Nonlinear pricing in spatial oligopoly

E L S E V I E R Economic Design 2 (1997) 379-397

Economic Des ign

Nonlinear pricing in spatial oligopoly 1

Jonathan H. Hamilton a, Jacques-Francois Thisse b,c,. a Department of Economics, Warrington College of Business Administration. University of Florida

Gainesville FL 32611, USA b CORE, Universit~ Catholique de Louvain, 34 voie du Roman Pays, 1348 Louvain-la-Neuve, Belgium

and c CERAS, Ecole Nationale des Ponts et Chausss 28 rue des Saints-P~res, 75343 Paris cedex 07,

France

Received 5 February 1996; accepted 29 November 1996

Abstract

A model of duopoly competition in nonlinear pricing when firms are imperfectly informed about consumer locations is analyzed. A continuum of consumers purchase a variable amount of a product from one of two firms located at the endpoints of the market. At the Nash equilibrium in quantity-outlay schedules, consumers buy the same quantities as they would from the same firm if it were a monopolist facing the same informational asymmetries, but they receive greater surplus. Hence, no efficiency gains result from competition. If consumers have the option to reveal their locations and have the firms deliver the goods, all consumers choose to reveal their locations in equilibrium. Thus, the inefficiencies from information asymmetries maY not arise because firms can deliver the good to consumers. In contrast, with a monopoly seller, consumers have no incentives to reveal their locations.

* Corresponding author. Fax: + 1-352-3922999; e-mail: [email protected]. l We thank Juan Martinez Legaz, Richard Romano, David Sappington, Larry Samuelson, Murat

Sertel, Steven Slutsky, an associate editor, and seminar audiences at Alicante, Barcelona, Florida, and Tilburg for helpful comments and suggestions. The first author thanks DGICYT (Spain) and the Public Utility Research Center and College of Business Administration (University of Florida) for financial support and the Institut d'Anhlisi Economica for their hospitality during part of this research. The second author thanks the Minist~re de la Recherche (France) and PIR-VILLE (CNRS) for financial support.

0928-5040/97/$17.00 Copyright �9 1997 Elsevier Science B.V. All rights reserved. PH S092 8 - 5 0 4 0 ( 9 6 ) 0 0 0 2 1 - 0

380 J.H. Hamilton, J.-F. Thisse / Economic Design 2 (1997) 379-397

JEL classification: D43; D82; C72; L42

Keywords: Competing principals; Nonlinear pricing; Spatial competition

1. Introduction

Price discrimination with respect to quantities purchased by consumers is a common policy used by firms. A monopolist who cannot identify individual consumers' demand curves must design a nonlinear price schedule that satisfies incentive compatibility constraints in order to induce each consumer to select the appropriate option. In this setting, Spulber (1981) shows that a monopolist who does not observe consumer locations can implement spatial price discrimination by offering quantity discounts. Maskin and Riley (1984) consider the general case of a monopolist facing incomplete information about consumer types.

Recent work extends the study of nonlinear pricing to the case of competing firms. Wilson (1993) discusses competitive nonlinear pricing in a variety of strategic environments. Coyte and Lindsey (1988) study differentiated finns, but confine their analysis to two-part tariffs. In a circular model of monopolistic competition, Spulber (1989) studies the oligopoly analogue to Maskin and Riley and shows that quantity discounts still arise. Stole (1995) analyzes several general models of nonlinear pricing in oligopoly. We study a more restrictive version of the model in Section 3.1 of Stole in order to answer additional questions.

We consider a model of two firms located at the endpoints of a line segment along which consumers are located and, therefore, demand different quantities of the product sold by the firms. Since the duopolists have different locations, they differ in their relative appeal to different consumers. Each consumer chooses which firm to patronize and which option in the nonlinear pricing menu to take from that firm. Therefore, firms face two problems in designing their contracts. First, the nonlinear pricing contract must attract consumers away from the rival. Second, the contract must also give consumers the incentive to choose the appropriate option from those offered by the firm.

As will be shown, when firms do not know consumers' locations, equilibrium results in complete segmentation of the market between the firms, each consumer transacting in the end with a single firm. Furthermore, in equilibrium each firm offers the same quantity to each type of consumer as it would as a monopolist. One striking consequence is that mergers between duopolists may result in neither efficiency gains or losses, but only changes the distribution of surplus between firms and consumers. Since competition occurs on the fringe between market areas, firms must give additional surplus to the marginal consumers (those indifferent between purchasing from either firm) which, in turn, benefits inframarginal consumers equally. In other words, eliminating competition benefits firms at the expense of consumers, but aggregate welfare remains unchanged.

J.H. Hamilton, J.-F. Thisse / Economic Design 2 (1997) 379-397 381

So far, research has focused on both the full-information and the asymmetric- information equilibria, but no one has made this feature an equilibrium outcome. In monopoly, consumers have no incentive to reveal a priori their private information because the firm could extract all consumer surplus. This is not the case under oligopoly because competition restricts each firm's ability to extract surplus. In a location model, consumers can credibly reveal their types (locations) by having the firms deliver the product to them. We study two games in which consumers choose whether or not to request delivery before or after these price schedules are announced by firms. In both cases, a consumer near the market boundary can always get greater surplus by reporting his location to both firms. This induces an unraveling process which results in all consumers requesting delivery in subgame perfect equilibrium. This may explain why delivery, instead of pick-up, is customary in the real world (for example, Greenhut (1981) observes that almost three-quarters of firms surveyed in the U.S., Germany, and Japan deliver their products). A full-information solution only emerges when there is competition for consumers. Furthermore, this equilibrium corresponds to a first-best solution in our model. Finally, note that when locations are unobservable firms practice second-degree price discrimination while, in the case of observable locations, nonlinear pricing takes the form of a combination of first- and third-degree price discrimination (the surplus a firm extracts from a consumer is limited by the rival's proximity and depends on the consumer's location). Thus, the revelation of locations by consumers introduces a drastic change in the structure of contracts.

