A Unified Model of Spatial Price Discrimination

Transcription

1 MPRA Munich Personal RePEc Archive A Unified Model of Spatial Price Discrimination Konstantinos Eleftheriou and Nickolas Michelacakis University of Piraeus 9 June 6 Online at MPRA Paper No. 76, posted 9 June 6 7:59 UTC

2 A Uni ed Model of Spatial Price Discrimination Konstantinos Eleftheriou ;y and Nickolas J. Michelacakis Abstract We present a general model of n rms with di erentiated production costs competing in a linear market within the framework of spatial price discrimination. We prove that the Nash equilibrium locations of rms are always socially optimal irrespective of the number of competitors, rm heterogeneity regarding marginal production costs, the level of privatization, the form of the transportation costs and the number and/or the varieties of the produced goods. An immediate implication of this result is that this form of competition is preferable from a welfare point of view. We also argue that (i) when rms are homogeneous regarding their marginal production costs, there always exists a unique Nash equilibrium, regardless of the form of the transportation cost function (ii) when rms are heterogeneous and transportation costs are linear, there is a unique Nash equilibrium which depends only on the relative mutual di erences of the marginal production costs. JEL classi cation: L3; L3; L33; R3 Keywords: Mixed oligopoly; Social optimality; Spatial competition; Di erentiated goods Introduction Whenever we make online purchases from the web, we are witnessing a form of market segmentation due to discriminatory pricing dependent on geographical location. This pricing practice is called spatial price discrimination (Cabral, ). However, this is not the only market where this type of pricing is common. Spatial price discrimination manifests itself in markets in which rms are geographically di erentiated such as the markets of cement and steel or markets of customer-tailored goods. The wide application of this pricing strategy together with the fact that it is forbidden by some countries when it cannot be justi ed on the grounds of transportation/delivery costs (e.g., Robinson-Patman Act, 936 in the US), makes the investigation of spatial price discrimination of great interest for both academics and policymakers. The main goal of the current paper is to examine the welfare properties of the equilibrium in a market where operating rms exercise spatial price discrimination. To this purpose, we develop an integrated model Department of Economics, University of Piraeus, 8 Karaoli & Dimitriou Street, Piraeus 85 34, Greece. kostasel@otenet.gr (Eleftheriou); njm@unipi.gr (Michelacakis). y Corresponding author. Tel: ; Fax: Greenhut (98) provides evidence that spatial price discrimination is apparent in cases where transportation cost represents at least 5% of total costs. For a review about the history of the enforcement of competition law against spatial discriminatory pricing, see Scherer and Ross (98).

3 giving new insight into the structure of customer-speci c pricing markets. The existing literature adopts at least one of the following ve assumptions: (i) the number of rms in the market does not exceed two (ii) all rms are privately owned (iii) transportation costs are linear (iv) only one homogeneous good is traded and (v) rms have common marginal production costs. We relax all the above assumptions by assuming a market with an arbitrary number of heterogeneous competitors, an arbitrary level of privatization for each rm, a general transportation cost function and an arbitrary number and/or varieties of traded goods. Firm heterogeneity is re ected by assuming di erent marginal costs of production. It should be emphasized that this heterogeneity is rm-speci c and not product-speci c (i.e., the marginal cost of production di ers across rms but remains the same for all goods/varieties produced by the same rm). The e ect of the above mentioned relaxation on the existence and the properties of the equilibrium is not clear. For example, d Aspremont et al. (979) showed that in the traditional Hotelling s model (Hotelling, 99), the nature of travel costs is important for the existence of an equilibrium. Surprisingly enough, they showed that an equilibrium exists when transportation costs are proportional to the square of distance while it doesn t when the travel costs are linear. On the other hand, Cremer et al. (99) highlighted the importance of the number of competing rms on the welfare properties of the equilibrium. The present paper contributes to the existing literature in manifold ways. We show that in a model of spatial price discrimination, where the produced goods have the same reservation price for the buyers, the market outcome will be socially optimal, and this result is independent of the number of rms in the market, rm heterogeneity regarding marginal production costs, the level of privatization of each rm, the form of the transportation cost function and the number and/or the varieties of the goods o ered by each competitor. We further argue that (i) when rms are homogeneous regarding their marginal production costs, there always exists a unique Nash equilibrium, regardless of the form of the transportation cost function extending a well-known similar result (e.g. Heywood and Ye, 9a) restricted to linear transportation costs and (ii) when rms are heterogeneous and transportation costs are linear, there is a unique Nash equilibrium which does not depend on the distribution of the marginal production costs. The driving force behind our welfare result is the same as in Lederer and Hurter (986); a rm can increase its pro t by opting for a production location so that the market is serviced with minimal total cost. 3 However, in Lederer and Hurter (986) the discussion is restricted to only two exclusively privately owned rms o ering the same good leaving untouched mixed markets with many competitors and multiple goods and the ensuing welfare questions. Moreover, our proof is completely di erent to the one found in Lederer and Hurter (986) allowing for direct generalization. 3 This implies the alignment of the social and private optima.

