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Duopoly with spatial and quantity- dependent price discrimination
Regional Science and Urban Economies 22 (1992) 175--185. North-Holland
Duopoly with spatial and quantitydependent price discrimination* Jonathan H. Hamilton University of Florida, Gainesville FL, USA
Jacques-Francois Thisse CORE, Universitb Catholique de Louvain, Louvain-la-Neuve, Belgium Virginia Polytechnic Institute, Blacksburg VA, USA Received June 1989, final version received October 1990
Spatial price discrimination with downward-sloping demands is extended to allow for non-linear pricing by competing firms. With linear discriminatory pricing only, firms' locations are not efficient, but only minimize transport costs given the firms' sales distributions. Furthermore, the quantities sold are less than the efficient quantities. With non-linear delivered pricing, at the subgame perfect Nash equilibrium, quantities are identical to those under marginal cost pricing. The socially optimal locations are an equilibrium of the firms' location game.
1. Introduction
Spatial competition models have recently focused on a variety of pricing strategies used by firms, d'Aspremont et al.'s (1979) proof of the nonexistence of a pure strategy price equilibrium for some location pairs in the Hotelling (1929) model has led many researchers to consider alternative models of price formation. Spatial discriminatory pricing has almost as long a history as Hotelling's mill (f.o.b.) pricing, dating back at least to Hoover (1937). Discrimination by location was studied for a long time only in monopoly settings, but recent work has examined spatial discrimination in oligopoly. Hurter and Lederer (1985) prove the existence of pure strategy Nash equilibria in a Hotelling-type model where firms first choose locations and then price schedules. Thisse and Vives (1988) demonstrate that if firms can choose between mill and discriminatory pricing, discrimination arises as a Prisoners' Dilemma Correspondence: Prof. Jonathan Hamilton, College of Business Administration, Department of Economics, University of Florida, Gainsville, FL 32611-2017, USA. *We thank two referees for their useful comments. Financial support from the University of Florida College of Business Administration and Public Policy Research Center. Virginia Polytechnic Institute Department of Economics, and C.I.M. (Belgium) is also acknowledged. 0166-0462/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved
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outcome: both firms do better if they use mill pricing. Hamilton, MacLeod and Thisse (1991) demonstrate that, with perfectly inelastic demands, mill pricing arises in equilibrium when firms have the ability to offer general contracts but customers' locations are unknown to them. Thus, it is primarily an information problem that limits firms' use of spatial discrimination. Spatial price discrimination has often been critically viewed as a means of exercising monopoly power, yet the recent work on spatial discrimination in oligopoly reveals almost the opposite result - firms compete vigorously in distant markets and consumers benefit overall from discrimination. 1 Thus, many economists have relaxed or even reversed their support for antitrust legislation which attempts to limit price discrimination by location. 2 Another form of price discrimination - non-linear pricing with respect to quantity - has been studied in spatial models by Spulber (1981, 1984). An interpretation of this approach is that, if consumers have price-sensitive demands and firms do not observe consumer locations, quantity-dependent pricing can lead to some discrimination by location. His models do not have firms competing for the same customers in setting price schedules. Rather, they are local monopolists, possibly facing competition on the fringe of their market areas. One of our objectives in this paper is to combine both types of discrimination in a model which includes location choice by firms) Many papers consider spatial discrimination only since they analyze models with inelastic consumer demands. If consumers buy either zero or one unit of the good, non-linear pricing is impossible. Hamilton, Thisse and Weskamp (hereinafter HTW) (1989) have analyzed linear spatial discrimination with price-sensitive demand functions. In much oligopoly analysis, it is irrelevant whether demand functions reflect individual consumers' responses to price or marginal consumers' decisions to buy or not to buy a single unit of the good. This is no longer true here. Absent legal prohibitions on quantity-dependent pricing, the HTW solution is an equilibrium only in models where the downward-sloping demand function at each location results from more consumers buying at lower prices. With downward-sloping demands for individual consumers, we derive the equilibrium allowing general pricing contracts. We also show that the equilibrium corresponds to the allocation in the core of the market game among producers and consumers for given locations that is most favorable to producers. With spatial and quantity-based discrimination, this allocation delivers each 1See Phlips (1983) for a discussion of the history of spatial discriminatory pricing and a discussion of its welfare implications.
2See, for example, Greenhut, Norman and Hung (1987). 3In the United States, the Robinson-Patman Act is the principal antitrust legislation on both spatial and non-linear price discrimination. It allows a variety of defenses for price discriminating firms, including cost justification and 'meeting the competition'. Even compared to other antitrust legislation, it has often been attacked for inhibiting rather than enhancing efficiency [see, for example, Scherer (1980)].
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consumer the same surplus as with only spatial discrimination and is also a first-best allocation conditional on locations. We then solve for the subgame perfect Nash equilibrium locations with non-linear spatial discrimination and show that the socially optimal locations are an equilibrium of the firms' location game.
2. The model Our model extends the model of Hurter and Lederer (1985) to allow for downward-sloping demand functions for each consumer. A consumer is located at a point x ~ ~ which is a compact subset of the plane with a nonempty interior. Let p(x) be the density of consumers at x. A consumer at x has a demand function for a homogeneous good D(p;x) which is strictly decreasing in p; we also allow the demand function to vary with the consumer's location. There are two firms producing the good, 1 and 2, which are located at points xl and x 2 e ~ r. Firm i will incur a cost t~(x)q to deliver q units to a consumer located at x. This includes both production and transport costs. We assume that t~(x) is a strictly increasing function of the distance between x and x i and ti(x~)>0. The firms deliver the good to the consumers and thus can identify consumer locations; they are also assumed to know the demand function of consumers at each location. Therefore, nonlinear pricing depending on consumer locations can be implemented. Profits for each firm are the excess of revenues collected from consumers over total delivery costs incurred. We study the subgame perfect Nash equilibria of the game in which both firms simultaneously choose locations and then, given the pair of locations, choose simultaneously non-linear pricing schedules that depend on location.
