European Regional Science Association 36th European Congress ETH Zurich, Switzerland 26-30 August 1996
Gilles Duranton London School of Economics London, UK [email protected] Stéphane Déo CEME, Université Paris I Paris, France [email protected]
FINANCING PRODUCTIVE LOCAL PUBLIC GOODS
Abstract: Local public economics typically assumes that local public goods only affect the utility of consumers. We assume on the contrary that local public goods are purely productive. The implications of this assumption are analyzed within standard dynamic growth models. Investment in the public good is enhancing productivity only in the jurisdiction where it takes place. Capital as well as people are perfectly mobile. After characterizing the first-best equilibrium, we show that its decentralization is more demanding than with consumptive local public goods. In particular, efficient decentralization cannot be obtained with competing land developers providing the public good through a simple land capitalization scheme. I. INTRODUCTION
How can an economy achieve optimal provision of local public goods (LPG)? Traditional public economics, originating from Samuelson [1954], stresses the difficulty of the question. Basically, a decentralized scheme cannot be implemented because of a standard free-rider problem. Moreover, the first order conditions for the 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 [1956] came with a clever alternative solution. His answer to the pivotal question raised above was to rely on the local aspects of some public goods. The idea is to use competition among the jurisdictions. 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 equilibrium attained in a perfectly competitive framework is the first-best.
Yet the assumption of an ex-ante optimal number of jurisdictions and the possibility of head- taxes are rather restrictive. However, the subsequent literature (see Wildasin [1987] or Mieszkowski and Zodrow [1989] for complete surveys on this issue) managed to relax them elegantly1. The Tiebout idea relies on the strong analogy between fiscal competition and the competition to supply private goods. Consequently free-entrance of developers leads to the first-best just as free-entrance of producers can achieve the first welfare theorem for private goods. As for the head-tax, the idea is to replace it by using the land market. Implementation of the first-best just requires the developer to be able to take advantage of the differential land rent in her jurisdiction since public spendings are capitalized in the land value. This differential2 land rent stems from the agents' competition for space.
The profit function of each developer in the most direct case is equivalent to total differential land rent (TDR) minus expenses for the public good (G). Then an immediate implication of the zero-profit condition is TDR = G. This result is known as the Henry George Theorem (George [1879]). 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 [1974], Vickrey [1977] or Arnott and Stiglitz [1979] among others. Of course it relies on strong assumptions. It is not valid with: - Imperfect competition, see Scotchmer [1986] and Duranton [1996]. - Imperfect geography (if the land-rent is not well-defined at the border of the city), see Arnott and Stiglitz [1979] and Pines [1991]. - Congestion, see Scotchmer [1986] and Fujita [1989].
1 - Imperfect taxation (if only a property tax is available instead of a land tax), see Mieszkowski [1972] or Hoyt [1991]. - Imperfect mobility (if agents cannot vote with their feet at zero cost). - Economy-wide externality, see Henderson and Abdel-Rahman [1991].
Despite all this, Tiebout's idea seems to receive some empirical support: Oates [1969], Edel and Sclar [1974], Hamilton [1976], Meadows [1976] or more recently Wassmer [1993] all draw favorable conclusions. Moreover, some important projects are explicitly funded by land capitalization schemes. One can think for instance of the railways in Tokyo (Kanemoto [1984], Kanemoto and Kiyono [1993], Midgley [1994]) or the public transports in Hongkong (Midgley [1994]).
Strange as it might be, most potential applications of the capitalization hypothesis (i.e., financing the public goods with the differential land rent) are dealing with infrastructures (which are 'productive public goods'3), whereas the theoretical analysis seems to focus mainly on 'consumptive' public goods (which enter directly the utility function). It is possible to stress that user-charges can usually provide convenient albeit not always simple pricing schemes for infrastructures. A general synthesis on these aspects is provided by Laffont and Tirole [1993]. But, the stylized fact we want to put forward here is that public infrastructures do matter and do matter locally. So, the main problem is that for most infrastructure, distances seem to matter (be it a road network, an airport facility or even electricity or water distribution), whereas perfect spatial discriminatory pricing is seldom available. Consequently producers tend to favor some localizations at the expense of others (see World Bank [1994]). Hence, the local aspect is hard to ignore. Our aim in what follows is to assess in what measure the land market can contribute to the financing of productive local public goods.
Another possible perspective is to consider that most local public goods usually treated as consumptive public goods also have a productive aspect. Indeed, only purely recreational public goods such as parks and museums can be considered as purely consumptive public goods. Even in these extreme cases, it may be argued that they have some productive aspects: museums can improve education, whose productive role is obvious4. The quality of leisure has also an impact on production. A last motivation is that local public economics typically recommends that local expenses should be financed through land taxation or property taxation. However property taxes represent only a fraction of local public finances (see Prud'Homme [1987] and Henderson [1995] for evidences). The problem is to know why non land-based instruments are used at the public level with such a pervasive insistence.
For consumptive public goods, the economic analysis is one of partial equilibrium by nature.
2 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. On the contrary, in our case of productive public good, 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 the production at the next period. A dynamic general equilibrium model is thus a natural tool. In other words, the analysis of productive public goods is inherently dynamic, because public capital can be accumulated and influences further production.