In designing institutions, such as antitrust authorities, one needs to take account of incentive constraints as well as resource constraints. Our paper sheds new light on the type of institutions one should use to regulate markets where firms use quantity discounts. We show that market structure may have no major impact on efficiency, so the antitrust authority's main concern should be distributional objectives in some circumstances. We also show that customers have incentives to make their locations observable to suppliers, thus giving suppliers an incentive to shift to another pricing scheme which leads to the first-best solution. These results suggest a reconsideration of some of the antitrust guidelines which are usually accepted in regulating spatial pricing.

Section 2 presents the basic model. In Section 3, we characterize the monopoly contracts both for construction of the equilibrium contracts and for later compari- son. Section 4 characterizes the equilibrium contracts. In Section 5, we analyze two games with delivery options. Section 6 illustrates our results using linear demands. Section 7 contains our conclusions.

2. The model

Consider a population of consumers who are identical except that each has a different location, x ~ [0, 1]. We refer to the consumer located at x as consumer


x. Each consumer obtains a gross surplus /~(q) from consuming q units of a good which two firms sell. The surplus, net of any payments, consumer x receives when buying from firm i is:

Si( q, x ) = i f ( q ) - t l x - x i lq - P i

where tl x - xil q is the transport cost from firm i to the consumer 's location and Pi is the amount of money paid to this fn'm. The two firms are identical except for the locations where they produce the good. Firm 1 is located at x I = 0 and firm 2 at x z = 1. For each firm, the marginal cost of producing the good is equal to c.

Our technical assumptions are that i f ( q ) is twice continuously differentiable, strictly increasing for all q less than some given threshold 7 t, and concave. Furthermore, the population of consumers is uniformly distributed on the interval [0, 1] with density one.

Suppose a consumer buys q units from firm i for a total outlay of Pi dollars. Then the firm earns a profit of P i - cq on the sale to that consumer. Let F ( q ) = i f ( q ) - cq be the surplus net of production costs. Substituting in Si, we can write:

7ri( x ) = F ( q ) - t l x - x i ] q - L ( q, x ) .

as the profit earned on the sale to consumer x. The firms know the surplus function and the distribution of consumers, but they

are unable to identify an individual consumer 's location: x is private information for each consumer if the firms do not deliver the product. Thus, they cannot practice perfect (first-degree) price discrimination according to location. However, since consumers at different locations prefer to purchase different quantities, the firms can practice second-degree pr ice discrimination through the use of nonlinear pricing. Obviously, to enforce such discrimination, a firm needs to be able to prevent resale by consumers. Following Maskin and Riley (1984), we model this problem by viewing each firm as offering a set of quantity-outlay pairs (q, P ) and letting each consumer choose his most preferred pair from those offered. In our analysis below, we assume that the schedules are differentiable and that second- order conditions are satisfied. Given our assumptions on the distribution of consumers and the surplus function, we can apply results from Maskin and Riley to guarantee that these properties are satisfied.

Let (q1(x ) , P l ( x ) ) denote the quantity-outlay pair firm 1 intends to provide to consumer x. Define Sl(x, .~) as the surplus obtained by consumer x who chooses the quantity-outlay pair designed for consumer ft. Then:

SI( x, .~) = F ( q l ( . ~ ) ) -- t q , ( ~ ) x - "rr,(~).

Incentive compatibility requires that S~ is maximized with respect to ~ (the


consumer's report of his type) at x (the true value of the consumer's type). The first-order condition for incentive compatibility is:

031 ~=x= F, d q l ( x ) d q l ( x ) d~rl (x) t x - - = 0 , ( 1 )

Off d2 dff d2

where F' = d F / d q . When consumers are induced to report types truthfully, we can write surplus as a function of the consumer type only:

S I (x ) = F ( q l ( x ) ) - txql ( x ) - "n'l(x). (2)

Using Eq. (1), surplus then changes as a function of x according to: 2

S]( x ) ~ d S l / d X = - tql ( x ) (3)

Similar analysis for firm 2 shows that:

S'z( x ) = tq2 ( x ) . (4)

The differential Eqs. (3) and (4) are constraints on the quantity-outlay functions chosen by the firms.

3. The monopoly schedule

Suppose that firm I is the only firm in the market. Then, it chooses a pair of functions ql (x) and Sl(x) to maximize:

7"I" 1 = fx ~L( x) dx 1

where XL denotes the set of consumers served by firm 1. With the surplus function above, the set of consumers served by firm 1 will be an interval [0, 2] where 2 _< 1. This follows immediately from the fact that, for any arbitrary quantity-outlay pair, if consumer x' is served by firm 1, any consumer x < x' obtains at least as much surplus from firm 1 and is also served by this firm.

