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Financing Productive Local Public Goods
Journal of Urban Economics 45, 264]286 Ž1999. Article ID juec.1998.2097, available online at http:rrwww.idealibrary.com on
Financing Productive Local Public Goods* Gilles Duranton Department of Geography, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom E-mail: [email protected]
and Stephane Deo ´ ´ Goldman Sachs, 2, rue de Than, 75017 Paris, France Received July 2, 1997; revised March 6, 1998 Public economics typically assumes that local public goods only affect the utility of consumers. We analyze the case of purely productive local public goods within standard growth models. Investment in the public good enhances productivity only in the jurisdiction where it takes place. Capital, as well as people, is perfectly mobile. After characterizing the first-best equilibrium, we show that its decentralization to fiscally independent jurisdictions is more demanding than with local public consumer goods. In particular, efficient decentralization cannot be obtained with competitive land developers providing the public good through a simple land capitalization scheme. Q 1999 Academic Press Key Words: Local public goods; infrastructure; capitalization hypothesis.
I. INTRODUCTION How can an economy achieve optimal provision of local public goods ŽLPG.? Traditional public economics, originating from Samuelson w31x, stresses the difficulty of the provision of public goods in a general context. Basically, a decentralized scheme cannot be implemented because of a standard free-rider problem. Moreover, the first-order conditions for a central planner to achieve first-best require him to observe the consumers’ marginal rates of substitution. So, even with a benevolent planner, the optimum is difficult to reach. However, Tiebout w33x devised a clever alternative solution. His answer to the pivotal question raised above was to *We thank Charlie Bean, Paul Cheshire, Toni Haniotis, Vernon Henderson, Gianmarco Ottaviano, Kevin Roberts, Jacques Thisse, David Wildasin, an anonymous referee, and especially the Editor for helpful discussions, comments, and suggestions. 264 0094-1190r99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
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rely on the local aspects of some public goods and to use competition among the jurisdictions. His proposal was to decentralize the provision of the public good, whereby decentralization means that the jurisdictions are autonomous for both taxes and public expenditures. In his original paper, assuming an optimal number of jurisdictions led by profit-maximizing developers, people, by voting with their feet, can choose between different levels of provision for the public good and the associated head-taxes. Then the competitive equilibrium is a first-best situation. Yet the assumption of an ex-ante optimal number of jurisdictions and the possibility of head-taxes are rather restrictive. However, the subsequent literature Žsee Wildasin w36x or Mieszkowski and Zodrow w25x for complete surveys on this issue. managed to relax them elegantly.1 The Tiebout idea relies on the strong analogy between fiscal competition and the competition to supply private goods. Consequently free-entry of developers leads to the first-best just as free-entry of producers can achieve the First Welfare Theorem for private goods. As for the head-tax, the idea is to replace it by using land-based instruments. Implementing the first-best just requires the developer to be able to take advantage of the differential land rent in her jurisdiction since public spending is capitalized into the land value.2 The profit function of each developer in the most direct case is equivalent to total differential land rent ŽTDR . minus expenses for the public good Ž G .. Then an immediate implication of the zero-profit condition is TDR s G. This result is known as the Henry George Theorem ŽGeorge w13x.. Then the only informational requirement to implement the first-best in a decentralized way is the observation of the land market. This result appears in Flatters, Henderson and Mieszkowski w11x, Vickrey w34x, and Arnott and Stiglitz w1x among others. Of course it relies on strong assumptions. It is not valid with: }Imperfect competition; see Scotchmer w32x }Imperfect geography Žif the land-rent is not well-defined at the border of the city.; see Arnott and Stiglitz w1x or Pines w28x }Congestion; see Scotchmer w32x and Fujita w12x }Imperfect taxation Žif only a property tax is available instead of a land tax.; see Mieszkowski w24x or Hoyt w18x }Imperfect mobility Žif agents cannot vote with their feet at zero . cost 1
To insure existence, some restrictions on the utility functions are still necessary. See Fujita w12x for developments. 2 By differential land rent we mean the share of land rent created by the action of the developer. In the paper, without any land development, land rent is equal to zero so that total land rent and differential land rent are equal.
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Despite all this, Tiebout’s idea seems to be empirically relevant: Oates w27x, Edel and Sclar w10x, Hamilton w15x, Meadows w22x or, more recently, Wassmer w35x all draw favorable conclusions and show that mobility influences the mix of public goods offered at the local level. Moreover, some important projects are explicitly funded by land capitalization schemes. One can think, for instance, of the railways in Tokyo ŽKanemoto w19x, Kanemoto and Kiyono w20x, Midgley w23x. or public transport in Hong Kong ŽMidgley w23x.. Strange as it might seem, most potential applications of the capitalization hypothesis Ži.e., public good finance through the differential land rent. concern infrastructures Žwhich are ‘‘productive public goods’’., whereas the theoretical analysis focuses entirely on ‘‘public consumer goods’’ Žwhich directly enter the utility function..3, 4 For infrastructure, it is possible that user-charges can provide convenient, albeit not always simple, pricing schemes; a general synthesis on these aspects is provided by Laffont and Tirole w21x. In particular, perfect spatial discriminatory pricing is seldom available, although most infrastructures have an important geographic dimension. For road networks, airports, or even electricity or water distribution, location matters. In other words, public infrastructures are a crucial part of the production process at the local le¨ el Žsee Aschauer w2x and World Bank w38x.. Apart from user-charges, another alternative for financing public infrastructure is capital taxation at the local level ŽWildasin w37x.. The literature on this shows the existence of a fiscal externality leading jurisdictions to implement suboptimal levels of capital taxation because of factor mobility. This literature usually takes public expenditures as given and ignores any linkage between taxation and the marginal productivity of capital. In other words, it again assumes implicitly that the services public goods are consumed rather than productive. However, fiscal externalities are likely to persist and even worsen with productive local public goods. So, as it seems difficult to finance productive public goods with either user-charges or capital taxation, our aim in what follows is to examine to what extent the land market can contribute to the financing of productive local public goods as it can for public consumer goods. More broadly, the question is to assess what are the ‘‘minimal’’ instruments needed to insure an efficient pro¨ ision of producti¨ e local public goods. 3 Brueckner and Wingler w6x, Richter and Wellisch w30x, and Zou w39x are notable exceptions. 4 The concepts of productive public good, public capital, and infrastructure are equivalent in the analysis that follows. For capitalization estimates on public consumer goods, see Cheshire and Sheppard w7x.
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Another possible perspective is to consider that many local public consumer goods also have a productive aspect. Indeed, only purely recreational public goods such as parks and museums can be considered as pure public consumer goods with no productive role. Even in these extreme cases, it may be argued that they have some productive aspects: museums can improve education, whose productive role is obvious.5 The quality of leisure also has an impact on production. A final motivation is that public economics typically recommends that local expenditure should be financed through the taxation of the differential land rent. However, land taxes represent only a fraction of local public finances Žsee Prud’Homme w29x and Henderson w16x for evidence.. We offer here a new explanation for this stylized fact. For public consumer goods, a partial equilibrium analysis is sufficient. We just need to consider an exogenous income, which generates a demand for the public good mediated by the land markets. Then, the problem is to see how the public good can be financed through the land market. By contrast, in the case of productive public goods, the initial income generates a demand for private goods, for land and for savings. Using the demand for land, it is possible to finance local public goods but the story does not end here, since the savings and the amount of public good determine future production. A dynamic general equilibrium analysis is thus required.6 In other words, the analysis of productive public goods is inherently dynamic, because public capital can be accumulated and influences further production. Moreover, assume for convenience that private producers operate with constant returns to scale. Then the introduction of public capital, which is assumed to increase the marginal productivity of private capital, implies increasing returns at the aggregate level. For these reasons, our model also embodies increasing returns. Using such a framework, we explore whether the classical results associated with Tiebout and George are still valid when the productive aspect of public goods is taken into account. It is shown below that, although Tiebout-style Ži.e., first-best . results can be obtained, they are very demanding and must rely on strong assumptions. The Henry George Theorem does not hold in the usual sense. Simple capitalization schemes do not work because the marginal product of public investment in one jurisdiction benefits both the workers who live in the jurisdiction and also capital holders who need not live where their savings 5
Education presents some specific features that justify a separate analysis with different w5x. assumptions. For instance, see Benabou ´ 6 Zou w39x uses a framework close to ours, except that he does not consider any land market, which is the main focus of this paper. He also provides alternative justifications for the use of growth models in local public finance analysis.
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are invested. As a consequence, land rents capitalize public investment only up to the increase in wages it causes multiplied by the share of housing in expenditures. In Section II, we propose a simple framework of a competitive production economy with productive local public goods. In Section III, we analyze various decentralized frameworks. Finally, the last section ends the analysis with some considerations concerning the provision of local public goods. II. ECONOMIES WITH PRODUCTIVE PUBLIC GOODS: THE FIRST-BEST We consider a large economy composed of S ‘‘islands,’’ each of area normalized to unity. The total population size N is large but such that N - SL*, where 1rL* is the minimal amount of land to be consumed per capita in order to enjoy positive utility. A given island may or may not be populated. The representative agent is infinitely lived Žor finitely lived with a dynastic utility function., supplies continuously one unit of labor inelastically, and we assume that his initial wealth is strictly positive. His utility function is Us with
q`
H0
Ž ut .
1y s
ey r t dt, 1ys u t s z t stx if st G 1rL*, ut s 0 if st - 1rL*, s ) 1.
Ž 1.
Instantaneous utility at time t, u t , is thus a Cobb]Douglas function of z t , the per capita consumption of the final good and st , the per capita quantity of land Žwhich is a proxy for housing consumption., with a minimum consumption of land. This specification was chosen for analytical convenience. Since the elasticity of substitution is constant, a simple Cobb]Douglas specification may drive us to a meaningless corner solution. Our minimal land requirement is here to avoid this unappealing implication. Note that this specification is more general than the usual assumption of constant consumption of land that would be equivalent to x s 0. There is a single consumption good, which can be consumed, used as public investment, or transformed into private capital. Each island is a separate unit of production. In each island, the production function combines public capital Ž G ., private capital Ž K . and labor Ž L. along a a Yi , t s AGib, t K i1y , t Li , t
with
a G b ) x, i s 1, . . . , S.
Ž 2.
