Symmetric and Asymmetric Equilibria in a Spatial Duopoly

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1 This version: February 003 Symmetric and Asymmetric Equilibria in a Satial Duooly Marcella Scrimitore Deartment of Economics, University of Lecce, Italy Jel Classification: L3, R39 Abstract We develo a model of horizontal di erentiation with a non-uniform symmetric distribution of consumers' references. Through a simle arametrization, we solve the rice-location game as a function of the degree of consumers' concentration towards the middle. A symmetric equilibrium is shown to exist for all values of the concentration arameter, rovided that the consumers' density is di erentiable at the centre of the suort of the distribution. At this equilibrium, an increase in consumer concentration is associated to a reduction in roduct di erentiation. When the distribution becomes su±ciently concentrated, two asymmetric equilibria arise which coexist with the symmetric one. Address for corresondence: Deartment of Economics, University of Lecce, Ecotekne, via er Monteroni, I-7300 Lecce (Italy) mscrimitore@economia.unile.it

2 Introduction The results achieved in recent years by the theory of roduct di erentiation may well exlain its increasing relevance in the analysis of industrial organization and in the study of the sources of market ower. The theory rests on the idea that in the resence of a di erentiated demand, the strategic interaction among rms develos along two lines: the rices charged and the characteristics chosen by a rm and its cometitors. One of the most investigated toics in this eld is the analysis of locational equilibria in a horizontally di erentiated market. The horizontal Hotelling model has been widely used in order to discuss roblems related to the satial rice cometition, the otimal roduct attributes, the otimal lant location, etc., and has found alications in the satial economics literature, as well as in trade and banking theory. These models rimarily focus on the existence of a Princile of Maximal or Minimum Di erentiation (Economides 986). This existence roblem amounts to asking whether the interlay between the structure of consumers' references for the di erentiated roduct and the otimal strategic behaviour of rms results into too little or too much roduct diversity. Following Hotelling (99), D'Asremont, Gabszewicz and Thisse (979) established a Princile of Maximal Di erentiation, by assuming that the intensity of consumer references for their ideal roduct may be reformulated in a locational setu in terms of quadratic transortation costs: in a duooly market, the rms try to set u aart from each other - di erentiate at most their roduct - in order to relax rice cometition. This nding sharly contrasts with the acclaimed Princile of Minimum Di erentiation of the original Hotelling model where rms,intheresenceoflineartransortationcosts,choosetoclusterin the roduct sace. Examles of the tendency for cometitors to reduce di erences in distance or in the roduct characteristics sace can be easily found in the real world. Conversely, examles of maximal di erentiation can be identi- ed in a truly locational ersective - e.g. the attitude for shoing centers and suermarkets to locate outside the urban center - but it is much more di±cult to observe maximal di erentiation in the characteristics sace. The existence of a greater or a lower di erentiation clearly deends on the interlay of a rice cometition e ect and a demand e ect: when the latter revails, the rms' strategies are found to exhibit a strong tendency towards agglomeration in the middle; by contrast, when the demand e ect is outweighed by the rice cometition e ect, moving away is the otimal behaviour. Recently, this debate has been extended to cover situations with 'low' demand (Hinlooen and van Marrewijk 999, Chirco, Lambertini and Zagonari, 000) and to multi-dimensional models (Calin and Nalebu 986, Neven and Thisse 990, Tabuchi 994). One common roerty of traditional locational models is the assumtion that consumers are uniformly distributed over the characteristics sace; with a few excetions, the situations in which the consumers' references are concentrated on a subsection of the available varieties have been neglected. In these cases one would exect that cometitors roduce fairly similar tyes of roducts, in order to better match the tastes of the relatively largest share of consumers (Beath and