Following Maskin and Riley (1984), we set up an optimal control problem in which q1(x) is the control variable, Sl(x) the state variable, and Eq. (3) the transition equation. The problem for firm 1 is:

2 m a x / ~rx(x) dx s.t. S] (x ) = - t q l ( x ) and Sx(x ) > S > 0 . q l ( x ) a O

2 This constraint is an equality since consumers are free to take contract options designed for consumers with higher or lower types. Consumer x taking the option for consumer x' > x incurs a smaller utility loss from consuming the variant 0 than consumer x' for whom the contract option had been intended. Hence, incentive compatibility in one direction implies dS l / dx >_- tql(x); in the other direction dS l / d x < - tql(x). Hence, equality must hold (almost) everywhere.


The second constraint is the individual rationality constraint, where S is the minimum surplus available to a consumer from buying nothing from the firm. 3

The necessary conditions for an optimal schedule imply:

OTr l Oql( x ) = F'( q1( x) ) - tx - tA,(x) = 0 (5)

and

A'l(x) = 1, (6)

where hi(x) is the costate variable and h i = d A J d x . The endpoint conditions for the differential equations are:

a , ( 0 ) = 0 and S , ( 2 ) = S .

The solution to Eq. (6) with A~(0)= 0 is simply A~*(x)= x. This has a simple interpretation: the cost to firm 1 of giving an additional unit of surplus to consumer x is x. If the surplus given to consumer x' is increased, all consumers x < x' must also receive the same additional surplus. Substituting this solution into Eq. (5), we obtain:

F'( qL( x) ) = 2tx. (7)

Define G( z) =- max{O, [F ' ] - I (z)} . Then, we can write the solution to this equation as:

q; ( x ) = G(2 tx ) , (8)

where G is decreasing over its support. Therefore, the quantity chosen by a consumer decreases as his location is further from firm 1. Solving F'(q) = 0 by Eq. (7), we find f f ' (q )= c so that the quantity provided to consumer 0 is the quantity for which marginal willingness to pay equals marginal cost. Given Eq. (8), for any other consumer that firm 1 serves, qi(x) is strictly less than the efficient or first-best quantity (the solution to f f ' (q t (x)) = c + tx).

Since S l ' ( 2 ) = S, the solution to Eq. (3) is:

$I* ( x ) = S + t f f G ( 2 t y ) dy. (9)

Clearly, surplus decreases as the distance from firm 1 to the consumer increases. From the q ( ( x ) and S t ( x ) schedules, one can easily derive the profit-maxi-

mizing contract in terms of the (ql, P1) pairs offered to consumers, using Eq. (2). In particular, Spulber (1981) and Maskin and Riley (1984) have shown that the firm offers quantity discounts and a different quantity-outlay pair to each type of consumer who buys from the firm. Furthermore, the optimal quantity schedule

3 In the monopoly problem, S is generally equal to zero. In contrast, in the duopoly problem, S is endogenous.


ql* (x) is independent of 2. Finally, for any values of ~ and S, the optimal S~* (x) function is vertically parallel to the Sl* (x) function for any other values of 2 and S. This follows immediately from the previous property and from Eq. (3).

If firm 1 can choose which consumers to serve, what is the optimal value of 2? The monopolist's tradeoff is that more surplus must be given to the inframarginal consumers to serve more consumers, while surplus is extracted from more consumers in an expanded market. Since the quantity sold to a consumer at the optimal schedule is independent of the surplus given to other consumers, the problem is relatively simple. The entire set of consumers will be served if enough surplus can be extracted from consumer h

[ F ' ] - l ( 2 t ) > 0 (10)

Thus, for F'(0) large enough, the monopolist located at 0 serves the entire market. One can interpret this condition as a statement that either the gross surplus is sufficiently large or the transportation rate is sufficiently small.

When firm 2 is the monopolist, the incentive compatibility constraint has the opposite sign because consumers with higher types will patronize firm 2 if consumers with lower types do so. Thus, firm 2 serves consumers in the interval [2, 1], and the solution is:

q2* ( x ) = G(2t (1 - x) ) (11)

and

S; ( x) = S + t fXG(2 t (1 - y ) ) dy. (12)

4. The duopoly equilibrium schedules

We analyze competition between the firms in the following game. Both firms simultaneously announce their (q , P) schedules. Each consumer then chooses which, if any, firm to patronize and a quantity-outlay pair. The monopoly solution entails quantity discounts, so consumers never wish to select more than one option. With competition, a consumer has the additional possibility of selecting options from different firms. We do not explicitly rule out this possibility in order to maintain the complete spirit of private information held by consumers. If firms cannot identify consumer types, presumably they are also unable to discover whether a consumer also patronizes the other firm. We thus allow consumers to have the option to purchase the good from both firms, but we find that they never avail themselves of this option.

