The Beijer Institute of Ecological Economics

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1 The Beijer Institute of Eologial Eonomis DISCUSSION PAPER Beijer Disussion Paper Series No. 247 Atmospheri Pollution in Rapidly Growing Urban Centers: Spatial Poliies and Land Use Patterns Efthymia Kyriakopoulou and Anastasios Xepapadeas. 24. The Beijer Institute of Eologial Eonomis The Royal Swedish Aademy of Sienes Box 55 SE-4 5 Stokholm Sweden Phone: Fax: beijer@beijer.kva.se

2 Atmospheri Pollution in Rapidly Growing Urban Centers: Spatial Poliies and Land Use Patterns 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, 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. The methodologial approah followed in this paper allows for endogenous determination of land use patterns and is shown to provide more preise results ompared to previous studies. JEL lassi ation: R4, R38, H23. Keywords: Spatial poliies, agglomeration, land use, atmospheri 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

3 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 used in order to explain why eonomi agents are not uniformly distributed aross the globe. In this ontext, it has been established that rms bene t from operating loser to other rms beause of di erent soures of urban agglomeration eonomies, suh as labour market interations, linkages between suppliers of intermediate and nal goods and knowledge spillovers. 2 All those soures have been shown to boost produtivity and promote the formation of business lusters. 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. Atmospheri pollution is unambiguously onsidered a signi ant fator of onern to both industries and onsumers when taking loation deisions. 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 stemming from ommuting osts 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 Papageorgiou and Smith (983) provide an early attempt to determine the irumstanes under whih positive and negative externalities indue agglomeration. 2 See Duranton and Puga (23) for a review of the theoretial literature of agglomeration eonomies. 2

4 mixed areas. Sine pollution problems, espeially in growing urban enters, are getting inreasingly serious, it is easy to understand why pollution externalities should be studied in a spatial ontext. The interation between industrial pollution and residential areas has attrated a lot of interest and has often been identi ed as a reason for government intervention. Mostly in ountries that experiene rapid development, environmental degradation auses major problems. The best example to illustrate this problem is China s urban growth. Over the last two deades, China has ahieved high industrial growth rates whih have reated numerous environmental problems. Aording to China s Energy Statistis Yearbook, in 2 the industrial setor onsumed the 89. perent of the total energy. Air and water pollution is highly onneted to industrial ativity in urban areas and urrently a lot of ities in China are faing extremely high pollution levels. Only one perent of the population that lives in Chinese ities enjoys air quality that meets the EU s standards (World Bank 27). On the other hand, the rapid development of eonomi ativity has attrated a lot of people who moved from rural areas to the urban areas of China. Millions of rural households are trying to take advantage not only of better employment opportunities in urban areas but also of other kinds of bene ts suh as better eduation levels and higher quality of life. This trend, however, annot prevent residents from loating loser to the polluted, urban industrial areas whih learly highlights the need for government intervention with the aim of reduing the negative pollution externalities. Apart from China, ountries that experiene a similar stage of newly advaned eonomi development, suh as Brazil, Russia and India, are expeted to grow by 46 perent from 25 to 23, whih will lead to signi ant levels of environmental degradation (OECD Environmental Outlook to 23, 28). 3 Thus, urban pollution alls for immediate ation that needs to be taken at loal level and whih learly points to the spatial aspet of the problem. 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 3 3

5 that will result from this kind of poliy ould enourage people to loate even loser to industrial areas reduing their ommuting ost. The objetive of this paper is to further analyze the trade-o between the positive (agglomeration fores) and negative (pollution) externalities that take plae in a ity ontext and have long attrated the interest of urban and environmental eonomis (Henderson, 974, Glaeser, 998, Arnott et al., 28, Zheng and Kahn, 23). 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. 4 The trade-o between shorter ommute and worse air quality (also modeled in Arnott et al., 28) is very relevant to highly polluted ities in the developing world. What has been added in this ontext by the present paper is the e et of the agglomeration eonomies whih is the main fore that drives the onentration of industries in spatial lusters. This fore omes from the existene of interations among rms whih failitate the mathing between rms and inputs. These inputs ould be either workers, or intermediate goods, or even ideas that stem from the exhange of information and knowledge between rms. These interations reate some bene ts for rms and boost their produtivity, whih means that, other things being equal, eah rm has an inentive to loate loser to the other rms, forming industrial or business areas. The introdution of agglomeration eonomies ombined with di used atmospheri pollution in this paper along with the traditional fator of ommuting ost provides new insights regarding the optimal urban strutures. More spei ally, we show that, ontrary to the monoentri ity result of the traditional land use models, the addition of environmental externalities promotes the formation of multi-enter ities at the optimum. We haraterize in partiular optimal and equilibrium land uses and we show that the derived market alloations di er from the optimal ones due to the assumed externalities 4 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. 4

6 in the form of positive produtivity spillovers and pollution di usion. We use the spatial model to de ne site-spei poliies that will improve the e ieny in the given region. More preisely, we show that the joint enforement of a site-spei pollution tax and a site-spei labor subsidy reprodues the optimal alloation as a market outome. Numerial experiments illustrate the di erenes between the two solutions and show that industrial areas are onentrated in smaller intervals in the optimal solution. Also, mixed areas emerge in the market alloation but not in the optimal one. More spei ally, using a general equilibrium model of land use 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 both rms and workers. Regarding rms, it implies implementation of environmental poliy in the form of a site-spei tax, that imposes extra osts, and at the same time it dereases labor produtivity. Regarding workers, atmospheri pollution disourages them from loating in polluted sites and imposes on them additional ommuting osts. 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. However, the higher the number of rms that loate in a spatial interval, the more polluted this interval will be, whih implies a higher environmental tax. Thus, if rms deide to loate lose to eah other so as to bene t from o-loated rms, 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 5

