Tidiane Ly
email: tidiane.ly@usi.ch.

Pierre-Philippe Combes

Florence Goette-Nagot

William Hoyt

Étienne Lehmann

Thierry Madiès

Steeve Mongrain

Raphaël Parchet

Sonia Paty

Stéphane Riou

Kurt Schmidheiny

Elisabet Viladecans- Marsal

John D Wilson

Taxes, trac jam and spillover in the metropolis

Keywords: Tax competition, Urban economics, Trac congestion, Public goods, Mobility JEL: H71, H72, R50, R51

published or not. The documents may come

Taxes, traffic jam and spillover in the metropolis

1. Introduction

Introduction

By 2100, 85% of the world's population is expected to live in metropolitan areas (MAs) (UN, 2013).

1 Today, this gure already reaches more than 50% of the world's population and roughly 80% of the OECD countries' population. Meanwhile, the economic strength of numerous MAs is comparable to that of countries. MAs cover only 4% of the land area in the OECD countries but account for 55% of their GDP (OECD, 2015). Because MAs are areas of residence for most people and are creators of substantial economic wealth, their governance is a paramount concern for policy makers worldwide. The French President Nicolas Sarkozy's statement is illustrative: The Greater Paris is at the core of our strategy of attractiveness and economic recovery. This project concerns all elected representatives.

2

The gouvernance of MAs has two outstanding features compared to country governance.

First, MAs are characterized by competition among many sub-metropolitan or local governments (e.g. counties, municipalities, districts, and townships) for mobile jobs and mobile residents. The governance of MAs is split among numerous local governments which often have signicant autonomy to raise local revenue from residential, business and property taxes for example, to nance a range of public services such as schools, amenities (e.g. public parks) and safety. 3 Mobility of private agents and autonomy of local public policy are a fertile ground for competition among local governments.

Second, MAs are characterized by strong agglomeration of jobs in MA centers and high mobility of households as residents and workers. Typically, MAs are composed of a large central city surrounded by numerous small suburban towns. Many workers are attracted by access to jobs and high wages in the central cities which cluster most activities. This concentration of workers (and often also residents) in the central cities creates negative agglomeration externalities such as congestion on amenities and public goods/services and various urban disturbances (noise, waste, etc.). Expensive housing in the city spurs many residents to settle in suburban towns where rents are more aordable. 4 These suburbanites are often obliged to undertake long commutes to work in the city, which generate negative commuting externalities such as trac congestion and air pollution.

Introduction

The main results of the paper are summarized below.

1. If towns can retain part of their workers using labor subsidies, the decentralized equilibrium is Pareto-ecient. As intermediate outputs, the resulting original rst-best local policies are characterized: public good provision optimally internalizes residents' and commuters' preferences; residential taxes guarantee optimal access to business districts; capital is not taxed.

This result highlights the key role of local labor subsidies in the regulation of trac congestion in MAs. However, in practice, this instrument is generally not available to municipalities for three main reasons. First, subsidies are often considered violations of competition rules. Therefore, legislation in the European Union, Canada, and Australia for example, heavily limits use of subsidies (OECD, 2010;[START_REF] Thomas | Investment incentives and the global competition for capital[END_REF]. 9

Second, subsidies are expensive. On the contrary, local labor taxation is often enacted as a budgetary tool to supplement other local revenue sources. For example, the introduction of payroll taxes in San Francisco in 1970 followed this logic [START_REF] Sherwood-Call | The labor tax as an alternative revenue source[END_REF].

10 Third, competition among local governments can lead to a subsidy race leading to rentseeking behavior by rms [START_REF] Tannenwald | Are state and local revenue systems becoming obsolete?[END_REF].

2. If towns are not allowed to subsidize labor, trac congestion in the city becomes exessive, and local governments respond by setting inecient second-best public policies: (i) the city over-taxes capital which leads to too little capital in the city; (ii) the city undertaxes residents which leads to too many residents in the city; (iii) in the presence of commuting-induced spillovers, local public goods are under-provided by the city and over-provided by the towns.

In sum, if towns cannot retain their workers through the use of subsidies, the central business district (CBD) attracts too many workers from the suburbs, leading to severe trac congestion in the city. The municipalities then use the other policy instruments to dampen this excessive congestion: the city limits the size and attractiveness of the 9 In several US states, local governments are allowed to provide rms with development subsidies such as tax breaks, low-cost loans, grants, infrastructure support and low-cost land [START_REF] Warner | Business incentive adoption in the recession[END_REF].

Development subsidies dier from the at rate subsidies considered in this paper which are based on the municipality's total workforce. Development subsidies are bilateral commitments involving a local government and a rm. Since they are at the discretion of local politicians, they tend to favor big and highly visible projects at the expense of less visible ones [START_REF] Elkin | City and regime in the American republic[END_REF]. Moreover, since these subsidies are based on ex-ante agreements, once established in the municipality, the rm might not fulll its promises related to job creation or other performance criteria [START_REF] Sullivan | Business subsidies ad municipal controls[END_REF].

10 See [START_REF] Sjoquist | Local revenue diversication: User charges, sales taxes, and income taxes[END_REF] for an exhaustive list of the US states whose local governments use labor taxation. In France, labor was taxed by municipalities from 1975 to 1999 [START_REF] Bouvier | Les nances locales[END_REF]. [START_REF] Peralta | Political support for tax decentralization[END_REF] reports on cases in Australia, Austria, Greece, Korea and Mexico.

CBD, while the towns make their secondary business districts (SBDs) more attractive.

This result highlights in particular, the detrimental role of commuting-induced spillovers which amplify the damage caused by trac congestion by spurring the municipalities to engage in inecient public good provision.

The paper updates our understanding of the governance of MAs in the context of typical negative externalities such as trac congestion, pollution and other disturbances due to the agglomeration of jobs in MA centers and household mobility. The analysis suggests that local governments may be using many of their policy instruments to help to tackle these externalities. However, national governments often fail to give local authorities an important role to address these externalities; most political solutions so far are either national e.g. road tolls and gasoline/green taxation [START_REF] Pigou | Wealth and welfare[END_REF][START_REF] Walters | The theory and measurement of private and social cost of highway congestion[END_REF][START_REF] Vickrey | Pricing in urban and suburban transport[END_REF] or regional e.g. integrated public transport provision [START_REF] Preston | Integration for seamless transport[END_REF]. Further decentralization of the ght against MA negative externalities could be fruitful. Trac jams and CO2 emissions, for instance, can vary spatially to a signicant extent within a single MA. Thus, local governments which are better informed about voters' preferences and habits could be more exible and would be able to better target these externalities than national or regional governments. This echoes [START_REF] Oates | The Metropolitan Century: Understanding urbanisation and its consequences[END_REF] decentralization theorem.

The main policy recommendation to emerge from the analysis is that suburban towns should be allowed more leeway to retain their workers which might help to unclog MA roads and reduce externalities such as air pollution. Local labor subsidies which are usually missing in local policy instrument sets, could be an appropriate tool for this task. More generally, the paper suggests that addressing the externalities typical of MAs will require the development of truly polycentric MAs by encouraging the growth of strong secondary economic poles.

This recommendation is in line with [START_REF] Cavailhes | Trade and the structure of cities[END_REF]; our paper stresses that local governments could play an important role in this reshaping of MAs.

The paper is organized as follows. Section 2 reviews the related literature. Section 3 introduces the model. Section 4 presents the optimal public policy rules for a city and its surrounding towns with and without access to labor taxes/subsidies. Section 5 provides a numerical illustration. Section 6 discusses the results. Section 7 concludes.

Related literature

This paper contributes to several literature streams. It is novel in proposing the inclusion of a spatial urban structure (e.g. [START_REF] Fujita | Urban economic theory: land use and city size[END_REF] in the standard capital tax competition model developed by [START_REF] Zodrow | Pigou, Tiebout, property taxation, and the underprovision of local public goods[END_REF] and [START_REF] Wilson | A theory of interregional tax competition[END_REF]. Thus, it links the local public 2. Related literature economics (LPE) and the urban economics (UE) literatures.

11 ,12 The proposed model is also one of the rst to include both inter-jurisdictional residential mobility and commuting in a capital tax competition framework à la [START_REF] Zodrow | Pigou, Tiebout, property taxation, and the underprovision of local public goods[END_REF]. 13 It links capital tax competition models with residential mobility which exclude commuting (e.g. [START_REF] Wilson | Mobile Labor, Multiple Tax Instruments, and Tax Competition[END_REF][START_REF] Brueckner | A Tiebout/tax-competition model[END_REF] to models that include commuting but ignore residents' mobility across jurisdictions (e.g. [START_REF] Braid | Symmetric Tax Competition with Multiple Jurisdictions in Each Metropolitan Area[END_REF][START_REF] Braid | A spatial model of tax competition with multiple tax instruments[END_REF][START_REF] Peralta | Political support for tax decentralization[END_REF]. Finally, our study of local government responses to trac congestion links the LPE literature and the transport economics literature (e.g. [START_REF] Parry | Comparing the eciency of alternative policies for reducing trac congestion[END_REF][START_REF] Rouwendal | Basic economic principles of road pricing: From theory to applications[END_REF] which tends to consider trac congestion from a national/regional government perspective.

Recent work on LPE reveals a trend towards inclusion of more real-life features of MAs in the tax competition model à la [START_REF] Zodrow | Pigou, Tiebout, property taxation, and the underprovision of local public goods[END_REF]. First, following [START_REF] Braid | A spatial model of tax competition with multiple tax instruments[END_REF], some papers (e.g. [START_REF] Peralta | Political support for tax decentralization[END_REF][START_REF] Legras | Commuting in happyville: Taxation with interjurisdictional commuting and pollution[END_REF] include costly commuting in the tax competition model.