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This specification makes public capital a necessary input. In this respect we follow the empirical literature Žsee Gramlich w14x. and we do not consider the case of publicly provided private goods. Note that whenever land consumption is equally divided between inhabitants of the same island i, we observe si s 1rL i since we suppose that people should live and work on the same island. For the sake of simplicity we ignore depreciation and assume that any unit of public capital, private capital, or labor can be used only in one island. Capital and labor are perfectly mobile across islands. For simplicity, we assume that the good can be transformed into capital and shipped elsewhere, but then it cannot be consumed anymore. In short, what consumers eat must come from their residential area. This assumption may seem rather unappealing but it makes sense for nontradable goods like most personal services. Furthermore it leads to considerable technical simplification.7 Note also that only consumers use space. This simplifying assumption allows us to neglect the issue of competition for space between different categories of agents Žproducers and consumers.. Finally, the analysis that follows concentrates on the steady-state behavior of our economy. There is no great loss of generality: in this model the analysis of the transition dynamic is either straightforward when b - a since the problem reduces to a neoclassical model of growth or trivial when b s a since there is no transition dynamics. A tendency to agglomeration is driven by the increasing returns present in the production function. Reducing the number of populated Žor developed. islands increases production. Production becomes very large when only one island is developed. Acting against this, the taste for land acts as a dispersion force. We assumed in Eq. Ž2. that the preference for housing was not too strong Ž x - b . so that the dispersion force does not always dominate the agglomeration force. The balance between agglomeration and dispersion is optimally solved in the following lemma. LEMMA 1. The first-best land use where indi¨ iduals are treated symmetrically is such that NrL* islands are populated by L* inhabitants each. See proof in Appendix. As often in the literature, we use a rawlsian Žor equal treatment . concept for our first-best. This assumption is here to avoid a first-best for which consumers would not be treated symmetrically as it sometimes happens in public or urban economics. ŽSee Mirrlees w26x for an example where utilitarian welfare leads to unequal treatment.. The intuition for 7
In particular, it will simplify our first-best land use Žland is used with the same intensity wherever it is used.. Otherwise, we may be led to complex patterns of land use Žthe density in inhabited islands need not be the same everywhere at the optimum..
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this lemma is the following. As long as the marginal productivity of public capital multiplied by the marginal utility of consumption is higher than the marginal utility of the consumption of space, it is socially worthwhile to reduce the consumption of space to increase production in order to avoid the spatial dilution of public capital. Note that this first-best land use is such that many islands are populated and that the number of populated islands is proportional to N. PROPOSITION 1. The first-best is such that mobile factors are symmetrically allocated across populated islands. If b - a , the long-run le¨ el of capital accumulated is K s ŽŽ1 y a .1y bA b b L* arr .1rŽ ay b . in each island. If b s a , public capital, pri¨ ate capital and consumption grow at the rate ŽŽ1 y a .1y aa aAL* a y r .rs . Proof. Thanks to Lemma 1, we can restrict our attention to symmetric situations with constant consumption of land. It is then possible to write the social planner’s program independently of land consumption Max U s
z i , Gi , K i
q`
H0
s z i1y ,t
1ys
ey r t dt,
Ž SP.
˙i, t q K˙i, t , which subject to the island budget constraint: Yi, t s L*z i, t q G states that within each island, production is devoted either to consumption or to capital accumulation.8 We also face the following transversality condition: lim eyr tl t K t s 0.
tªq`
Ž 3.
Since s ) 1, intertemporal utility is always well defined. Thus this transversality condition does not play an important role in our analysis. The first-order conditions of SP are
¡z e s l ~lŽ 1 y a . AG ¢lb AG K ys y r t
by1
b
˙ Ky a L* a s yl
1y a
Ž 4.
a
˙ L* s yl
These first-order conditions are sufficient since, thanks to Lemma 1, labor can be treated parametrically so that the production function within each island is weakly concave. From this, we can state Gs 8
bK 1ya
We neglect the subscripts whenever there is no possible confusion.
Ž 5.
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and
˙z z
s
Ž1 y a .
1y b
A b b K by a L* a y r
s
.
Ž 6.
Depending on the value of b , we distinguish two different cases: v
b - a ŽSolow case.. There is a long-run steady state for which
˙zrz s 0:
Ks
ž
Ž1 y a .
1y b
A b b L* a
r
1r Ž a y b .
/
.
Ž 7.
In this case, we have decreasing returns to the reproducible factors. Consequently the economy converges towards its long-run equilibrium level K. v
b s a ŽBarro]Romer case.. From the previous equations, we easily
get «
˙z z
s
K˙ K
s
˙ G G
s
Ž1 y a .
1y a
a aAL* a y r
s
.
Ž 8.
Here, given that returns to the reproducible factors are precisely one, endogenous steady growth is possible. The equilibrium growth path is such that consumption, public, and private capital all increase at the same rate. The initial program involved a dynamic optimization over two choice variables with two state variables. However, thanks to Lemma 1, the consumption of space remains constant. This allows us to get rid of one choice variable and to return to convex analysis. Then, at the optimum the marginal productivity of public capital should be equal to that of private capital which reduces our problem to a standard dynamic programming problem. Depending on the value of b , two different results are obtained.9 These two different cases underline two different conceptions of the role of public spending in the growth process. In the first case, growth is taken as exogenous Žto the model.; that is to say, infrastructure does not act as the engine of growth, although it may intervene crucially in the productive process. This vision is defended, for instance, by Gramlich w14x. It states that infrastructure is necessary for production to occur, but that it does not influence the long-run growth rate. The other conception states that public 9 As usual in growth models, the case b ) a does not make much sense economically since it implies explosive growth.
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spending can be considered as a major engine of growth. Theoretically, this hypothesis was put forward by Barro w3x and Barro and Sala-i-Martin w4x. Empirically, it is defended by De Long and Summers w8x, w9x. It is not our purpose here to take sides in this controversy but to show that for both specifications our results are similar. Under centralized provision of public goods, it can be shown easily that the first-best solution in these two programs can be decentralized partially. Competitive firms hiring labor and private capital will ensure efficiency. Indeed, these firms face a well-behaved production function with constant returns to scale in private inputs. Public capital can be financed through a tax on production. But, the efficiency of this tax rests on three strong assumptions, which are i. observability of private production, ii. no labor supply distortion, and iii. immobility of factors. In a competitive equilibrium, consumers receive an interest rate rt for their savings and a wage wt for their work. Each factor is paid its marginal product and firms make zero profits. Moreover, consumers have to pay a rent R t for each unit of land which can be reinjected in the economy through public land property. Our concern in what follows is then to explore whether it is possible to decentralize not only the allocation of labor and private capital but also the provision of public infrastructure. III. DECENTRALIZATION OF THE FIRST-BEST The Institutional Regimes and Equilibrium Concepts We suppose now that the development of islands is undertaken by competing developers. So by decentralization, we mean that the provision of public goods and the authority to tax are delegated to competitive land developers. The competitive process is the following. On any undeveloped island, a developer can create a jurisdiction where she will provide the public good. Consumers then make their location and saving decisions with both labor and capital being perfectly mobile. Within this broad framework, decentralization can be established through various institutional settings. The first alternative we consider is one where the developer has complete control over everything in her jurisdiction Ž Regime 1.. All possible fiscal and financial instruments are available and can be implemented. Regime 1 can be viewed as a benchmark case, but it may also be applicable to the US where land developers can have extensive powers Žsee Henderson and Mitra w17x.. It also strongly reminds us of traditional factory-towns. More formally we define as equilibrium for Regime 1 a situation whereby: }Consumers maximize their utility with respect to their location and the allocation of their income between consumption and savings.
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}Competitive land developers maximize their profit with respect to the amount of public and ‘‘private’’ capital, land rent, population, and wages in their jurisdiction. They take the behavior of the other land developers and the interest rate as given. }Land developers make zero profit Žfree-entry .. }All markets clear. We also consider Regime 2, where the jurisdictions can only implement a residential tax and a pure land tax, retaining also some power over population control. Given perfect factor mobility, using a residential tax seems reasonable. Other fiscal instruments may be more difficult to use. This regime may be relevant for countries with a decentralized system of government like Switzerland. Decentralization takes place in this framework at two levels since, on the one hand, the provision of public goods is left to private developers and, on the other hand, production of the consumption good is realized within the jurisdictions by competitive firms. Thus, we define as equilibrium for Regime 2 a situation whereby: }Consumers maximize their utility with respect to their location and the allocation of their income between consumption, land, and savings. }Competitive land developers maximize their profit with respect to the amount of public capital they provide, their population and a residential land tax, taking everything else as given. }Competitive firms in each jurisdiction maximize their profits with respect to private capital and labor, taking all prices as given. }Land rent is determined competitively in each jurisdiction. Developers and firms make zero profit. }All markets clear. Finally, and this may be more relevant to describe the countries with a centralized system of government Žbut where local public goods remain locally provided., we consider an environment in which the land developer cannot use any fiscal instrument but a pure land tax. Her sole source of income is thus the land rent, which is competitively determined Ž Regime 3 .. Of course, although the developer cannot set land rent directly in her jurisdiction, she can manipulate it indirectly by public expenditure actions. Besides, since the Henry George Theorem is often valid with public consumer goods, the natural temptation is to restrict the jurisdictions to use only land rent to finance the productive public goods Ži.e., to use a simple capitalization scheme.. If this arrangement enables us to attain first-best, then its practical implementation should be simpler than in the two previous cases since it would just require land prices to be observed.
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Thus, we define as equilibrium for Regime 3 a situation whereby: }Consumers still maximize as previously. }Competitive land developers maximize their profit with respect to the amount of public capital they provide and the population in their island, taking everything else as given. Their sole source of income is the competitively determined land rent in their jurisdiction. }Competitive firms behave as previously. }Land developers and firms make zero profit. }All markets clear. Regime 1 In that case, we get the following proposition, which states that if developers have enough degrees of freedom, the first-best can be reached. PROPOSITION 2. The equilibrium under Regime 1 is the equal treatment first-best. Proof. Developers maximize the sum of their discounted profits, that is, production and land rent minus the cost of factors. Due to perfect factor mobility this objective is equivalent to the maximization of instantaneous profit at each point in time. Thus, each developer faces the problem: Max p t s Yt q R t y wt L t y rt K t y rt Gt , s.t. u t G U.
Ž DP1.
The developer can maximize profits without any restriction on the instruments she uses. Her only constraint is to offer a level of utility as high as can be obtained in other islands. Given the concavity of the utility function, each developer will divide her land equally among her consumers. Wages and rents are redundant since labor supply is inelastic and land is equally divided. Without loss of generality we can therefore normalize R s 0. Our program becomes Max
w, ˆ G, K , L
p t s Yt y w ˆt L t y rt K t y rt Gt ,
Ž 9.
s.t. u t G U.
where w ˆ is the wage distributed when land rent is normalized to zero. Before solving the developers’ program, we must consider the consumers’ program in jurisdiction i: Max U s z
q`
H0
Ž z t six, t .