7 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. They show that in a spatial ontext, in order to ahieve the global optimum, a spatially di erentiated added-damage tax is needed. As mentioned above, 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 expliitly 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 produtivity spillovers. This interation is fundamental in determining 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. In ontrast to Kyriakopoulou and Xepapadeas (23), the present paper does not inlude a natural advantage site or any other form of inhomogeneous spae, but inludes ommuting ost. This allows a stronger fous on the endogenous loation deisions of eonomi agents. The methodologial approah followed in this paper, that was rst introdued in Kyriakopoulou and Xepapadeas (23), 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 6

8 onstitutes an advane ompared to the previous studies exploring the internal struture of ities, 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). 5 We believe that the spatial poliies derived here, whih an be alulated using the 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 ity 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. 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 5 In Kyriakopoulou and Xepapadeas (23), this approah was used to determine the distribution of eonomi ativity aross inhomogeneous spae without expliitly de ning any residential areas. In this paper, the same approah is used in order to study the ompetition between residential and industrial loation deisions that will nally determine the di erent land uses. 7

9 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 whih an be explained by Marshallian agglomeration eonomies bene ting o-loated rms. 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. The trade-o between these two fores de nes the industrial areas in our spatial domain. Assumption 2. Positive produtivity spillovers Firms are positively a eted by loating near other rms beause of externalities in prodution that take several forms. Here we assume that the role of the agglomeration fore is to failitate the mathing between rms and inputs. These inputs an be workers, intermediate goods or even ideas. More spei ally, in this model, rms bene t if they loate in areas with higher employment densities. The positive prodution externality is assumed to be linear and to deay exponentially at a rate with the distane between (r; s): 8

10 z(r) = e (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) = e (r 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: As explained above, this kind of prodution externality relates the prodution at eah spatial point with the employment density in nearby areas. In this ontext, in order to apture the importane of proximity among o-loated rms, we assume that higher employment densities in a spei site imply higher bene ts for the rms that will deide to loate loser to this site. This assumption has been used extensively in urban models of spatial interations and omprises one of the driving fores of business agglomeration. 6 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. As stated above, the generation of emissions during the prodution of the output 6 Similar theoretial modeling has been applied in Luas (2), Luas and Rossi-Hansberg (22), and Kyriakopoulou and Xepapadeas (23). 9

11 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). 7 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) = e (r s)2 (s) ln E(s)ds; with the normal dispersal kernel equal to k(r; s) = e (r 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. 8 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., wood res, ars, and fatories), biomass burning, and industrial proesses with inomplete 7 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. 8 See, e.g., Williams (22) and Bruvoll et al. (999).

12 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. Aordingly, they tend to loate far from the industrial rms to avoid polluted sites. The

13 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 : 9 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 High onentrations of workers 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 in order to bene t from the higher onentrations of workers, 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 and dispersion fores and in designing optimal 9 This agent will spend k jr sj e kjr sj units of time working. 2

14 poliies. The 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: 3

15 e (r s)2 "(s)ds + g (r) = y(r) (6) e (r s)2 y(s)ds + ( b) + bk e (r 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 obtain the optimal amount of inputs L (r) and E (r); whih will 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. 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) 4

16 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 determine the residential loation deisions, 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) #D (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 in line with the hedoni valuation literature aording to whih nonmarket assets suh as air quality and environmental amenities in general are apitalized in property values. As 5

17 an example of this literature, Bayer et al. (29) estimate the elastiity of willingness to pay with respet to air quality to be Finally, assuming that the land density is ; we an de ne the optimal population 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 o-loated rms. 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 6

18 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 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; where (r) 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 positive produtivity spillover e et 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 7

19 the industrial land rents: e ^R z Ab b b I (z; w; ) = ( b ): (8) w b In the expliit solution for L; E; and R I presented above, there are two integral equations: one desribing the bene ts from the higher employment densities and the other desribing the onentration of pollution at eah spatial point. Most authors who have studied the e et of the 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, 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 e (r ( ) bk S s)2 y(s)ds + e (r s)2 "(s)ds + g (r) = y(r) (9) b b Z S Z e (r ( b)( ) bk S s)2 y(s)ds + e (r 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 There are kernels in the right-hand side of equations 6-8 (see the de nition of z(r); A(r); and (r) above). 8

20 of inputs, (L; E) and output (q) exists. The proof of existene and uniqueness of both the optimum and the equilibrium is presented in the following steps: 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. See Moiseiwitsh (25) for more detailed de nitions. 9

21 If k (r; s) is an L 2 -kernel and generates a ompat operator W; then the integral equation Y a b W Y = G (2) 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 Patterns 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. 2

22 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 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 higher employment densities. 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) 2

23 This is the wage arbitrage ondition that implies that no one an gain by hanging her job loation. 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. 22

24 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 } positive produtivity 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 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 subsidy of the form v (r) = pe z(s) X(s) b L(s) b ds: Thus, rms would have to pay a lower labor ost, w(r) v (r); employ more labor, bene t from the stronger positive 23

25 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 positive produtivity spillover e et and the spatial pollution e et in equation (32). Thus, an optimal tax, instead of imposing (r) = MD (r) = SR h i X (s) ; 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 damage 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 the 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, 24

26 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. Sitespei subsidies should be given in every industrial loation and must equal the positive produtivity e ets aused by the onentration of workers in nearby loations. 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. 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 : 25

27 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 = :: 2 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 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 2 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). 26

28 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 produtivity 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 spillovers that boost produtivity (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 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 27