Regional Science and Urban Economics 38 (2008) 408–423

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Regional Science and Urban Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / r e g e c

Anti-sprawl policies in a system of congested cities ☆ Alex Anas a,⁎, David Pines b,1 a b

Department of Economics, State University of New York at Buffalo, Amherst, New York, 14260, United States of America The Eitan Berglass School of Economics, Tel-Aviv University, Ramat-Aviv, Israel

a r t i c l e

i n f o

Available online 10 May 2008 JEL classification: D61 D62 H23 H44 R13 R14 R41 R48 R52 Keywords: Sprawl Congestion Congestion tolls Urban growth boundaries

a b s t r a c t Armed with recurring analyses since the mid 1960s, economists believe that the under-pricing of traffic congestion in urban areas causes not only excessive travel but also excessively low land use densities and excessively spread out cities, a condition popularly known as urban sprawl. This conclusion is derived from analyses of a single monocentric city. We extend the analysis to a system of two asymmetric monocentric cities closed in aggregate population, differing by their amenity. In this setup, we analyze the effect of optimally tolling traffic congestion, or of optimally determined urban growth boundaries (UGBs), a constrained optimum regime that can also be implemented by taxes and subsidies on land. We prove that either policy regime may expand aggregate urban land use relative to laissez-faire. This is certainly true when the elasticity of substitution between lot size and other goods is sufficiently small and/or the cities are sufficiently asymmetric in their amenities. In both cases, the inter-city expansive effect of tolling, or of the UGB regime on aggregate urban land use outweighs the contractive intra-city effect (which is the only effect considered in earlier studies). Only when the elasticity of substitution is sufficiently large and/or the cities are sufficiently symmetric, the intra-city contractive effects of tolling or of the UGBs on aggregate land use, dominate the inter-city expansive effect, validating the earlier belief. These, properties are illustrated in simulations which supplement our proofs. © 2008 Elsevier B.V. All rights reserved.

1. Introduction In recent years, American urban planners have been deeply concerned with urban sprawl, a poorly articulated concept that is, however, generally understood to mean that urbanized areas are spread out at low densities, making excessive or wasteful use of land. A variety of ills, real or perceived, are blamed on this vaguely defined but strongly felt condition. Foremost among the complaints is that urban expansion encourages excessive use of the automobile, creating too much travel, pollution, congestion and vanishing urban open spaces. Low densities resulting from urban expansion are blamed for the death of traditional neighborhoods that can be walked, increasing obesity trends, reduced social interaction, and the depopulation of central cities. There are several controversies associated with urban sprawl. First, what is its definition and, given the definition, how can we measure it? Once we agree on its definition and measure, what are its causes and consequences, and should policy be directed towards the underlying causes of which sprawl is a symptom, or also towards the sprawl itself?

☆ The paper was presented at the 53rd annual meetings of the North American Regional Science Association, held during November 16–18, 2006 in Toronto, Canada, and the final manuscript was presented at a conference on “Residential sprawl and segregation” hosted by INRA–CESEAR at Dijon, France, October 22–23, 2007. A lecture partially based on the paper titled “The economics of traffic congestion and urban sprawl” was presented at the 2nd Kuhmo-Nectar summer school on “Urban transport: network and spatial interactions” held during July 9–11, 2007 at the Faculty of Economics and Business, University of Urbino, Urbino, Italy. A keynote plenary lecture also partially based on the paper, titled “New results in the theory of urban sprawl” was presented at the Conference of Urban and Regional Economics, hosted by National Taipei University, Taipei, Taiwan held during December 2,3 2007. Three anonymous referees and the editor of this special issue made helpful comments on an earlier draft. ⁎ Corresponding author. Tel.: +1 716 688 5816; fax: +1 801 749 7805. E-mail addresses: [email protected] (A. Anas), [email protected] (D. Pines). 1 Tel.: +972 3 640 9904; fax: +972 3 640 9908. 0166-0462/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.regsciurbeco.2008.05.001

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Regarding the definitions, the term urban sprawl is sometimes used to describe just an expansion of urban land areas on the aggregate, and sometimes the discontinuous pattern of land development or the absence of compactness. To distinguish between the two, the second is often referred to as “leap-frog sprawl”. In the present study we abstract from leap-frogging, concentrating on overall urban area expansion, which we call geographic sprawl (GS). We explore the relationship between geographic sprawl and the economic cost of human interactions in urban areas which we call economic sprawl (ES) and measure it as aggregate commuting costs. We show that sometimes a policy designed to reduce the economic sprawl and thus enhance welfare, may be associated with an increase in geographic sprawl.2 In discussing alternative measures of suburbanization (GS, in our terms), Mills (1972b, first edition, pp. 92–102) suggested using the reciprocal of the gradient of the exponential population density–distance function, because a decline in this gradient indicates an increase in the share of the metropolitan population living beyond a cordon line put at any distance around the city center. As it turns out, 2 divided by this gradient is approximately the average distance of residence locations from the city center, reflecting a distance measure of travel (Ashenfelter, 1976; White, 1977). Hence, if aggregate transport cost were positively correlated with the average distance from the city center and if the exponential density function accurately reflected the variation of residential density from city centers, then Mills' measure of suburbanization is similar to our measure of economic sprawl (ES) for a single city, and is positively correlated with its geographical sprawl (GS). There has been persistent and widespread empirical evidence on the validity of the log of density being linear in distance from the city center.3 Once we accept that the density function declines with distance from the center at a constant relative rate, we can evaluate the gradient even from fragmented data. It then follows that a process of suburbanization of both employment and population from the end of the 19th up to at least the second half of the 20th century in many metropolitan areas, was reflected in a continuous decline in the density gradient (e.g., see Clark, 1951; Muth, 1961, 1969; Mills, 1972a, 1972b). Although there are many causes for urban expansion (such as an increase in incomes, Wheaton, 1974), economists and planners agree that the automobile is a major contributor but perceptions differ widely. On the one hand, the website of the Sierra Club, second only to Gore (1993), laments automobile dependence as if it were a universal evil. On the other hand, sober and measured assessments of the effects of automobiles and highways on sprawl and land use have been given by Mills (1972b), Downs (1992), Dunphy (1997), Glaeser and Kahn (2004). Nechyba and Walsh (2004) put it as follows: “It is difficult to imagine large increases in suburbanization without this rise of the automobile, even if other causes have contributed to the sprawling of cities in the presence of the automobile.” (p. 182) There is a fundamental inefficiency in the use of automobiles in urban travel, arising from the fact that congestion is not priced. Unpriced congestion is a negative externality that arises because drivers do not take into account that they delay other drivers. The underpricing makes the private average cost of travel cheap relative to its social marginal cost and thus causes too much travel. This was recognized by economists more than four decades ago. The use of marginal cost pricing to reduce highway congestion and improve efficiency was advocated by Walters (1961) and Vickrey (1965). With practical and political difficulties to implement marginal cost pricing, two questions that arise bear on the popular concerns about sprawl. How does unpriced congestion and, consequently, excessively cheap road travel affect urban form? What can be done to improve the resource misallocation when marginal cost pricing is not applied? To answer these questions, economists use the monocentric model which assumes that all jobs are located in a city's center, while commuters are distributed in residences all around the center. The first rigorous analysis of congestion based on the monocentric model was that of Strotz (1965) who examined how land should be allocated between roads and residential uses. Since Strotz, the monocentric setup has been applied repeatedly to the study of criteria for allocating land to roads, starting with the first-best regime (e.g., Dixit, 1973) and continuing with second-best issue, that is optimally allocating land to roads when congestion is not priced (e.g., Solow, 1972; Kanemoto, 1977; Arnott, 1979; Pines and Sadka, 1985). One of the robust analytical findings that emerged was that the shadow price of land is above the market rent close to the center and below the market rent close to the boundary with agriculture. This finding implies that, given unpriced congestion, cities are too spread out and that it is advantageous to reduce the city area by implementing an urban growth boundary (UGB). More recently, Wheaton (1998) compared the unpriced congestion regime with first-best allocation regime of congestion tolls in a monocentric analysis and concluded that: “Because driving and location are equivalent, tolling congestion is the same as regulating density…Simulations suggest optimal cities should have densities that are orders of magnitude greater than market cities.” (p. 258, in Abstract) Wheaton also showed that a subsidy to high density land development, if properly calibrated, would be another means of realizing the first-best optimum. Thus forty years of persistent economic analysis has lent a helping hand to the idea that compact cities are desirable as a planning goal. This is exactly what anti-sprawl planners have advocated as noted by Ewing (1997).

2 The Random House Dictionary defines sprawl as “to be stretched or spread out in an unnatural or ungraceful manner” and www.dictionary.com defines urban sprawl as “haphazard growth or extension outward, especially that resulting from real estate development on the outskirts of a city”. In their monumental empirical study, Burchfield et al. (2006) measure sprawl by the average share of the undeveloped land in the adjacent square kilometer surrounding each of the 8.7 billion 30 × 30 m cells of residential development in the U.S. and the change of this measure over time. 3 Clark (1951) was the first who presented estimates of exponential population density functions and their variation across time for cities in the U.S., Europe, and elsewhere, illustrating the continuous decline of their gradients (Clark, 1967, pp. 342–354). Muth (1961, 1969) provided a theoretical derivation of the exponential density function from primitives and careful estimates of its parameters for forty-six large urbanized areas.

