Hotelling Meets Weber Fu-Chuan Lai y and Takatoshi Tabuchi z 1 September 011 Abstract In this paper, spatial competition between two sellers in a market (Hotelling, 199) and total transportation costs minimization (Weber, 1909) are combined, and equilibrium and optimum locations of rms are analyzed along with the consequent policy implications. We show that when the output prices are xed and equal, both rms agglomerate at the market center, irrespective of the distribution of inputs. Further, we also show that when output price is endogenous, the middle point of rm locations in Hotelling s model is identical to the Weber point. Finally, we show that the locations of Hotelling s rms are far from the socially optimal location. Keywords: Hotelling, Weber, location competition, social welfare. JEL Classi cation: D43, L13 We thank Weng-Jung Liang, Yves Zenou, anonymous referees, and participants of the seminar at National Taiwan University for valuable comments and suggestions. All remaining errors are ours. The nancial supports by Japan Society for the Promotion of Science (Scienti c Research B 330075) and the National Science Council in Taiwan (98-415-H-001-094) are deeply appreciated. y Research Center for Humanities and Social Sciences, Academia Sinica, 18 Academia Rd., Sec., Nankang, Taipei 1159, Taiwan, and Department of Public Finance, National Chengchi University, Taipei, Taiwan. z Faculty of Economics, University of Tokyo, Hongo 7-3-1, Bunkyo-ku, Tokyo 113-0033, Japan.
1 Introduction In their respective economic theories, Weber (1909) considered the location of a monopolist s manufacturing rm, whereas Hotelling (199) considered the location of duopolists retail rms. It is interesting to imagine what would have happened if they met early in the early 0th century. Weber examined the least shipping cost location of a manufacturing rm with respect to the locations of input materials and consumer demand. In relation to this, Weber established the Weber triangle or the locational triangle. In this triangle, there is one product and two sources of raw material and the main interest of a monopolist is to minimize the transportation costs of two inputs to produce one output. On the other hand, Hotelling dealt with inter rm competition between duopolists in location and price of an identical good. Each duopolist attempts to maximize their pro t by selling an identical good to consumers who are equally distributed over a line segment. Hotelling ignored the input materials and focused only on the output market, while Weber took both inputs and output into account. Weber (1909) is more realistic in terms of the fact that manufacturing rms use inputs in producing a nal product. However, Hotelling (199) is more realistic in terms of the fact that there is competition between rms. The purpose of this paper is to combine the two classical location theories. Speci - cally, we consider a situation in which two rms use input materials that involve transportation costs a la Weber (1909), and compete in location and price a la Hotelling (199). This is a generalization of Weber s transportation cost minimization and Hotelling s pro t maximization. In this paper, it is shown that Weber s rm location coincides with the midpoint of Hotelling s rm locations. The coincidence implies the similarity between these two classical location theories, the establishment of which is the main objective of this study. To the best of our knowledge, there is no existing literature that has attempted such a combination except Hansen et al. (1986) and Birkin and Wilson (1986a, b). Hansen et al. (1986) indicated that the Hotelling s duopolists and the Weber s monopolist will always choose to be located at the place of median demand when consumers are distributed at vertices of a symmetric network. However, their model does not involve 1
input transportation costs and price competition. On the other hand, in this paper, rms consider the transportation of not only the output but also inputs and compete in terms of not only location but also price. Birkin and Wilson (1986a, b) made an ambitious attempt to unify location models including Weber (1909) and Hotelling (199). However, they did not consider strategic behavior of rms and focused more on geography rather than economics. The remainder of this paper is organized as follows. Section considers Weber s location problem for an arbitrary number of input materials (Hansen, et al. 1987). Section 3 studies a location game and a location-then-price game in Hotelling s spatial competition (Gabszewicz and Thisse, 1986; Anderson, et al. 199). Section 4 discusses the social welfare by analyzing the rst-best and second-best optima. The nal section presents the concluding remarks. Weber s monopolist location A Weberian monopolist establishes a rm (= plant) at location x R, uses n raw materials (= inputs) to produce nal products, and distributes them to all consumers. Suppose that inputs are distributed discretely and consumers are distributed continuously on a line segment. Speci cally, input i(= 1; ; : : : ; n) is located at m i R and consumers are uniformly distributed over a unit line segment [0; 1], as illustrated in Figure 1. Each consumer buys only one unit of the nal product from the monopolist. The objective of the monopolist is to minimize the sum of the input and output transportation costs by choosing his plant location. It is assumed that the transportation costs of inputs and outputs are proportional to the square of the distance, 1 and the unit transportation rates of inputs and outputs are given by t and 1, respectively. Further, assume that producing one unit of the output 1 We consider quadratic transportation costs rather than linear ones throughout the paper. This is because the existence of price equilibrium is guaranteed only when the output transportation costs are quadratic (d Aspremont, Gabszewicz, and Thisse, 1979). Otherwise, we cannot obtain SPNE for any pair of rm locations in Hotelling s spatial competition in Section 3.. In reality, most transportation costs are often concave in terms of distance. However, when the time costs of transportation are included, the total transportation costs can become convex in distance.