Since we can reasonably assume that private producers operate with constant returns to scale, taking into account public capital which increases their productivity amounts to considering increasing returns at the aggregate level. For these reasons, our dynamic model embodies increasing returns. Then we test whether the classical results are still valid as the productive aspect of all public goods is taken into account. It is shown that although Tiebout style (i.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 benefits to both the workers who live in the jurisdiction and also to capital holders who need not live where their savings are invested. Land rent capitalizes the increase in wages caused by public investment only up to the share of housing in expenditures. On the contrary, increased public investments raise the interest rate and thus the income of everyone and the demand for land in all jursidictions.
In section II, we propose a simple framework of a competitive production economy with productive 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 of surface S populated by individuals of mass N such that S > N L* , where 1 L* is the minimal amount of land to be consumed in order to enjoy a positive instantaneous utility. According to standard assumptions, we consider that each unit of land can be considered as a separate island. Our individuals are infinitely-lived or finitely- lived with a dynastic utility function and they supply continuously one unit of labor inelastically :
3 1−σ +∞ (u ) −ρ U = ∫ t e tdt 0 1−σ = x ≥ * with ut zt.st if st 1 L (1) = < * ut 0 if st 1 L σ > 1
Instantaneous utility is a Cobb-Douglas function5 of z, the consumption of the good and s, the quantity of land. For each 'island' of surface one, we face the production function:
β 1−α α α ≥β> Yt = A.Gt . Kt .Lt with x (2)
Note that s = 1 L since we suppose that people should live and work on the same unit of land. For the sake of simplicity any unit of capital (K), labor (L) or public spending (G) can be used only in one island (there is no overlap6) and we ignore depreciation. As assumed by equation (2), there is only one good, which can be consumed, used as public investment or transformed into private capital. Finally capital and labor are perfectly mobile across islands. (The good can be transformed into capital and shipped elsewhere but it cannot be consumed anymore or, in other words, what consumers eat must come from their residential area.) Note that only consumers use space. This simplifying assumption allows us to neglect the issue of competition for space between different categories of agents (producers and consumers)7.
The agglomerative force is given by the increasing returns since the production function is homogeneous of degree more than one. Reducing the surface occupied by labor increases production. Production tends to infinity when the surface tends to zero (or at least it becomes very large when only one jurisdiction is developed). On the contrary, the taste for land acts as a dispersion force. The tension between agglomeration and dispersion is solved by the following lemma:
Lemma 1 If: i/ x <β ii/ Inter-jurisdiction transfers for consumptive motives are impossible. iii/ The planner uses of a Rawlsian welfare metric8. Then the centripetal force dominates the centrifugal force if s ≥1 L* and the area developed per consumer is equal to s = 1 L* .
See proof in Appendix 1.
4 Proposition 1 : The first-best is such that mobile factors are symmetrically allocated across existing jurisdictions. If β<α , long-run level of capital accumulated is described by equation (7). If β= α, the growth rate is given by equation (8).
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.
−σ +∞ z1 −ρ Max U = ∫ t e tdt z,G,K 0 1 −σ (SP)
= * + ú + ú Subject to the "representative island" budget constraint: Yt L . zt Gt Kt
−ρ 9 t λ = and to the transversality condition : lim e . t.Kt 0 (3) t −>+∞
The first order conditions of (SP) are:
−σ −ρ z e t = λ β −α α λ(1−α)A.G .K .L* = −λú (4) α β−1 1−α * ú λ.β.A.G .K .L = −λ
From this, we can state:
β.K G = (5) 1−α
−β β β−α α úz (1 −α)1 A.β .K .L* −ρ = (6) z σ
Depending on the value of β, we distinguish two different cases:
• β<α (Solow case). There is a long-run steady state (for which zú z =0):
(α−β ) −β β α 1 (1−α)1 A.β .L* => K = (7) ρ
In this case, we have overall increasing returns but decreasing returns to the accumulative
5 factor. Consequently endogenous growth is impossible and the economy converges towards its long-run equilibrium level K .
• β = α (Barro-Romer case). From the previous equations, we get:
−α α α úz Kú Gú (1 −α)1 α A.L* −ρ => = = = (8) z K G σ
Here given that returns to the accumulative factor are precisely one, endogenous steady growth is possible. The equilibrium growth path is such that consumption, public and private capital are all increasing at the same rate.
These two different cases underline two different conceptions of the role of public spendings in the growth process. In the first case, growth is taken as exogenous (to the model), that is to say infrastructures do not act as the engine of growth. This vision is defended for instance by Gramlich [1994]. It states that infrastructures do matter for production to occur but that they do not influence the dynamics of growth. The other conception states that public spendings can be considered as a major engine of growth. Theoretically, this vision was put forward by Barro [1990] and Barro and Sala-i-Martin [1992]. Empirically, this conception is defended by De Long and Summers[1991] and [1993].
It can be shown easily that the first-best solution in these two programs can be decentralized, as long as only production is concerned. In the Solow case, the government just need to provide the optimal amount of public infrastructure in each jurisdiction. Firms then face constant returns to scale so that no problem arises. The steady-state is such that no public spending is required after it is reached. In case of exogenous growth or endogenous growth (in the Barro-Romer case), public spendings can be financed easily through a tax levied on production10. Barro [1990] shows that the government's optimal rate of taxation is the one that maximizes the growth rate. In case of (decentralized) competitive production, consumers receive an interest rate rt for their savings and a wage wt for their work. Each factor is paid at its marginal productivity and firms make zero profit. Moreover consumers have to pay a rent Rt for each unit of land (and this rent is re-injected through public land property). Our concern in what follows is to decentralize both the provision of public infrastructure and the production of goods considering that all factors are mobile.
