How to make the metropolitan area work? Neither big ...

How to make the metropolitan area work? Neither big government, nor laissez-faire.∗ Carl Gaigné†, Stéphane Riou‡ and Jacques-François Thisse§ November 21, 2014

Abstract We study how administrative boundaries and tax competition among asymmetric jurisdictions interact with the labor and land markets to determine the economic structure and performance of metropolitan areas. Contrary to general belief, political fragmentation and cross-border commuting need not be welfare-decreasing. Tax competition implies that the central business district is too small and prevents public policy enhancing global productivity to produce their full impact. Although our results support the idea of decentralizing the provision of local public services by independent jurisdictions, they highlight the need of coordinating tax policies and the importance of the jurisdiction sizes within metropolitan areas. Keywords: metropolitan area, tax competition, local labor markets, job decentralization, administrative boundary JEL classification: H41, H71, R12

∗

We thank M. Berliant, R. Braid, J. Hamilton, V. Henderson, R. Inman, S. Kok, F. Moizeau, H. Overman,

C. Perroni, R. van der Ploeg, as well as participants to seminars at the Paris School of Economics, London School of Economics, University of Warwick, and the Public Policies and Spatial Economics Workshop (LyonSaint-Etienne), for their useful comments and suggestions. S. Riou and J.-F. Thisse gratefully acknowledge the financial support from, respectively, the Région Rhônes-Alpes and the Government of the Russian Federation under the grant 11.G34.31.0059. † INRA, UMR1302, SMART Rennes (France) and CREATE, Laval University, Quebec (Canada). ‡ Université de Lyon, Lyon, F-69007, France ; Université Jean Monnet, Saint-Etienne, F-42000, France ; CNRS, GATE Lyon Saint-Etienne, Ecully, F-69130, France. § CORE, Université catholique de Louvain (Belgium), Higher School of Economics (Russia) and CEPR.

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1

Introduction

According to Alain Juppé, a former prime minister of France and mayor of the city of Bordeaux, “governments are too small to deal with the big problems and too big to deal with the small problems” within today’s political limits. Bruce Katz, a vice president at the Brookings Institution, went one step further when he said that “metro governance is almost uniformly characterized by fragmentation and balkanization, by cultures of competition rather than one of collaboration.”1 Empirical works confirm the idea that the institutional structure of a metropolitan area has a significant impact on both the efficiency of its local public services and on the welfare of its residents by influencing the distribution of jobs and the level of housing and commuting costs (Glaeser and Kahn, 2001; Cheshire and Magrini, 2009). Since metropolitan areas also produce a sizable and growing share of the wealth of nations, we may safely conclude that there is a need for a sound economic analysis of those entities.2 When jurisdictions compete for tax revenue to finance the public services provided to their residents, the political fragmentation of the metropolitan area affects firms’ and consumers’ locations. Conversely, the locational choices made by those agents determine the jurisdictions’ fiscal bases. The complexity of such an environment leads policy-makers to stress the need for coordinating the actions of local governments. To seriously assess the desirability and scope of such a move, we need to understand how the institutional design of the metropolitan area affects the various channels that link the local governments, the labor market and the land market. The purpose of this paper is to develop a model with one central city and several suburban jurisdictions, in which the tax competition process among jurisdictions interacts with the labor and land markets to shape the economic structure and performance of the metropolitan area. The standard approach to jurisdiction/club formation is to focus on the trade-off between the crowding effect of public services, which increases with jurisdiction size, and the unit cost of public services, which decreases with population size. We contend that the problem may be tackled from a different, but equally important, angle by recognizing that workplaces and residences do not necessarily belong to the same jurisdiction. In practice, the central city attracts a large number of workers who live in adjacent but independent areas, thus giving rise to a substantial amount of “cross-border” commuting. So workers face a second trade-off. They can earn a high wage in centrally located firms and bear high commuting costs. Or, they can receive lower pay in firms located in secondary business centers and pay less for commuting. 1

According to Brülhart et al. (2015), the metropolitan areas in OECD countries with more than 500,000

inhabitants are on average divided into 74 jurisdictions, but up to 280 in France. The metropolitan area of New York is divided into 356 jurisdictions. 2 For example, the estimated GDP of the metropolitan area of Tokyo or New York in 2006 is similar to those of Canada or Spain

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By combining these two trade-offs within a unifying framework, we distinguish between the administrative and economic limits of the central city, a distinction that has not attracted much attention in the literature (Glaeser, 2013; Brülhart et al., 2015). This is not yet the end of the story. It is well documented that the productivity of labor is higher in larger cities (Ellison et al., 2010; Jofre-Monseny et al., 2011; Combes et al., 2012). The reason for this lies in the existence of various mechanisms, generically nicknamed “agglomeration economies,” which have the nature of external effects because the productivity of workers varies with the overall distribution of firms, which is itself endogenous. Another major trend shaping modern metropolitan areas is the decentralization of jobs toward secondary employment centers, which is fostered by the development of new information technologies (Glaeser and Kahn, 2004). Any serious attempt that aims to study how the metropolitan area works must account for these two forces. To capture them in a parsimonious way, we consider a modeling strategy in which the intensity of agglomeration economies in an employment center increases with the number of firms set up therein. Owing to distance-decay effects, the intensity of agglomeration economies rises less than proportionally with the number of firms located in other centers. Since the central city has a better access to the metropolitan pool of firms and workers, the jurisdictions are asymmetric in a way that differs from the standard modeling approaches used in the tax competition literature. In particular, agglomeration economies tend to be stronger in the central city than in the edge-cities. For our purpose, a polycentric metropolitan area, in which only a fraction of jobs is located in the central city, is the relevant configuration (Glaeser and Kahn, 2001). Although the volume of employment in the central city may be higher or lower than in the suburban districts, we show that the central city’s employment center is bigger than any suburban center. Therefore, once it is recognized that the process of tax competition unfolds within the urban space, the tax game becomes more involved than in existing models of fiscal competition. Indeed, the welfare of a suburbanite working in the central city is affected by the agglomeration economies prevailing in this city, whereas the welfare of a suburbanite working in a secondary center is affected by the agglomeration economies arising therein. Similarly, the welfare of a consumer residing and working in the central city varies with the number of suburbanites who commute to the central city. For such a rich pattern of interactions to be studied, we need a setting that combines both urban economics and local public finance. To carry out our study, we develop a simple general equilibrium model in which consumers and firms are free to choose their location within the metropolitan area, while local governments act strategically to attract firms. Given the foregoing discussion, the following features cannot be ignored: the interactions between taxation and the labor and land markets, the existence of cross-border commuting, the presence of agglomeration economies that are endogenous and

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unevenly distributed within the metropolitan area, and the difference in population size between the center and the suburbs. Our model is the first that encompasses those various features within a unified framework that combines ideas and concepts from local public finance and urban economics (Brülhart et al., 2015). The flip side of the coin is the need to reduce the complexity of the problem. This is why we consider a simple setting that takes all the abovementioned features into account while remaining tractable. Our main findings may be organized in three distinct, but complementary, categories. 1. We study the first-best outcome, which we use later on as a benchmark. The planner, who aims to maximize welfare within the whole metropolitan area, determines the areas providing the public services and the employment centers by choosing where consumers live and work. First, it is not desirable to amalgamate the suburban areas with the central one. Moreover, the economic boundary of the central city always encompasses its administrative boundary. Put together, these results imply that the optimal political and economic boundaries of the central city do not coincide, a result that clashes with the general belief that these boundaries should be the same (OECD, 2006). This difference stems from the trade-offs that determine each type of boundary. First, the optimal provision of local public goods equalizes the marginal congestion cost borne by the users and the marginal cost of supplying those goods. Therefore, decentralizing the provision of local public goods is socially desirable as long as the degree of increasing returns in producing these goods is not too high. By contrast, the optimal size of central and secondary business centers depends on the interplay between commuting and agglomeration economies, that is, parameters that do not enter the above trade-off. To be precise, when commuting costs are not too low, agglomeration economies are not too strong, or both, jobs are shared between the central and suburban areas. Otherwise, all jobs are located in the central district. 2. We then study the decentralized outcome when the number of independent jurisdictions and their administrative boundaries are exogenously given when commuters travel from suburbia to the central city. There are three types of players: a large number of consumers (formally, a continuum), a large number of firms (formally, a continuum), and a finite number of local governments. Consumers choose a residence and a workplace. Firms choose a location and the wages paid to their employees. Jurisdictions supply public services. To finance them, local governments choose non-cooperatively a property tax levied on their residents and a business tax paid by the firms located in their jurisdiction. We show that the central city always levies a higher business tax than suburban governments, a result that is backed up by the empirical literature (see Brülhart et al., 2015, for references). The reason for this is that consumers working in the central city need not reside therein, which incentivizes the central city’s government to practice tax exporting, the extent of which depends on the relative size of jurisdictions and the

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intensity of agglomeration economies. This in turn reduces the wage gap between consumers working in the central and secondary business centers, and thus affects the overall productivity of the metropolitan area. Furthermore, the equilibrium size of the central business center is too small, which implies that the potential provided by the existence of agglomeration economies is underexploited. In other words, tax competition reduces the global productivity of the metropolitan area through an excessive decentralization of jobs. This result shows the importance of using a spatial setting in which the commuting pattern is endogenous through the location choices made by firms and consumers. What is more, under corporate tax competition, when the population size of the central city is optimal, the central business district is too small, whereas the former is too large when the size of the latter is optimal. Therefore, redrawing the limit of the central city is not the remedy to correct the misallocation of jobs within the metropolitan area. This tension stems from the fact that the distribution of jobs is governed by a system of forces that overlaps imperfectly with that taken into account by the local governments. As a consequence, there is no reason to expect the two types of boundaries to coincide. It should be stressed, however, that the misallocation of jobs is exacerbated when the relative population size of the central city is small. Furthermore, although higher agglomeration economies, lower commuting costs, or both raise the global efficiency of the metropolitan area, the gap between the optimal and equilibrium central business districts grows. Thus, tax competition prevents public policy enhancing the global productivity of the metropolitan area to produce their full impact. 3. Once it is recognized that suburbanites commuting to the central business district may consume the public services supplied by the central city, the tax gap widens because the central city sets an even higher tax rate to reduce the production costs borne by its residents. All in all, the central city residents are hurt twice by the suburbanites’ free-riding behavior: they end up bearing higher provision costs for their public services and earning lower wages. This concurs with Katz for whom the culture of competition that prevails in many metropolitan areas is damaging to the central city. Our analysis suggests that neither the amalgamation nor the decentralization among competing jurisdictions is the best way to govern large metropolitan areas. Instead, combining a multi-jurisdictional political system with an economic government of the metropolitan area or a deep inter-jurisdictional cooperation seems to be a more efficient way to solve the various distortions inherent to the working of a metropolitan economy. In particular, our findings point to the need for a common governance in local tax policies within MAs. Such a recommendation has been implemented in a few European countries under the concrete form of fiscal coordination (OECD, 2006). In the United States, the tax-base sharing program implemented in Minneapolis-Saint Paul has decreased incentives for local governments to compete for a larger

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tax base (Inman, 2009). As for the transportation policy, different institutional systems prevail, ranging from the local to the federal government. A last comment is in order. The legal environment in which metropolitan areas operate vastly differs across countries. The model presented in this paper is context-free in that it focuses on (some of) the fundamental characteristics common to most metropolitan areas and disregards specific and idiosyncratic issues that are important in some countries but not in others. Related literature. Ever since Tiebout (1956), it is widely acknowledged that a wide portfolio of local jurisdictions allows consumers to live in the locale supplying the tax/service package that fits best their preferences. However, once it is recognized that the provision of many public services is governed by increasing returns, Hochman et al. (1995) showed that political fragmentation may generate a substantial waste of resources. Rhode and Strumpf (2003) showed that Tiebout mechanisms are not a dominant factor in the long-run residential choices within the Boston Metropolitan Area, even though this metropolitan area is often presented as the archetype of the Tiebout model. Only a handful of papers have studied the economic organization of a metropolitan area. Hoyt (1991) and Noiset and Oakland (1995) did not account for the fact that jobs may be located outside the central city, while Braid (2002) disregarded tax competition. When local government compete for capital, consumers/workers live and work within the same jurisdiction in equilibrium (Braid, 1996; Epple and Zelenitz, 1981; Janeba and Osterloh, 2013). Braid (2010) focussed on the distances between jurisdictions that choose to offer, or not to offer, a local public good that may be consumed by non-residents. He did not study, however, the interactions between the provision of public services and the labor and land markets. In the next section, we provide a detailed description of the model, which is non-standard. In Section 3, we study the socially optimal organization of the metropolitan area, whereas Section 4 focusses on the decentralized outcome. In Section 5, we consider the second-best outcome in which a planner chooses the welfare-maximizing limit of the central city, while local governments, firms and residents pursue their own interest. In Section 6, we check the robustness of our main findings when suburbanites working in the central business district consume the public services provided by the central city, and when the central city supplies a broader array of public services than the suburban jurisdictions. Section 7 concludes.

2

The model and preliminary results

The metropolitan economy is populated by L consumers who are free to choose their residential location and workplace. There are three consumption goods: (i) land, which is used as a 6

proxy for housing, (ii) a local public good financed by jurisdictions through taxes, and (iii) a homogeneous good, the numéraire, produced by profit-maximizing firms whose locations are endogenous. Our setting can easily been extended to the case of workers producing a differentiated good under monopolistic competition. Note also that the numéraire can be used to import other goods produced in cities specialized in the production of these goods.

2.1

Jurisdictions and the provision of public goods

The metropolitan area (MA) is formed by m+1 jurisdictions. It is endowed with a hub-and-spoke transportation network, which means that the m ≥ 2 suburban jurisdictions are connected only to the central city, which has a direct access to all suburbanites. Formally, the MA is described by m one-dimensional half-lines sharing the same initial point x = 0. Distances and locations are expressed by the same variable x measured from 0. Figure 1 provides a bird-eye view of the MA.3 The central city hosts the central business district (CBD) of the MA at x = 0, while each suburban jurisdiction may, or may not, accommodate a secondary business district (SBD). The internal economic structure of the MA is endogenous in that the CBD and SBD sizes are variable and determined at the equilibrium. When a suburban jurisdiction accommodates a SBD, we call it an edge city to differentiate it from the central city. In this case, the MA is polycentric; otherwise, it is monocentric. The most interesting case to study involves edge cities as job decentralization appears to be a powerful trend in many MAs. Figure 1 about here The central city shares the same boundary b with each suburban jurisdiction, while all suburban jurisdictions have the same outer limit B. The economic limit yi of the central city along the ith spoke is given by the location of the individual who is indifferent between working in the CBD or in the ith SBD located at a distance xsi from the CBD. Therefore, all workers located in the interval [0, yi ] work in the CBD while those in (yi , B] work in the ith SBD. Unless explicitly mentioned, we delete the subscript i because we consider a symmetric equilibrium (yi = y and xsi = xs ). Note that the economic limit y need not coincide with its administrative limit b, which implies the existence of cross-border commuting. Each jurisdiction supplies the same amount of public good G to its residents (e.g. schools, daily care clinics, recreational facilities). Only the local residents use the public services supplied 3

We could use a more general setting in which the suburban jurisdictions are directly connected to their

neighbors. However, doing this increases the complexity of the formal developments while the gains in results are limited. What matters for most of our results is that the central city has more direct connections to the other jurisdictions than each suburban jurisdiction.

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by their jurisdictions. That G is exogenous and the same across jurisdictions is in accordance with what is observed in several European countries where the level of public good is determined by constitutional rules that impose the equal treatment of people in the access to some public services. However, this does not apply to more decentralized countries, such as the United States or Switzerland, where jurisdictions are free to choose the amount of public good they supply to their residents. In Section 6, we will allow the cross-border commuters to consume the central city public services and the central city to supply a broader range of public services than the suburban jurisdictions. We have chosen this order of presentation because the case where G is exogenous allows us to isolate the impact of tax competition on the structure of the MA from that of competition in public good. When the supplied population is l, the cost of providing the public good G is given by c C(l) = F + l2 , 2 where F stands for a jurisdiction’s investment outlays and cl2 /2 > 0 the variable production cost, which increases with the population size l of the jurisdiction. This specification features two characteristics that are encountered in the provision of local public goods: (i) the need to build infrastructures having a minimum size and (ii) the congestible nature of these infrastructures, so that supplying a rising number of users requires growing expenditures. In other words, from the local government’s point of view the extra cost generated by a bigger population is reflected in the provision cost of the public goods.4 Each jurisdiction must balance its budget. This is achieved by allowing a jurisdiction to use two instruments, that is, a property tax levied on the land rent prevailing in the jurisdiction and a business tax paid by the firms located therein. To keep the formal analysis simple, we assume that the business tax has the nature of a lump-sum tax.

2.2

Consumers and land rents

Consumers use a lot having the same fixed size. The units of land is chosen for this parameter to be normalized to one. The assumption of a fixed lot size does not allow replicating the welldocumented fact that the population density is higher in the central city than in the suburbs. It is widely used, however, in models involving an urban economics building block because it captures the basic trade-off between long/short commutes and low/high land rents. Therefore, the central city population (ℓ0 ) and a suburban jurisdiction population (ℓ) are, respectively, given by ℓ0 = mb 4

ℓ = B − b,

Alternatively, we could assume that the public services are congestible (G − cl), but supplied at a zero

marginal cost.

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where B ≡ L/m is assumed to be larger than b. The unit commuting cost τ > 0 borne by consumers is the same in both the central city and the suburban jurisdictions, perhaps because transportation infrastructures are planned at the level of the entire MA. Therefore, commuting costs are equal τ x or τ |x − xs | according to the location of her employment center. Each jurisdiction owns its land and the aggregate land rent is evenly redistributed among the residents.5 In what follows, we consider the following three patterns: (i) when b < y < B, there are m + 1 employment center; (ii) when b < y = B, there is a single employment center but m + 1 jurisdictions; and (iii) when b = y = B, there is a single city. Hence, we focus on the case where y ≥ b, which means that some workers commute from their suburban residences to the central city.6 Because of cross-border commuting, land is used on both sides of the boundary between the central and suburban jurisdictions. As a result, there is not vacant land within the MA. When a consumer lives and works in the central city, her indirect utility is given by V0 (x) = w0 − (1 + t0 ) R0 (x) − τ x + G +

ALR0 , ℓ0

(1)

where R0 (x) is the land rent at a distance x from the CBD, while w0 is the wage paid by the firms located in CBD, t0 the property tax levied by the central city government, while ALR0 = m

b R (x)dx 0 0

is the aggregate land rent in the central city. When a consumer lives in a suburban jurisdiction and works in the central city, her indirect utility becomes

ALR , (2) ℓ where R(x) and t are, respectively, the land rent and property tax in a suburban jurisdiction, V0s (x) = w0 − (1 + t)R(x) − τ x + G +

while ALR =

B R(x)dx. b

By contrast, when a consumer lives and works in a suburban jurisdiction, her indirect utility is V s (x) = w − (1 + t)R(x) − τ |x − xs | + G + 5

ALR , ℓ

(3)

Instead one could think of using the aggregate land rent to finance the local public good. The Henry

George Theorem holds when each jurisdiction reaches its optimal size. This condition can hardly be satisfied here because the total population size and the number of jurisdictions are given. Furthermore, the land rent capitalizes several effects in our setting. 6 Since reverse commuting flows stand for a small fraction of commuters, the case y < b is empirically less relevant than that where y > b (Baum-Snow, 2010). The case of reverse commuting is addressed in Appendix A.

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where w is the wage rate paid in a SBD. Without loss of generality, we assume that the opportunity cost of land is zero. The land rent at each location in the central city is as follows. Given V0 (x), the bid rent at x of a worker living and working in the central city (Φ0 (x)) must solve ∂V0 (x)/∂x = 0 or, equivalently, (1 + t0 )(∂Φ0 /∂x) + τ = 0 whose solution is Φ0 (x) = r0 −

τ x, 1 + t0

(4)

where r0 is a constant that will be determined in Section 3 and t0 the property tax set in the central city. The land rent prevailing in the central city is given by R(x) = max {Φ0 (x), 0}. The land rent prevailing in a suburban jurisdiction is given by R(x) = max {Φs0 (x), Φs (x), 0} ,

(5)

where Φs0 (x) (Φs (x)) is the bid rent at x of a worker living in a suburban jurisdiction and working in the central city (an edge city). Given V0s (x) and V s (x), the equilibrium land rent is such as ∂V0s (x)/∂x = ∂V s (x)/∂x = 0. As a consequence, the bid rents are Φs0 (x) = r0s −

1 x 1+t

Φs (x) = rs −

1 |x − xs | , 1+t

where both r0s and rs will be determined in Section 3; t is the property tax set in a suburban jurisdiction. Thus, in each jurisdiction, the slope and intercept of the land rent profile are endogenous. Note also that the redistribution of the land rent is jurisdiction-specific. Assuming that the total land rent within the MA is shared among all consumers is not consistent with the existence of independent and competing jurisdictions. In addition, our assumption allows us to ignore the external effect stemming from the strategic manipulation by jurisdictions of the metropolitan land rent.

2.3

Firms and wages

Labor is the only production factor. When a worker stands alone, he produces one unit of an homogeneous good, which is chosen as the numéraire. We may then use indifferently the terms worker and firm, so that the total number of firms N established in the MA is given by N = L. The economic size of the central city is equal to my, while the economic size of an edge city is B − y. Given that workers distribute themselves symmetrically around their employment center in equilibrium, a SBD is located at the center of the interval [y, B]: xs = y +

10

B−y . 2

(6)

In the introduction, we have seen that agglomeration economies may stem from different sources. In what follows, we choose not to privilege one type of economies against the others, the reason being that the results we will obtain could well depend on the particular type considered. Rather, we prefer to use a formulation capturing the main features that are typically associated with the presence of agglomeration economies. Where do agglomeration economies pop up in an MA? Whereas Glaeser and Kahn (2004) argue that, due to the development of new information and communication technologies, the scope of spillovers has spread within the MA, Baum-Snow (2014) finds evidence that “spillovers across firms within central cities represent a large fraction of metropolitan area wide agglomeration economies.” We assume that a firm benefits from agglomeration economies if it chooses to set up in the CBD or in a SBD, the reason being that firms need specific local public goods and private facilities to benefit from the spillovers generated by other firms. As an isolated firm gets its stand-alone productivity, firms located in the central city choose to gather in the CBD, while those located in an edge city form a SBD. All firms (or workers) benefit from agglomeration economies regardless of the center where they are located, but they do so with different levels of intensity according to the type of center in which they are established. Let n0 be the number of firms set up in the CBD and ni the number of firms in the ith SBD. Since we focus on a symmetric configuration, we may set ni = n. The number of firms located in the central city being equal to the number of workers commuting to the CBD, we have n0 = my and, therefore, n = B − y. When a firm sets up in the CBD, the production of a worker increases by the amount E0 = α(n0 + λmn),

(7)

where α ≥ 0 is a non-negative constant while λ ∈ (0, 1) measures the intensity of spillovers between the CBD and a SBD. In other words, a CBD-worker produces 1 + E0 units of the numéraire, where E0 rises with the number n0 of firms located in the CBD. Such a firm also benefits from spillovers generated in the each SBD, the intensity of which depends on the number of firms located therein. Nevertheless, even at the Age of Internet information is subject to distance-decay effects, which we take into account by assuming λ < 1. As for the production of a worker in a SBD, it rises by the amount E = α (n + λn0 ) ,

(8)

which increases with the size n of the SBD. As shown by (7) and (8), the central city benefits from spillovers brought about by all SBDs, whereas an edge city benefits from the spillovers generated by the CBD only. This reflects the comparative advantage of the CBD. In the limit, the MA may be monocentric with a single 11

employment center. Depending on where firms choose to set up, the sizes of the CBD and SBDs, as well as the spillover effects within and between the employment centers are endogenous. In particular, agglomeration economies in the central city are stronger than those in the edge cities (E0 > mE) if and only if

n0 1−λ > , N − n0 1 − λm

which holds if λ > 1/m. We assume throughout the paper that λ > 1/m. In order to understand how agglomeration economies affect the production level in the MA, we find it illuminating to consider the production function of each type of center. The production function of the CBD is given by the sum of the quantities produced by the firms located therein: F0 (n0 ) = n0 (1 + E0 ) = n0 [1 + α (n0 + λmn)] . This expression is increasing and convex in n0 . As for the production function of a SBD, it is similarly given by F (n) = n(1 + E) = n [1 + α(n + λn0 )] , which is also increasing and convex in n. As a consequence, the marginal productivity of labor in the CBD or in a SBD increases in response to a hike in the size of the corresponding center. The morphology of the MA and the hierarchy between employment centers are thus the equilibrium outcome of interactions between centers whose productivity level depends on the whole distribution of the production sector across the MA. Admittedly, our description of agglomeration economies is not as rich as in Fujita and Ogawa (1982) and Lucas and Rossi-Hansberg (2002). However, we want to stress that these authors appeal to numerical methods to characterize the equilibrium outcome. In this paper, we need the explicit form of the equilibrium pattern of activities to study how tax competition affects the structure of the MA. Although simple, (7) and (8) capture the main features of the formation and diffusion of agglomeration economies studied by the above authors. Let Π0 (Π) be the profits earned by a firm set up in the central city (an edge city). A firm located in the CBD earns profits equal to Π0 = 1 + E0 − w0 − T0 ,

(9)

where 1 + E0 denotes the firm’s revenue, w0 the income earned by the corresponding worker and T0 the tax rate levied in the central city. Because our setting is symmetric, all suburban jurisdictions charge the same business tax rate T and firms pay the same wage w. Thus, when a firm sets up in an edge city, its profit function becomes: Π = 1 + E − w − T. 12

(10)

In each employment center, the equilibrium wage is determined by a firm’s revenue net of tax. Setting (9) and (10) equal to zero and solving, respectively, for w0 and w, we get w0 = 1 + E0 − T0

w = 1 + E − T.

(11)

Thus, wages are endogenous and different between the CBD and the SBDs. Thus, they depend on the whole distribution of firms within the MA through the intensity of agglomeration economies, as well as on the business tax rate. Furthermore, if business taxes alleviate residents’ tax burden, they also reduce the wages earned by workers. When consumers work and live within the same jurisdiction, both effects are washed out. This is no longer true, however, when they work and live in different jurisdictions. As a consequence, the property tax paid in the jurisdiction where the consumer lives and the business tax paid in the jurisdiction where she works affect her utility level, whence her residence and workplace. By implication, both types of taxes affect the distribution of activities.

3

The optimal metropolitan area

In this section, we assume that a benevolent planner maximizes the total welfare in the MA by choosing the size and number of areas supplying public services, as well as consumers’ and firms’ locations, hence the commuting pattern. To be precise, the planner must choose to centralize (or decentralize) the provision of publics goods with the aim of minimizing investment costs (or operating costs), as well as to concentrate firms and jobs in the CBD (disperse firms and jobs through the CBD and SBDs) with the aim of maximizing agglomeration economies (or minimizing commuting costs). Owing to symmetry, b, y and B are the same along each spoke. There are three types of commuting patterns: (i) a consumer lives and works in the central city; (ii) a consumer lives in a suburban service area but works in the central city; and (iii) a consumer resides and works in the same suburban service area. Individual utilities being linear, the total welfare WT within the MA is defined, up to a positive constant, by the total surplus: WT = GDP − CC + G − P C,

(12)

which involves two terms that depend the marginal worker y. (i) Using (7) and (8) shows that the gross domestic product (GDP) of the MA is given by GDP = n0 (1 + E0 ) + mn(1 + E). Because of the firm population constraint N = n0 + mn, when n0 increases (decreases), a few firms are reallocated from the SBDs to the CBD (from the CBD to the SBDs). Since 13

n0 = my and n = B − y, we may rewrite the production functions F0 and F in terms of y only: GDP = my [1 + α (my + λm(B − y))] + m(B − y) [1 + α (B − y + λmy)] . As expected, the metropolitan GDP increases with α and λ. Moreover, regardless the value of λ, the metropolitan GDP grows with y and reaches its maximum value when the MA is monocentric: y = B. Even though the development of SBDs generates efficiency gains, the GDP of the MA is maximized when all jobs are concentrated at the CBD, that is, the MA has a unique and integrated labor market. This is because the agglomeration economies are not entirely local, so that the GDP generated by the sole SBDs does not necessarily grow with the number of firms located therein. Indeed, when λ is high enough, it is readily verified that the GDP produced in the SBDs is maximized at y = [Bα (mλ − 2) − 1]/2α (mλ − 1), which is both positive and smaller than B. (ii) The total commuting costs borne by the individuals working in the CBD and in the SBDs are given by y

B

CC = m τ xdx + m τ x − 0

y

B+y y2 dx = mτ + 2 2

B −y 2

2

.

Minimizing workers’ commuting costs within each edge city implies that it is optimal to locate the SBDs at the middle point xs = (y + B)/2. It is then readily verified that CC is minimized when y = B/3, that is, the size of central city’s labor market is equal to mB/3, while that of an edge city is 2B/3. As a result, the population working in the central city is smaller than the total labor force in the edge-cities as a whole. This is sufficient to show the existence of a trade-off between the maximization of the metropolitan GDP and the minimization of commuting costs within the MA. (iii) The total cost of providing public goods G in the entire MA is given by c c P C = (m + 1)F + (mb)2 + m (B − b)2 . 2 2

3.1

The optimal size of service areas and labor pools

Assume that the number of service areas is given (m + 1). By choosing the boundary b of the central city, the planner determines the population size in each service area. Clearly, a marginal expansion of the central city (a higher b) reduces the number of residents in all suburban service areas. As a consequence, the cost of supplying public goods decreases therein, whereas it rises in the central city. Differentiating WT with respect to b yields ¯b =

B < B. m+1 14

(13)

Thus, regardless of the values of L and m it is always optimal to decentralize the provision of public goods into m + 1 service areas. The optimal size of a suburban service area is equal to

L >0 m+1 while the optimal size of the central city is equal to ℓ¯ = m¯b. Thus, the central city and the ℓ¯ = B − ¯b =

suburban service areas have the same population size. As a result, production costs in public goods are equalized across all service areas. Note also that, at the optimum, the total number of suburbanites exceeds the number of the central city residents. Furthermore, by choosing y, the planner determines the size of the CBD (my), hence that of each SBD (B − y). As mentioned above, total commuting costs reach their lowest value when the average traveled distance is minimized, i.e., y = B/3 ≥ ¯b. Because y does not affect the production cost of public goods, the optimal economic boundary of the central city is the outcome of the trade-off between commuting costs and agglomeration economies. By implication, the optimal value of y must belong to the interval [B/3, B]. When α>

τ , 2m(1 − λ)

(14)

the social welfare function is either increasing or convex over the interval [¯b, B] with W (¯b) < W (B). As a consequence, the optimal boundary is given by y¯ = B. In other words, when the agglomeration economies are strong, commuting cost are low, or both, the social optimum involves the agglomeration of firms in the CBD, for otherwise the productive efficiency losses would be too high. Thus, whereas the supply of public goods is decentralized within the MA, the labor market is integrated. To put it differently, the decentralization of the provision of public goods within the MA and the agglomeration of firms and jobs in the CBD do not necessarily conflict. When (14) does not hold, differentiating WT with respect to y shows that the optimal economic boundary of the central city is given by y¯ =

B [τ + 4α(λm − 1)] , 3τ − 4α(m + 1 − 2λm)

(15)

where B/3 < y¯ < B because m ≥ 2. It is readily verified that y¯ increases in α and decreases in τ , thus confirming that the agglomeration economies push toward the concentration of jobs in the CBD, whereas the commuting costs act as a dispersion force that fosters the growth of the SBDs. Moreover, when the agglomeration economies are not too weak, the presence of spillovers between the CBD and the SBDs fosters the dispersion of firms within the MA at the optimum: d¯ y /dλ < 0.

15

3.2

The optimal number of service areas

The planner may also choose the degree of decentralization in the provision of public goods through the variable m. Since L = mB, choosing m amounts to choosing the spatial extent of the MA. To highlight the main forces at work, we assume that there is no agglomeration economies (α = 0). Two cases may arise. In the first one, it is optimal to concentrate firms and jobs in the CBD. Differentiating WT with respect to m at ¯b and setting y¯ = B leads to the following equilibrium condition: τ L2 c + 2 2m 2

L m+1

2

− F = 0.

(16)

Note that (16) includes the trade-off between the fixed cost of launching a new service area and the cost saved on the functioning of the existing service areas when a new one is added to the MA. Indeed, we have dm/dF ¯ < 0 and dm/dc ¯ > 0. It is also readily verified that dm/dτ ¯ > 0 and dm/dL ¯ > 0. Lower commuting costs, a less populated MA, or both lead to a smaller number of service areas. As a consequence, the optimal structure of the MA is governed by the trade-off between commuting costs and the cost of providing public goods. In particular, if F keeps on a higher value, the planner provides the public goods through a smaller number of service areas. In the second case, it is optimal to break up the MA into several employment centers. The optimal value of m is now implicitly given by c τ L2 + 2 6m 2

L m+1

2

−F =0 .

The impact of τ , L, F and c is the same as in the first case. In the presence of agglomeration economies (α > 0), (15) implies that more workers are allocated to the CBD, which increases the total commuting costs. As a consequence, the planner has an incentive to increase m to reduce these costs. Proposition 1 comprises a summary. Proposition 1. Assume a planner maximizing total welfare within the metropolitan area. Then, unless the provision of public goods is subject to strong increasing returns, the optimal metropolitan area involves several public goods providers, as well cross-border commuting from the suburban areas to the central city. Thus, cross-border commuting and the fragmentation of the MA into several public goods’ providers need not be wasteful. Even in the absence of agglomeration economies (α = 0), the position of the CBD in the transportation network makes cross-border commuting socially desirable when m > 2 because y¯ = B/3 > ¯b, while y¯ = ¯b when m = 2.

16

4

Tax competition and the metro structure

We now consider a decentralized tax setting in which the institutional environment, i.e., the number of suburban jurisdictions, m, and the administrative boundary, b, between these jurisdictions and the central city are given. Our purpose is to find how the institutional parameters b and m, as well as the main economic parameters, affect the tax policies and the location of firms and jobs.7 The spatial structure of the MA implies that competition among jurisdictions is strategic: each suburban jurisdiction competes directly with the central city only whereas the central city competes with every suburban jurisdiction. The interactions between local governments and market forces are described by a three-stage game that blends atomic and non-atomic players. There are three groups of players: a continuum of consumers, a continuum of firms, and m + 1 local governments. Consumers choose where to live and where to work; firms choose where to locate and the wage to pay to their employees; and local governments choose a business tax and a property tax. In the first stage, consumers choose the jurisdiction they want to join and their location therein, anticipating the property tax they will pay and the wage they will earn. Therefore, in equilibrium consumers will reach the same utility level. In the second stage, the population in all jurisdictions has already been determined, so that local governments choose simultaneously and non-cooperatively a business tax and a property tax to maximize the total welfare of their residents. Last, firms choose their profit-maximizing locations and consumers their workplace, while land and labor markets clear. The locations and sizes of the SBDs are determined when firms choose their locations in the third stage. Once consumers are mobile, the specification of governments’ objective is known to be a controversial issue (Cremer and Pestieau, 2004). Our three-stage game obviates this difficulty because governments know who their residents are, and thus may determine the total welfare to maximize. Moreover, the relationship between jobs and people having often the nature of an “egg-and-chicken” problem, firms choose their locations and consumers their workplaces simultaneously. We seek a subgame perfect Nash equilibrium. As usual, the game is solved by backward induction. In what follows, we focus on a symmetric equilibrium: Ti = T and ti = t for i = 1, ...m. In this event, wages paid in the SBDs are the same: wi = w for i = 1, ..., m. Since there is no vacant land, we have B = L/m. Since wi = w, it must be that yi = y. Note that considering a symmetric outcome vastly simplifies the comparison between the equilibrium and 7

Hoyt (1992) developed a setting in which the central city’s government influences the land rent in suburban

jurisdictions, whereas the tax policy of the government of a suburban jurisdiction has no impact on the central city’s land rent because its population share is negligible. Like us, Hoyt showed that the property tax is higher in the central city. However, unlike us, he treated households’ residential locations and workplaces as exogenous.

17

optimum. Because characterizing the equilibria of all subgames is long and tedious, we find it convenient to restrict ourselves to the equilibrium path. In particular, consumers being mobile and identical, they anticipate that they reach the same (indirect) utility level V ∗ at the end of the game. Thus, we have V ∗ = V0 (x) for 0 ≤ x ≤ b, V ∗ = V0s (x) for b < x ≤ y, and V ∗ = V s (x) for y < x ≤ B. In what follows, we call the economic boundary of the central city the limit y of the area that includes all the individuals working in the CBD. Since all the tax rates are given, (1) and (11) imply that the (indirect) utility of a consumer residing in the central city is given by ALR0 (17) ℓ0 for 0 ≤ x ≤ b. There are two groups of suburbanites living in a suburban jurisdiction: those V0 (x) = 1 + E0 − T0 − (1 + t0 )R0 (x) − τ x + G +

who work in the CBD and pay the land rent R0s , and those who work in their SBD and pay the land rent Rs . Using (2) and (11) shows that the utility of a consumer belonging to the first group is

ALR (18) ℓ with b < x ≤ y, while using (3) and (11) implies that the utility of a consumer belonging to V0s (x) = 1 + E0 − T0 − (1 + t)R0s (x) − τ x + G +

the second group is V s (x) = 1 + E − T − (1 + t)Rs (x) − τ |x − xs | + G +

ALR ℓ

(19)

with y < x ≤ B.

4.1

Labor and land market equilibrium

In the third stage, firms and consumers observe the tax rates chosen by the local governments. Then, firms select a location as well as the wage they pay while consumers choose their working places. Because consumers are mobile, they accurately anticipate in the first stage that the equilibrium land rent equalizes utility across mobile individuals. 4.1.1

Job location

Wages being given by (11), it remains to determine the distribution of jobs within the MA. For this, we must find the location y of the marginal worker, which is the same along all spokes. We assume throughout this section that y exceeds b and will determine the conditions for this to hold in equilibrium. Using (6) and (11), the worker at y is indifferent between the CBD or the SBD if and only if V0s (y) = V s (y) or, equivalently, w0 (y) − w(y) = τ y − τ (xs − y) = τ 18

3y − B . 2

(20)

In other words, CBD- and SBD-workers do not earn the same wage, the difference between wages compensating workers for the difference between commuting costs along the same spoke. Moreover, given T0 and T , a higher value of y strengthens the agglomeration economies arising in the CBD and weakens those in the SBDs. As a consequence, when more firms locate in the central city, the CBD-wage and the wage gap between the CBD and the SBDs rise. This is because the central city’s firms must pay a higher wage to attract more distant workers. In sum, stronger agglomeration economies allows the CBD-firms to pay a higher wage that compensates the CBD-workers for their higher commuting costs. The reverse holds in the SBD. Substituting (11) into (20) yields the equilibrium economic boundary of the central city: y ∗ (T0 , T ) =

B[τ + 2α (λm − 1)] − 2 (T0 − T ) , 3τ − 2α(m + 1 − 2λm)

(21)

Because of the presence of externalities, we must focus on stable equilibria. Since V0s (y) and V s (y) are linear in y, this is so if and only if V s (y)− V0s (y) decreases with y and V0s (B) < V s (B). The first condition is equivalent to α B[αm(1 − λ) − τ ].

(23)

Consequently, the economic boundary of the central city expands (shrinks) with T (T0 ) because the CBD becomes relatively more (less) attractive. Moreover, stronger agglomeration economies (higher α) yield a bigger CBD while lowering commuting costs (lower τ ) have the same effect. Even though agglomeration economies may occur in all business districts, the CBD grows at the expense of the SBDs. Similarly, y ∗ increases with m as long as b < y ∗ < B. In this case, a more fragmented MA has smaller SBDs. Last, stronger spillovers between the CBD and the SBDs (higher λ) moderate the concentration of firms in the central city when α > τ /2 (m − 1). This is because the intensity of agglomeration economies is sufficiently high to strengthen the attractiveness of the central city. Otherwise, stronger spillovers exacerbate the agglomeration of firms. 4.1.2

Land rent

We now turn to the determination of the equilibrium land rents. Using the equilibrium conditions V0 = V0s = V s = V ∗ , we are now equipped to determine the value of r0 for the central 19

city as well as the values of r0s and rs for the suburban jurisdictions. At x = B, the land rent equals the opportunity cost of land, which is zero. At x = y ∗ , the land rent must be equal to 0 for V (y ∗ ) = V (B) to hold. Indeed, if a consumer offers a positive bid to reside at y ∗ , her utility is given by V (y ∗ ) < V (B). The results in turn imply rs = which yields Rs (x) =

τ (B − y ∗ ) , 2(1 + t)

τ (B − y ∗ ) τ − |x − xs |. 2(1 + t) 1+t

Since Rs (y ∗ ) = 0, repeating the above argument leads to r0s =

τ ∗ y , 1+t

and thus

τ (y ∗ − x). 1+t Using the above two expressions for the land rents, we get the aggregate land rate in any R0s (x) =

edge city: ALR =

τ (y ∗ − b)2 (B − y ∗ )2 + , 1+t 2 4

(24)

which decreases with the property tax rate t. Using the equilibrium condition V0 (b) = V0s (b) and (4), we get r0 =

1 ALR0 ALR τ (y ∗ − b) + − , 1 + t0 ℓ0 ℓ

(25)

which shows that the central city land rent capitalizes the differences in congestion costs and in the aggregate land rent redistributed across local residents. For any given value of t0 , whence of r0 , we have R0 (x) = r0 −

τ x. 1 + t0

Consequently, b

ALR0 = m R0 (x)dx = 0

ℓ0 τ t0

y∗ −

b 2

−

ALR , ℓ

(26)

which also decreases with the property tax set in the central city. Plugging this expression in (25) shows that r0 depends on the two property tax rates, whereas rii and ri0 are independent of t0 . Figure 2 provides a side view of the land rent profile. It shows that the land rent is not continuous at the boundary b because consumers just inside and outside that boundary face different property taxes and live in jurisdictions with different costs per capita for the public 20

services and different aggregate land rent per capita. The above expressions show that the land rent profile varies with the economic boundary of the central city. Figure 2 about here Furthermore, the equilibrium land rents R0 (x), R0s (x) and Rs (x) fully capitalize the property tax levied by the jurisdiction containing the location x. Hence, when the local tax increases, the land rent is shifted downward. Note, however, that while Rs (x) and R0s (x) do not depend on the central city property tax, the tax policy of the suburban jurisdictions generates a tax externality capitalized in the land rent paid in the central city. Indeed, (4) and (26) imply that R0 (x) rises with t. Moreover, our framework allows determining the costs and benefits that are capitalized and where the capitalization arises (Starrett, 1981). There is “externality capitalization” in the central city because workers move from the suburban jurisdictions to the CBD. Before proceeding, note also that the full price of land in the central city, defined by (1 + t0 )R0 (x), decreases (increases) with t0 (t) through a pure land capitalization effect of the property tax rates captured by r0 . By contrast, the full price of land (1+t)Rs (x) and (1+t)Rs0 (x) that prevail in the suburban jurisdictions are independent of the property tax. This property crucially depends on the assumption of symmetric suburban jurisdictions.

4.2

Tax competition between the central and suburban jurisdictions

Business and property taxes allow each local government to finance the local public good provided to its residents. Hence, the budget constraint of jurisdiction i = 0, 1, ..., m is given by F +c

ℓ2i = Ti ni + ti ALRi , 2

where Ti is the business tax and ti the property tax levied in jurisdiction i. One appealing feature of our tax game is that we may determine the business tax rates independently of the property tax rates. 4.2.1

Business tax

Local governments set non-cooperatively their business tax rates or subsidies with the aim of maximizing the welfare of their residents. Specifically, the central city maximizes W0 with respect to T0 ∈ R, while every suburban jurisdiction maximizes W with respect to T ∈ R. Since firms choose their locations in the third stage, governments anticipate the consequences of their choices on the size of their business districts. Depending on the impact of tax competition on firms’ locations, the metro is polycentric or monocentric. 21

The social welfare functions of the local governments. (i) The total welfare in the central city is given by b

W0 = m V (x)dx = ℓ0 (1 + E0 − T0 + G) − t0 ALR0 − τ 0

mb2 , 2

where we have used (17). Substituting the budget constraint F + cℓ20 /2 = T0 n0 + t0 ALR0 and the labor market balance condition n0 = ℓ0 + m(y − b) into (44), we obtain W0 = ℓ0 (1 + E0 + G − T0 ) + T0 my ∗ − F − c

ℓ20 τ ℓ20 − , 2 2m

(27)

where y ∗ is given by (21). Raising the business tax T0 gives rise to two opposing effects. First, a higher business tax tends to generate a higher tax revenue, but it yields a smaller fiscal basis. Moreover, y ∗ > b implies that there is cross-border commuting toward the central city. Therefore, because a fraction of the CBD workers lives outside the central city, there is tax exporting and its extent is equal to T0 my ∗ − T0 ℓ0 = T0 m(y ∗ − b). Second, a higher T0 reduces the wage paid to the CBD workers (see (11)). In addition to this direct effect, (21) shows that fewer individuals work in the CBD, which further lowers the net wage paid to the CBD workers through weaker CBD-agglomeration economies (lower E0 ). The equilibrium corporate tax in the central city is the outcome to this trade-off. Note that a marginal increase in T0 has no impact on the commuting costs within its jurisdiction because all the residents work in the CBD. However, by reducing the number of CBD firms, increasing T0 affects the commuting costs paid by the suburban consumers, an effect not internalized by the central city government. Maximizing W0 with respect to T0 yields: dW0 ∂y ∗ ∂y ∗ = m y ∗ − b + T0 + αmb(1 − λ) = 0, dT0 ∂T0 ∂T0

(28)

where d2 W0 /dT02 < 0 holds for any polycentric configuration. The first three terms between brackets capture the tax exporting effect described above while the last term pins down the negative impact of the business taxation on the CBD-agglomeration economies capitalized into the wage of the central city’s residents. As expected, the capitalization effect decreases in presence of stronger spillovers (higher λ) because wages become less dependent on the spatial distribution of firms. As noticed above, this is at the origin of a new externality because the central city’s government does not internalize the impact of their tax policy on the wage earned by the suburbanites working in the CBD. (ii) An edge city includes two types of workers, those who work in the CBD and those who work in their own SBD. Since we focus on the choice made by single jurisdiction, we index the 22

variable specific to this jurisdiction by the subscript i. Using (18) and (19) and the budget constraint F + cℓ2 /2 = T ni + tALRi where ni = B − yi∗ , the total welfare in the ith edge city is given by Wi = (yi∗ − b)(1 + E0 + G − T0 ) + (B − yi∗ )(1 + Ei + G − Ti ) + Ti (B − yi∗ ) yi∗

ℓ2 −F −c , 2

B

τ xdx + τ |x − xs | dx

−

yi∗

b

(29)

while the total commuting costs borne by the residents of this edge city are given by yi∗ b

(B − yi∗ )2 (yi∗ )2 − b2 τ xdx + τ |x − x | dx = τ + . 2 4 yi∗ B

s

The total welfare in the ith edge city can be rewritten as follows: Wi = (yi∗ − b)(E0 − T0 ) + (B − yi∗ )Ei − τ

Byi∗ 3(yi∗ )2 − + constant. 4 2

(30)

Because the business tax policy induces the redistribution of firms within the MA, it affects the intensity of the agglomeration economies in both the CBD and the SBDs, and thus the welfare of the two groups of workers residing in the suburban jurisdiction. In a polycentric MA, it is readily verified that the increased agglomeration in the CBD that follows a hike in Ti boosts the productivity in the CBD (dE0 /dTi > 0) while being damaging to the SBD (dEi /dTi < 0). In addition, a higher Ti lowers the share of jobs in the SBD, and thus increases commuting costs within the ith edge city. Unlike the central city government, a suburban government cares about the trade-off between commuting costs and agglomeration economies, but it does so within its own jurisdiction only. Differentiating Wi with respect to Ti yields the equilibrium condition: dWi dy ∗ dE0 ∂y ∗ dE ∂y ∗ = −Ti + (y ∗ − b) ∗ i + (B − y ∗ ) ∗ i = 0, dTi dTi dyi ∂Ti dyi ∂Ti

(31)

¯. where we have used (20) to simplify the first term. Moreover, d2 Wi /dTi2 < 0 for all α < α To distinguish between the various effects at work, we first discuss the case in which there are no agglomeration economies (α = 0). We then turn our attention to the more relevant case of positive agglomeration economies. The case of no agglomeration economies. To start with, we study the tax equilibrium in the limiting case of no agglomeration economies (α = 0). In this case, (31) implies that T ∗ = 0. By plugging this value into (28), we obtain the equilibrium business tax set in the central and edge cities: T0∗ =

τ (B − 3b) 4 23

and T ∗ = 0.

(32)

Thus, the suburban governments neither tax nor subsidize firms at the equilibrium (T ∗ = 0) while the business tax set by the central city is positive if its size within the MA is not too big. Note that, at the tax rates (32), the economic boundary of the central city is given by y∗ =

B b + , 6 2

(33)

where y ∗ > b if and only if B > 3b.8 In this case, we have T0∗ > T ∗ = 0. Thus, even in the absence of agglomeration economies, the central city’s government sets a business tax rate higher than the edge cities. This is due to the central position of the CBD in the transportation network. The intuition behind this finding is easy to grasp. As noticed by Baldwin and Krugman (2004), a locale with a comparative advantage can set a higher corporate tax rate because more firms want to locate there. As in Proposition 2, there is tax exporting when the population of the MA is sufficiently larger than the central city population (L > 3mb). Observe that T0∗ decreases with b. Indeed, when the limit of the central city expands, (21) implies that y ∗ remains the same as long as T0 and T are given. Therefore, the extent of tax exporting y ∗ − b shrinks. This incentivizes the central city government to lower its tax rate to restore its fiscal base. By contrast, when b rises and exceeds B/3, (27) and (30) no longer describe the social welfare functions of the local governments, the reason being that b > y ∗ . In this case, we show in Appendix A that all jurisdictions set a positive corporate tax. Since the suburban jurisdictions now benefit from tax exporting, they levy a higher tax than the central city. Nevertheless, the CBD and SBDs sizes are unchanged because the tax wedge remains equal to what we obtain in the case where b < B/3. Finally, as in Hoyt (1991), but in a different context, the equilibrium tax rate T0∗ decreases with the number of suburban jurisdictions. Indeed, reducing the number of suburban jurisdictions softens tax competition and allows the central city to set a higher business tax. Similarly, lowering commuting costs leads to a smaller business tax rate in the central city because the location of jobs is more sensitive to a change in the spatial difference in business tax rates for low commuting costs. Put differently, the elasticity of the tax base with respect to the tax rate increases when τ falls. The case of positive agglomeration economies. We now assume that α > 0. Solving (28) with respect to T0 yields the best-reply function of the central city’s government: T0∗ (T ) =

T α τ + (λm − 1)(B − b) + (B − 3b), 2 2 4

(34)

which is linear and upward sloping. In this expression, everything being equal the second term of the right-hand side shows how the central city’s government capitalizes on the agglomeration 8

Note that (33) is equivalent to (37) in which α = 0.

24

economies to charge a higher tax rate when α rises. However, we will see below that T ∗ depends on α in a non-monotone way, thus suggesting that the global impact of α on T0∗ is a priori ambiguous. Similarly, using (7), (8) and (21) and solving (31) with respect to T , we obtain the best reply of a suburban government: T ∗ (T0 ) = −

4α α T0 + [2αm(B − (2λ − 1)b) − 2α(B − b) − τ (B + 3b)] , D D

(35)

where

3τ − 6α − 2αm + 4αλ + 4αλm 1−λ is positive because α < α ¯ (see (22)). Therefore, the best reply function of a suburban government D≡

is linear and downward sloping. When α = 0, the foregoing expressions boil down to T ∗ = 0 and T0∗ = τ (B − 3b)/4. Since the slopes of the best reply curves have opposite signs, the two curves intersect in the plane, and there always exists a unique tax equilibrium. Given that the central city’s government sets a positive tax when α = 0, we expect T0∗ to be positive when there are agglomeration economies. We show in Appendix B that T0∗ is positive for all α ≥ 0. Thus, regardless of the intensity of agglomeration economies, the central city’s government always sets a positive business tax. Last, the two taxes rates are interdependent if and only if α > 0, that is, when there are agglomeration economies. As shown by (35), the magnitude of the agglomeration economies has an impact on the extent to which the tax policy in the suburbs reacts to the central city’s tax policy. More precisely, stronger agglomeration economies makes T ∗ more sensitive to T0∗ . As a consequence, the presence of agglomeration economies erodes the fiscal autonomy of the suburban jurisdictions. Regarding T ∗ , observe that (31) may be rewritten as follows: T ∗ = α(1 − λ) [(y ∗ (T0∗ , T ∗ ) − b) − (B − y ∗ (T0∗ , T ∗ ))] .

(36)

Therefore, the sign of T ∗ depends on the relative numbers of individuals working in the CBD. If y ∗ − b > B − y ∗ , i.e., a majority of the suburban residents work in the CBD, then T ∗ > 0. This is because the overall welfare of the suburban residents is less dependent on the attractiveness of their employment centers. As consequence, the suburban governments have less incentives to grant subsidies to firms while the positive business CBD-tax rate allows the cross-border commuters to enjoy a higher wage through stronger agglomeration economies in the CBD. In contrast, if the majority of residents work in their SBD, the suburban governments subsidies firms to boost the productivity of their employment centers, which in turn increases

25

the wage of the residents working therein. In sum, the sign of T ∗ depends on where the median resident of a suburban jurisdiction works. Using (36), the suburban workers are equally shared between the CBD and the SBD when α solves the equation y ∗ (α) = (B + b)/2, that is, α0 ≡

τB (1 − λ)m(b + B)

where b < y ∗ (α0 ) < B. Since (36) is convex, when 0 < α < α0 , T ∗ is negative and decreasing, then, increases with α. This tax rate becomes positive when α > α0 , that is, when the suburbanites commuting to the CBD are more numerous than those working in their SBD. On the other hand, (34) implies that the relationship between α and T0∗ is ambiguous as long as the agglomeration economies have not reached the value that minimize T ∗ . Once α exceeds this threshold, T0∗ grows with α. Symmetrizing (31) and plugging (28)-(31) into (21) implies y∗ =

1 τ (B + 3b) + 2α(λm + λ − 2)(B + b) , 2 3τ − 2α(m + 1 − 2λm) − 2α(1 − λ)

(37)

which always increases with α and decreases with τ . In other words, despite the positive tax differential, stronger agglomeration economies render the CBD more attractive through a higher number of firms located therein. In presence of agglomeration economies, both T0∗ and T ∗ depend on the central city size. How the tax policy of the suburban jurisdictions is affected by b is a priori ambiguous. Indeed, T ∗ depends on the relative numbers of suburbanites working in the CBD, which varies with b. Since y ∗ first increases with b, the number of suburbanites working in the SBD (B − y ∗ ) fall when the limit of the central city expands. On the other hand, the effect on the number of suburbanites working in the CBD (y ∗ − b) is a priori unclear because it depends on the intensity of the agglomeration economies. Using (37), it is straightforward to show that ∂ (y ∗ − b) 3τ ≷ 0 when α ≷ 0, ∂b 3τ − 2α(m + 1 − 2λm) − 2α(1 − λ) which means that a bigger central city always leads the suburban jurisdictions to set a higher business tax rate. 26

As for the tax differential, it is given by T0∗ − T ∗ = −

T∗ α τ + (λm − 1)(B − b) + (B − 3b), 2 2 4

(38)

which is positive for all α and τ if the right-hand side of this expression is positive when T ∗ takes on its highest value. It follows from (36) that the largest value of T ∗ is reached when y ∗ = B, ∗ ∗ ≡ α(1 − λ)(B − b). Plugging Tmax into (23) shows that T0∗ > T ∗ if λ > 2/(m + 1) so that Tmax

and B > 3b.9 The former inequality is slightly more stringent than our condition λ > 1/m while the latter means that the population of the central city (mb) is less than one third of the total population (L) of the MA. Note that these two conditions are sufficient but not necessary for T0∗ to exceed T ∗ . The next proposition comprises a summary. Proposition 2. Assume a polycentric MA. Then, the central city’s government always sets a positive business tax and the suburban governments subsidy (tax) firms when the agglomeration economies are weak (strong). Furthermore, the business tax set in the central city exceeds that set in the edge cities when (i) the city population size is not too large relative to the population of the MA and (ii) the spillovers between the CBD and the SBDs are not too weak. It remains to check that the MA is polycentric (y ∗ < B) at the tax equilibrium. This is so if and only if

α 1 5B − 3b < , τ 2 2[B(m + 1) − b] − λ[2mB + (m + 1)(B − b)]

(39)

which is positive if λ
3b is necessary and sufficient for tax-exporting to arise when α = 0.

27

When y ∗ (T0∗ , T ∗ ) = B, we have T ∗ = α(1 − λ)(B − b) where T ∗ has the nature of a shadow tax since there is no SBD. Plugging the highest value of τ compatible with (39) shows that (40) is the highest tax rate in the MA such that y ∗ (T0∗ , T ∗ ) = B. Therefore, the central city behaves as if it were a monopolist setting the limit-tax that deters the emergence of SBDs. Put differently, agglomeration economies are now sufficiently strong for the central city government to choose a tax rate that blockades the entry of SBDs. Note (40) is the highest rate that satisfies this property. 4.2.2

Property tax

It remains to determine the equilibrium property taxes. Once T ∗ is determined, the property tax revenue of a suburban jurisdiction must be such that F +c

ℓ2 = T ∗ (B − y ∗ ) + t∗ ALR. 2

Using this expression and (24), we obtain the equilibrium property tax given by 2

F + c ℓ2 − T ∗ (B − y ∗ )

∗

t = τ

(y∗ −b)2 2

+

(B−y∗ )2 4

2

− F − c ℓ2 + T ∗ (B − y ∗ )

.

(41)

This highlights how the property tax set in the edge cities interact in a complex way with the pattern of agglomeration economies. Indeed, t∗ is affected by a higher α through three main channels: (i) negatively through the aggregate land rent, (ii) positively through the business tax basis, and (iii) positively or negatively through the level of business tax. For this reason, it is hard to make any prediction about the impact of agglomeration economies on t∗ . The property tax may also increase or decrease with the central city size. Indeed, whereas ∗

t may increase with b because the business tax basis of a suburban jurisdiction shrinks, the opposite effect may arise because, the suburban jurisdiction becoming less populated, the public services are less costly to provide. Obviously, the latter effect dominates the former when c is high enough. Moreover, as demonstrated above, the number of cross-commuters from the suburbs (y ∗ − b) and, then, the aggregate land rent may increase or decrease with b depending on the value of α. As for the central city, its budget constraint implies that the equilibrium property tax satisfies the relationship:

F + cℓ20 /2 − T0∗ my ∗ = ALR0 ∗ where ALR0 depends on t0 . There is no need to solve this equation, however. Indeed, once t∗0

T0∗ my ∗ is determined, the value of t0 ALR0 is constant regardless of the value of t0 (see (26)). Put differently, once the business tax is chosen, the budget constraint is satisfied through the adjustment of the land tax base ALR0 only. Thus, there is a continuum of property tax 28

equilibria. Note, however, that the actual value of t0 has no impact on the common utility level in the MA, so that any equilibrium value may be chosen.

5

On the efficiency of the decentralized outcome

Being given by a Nash equilibrium, the tax competition outcome is inefficient. However, we do not know how this inefficiency manifests itself and how to correct it.

5.1

Is the CBD too large or too small?

Consider first the case of no agglomeration economies. Comparing (15) and (37) when α = 0 yields

B − 3b < 0. (42) 6 Thus, the inefficiency of tax competition takes the form of too small a CBD. In other words, y ∗ − y¯ = −

the business tax rate set in the central city is too high when compared with the rate set in the edge cities. Any common business tax rate T¯ set by all the local governments would imply that y ∗ = y¯. In this case, the central city would benefit from a high business rate because the tax exporting effect T¯m(¯ y − b) increases linearly with T¯ . In this case, it follows from (27) that consumers in the central city are better off when T¯ is larger. Conversely, because (29) decreases with T¯ , residents in the suburban jurisdictions are worse off. As a consequence, even in the absence of agglomeration economies, the central city and the edge cities have opposing interests when they strive to choose a common tax policy. In the presence of agglomeration economies, tax harmonization ceases to be optimal. To show it, assume that all jurisdictions set the common tax rate Tˆ. The economic boundary of the central city is then given by y ∗ (Tˆ ) =

B[τ + 2α (λm − 1)] . 3τ − 2α(m + 1 − 2λm)

Since y ∗ (Tˆ ) < y¯, tax harmonization yields a spatial distribution of firms that remains inefficiently dispersed when α > 0. In other words, the equilibrium size of the CBD remains smaller than its optimal size, whereas the SBDs are too large. For y ∗ (T0 , T ) = y¯ to hold, the tax differential must be equal to ∆O =

ατ B (m + λm − 2) . 4α[1 + m(1 − 2λ)] − 3τ

(43)

It is readily verified that ∆O < 0 for all y¯ < B. Hence, in a polycentric MA, for the distribution of jobs to be socially optimal the business tax rate in the suburban jurisdictions must exceed the rate set in the central city. Recall that tax competition leads to the opposite outcome. 29

To summarize, we have: Proposition 3. Assume a polycentric MA. Then, corporate tax competition leads to an insufficient concentration of jobs in the CBD. Thus, tax competition between jurisdictions makes the MA less productive by preventing a better exploitation of agglomeration economies through a higher concentration of firms and jobs. Furthermore, besides the laboriousness of gathering the information needed to compute ∆O , the central city and the suburban jurisdictions still have conflicting interests in choosing specific tax rates. This points once again to the difficulty of making the MA work when this one is fragmented in competing jurisdictions. To illustrate the nature of the difficulty, let us consider the following argument. Differentiating W0 + mW with respect to T0 , using the first-order condition (31) and simplifying yields the following expression: m

dW dT0

= αm (m − 1) [(y ∗ − b) (1 − λ) + (B − y ∗ ) λ] T0 =T0∗ ,Ti =T ∗

dy − m(y ∗ − b) < 0. dT0

(44)

Since W0 is concave, the above expression implies that, given T ∗ , the equilibrium tax T0∗ exceeds the tax rate that maximizes W0 +mW . Similarly, it follows from (27) that the derivative of W0 + mW with respect to Ti is equal to m (y ∗ − b) > 0. Given T0∗ , this implies that T ∗ is lower than the tax rate maximizing total welfare. Unlike the tax policy of the central city, the suboptimality of T ∗ does not depend on the intensity of the agglomeration economies. Since a fraction of the suburbanites work in the central city, suburban governments internalize the impact of their tax policy on the agglomeration economies at work in the CBD.

5.2

Does redrawing the central city limit remedy the misallocation of jobs?

Since tax coordination is hard to implement and since the central city’s population size influences the extent of the distortion created by tax competition within the metro, it is natural to ask whether changing the central city limit prior to the tax game is a desirable strategy. From (42), the productive efficiency loss generated by the misallocation of jobs (¯ y − y∗) decreases when the central city population rises (d(¯ y − y ∗ )/db < 0). Hence, the productive efficiency loss generated by the misallocation of jobs decreases. Indeed, when b increases the central city chooses to set a lower business tax because the fraction of workers residing outside its limit decreases, that is, the extent of tax exporting shrinks, which leads more firms to set up in the CBD. The following proposition is a summary. Proposition 4. Assume a polycentric metropolitan area. Then, the productive efficiency loss decreases when the relative size of the central city population rises. 30

This proposition suggests the following question: how can the planner choose the central city limit? The boundary b at which the misallocation of jobs vanishes under tax competition is given by the solution ˜b to the equation y ∗ (b) = y. When α = 0, it is readily verified that the solution to y ∗ (b) = y¯ is equal to y¯. In this case, there is no cross-border commuting, hence no tax exporting. However, in doing so, the planner does not choose the optimal boundary because ˜b = y > b. When α > 0, it is readily verified that ˜b exceeds y¯. As a result, we have ˜b > b. Therefore, for all α ≥ 0 the central city size exceeds its optimal size, whereas the suburban jurisdictions are too small. Conversely, choosing b for the limit of the central city leads to an insufficient concentration of jobs in the CBD because y ∗ (b) < y. In sum, under corporate tax competition, we have: Proposition 5. Assume a polycentric metropolitan area. When the population size of the central city is optimal the CBD is too small, whereas the central city is too large when the size of the CBD is optimal. The above proposition suggests a second-best approach in which the planner chooses the central city limit that maximizes the total welfare within the MA. In other words, the planner first chooses the welfare-maximizing value of b, and then lets consumers, firms and local governments to pursue their own interest. In what follows, we focus on the case where α = 0. The central city limit maximizing total welfare when T0 and T are given by the corresponding equilibrium tax rates is given by b∗ =

B (8c + τ ) 3τ + 8c (m + 1)

where y ∗ (b∗ ) > b∗ > b for m ≥ 2 and B > b∗ if B > y ∗ . Hence, as in the first-best solution, the second-best outcome under tax competition involves political fragmentation and a decentralized supply of public services. However, unlike the first-best solution, suburban jurisdictions are now smaller than the central city. Because tax competition favors the edge cities at the expense of the central city, the second-best approach aims to reduce this distortion by launching a bigger central city. The planner may also determine the degree of fragmentation of the MA maximizing the total welfare WTs = W0 + mW , which is implicitly given by the equilibrium condition: ∂WTs 3τ = −F + ∂m 16

L m

2

− b2 +

c 2

L m

2

− b2 (2m + 1) = 0

(45)

with ∂ 2 WTs /∂m2 < 0. As in the first-best analysis, for a given administrative boundary b, the second-best number of suburban jurisdictions increases with the population size and the cost parameter capturing the crowding effect of public services, whereas it decreases with investment outlays. 31

Tax competition incentivizes the planner to establish a more fragmented MA than in the first-best configuration. Indeed, the optimality condition (45) evaluated at b = b and m = m implies that ∂WTs ∂m

= m=m,b=b

3τ 16

L 3m

2

2

−b

,

which is positive for all m ≥ 2. Therefore, the planner raises the number of edge cities, that is, reduces their population size, to alleviate the efficiency loss associated with the insufficient concentration of firms in the CBD. Last, when α = 0, it is straightforward to show that a larger number of suburban jurisdictions leads the central city government to decrease its tax rate in higher proportion than the authorities of the edge cities, thus reducing the misallocation of jobs. In this case, the central city competes with more rivals. Therefore, like in standard oligopoly theory, a larger number of suburban jurisdictions exacerbates tax competition and reduces the misallocation of jobs. To sum up, we present the next proposition. Proposition 6. If the planner chooses the welfare-maximizing limit of the central city, the second-best central city is bigger than the first-best central city. Furthermore, the resulting metropolitan area involved several jurisdictions while there is commuting from the suburbs to the center. Equally important, the planner can improve the efficiency of the distribution of firms by investing in transport infrastructure within the whole MA. Indeed, lowering commuting costs entices the central city government to decrease its business tax rate. This in turn leads more firms to set up in the CBD, which reduces the productivity losses generated by political fragmentation and tax competition. In this case, the choice of the boundary of the central city becomes less critical.

6

The consumption of public goods

Thus far, we have neglected (i) the possibility for the suburbanites working in the CBD to consume the public services provided in the central city and (ii) the fact that the central city provides a broader range of public services than the suburban jurisdictions. In this section, we discuss what our main findings become in each of these two cases.

6.1

Public good spillovers

That suburbanites consume the public services supplied in the central city is known to be a major source of distortion in the allocation of public resources within a MA. For this reason, we assume here that suburbanites working in the CBD benefit from both the public services 32

provided in their own jurisdiction and in the central city. These consumers enjoy a utility level equal to 2G from consuming all the public services. What do our main findings become in the presence of those spillovers? For conciseness, we consider the simple case where α = 0. The optimal metropolitan area. In presence of public good spillovers, the analysis of Section 3 is no longer valid because the planner faces an additional trade-off. On the one hand, the suburbanites enjoy an additional utility gain given by m (y − b) G; on the other hand, the cost of the public services provided by the central city is raised by c (my)2 /2− c (mb)2 /2. If the total utility change is non-positive, that is, G ≤ cm (b + y) /2, the analysis of Section 3 holds true because consuming the public services supplied in the central city makes the suburbanites worse-off. When G > cm (b + y) /2, the social welfare function becomes WT = mBG + m (y − b) G −

mτ 2 y − mτ 2

B−y 2

2

c cm (B − b)2 − (m + 1)F. − (my)2 − 2 2

(46)

Solving the first-order conditions for welfare maximization with respect to b and y yields ¯bsp = B − G < B c

sp

y¯ =

2G + Bτ 3τ + 2cm

(47)

where ¯b > 0 if and only if cB exceeds G; otherwise, ¯b = 0. sp

sp

The question addressed here is to figure out how public good spillovers affect the optimal sp sp organization of the MA. One solution consists in comparing y¯ with y¯ and ¯b with ¯b. The expression (47) shows that the central city population size shrinks whereas its economic size expands with G. More precisely, when G is large enough for G > cm ¯bsp + y¯sp /2 to hold, we always have ¯bsp < ¯b while y¯sp > y¯ if and only if G > cmB/3. Under these circumstances, the presence of public good spillovers leads the planner to choose a smaller central city but a larger CBD, the reason being that the value of m (y − b) G in (46) is high. As a result, the sp sp discrepancy between y¯ and ¯b gets wider. In sum, when the utility of the public services is sufficiently high, cross-border consumption is always socially desirable. The decentralized outcome. The indirect utility of a consumer living and working in an edge city remains unchanged, but the indirect utility of a suburbanite working in the central city is now given by vi0 (x) ≡ Vi0 (x)+G. Equalizing these indirect utilities yields the equilibrium economic boundary of the central city: sp

y (T0 , T ) =

B 2[G − (T0 − T )] + , 3 3τ

which increases with G because the central city is more attractive to the suburbanites. The sp

sp

sp

objective function of the central city government is given by W0 = W0 (y ) − c(my )2 /2 + 33

cℓ20 /2 while the suburban governments maximize W sp = W (y sp ) + (y sp − b) G. The first order conditions are given, respectively, by sp

dW0 =m dT0 which implies sp

T0 =

2T0 2cm y −b − + 3τ 3τ sp

sp

dW dT

(3τ + 2cm) (Bτ + 2G) − 9bτ 2 4(3τ + cm)

=−

2T , 3τ

T sp = 0.

(48)

sp

Standard calculations reveal that dW0 /dT0 > 0 when T0 = T0∗ . As a result, the business tax set by the central city is higher in the presence than in the absence of public good spillovers. The reason is easy to grasp. More workers lure the CBD because they can enjoy the public services provided by the central city. This entices more firms to set up there. The extent of tax exporting thus increases, whereas the provision cost of public services is higher. Both effects lead the central city to increase its business tax. In the suburban jurisdictions, the effects are opposite. On the one hand, fewer firms locate in a SBD, so that the corresponding local government collect less income from firms. On the other, a share of the residents enjoy the consumption of more public services. The two effects cancel each other, so that the tax rate set by the suburban jurisdictions does not change (T sp = T ∗ = 0). The existence of public good spillovers has both expected and unexpected redistributional implications for consumers living in the central and peripheral jurisdictions. First, the outcommuting suburbanites benefit from more public services whereas the central city residents bear a higher provision cost for the public good. What is more, because the business tax paid sp

by the CBD firms is higher (T0 > T0∗ ), the central city workers get a lower pay. In contrast, the SBD workers earn the same wage because T

sp

= T ∗ = 0. As a consequence, the central city

residents are hurt twice through an externality effect and an income effect. Therefore, the freeriding problem between the central city and the suburban jurisdictions has implications that go beyond the standard consumption effects generated by spillovers. This makes the cooperation between the central and edge cities even more compelling for the MA to be efficient. The economic boundary of the central city y sp

y −y

sp

=

sp

is now such that

3τ + cm sp y −b . 3τ

sp

sp

sp

Hence, when y −b > 0, there is again insufficient concentration of jobs in the CBD (¯ y > y ). However, it is not clear whether the presence of spillovers exacerbates the misallocation of jobs within the MA. Indeed, although the consumption of the central city public services by suburbanites tends to generate more jobs in the CBD, the higher business tax set by the central city deters firms to locate in the CBD. The ultimate impact depends on the parameter values. To summarize, 34

Proposition 7. Assume a polycentric MA. If the suburbanites working in the CBD consume the central city’s public goods, the tax differential between the central city and the edge cities rises. Furthermore, the size of CBD remains too small.

6.2

The central city as a bigger provider of public goods

In the real world, the central city often supplies a broader range of public services than the suburban jurisdictions. It is, therefore, legitimate to ask what the above findings become in such a context. To show it, we assume that the central city provides a public good of size βG with β > 1. Alternatively, if G is a CES-bundle of differentiated public services, βG represents a range of more differentiated services. The optimal population size of the central city increases with β because of the higher utility stemming from the consumption of the public services: ¯bβ = ¯b + β (G − 1) > ¯b. c (m + 1) As a result, the institutional structure of the MA now depends on the relative provision of public services between the central city and the suburban jurisdictions, β. The optimal size of the central city increases with the range of public services it provides, whereas the suburban jurisdictions shrink. In the limit, the planner chooses to have a single jurisdiction if (β − 1) G/cm > B, a condition that is unlikely to hold in a large MA. Supplying a wider range of public services in the central city does not affect its optimal economic boundary as long as the suburbanites do not consume these services. Thus, the optimal MA may be institutionally fragmented while having an integrated labor market or may involve a single jurisdiction together with several employment centers. Moreover, since the business tax competition process does not depend on G, the asymmetry in the provision of public services has no impact on the jurisdictions’ business taxes, and thus Propositions 2 and 3 remain valid. When suburbanites working in the CBD consume the central city public services, the central city becomes even more attractive. In this case, the discrepancy between the optimal administrative and economic boundaries is exacerbated. Indeed, the limit given by (47) is unaffected because the planner has no reason to differentiate across CBD-workers. On the contrary, y¯sp increases with β because more consumers are able to enjoy the wider array of public services provided by the central city. Furthermore, as shown by (48) in which G is replaced with βG, the central city’s government increases its business tax. As a consequence, the tax differential widens and Proposition 3 holds true.

35

7

Concluding Remarks

Metropolitan areas are non-legal entities that play a key role in the economic development of emerging and developed countries alike. This probably explains why political scientists have long been interested in issues related to metropolitan governance. The earliest approach that we are aware of - the regionalism approach that continues to shape the political debates - views the multiplicity of political jurisdictions as inherently inefficient. Political fragmentation would limit the ability to deal with area-wide urban problems that transcend local jurisdictions. The prescription is then to promote metropolitan governments and a better correspondence between political and functional or economic areas (Hochman et al.,1995). In contrast to this view, the public choice approach, based on Tiebout (1956) and Ostrom, Tiebout, and Warren (1961), does not see systematic inefficiency in the polycentric political organization of a MA. Similar to market economies where firms compete to offer the best good at the best price, political fragmentation would allow residents to select the jurisdiction that offers them the best package. Our model delivers a clear-cut message that strongly suggests an intermediate approach. Indeed, both the first-best and second-best solutions involve a certain degree of decentralization of public services within the MA as well as an economic boundary of the central city that encompasses its administrative boundary. In addition, redrawing the boundary between the central and suburban jurisdictions does not allow reaching the first best, but such a policy may damper the inefficient allocation of firms and jobs across employment centers. This points to the need for multifunctional governance: “small” things should be managed by local jurisdictions, and “big” things by a metropolitan government. Labor and transport issues in particular should be handled at the metropolitan level. Although derived from a simple model, these conclusions are sufficient to show that policy recommendations based on the regionalism and public choice approaches are unwarranted. Although we recognize that political fragmentation is not bad per se, the tax competition process leads to an inefficient distribution of firms and jobs. This leads us to formulate some policy recommendations in the spirit of what is known as “New Regionalism” (Savitch and Vogel, 2000) - mixing a polycentric political system with inter-municipal cooperation to solve mutual problems. Our analysis also shows that some policies must be conducted at the level of the MA as a whole. For example, an integrated transportation policy that aims to lower commuting costs will increase the overall productivity of the MA. Simultaneously, it will also increase the efficiency loss generated by tax competition. In this respect, our results give credence to the large transportation projects that are being developed in Greater London (Crossrail) and Greater Paris (Grand Paris Express), but it suggests supplementing these projects with other instruments to magnify their potential positive impact.

36

Our framework can also serve to address more controversial issues in local public finance and transportation economics. According to Inman (2009), rethinking the governance of the MAs through Business Improvement Districts (BIDs) and Neighborhood Improvement Districts could be a way to improve the fiscal performance of a large MA. BIDs are business associations and can be considered self-financing private governments that offer supplemental services to their members. By restoring market-driven incentives in location choices, the development of BIDs within the central city would make it more attractive, thus strengthening agglomeration economies. Helsley and Strange (1998) analyze such organizations and show that their welfare effects on consumers are ambiguous and complex. However, their analysis should be extended to the case of a genuine urban framework with the aim of determining the impact of private governments on the spatial distribution of firms and consumers. Our analysis supports what seems to be the minimal set of requirements needed to promote more efficient MAs. Yet, it is fair to say that our findings have been obtained under several simplifying assumptions; thus care is needed in interpreting them. First, we did not address competition in public goods, an issue that is notoriously difficult, especially because many models are plagued with the non-existence of a Nash equilibrium. In this respect, it is worth noting that our model may be interpreted as one in which jurisdictions avoid the damaging effects of a race to the bottom by coordinating their supply of public services. Therefore, even in the absence of such distortions, our analysis has unveiled new sources of market failure. Moreover, it is well known that one political and social difficulty encountered within a MA stems from the heterogeneity of households that cluster in specific neighborhoods, which in turn generates spatial discrimination across socioeconomic groups. This issue has been tackled in the monocentric city model of urban economics but has not been explored in the context of a polycentric MA. Lastly, we did not allow consumers to choose a variable lot size by trading the homogeneous good against land. Several of our results remain valid when the population density is not uniform anymore but the determination of the equilibrium land rent within each jurisdiction is a more delicate issue.

References Baldwin, R.E. and P.R. Krugman (2004) Agglomeration, integration and tax harmonization. European Economic Review 48: 1-23. Baum-Snow, N. (2010) Changes in transportation infrastructure and commuting patterns in U.S. metropolitan areas, 1960-2000. American Economic Review. Papers & Proceedings 100: 378-82. Baum-Snow, N. (2014) Urban transport expansions, employment decentralization, and the

37

spatial scope of agglomeration economies. Mimeo. Braid, R. (1996) Symmetric tax competition with multiple jurisdictions in each metropolitan area. American Economic Review 86: 1279-90. Braid, R. (2002) The spatial effects of wage or property tax differentials, and local government choice between tax instruments. Journal of Urban Economics 51: 429-45. Braid, R. (2010) Provision of a pure local public good in a spatial model with many jurisdictions. Journal of Public Economics 94: 890-7. Brülhart, M., S. Bucovetsky and K. Schmidheiny (2015) Taxes in cities. In: G. Duranton, J.V. Henderson and W. Strange (eds.) Handbook of Regional and Urban Economics. Volume 5, forthcoming. Cheshire, P. and S. Magrini (2009) Urban growth drivers in a Europe of sticky people and implicit boundaries. Journal of Economic Geography 9: 85-116. Combes, P.-P., G. Duranton, L. Gobillon, D. Puga and S. Roux (2012) The productivity advantages of large cities: Distinguishing agglomeration from firm selection. Econometrica 80: 2543-25. Cremer, H. and P. Pestieau (2004) Factor mobility and redistribution. In: J.V. Henderson and J.-F. Thisse (eds.). Handbook of Regional and Urban Economics. Volume 4. Amsterdam: Elsevier, 2529-60. Ellison, G., E.L. Glaeser and W.R. Kerr (2010) What causes industry agglomeration? Evidence from coagglomeration patterns. American Economic Review 100: 1195-213. Epple, D. and A. Zelenitz (1981) The implications of competition among jurisdictions: Does Tiebout need politics? Journal of Political Economy 89: 1197-217. Fujita, M. (1989) Urban Economic Theory. Land Use and City Size. Cambridge: Cambridge University Press. Fujita, M. and H. Ogawa (1982) Multiple equilibria and structural transition of non-monocentric urban configurations. Regional Science and Urban Economics 12: 161-96. Glaeser, E. L. (2013) Urban public finance. In: A.J. Auerbach, R. Chetty, M. and E. Saez (eds.). Handbook of Public Economics. Volume 5. Amsterdam: Elsevier, 195-256. Glaeser, E.L. and J.D. Gottlieb (2009) The wealth of cities: agglomeration economies and spatial equilibrium in the United States. Journal of Economic Literature XLVII: 983-1028. Glaeser, E.L. and M.F. Kahn (2001) Decentralized employment and the transformation of the American city. Brookings-Wharton Papers on Urban Affairs (2): 1-47. Glaeser, E.L. and M.E. Kahn (2004) Sprawl and urban growth. In: J.V. Henderson and J.-F. Thisse (eds.). Handbook of Regional and Urban Economics. Volume 4. Amsterdam: Elsevier, 2481-527. Haughwout, A. and R. Inman (2009) How should the suburbs help their central city?

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Growth- and welfare-enhancing intrametropolitan fiscal distributions. Annals of the American Academy of Political and Social Science 621: 39-52. Helsley, R.W. and W.C. Strange (1998) Private government. Journal of Public Economics 69: 281-304. Hochman, O., D. Pines and J.-F. Thisse (1995) On the optimal structure of local governments. American Economic Review 85: 1224-40. Hoyt, W.H. (1991) Property taxation, Nash equilibrium and market power. Journal of Urban Economics 30: 123-31. Inman, R. (2009) Making Cities Work. Princeton (NJ): Princeton University Press. Janeba, E. and S. Osterloh (2013) Tax and the city - A theory of local taxcompetition. Journal of Public Economics 106: 89-100. Lucas, R.E. and E. Rossi-Hansberg (2002) On the internal structure of cities. Econometrica 70: 1445-76. Noiset, L. and W. Oakland (1995) The taxation of mobile capital by central cities. Journal of Public Economics 57: 297-316. OECD (2006) Competitive Cities in the Global Economy. OECD Territorial Reviews. Ostrom, V., C. Tiebout and R. Warren (1961) The organization of government in metropolitan areas: a theoretical inquiry. American Political Science Review 55: 831-42. Rhode, P.W. and K.S. Strumpf (2003) Assessing the importance of Tiebout sorting: local heterogeneity from 1850 to 1990. American Economic Review 9: 1648-77. Savitch H.V and R. Vogel (2000) Paths to New Regionalism. State and Local Government Review 32: 158-68. Starrett D. (1981) Land value capitalization in local public finance. Journal of Political Economy 89: 306-27. Tiebout, C.M. (1956) A pure theory of local public expenditures. Journal of Political Economy 64: 416-24.

Appendix A. Assume α = 0. If y ∗ < b, the total welfare in a suburban jurisdiction becomes W = (1 − T )(B − b) − τ − 2

B − y∗ 2

2

τ 2

B + y∗ −b 2

2

ℓ2 + T (B − y) − F − c , 2

where y ∗ is still given by (21).

39

As for the total welfare in the central city, it is now given by the following expression: W0 = (1 − T0 )my ∗ + m(b − y ∗ ) (1 − T ) + T0 my ∗ −mτ

(y ∗ )2 − mτ 2

B − y∗ 2

2

−

B + y∗ −b 2

2

ℓ2 − F − c 0. 2

In this case, the equilibrium tax rates are as follows: T0∗ =

τ (b − B/3) 4

T ∗ = τ (b − B/3) ,

(A.2)

where T ∗ > T0∗ > 0 for all b > B/3. Observe that the tax wedge is identical to that obtained in our benchmark case (b < B/3). Therefore, for all b > B/3 the equilibrium economic boundary of the central city is still equal to y ∗ = B/6 + b/2 when y ∗ < b. B. The Nash tax equilibrium is derived from the two reaction functions (34) and (35). Solving these two equations yields T0∗ =

B 2α [2α(m − 1) − τ ] + D [τ + 2α(λm − 1)] 4 2α + D b 2α [3τ − 2αm(m + 1) + 4αλm] + D [3τ + 2α(λm − 1)] − , 4 2α + D

and

2αB [τ − αm(1 − λ)] (1 − λ) 2α2 bm + . 2α + D 2α + D To show that T0∗ > 0, consider T as the abscise and T0 as the ordinate of the plane. The T∗ = −

expression (34) shows that T0 = 0 when T1 = −α(λm − 1)(B − b) − τ (B − 3b)/2 < 0. The best reply of a suburban government (35) yields T2 =

8α [αm(B − (2λ − 1)b] − 2α(B − b) − τ (B + 3b) D

(49)

when T0 = 0. Since T2 > T1 when α = 0 and T2 grows while T1 decrease with α, it is straightforward that T0∗ (T ) and T ∗ (T0 ) always intersect in the positive half-plane T0 > 0.

40

Figure 1. The spatial pattern of the metropolitan area (m=8)

SBD

CBD b

y

L/m

Suburban city Central city

Figure 2. Equilibrium Land Rent Land Rent

R0 (x )

Rii (x ) Ri0 (x )

x

b

y

x

s

L m