requires (r 1 ; r ; : : : ; r n ) units of inputs. Without loss of generality, all input prices are zero; however, distance-related transportation costs are incurred. Then, the total cost is expressed as min x TC(x) = nx t i (m i x) + i=1 Z 1 0 (x z) dz = S Sx + T x + x x + 1 3 ; where x is the monopolist s location, t i tr i, T t P n i=1 r i, S t P n i=1 r im i, and S t P n i=1 r im i. Solving the rst-order condition yields the following optimal location: x W = S + 1 (T + 1) ; which is the so-called Weber point. The second-order conditions are obviously satis ed. When the input transportation rate t is rather low, the Weber point approaches the median point of the consumer distribution, i.e., the market center: lim x W = 1 t!0 : When the input transportation rate reaches in nity, the Weber point is equal to the center of gravity of the input distribution: lim t!1 xw = S P T = Pi t im i i t : i This is nothing less than the principle of median location (Alonso, 1975) because the median location of the inputs and output are always somewhere between the market center and the center of gravity. (1) 3 Hotelling s spatial competition 3.1 Location competition with inputs In the original Hotelling s (199) spatial competition, two retail rms sell products to consumers without using any input materials. Consumers are assumed to be uniformly 3
distributed over a line segment [0; 1], visit either one of the rms that provides the lowest full price of a good, and purchase exactly one unit of the product. The full price is the sum of the mill price of a product (= output) and the transportation cost of the product to the rm s location. As done in the previous section, it is assumed that the transportation cost of the output is equal to the square of the distance. The utility of a consumer who resides at x and purchases one unit of the product from rm k (= 1; ) is given by u k = y p k (x k x) ; () where y is the consumer s income, p k is the mill price of a product sold by rm k, and x k is the location of rm k. Further, it is assumed that rms cannot be situated at the same location: x 1 6= x. Let ^x be the location of marginal consumers who are indi erent in terms of purchasing from either rm. Because the full prices are equalized at ^x, p 1 + (^x x 1 ) = p + (^x x ) (3) should hold. In this section, there is no price competition; thus, the mill prices are xed and equal to p. Solving (3) yields the location of the marginal consumers, which is the market boundary between the two rms: ^x = x 1 + x : In this sequence, we introduce the transportation costs of input materials into the Hotelling s spatial competition model in order to combine Weber (1909) and Hotelling (199). As in the previous section, producing one unit of the output requires (r 1 ; r ; : : : ; r n ) units of inputs. Although n = in Weber s triangle, n can be an arbitrary number. When n = 0, it degenerates to the original Hotelling s spatial competition. In order to ship one unit of input i, rms must incur the transportation cost of t i x, where t i > 0 and x is the distance. Then, the pro t of each rm is given by 1 = p = p P i t i (m i x 1 ) ^x; P i t i (m i x ) (1 ^x): (4) 4
Since location is the only strategic variable, each rm selects its location in order to maximize its pro t. We assume that p > X i 1 t i m i (5) so that the pro t of each rm is positive when both rms locate at the market center. In the sequel, we rst show that the rms locate back-to-back, and then show that the only possible equilibrium location is the market center. First, assuming that x 1 < x, the following three cases may arise. (i) If x 1 S=T x, then it is obvious that rm 1 moves to the right insofar as x 1 S=T and rm moves to the left insofar as S=T x, and thus they locate back-to-back at the center of gravity. (ii) If x 1 < x S=T, then rm 1 moves to the right given the xed location of x. If rm moves to the right, rm 1 follows and also moves to the right until they locate back-to-back somewhere at x S=T insofar as their pro ts are positive. (iii) If S=T x 1 < x, we can apply the same argument as case (ii). Hence, both rms locate back-to-back in any case. Next, if the back-to-back locations are not at the market center, then their revenues are unequal but the input transportation costs are the same, so the rm closer to the market center will do better than the other rm. That