1y s
1ys
s.t. w ˆi , t q rt Zt s z t q Z˙t ,
ey r t dt,
Ž CP1 .
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and si G 1rL*, where Z is individual wealth. Assuming temporarily a constant lot size proposed by the developers, it is possible to take the term in s in the utility function out of the integral. The first-order conditions for CP1 are
½
zys eyr t s l , ˙. l r s yl
Ž 10 .
After simplification, Eq. Ž10. and the budget constraint imply w ˆt q
r s y 1. q r
žŽ
s
/
Zt s z t .
Ž 11 .
Let us return to the developers’ program Ž9.. Since, the utility function in the constraint of Eq. Ž9. cannot be differentiated everywhere, assume first that there is an optimal L such that L - L*. We can inject Eq. Ž11. in the first constraint of Eq. Ž9.. If we denote by m the Lagrange multiplier associated with this constraint, the first-order conditions imply
¡ Y s r , K Y
~ G Y L
s r, yw ˆsx
¢L s m L
1 x
m Lxq1
w ˆq
ž ž
r Ž s y 1. q r
s
Ž 12 .
//
Z ,
.
Then, simple manipulations show that this system can be satisfied only for L ª q` which violates the assumption made above. ŽDevelopers have an incentive to use as much as possible of the three factors.. Thus we cannot have L - L* which implies immediately that L s L*. Combining L s L* with the first-order conditions in K and G, which remain unchanged, we obtain
¡G s b K , ~ 1ya ¢r s Ž 1 y a .
1y b
Ž 13 . b
b AK
by a
a
L* .
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Then, using the zero-profit condition, we find w ˆs
ayb L
Y.
Ž 14 .
In the Solow case, consumers save until the steady-state is reached. This steady-state is such that r s r . We find the long-run level of capital to be K as in Eq. Ž7.. Consequently, our decentralization scheme enables us to implement the first-best in the case of decreasing returns to the reproducible factor. As for the Barro]Romer case, the first-order and zero-profit conditions imply
¡G s a K , ~ 1ya ¢r s Ž 1 y a .
1y a
Ž 139. a
a
a AL* ,
and w ˆ s 0.
Ž 14’.
From this we can easily show that the economy enjoys a growth rate which is the same as in Eq. Ž8.. Thus, it is also possible to decentralize our first-best solution if we allow for constant returns to the reproducible factors. In conclusion, for both cases, land use, the allocation of factors, and their distribution is the same as with the equal treatment first best. Note that heavy intervention is required in our ‘‘factory-islands’’ since the developers are responsible for both the wage policy and the spatial allocation of consumers in their island. Moreover, the Henry George Theorem becomes meaningless since the first-best solution can be obtained for any level of land rent. ŽActually, land markets can be completely ignored and developers can allocate their consumers directly to their housing lot.. Note also that the net wage is zero in the Barro]Romer case. The reason is that, on the one hand, reproducible factors do not use any land and are thus paid their marginal product. On the other hand, to get self-sustaining growth, we need the sum of the shares of public and private capital to be one in the production function. Consequently, the product is exhausted on public and private capital income. To get more intuition concerning the working of the model, we need to explore the other two regimes. Regime 2 We now restrict the competencies of the developers since it may be argued that having land developers fixing lot size, producing private goods, and providing public capital may not be realistic. So in this section we
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analyze the case where land developers provide public capital which is financed through land rents and a residential head-tax. PROPOSITION 3. The equilibrium in Regime 2 is the equal treatment first-best. De¨ elopers set a Ž steady-state. residential head-tax equal to: }T s Ž b y x . r KrŽ1 y a . in the Solow case. }Tt s a YtrL t y x ŽŽ r Ž s y 1. q r .rs . Zt in the Barro]Romer case. Proof. The consumers’ program in jurisdiction i is Max U s z, s
q`
H0
Ž z t six, t .
1y s
1ys
ey r t dt,
Ž CP2 .
s.t. wi , t q rt Zt y Z˙t y Ti , t s R i , t si , t q z t , and si , t G 1rL*, where Ti, t is the taxation in jurisdiction i. The wage w is offered by independent competing producers and R is competitively determined. As previously, assume temporarily that s is constant over time; the first-order conditions imply after simplification
¡˙z r y r , ¢Rs s xz.s
~zs
Ž 15 .
Again, because of the mobility of the productive factors, the developers’ program reduces to: Max p t s R t q L t Tt y rt Gt ,
G, T , L
s.t. u t s U,
Ž DP2.
where T is the residential head-tax. Instead of solving this program, note that we can write wt y Tt y R t st s w ˆt .
Ž 16 .
Moreover, competitive producers within each jurisdiction will induce wt L t s Yt y rt K t . We can then re-write the developer’s program: Max p t s Yt y w ˆt L t y rt Gt y rt K t ,
G, w ˆ, L
s.t. u t s U
Ž 17 .
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Then assume again an optimal L such that L - L*. The first-order conditions of Eq. Ž17. imply
¡ Y s r , ~
G Y L
yw ˆsx
¢L s m L
1 x
m L
xq1
r s y 1. q r
Ž ž ž w ˆq
s
//
Z ,
Ž 18 .
.
The competitive behavior of firms within each island implies that
Yr K s r .
Ž 19 .
Equations Ž18. and Ž19. together are equivalent to Ž12.. Thus, the same argument as in Proposition 2 applies, leading to L s L*. Then combining the first condition of Eq. Ž18. with Eq. Ž19. when L s L*, we obtain Eq. Ž13. again. Using the zero-profit condition for the developers leads to Eq. Ž14. where w ˆ s Ž a y b .YrL. We can now plug Eq. Ž15. in Eq. Ž16.. We find that Tt s w ˆt y wt y xz t . In the Solow case, using Ž11., Ž14., the market clearing condition on the capital market, ZL* s K q G, and the steadystate condition for which r s r . We get simply T s Ž b y x . r Kr Ž 1 y a . s Ž b y x . Y Ž K , L* . .
Ž 20 .
In the Barro]Romer situation, we have Tt s a YtrL* y x
r Ž s y 1. q r
s
Zt ,
Ž 21 .
with r as in Ž139.. The first-best can be obtained if we replace the developer’s direct intervention in production by a residential head-tax. Thus, the outcomes under Regime 1 and Regime 2 are similar. ŽThe main difference between DP1 and DP2 rests with the provision of private capital. But, since it is competitive in both cases, the outcome is the same.. The only difference between Regime 1 and Regime 2 is that direct wage setting is replaced by a residential head-tax. This residential tax may seem at first difficult to implement since it requires the knowledge of all the parameters in the model Žas in most public good provision problems.. But, because of the symmetry across individuals, we just need the developer to be able to observe the land rent and to set a local residential head-tax to maximize
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her total profit. As we remarked above, both reproducible factors are paid at their marginal productivity Žsee the first condition of Eq. Ž18. and Eq. Ž19... The reason for this is that private capital and public capital are perfectly mobile and do not consume any amount of land. Labor, by contrast, consumes land. Any marginal consumer in a jurisdiction lowers both the average productivity of labor due to decreasing returns and the amount of land available for housing purpose. The developer, however, benefits from his arrival since he also generates a positive pecuniary externality for the returns to private and public capital and since his demand for land increases land rent. This head-tax is thus not the usual congestion tax, which is often needed in public good finance analysis Že.g., Fujita w12x, Scotchmer w31x, or Wildasin w35x.. The reason for this instrument is really because private capital benefits from public investment without having to pay for it through the land market. The fiscal instruments we need here to implement the first-best are a pure land tax and a residential head-tax. As a corollary, the Henry George Theorem breaks down because of mobile factors that benefit from the public good without having to use any land. Of course, it is immediate that the same efficient outcome could be achieved by any other non-distortionary tax like a tax on labor, on local earnings, or on consumption. What really matters is that this tax must be levied on one factor labor and not on public nor private capital to ensure efficiency Žabstracting from distribution issues here.. Why focus on the land market then? The answer we suggest is that head-taxes may be more easy to implement on the land market than anywhere else. In order to be of practical use, the model calls for further work. Firstly, in most densely populated areas, land is used to a large extent for residential purpose. But, non-agricultural productive activities also use land as a factor of production. This is likely to imply a tax levied on capital. Secondly, if only distortionary instruments are available, we will need to derive the optimal set of instruments in a second-best environment. We leave these two extensions for future work. In the next section, we concentrate on the more restricted case where only a pure land tax is available. Regime 3 Since the first-best requires a non-zero taxrsubsidy, we can infer easily that it will be impossible to obtain efficiency under Regime 3. However, analysis of Regime 3 is revealing because we can explore the inefficiencies resulting from a more constrained regime. This also strengthens the intuition concerning our previous results.
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PROPOSITION 4. The equilibrium in Regime 3 leads to a sub-optimal pro¨ ision of public good. Proof. The consumers’ program in jurisdiction i is
Max U s z, s
q`
H0
Ž z t six, t .
1y s
ey r t dt,
1ys
Ž CP3 .
s.t. wi , t q rt Zt y Z˙t s R i , t si , t q z t , and si , t G 1rL*. Let us use the same approach as with the previous propositions. Assume temporarily that the consumption of land is constant over time. We find the first-order conditions of CP3 to be Rs s xz and ˙ zrz s Ž r y r .rs , ˙ Ž . which implies Zrz s r y r rs along the balanced growth path. ŽThe first-order conditions of CP3 are the same as those of CP2. This not surprising since the only difference between CP2 and CP3 is the tax T in CP2 which is taken as given by the consumers.. Plugging the previous results in the budget constraint in CP3, we obtain Ri s
x xq1
ž
L i wi q ZL i r y
ž
ryr
s
//
.
Ž 22 .
As for the developers’ program, we have Max p t s R t y rt Gt , G, L
Ž DP3.
s.t. u t s U.
Then suppose again that there is an optimal L such that L - L*. Since u s zs x and Rs s xz, we can write u s RrŽ xL1q x .. Then note that competition between producers implies wL s Y y rK. Injecting this in Eq. Ž22. and replacing in DP3, we obtain x
Max p t s
xq1
G, L
s.t.
x xq1
ž
ž
Yt y rt K t q Zt L r y
ž
Yt y rt K t q Zt L r y
ž
ryr
s
ryr
//
s
//
y rt Gt ,
1q x
Ž xL
Ž 23 . . s U.