3. The equilibrium contracts We first analyze the second stage of the game after firms have chosen locations xl and x2. If firms are allowed to offer any sort of contract to consumers, then one imagines that they will offer the contract most favorable to themselves, subject to competitive pressures from rivals. Rather than considering non-linear pricing explicitly, we need only examine outlayquantity pairs offered to consumers at each location [see Willig (1978) for a similar formulation]. If firms are free to offer contracts to consumers, in equilibrium no consumer can obtain a better contract from the other firm. When a consumer is offered equal surplus from both firms, he chooses to contract with the lower cost firm. ~ With contracts tied to locations, ~l'his is so because the lower cost firm can always offer the consumer a strictly better contract. O u r assumptions allow us to avoid the technicality of an e-equilibrium of the game. See Hurter and Lederer (1985) for a similar assumption.
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concessions offered to one set of consumers need not be offered to anyone else, given the absence of arbitrage possibilities) Thus firms will offer the best possible contracts given their costs. The lower cost firm will supply the good, and the only effective constraint on that firm is that it offers consumers at least as much surplus as the higher cost firm. The optimal contract combines features of first- and third-degree price discrimination. Because customers differ with respect to location, the contracts offered them must depend on location as well as allowing for nonlinear pricing. The firm can capture all gains from non-linear pricing, since the ability of the higher cost firm to offer non-linear pricing leaves unchanged that firm's best offer. Let Oi(x)=.{q~(x), Ri(x)} be the contract offered by firm i to consumers at location x, where q~ is the quantity delivered and R~ is the payment from the consumer.
Proposition 1. I f firms can perfectly discriminate with respect to both quantity and location, then the equilibrium contracts are: q*(x) = D(ti(x); x)
where
ti(x) <=tj(x),
and t ~(x) R~(x) = ti(x)qt(x) + I D(v; x) dr. ti(x)
Proof. The best schedule for the consumer at x that firm j, j ~ i, could offer without suffering a loss is q~(x)= q(t~(x); x) and Rj(x)= t~(x)q~(x). Firm i offers the consumer at x the same a m o u n t of surplus using the above contract. The m a x i m u m quantity for which a consumer's marginal willingness to pay is at least as high as the delivery cost determines the quantity offered. Any smaller quantity would mean firm i gives up potential profit. Q.E.D. The optimal contract is illustrated in fig. 1. The area to the left of the demand curve above the higher delivery cost is the surplus the consumer actually receives. The area to the left of the demand curve between the two delivery costs is the surplus captured by the lower cost firm and equals the profit that this firm earns with the optimal non-linear contract. In the special case of perfectly inelastic demands, the equilibrium contracts SThe contracts in equilibrium do not violate the constraint that consumers cannot choose to buy the good elsewhere and transport it back to their original locations. Observe that marginal delivered cost increases at the same rate as transport cost. Hence, for convenience, we can ignore the no-arbitrage constraint in the presentation.
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179
p(x)
~(x)
ti(x)
D(~(x);x)
D(ti(x);x) = qi*(x)
q(x)
Surplus to Consumer = s*(x) Profit to Firm i = Ri*(x ) - qi*(x) ti(x )
Fig. 1
reduce to q*(x)= 1 and R*(x)= tj(x), for i, j = 1, 2 and j ~ i. Of course, since demands are perfectly inelastic, there is no room for first-degree price discrimination, so the equilibrium contracts involve only third-degree (spatial) price discrimination. Given locations, we can also analyze the efficiency of this equilibrium. An allocation in this economy can be described using the quantity of the good each consumer obtains, the outlay of the numeraire given to the firm, and the corresponding delivery cost; each firm's profit is the excess of outlays collected over total delivery costs. If an allocation lies in the core of the market game among firms and consumers, it is efficient. Let N = S f u {1,2} be the set of agents in the economy (consumers and the two firms). Let S___N be an arbitrary coalition and A(S) be the set of feasible allocations for coalition S. An allocation Z E A(N) is in the core iff there does not exist a coalition S ~ N and an allocation Z'~A(S) which makes all members of S better off. Thus, if profits to firms and surplus to consumers in the coalition can be increased, the current allocation is not in the core. Consumers have no trades to make with each other; similarly, firms only gain by trading the good with consumers and have nothing to trade with each other. Hence, a
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J.H. Hamilton and J.-F. Thisse, Duopoly and price discrimination
blocking coalition must contain a (non-negligible) subset of consumers and at least one firm. The core of the market game for given locations also contains the contracts described in Proposition 1. Since the allocation in Proposition 1 has no constraints on the form of contracts, it is perhaps not surprising that it is also in the core.
Proposition 2. The allocation described in Proposition 1 is in the core of the market game and is also the most favorable allocation in the core for the firms. Proof. At each x E•, the outlay-quantity contract was constructed to maximize profit for the selling firm by extracting all surplus in excess of the surplus the consumer could obtain from the rival firm. Thus, no other contract could make the supplier and the consumer better off, and no better contract for the consumer could be profitably provided by the other firm. Since some consumers must agree to join the firms in any blocking coalition, the two firms cannot block this allocation by offering consumers less surplus. Q.E.D. Notice also that the quantities obtained by the consumers are the first-best quantities since they correspond to those which would be purchased at prices equal to marginal cost (the lower of the delivery costs).
