Spatial Policies and Land Use Patterns: Optimal and Market Allocations

Transcription

1 Spatial Poliies and Land Use Patterns: Optimal and Market Alloations Efthymia Kyriakopoulou University of Gothenburg and Beijer Institute of Eologial Eonomis Anastasios Xepapadeas y Athens University of Eonomis and Business and Beijer Fellow Abstrat We study the optimal and equilibrium distribution of industrial and residential land in a given region. The trade-o between the agglomeration and dispersion fores, in the form of pollution from stationary fores, environmental poliy, prodution externalities, and ommuting osts, determines the emergene of industrial and residential lusters aross spae. In this ontext, we de ne two kinds of spatial poliies that an be used in order to lose the gap between optimal and market alloations. More spei ally, we show that the joint implementation of a site-spei environmental tax and a site-spei labor subsidy an reprodue the optimum as an equilibrium outome. We also propose a novel approah that allows for endogenous determination of land use patterns and provides more preise results ompared to previous studies. JEL lassi ation: R4, R38, H23. Keywords: Spatial poliies, agglomeration, land use, pollution, environmental tax, labor subsidy. Corresponding Author, Department of Eonomis, University of Gothenburg, Box 64, Vasagatan, SE 45 3, Gothenburg, Sweden. Telephone: e.kyriakopoulou@eonomis.gu.se y Department of International and European Eonomi Studies, Athens University of Eonomis and Business and Beijer Fellow xepapad@aueb.gr

2 Introdution The formation of residential and industrial lusters in a ity or region re ets the existene of fores that drive the observed spatial patterns. Agglomeration and dispersion fores have been extensively analyzed in the literature of urban eonomis and have played an important role in explaining the initial formation and the further development of ities. Positive and negative aspets of spatial interation have been identi ed in order to explain why eonomi agents are not uniformly distributed aross the globe. In this ontext, it has been proven that rms bene t from operating loser to other rms sine it gives them aess to a pool of knowledge and the possibility of exhanging ideas is likely to boost produtivity. And this is where workers ome into the piture, as rms have to ompete not only with the rest of the rms when they hoose their loation, but also with workers. Sine ommuting always implies extra osts, whih inrease with distane, workers prefer to loate loser to their workplaes. Thus, even though in most regions of the globe there is exess supply of heap land, eonomi agents are willing to pay high land rents in order to loate in large enters. Apart from the above fores, whih are well-known from both the theoretial and empirial literature, there are additional determinants of the loation deisions of eonomi agents that need to be studied in a formal framework. Air pollution is onsidered an unambiguously signi ant fator of onern to both industries and onsumers in many ways. Industries generate emissions, and sine workers are negatively a eted by pollution they try to avoid loating near them. However, the spatial interdependene of industries and workers explained above makes the problem of air pollution even bigger. If industries were loated in pure business areas with no residents around, then the damage from the generation of emissions would be muh lower ompared with the ase of industries being loated lose to residential or mixed areas. As regards urban areas, where pollution problems are getting inreasingly serious, it is easy to understand that pollution externalities should be studied in a spatial ontext. Papageorgiou and Smith (983) provide an early attempt to determine the irumstanes under whih positive and negative externalities indue agglomeration. 2

3 The interation between industrial pollution and residential areas has often been identi ed as a reason for government intervention. Mostly in developed ountries, governments impose high taxes or di erent kinds of environmental poliy on polluting ativities or fore industries to take abatement measures to redue the total level of pollution. In the EU, for example, air pollution has been one of the main politial onerns sine the late 97s. In this ontext, one of the objetives of the Sixth Environment Ation Programme (2) is to ahieve air quality levels that do not give rise to unaeptable impats on human health and the environment. Di erent ations need to be taken at loal, national, European, and international level, whih learly points to the spatial aspet of the problem. 2 In this way, the role of environmental poliy is ruial in the development of residential and industrial lusters, as strit environmental measures an disourage rms from operating in spei areas, while the redued pollution levels that will result from this kind of poliy ould enourage people to again loate lose to business areas. This paper ontributes to the literature by extending the general equilibrium models of land use by inorporating environmental externalities. More spei ally, we study how pollution from stationary soures whih a et workers negatively and make governments impose environmental regulations ombined with other agglomeration fores suh as externalities in prodution and ommuting ost will nally determine the internal struture of a region. One the optimal and equilibrium land uses are spei ed, we haraterize two kinds of spatial poliies that an be used in order to implement the optimal alloations as equilibrium outomes. In partiular, the derived market alloations di er from the optimal ones due to the assumed externalities in the form of positive knowledge spillovers and pollution di usion. Thus, we use the spatial model to de ne site-spei poliies that will improve the e ieny in the given region. We show that the joint enforement of a site-spei pollution tax and a site-spei labor subsidy will reprodue the optimal alloation as a market outome. Numerial experiments will illustrate the di erenes between the two solutions and will show that industrial areas are onentrated in smaller intervals in the optimal solution. Also, mixed areas emerge in the market alloation but 2 Information on the environmental poliy enfored by the EU an be found at 3

4 not in the optimal one. The reason we use the existene of interations among rms as the basi agglomeration fore of the model lies in the fat that they have been proven to be the basi fore for the lustering of eonomi ativity. 3 These interations failitate exhange of information and knowledge between rms, whih means that, other things being equal, eah rm has an inentive to loate loser to the other rms, forming industrial or business areas. On the other hand, the formation of a pure business enter inreases the average ommuting ost of workers and gives rise to higher wages and land rents in the area surrounding the luster. This proess ats as a entrifugal fore that impedes further agglomeration of rms. The trade-o of prodution externalities and ommuting osts has been explained extensively in a lot of studies, suh as in Luas and Rossi-Hansberg (22), Rossi-Hansberg (24) and Fujita and Thisse (22) (Chapter 6). In an earlier paper, Fujita and Ogawa (982) presented a model of land use in a linear ity, where the population was xed and rms and households would ompete for land at the di erent spatial points. In this paper, using a general equilibrium model of land use and following Luas and Rossi-Hansberg (22) in the modeling of knowledge spillovers, we examine how pollution reated by emissions, whih are onsidered to be a by-produt of the prodution proess, determines the residential and industrial loation deisions and hene a ets the spatial struture of a region. Aordingly, pollution a ets negatively the produtivity of labor, implies implementation of environmental poliy in the form of a site-spei tax, and disourages workers from loating in polluted sites. An important point here is that pollution omes from a stationary soure yet di uses in spae, reating uneven levels of pollution at di erent spatial points. As far as the poliy is onerned, rms will be obliged to pay a site-spei pollution tax, the size of whih will depend on the marginal damage of pollution at the site where they will deide to loate. However, the higher the number of industries that loate in a spatial interval, the more polluted this interval will be. Thus, if rms deide to loate lose to eah other so as to bene t from positive knowl- 3 For empirial studies on rming the role of knowledge spillovers in the loation deisions of rms, see Keller (22), Bottazzi and Peri (23) and Carlino et al.(27). 4