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In practice, planning experiments have sought to limit urban expansion by attempting to raise central densities, although the mispricing of traffic congestion is not necessarily the reason why planners favor such policies. London had a greenbelt in the late 1930s, while Moscow, New Delhi, Ottawa and Tianjin have followed the same practice. Zoning and greenbelt policies with similar effects are common in the United Kingdom's Town and Country Planning System (Cheshire and Sheppard, 2002), and in other European countries, and have also been implemented in Korea (Lee and Linneman, 1998; Son and Kim, 1998). Portland, Oregon with its strict urban boundary is well-known. The idea of growth boundaries are part of Oregon State land use law. Boulder, Colorado, is another example in the United States. Sprawl is now considered so important an issue that states such as Maryland and New Jersey have been implementing a variety of aggressive land use policies to curb future urban expansion. In New Jersey which is the most urbanized American state, the government has favored buying land to keep it from being developed in the future. The view that a UGB is an effective second-best policy to congestion tolls has been challenged only recently by examining what might happen if the unrealistic assumption of monocentric land use were to be relaxed. Anas and Rhee (2007) presented a numerically solvable theoretical model in which jobs are free to locate between a core job center and a suburban jobs center. Using such a model they showed some surprising results. First, allowing jobs to be mobile dramatically changes the effects of tolls on urban form. Because tolls fall heavily on those with the longest commutes, suburban residents who work in the core acquire an incentive, ruled out by the monocentric modelers, to move their jobs to the suburbs. This effect can, in many cases, dominate the effect, observed in monocentric models, of moving residence closer to the core's jobs. As a result, congestion tolls can increase suburban population, decrease central city densities and make urban land use more not less sprawled. The authors also show that this type of increased geographic sprawl, involves less not more travel expenditure (less economic sprawl) and is therefore efficient. In the context of Anas and Rhee (2007) with mobile jobs, placing a restrictive UGB around a congested urban area that should expand with tolls is not always a second-best policy. Rather, the authors note that an expansive UGB would be required. This is especially true in urban areas where the level of congestion near the core is high and, therefore, the response of jobs to suburbanize when congestion tolls are levied is also high.4 In Anas and Rhee (2006, 2007), sprawl is defined not as a land use measure but rather as the total travel time per person, an economic measure reflecting an important cost of human interaction. The authors showed, in their 2006 paper that as production decentralized out of the center, the density gradient and the average commuting time decreased, while discretionary travel time (e.g. shopping) increased. In the current article, we extend the study of congestion toll and UGB policies to the population distribution and allocation of resources among cities. More specifically, we study how such policies affect the aggregate use of urban land which we defined earlier as geographic sprawl and the aggregate commuting cost which we defined as economic sprawl. In Section 2 we present a simple urban system consisting of two congested cities with different amenities and, consequently different population sizes. Each city is monocentric and the residential area is perfectly homogeneous. Using this simple setup, we compare the laissez-faire equilibrium to two planning regimes: the first-best (which uses congestion tolls to internalize the externality) and the second-best (which uses UGBs or an equivalent land taxation scheme). We show that, unambiguously, the first-best policy (tolling), reduces aggregate commuting cost thus alleviating economic sprawl, but this comes at the expense of increasing geographic sprawl, because the optimal policy generates a larger aggregate area for the two cities than does the laissezfaire equilibrium. The effect of the second-best policy (UGB) on the aggregate area of the two cities is ambiguous. The area of the small city expands but the change in the large city's area is unclear. We are able to prove, however, that if the elasticity of substitution between housing and other consumption (σ, hereinafter) is zero, regardless of the other parameters, the aggregate area of the two cities increases in this second-best case also. We reason that by continuity the result should extend to the case of sufficiently small elasticity of substitution. In Section 3, we extend our model to allow heterogeneity of the residential areas in terms of accessibility to jobs. Accordingly, each city consists of a core area which has all the jobs and where residents can also locate and has no congestion, and a suburban area which also has no congestion internally but is connected to the core by a congested bridge. Using this extended setup, we explore the effect of first-best tolling, and the urban growth boundaries, which correspond to a lower-best constrained optimum. In accordance with the inter-city allocation effects of Section 2, first-best tolling and UGB have an expansive effect on GS, not only when σ = 0 but also when σ is sufficiently small. In contrast to Section 2, however, in the setup of Section 3, both policies have intracity allocation effects as well which are contractive on GS and which are stronger with larger σ, making the two effects move in opposite directions when σ is large enough. The interplay of the inter-city and intra-city effects of tolling and UGB on GS is further explored by simulations. In one set of simulations, we keep the importance of the amenity in the utility function (or, equivalently, the dissimilarity in the amenities of the two cities) constant but we vary σ. These simulations illustrate that there exists some critical value of σ such that below it the inter-city expansive effect of tolling on GS (examined in Section 2) outweighs the intra-city contractive effect (examined in the traditional literature), whereas above the critical level this relationship is reversed. A critical level of σ also applies to the UGB regime, implemented by taxing and subsidizing suburban land. Likewise, given any σ, the simulations illustrate that there exists a critical level of amenity importance in the utility function (or, equivalently, a differential between the amenities of the two cities)

4 In a related paper, Anas and Rhee (2006), a fully general equilibrium model is used in which jobs and residences are dispersed and intermixed throughout the urban area and in which commuting as well as non-work travel patterns are endogenously determined as consumers sort among communities and commuting arrangements, choosing where to reside, where to work, where to shop, how big a lot to rent and where and how much labor to supply. Simulations with the model also show that UGBs that restrict the urban area are not second-best policies.

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such that below it, the intra-city effect of tolling which is contractive on GS, dominates the inter-city effect which is expansive on GS, and aggregate urban land use contracts, whereas above the critical level, the inter-city expansive effect dominates the intra-city contractive effect and tolling increases aggregate urban land use. Such a critical level also applies to the lower-best UGB regime. Section 4 concludes with a remark on an extension and the implications of our results for policy-making. 2. Inter-city allocation effects In this section we highlight the inter-city allocation effects of congestion alleviation policies, an aspect completely ignored in the urban economics literature. To that end, we use a very simple model of a monocentric city where the intra-city locational heterogeneity of land is ignored. Accordingly, we abstract from intra-city allocation effects, which were intensively studied within the monocentric literature. These effects will be added in Section 3 by modifying the setup. 2.1. The basic setup There are many reasons for the existence of cities with different population sizes. One which we use here to explain size variation is a non-reproducible natural or man-made amenity which varies among the sites where cities are located.5 Natural amenities include climate, topography, the proximity of attractive landscapes, location with a unique economic advantage or combinations of these (first nature). Man-made amenities include historically significant buildings and monuments and the cultural wealth embodied in them (second nature). When, neither natural nor man-made amenities can be replicated, the population of a city grows to a level that reflects the demand for the specific amenities the city commands and city sizes differ according to how these non-replicable amenities are distributed among potential city sites. Therefore, the treatment of a system of cities with exogenous amenities is a highly appropriate framework for economic analysis.6 We consider a closed urban system of just two open cities which accommodate an exogenously given number of n residents who are identical in both preferences and initial endowments. The residents can migrate freely between the two cities but can only work in the city in which they reside. The cities differ by an exogenous amenity that can only be enjoyed by living in that city. The amenity level in city i = 1, 2 is Ai (A1 N A2). Each city has a Central Business District (CBD) which is a landless jobs center, and a suburb to which the supply of land from farming is perfectly elastic. All workers reside in the suburb of their city where they rent a plot of land, and commute to the CBD daily to supply one unit of labor. Each unit of labor produces one unit of a composite commodity, the numeraire good. Thus, all workers together produce n units. The composite good can be consumed and used to pay for land and commuting. The CBD is connected to its suburb by a bridge of limited capacity. Commuting travel within the suburban area is costless, but crossing the bridge into the CBD costs t(ni) units of the composite good. This congestion cost increases with ni, the number of commuters, at an increasing rate, that is, t′(ni) N 0, t″(ni) N 0 The distribution of the population is endogenous with n ≥ n1 N0 resident-workers in city 1 and n2 = n − n1 ≥ 0 in city 2. If the aggregate population is not large enough, everyone will residework in the higher-amenity city 1. We assume that the aggregate population is large enough so that both cities are occupied: n N n1 N0, n2 = n − n1 N0. The utility function of a resident-worker is u(Ai,xi,hi) where xi is the numeraire composite good consumed and hi is the lot size in the suburb of city i rented for unit rent, Ri. The utility function is strictly increasing and strictly concave in each of the three Þ Þ arguments. A, x, h are each normal and A and h, and A and x are pair wise net substitutes, that is ∂hð A;R;u b0 and ∂xðA;R;u b0. h(A,R,u) ∂A ∂A and x(A,R,u) are the compensated demand functions for h and x respectively, and E(A,R,u) ≡ x(A,R,u) + Rh(A,R,u) is the minimum expenditure function. The aggregate area of the suburb in city i used for residences is denoted by Hi. One unit of land costs r units of the composite good to acquire for rental to city residents, where r is exogenous and can be a payment to farmers who forgo agricultural use of the land or, alternatively, it can be the cost of converting a unit amount of land from raw form to usable urban land. As explained in the Introduction, we will utilize precise definitions of economic and geographic sprawl. Economic sprawl is total commuting cost, ESu∑i¼1;2 ni t ðni Þ and geographic sprawl is the total urbanized land area GSu∑i¼1;2 Hi ¼ ∑i¼1;2 ni hðAi ; Ri ; uÞ. Although planners have focused on GS, it is ES which is more directly related to the cost of human interaction. We define an economic regime as an allocation and a supporting price/tax system, and common utility level {u,n1,R1,R2,H1,H2;τ1, τ2,s1,s2}. τi is the bridge toll paid (if positive) or the bridge subsidy received (if negative) by each resident of city i and, si is a tax (if positive) or a subsidy (if negative) on each land unit of city i. We define an equal-treatment equilibrium regime as one that satisfies the following conditions: n1 ðxðA1 ; R1 ; uÞ þ hðA1 ; R1 ; uÞr þ t ðn1 ÞÞ þ n2 ðxðA2 ; R2 ; uÞ þ hðA2 ; R2 ; uÞr þ t ðn2 ÞÞ−n ¼ 0