14

Commuting costs are purely distance costs and do not include trac congestion; commuters do not inict externalities on one another. Second, eorts have been devoted to include a core-periphery structure in tax competition models [START_REF] Janeba | Tax and the city a: A theory of local tax competition[END_REF][START_REF] Gaigné | How to make the metropolitan area work? neither big government, nor laissez-faire[END_REF], accounting for the fact that MAs are typically composed of a central city surrounded by suburban towns. Trac congestion stands at the crossroads of these two approaches: it involves commuting costs typically incurred by suburbanites commuting to the MA center.

This paper allows commuters to consume local public goods at their workplace.

15 Since [START_REF] Gordon | An optimal taxation approach to scal federalism[END_REF], the standard way to model local public good spillovers is to assume that public goods provided in a jurisdiction benet indiscriminately all residents from neighboring jurisdictions (e.g. [START_REF] Wellisch | On the decentralized provision of public goods with spillovers in the presence of household mobility[END_REF]. In practice, commuters benet more than non-commuters 11 The LPE literature might be viewed by some to be a strand of the UE literature. However, these two bodies of work are complementary. While LPE models consider endogenous local governments and generally ignore the spatial dimension of MAs, UE models consider local government choices as exogenous but pay particular attention to the spatial features of MAs.

12 [START_REF] Gaigné | How to make the metropolitan area work? neither big government, nor laissez-faire[END_REF] proposes a tax competition model with a spatial urban structure. Our model is closer to the standard tax competition models in that it includes endogenous public good provision and taxable mobile capital which are probably the most prevalent features in the tax competition literature.

13 To our knowledge, only three recent papers account for capital, residents' and workers' mobility: Ly (2018a), [START_REF] Ly | Local taxation and tax base mobility: Evidence from France[END_REF] and [START_REF] Agrawal | Tax competition with mobile labor, residents, and capital[END_REF]. The paper by Agrawal et al. is particularly important and describes how scal externalities and strategic behavior emerge when jurisdictions are atomistic in the capital market but have market power in the labor market.

14 Since [START_REF] Braid | Symmetric Tax Competition with Multiple Jurisdictions in Each Metropolitan Area[END_REF], costless commuting is more common in tax competition models. See e.g. [START_REF] Kächelein | Asymmetric capital tax competition and choice of tax rate[END_REF], [START_REF] Sas | Can scal equalisation mitigate tax competition? ad valorem and residencebased taxation in a federation[END_REF] and Ly (2018a).

taxes, traffic jam and spillover from neighboring jurisdictions' public goods. Our modeling of spillovers induced by commuting reects this. Moreover, workers' consumption of public goods also reduces the traditional divide between public goods consumed by households and public inputs which benet rms as is the focus in [START_REF] Matsumoto | A note on tax competition and public input provision[END_REF][START_REF] Matsumoto | A tax competition analysis of congestible public inputs[END_REF].

The model

This section presents the model used for the analysis. Subsection 3.1 describes the model basic economic and spatial structure and introduces trac congestion, the central externality in this paper. Subsection 3.2 describes households' behavior and introduces the commuting spillover externality. Subsection 3.3 describes how rms choose their inputs and compete with households for land. Subsection 3.4 derives the location system which characterizes the distribution of capital, residents and workers in the MA resulting from subsections 3.1 to 3.3.

Metropolitan structure and commuting costs

The economy consists of a MA composed of n + 1 municipalities: one central city c and n symmetric suburban towns s i with i = 1, . . . , n. Residents, workers and business capital are mobile across municipalities. City and towns dier in two respects.

First, the MA is endowed with a hub-and-spoke commuting transportation network [START_REF] Gaigné | How to make the metropolitan area work? neither big government, nor laissez-faire[END_REF] which means that the n towns s i are connected only to the city c, while the city has direct access to all of the towns in the MA. Therefore, having chosen a residence location, an individual living in town s i can work in her home town s i or in the city c. All residents living in c work in c. This MA structure is aimed at capturing the fact that suburbanites usually hesitate between working in the city or in their home town. Commuting from the city to the suburb or from one suburban town to another similar suburban town is less frequent.

Second, city and towns dier in size as specied in the following assumption:

Assumption 1 (Size).

(i) The city is suciently large to directly inuence all the town variables.

(ii) The towns are atomistic; they cannot inuence the variables in other municipalities.

Assumption 1 is in line with patterns observed in most MAs which are characterized by strong size asymmetry between city and towns as documented in [START_REF] Brülhart | Chapter 17 -taxes in cities[END_REF]. 16 The paper adopts the following notations, for each municipality j, j ∈ {c; s i }:

16 [START_REF] Brülhart | Chapter 17 -taxes in cities[END_REF] report that Paris, Hamburg, Vienna and New York for example, have respectively 268, 229, 199 and 179 times more population than the average populations of the municipalities in their MAs.

3. The model .

• R j : number of residents living in j, .

• W j : number of workers employed in j, .

• W jj : number of residents living in j and working in j , .

• K j : amount of business capital used in j, .

• L j : amount of business land used in j, .

• L j : land endowment of municipality j, .

• P : population of the MA, .

• K : capital endowment of the MA, where R j , W j , W jj K j and L j are endogenous variables of the model, while L j , P and K are exogenously xed. The capital resource constraint is:

17

K c + n i=1 K s i = K. (e.
R s i = W s i c + W s i s i , W s i = W s i s i . ( 1
)
Each individual is assumed to consume a single unit of land, so that local land market clearing for each municipality j ∈ {c; s i } requires:

R j + L j = L j .

(2)

The numbers of residents and workers may not coincide within a municipality but they do coincide at the level of the MA since each individual chooses a workplace within the MA. The population and workforce resource constraints are:

R c + n i=1 R s i = P (e.2)
17 For ease of reference, the key equilibrium conditions are numbered (e.1), (e.2), etc..

18 The sources of this market power are various. While in practice not all capital in a MA or region is xed, a non negligible share of the savings of the MA citizens is channeled to regional banks which invest primarily in regional rms [START_REF] Janeba | Tax and the city a: A theory of local tax competition[END_REF]. In addition, some cities such as New-York, Paris and Singapore are international nancial centers and concentrate a signicant share of the global capital stock.

taxes, traffic jam and spillover

W c + n i=1 W s i = P. (e.3)
Jobs are concentrated in the CBD of the MA, as postulated in:

Assumption 2 (Concentration). Assume that the city employs at least one-third of the MA workers: A ≡ R c n .

W c > P 3 , which is equivalent to W c > n i=1 W s i /
(

) 3
The CBD is the union of the intervals [-L c /n, -A]. It represents the business land used by rms located in the city whose total area is

L c = L c -R c .
Each town is represented spatially by the interval ]0, L s ]. For each town s i , this interval is divided into four subintervals:

• ]0, B i ] of length W s i c
, is occupied by the residents of s i who commute to the CBD.

• ]B i , C i ] (resp. ]D i , L s ]) of length W s i s i /2, is occupied by half of the residents of s i who work in s i . 22 • ]C i , D i ] of length L s i , is the SBD of s i .
Using (1) and ( 2), the following explicit expressions for B i , C i and D i result:

B i ≡ W s i c = R s i -W s i , (4)
C i ≡ W s i c + W s i s i 2 = R s i - W s i 2 , (5)
D i ≡ W s i c + W s i s i 2 + L s i = L s - W s i 2 . (6)
Next, we introduce commuting costs. An individual who lives at location l and works at location l incurs commuting cost: 23 21 Notice that A does not depend on i which means that the amount of business land used by rms is the same on each strip of the MA. This is because the rms in the SBD pay the same land rent per business land unit regardless of which strip they are located on (see subsection 3.3).

22 At equlibrium, there are only two possible locations for the SBD: (i) central to the distribution of the town's residents-workers; (ii) at the extremity of the town. We assume that the former case holds, e.g. for historical reasons. [START_REF] Gaigné | How to make the metropolitan area work? neither big government, nor laissez-faire[END_REF] argue that a central location would be the optimal choice for the municipalities.

Our results would be qualitatively similar with SBDs at the extremity of towns.

23 This modeling of commuting costs with trac congestion is in line with [START_REF] Fujita | Urban economic theory: land use and city size[END_REF]. For tractability, we assume that c(l) is linear, and that the marginal congestion costs are same in the city and in the suburbs. The linerarity of our trac congestion function is consistent with structural models of trac congestion [START_REF] Arnott | A structural model of peak-period congestion: A trac bottleneck with elastic demand[END_REF]. All the results derived in the paper can easily be generalized to any non-linear function of the form:

T (l, l ) ≡    T (l -l) + l l c z -W + l dz if l < l , T (l -l ) + l l c W + l -z dz if l > l , with c(z) ≡ a z b ,
for any dierentiable function T (•) and scalars a and b.

T (l, l ) ≡      l l c z -W + l dz if l < l , l l c W + l -z dz if l > l , with c(z) ≡ t + a z, (7)
where W + l is the residence location of the individual working at l living the farthest from l . c(l) is the marginal transportation cost at location l. Parameter a > 0 (resp. t > 0) captures trac congestion costs (resp. distance costs), as explained below. We assume that workers incur no commuting costs within business districts: for example, an individual working in the CBD pays only for commuting from her residence location to the boundary of the CBD, -A. 24To understand commuting costs (7), consider an individual working in the CBD, that is at location l = -A, and living at location l ∈] -A, B i ] on one of the strips depicted in Figure 1. The individual's commuting costs ( 7) can be written as:

T (l, -A) = t(l + A) + a l -A (B i -z)dz, (8)
The rst part of (8), t(l + A), is the traditional distance commuting costs considered in the literature (e.g. [START_REF] Braid | A spatial model of tax competition with multiple tax instruments[END_REF][START_REF] Peralta | Political support for tax decentralization[END_REF][START_REF] Gaigné | How to make the metropolitan area work? neither big government, nor laissez-faire[END_REF]. Parameter t is the cost per unit of distance. 25 These costs (fuel, transport tickets, etc.) would be incurred by the commuter even were she the only individual on the road.