4. The equilibrium locations
In the first stage of the game, the firms choose locations looking ahead to the second-stage outcome. At the equilibrium contract, the consumer at x obtains surplus s*(x)=-s(0*(x); x) for x such that ti(x)< tj(x) which is given by
s*(x) =
S
D(v; x) dv + p(q?(x); x)q'~(x) - R.*,(x),
p(~*{x);x)
where p(q; x) is the inverse of the demand function D(p; x). At the equilibrium contracts, profit for firm i equals 6 U*(xl, x~) = rl,(x~, x j, o.?(x), o* (x))
=
S
[R*(x)-- q*(x)ti(x)]p(x) dx.
ti(x)
Total surplus for the economy is 6For any locations where the two firms have equal delivery costs, no profits are earned on contracts at that location. Hence, we m a y disregard ties in delivery costs.
J.H. Hamilton and J.-F. Thisse, Duopoly and price discrimination
181
S*(X1, X2) = J" S*(X)p(X) d x "Ji-H ~ ( x 1 , x2) + I'I~(Xl, x2) ~f
which can be simplified to S*(Xl,X2)= ~
~
minti(x)
D(v;x) dvp(x)dx.
In the case of perfectly inelastic demands, the maximization of total surplus with respect to locations reduces to the minimization of total transportation costs as in Hurter and Lederer (1985). With downwardsloping demands, changes in total surplus must also incorporate changes in delivered quantities that result from changes in locations. Our next proposition shows the efficiency of a location equilibrium.
Proposition 3. Let (xl,x:) be a location pair which maximizes total surplus. Then (xl, x2) is a location equilibrium. Proof. We can rewrite total surplus as S*(x,,x2)=~[max
('~'
x)dv, O}
,jix D(v;
Observe now that tj(x)
rt*(x,, x2) =
~
~ O(v; x) dvp(x) dx
t,(x)
=~
{ti(x), tj(x)}
~
O(v; x) dvp(x) dx
ti(x)
Since profit for firm i can be written as
H~(xl, x2) = S*(x. x~) - I ~ D(v; x) dvo(x) dx, ~f tj(x)
and since the second term in the right-hand side of this expression is
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J.H. Hamilton and J.-F. Thisse, Duopoly and price discrimination
independent of x~, for any maximizer (x*, x~') of S*, x* maximizes /-/i with respect to xi given x*. Thus, (x*,x*) is an equilibrium of the location game. Q,E.D. It follows directly from this proposition that the set of location equilibria coincides with the set of points maximizing S* with respect to xi given xj, for i , j = 1,2 and i ~ j . 7 We have not directly proven the existence of an equilibrium, yet it suffices to establish that there always exists a pair of locations that maximizes social surplus. If S* is upper-semicontinuous in xl and Xz, then a maximum of total surplus always exists since :Y is compact. Continuity of delivery costs with respect to x~ is sufficient to guarantee this. It cannot be shown that any location equilibrium is a maximizer of total surplus. Even with perfectly inelastic demands, Hurter and Lederer (1985, p. 549) present an example which demonstrates that additional equilibria may exist. However, we have shown that the efficient locations can be sustained as a Nash equilibrium of the first-stage game. Even though the distribution of consumer surplus is not symmetric about firms' locations, the delivered quantities are. Thus, maximizing the surplus extracted from consumers within the market area drives firms to the efficient locations. Consumer surplus at every location is identical to that obtained under linear spatial price discrimination. However, since the equilibrium locations differ, the distribution of surplus across consumer locations differs. Our solution complements results of Spence (1976) who demonstrates that first-degree price discrimination is necessary for the optimal set of products to be offered in a representative consumer model. Spence explicitly considers product differentiation models which are one possible interpretation of spatial models. In our model, the quantity-dependent price schedules must also depend on customer locations. In effect, firms combine first- and thirddegree price discrimination. Third-degree price discrimination is adequate to control for differences across consumers with respect to location, while firstdegree discrimination is necessary to provide firms the incentive to locate efficiently.
5. Example: A linear model To illustrate the solution presented in the last two sections, let us now consider the case where all consumers have identical linear demand curves, D(p;x)= I - p ( x ) , and both firms have delivery costs which are linear in distance from the firm to the consumer, t i ( x ) = t l x - x ~ l . Let ~r=[0,1], a VThisresultsis reminiscentof Proposition 1 in Spence(1976).
J.H. Hamilton and J.-F. Thisse, Duopoly and price discrimination
183
compact subset of the real line (instead of the plane). We further assume a uniform distribution of consumers, i.e., p(x) = 1 for all x ~ [0, 1]. Using the formulas presented above in Proposition 1, the equilibrium contracts for given locations are:
ql'(x) = 1 -tlx-xa 1, R*(x) =(1 - t Ix - xl) /2 - ( 1 - t I x - x2 I)2/2
+tlX-Xxl(a-tlx-x,I),
xEE0,x-],
and
q~(x) = 1 -- t lx-- x 21, R*(x) = ( 1 - t l x - - x 2 1 ) 2 / 2 - ( 1 - t[x--xx I)2/2
+tlx-x=l(1-tlx-x2[),
1],
where £ is the location of the consumer equidistant between the firms (£=(x~ +x2)/2) and the consumer at £ is assigned to firm 1, without loss of generality, s Having found the equilibrium contracts, we can now determine equilibrium locations for the firms. Profits at equilibrium contracts for the two firms can be written as /-/~(XI, X2) = ~ I-R~(x) - - q*(x)t o
Ix-
xll] d x
and 1
II~(Xl, X2) = ~ I-II~(X) - - q*(x)t
Ix-- xx I] dx.