6 III. DECENTRALIZATION OF THE FIRST-BEST
1 DESCRIPTION OF SOME POSSIBLE INSTITUTIONAL REGIMES AND PRELIMINARY RESULT
Decentralization can be established along various institutional regimes. The first reason to consider different regimes is the way of financing jurisdictions. Since the Henry George Theorem is usually valid with consumptive public goods, the temptation is natural to restrict the jurisdictions only to the use of land rent to finance the productive public goods. If this arrangement enables us to attain first-best, then its practical implementation should be very simple. Since in practice it might be hard to prevent jurisdictions from using the financial markets, we also give them this opportunity. We call this framework 'restricted financing'. 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 is realized within the jurisdictions by independent private producers. The other alternative we consider is the one for which the developer has complete control over everything in her jurisdiction. We call this situation 'free financing'. Notice that in this latter case the decentralization of production is not complete due to the intervention of developers in the production process. Note also that this last framework strongly reminds us of traditional factory-towns.
The second dimension for which institutions may matter is the bidding process on the land market. The first institutional regime is the one for which the land market is competitive on the supply as well as on the demand-side (referred to as 'competitive land market'). Consumers are bidding on a given quantity taking the price of land as given. The developer also takes land rent as given (although she can manipulate it indirectly by an action on public spendings). In this case the capitalization process is easy to set-up since one just need to tax all land rent (or a fixed fraction of it). Another possible type of land market is the one for which the developer has some 'market power' on the supply of land in her jurisdiction. In this case, the developer can implement some form of taxation attached to the residence in her jurisdiction (head tax) taking only the instantaneous utility level as given (we speak here of 'locally monopolistic land market').
In the description of these two settings, we implicitly assumed that lot sizes were not constrained, or that consumers could bid freely for land. They could use as much land as they liked given its price (land rent). This situation is the one observed only under perfect divisibility of land. But this latter assumption is rather restrictive. Thus the divisibility of land is the third dimension we must consider because it is also possible to assume that lot sizes are fixed. This other alternative is justified in particular because the land market is widely acknowledged to be imperfect. Fixed lot-size can be seen as a major form of market
7 imperfection (Arnott [1987]). Since in real life people bid for land parcels at different times, the available lots are of fixed (but different) sizes due to the presence of surrounding neighbors. Marginal decision is replaced by discrete choice, since in general no-one can bid for his ideal lot-size given land rent. We will thus explore two polar cases: free lot-size and fixed lot-size.
Our aim is thus to see how we can replicate the first-best equilibrium with the least demanding framework. For instance, it would be preferable to be able to get the first-best equilibrium even with some imperfection on the land market (fixed lot-size). Moreover, we may prefer a priori a situation for which the developer should not be allowed to intervene too much in the local economy with her role restricted to the provision of the public good. Finally, the competitive land market may be more desirable since it does not require any direct intervention of the developer on the land market. Our analysis starts from the most demanding framework, that is the one for which the developer can intervene extensively in the local economy (fixation of land rent, wages, public spendings, capital investments, possible borrowings on the capital markets...). Then we examine the effects of various restrictions.
From the previous discussion, 8 regimes are possible but there is some redundancy among them. For instance in our model, a residential head-tax is equivalent to a wage policy set by the developer since labor supply is inelastic. This means that the assumption of free financing renders all distinctions void on the land market since the developer has complete power within her jurisdiction. The possible regimes, that are analyzed further, are: - Free financing. - Constrained financing with locally monopolistic land markets. - Constrained financing with competitive land markets and free lot-size. - Constrained financing with competitive land markets and fixed lot-size.
In all cases, the competitive process remains the same. Developers can enter freely and create jurisdictions. The financing of the public goods depends on the regime considered. With free financing for instance, developers can compete on wages, land rent and the amounts of private and public capital they use. On the contrary, with constrained financing, competitive land markets and fixed lot-size, they just compete on the amount of public capital they provide. Consumers make their location and saving decisions with both labor and capital being perfectly mobile.
Before going into the detailed analysis of these institutions, we can prove the following lemma.
8 Lemma 2: Whatever the institution one considers, any stable competitive equilibrium is such that the area developed per consumer is equal to s = 1 L* .
− − − Proof: For L < L* and free lot-size, we observe that ∂u ∂L = L x ∂z ∂L − x.z.L x 1. An equilibrium must be such that no developer should have an incentive to change her strategy. In case, an individual should move, suppose that all the surplus created is distributed to consumers. Assuming any extra production would be immediately consumed, such a movement to raise one's instantaneous utility would be profitable since ∂u ∂L =αY −x.L.z > 0 (remember that α > x and Y ≥ Lz ). This is a sufficient condition for an increase in total utility since it does not modify the state variable. If all the surplus created is not distributed to consumers, the developer can increase her profits with such a movement. Of course, L > L* would induce the entrance of new developers. Thus, any symmetric equilibrium must be such that L = L* .
2 FREE FINANCING
In that case, we get the following proposition, which unsurprisingly states that if developers have enough degrees of freedom, the first-best can be reached.
Proposition 2 : The first-best can be decentralized with free-financing.