The Nash equilibrium contracts share several properties with the monopoly contracts derived in Section 3. First, the individual rationality constraint binds only for a single consumer 2 for whom Sl(2) = $2(2). Second, the incentive compati-


Sl(X) / ~ . ' ~ 1 "" t ql (x) /S2(X)

S ~ ( x ~ ~ $1 " "tql(x)

I I

I I

0 x*

Fig. 1. Sl(x) is the surplus from (ql(x), Pl(x)) for x~[O, 2]. Consumers in (x ~ 2) also select (ql(x * ), Pl(x * )) and receive surplus as given by the dashed line. Sl* is the surplus from the optimal contract satisfying Sl(2) = $2(2), which yields higher profit for finn 1.

bili ty constraints cont inue to bind for all cus tomers o f a firm. 4 To establish these

results, we need two lemmas. The first shows that bunching (giving an interval of

types o f consumers the same quant i ty-out lay pair) is never optimal, whether or not

the r iva l ' s contract entails bunching. (Al l proofs are conta ined in Append ix A.)

Lemma 1. Suppose that f i rm 1 serves all consumers in the interval [0, 2]. Then, f irm 1 always strictly prefers to design contracts with q i ( x ) = G(2tx) for all consumers in [0, 21.

Thus, the f i rm can always increase its profi t f rom sales in the interval [ x *, 2]

by of fer ing a contract tai lored to each consumer . In doing so, it can also reduce the

surplus it g ives all consumers in [0, x *] and further increase its profit . Fig. 1

il lustrates the gain f rom serving all consumers who patronize f i rm 1 with a

contract ta i lored to them.

I f t ransport costs were high enough, the f irms wou ld be monopol i s t s with

disjoint sets o f customers. The second l e m m a states condi t ions such that the f irms

act ively compe te for the marginal consumer .

4 We assume that finn 1 offers to provide a consumer at x with G(2tx) (and finn 2 offers to provide G(2t(1 - x))) even if the consumer prefers to patronize the rival firm. This is not strictly necessary since the offer to a consumer at 2 by firm 1 is the best offer to any consumer in (2, 1]. The equilibrium is unique with respect to the resulting allocation, but various offers to consumers of the other firm are possible contracts.

J.H. Hamilton, J.-F. Thisse / Economic Design 2 (I 997) 379-397 387

Lemma 2. Assume that each f irm would serve at least one-half o f the market i f it were a monopolist. Then, all consumers in [0, 1] will be offered a contract by at least one firm.

The existence of equilibrium contracts has been proven by Spulber (1989) and Stole (1995) for similar models. 5 We offer an independent proof rather than translating their arguments for our model.

Proposition 3. Assume that either f i rm as a monopolist would serve at least hal f o f the market. Then, there exists a unique equilibrium. The equilibrium quantities and surplus levels are given by Eqs. (8), (9), (11) and (12) with .~ = Yc = 1 / 2 .

Several points deserve mention here. First, consumers are free to buy the product from both firms, but no one chooses to do so because firms offer quantity discounts. The marginal consumer is indifferent between either firm, but he would be strictly worse off if he were to buy from both firms. Second, our approach implicit ly uses the revelation principle. Since the incentive compatibil i ty constraint facing each firm is identical to that which it faces as a monopolist , a consumer need only report his type to the firm he patronizes. The consumer would have no gain from making any other report to the other firm. Thus, we can apply the standard approach to asymmetric information problems. Last, since the equilibrium quantity offered to a consumer depends only upon the distance between the consumer and the firm, the quantity schedules are identical to those which monopolists would offer. Thus, we obtain the following result.

Corollary 4. Competition for customers raises consumer surplus by an equal amount for all customers who do not switch suppliers.

In other words, the marginal and inframarginal consumers benefit equally from the presence of competition. Even though the marginal consumer is in a better posit ion to switch suppliers, the binding incentive compatibil i ty constraints insure that all customers benefit to the same extent from competition. 6

Suppose now that the duopolists merge into a single firm operating two plants. Under competit ion, all consumers patronize their lower-cost supplier, so the merged firm will continue to serve each customer from the same source as before. Hence, we obtain the following result.

5 Stole's results for a model with quality differences are similar to Proposition 3, but further assumptions (which we have made) are needed when consumers purchase different quantities.

6 When individual demands are perfectly inelastic, as in Hotelling's model, nonlinear pricing reduces to linear pricing. In this case, when a firm cuts its price to gain more customers, all its customers also obtain the same additional surplus which amounts here to the price cut.


Proposition 5. I f the two firms merge into a monopolist operating two plants serving the entire market, then no consumer switches suppliers and there are no efficiency gains or losses from merger.

Thus, under asymmetric information, mergers may have no impact on welfare. However, from Corollary 4, all consumers equally lose from the merger because the monopolist extracts more surplus from all of them.

Another possible change to consider is from a pair of monopolies with a market split at a point other than the center. For example, suppose the two firms served markets in neighboring countries and trade between them was liberalized. In this case, duopoly competition would yield efficiency gains because some consumers near the center would switch suppliers.

Finally, we can compare the equilibrium above with the nonlinear pricing equilibrium under full information (Hamilton and Thisse, 1992). With full information, each consumer buys the efficient quantity and receives the maximum surplus that the rival could offer. Under asymmetric information, the quantity schedule chosen by each firm declines as if demand for the product fell twice as fast with a change in consumer type as it actually does (to see this, set At(x) = 0 for all x in Eq. (5)). The schedules of consumer surplus as a function of consumer locations slope down as we move away from the firms under asymmetric information, while they slope up under full information. In particular, the marginal consumer gets the smallest surplus under asymmetric information, but the largest one under full information.