281
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The first-order conditions are
¡ ~
¢
x
Y
x q 1 G x Y xq1 sm
ž
x
yrsm
L
q
x xq1
ž
ž
Y
x q 1 G r Ž s y 1. q r
,
s
Y L
q
ž
// Z
Ž 24 .
r Ž s y 1. q r
s
Z y m x Ž x q 1 . ULx .
//
After simplification, the first condition implies m s 1 y Ž x q 1. rGrŽ x b Y .. Injecting this in the second condition of Eq. Ž24. leads to a contradiction since the term on the rhs is always strictly superior to the term on the lhs Ži.e., developers are always willing to increase their population.. Again, the incentive to increase the population in each island leads to L s L*. The zero-profit condition for the developers implies Gs
xL*
Ž x q 1. r
ž
wqZ ry
ž
ryr
s
//
.
Ž 25 .
Competition between firms within each island implies again that Yr K s r. Market clearing on the capital market leads to ZL* s K q G. Combining the last three equations, we find Gs
1r Ž 1 y a . y Ž r y r . rrs
Ž 1rx . y Ž r y r . rrs
K.
Ž 25 .
In the Solow case, this implies Gs
x 1ya
K,
Ž 26 .
instead of G s brŽ1 y a . K in the first-best ŽEq. Ž5... This leads to the following steady-state:
Ks
ž
Ž1 y a .
1y b
r
Ax b L* a
1r Ž a y b .
/
.
Ž 27 .
Since x - b , we obviously have K - K. In the Barro]Romer case, a straightforward comparison between Eqs. Ž26. and Ž5. also shows there is also under-provision of public capital.
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Consumers want to live where they receive the highest wages. Since land is scarce, competition for land in developed islands creates a land rent that enables the financing of the local public goods. Note that if x s 0 Žno preference for land above the minimum required., the public good does not get provided at all. In the traditional case of public consumer goods, there is a direct relation between public expenditure and the utility of consumers. In the case of productive public goods, the relation is not as direct: more public expenditures induce higher production and only a share of the marginal production is received by the workers Ži.e., a fraction a .. Furthermore, the marginal increase of wages generated by the marginal public investment is used only partially for housing expenditures Žthe share of housing is xrŽ x q 1. in total expenditure.. This appears in Eq. Ž24. where the marginal value of public investment is capitalized only through the share of wages in the production multiplied by the share of housing in consumer expenditures. By contrast, in a model without mobile capital and without intertemporal consumption, but nonetheless productive public goods, Brueckner and Wingler w6x reach an efficiency result. Another efficiency result is reached by Richter and Wellisch w30x in a static framework with exogenous but mobile firms and without a housing market. However, land rent in a given jurisdiction partly capitalizes public investments made in all other jurisdictions. This happens because the demand for land Žand thus land rent. depends not only on current income, but also on total wealth Žsee Eq. Ž24. again.. Higher public investments in jurisdiction j imply a higher interest rate for private capital in the economy, and thus a higher wealth for consumers in jurisdiction i. This second effect prevents the Henry George Theorem from failing too spectacularly. Capitalization rates are very low Žaround 20% at most if we try to calibrate the model. but the optimal tax in the Solow case at the symmetric equilibrium is just the difference between the share of public capital in the production and the share of housing in the expenditure Žsee Eq. Ž20... To put it differently, note that in the case of public consumer goods, efficiency is obtained with the equalization of marginal rates of substitution. Here efficiency is reached with the equalization of marginal rates of return: the marginal productivity of private capital should be equal to that of public capital. The problem we have is that the developer’s marginal revenue for public investment is given by the marginal increase of land rent, which is driven by a preference parameter. Indeed, investment in public capital is driven by consumers equalizing their marginal rates of substitution, which has nothing to do with the equalization of marginal rate of transformation. ŽNote that the optimal tax in Eq. Ž20. is a linear function of the difference between the share of public capital and the preference for land.. Thus, with simple capitalization, we introduce a
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preference parameter in a technological relation so that at the ŽSolow. symmetric equilibrium we have G s xYrr instead of G s b Yrr . This analysis bears some implications for economic policy. It suggests that fiscal decentralization should be either minimal or fairly extensive in the sense that both land rents and a land tax are needed. From a more positive perspective, fiscal autonomy is likely to be more efficient when the local authorities can use a wide range of instruments and do not have to rely only on simple capitalization schemes Žor a pure land tax.. IV. CONCLUDING REMARKS ON THE PROVISION OF LOCAL PUBLIC GOODS We have shown in this paper that considering the productive aspects of local public goods seriously complicates the decentralization of the provision of infrastructure. The first-best can be decentralized, but the local authority must be able to intervene at least through a residential head-tax Žindexed on production and wealth.. With heterogeneous agents, the required residential tax could be even more difficult to implement: the optimal tax must be indexed on variables that are clearly difficult to observe at the local level Že.g., wealth in the upper-tail of the distribution.. Without any tax, direct internalization of the land rent is not efficient. Marginal local public investments are capitalized in local land values only through the share of housing multiplied by the share of wages in local production. Empirically, this should give capitalization rates below 20%. Conversely, land values within one jurisdiction also partially capitalize public investments made in all the economy because of the mobility of capital Žhigher interest rates induce a higher demand for land.. However, despite our apparently rather negative results, Tiebout and George’s ideas may deserve more attention than they are presently given. Our interpretation of existing land capitalization schemes is that, although they may not be adequate as main instruments, they may constitute useful additional instruments Žcapitalization occurs up to the share of wage multiplied by that of housing.. For instance, the Hong Kong underground network was partly financed through a land capitalization scheme Žsee Midgley w23x for more details.. Before the construction of the network, the operator was able to buy large parcels surrounding the future stations. Now the operator receives a majority of its income through the fares, but the profit made on land operations Ždevelopment of shopping areas, commercial real estate, and residential buildings. is by no means negligible, at around 15% of the construction cost. As a consequence, the Hong Kong underground is the only profitable underground network in the world.
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APPENDIX: PROOF OF LEMMA 1 After some straightforward manipulations Ži.e., broadly similar to the ones performed through Eqs. Ž2. to Ž6.., the optimal consumption path for a fixed amount of developed land is z s Y y g Ž K q G.
gs
with
Ž1 y a .
1y b
A b b K by a sy a y r
s
, Ž A1.
where g is the growth rate of the economy. Then we can write zs
AG b K 1y a sb
y
Ž1 y a .
1y b
A b b K by a sy a y r 1 y a q b
ž
s
1ya
/
K . Ž A2.
This leads to u s s xy aAK 1y aq b
b
b
žž / 1ya
s ay b y
Ž1 y a .
q
1y b
Ab b
s r a
s s AK
by a
ž
ž
1yaqb 1ya
1yaqb 1ya
//
/
. Ž A3.
After simplifications, one can check that for any z ) 0 x-b«
u s
F 0.
Ž A4.
So for a given accumulation path Ži.e., some existing G, K and savings at each date., from x - b , the instantaneous utility of a typical consumer always increases by having him consuming less space until the constraint s G 1rL* is binding. ŽWe did not consider here explicitly the possibility of asymmetric equilibria. But in this respect, note that, at the equal-treatment first-best, instantaneous utility must be the same for everybody. By assumption, we also need production to take place on the island where one lives. Thus having transfers coming in or out in island i amounts to modifying A i , which drives us back to the argument above.. REFERENCES 1. R. Arnott and J. Stiglitz, Aggregate land rents, expenditure on public goods, and optimal city size, Quarterly Journal of Economics, 92, 471]500 Ž1979.. 2. D. Aschauer, Is public expenditure productive?, Journal of Monetary Economics, 23, 177]200 Ž1989..
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3. R. Barro, Government spendings in a simple model of endogenous growth, Journal of Political Economy, 98, S103]S125 Ž1990.. 4. R. Barro and X. Sala-i-Martin, Public finance in models of economic growth, Re¨ iew of Economic Studies, 59, 645]662 Ž1992.. 5. R. Benabou, Workings of a city: location, education and production, Quarterly Journal of ´ Economics, 108, 619]652 Ž1993.. 6. J. K. Brueckner and T. Wingler, Public intermediate inputs, property values, and allocative efficiency, Economics Letters, 14, 245]250 Ž1984.. 7. P. Cheshire and S. Sheppard, On the price of land and the value of amenities, Economica, 62, 247]267, Ž1995.. 8. B. De Long and L. Summers, Equipment investment and economic growth, Quarterly Journal of Economics, 106, 445]502 Ž1991.. 9. B. De Long and L. Summers, How strongly do developing economies benefit from equipment in investment?, Journal of Monetary Economics, 32, 395]415 Ž1993.. 10. M. Edel and E. Sclar, Taxes, spending, and property values: supply adjustment in a Tiebout-Oates model, Journal of Political Economy, 82, 941]954 Ž1974.. 11. F. Flatters, J. V. Henderson, and P. Mieszkowski, Public goods, efficiency and regional fiscal equalization, Journal of Public Economics, 3, 99]112 Ž1974.. 12. M. Fujita, ‘‘Urban Economic Theory, Land Use and City Size,’’ Cambridge University Press, Cambridge, United Kingdom Ž1989.. 13. H. George, ‘‘Progress and Poverty,’’ Doubleday, New York Ž1879, reedition 1907.. 14. E. Gramlich, Infrastructure investment: a review essay, Journal of Economic Literature, 32, 1176]1196 Ž1994.. 15. B. Hamilton, The effects of property taxes and local public spending, on property values: a theoretical comment, Journal of Political Economy, 84, 647]650 Ž1976.. 16. J. V. Henderson, Will homeowners impose property taxes?, Regional Science and Urban Economics, 25, 153]181 Ž1995.. 17. J. V. Henderson and A. Mitra, The new urban landscape: Developers and Edge cities, Regional Science and Urban Economics, 26, 613]643 Ž1996.. 18. W. Hoyt, Competitive jurisdictions, congestion, and the Henry George Theorem: when should property be taxed instead of land, Regional Science and Urban Economics, 21, 351]370 Ž1991.. 19. Y. Kanemoto, Pricing and investment policies in a system of competitive commuter railways, Re¨ iew of Economic Studies, 51, 665]682 Ž1984.. 20. Y. Kanemoto and K. Kiyono, Regulation of commuter Railways and spatial development, mimeo Ž1993.. 21. J.-J. Laffont and J. Tirole, ‘‘A Theory of Incentives in Procurement and Regulations,’’ MIT Press, Cambridge, MA Ž1993.. 22. G.-R. Meadows, Taxes, spending, and property values: A comment and further results, Journal of Political Economy, 84, 869]880 Ž1976.. 23. P. Midgley, Urban transport in Asia, World Bank Technical Paper 224 Ž1994.. 24. P. Mieszkowski, The property tax: an excise tax or a profit tax, Journal of Public Economics, 1, 415]435 Ž1972.. 25. P. Mieszkowski and G. Zodrow, Taxation and the Tiebout model: the differential effects of head taxes, taxes on land rent, and property taxes, Journal of Economic Literature, 27, 1098]1146 Ž1989.. 26. J. Mirrlees, The optimum town, Swedish Journal of Economics, 74, 114]135 Ž1972.. 27. W. E. Oates, The effects of property taxes and local spending on property values: an empirical study of tax capitalization and Tiebout hypothesis, Journal of Political Economy, 77, 957]971 Ž1969..