Proof: The developers try to maximize the sum of their discounted profits, but due to perfect mobility of all factors this objective is equivalent to the maximization of instantaneous utility at each point in time. Thus, each developer faces:
π = + − − − Max t Yt Rt wt.Lt rt.Kt rtGt w,R,G,K,L (DP1) ≥ s.t. Ut U
Given the concavity of the utility function, each developer divides its land equally among its consumers; she also sets wages and investment for both K and G in her jurisdiction. Without loss of generality we can then normalize R=0 given that wages and rents are redundant since labor supply is inelastic. If we find an equilibrium for this situation, there is also a similar equilibrium for each level of rent fixed by the developer (it suffices to tax or subsidize labor since land consumption in a jurisdiction as well as labor supply is taken as fixed by the consumers). Our program becomes:
9 π = − − − Max t Yt wˆ t .Lt rt.K t rtGt wˆ ,G,K,L (9) = s.t. Ut U where wˆ is the wage distributed when land rent is normalized to zero. Before solving the developers' program, we must consider the consumers' program:
−σ x 1 +∞ (z .s ) = t t −ρt Max U ∫ e dt z 0 1 −σ (CP1) + = + ú s.t. wˆ t rt. Zt zt Z t and s ≥ 1 L* where Z is the individual wealth. Taking this equilibrium as given, consumers face available lots that all have the same size and this size is constant over time (Lemma 2). Consequently, it is possible to drop the term in s in the utility function.
The first order conditions of (CP1) are:
−σ −ρ z .e t =λ (10) λ.r = −λú
For r constant over time, we find that:
r(σ −1) +ρ wˆ + Z = z (11) t σ t t
Back to the developers' program and using (11), the first order conditions are:
r(σ −1) +ρ wˆ + Z ∂Y ∂Y σ = r − wˆ = −x.λ + ∂K ∂L Lx 1 (12) ∂Y 1 = r − L =λ ∂G Lx
Using lemma 2 and the zero profit condition, we obtain in the Solow case a symmetric equilibrium characterized by:
10 (1 −α+β)Y =(G+K).r β.K G = 1−α α−β (13) wˆ = Y L −β β β−α α r = (1−α)1 β A.K L*
Consumers then save until the steady-state is reached. This steady-state is such that r = ρ. We find the long-run level of capital being:
(α−β) −β β α 1 (1−α)1 A.β .L* K = (14) ρ
Consequently, our decentralization scheme enables us to implement the first-best in the Solow case. As for the Barro-Romer case, the first order and zero profit conditions read:
Y = (G + K).r α.K G = 1−α (15) wˆ = 0 −β β α r = (1 −α)1 β A.L*
From this we can easily obtain:
−α α α úz Kú Gú (1 −α)1 α A.L* −ρ = = = (16) z K G σ
Thus, it is also possible to decentralize our first-best solution if we allow for endogenous growth.
Note that heavy intervention is required in our 'factory-island' since the developer is responsible for both the wage policy and the spatial allocation of consumers. Moreover the Henry George Theorem becomes meaningless since the first-best solution can be obtained for any level of land rent.
11 3 CONSTRAINED FINANCING WITH LOCALLY MONOPOLISTIC LAND MARKETS
If the developer can intervene on the land market through a lump-sum taxation, we can show that the first-best equilibrium can be decentralized without her direct intervention in the production process.
Proposition 3 : The first-best can be decentralized if the developers can set a residential head-tax in their jurisdictions. The head tax is equal to: • = − ( + )(α −β) − ρ Tt (1 x 1 )wt x. .Zt in the Solow case. r(σ −1) +ρ • T = w − x Z in the Barro-Romer case. t t σ t
See Appendix 2 for a proof.
This tax may seem at first really difficult to implement since it requires the observation of all the parameters in the model. In fact it is not. The main reason is that we just need the developer to be able to observe the land rent and to set a head-tax or subsidy to maximize her total profit. However, two other difficulties cannot be escaped as easily. First, a head-tax is not a usual instrument for local taxation and it may be hard to implement in practice. For instance, we may face households of different sizes or with different levels of initial wealth. Since the optimal head-tax depends on the wealth of the residents, different levels of wealth may then be synonymous with distortive capital taxation. Moreover, this is difficult to implement in practice at a local level, since privately held capital is invested in many jurisdictions and is not easily observable. The tax also depend on the wages. In a more general framework, labor supply at the individual level cannot be exogenous as it is here, so that a tax on labor is also distortive...
Besides, our model can be related to the recent literature on the choice of instruments. Henderson [1994] and [1995] argues that imperfect competition can drive jurisdiction to use non land-based instruments. Here, decentralization leads jurisdiction to use non land-based instruments in a competitive model due to increasing returns at the local level. Note also that the taxes in proposition 3 can be interpreted partly as 'congestion' taxes. The idea is that any marginal consumer in a jurisdiction lowers the average productivity of labor due to decreasing returns. On the contrary, the developer benefits from his arrival since he is also a positive pecuniary externality for the returns to private and public capital and since his demand for land increases land rent.
Finally, the idea that people should consume a minimum amount of land causes an incentive for labor to spread. Otherwise due to increasing returns, the whole economy would agglomerate in one point (or at least in one jurisdiction). When only capital 'consumes' land 12 (through the production function), the incentive can be for capital to spread instead of labor. Using only this second assumption, we will observe a complete utilization of land if the share of land in the production function is sufficiently important. In that case, all private factors are paid at their marginal productivity (supposing that the sum of the share of land plus the share of labor plus the share of capital is one). Then there is no reason for the marginal productivity of land to be equal to the marginal productivity of the public good. Thus, again, total differential land rent is not in general equal to public expenditures when capital consumes land. The result is straightforward. Again, the Henry George Theorem does not hold and the first-best cannot be decentralized in a simple way (see Appendix 3 for technical details).