5 edge spillovers, they will have to pay a higher pollution tax and su er some loss in the form of dereased labor produtivity due to pollution. Thus, pollution disourages the agglomeration of eonomi ativity. As for the onsumers, they are negatively a eted by pollution and prefer to loate in lean areas. Yet this means that they will have to move further away from the rms, whih implies higher ommuting osts. The balane among these opposite fores, as well as the use of land for both prodution and residential purposes, will nally de ne the industrial and residential areas. The rst models of spatial pollution (e.g., Tietenberg, 974, Henderson, 977) assumed a pre-determined loation for housing and industry, without giving the possibility to workers to loate in an area that is already haraterized as industrial and without allowing for a hange in the spatial patterns. The paper that is losest to the present one in the modeling of pollution is Arnott et al. (28), who assume non-loal pollution in order to investigate the role of spae in the ontrol of pollution externalities. More analytially, they study how the trade-o between pollution osts and ommuting osts a et the loation ombinations of housing and industry around a irle. They show that in a spatial ontext, in order to ahieve the global optimum, a spatially di erentiated added-damage tax is needed. The di erene between the present paper and Arnott et al. (28) (apart from the methodologial part, whih will be explained below) is that we examine how pollution di usion interats with the fore that has been identi ed to explain most of the spatial industrial onentration in lusters, i.e., the positive prodution externalities. This interation determines the equilibrium and optimal land uses and help us haraterize spatial poliies in the form of environmental taxes and labor subsidies that reprodue the optimum as equilibrium outome. Another form of interation between pollution di usion and a natural ost-advantage site, as well as its e ets on the distribution of prodution aross spae, are analyzed in Kyriakopoulou and Xepapadeas (23). Their results suggest that in the market alloation, the natural advantage site will always attrat the major part of eonomi ativity. However, when environmental poliy is spatially optimal, the natural advantage sites lose their omparative advantage and do not at as attrators of eonomi ativity. 5

6 The methodologial ontribution of this paper lies in the use of a novel approah that allows for endogenous determination of land use patterns through endogenization of the kernels desribing the two externalities. This approah is based on a Taylor-series expansion method (Maleknejad et al., 26) and helps us solve the model and provide an aurate solution for the level of the residential and industrial land rents, whih will nally determine the spatial pattern of our region. The method also helps in the determination of the site-spei poliies studied here, whih an be used to reprodue the optimal struture as a market outome. We believe that this onstitutes an advane ompared to the previous studies, where arbitrary values were assigned to the funtions desribing the spillover e ets (as in Luas and Rossi-Hansberg, 22) or there is not an expliit endogenous solution of the externality terms (as in Arnott et al., 28). We believe that the spatial poliies derived here, whih an be alulated using the novel approah desribed above, provide new insights and an ontribute to the improvement of e ieny in the internal of a region. The rest of the paper is organized as follows. In Setion 2 we present the model and solve for the optimal and market alloations. In Setion 3 we desribe the spatial equilibrium onditions, while in Setion 4 we derive the optimal, spatial poliies whih an be used to lose the gap between e ient and equilibrium alloations. In Setion 5 we present the numerial algorithm that is used to derive the di erent land use patterns, and then we show some numerial experiments. Setion 6 onludes the paper. 2 The Model 2. The region We onsider a single region that is losed, linear, and symmetri. It onstitutes a small part of a large eonomy. The total length of the region is normalized to S and and S are the left and right boundaries, respetively. The whole spatial domain is used for industrial and residential purposes. Industrial rms and households an be loated anywhere inside the region. Land is owned by absent landlords. 6

7 2.2 Industrial Firms There is a large number of industrial rms operating in the internal of our region. The loation deisions of these rms are determined endogenously. Assumption. Prodution All rms produe a single good that is sold at a world prie, and the world prie is onsidered exogenous to the region. The prodution is haraterized by a onstant returns to sale funtion of land, labor L(r); and emissions E(r): Prodution per unit of land at loation r is given by: q(r) = g(z(r))x(a(r); L(r); E(r)); () where q is the output, L is the labor input, and E is the amount of emissions generated in the prodution proess. Also, prodution is haraterized by two externalities: one positive and one negative. Hene, A is the funtion that desribes the negative externality, whih is basially how pollution at spatial point r a ets the produtivity of labor at the same spatial point. z desribes the positive prodution externality in the form of knowledge spillovers. In the numerial simulations, the funtions g and x are onsidered to be of the form: g(z(r)) = e z(r) x(a(r); L(r); E(r)) = (A(r)L(r)) b E(r) : The two opposing fores that will be shown to a et the loation deisions of rms are assoiated with the two kinds of prodution externalities mentioned above. More spei ally, the main fore of agglomeration is related to the positive knowledge spillovers, while the dispersion fore omes from the negative onsequenes of pollution. The tradeo between these two fores de nes the industrial areas in our spatial domain. Assumption 2. Positive knowledge spillovers Firms are positively a eted by loating near other rms beause of externalities in 7

8 prodution, namely positive knowledge spillovers. The positive prodution externality is assumed to be linear and to deay exponentially at a rate with the distane between (r; s): z(r) = s)2 (s) ln L(s)ds: Note that (r) is the proportion of land oupied by rms at spatial point r; and (r) is the proportion of land oupied by households at r. The funtion k(r; s) = s)2 is alled normal dispersal kernel, and it shows that the positive e et of labor employed in nearby areas deays exponentially at a rate between r and s: This kind of prodution externality relates the prodution at eah spatial point with the employment density in nearby areas. In this ontext, rms bene t from the interation with the other rms if they loate in areas with a high density of industries. This assumption has been used extensively in urban models of spatial interations and omprises one of the driving fores of business agglomeration. 4 Assumption 3. Pollution The prodution proess generates emissions that di use in spae and inrease the total onentration of pollution in the ity. This is reinfored in areas with a high onentration of eonomi ativity, where a lot of rms operate and pollute the environment. The use of emissions in the prodution and the negative onsequenes that follow require enforement of environmental regulation. Sine emissions, as well as the onentration of pollution, di er throughout the spatial domain, environmental regulations will be site-spei. In partiular, environmental poliy is striter in areas with high onentrations of pollution and laxer elsewhere. This means that it is more ostly for rms to loate at spatial points with high levels of pollution. However, apart from the ost of pollution in terms of environmental poliy, rms avoid loating in polluted sites sine pollution a ets the produtivity of labor negatively. As a result, pollution works as a entrifugal fore among rms. 4 For empirial studies that on rm the signi ane of this fore, see Footnote 2. For the theoretial modeling of knowledge spillovers, see Luas (2), Luas and Rossi-Hansberg (22), and Kyriakopoulou and Xepapadeas (23). 8

9 As stated above, the generation of emissions during the prodution of the output damages the environment. The damage funtion per unit of land is given by D(r) = X(r) ; (2) where D is the damage per unit of land and ; D (X) > ; D (X). 5 Aggregate pollution, X; at eah spatial point r is a weighted average of the emissions generated in nearby industrial loations and is given by: ln X(r) = s)2 (s) ln E(s)ds; with the normal dispersal kernel equal to k(r; s) = s)2 : Using similar interpretation with the kernel desribing the prodution externality, emissions in nearby areas a et the total onentration of pollution at the spatial point r; while this e et delines as the distane between the di erent spatial points r and s inreases. is a parameter indiating how far pollution an travel; it depends on weather onditions and the natural landsape. Finally, the negative e et of pollution on the produtivity of labor is given by A(r) = X(r) ; where 2 [; ] determines the strength of the negative pollution e et. = implies that there is no onnetion between aggregate pollution and labor produtivity, while a large value of means that workers beome unprodutive due to the presene of pollution. The negative e ets of pollution on the produtivity of labor are usually explained through their onnetion with health e ets. 6 The air pollution in China an be thought of as an example of this. In 22, the China Medial Assoiation warned that air pollution was beoming the greatest threat to health in the ountry, sine lung aner and ardiovasular disease were inreasing due to fatory- and vehile-generated air pollution. More preisely, a wide range of airborne partiles and pollutants from ombustion (e.g., 5 In order to model the damage funtion, we follow Koldstad (986), who de nes damages at a spei loation as a funtion of aggregate emissions of the loation. We do not diretly relate damages to the number of people living in that loation, so as to avoid the potential ontradition of assigning very low damages to a heavily polluted area that laks high residential density. 6 See, e.g., Williams (22) and Bruvoll et al. (999). 9

