Optimal Zoning in the Circular and Linear City under different Regulatory Approaches

1 Optimal Zoning in the Circular and Linear City under different Regulatory Approaches Hamid Hamoudi * , Isabel Rodriguez Departamento Fundamentos del Análisis Económico, Universidad Rey Juan Carlos, Paseo de los Artilleros s/n, 28032 Madrid, Spain. ____________________________________________________________________ Abstract: In this article we study zoning in a circular and linear city model where firms are not allowed to locate in certain areas. A biased regulator is then introduced in a Bertrand spatial Competition framework. We prove that city zoning depends on the regulator bias towards consumers or firms. Both city models show a formal equivalence in the results: a consumerbiased regulator pushes for strong competition whereas a firm-biased regulator induces weak competition. Keywords: Circular and Linear model, Equilibrium, Zoning,. JEL Classification: C72, D43, L13 * Corresponding author E-mail address : [email protected] 2 1. INTRODUCCIÓN Life in the cities is full of externality effects caused by firms which are detrimental to the population. Public intervention can then become an internalization mechanism for firms that impose social costs on the rest of society. Authorities need to find regulating instruments which enable them to reduce any harmful external factors in the best possible way. Some of the possible tools to prevent negative externalities include tax related policies or urban planning. Authors like Mills (1989), Henderson (1991), Miceli (1992) and Wheaton (1993) have shown that zoning is a very popular urban planning policy. We analyse a regulation design in a duopolistic framework á la Hotelling under alternative political profiles. We consider a regulator in charge of the design of an urban city space divided in two different zones: an exclusively residential area where consumers live and a mixed area where consumers and firms locate. The aim of the regulator when restricting certain areas is to provide a high-quality environment, reduce trouble and prevent delinquencies. One of the advantages from this approach is that it can be analysed in terms of industrial policies 1 . In the context of regulation it is well known that authorities try to satisfy consumers and firms. Consequently, it is common to define a welfare function as a sum of consumer´s and producer´s surplus. However, in this paper, we introduce the social welfare function as a linear combination of firm´s profit and consumer´s utility 2 , in a similar way to Hamoudi and Risueño (2012). The interpretation for this welfare function is related to the political profile of the public authority. A “social” government or planner has more incentives to overvalue consumer surplus whereas a “liberal authority” values the surplus of firms more highly. In between the two types, a “centre or neutral oriented” government assigns similar weights to both firms and consumers. 1 The analogies between location and differentiation models are well-known: geographic space reflects characteristic space; consumer location expresses the preferred variety; location of firms is associated to firms´ offered product variety and transportation cost is taken as a disutility. 2 Such weighted approach was defined by Baron and Myerson (1982) and used in welfare function formulation with several regulated models by Armstrong et al. (1994). 3 An important amount of studies on zoning regulation can be found in the spatial competition literature. This research focuses mainly on optimal firm location, regulated zone dimensions, pricing, land usage 3 and social welfare effects. In this regard, Lai y Tsai (2004) examine Hotelling´s linear city model with restrictions on the location of firms. They show that maximum differentiation holds under Bertrand competition and social welfare is improved. Tsai et al. (2006) analyse how zoning affects firm´s location and land rents. Chen and Lai (2008) investigate the effects of symmetric zoning in the linear city and prove that firms locate in equilibrium at the extremes of the zoning area under Cournot competition. They conclude that introducing a regulated zoning area can be welfare improving. Matsumura and Matsushima (2011) study a duopoly model with restrictions on the location of firms. Their objective is to