4 CONSTRAINED FINANCING WITH COMPETITIVE LAND MARKETS AND FREE LOT-SIZE
Since the first-best requires a non-zero tax/subsidy, it is impossible to obtain it here. However the analysis is worth it. We can explore the inefficiencies resulting from constrained financing in a second-best framework. This also gives us more intuitions concerning our previous results.
Proposition 4 Restricting the financing of jurisdictions to the sole use of land rent leads to under-investment in public and private capital. The equilibrium is such that jurisdictions can run positive profits. Profit increases with the preference for land when x is small.
See proof in Appendix 4.
From this result, we can see that if the preference for land is strong (high x), jurisdictions will run positive profits. The conclusion of this analysis is that the market equilibrium in this framework induces under-accumulation of both public and private capitals. The intuition of this result rests with the way of financing public infrastructure. Consumers want to live where they receive the highest wages. Since land is scarce, competition for land creates a land rent that enables the financing of the local public goods. In the traditional case of consumptive public goods, there is a direct relation between public expenditures and the utility of consumers. In the case of productive public goods, the relation is not as direct: more public expenditures induce a higher production and only a share of the marginal production is received by workers (i.e., a fraction α). And the marginal increase of wages generated by the marginal public investment is used only partially for housing expenditures (the share of housing is x (x +1) in the total expenditures). So the marginal value of public investment is capitalized only through the share of wages in the production multiplied by the share of housing in the consumers' expenditures11.
13 Alternatively, note that in the case of consumptive public goods, efficiency is defined by the equalization of marginal rates of substitutions. Here, on the contrary, efficiency is defined by the equalization of marginal rates of returns. The problem 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. In so doing, we introduce a preference parameter in a technological relation.
On the other hand, land rent in a given jurisdiction capitalizes partly public investments made in all other jurisdictions, since demand for land is increased by higher financial incomes. If this effect is sufficiently strong, land developers make positive profits.
5 CONSTRAINED FINANCING WITH COMPETITIVE LAND MARKETS AND FIXED LOT-SIZE
The lot-size can be set ex-ante in the jurisdictions for exogenous reasons. To simplify, we assume that only one type of lot with constant size is available. In that case, we assume that this lot-size is equal to 1/L*.
Proposition 5 Within this institutional framework, the public good is not provided.
Proof: Suppose a symmetric Nash equilibrium for which all developers provide a given quantity G of LPG. Consumers will bid (and make their residential choice) on the difference w - R to maximize their net income coming from their jurisdiction at each point of time. Since lot-size is fixed, people cannot bid for more space. Consequently, they will just bid to live in a jurisdiction with more LPG (i.e., higher wages).
The wage w is equal to the marginal productivity (this results from the first order condition faced by the competing producers within the jurisdictions). In case of a symmetric equilibrium, wages are the same everywhere so that land rent is equal to zero in all jurisdictions. So, whatever the amount of LPG provided by the developers, the latter will run a deficit. The only symmetric equilibrium is then such that G = 0.
We face here a strong inefficiency since the public good is not provided at all in this extreme case. It means that imperfections on the land market are likely to create even more difficulties to implement the first-best.
14 V. CONCLUDING REMARKS ON THE PROVISION OF LPG
We have shown in this paper that considering the productive aspects of local public goods amounts to complicating seriously the decentralization of the provision of infrastructures. The first-best can be decentralized, provided non-land-based instruments can be used. The local authority should be able to intervene either through taxes (or subsidies) at the production level or through head-tax for residents. This clearly fits empirical evidences. However the first-best taxation should be indexed on variables that are clearly difficult to observe at the local level (e.g., wealth). 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 wage in the local production. Conversely, land values within one jurisdiction also capitalize public investments made in all the economy because of the free mobility of capital (higher interest rates induce a higher demand for land).
Moreover we must distinguish different levels of applications concerning infrastructures: local, regional and national (and even supra-national in the European case), whereas our model just assumed only one level of decision (the jurisdiction of size one). The existence of multiple levels of decision would drive us into more complications. Besides it is hard to ignore the overlapping aspect of numerous infrastructures (e.g., roads). All these problems induce the necessity of a central planner in some cases and/or the necessity of the large actors to be large enough and to be able to decide the mix of all infrastructures (our G is a composite of many different elements). Otherwise, if for a same point in space, different actors provide different infrastructures, strategic problems are likely to arise.
However, despite our apparently rather negative results, Tiebout and George's ideas12 may deserve more attention than they are presently given. Our interpretation of existing land capitalization schemes is that they can constitute good additional instruments. For instance the Hongkong underground network was partly financed through a land capitalization scheme (see Midgley [1994] 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 (development of shopping areas, commercial real estate and residential buildings) is not negligible. It is estimated at around 15% of the construction cost. As a consequence the Hongkong underground is the only profitable underground network in the world.
We thank Toni Haniotis, Vernon Henderson, Kevin Roberts, Jacques Thisse and David Wildasin for helpful discussions, comments and suggestions.