10 wood res, ars, and fatories), biomass burning, and industrial proesses with inomplete burning reate the so-alled "Asian brown loud", whih is inreasingly being renamed the "Atmospheri Brown Cloud" sine it an be spotted in more areas than just Asia. The major impat of this brown loud is on health, whih explains the need for a positive parameter above. 2.3 Households A large number of households are free to hoose a loation in the interval of the given region. The endogenous formation of residential lusters is determined by two fores that a et households loation deisions: ommuting osts and aggregate pollution. Assumption 4. Utility maximization. Consumers derive positive utility from the onsumption of the good produed by the industrial setor and the quantity of residential land, while they reeive negative utility from pollution. Thus, a household loated at the spatial point r reeives utility U((r); l(r); X(r));where is the onsumption of the produed good and l is residential land. To obtain a losed-form solution, we assume that the utility U is expressed as U(r) = (r) a l(r) a X(r) ; (3) where < a < and : As explained above, the residential loation deisions are determined by two opposing fores. The rst one is related to ommuting osts, whih are modeled below. This is a fore that impedes the formation of pure residential areas sine workers have an inentive to loate lose to their workplae so as not to spend muh time/money ommuting. As a result, ommuting osts promote the formation of mixed areas where people live next to their workplaes. The seond fore is a fore that promotes the onentration in residential lusters and omes from the fat that the onsumers reeive negative utility from pollution. A-

11 ordingly, they tend to loate far from the industrial rms to avoid polluted sites. The pollution levels at eah spatial point, whih are determined by the loation and prodution deisions of industrial rms, are onsidered as given for onsumers. Assumption 5. Commuting osts Consumers devote one unit of time working in the industrial setor, part of whih is spent ommuting to work. Agents who work at spatial point r; but live at spatial point s; will nally reeive w(s) = w(r)e kjr sj : 7 This equation orresponds to a spatially disounted aessibility, whih has been used extensively in spatial models of interation. Now, if a onsumer lives at r and works at s; the wage funtion beomes w(s) = w(r)e kjr sj : If r is a mixed area, people who live there work there as well, and w(r) denotes both a wage rate paid by rms and the net wage earned by workers. 2.4 Agglomeration fores The entripetal and entrifugal fores explained above are summarized in the following table. Fores promoting: Industrial Firms Households Conentration in lusters Strong knowledge spillovers High pollution levels Dispersion High pollution levels High ommuting osts To summarize the e et of the agglomeration fores assumed in this paper, industrial rms onentrate in lusters under the presene of strong knowledge spillovers, while high pollution levels work in the opposite diretion sine they imply a double negative e et for the same rms. Moreover, high pollution levels promote the formation of residential lusters, sine residents try to avoid the industrial polluted areas. However, this tendeny is moderated in the ase where these agents have to pay high ommuting osts. The use of land for industrial and residential purposes prevents the two parts from loating around a unique spatial point. The objetive of this paper is in examining the optimal and equilibrium patterns of land use under the above agglomeration fores and in designing optimal poliies. The 7 This agent will spend k jr sj e kjr sj units of time working.

12 trade-o between the above fores will de ne residential, industrial, or mixed areas in the internal of the region under study. 2.5 The Endogenous Formation of the Optimal Land Use We assume the existene of a regulator who makes all the industrial and residential loation deisions aross the spatial interval [; S]: The objetive of the regulator is to maximize the sum of the onsumers and produers surplus less environmental damages in the whole region. Thus, if we denote by p = P (q) the inverse demand funtion, the optimal problem beomes: max L;E 2 Z 4 q(r) P (v)dv w(r)l(r) D(r) 5 dr: 3 The FONC for the optimum are: = w(r) @E(r) or pbe z(r) X(r) b L(r) b E(r) + pe z(s) X(s) b L(s) b ds = w(r) pe z(r) X(r) b L(r) b E(r) pbe z(s) X(s) b L(s) b E(s) + ds = : After making some transformations that are desribed in detail in Appendix A, we get the following system of seond kind Fredholm linear integral equations with symmetri kernels: 2

13 s)2 "(s)ds + g (r) = y(r) (6) s)2 y(s)ds + ( b) + bk s)2 "(s)ds + g 2(r) = "(r); (7) where y(r) = ln L(r) and "(r) = ln E(r); while g(r) and g2(r) are some known funtions. In order to determine the solution of the system (6) - (7), we use a Taylor-series expansion method (Maleknejaket et al., 26), whih provides aurate, approximate solutions of systems of seond kind Fredholm integral equations. Following this tehnique, we get the optimal amount of inputs L (r) and E (r); whih an determine the optimal level of prodution at eah spatial point, q (r): The optimal emission level will nally de ne the total onentration of pollution at eah spatial point r; X (r); as well as the damage, D (r): The optimal land use is determined in two stages. In the rst stage, we derive the optimal industrial land rent, whih is the rent that rms are willing to pay at eah spatial point in order to settle there. Using the above optimal values, we an de ne the optimal industrial land-rent as follows: R I(r) = pq (r) w(r)l (r) D (r): (8) In the seond stage, we derive the optimal residential land-rent funtion, i.e., the maximum amount of money that agents are willing to spend in order to loate at a spei spatial point. Thus, total revenues, w(r); are spent on the land they rent at a prie R H (r) per unit of land and on the onsumption of the good, (r); whih an be bought at a prie p: So, onsumers minimize their expenditures: w(r) = R H (r)l(r) + p(r) = min l; [R H(r)l + p] (9) 3

14 subjet to U(; l; X) u () so that no household will have an inentive to move to another spatial point inside or outside the region. To solve for the equilibrium, we assume that a onsumer living at site r onsiders the amount of aggregate pollution X(r) at the same spatial point as given. This is atually derived above, so here we use the optimal value X (r): Using equation (3), we form the Lagrangian of the problem as follows, L = R H (r)l(r) + p(r) + $[u a l a + D (r)]; () and obtain the following rst order onditions (FONC): R H (r) = ( a)$l a a (2) p = a$ a l a : (3) Solving the FOC and making some substitutions, we get the optimal residential land rent at eah spatial point: R H(r) = " w(r) (u + D (r))( ) # ; where w(r) = w(s)e kjr sj is the net wage of a worker living at r and working at s: Also, RH (r) is the rent per unit of land that a worker bids at loation r while working at s and enjoying the utility level u: We observe that #R H (r) #X (r) < : This means that residential land rents are lower in areas with high pollution onentrations. In other words, people are willing to spend more money on areas with better environmental amenities. This is supported by the fat that the highest residential rents in the real world are observed in purely residential areas in the suburbs of the ities, far from the polluted business enters. Finally, assuming that the land density is ; we an de ne the optimal population 4