analyse the effects on consumer’s welfare. The model shown is related to the issue of urban sprawl in order to determine the allowed dimension of economic activities. Lastly, Hamoudi and Risueño (2012) consider the effect of zoning regulation in duopolistic circular model with Bertrand competition where consumers and firms are situated in different city regions. They show that the optimal size of the shopping area depends on the regulator’s political profiles. The present research paper uses two standard spatial competition models: the circular and the linear city. Both models lead to similar results for Bertrand competition as in D’Aspremont et al (1979) and De Frutos et al.(1999) but different results for competition á la Cournot as in Hamilton et al. (1989), Anderson and Neven (1991) or Pal (1998)). Here, we intend to examine similarities and formal differences between both types of zoning models as well as highlighting the implications for urban and industrial policies. The results obtained for the circular and the linear case show that a firm-biased regulator favours dispersion in location and maximum differentiation in terms of the offered good. On the other hand, a consumerbiased regulator produces opposite results: agglomeration and minimum differentiation. The rest of this article is organized as follows: part one analyses the circular model and part two the linear model. In both cases we describe the model first and then determine price 3 See Fujita and Thisse´s model (1986). 4 and location equilibrium as well as optimal zoning policies. Each part concludes with the most relevant remarks on the effects of regulation on competition. 2. THE CIRCULAR MODEL 2.1 THE MODEL. We study a location model in which a regulating authority plans the design of a city in a unitary length circular space. Any point in the circle corresponds to a number from the interval [0,1]. The southeasters’ point is 0 and we move anti clock wise from there. Points 0 and 1 will therefore coincide. The planner divides the circle in two regions: the first one is a commercial area bounded by points 21 ,vv such that 2/10 21 ≤≤≤ vv where firms and families locate. The second one is the residential area where only families locate (see figure 1). There are two firms located at 1 xand 2 x such that 21 xx ≤and ],[, 2121 vvxx ∈selling the same good in the commercial area at prices 1 pand 2 p respectively. Figure 1: Circular Market 5 A continuum of consumers spread uniformly along the city. Each consumer buys one unit of good and pays the cost of transporting from the location of the firm from which it is bought to his/her own location. The transportation cost incurred by consumer is assumed to be a quadratic function of distance. Specifically, the function is taken as: 2,1,0),())(( 2 =>= ibxdbxdc ii , where the distance d i (x) = │x – x i │ between location consumer x and the location firm x i , is defined as the shortest distance on the circle between the two points x, x i . Let s be the gross surplus for an arbitrary consumer, x. We assume s is large enough )0( >> sto allow all consumers to buy. Utility for consumer x from buying the good from firm i is, therefore, given as: ))(()( xdcpsxu iii − − = . A consumer purchases the product from firm i when jiixuxu ji ≠ = < ,2,1),()( . The model is then formalized as a three-stage game. In the first stage, the regulator chooses the optimum size of the commercial area; in the second stage firms choose their locations simultaneously; in a third stage firms decide on their prices at the same time. The game is solved by backward induction. First, we discuss the equilibrium outcomes given the size of the commercial area. 2.2 PRICE AND LOCATION EQUILIBRIUM The fact that the location space for firms is restricted does not affect in any way the location of the indifferent consumers 4 . Thus, the demand function expression remains the same as for the unrestricted space case (see de Frutos et al (1999)). We can then make the following change of variable: z = x 2 – x 1, where z represents the distance between both firms, that is, the difference between the chosen characteristics. The demand function is represented as follows: 4 An indifferent consumer, α buys from firm 1 or 2 so that; )()( 21 α α uu = . 