDxo~~~~~~32nTr

Proof of proposition 5. Suppose that one public firm locates at xo = 0 and theother public firm locates at xp(7 0). In this case, lemmas 1 and 2 hold.Consumer surplus is

( 2-(a -2m2 (a - Tr(m-cs 2 x ( 2b dm + 2b Pdm)

12a2 - 6(1 - 2xp + 2x2)ar + (1 - 3xp + 3x2)T2_~~~~~~~ X24b

In this case, lemma 5 holds. Let R denote the set of private firms that locate atxi E [0, xp] in equilibrium. Let L denote the set of private firms that locateat xi E (xp, 1] in equilibrium. Let I denote the number of firms belonging to L.The profit of each private firm i E R is

iiR = -i (T(2m - xi)) [2 (Trxi) 2 (T(xp + xi - 2m))b dm + bdm + b dm

2JX

(3xp - 4xi)x?22

6b



23/27



Xi=

2P (37)

The profit of each private firm i E L is

F (-r(2m - x, - xi))2 dm ? (T(1 -xi)) dxp+Xi b I b2

(T(1 + xi-2m))2Jx+ dm

(4xi - 3xp - 1)(1I-6b


xi +? (38)2

Given the locations of the private firms, social welfare is

csZ i~?R iiLS +I:

, +I: i

I

iER iEL


T(4a - T)(1 - 2xp) - T2x2 - x;) = (39)8b iE 2b ZiE 2b

Substituting xi = xp/2 in (37) for i E R and xi = (1 + xp)/2 in (38) for i E L into(39) yields, if 1j n - 1,

(4a- (1 + )T) -/(4a-(n- + 1)T)(4a-(1? 1)T) (40)XA (n - 21)Tr(0

If l = n-1, we obtain

Xi-(i E R), x=4(iEL),xp=, . (41)

The equality n - 1 = 1 holds only if n is even. If 1= n - 1 = n/2, the outcomeabove becomes symmetric; thus we obtain 1R = Hfr. Accordingly, 1= n/2 canbe an equilibrium.



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Suppose that / t n/2. We obtain

(4a-~(l

+1l)T) UnrR = (/)(2

24b-i- (42)

11( 4a-(n-l+ 1)T) Ui 24b-r (43)

where

(4~a - (I + I),T V4a - (n - I + 1)Tr3

.(4

We find that LR is greater than HL res. EL is greater than HR), if /< n/2(res. I> n/2). Thus, if I< n/2 (res. I> n/2), I must be zero (res. n). Given /= n,the equilibrium location pattern is

=-(4a -(2n + I)Tr)+ (4-45a-n)lTXa2 (45)

_ -(4a - (n + l)'r) + vr(4a - Tr)(4a - (n + 1)T) (46)xp-- lT . (46)

Proof of proposition 6. In the price-setting model, each firm has only two realcompetitiors, namely, the two surrounding it. We now show that for bothpublic and private firms the optimal location is the centre of the two rivals.

Suppose that the public firm adopts the strategy of, at each point, settingprice at its marginal cost (transportation cost) at that point. Consider the casein which the public firm's neighbours locate at 0 and 1 on the Hotelling line. Ifthe public firm's optimal location is 1/2, the equidistant result holds. The

public firm's optimal location choice is

max T [(a - T(Xo - m)) + (a - T(Xo - m))(X(xo m) - Tm)] dm

Xo ( aT( Mn))2 _____ ___ X___ __ 2+ a

- r(xo dm + 2- (a -T(mn- d))2b ]xO 2b

+J [(a- T(_m- xo))2 + (a - T(m - xo))(T(m - xO) - T( - m))] dm.


r(4a +Tr)(1 -2xo) 0=8b

The optimal location is 1/2.



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Consider he case in which private irm's neighbours ocate at 0 and 1 on theHotelling line. We have assumed that the public firm adopts marginal costpricing. Therefore, whether he neighbour s a public or a private firm, if theprivate irm i is the most efficient at a particular oint, that firm can supply atthat point. The firm i's optimal ocation choice is

fax (a - m)(m-(x i-m)) dm + (a - Tm)(Tm T(m xi)) dm

I+x-

? (a - -r(1 OXTO-

T(m - xi)) dm., ~~~~b

The first order condition s

'r(4a - f - 4'rxi)(I - 2xi)8b

The optimal location is 1/2. Therefore, he equidistant ocation pattern s anequilibrium utcome. QED

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Matsushima & Matsumura 2003 - Mixed Oligopoly and Spatial Agglomeration