15 density N at eah spatial point r; N (r) = N (r)l(r) = =) N(r) = l(r) (u + D (r)) (w(r)) a a a ( a a ) a a ( a ) a a It is obvious that the population distribution moves upward when the net wage inreases and when the onentration of pollution at the same spatial point dereases. The omparison between the RI (r) and the R H (r) at eah spatial point provides the optimal land uses. : 2.6 The Endogenous Formation of the Equilibrium Land Use Equilibrium and optimal land uses will di er beause of the existene of externalities. On the one hand, the deisions about the amount of emissions generated by eah rm a et the total onentration of pollution in the internal of our region. However, in equilibrium, when rms hoose the amount of emissions that will be used in the prodution proess, they do not realize or do not take into aount that their own deisions a et aggregate pollution, whih atually desribes their myopi behavior. When, for instane, a rm inreases the amount of generated emissions at site r, aggregate pollution is inreased not only at r; but also in nearby plaes through the di usion of pollution. These higher levels of aggregate pollution a et rms in two ways: rst, they inrease the ost of environmental poliy. Seond, they make the negative pollution e et on the produtivity of labor stronger. Finally, rms in equilibrium do not onsider the fat that their own loation deisions a et the produtivity of the rest of the rms through knowledge spillovers. For instane, they do not realize the fat that employing one extra worker will not only inrease their produtivity but also the produtivity of nearby rms. Therefore, equilibrium loation deisions do not internalize fully the above e ets, whih distorts the optimal land uses studied above and makes them di er from the equilibrium ones. To derive the equilibrium solution, we assume that a rm loated at spatial point r 5

16 hooses labor and emissions to maximize pro ts: R I (r) = max L;E fpez(r) (A(r)L(r)) b E(r) w(r)l(r) (r)e(r)g; wher) is the environmental tax enfored by the government. The tax here is assumed to be a site-spei environmental poliy instrument, whih is equal to the marginal damage of emissions, i.e., (r) = MD (r): The solution will be a funtion of (z; A; ; p; w): L = ^L(z; A; ; p; w) and E = ^E(z; A; ; p; w): The maximized pro ts at eah spatial point ^R I (z; A; ; p; w) an also be interpreted as the business land rent, whih is the land rent that a rm is willing to pay so as to operate at this spatial point. Following the disussion at the beginning of this setion, a rm loated at site r treats the onentration of pollution X(r); the negative pollution e et on the produtivity of labor A(r); and the e et of knowledge spillovers in the prodution proess z(r) as exogenous parameter X e ; A e ; and z e respetively. This assumption implies that the tax (r) is also treated as a parameter at eah spatial point. The rst order neessary onditions (FONC) for pro t maximization are: pbe z(r) X(r) bk L(r) b E(r) = w(r) (4) pe z(r) X(r) bk L(r) b E(r) = (r): (5) So, we solve expliitly for: ^L(z; b w; ) = w Ae z b (6) ^E(z; w; ) = b b b Ae z b w b b : (7) Substituting (6) and (7) into the maximized pro t funtion, we solve expliitly for the industrial land rents: e ^R z Ab b b I (z; w; ) = ( b ): (8) w b 6

17 In the expliit solution for L; E; and R I presented above, there are two integral equations: one desribing knowledge spillovers and the other desribing the onentration of pollution at eah spatial point. 8 Most authors who have studied knowledge spillovers of this form use simplifying assumptions about the values that the kernels take at eah spatial point. However, this approah fores rms to loate around the sites that orrespond to the highest assumed arbitrary values of knowledge spillovers, and hene we do not take into aount that L(s) and E(s), s 2 S, appear in the right-hand side of (6)-(7) and therefore these equations have to be solved as a system of simultaneous integral equations. Instead of following this approah, we hoose to use a novel method of solving systems of integral equations, whih was also implemented in Kyriakopoulou and Xepapadeas (23). More spei ally, if we take logs on both sides of equations (4)-(5) and do some transformations that are desribed in Appendix B, the FONC result in a system of seond kind Fredholm integral equations with symmetri kernels: Z S Z ( ) bk S s)2 y(s)ds + s)2 "(s)ds + g (r) = y(r) (9) b b Z S Z ( b)( ) bk S s)2 y(s)ds + b b s)2 "(s)ds + g 2 (r) = "(r); (2) where y(r) = ln L(r); "(r) = ln E(r) and g (r); g 2 (r) are some known funtions. Proposition Assume that: (i) the kernel k(r; s) de ned on [; S] [; S] is an L 2 - kernel that generates the ompat operator W; de ned as (W ) (r) = R S k (r; s) (s) ds; s S; (ii) b is not an eigenvalue of W ; and (iii) G is a square integrable funtion. Then a unique solution determining the optimal and equilibrium distributions of inputs and output exists. The proof of existene and uniqueness of both the optimum and the equilibrium is 8 There are kernels in the right-hand side of equations 6-8 (see the de nition of z(r); A(r); and (r) above). 7

18 presented in the following steps: 9 A funtion k (r; s) de ned on [; S][; S] is an L 2 -kernel if it has the property that R S jk (r; s)j2 drds < : R S The kernels of our model have the formulation e (r s)2 with = ; (positive numbers) and are de ned on [; ] [; ] : We need to prove that R R e (r 2 s)2 drds < : Rewriting the left part of inequality, we get R R e (r s)2 2 drds: The term e (r s)2 takes its highest value when e (r s)2 is very small. Yet the lowest value of e (r s)2 is obtained when either = or r = s and in that ase e = : So, < e < : When R R (r s)2 e = and S = ; then 2 (r s)2 e drds = (r s)2 < : Thus, the kernels of our system are L 2 -kernels. If k (r; s) is an L 2 -kernel, the integral operator (W ) (r) = k (r; s) (s) ds ; s S that it generates is bounded and kw k jk (r; s)j 2 2 drds : So, in our model the upper bound of the norm of the operator generated by the n R S R o S L 2 -kernel is kw k jk (r; s)j2 2 R R 2 2 drds = e drds : i (r s)2 If k (r; s) is an L 2 -kernel and W is a bounded operator generated by k; then W is a ompat operator. If k (r; s) is an L 2 -kernel and generates a ompat operator W; then the integral equation 9 See Moiseiwitsh (25) for more detailed de nitions. Y a b W Y = G (2) 8

19 has a unique solution for all square integrable funtions G if ( b ) is not an eigenvalue of W (Moiseiwitsh, 25): If ( b ) is not an eigenvalue of W; then I b W is invertible. As we show in Appendix C, both systems (6)-(7) and (9)-(2) an be transformed into a seond kind Fredholm Integral equation of the form (2). Thus, a unique optimal and equilibrium distribution of inputs and output exists. To solve systems (6)-(7) and (9-2) numerially, we use a modi ed Taylor-series expansion method (Maleknejad et al., 26). More preisely, a Taylor-series expansion an be made for the solutions y(s) and "(s) in the integrals of systems (6)-(7) and (9-2). We use the rst two terms of the Taylor-series expansion (as an approximation for y(s) and "(s)) and substitute them into the integrals of (6)-(7) and (9-2). After some substitutions, we end up with a linear system of ordinary di erential equations. In order to solve the linear system, we need an appropriate number of boundary onditions. We onstrut them and then obtain a linear system of three algebrai equations that an be solved numerially. The analytial solution of the optimal and equilibrium model is provided in Appendies A and B. 3 Land Use Strutures Having studied the optimal and equilibrium problems, we are able to de ne the di erent land uses in eah ase. The region under study is stritly de ned in the spatial domain [; S] and rms and households annot loate anywhere else. Thus, a spatial equilibrium is reahed when all rms reeive zero pro ts, all households reeive the same utility level u; land is alloated to its highest values, and rents and wages lear the land and labor markets. Consumers dislike pollution, whih means that they have an inentive to loate far from industrial areas. On the other hand, onsumers work at the rms and if they loate far from them, they will su er higher ommuting osts, whih promotes the formation of 9