The equilibrium locations can easily be shown to be interior and distinct. Hence, they must solve the first-order conditions, dlI*/Oxl = 0 and aFl*/Ox2 = 0. At the equilibrium contracts +17" x, .+,+ Oxl = - t I [l - tx I + tx] dx + t J [1 + txt - tx] dx = O, 0 xl aWhile we have assumed that firm 1 locates to the left of finn 2, this is not imposed on the equilibrium. Should firm 1 locate to the fight of firm 2, firm l's profits can be found from ll2(xl,x2). The functional forms for profits assume that firm 1 locates on the left.
R.S+U.E.--B
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J.H. Hamilton and J.-F. Thisse, Duopoly and price discrimination
and a similar expression holds for firm 2. The expressions within the integrands are ql(x), so the equilibrium locations are the medians of the sales distributions. Given the rival firm's location, maximizing profits is equivalent to minimizing transport costs. Since the distribution of delivered quantities is symmetric about each firm's location under the equilibrium contracts, the unique location equilibrium is xl = 1/4 and x2 = 3/4, which is obviously the efficient pair of locations. If the firms were constrained to use linear spatial discriminatory pricing, as in H T W (1989), the equilibrium quantities would be given by the quantities demanded at the higher of the delivery costs and thus less than the first-best quantities. Regarding location choice, the firms minimize own delivery costs for the resulting distribution of sales. These points are the medians of the sales distributions and are given by
X 1 ~--- 1 - - X 2 :
lOt - 8 + / ( l O t - 8) ~ + 24(4- 3 ~ 24t
Since the distribution of sales is no longer symmetric about a firm's location, firms choose to locate closer to the center than the middle point of their market areas. Thus, we see that the Hurter-Lederer efficient location result relies on perfectly inelastic consumer demands. For firms to locate efficiently when demands are price-sensitive, they need more flexibility in pricing, as we have shown in section 4.
6. Concluding remarks We have demonstrated the necessity for discriminatory pricing in competition to depend both on location and quantities if efficiency is to be obtained. Thus, the level of information and contractual complexity required to sustain efficient outcomes is quite high. Linear spatial price discrimination is only efficient when consumers buy zero or one unit. With individual downward-sloping demands, the distribution of consumer surplus that can be captured causes firms to move away from the efficient locations. Linear spatial discrimination is inadequate to generate efficient monopolistically competitive outcomes, as in Spence (1976). A single non-linear pricing schedule for each firm would also lead to inefficient outcomes. If firms cannot observe customer locations, but they can offer arbitrary contracts to those customers, the problem for each firm would involve incentive compatibility constraints which link together the contracts targeted at different locations. A menu of outlay-quantity pairs will be offered by each firm and consumers will choose the best of all available contracts. Consumers would pay transport costs since they do not reveal locations to the firms. Such a
J.H. Hamilton and J.-F. Thisse, Duopoly and price discrimination
185
model, which is an example of a principal-agent problem with competing principals (the firms), would generate inefficient non-linear pricing since the schedules would not differ across a relevant consumer characteristic-location. In this paper, we have followed the standard approach taken in spatial competition by assuming a non-atomic distribution of consumers over some subset of R z. There is an alternative framework used in location theory, i.e., the network model. In this model, consumers are located at a finite number of points corresponding to vertices of the network. Vertices are linked by arcs that represent transportation routes, and the distance between any two points of the network is equal to the length of the shortest path connecting these points. It is worth noting that our results can be extended to the network model [see Lederer and Thisse (1990) for the analysis of linear spatial price discrimination on a network with perfectly inelastic demands]. In particular, it can be shown that there exist equilibrium locations in the set of vertices when transportation costs are concave in distance. As a final remark, we wish to point out that our results remain valid with n firms where the second-lowest delivery cost determines equilibrium consumer surplus. We could also expand the model to allow a prior stage of entry choices by firms which incur sunk costs upon entry.
References d'Aspremont, C., J.J. Gabszewicz and J.-F. Thisse, 1979, On Hotelling's 'Stability in competition', Econometrica 47, 1145-1150. Greenhut, M., G. Norman and C.-S. Hung, 1987, The economics of imperfect competition: A spatial approach (Cambridge University Press, Cambridge). Hamilton, J., W.B. MacLeod and J.-F. Thisse, 1991, Spatial competition and the core, Quarterly Journal of Economics 106, forthcoming. Hamilton, J., J.-F. Thisse and A. Weskamp, 1989, Spatial discrimination: Bertrand vs. Cournot in a model of location choice, Regional Science and Urban Economics 19, 87-102. Hotelling, H., 1929, Stability in competition, Economic Journal 39, 41-57. Hoover, E., 1937, Spatial price discrimination, Review of Economic Studies 4, 182-191. Hurter, A. and P. Lederer, 1985, Spatial duopoly with discriminatory pricing, Regional Science and Urban Economics 15, 541-553. Lederer, P. and J.-F. Thiss¢, 1990, Competitive location on networks under delivered pricing, Operations Research Letters 9, 14%153. Phlips, L., 1983, The economics of price discrimination (Cambridge University Press, Cambridge). Scherer, F., 1980, Industrial market structure and economic performance, 2nd ed. (Rand McNally, Chicago, IL). Spence, A.M., 1976, Product selection, fixed costs, and monopolistic competition, Review of Economic Studies 43, 217-235. Spulber, D., 1981, Spatial nonlinear pricing, American Economic Review 71, 923-933. Spulber, D., 1984, Competition and multiplant monopoly with spatial nonlinear pricing, International Economic Review 25, 425-439. Thisse, J.-F. and X. Vires, 1988, On the strategic choice of spatial price policy, American Economic Review 78, 122-137. Willig, R., 1978, Pareto-superior nonlinear outlay schedules, Bell Journal of Economics 9, 56-69.