The second part of (8), a l -A (B i -z)dz, is the continuous sum of the trac congestion costs that this individual incurs during the journey to work. At each location l 0 , this commuter is on the road with B i -l 0 other commuters and pays the trac congestion cost a(B i -l 0 ).

Denition 1 (Trac jam). Parameter a > 0 is trac jam intensity. It measures the marginal trac congestion cost induced by an additional commuter.

The key parameter a can be interpreted as the inverse of the transportation supply at each location l. Denition 1 highlights the rst key externality in this paper. Trac congestion is not novel in the UE literature (see e.g. [START_REF] Fujita | Urban economic theory: land use and city size[END_REF]; however, to our knowledge, there are no studies of such externalities in a context of competition among sub-central governments.

The study of trac congestion in a tax competition context is allowed by the spatial urban structure introduced in this model and by the identication of commuters as residents.

3. The model

Households

All households are identical and composed of a single individual. An individual living at location l in municipality j ∈ {c; s i } and working at l in j ∈ {c; s i } consumes x jj units of a private numeraire good, and one unit of land paying land rent ρ i (l), supplies inelastically one unit of labor in j receiving wage w j , pays the commuting costs T (l, l ) and the local residential head tax τ R j .

Each household is endowed with k ≡ K/P units of capital and j ≡ L j /P units of each municipality j's land: the MA's capital endowment K and each municipality j's land endowment L j are evenly distributed among all the households in the MA. Since k is invested in the municipality providing the highest return, the equilibrium capital return r is identical in all municipalities. The landowner of j supplies it to the household or the rm making the highest bid and pays the land tax τ L j j to municipality j. 26 The budget constraint of a household who lives at location l in municipality j and works at location l in municipality j is:

x jj + ρ i (l) = w j -T (l, l ) -τ R j + rk + Γ. ( 9
)
where Γ is the individual land income net of land taxation, whose explicit form can be found in Appendix A. This individual's utility is:

x jj + U j (G j ) + U jj (G j ) ≡ x jj + U j + U jj , (10)
where G j is the local public good provided by government j. The utility derived from public good consumption is increasing and concave, that is U j G , U jj G > 0 and U j GG , U jj GG < 0, where the subscripts denote derivatives.

Denition 2 (Commuting spillovers). Function U jj measures benets earned by workers from local public services and amenities (safety, road conditions, etc.) provided at their workplace.

Function U jj captures utility spillovers in local public good consumption induced by commuting. Local authorities can make their municipality more attractive to workers by improving these services and amenities.

27 Typically, a resident of town s i who considers commuting to the city compares the sub-utilities of working in her home town, x s i s i +U s i s i (G s i ), and working in the city, x s i c + U s i c (G c ). Her decision depends not only on private consumption x jj but also on public good provision both in the city and in her residence town, captured by U jj .

26 The competition for land between households and rms is in line with that introduced in Ly (2018a).

taxes, traffic jam and spillover Denition 2 highlights the second key externality of this paper. To our knowledge, this model is the rst to investigate local public good provision in the presence of commuting spillovers. This is enabled by the fact that each commuter is identied as a resident whereas in previous tax competition models commuters are pure private inputs, thus U jj = 0 (e.g. [START_REF] Braid | Symmetric Tax Competition with Multiple Jurisdictions in Each Metropolitan Area[END_REF][START_REF] Braid | A spatial model of tax competition with multiple tax instruments[END_REF][START_REF] Peralta | Political support for tax decentralization[END_REF][START_REF] Kächelein | Asymmetric capital tax competition and choice of tax rate[END_REF]Ly, 2018a).

The indirect utility of a household residing in j at location l and working in j at location l is:

V jj (l) ≡ w j -ρ i (l) -T (l, l ) -τ R j + rk + Γ + U j + U jj , (11)
where x jj has been substituted in (10) using ( 9). Residential mobility implies that every resident in the MA obtains the same level of utility u at equilibrium. Equating the indirect utility function ( 11) with the prevailing utility level u, and solving for ρ i (l) allows us to determine the residential bid land rent:

ψ i (l) = w j -T (l, l ) + Λ jj -u (12)
where Λ jj ≡ rk + Γ + U j + U jj -τ R j , with j, j ∈ {c; s i }. The bid land rent ψ i (l) is the maximum rent that a household working in j is willing to pay to live at location l.

The marginal resident of town s i who is indierent between working in the CBD and working in the SBD of s i resides at location l = B i which is characterized by:

V s i c (B i ) = V s i s i (B i ).
(13)

Firms

In municipality j ∈ {c; s i }, the private good is produced combining labor W j , capital K j and business land L j according to the production function F j ≡ F j (W j , K j , L j ) which represents all local rms in municipality j. F j exhibits constant returns to scale and satises F j X > 0, F j XX < 0 and F j XY > 0 for all X, Y ∈ {K j ; W j ; L j }, which means that F j exhibits positive but decreasing marginal returns with respect to each factor, and that factors are technological complements. 28 The prot of the rms in municipality j is:

Π j ≡ F j (W j , K j , L j ) -(w j + τ W j )W j -(r + τ K j )K j -ρ L j L j , (14)
28 Our main results hold whenever F j is homogeneous whatever its degree of homogeneity. Increasing returns to scale often interpreted as agglomeration economies [START_REF] Burbidge | Capital tax competition and returns to scale[END_REF] were included in a preliminary version of this model (see [START_REF] Ly | Tax competition within Metropolitan areas[END_REF]. However, constant returns to scale do not rule out agglomeration economies. They can be included by assuming that factors are more productive in the city than in towns, that is ∀ (W, K, L), F c (W, K, L) > F s (W, K, L), as assumed in the numerical example of section 5.

3. The model where w j is the wage prevailing in municipality j, r is the MA return to capital, ρ L j is the municipal business land rent, τ W j is the tax rate on labor levied by municipality j, 29 and τ K j is its capital tax rate. For each municipality j ∈ {c; s i }, optimal demand for labor W j and capital K j maximize Π j and are characterized by:

F j W = w j + τ W j , (15)
F j K = r + τ K j , (16)
where F j 's subscripts stand for derivatives. Firms make no prot at equilibrium, so that for each municipality j ∈ {c; s i }, Π j = 0. Solving the zero-prot condition for ρ L j determines the business bid land rent:

ψ L j = F j L . ( 17
)
which is obtained by inserting ( 15), ( 16) and Euler's identity F j = F j W W j + F j K K j + F j L L j into the zero-prot condition. The bid land rent ψ L j is the maximum rent that rms are ready to pay for land L j . Firms compete with households for land which implies that in each municipality, the business bid land rent is equal to the residential bid land rent oered by the resident who lives the closest to the business district:

30 ψ L c = ψ i (-A) , ψ L s i = ψ i (C i ) = ψ i (D i ) . (18)
Eliminating ψ i (l), l ∈ {-A; C i ; D i } using (12) and ψ L j using ( 17) from ( 18), for each munici- pality j ∈ {c; s i } we obtain:

F j L = w j + Λ jj -u, (19)
recalling that the individual residing at locations -A, C and D pay no commuting costs. Condition (19) characterizes the distribution of land between residents and rms in municipality j.

Location system

The previous subsections described the behavior of individual households and rms. At the metropolitan level, these decentralized behaviors entail a specic distribution of capital, residents and workers among municipalities. This subsection characterizes this distribution by deriving the so-called location system which is of particular importance to local governments.

29 Notice that assuming alternatively that τ W j is paid directly by workers would lead to strictly identical results.

However, it is simpler from a modeling viewpoint, to introduce τ W j in rms' prot.

taxes, traffic jam and spillover

Since the return to capital r is the same throughout the MA, r can be eliminated from the condition for capital input demand ( 16) so that:

F c K -τ K c = F s i K -τ K s i (e.4)
Conditions (e.1) and (e.4) characterize the distribution of capital {K c ; K s 1 ; . . . ; K sn } in the MA. Similarly, the same utility level u prevails throughout the MA, so that u can be eliminated from the marginal renter condition ( 19):

F c W -τ W c -F c L + U c + U cc -τ R c = F s i W -τ W s i -F s i L + U s i + U s i s i -τ R s i (e.5)
where the denition of Λ jj and the optimal condition for labor and land input demand, ( 15) and ( 17), have been inserted. Conditions (e.2) and (e.5) characterize the distribution of residents {R c ; R s 1 ; . . . ; R sn } in the MA. The marginal worker condition ( 13) can be written explicitly as:

F c W -τ W c + U s i c -T (B, -A) = F s i W -τ W s i + U s i s i -T (B, C), (e.6)
using the expression for the indirect utility (11) and replacing the wages from the optimal condition for labor input demand (15). Conditions (e.3) and (e.6) characterize the distribution of labor {W c ; W s 1 ; . . . ; W sn } in the MA.

In sum, the (3n + 3)-equation location system (e.1)(e.6) in which L j , j ∈ {c; s i } has been substituted in the production function using (2) implicitly denes the 3n + 3 variables {K j ; R j ; W j } j∈{c;s i } as a function of the local governments' policy instruments {G j ; τ K j ; τ R j ; τ W j ; τ L j } j∈{c;s i } . As will be seen in the next section, the location pattern characterized by system (e.1)(e.6) is the main driving force behind municipality's policies.

Decentralized policies

Pareto eciency is characterized in the online appendix . In this section, we characterize decentralized local public policies chosen by municipalities when local labor taxation/subsidy is allowed (subsection 4.1) and when it is proscribed (subsection 4.2). Since all towns s i are identical, we drop index i in the remainder of the paper.