20 mixed areas. The trade-o between these two fores will de ne the residential loation deisions. Firms have a strong inentive to loate lose to eah other in order to bene t from the positive knowledge spillovers. However, if all rms loate around a spei site, this site will beome very polluted, whih will inrease both the ost of environmental poliy and the negative produtivity e et. Thus, if all rms deide to loate in one spatial interval, then they will be obliged to pay a higher environmental tax and su er from the negative pollution e ets. In other words, high pollution levels impede the onentration of eonomi ativity. The trade-o between these fores will de ne the size of the industrial areas. The onditions determining the land use at eah spatial point are desribed in the following steps:. Firms reeive zero pro ts. 2. Households reeive the same level of utility U(; l; X) = u: 3. Land rents equilibrium: at eah spatial point r 2 S; R(r) = maxfr I (r); R H (r); g (22) R I (r) = R(r) if (r) > and R I (r) > R H (r) (23) R H (r) = R(r) if (r) < and R H (r) > R I (r): (24) 4. Commuting equilibrium: at eah spatial point r 2 S; w(r) = w(s)e kjr sj = max s2s [w(s)e kjr sj ]: (25) As people hoose s to maximize their net wage, this means that in equilibrium w(s)e kjr sj w(r) w(s)e kjr sj (26) This is the wage arbitrage ondition that implies that no one an gain by hanging his job loation. 2

21 5. Labor market equilibrium: for every spatial point r 2 S; ( (s))n(s)ds = (s)l(s)ds: (27) 6. Industries and households population onstraints: ( (s))n(s)ds = N (28) (s)l(s)ds = L; (29) where N is the total number of residents and L the total number of workers. 7. Land use equilibrium: at eah spatial point r 2 S; (r) (3) (r) = if r is a pure industrial area (r) = if r is a pure residential area < (r) < if r is a mixed area. Equations (22)-(24) mean that eah loation is oupied by the agents who o er the highest bid rent. Condition (25) implies that a worker living at r will hoose her working loation s so as to maximize her net wage. Condition (27) ensures the equality of labor supply and demand in the whole spatial domain. This ondition will determine the equilibrium wage rate at eah spatial point, w (r): Finally, onditions (28)-(29) mean that the sum of residents in all residential areas is equal to the total number of residents in the ity and that aggregate labor in all industrial areas equals the total number of workers in the ity. 2

22 4 Optimal Poliies: Labor Subsidies and Environmental Taxation Using the optimal values for L ; E ; z ; A ; X ; N ; and ; we an determine the wages and the level of the tax that would make rms and households in the equilibrium to make the same deisions as in the optimum. Thus, we would be able to implement the optimum as an equilibrium outome. From the rst-order onditions for the optimum (for (r) = ); w(r) = pbe z(r) X(r) b L(r) b E(r) + pe z(s) X(s) b L(s) ds {z } knowledge spillover e et (3) and pe z(r) X(r) b L(r) b E(r) 2 3 4pbe z(s) X(s) b L(s) b E(s) + X(s) {z ds = : (32) labor produtivity e et {z } {z} spatial pollution e et If the environmental tax enfored by the government is a site-spei environmental poliy equal to the marginal damage of emissions, (r) = MD (r) = X (s) ; then the di erenes between the optimum and the equilibrium are shown by the bold terms above. Let us analyze the rst-order ondition with respet to labor input. Firms here seem to internalize the externality that is related to the knowledge spillover e et taking into aount the positive e et of their own deisions on the produtivity of the rest of the rms, loated in nearby areas. Sine the di erene between the optimal and equilibrium FOC omes from the knowledge spillover e et in equation (3), the poliy instrument that would partly lead the equilibrium to reprodue the optimal distributions would be a SR subsidy of the form v (r) = pe z(s) X(s) b L(s) b ds: Thus, rms would to pay a lower labor ost, w(r) v (r); employ more labor, bene t from the stronger 22

23 knowledge spillovers, and produe more output. As far as the seond FOC wrt emissions is onerned, given that rms in equilibrium pay a tax equal to the marginal damage, as stated above, the di erene between the two ases is presented by the labor produtivity e et and the spatial pollution e et in equation (32). Thus, an optimal tax, instead of imposing (r) = MD (r) = X (s) ; SR h i should be of the form (r) = pbe z(s) X(s) b L(s) b E(s) + ds: It is obvious that the optimal taxation, (r); is higher than the equilibrium one, (r); at eah spatial point in the internal of our ity or region. The reason is that, rst, the optimal taxation takes into aount the extra damage aused in the whole region by emissions generated at r (spatial pollution e et). However, apart from this e et, the optimal taxation aptures the fat that inreased emissions in r mean lower produtivity for rms loating in r and in nearby areas (labor produtivity e et spatial pollution e et). This negative produtivity e et is now added to the ost of taxation, and the full aused by the generation of emissions during the prodution proess is internalized. Theorem 2 A labor subsidy of the form v (r)= and an environmental tax of the form pe z(s) X(s) b L(s) ds (r) = h pbe z(s) X(s) b L(s) b E(s) + ds will implement the optimal distributions as equilibrium ones. Proof. In equilibrium, rms will maximize their pro ts, households will minimize their expenditures given a reservation utility, land is alloated to its highest value, the wage no arbitrage ondition is satis ed, and all workers are housed in the internal of our region. Sine all the above are in line with the optimal problem as well, the only thing we need to do in order to impose the optimal alloation as an equilibrium one is to use the optimal poliy instrument desribed in Theorem 2. Thus, the joint enforement of a labor subsidy, 23

24 whih will derease the labor ost for the rms, and a higher environmental tax will lose the gap between the equilibrium and optimal alloations. Proposition 3 E ieny in a market eonomy an be ahieved by using the site-spei poliy instruments desribed in Theorem 2. Uniform taxes or subsidies, whih produe the same revenues or expenses, do not lead to optimal alloations. Proof. An industry, paying (r) for generating E (r) emissions, reeiving v (r) for employing L (r) workers and paying w(r) wages for the same number of workers and RI (r) as land rents, will reeive zero pro ts in equilibrium. Having proved the uniqueness of the equilibrium, any other level of taxes or subsidies will not satisfy the zero pro t ondition for the same amount of emissions and labor, and will not onstitute an equilibrium outome. Site-spei taxes should be enfored in every industrial loation and must equal the added damages aused by the emissions generated from this unit of land. Site-spei subsidies should be given in every industrial loation and must equal the positive e ets aused by the di usion of knowledge oming from this industrial loation and a eting the rest of the industries. 5 Numerial Experiments Numerial simulations will help us obtain di erent maps explaining the residential and the industrial lusters formed in our ity. To put it di erently, the optimal and equilibrium spatial distributions of residential and industrial land rents will determine the loation of rms and households in our domain. The numerial method of Taylor-series expansion, desribed above, will give us the optimal and equilibrium values of land rents. We solve the system of integral equations using Mathematia. The numerial algorithm to haraterize the optimal and equilibrium land use patterns onsists of the following steps: Step. We give numerial values to the parameters of the model. 24