4 Our ndings about the characterization of the equilibrium are also in line with Vogel (). Vogel () proves the existence of a unique equilibrium for an arbitrary number of heterogeneous rms located in a circular market competing within the framework of spatial price discrimination bearing linear transportation costs, regardless of the distribution of their marginal production costs. We investigate the properties of the equilibrium, should it exist, when an arbitrary number of heterogeneous rms with various degrees of privatization o ering multiple goods compete in a linear market under spatial price discrimination. In line with Vogel (), we prove the existence of a unique equilibrium when transportation costs are linear. We succeed, however, in establishing the social optimality of the Nash equilibrium in all cases. Moreover, the fact that our model imposes no constraints on the level of privatization extends our contribution to the theory of mixed oligopoly under spatial price discrimination. The studies which are closest to ours are those of Heywood and Ye (9a) and Heywood and Ye (9b). Heywood and Ye (9a) assume a market with an arbitrary number of homogeneous rms having binary ownership status (private or public) and focus on the role of the public rm in the Stackelberg equilibrium where the leader is a private or a public rm. They extend their model accounting for the existence of foreign rms in Heywood and Ye (9b). The aforementioned papers impose too many restrictions in their modeling structure. Speci cally, the framework of Heywood and Ye (9a) and Heywood and Ye (9b) imposes restrictions on the form of the transportation cost function, the degree of privatization and the attributes/number of goods in the market. In addition, a fundamental restriction of the aforementioned papers lies in the assumption of common marginal production costs. Our paper is part of a wide literature on the welfare implications of spatial price discrimination; see, e.g., Greenhut and Ohta (97), Holahan (975), Thisse and Vives (988), Hamilton et al. (989), Hamilton et al. (99), MacLeod et al. (99), and Braid (8). Building on this literature, we make inroads into the theory of mixed oligopoly when rms have di erentiated marginal production costs. The implications of our results can be summarized as follows: (i) A spatial price discriminatory market can serve as a typical example of how a laissez-faire economy can lead to social optimality, (ii) the social optimality of the equilibrium is independent of rm heterogeneity and (iii) the dependence of the equilibrium locations on the relative di erence of the marginal production costs has important policy implications for government intervention. In the example of subsection 3. the government can (pre)determine the locations of rms through intervention on the marginal production costs (e.g., tancentives and subsidies for rms located in low-populated areas etc.) without a ecting social optimality. The rest of the paper is structured as follows. The next section presents the benchmark model where an arbitrary number of rms characterized by a di erent degree of privatization o er a homogeneous 3

5 good. Two cases are examined: (a) rms have common marginal production costs and (b) rms are heterogeneous. The market is represented by a unit interval with the consumers uniformly distributed along it. A three-stage game of complete information is played by rms and consumers. More speci cally, in the rst stage rms simultaneously choose their locations. In the second stage, after observing their competitors locations, rms engage in Bertrand competition à la Hoover (937) and Lerner and Singer (937). In other words, rms set their prices simultaneously and are allowed to price discriminate by charging a di erent price for di erent locations. Finally, in stage three, consumers make their purchasing choices to clear the market. After presenting our theoretical construct, we solve the game and characterize the Nash equilibrium. Section 3 generalizes the ndings of section for the case of multiple goods (or di erent varieties of the same good), section 4 concludes. The benchmark model We consider a market consisted of n private rms and a continuum of consumers uniformly distributed over the unit interval [; ] representing a linear country. 4 Let, i = ; :::; n, denote the location of rm i in the interval [; ] with x < x < ::: <. All rms produce and sell the same homogeneous good. Each consumer buys one unit of the good from the lowest price rm, providing that this price is lower or equal to her reservation price (i.e., the maximum price that the consumers are willing to pay for the good), m >. The marginal production cost of rm i is c i. Spatial price discrimination à la Hoover (937) and Lerner and Singer (937) is assumed. Speci cally, the price charged for the good by the rm the consumer chooses to buy from, is equal to (or in nitesimally less than) the delivered cost of the remaining rms. Delivered costs are equal to the sum of transportation and production costs. Let f(d(x ; x )) := jf (x ) F (x )j evaluate the transportation cost between points x and x, where d denotes the shipped distance and F be any function with F positive and continuous on [; ]. Firms are located such that f(d(+j ; )) > jc i+j c i j (non-negative pro t condition) 5 for any j > with i; i + j f; :::; ng and c i+j, c i the corresponding marginal costs of rms i + j and i located at points +j and respectively. Let s i;i+j denote the locations of the indi erent consumer with respect to rms i and i + j. Lemma. (i) < s i;i+j < +j and F (s i;i+j ) = F (+j)+f ( ) + c i+j c i and (ii) if j < j then (a) s i;i+j < s i;i+j and (b) provided j < j < j, s i+j ;i+j < s i+j ;i+j. 4 By uniformly distributed, we mean that the proportion of consumers buying the good remains the same, regardless of the subinterval of [; ]. 5 In the opposite case if, f(d(+j ; )) jc i+j c i j, the total sales of either rm i or rm i + j drop to zero. 4