Duopoly with spatial and quantitydependent price discrimination* Jonathan H. Hamilton University of Florida, Gainesville FL, USA
Jacques-Francois Thisse CORE, Universitb Catholique de Louvain, Louvain-la-Neuve, Belgium Virginia Polytechnic Institute, Blacksburg VA, USA Received June 1989, final version received October 1990
Spatial price discrimination with downward-sloping demands is extended to allow for non-linear pricing by competing firms. With linear discriminatory pricing only, firms' locations are not efficient, but only minimize transport costs given the firms' sales distributions. Furthermore, the quantities sold are less than the efficient quantities. With non-linear delivered pricing, at the subgame perfect Nash equilibrium, quantities are identical to those under marginal cost pricing. The socially optimal locations are an equilibrium of the firms' location game.
1. Introduction
Spatial competition models have recently focused on a variety of pricing strategies used by firms, d'Aspremont et al.'s (1979) proof of the nonexistence of a pure strategy price equilibrium for some location pairs in the Hotelling (1929) model has led many researchers to consider alternative models of price formation. Spatial discriminatory pricing has almost as long a history as Hotelling's mill (f.o.b.) pricing, dating back at least to Hoover (1937). Discrimination by location was studied for a long time only in monopoly settings, but recent work has examined spatial discrimination in oligopoly. Hurter and Lederer (1985) prove the existence of pure strategy Nash equilibria in a Hotelling-type model where firms first choose locations and then price schedules. Thisse and Vives (1988) demonstrate that if firms can choose between mill and discriminatory pricing, discrimination arises as a Prisoners' Dilemma Correspondence: Prof. Jonathan Hamilton, College of Business Administration, Department of Economics, University of Florida, Gainsville, FL 32611-2017, USA. *We thank two referees for their useful comments. Financial support from the University of Florida College of Business Administration and Public Policy Research Center. Virginia Polytechnic Institute Department of Economics, and C.I.M. (Belgium) is also acknowledged. 0166-0462/92/$05.00 © 1992--Elsevier Science Publishers B.V. All rights reserved
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J.H. Hamilton and J.-F. Thisse, Duopoly and price discrimination
outcome: both firms do better if they use mill pricing. Hamilton, MacLeod and Thisse (1991) demonstrate that, with perfectly inelastic demands, mill pricing arises in equilibrium when firms have the ability to offer general contracts but customers' locations are unknown to them. Thus, it is primarily an information problem that limits firms' use of spatial discrimination. Spatial price discrimination has often been critically viewed as a means of exercising monopoly power, yet the recent work on spatial discrimination in oligopoly reveals almost the opposite result - firms compete vigorously in distant markets and consumers benefit overall from discrimination. 1 Thus, many economists have relaxed or even reversed their support for antitrust legislation which attempts to limit price discrimination by location. 2 Another form of price discrimination - non-linear pricing with respect to quantity - has been studied in spatial models by Spulber (1981, 1984). An interpretation of this approach is that, if consumers have price-sensitive demands and firms do not observe consumer locations, quantity-dependent pricing can lead to some discrimination by location. His models do not have firms competing for the same customers in setting price schedules. Rather, they are local monopolists, possibly facing competition on the fringe of their market areas. One of our objectives in this paper is to combine both types of discrimination in a model which includes location choice by firms) Many papers consider spatial discrimination only since they analyze models with inelastic consumer demands. If consumers buy either zero or one unit of the good, non-linear pricing is impossible. Hamilton, Thisse and Weskamp (hereinafter HTW) (1989) have analyzed linear spatial discrimination with price-sensitive demand functions. In much oligopoly analysis, it is irrelevant whether demand functions reflect individual consumers' responses to price or marginal consumers' decisions to buy or not to buy a single unit of the good. This is no longer true here. Absent legal prohibitions on quantity-dependent pricing, the HTW solution is an equilibrium only in models where the downward-sloping demand function at each location results from more consumers buying at lower prices. With downward-sloping demands for individual consumers, we derive the equilibrium allowing general pricing contracts. We also show that the equilibrium corresponds to the allocation in the core of the market game among producers and consumers for given locations that is most favorable to producers. With spatial and quantity-based discrimination, this allocation delivers each 1See Phlips (1983) for a discussion of the history of spatial discriminatory pricing and a discussion of its welfare implications.
2See, for example, Greenhut, Norman and Hung (1987). 3In the United States, the Robinson-Patman Act is the principal antitrust legislation on both spatial and non-linear price discrimination. It allows a variety of defenses for price discriminating firms, including cost justification and 'meeting the competition'. Even compared to other antitrust legislation, it has often been attacked for inhibiting rather than enhancing efficiency [see, for example, Scherer (1980)].
J.H. Hamilton and J.-F. Thisse, Duopoly and price discrimination
177
consumer the same surplus as with only spatial discrimination and is also a first-best allocation conditional on locations. We then solve for the subgame perfect Nash equilibrium locations with non-linear spatial discrimination and show that the socially optimal locations are an equilibrium of the firms' location game.
2. The model Our model extends the model of Hurter and Lederer (1985) to allow for downward-sloping demand functions for each consumer. A consumer is located at a point x ~ ~ which is a compact subset of the plane with a nonempty interior. Let p(x) be the density of consumers at x. A consumer at x has a demand function for a homogeneous good D(p;x) which is strictly decreasing in p; we also allow the demand function to vary with the consumer's location. There are two firms producing the good, 1 and 2, which are located at points xl and x 2 e ~ r. Firm i will incur a cost t~(x)q to deliver q units to a consumer located at x. This includes both production and transport costs. We assume that t~(x) is a strictly increasing function of the distance between x and x i and ti(x~)>0. The firms deliver the good to the consumers and thus can identify consumer locations; they are also assumed to know the demand function of consumers at each location. Therefore, nonlinear pricing depending on consumer locations can be implemented. Profits for each firm are the excess of revenues collected from consumers over total delivery costs incurred. We study the subgame perfect Nash equilibria of the game in which both firms simultaneously choose locations and then, given the pair of locations, choose simultaneously non-linear pricing schedules that depend on location.