3 Katsoulacos 99). The roblem of the otimal rices and locations has been exlicitly solved by Tabuchi and Thisse (995) with a triangular and symmetric distribution. They show that, given that distribution, any symmetric location around the middle cannot be an equilibrium. Indeed, two asymmetric equilibria arise, characterized by strong roduct di erentiation between the rms, with one of them locating outside the suort of the customer distribution.their results, however, heavily deend on the non di erentiability of the consumers density function, which generates a discontinuity of the reaction functions in corresondence of any symmetric location. In this aer, we aim at extending Tabuchi and Thisse analysis in two directions. We o er a simle arametrization of the degree of consumers' concentration around the middle - which include the uniform and the triangular distribution as limit cases. This allows us to solve the rice-location roblem as a function of the degree of consumers concentration. Within this setu, we are able to show that a symmetric equilibrium exists, rovided the density is di erentiable at the center of its suort. Moreover, we are able to give some theoretical suort to the idea that a higher concentration of consumers around the center induces rms to reduce the otimal roduct di erentiation. Finally, we nd that the asymmetric equilibria identi ed by Tabuchi and Thisse may arise for a lower degree of consumers concentration than that imlied by the triangular distribution and that these asymmetric equilibria may coexist with asymmetricone. The aer is organized as follows. In section we describe the basic model and discuss the simle arametrization of consumers' concentration adoted in the sequel. The exlicit solution of the rice-location roblem is resented in section 3. Some comments and concluding remarks are rovided in Section 4. The model Let us consider a market for a horizontally di erentiated roduct, where the oulation of consumers is normalized to. Consumers, indexed with x, are distributed over the interval [0; ] ; accordingtoadensityf (x; w), where the arameter w that can be viewed as a concentration index of the consumers' tastes. More recisely, the density f (x; w) is characterized as follows: f (x; ) = ; for x [0; ] f (x; 0) = jx j ; for x [0; ] 4 w for 0 <w< f (x; w) = x for x<, w f (x; w) = w for x ; +w, +w 4 f (x; w) = ( x) for x<+w w 3

4 Figure : The density function for di erent values of the concentration arameter AsshowninFigure, f (x; w) is symmetric around x ==; for w =it describes a uniform distribution while, as w decreases it concentrates towards the middle becoming traezoidal and collasing to a triangle for w = 0. Roughly seaking, our density is a traezoid, with longest base equal to, shortest base equal to w and altitudo equal to.foragivenw (0; ], we de ne 'central +w w interval' the interval x ; +w, while we call 'left external' and 'right external' interval resectively the intervals x 0; w µ +w and x. ;. In this framework we consider a duooly model in which both rms, rm and rm, roduce a di erentiated roduct at a constant and equal to zero marginal cost. The location x chosen by each rm reresents the good it decides to roduce: the ideal consumer's roduct may match with the roduct o ered, otherwise consumers choose to buy a "less than ideal" roduct aying a transortation cost that we consider quadratic in distance. Each consumer takes at most one unit of the roduct, so that total demand for the good o ered by the rm located in x is given by the number of customers it atronizes. In the sequel we shall assume full market coverage. Let us denote with a the distance of rm from the origin, while b is the distance of rm. In order to exclude the ossibility of leafrogging by either rms we assume a<b-wherea ( ; ) andb ( ; ) - and marginal 4

5 consumer lying between the two rms. As is well known, the rice-location roblem is a two-stage game in which at the rst stage the rms choose their location and at the second stage choose their rices. The game is simultaneous. The otimal rms' behaviour obviously di ers according to the value of w: The results in terms of otimal locations are well known in the literature when w = and when w = 0: in the unconstrained Hotelling game with a uniform distribution of consumers the rms maximize ro ts by locating at =4 e5=4 (Lambertini, 994); moreover, Tabuchi and Thisse (995) demonstrate that with a triangular distribution two asymmetric equilibria arise, 6=9; 5 6=8 and 5 6=8; + 6=9. The following analysis will focus on the rice-location equilibria for intermediate values of the arameter w, i.e. becomes traezoidal. when the density 3 Consumer concentration and equilibrium rices and locations We look for a subgame erfect equilibrium through backward induction, solving rst for the rices and then for the locations as a function of the exogenous arameter w and the otimal rices determined in the rst stage. Notice that if rm and set a rice resectively equal to and being located resectively in a and b, the above hyotheses on transortation costs, unit demand and full market coverage imly that the marginal consumer's location is z = µ b a + b a + a Clearly, given the shae of our density, the rms' reaction functions in both stages of the game will be di erent according to the fact that the rms know that their behaviour imlies that the marginal consumer lies in the 'central interval' w or in the two external intervals, i.e. z ; +w - central interval - or z 0; w µ +w and z. ; - external interval. We solve the model under both conjectures and verify under which conditions one or more equilibria exist in which conjectures are ful lled. Notice that, given the simmetry of the density, the ossible existence of a subgame erfect equilibrium such that z 0; w imlies the existence of a secular equilibrium, with the marginal µ +w consumer lying in a secular osition within the interval ;. This allows to restrict the analysis to one external area only. 3. The marginal consumer lies in the central interval Given the hyothesis of unit consumers' demand and given our normalization, the market demand for each good corresonds to its market share. Therefore, () 5