is, any location di erent from the market center cannot be an equilibrium since one or other rm will want to relocate to the closer side of the market center. Thus, we have shown the following proposition. Proposition 1. Both rms locate at the market center in the Hotelling s location competition with a xed equal price and an arbitrary number of inputs when the output price is su ciently high so that (5) holds. It is worth noting that the outcome of the Hotelling s location competition is invariant to the number and location of input materials. That is, while the output transportation cost is an important determinant of rm location, the input transportation costs are not in the Hotelling s location competition. However, this proposition is true only when the assumption (5) holds. What if the assumption (5) is violated? For simplicity, assume that rms exit from the market when they cannot earn nonnegative pro ts. We show below that two rms do not coexist in the market and Proposition 1 is no longer true. 5
The bracketed term in each equation of (4) is shown to be decreasing as x i (i = 1; ) changes from 1= to S=T. Two cases may arise. (i) If the output price is intermediate ( P i t i (m i S=T ) p P i t i (m i 1=) ), then one rm exits from the market. This is because the incumbent rm earns zero pro t at location x, which is a solution of p = P i t i (m i x) and two rms cannot be situated at the same location. As p decreases from P i t i (m i 1=) to P i t i (m i S=T ), the location x is moves from 1= to S=T, which di ers from the central location shown in Proposition 1. Note that in the degenerate case of S=T = 1=, one rm locates at the market center with zero pro t. (ii) If the output price is low ( P i t i (m i S=T ) > p), then no rm enters the market. 3. Location-then-price competition with inputs In this section, we introduce price competition to the model presented in the previous section. In other words, the two rms simultaneously select their locations x 1 and x in the rst stage, and then simultaneously choose their mill prices p 1 and p in the second stage. Solving (3) yields the market boundary between the two rms: ^x = p 1 p + x 1 x : (6) (x 1 x ) In order to obtain the SPNE, we solve the second-stage price equilibrium rst. Each rm maximizes its pro t : 1 = [p 1 Pi t i(m i x 1 ) ] ^x; = [p Pi t i(m i x ) ] (1 ^x) (7) with respect to its price. Solving the rst-order conditions yields the following unique equilibrium prices: p H 1 = 1 3 [(x x 1 )( + x 1 + x ) + P i t i [(m i x 1 ) + (m i x ) ]] ; p H = 1 3 [(x x 1 )(4 x 1 x ) + P i t i [(m i x 1 ) + (m i x ) ]] : (8) The second-order conditions are also satis ed. Including the prices (8) into the pro ts (7) and manipulating, we obtain pro ts as functions of locations x 1 and x : 1 = (x x 1 )^x ; = (x x 1 ) (1 ^x) : (9) 6
Next, we solve the rst-stage location equilibrium. Di erentiating (9) with respect to x 1 and x, we get the rst-order conditions. Solving them yields the unique SPNE equilibrium locations pair: S 1=4 x H 1 ; x H = T + 1 ; S + 5=4 : (10) T + 1 The second-order conditions are also satis ed. The SPNE locations (10) in Hotelling s location-then-price competition di er from x HL 1 ; x HL = (1=; 1=) in Hotelling s location competition presented in the previous section. Further, we de ne x H 1 + x H = S + 1= T + 1 as the Hotelling midpoint. Then, we have x H 1 + x H 1 = T S 1 : (1) T + 1 T This sign is positive if S=T > 1=. That is, if the center of gravity of input materials is to the right of the market center, then the Hotelling midpoint is also located to the right of the market center. This inequality is true when a majority of the input locations m i are to the right of the market center. Plugging (10) into (8), we get p H 1 p H 3T S = (T + 1) T 1 : Since the sign of this equation is identical to that of (1), it may be stated that if the Hotelling midpoint is also to the right of the market center 1=, then rm 1 locates closer to the market center than rm, and rm 1 charges a higher price than rm. as Substituting (8) and (10) into (6), the location of marginal consumers is computed ^x H = 1 : Hotelling concludes that both rms tend to agglomerate at the market center in location-then-price competition with a linear transportation cost. However, d Aspremont et al. (1979) indicated a aw in Hotelling s proof and suggested a quadratic transportation cost in order to ensure the existence of SPNE. 7 (11)