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28. D. Pines, Tiebout without politics, Regional Science and Urban Economics, 21, 469]489 Ž1991.. 29. R. Prud’homme, Financing urban public services, in ‘‘Handbook of Regional and Urban Economics’’ ŽE. Mills, Ed.., Elsevier Science Publishers, Amsterdam, Vol. II Ž1987.. 30. W. Richter and D. Wellisch, The provision of local public goods and factors in the presence of firm and household mobility, Journal of Public Economics, 60, 73]93 Ž1996.. 31. P. Samuelson, The pure theory of public expenditures, Re¨ iew of Economics and Statistics, 36, 387]389 Ž1954.. 32. S. Scotchmer, Local public good in an equilibrium, Regional Science and Urban Economics, 16, 463]481 Ž1986.. 33. C. Tiebout, A pure theory of public expenditures, Journal of Political Economy, 64, 416]424 Ž1956.. 34. W. Vickrey, The city as a firm, in ‘‘The Economics of Public Services’’ ŽM. Feldstein and R. Inman, Eds.., MacMillan, London Ž1977.. 35. R. Wassmer, Property taxation, property base, and property values: an empirical test of the ‘‘new view,’’ National Tax Journal, 46, 135]159 Ž1993.. 36. D. Wildasin, Theoretical analysis of local public economics, in ‘‘Handbook of Regional and Urban Economics’’ ŽE. Mills, Ed.., Elsevier Science Publishers, Amsterdam, Vol. II Ž1987.. 37. D. Wildasin, Interjuridictional capital mobility: fiscal externality and corrective subsidy, Journal of Urban Economics, 25, 193]212 Ž1989.. 38. World Bank, ‘‘World Development Report,’’ Oxford University Press, London Ž1994.. 39. H.-F. Zou, Taxes, federal grants, local public spending, and growth, Journal of Urban Economics, 39, 303]317 Ž1996..
Financing Productive Local Public Goods* Gilles Duranton Department of Geography, London School of Economics, Houghton Street, London WC2A 2AE, United Kingdom E-mail: [email protected]
and Stephane Deo ´ ´ Goldman Sachs, 2, rue de Than, 75017 Paris, France Received July 2, 1997; revised March 6, 1998 Public economics typically assumes that local public goods only affect the utility of consumers. We analyze the case of purely productive local public goods within standard growth models. Investment in the public good enhances productivity only in the jurisdiction where it takes place. Capital, as well as people, is perfectly mobile. After characterizing the first-best equilibrium, we show that its decentralization to fiscally independent jurisdictions is more demanding than with local public consumer goods. In particular, efficient decentralization cannot be obtained with competitive land developers providing the public good through a simple land capitalization scheme. Q 1999 Academic Press Key Words: Local public goods; infrastructure; capitalization hypothesis.
I. INTRODUCTION How can an economy achieve optimal provision of local public goods ŽLPG.? Traditional public economics, originating from Samuelson w31x, stresses the difficulty of the provision of public goods in a general context. Basically, a decentralized scheme cannot be implemented because of a standard free-rider problem. Moreover, the first-order conditions for a central planner to achieve first-best require him to observe the consumers’ marginal rates of substitution. So, even with a benevolent planner, the optimum is difficult to reach. However, Tiebout w33x devised a clever alternative solution. His answer to the pivotal question raised above was to *We thank Charlie Bean, Paul Cheshire, Toni Haniotis, Vernon Henderson, Gianmarco Ottaviano, Kevin Roberts, Jacques Thisse, David Wildasin, an anonymous referee, and especially the Editor for helpful discussions, comments, and suggestions. 264 0094-1190r99 $30.00 Copyright Q 1999 by Academic Press All rights of reproduction in any form reserved.
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rely on the local aspects of some public goods and to use competition among the jurisdictions. His proposal was to decentralize the provision of the public good, whereby decentralization means that the jurisdictions are autonomous for both taxes and public expenditures. In his original paper, assuming an optimal number of jurisdictions led by profit-maximizing developers, people, by voting with their feet, can choose between different levels of provision for the public good and the associated head-taxes. Then the competitive equilibrium is a first-best situation. Yet the assumption of an ex-ante optimal number of jurisdictions and the possibility of head-taxes are rather restrictive. However, the subsequent literature Žsee Wildasin w36x or Mieszkowski and Zodrow w25x for complete surveys on this issue. managed to relax them elegantly.1 The Tiebout idea relies on the strong analogy between fiscal competition and the competition to supply private goods. Consequently free-entry of developers leads to the first-best just as free-entry of producers can achieve the First Welfare Theorem for private goods. As for the head-tax, the idea is to replace it by using land-based instruments. Implementing the first-best just requires the developer to be able to take advantage of the differential land rent in her jurisdiction since public spending is capitalized into the land value.2 The profit function of each developer in the most direct case is equivalent to total differential land rent ŽTDR . minus expenses for the public good Ž G .. Then an immediate implication of the zero-profit condition is TDR s G. This result is known as the Henry George Theorem ŽGeorge w13x.. Then the only informational requirement to implement the first-best in a decentralized way is the observation of the land market. This result appears in Flatters, Henderson and Mieszkowski w11x, Vickrey w34x, and Arnott and Stiglitz w1x among others. Of course it relies on strong assumptions. It is not valid with: }Imperfect competition; see Scotchmer w32x }Imperfect geography Žif the land-rent is not well-defined at the border of the city.; see Arnott and Stiglitz w1x or Pines w28x }Congestion; see Scotchmer w32x and Fujita w12x }Imperfect taxation Žif only a property tax is available instead of a land tax.; see Mieszkowski w24x or Hoyt w18x }Imperfect mobility Žif agents cannot vote with their feet at zero . cost 1
To insure existence, some restrictions on the utility functions are still necessary. See Fujita w12x for developments. 2 By differential land rent we mean the share of land rent created by the action of the developer. In the paper, without any land development, land rent is equal to zero so that total land rent and differential land rent are equal.
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Despite all this, Tiebout’s idea seems to be empirically relevant: Oates w27x, Edel and Sclar w10x, Hamilton w15x, Meadows w22x or, more recently, Wassmer w35x all draw favorable conclusions and show that mobility influences the mix of public goods offered at the local level. Moreover, some important projects are explicitly funded by land capitalization schemes. One can think, for instance, of the railways in Tokyo ŽKanemoto w19x, Kanemoto and Kiyono w20x, Midgley w23x. or public transport in Hong Kong ŽMidgley w23x.. Strange as it might seem, most potential applications of the capitalization hypothesis Ži.e., public good finance through the differential land rent. concern infrastructures Žwhich are ‘‘productive public goods’’., whereas the theoretical analysis focuses entirely on ‘‘public consumer goods’’ Žwhich directly enter the utility function..3, 4 For infrastructure, it is possible that user-charges can provide convenient, albeit not always simple, pricing schemes; a general synthesis on these aspects is provided by Laffont and Tirole w21x. In particular, perfect spatial discriminatory pricing is seldom available, although most infrastructures have an important geographic dimension. For road networks, airports, or even electricity or water distribution, location matters. In other words, public infrastructures are a crucial part of the production process at the local le¨ el Žsee Aschauer w2x and World Bank w38x.. Apart from user-charges, another alternative for financing public infrastructure is capital taxation at the local level ŽWildasin w37x.. The literature on this shows the existence of a fiscal externality leading jurisdictions to implement suboptimal levels of capital taxation because of factor mobility. This literature usually takes public expenditures as given and ignores any linkage between taxation and the marginal productivity of capital. In other words, it again assumes implicitly that the services public goods are consumed rather than productive. However, fiscal externalities are likely to persist and even worsen with productive local public goods. So, as it seems difficult to finance productive public goods with either user-charges or capital taxation, our aim in what follows is to examine to what extent the land market can contribute to the financing of productive local public goods as it can for public consumer goods. More broadly, the question is to assess what are the ‘‘minimal’’ instruments needed to insure an efficient pro¨ ision of producti¨ e local public goods. 3 Brueckner and Wingler w6x, Richter and Wellisch w30x, and Zou w39x are notable exceptions. 4 The concepts of productive public good, public capital, and infrastructure are equivalent in the analysis that follows. For capitalization estimates on public consumer goods, see Cheshire and Sheppard w7x.
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Another possible perspective is to consider that many local public consumer goods also have a productive aspect. Indeed, only purely recreational public goods such as parks and museums can be considered as pure public consumer goods with no productive role. Even in these extreme cases, it may be argued that they have some productive aspects: museums can improve education, whose productive role is obvious.5 The quality of leisure also has an impact on production. A final motivation is that public economics typically recommends that local expenditure should be financed through the taxation of the differential land rent. However, land taxes represent only a fraction of local public finances Žsee Prud’Homme w29x and Henderson w16x for evidence.. We offer here a new explanation for this stylized fact. For public consumer goods, a partial equilibrium analysis is sufficient. We just need to consider an exogenous income, which generates a demand for the public good mediated by the land markets. Then, the problem is to see how the public good can be financed through the land market. By contrast, in the case of productive public goods, the initial income generates a demand for private goods, for land and for savings. Using the demand for land, it is possible to finance local public goods but the story does not end here, since the savings and the amount of public good determine future production. A dynamic general equilibrium analysis is thus required.6 In other words, the analysis of productive public goods is inherently dynamic, because public capital can be accumulated and influences further production. Moreover, assume for convenience that private producers operate with constant returns to scale. Then the introduction of public capital, which is assumed to increase the marginal productivity of private capital, implies increasing returns at the aggregate level. For these reasons, our model also embodies increasing returns. Using such a framework, we explore whether the classical results associated with Tiebout and George are still valid when the productive aspect of public goods is taken into account. It is shown below that, although Tiebout-style Ži.e., first-best . results can be obtained, they are very demanding and must rely on strong assumptions. The Henry George Theorem does not hold in the usual sense. Simple capitalization schemes do not work because the marginal product of public investment in one jurisdiction benefits both the workers who live in the jurisdiction and also capital holders who need not live where their savings 5
Education presents some specific features that justify a separate analysis with different w5x. assumptions. For instance, see Benabou ´ 6 Zou w39x uses a framework close to ours, except that he does not consider any land market, which is the main focus of this paper. He also provides alternative justifications for the use of growth models in local public finance analysis.