3. The equilibrium contracts We first analyze the second stage of the game after firms have chosen locations xl and x2. If firms are allowed to offer any sort of contract to consumers, then one imagines that they will offer the contract most favorable to themselves, subject to competitive pressures from rivals. Rather than considering non-linear pricing explicitly, we need only examine outlayquantity pairs offered to consumers at each location [see Willig (1978) for a similar formulation]. If firms are free to offer contracts to consumers, in equilibrium no consumer can obtain a better contract from the other firm. When a consumer is offered equal surplus from both firms, he chooses to contract with the lower cost firm. ~ With contracts tied to locations, ~l'his is so because the lower cost firm can always offer the consumer a strictly better contract. O u r assumptions allow us to avoid the technicality of an e-equilibrium of the game. See Hurter and Lederer (1985) for a similar assumption.
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J.H. Hamilton and J.-F. Thisse, Duopoly and price discrimination
concessions offered to one set of consumers need not be offered to anyone else, given the absence of arbitrage possibilities) Thus firms will offer the best possible contracts given their costs. The lower cost firm will supply the good, and the only effective constraint on that firm is that it offers consumers at least as much surplus as the higher cost firm. The optimal contract combines features of first- and third-degree price discrimination. Because customers differ with respect to location, the contracts offered them must depend on location as well as allowing for nonlinear pricing. The firm can capture all gains from non-linear pricing, since the ability of the higher cost firm to offer non-linear pricing leaves unchanged that firm's best offer. Let Oi(x)=.{q~(x), Ri(x)} be the contract offered by firm i to consumers at location x, where q~ is the quantity delivered and R~ is the payment from the consumer.
Proposition 1. I f firms can perfectly discriminate with respect to both quantity and location, then the equilibrium contracts are: q*(x) = D(ti(x); x)
where
ti(x) <=tj(x),
and t ~(x) R~(x) = ti(x)qt(x) + I D(v; x) dr. ti(x)
Proof. The best schedule for the consumer at x that firm j, j ~ i, could offer without suffering a loss is q~(x)= q(t~(x); x) and Rj(x)= t~(x)q~(x). Firm i offers the consumer at x the same a m o u n t of surplus using the above contract. The m a x i m u m quantity for which a consumer's marginal willingness to pay is at least as high as the delivery cost determines the quantity offered. Any smaller quantity would mean firm i gives up potential profit. Q.E.D. The optimal contract is illustrated in fig. 1. The area to the left of the demand curve above the higher delivery cost is the surplus the consumer actually receives. The area to the left of the demand curve between the two delivery costs is the surplus captured by the lower cost firm and equals the profit that this firm earns with the optimal non-linear contract. In the special case of perfectly inelastic demands, the equilibrium contracts SThe contracts in equilibrium do not violate the constraint that consumers cannot choose to buy the good elsewhere and transport it back to their original locations. Observe that marginal delivered cost increases at the same rate as transport cost. Hence, for convenience, we can ignore the no-arbitrage constraint in the presentation.
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179
p(x)
~(x)
ti(x)
D(~(x);x)
D(ti(x);x) = qi*(x)
q(x)
Surplus to Consumer = s*(x) Profit to Firm i = Ri*(x ) - qi*(x) ti(x )
Fig. 1
reduce to q*(x)= 1 and R*(x)= tj(x), for i, j = 1, 2 and j ~ i. Of course, since demands are perfectly inelastic, there is no room for first-degree price discrimination, so the equilibrium contracts involve only third-degree (spatial) price discrimination. Given locations, we can also analyze the efficiency of this equilibrium. An allocation in this economy can be described using the quantity of the good each consumer obtains, the outlay of the numeraire given to the firm, and the corresponding delivery cost; each firm's profit is the excess of outlays collected over total delivery costs. If an allocation lies in the core of the market game among firms and consumers, it is efficient. Let N = S f u {1,2} be the set of agents in the economy (consumers and the two firms). Let S___N be an arbitrary coalition and A(S) be the set of feasible allocations for coalition S. An allocation Z E A(N) is in the core iff there does not exist a coalition S ~ N and an allocation Z'~A(S) which makes all members of S better off. Thus, if profits to firms and surplus to consumers in the coalition can be increased, the current allocation is not in the core. Consumers have no trades to make with each other; similarly, firms only gain by trading the good with consumers and have nothing to trade with each other. Hence, a
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blocking coalition must contain a (non-negligible) subset of consumers and at least one firm. The core of the market game for given locations also contains the contracts described in Proposition 1. Since the allocation in Proposition 1 has no constraints on the form of contracts, it is perhaps not surprising that it is also in the core.
Proposition 2. The allocation described in Proposition 1 is in the core of the market game and is also the most favorable allocation in the core for the firms. Proof. At each x E•, the outlay-quantity contract was constructed to maximize profit for the selling firm by extracting all surplus in excess of the surplus the consumer could obtain from the rival firm. Thus, no other contract could make the supplier and the consumer better off, and no better contract for the consumer could be profitably provided by the other firm. Since some consumers must agree to join the firms in any blocking coalition, the two firms cannot block this allocation by offering consumers less surplus. Q.E.D. Notice also that the quantities obtained by the consumers are the first-best quantities since they correspond to those which would be purchased at prices equal to marginal cost (the lower of the delivery costs).