ð1Þ

ni hðAi ; Ri ; uÞ−Hi ¼ 0; i ¼ 1; 2; n2 ¼ n−n1 ;

ð2Þ

5 Heterogeneous city system may also emerge from market forces even on completely homogeneous initial locational conditions. One such force is the advantage of specialization (e.g. see Wilson, 1987). 6 Under the alternative and commonly used setup of replicable cities, there would be a system of identical cities. Then introducing congestion tolls or any other policy in each city would cause the number of cities to change and changes to occur within each city. In another study, we are examining the same urban sprawl issues in such a system of replicable cities.

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Ri ¼ r þ si ; i ¼ 1;2

ð3Þ

EðA1 ; R1 ; uÞ þ t ðn1 Þ þ τ 1 −ðEðA2 ; R2 ; uÞ þ t ðn2 Þ þ τ2 Þ ¼ 0;

ð4Þ

Eq. (1) is the market clearing condition of the composite good in the closed urban system, where n is the supply and the rest of the equation is the demand expressed as the sum of the real resources required to accommodate the total population in the two cities;7 Eq. (2) shows the market clearing conditions for land in the two cities; Eq. (3) shows the arbitrage conditions that say no after-tax economic profits accrue to investors renting land from farmers and re-leasing it to consumers; and Eq. (4) is the inter-city migration equilibrium. Eqs. (1) and (4) together are consistent with, EðAi ; Ri ; uÞ þ t ðni Þ þ τ i ¼ 1 þ

∑i¼1;2 ni τ i þ ∑i¼1;2 si Hi ; i ¼ 1;2; n

ð5Þ

the budget constraint of a worker-resident of city 1 and city 2 respectively. Eq. (5) says that each consumer receives his marginal product (equal to 1) as private income and an equal share of the planner's aggregate net tax revenue as public income. Note also that it can be verified from Eq. (4) that only τ1 − τ2 matters. In other words, given any two {τ1,τ2,s1,s2} with the same τ1 − τ2, solving Eqs. (1)–(4) generates the same {u,n1,R1,R2,H1,H2}. Given {τ1,τ2,s1,s2}, the solution procedure can be simplified to three sequential steps. In the first step, {R1,R2} is determined from Eq. (3) given {s1,s2}. In the second step, {u,n1} is determined by solving Eqs. (1) and (4) given {R1,R2}. In the third step, {u,n1,R1,R2} is used to find {H1,H2} from Eq. (2). In this article, we will study three basic regimes: 1) Laissez-faire: {u,n1,R1,R2,H1,H2} is fully determined by the markets and is found by solving Eqs. (1)–(4), given that τ1 = τ2 = s1 = s2 = 0. The planner plays no role, since there are no policy instruments and no revenue to be distributed. 2) First-best planning: the planner chooses the instruments {s1,s2,τ1,τ2} that maximize u subject to Eqs. (1)–(4). That is, all four policy instruments are available to the planner who decides which of them to apply and at what quantitative level. 3) Second-best planning: instruments τ1,τ2 are not available and the planner chooses only s1,s2 to maximize u subject to Eqs. (1)–(4) given that τ1 = τ2 = 0. Observe first that the two planning regimes represent alternative mixed economies where the consumers, who are ex-ante identical in preferences and initial endowment, take amenities, commuting conditions (congestion function), the (after-tax) rents R1,R2, congestion tolls τ1,τ2 and their income (right sides of Eq. (5)) as given. Then they maximize their utility by choosing in which city to reside-work (i.e. which amenity to enjoy and which congestion level to suffer), their lot sizes and their consumptions of the composite good. We will also discuss alternative implementation of the two planning regimes, where instead of using taxes/ subsidies as instruments, the planner directly determines {u,n1,R1,R2,H1,H2} in the first-best regime, and only H1,H2 to maximize u in the second-best regime, leaving {n1,R1,R2} to the markets. 2.2. The laissez-faire regime With τ1 = τ2 = s1 = s2 = 0, we find R1 = R2 = r by the first step of the solution procedure. By the second step, {u,n1} are solved from Eqs. (1) and (4) where R1 = R2 = r are used. We then get n2 = n −n1, and Hi = nih(Ai,r,u), i = 1,2, by the third step. It is useful to think about how an exogenous change in the amenities would affect {u,n1} in the laissez-faire regime. Suppose that A1 increased. A new equilibrium would be established by resident-workers moving from city 2 to city 1. The new equilibrium utility would be higher since in city 2 congestion decreases, but A2 and r are unchanged. Thus, du / dA1 N0, dn1 / dA1 N0. If we improve A2, du / dA2 N 0, dn1 / dA2 b 0.8 Proposition 1. In a laissez-faire equilibrium, (a) n1 N n2 and (b) h2 N h1 (c) x2 N x1. Proof. (a) Because utility is increasing in A, E(A,r,u) is decreasing in A and since A1 N A2 by assumption, it follows that E(A2,r,u) − E(A1,r,u) N 0 (Fig. 1). Therefore, from Eq. (4), t(n1) − t(n2) N 0 and since t′(ni) N 0, it follows that n1 N n / 2 N n2. (b) Since A1 N A2 and since, by assumption, A and h are net substitutes, it follows from h(A1,r,u), h(A2,r,u) that h2 N h1 (Fig. 1). (c) Similarly, since A and x are net substitutes, it follows from x(A1,r,u), x(A2,r,u) that x2 N x1 (Fig. 1).9 □ The equilibrium is depicted in Fig. 1, where the two indifference curves correspond to the same equilibrium utility level, u but the lowered curve belongs to the higher-amenity city. The third and exogenous good, the amenity Ai, is a shift variable in the indifference map drawn in (x,h)-space. The lower-amenity city entails a higher expenditure level compensating for the loweramenity, and the difference between the expenditures equals that between the commuting costs: E(A2,r,u) −E(A1,r,u) =t(n1) −t(n2) N 0.

7 In nominal resources, using abbreviated notation, aggregate receipts by all consumers, including tax revenues, equal aggregate expenditures: n þ ∑i¼1;2 ni τ i þ ∑i¼1;2 Hi si ¼ ∑i¼1;2 ni ðxi þ Ri hi þ ti þ τ i Þ. Using Eqs. (2) and (3) to substitute out Hi and si, and then canceling terms, this equation becomes Eq. (1). 8 The formal proof, in a supplementary appendix, is available from the authors. 9 We have also proved that the city that is smaller in population can have a larger land area than the city that is larger in population. The intuition is that lot size and the amenity are strong net substitutes. The formal proof is in the supplementary appendix, available from the authors.

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Fig. 1. Laissez-faire inter-city equilibrium with A1 N A2.