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are invested. As a consequence, land rents capitalize public investment only up to the increase in wages it causes multiplied by the share of housing in expenditures. In Section II, we propose a simple framework of a competitive production economy with productive local public goods. In Section III, we analyze various decentralized frameworks. Finally, the last section ends the analysis with some considerations concerning the provision of local public goods. II. ECONOMIES WITH PRODUCTIVE PUBLIC GOODS: THE FIRST-BEST We consider a large economy composed of S ‘‘islands,’’ each of area normalized to unity. The total population size N is large but such that N - SL*, where 1rL* is the minimal amount of land to be consumed per capita in order to enjoy positive utility. A given island may or may not be populated. The representative agent is infinitely lived Žor finitely lived with a dynastic utility function., supplies continuously one unit of labor inelastically, and we assume that his initial wealth is strictly positive. His utility function is Us with
q`
H0
Ž ut .
1y s
ey r t dt, 1ys u t s z t stx if st G 1rL*, ut s 0 if st - 1rL*, s ) 1.
Ž 1.
Instantaneous utility at time t, u t , is thus a Cobb]Douglas function of z t , the per capita consumption of the final good and st , the per capita quantity of land Žwhich is a proxy for housing consumption., with a minimum consumption of land. This specification was chosen for analytical convenience. Since the elasticity of substitution is constant, a simple Cobb]Douglas specification may drive us to a meaningless corner solution. Our minimal land requirement is here to avoid this unappealing implication. Note that this specification is more general than the usual assumption of constant consumption of land that would be equivalent to x s 0. There is a single consumption good, which can be consumed, used as public investment, or transformed into private capital. Each island is a separate unit of production. In each island, the production function combines public capital Ž G ., private capital Ž K . and labor Ž L. along a a Yi , t s AGib, t K i1y , t Li , t
with
a G b ) x, i s 1, . . . , S.
Ž 2.
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This specification makes public capital a necessary input. In this respect we follow the empirical literature Žsee Gramlich w14x. and we do not consider the case of publicly provided private goods. Note that whenever land consumption is equally divided between inhabitants of the same island i, we observe si s 1rL i since we suppose that people should live and work on the same island. For the sake of simplicity we ignore depreciation and assume that any unit of public capital, private capital, or labor can be used only in one island. Capital and labor are perfectly mobile across islands. For simplicity, we assume that the good can be transformed into capital and shipped elsewhere, but then it cannot be consumed anymore. In short, what consumers eat must come from their residential area. This assumption may seem rather unappealing but it makes sense for nontradable goods like most personal services. Furthermore it leads to considerable technical simplification.7 Note also that only consumers use space. This simplifying assumption allows us to neglect the issue of competition for space between different categories of agents Žproducers and consumers.. Finally, the analysis that follows concentrates on the steady-state behavior of our economy. There is no great loss of generality: in this model the analysis of the transition dynamic is either straightforward when b - a since the problem reduces to a neoclassical model of growth or trivial when b s a since there is no transition dynamics. A tendency to agglomeration is driven by the increasing returns present in the production function. Reducing the number of populated Žor developed. islands increases production. Production becomes very large when only one island is developed. Acting against this, the taste for land acts as a dispersion force. We assumed in Eq. Ž2. that the preference for housing was not too strong Ž x - b . so that the dispersion force does not always dominate the agglomeration force. The balance between agglomeration and dispersion is optimally solved in the following lemma. LEMMA 1. The first-best land use where indi¨ iduals are treated symmetrically is such that NrL* islands are populated by L* inhabitants each. See proof in Appendix. As often in the literature, we use a rawlsian Žor equal treatment . concept for our first-best. This assumption is here to avoid a first-best for which consumers would not be treated symmetrically as it sometimes happens in public or urban economics. ŽSee Mirrlees w26x for an example where utilitarian welfare leads to unequal treatment.. The intuition for 7
In particular, it will simplify our first-best land use Žland is used with the same intensity wherever it is used.. Otherwise, we may be led to complex patterns of land use Žthe density in inhabited islands need not be the same everywhere at the optimum..
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this lemma is the following. As long as the marginal productivity of public capital multiplied by the marginal utility of consumption is higher than the marginal utility of the consumption of space, it is socially worthwhile to reduce the consumption of space to increase production in order to avoid the spatial dilution of public capital. Note that this first-best land use is such that many islands are populated and that the number of populated islands is proportional to N. PROPOSITION 1. The first-best is such that mobile factors are symmetrically allocated across populated islands. If b - a , the long-run le¨ el of capital accumulated is K s ŽŽ1 y a .1y bA b b L* arr .1rŽ ay b . in each island. If b s a , public capital, pri¨ ate capital and consumption grow at the rate ŽŽ1 y a .1y aa aAL* a y r .rs . Proof. Thanks to Lemma 1, we can restrict our attention to symmetric situations with constant consumption of land. It is then possible to write the social planner’s program independently of land consumption Max U s
z i , Gi , K i
q`
H0
s z i1y ,t
1ys
ey r t dt,
Ž SP.
˙i, t q K˙i, t , which subject to the island budget constraint: Yi, t s L*z i, t q G states that within each island, production is devoted either to consumption or to capital accumulation.8 We also face the following transversality condition: lim eyr tl t K t s 0.
tªq`
Ž 3.
Since s ) 1, intertemporal utility is always well defined. Thus this transversality condition does not play an important role in our analysis. The first-order conditions of SP are
¡z e s l ~lŽ 1 y a . AG ¢lb AG K ys y r t
by1
b
˙ Ky a L* a s yl
1y a
Ž 4.
a
˙ L* s yl
These first-order conditions are sufficient since, thanks to Lemma 1, labor can be treated parametrically so that the production function within each island is weakly concave. From this, we can state Gs 8
bK 1ya
We neglect the subscripts whenever there is no possible confusion.
Ž 5.
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and
˙z z
s
Ž1 y a .
1y b
A b b K by a L* a y r
s
.
Ž 6.
Depending on the value of b , we distinguish two different cases: v
b - a ŽSolow case.. There is a long-run steady state for which
˙zrz s 0:
Ks
ž
Ž1 y a .
1y b
A b b L* a
r
1r Ž a y b .
/
.
Ž 7.
In this case, we have decreasing returns to the reproducible factors. Consequently the economy converges towards its long-run equilibrium level K. v
b s a ŽBarro]Romer case.. From the previous equations, we easily
get «
˙z z
s
K˙ K
s
˙ G G
s
Ž1 y a .
1y a
a aAL* a y r
s
.
Ž 8.
Here, given that returns to the reproducible factors are precisely one, endogenous steady growth is possible. The equilibrium growth path is such that consumption, public, and private capital all increase at the same rate. The initial program involved a dynamic optimization over two choice variables with two state variables. However, thanks to Lemma 1, the consumption of space remains constant. This allows us to get rid of one choice variable and to return to convex analysis. Then, at the optimum the marginal productivity of public capital should be equal to that of private capital which reduces our problem to a standard dynamic programming problem. Depending on the value of b , two different results are obtained.9 These two different cases underline two different conceptions of the role of public spending in the growth process. In the first case, growth is taken as exogenous Žto the model.; that is to say, infrastructure does not act as the engine of growth, although it may intervene crucially in the productive process. This vision is defended, for instance, by Gramlich w14x. It states that infrastructure is necessary for production to occur, but that it does not influence the long-run growth rate. The other conception states that public 9 As usual in growth models, the case b ) a does not make much sense economically since it implies explosive growth.
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spending can be considered as a major engine of growth. Theoretically, this hypothesis was put forward by Barro w3x and Barro and Sala-i-Martin w4x. Empirically, it is defended by De Long and Summers w8x, w9x. It is not our purpose here to take sides in this controversy but to show that for both specifications our results are similar. Under centralized provision of public goods, it can be shown easily that the first-best solution in these two programs can be decentralized partially. Competitive firms hiring labor and private capital will ensure efficiency. Indeed, these firms face a well-behaved production function with constant returns to scale in private inputs. Public capital can be financed through a tax on production. But, the efficiency of this tax rests on three strong assumptions, which are i. observability of private production, ii. no labor supply distortion, and iii. immobility of factors. In a competitive equilibrium, consumers receive an interest rate rt for their savings and a wage wt for their work. Each factor is paid its marginal product and firms make zero profits. Moreover, consumers have to pay a rent R t for each unit of land which can be reinjected in the economy through public land property. Our concern in what follows is then to explore whether it is possible to decentralize not only the allocation of labor and private capital but also the provision of public infrastructure. III. DECENTRALIZATION OF THE FIRST-BEST The Institutional Regimes and Equilibrium Concepts We suppose now that the development of islands is undertaken by competing developers. So by decentralization, we mean that the provision of public goods and the authority to tax are delegated to competitive land developers. The competitive process is the following. On any undeveloped island, a developer can create a jurisdiction where she will provide the public good. Consumers then make their location and saving decisions with both labor and capital being perfectly mobile. Within this broad framework, decentralization can be established through various institutional settings. The first alternative we consider is one where the developer has complete control over everything in her jurisdiction Ž Regime 1.. All possible fiscal and financial instruments are available and can be implemented. Regime 1 can be viewed as a benchmark case, but it may also be applicable to the US where land developers can have extensive powers Žsee Henderson and Mitra w17x.. It also strongly reminds us of traditional factory-towns. More formally we define as equilibrium for Regime 1 a situation whereby: }Consumers maximize their utility with respect to their location and the allocation of their income between consumption and savings.