5. Revelation of locations by consumers

The equilibrium in Section 4 arises when no consumer locations are known by the firms. In the case of monopoly, no consumer has an incentive to reveal his location to the firm before accepting a contract, because the monopolist will then offer the consumer a contract particular to him with zero surplus. Under oligopoly, the presence of competition restricts the possibility for a firm to offer a contract with zero consumer surplus when the firm knows a customer's location, since the rival can always offer a contract with positive surplus. Thus, it is of interest to study endogenous revelation of private information under competition. While in some contexts it may be difficult for consumers to reveal credibly their private information, delivering the good to consumers permits firms to learn consumers' locations. Therefore, the equilibrium with all consumers having firms deliver the product is the full information one described in the previous section.

Would consumers gain by revealing locations to both firms and have the good delivered to them? In the full-information equilibrium, the consumer at 1 /2 receives the maximum surplus and firms make zero profit there. In the asymmetric-information equilibrium, the farm earns positive profit on the sale to the


consumer at 1/2, so this consumer must obtain less surplus than in the full-information equilibrium. The consumers at 0 and 1 obtain the least surplus in the full-information equilibrium, but the most surplus in the asymmetric-information equilibrium. Consequently, we cannot determine in general which equilibrium consumers prefer.

Delivery of the good imposes a different incentive compatibility requirement for the equilibrium contracts. No consumer should prefer to take delivery of the good at another location and transport the good back to his true location. This is always satisfied in the full-information equilibrium. The additional costs of transporting the good from the reported location to the true location are actually greater than the costs of misreporting type in the private information solution because the quantities consumed are greater in the full-information solution. Thus, the surplus gradient for reporting a location farther away and taking delivery there is steeper and of the opposite sign from S' i.

Now consider the game where, first, firms choose nonlinear price schedules for pick-up and, then, consumers choose whether to buy at the offered price or reveal their locations to both firms to obtain a better deal. Firm 1 offers a set of contracts for pick-up subject to the incentive compatibility constraint Eq. (3) to consumers in an interval [0, 2). Competition for those consumers who reveal locations implies that they get exactly their full-information equilibrium allocations. Since the price schedules for pick-up have already been set, the firms make the same offers for delivery as they do in the full-information case (see Hamilton and Thisse, 1992).

Suppose all consumers who do better in the full-information equilibrium (2 < x < 1 - 2 ) reveal their locations, and all consumers who do better in the asymmetric-information solution (x < 2 and x > 1 - 2 ) do not report locations. Consumers in [2, 1/2] get surplus equal to S ~ ( x ) = F ( G ( t ( 1 - x ) ) ) - t ( 1 - x )G( t (1 - x)) , while consumers in [0, 2] get surplus equal to:

^

Sl(x) -- S ( ( 2 ) + t fXG(2ty) dy (13)

(recall Eq. (9)). The consumer at 2 is indifferent between reporting and not reporting his location, so SI(2) equals S((2) .

We can now write profit for firm 1 as a function of 2, the consumer who is indifferent about reporting and not reporting location, as follows:

< ( 2 ) = f0*[ F( G ( 2 t x ) ) - t x G ( 2 t x ) ) - Sx(x)] dx + f l/2[ F ( G ( t x ) )

- t xG( t x ) - S F ( x ) ] dx. (14)

Thus, firm 1 chooses a price schedule which will induce all consumers in (2, 1/2] to reveal their types and obtain the full-information contract, while offering the type of contract described in Section 3 to those consumers who prefer not to


reveal their types. Since the profit and surplus under full information is the corresponding Nash equilibrium solution, all the impact of competition from the rival firm is embedded in this expression.

The next proposition shows that, when consumers are free to report their locations to both firms and opt out of the asymmetric-information pricing contract, all consumers choose to do so.

Proposition 6. Assume that each f irm would serve the whole market i f it were a monopolist and allow consumers to request delivery. In the game where, first, f irms set price schedules for pick-up and, then, consumers decide whether to request delivery, all consumers have the product delivered to them at the subgame perfect equilibrium.

In other words, if firms correctly predict which consumers will report locations when designing the contracts, all consumers will take the delivery option.

Consider another game of endogenous revelation in which, first, consumers decide whether to reveal their types to both firms and, then, each firm chooses two nonlinear price schedules for consumers who have and have not revealed types. Here, consumers anticipate the offers firms will make for both pick-up and delivery. Clearly, if a firm delivers the good, it then offers the full-information equilibrium contract. What price schedule will a firm offer consumers who did not reveal? It is reasonable to suppose that nonrevealers can still request delivery at any location where consumers do request delivery. This would be the case if firms offer all opportunities to all consumers, as might be required by antitrust laws which attempt to limit price discrimination.

Proposition 7. Assume that each f irm would serve the whole market i f it were a monopolist and allow consumers to request delivery. In the game where consumers choose whether or not to request delivery and then f irms set price schedules for pick-up and delivery, i f f irms do offer any consumers delivery, all consumers have the product delivered to them at the subgame perfect equilibrium.

In this game, the firms will not necessarily compete vigorously for the consumers who reveal their types. Indeed, this competition has an adverse effect on the profit earned from the nonrevealers. However, if the consumer at 1 /2 prefers the option for pick-up at firm 1 to the offer by this firm for delivery to the most distant consumer requesting delivery (beyond 1/2), then firm 1 will make offers for delivery and, therefore, will compete for those distant customers. When any customer is offered delivery, all customers will request delivery.