Optimal local public policies with labor taxes/subsidies: First-best

This subsection considers the case where local labor taxes/subsidies τ W j , j ∈ {c; s} are available so that each local government j ∈ {c; s} nances its local public good provision G j by levying a head tax on its residents τ R j , a unit tax on capital τ K j , a unit tax on labor τ W j and a unit 4. Decentralized policies tax on land property τ L j . Local government j's budget constraint is:

τ R j R j + τ K j K j + τ W j W j + τ L j L j = C j , (20)
where C j ≡ C j (G j ) is the increasing cost function of public good provision, so that C j (G j ) ≡ C j G > 0. 31 Municipality j chooses freely τ R j , τ K j , τ W j and G j , and adjusts the land tax τ L j to clear its local budget constraint (20). 32 Local government j maximizes the utility of its residents. Residential mobility in j implies that all residents of j have the same equilibrium utility level. Then, government j's objective is reduced to maximizing the utility of a representative resident, say the resident living the closest to municipality j's business district that is, at l = -A for the city and at l = C or l = D for a town.

When making its policy choices, government j considers as given the choices of the other governments. Additionally, government j does not directly control for the location of capital, residents and workers, but it rationally takes into account their location responses to its policies. Due to their size dierence (Assumption 1) city and towns do not account for the same set of location responses.

Since town s is atomistic , it regards its decisions as having no eect on the variables in the city and the other towns. It accounts for the fact that its population R s , workforce W s and capital K s are dened implicitly as functions of its policy instruments {τ R s ; τ K s ; τ W s ; G s } by the location system (e.4), (e.5) and (e.6). Therefore, government s's problem is to maximize V ss (C), choosing τ R s , τ K s , τ W s and G s , and subject to the location system (e.4)(e.6). Unlike the towns, the size of the city is signicant; the city accounts for the responses of all economic agents in the MA to its policy decisions. It accounts for the fact that the city's and the towns' population (R c and R s ), workforce (W c and W s ) and capital (K c and K s ) are dened implicitly as functions of its policy instruments {τ R c ; τ K c ; τ W c ; G c } by the location system (e.1) (e.6). Government c's problem therefore, is to maximize

V cc (-A), choosing τ R c , τ K c , τ W c and
G c , subject to the full location system (e.1)(e.6).

Solving the municipalities' maximization problems, the following result can be derived:

Result 1. The optimal public policy rules followed by the benevolent local government of municipality j ∈ {c; s} when nancing its public good provision G j using the tax instrument 31 For simplicity, we consider nonrival public goods, while most empirical studies nd evidence of rivalry (e.g. [START_REF] Bergstrom | Private demands for public goods[END_REF]. See [START_REF] Ly | Tax competition within Metropolitan areas[END_REF] for a version of the model with this rival public goods; the results are in line with those in this paper.

32 Formally, τ L j is eliminated from the net land income Γ using (20) in the remainder of the paper (see Appendix A.2). Note that whatever the other instrument (i.e. τ R j , τ K j , τ W j or Gj) the modeler chooses to account for budget constraint (20), the results would be unchanged: τ L j would still be used indirectly as the budgetclearing instrument by municipalities since Lj is x, contrary to Rj, Kj, Wj.

taxes, traffic jam and spillover set {τ R j ; τ K j ; τ W j ; τ L j } are:

τ W s = -Φ, (21a) τ R s = t W s 2 + a W s 2 2 + Φ, (22a)
τ K s = 0,

(23a) R s U s G + W s U ss G = C s G , (24a)
τ W c = 0,

(21b) τ R c = t W c n + a W c n 2 , ( 22b
) τ K c = 0, (23b) R c (U c G + U cc G ) + nW sc U sc G = C c G , (24b)
where τ L j , j ∈ {c; s} allows to clear the budget constraint (20). And:

Φ ≡ a 2 W c n 2 - W s 2 2 > 0. ( 25
)
whose positive sign stems from Assumption 2.

Proof. See Appendix B.

Result 1 implies that:

Corollary 1. The decentralized equilibrium levels of τ R c -τ R s , τ K c -τ K s , R s , K s and W s , when the local tax instrument set is {τ W j ; τ R j ; τ K j ; τ L j }, j ∈ {c; s} are characterized by:

τ W c -τ W s = Φ (26) τ R c -τ R s = t W c n - W s 2 + Φ (27) τ K c -τ K s = 0 (28) x cc + F c L = x sc + F s L (29) F c K = F s K (30) F c W -x sc -t W c n -a W c n 2 = F s W -x ss -t W s 2 -a W s 2 2 . ( 31
)
and R c , K c and W c are dened by the constraints (e.1)(e.3).

Proof. See the online appendix .

Corollary 1 reveals that municipalities choose their policy instruments in accordance with a central planner's eciency rules.

4. Decentralized policies public goods G j using the tax instrument set {τ R j ; τ K j ; τ W j ; τ L j } are provided indirectly with incentives to implement the ecient allocation of resources in the MA.

Labor taxation. Conditions (21a) and (21b) characterize the setting of local labor taxes.

Condition (21a) indicates that town s uses its labor tax as a subsidy. The rationale behind this central result is as follows. Given the population R s , an additional worker in the SBD of s is a resident of s who initially was commuting to the CBD. This can be depicted as a one unit shift of point B i towards 0 in Figure 1. Then, the total trac congestion cost decreases by 0.5a(W c /n) 2 in the city, and increases by 0.5a(W s /2) 2 in a town. Assumption 2, implies that due to the high concentration of workers in the CBD (W c /n > W s /2), the additional worker in the SBD of s reduces the net trac congestion cost incurred by the residents of s. This gain is internalized by government s by a labor subsidy as stated in (21a).

Intuitively, condition (21a) means that since trac congestion is higher in the city than in the suburbs, the town governments subsidize their workers to prevent to many from commuting to the CBD. Importantly, if there were no trac congestion (a = 0), towns would have no incentive to use the labor subsidy (i.e. τ W s = 0, since Φ = 0). Trac congestion requires that the towns' tax instrument sets include local labor taxes to achieve a Pareto ecient allocation of labor in the MA.

According to condition (21b), the city does not use its labor tax. This indicates that the eorts made by the self-interested atomistic towns to retain their workers are sucient to ensure an optimal level of trac congestion in the city. 34 Each of the Wc/n CBD workers located on the strip on which the new resident settles incurs one more space unit of commuting. 35 Notice that if commuting is not costly that is t = a = 0, local governments do not use their tax on residents.

In this case, local authorities know that workers are perfectly mobile so that a jurisdiction's population and workforce are not connected directly. τ R cannot be used to control access to the business district. taxes, traffic jam and spillover Like condition (22b), condition (22a) indicates that the residential tax τ R s internalizes the additional congestion cost induced by a new resident: tW s /2 + a(W s /2) 2 . Indeed, a new resident living and working in s crowds out one unit of business land and increases the commuting distance to the SBD of s by W s /2 space units which generates the additional commuting costs mentioned above. 36 ,37 Compared to the city, an additional cost is internalized by the residential tax in town s. This cost is the subsidy -τ W s = Φ oered to the new residentworker settling in s, according to condition (21a).

Capital taxation. Conditions (23a) and (23b) which characterize the capital taxation rules, state, as standard, that municipalities have no incentive to tax capital. Capital entails no externality to be internalized. Public good provision. Conditions (24a) and (24b) characterize the public good provision rules. They are Samuelson eciency rules extended to an economy in which mobile workers also benet from local public goods. They state that municipality j ∈ {c; s} chooses its public good provision G j so as to equalize the sum of the marginal willingness to pay for its public good to the marginal cost of providing the public good, C j G .

Condition (24a) indicates that town s accounts for the satisfaction of all its residents R s U s G and especially its residents-workers W s U ss G . Interestingly, condition (24b) reveals that the city not only accounts for the benets of its own residents R c (U c G + U cc G ) but also internalizes the public good spillovers induced by workers' mobility from the suburb to the city nW sc U sc G . 38

The main result of this subsection can be summarized in the following proposition: Proposition 1. If local labor taxation/subsidy is allowed, a decentralized equilibrium leads to Pareto-eciency, and:

(i) towns subsidize labor to internalize the trac jam cost dierential with the city.

(ii) residential taxation guarantees optimal access to business districts.

(iii) capital taxation is not used.

(iv) public good provision follows the Samuelson rule extended to commuting spillover.

36 We assume in this interpretation that the new resident works in the SBD. We could also transform (22a) using (e.6) to describe the case of a new resident commuting to the CBD. The resulting condition is less easy to interpret even if the same forces are at stake. 37 Contrary to τ R j , τ W j does not internalize the commuting cost due to distance tW + j , where W + j is as dened in (7). The reason for this is that given the number of workers, new residents increase the overall commuting distance in j by crowding out business land. However, given the number of residents, new workers do not aect the overall commuting distance in j.

Decentralized policies

Proposition 1 presents an original picture of local government behavior compared to the LPE literature which is further discussed in section 6.

Optimal local public policies absent labor taxes/subsidies

The above analysis assumes that the local tax instrument set includes labor taxation/subsidy. Therefore, towns are able to internalize trac congestion by using labor subsidies as a Pigouvian policy instrument. However, in OECD countries labor subsidies are rarely available at the local level (section 1). We now consider the more common case where municipalities have no access to local labor taxes/subsidies. Solving the same municipality maximization problems as in subsection 4.1, with τ W j = 0, the following result can be derived:

Result 2. The optimal public policy rules followed by the benevolent local government of municipality j ∈ {c; s} when nancing its public good provision G j using the tax instrument

set {τ R j ; τ K j ; τ L j } are: τ R s = t W s 2 + a W s 2 2 + Φ + d R s , (32a)
τ K s = -d K s , (33a)
R s U s G + W s U ss G = C s G -d G s , (34a) τ R c = t W c n + a W c n 2 + d R s -d R c (32b) τ K c = -d K s + d K c (33b) R c (U c G + U cc G ) + nW sc U sc G = C c G + d G c (34b)
and τ L j clears the budget constraint (20) with τ W j = 0. The distortion terms are dened as: 39

d R s ≡ (F s W L -t -aW c /n) Φ D s , d K s ≡ F s W K Φ D s > 0, d G s ≡ U ss G Φ D s > 0, d R c ≡ (nF c W L + F s W L ) Φ D c > 0, d K c ≡ (nF c W K + F s W K ) Φ D c > 0, d G c ≡ nU sc G Φ D c > 0,
where Φ is dened in ( 25) and Φ > 0 under Assumption 2, and:

D s ≡ 3 2 t + a W c n + W s 4 -F s W W > 0, D c ≡ D s -nF c W W > 0.
Proof. See Appendix C.