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}Competitive land developers maximize their profit with respect to the amount of public and ‘‘private’’ capital, land rent, population, and wages in their jurisdiction. They take the behavior of the other land developers and the interest rate as given. }Land developers make zero profit Žfree-entry .. }All markets clear. We also consider Regime 2, where the jurisdictions can only implement a residential tax and a pure land tax, retaining also some power over population control. Given perfect factor mobility, using a residential tax seems reasonable. Other fiscal instruments may be more difficult to use. This regime may be relevant for countries with a decentralized system of government like Switzerland. Decentralization takes place in this framework at two levels since, on the one hand, the provision of public goods is left to private developers and, on the other hand, production of the consumption good is realized within the jurisdictions by competitive firms. Thus, we define as equilibrium for Regime 2 a situation whereby: }Consumers maximize their utility with respect to their location and the allocation of their income between consumption, land, and savings. }Competitive land developers maximize their profit with respect to the amount of public capital they provide, their population and a residential land tax, taking everything else as given. }Competitive firms in each jurisdiction maximize their profits with respect to private capital and labor, taking all prices as given. }Land rent is determined competitively in each jurisdiction. Developers and firms make zero profit. }All markets clear. Finally, and this may be more relevant to describe the countries with a centralized system of government Žbut where local public goods remain locally provided., we consider an environment in which the land developer cannot use any fiscal instrument but a pure land tax. Her sole source of income is thus the land rent, which is competitively determined Ž Regime 3 .. Of course, although the developer cannot set land rent directly in her jurisdiction, she can manipulate it indirectly by public expenditure actions. Besides, since the Henry George Theorem is often valid with public consumer goods, the natural temptation is to restrict the jurisdictions to use only land rent to finance the productive public goods Ži.e., to use a simple capitalization scheme.. If this arrangement enables us to attain first-best, then its practical implementation should be simpler than in the two previous cases since it would just require land prices to be observed.
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Thus, we define as equilibrium for Regime 3 a situation whereby: }Consumers still maximize as previously. }Competitive land developers maximize their profit with respect to the amount of public capital they provide and the population in their island, taking everything else as given. Their sole source of income is the competitively determined land rent in their jurisdiction. }Competitive firms behave as previously. }Land developers and firms make zero profit. }All markets clear. Regime 1 In that case, we get the following proposition, which states that if developers have enough degrees of freedom, the first-best can be reached. PROPOSITION 2. The equilibrium under Regime 1 is the equal treatment first-best. Proof. Developers maximize the sum of their discounted profits, that is, production and land rent minus the cost of factors. Due to perfect factor mobility this objective is equivalent to the maximization of instantaneous profit at each point in time. Thus, each developer faces the problem: Max p t s Yt q R t y wt L t y rt K t y rt Gt , s.t. u t G U.
Ž DP1.
The developer can maximize profits without any restriction on the instruments she uses. Her only constraint is to offer a level of utility as high as can be obtained in other islands. Given the concavity of the utility function, each developer will divide her land equally among her consumers. Wages and rents are redundant since labor supply is inelastic and land is equally divided. Without loss of generality we can therefore normalize R s 0. Our program becomes Max
w, ˆ G, K , L
p t s Yt y w ˆt L t y rt K t y rt Gt ,
Ž 9.
s.t. u t G U.
where w ˆ is the wage distributed when land rent is normalized to zero. Before solving the developers’ program, we must consider the consumers’ program in jurisdiction i: Max U s z
q`
H0
Ž z t six, t .
1y s
1ys
s.t. w ˆi , t q rt Zt s z t q Z˙t ,
ey r t dt,
Ž CP1 .
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and si G 1rL*, where Z is individual wealth. Assuming temporarily a constant lot size proposed by the developers, it is possible to take the term in s in the utility function out of the integral. The first-order conditions for CP1 are
½
zys eyr t s l , ˙. l r s yl
Ž 10 .
After simplification, Eq. Ž10. and the budget constraint imply w ˆt q
r s y 1. q r
žŽ
s
/
Zt s z t .
Ž 11 .
Let us return to the developers’ program Ž9.. Since, the utility function in the constraint of Eq. Ž9. cannot be differentiated everywhere, assume first that there is an optimal L such that L - L*. We can inject Eq. Ž11. in the first constraint of Eq. Ž9.. If we denote by m the Lagrange multiplier associated with this constraint, the first-order conditions imply
¡ Y s r , K Y
~ G Y L
s r, yw ˆsx
¢L s m L
1 x
m Lxq1
w ˆq
ž ž
r Ž s y 1. q r
s
Ž 12 .
//
Z ,
.
Then, simple manipulations show that this system can be satisfied only for L ª q` which violates the assumption made above. ŽDevelopers have an incentive to use as much as possible of the three factors.. Thus we cannot have L - L* which implies immediately that L s L*. Combining L s L* with the first-order conditions in K and G, which remain unchanged, we obtain
¡G s b K , ~ 1ya ¢r s Ž 1 y a .
1y b
Ž 13 . b
b AK
by a
a
L* .
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Then, using the zero-profit condition, we find w ˆs
ayb L
Y.
Ž 14 .
In the Solow case, consumers save until the steady-state is reached. This steady-state is such that r s r . We find the long-run level of capital to be K as in Eq. Ž7.. Consequently, our decentralization scheme enables us to implement the first-best in the case of decreasing returns to the reproducible factor. As for the Barro]Romer case, the first-order and zero-profit conditions imply
¡G s a K , ~ 1ya ¢r s Ž 1 y a .
1y a
Ž 139. a
a
a AL* ,
and w ˆ s 0.
Ž 14’.
From this we can easily show that the economy enjoys a growth rate which is the same as in Eq. Ž8.. Thus, it is also possible to decentralize our first-best solution if we allow for constant returns to the reproducible factors. In conclusion, for both cases, land use, the allocation of factors, and their distribution is the same as with the equal treatment first best. Note that heavy intervention is required in our ‘‘factory-islands’’ since the developers are responsible for both the wage policy and the spatial allocation of consumers in their island. Moreover, the Henry George Theorem becomes meaningless since the first-best solution can be obtained for any level of land rent. ŽActually, land markets can be completely ignored and developers can allocate their consumers directly to their housing lot.. Note also that the net wage is zero in the Barro]Romer case. The reason is that, on the one hand, reproducible factors do not use any land and are thus paid their marginal product. On the other hand, to get self-sustaining growth, we need the sum of the shares of public and private capital to be one in the production function. Consequently, the product is exhausted on public and private capital income. To get more intuition concerning the working of the model, we need to explore the other two regimes. Regime 2 We now restrict the competencies of the developers since it may be argued that having land developers fixing lot size, producing private goods, and providing public capital may not be realistic. So in this section we
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277
analyze the case where land developers provide public capital which is financed through land rents and a residential head-tax. PROPOSITION 3. The equilibrium in Regime 2 is the equal treatment first-best. De¨ elopers set a Ž steady-state. residential head-tax equal to: }T s Ž b y x . r KrŽ1 y a . in the Solow case. }Tt s a YtrL t y x ŽŽ r Ž s y 1. q r .rs . Zt in the Barro]Romer case. Proof. The consumers’ program in jurisdiction i is Max U s z, s
q`
H0
Ž z t six, t .
1y s
1ys
ey r t dt,
Ž CP2 .
s.t. wi , t q rt Zt y Z˙t y Ti , t s R i , t si , t q z t , and si , t G 1rL*, where Ti, t is the taxation in jurisdiction i. The wage w is offered by independent competing producers and R is competitively determined. As previously, assume temporarily that s is constant over time; the first-order conditions imply after simplification
¡˙z r y r , ¢Rs s xz.s
~zs
Ž 15 .
Again, because of the mobility of the productive factors, the developers’ program reduces to: Max p t s R t q L t Tt y rt Gt ,
G, T , L
s.t. u t s U,
Ž DP2.
where T is the residential head-tax. Instead of solving this program, note that we can write wt y Tt y R t st s w ˆt .
Ž 16 .
Moreover, competitive producers within each jurisdiction will induce wt L t s Yt y rt K t . We can then re-write the developer’s program: Max p t s Yt y w ˆt L t y rt Gt y rt K t ,
G, w ˆ, L
s.t. u t s U
Ž 17 .
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Then assume again an optimal L such that L - L*. The first-order conditions of Eq. Ž17. imply
¡ Y s r , ~
G Y L
yw ˆsx
¢L s m L
1 x
m L
xq1
r s y 1. q r
Ž ž ž w ˆq
s
//
Z ,
Ž 18 .
.
The competitive behavior of firms within each island implies that
Yr K s r .
Ž 19 .
Equations Ž18. and Ž19. together are equivalent to Ž12.. Thus, the same argument as in Proposition 2 applies, leading to L s L*. Then combining the first condition of Eq. Ž18. with Eq. Ž19. when L s L*, we obtain Eq. Ž13. again. Using the zero-profit condition for the developers leads to Eq. Ž14. where w ˆ s Ž a y b .YrL. We can now plug Eq. Ž15. in Eq. Ž16.. We find that Tt s w ˆt y wt y xz t . In the Solow case, using Ž11., Ž14., the market clearing condition on the capital market, ZL* s K q G, and the steadystate condition for which r s r . We get simply T s Ž b y x . r Kr Ž 1 y a . s Ž b y x . Y Ž K , L* . .
Ž 20 .
In the Barro]Romer situation, we have Tt s a YtrL* y x
r Ž s y 1. q r
s
Zt ,
Ž 21 .
with r as in Ž139.. The first-best can be obtained if we replace the developer’s direct intervention in production by a residential head-tax. Thus, the outcomes under Regime 1 and Regime 2 are similar. ŽThe main difference between DP1 and DP2 rests with the provision of private capital. But, since it is competitive in both cases, the outcome is the same.. The only difference between Regime 1 and Regime 2 is that direct wage setting is replaced by a residential head-tax. This residential tax may seem at first difficult to implement since it requires the knowledge of all the parameters in the model Žas in most public good provision problems.. But, because of the symmetry across individuals, we just need the developer to be able to observe the land rent and to set a local residential head-tax to maximize
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her total profit. As we remarked above, both reproducible factors are paid at their marginal productivity Žsee the first condition of Eq. Ž18. and Eq. Ž19... The reason for this is that private capital and public capital are perfectly mobile and do not consume any amount of land. Labor, by contrast, consumes land. Any marginal consumer in a jurisdiction lowers both the average productivity of labor due to decreasing returns and the amount of land available for housing purpose. The developer, however, benefits from his arrival since he also generates a positive pecuniary externality for the returns to private and public capital and since his demand for land increases land rent. This head-tax is thus not the usual congestion tax, which is often needed in public good finance analysis Že.g., Fujita w12x, Scotchmer w31x, or Wildasin w35x.. The reason for this instrument is really because private capital benefits from public investment without having to pay for it through the land market. The fiscal instruments we need here to implement the first-best are a pure land tax and a residential head-tax. As a corollary, the Henry George Theorem breaks down because of mobile factors that benefit from the public good without having to use any land. Of course, it is immediate that the same efficient outcome could be achieved by any other non-distortionary tax like a tax on labor, on local earnings, or on consumption. What really matters is that this tax must be levied on one factor labor and not on public nor private capital to ensure efficiency Žabstracting from distribution issues here.. Why focus on the land market then? The answer we suggest is that head-taxes may be more easy to implement on the land market than anywhere else. In order to be of practical use, the model calls for further work. Firstly, in most densely populated areas, land is used to a large extent for residential purpose. But, non-agricultural productive activities also use land as a factor of production. This is likely to imply a tax levied on capital. Secondly, if only distortionary instruments are available, we will need to derive the optimal set of instruments in a second-best environment. We leave these two extensions for future work. In the next section, we concentrate on the more restricted case where only a pure land tax is available. Regime 3 Since the first-best requires a non-zero taxrsubsidy, we can infer easily that it will be impossible to obtain efficiency under Regime 3. However, analysis of Regime 3 is revealing because we can explore the inefficiencies resulting from a more constrained regime. This also strengthens the intuition concerning our previous results.