The right side is the difference between the costs of congestion, and the left the premium consumers pay to live in the higher-amenity city. The laissez-faire inter-city population distribution is inefficient, because a commuter pays only the average cost t(ni), rather than the marginal cost t(ni) + nit′(ni). The unpriced difference, nit′(ni), is the negative externality. The population is larger in city 1 (Proposition 1) and, therefore, so is the negative externality. The marginal social cost saving from relocating population from city 1 to 2 is therefore n1t′(n1) − n2t′(n2) N 0. 2.3. First-best planning regime (congestion tolls) In this case, the planner is free to choose the full instrument menu {s1,s2,τ1,τ2}. Relying on the sequential solution procedure, the planner chooses the set {s1,s2,τ1,τ2} that maximizes u subject (1), (3), and (4) and, thus, determines the first-best {u,n1,R1,R2}. Then, the solution for n2 = n − n1, and Ri are used to determine Hi by Eq. (2). The normalized Lagrangian of the planner's first step is: I ¼ u=μ−n1 ðxðA1 ; r þ s1 ; uÞ þ hðA1 ; r þ s1 ; uÞr þ t ðn1 ÞÞ−n2 ðxðA2 ; r þ s2 ; uÞ þ hðA2 ; r þ s2 ; uÞr þ t ðn2 ÞÞ þ n−θðEðA1 ; r þ s1 ; uÞ þ t ðn1 Þ þ τ1 −ðEðA2 ; r þ s2 ; uÞ þ t ðn2 Þ þ τ2 ÞÞ;

ð6Þ

where θ is the normalized shadow price of Eq. (4). The first-order conditions with respect to either τ1 or τ2 implies that θ = 0. Hence, the first-order conditions with respect to si, using θ = 0 are: si :

−ni ð∂xðAi ; Ri ; uÞ=∂Ri þ r∂hðAi ; Ri ; uÞ=∂Ri ÞjRi ¼rþsi ¼ 0;

i ¼ 1; 2:

ð7Þ

Recall that the derivative property of the expenditure function is, 10

ð∂xðAi ; Ri ; uÞ=∂Ri þ Ri ∂hðAi ; Ri ; uÞ=∂Ri ÞjRi ¼rþsi ¼ 0; i ¼ 1;2:

ð8Þ

Eqs. (7) and (8), together imply, as expected, that Ri ¼ rZsi ¼ 0;

i ¼ 1; 2:

ð9Þ

Eq. (9) says that the urban rent on land is its alternative cost in farming. Observe, however, that, in contrast to laissez-faire, where by definition, si = 0, under the first-best planning, this is derived. The first-order condition with respect to n1, after setting s1 = s2 = 0, becomes xðA1 ; r; uÞ þ hðA1 ; r; uÞr þ t ðn1 Þ þ n1 t 0ðn1 Þ ¼ xðA2 ; r; uÞ þ hðA2 ; r; uÞr þ t ðn2 Þ þ n2 t 0ðn2 Þ

ð10Þ

Comparing (4) and (10), it is implied that τuτ1 −τ2 ¼ n1 t 0ðn1 Þ−n2 t 0ðn2 Þ > 0:

ð11Þ

Therefore, the distortion in the laissez-faire can be corrected by levying per-capita congestion tolls τi = nit′(ni) in each city, or by a net toll (subsidy) τ = n1t′(n1) − n2t′(n2) N 0, levied in city 1 (city 2) only, or by any combination of tolls and subsidies, so that the sum of the toll in city 1 and the subsidy in city 2 add up to τ.

10

From Shephard's Lemma

∂E ∂R

∂x ∂h ¼ h ¼ ∂R þ R ∂R þ h Z ð8Þ.

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Proposition 2 characterizes the differences between the laissez-faire and the first-best planning regimes. Without causing ambiguity, we use abbreviated notation. Proposition 2. First-best planning improves the utility level of the laissez-faire allocation by shifting population from the high-amenity, more congested city to the lower-amenity, less congested city. This causes (a) the per-capita composite good and lot size to increase in each city: xifb N xilf, hifb N hilf; (b) the aggregate land area of the lower-amenity, lower-congestion city to increase: H2fb N H2lf; (c) the fb lf lf fb lf aggregate consumption of the composite good and the land area (geographic sprawl) to increase: ∑i nfb i xi > ∑i ni xi ; ∑i Hi > ∑i Hi ; (d) aggregate transportation cost (economic sprawl), ∑i ni t ðni Þ; to decrease and (e) average income to increase. Proof. (a) Recall that the laissez-faire allocation is a degenerate version of the first-best planner's optimization problem in which all four policy instruments are constrained to be zero. Therefore ulf ≤ ufb. By Proposition 1 n1lf N n2lf and as we saw, the negative externality is higher in city 1. Therefore, there is a utility increase under the first-best from shifting population to city 2. Hence, lf fb lf ufb N ulf, n1fb b n1lf, n2fb N n2lf. It follows that xfb i N xi and hi N hi for i = 1,2, because both goods are normal, the unit rent is r in both regimes and because ufb N ulf. (b) Since first-best planning increases both lot sizes and population in city 2, it unambiguously expands in land area, that is, H2fb N H2lf. (c) By part (a), both x and h increase for those not changing city. For those moving from city 1 to city 2, x and h increase for three reasons: (i) because they are normal goods and u has increased, (ii) because they are net substitutes with amenity and amenity has decreased, and (iii) by Proposition 1, x and h are initially higher in city 2. (d) It follows from the market clearing condition (1) that because aggregate land and aggregate composite good consumption have each increased by (c), then aggregate private transport costs must have decreased. (e) It follows from Eq. (5) that first-best per-capita income is given by E1 + t(n1) + τ = E2 + t(n2) = 1 + n1τ / n, higher than 1, the laissez-faire income, by τ N 0. □ Although in the setup of this section, intra-city effects of congestion do not exist because intra-city geographic heterogeneity has been suppressed, in Section 3 we will study conditions under which the inter-city effect, identified in Proposition 2(c) indeed increases geographic sprawl more than offsetting the intra-city effect which reduces geographic sprawl. Two comments are in order. First, observe that the effect of first-best planning on the land area of the larger city remains ambiguous in our analysis so far because, by Proposition 2, on the one hand, the per-capita demand for lot size has increased in city 1, while, on the other hand, the population has decreased. Second, observe that the policy instruments of the planner were the taxes. We can also pose the first-best as a direct planning problem where an omnipotent and benevolent planner directly determines all allocations, completely ignoring the markets. Such a planner maximizes u by choosing {n1,x1,x2,h1,h2,H1,H2,u} and subject to: n1 ðx1 þ h1 r þ t ðn1 ÞÞ þ n2 ðx2 þ h2 r þ t ðn2 ÞÞ−n ¼ 0;

ð12Þ

n1 h1 −H1 ¼ 0;

ð13Þ

n2 h2 −H2 ¼ 0;

u−uðAi ; xi ; hi Þ ¼ 0;

i ¼ 1; 2:

ð14Þ

Forming the corresponding Lagrangian and calculating the first-order conditions, we obtain a {n1⁎,x1⁎,x2⁎,h1⁎,h2⁎,H1⁎,H2⁎,u⁎} identical to the first-best values given by the mixed regime. Then, the planner cannot achieve a higher utility than can be reached by using the single policy tool τfb, of the mixed regime, leaving the allocations and prices to be determined by market forces. 2.4. Second-best planning regime (land tax/subsidy or UGB) In this case the planner can only use {s1,s2}, the unit taxes/subsidies on land in each city. This regime is less constrained than laissez-faire but more constrained than the first-best regime. It is a second-best policy as we will prove formally. The Lagrangian is same as Eq. (6) but with τ1 = τ2 = 0. The first-order conditions with respect to n1,s1,s2 in abbreviated notation are: n1 : −x1 −rh1 −t ðn1 Þ−n1 t 0ðn1 Þ þ x2 þ rh2 þ t ðn2 Þ þ n2 t 0ðn2 Þ−θðt 0ðn1 Þ þ t 0ðn2 ÞÞ ¼ 0;
s1 : −n1
s2 : −n2

 ∂x1 ∂h1 þr −θh1 ¼ 0; ∂R1 ∂R1 ∂x2 ∂h2 þr ∂R2 ∂R2

ð15Þ

ð16Þ

 þ θh2 ¼ 0:

ð17Þ

We set τ1 = τ2 = 0 and Ei = xi + Rihi in Eq. (4)), and then we add its left side to the left side of Eq. (15). The sum remains zero and after simplifying, becomes: h1 ðR1 −r Þ−h2 ðR2 −r Þ−ðn1 þ θÞt 0ðn1 Þ þ ðn2 −θÞt 0ðn2 Þ ¼ 0:

ð18Þ

A. Anas, D. Pines / Regional Science and Urban Economics 38 (2008) 408–423

415

Since by Eq. (8), ∂xi / ∂Ri + Ri∂hi / ∂Ri = 0 we can multiply it by ni and add the result to Eq. (16) for i = 1 and to Eq. (17) for i = 2 respectively without violating their equalities. Rearranging the results and using the arbitrage conditions, Eq. (3), we get: s1 ¼ R1 −r ¼

θh1 ; n1 ð∂h1 =∂R1 Þ

s2 ¼ R2 −r ¼ −

ð19Þ

θh2 : n2 ð∂h2 =∂R2 Þ

ð20Þ

Note that Eqs. (18)–(20) constitute three equations in s1,s2 and θ. Solving these for θ, we obtain: θ¼

n1 t 0ðn1 Þ−n2 t 0ðn2 Þ h21 =n1 ∂h1 =∂R1

h2 =n

þ ∂h2 =∂R2 −ðt 0ðn1 Þ þ t 0ðn2 ÞÞ 2

b0:

ð21Þ

2

The sign of θ follows because the numerator is positive since n1 N n2 and t″(ni) N 0, and the denominator is negative because ∂hi / ∂Ri b 0. Then, from Eqs. (19) and (20), s1 = R1 − r N 0 and s2 = R2 − r b 0. The intuition for this result is straightforward. Under laissezfaire, n1 N n2 and the externality generated by a commuter is larger in city 1, as we saw. Hence, the second-best planner has to mimic the action of the first-best planner and induce population to migrate from the larger to the smaller city. In the case of firstbest this was done by tolling the residents of the high-amenity city. This useful tool is not available under second-best planning. In this case, inducing migration from the high to the low-amenity city is done by the inferior instruments available to the planner: taxing land in the larger city and subsidizing it in the smaller city. The effective laissez-faire unit rent on land is raised in the larger and lowered in the smaller one. Note an important difference between the first-best tolls policy (τ1,τ2) and the second-best tax/subsidy policy (s1,s2). Whereas, in the case of the former only one (differential) toll τ1 = τ or τ2 = −τ is required to realize the first-best policy, in the case of the second-best policy, both instruments, s1 N0, s2 b 0 must be used. The intuition is that the tax or the subsidy are each designed to induce migration from the high to the low-amenity city and thus save resources that are otherwise used up in excessive transportation. Achieving these savings, however, is associated with a dead-weight loss, because of the deviation from first-best marginal cost pricing. The cost of the deviation is minimized by using a relatively low tax on the larger and a relatively low subsidy on the smaller city, rather than using a high level of only one of them in only one city. The dead-weight loss is minimized by a version of the rule of Ramsey (1927), which is verified by dividing Eqs. (19) by (20) and multiplying the result across, by R2 / R1, getting s1 =R1 ðn2 =nÞη2 ¼− ; s2 =R2 ðn1 =nÞη1

ð22Þ

where ηi = (∂hi / ∂Ri) / (hi / Ri) is the own-price elasticity of compensated demand.11 Eq.(22) says that the ratio of the land tax/subsidy rates in the two cities, is proportional to the inverse of the ratio of the compensated demand elasticities, weighted by the share of population residing in each city. When the compensated elasticity is constant (e.g., Cobb Douglas utility), s1 / R1 b |s2 / R2| since n1 N n2. For s1 / R1 N |s2 / R2| to be possible, |η1| must be sufficiently smaller than |η2|. The discussion extends to cases where land taxation can be applied in only one of the two cities. Then, one of these policies, using either s1 N0 or s2 b 0 that yields higher utility is a third-best, and the other a fourth-best. The effects of these second- and lower-best policies are as follows. Proposition 3. (1) Optimally taxing land in the high-amenity city and, simultaneously, subsidizing land in the low-amenity city, or (2) optimally taxing only the land in the high-amenity city, or (3) optimally subsidizing only the land in the low-amenity city, have the following effects relative to laissez-faire: (a) utility is increased, (b) population is reallocated from the high to the low-amenity city, (c) composite good consumption by the consumer in the high-amenity city is increased, (d) lot size in the low-amenity city is increased, (e) aggregate land area in the low-amenity city is increased, and (f) if the elasticity of substitution between the composite good and housing (σ) is zero, lot size and composite good consumption per consumer in each city is increased, increasing the aggregate area of both cities together (geographic sprawl), and the aggregate composite good consumption, but decreasing the aggregate transport cost (economic sprawl). Proof. (a) It follows from Eqs. (19)–(21) that s1 N0, s2 b 0. This implies that the constraints of laissez-faire, s1 = 0, s2 = 0 are binding. This proves (a) for case 1. In case 2, Eq. (20) and the second term of the denominator of Eq. (21) disappear. Then, Eq. (19) and the 11 As in the Ramsey rule, only the own price elasticity with respect to land appears in Eq. (22). This is justified in our case because the price of land in one city does not directly affect the utility of a consumer who resides in the other, and the supply of land in one city is independent of the supply in the other. Were there two goods in addition to land, say x and z where each is produced by labor, we would obtain the more general version of the Ramsey rule (e.g. Atkinson and Stiglitz, 1980) in which cross-elasticities appear. Eq. (22), however, is different from the original version of the Ramsey rule in that, although in our case all consumers are ex-ante identical, ex-post they have different utility functions according to their city of residence , since they can consume land only in their city of residence. That is why in Eq. (22) the deadweight loss resulting from the deviation from first-best marginal cost pricing is weighted by the relative population size of each city.

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modified Eq. (21) imply that s1 is positive. Hence, the laissez-faire constraint s1 = 0 is binding. This proves (a) for case 2. Symmetrically, it can be shown that, in case 3 s2 is negative, implying that the laissez-faire constraint s2 = 0 is binding, which proves (a) for case 3. (b) The proof is in the Appendix A. (c) Since utility increases by (a), there is a positive income effect in both cities because the composite good is a normal good. Because the after-tax cost of land rises in the larger city, the substitution effect works in the same direction as the income effect in the case of the composite good. Hence, per-capita composite good consumption increases in the larger city. (d) Each consumer in the smaller city increases lot size because utility has increased creating a positive income effect since lot size is normal, while the after-tax price of land has decreased creating a non-negative substitution effect. (e) Aggregate land consumption in the smaller city increases because both its lot size and its population have increased as proved under (d) and (b) respectively. (f) Consider the case of σ = 0, that is u = f(Ai) min{axi,bhi}, a,b N 0, f′(Ai) N 0. xi = u / af(Ai), hi = u / bf(Ai). Then, total composite good and total land consumption increase by the positive utility effect (usb N ulf) since (a) remains valid, and by normality and because population migrates from the larger city with initially low lot size and low per-capita composite good consumption to the smaller city with initially larger lot size and larger per-capita composite good consumption. Since aggregate output is unchanged (equals n), it follows from Eq. (1) that aggregate travel cost decreases. □ It can be conjectured that part (f) of Proposition 3 extends to cases where the elasticity of substitution between the composite good and lot size for any amenity level is not zero but sufficiently small, provided the relevant functions of the elasticity are continuous. The validity of this conjecture is illustrated by our simulations in Section 3, where the utility function is C.E.S. The second- and lower-bests regimes can also be implemented by directly controlling H1 and H2 (for the second-best) or only one of them (for the third- or fourth-bests) as the policy instruments, letting the markets determine {u,n1,R1,R2}, and confiscating and equally redistributing as public income, the total differential land rent, ∑i = 1,2(Ri − r)Hi. Such a policy is known as the urban growth boundary (UGB) policy and, as explained in the Introduction, the extant literature has examined this policy only for a single city.12 Proposition 4. A second-best optimal allocation {usb,n1sb,R1sb,R2sb} can be implemented either by choice of the policy instruments {s1 N0, s2 b 0}, or by direct choice of {H1,H2}. The lower-best optimal solutions can be implemented either by choice of {s1 N0, s2 = 0} or {s1 = 0, s2 b 0}, or by direct choice of {H1} or {H2} respectively. Proof. A second-best regime can be determined by choosing {s1,s2} to maximize u subject to the constraints Eqs. (1)–(4). Let SB ≡ {usb,n1sb,R1sb,R2sb,H1sb,H1sb} be such an optimal solution. Now consider a regime, where {H1,H2} is chosen to maximize u subject to Eqs. (1)–(4). Denote this by UGB ≡ {uugb,n1ugb,R1ugb,R2ugb,H1ugb,H1ugb}. Since SB is a feasible solution under the UGB regime and UGB is feasible under SB regime, it follows that neither SB nor UGB can do better than the other regime. Then, uugb = usb. □ 3. Inter- and intra-city allocation effects 3.1. Discussion In Section 2, we proved that the inter-city effect of executing first-best planning is always expansive on the aggregate urban land use, and conjectured that the inter-city effect of the land tax/subsidy or UGB policy may be expansive depending on the magnitude of σ. This in itself improves on the prevailing view (e.g. Brueckner, 2000; Bento et al., 2006), based only on intra-city effects, as derived over several decades from repeated analyses of a single monocentric city. In order to derive our results about inter-city allocation effects in Section 2, we assumed away geographic heterogeneity within cities. In this section we extend the simple setup of Section 2 to allow a simultaneous exploration of both the intra-city and the intercity effects of the first-best congestion tolling and the UGB policies by relaxing the intra-city spatial homogeneity assumption. Accordingly, the landless CBD still contains all the jobs which do not require land, but is now surrounded by a residential area defined as a core, connected to the suburban residential area by congested bridges. Costs of travel within the core or the suburb are zero and all travel costs are incurred at the congested bridges which suburban residents must cross to reach the core and their jobs in the CBD. The number of residents of a district is denoted by nij, where j denotes the district (j = 1 for core and j = 2 for suburb) while i = 1,2 denotes the city. The same city-wide amenity is enjoyed whether one resides in the core or in the suburb of the same city. Suppose that the land area of each core is an exogenous H0, so that the aggregate land size of city i is H0 + Hi, and total urbanized area (geographic sprawl) is then 2H0 + H1 + H2. Land to each suburban area is perfectly elastically supplied at rent r, as in Section 2, while the total population, n, is large enough so that both cores are completely occupied and at least some reside in each suburb. Then, the laissez-faire equilibrium rent on land in city i's core is Ri1 N r, while in the suburbs Ri2 = r. By residing in the core the congestion on the bridge is completely avoided, and the core's rent premium Ri1 − r N 0 prevents relocation between the core 12 In a policy-oriented article on sprawl, Brueckner (2000) extrapolated the finding of standard single-city, monocentric analysis and claimed broad generality for the result that the UGB is a second-best policy tool for reducing geographic sprawl. As explained in the Introduction, however, the UGB often fails to be a second-best policy when jobs are not pinned in the CBD and can move to the suburbs (Anas and Rhee, 2007), or when employment is dispersed and intermixed with population (Anas and Rhee, 2006) or when there are more than one monocentric cities (this article). More recently, Bento et al. (2006) simulated some of the effects of realistic policies such as the UGB, the development tax, and the gasoline tax in a single monocentric city, without presenting analytical proofs. Their context additionally differs from ours in several respects but most importantly in that: (1) their city is open to population and the level of utility is fixed; (2) efficiency is measured by the increase in aggregate land values which are not redistributed but accrue to absentee landlords; (3) the planner redistributes development tax revenues among the absentee landowners and the farmers, but gasoline tax revenues are redistributed among the consumers, while under the UGB policy, there is no taxation of land rents and, therefore, aggregate rent changes accrue to the absentee landlords.