Result 2 implies that:

39 The signs are obtained recalling that Φ ≥ 0, FXX < 0 and FXY > 0 for all X, Y ∈ {K; W ; L}.

taxes, traffic jam and spillover Corollary 2. The decentralized equilibrium levels of

τ R c -τ R s , τ K c -τ K s , R s , K s and W s , when the local tax instrument set is {τ R j ; τ K j ; τ L j }, j ∈ {c; s} are characterized by: τ R c -τ R s = t W c n - W s 2 + Φ + d R c (35) τ K c -τ K s = d K c (36) x cc + F c L + d R c = x sc + F s L (37) F c K -d K c = F s K (38) F c W -x sc -t W c n -a W c n 2 + Φ = F s W -x ss -t W s 2 -a W s 2 2 . ( 39
)
and R c , K c and W c are dened by the constraints (e.1)(e.3).

Proof. See the online appendix .

Comparing Corollary 2 to the Pareto-ecient outcome of Corollary 1 shows that municipality behaviors are distorted in a sub-ecient way, as shown by the distortion terms d x j , j = {c; s} and x ∈ {R; K; G}. 40 The only cause of these distortions is the impossibility for towns to regulate trac congestion using labor subsidies. 41 Since towns cannot subsidize workers, there are too many workers in the city and too few in the suburbs. This labor misallocation can be seen in the presence of Φ > 0 in condition (39) which was absent from eciency condition (31). Then, town (resp. city) authorities distort their remaining instruments {τ R j ; τ K j ; G j } to be more (resp. less) attractive to workers.

Residential taxation. Condition (32a) depicts how town s distorts its tax on residents to overcome the absence of labor tax. The sign of the distortion d R s is ambiguous. To understand this, consider two polar cases.

First, suppose that t > 0, a > 0 and F s W L ≈ 0. Then, d R s < 0 that is, the towns under-tax their residents compared to (22a). Imperfect worker mobility (i.e. costly commuting) spurs the towns to set a low residential tax to attract residents-workers and hence alleviate the trac congestion incurred by their residents working in the CBD.

Second, suppose that t ≈ 0, a ≈ 0 and F s W L > 0. Then, d R s > 0 that is, the towns overtax their residents. Local authorities know that population and workforce are not linked by imperfect worker mobility since commuting is not costly. However, they can attract workers by increasing the amount of land available to rms. Indeed, technological complementarity 40 See Figure OA.1 in the online appendix for an illustration of the misallocation implied by conditions (34a), (34b), and (37)(39).

41 Were there no trac congestion, local governments would not distort their policy instruments. Indeed, a = 0 implies d x j = 0, so that Result 2 and Corollary 2 become Result 1 and Corollary 1.

F s W L > 0 implies that more business land increases local wage and attracts workers. To increase the land size of their SBD, the towns over-tax residents to reduce their population and make room for business land.

The above two polar cases show that towns can under-tax or over-tax their residents, depending on the relative importance of commuting costs and technological complementarity between labor and business land.

Condition (32b), which characterizes the city's residential tax policy, can be better interpreted by considering condition ( 35) in which d R s is eliminated using (32a). 42 Comparing condition ( 35) to the rst-best condition ( 27), indicates that given the towns' residential tax rate, the city chooses an ineciently low residential tax rate, since -d R c < 0.

This too low residential tax is a response to the excessive number of workers inducing trac congestion in the city (condition ( 39)). The city cuts its residential tax to attract new residents, which limits the land size of the CBD and thus avoids some commuting ows.

However, this attractive city residential tax policy entails a misallocation of residents in the MA. Condition (37) shows that the number of residents in the city (resp. towns) is ineciently high (resp. low) compared to the Pareto-ecient condition (29).

Capital taxation. Condition (33a), shows that the towns now subsidize capital, since -d K s < 0. Compared to the rst-best condition (23a), the towns now under-tax capital.

Since they cannot subsidize labor, they subsidize capital to attract workers, exploiting the technological complementarity between capital and labor (F s W K > 0).

Condition (33b) indicates that the specic distortion of the city's capital tax is positive, d K c > 0. This distortion can be better understood by comparing (36) to the rst-best condition (28).

43 Condition (36) shows that the city chooses an ineciently high capital tax rate, given the towns' capital tax rate, since d K c > 0.

In response to the excessive number of costly workers in the CBD, the city authorities exploit the technological complementarity between capital and labor, by increasing their capital tax. By doing so, they discourage capital, and thus indirectly discourage workers from locating in the city. Condition (38) shows that the amount of capital in the city (resp. towns) is ineciently low (resp. high) compared to the Pareto-ecient condition (30).

42 Notice that the city's residential tax τ R c internalizes the distortion of the towns d R s , as can be seen in (32b). This is because residents' mobility links the welfare of city residents to the welfare of suburbanites; all households have the same level of utility at equilibrium. 43 The city's capital tax τ K c internalizes the distortion of the towns d K s . The reason is the same as for τ R c . See footnote 42. taxes, traffic jam and spillover Public good provision. Condition (34a) shows that, since the towns cannot subsidize labor, the towns over-provide public goods compared to the ecient extended Samuelson rule (24a), since -d G s < 0. By increasing their public good provision, the towns are able to encourage more workers to work in their SBD, since U ss G > 0.

Condition (34b) indicates that the city behaves in an opposite way by providing an ineciently low amount of public good, since d G c > 0. Since towns cannot keep a sucient number of workers, the city controls trac congestion by decreasing its public good supply which discourages some suburbanites from commuting to the CBD, since U sc G > 0.

The distortions in the local policy setting caused by the existence of trac jams in the MA are summarized in the following proposition:

Proposition 2. If towns are not allowed to subsidize workers, a decentralized equilibrium does not lead to Pareto-eciency, and under Assumption 2:

(i) the city under-taxes residents compared to the towns.

(ii) the city over-taxes capital compared to the towns.

(iii) the city (resp. towns) under-provides (resp. over-provide) local public goods.

Proposition 2, which shows the eects of trac congestion on local public policy in a MA, is novel in the literature. The main new insight is the following. If the city concentrates an important share of the workforce of the MA (Assumption 2), the impossibility for towns to retain workers by employing subsidies entails an overow of workers in the CBD. To prevent excess trac congestion incurred by households working in the CBD, city and towns engage in inecient policies. Notably, part (iii) indicates that local governments distort their public good provision accounting for local public good spillovers induced by commuting. The city (resp. towns) reduces (resp. increase) its (resp. their) public good provision to be less (resp. more) attractive to workers. To our knowledge, our model is the rst to consider commuting spillovers and to depict how they can induce distortion in local governments' choices.

We have seen also that the distortions in the local policy setting induced by the excessive level of trac congestion in the city relative to the towns (i.e. too many workers in the city) lead to a misallocation of residents and capital in the MA. This is summarized in the following proposition.

Proposition 3. If towns are not allowed to subsidize workers, under Assumption 2, a decentralized equilibrium leads to:

(i) an ineciently high number of workers in the city, compared to the towns.

(ii) an ineciently high number of residents in the city, compared to the towns. 22 5. Numerical example (iii) an ineciently low amount of capital in the city, compared to the towns.

Numerical example

To gain general equilibrium insights into the results of section 4, this section provides a numerical illustration. The purpose is to shed a light on the distortions caused by various trac jam intensities a. To this end, we assume that the cost function of public good provision is C j (G) = G 2 , that the utility function is x j + U j (G j ) + U jj (G j ) = x j + G γ j + G j and that the production is carried out by the Cobb-Douglas function The numeric results in Figure 2 are in line with Proposition 2 and Proposition 3. The simulations conrm that without trac congestion a = 0, Pareto-eciency is achieved in the decentralized equilibrium without labor subsidy: ∆X = 0 for all variables. Figure 2g indicates that an excessive number of workers in the CBD (∆W c > 0) arises when there is trac congestion (a > 0), and that this excess of workers gets worse as trac congestion intensies (a increases). Figures 2a2d show how municipalities respond to the overcrowded CBD by increasingly distorting their policy instrument setting as a increases.

F j (W j , K j , L j ) = A j K α j W β j L 1-α-β j . 44 (
First, Figure 2a indicates that the city sets a too high capital tax relative to the towns

(∆(τ K c -τ K s ) > 0)
. This strategy allows the city to reduce the size of the CBD by exerting excessive outows of capital towards the towns, as illustrated by Figure 2e (∆K c < 0).

Second, Figure 2b highlights another way to cut the size of the CBD. The city sets an ineciently low tax on residents compared to the towns (∆(τ R c -τ R s ) < 0). This allows the city to attract a higher number of residents as suggested by Figure 2f (∆R c > 0), and to crowd out business land.

Finally, public good spillovers due to commuting allow municipalities to directly aect the workers ow. According to Figure 2c, to reduce its attractiveness to workers the city provides an ineciently low amount of public good (∆G c < 0). However, Figure 2d shows that since towns want to retain their workers, they do the opposite and over-provide public goods (∆G s > 0).

Discussion

The model developed in this paper sheds new light on several important and often longstanding LPE debates. This section discusses some of them. Subsection 6.1 highlights the original role played by residential taxation in MAs. Subsection 6.2 questions the standard result that cities charge higher tax rates than towns. Subsection 6.3 discusses how household mobility calls for a reconsideration of externalities at the local level. Subsection 6.4 outlines that labor taxation should not be regarded as a second-class policy in MAs.