15 REFERENCES
ARNOTT R., 1987, "Economic Theory and Housing", Handbook of Regional and Urban Economics vol II, 959-1021, dir. Mills, Elsevier Science Publishers (North-Holland). ARNOTT R., AND STIGLITZ J., 1979, "Aggregate Land Rents, Expenditure on Public Goods, and Optimal City Size", Quarterly Journal of Economics 92, 471-500. BARRO R., 1990, "Government Spendings in a Simple Model of Endogenous Growth", Journal of Political Economy 98, S103-S125. BARRO R., AND SALA-I-MARTIN X., 1992, "Public Finance in Models of Economic Growth", Review of Economic Studies 59, 645-662. BÉNABOU R., 1993, "Workings of a city: Location Education and Production", Quarterly Journal of Economics 108, 619-652. BERTOLA G., 1993, "Factor Shares and Savings in Endogenous Growth", American Economic Review 83, 1184-1198. DE LONG B., AND SUMMERS L., 1991, "Equipment Investment and Economic Growth", Quarterly Journal of Economics 106, 445-502. DE LONG B., AND SUMMERS L., 1993, "How strongly do developing economies benefit from equipment in investment ?", Journal of Monetary Economics 32, 395-415. DURANTON G., 1996, "Land Policy, Urbanization and Growth", mimeo. EDEL M., AND SCLAR E., 1974, "Taxes, Spending, and Property Values: Supply Adjustment in a Tiebout-Oates Model", Journal of Political Economy 82, 941-954. FLATTERS F., HENDERSON J.-V., AND MIESZKOWSKI P., 1974, "Public goods, efficiency and regional fiscal equalization", Journal of Public Economics 3, 99-112. FUJITA M., 1989, "Urban Economic Theory, Land Use and City Size", Cambridge University Press. GEORGE H., 1879, "Progress and Poverty", NY. GRAMLICH E., 1994, "Infrastructure Investment: A Review Essay", Journal of Economic Literature 32, 1176-1196. HAMILTON B., 1976, "The Effects of Property Taxes and Local Public Spending, on Property Values: A Theoretical Comment", Journal of Political Economy 84, 647-650. HENDERSON J.-V., 1994, "Community Choice of Revenue Instruments", Regional Science and Urban Economics 24, 159-184. HENDERSON J.-V., 1995, "Will homeowners impose property taxes?", Regional Science and Urban Economics 25, 153-181. HENDERSON J.-V., AND ABDEL-RAHMAN H., 1991, "Urban Diversity and Fiscal Decentralization", Regional Science and Urban Economics 21, 491-509. HOCHMAN O., PINES D., AND THISSE J.-F., 1995, "On the Optimal Structure of Local Governments", American Economic Review 85, 1224-1240.
16 HOYT W., 1991, "Competitive Jurisdictions, congestion, and the Henry George Theorem: When should property be taxed instead of land", Regional Science and Urban Economics 21, 351-370. KANEMOTO Y., 1984, "Pricing and Investment Policies in a System of Competitive Commuter Railways", Review of Economic Studies 51, 665-682. KANEMOTO Y., AND KIYONO K., 1993, "Regulation of Commuter Railways and Spatial Development", mimeo. LAFFONT J.-J., AND TIROLE J., 1993, A Theory of Incentives in Procurement and Regulations, MIT Press (Cambridge, Mass.). MEADOWS G.-R., 1976, "Taxes, Spending, and Property Values: A Comment and Further Results", Journal of Political Economy 84, 869-880. MIDGLEY P., 1994, "Urban Transport in Asia", World Bank Technical Paper 224. MIESZKOWSKI P., 1972, "The property tax: an excise tax or a profit tax", Journal of Public Economics 51, 415-435. MIESZKOWSKI P., AND ZODROW G., 1989, "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. MIRRLEES J., 1972, "The Optimum Town", Swedish Journal of Economics 74, 114-135. OATES W.E., 1969, "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. OLSEN E., 1987, "The demand and supply of housing service: a critical survey of the empirical literature", Handbook of Regional and Urban Economics vol II, 989-1022, dir. Mills, Elsevier Science Publishers (North-Holland). PINES D., 1991, "Tiebout without Politics", Regional Science and Urban Economics 21, 469-489. PRUD'HOMME R., 1987, "Financing Urban Public Services", Handbook of Regional and Urban Economics vol II, 1179-1206, dir. Mills, Elsevier Science Publishers (North- Holland). SAMUELSON P., 1954, "The Pure Theory of Public Expenditures", Review of Economy and Statistics 36, 387-389. SCOTCHMER S., 1986, "Local Public Good in an Equilibrium", Regional Science and Urban Economics 16, 463-481. TIEBOUT C., 1956, "A Pure Theory of Public Expenditures", Journal of Political Economy 64, 416-424. VICKREY W., 1977, "The city as a firm", in Feldstein M., et Inman R., The Economics of Public Services, MacMillan, London. WASSMER R., 1993, "Property taxation, property base, and property values: An Empirical test of the 'new view'", National Tax Journal 46, 135-159.
17 WILDASIN D., 1987, "Theoretical Analysis of Local Public Economics", Handbook of Regional and Urban Economics vol II, 1131-1178, dir. Mills, Elsevier Science Publishers (North-Holland). WORLD BANK, 1994, World Development Report.