A. Anas, D. Pines / Regional Science and Urban Economics 38 (2008) 408–423

417

and the suburb of the same city in equilibrium. Aggregate population is assumed large enough so that both suburbs are occupied: ni1 b ni b n; i = 1,2. A regime is defined by {u,n11,n12,n21,n22,R11,R12,R21,R22,H1,H2} subject to the policy instruments {τ1,τ2,s1,s2,T}, where T is a head tax/subsidy to the residents of the core of city 2. The policy instrument, T, did not appear in Section 2 because in that case there were no cores. The differential congestion toll τ ≡ τ1 − τ2 was, in fact, equivalent to a head tax on the residents of city 1 or a head subsidy on those of city 2. In the present model, cores are present and thus an additional relocation margin exists by residents moving from the core of city 1 to the core of city 2. Therefore, we introduce T as an additional policy instrument that controls migration at this margin. Any equilibrium regime satisfies: ∑ i¼1;2

!      ∑ nij x Ai ; Rij ; u þ rh Ai ; Rij ; u þ ni2 t ðni2 Þ −n ¼ 0;

ð23Þ

j¼1;2

ni1 hðAi ; Ri1 ; uÞ−H0 ¼ 0; i ¼ 1; 2;

ð24aÞ

ni2 hðAi ; Ri2 ; uÞ−Hi ¼ 0;

ð24bÞ

Ri2 ¼ r þ si ;

i ¼ 1; 2;

i ¼ 1; 2;

ð25Þ

EðA1 ; R11 ; uÞ−EðA2 ; R21 ; uÞ−T ¼ 0;

EðAi ; Ri1 ; uÞ−ðEðAi ; Ri2 ; uÞ þ t ðni2 Þ þ τi Þ ¼ 0;

n11 þ n12 þ n21 þ n22 −n ¼ 0:

ð26Þ

i ¼ 1; 2;

ð27Þ

ð28Þ

Eqs. (23)–(25) correspond to Eqs. (1)–(3). Eq. (26) replaces Eq. (4) as the inter-city migration equilibrium; and Eq. (27) is the intra-city location equilibrium; Eq. (28) is the conservation of population. The laissez-faire equilibrium is obtained when we solve Eqs. (23)–(28) after imposing the restrictions τi = si = T = 0 for i = 1,2 and solving for the eleven unknowns. The first-best is obtained when {τ1,τ2,s1,s2,T} and the eleven unknowns are chosen to maximize u fb fb subject to Eqs. (23)–(28). It is easy to show that this optimization implies sifb = T fb = 0; Ri2 = r and τifb = ni2 t′(ni2fb) for i = 1,2. The second-best is obtained when u is maximized subject to Eqs. (23)–(28) and, in addition, τi = 0 is imposed for i = 1,2. To make things a bit simpler, however, in this section, we will limit our attention to a lower-best regime where the second-best is further constrained by T = 0, that is the gross income is constrained not to vary across cities. We refer to this constrained optimum regime as the UGB regime. This UGB-constrained optimum regime is found by maximizing u subject Eqs. (23)–(28) and the additional constraints T = τ1 = τ2 = 0. This can be done by solving for the eleven unknowns given any s1,s2 and then searching over all admissible values of s1,s2 until u is maximized. The Ramsey rule, Eq. (22), where subscript 1 is now replaced by 12 and the subscript 2 by 22, holds again as it did in the case of cities without cores (Section 2). Recalling that σ ≥ 0 denotes the elasticity of substitution between the composite good and lot size, the following proposition summarizes our results with respect to ES (economic sprawl) and GS (geographic sprawl). Proposition 5. (a) The UGB is always restrictive in the larger (high-amenity) city, that is, s1 N0. (b) The UGB can be either restrictive (s2 N 0) or expansive (s2 b 0) in the smaller (low-amenity) city, depending on the variation of the amenity effect on utility between the two cities13 and on σ. In particular, (i-1) Suppose that σ N 0. If A1 = A2 or, equivalently, if A1 N A2, and consumers do not care about the amenity, then the two cities are identical and the FB and UGB policies in both cities are restrictive (s1 = s2 N 0). In this case both the FB and the UGB always generate lower GS and ES than does the LF, consistent with the findings of the conventional analysis based on a single monocentric city. (i-2) If σ N 0, A1 N A2, and the residents care about the amenity, then the cities are no longer identical with the lower-amenity city being smaller. Then, the UGB policy in the smaller city can be expansive (s2 b 0). In this case, the effect of the expansive UGB on the small (low-amenity) city can more than offset the effect of the restrictive UGB on the large (high-amenity) city such that GS under UGB and FB can be larger than under LF, just as we had found in the simpler model of Section 2 which had no cores. (ii-1) The result of (i-2) also holds when σ = 0, A1 N A2, and the residents care about the amenity. Then, the FB and UGB policies are identical, and under either UGB or FB, ES is smaller whereas GS is larger than under LF. (ii-2) Finally, if σ = 0 and A1 = A2, then the two cities are symmetric and the two policies are again identical but cannot improve on the LF. 13 The variation of the amenity effect on utility between the cities depends on both the difference A1 − A2 ≥ 0 between the cities and on the importance of the amenity's effect on utility, or the marginal utility of the amenity.

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Table 1 Effects of the policy regimes under various σ σ

Regime

u (utils)

H1 (mi2)

H2 (mi2)

GS (mi2)

ES ($/year/capita)

R11 ($/ac)

R21 ($/ac)

R12 ($/ac)

R22 ($/ac)

1/9

LF TOLLS UGB LF TOLLS UGB LF TOLLS UGB LF TOLLS UGB LF TOLLS UGB

61.631 61.754 61.752 88.236 88.417 88.407 107.247 107.502 107.460 77.860 78.199 78.061 67.051 67.376 67.259

11,253 10,521 10,520 9562 8959 9022 5989 5576 5546 1675 1470 1425 1065 886 846

9529 10,331 10,331 8084 8778 8704 4956 5411 5431 1163 1314 1315 620 728 728

22,000 22,111 22,110 18,904 18,995 18,984 12,204 12,245 12,236 4096 4043 3999 2944 2872 2832

8557 8394 8397 8364 8193 8213 7621 7409 7446 4135 3613 3848 2725 2168 2357

27,678 36,203 27,864 28,868 38,726 29,205 33,304 48,103 34,069 50,719 81,407 57,088 55,372 86,033 65,262

25,019 32,785 25,190 25,753 34,719 26,060 28,358 41,739 29,049 34,996 61,957 40,552 33,535 59,949 41,983

20,000 20,000 20,824 20000 20000 20,999 20,000 20,000 21,936 20,000 20,000 29,721 20,000 20,000 34,910

20,000 20,000 19,641 20,000 20,000 19,758 20,000 20,000 19,689 20,000 20,000 22,372 20,000 20,000 24,246

1/8

1/7

1/6

1/5.8

α = 0.5; a = 1 × 1013; b = 1 × 10− 13; A1 = 1.20372; A2 = 1.0; γ = 1.87 × 10− 5; δ = 1.30. y = $40,000; n = 10,000,000; r = $20,000/ac; H0 = 629.22 mi2. LF: laissez-faire; TOLLS: first-best optimum; UGB: lower-best (urban growth boundaries). The numbers in bold in the last two columns are the suburban rents under the UGB regime. In the first three cases, they indicate that the UGB is restrictive in city 1 and expansive in city 2. In the last two cases, it is restrictive in both cities.