6        −≤− −≤−≤−−        + − − −−≤− = 21 21 12 21 21211 )1(0 )1()1( 2 1 )1(2 )1(1 ),,,,( ppzbz zbzppzbz zbz pp zbzpp vxxppD In addition, demand for firm 2 is: D 2 = 1 – D 1 . Once consumer demands are found, the profit functions can be calculated using the following relation: jisiendojiparappDpppB jiiijii ≠=== 2,12,1),,(),( The existence of Nash equilibrium in prices is guaranteed for any size of the market v and any value of z, since profit functions are strictly concave (see De Frutos 1999). The solution corresponds to: ).1()( 2 )( 1 zbzz N pz N p−== Consequently, demand and profit functions can be written as: )1( 2 1 )()(, 2 1 2121 zzbzBzBDD NNNN −==== Proposition 1: There is a unique Nash location equilibrium for any commercial area given by: [v 1 ,v 2 ], 2211 ,vxvx NN == Demonstration: (See Appendix). ■ The perfect equilibrium subgame expressions for prices, demand and profits are: 12 ,2,1),1( 2 1 )(, 2 1 ),1()( vvvivbvvBDvbvvp N i N i N i −==−==−= Remarks In the circular model under zoning regulation, the location pattern satisfies the maximum differentiation principle: 2211 ,vxvx NN == . Notice than if 2/1 12 =− vv , the location equilibrium remains the same as in the circular model without zoning (see De Frutos et al. (1999)). Therefore, zoning half of the circular market does not affect the location strategies 7 of firms. The intuition behind the maximum differentiation result is straightforward: firms locate at 2211 ,vxvx NN == in order to avoid competition and reap some spatial monopoly. The demands are equal and independent of the size v. Thus, zoning does not affect the structure of demand in the location equilibrium. Prices and profits are also equal to each other and they are increasing to respect v. 0)21( 21 ≥−= ∂ ∂ = ∂ ∂vb v p v p NN ,       ∈∀≥−= ∂ ∂ = ∂ ∂ 2 1 ,00)21( 2 1 21 vforvb v B v B NN . If the regulator chooses a large value for v, competition decreases. Therefore, if firms were able to decide on their location, they will always be interested in the commercial area to be as large as possible in which case, 2/1 = v. On other hand, the regulator can force both firms to undergo more competition which could yield to zero profits for them. Indeed, if v tends to zero, prices and benefits will tend to zero, which mean that both firms engage in Bertrand competition. Subsequently, zZoning regulation can be seen as an industrial instrument to limit firms’ monopoly power. Given that the location pattern in this zoning model still satisfies the maximum differentiation principle, the purpose of a regulator is to find the dimensions for the commercial area and the exclusively residential area. 2.3 OPTIMAL ZONING Planners take their decisions according to the interests of firms and consumers and for this reason the objective function is usually defined as the sum of firm´s profit and consumer´s utility. Instead, at this point, we use an objective function for the regulator described as a linear combination from profits (firms) and utility (consumers). Consequently, we introduce the possibility to formally represent the regulator´s preferences in terms of the weight attached to profits or consumers. The welfare function can now be written as: )()1()()( vUvBvW N λλ −+= . Where: 8 • λ is the weight given by the regulator to firms. Thus, (1 – λ) accounts for the weight given to consumers. • )()()( 21 vBvBvB NNN += is firms’ profit. •      +−= )()()( v T Cv N BSvU , is the utility for all consumers. • S is consumer´s gross surplus. • )(vC T stands for total transport cost paid by consumers. Total transport cost is then formalized as: 21 )( IIvC T + = - I 1 corresponds to total transport cost paid by consumers when they buy the good from seller 1. - I 2 is the total transport cost paid by consumers when they buy the product from seller 2. [ ] [ ] dxxxbdxxxbI N N NN 2 1 1 0 2 11 2 1)1()( ∫∫ +−+−= α α ; [ ] dxxxbI N N N ∫ −= 2 1 2 22 )( α α ,where NN 21 , αα are indifferent consumers in terms of buying from firm 1 or firm 2.5 We can then represent )(vC T as follows: )133( 12 )( 2+−= vv b vCT. • )(vC T decreases as the size of v increases. The mixed location area for consumersfirms is: [ ] 2/1,0,0)12( 4 ∈∀≤−= ∂ ∂vv b v C T . When the regulator´s objective is to minimize the total transport cost for consumers, 2/1 = λ in the objective function. The optimal size for this case is 2/1 = v . Utility for the total of consumers is given by: ( ) .199 12 )( 2 ++−−= vv b SvU • )(vU is decreasing and reaches a maximum for v equal zero. In contrast to firms consumers are interested in a minimum size for the mixed consumers-firms area which then turns into a single point and 0 = v . The price of the good is equal to zero for this 5 Indifferent consumers . 