25 Step 2. We solve for the optimal (and equilibrium) distributions L ; E ; q ; N ; ; z ; X (^L; ^E; ^q; ^N; ^; ^z; ^X) at every spatial point as a funtion of : Step 3. We derive the optimal (and equilibrium) distributions of residential and industrial land rents R I ; R H ( ^R I ; ^R H ) and plot them in graphs so as to haraterize the areas as residential, industrial, or mixed. Then, we determine the value (see below). Step 4. We alulate the total number of residents and workers in the region. The aim is to have equal numbers of residents and workers, whih will satisfy the ondition that all workers should be housed inside the region. Step 5. If the number of residents does not equal the number of workers, then the level of the wage hanges and we start solving the problem again (bak to Step 2). We follow this proess until we obtain equal numbers of residents and workers. An iterative approah is used sine a hange in the wage level will also hange the demand for the seond input (emissions), whih in turn will a et the aggregate pollution. However, aggregate levels of pollution hange the level of environmental tax and a et both the produtivity of labor and the residential loation deisions. Step 6. The value for eah spatial point is nally determined. If an interval is purely residential or industrial, whih means that one of the land rents is always higher than the other, then is either or ; respetively. When land rents are equal in a spei interval, we alulate a value of < < suh that the numbers of residents and workers are equal. The ex-post alulation of allows the expliit endogenous solution of the externalities of the model, and we onsider this to be an advantage of this approah over previous solutions where the spatial kernels were arbitrarily hosen. The results of this numerial algorithm are presented below. Figure shows the optimal distributions of labor, emissions, output, and land rents, assuming the following values for the parameters: = 2; = :5; = : and k = :: The distribution of workers, emissions, and output is higher around two spatial points (r = :6; 8:4): This happens beause at the optimum all the externality e ets are internalized by the The results presented here are fairly robust in parameter hanges. For a disussion on these parameter values, see Kyriakopoulou and Xepapadeas (23) and Luas and Rossi-Hansberg (22). 25

26 regulator. Thus, high levels of pollution that ome from the prodution proess inrease the per unit damage of emissions at polluted sites, as well as the negative e et on the produtivity of labor. This prevents industrial onentration around one spatial point, as it is predited by models onsidering only the positive spillover e ets. In other words, the rst reason industrial ativity at the optimum onentrates around two spatial points is that it aptures bene ts from the positive knowledge spillovers, whih are higher in areas with high employment density. The seond one is that by avoiding reating highly polluted areas, it keeps the produtivity loss assoiated with aggregate pollution at a lower level. Studying households loation deisions, we an observe in the last part of Figure (d) that residents are willing to pay higher land rents in less polluted areas, i.e., in the enter of our region and lose to the two boundaries. It is also very obvious that in the spatial intervals preferred by the industries, the residential land rents are very low. Note that the gap between the levels of the two land rents is represented by the blak areas. As a result, we ould argue that the optimal land use struture inludes two industrial areas and three residential areas in between. At this point it is of great interest to study the market alloations using the same parameter values. In Figure 2, we an see the same plots, i.e., labor, emissions, output, and land rents distribution. Without the assumption of pollution di usion, whih implies the enforement of environmental poliy, rms would onentrate around a entral loation in order to bene t from positive knowledge spillovers (see Kyriakopoulou and Xepapadeas, 23). However, the trade-o between these spillovers and the ones assoiated with the environmental externalities make rms move further from the entral area, whih results in higher distributions of labor, emissions, and output lose to the boundaries. The opposite is true for households, who prefer to loate in the rest of the region in order to avoid the polluted industrial sites. The omparison between residential and industrial land rents, under the ondition that all agents should work and be housed in the region under study, leads to a mixed area at the ity enter, surrounded by two residential areas, whih are followed by two industrial areas lose to the boundaries. There are two peaks 26

27 in the residential areas, whih an be explained as follows: In these areas workers are willing to pay higher land rents to avoid the high ommuting osts that would result from loating further away, yet as we move lose to the boundary, i.e., to industrial areas, the pollution disourages workers from paying high land rents. In the mixed areas we also need to speify the value so as to have the same number of residents and workers. In this numerial example, = :35; i.e., the 35% of the interval where agents and industries oexist is overed by the industrial setor and the remaining 65% by the residential setor. The most apparent di erene between the optimal and the equilibrium land use patterns is that, while mixed areas an emerge as an equilibrium outome, a similar emergene of mixed areas at the optimum does not seem possible within our parameter range. This result is in line with previous literature studying optimal ity patterns, suh as Rossi- Hansberg (24), who proves that the optimal land use struture has no mixed areas. What we an also observe is the fat that industries operate in a muh smaller interval overing 25% of the region in this numerial example, while in the market outome rms operate in 4% of the given area. The full endogenization of the external e ets at the optimum impedes rms from loating in entral areas, whih would be the expeted result and seems to be the ase in the market alloation. Contrary to this, the optimal solution seems to be a onentration of rms in small, spatial intervals, reating pure industrial lusters and hene restriting the di usion of pollution aross the region, whih will redue the damage to the residential areas. Some omparative analysis will help us understand whih alloation is the most e ient in terms of the amount of generated emissions per unit of output alulated in the whole region. In the numerial experiment presented above, the optimal emissions per output equal :99 while the equilibrium rate is :36: Implementing the optimal poliy instruments and deriving the optimum as an equilibrium outome will signi antly improve the generated emissions per unit of output by dereasing this rate by 27%: 27

28 6 Conlusion This paper studies the optimal and market alloations in a spatial eonomy with pollution oming from stationary soures. It ontributes to the literature by ombining the assumption of pollution di usion with two other fores that have been proven to signi - antly a et the spatial patterns: ommuting osts and externalities in produtions. The seond di erene ompared with previous literature lies in the use of a novel approah of solving spatial models, whih allows the full endogenization of the assumed external e ets, i.e., the pollution and prodution externalities. In order to model the above agglomeration and dispersion fores, we use a linear region where households and rms are free to hoose where to loate. Firms produe by using land, labor, and emissions, enjoy positive knowledge spillovers, and pay an extra ost in the form of environmental taxation. Households work in the industrial setor, ommute to work, onsume the produed good and housing servies, and derive negative utility from pollution. The optimal and the equilibrium spatial patterns are derived when onsidering the trade-o between the externalities in prodution, workers ommuting ost, and the onsequenes of aggregate pollution in terms of environmental poliy and pollution damages. A rst onlusion that omes from the inorporation of environmental issues in a general equilibrium model of land use is that the monoentri ity result does not exist anymore. We show that rms have an inentive to reate lusters in more than one loation so as not to inrease the ost of environmental poliy even further by making a site very polluted. Also workers inentive to loate lose to rms to avoid high ommuting osts has now hanged, sine pollution works to enourage them to loate in pure residential areas. However, the most important result is that under the existene of pollution and prodution externalities, the optimal and equilibrium land uses di er a lot. This model allows us to identify the di erent alloations and suggest spatial poliies that will lose the gap between e ient and equilibrium outomes. More spei ally, we show that the joint implementation of a site-spei labor subsidy and a site-spei environmental tax 28