6 Proof. By de nition of s i;i+j, jf (s i;i+j ) F ( )j + c i = jf (+j ) F (s i;i+j )j + c i+j By assumption F is an increasing function, so if s i;i+j [; ] [ [+j ; ] then f(d(+j ; )) = F (+j ) F ( ) = jc i+j c i j a contradiction, thus, < s i;i+j < +j and F (s i;i+j ) F ( ) + c i = F (+j ) F (s i;i+j ) + c i+j i.e., which proves (i). F (s i;i+j ) = F (+j) + F ( ) + c i+j c i To prove (ii) (a) we argue by contradiction. Assume s i;i+j s i;i+j, then F (+j ) + F ( ) + c i+j c i F (+j ) + F ( ) + c i+j c i, F (+j ) F (+j ) c i+j c i+j equivalently f(d(+j ; +j )) jc i+j c i+j j a contradiction. The proof of (ii) (b) follows similar lines. Consumers and rms engage in a three-stage game of complete information. In stage one, rms simultaneously decide their location. Having observed the location of their competitors, rms simultaneously choose delivered price schedules in the second stage. In the nal stage, consumers take their purchasing decisions. The rest of the section is structured as follows. Subsection. examines the properties of the equilibrium in the case of privately owned rms with common marginal costs. The case of mixed ownership with homogeneous rms is analyzed in subsection.. The corresponding analyses for heterogeneous rms are presented in subsections.3 and.4, respectively. 5

7 . Homogeneous rms In this subsection, we examine the case where all rms have the same marginal cost of production, i.e. c = c = ::: = c n. The existence and social optimality of a symmetric equilibrium (see Proposition below) provided that transportation costs are linear is known to exist and can be found in various references in the literature (e.g. Heywood and Ye, 9a). We provide a di erent proof as an intermediate result, relaxing the assumption of linear transportation costs. In the case of linear transportation costs uniqueness follows from Proposition 3. To simplify our analysis and without loss of generality, we set c i =. The aggregate shipping cost 6 for all locations z of consumers who buy from any of the n rms is equal to T H (x ; :::; ) = nx i= T H i (x ; :::; ) () where R x [F (x ) F (z)]dz + R x +x x [F (z) F (x )]dz A for i = T H i (x ; :::; ) = R xi + [F ( ) F (z)]dz + R ++ [F (z) F ( )]dz A for < i < n () R xn +xn [F ( ) F (z)]dz F ( )]dz + R [F (z) A for i = n is the total transportation cost for those consumers buying from rm i. As de ned in Lederer and Hurter (986) the social cost is the total supply cost when rms behave in a cooperative, cost minimizing manner. The fact that delivered costs coincide with transportation costs implies that the aggregate transportation cost represents the social cost. Hence, the socially optimal locations can be derived by minimizing the social cost with respect to each location. 7 Firm i is selling its product at a price matching (or which is in nitesimally less than) the delivery cost of its direct competitor which is the rm nearest to its location. The indi erent consumer between rms i 6 The terms delivered cost and shipping cost are used interchangeably hereafter. 7 In other words, social welfare is de ned as the total consumer s willingness to pay less the aggregate transportation and production costs. 6

8 and i +, according to Lemma, is located at s i;i+ = ++. Thus, the pro t function of rm i is R x [F (x ) F (z) F (x ) + F (z)] dz + R x +x x [F (x ) F (z) F (z) + F (x )] dz A if i = >< H i (x ; ::; ) = R xi + [F (z) F ( ) F ( ) + F (z)] dz + R ++ [F (z) F ( ) F (z) + F ( )] dz + R xi++ ++ [F (+ ) F (z) F (z) + F ( )] dz if < i < n and ++ R ++ + [F (z) F ( ) F ( ) + F (z)] dz + R ++ [F (+ ) F (z) F ( ) + F (z)] dz + R ++ [F (+ ) F (z) F (z) + F ( )] dz if < i < n and ++ C A C A (3). R xn +xn [F (z) F ( ) F ( ) + F (z)] dz + R [F (z) F ( ) F (z) + F ( )] dz A if i = n Lemma. When rms are homogeneous, the marginal aggregate transportation cost with respect to the location of rm i, i = ; :::; n, is 8 H = = >: F (x ) F ( x x ) for i = F ( ) F ( + ) for < i < n F ( xn ) F ( ) for i = n : Proof. Let G(y) := R F (y)dy, then for all i, < i < n, T H i (x ; :::; ) = [ G( z)] G() + G( ) + G( + + +[G(z )] ++ = ). appears in the expression of the aggregate transportation cost only 7

9 in Ti H ; Ti H and T H i+ Z xi T H (x ; :::; ) = T H (x ; :::; ) + ::: + [F (x + ) F (z)]dz + [F (z) F ( )]dz i x i {z } +T H i (x ; :::; ) + Z xi+ Z + ++ [F (x ++ ) F (z)]dz + [F (z) F (+ )]dz + ::: + Tn H (x ; :::; ) i+ x i+ {z } T H i+ T H i Z +!! Hence, we H i H i H i+ H = F ( ) + F ( ) F ( + ) F ( + ) = F ( ) F ( + ) for < i < n. The cases i = and i = n are treated similarly to H =@x = F (x ) F ( x x ) H =@ = F ( xn ) F ( ) completing the proof of the Lemma. Proposition. When rms are homogeneous, the marginal aggregate transportation cost with respect to the location of rm i, i = ; :::; n, is opposite to the marginal pro t of rm i, H = H i = : Proof. Letting G(y) := R F (y)dy, for < i < n, (3) becomes H i (x ; :::; ) = G( + Di erentiating the above expression, we get ) G( ) G( + ) + H i = F ( ) + F ( + ) For i = and i = n, we get H =@x = F (x ) + F ( x x ) H n =@ = F ( xn ) + F ( ). Lemma completes the proof of the proposition. Following our discussion above, the socially optimal locations are derived by minimizing () with respect to each rm s location. Hence, the socially optimal locations satisfy the system: 8