2 1 2 , 2 21 2 21 1 + + = + =vvvv NN αα 9 value and transport cost reaches a maximum. Consumers pay a high price for transport cost but are compensated by a zero cost for the good. The intuition for this result is illustrated by the fact that consumers travel massively (ignore collateral costs) when a free good is offered. Given the expression for the welfare function W(v) the following can be highlighted: - When λ >1/2, the regulator weights the interest of firms more than consumer´s. In this case we define the planner as conservative. - However, when λ <1/2, the opposite happens and we define the biased regulator as liberal. - When λ =1/2, we have a neutral case in which the same weight is given to both groups: consumers and firms. Furthermore, the welfare function for this case is depicted as: )]([2/1)( vCSvW T −= The social welfare W(v) is defined as the sum of the firms’ profit and the utility for all consumers. We now focus on the study of the welfare function when equilibrium in prices and location is taken into account. The optimal strategy for the regulator is given by: )2/1(0.. ),( ≤≤ = vts vWMaxArgv v O By substituting the expression for )()( vCyvB T N in the objective function we obtain the following:      +−−−+−−= )133( 12 )1()1()12()( 2 vv b SvbvvW λλ 16 If the regulator´s bias (λ) takes some value between 3/7 and 5/9; the optimal size ∗ L v reaches a minimum in the lower end of the interval for ,7/3 = λ and a maximum in the upper end for ,9/5 = λ i.e for 0)7/3( * = L v, 1)9/5( * = L v. In this case 10 * ≤≤ L v is increasing so that when λ is higher the value of v grows. We then move from agglomeration to dispersion (from minimum differentiation to maximum differentiation). When the social planner assigns a value of ,2/1 = λ the optimal size of the commercial area is reached for 2/1 * = L v. In this case, )0(U , )0(Band )0(Ware given by: i) utility for the consumers is: ,)234625( )1(48 ) )1(2 37 ( 2 2 −+ − −= − − λλ λλ λ b SU ii) profit function of firms is : )1(2 37 ) )1(2 37 ( λ λ λ λ − − = − −bB iii) welfare function is: )23118143( )1(48 )1() )1(2 7 ( 2 +− − −−= − − λλ λ λ λ λ b SW . - Finally, if the regulator is firms-biased, 55,09/5 ≈ ≥ λ , then ,1= ∗ L v or identically, 1,0 2211 ==== ∗∗ L N L N vxvx . The mixed consumers-firms area corresponds to the interval [ ] 1,0 ; firms can locate in the whole market and they choose maximum differentiation in terms of product variety. This corresponds to agglomeration when interpreted in terms of location patterns. Now, )0(U, )0(Band )0(Wcorrespond to: i) the utility for the total of consumers: ,12/13)1( bSU − = ii) profit of firms is: ,)1( bB = iii) welfare function is: )1325()12/()1()1( − + − = λ λ bSW . Similar to the circular model when the regulator is firms-biased meaning that, ),17/3( ≤ ≤ λ the total utility for consumers decreases and )0()2/1( UU < , firms profits improve, )0()2/1( BB > . 17 We can easily represent these analytical results in the following figure: Figure 4: Linear Model. Notably, the results for the optimal size of the mixed commercial-residential are the same for the linear and the circular model. 4. CONCLUSIONS In this article we analyse spatial competition in a regulated market in which consumers locate freely along the market space, whereas firms are obliged to locate in a restricted area. First, we study the influence of regulation on competition in a circular space and the same question is then analysed for the circular space. The contribution from the present article is the analysis of a regulator with different biases (weights in the objective function). Under these premises we obtain agglomeration or dispersion results which can be interpreted in terms of urban policies. Moreover, these results can also be scrutinized from an industrial policy perspective as minimum or maximum differentiation cases. On the other hand, the question of the influence from the regulators´ bias on competition is also investigated. In this respect it can be stated that strong competition is triggered in the consumer-biased regulator case whereas 18 weak competition arises in the case of a firms-biased regulator. For a neutral regulator we find moderate competition. Lastly, we find no relevant differences between the results for the circular zoned market and the linear one. REFERENCES Anderson, SP., Neven, DJ (1991).Cournot competition yields spatial agglomeration. International Economic Review 32 (4):pp 793-808. Armstrong, M., Cowen, S., Vickers, J. 1994. Regulatory Reform: Economic Analysis and the British Experience. MIT Press. Baron, DP., Myerson, RB. 1982. Regulating a Monopolist with Unknown Costs. Econometrica, Vol. 50, Nº. 4, pp. 911-930. Chen, Ch.