6 the demand for the two rms are resectively: q = F (z;w) q = F (z;w) w where F is the cumulative function of f. As long as z ; +w, q (z; w) = z ( + w) +w so that, by substituting (), we get: à b a q = + b + a +! (w ) +w The demand accruing to the rm will be: " à b a q = + b + a +!# (w ) +w () (3) Since there are no roduction costs, the ro t functions of the two rms are: " à b a ¼ = + b + a +!# (w ) (4) +w à b a ¼ = " + b + a +!# (w ) (5) +w 3.. The rice stage By di erentiating the rms' ro t functions and solving the rst order condition with resect to rices, we nd the following reaction functions: = + b a (a + b + w (b a)) = b + a (3 (a + b)+w (b a)) The Nash equilibrium in rices is therefore: = 3 = 3 b a + 6 (b a)+ w (b a) (6) a b (b a)+ w (b a) (7) 6

7 3.. The location stage Substituting the otimal rices in (4) and (5), ro ts are exressed as a function of locations and w: ¼ = b a + a) (b + a)++3w (b a)+ w (b +w ¼ = (b a)+ w (b a) b 5+3w (a + b) a 3 +w The rst and second order condition for ro t maximization are satis ed for a = 6 w + 3 b (8) b = 3 a w (9) The solution of the system (8) and (9) gives the otimal symmetric locations a = w e b = w. If rms locate in a and b, their otimal rices are = = w w and the indi erent consumer is located in :the conjecture that the indi erent consumer lies in the central area is ful lled. We can therefore establish the following roosition: Proosition For all values of w (0; ] there exist a subgame erfect symmetric Nash equilibrium in rices and locations. Notice that the otimal locations coincide with those identi ed in Lambertini (994), a = 4 e b = 5 4 ; when w =. The otimal rices are increasing in w: a higher degree of concentration around the midlle (lower w) induces rms to move inwards in order to match the tastes of a growing share of consumers: the more concentrated is the consumer distribution, the less the rms di erentiate their roducts.. This reduced di erentiation strenghten rice cometition. The overall equilibrium shows clearly a dominance of the demand e ect: the advantage of acquiring the consumers in the central area dominates the advantage of softening cometition through a large roduct di erentiation. 3. The marginal consumer lies in one of the external intervals Now we want to verify whether there exist subgame erfect equilibria, such that the marginal consumer falls in the left external interval 0; w.inthisinterval 4 the density function's sloe is. Hence, as z 0; w w, the demand for rm is: q (z;w) = z ( + w)( w) 7

8 Substituting () in the above exression we obtain the following demand functions in terms of the locations and rices: 0µ b a + b a + a q = B ( + w)( w) A q = 6 4 µ b a + b a + a ( + w)( w) The ro t functions are: 0 ¼ = ¼ = 6 4 µ b a + b a + a ( + w) w µ b a + b a + a ( + w) w C A The rice stage The rst and second order conditions for ro t maximization with resect to rm 's rice are satis ed by the following reaction function: = + b a (0) 3 As far as rm is concerned, the rst and second order conditions are satis ed by the reaction function: = 3 b + 3 a q [ +(a b )] +[6(b + a ) ba]( w ) () The rst order condition is satis ed also by = + b a. However, at this solution the second order condition for a maximum is not satis ed for w<. Again, we have two solutions satisfying the FOC. The other solution = 3 b + 3 a + 3 q [ +(a b )] +[6(b + a ) ba]( w ) 3 does not satis es the second order condition. 8

9 The solution of the system (0) and () gives the following Nash equilibrium in rices: = ½ (b a) = 3½b 3½a b + a where ½ is a root of the olynomial 8x (b + a) x + w. The existence of two solutions demonstrates that the reaction functions intersect twice. Since the two roots of the olynomial are x = q 8 (a + b)+ (a + b) +8( w 8 ) x = 8 (a + b) q (a + b) +8( w 8 ) we may establish that these intersections occur at the following two rice coules: = q µa 8 (a b) + b (a + b) +8( w ) () = 3 µ q a + b (a + b) +8( w 8 ) (a b)+a b (3) using solution x,or = q µa 8 (a b) + b + (a + b) +8( w ) (4) = 3 q µa 8 (a b) + b + (a + b) +8( w ) + a b (5) using solution x. It must be noticed, however, that only (4) and (5) entail ositive rices at equilibrium for both rms. Therefore this is the only economically meaningful economic solution to the rice game. 3.. The location stage The ro t functions calculated at the otimal rices are: µ q 3 ¼ = a + b + (a + b) +8( w ) (a b) 3 ( + w)( +w) µ q ¼ = 3 a + b + (a + b) +8( w 8 ) (a b)+a b 0 h 3 8 a+b+ (a+b) +8( w ) (a b)+a b i h + 8 ³a+b+ 4 (a b) b a 6 4 ( w ) i 3 (a+b) +8( w ) +b a5 +aa

10 By di erentiating rm 's ro ts with resect to its location, we get the following rst order condition: q µ q 3 3 a + b + (a + b) (a + b) +8( w )+3(a b) +8( w ) q =0 (w ) (a + b) +8( w ) which gives the otimal location: 3 a = 5 4 b 4 9b +6( w ) (6) If we now di erentiate rm 's ro ts with resect to its location, and substitute the reaction function (6), tedious calculations (see the Aendix) show that we can identify the following accetable solution: Using (7) into (6) we have b = 5 8 6( w ) (7) a = 9 6( w ) (8) Therefore, equations (7) and (8) give the otimal locations as a function of w, under the conjecture that the marginal consumer lies in the interval 0; w. We now have to verify whether there is a range of w such that this conjecture is actually ful lled. We rst notice that when w = 0 - i.e. when the density describing the consumers' references is a symmetric triangle - the equations (7) and (8) collase to a = 9 6andb = 5 8 6, that corresond exactly to Tabuchi and Thisse's solutions. In general, when evaluated at the otimal locations (7) and (8), the rice equations (4) and (5) become resectively: = 7 w (9) 8 = 7 w (0) 9 By substituting in (7)-(0) into (), we nd the marginal consumer's location as a function of w: z = w 6( w ) + 6( w ) 3 The above FOC has two solutions. The other is a = 5 4 b + 4 9b +6( w ), which does not satisfy the condition a<b. 0

11 This allows us to establish that the rms' conjectures generating the asymmetric equilibrium (7)-(0) are ful lled if w 6( w ) + 6( w ) < w i.e., if w< 5. By a similar reasoning, it can be roved that, under the same condition on w, a secular asymmetric equilibrium exists, with the marginal consumer lying in the right external interval, with rms located resectively at a = 5 6( w )=8 and b = 6( w )=9. We can therefore establish the following roosition: Proosition For 0 <w< 5 there exist three subgame erfect Nash equilibria in rices and locations, a symmetric equilibrium and two asymmetric ones. Notice that in the asymmetric equilibria one rm locates outside the market area, while the other locates in the external interval oosite to that in which lies the marginal consumer. As w increases in the admissible range 0; 5,both rms move inwards. Given w, the rm locating within the market area may charge higher rices and enjoy higher ro ts. It may be interesting to ask what haens when w = 5. In this case, the asymmetric equilibria de ned above make the marginal consumer fall in 5,or secularly in 3, i.e. in corresondence of the hedges of the density function. 5 This is a situation similar to that Tabuchi and Thisse describe with resect to a ossible symmetric equilibrium: since the density is not di erentiable, the reaction functions are indeed discontinuous. Let us assume that the solution (7)-(0) holds for w = 5. Then the following results would aly: a = 4 5, b = 3 = 8 75, = 56 75, z = 5 ¼ = 8 5, ¼ = 5 In order to ensure that it is indeed an equilibrium, we must exclude the rofitability of unilateral deviations from the candidate equilibrium location, in corresondence of the admissible rices for such a location. Let us de ne the alternative location for rm : a 0 = ² If rm locates in a 0, while rm locates in 3, the marginal consumer lies in the central interval, and the rice rules (6)-(7) aly, so that µ 4 = ² 3 = ² ² µ ²

12 When evaluated at these rices, and at a 0 = ², andb = 3, the ro ts of rm turn out to be à ² µ 45!à ² 35 3 ² ²! 4 5 ² + ² > 8 5 for arbitrarily small ositive values of ². Thisisenoughtorovethat,for w = 5, the solutions (7)-(0) are not subgame erfect equilibria and allows us to establish that for w ==5 there exists only a subgame erfect symmetric Nash equilibrium in rices and locations, de ned by equations (6)-(9). 4 Remarks and conclusions In this aer we have analysed the e ects of the consumers' concentration towards the middle of the sace of roduct characteristics, in a a model of horizontal di erentiation with quadratic transortation costs. The consumers' density is assumed to be symmetric and traezoidal; if the size of the market is normalized to, this allows to consider the lenght of the shortest base as a mean reserving sread of consumers' references. Clearly, the traditional uniform distribution and a symmetric triangular distribution can be nested into this setu as limit cases. We have roved that as far as the shortest base is ositive - i.e. the distribution is di erentiable at = - a symmetric subgame erfect Nash equilibrium exists in the two stage rice-location game. The result we achieve is rather intuitive: starting from the otimal solution obtained under the standard uniform distribution, as references become more concentrated around the middle, both rms move inwards and reduce the degree of roduct di erentiation. This clearly reinforces rice cometition and results in lower equilibrium rices. This result is consistent with a more general intuition that homogeneity of consumers might have imortant imlications in terms of reducing the rms'market ower (Benassi, Chirco, and Scrimitore, 00). Moreover, our discussion shows that the asymmetric equilibria identi ed by Tabuchi and Thisse may coexist with the above symmetric equilibrium. For a relevant range of values of our mean reserving sread arameter - when references become su±ciently concentrated - two asymmetric subgame erfect equilibria aear, with one rm roducing a relatively 'average' roduct, and the other rm choosing to locate outside the characteristics sace. Once one rm decides to roduce a roduct which meets the taste of the large share of consumers located around the middle, the other rm nds it otimal to avoid a destructive rice cometition by choosing a roduct with 'extreme' and 'out of market' characteristics. However, this eculiar location choice requires that a low rice is charged, in order to cature at least the consumers located at the nearest tail of the distribution. This solution is such that as w increases within its admissible range - the distribution becomes more disersed - both

13 rms locate inwards and decrease their rice. As the relative weight of the tails increases, the rm roducing outside the market area erceives an incentive to make its roduct more attractive for the growing share of consumers it may atronize - those located at its nearest tail. The rm roducing inside the market area, erceiving no cometition at the other tail, challenges its rival by locating further towards the middle. These movements result in a tougher rice cometition. Whilethesimlesetudiscussedinthisaerallowsforanexlicitgeneral solution which covers the situations reviously discussed in the literature, it is nevertheless clear that the relation between any concentration index of the consumers' references and the roerties of equilibria should be framed in a more general setting, indeendently of the ossibility of de ning analytical solutions. This is an imortant issue of the research agenda on roduct di erentiation. 3

14 Aendix By solving with resect to b the rst order condition for rm 's ro t maximization at the location stage, we obtain the following critical values ² b = a + 3 ( w ) ² b = a 3 ( w ) ² b = ½, where½ is a root of the olynomial: 4x 4 +6x 3 a + 30w +30 x + + a 3 +aw a x a +a w w 8w 4 (A) Let us consider these solutions. ² Consider rst the solution b = a + 3 ( w ). Given the reaction function of rm, we have to solve the following system in order to discuss the candidate otimal locations of rm and rm : a = 5 4 b 4 (9b +6( w )) b = a + 3 ( w ) Again we have two solutions: the coule of locations a = 6 7 ( w ) and b = 3 ( w ), and the coule a = 3 ( w ), and b = 7 6 ( w ). The rst coule imlies a value for b lower than a. Thissolution is therefore unaccetable. However, if eqts (4) and (5) were evaluated at the second coule, the marginal consumer would lie at 3 4 ( w ). Since the disequation 3 4 ( w ) > w is always satis ed, at this solution the rms' conjectures would not be ful lled. ³ ² It is easy to check that, if b = a + 3 ( w ),thevaluesofb that solve the system of the two reaction functions are both smaller than a. This contradicts the assumtion a<b. We now consider the olynomial (A). Its solutions obtained by substituting rm 's otimal rely are: x = (w ) x = 5 6( w ) 8 x = 8 5 6( w ) 4

15 ² We can immediately rule out the comlex solution x = (w ). ² If b = 5 8 6( w ), the otimal location of rm is a = ( w ). In this case a<b, but both otimal solutions are negative and this contrasts again with the conjectures about the location of the marginal consumer. ² The last solution is indeed the only accetable one. If b = 8 5 6( w ), then a = 9 6( w ): Using these otimal locations into (4) and (5) we may verify that the marginal consumers is in the left external interval for w< 5. Notice that this solution collases to that obtained by Tabuchi and Thisse by setting w =0. 5

16 References [] J. Beath and Y. Katsoulacos (99), The Economic Theory of Product Di erentiation, Cambridge University Press, Cambridge. [] C. Benassi, A. Chirco and M. Scrimitore (00), "Income Concentration and Market Demand", Oxford Economic Paers, 54, [3] A. Calin and B. Nalebu (986), "Multi-dimensional Product Di erentiation and Price Cometition", Oxford Economic Paers, 38, [4] A. Chirco, L. Lambertini and F. Zagonari (000), "How Demand A ects Otimal Prices and Product Di erentiation", Aunti, Note e Ricerche, University of Bologna - Rimini Center. [5] C. d'asremont, J.J. Gabszewicz and J.-F. Thisse (979), On Hotelling's `Stability in cometition', Econometrica, 47, [6] N. Economides (986), "Minimal and Maximal Product Di erentiation in Hotelling's Duooly", Economics Letters,, [7] J. Hinlooen and C. van Marrewijk (999), "On the limits and Possibilities of the Princile of Minimum Di erentiation", International Journal of Industrial Organization, 7, [8] H. Hotelling (99), "Stability in Cometition", Economic Journal, 39, [9] L. Lambertini (994), "Equilibrium Locations in the Unconstrained Hotelling Game", Economic Notes, 3, [0] D. Neven and J-F.Thisse (990), "Quality and Variety Cometition" in J.J. Gabszewicz, J.-F Richard and Wolsey L., Economic Decision Making: Games, Econometrics and Otimization, Contributions in Honour of Jacques Drµeze, Amsterdam, North-Holland, [] T. Tabuchi (994), "Two-stage Two-dimensional Satial Cometition between two Firms", Regional Science and Urban Economics, 4, [] T. Tabuchi and J.-F. Thisse (995), Asymmetric equilibria in satial cometition", International Journal of Industrial Organization, 3,