10 Proposition shows that this is equivalent H = = ; i = ; :::; n: H i = = ; i = ; :::; n: (5) which proves Proposition. In models of spatial price discrimination, where rms are homogeneous, o er the same good and the market is represented by a uni-dimensional interval, the Nash equilibrium locations of rms are socially optimal. The next step is to check the existence and uniqueness of the Nash equilibrium. Proposition 3. In models of n homogeneous rms selling a good to consumers uniformly distributed along a linear country at prices conditional to consumer location, there always exist a unique Nash equilibrium which is socially optimal and does not depend on the form of the transportation cost function. Proof. According to Proposition, it su ces to exhibit either a socially optimum or a Nash equilibrium. We test t of the unique Nash equilibrium when transportation costs are linear, i.e. x i = i n i = ; :::; n.8 By Lemma, the above solution must satisfy the system OT H (x ; :::; ) =. Indeed, for every < i < n, we must have F ( ) F ( + ) =. But this is equivalent to F ( n ) F ( n ) = which is true regardless of the transportation cost function f. The border cases are similarly veri able. The uniqueness of the equilibrium can be proved as follows. The equilibrium satis es (by Lemma ) F (x ) F ( x x ) = F ( x x ) F ( x 3 x ) =. F ( xn ) F ( ) = 8 For the derivation of the Nash equilibrium locations under linear transportation costs, see Heywood and Ye (9a).. 9

11 which implies F (x ) = ::: = F ( ) = r. Since F is, there is a unique D = F (r) with F (x ) = q, x = D F ( x x ) = q () x x = D, x = 3D F ( x 3 x ) = q () x 3 x = D, x 3 = 5D. F ( xn ) = q, xn = D, = (n )D F ( ) = q, = D, = D The last two equations give D = nd D () D = n completing the proof.. Mixed oligopoly with homogeneous rms In our analysis so far, all rms are privately owned. Let us now assume that single rm l, l = f; :::; ng is partly privately owned and partly publicly owned in proportions a l and a l (in other words a l can be considered as the degree of privatization), respectively with a l [; ]. In such a case, rm l will decide about its optimal location by maximizing the weighted average of its own pro ts and social welfare with weights a l and a l, respectively. Social welfare is equal to the sum of the aggregate pro ts (the pro t of all rms) and consumers surplus. The consumers surplus is given by CS H (x ; :::; ) = nx CSi H (x ; :::; ) (6) i= where CS H i (x ; :::; ) is the consumer surplus generated for the consumers buying from rm i, therefore,

12 8 R x [m F (x ) + F (z)]dz + R x+x x [m F (x ) + F (z)]dz for i = >< CSi H (x ; :::; ) = R xi + R ++ + [m F (z) + F ( )]dz + R ++ [m F (z) + F ( )]dz + R xi++ ++ [m F (+ ) + F (z)]dz for ++ and < i < n [m F (z) + F ( )]dz + R ++ + R ++ [m F (+ ) + F (z)]dz for ++ and < i < n [m F (+ ) + F (z)]dz (7) >: R xn +xn [m f(z )]dz + R [m f(z )]dz for i = n Direct calculation proves 8 R x +x mdz T H (x ; :::; ) for i = Lemma 3. H i (x ; :::; )+ CS H i (x ; :::; ) = >< R ++ + mdz Ti H (x ; :::; ) for < i < n >: R xn +xn mdz T H n (x ; :::; ) for i = n Summing up over all rms one gets the following Proposition which could be viewed as the second main result of this subsection. Proposition 4. nx H i (x ; :::; ) + CS H (x ; :::; ) = m T H (x ; :::; ) i= Proof. Straightforward calculations. The pro t function of the partly publicly owned rm l will be " # X H l (x ; :::; ) = H l (x ; :::; ) + ( a l ) H i (x ; :::; ) + CS H (x ; :::; ) i6=l (8) where H l would be the pro t function of rm l if it was fully privately owned.

13 Proposition 5. When rms are homogeneous, Nash equilibria remain socially optimal regardless of the degree of privatization of the individual rms l; l n. Proof. Fix a random l; l n. Using Proposition 4 and (8), we get H l (x ; :::; ) = H l (x ; :::; ) + ( a l ) m T H (x ; :::; ) H l (x ; :::; ) From H =@x l H l =@x l H =@x H l =@x l = which implies H l =@x l H l =@x l. Induction on i completes the proof..3 Heterogeneous rms In this subsection, we study the case of n rms producing with di erent marginal production costs. In consistency with our notation above, let s i;i+j denote the locations of the indi erent consumer. Then the pro t of rm i, N i (x ; :::; ), is given by N (x ; :::; ) = Z s; [f(d(x ; z)) + c f(d(x ; z)) c ]dz N i (x ; :::; ) = Z si ;i+ + s i ;i [f(d(z; )) + c i f(d(z; )) c i ]dz Z si;i+ s i ;i+ [f(d(+ ; z)) + c i+ f(d(z; )) c i ]dz N n (x ; :::; ) = Z s n ;n [f(d(z; )) + c n f(d(z; )) c n ]dz The next Lemma relates the pro t of rm i in this case, N i (x ; :::; ), to the corresponding pro t of the same rm, H i (x ; :::; ), if all marginal costs were equal. Lemma 4. N (x ; :::; ) = H (x ; :::; ) + (c c ) x +x + R s ; x +x [F (x ) + F (x ) F (z) + c c ]dz

