A SINTHESIS OF LOCATION MODELS

Transcription

1 A SINTHESIS OF LOCATION MODELS Hamid Hamoudi* and Marta Risueño** 7 / Februar / 006 *Hamid Hamoudi Dep. Economía Universidad Europea de Madrid C/ Tajo s/n Villaviciosa de Odón 8670 Madrid tf: hamid.hamoudi@uem.es Corresponding autor: ** Marta Risueño (autor de contacto) Dep. Econ. Aplic. II Fund. Anal. Económico Universidad Re Juan Carlos Paseo de los Artilleros s/n 80 Madrid tf: marta.risueno@urjc.es

2 Abstract: This article considers a model of spatial competition where firms and consumers are located in a semicircular space rather than in the whole circle (Salop s model) or the linear cit (Hotelling s model), under the assumptions of both, convex and concave, transportation costs. The paper tries to generalize the results of the two previous models. We find that the existence of a price equilibrium is warranted ever firms location when the length of the semicircular space is greater than /. Resumen: El artículo considera un modelo de competencia espacial donde las empresas los consumidores se localizan en un espacio semi-circular en lugar del círculo completo (modelo de Salop) o la ciudad lineal (modelo de Hotelling), bajo el supuesto de costes de transporte cuadráticos. El trabajo intenta generalizar los resultados de los dos modelos anteriores. Encontramos que la existencia de un equilibrio en precios está garantizada para cualquier localización de las empresas cuando la longitud del espacio semi-circular es maor que /. Ke words: urban design, regulator, spatial competition, transportation costs, segmented market JEL Classification: C7, D.

3 .-Introduction In most urban configurations, we observe the existence of portions of land devoted to non-residential purposes. We can find environmentall protected areas, parks, recreational facilities, etc. When considering urban design, regulators have to decide whether the should leave some part of urban land this tpe of recreational activities, and if the do so, what is the optimal size of these non-residential areas. These zones represent a discontinuit in the market since neither consumers nor firms can be located within them; however, it is possible to find consumers and firms adjacent to both ends of these restricted areas. In order to stud the implications of this tpe of urban configurations, we consider a circular spatial model, where there is a segment where people live and firms locate, that we shall refer to as the market, and another segment that can onl be used non-residential purposes. (See figure ) In this context, a three-stage game can be considered in which in the first stage the regulator chooses the size of the market, in the second stage firms choose locations and in the third stage firms compete in prices. We will consider that the regulator chooses the length of the market in a nonstrategic manner; theree, the model we present, once market size is given, can be reduced to a two-stage game in which firms first choose location, and then compete in prices. This model can be thought of as a snthesis of the circular and linear spatial models. If the size of the market is half of the circumference, l/, or less, the model is equivalent to Hotelling s (99) linear cit, while if the size of the market is the whole circle, l, we are in Salop s (979) configuration. We analze the existence of a price equilibrium when the market segment, h, is restricted to be less than l.

4 Together with market space configuration, industr structure, demand distribution and the tpe of transportation costs function assumed are fundamental in defining a spatial model, and these choices will determine the existence or nonexistence of equilibrium. The so-called existence problem of equilibrium has produced a large number of academic works in which different assumptions about number of firms, consumer s densities, transportation costs functions, etc., have been made. For a revision of the seminal papers of this literature see Gabszewicz and Thisse (986) and the references cited within it. As mentioned above, an important line of research has dealt with the tpe of transportation costs functions that can ensure equilibrium existence in both, the linear and circular models. In the linear model, D Aspremont et al. (979) used a quadratic transportation costs function and showed the existence of a maximum differentiation equilibrium. Economides (986) studied the function C(d) = d α, α and determined the values of α which price equilibrium exists. Gabszewicz and Thisse (986) and Anderson (988) have studied the transportation costs functions C(d) =ad+bd, a>0, b>0, and found that a perfect equilibrium does not exist ever possible firms location; while Hamoudi & Moral (005) showed that, with concave transportation costs, equilibrium cannot be attained. In the circular model, Anderson (986) studied equilibrium existence convex transportation costs functions. De Frutos et al. (999) have analzed convex and concave transportation costs functions and the have proven the existence of equilibrium an location of firms the functions: C(d) = k(d-d ) and C (d) = kd. We will make the standard assumptions of two firms selling a homogeneous product, consumers evenl distributed a long the market segment, and we will stud the existence problem in our model using the two linear quadratic functions that ensure the existence of a perfect equilibrium in pure strategies in the circular model.