4. The equilibrium locations
In the first stage of the game, the firms choose locations looking ahead to the second-stage outcome. At the equilibrium contract, the consumer at x obtains surplus s*(x)=-s(0*(x); x) for x such that ti(x)< tj(x) which is given by
s*(x) =
S
D(v; x) dv + p(q?(x); x)q'~(x) - R.*,(x),
p(~*{x);x)
where p(q; x) is the inverse of the demand function D(p; x). At the equilibrium contracts, profit for firm i equals 6 U*(xl, x~) = rl,(x~, x j, o.?(x), o* (x))
=
S
[R*(x)-- q*(x)ti(x)]p(x) dx.
ti(x)
Total surplus for the economy is 6For any locations where the two firms have equal delivery costs, no profits are earned on contracts at that location. Hence, we m a y disregard ties in delivery costs.
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181
S*(X1, X2) = J" S*(X)p(X) d x "Ji-H ~ ( x 1 , x2) + I'I~(Xl, x2) ~f
which can be simplified to S*(Xl,X2)= ~
~
minti(x)
D(v;x) dvp(x)dx.
In the case of perfectly inelastic demands, the maximization of total surplus with respect to locations reduces to the minimization of total transportation costs as in Hurter and Lederer (1985). With downwardsloping demands, changes in total surplus must also incorporate changes in delivered quantities that result from changes in locations. Our next proposition shows the efficiency of a location equilibrium.
Proposition 3. Let (xl,x:) be a location pair which maximizes total surplus. Then (xl, x2) is a location equilibrium. Proof. We can rewrite total surplus as S*(x,,x2)=~[max
('~'
x)dv, O}
,jix D(v;
Observe now that tj(x)
rt*(x,, x2) =
~
~ O(v; x) dvp(x) dx
t,(x)
=~
{ti(x), tj(x)}
~
O(v; x) dvp(x) dx
ti(x)
Since profit for firm i can be written as
H~(xl, x2) = S*(x. x~) - I ~ D(v; x) dvo(x) dx, ~f tj(x)
and since the second term in the right-hand side of this expression is
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J.H. Hamilton and J.-F. Thisse, Duopoly and price discrimination
independent of x~, for any maximizer (x*, x~') of S*, x* maximizes /-/i with respect to xi given x*. Thus, (x*,x*) is an equilibrium of the location game. Q,E.D. It follows directly from this proposition that the set of location equilibria coincides with the set of points maximizing S* with respect to xi given xj, for i , j = 1,2 and i ~ j . 7 We have not directly proven the existence of an equilibrium, yet it suffices to establish that there always exists a pair of locations that maximizes social surplus. If S* is upper-semicontinuous in xl and Xz, then a maximum of total surplus always exists since :Y is compact. Continuity of delivery costs with respect to x~ is sufficient to guarantee this. It cannot be shown that any location equilibrium is a maximizer of total surplus. Even with perfectly inelastic demands, Hurter and Lederer (1985, p. 549) present an example which demonstrates that additional equilibria may exist. However, we have shown that the efficient locations can be sustained as a Nash equilibrium of the first-stage game. Even though the distribution of consumer surplus is not symmetric about firms' locations, the delivered quantities are. Thus, maximizing the surplus extracted from consumers within the market area drives firms to the efficient locations. Consumer surplus at every location is identical to that obtained under linear spatial price discrimination. However, since the equilibrium locations differ, the distribution of surplus across consumer locations differs. Our solution complements results of Spence (1976) who demonstrates that first-degree price discrimination is necessary for the optimal set of products to be offered in a representative consumer model. Spence explicitly considers product differentiation models which are one possible interpretation of spatial models. In our model, the quantity-dependent price schedules must also depend on customer locations. In effect, firms combine first- and thirddegree price discrimination. Third-degree price discrimination is adequate to control for differences across consumers with respect to location, while firstdegree discrimination is necessary to provide firms the incentive to locate efficiently.
5. Example: A linear model To illustrate the solution presented in the last two sections, let us now consider the case where all consumers have identical linear demand curves, D(p;x)= I - p ( x ) , and both firms have delivery costs which are linear in distance from the firm to the consumer, t i ( x ) = t l x - x ~ l . Let ~r=[0,1], a VThisresultsis reminiscentof Proposition 1 in Spence(1976).
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compact subset of the real line (instead of the plane). We further assume a uniform distribution of consumers, i.e., p(x) = 1 for all x ~ [0, 1]. Using the formulas presented above in Proposition 1, the equilibrium contracts for given locations are:
ql'(x) = 1 -tlx-xa 1, R*(x) =(1 - t Ix - xl) /2 - ( 1 - t I x - x2 I)2/2
+tlX-Xxl(a-tlx-x,I),
xEE0,x-],
and
q~(x) = 1 -- t lx-- x 21, R*(x) = ( 1 - t l x - - x 2 1 ) 2 / 2 - ( 1 - t[x--xx I)2/2
+tlx-x=l(1-tlx-x2[),
1],
where £ is the location of the consumer equidistant between the firms (£=(x~ +x2)/2) and the consumer at £ is assigned to firm 1, without loss of generality, s Having found the equilibrium contracts, we can now determine equilibrium locations for the firms. Profits at equilibrium contracts for the two firms can be written as /-/~(XI, X2) = ~ I-R~(x) - - q*(x)t o
Ix-
xll] d x
and 1
II~(Xl, X2) = ~ I-II~(X) - - q*(x)t
Ix-- xx I] dx.