14 R si ;i + + R s i;i+ ++ R si ;i+ ++ N i (x ; :::; ) = H i (x ; :::; ) [F (z) F ( ) + c i F ( ) + F (z) c i ]dz [F (+ ) F (z) + c i+ F (z) + F ( ) c i ]dz [F (+ ) F (z) + c i+ F (z) + F ( ) c i ]dz +(c i c i ) + + (c i+ c i ) R sn ;n +xn N n (x ; :::; ) = H n (x ; :::; ) [F (z) F ( ) F ( ) + c n c n ] dz +(c n c n ) xn +(c n c n )( ) Proof. N (x ; :::; ) = R s ; [f(d(x ; z)) + c f(d(x ; z)) c ] dz = R x [F (x ) F (z) + c (F (x ) F (z) + c )] dz + R x +x x [F (x ) F (z) + c (F (z) F (x ) + c )] dz + R s ; x +x [F (x ) F (z) + c (F (z) F (x ) + c )] dz = R x [F (x ) F (z) F (x ) + F (z)]dz +(c c )x + R x +x x [F (x ) F (z) F (z) + F (x )]dz + R s ; x +x + R s ; x +x +(c c ) x x [F (x ) + F (x ) F (z) + c c ]dz = H (x ; :::; ) +(c c ) x +x [F (x ) + F (x ) F (z) + c c ]dz which settles the proof for N (x ; :::; ). To prove the Lemma for the N i (x ; :::; ), without any real loss of generality, we consider the case < s i ;i < + < < ++ < ++ < s i ;i+ < s i;i+ < +. 3

15 Then, N i (x ; :::; ) = R s i ;i [f(d( ; z)) + c i f(d( ; z)) c i ]dz + R s i ;i+ [f(d( ; z)) + c i f(d( ; z)) c i ]dz + R s i;i+ s i ;i+ [f(d(+ ; z)) + c i+ f(d( ; z)) c i ]dz = R + s i ;i [F (z) F ( ) + c i (F ( ) F (z) + c i )]dz + R + [F (z) F ( ) + c i (F ( ) F (z) + c i )]dz + R ++ [F (z) F ( ) + c i (F (z) F ( ) + c i )]dz + R s i ;i R xi R s i;i+ ++ R si ;i+ ++ = H i (x ; :::; ) + R s i;i+ ++ R si ;i+ ++ [F (z) F ( ) + c i (F (z) F ( ) + c i )]dz [F (+ ) F (z) + c i+ (F (z) F ( ) + c i )]dz [F (+ ) F (z) + c i+ (F (z) F ( ) + c i )]dz [F (+ ) F (z) + c i+ (F (z) F ( ) + c i )]dz R si ;i + Finally, to prove the Lemma for i = n consider [F (z) F ( ) F ( ) + c i c i ]dz [F (+ ) + F ( ) F (z) + c i+ c i ]dz [F (+ ) + F ( ) F (z) + c i+ c i ]dz +(c i c i ) + + (c i+ c i ) : N n (x ; :::; ) = R s n ;n [f(d(z; )) + c n f(d(z; )) c n ]dz + R = R s n ;n [F (z) F ( ) + c n (F ( ) F (z) + c n )]dz + R [F (z) F ( ) + c n (F (z) F ( ) + c n )]dz = R s n ;n +xn +xn [F (z) F ( ) F ( ) + c n c n ]dz [F (z) F ( ) F ( ) + F (z)]dz + (c n c n ) xn + R [F (z) F ( ) F (z) + F ( )]dz + (c n c n )( ) = H n (x ; :::; ) R sn ;n which completes the proof of the Lemma. +xn [F (z) F ( ) F ( ) + c n c n ]dz +(c n c n ) xn + (c n c n )( ) We now calculate the total shipping cost function T N (x ; :::; ). T N (x ; :::; ) = R s ; [f(d(z; x )) + c ]dz + ::: + R s i;i+ s i ;i [f(d(z; )) + c i ]dz +::: + R s n ;n [f(d(z; )) + c n ]dz 4