45 For space reasons, we do not report the graphs for ∆Ks = -∆Kc/n, ∆Rs = -∆Rc/n and ∆Ws = -∆Wc/n which are symmetric to ∆Kc, ∆Rc and ∆Wc with respect to the x-axis.

6. Discussion

A new role for the residential tax: job accessibility

The paper sheds new light on the role of local residential taxes. In the literature, the essential motive for taxing mobile residents is to internalize crowding costs induced by their consumption of a rival local public good. This role has been conrmed for most types of residence based taxes such as head taxes [START_REF] Wildasin | Locational eciency in a federal system[END_REF][START_REF] Wildasin | Urban public nance[END_REF], housing taxes [START_REF] Hoyt | Competitive jurisdictions, congestion, and the Henry George Theorem : When should property be taxed instead of land?[END_REF][START_REF] Krelove | The persistence and ineciency of property tax nance of local public expenditures[END_REF] and wage taxes [START_REF] Wilson | Mobile Labor, Multiple Tax Instruments, and Tax Competition[END_REF]. 46 In all these cases, without crowding costs, local governments do not tax their residents if they can nance public goods via another undistortive tax instrument.

In this paper, despite the nonrivalry of local public goods and the availability of an undistortive tax on land, municipalities use residential taxes even without trac congestion to control access to their business district: 47 they raise their residential tax to discourage too many residents-workers from choosing their business district as a workplace. To our knowledge, the link between residential taxes and job accessibility has not previously been demonstrated.

6.2. Who sets the highest tax rate in the MA?

The paper also provides new insights into the longstanding question about which, city or towns, sets the highest taxes on capital and residents in a MA? The traditional tax competition literature is unambiguous: the city charges higher tax rates than the towns on both capital [START_REF] Bucovetsky | Asymmetric tax competition[END_REF][START_REF] Wilson | Tax competition with interregional dierences in factor endowments[END_REF][START_REF] Bucovetsky | An index of capital tax competition[END_REF] and residents [START_REF] Epple | The Implications of Competition among Jurisdictions: Does Tiebout Need Politics?[END_REF][START_REF] Hoyt | Market power of large cities and policy dierences in metropolitan areas[END_REF]. 48 The basic reason for this is that due to its larger population, a city has more inuence on capital and housing prices than towns do, and therefore, can set higher tax rates on capital and housing.

These results have been conrmed by several theoretical contributions, especially for capital taxation. According to the new economic geography literature (e.g. [START_REF] Baldwin | Agglomeration, integration and tax harmonisation[END_REF], agglomeration economies lock mobile capital into the largest jurisdictions which benet from this agglomeration rent and can levy a high capital tax rate on this quasi-xed tax base. 49 In a two-jurisdiction tax competition model with commuting, [START_REF] Kächelein | Asymmetric capital tax competition and choice of tax rate[END_REF] shows that due to the limited geographic mobility of labor, the more populated jurisdiction sets a higher capital tax rate. Gaigné et al. (2016, p.108) nd that the central city sets a higher 46 The distortions arising when a residence-based tax is not available and local public goods are rivals are studied in [START_REF] Wellisch | A Second-Best Theory of Local Government Policy[END_REF]. 47 Formally, this can be seen by assuming that a = 0 and t > 0 in (22a) and (22b).

48 See [START_REF] Wilson | Theories of Tax Competition[END_REF] and [START_REF] Brülhart | Chapter 17 -taxes in cities[END_REF] for surveys of the literature on asymmetric tax competition.

49 Empirical evidence of the existence of taxable agglomeration rents is less clear. While [START_REF] Paty | Market access eect and local tax setting: evidence from french panel data[END_REF] for example provide results supporting this nding for French municipalities, [START_REF] Luthi | The eect of agglomeration size on local taxes[END_REF] using panel data of Swiss municipalities nd no evidence of a positive impact of agglomeration forces on capital tax rates.

taxes, traffic jam and spillover business tax rate than suburban municipalities, due to the central position of the CBD in the transportation network (see subsection 6.4 below).

50

Yet, the theoretical consensus stating that cities tax more heavily than towns does not always coincide with basic empirical facts. For instance, in 2009 among the ten most populated French MAs, 54% of suburban towns set a higher tax rate on capital than the rate set by the city in their MA, and 41% of towns set a higher housing tax rate [START_REF] Ly | Tax competition within Metropolitan areas[END_REF]. 51 This paper provides some insights which help to explain these observations. The model shows that whether the city or the towns set higher tax rates on capital and residents depends on the concentration of workers in the city. 52 If the city concentrates a low share of the MA workforce, the city sets lower tax rates than the towns on both capital and residents. In other words, in MAs where jobs are not overly concentrated in the city, we should observe a more signicant number of towns setting higher taxes on capital and residents than the city. The intuition is as follows. If the towns accounts for a high share of the workforce, then (i) the city taxes its capital less than the towns to reduce trac congestion in the suburbs, and (ii)

the city taxes its residents less than the towns to guarantee a sucient level of access to jobs in SBDs.

Externalities and household mobility

Frequently, inter-jurisdictional externalities are considered as the main source of ineciencies in a context of decentralized government activities. A local government, by accounting exclusively for the welfare of its own residents, ignores the benets/costs of its policy choices for non-residents, and thus implements inecient policies. This conclusion is based on the assumption of immobile residents which is not satisfactory in the case of sub-metropolitan jurisdictions such as municipalities. Residents' inter-jurisdictional mobility forces local governments to account for the welfare of non-residents such that at the local level, inter-jurisdictional externalities are minor. Our paper reveals that an important source of ineciencies in MAs is the inter-individual externality of trac congestion. This involves distortions which can 50 A notable exception is [START_REF] Janeba | Tax and the city a: A theory of local tax competition[END_REF] where cities compete not only with the towns of their MA but also with cities in other MAs, while towns compete only with the municipalities in their particular MA. When the number of MAs is suciently large, the competition faced by a city is so high compared to that faced by the towns in its MA that the city sets a lower capital tax rate than that set by the towns.

51 These mitigated results are noted by [START_REF] Brülhart | Chapter 17 -taxes in cities[END_REF]Brülhart et al. ( , p. 1151)): empirical estimates provide conicting evidence for the eect of population size on jurisdictions' tax rates.

52 Formally, τ R c ≷ τ R s and τ K c ≷ τ K s if Wc ≷ P/3. That is, the ordering of the tax rates depends on whether Assumption 2 holds or not. Proof. Algebraic manipulations show that condition (35) can be written as:

τ R c -τ R s = t Wc n -Ws 2 + Φ Dc 3t 2 + a Wc n + Ws 4
, so that sign(τ R c -τ R s ) = sign(Wc/n -Ws/2) = sign(Wc -P/3). From condition (36), we directly have sign(τ K c -τ K s ) = sign(Wc -P/3).

Discussion

work in the opposite direction to distortions induced by inter-jurisdictional externalities. This applies particularly to two types of externality.

The rst is the scal externality induced by capital mobility. The standard tax competition result is that capital inter-jurisdictional mobility spurs local governments to set ineciently low taxes on capital [START_REF] Wilson | A theory of interregional tax competition[END_REF][START_REF] Zodrow | Pigou, Tiebout, property taxation, and the underprovision of local public goods[END_REF]. This ineciency is due to a positive inter-jurisdictional externality: jurisdictions overlook the fact that increasing their own taxes will benet other jurisdictions receiving capital inows [START_REF] Wildasin | Interjurisdictional capital mobility: Fiscal externality and a corrective subsidy[END_REF]. Residential mobility eliminates this externality since tax competition does not matter in the case of household mobility, since each region considers the welfare of nonresidents by taking the migration equilibrium into account (Wellisch, 2006, p. 115). The present paper newly shows that the distortion threat in a MA with capital and household mobility is over-taxation of capital by the city which is aimed at preventing excessive trac congestion in the CBD.

The second externality is the public good spillover or its consumption by non-residents. A well-known result is that these spillover eects drive jurisdictions to under-provide local public goods, since in decentralized decision-making, the value of local public services to nonresidents is ignored (Gordon, 1983, p. 578). This result also relies on the assumption of immobile residents. As shown in [START_REF] Wellisch | On the decentralized provision of public goods with spillovers in the presence of household mobility[END_REF], relaxing this assumption leads to ecient local public good provision, since local governments consider the welfare of mobile non-residents. Our analysis shows that despite residents' mobility, local public good spillovers lead local governments to distort their public good provision in order to prevent excessive trac congestion.

However, under-provision occurs only in the city; towns over-provide public goods.

Local labor tax/subsidy: a rst-class policy instrument

This paper shows that labor taxation/subsidy is a rst-best tax instrument. However, most work on tax competition with commuting and source-based taxation on labor considers it, a second-best instrument. In this literature, two main reasons explain why local governments make use of a labor tax/subsidy.

First, a labor tax might be imposed for budgetary reasons in the unavailability of any other tax relying on a less mobile tax base. This requires that labor mobility is limited, usually by commuting costs. 53 For example, in a model with symmetric jurisdictions, [START_REF] Braid | A spatial model of tax competition with multiple tax instruments[END_REF] shows that if local public goods are nanced by a labor tax and a capital tax, only the former is imposed in the presence of commuting costs. 54 In a model with a city and towns, Gaigné et al.

53 An exception is [START_REF] Braid | Symmetric Tax Competition with Multiple Jurisdictions in Each Metropolitan Area[END_REF] in which labor limited mobility results from the xity of business land.

54 If commuting costs decrease the labor tax also decreases, until commuting costs reach zero when labor is no longer taxed and no public goods are provided. See proposition 3 and 4 in [START_REF] Braid | A spatial model of tax competition with multiple tax instruments[END_REF].

taxes, traffic jam and spillover (2016) nd a related result which shows that the city exploits labor limited mobility induced by commuting costs to impose a positive business tax, while towns set a zero tax. 55 The link between labor taxation and commuting costs is however broken if a less distortive tax allows the nancing of the public good.