The equilibrium locations can easily be shown to be interior and distinct. Hence, they must solve the first-order conditions, dlI*/Oxl = 0 and aFl*/Ox2 = 0. At the equilibrium contracts +17" x, .+,+ Oxl = - t I [l - tx I + tx] dx + t J [1 + txt - tx] dx = O, 0 xl aWhile we have assumed that firm 1 locates to the left of finn 2, this is not imposed on the equilibrium. Should firm 1 locate to the fight of firm 2, firm l's profits can be found from ll2(xl,x2). The functional forms for profits assume that firm 1 locates on the left.
R.S+U.E.--B
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J.H. Hamilton and J.-F. Thisse, Duopoly and price discrimination
and a similar expression holds for firm 2. The expressions within the integrands are ql(x), so the equilibrium locations are the medians of the sales distributions. Given the rival firm's location, maximizing profits is equivalent to minimizing transport costs. Since the distribution of delivered quantities is symmetric about each firm's location under the equilibrium contracts, the unique location equilibrium is xl = 1/4 and x2 = 3/4, which is obviously the efficient pair of locations. If the firms were constrained to use linear spatial discriminatory pricing, as in H T W (1989), the equilibrium quantities would be given by the quantities demanded at the higher of the delivery costs and thus less than the first-best quantities. Regarding location choice, the firms minimize own delivery costs for the resulting distribution of sales. These points are the medians of the sales distributions and are given by
X 1 ~--- 1 - - X 2 :
lOt - 8 + / ( l O t - 8) ~ + 24(4- 3 ~ 24t
Since the distribution of sales is no longer symmetric about a firm's location, firms choose to locate closer to the center than the middle point of their market areas. Thus, we see that the Hurter-Lederer efficient location result relies on perfectly inelastic consumer demands. For firms to locate efficiently when demands are price-sensitive, they need more flexibility in pricing, as we have shown in section 4.
6. Concluding remarks We have demonstrated the necessity for discriminatory pricing in competition to depend both on location and quantities if efficiency is to be obtained. Thus, the level of information and contractual complexity required to sustain efficient outcomes is quite high. Linear spatial price discrimination is only efficient when consumers buy zero or one unit. With individual downward-sloping demands, the distribution of consumer surplus that can be captured causes firms to move away from the efficient locations. Linear spatial discrimination is inadequate to generate efficient monopolistically competitive outcomes, as in Spence (1976). A single non-linear pricing schedule for each firm would also lead to inefficient outcomes. If firms cannot observe customer locations, but they can offer arbitrary contracts to those customers, the problem for each firm would involve incentive compatibility constraints which link together the contracts targeted at different locations. A menu of outlay-quantity pairs will be offered by each firm and consumers will choose the best of all available contracts. Consumers would pay transport costs since they do not reveal locations to the firms. Such a
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model, which is an example of a principal-agent problem with competing principals (the firms), would generate inefficient non-linear pricing since the schedules would not differ across a relevant consumer characteristic-location. In this paper, we have followed the standard approach taken in spatial competition by assuming a non-atomic distribution of consumers over some subset of R z. There is an alternative framework used in location theory, i.e., the network model. In this model, consumers are located at a finite number of points corresponding to vertices of the network. Vertices are linked by arcs that represent transportation routes, and the distance between any two points of the network is equal to the length of the shortest path connecting these points. It is worth noting that our results can be extended to the network model [see Lederer and Thisse (1990) for the analysis of linear spatial price discrimination on a network with perfectly inelastic demands]. In particular, it can be shown that there exist equilibrium locations in the set of vertices when transportation costs are concave in distance. As a final remark, we wish to point out that our results remain valid with n firms where the second-lowest delivery cost determines equilibrium consumer surplus. We could also expand the model to allow a prior stage of entry choices by firms which incur sunk costs upon entry.
References d'Aspremont, C., J.J. Gabszewicz and J.-F. Thisse, 1979, On Hotelling's 'Stability in competition', Econometrica 47, 1145-1150. Greenhut, M., G. Norman and C.-S. Hung, 1987, The economics of imperfect competition: A spatial approach (Cambridge University Press, Cambridge). Hamilton, J., W.B. MacLeod and J.-F. Thisse, 1991, Spatial competition and the core, Quarterly Journal of Economics 106, forthcoming. Hamilton, J., J.-F. Thisse and A. Weskamp, 1989, Spatial discrimination: Bertrand vs. Cournot in a model of location choice, Regional Science and Urban Economics 19, 87-102. Hotelling, H., 1929, Stability in competition, Economic Journal 39, 41-57. Hoover, E., 1937, Spatial price discrimination, Review of Economic Studies 4, 182-191. Hurter, A. and P. Lederer, 1985, Spatial duopoly with discriminatory pricing, Regional Science and Urban Economics 15, 541-553. Lederer, P. and J.-F. Thiss¢, 1990, Competitive location on networks under delivered pricing, Operations Research Letters 9, 14%153. Phlips, L., 1983, The economics of price discrimination (Cambridge University Press, Cambridge). Scherer, F., 1980, Industrial market structure and economic performance, 2nd ed. (Rand McNally, Chicago, IL). Spence, A.M., 1976, Product selection, fixed costs, and monopolistic competition, Review of Economic Studies 43, 217-235. Spulber, D., 1981, Spatial nonlinear pricing, American Economic Review 71, 923-933. Spulber, D., 1984, Competition and multiplant monopoly with spatial nonlinear pricing, International Economic Review 25, 425-439. Thisse, J.-F. and X. Vires, 1988, On the strategic choice of spatial price policy, American Economic Review 78, 122-137. Willig, R., 1978, Pareto-superior nonlinear outlay schedules, Bell Journal of Economics 9, 56-69.