16 The following Lemma holds true. Lemma 5. Proof. T N (x ; :::; ) = T H (x ; :::; ) + c x x +c x x + ::: + c i x i + c i+ x i + ::: + c n n + c n ( ) + R + R s i;i+ + R + + R s ; ++ + P n i= ++ + R + R si;i+ ++ [F (z) F ( ) F (+ ) + c i c i+ ]dz T N (x ; :::; ) = R x [F (x ) F (z) + c ]dz + R x +x x [F (z) F (x ) + c ]dz + R s ; x +x R si ;i + [F (z) F (x ) + c ]dz + ::: [F ( ) F (z) + c i ]dz [F ( ) F (z) + c i ]dz + R ++ [F (z) F ( ) + c i ]dz R si;i+ [F (z) F ( ) + c i ]dz ++ [F (+ ) F (z) + c i+ ]dz [F (+ ) F (z) + c i+ ]dz + R [F (z) F (+ ) + c i+ ]dz +xn + R s i+;i R sn ;n +xn [F (z) F (+ ) + c i+ ]dz + ::: [F ( ) F (z) + c n ]dz [F ( ) F (z) + c n ]dz + R [F (z) F ( ) + c n ]dz = R x [F (x ) F (z)]dz + c x + R x+x x x [F (z) F (x )]dz + c x x +x + R [F (z) F (x ) F (x ) + c c ]dz + ::: + R x +c i i + R xi+xi+ x [F (z) F ( )]dz + c i+ i + R s i;i R s n ;n +xn completing the proof of the Lemma. [F (z) F ( ) F (+ ) + c i c i+ ]dz + ::: [F (z) F ( ) F ( ) + c n c n ]dz + [F ( ) F (z)]dz x +xn [F ( ) F (z)]dz + c n n + R [F (z) F ( )]dz + c n ( ) Proposition 6. The marginal aggregate shipping cost with respect to the location of rm i, i = ; :::; n, is opposite to the marginal pro t of rm i, N (x ; :::; )= N i (x ; :::; )= Proof. We prove the Proposition for i, < i < n; the border cases, for i = and i = n, being very similar. 5

17 According to Lemma N i (x ;:::;) i (x ;:::;) R si i + R si;i+ R si ++ ;i+ ++ (c i c i ) + (c i+ c i ) [F ( ) + F ( ) F (z) + c i c i ]dz [F (+ ) + F ( ) F (z) + c i+ c i ]dz [F (z) F (+ ) F ( ) + c i c i+ ]dz (9a) On the other hand, from Lemma 5, we get Pn N (x ;:::;) H (x ;:::;) + c i + c i c i c i+ R sj;j+ x j +x j+ [F (z) F (x j ) F (x j+ ) + c j c j+ ]dz (9b) For all j 6= i ; i turning (9a) into and (9b) into " Z sj;j+ [F (z) F (x x j +x j ) F (x j+ ) + c j c j+ ]dz = j+ R si N i (x ;:::;) i (x ;:::;) c i + c i+ + R si;i+ ++ R si i [F ( ) + F ( ) F (z) + c i c i ]dz [F (+ ) + F ( ) F (z) + c i+ c i N (x ;:::;) H (x ;:::;) + c i + R si;i+ ++ c i+ [F (z) F ( ) F ( ) + c i c i ]dz [F (z) F ( ) F (+ ) + c i c i+ ]dz (a) (b) Proposition of the homogeneous case ensures H (x ;:::;) sides of (a) and (b) are equal proving the Proposition. i (x ;:::;), therefore the right-hand From the analysis so far, we get that socially optimal locations satisfy the system whereas Nash equilibrium locations N (x ; :::; )= =, i = ; :::; n: () 6

18 @ N i (x ; :::; )= =, i = ; :::; n: () Proposition 7. In models of spatial price discrimination, where rms, o er the same good and the market is represented by a closed interval, the Nash equilibrium locations of rms are socially optimal. Proof. By a linear transformation any closed interval can be mapped in a one-to-one way onto [; ]. The rest of the proof follows by Proposition 6 which ensures that the system of () and () are equivalent and therefore have the same solution..4 Mixed oligopoly with heterogeneous rms The consumer surplus generated for the consumers buying from rm i is 8 R s; [m f(d(x ; z)) c ]dz for i = >< CSi N (x ; :::; ) = R si ;i+ s i ;i [m f(d(z; )) c i ]dz + R s i;i+ s i ;i+ [m f(d(+ ; z)) c i+ ]dz for < i < n >: R s n ;n [m f(d(z; )) c n ]dz for i = n Following a similar analytical reasoning with section., we get the following proposition Proposition 8. nx N i (x ; :::; ) + CS N (x ; :::; ) = m T N (x ; :::; ) i= P where CS N (x ; :::; ) = n CSi N (x ; :::; ) is the total consumers surplus when rms are heterogeneous. i= Proof. Straightforward calculations. Similar to section., the pro t function of the partly publicly owned rm l when marginal costs of production are di erent will be " # X N l (x ; :::; ) = N l (x ; :::; ) + ( a l ) N i (x ; :::; ) + CS N (x ; :::; ) (3) i6=l where N l would be the pro t function of rm l if it was fully privately owned. 7