APPENDIX 1
After some straightforward manipulations (i.e., broadly similar to the ones performed through equations (2) to (6)), the optimal consumption path given a fixed amount of developed land is:
−β β β−α −α (1−α)1 A.β .K .s −ρ z = Y − g(K + G) with g = (A.1) σ where g is the growth rate of the economy. Then we can write:
β −α −β β β−α −α AG K1 (1−α)1 A.β .K .s −ρ 1−α+β z = β − K (A.2) s σ 1−α
This leads to:
β β ( −α)1−β ββ −α+β ρ −α+β = x −α 1−α+β α−β − 1 A. 1 + 1 u s A.K s α β−α 1 −α σ 1−α s σ.A.K 1−α (A.3)
After simplifications, one can check that for any z >0:
∂u x <β ⇒ ≤ 0 (A.4) ∂s
So for a given accumulation path (i.e., some existing G, K and savings at each date), from i/, a typical consumer can always increase his instantaneous utility by consuming less space until the constraint s ≥1 L* is binding.
Suppose now that there exists an asymmetric equilibrium (with some jurisdictions that are less populated; given their smaller productivity, a higher share of their production must be consumed and savings are made from capital investments made in "populated" jurisdictions). Then, from iii/, we require instantaneous utility to be the same for everybody. From ii/, we 18 also need production to take place in the jurisdiction where one lives. For inhabitants in the less populated jurisdictions, instantaneous utility is maximized when all the production is consumed. After simplifications, we can write:
β −β −α+β β u = sx A.K1 (A.5) 1 −α
Consequently, even if all the production is consumed whatever K and G, the per capita consumption of land is s = 1 L* . The first-best is then symmetric and s = 1 L* .
APPENDIX 2
The consumers' program writes:
−σ x 1 +∞ ( ) zt.st −ρt Max U = ∫ e dt (CP2) z,s 0 1 −σ + − ú − = + = ( ) s.t. wt rt.Zt Z t Tt Rt.st zt E wt ≥ * and st 1 L ( ) E wt is the expenditure of the consumer at date t and Tt is the taxation in the jurisdiction. The wage w is offered by independent competing producers.
The first order conditions are:
−σ ( −σ ) −ρ z .sx 1 .e t = λ −σ ( −σ)− −ρ x.z1 .s x 1 1.e t = λR (A.6) ú λ.r = −λ
After simplifications, and given that the equilibrium is such that s is constant over time, we obtain:
−ρ zú = r z σ R.s = x.z (A.7) s = 1 L
19 Again, because of the mobility of the productive factors, the developers' program reduces to:
π = + − Max t Rt Lt.Tt rt.Gt G,T,L (DP2) = s.t. Ut U
Instead of solving this program, note that we can write:
− − = wt Tt Rt .st wˆ t (A.8)
One can write again the developer's program:
π = − − Max t wt.Lt wˆ t .Lt rt.Gt G,wˆ ,L (A.9) = s.t. Ut U
Moreover competing producers within each jurisdiction will induce:
= β = ( −β) wt .Lt .Yt and rt .Kt 1 .Yt (A.10)
Equations (A.9) and (A.10) are equivalent to (DP1) after using the first order condition in K = ( −β) (remember that equation (12) states that rt .Kt 1 .Yt and is thus equivalent to (A.10)). Then, using the zero profit condition and (A.7), we obtain (13) again. This means that the first-best can be obtained if we replace the developers' direct intervention in the production by a head-tax. This result is not very surprising. Due to capital mobility, the first order condition with respect to K is the same if the developer decentralizes production. Direct fixation of wage is replaced by a head-tax levied on a gross wage being equal to the marginal productivity of labor. This head-tax can be characterized easily, in the Solow case, using (13), (A.7) and the steady-state condition for which Zú = 0.
= − ( + )(α −β) − ρ Tt (1 x 1 )wt x. .Zt (A.11)
In the Barro-Romer situation:
r(σ −1) +ρ T = w − x Z (A.12) t t σ t with r as in (15).
20 APPENDIX 3
Suppose that now utility is (land has only a productive role):
1−σ +∞ (z ) −ρ U = ∫ t e tdt with σ > 1 (A.13) 0 1 −σ
The individual production function is:
β 1−α−γ γ α γ ≥ β α > yt = A.Gt .kt .st .lt with and 0 (A.14)
So in each jurisdiction:
β 1−α−γ α Yt = A.Gt . Kt .Lt (A.15)
Since γ > β, all land is developed since the marginal productivity of land is superior to the marginal productivity of the localized public good (population is then L** in each jurisdiction). Due to the decentralization of the production to competing firms, each factor is paid at its marginal productivity. In the Solow case, when developers are not constrained, the equilibrium level of public good is:
(α+γ−β) (α+γ) (1−α−γ) 1 A.β (1−α−γ ) α G = L** with r = ρ (A.16) ρ
But this level of public capital cannot be accumulated using only land rent since total differential land rent equals:
γ β 1−α−γ α Rt = . A.Gt .Kt .Lt (A.17)
1 (α+γ−β) β α (1−α−γ) β => R = γ (1 −α−γ ) A.L (A.18) ρ(1+β−α−γ)
We observe L = L** and, if we assume that all the land revenue is used to finance the public good, we get:
(α+γ−β) β α 1 (1−α−γ ) β R =ρ.G ⇔ G = γ (1−α−γ ) A.L** (A.19) ρ
21 γ R = G (A.20) β
Consequently, straightforward land capitalization does not induce first-best. Another instrument is necessary.