29 an reprodue the optimum as an equilibrium outome. The numerial approah employed in this paper an be used to investigate further the role of pollution in spatial models of land use and provide insights on optimal spatial poliies. The idea of two kinds of industries polluting and a non-polluting ones ould be studied using the numerial tools presented here. Another possible extension of this model is to assume that pollution omes from non-stationary soures, like the transport setor, whih is atually the ase in modern ities. We leave these issues for future researh. Aknowledgments Efthymia Kyriakopoulou aknowledges support from the FORMAS researh program COMMONS, as well as from the Swedish Government as part of the Sustainable Transport Initiative in ooperation between the University of Gothenburg and Chalmers University of Tehnology. Anastasios Xepapadeas aknowledges support from the European Union (European Soial Fund ESF) and Greek national funds through the Operational Program "Eduation and Lifelong Learning" of the National Strategi Referene Framework (NSRF) Researh Funding Program: Thalis Athens University of Eonomis and Business Optimal Management of Dynamial Systems of the Eonomy and the Environment: The Use of Complex Adaptive Systems. Referenes Arnott, R., Hohman, O., Rausser, G.C., 28. Pollution and land use: optimum and deentralization. J. Urban Eon. 64, Bottazzi, L., Peri, G., 23. Innovation and spillovers in regions: evidene from European patent data. Europ. Eon. Rev. 47, Bruvoll, A., Glosrød, S., Vennemo, H., 999. Environmental drag: evidene from Norway. Eol. Eon. 3, Carlino, G.A., Chatterjee, S., Hunt, R.M., 27. Urban density and the rate of invention. J. Urban Eon. 6,

30 Fujita, M., Ogawa, H., 982. Multiple equilibria and strutural transition of nonmonoentri urban on gurations, Reg. Si. Urban Eon. 2, Fujita, M., Thisse, J.-F., 22. Eonomis of Agglomeration: Cities, Industrial Loation and Regional Growth, rst ed. Cambridge University Press, Cambridge. Henderson, J.V., 977. Externalities in a spatial ontext: The ase of air pollution. J. Publi Eon. 7, 89-. Keller, W., 22. Geographi loalization of international tehnology di usion. Amer. Eon. Rev. 92, Kolstad, C.D., 986. Empirial properties of eonomi inentives and ommand-andontrol regulations for air pollution ontrol. Land Eon. 62, Kyriakopoulou, E., Xepapadeas, A., 23. Environmental poliy, rst nature advantage and the emergene of eonomi lusters. Reg. Si. Urban Eon. 43, -6. Luas Jr., R.E, 2. Externalities and ities. Rev. Eon. Dynam. 4, Luas Jr., R.E., Rossi-Hansberg, E., 22. On the internal struture of ities. Eonometria 7, Maleknejad, K., Aghazadeh, N., Rabbani, M., 26. Numerial solution of seond kind Fredholm integral equations system by using a Taylor-series expansion method. Appl. Math. Comput. 75, Moiseiwitsh, B.L., 25. Integral Equations, rst ed. Dover Publiations, New York. Papageorgiou, Y.Y., Smith, T.R., 983. Agglomeration as loal instability of spatially uniform steady-states. Eonometria 5, 9-9. Polyanin, A.D., Manzhirov, A.V., 998. Handbook of Integral Equations, rst ed. Chapman & Hall/CRC Press, Boa Raton. Rossi-Hansberg, E., 24. Optimal urban land use and zoning. Rev. Eon. Dynam. 7, Tietenberg, T.H., 974. Derived demand rules for pollution ontrol in a general equilibrium spae eonomy. J. Environ. Eon. Manage., 3-6. Williams, R.III, 22. Environmental tax interations when pollution a ets health or produtivity. J. Environ. Eon. Manage. 44,

31 Appendix A We use the modi ed Taylor-series expansion method in order to solve a system of seond kind Fredholm integral equation with symmetri kernels, and derive the optimal land use patterns. The FONC for the optimum are given by (4) and (5). The FONC with respet to L(r) is: pbe z(r) X(r) b L(r) b E(r) + pe z(s) X(s) b L(s) b ds = where z(r) = s)2 (s) ln(l(s))ds For di erent values of r; s the integral an be written as: fln L() + e ( r)2 ln L(r) + e ( S)2 ln L(S) j r= +::::: + )2 ln L() + ln L(r) + S)2 ln L(S) j r=r +:::::+ +e (S )2 ln L() + e (S r)2 ln L(r) + ln L(S) j r=s g So, z(s) L(r) = L(r) [e ( r)2 + ::: + + ::: + e (S r)2 ] = L(r) s)2 ds: For the numerial analysis, we approximate the value of the integral that expresses the aggregate impat on all sites from a hange in site r, by valuing the aggregate impat with the marginal valuation at site r: Then the FONC wrt L(r) beomes: bpe z(r) X(r) b L(r) b E(r) + pe z(r) X(r) b L(r) b E(r) L(r) s)2 ds = w so pe z(r) X(r) b L(r) b E(r) (b + s)2 ds) = w: At this step, we assume that (s) = for all s 2 S: 3

32 Taking logs, ln p + s)2 ln(l(s))ds b s)2 ln(e(s)) ds + (b ) ln L(r) + ln E(r) + ln(b + s)2 ds) = ln w: Next, we di erentiate with respet to E(r): pe z(r) X(r) b L(r) b E(r) pbe z(s) X(s) b L(s) b E(s) ds = Aggregate pollution, X(r), is desribed by: ln X(r) = R S s)2 ln(e(s)) ds or e ln X(r) = e R S X(r) = e s)2 ln(e(s)) ds or SR h i s)2 ln E(s) ds : For di erent values of r; s the exponential term an be written as: e [ln E()+e ( r)2 ln E(r)+e (S)2 ln E(S)] p r= +:::::: + e [)2 ln E()+ln E(r)+ S)2 ln E(S)] p r=r +::::::+ +e [e (S)2 ln E()+e (S r)2 ln E(r)+ln E(S)] p r=s : So, di erentiating this expression wrt E(r), we have: SR X(s) = e ( r)2 + :::: + e (S r) 2 + :::: + e E(r) E(r) E(r) E(r) SR e E(r) SR e E(r) h h i s)2 ln E(s) i s)2 ln E(s) ds R S h i s)2 ln E(s) ds = ds e ( r)2 + :::: + + :::: + e (S r)2 = e (s r)2 ds: For the numerial analysis, we approximate the value of the integral that expresses the aggregate impat on all sites from a hange in site r by valuing the aggregate impat with the marginal valuation at site r: Then the FONC wrt E(r) beomes: 32