19 Proposition 9. When rms are heterogeneous, Nash equilibria remain socially optimal regardless of the degree of privatization of the individual rms l; l n. Proof. Fix a random l; l n. Using Proposition 8 and (3), we get N l (x ; :::; ) = N l (x ; :::; ) + ( a l ) m T N (x ; :::; ) N l (x ; :::; ) From Proposition N =@x l N l =@x l N =@x N l =@x l = which implies N l =@x l N l =@x l. Induction on i completes the proof. 3 The case of multiple goods with heterogeneous rms 3. Private rms We now assume the existence of L di erent goods or di erent varieties of the same good or both. Let k j denote the number of rms producing good j, j = ; :::; L with k j n. Let T N;j denote the aggregate transportation cost related to the provision of good j and N;j i the corresponding pro t per consumer of rm i from selling good j with N;j i := if good j is not produced by rm i. The fraction of consumers P buying product j is now denoted by h j (; ] uniformly spread over [; ] with L h j = ; hence, there will be buyers for all available products. In the case where good j is produced by only one rm, then this rm enjoys monopoly privileges and charges a price equal to, or in nitesimally smaller than, the reservation price m j, i.e. the maximum price the consumer is willing to pay for good j. A fundamental assumption in this multi-good setting is that m = ::: = m L = m (i.e. the reservation price of all goods is identical). 9 Let ~T N denote the aggregate shipping cost for all products and ~ N i the total pro t of rm i for all products it produces. Proposition. The marginal aggregate shipping cost with respect to the location of rm i is opposite to the marginal pro t of rm i, T ~ N = ~ N i =. Proof. By de nition T ~ P N = L h j T N;j and ~ P N i = L h j N;j. Applying Proposition 6 for every single traded product j we get ~ T N = = j= LX h N;j = = j= i LX j= j= h N;j i = ~ N i = 9 It should be noted that this assumption is more realistic in the case of the di erent varieties of the same good and less in the case of di erent goods. 8

20 Theorem. In models of spatial price discrimination, where rms have di erent marginal production costs, produce di erent combination of goods, consumers are distributed uniformly along a linear city of unit length and have the same reservation price for all goods, the Nash equilibrium locations of rms are socially optimal. Proof. To derive the socially optimal locations we have to minimize T ~ N with respect to each rm s location. Hence, the socially optimal locations satisfy the following system of ~ T N = = ; i = ; :::; n: (4) On the other hand, the Nash equilibrium locations are given by the solution of the following ~ N i = = ; i = ; :::; n: (5) Because of Proposition, systems (4) and (5) are equivalent and hence they have the same set of solutions. 3. Mixed oligopoly Let s now turn to the case where some rm, say rm l is partly privately owned and partly publicly owned. Keeping the notation the same as in subsections.4 and 3., P N l = L h j N;j l where h j N;j l is the pro t of the partially privatized rm l from selling good j. It is understood that N;j l by rm l. j= = if good j is not produced Theorem. The degree of privatization does not a ect the socially optimal Nash equilibrium locations. Proof. From the proof of Proposition 9, we have that for every single product N;j l =@x l N;j l =@x l : N l =@x l = LX j= h j (@ N;j l =@x l ) = LX j= h N;j l =@x l = LX N;j l =@x l ~ N i = 9

21 Theorem 3. In a mixed oligopoly of n rms, with n 3, producing di erent combinations of goods with di erentiated marginal production costs and linear transportation costs, there exists a unique socially optimal Nash equilibrium of locations for any (c ; :::; c n ) in the non bounded subset C, of the positive orthant R n +, de ned by the inequalities (n )c i + (n )c i+ nx cj j=;j6=i;i+ < (6) nx (n )c c j < (7) j= and Xn c j + (n )c n < (8) j= for i = ; :::; n. Further any two marginal cost vectors (c ; :::; c n ) and (c ; :::; c n) in the subset C, such that c i = c i + u lead to the same equilibrium locations. Proof. We prove Theorem 3 in the simplest possible setting that of private rms producing only one common good assuming the per distance transportation cost, t, equal to one. The general case for the mixed oligopoly with multiple goods and t 6= can then be proved along similar to the analysis above lines. According to Proposition 7, the optimal locations must satisfy the system 3x x = c +c x +x x 3 = c +c 3 x +x 3 x 4 = c +c = c n +c n +.

22 It is straightforward to check that the above system is row equivalent to 3x x = c +c 5 x 4 3 x 3 = c 3 + c 3 +c 3 where the i-line < i < n is given by 7 x x 4 = c 5. 4n x 4 n n = c n. 4 5 c c 3+c 4 4 c 4 n ::: c 4(n ) n n + c n n+ i + i + = 4 i c 4 i c ::: 4 i c i + i 3 i c i + c i+ : Solving for we get where i 3 = 4 +, < < 4. = i i + [+ + 4 i (c i c ) + ::: + 4 i (c i c ) + i (c i c + ) + 4 i (c i+ c + ) + 4 i (c i+ c + ) + ::: + 4 i (c i+ c i )] Inherent to the discussion leading to Proposition 7 was the assumption that x < x < ::: <. It is a straightforward, albeit tedious, calculation to show that < + () (n )c i + (n )c i+ nx cj j=;j6=i;i+ < : Further, we get and nx < x () (n )c c j < j= Xn < () c j + (n )c n < : j= To prove that the domain, C, de ned by the above set of inequalities is not bounded it su ces to consider all n-tuples (c ; :::; c n ) with c = ::: = c n (homogeneous case) on the positive part of the main diagonal of R n +.