The formal proofs of Proposition 5 are in the supplementary appendix, available from the authors. The truth of the statements, however, can be conjectured from intuitive underpinnings. (i-1), of (b) follows from the monocentric literature. Since A1 = A2, by symmetry, the two cities are identical under each regime. Then, only intra-city effects exist because under the symmetry, efficiency can be improved only by reallocating consumers from suburbs to cores. Therefore, both ES and GS are lower under UGB and FB, than under LF.14 Under (i-2), when the two cities are considerably asymmetric so are the population distributions and the distortive externalities, ni2t′(ni2). Then, redistributing the population in favor of the smaller city reduces the distortion in the resource allocation. Hence, subsidizing the suburban land in the smaller city's suburb (or installing an expansive UGB) while taxing it in the larger city's suburb (or installing a restrictive UGB), induces the desired population redistribution at the lowest cost associated with the land rent's deviation from the opportunity cost (according to the Ramsey pricing rule). Under (ii-1) and (ii-2) of (b), since σ = 0, there are no substitution effects (consumers are insensitive to rents). As shown in the supplementary appendix, the allocations of FB and UGB are the same and so is the utility level of FB and UGB. Because in each of these two regimes utility increases from laissez-faire, the normality of lot size causes lot size to increase for each consumer who either stays in the core or moves from the core to the suburb of his own city. Furthermore, we show in the supplementary appendix that the larger city loses population, implying that, indeed, under FB and UGB, each resident occupies a larger lot. By the same reasoning, aggregate composite commodity demand is also higher than under LF, while output, n, is unchanged. Thus, from Eq. (23) ES must decrease. This decrease is achieved by the reduction in the excessively asymmetric population distribution which prevails under laissez-faire. Intuitive conjectures allow us to extend the domains over which Proposition 5 applies. First, given σ N 0, (i-1) is also valid for sufficiently small deviation of A1 from A2, provided the effect of the above two regimes on GS and ES is continuous in A1. Likewise, provided the effects of the FB and UGB regimes on GS and ES are continuous in σ, (ii-1) is valid not only for σ = 0, but also for sufficiently small σ. Furthermore, for sufficiently large σ, the effect of the two regimes on GS under (ii-1) is likely to change sign because, with a larger σ, it becomes easier to accommodate more people in the cores with the resources released, by transferring people from the suburbs to the cores. Thus, the intra-city effect of the above two regimes becomes more pronounced than otherwise relative to their inter-city effect. We, therefore, conjecture that, when A1 N A2, a sufficiently large σ results in GS under both FB and UGB to be smaller than under LF. Regarding the deviation of A1 from A2 under 5(a), keeping σ N 0, the larger is the deviation of A1 from A2, the more is the laissezfaire population skewed in favor of city 1, the larger is the gap between the private and the social commuting costs, and the larger is the benefit from transferring people from city 1 to city 2. Consequently, the larger is the deviation of A1 from A2, the more pronounced is the inter-city effect of the above two regimes relative to the intra-city effect and the sign of the combination of these two effects on GS may change. We now turn to the simulations results that fully demonstrate the validity of all of the above proved results and conjectures. We will limit ourselves to laissez-faire, the first-best regime of congestion tolling and the lower-best regime of two UGBs, one in each city, ignoring the second-best regime and thus setting T = 0 in Eq. (26), as already explained above.

14 Furthermore, the result holds for any exogenously given number of symmetric monocentric cities with cores, among which the given population must be allocated.

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419

Fig. 2. Effect of σ on excess geographic and economic sprawl.

3.2. Simulations  σ−1 σ σ−1 σ −1   In the simulations we used the C.E.S. utility u Ai ; xij ; hij ¼ Aαi axijσ þ bhijσ , α ≥ 0, a,b N 0 and 0 b σ b ∞. α, the elasticity of utility with respect to the amenity, measures the importance of the amenity in consumer preferences. As σ → 0, u(Ai,xij,hij) =Aiα min(αxij,bhij) and xij,hij become perfect complements. In this section, we let y denote the marginal product of labor (up to here it was set to one).     Then, V Ai ; Rij ; mij ¼ Aαi aσ þ bσ R1−σ mij is the indirect utility, where m11 =m21 =y + (ADR+ζ × TOLLS) /n, mi2 =mi1 −t(ni2) −ζni2t′(ni2); i = 1,2 ij and ζ = (0,1). ADRu∑i¼1;2 ½H0 ðRi1 −rÞ þ Hi ðRi2 −r Þ is the aggregate differential rent and toll revenue is TOLLSu∑i¼1;2 n2i2 t 0ðni2 Þ. The congestion function is t(ni2) =γ(ni2)δ, γ N 0, δ N 1. When ζ = 1 optimal tolls are used (the first-best regime) and when ζ = 0 there are no tolls (the laissez-faire or UGB regimes). As shown by the mij expressions, ζ × TOLLS plus ADR are equally distributed among all residents. We present three sets of numerical simulation results.15 The first set of results explore the effect of varying σ, to validate Proposition 5(b) (ii), which holds for σ = 0, and its intuitive extension to positive values of σ, keeping all other parameters constant at the values shown in the bottom of Table 1. Note, in particular, that A1 N A2and α = 0.5 so that the amenity differential matters to the consumer. A higher σ makes consumers more sensitive to rents. Then, under laissez-faire, consumers more strongly substitute composite good for lot size, and are thus more willing to avoid congestion tolls by moving to the higher-rent cores than to the suburb of the smaller city. This reflects the intracity effect of a higher σ. The inter-city effect of a higher σ is that consumers are, for the same reason, more willing to move to the larger and higher-rent city where they enjoy the higher amenity. A higher σ, therefore, increases concentration in the larger city and decreases geographic sprawl (GS). An extreme reflection of this in Fig. 2, is that when σ is somewhere between 1/6 and 1/5, the suburban area of the smaller city is unoccupied (and at higher values of σ, the smaller city and/or the suburb of the larger city would also become unoccupied). At the other extreme, as σ decreases toward zero (reaching 1/9 in Fig. 2), consumers become insensitive to rents and only income effects are present. Then, policies are relatively ineffective in inducing significant 1 σ −1

15

We wrote the codes in Maple. These are available from the authors upon request.

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changes in behavior. Indeed, in Fig. 2, for values of σ b 1/9, there is no discernible difference between the first-best optimum and the lower-best optimal UGB allocation. Fig. 2, where excess sprawl is measured as laissez-faire GS minus optimal (or UGB) GS shows that for low enough values of σ there is insufficient geographic sprawl under laissez-faire (excess sprawl is negative). In such cases, the inter-city effects of congestion tolls are dominant and economic sprawl is reduced most efficiently by reallocating population to the smaller city, much as in the simpler model of Section 2 where the intra-city effects were assumed away. The UGB policy requires setting an expansive boundary in the smaller city (R22 b r) and a restrictive one in the larger city (R12 N r). For large enough values of σ, laissez-faire geographic sprawl is excessive and the intra-city effects of tolls are dominant. In such cases, economic sprawl is most efficiently reduced by reallocating population to the cores. In such cases of the UGB policy with large enough σ, both boundaries must be restrictive (R12 N r, R22 N r). Table 1 shows the detailed results corresponding to points in Fig. 2. A second set of results are presented in Fig. 3. These explore cases where σ is kept constant but the importance of the amenity in the utility function measured by the parameter α ≥ 0 on the horizontal axis is varied, while all other parameters are constant at values shown in the bottom of the table. Note that varying α ≥ 0 is perfectly equivalent to keeping A2 = 1 constant and varying A1 N A2 = 1. In Fig. 3, we plot not the excess but the absolute levels of economic or geographic sprawl in the vertical axes. When α = 0, the two monocentric cities are identical, since consumers do not care about any amenity differential. Then, by symmetry, inter-city allocation effects cancel each other and by the intra-city effects, the reduction of economic sprawl requires inducing consumers to move from the suburbs to the cores of their cities. In the optimum this is achieved by tolling and in the case of the UGB policy by placing restrictive UGBs around both cities. Hence, both regimes reduce the excess economic and geographic sprawl that exists under laissez-faire. As α is gradually increased from zero, however, the amenity becomes increasingly important in preferences and more consumers locate in the higher-amenity larger city under laissez-faire, increasing the aggregate congestion externality and the difference between the congestion levels of the two cities. This makes the core rents in the larger city increasingly higher than the core rents in the smaller one. Congestion tolls reallocate population not only from the suburbs to the cores (the intra-city effect), which was true at α = 0, but also from the larger to the smaller city (the inter-city effect). Somewhere around α ≈ 0.9, the inter-city expansive effect of tolls on GS exactly cancels the intra-city contractive effect. We can see that for all α N 0.9, the optimal policy increases geographic sprawl as it decreases economic sprawl

Fig. 3. Effects of α on geographic and economic sprawl.

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Table 2 Laissez-faire, optimum and UGB under low and high congestion δ = 1.30: tolls, UGB are expansive on laissez-faire geographic sprawl

δ = 1.45: tolls, UGB are contractive on laissez-faire geographic sprawl

Laissez-faire (LF)

First-best optimum (FB) (tolls)

Constrained lower-best (UGB)

Laissez-faire (LF)

First-best optimum (FB) (tolls)

Constrained lower-best (UGB)

u (utils) Income ($/year) Average differential rent R11 ($/ac) R21($/ac) R12 ($/ac) R22 ($/ac) t(n12) t(n22) Average externality  2 0 n12 t ðn12 Þ þ n222 t 0ðn22 Þ =n

82.6819 $42,094 $2094 $65,193 $26,810 $20,000 $20,000 $10,000 $2020 $7122

83.9728 $49,458 $4356 $97613 $50568 $20,000 $20,000 $7332 $3808 $5102

83.7526 $43,847 $3847 $70924 $30939 $35,838 $18,009 $7607 $3866 $5398

58.2628 $55,719 $15,719 $257,348 $173,009 $20,000 $20,000 $33,104 $27,480 $19,819

58.9172 $94,867 $37,348 $558,266 $409,184 $20,000 $20,000 $29,387 $27,066 $17,519

58.3930 $62,170 $22,170 $306,245 $211,128 $75134 $47,430 $30,989 $28,374 $19,075

Average toll n11 n12 n21 n22 n11 + n12 n21 + n22 H1 (mi2)