33 p e z(r) X(r) b L(r) b E(r) bkpe z(r) X(r) b L(r) b E(r) E(r) e SR h i s)2 ln E(s) ds e (s r)2 ds X(r) E(r) e SR h i s)2 ln E(s) ds e (s r)2 ds = ) p e z(r) X(r) b L(r) b b e (s r)2 ds A = X(r) E(r) e (s r)2 ds: Taking logs, ln p+ s)2 ln(l(s))ds b R S s)2 ln(e(s)) ds+b ln L(r)+( ) ln E(r) = ln + R S s)2 ln(e(s)) ds ln E(r)+ln S R e (s r)2 ds ln SR b e (s r)2 ds ) ln p + s)2 ln(l(s))ds + b ln L(r) + ln E(r) = SR ln + ( + b) h i R S s)2 ln E(s) ds + ln e (s r)2 ds ln SR b e (s r)2 ds : So, the rst-order onditions are: ln p + s)2 ln(l(s))ds b s)2 ln(e(s)) ds + (b ) ln L(r) + ln E(r) + ln(b + s)2 ds) = ln w and 33

34 ln p + s)2 ln(l(s))ds + b ln L(r) + ln E(r) = ln + ( + b) h i s)2 ln E(s) ds + e (s r)2 ds A b e (s r)2 ds A : Setting ln L = y and ln E = "; we obtain the following system: s)2 y(s)ds b s)2 "(s) ds+(b )y(r)+"(r) = ln w ln p ln(b+ s)2 ds) s)2 y(s)ds + by(r) + "(r) ( + b) = ln ln p + e (s r)2 ds A b e (s r)2 ds s)2 "(s) ds A : We need to do the following transformation in order to obtain a system of seond kind Fredholm integral equations with symmetri kernels: 8 (r s) 2 b e y(s)ds C A b (r s) B 2 e "(s)ds A >< B (r s) 2 B C + e dsa + ln p ln w (s r) 2 B B C ln p ln e dsa + b e >: (s + r) 2 ds = CC AA 34

35 b B B y(r) C b A "(r) {z } {z } A Z B = AZ A B = Z; where A = b b C A b b ln (b + R S ln p ln ln 8 >< C B b C A >: b e (s Z S s)2 ds)+ ln p ln w r)2 ds + ln ( b)+b C A Z S SR b e (s r)2 ds s)2 y(s)ds s)2 "(s)ds s)2 y(s)ds s)2 "(s)ds C A + 9 >= C = A >; C A + y(r) "(r) C A ) ln (b + b 2 e b (r s) 2 ds)+ ln p ln w ln p+ ln + ln ( 2 ln ( 6 4ln (b + 6 4ln p ln ln ( b e e (s r) 2 ds) (r s) 2 7 ds)+ ln p ln w5 e (s r)2 ds)+ ln ( b 3 e e (s (s r)2 ds) 3 r) 2 7 ds) 5 C A 35

36 = y(r) "(r) C A So, the system of seond kind Fredholm integral equations is: s)2 "(s)ds + g (r) = y(r) (A) s)2 y(s)ds + ( b) + b s)2 "(s)ds + g 3(r) = "(r); (A2) where g (r) = ln (b + s)2 ds)+ ln p ln w ln p+ ln + e (s r)2 ds A(A3) b e (s r)2 ds A g 2(r) = 2 3 b 4ln (b + s)2 ds)+ ln p ln w5 (A4) 2 3 b 4ln p ln e (s r)2 dsa + b e (s r)2 dsa5 : We use a modi ed Taylor-series expansion method for solving Fredholm integral equations systems of seond kind (Maleknejad et al., 26). 2 So, a Taylor-series expansion an be made for the solutions y(s) and "(s) : 2 K. Maleknejad, N. Aghazadeh, and M. Rabbani, Numerial solution of seond kind Fredholm integral equations system by using a Taylor-series expansion method, Appl. Math. Comput. 75, (26). 36

37 y(s) = y(r) + y (r)(s r) + 2 y (r)(s r) 2 "(s) = "(r) + " (r)(s r) + 2 " (r)(s r) 2 : Substituting them into (), (2), and (3): s)2 f"(r) + " (r)(s r) + 2 " (r)(s r) 2 g ds + g (r) = y(r) s)2 fy(r) + y (r)(s r) + 2 y (r)(s r) 2 g ds+ ( b) + b s)2 f"(r) + " (r)(s r) + 2 " (r)(s r) 2 g ds + g 2(r) = "(r): Rewriting the equations, we have: y(r) s)2 ds "(r) (A5) s)2 (s r)ds " (r) Z S 2 s)2 (s r) 2 ds " (r) = g (r) s)2 ds Z S y(r) + s)2 (s r)ds y (r)+ 2 s)2 (s r) 2 ds y (r) + ( b) + b s)2 ds "(r) (A6) 37

38 ( b) + b s)2 (s r)ds " (r) 2 ( b) + b s)2 (s r) 2 ds " (r) = g 2(r): If the integrals in equations (3)-(4) an be solved analytially, then the braketed quantities are funtions of r alone. So (3)-(4) beome a linear system of ordinary di erential equations that an be solved if we use an appropriate number of boundary onditions. To onstrut boundary onditions, we di erentiate (), (2): y (r) = 2 (r s) s)2 "(s) ds + g (r) (A7) y (r) = (r s) 2 s)2 "(s) ds + g (r) (A8) " (r) = 2 (r s) s)2 y(s) ds+ (A9) ( b) + b 2 (r s) s)2 "(s) ds + g 3 (r) " (r) = (r s) 2 s)2 y(s) ds+ (A) ( b) + b (r s) 2 s)2 "(s) ds + g 3 (r): We substitute y(r); "(r) for y(s); "(s) in equations (5)-(8): y (r) = 2 (r s) s)2 ds "(r) + g (r) (A) 38

39 y (r) = (r s) 2 s)2 ds "(r) + g (r) (A2) " (r) = 2 (r s) s)2 ds y(r)+ (A3) ( b) + b 2 (r s) s)2 ds "(r) + g3 (r) " (r) = (r s) 2 s)2 ds y(r)+ (A4) ( b) + b (r s) 2 s)2 ds "(r) + g3 (r): From equations (A)-(A4), y (r); y (r); " (r); " (r) are funtions of y(r); "(r); g (r); g (r); g 3 (r); g 3 (r): Substituting them into (A5) and (A6), we have a linear system of two algebrai equations that an be solved using Mathematia. Appendix B The same method of modi ed Taylor-series expansion was used in order to solve for the market alloations. We take the logs of the system (4) and (5) and follow the same proess as the one desribed in Appendix A. Appendix C Transformation of the system of equations (6)-(7) to a single Fredholm equation of 2nd kind (Polyanin and Manzhirov, 998). We de ne the funtions Y (r) and G(r) on [; 2S], where Y (r) = y i (r (i )S) and G(r) = g i (r (i )S) for (i )S r is: 3 Next, we de ne the kernel (r; r) on the square [; 2S] [; 2S] as follows: (r; s) = k ij (r (i )S; r (j )S) for (i )S r is and (j )S r js: 3 We assume that y = y and y 2 = "; so as to follow the notation of our model. 39

40 (a). Labor (b). Emissions (). Output (d). Land Rents Figure : Optimal Densities So, the system of equations(6)-(7) an be rewritten as the single Fredholm equation R 2S Y (r) (r; s) Y (s) ds = G(r), where r 2S: b If the kernel k ij (r; s) is square integrable on the square [; S] [; S] and g i (r) are square integrable funtions on [; S], then the kernel (r; s) is square integrable on the new square: [; 2S] [; 2S] and G(r) is square integrable on [; 2S]: Funtions g i (r); as desribed in Appendix A by equations (A3)-(A4) are square integrable. 4