APPENDIX 4
The consumers' program is:
−σ x 1 +∞ (zt.st ) −ρ Max U = ∫ e t dt (CP3) z,s 0 1 −σ + − ú = + = ( ) s.t. wt rt.Zt Z t Rt.st zt E wt ≥ * and st 1 L
After simplifications, first order conditions can be expressed by:
E(w) R = s(1 +1 x ) (A.21) r −ρ Zú = Z σ
As for the developers' program, we have:
π = − Max t Rt rtGt G,L (DP3) = s.t. Ut U
Using (A.21), the first order conditions of the developers are:
x+1 L = −λ(x +1) ∂ ∂ (A.22) w L − = λ w 1 . r . x ∂G (1+1 x) ∂G L (1 +1 x)
∂w x(x +2) ⇒ r = L (A.23) ∂G (x+1)2
Moreover, equalization of the marginal rates of productivity between private and public
22 investment implies:
1 −α K = G (A.24) β
Wages offered by private firms are then:
−α β α− w =α.A.K1 .G .L 1 (A.25)
If we plug the derivative of (A.25) in (A.23), we find after simplifications:
β −α β−α α x(x + 2) r = α.β .(1 −α)1 .A.G .L (A.26) (x + 1)2
Of course, as in the previous cases, the equilibrium is such that L = L* (Lemma 2).
− β<α (Solow case). The steady-state is reached when r = ρ. The long-run level of accumulated capital is then:
1 (α−β ) −β β α x(x+2) α.(1 −α)1 A.β .L* (x +1)2 K = (A.27) ρ
After some manipulations, we can observe that:
1 (α−β ) x(x + 2) K = α. K ⇒ K ≤ K (A.28) (x +1)2
Note also that free-entry does not drive profits to zero because of the increasing returns (see Lemma 2 again). The steady-state is such that all instantaneous income is spent. Consequently, the steady-state land rent is:
Y R = (A.29) L*(1+1 x)
Then, the long-run level of profit is given by:
β β 1+β−α x π = A K L* −α.β (A.30) 1−α x +1 23 Profits can be either positive or negative. It stems essentially from the passive role played by the developers (they just maximize over G). We do not really bother about negative profits and we can assume that they can be financed through lump-sum transfers. Note that profits go up with the increase in the preference for land.
- β= α (Barro-Romer case). In this case, the rationale is not really different. Using (A.26) and the symmetric equilibrium condition L = L* , we obtain:
β −α β−α α x(x + 2) r = α.β .(1 −α)1 .A.G .L* (A.31) (x +1)2 and the corresponding growth rate:
β −α β−α α x(x + 2) α.β .(1 −α)1 .A.G .L* −ρ úz (x +1)2 = (A.32) z σ
By a straightforward comparison with equation (6), we can see that, due to the terms in a and x, the growth rate is below the optimal one. Note again that the zero-profit condition does not appear up to know. It means again that despite perfect competition, developers make either profits or losses. After simplifications, the profit is expressed by:
α 2+α α α α * x(x + 2) A −α − α .A.L ( −α)α *1 (1−α) (x +1)2 1 L π = x.K +α −α α x(x + 2) (A.33) x +1 α1 .(1−α)1 .A.L* −ρ ( + )2 − 1 x 1 (1−α)L* σ
For instance, if we restrict our attention to the case where α = 0.5, L* =1, σ = 2, ρ= 1 and A = 8 , we obtain:
xK (2x +1)(x + 2) π = 9 − 2 (A.34) x +1 (x +1)2
It is clear in our example that for low values of x (weak taste for housing), the jurisdictions will make small profits, whereas for high values of x (strong taste for housing), they will be very profitable whatever their number.
24
1 To insure existence, some restrictions on the utility functions are also necessary. Substitutability between demand for land and demand for the public good is usually required, see Fujita [1989] 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 normalized to zero so that total land rent and differential land rent are equal. 3 The concepts of productive public good, public capital and infrastructure are equivalent in the analysis that follows. 4 Education presents some specific features that justify a separate analysis with different assumptions. See for instance Bénabou [1993]. 5 As usual, dynamic results can be obtained only for heavily specified models. Our instantaneous utility function may seem ad-hoc. It allows however to consider the inferior good aspect of land. A Stone-Geary specification would be more elegant but it is not tractable in a dynamic context. 6 See Hochman, Pines and Thisse [1995] for the implications of this problem. 7 See discussion below in III.3. 8 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. See Mirrlees [1972] for an example where utilitarian welfare leads to unequal treatment. 9 Since σ > 1, the intertemporal utility is always well defined. Thus this transversality condition does not play an important role in our analysis. 10 The efficiency of this tax rests on two strong assumptions, which are 1/ observability of private production and 2/ no distortion of labor supply. Financing infrastructure through the land market avoids these two shortcomings. 11 In the Barro-Romer case, for the first-best, the wage is taken by the developer to finance the local public good (see the third part of (15)). So the only income obtained by the consumers is given by the financial returns as if the share of labor was used completely to finance the land rent. Here, the wage is positive and used to finance housing expenses and consumption. Since the share of housing in the expenditure function is not one, only a fraction of the wages is spent on the land rent and is then directed towards the financing of the public good. Moreover, as in the previous case, only a fraction of marginal public investment appears in the wages, which adds to under-investment. Then the public expenditures are not sufficient (see Bertola [1993] for a more complete discussion of distribution effects in growth models). 12 Respectively introduce competition at the local level and use information available on the land market.
25