Assume, instead, that the consumers who have not revealed their locations can only pick up the good at one of the firms. Then, the firms can offer less surplus to the nonrevealers than in the equilibrium of Proposition 3. Consequently, for any ~, some consumers in [0, ~) would have preferred to reveal. The solution in which


those closer than a2 do not reveal thus unravels, so that all consumers will reveal in equilibrium.

Surprisingly, the equilibria with asymmetric information about consumer types cannot survive when consumers have an option to report their types credibly to both f i rms and obtain the full-information outcome. In location models, the possibility of arranging for delivery makes this revelation credible. Who gains and loses from revelation is not clear. Certainly, the consumers with central locations gain. Since the delivery equilibrium is efficient, it is possible that all groups gain. However, we show below in an example that consumers in the aggregate may lose surplus and the firms may gain profits in the delivery equilibrium relative to the pick-up equilibrium.

Our results indicate that modelling f i rms as uninformed about consumer types may not be appropriate i f consumers have opportunities to reveal their types to competing firms. This is in stark contrast to models of private information with monopoly firms. The monopolist gives the marginal consumer less than the efficient quantity and zero surplus; all other consumers receive positive surplus. I f all consumers took delivery, the firm could give zero surplus to all of them. No individual consumer gains by requesting delivery and, therefore, consumers do not reveal their locations to the firm. However, when firms compete for customers, the incentive compatibility constraint is paramount in determining what contract a consumer is offered. Revealing one 's location makes crucial the contract the rival offers the consumer. It is this fact that creates the incentive to reveal locations, thus making pick-up less likely than delivery.

This is not the only possible model of pick-up and delivery. Our firms have the ability to use nonlinear pricing in both information structures. If the nonlinear pricing were only feasible for pick-up, but delivery required linear pricing with respect to quantity or freight absorption (such as uniform delivered pricing or zone pricing), then pick-up might be more viable. 7

6. Example: Linear demands

Suppose that consumer surplus is quadratic: / 6 ( q ) = a q - q2 /2 . Then the demand curve for the product is linear: q(p , x ) = a - tl x - xi[ - p. Let a --- a - c be the intercept of the demand curve net of marginal production costs. Firm l ' s profits from selling q to consumer x equal:

rq ( x ) = [ a - tx]q( x ) - [ q ( x ) ] 2 / 2 - Sl( x ) .

7 Furlong and Slotsve (1983) show that pick-up and delivery (with uniform delivered pricing) options can be used together by a monopolist to increase profit when the monopolist does not discriminate with respect to quantity.


It follows from Proposition 3 that:

q t ( x ) = a - 2tx

and

Sl* ( x ) = 7S - atx + t2x 2 + atYc -- t2.~ 2

where firm 1 serves consumers in [0, 2] and S = Sl(2). 8 Since demands are linear, it is easy to see that the assumption of Lemma 2

implies that a > t. (If a > 2 t, then each firm is willing to serve the entire market as a monopolist.) Thus, the two firms will have the same marginal consumer, denoted by ~ = 1/2 . The profit earned by firm 1 from a consumer located at x is:

( a - 2 t x ) 2 "Tr l ( X ) = ( a - - t x ) ( a - - 2 t x ) 2 S l (X)"

Eq. (13) becomes:

a 2 / 2 - 2atYc + 2t22(1 - 2) - S I ( 2 ) = 0.

Substituting in ,~ = 1/2 , we find that:

Sl* ( 1 / 2 ) = a2 /2 -- at + t 2 / 2 ,

and hence:

SI* ( x ) = a2 /2 - a t / 2 + t 2 / 4 - atx + tXx 2 . (15)

It is instructive to compare this equilibrium to that under linear pricing. If each firm sells all output at a single mill price, the Nash equilibrium is:

3 t + 2 a (52t 2 - 16at+ 16a2) 1/2

P1* 4 8 (16)

Substituting into the demand function, we see that the difference between con- sumption in the two equilibria is:

q ~ ( x ) - q ~ ( x ) = - p ; + t x , f o r x ~ [ 0 , 1 / 2 ) ,

which is always negative at x = 0 since that consumer obtains the efficient quantity with nonlinear pricing. The marginal consumer and, therefore, all consumers obtain more of the product under nonlinear pricing as long as a > 3 t / 2 (see Eq. (16)). Surplus for a consumer under linear pricing equals:

S ~ ( x ) = ( a - t x - p , ) 2 / 2 , f o r x ~ [ 0 , 1 / 2 ) .

Integrating S 1 and S~ over firm l ' s market area, we find that:

fo1/2[ $1(x) - S~( x)] dx = - a t ~ 4 + 7 t2 /48 + p t ( a / 2 - t / 8 - p l / 4 ) .

This is positive for t < 0,836a. Thus, contrary to common intuition about monopoly

8 In this section, we only report results for firm 1 and consumers in [0, 1/2], since the results for firm 2 and consumers in [1/2, 1] can be found by symmetry.

,I.H. Hamilton, J.-F. Thisse / Economic Design 2 (1997) 379-397 393

price discrimination, under oligopoly, consumers as a whole prefer nonlinear price discrimination to linear mill pricing unless t is nearly as large as a.

Using Eq. (15) and integrating over firm l ' s market area, we f ind that equilibrium profits for firm 1 with nonlinear pricing are:

I ~ 1 = at~4 - t 2 / 8 - t2/24.