That is, the equilibrium market share is always the same for any asymmetric locations of input materials. The higher price of the good sold by the rm located closer to the market center causes the demand to be reduced relative to the demand for the good sold by the other rm. However, since this e ect is mitigated by the locational advantage of the closer rm, the market shares of the both rms come out even. Furthermore, plugging (8) and (10) into (7), we have the same pro t for both rms: H 1 = H = 3 4(T + 1) : In light of the above ndings, it may be stated that even in the case of asymmetric con gurations of input materials, we have the following result. Proposition. Given an asymmetric distribution of an arbitrary number of inputs, one rm locates closer to the market center and charges a higher price. Nevertheless, the market share and the pro ts of both rms are identical in SPNE. Although the rm that is located closer to the market center earns the higher revenue due to the higher price with the equal market share, it incurs higher transportation costs of input materials that are located farther. The higher revenue is compensated by the higher input transportation costs, thereby equalizing the pro ts of the two rms. Finally, comparing the Hotelling midpoint (11) with the Weber point (1), we immediately obtain coincidence between them. Thus, we have established the following proposition that relates Weber (1909) and Hotelling (199). Proposition 3. The Hotelling midpoint of duopolists locations is equal to the Weber point of monopolist location. 3 The above proposition indicates a similarity between the two classical location theories. The Hotelling midpoint implies ltering out the inter rm competition e ect; hence, it does not di er from the Weber point of cost minimization. It is evident from this proposition that the two rms under Hotelling s spatial competition always locate symmetrically around the Weber point, irrespective of the locations of the input materials, m i for i = 1; ; : : : ; n. 3 This can also be shown in the case of triopoly. 8
Figure depicts the situation in which one input material is located at a considerable distance from the market with the parameter values of (m 1 ; t; r 1 ) = (5; 1; 1). Observe that the rm locations of Hotelling and Weber are midway between the consumer distribution [0; 1] and the input location m 1. If the transportation rate t of inputs is su ciently low relative to the transportation rate of outputs, then the rms tend to locate close to the consumer distribution like Hotelling s rms. If the transportation rate of inputs is su ciently high, then the rms approach the center of the gravity of input materials like Weber s rm. The similar observations can be made for Figure 3, where the input location is inside the consumer distribution. This may account for location of restaurants. The input materials (ingredients) come from a port or a station located at m 1, and the outputs (dishes) are available for consumers by visiting restaurants. It is often observed in the world that seafood and sushi restaurants are concentrated near ports in comparison with consumer distributions because access to fresh shes is important in serving fresh seafood dishes. 4 Welfare considerations In this section, we rst consider the rst-best optimum, in which the government is able to control the locations and prices of all rms in order to maximize the social welfare. Thereafter, we investigate the second-best optimum, where the policy variables of the government are limited to the taxes on the transportation of inputs and outputs. For example, they are gasoline taxes when rms and consumers use automobiles in the transportation of input materials and products. 4.1 First-best optimum Unlike the previous section, market boundary ^x is determined by the social planner rather than consumers in the rst-best world. The social planner determines the locations of the two rms (= plants), x 1 and x, and the market boundary ^x in order to maximize social welfare. Since utility is quasilinear, the sum of the consumers expenditure is equal to the sum of the rms revenues. This implies that maximization of social 9
welfare is equivalent to minimization of the sum of the transportation costs of inputs and outputs. Therefore, the objective function of the social planner is de ned by min x 1 ;x ;^x TC = X i t i (m i x 1 ) ^x+ X i t i (m i x ) (1 ^x)+ Z ^x (x 1 z) dz+ Z 1 0 ^x (x z) dz: Solving the rst-order conditions @TC=@x 1 = @TC=@x = @TC=@^x = 0, it is easily veri ed that there is a unique minimizer given by S + 1=4 (x o 1; x o ; ^x o ) = T + 1 ; S + 3=4 T + 1 ; 1 : Observe the same properties as Hotelling s price-then-location competition in section 3.. First, the midpoint of the social optimum locations, (x o 1 + x o )=, coincides with the Weber point. Second, the market boundary of the social optimum locations, ^x o, also coincides with the market center 1=. 4 That is, the market boundary determined by the social planner turns out to be the same as that by consumers who choose a store o ering a lower full price in a market economy. Finally, comparing Hotelling s price-then-location competition, we can show that x H 1 < x o 1 and x o < x H This means that in order to relax price competition, rms in Hotelling s spatial competition tend to further from the socially optimum locations. The above account may be summarized in the form of the following proposition: Proposition 4. The socially optimum midpoint of rm locations is equal to the Weber point. Furthermore, rms tend to locate outside of the socially optimum locations in Hotelling s spatial competition. Consider two polar cases as done in the previous section. When the input transportation rate t becomes zero, the optimal locations of rms are given by 1 lim (x o t!0 1; x o ) = 4 ; 3 : 4 This coincides with the social optimum con guration in the absence of input transportation. 4 Qualitatively similar results hold in the linear transportation cost. 10
On the other hand, when the input transportation rate t reaches in nity, the optimal locations of rms are given by S lim t!1 (xo 1; x o ) = T ; S : T That is, two rms approach the center of the gravity of input materials, which is the similar to the nding in Section. If the input and output transportation rates are nite, then the optimum locations are between the two extreme cases. 4. Second-best optimum Although the rst-best optimum is the most desirable for society as a whole, it may not be possible to achieve it in a market economy. In fact, the social planner cannot enforce the locations of two rms in a democratic society. However, the social planner can impose gasoline taxes. Thus, the second-best optimum may be examined by considering the following three-stage game. In the rst stage, the social planner determines the ad valorem gasoline tax rates on transporting inputs and outputs, whose rates can be di erent. In the second stage, the two rms simultaneously decide their locations x 1 and x. In the third stage, they simultaneously choose their product prices p 1 and p. The analysis of the last two stages presented in section 3. can be utilized with the addition of the gasoline taxes. The inclusion of the gasoline tax for outputs, g, leads to the following modi cation of consumer utility () at x visiting rm k u k = y p k (1 + g)(x k x) ; and the market boundary between the rms ^x = p 1 p + x 1 x (1 + g)(x 1 x ) ; (13) With the gasoline tax for inputs, g t, the pro ts are rewritten as 1 = [p 1 Pi t(1 + g t)(x 1 s) ] ^x; = [p Pi t(1 + g t)(x s) ] (1 ^x): (14) 11
Solving the rst-order conditions @ 1 =@p 1 = @ =@p = 0, plugging them into (14), and solving them for the rst-order conditions yield the following unique SPNE location: S(1 + (x s 1;x s ; ^x s gt ) (1 + g)=4 ) = T (1 + g t ) + (1 + g) ; S(1 + g t) + 5(1 + g)=4 ; 1 : (15) T (1 + g t ) + (1 + g) The second-order conditions for x 1 and x are also satis ed. The market boundary of the second-best optimum locations, ^x s, also coincides with the market center 1=. The objective of the social planner is to maximize the social welfare, which comprises the consumer surplus, producer surplus, and tax revenues. Since the consumer utility is quasilinear, it is transferable and can be added. Therefore, the aggregate consumer surplus is de ned as: CS Z ^x 0 u 1 dx + Z 1 ^x u dx; where ^x is given by (13). The aggregate producer surplus is de ned by PS 1 + : The total tax revenue is de ned by TR = g t " X i t i (x 1 m i ) ^x + X i t i (x m i ) (1 ^x) # +g Z ^x 0 (x 1 x) dx + Z 1 ^x (x x) dx : Substituting equilibrium prices and (15) into the RHS of CS, PS and TR, the social welfare W CS + PS + TR is expressed as a function of g t and g. Solving @W=@g t = @W=@g = 0 yields the two identical equations given by g s = (16S 16ST + 7T )gt s 6T (T + 1) 16 (S T=) : (16) + 9T + 6T This suggests that there is a continuum of optimal combinations of the two tax rates g s t and g s. Stated di erently, the second-best optimum is guaranteed as far as (16) is 1
satis ed. 5 Substituting the optimal tax rates into (15), the second-best locations of rms may be expressed as a function of the primitives S and T. The rm locations of the rst-best, the second-best, and Hotelling s location-thenprice competition are illustrated in Figure 3 with (m 1 ; t; r 1 ) = (7=10; 1; 1). It must be observed that the rst-best locations of rms are inside the Hotelling s ones as shown by Proposition 4. However, it must be noted that it cannot be generally stated whether or not the second-best locations are too far apart in comparison with the rst-best locations and the Hotelling s ones. Finally, comparing the social welfare of the rst-best optimum, second-best optimum, and Hotelling s spatial competition, it may be indicated that the social welfare of the rst-best is larger than that of the second-best, and that of the second-best is larger than that of Hotelling s spatial competition. From the example presented in Figure 3, they are 0.98, 0.90, and 0.803, respectively. 5 Conclusions The main objective of this paper is to combine the two classical location theories: Weber s (1909) total transportation cost minimization in a monopolistic market and Hotelling s (199) spatial duopoly competition in a duopolistic market under assumptions of discrete input distribution and continuous consumer distribution. First, we showed that the Weber point is located at the center of gravity. Thereafter, solving the SPNE, we showed that Hotelling rms agglomerate at the market center, irrespective of the distribution of inputs with a xed price of the output. Further, we showed that the Weber point is identical to the Hotelling midpoint with and without price competition. Finally, the analysis revealed that the rst- and the second-best locations are di erent from the equilibrium locations. 5 For example, we can determine the optimal tax rates g o t = 0 and g o = 6T (T + 1) 16 (S T=) + 9T + 6T < 0: This implies zero gasoline tax for rms and a gasoline subsidy for consumers. 13
References [1] W. Alonso (1975) Location theory, in J. Friedmann and W. Alonso (eds.), Regional Policy: Readings in Theory and Applications, Cambridge, (Mass.): MIT Press. [] S.P. Anderson, A. de Palma, J.-F. Thisse (199) Discrete Choice Theory of Product Di erentiation, Cambridge, (Mass.): MIT Press. [3] M. Birkin, A. G. Wilson (1986a) Industrial location models 1: a review and an integrating framework, Environment & Planning A, 18, 175-05. [4] M. Birkin, A. G. Wilson (1986b) Industrial location models : Weber, Palander, Hotelling, and extensions within a new framework, Environment & Planning A 18, 93-306. [5] C. d Aspremont, J.J. Gabszewicz, J.-F. Thisse (1979) On Hotelling s stability in competition, Econometrica 47, 1145-50. [6] J.J. Gabszewicz, J.-F. Thisse (1986) Spatial competition and the location of rms, Fundamentals of Pure and Applied Economics 5, Harwood Academic Publishers, 1-71. [7] P. Hansen, M. Labbé, D. Peeters and J.-F. Thisse (1987) Facility location analysis, Fundamentals of Pure and Applied Economics, Harwood Academic Publishers, 1-70. [8] P. Hansen, J.-F. Thisse, and R.E. Wendell (1986) Equivalence of solutions to network location problems, Mathematics of Operations Research 11, 67-678. [9] H. Hotelling (199) Stability in competition, Economic Journal 39, pp.41-57. [10] A. Weber (1909) Über den Standort der Industrie, English translation: Theory of the Location of Industries, Chicago: The University of Chicago Press, 199. 14
m 1 m m i j j 0 1 m n - x Figure 1: The discrete distribution of input materials (m 1 ; : : : ; m n ) and continuous distribution of consumers (x [0; 1]) on a line x HL 1 = x HL x # H 1 +xh m 1 j j j j j j 1 0 1 5 x H 1 x W x H Figure : The Weber point (x W ) and the locations of Hotelling s rms (x H 1 ; x H ) x HL 1 = x HL # j j j j j j j j j j 1 0 x H 1 x o 1 x s x W x o 1 1 x s x H Figure 3: The locations of rst-best (x o 1; x o ), second-best (x s 1; x s ), and Hotelling s competition (x H 1 ; x H ) m 1 15