-Sh., Lai, F.-Ch.. 2008. Location Choice and Optimal Zoning under Cournot Competition. Regional Science and Urban Economics 38, pp. 119-126. D’Aspremont, C., Gabszewicz, J.J. Thisse, J-F. 1979. On Hotelling’s Stability in Competition” Econometrica 47,pp1145-1150. De Frutos, M.A., Hamoudi, H. Jarque, X. 1999. Equilibrium Existence in the Circle model with linear quadratic transport cost. Regional Science and Urban Economics 29, pp 605- 615. Hamilton, JH., Thisse JF, Weskamp, A. 1989. Spatial discrimination: Bertrand versus Cournot in a model of location choice. Regional Science and Urban Economics 59(3), pp 87- 615. Hamoudi, H. and Risueño, M. 2012. The effect of zoning in Spatial Economics. Journal of Regional Science Vol 52, Nº2, pp 361-374 Henderson, J. 1991. Optimal regulation of land development through price and fiscal controls. Journal of Urban Economic 30: pp. 64-82. Lai, F.-Ch. and Tsai, J.-F. 2004. Duopoly Location and Optimal Zoning in a Small Open City. Journal of Urban Economics 55: pp.614-626. 19 Matsumura, T., Matsushima, N. Article first published online: 3 aug 2011. Locating Outside a Linear City Can Benefit Consumers. Journal of Regional Science Miceli, T., 1992. Optimal Fiscal Zoning when the Local Government is a Discriminating Monopolists. Regional Science and Urban Economics Vol, Nº 22, pp. 579-504. Mills, D.,1989. Is zoning a negative –sum game? Land Economics Vol, Nº 65, pp. 490-596. Pal , D., 1998. Does Cournot competition yields spatial agglomeration? Economics Letters 60 (1) :pp 49-53. Tsai, Jyh-Fa, Peng, Shin-Kun, Lai, Fu-Chuan. 2006. Spatial Duopoly with Zoning. Ann Reg Sci 40: pp. 515-530 Wheaton, W. 1993. Land Capitalisation, Tiebout Mobility, and the Role of Zoning Regulation. Journal of Urban Economics 34: pp. 102-117. APPENDIX. Demonstration of Proposition 1: Given the expressions for the price equilibrium profit functions; the Nash equilibrium locations can be calculated by using the first order condition: ,0,0 2 2 1 1 = ∂ ∂ = ∂ ∂ x B x B We obtain that 2 1 12 =−xx Since ,2/10 2211 ≤≤≤≤≤ vxxv for any value of 21 ,xx , we then have that 2/1 12 ≤− xx , so that the necessary condition ,0,0 2 2 1 1 = ∂ ∂ = ∂ ∂ x B x B is only fulfilled for locations: 2211 ,vxvx NN == .■ Demonstration of Proposition 2: For clarity reasons, the welfare function is rewritten as follows: 20 ) 12 )(1()1()37( 4 )( b Svv b vW −−+−−= λλ By using the first order condition we find that: )37)(21( 4 −−= ∂ ∂ λ v b v W. Considering that this condition depends on the value of parameter λ, in order to determine the maximum a second order condition is needed: 2 2 v W ∂ ∂. Taking into account that: 0)37( 2 2 2 =−−= ∂ ∂ λ b v W , we can deduct the following results: 0 7 3 2 2 ≤ ∂ ∂ ⇒ ≥− v W If λ the social welfare function is concave and reaches a maximum for 2 1 3== ∗ c vv 0 7 3 2 2 = ∂ ∂ ⇒ =− v W If λ the social welfare is constant which means it takes the same value for any value of v between 0 y ½. This means the maximum is reached for ∗ = 2c vv so that . 2 1 0 2 ≤≤ ∗ c v 0 7 3 2 2 ≥ ∂ ∂ ⇒ ≤− v W If λ , in this case the social welfare function is convex and the solution for the first order condition corresponds to a minimum, so that a maximum is obtained for 0 1 == ∗ c vv . ■ Demonstration of Proposition 3: By using the first order condition, [ ] 0))1(2)37( 4 =−−−= ∂ ∂ λλ v b v W and for 1 ≠ λ , it is found that )1(2 37 * λ λ − − = L v . Assuming that the second order condition is: 21 0)1( 2 2 2 <−−= ∂ ∂ λ b v W . Given that: 10 * ≤≤ L v, * L v is the maximum of the objective function iif: . 9 5 7 3≤≤ λ - If ⇒≤ 7 3 λ [ ] 1,000 *∈∀≤ ∂ ∂ ≤vpara v W yvL , el máximo se alcanza, para 0 *= L v. - si 9 5 ≥ λ [ ] 1,001 *∈∀≥ ∂ ∂ ≥⇒vfor v W andvL, the maximum is reached for 1 *= L v .■