Equilibrium profits for firm 1 with linear pricing are:

Fix L = Pl( a / 2 - t / 8 - P l / 2 ) .

From Eq. (16), we find that profits are greater with linear pricing for t < 0.538 a. Thus, firms prefer linear pricing for low values of t. However, when nonlinear pricing is feasible (no arbitrage among consumers), the firms cannot avoid nonlinear pricing.

Finally, observe that welfare, defined as the sum of consumer surplus and profits, is always higher under nonlinear pricing. We can also make further comparisons with the full-information equilibrium. Surplus for each consumer under full information equals:

S F ( x ) = ( a - - t + t x ) 2 / 2 , f o r x ~ [ 0 , 1 /2) .

Integrating S~" and S F over firm l ' s market area, we find that:

f01/2 [ $I* ( x ) - sF( x)] dx = t2/48,

so consumers in the aggregate always prefer the equilibrium outcome under asymmetric information to the full information outcome. Since the full-information equilibrium is efficient, if consumers are better off in the asymmetric-information equilibrium, the firms must earn higher profits in the full-information equilibrium. Thus, when the firms induce consumers to take delivery in Proposition 6, the f i rms gain from revelation relative to the case where no consumers take the delivery option.

7. Conclusions

We constructed the Nash equilibrium in nonlinear pricing schedules for spatial duopolists when they cannot identify consumer locations. In equilibrium, consumers obtain the same quantities as they would purchase if their supplier were a monopolist in the same information environment. The individual rationality constraint for each firm binds only with respect to the marginal consumer. Since the quantities consumed do not change with the introduction of duopoly competition, equal monopoly markets and an integrated duopoly market are equally efficient. All consumers gain an equal amount of surplus in the shift to competition, which is surprising since consumers have different relative rankings of the two suppliers.


In our model, efficiency gains only arise from duopoly competition if the separate monopoly markets differ in size. Similarly, if production costs differed across firms, a two-plant monopolist would serve some customers from different plants after merger, thus yielding an efficiency gain.

While retail consumers often pick up the good and thereby avoid credibly revealing their locations, delivery by fin'ms permits credible revelation of locations and is pervasive in wholesale trade. In the latter case, the resulting equilibrium is efficient and the consumers located near the market division receive the most surplus. 9 We find that, if consumers have the option to request delivery from the firm, then all consumers have the firm deliver the product in equilibrium. However, consumers have no incentive to request delivery from a monopoly supplier. Competitive nonlinear pricing thus provides a rationale for behavior frequently observed in the real world (see Greenhut (1981) and Phlips (1983)). Furthermore, when transport costs are not too high, the example of Section 6 shows that firms might well be better off with delivery than with pick-up, thus suggesting another justification for the practice.

An important extension would be to endogenize firm locations. The main difficulty this creates is that the density function of consumers with respect to their distance from firms becomes discontinuous. In turn, the quantities in the nonlinear pricing schedule become discontinuous and adding additional customers at the margin changes the optimal quantity schedule for inframarginal consumers.

Appendix A

Proof of lemma 1. Suppose that firm 1 deletes part of its contract offer on some subinterval of [0, 2]. Without loss of generality, assume that firm 1 offers contracts only to consumers in [0, x * ] where x * < 2. Thus, consumers in (x *, 2] obtain a surplus equal to S l ( x * ) - t q l ( x * ) ( x - x*). Given that Sl(X) is strictly convex by Eq. (9), this surplus is strictly less than Sl(x). Since Sl(X) >_ S2(x) for x ~ (x *, ~], firm 1 can increase its profit by offering a contract to each consumer in (x* , 2] which satisfies ql(x) = G(2tx).

Note that this argument holds for any S2(x) function which satisfies firm 2's incentive compatibility constraint, whether or not firm 2 bunches consumers. []

Proof of lemma 2. Suppose that neither firm serves consumers with types in the interval (x ' , x"). Then, since the contracts maximize profit for each firm given the set of consumers served, S~(x') --- S2(x") = 0. But, as shown in Section 3, each firm would do strictly better to serve additional consumers. []

9 Consumers at the center of the market also do best under linear delivered pricing (see Thisse and Vives, 1988).


Proof of proposition 3. Consider firm l ' s choice of its best reply. The only relevant element of firm 2's contract is Sz(x). Given the S2(x) function, firm 1 chooses 2 together with a quantity-surplus schedule which satisfies Sl(2) = $2(}). The optimization problem for firm 1 given Sz(x) is to maximize:

H 1 = 7rx(x) dx s.t. S,(2) = $ 2 ( 2 ).

The only choice variable in this problem is 2, and the FOC is:

OHl* ~ Orrl( x) = rr~(2) + fo O----~-x dx = O. 02

From Section 3, ql" (x) is independent of 2, and S'1(x) is independent of S1(2). Hence, using Eq. (2):

07/'1(X) OSl( x ) OSl( 2) . . . . . . . s i ( 2 ) ,

02 as,(2) 02

where the last equality holds because S'2(2) is the marginal surplus firm 1 must give for a marginal increase in 2. We then obtain:

OH 1 * 02 ~-- q/'l(2) - 2Si(2) = 0, (A.1)

as a necessary condition for the best reply. Under the assumptions of Lemma 2, the two firms serve all consumers in any

candidate equilibrium, so 2 is the same for both firms. Given Sl(x), firm 2 maximizes:

u ; s.t. S2(2)=S,(2).