0 1,931,231 5,173,193 1,384,054 1,511,522 7,104,424 2,895,576 1962 716

Geographic sprawl 2H0 + H1 + H2 Economic sprawl

3936

0 1,924,524 4,191,883 1,392,867 2,490,726 6,116,406 3,883,594 1488 (−24.2%) 1213 (+69.4%) 3959 (+0.6%) $4152 (−24.2%)

0 3,101,747 2,386,565 2,412,678 2,099,009 5,488,312 4,511,688 637

H2 (mi2)

$5102 1,985,067 4,074,664 1,478,444 2,461,824 6,059,731 3,940,269 1569 (−20.0%) 1183 (+65.2%) 4011 (+1.9%) $3925 (−28.3%)

$17,519 3,199,623 2,198,361 2,524,917 2,077,099 5,397,984 4,602,016 594 (−6.8%) 701 (+0.1%) 2553 (−1.7%) $12,082 (−11.6%)

0 3,129,403 2,280,362 2,444,420 2,145,815 5,409,765 4,590,235 516 (−19.0%) 640 (−8.6%) 2415 (−7.0%) $13,155 (−3.8%)

½n12 t ðn12 Þ þ ðn22 Þt ðn22 Þ=n

$5478

700 2596 $13,669

α = 1.2; a = 1 × 1013; b = 1 × 10− 13; σ = 1/6; A1 = 1.20372; A2 = 1.0; γ = 1.87 × 10− 5; y = $40,000; n = 10,000,000; r = $20,000/ac; H0 = 629.22 mi2.

relative to laissez-faire, because the inter-city effect dominates. When α = 2.0, optimal allocation requires geographic sprawl to be 11.2% larger than under laissez-faire, while economic sprawl becomes 47% lower than under laissez-faire. Thus, to save 47% in congestion costs an 11.2% increase in aggregate land used is required. In the optimum, as the dominant inter-city effects of tolls shift population to the smaller city where congestion and core rents are lower, substitution and income effects (higher utility) cause larger lots to be rented in the smaller city. Meanwhile, lot sizes in the larger city also expand since utility is increased and core rents are reduced, but since population is also reduced the effect on aggregate land use in the larger city appears ambiguous. However, in all the simulations, we found that the larger city is reduced in land area relative to the laissez-faire. The corresponding optimal UGB policy when α N 0.9, requires that the UGB in the large city be restrictive on land use while the one in the smaller city be expansive (R12 N r, R22 b r), which is what we determined in Section 2 where intra-city effects were nonexistent by assumption. The third set of results is presented in Table 2. This table presents the three regimes for two cases only. In both cases, all parameters are at values shown in bottom of the table (α = 1.2 and the amenity differential matters and σ = 1/6), except that the exponent of the congestion function δ is increased from 1.3 to 1.45 in one step. In this case, Table 2 provides a lot of detail about each regime so that interested readers may (on their own) study in detail how the regimes change the population allocations, the land use, the congestion costs, the externality, the rents and the composition and level of the public part of per-capita income. At δ = 1.3 congestion is relatively low and aggregate geographic sprawl, 2H0 + H1 + H2, increases by 1.9% when optimal tolls are levied and by 0.6% under the constrained optimal UGB policy which requires a restrictive (expansive) boundary around the larger (smaller) city. When δ = 1.45 congestion is significantly higher, and the aggregate geographic sprawl decreases by 1.65% when optimal tolls are levied and by 7% by placing a restrictive UGB around each city. The intuition for the much more restrictive aggregate land use under UGB (i.e. 7% instead of 0.6%) when congestion is higher is that the two UGBs reduce congestion by acting in the inefficient margin (i.e. the land market). The higher the congestion is, the less effective it becomes to reduce it by working in the inefficient margin. Thus, a much more restrictive UGB policy is required when congestion is higher. The results also demonstrate how both policies reduce economic sprawl regardless of δ and how the tolls policy is much more effective in reducing economic sprawl than is the UGB policy, especially when congestion is high. Although varying δ was not part of Proposition 5, the intuition behind it is that under higher congestion, laissez-faire population is reallocated more to city 2 and the two cities are closer to being symmetric. Therefore, just as in Proposition 5 and its extension by continuity, sufficient symmetry of the two cities, on account of higher congestion, again causes the two policies to be contractive on GS.

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4. Concluding remarks We focused only on the negative externality of traffic congestion, intentionally ignoring positive externalities from production. If such a positive externality existed, the positive gap between marginal and average productivity does not necessarily increase monotonically with city population. This is in contrast to congestion, where the negative externality, ni2t′(ni2), strictly increases with population in each city as does the total externality, ∑i¼1;2 n2i2 t 0ðni2 Þ;. Therefore, extension of our models to the interplay between positive agglomeration economies and the negative congestion externality need not reverse the results obtained in this article. It is remarkable that in a system of monocentric cities which vary in population, geographic and economic sprawl are not necessarily positively correlated. We showed that planners who seek to eliminate excess economic sprawl can do so either by tolling congestion or, in a second-best sense, by taxing or by applying equivalent UGB policies. To achieve a lower measure of economic sprawl, we showed that planners must often accept a higher level of geographic sprawl, and should implement diametrically opposed zoning policies in large versus small cities. Thus, it would be more effective to pursue land use policy as national policy rather than as local or regional policy. In the United States, land use policy falls within the jurisdictions of the states, and it is usually relegated down the hierarchy to the level of local jurisdictions. Some states such as Oregon have exercised it at the state level and others such as Maryland and New Jersey have begun to do so. If local planners are responsible for controlling geographic sprawl in their areas, their anti-sprawl policies which are not coordinated with other local planners will end up shuffling population among cities and very possibly inefficiently increasing overall sprawl. The most dramatic consequence might be that planners acting independently might exercise restrictive land use policies in large cities, neglecting expansive land use policies in smaller cities. Appendix A. Proof of Proposition 3(b) We want to prove that n1sb b n1lf. Suppose to the contrary that n1sb ≥ n1lf. We know from Eq. (5), that the per-capita income in the second-best regime is msb u1 þ

    nsb   nsb sb sb 1 2 Rsb Rsb h A1 ; Rsb h A2 ; Rsb 1 ;u 1 −r þ 2 ;u 2 −r : n n

ðA1Þ

Therefore, at the second-best optimum, the budget constraint in city 1 is       sb sb sb þ h A1 ; Rsb Rsb ¼ msb : x A1 ; Rsb 1 ;u 1 ;u 1 þ t n1

ðA2Þ

Adding and subtracting h(A1,R1sb,usb)r from the left side of Eq. (A2), it becomes,          sb sb sb sb þ h A1 ; Rsb r þ h A1 ; Rsb Rsb ¼ msb x A1 ; Rsb 1 ;u 1 ;u 1 ;u 1 −r þ t n1         sb  sb sb sb x A1 ; Rsb þh A1 ; Rsb r þ t nsb ¼ msb −h A1 ; Rsb R1 −r 1 ;u 1 ;u 1 1 ;u     nsb   nsb sb sb sb sb sb sb 1 −n 2 R1 −r þ R2 −r h A 2 ; R2 ; u h A 1 ; R1 ; u ¼1þ n n  nsb     nsb  sb sb 2 h A2 ; Rsb Rsb Rsb ¼ 1− 2 h A1 ; Rsb 1 ;u 1 −r þ 2 ;u 2 −r < 1: n n

ðA3Þ

That the right side of the last inequality is less than one follows because the middle term is negative since R1sb N r and the last is also negative since R2sb b r. Note now that             lf lf þ h A1 ; Rsb r þ t nlf1 : ðA4Þ x A1 ; r; ulf þ h A1 ; r; ulf r þt nlf1 b x A1 ; Rsb 1 ;u 1 ;u |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ¼EðA1 ;r;ulf Þ The sign of the inequality in Eq. (A4) follows, since utility is strictly concave and strictly increasing in its arguments x and h. The sum of the first two terms on the left side of Eq. (A4) is the minimum expenditure, based on the unique expenditure minimizing demands x(A1,r,ulf), h(A1,r,ulf). Any other demands, like those appearing on the right side of Eq. (A4) entail a higher expenditure than that on the left side, at the same utility level ulf and rent on land r. We also know that,             lf lf sb sb þ h A1 ; Rsb r þ t nlf1 bx A1 ; Rsb þ h A1 ; Rsb r þ t nsb x A1 ; Rsb 1 ;u 1 ;u 1 ;u 1 ;u 1 b1:

ðA5Þ

The first inequality in Eq. (A5 follows because both x and h are normal, and therefore demands increase as usb N ulf, and because t(n1sb) ≥ t(n1lf) by n1sb ≥ n1lf, the premise supposed to be true for this proof. The second inequality in Eq. (A5) was established by Eq. (A3). Hence, by Eqs. (A4) and (A5): E(A1,r,ulf) +t(n1lf) b 1. But this contradicts that the laissez-faire value of income is one: E(A1,r,ulf) + t(n1lf) = 1. Therefore, the supposed premise n1sb ≥n1lf cannot be true and, instead, n1sb b n1lf must be true. □

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