23 Remark. The case n = is treated thoroughly in subsection A duopolistic model of heterogeneous rms - Policy implications To highlight the policy implications of our ndings in subsection 3., we present an application for a duopoly with linear transportation costs. Let C i denote the marginal production cost of rm i. There are three varieties of a di erentiated product o ered to consumers, U and W from rm and V and W from rm. Let also the fraction of consumers buying only good U equal the fraction of consumers buying good V, with both set equal to c. Product W is bought by a fraction b of consumers. Transportation costs are linear and equal to td, where t is a positive scalar and d is the distance shipped. The locations of rm and over the interval [; ] are x and x, respectively (without loss of generality x < x ). Keeping the structure of the game and the rest of the notation as above, the pro t functions of rms and when both are privately owned are: ct ~ = c(m C ) [x + ( x ) ] R x b[t(x x ) + C C ]dz + R A ( x +x + C C t ) x b[t(x + x z) + C C ]dz ct ~ = c(m C ) [x + ( x ) ] R x ( x +x + C C t ) b[t(z x x ) + C C ]dz + R x b[t(x x ) + C C ]dz A (9) () with C C t x x. The location s of the indi erent consumer for good W is determined by equating the two delivered costs in regard to the common good W : t(x s)+c = t(s x )+C ) s = x +x + C C t. Having evaluated the integrals, (9) and () become C C x x If t > ~ = c(m C ) ct [x + ( x ) ] +bx [t(x x ) + C C ] (9b) + b [t(x 4t x ) + C C ] ~ = c(m C ) ct [x + ( x ) ] +b( x )[t(x x ) + C C ] + b 4t [t(x x ) + C C ] (b), both rms are reduced to spatial-price discriminating monopolists where the common good W is now provided only by rm. We consider this case trivial and focus only on the case where C C x x t.

24 Firm chooses x to maximize (9b), and rm chooses x to maximize (b), leading to the following Nash equilibrium locations where! = b(c C ) t(b+c) and A = b 4(b+c). (x ; x ) = The total shipping cost will be equal to A +!; + A +! () R x + ~T = ct [x + ( x ) ] + ct [x + ( x ) ] + cc + cc b[t(x z) + C ]dz + R ( x+x + C C t ) x b[t(z x ) + C ]dz R x ( x +x + C C t ) b[t(x z) + C ]dz + R x b[t(z x ) + C ]dz A () Maximizing () with respect to x and x gives the socially optimal locations A +!; + A +! (3) We now turn to the case where rm is partly privately owned and partly publicly owned in proportions a and a, respectively with a [; ]. In this case, the pro ts of rm will be = c(m C ) ct [x + ( x ) ] +b( x )[t(x x ) + C C ] (4) + b 4t [t(x x ) + C C ] + ( a )g(x ; x ) where ct g(x ; x ) = c(m C ) [x + ( x ) ] R x b[t(x x ) + C C ]dz + R ( x +x x b[t(x + x z) + C C ]dz + R ( x+x + C C t ) b[m t(x z) C ]dz + C C t ) + R ( x +x + C C t ) b[m t(z x ) C ]dz = (b+c) [tx ( x ) + m t C ] A (5) It is straightforward to show ; x )=@x = showing that the equilibrium remains intact irrespective of the degree, a, of privatization. Furthermore, the distance between the optimal locations, x and x, is independent of marginal production costs and t and equals A. It follows that anybody who wishes to in uence the location x of 3

25 either rm, with x (; =), or the location of rm, x, with x (=; ), can do so by intervening on the marginal cost relative di erence, C to choose C < C in such a way that C given X. C. For example, given an a priori X (; =), it su ces C = t(b+c) (X + A) for rm to locate optimaly on the b 4 Conclusion We have proved that when rms exercise spatial price discrimination, the equilibrium outcome is socially optimal and independent of the underlying assumptions on the number of rms, rm heterogeneity, the nature of transportation costs, the number or the varieties of the provided goods and the degree of privatization. Even though we expect our ndings to hold when consumers are non-uniformly distributed, we have intentionally opted for the less technically demanding setting of consumer uniform distribution to showcase the essence of our ideas. The technical requirements in relation to the non-uniformly distributed case are thoroughly presented in Lederer and Hurter (986). To the best of our knowledge, our analysis is the rst attempt to present an holistic view of models of spatial price discrimination. Moreover, our ndings verify the robustness of the laissez-faire doctrine and can be easily applied to the case of vertically related markets (see Eleftheriou and Michelacakis, 6). It is also not hard to deduce the validity of our ndings for discontinued markets where speci c locations are ruled out. Possible extensions could investigate strategic delegation e ects and spatial two dimensional markets. Acknowledgement. We would like to thank Costas Arkolakis, Anil Arya, Sofronis Clerides, Jacques Thisse, Dimitrios Xefteris, Anastasios Xepapadeas and Nicholas Ziros for their valuable comments, which have helped to substantially improve the quality of this paper. All remaining errors are ours. References Braid, R. (8). Spatial price discrimination and the locations of rms with di erent product selections or product varieties. Economics Letters 98, Cabral, L. (). Introduction to Industrial Organization, MIT Press, Cambridge, MA. Cremer, H., Marchand, M. and Thisse, J. F. (99). Mixed oligopoly with di erentiated products. International Journal of Industrial Organization 9, x (; =) is implied by the fact that C C x x t (non-negative pro t condition) and x x = A. 4

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