The FOC with respect to 2 for firm 2 is:

OH2* 1 077"2(X) 02 = - ' r r 2 (2 ) + ~ 0 x d x = 0 "

Since O~r2(x)/02 = -S~(2), this simplifies to:

OH2* 02 = - rr2(2) - (1 - 2 )S i (2 ) = 0. (A.2)

Since S](x)< 0, and S'2(x)> 0, Eqs. (A.1) and (A.2) may have a common solution. From inspection of Eqs. (A.1) and (A.2), when 2 = 1/2, we have rrl(2) = ~r2(2) and S] (2)= -S'2(2). Hence, there exists a common value Si(2) >__ 0 such that the two equations have a solution. To show uniqueness, we solve for the gradients of surplus and find that:

F( q1( 2) ) - t2ql( 2 ) - Sl(X ) - 2 t q 2 ( 2 ) = 0 ( a . 3 )


and

- V ( q z ( 2 ) ) + t ( 1 -2)qz(Yc ) + $ 2 ( 2 ) + (I - 2 ) t q l ( 2 ) = 0 . (A.4)

In equilibrium, $1(2) = $1(2). Define Hi(x) -- F(ql(x)) - txql(x) and Ha(x) =- F(qa(X)- t(1 -x)qz(x) . From the properties shown above, Hi(x) is decreasing in x and Ha(x) is increasing in x. Eliminating Si(2) from Eqs. (A.3) and (A.4), we obtain:

H l (~ ) + (1 --2)tql(Yc ) = H 2 ( 2 ) + 2tq2(~ ).

As 2 increases, the LHS becomes smaller since Gl(2), 1 - 2 , and ql(x) all fall, while the RHS becomes larger since G2(2), 2, and q2(2) all rise. Thus, there is only one value of 2 which solves the FOCs, and we have already shown that to be 2 = 1 / 2 . []

Proof of corollary 4. The gradients of the surplus function are the same as those which hold under monopoly provision. So an increase in Si(2) leads to the same benefit for all consumers of firm i. []

Proof of proposition 5. Inspection of the monopoly and duopoly equilibrium quantity schedules shows that qi(x) depends only upon Ix i -x l . Thus, the monopoly and duopoly equilibrium quantity schedules will be identical for all consumers who patronize the same firm in both cases. Since 2 = 1 /2 and the monopolist makes the same market division, no one switches and the quantity consumed by every consumer remains the same. Any change in the payment schedule alone leaves the sum of profits plus consumer surplus unchanged, so no efficiency gains result in that case. []

Proof of proposition 6. Differentiating Eq. (14), we find:

d_F/1 =X(G(Zt2)) - t2G(Zt2) - S , ( 2 ) + fo - - d x - [X(G(t2))

d2 02

- t2G(t~) - SlY( 2)] .

The two surplus terms cancel out and F(G(tx)) - txG(tx) > F(G(2 tx)) - txG(2 tx) for all x as long as each firm would supply the whole market as a monopolist. The term under the integral is strictly negative, since an increase in 2 requires giving more surplus to all inframarginal consumers. Thus, the sum of these terms is strictly negative. The ftrm will therefore always prefer to have more consumers reveal their types and take delivery at their own locations. []

Proof of proposition 7. Assume that some consumers reveal types. Then the problem facing farm 1 in constructing the contract for pick-up is identical to the monopoly problem of choosing a price schedule for x < ff subject to the constraint


that S 1 ( ~ ) = sF(ff). This is not the same as the monopoly problem of Section 3 since firm 1 earns positive profits from consumers who opt for delivery at ~, the nearest location where delivery is offered. Thus, by setting ff < 2, the firm can decrease Sl(x) for every consumer who picks up the good at the firm. But then, some consumers in [0, ~) would prefer to report their locations. Since they anticipate that this will happen, all consumers must reveal their types in any subgame perfect equilibrium. []

References

Coyte, P. and C. Lindsey, 1988, Spatial monopoly and spatial monopolistic competition with two-part pricing, Economica 55, 461-477.

Furlong, W. and G. Slotsve, 1983, "Will that be pickup or delivery?": An alternative spatial pricing strategy, Bell Journal of Economics 14, 271-274.

Greenhut, M.L., 1981, Spatial pricing in the USA, West Germany, and Japan, Economica 48, 79-86. Hamilton, J. and J.-F. Thisse, 1992, Duopoly with spatial and quantity-dependent price discrimination,

Regional Science and Urban Economics 22, 175-185. Maskin, E. and J. Riley, 1984, Monopoly with incomplete information, Rand Journal of Economics 15,

171- 196. Phlips, L., 1983, The Economics of Price Discrimination (Cambridge University Press, Cambridge). Spulber, D., 1981, Spatial nonlinear pricing, American Economic Review 81,923-933. Spulber, D., 1989, Product variety and competitive discounts, Journal of Economic Theory 48,

510-525. Stole, L., 1995, Nonlinear pricing and oligopoly, Journal of Economics and Management Strategy 4,

529-562. Thisse, J.-F. and X. Vives, 1988, On the strategic choice of spatial price policy, American Economic

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Nonlinear pricing in spatial oligopoly