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PROPOSITION 4. The equilibrium in Regime 3 leads to a sub-optimal pro¨ ision of public good. Proof. The consumers’ program in jurisdiction i is
Max U s z, s
q`
H0
Ž z t six, t .
1y s
ey r t dt,
1ys
Ž CP3 .
s.t. wi , t q rt Zt y Z˙t s R i , t si , t q z t , and si , t G 1rL*. Let us use the same approach as with the previous propositions. Assume temporarily that the consumption of land is constant over time. We find the first-order conditions of CP3 to be Rs s xz and ˙ zrz s Ž r y r .rs , ˙ Ž . which implies Zrz s r y r rs along the balanced growth path. ŽThe first-order conditions of CP3 are the same as those of CP2. This not surprising since the only difference between CP2 and CP3 is the tax T in CP2 which is taken as given by the consumers.. Plugging the previous results in the budget constraint in CP3, we obtain Ri s
x xq1
ž
L i wi q ZL i r y
ž
ryr
s
//
.
Ž 22 .
As for the developers’ program, we have Max p t s R t y rt Gt , G, L
Ž DP3.
s.t. u t s U.
Then suppose again that there is an optimal L such that L - L*. Since u s zs x and Rs s xz, we can write u s RrŽ xL1q x .. Then note that competition between producers implies wL s Y y rK. Injecting this in Eq. Ž22. and replacing in DP3, we obtain x
Max p t s
xq1
G, L
s.t.
x xq1
ž
ž
Yt y rt K t q Zt L r y
ž
Yt y rt K t q Zt L r y
ž
ryr
s
ryr
//
s
//
y rt Gt ,
1q x
Ž xL
Ž 23 . . s U.
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The first-order conditions are
¡ ~
¢
x
Y
x q 1 G x Y xq1 sm
ž
x
yrsm
L
q
x xq1
ž
ž
Y
x q 1 G r Ž s y 1. q r
,
s
Y L
q
ž
// Z
Ž 24 .
r Ž s y 1. q r
s
Z y m x Ž x q 1 . ULx .
//
After simplification, the first condition implies m s 1 y Ž x q 1. rGrŽ x b Y .. Injecting this in the second condition of Eq. Ž24. leads to a contradiction since the term on the rhs is always strictly superior to the term on the lhs Ži.e., developers are always willing to increase their population.. Again, the incentive to increase the population in each island leads to L s L*. The zero-profit condition for the developers implies Gs
xL*
Ž x q 1. r
ž
wqZ ry
ž
ryr
s
//
.
Ž 25 .
Competition between firms within each island implies again that Yr K s r. Market clearing on the capital market leads to ZL* s K q G. Combining the last three equations, we find Gs
1r Ž 1 y a . y Ž r y r . rrs
Ž 1rx . y Ž r y r . rrs
K.
Ž 25 .
In the Solow case, this implies Gs
x 1ya
K,
Ž 26 .
instead of G s brŽ1 y a . K in the first-best ŽEq. Ž5... This leads to the following steady-state:
Ks
ž
Ž1 y a .
1y b
r
Ax b L* a
1r Ž a y b .
/
.
Ž 27 .
Since x - b , we obviously have K - K. In the Barro]Romer case, a straightforward comparison between Eqs. Ž26. and Ž5. also shows there is also under-provision of public capital.
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Consumers want to live where they receive the highest wages. Since land is scarce, competition for land in developed islands creates a land rent that enables the financing of the local public goods. Note that if x s 0 Žno preference for land above the minimum required., the public good does not get provided at all. In the traditional case of public consumer goods, there is a direct relation between public expenditure and the utility of consumers. In the case of productive public goods, the relation is not as direct: more public expenditures induce higher production and only a share of the marginal production is received by the workers Ži.e., a fraction a .. Furthermore, the marginal increase of wages generated by the marginal public investment is used only partially for housing expenditures Žthe share of housing is xrŽ x q 1. in total expenditure.. This appears in Eq. Ž24. where the marginal value of public investment is capitalized only through the share of wages in the production multiplied by the share of housing in consumer expenditures. By contrast, in a model without mobile capital and without intertemporal consumption, but nonetheless productive public goods, Brueckner and Wingler w6x reach an efficiency result. Another efficiency result is reached by Richter and Wellisch w30x in a static framework with exogenous but mobile firms and without a housing market. However, land rent in a given jurisdiction partly capitalizes public investments made in all other jurisdictions. This happens because the demand for land Žand thus land rent. depends not only on current income, but also on total wealth Žsee Eq. Ž24. again.. Higher public investments in jurisdiction j imply a higher interest rate for private capital in the economy, and thus a higher wealth for consumers in jurisdiction i. This second effect prevents the Henry George Theorem from failing too spectacularly. Capitalization rates are very low Žaround 20% at most if we try to calibrate the model. but the optimal tax in the Solow case at the symmetric equilibrium is just the difference between the share of public capital in the production and the share of housing in the expenditure Žsee Eq. Ž20... To put it differently, note that in the case of public consumer goods, efficiency is obtained with the equalization of marginal rates of substitution. Here efficiency is reached with the equalization of marginal rates of return: the marginal productivity of private capital should be equal to that of public capital. The problem we have is that the developer’s marginal revenue for public investment is given by the marginal increase of land rent, which is driven by a preference parameter. Indeed, investment in public capital is driven by consumers equalizing their marginal rates of substitution, which has nothing to do with the equalization of marginal rate of transformation. ŽNote that the optimal tax in Eq. Ž20. is a linear function of the difference between the share of public capital and the preference for land.. Thus, with simple capitalization, we introduce a
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283
preference parameter in a technological relation so that at the ŽSolow. symmetric equilibrium we have G s xYrr instead of G s b Yrr . This analysis bears some implications for economic policy. It suggests that fiscal decentralization should be either minimal or fairly extensive in the sense that both land rents and a land tax are needed. From a more positive perspective, fiscal autonomy is likely to be more efficient when the local authorities can use a wide range of instruments and do not have to rely only on simple capitalization schemes Žor a pure land tax.. IV. CONCLUDING REMARKS ON THE PROVISION OF LOCAL PUBLIC GOODS We have shown in this paper that considering the productive aspects of local public goods seriously complicates the decentralization of the provision of infrastructure. The first-best can be decentralized, but the local authority must be able to intervene at least through a residential head-tax Žindexed on production and wealth.. With heterogeneous agents, the required residential tax could be even more difficult to implement: the optimal tax must be indexed on variables that are clearly difficult to observe at the local level Že.g., wealth in the upper-tail of the distribution.. Without any tax, direct internalization of the land rent is not efficient. Marginal local public investments are capitalized in local land values only through the share of housing multiplied by the share of wages in local production. Empirically, this should give capitalization rates below 20%. Conversely, land values within one jurisdiction also partially capitalize public investments made in all the economy because of the mobility of capital Žhigher interest rates induce a higher demand for land.. However, despite our apparently rather negative results, Tiebout and George’s ideas may deserve more attention than they are presently given. Our interpretation of existing land capitalization schemes is that, although they may not be adequate as main instruments, they may constitute useful additional instruments Žcapitalization occurs up to the share of wage multiplied by that of housing.. For instance, the Hong Kong underground network was partly financed through a land capitalization scheme Žsee Midgley w23x for more details.. Before the construction of the network, the operator was able to buy large parcels surrounding the future stations. Now the operator receives a majority of its income through the fares, but the profit made on land operations Ždevelopment of shopping areas, commercial real estate, and residential buildings. is by no means negligible, at around 15% of the construction cost. As a consequence, the Hong Kong underground is the only profitable underground network in the world.
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APPENDIX: PROOF OF LEMMA 1 After some straightforward manipulations Ži.e., broadly similar to the ones performed through Eqs. Ž2. to Ž6.., the optimal consumption path for a fixed amount of developed land is z s Y y g Ž K q G.
gs
with
Ž1 y a .
1y b
A b b K by a sy a y r
s
, Ž A1.
where g is the growth rate of the economy. Then we can write zs
AG b K 1y a sb
y
Ž1 y a .
1y b
A b b K by a sy a y r 1 y a q b
ž
s
1ya
/
K . Ž A2.
This leads to u s s xy aAK 1y aq b
b
b
žž / 1ya
s ay b y
Ž1 y a .
q
1y b
Ab b
s r a
s s AK
by a
ž
ž
1yaqb 1ya
1yaqb 1ya
//
/
. Ž A3.
After simplifications, one can check that for any z ) 0 x-b«
u s
F 0.
Ž A4.
So for a given accumulation path Ži.e., some existing G, K and savings at each date., from x - b , the instantaneous utility of a typical consumer always increases by having him consuming less space until the constraint s G 1rL* is binding. ŽWe did not consider here explicitly the possibility of asymmetric equilibria. But in this respect, note that, at the equal-treatment first-best, instantaneous utility must be the same for everybody. By assumption, we also need production to take place on the island where one lives. Thus having transfers coming in or out in island i amounts to modifying A i , which drives us back to the argument above.. REFERENCES 1. R. Arnott and J. Stiglitz, Aggregate land rents, expenditure on public goods, and optimal city size, Quarterly Journal of Economics, 92, 471]500 Ž1979.. 2. D. Aschauer, Is public expenditure productive?, Journal of Monetary Economics, 23, 177]200 Ž1989..
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