56

Second, a labor tax/subsidy can be used also for tax exporting purposes that is, shifting the tax burden to nonresidents. This case is illustrated in [START_REF] Peralta | Political support for tax decentralization[END_REF] which shows that the labor importer jurisdiction taxes labor while the labor exporter subsidizes it.

57

For tax exporting also, local governments prefer an undistortive instrument such as a land tax to a distortive tax on labor.

In the present paper, the availability of an undistortive tax on land allows for both local budget balancing and tax exporting. Towns subsidize labor for a dierent reason that of internalizing trac congestion costs. This can be considered the rst-best or natural role of local labor taxation/subsidy in a MA.

Conclusion

This paper investigates how local governments such as municipalities address one of the biggest MA challenges: trac congestion. Trac congestion is a typical MA negative externality, similar to air, waste and noise pollutions, which results from the spatial agglomeration of jobs and household mobility. It is also considered a major problem by policy makers and economists due to its signicant costs (e.g. fuel, working time wasted, stress, maintenance of congested roads, greenhouse gas emissions).

The study develops a new urban tax competition model which links local public economics and urban economics models. The analysis shows that common local scal instruments may be used by municipalities to curb overwhelming trac congestion in MA centers. Specically, by charging high capital tax rates and low residential tax rates, the central city can make its business district less attractive to workers which reduces trac congestion. Since many public goods (e.g. safety, public hygiene and roads) are consumed not only by the municipality's residents but also by its workers, local public good provision can also be employed. To free up the MA's roads, the central city and the suburban towns respectively can reduce and increase their public good provision. The aforementioned instruments are however only second-best 55 See proposition 2 in [START_REF] Gaigné | How to make the metropolitan area work? neither big government, nor laissez-faire[END_REF]. Notice that Gaigné et al. consider a poll tax on rms. However, since labor is the only input, the economic incidence of this tax is essentially similar to that of a labor tax.

Conclusion

tools; the paper shows that the most ecient way to regulate trac congestion is to allow suburban towns to provide rms with local labor subsidies.

Decentralized local labor subsidies could protably complement national (e.g. lane tolls, gasoline taxes) and regional (e.g. integrated public transport) solutions. These solutions often receive poor public support. 58 Local labor subsidies would be more acceptable; unlike congestion charges, they are perceived as a rewarding rather than a punitive instrument.

Moreover, local governments are better informed than higher level authorities about their residents' needs and can adapt their policies to their preferences. For instance, in a town where voters complain hugely about time spent every day stuck in trac jams, the local mayor will be nudged to develop the town's business district through the provision of labor subsidies.

In practice, local labor subsidies are critically missing, mainly because they are at odds with competition policy. Subsidies are strictly controlled by supranational authorities such as the European Commission, the World Trade Organization and the West African Economic and Monetary Union (OECD, 2010). 59 In view of the signicant market failures in MAs, these local level restrictions would seem worthy of further consideration.

The easiest way to implement the subsidies recommended in this paper would be to let the residents of each town vote on the level of monetary subsidy per worker, and to provide the subsidy in the form of a tax credit to all local rms based on the number of their employees.

Notice that this scal instrument diers from most existing local development subsidies and tax credits, for instance in the US, that are often arbitrarily implemented by local governments and are based on multiple performance criteria rather than focusing on the number of employees.

60 An alternative solution is to subsidize workers rather than rms. For instance, town authorities could provide local workers with direct monetary subsidies or in-kind advantages such as luncheon vouchers, free parking places or public transport rebates.

The constantly growing economic signicance of MAs worldwide means that many important challenges hitherto considered national will need to be rethought at the metropolitan level. The paper contributes to this overall discussion by inviting national policy makers to reshape over-concentrated MAs to achieve greater polycentrism.

A.2. Equilibrium land income

We can now derive the explicit form of the individual land income in equilibrium. To do so, we need to insert (A.2)(A.2) and (A.8)(A.11) into (A.1) and collect terms, as described in this subsection. Since land rent is constant within business districts and the land rent paid by suburban residents working where they live is symmetric on both side of the SBD (see

0 -Lc n ρ i (l) -τ L c dl = 0 -A ρ i (l) -τ L c dl + (F L -τ L c ) L c n Ls 0 ρ i (l) -τ L s i dl = B i 0 ρ i (l) -τ L s i dl + 2 C i B i ρ i (l) -τ L s i dl + (F s i L -τ L s i )L s i
Then, from the aggregate land rent denition (A.1), we obtain:

Γ = P -1 n i=1 I c i + I s i + n -1 (F c L L c -τ L c L c ) + (F s 1 L L s 1 -τ L s 1 L s ) (A.
I c i = t + a(R s i -W s i ) 2 R c n 2 - a 3 R c n 3 + F c L R c n (A.14) I s i = aR 3 s i 6 + R 2 s i (t -aW s i ) 2 + W 2 s i (6t -aW s i ) + 3R s i W s i (aW s i -4t) 8 + F s i L R s i (A.15)
Then, replacing τ L c and τ L s i in (A.12) using the local budget constraints (20), and replacing L c and L s i using the land occupation condition (2), we obtain: .16) which is the explicit expression of the individual land income Γ in equilibrium, with I c i and I s i dened in (A.14) and (A.15).

Γ = P -1 n i=1 I c i + I s i + F s i L (L s i -R s i ) + τ R s i R s i + τ K s i K s i + τ W s i W s i -C s i + n -1 (F c L (L c -R c ) + τ R c R c + τ K c K c + τ W c W c -C c ) , (A
taxes, traffic jam and spillover and her budget constraint is:

x ss + ρ(C) = w s + rk + Γ -τ R s ,

with ρ(C) = F s L . Replacing w s using (15), it follows that the municipalities' program is to maximize:

-F s L + F s W -τ W s + rk + Γ + U s + U ss -τ R s , choosing τ R s , τ K s , G s , W s , R s , K s , subject to constraints (A.19), (A.20) and (A.21) to which we respectively associate the Lagrange multipliers λ 1 , λ 2 and λ 3 . The variables r, k and the variables of the city X c and those of the other towns X s j are exogenous from the atomistic representative town's viewpoint especially, in the aggregate land rent Γ dened in (A.16).

The business land L s has been replaced into the production function using the land occupation condition (2).

From the above program, we can derive the rst-order conditions. The rst-order conditions with respect to τ R s , τ K s , τ W s and G s are respectively: (A.28)

The rst-order condition with respect to R s is:

τ R s P + t P R s - 3W s 2 -Pλ 2 + a 2P R 2 s + 3W 2 s 4 - R 2 c n 2 -2R s W s + W s - R c n -R s 2Pλ 2 - L s P + λ 3 -1 F s LL + λ 1 F s LK + (λ 2 + λ 3 -1)F s LW = 0.
(A.29)

The rst-order condition with respect to K s is:

τ K s P + L s P + λ 3 -1 F s KL -λ 1 F s KK -(λ 2 + λ 3 -1)F s KW = 0.
(A.30)

And, the rst-order condition with respect to W s is:

τ W s P - 3t 2P (R s -W s -λ 2 P) + a 2P R 2 c n 2 + 3R s W s 2 -R 2 s - 3W 2 s 4 + R c n + R s - 3 4 W s 2λ 2 P

  Residential taxation. Conditions (22a) and (22b) characterize the residential taxation rules. Condition (22b) indicates that the scal benet τ R c of a new resident in c covers the additional cost of commuting to the CBD induced by this new resident:tW c /n + a(W c /n) 2 .By crowding out one unit of business land (shift of -A towards L c /n in Figure1), the new resident increases the overall distance to the CBD by W c /n.34 This implies an additional commuting cost of tW c /n due to distance and a(W c /n) 2 due to trac congestion.Intuitively, a new resident in c makes the CBD less accessible by increasing the cost of commuting to work in c. In order to safeguard an optimal level of access to the CBD, local government c uses its residential tax τ R c to control its population size. 35 characterized in the online appendix . The economic interpretations of conditions (26)(31) can be found in the online appendix .

  a) Capital tax rate gap. (b) Resident tax rate gap. empty (c) Public good in the city. (d) Public good in town s.empty (e) Capital in the city.(f ) Residents in the city.(g) Workers in the city.

Figure 2 .

 2 Figure2. Distortions with respect to trac jam intensity a. For each variable X, ∆X ≡ X -X where X is the rst-best (Pareto-ecient) equilibrium value of X resulting from Result 1, and X is its second-best equilibrium value resulting from Result 2.

Figure 2

 2 Figure 2 reports the simulation results which depict the distortions of the key variables of the model as a function of trac jam intensity a. The distortions ∆X with X ∈ {τ K c -τ K s ; τ R c -

  taxes, traffic jam and spillover τ R s ; G c ; G s ; K c ; R c ; W c } 45 are measured as the dierence between the equilibrium value of X without local labor subsidies (Result 2 and Corollary 2) and the Pareto-ecient level of X(Result 1 and Corollary 1).