5 We find that there exists a price equilibrium an location of firms provided that the length of the market segment is greater than approximatel. Furthermore, this equilibrium is unique and implies firms locating opposite to each other. However, unlike the results obtained in the circular model, the equilibrium is not smmetric since one of the firms endures a competitive disadvantage due to the market discontinuit. If h is smaller than, we find that, certain values of h and firms locations, there is a strip where no price equilibrium ma exist. Nevertheless, there are man combinations of market sizes and firm locations which equilibrium can be obtained. When we compare the intensit of competition, given the size of the market, we find that competition is more intense low values of h, and the equilibrium region is smaller. The paper is organized as follows, in section we present the model under the hpothesis of both concave and convex transportation costs, in section we stud the existence of equilibrium, section contains the conclusions, and finall major proofs and graphs can be found in the appendix..- The Model We consider a circular cit of length l where the regulator chooses the size of the market, h, so that firms and consumers can onl be located on a certain segment of the circle h l. There are two firms selling a homogeneous product, with zero production costs, located at x and, with0 x h. Consumers are evenl distributed along h, and each consumer bus a single unit of this product per unit of time, irrespective of its price. Since the product is homogeneous, consumers will bu from the firm who offers the least delivered price, that is, the mill price plus transportation costs. Let p, p, denote the mill prices charged b firms located at x and, respectivel. The distance between consumer z and firm i is given b d i = z i, i = x,. We will consider two possible tpes of transportation costs: concave and convex. In particular, we will assume the onl two cost functions from the 5

6 linear quadratic famil: C(d i ) = k(d i -d i ) and C (d i ) = kd i, that have been shown to ensure existence of a perfect equilibrium in pure strategies in the circular model. (See De frutos et all., 999). These two cost functions are strategicall equivalent, and theree, the equilibrium results obtained in one case can be immediatel translated to the other b an appropriate change of variables, (See De Frutos et al, 00) Although firms and consumers can onl be located within h, consumers can travel along the whole circle and the will alwas take the direction that implies the shorter distance to the chosen firm. The model described above, given that the regulator behaves in a non-strategic manner, does give rise to a two-stage game in which firms first decide simultaneousl their location and then simultaneousl choose prices. It turns out that the solution to this game depends criticall on the length of the market segment, h. In order to determine the market boundaries and derive the demands faced b each firm, we will have to find the indifferent consumers. A consumer is indifferent to buing from one firm or the other if and onl if: p + C d ) = p + C( ). ( d That is, a consumer is indifferent if she faces the same full price from the two firms. We find that, depending on the length of h considered, one or two indifferent consumers ma exist simultaneousl, and different price intervals, I i j, which demand is defined, will be obtained. We will onl present the demand firm, since demand firm, is alwas given b: D = h D..-Concave Transportation Costs The transportation costs function faced b consumers is given b: C(d i ) = k(d i -d i ). We assume, without loss of generalit that l = k =, x = 0 and 0 /. 6

7 When the market length considered is the whole circle (See figure ), four possible indifferent consumers are found, each one belonging to a different segment of the circumference given b: m [0, ], m [, ½], m ' [½, +½], m '' [+½, ], however m =m =m. Theree, onl two distinct indifferent consumers ma exist: an indifferent consumer m [ 0, ] and, an indifferent consumer, that we will denote, m, m [, ], given b the following expressions: p p m = + ( ) () m + = + () Both belonging to the prices interval: ) ( ) ( If we restrict the length of the market to h, h, there are two distinct cases to consider: Either m h, and there exist two indifferent consumers, m [ 0, ], belonging to m, h, belonging to the full prices interval, as above, and an indifferent consumer [ ] the price interval ) (h ) (. On the other hand, ifh m, there is onl one indifferent consumer, m, the prices interval h ) ( ) (. Ever consumer m m and ever consumer m m will bu from firm. Demand firm is given b: h m + ( h m) D = m 0 Where: I I = [, ( )], I = [ ( ), (h ) ], I [ (h ), ( )] [ ( + ] = ), =, 7

8 ..- Convex Transportation Costs As has been shown in De Frutos et all., (00), in the circular model, an convex transport cost function there exists a concave one such that the location-thenprice games induced b these functions are strategicall equivalent. In particular, the show that in a circular market, an consumer m facing a concave transportation cost function, there is another consumer n facing a convex transportation cost function whose utilit is a monotone transmation of the utilit of m, and theree, n will bu to the same seller as m. We use this result, and an appropriate transmation of variables (see figure ), to stud the convex transportation costs case. Let C(d i ) = kd i and that l = k =, x < and = /. In this case, and the whole circular market we obtained two tpes of indifferent consumers: n = n ' = n '' [ 0 x + ] and n [ x, ], +, given b: n p p = + ( x ( x) + ) () p p n ( = + + x) + ( + x) () Both defined the prices interval: x )( + x) ( x)( + x ). ( Ever consumer n n and ever consumer n n will bu from firm. When we restrict the length of the market to h, Eithern h, and there are two indifferent consumers, n [ 0, x + ] n [ x +, h] [ )( + x) ( + x)(h ( + x) ) ], defined the full prices interval, as above, and, belonging to the prices interval ( x, or h n and there is onl one indifferent consumer, n, defined in [ + x )(h ( + x) ) ( x)( + x )] ( Demand firm is given b: 8

9 h n D = n 0 Where: + ( h n ) I [ ( )( = x )], I = [ x)( + x),( + x)(h ( + x) ) ] I +, x ( [ )( ( ) ),( )( = + x h + x x )], I = [ x)( + ), + ] I + ( x ( x Note that the results obtained are identical to the concave case = + x). Since the demand functions are identical to those of the concave case, we will onl have to stud equilibrium one of the cost functions. (.- Equilibrium Given the size of the market, h, and using the usual approach a two-stage non-cooperative game in which firms select a position at the first stage and subsequentl set their prices, we stud the subgame perfect equilibrium. We recall that N a perfect price-location equilibrium is defined as a pair ( p,0), ( p N, N ) N N N N N N (i) p = p ( 0,, h) and p = p ( 0,, h), such that: N N N N N N N N N (ii) B 0,, h, p (0,, h), p (0,, h)) B (0,, h, p (0,, h), p (0,, )) ( h [ ] 0,. Where p N, p N ) is a Nash equilibrium in the price subgame when the locations choice ( is fixed. We will now compute the equilibrium of the price subgame. A Nash equilibrium in the price subgame will be a pair of prices p N, p N ), such that : ( N N N N p = ArgMaxB ( p p ), p = ArgMaxB ( p p ), P, P Since production costs are assumed to be zero, the profit functions firm i are given b B i = p i D i, i =,. Theree, the expressions the profit functions firms and are given b: 9

10 hp p B = p 0 ( ) ( ) (h ) ( + ) ( + ) I (5) B 0 p = p hp ( ) ( ) ( + ) (h ) ( + ) (6) If we look at the profit function of firm, it is obvious that no price equilibrium can exist in regioni, since firm could alwas benefit from lowering its price and capturing some demand from firm. The same reasoning applies to firm in region theree, in order to explore possible equilibria we can restrict ourselves to regions I and I. However, the profit functions are not concave in prices and the could exhibit different configurations. In particular, the ma exhibit one or two local maxima. Depending on the values of two cases ma arise, one in which the global optimum belongs to regioni, and another in which the global optimum belongs to I. I,..- Equilibrium in Region I For this prices interval two indifferent consumers exist. Computing the first order conditions the profit functions in I, we obtain: p N = ( )(h ) and p N = ( )(h + ) (7) and profits are given b: B ( p N, p N ) = ( )(h ) and 8 0

11 B ( p N, p N ) = ( )(h + ) (8) 8 N N Proposition : The pair ( p (0,, h), p (0,, )) constitutes a Nash equilibrium if and h onl if ( h, ) R, where: R + = ( h, ) h h, where, h( ) = Proof: See Appendix Proposition : There exists a subgame perfect price-location equilibrium if and onl if h 0,7and then, this equilibrium is unique and is given b: N =, p N (0, N, h) = h, p N (0, N, h) = h + Proof: From proposition, there exists a price equilibrium if h h ( ), however, simple h 9 calculations show that h( ) is increasing ( = > 0) and reaches a ( ) + maximum = (see figure ) and h ( ) = 0, 7, theree h 0, 7 there exist a price equilibrium an firms location. If we look at the optimal location of firm (given that the location of firm is fixed at 0) we find that h B N = 0 = = p N (0,, ) and p N (0,, ) we obtain p h N. Substituting b in the expressions of (0, N, ) = h h, (0,, ) = h + N N p h.?..- Equilibrium in Region I For this prices interval onl one indifferent consumer exists. Computing the first order conditions the profit functions in I, we obtain: p N = ( )(h + ) and p N = ( )(h ) (9)

12 and profits are given b: B ( p B ( p N N, p, p N N ) = ) = ( 8 ( 8 )(h + ) and h )( ) (0) N N Proposition : The pair ( p (0,, h), p (0,, )) constitutes a Nash equilibrium if and h (5 + ) onl if( h, ) R, Where: R = ( h, ) h h, where, h = () ( + ) Proof: See Appendix. Corollar: if h < 0, 7, there is no price equilibrium in region R defined b: { h, ) / h ( ) < h h ( )} R = < ( Proof: R is the intersection of the two complements of regions R and R. Figure 6 combines the two equilibrium regions depicted in the previous two figures. As can be seen from the graph, there is onl a narrow strip where no equilibrium exists.? In figure, the equilibrium area region I is depicted, all points in the shaded area are possible equilibria. Note that h 0, 7there exists a price equilibrium ever possible location of firm. That is, provided that the regulator does not devote an area greater than a quarter of the cit size to non-residential use, this tpe of urban configuration does not pose an equilibrium problems. When we look at the optimal location of firm in regioni, when there are two indifferent consumers, we find that firm benefits from moving awa from firm and locating at the opposite boundar of the market segment where m exists. The result is equivalent to what we obtain when we consider the complete circular model: there are two indifferent consumers and firms locate opposite to each other and equidistant to the two indifferent consumers. However, in our model the competitive situation of the two

13 firms is not the same. In the market segment [0, ½], where the indifferent consumer m exists, the situation of the two firms is smmetric and that is wh firm will choose the opposite extreme to firm. However, when we consider the left half of the circumference, firm is located at the edge of the non-residential area, and theree this side of its potential market is restricted, as a result, firm ma charge a larger price and obtain larger profits than firm. Figure 5 shows the equilibrium area regioni. All points belonging to the shaded area (above the line h = ), are equilibrium candidates. When we look at the optimal location of firm in this region, we find B that = (h ) 0 8 < and theree, firm will tend to move towards firm. In regioni there is onl one indifferent consumer and the market resembles the linear cit case, given that firm is fixed at 0, as firm moves closer to firm it squeezes the demand of firm. D N = If we look at the prices differences of the two equilibrium regions we obtain: p N p N ( )( h) N N N ( )( h ) = < 0, D = p p = < 0, We can see that, in both cases, the price of firm is smaller than that of firm, this can be explained, as mentioned above, in terms of market configuration since firm has more potential customers than firm. Also, when we compare the prices differences in both regions, D N and D N, we N N ( )( h( ) + ) find that: D D = > 0 all ( h, ), so N N D > D. This implies that the intensit of competition is stronger in the second case than in the first one. This is not surprising since market size is smaller in the second case than in the first and this must induce stronger competition. This can explain wh the equilibrium region of the second case is smaller than the first one.

14 .- Conclusions In most urban configurations, the existence of portions of land devoted to non- residential purposes is observed. In considering urban design, regulators have to decide what the optimal size of these non-residential areas is. In this article we propose a circular model of spatial competition in which the market is restricted in order to allow a non-residential area. Aside from this discontinuit in the market, we make standard assumptions and use the two linear quadratic functions that have been proven to ensure the existence of price equilibrium in the circular model. We find that, provided the regulator chooses the size of the market segment to be greater than approximatel ), ( h 0,7 there is a subgame perfect price-location equilibrium and this equilibrium is unique. However, this equilibrium will not be smmetric, since the firm located adjacent to the market discontinuit will endure a competitive disadvantage. On the other hand, if market size is chosen to be less than, certain values of h and firms locations, there is a narrow strip where no price equilibrium ma exist. As usual, equilibrium failure is due to the non-convexities exhibited b profit functions.

15 5.-Appendix Proof of Proposition : N N In order ( p, p ) to be a price equilibrium, the following conditions must be met: (i) p N N p I N N N (ii) B ( p, p ) B( p, p ) p N N N (iii) B ( p, p ) B( p, p) p Condition (i) Implies that: a) ( ) Which is alwas true, and N N N N b) (h ) Which holds if and onl if: h h, where ( + 5) h = ( + ) N N So ( p, p ) could be a Nash price equilibrium if and h belong to the set: R {( h, h h } = ) In order to verif condition (ii) we have to check, p N given, what is the maximum ** N p B ( p, p ) in ** N a) p = p + (h ) I which is reached at: N N N, in this case we have B ( p p ) > B ( p p ),, ** b) p = ( )(h + ), Where ** N firm is given b: ( p, p ) = ( ) ( + h) 8 N N N that B ( p, p ) B ( p p ), p > p N + (h ), and the profit function ** B [ ], some simple calculation then show + all ( h, ) such that h h, where h = In order to verif condition (iii) we have to carr out the same analsis firm : N Given, N p the profit function of firm, B ( p ) p [ o,+ ), that is reached at p =, p. has onl one maximum N N N N p, thus we have B ( p, p ) B ( p, p ) p. N N Theree, combining conditions i, ii, iii, we obtain that ( p, p ) is a Nash price equilibrium h h and h h. 5

16 N N However, we also verif that h > h >, (See figure ), theree ( p, p ) price equilibrium in the regionr defined as: R {( h, h h } = (See figure ) ) is a Nash Proof of Proposition : N N In order ( p, p ) to be a price equilibrium, the following conditions must be met: (i ) p N N p N N N (ii ) B ( p, p ) B( p, p ) p N N N (iii ) B ( p, p ) B( p, p) p Condition (i ) implies that N N (5 + ) a) (h ) p p Which holds if and onl if, h h where h = ( + ) N N b) ( ) Which is alwas true. N N So ( p, p ) could be a Nash price equilibrium if and h belong to the set: R {( h, h h } = ) In order to verif condition (ii ), N N Given p, and considering R, we verif that B( p, p ) admits a global maximum N in p = p. condition (iii ), N N Similarl, given p, and considering R, we verif that B( p, p) admits a global N maximum in p = p. N N Finall, ( p, p ) is a Nash price equilibrium in regionr defined as: R {( h, h h } = (See figure 5) ) 6

17 Figure Figure Figure Figure 7

18 Figure 5 Figure 6 8

19 6.- References Anderson, S. P., 986. Equilibrium existence in the circle model of product differentiation. London Papers in Regional Sciences Series 6, 9-9. Anderson, S. P., 988. Equilibrium existence in the linear model of spatial competition. Economica 55, D Aspremont C., Gabszewicz J. J., Thisse J. F., 979. On Hotelling s stabilit in competition. Econometrica 7, De Frutos M. A., Hamoudi H., Xarque X., 999. Equilibrium existence in the circle model with linear quadratic transport costs. Regional Sciences and Urban Economics 9(5), De Frutos M. A., Hamoudi H., Xarque X., 00. Spatial competition with concave transport costs. Regional Sciences and Urban Economics, Economides, N., 986. Minimal and maximal product differentiation in Hotelling s duopol. Economic Letters, Gabszewicz J. J., Thisse J. F., 986. Location Theor. Harwood Academic Publishers, Switzerland. Gabszewicz J. J., Thisse J. F., 986. On the nature of competition with differentiated products. The Economic Journal 96, Hamoudi H., Moral M. J., 005. Equilibrium existence in the linear model: concave versus convex transportation costs. Papers in Regional Sciences 8, 0-9. Hotelling, H., 99 Stabilit in Competition. The Economic Journal 9,

20 Salop S. C., 979. Monopolistic competition with outside goods. Bell Journal of Economics 0,