Figure

  Figure A.1), we have:

  .8)(A.10) into (A.13), replacing A, B i and C i by their denitions (3)(5), and integrating, we obtain:

  2 and at equilibrium W c /n > W s i /2 since the towns are symmetric.For convenience, the spatial structure of the MA is depcited in Figure1. The MA is composed of n strips of 1 space unit width and L c /n + L s space unit length. Thus, the total area of the MA is L c + nL s that is, the sum of the land endowments of the city and the n of the n towns, L s . The city is represented spatially by the union of the n intervals [-L c /n, 0]. Each of these intervals is divided between business land [-L c /n, -A] and residential land [-A, 0]. Since each household consumes one unit of land, we have:21

		3. The model
	extremity of one
	Assumption 2 is theoretically relevant for studying trac congestion and is probably the most
	common case in practice. 19 ,20 Subsection 6.2 discusses our results if this assumption does not
	hold.
		Ls
			Town's residents-workers
		D 1
			SBD
	Town s 1	C 1
		B 1
			Town's commuters
		0
			City's residents-workers
		-A
			CBD
		-Lc/n
	Town s 2	-A	-A	Town s 3
	0		0
	B 2		B 3
	C 2		C 3
	D 2		D 3
	Ls			Ls
	Figure 1. Graphical representation of the MA, with n = 3.
	towns. Each strip is an interval [-L c /n, L s ] which links the center of the city, -L c /n, to the
	19 For instance, in 2016, among the 100 most populated French MAs, the central city accounted for 51% of the
	MA's workforce on average; 87% of cities accounted for more than a third of their MA workforces. Source:
	data collected from the French National Institute of Statistics and Economic Studies.
	20 Gaigné et al. (2016) make a related assumption. They assume that more than a third of the MA's population
	is concentrated in the city. Our assumption is less demanding since central cities usually employ a higher
	number of workers than their number of residents.

  They would protably provide 58 Congestion charges are unpopular with members of the public. This is exemplied by the 2019 political crisis in France involving the `Gilets jaunes' which was triggered by a rise in gasoline taxes. Also, public transport solutions are not unanimously favored. EC (2013) surveyed 83 European municipalities and reports that 25% of individuals are not satised with their public transport and this gure rose to 75%, 63% and 48% respectively in Naples, Rome and Budapest. 59 All World Trade Organization members signed the Agreement on Subsidies and Countervailing Mechanisms following the Uruguay Round in 1995. 60 See footnote 9 for more details about existing development subsidies. 29 taxes, traffic jam and spillover local authorities with more means to foster the development of strong secondary economic poles.[t + a(L s -z)]dz + F s i L , l ∈ ]D i , L s ]

taxes, traffic jam and spillover

ρ i (l) = -l D i (A

.11)

In most models with commuting (e.g.[START_REF] Braid | Symmetric Tax Competition with Multiple Jurisdictions in Each Metropolitan Area[END_REF][START_REF] Braid | A spatial model of tax competition with multiple tax instruments[END_REF] Ly, 2018a), residents and workers are disconnected so that commuters are reduced purely to production factors which by nature, neither consume public goods nor pay taxes. In an extension of their analysis,[START_REF] Gaigné | How to make the metropolitan area work? neither big government, nor laissez-faire[END_REF] consider suburban commuters' consumption of public goods in the city, but public good provision is xed exogenously.

This is similar to the assumption inFujita (1989, chap. vii) andZenou (2009, chap. vi). Equivalently, we could assume that workers have the same random commuting cost whenever they enter a business district.

For simplicity, commuting costs are assumed to be monetary costs. This ignores time opportunity costs. We do not think that this alternative formulation would provide dramatically dierent results since agents are homogeneous and thus earn the same wage if working at the same place.

Local public goods limited to residents (e.g. local public schools) are not considered in the paper. General- izing the paper results to such goods is straightforward.

Suppose for instance that rms in town si bid more than residents at l = Ci, so that ψ L s i > ψi (Ci). In this case, some owners of residential land decide to supply their land to rms, business land increases in si and rms bid less for land (since F s i LL < 0) until ψ L s i = ψi (Ci).

This result echoes[START_REF] Tiebout | A Pure Theory of Local Expenditures[END_REF] result. Here, residential mobility forces the city to account for all household welfare when providing its public good.[START_REF] Wellisch | On the decentralized provision of public goods with spillovers in the presence of household mobility[END_REF] derives a similar results in a regional tax competition model.

This is shown in proposition 2 in[START_REF] Braid | A spatial model of tax competition with multiple tax instruments[END_REF] which indicates that if a tax on immobile residents is available, labor taxation is no longer applied.

See proposition 2 in[START_REF] Peralta | Political support for tax decentralization[END_REF]. Also, see[START_REF] Gordon | An optimal taxation approach to scal federalism[END_REF] for a similar early tax exporting result in a model with residents' mobility.

the IEB Workshop on Taxation and Social Expenditure (Barcelona), French

Competing local governments which can nance local 33 Conditions (24a), (24b) and (26)(31) are strictly equivalent to the eciency conditions (oa.5)(oa.12) Appendix A Individual land income According to the description made in section 3, the land income of an individual is:

The objective of this appendix is to derive the explicit expression of the individual land income Γ in equilibrium. To this aim, we rst determine the equilibrium land rent function (subsection A.1) from which we derive the equilibrium individual land income (subsection A.2).

A.1. Land rent function

In this subsection, we determine the land rent function in equilibrium: the land rent ρ i (l) paid by households or rms as a function of their location l on strip i. In business districts, the land rent is equal to the business bid land rent characterized in (17), so that:

In residential land, the land rent is equal to residents' bid land rent characterized in (12), so that:

where T (l, l ) has been replaced by its the denition (7). A typical land rent function characterized by (A.2 Using ( 19) and (e.6) to eliminate Λ s i s i , Λ s i s i and Λ s i s i , (A.4)(A.7) become:

(A.9)

Appendix

Since all towns are identical, the following appendices drop index i for convenience. However, notice that a given town's decision does not engage other towns.

Appendix B Proof of Result 1

The purpose of this section is to derive the municipalities' optimal policy rules when the tax instrument set includes a tax/subsidy on labor. Specically, we prove the towns' optimal policy rules (21a)(24a) in subsection B.1 and the city's optimal policy rules (21b)(24b) in subsection B.2. Before proceeding, recall that since F j is homogenous of degree 1, we have the following Euler's formulas for each X ∈ {W ; K, L}:

As explained in subsection 3.4, the equilibrium level of K c , K s , R c , R s , W c and W s are fully dened by the equations (e.1)(e.6) which are restated here for convenience:

where (A.20) is obtained by replacing T (l, l ), A, B and C by their denitions ( 7) and (3)( 5)

in (e.6).

B.1. Optimal policy rules of a town

In this subsection we prove the representative town's optimal policy rules (21a)(24a) when labor taxation/subsidy is available. The utility of the resident of s living the closest from her home town SBD is:

The rst-order conditions with respect to the tax instruments (A.25)(A.27) entail:

Then, using Euler's identities (A.18) and substituting L s from (2), we have:

And, from (e.2) and (e.3), we have:

Inserting(A.32) and (A.33) into the rst-order conditions with respect to public good provision (A.28), population (A.29), capital (A.30) and labor (A.31), substituting R c using (A.34) and collecting terms implies:

which proves conditions (22a)(24a) in Result 1.

B.2. Optimal policy rules of the city

In this subsection we prove the city's optimal policy rules (21b)(24b) when labor taxation/subsidy is available. Since the central municipality regards all towns as symmetric, the aggregate land rent becomes:

The utility of the resident of c living the closest from the CBD is:

taxes, traffic jam and spillover and her budget constraint is:

with ρ(-A) = F c L . Replacing w c and r using ( 15) and ( 16), it follows that the municipalities'

program is to maximize: into the production function using the land occupation condition (2).

From the above program, we can derive the rst-order conditions. The rst-order conditions with respect to τ R c , τ K c , τ W c and G c are respectively:

The rst-order condition with respect to R c is:

The rst-order condition with respect to R s is:

The rst-order condition with respect to K c is:

The rst-order condition with respect to K s is:

The rst-order condition with respect to W c is:

The rst-order condition with respect to W s is:

We can now prove the optimal behavior rules when labor taxes are available (21b)(24b).

The rst-order conditions with respect to the tax instruments (A.39)(A.42) imply:

which can also be written as:

using conditions (A.22)(A.24) and K = kP. Then, using Euler's identities (A.18) and substituting L j , j ∈ {c; s} from (2), we have:

Inserting (A.48)(A.51) into each of the rst-order conditions (A.42)(A.47) entails respectively:

which proves conditions (21b)(23b).

Appendix C Proof of Result 2

The purpose of this section is to derive the municipalities' optimal policy rules when the tax instrument set includes a tax/subsidy on labor. Specically, we prove the towns' optimal policy rules (32a)(34a) in subsection C.1 and the city's optimal policy rules (32b)(34b) in subsection C.2.

C.1. Optimal policy rules of a town

In this subsection we prove the representative town's optimal policy rules (32a)(34a) when labor taxation/subsidy is available. The proof is based on the rst-order conditions (A.25) (A.31). However, the rst-order condition with respect to the labor tax (A.27) is ignored and τ W s is replaced by 0 everywhere it appears. Inserting the rst-order conditions with respect to the tax instruments (A.25) and (A.26) into the rst-order condition with respect to labor (A.31), substituting L s from (2) and using Euler's theorem entails:

Appendix

The rst-order conditions with respect to the tax instruments (A.25) and (A.26), and condition (A.62) entail:

Then, using Euler's identities (A.18) and substituting L s from (2), we have:

which proves conditions (32a)(34a) in Result 2.

C.2. Optimal policy rules of the city

We now turn to the proof of the optimal behavior rules of the city when labor taxes are not available (32b)(34b). The proof is based on the rst-order conditions (A.39)(A.47), ignoring the rst-order condition with respect to the labor tax (A.41) is ignored and replacing τ W c and τ W s everywhere they appear by 0.

Inserting the rst-order conditions with respect to the tax instruments (A.25) and (A.26) into the rst-order condition with respect to labor respectively of the city (A.46) and the town (A.47), substituting L j , j ∈ {c; s} from (2) and using Euler's theorem entails:

where Φ is as dened in (A.63) and:

The rst-order conditions with respect to the tax instruments (A.39) and (A.40), and condition (A.62) entail:

which can also be written as:

using conditions (A.22)(A.24) and K = kP. Then, using Euler's identities (A.18) and substituting L j , j ∈ {c; s} from (2), we have: