The provision of local public goods and factors in the presence of firm and household mobility

JOURNALOF

PUBLIC Journal of Public Economics 60 (1996) 73-93

EI,SEVIER

ECONOMICS

The provision of local public goods and factors in the presence of firm and household mobility W o l f r a m F. R i c h t e r a, D i e t m a r W e l l i s c h b'* aUniversitiit Dortmund, WiSo-Fakultiit, Lehrstuhl VWL II, D-44221 Dortmund, Germany bTechnische Universitiit Dresden, Fakultiit Wirtschaftswissenschafi, D-01069 Dresden, Germany Received September 1993; revised version received September 1994

Abstract This paper studies the efficiency properties of allocations when firms and households are mobile and when local governments provide local public goods and local public factors. The analysis differentiates between immobile land owners and perfectly mobile workers and concludes that an efficient allocation is obtained if there is no outflow of land rents to absentee owners. If rents flow out, only local public goods are supplied in accordance with the Samuelson Rule. The provision of local public factors is inefficiently low, however, and jurisdictions tax mobile firms and mobile households inefficiently high in order to restrict the outflow of rents. An optimal intervention scheme is derived in this case. We also analyze the distortions that result when the non-availability of local head taxes makes it impossible to internalize crowding costs. Keywords: Local public goods and factors J E L classification: H70; H77

1. Introduction T h e r e n o w exists an extensive b o d y of literature analyzing the efficiency p r o p e r t i e s o f local public g o o d s provision w h e n households are mobile across jurisdictions. See, for example, B r u e c k n e r (1983), Wildasin (1986, * Corresponding author. 0047-2727/96/$15.00 © 1996 Elsevier Science S.A. All rights reserved SSDI 0047-2727(95)01515-9

74

W.F. Richter, D. Wellisch / Journal of Public Economics 60 (1996) 73-93

1987), and Krelove (1993). These models assume that all households are mobile and local governments maximize after-tax land rents. The conclusion is that a competitive equilibrium between jurisdictions is socially efficient (i) if jurisdictions can levy residence-based head taxes on mobile residents to internalize their crowding costs and (ii) if they can levy a source-based tax on the land rents originating in the jurisdiction. The objective of this paper is to extend this standard result in several respects. First, we not only consider household mobility, but also the mobility of firms. Firms locate so as to maximize their profits. ~ Hence, the additional question arises whether jurisdictions tax mobile firms in a socially efficient way. The second extension concerns the provision of local public services. We assume that local governments provide local public goods to residents as well as local public factors to local firms. This extension will prove to be useful since we demonstrate that there are cases in which the supply of local public goods is efficient while the provision of public factors is inefficient. Third, we divide local residents into two groups. There are mobile workers and immobile landowners. Immobile landowners may own land in their jurisdiction of residence as well as in other jurisdictions. Therefore, there is a potential outflow of land rents to absentee owners. We assume that jurisdictions are small and that they maximize the utility of immobile residents since they cannot affect the utility of mobile residents. This model structure discloses a source of inefficiency of local behavior which is non-existent when local governments maximize land rents. This source is the outflow of land rents, or more generally, the outflow of income accruing to absentee owners of immobile local production factors, e We will show that the outflow of land rents impedes the efficiency of decentralized government behavior. H o w e v e r , this is a second-best argument. If local governments have the power to tax away all land rents originating in the jurisdiction, they do so and have the incentive to behave efficiently. They provide local public goods and factors according to the Samuelson Rule and they tax mobile households and mobile firms at marginal crowding costs. This extends the efficiency results of Brueckner (1983) and Wildasin (1986, 1987) mentioned above to an environment which includes firm mobility and the provision of local public factors. By maximizing immobile landowners' utility, each jurisdiction effectively maximizes net land rents. More interesting results can be derived if we are in a second-best world. If 'Richter (1994) and Wellisch (1995) analyze the efficiency properties of decentralized decision-making in models of firm mobility. 2 Wildasin (1983, 1991) also differentiates between mobile workers and immobile landowners. However, he assumes that immobile residents own local land completely so that there is no rent outflow.

W.F. Richter, D. Wellisch / Journal of Public Economics 60 (1996) 73-93

75

jurisdictions are constrained in taxing land rents, then a central government must intervene and provide jurisdictions with incentives to act as if there were no rent outflow. If no central government intervention takes place, jurisdictions do not act efficiently. They only provide local public goods according to the Samuelson Rule. However, they tend to provide local public factors inefficiently low and they tax mobile households and firms inefficiently high in order to restrict the rent outflow. The point has been made before by Richter and Wellisch (1993). As a second source of inefficient local behavior, we finally consider the case where local governments have no tax on mobile households at their disposal. Such a scenario has been analyzed before by Hoyt (1991) and Krelove (1993). To isolate the inefficiencies associated with this deficiency, we assume that all rents remain within the jurisdiction. When mobile residents create crowding costs, local governments try to restrict immigration indirectly, thus generating inefficiencies. More specifically, local governments provide inefficiently low levels of local public goods to displace mobile residents. They also provide inefficiently low levels of public factors and they tax mobile firms inefficiently high in order to reduce the marginal product of labor which keeps mobile households out of the jurisdiction. To address all the points discussed above, the paper is organized as follows. In Section 2 we describe the model structure and we derive the necessary conditions for a socially efficient allocation. Section 3 demonstrates that decentralized government activities are socially efficient if regions can prevent an outflow of land rents by taxing them completely away. Otherwise, a central government must intervene and must provide incentives to jurisdictions to act as if there were no rent outflow. In Section 4, two sources of inefficient regional behavior are discussed. First, there is a rent outflow and no central government intervention takes place, and second, regions have no tax on mobile residents at their disposal. Conclusions are drawn in Section 5 and proofs are given in the appendix.

2. Efficient allocation 2.1. The model

Consider a federation consisting of I jurisdictions. In each jurisdiction there are immobile and mobile households. Let A indicate the immobile type and B the mobile one. For simplicity, we normalize the number of type-A households in each jurisdiction i, i = 1 . . . . . I, to one. n ~ denotes the number of type-B households. N B is the exogenously given total number of identical mobile households in the federation. We characterize both

76

W.F. Richter, D. Wellisch / Journal of Public Economics 60 (1996) 73-93

household types by their utility functions, U A i (x A i, zi) and U B ( x B~, Zi), where x ~ and x/B are the consumption levels of private goods and z~ denotes the local public good supply. We assume that there are no spillover effects in the provision of z i. Each mobile household is endowed with one unit of labor which is inelasticaily supplied in its jurisdiction of residence. Immobile households are not endowed with labor. The costs of providing local public goods are given by Ci(zi, n~).a These costs are expressed in units of the private numeraire good and vary with the public good level and with the n u m b e r of users. The specification is sufficiently flexible to allow for both increasing returns in the provision of zi, C~ =- o c i / o z i < C / z i , and increasi B ing returns with respect to the number of users, C i < C / n i. The local public good is said to be pure if there is no congestion, C,~ = 0. However, some positive marginal congestion is the empirically more relevant case. 3 T h e r e are M identical mobile firms in the federation. M is exogenous, which excludes the formation of new firms. We thus only model the locational choices of firms and not market entry, m~ is the number of mobile firms locating in jurisdiction i. Firms are said to be identical if they use the same technology, represented by the production function F(li, n i, &). Production makes use of three factors: l~ stands for the immobile factor land; n i for the mobile factor labor; and g, for the local public factor. Jurisdictions are endowed with a fixed amount of land, Li. We assume that there are no spillover effects in the use of the local public factor. The cost of providing the public factor are given by the function H~(gi, rn~). These costs are also expressed in units of the private good and depend on the level of public inputs as well as on the number of firms locating in the jurisdiction. The special case of pure public inputs, HI,, =--OH~/Orn~ = 0, is included. In general, public factors will be impure, H~ > 0, however. For simplicity, we treat n B i and rn i as real numbers, expressing the idea that households and firms are small relative to their markets. Previous studies 4 do not model firm mobility explicitly. This can be justified if production is characterized by linear homogeneity and if public inputs are quasi-private, i.e. if average costs, Hg/rn~, are linear in g~ and constant in m i. Only with these special assumptions is the locational pattern of firms indeterminate in a socially efficient state, so that it makes no sense to differentiate firms at the individual level. Following Richter (1994), we choose the more general model specification where the number of local firms is a non-trivial endogenous variable. The functions U A, U a, F, H i, and C i are twice differentiable and satisfy 3Empirical studies of Borcherding and Deacon (1972) and Bergstrom and Goodman (1973) find strong empirical evidence of congestion. In many cases, costs become nearly proportional to the population, once a minimum population size of 10,000-50,000 is reached. 4 For a comprehensive overview, see Wildasin (1986).

W.F. Richter, D. Wellisch / Journal o f Public Economics 60 (1996) 73-93

77

standard assumptions. We indicate derivatives by subindices. First partial derivatives have the usual signs. F is strictly concave in the private factors l/ and n i. F may be linear-homogeneous in Ii, n i and gi. For the results derived in this paper, we need not to be more specific about the degree of homogeneity. In order to achieve efficiency, a social planner would have to maximize UB(xB1, z l )

(1)

in the vectors (xA), (XB), (Zi) , (gi), (li), ( m i ) , (ni), (nai) subject to U A i(x A i,zi)=a

A ~ . . i .= . 1,

,I ,

i=2 ..... I,

UB(x~,z~)=UB(Xl,Zl), B n i - - m i n i = O ,

L i - rnil i = O ,

(2) (3)

i=1 .... ,I,

(4) (5)

i = 1, . . . , I , n B, = 0 ,

NB-

(6)

i

(/x): M - ~ m , = O ,

(7)

i

( A): ~'~ [miF(li, ni, gi)

A -

x

i

-

n

B

B

ix

i -

i

C (zi, nai ) - H i ( g , , m i ) ] = O .

i

(8)

T, ~, and A are Lagrange multipliers. We mentioned only those that are needed in what follows. According to (1), the social planner maximizes the utility of a representative mobile resident in jurisdiction 1, while fixing the utilities of all immobile residents at predetermined levels, (2). Condition (3) reflects costless migration of mobile households by ruling out jurisdictional utility differentials that would be incompatible with free locational choices. According to (4), the local labor supply is equally distributed among local firms. (5) requires the same for the fixed local endowment of land. (6) and (7) require that mobile households and firms have to locate somewhere in the federation. Finally, (8) is the global market clearing condition for the private good. Aggregate production has to meet households' consumption and the real costs of providing public goods and factors. 2.2.

First-order conditions

A solution to the planner's problem may well involve allocations for which no production takes place in some jurisdictions. We ignore this well-known problem in regional economics and focus instead exclusively on

W.F. Richter, D. Wellisch / Journal of Public Economics 60 (1996) 73-93

78

so-called interior solutions characterized by n B i, rni > 0, Vi. The first-order conditions turn out to be (for all i) uAi - -

Bi B

Uz

_

i

u A i + n i ~--x -- Cz '

(9)

miF ~ = Hi,

(10)

F~ - x ~ - C~ = - ~Y,

(11)

F-

t~ , liF ~ - niF i - H i = ---~

(12)

where we have made use of the abbreviations U Ag, U a~ and F i for i = A A 1 . . . . . I. For example, U Ai - U i (x ~, zi). (9) is the Samuelson Rule for the efficient supply of local public goods. (10) is the corresponding Samuelson Rule for the efficient provision of local public factors. Efficiency requires that the marginal product of local public factors must be equal to the marginal costs. The marginal product is obtained by summing over all firms locating in the region. (11) and (12) characterize efficient spatial allocations of households and firms. According to (11), the net social benefit of an additional mobile household to a region must be equal in all regions. The benefit for the marginal household is its marginal product, F,~. The costs consist of its consumption of private goods, x ~, and of congestion costs, C,~. Following (12), an efficient locational pattern of mobile firms is similarly achieved if the social net benefit of a firm to a region is interregionally equalized. The social net benefit of a firm is measured by its pure profit, F ' - l ~ F ~ - n~F',, minus marginal congestion costs, H~. If local public factors are quasi-private and if F is linear-homogeneous in l~, ni and gg, then the efficient locational pattern of mobile firms turns out to be indeterminate. Consider the case of an efficient allocation (n*, m*, g*). Then, any allocation (n~, mi, g~) satisfying n~mi = m * , n*, gim~ = g ~*m ~*, and ~i mg = M is equally efficient. It should be emphasized that (9)-(12) are necessary conditions. They need not be sufficient for an efficient allocation. The problem of lack of sufficiency is well known from the literature and will be ignored in the sequel. See, for example, Stiglitz (1977) and Richter (1994). This paper will only focus on institutional scenarios supporting allocations that satisfy the necessary conditions (9)-(12). 2.3. Efficient interregional resource distribution

A leading question in the literature is the extent to which interregional transfers of rent income are needed to sustain efficiency. Hercovitz and

W.F. Richter, D. Wellisch / Journal o f Public Economics 60 (1996) 73-93

79

Pines (1991) and Krelove (1992) among others make the remarkable point that rents must flow out of regions if market equilibria are to be efficient. The assertion is that the equality of local production, m~F i, and local utilization of goods, Yi ~ x A + ?1 B x B~ + C i + H ,i is in general incompatible with efficiency. It is important to note that the present model does not require interregional transfers to sustain efficiency. On the contrary, we demonstrate that any regional outflow of land rents interferes with regional incentives to ensure an efficient allocation. Obviously, such conflicting results have to be explained by differences in the model. In fact, there are major differences. Whereas the present model allows for mobile firms as well as mobile and immobile households, Krelove (1992) and others restrict attention to immobile production and perfectly mobile households. As it turns out, mobility of firms does not cause the different conclusions on interregional transfers. It is the presence of immobile households that introduces degrees of freedom in the efficient allocation of resources. T o show this, we solve (11) for x~ and (12) for F~. We utilize the resulting expressions and substitute them into Yi = X,A + n Bi x iB + C i + H i. The equality of local production and local absorption, m r ' = Yi, then turns out to be equivalent to A

x~ - R~ B

ni

y =--

(13)

where It£

B

i

R i =- L i F ~ + rni-~ - (C' - n i C . ) - (H / - m i l l i ) can be interpreted as the net rent generated in jurisdiction i. R~ is land rent plus the social marginal benefit of local firms minus non-congestion costs of providing local public goods and factors. (13) states that the excess of consumption by the immobile household over locally generated net rents is equated across regions on a per capita basis. Since there are mobile as well as immobile households, (13) is compatible with an efficient allocation, in general. T o see this, note that ( 2 ) - ( 1 2 ) include 81 + 2 conditions that must determine 9•+2 variables at an efficient allocation: the /-dimensional A B B vectors (x i ), (x i), (zi), (gi), (li), (mi), (ni), ( n , ) , (a A i ) and the Lagrange multipliers y/A and/x/A. Since there are I + 1 different household types, i.e. the group of identical mobile households and I immobile households, 1 variables can be chosen freely for distributional purposes at an efficient allocation. For instance, (x,A) can be set such that (13) is satisfied, i.e. x ia = Ri for all i, ensuring that m i F i =Yi is compatible with an efficient allocation. If there were only mobile households, (13) would reduce to

W.F. Richter, D. Wellisch / Journal o f Public Economics 60 (1996) 73-93

80

Ri B

-

Y

(14)

As Krelove (1992) emphasizes, this condition cannot be met at an efficient allocation, in general. Without immobile households, there are no degrees of freedom at an efficient allocation which can be used for distributional purposes. Since condition (2) is no longer relevant, the number of conditions that must be met at an efficient allocation reduces to 7I + 2, i.e. (3)-(12). The number of variables that must be determined from this system also reduces to 71 4-2 since the vectors (x A) and 0i A) drop out. Hence, if all households are mobile, interregional transfers are needed to sustain efficiency. However, if there are some immobile households in the jurisdictions, then the necessity of transfers becomes ambiguous. It is shown below that preventing any outflow of land rents is even necessary for jurisdictions to choose efficient allocations. Moreover, the entire absorption of land rents by jurisdictions does not violate any necessary condition for an efficient allocation.

3. Competitive equilibrium allocations

3.1. Private behavior To avoid problems caused through strategic interactions, we assume perfectly competitive behavior throughout the economy. This implies that jurisdictions are small. Before describing the behavior of local governments, we turn to decentralized decisions of households and firms. We assume that all land in the federation is in the hands of immobile residents. It will not be ruled out that part of local land belongs to absentee owners. Let r i denote the gross land rent in jurisdiction i, ti the proportional tax rate on land rents, and tr A the share of jurisdiction i's land endowment which is held by the immobile resident in i, 0 < tr A ~< 1. The immobile resident's net income from land is (1 -ti)trAriLi. I~ is his income from the ownership of land outside the jurisdiction, I ~ = Ej,, i aji(1 - trk)(1 -- tj)rjLj, El,, j olji = 1 for all j, where aj~(1- o"A) is the fraction of Lj held by the immobile resident living in jurisdiction i. The regional government levies a head tax, ~.A, on the immobile resident, z/A must be interpreted as a subsidy in the case of negative values. Total income is used to consume private goods: x A = I A + (1

-

t i ) o r A r i L i -- r A .

(15)

Mobile households supply one unit of labor inelastically in their region of residence. Wage income is wi. Aside from wage income, mobile residents

W.F. Richter, D. Wellisch / J o u r n a l o f P u b l i c E c o n o m i c s 60 (1996) 7 3 - 9 3

receive profit income denotes the c o m m o n is not dependent on residence-based head private consumption X ai : W i " ~

ia

81

from their ownership of firms, I B = ( M / N a ) ( r , where ff equilibrium profit level of all firms. By assumption, I a the jurisdiction where B resides. 5 The region levies a tax, z ~, on mobile residents. Total income is used for only:

--T

ai

(16)

.

Mobile households choose their residence so as to maximize utility. Since there are no migration costs, a migration equilibrium is achieved if mobile households achieve the economy-wide utility level, f i b U B (x i, zi) = fib .

(17)

Firms have to m a k e decisions on where to locate and on how much to produce. Let 7r i denote the after-tax profit of a firm locating in i. A locational equilibrium is achieved if profits are equalized across the federation: i

~r = ~ - .

(18)

The after-tax profit of a representative firm is i

7r = F ( l i , h i , g i ) - r J i -

F

wini - r i ,

(19)

where r i and w i are local factor prices and z v is a local poll tax on firms. We assume that firms maximize profits and that they take r i and w i as given. H e n c e , we obtain:

F; =

(20)

ri,

F~ = w i .

(21)

Local m a r k e t clearing requires (22)

L i = rail i

and B

n i = mini

•

(23)

T h e provision of local public goods and factors causes costs of i l~, i C (zi, n ~ ) + H (gi, m~). These costs must be covered by local tax revenues and by net transfers, S ~, provided by the central government. A leading question is whether some S ~ is in fact needed in order to ensure social efficiency. T h e local g o v e r n m e n t ' s budget constraint is 5The distribution of the ownership of firms and land between mobile and immobile households could be modelled differently without changing any of the basic conclusions. For example, we could assume that both immobile and mobile households own land and firms.

W.F. Richter, D. Wellisch / Journal of Public Economics 60 (1996) 73-93

82 A

B B

F

Si

"l" i q-hie" i q-mi,t- i + r i L i t i +

i

i

=C(zi,

n~)+n(gi,

mi) ,

(24)

and the central government's budget constraint reads as Z

Si

(25)

-~- 0 .

i

Competition between jurisdictions is expressed by the assumption that each single jurisdiction takes the equilibrium utility level of mobile households, ~7a, and the equilibrium profit level, ~, as given. This indicates the Walrasian type of equilibrium under consideration.6 Let us once again emphasize the meaning of all variables, r i and wi are endogenous prices and Ii, m i, n i, n ai , x A, and x/a are endogenous quantities. They are all determined within the jurisdiction, t,, ~'A F gi, and z i are i , "riB, "l'i, control variables of the local government, o"A, Li, I A, I B, fiB, and ~ are exogenous from the viewpoint of the individual jurisdiction. I B, fiB, and ~i" are independent of i. In order to analyze the efficiency properties of competitive equilibria, we focus on a representative jurisdiction. We can therefore drop the local index i from our notation without much loss of clarity.

3.2. Local government behavior The decisions of mobile households and mobile firms cannot be directly controlled by local governments. However, rational local governments must take into consideration the responses of n B and m to policy choices. The equilibrium conditions (17) and (18) allow us to derive n B and m as implicit functions of ~.B, ~.F, Z, and g:

V ( n B , m , ~ a ' rF ' z , g ) - - U a ( F n ( p(nB, m, B

F z, g) =__F ( L ,

,

L ' na m' g ) _~-a+l a,z ) =fa, na

)E(L

, m,g

-

na t

)L

' m 'g

nB - F , ~,~, --m--,g_ m _ ~.F = ~.. I/ L

nB

(26)

m

)

(27)

Here, use has been made of (16) and (19)-(23) to eliminate the endogenous variables x B, r, w, l, and n. Notice that (26) and (27) are independent of t 6Note also that the global market clearing conditions ( 6 ) - ( 8 ) are assumed to hold in equilibrium. More specifically, the equilibrium utility level of mobile households, ti B, and the equilibrium profit level, ~i-, are such that mobile households and firms are allocated across the jurisdictions. This ensures (6) and (7). Condition (8) is automatically satisfied due to Walras's law.

W.F. Richter, D. Wellisch / Journal of Public Economics 60 (1996) 73-93

and T

A

./A,

83

indicating that there are no migration responses to changes in t and

For later reference, we derive the responses of the marginal labor productivity, Fn, to variations in the local government's choice variables. For a given fiR, we obtain from (26): dF. d~-A

.

dFn dF. dF~ . . . dt dr F dg - 0 ,

dF. dr a

1,

dF~ dz

- -

UB . U xB

-

(28)

Let us assume that the local government maximizes the utility of the immobile resident when choosing its control variables since it takes the utility of mobile residents as given. This behavioral assumption describes the locally efficient behavior of jurisdictions. Without assuming such locally efficient behavior, we should not expect that jurisdictions will achieve a socially efficient allocation even if everything else were perfect. Adopting this assumption, the local government has to maximize UA(x A, z)

(29)

in its choice variables t, r B, r F, z and g, where x A has to satisfy X

A

----

i A

+[S-C(z, nB)-H(g,m)q-riB7 +m(F"B

Fn

--

--

(30) with 8, ~ (1 - t)(1 - o-A)L.

(31)

T o derive (30), we have solved (24) and (27) for ~.A and v F, respectively, and substituted the resulting expressions into (15). In the following analysis, 8, is a key variable. It denotes outside claims against local land. Land rents are completely appropriated within the jurisdiction if t = 1 or if orA = 1. In both cases we have 3, = 0 and there is not outflow of rents. A rational local government will consider how the centrally chosen net transfer, S, is affected by its own decisions. T o derive the optimal intervention scheme for a central government, we specify S as a function of B F ~- , r , z , a n d g : S = S(rB, ~.F, z, g) .

(32)

As t and ~.A do not affect n s and m, it is reasonable to assume that S does not depend on these instruments. Maximizing (29) with respect to t and taking into account that n a and m are not affected, we obtain:

84

W.F. Richter, D. Wellisch / Journal of Public Economics 60 (1996) 73-93 dU A 1 dt U A

(1 - o'A)LFt > 0

for tr A < 1.

(33)

H e n c e , we have L e m m a 1. I f local land is partly owned by non-residents, o ' A < 1, then jurisdictions choose the m a x i m u m feasible t. I f tr A = 1, then the optimal t is indeterminate. Jurisdictions are only interested in the welfare of inside residents. They ignore positive effects resulting from rents accruing to absentee land owners. T h e y will therefore stop any outflow of land rents by completely taxing them away. Inside owners of land are compensated by lump-sum transfers, z g. The other first-order conditions of an optimal local government behavior 7 are dU a 1 dz B U A

m

OS _ 6t dFt On B Om OrB d~."----ff+ ( r " - C.) "~--ff~s+ (TF -- Hm) "~--ffT B =0'

dU A 1 d7-F U A x

OS dFt On B Om oT.F -- a t - ~ TF + ('r B -- C n ) - ' ~ F + (7 F -- g i n ) -o-T F -- - 0

dU A

1

dz

U~

OS Oz

_ 6

dF~ +

' dz

(34)

,

(35)

On B (~.._C.)_.._~_z+(~V_Hm) Omoz

UA B + ~ A + n B ~U, azUx Ux

(36)

C~ = 0 '

dU A 1 0 S _ 6 dFt OnB dg u A -- -~gg ' dg + (~'a - c " ) --~g + (rv - H'') omOg + mFg - n . = o .

(37)

Evaluation of (34)-(37) gives us Proposition I. I f there is no outflow o f land rents, either because the local land endowment is completely owned by the immobile resident, tr A = 1, or because jurisdictions can tax land rents completely, t = 1, the competitive equilibrium is socially efficient without any central government intervention, S=-O.

7

See Section A.1 in the appendix for details.

W.F. Richter, D. Wellisch / Journal o f Public Economics 60 (1996) 73-93

85

If tr A = 1 o r if t = 1, t h e n 8, = 0. F o r S -= 0, t h e p a r t i a l d e r i v a t i v e s o f S in ( 3 4 ) - ( 3 7 ) v a n i s h . H e n c e , t h e l e f t - m o s t e x p r e s s i o n s in ( 3 4 ) - ( 3 7 ) a r e e q u a l to z e r o . I n this case, (34) a n d (35) s h o w t h a t an efficient l o c a t i o n a l p a t t e r n o f m o b i l e h o u s e h o l d s , 1" B = Co, a n d m o b i l e firms, r F = H , ~ , is a c h i e v e d since t h e m a t r i x with e l e m e n t s O n B / 8 ~ "B, 8 m / O z B, 8n~/O~'F a n d O m / O z ~ is o f full r a n k . This will b e p r o v e d in S e c t i o n A . 1 of t h e a p p e n d i x . I n s e r t i n g this r e s u l t into (36) d e m o n s t r a t e s t h a t t h e S a m u e l s o n R u l e for l o c a l p u b l i c g o o d s a p p l i e s . T h e S a m u e l s o n R u l e for local p u b l i c f a c t o r s is o b t a i n e d f r o m (37). A s m e n t i o n e d in f o o t n o t e 6, the r e s o u r c e c o n s t r a i n t s ( 6 ) - ( 8 ) a r e satisfied. H e n c e , all f i r s t - o r d e r c o n d i t i o n s o f a socially efficient allocation are met. 8 Q.E.D. Proof.

T h e b a s i c c o n c l u s i o n is t h a t j u r i s d i c t i o n s will b e h a v e efficiently if t h e y h a v e a sufficient set o f i n s t r u m e n t s at t h e i r d i s p o s a l a n d if t h e y act u n d e r c o n d i t i o n s o f p e r f e c t c o m p e t i t i o n . A sufficient set o f i n s t r u m e n t s m e a n s t h a t t h e y a r e a b l e to tax l a n d r e n t s c o m p l e t e l y . T h i s result can b e c o m p a r e d with p r e v i o u s o n e s . W i l d a s i n (1986) d e m o n s t r a t e s t h a t an efficient a l l o c a t i o n o f m o b i l e l a b o r a n d local p u b l i c g o o d s is a c h i e v e d by j u r i s d i c t i o n s a c t i n g i n d e p e n d e n t l y if t h e y a r e small a n d if t h e y m a x i m i z e l a n d rents. T h i s is c o n s i s t e n t w i t h P r o p o s i t i o n 1 since f o r o-A = 1 o r t = 1, j u r i s d i c t i o n s effect i v e l y m a x i m i z e l a n d r e n t s ( f o r given z). P r o p o s i t i o n 1 g o e s b e y o n d t h a t r e s u l t b y s a y i n g t h a t j u r i s d i c t i o n s also s u p p l y local p u b l i c f a c t o r s efficiently, a n d t h a t t h e y e s t a b l i s h an efficient l o c a t i o n a l p a t t e r n o f m o b i l e firms by i n t e r n a l i z i n g t h e i r c r o w d i n g COSTS.9

8 Full land rent absorption by jurisdictions does not necessarily mean that the production in each jurisdiction is equal to the utilization of the private good. t, = 1 and S' = 0 together with (16), (21), (30) and (31) imply y~ = m i F ' + [nB,M / N B - m,l~r. However, as was demonstrated in Subsection 2.3, there is a degree of freedom in choosing x7 at an efficient allocation. This freedom could be used by the central government to set S ~= S o = [m i - n a,M/Na]~r. Thus, the central government would ensure that y, = m,F ~ in each jurisdiction. Note that jurisdictions are still confronted with the correct incentives since they take So as exogenously given, and that (25) is satisfied, I~ S O= ~ I, [m, - n a,M / NB] = O. 9 It is interesting to analyze whether the conclusions drawn in Proposition 1 must be modified when only one group of identical mobile households exists, an assumption which has often been made before in the literature. Here, the requirement that land rents are not allowed to leave the jurisdiction would violate the condition for an efficient interregional allocation of resources. See Subsection 2.3. However, the requirement of full land rent absorption is only needed in our framework to guarantee that local governments are confronted with the correct incentives to choose an efficient allocation, i.e. to maximize after-tax land rents. It is no longer needed in a model without immobile residents since local governments cannot influence the utility of mobile households, and maximizing after-tax land rents would then be the natural behavioral assumption. Furthermore, if mobile households were equally endowed with land in all jurisdictions, the condition for an efficient interregional allocation of resources would be satisfied.

86

W.F. Richter, D. Wellisch / Journal of Public Economics 60 (1996) 73-93

3.3. Central government intervention In general, land is owned in part by households living outside the jurisdiction and constitutional barriers rule out complete taxation of land rents. An example is G e r m a n y where private property enjoys legal protection by the written constitution. According to a common understanding, this makes confiscatory taxation of land rents institutionally infeasible. A subtle question concerns the level up to which the rates of rent taxes can be seen not to violate private property rights. It may be difficult, if not impossible, to fix this level in real values. However, t = 1 is, no doubt, infeasible. Let us therefore assume that there is some [ ~ [0, 1) exogenously constraining the local government's scope of feasible taxation. According to L e m m a 1, jurisdictions will always exhaust the scope by setting t = [. Allocational distortions result which are analyzed in more detail in Subsection 4.1 below. Before moving on to Section 4, we examine the case of central government intervention. The justification for intervention relies on a second-best argument: the externalities resulting from institutional restrictions must be internalized. The objective of this intervention is to give jurisdictions incentives to act as if all local land rents accrue to inside residents. This idea is made more precise by

Proposition 2. Suppose t<~[ < l and o'A<(1. If the central government chooses S so that jurisdictions face the same incentives as if there were no ouOqow of rents, first-best allocations result. The transfer formula S = 8~FI + S Osatisfies this requirement, where jurisdictions take S O as exogenously given. Proof. Following L e m m a 1, jurisdictions choose t = [. If S = 87F~ + S o is chosen, then the left-most expressions in (34)-(37) vanish. Proposition 2 can now be proved as Proposition 1. Q . E . D . As emphasized in Subsection 2.3, there is a degree of freedom in choosing g X and therefore S o at an efficient allocation. It can be used to choose S O = - t S z F t, where &Ft denotes the average outflow of land rents in the federation.

Corollary 1. The transfer formula S = 8~Ft -87F t supports an efficient allocation and satisfies the central government budget constraint. & is consistent with the a priori requirement that S does not depend on t. q~he determination of S - - 8 7 F t - 8 F t also has the advantage of providing incentives for jurisdictions to u___nndertaketheir own tax efforts and to choose t = t. The formula S = 8,Ft - S t F t would also provide incentives for efficient

W.F. Richter, D. Wellisch / Journal of Public Economics 60 (1996) 73-93

87

local behavior. However, the value of t would be indeterminate, i.e. jurisdictions would have no incentives to tax local land. In the next section we analyze the distortions that occur if there is no central intervention.

4. Distortions in the absence of central interventions 4.1. Rent outflow

Let us have a closer look at the inefficiencies that are generated by local decisions when the tax rate t is constrained by an upper bound, t ~< i-< 1. Following L e m m a 1, jurisdictions will choose t = i-. No central government intervention takes place by assumption. By computing migration responses OnB/O'r B, OmlOz B, OnBlO'r F, and Om/O'r F using the locational equilibrium conditions (26) and (27) and by inserting them into (34)-(37), we obtain, as is shown in Section A.2 of the appendix. Proposition 3. Suppose t ~ < / < 1, tr A < 1, and S =- O. Then jurisdictions will choose

dE,

F,.

(i)

ra - C" = O~ dnB =

(ii)

dF~ IFll + nF1. ~.v _ H,, = 6 d---~= - 6 m '

(iii) (iv)

m '

rnFg - Hg = 6 Ftg , OA

B - z +n a U z UxA -77-ff, a = Cz • Ux

The results can be interpreted as follows. Jurisdictions try to restrict the outflow of rents. Since they are constrained in reaching this objective by direct taxation of land rents, they turn to indirect methods. They will tax mobile households inefficiently high, z a > C,, in order to displace them if Ft, > 0. By using this policy, jurisdictions reduce land rents and therefore the outflow of rents by simply deterring people from residing in the jurisdiction. According to Proposition 3, jurisdictions also tax mobile firms inefficiently high, ~F > H,,, if land rents decline with a lower number of local firms, d F J d m > 0 . In this case they can also restrict the rent outflow by pushing firms out of the jurisdiction. The same reasoning holds for the provision of local public factors. If an increased supply of local public factors increases land rents and therefore the outflow of rents, Ftg>O, local governments provide public factors inefficiently low relative to the Samuel-

88

W.F. Richter, D. Wellisch / Journal of Public Economics 60 (1996) 73-93

son Rule. While the costs of providing public factors are completely internalized within the jurisdiction, the benefits partly spill out with outflowing land rents. This results in an underprovision of local public factors. Interestingly enough, the provision of local public goods follows the Samuelson Rule. One may wonder why jurisdictions do not resort to inefficiently low levels of public goods when trying to restrict the outflow of rents. A low provision of public goods might help to deter wage earners from settling within the region which indirectly affects the productivity of land and the outflow of land. The reason is that jurisdictions can tax mobile households directly. Because of O * / O z = - ( U a ~ / U ~ ) / ( O * / O ~ B ) , *~ {n B, m} [see (A.12) and (A.13) in the appendix], any effect resulting from a variation of z can be achieved more directly by a variation of z B. 4.2. Non-availability o f local h e a d taxes

In this subsection we concentrate on another reason for inefficient local behavior: local governments cannot levy a head tax on mobile residents. For this case we derive the kind of distortions to be expected and we provide an intuitive explanation for the resulting inefficient local behavior. We assume that jurisdictions cannot tax mobile households since in many countries a residence-based tax on mobile households is not at the disposal of local governments. In Germany, for example, local governments are able to tax local firms, but they are unable to tax own residents. In order to isolate the distortions associated with this insufficient instrument set, we assume that there is no rent outflow, either because tr A = 1 or t = 1. We recall the first-order conditions for an optimal local policy: dU A

1

dT F U A

dU A

1

-

- C n Ona Om --+(~'~-Hm)-~TF=O

(38)

Om + U _ A z + n a U az _ - C - ~On - z B +(~'F--H,~)-~z uA ~ Cz=0

(39)

0,/.F

dU A 1 - C n On B Om + mFg dg U A -"~-g °t- ('r F -- Hm ) "~-g

-Hg = 0 .

(40)

The migration responses Ona/O* and a m ~ a , , • E {r F, z, g} are listed in Eqs. (A.8)-(A.13) of the appendix. Inserting these responses into (38)-(40) we get, after some straightforward manipulations P r o p o s i t i o n 4. S u p p o s e tr A = 1 o r t = 1, b u t jurisdictions h a v e no tax o n

W.F. Richter, D. Wellisch / Journal o f Public Economics 60 (1996) 73-93

89

mobile households at their disposal, r B = O. Then the competitive equilibrium is characterized by

(i)

m dF. lF.t + nF.. r v - H., = - C . F.. dm - C. F.. "

(ii)

~ ' Ax

(iii)

mFg - Hg = - C. ~

U A

n B U s,_ -77~B'-C~=C"u.

m dF. F.. dz m

m U~ - C . F.. V ~ "

F., .

Jurisdictions cannot internalize the congestion costs of mobile households by taxing them directly. Therefore, jurisdictions try to control the entry of mobile households by means of other policy instruments. This causes an inefficient allocation. According to Proposition 4, jurisdictions tax mobile firms inefficiently high whenever the marginal productivity of mobile labor increases with the number of firms locating in the jurisdiction, d F . / d m > O. Hence jurisdictions can restrict the immigration of mobile households by displacing mobile firms. A more direct way is to provide an inefficiently low level of local public goods. By this measure, mobile households can be discouraged to reside in the jurisdiction. Contrary to the case of a rent outflow, jurisdictions do not provide local public goods according to the Samuelson Rule. Finally, if the marginal product of mobile labor increases in the level of local public factors, dF./dg > 0, then the provision of public inputs will be inefficiently low relative to the Samuelson criterion. Jurisdictions discourage the immigration of mobile households by providing an inefficient infrastructure of production.

5. Conclusions

This paper analyzes decentralized government activities in a federation consisting of many small jurisdictions. In each jurisdiction, immobile as well as mobile households reside. Aside from the mobility of households, we explicitly model firm mobility. Firms locate so as to maximize their profits. Jurisdictions provide local public goods and local public factors and maximize the utility of immobile residents. Within this framework we have derived the following results. If jurisdictions act under conditions of perfect competition, if they can prevent any outflow of rents accruing to outside owners of fixed local factors, and if jurisdictions have a sufficient set of policy instruments at their disposal, then a competitive equilibrium is socially efficient. If there is a rent outflow, however, since jurisdictions cannot tax away all rents, then a central government must correct for inefficient local behavior. It has to provide the same incentives as if there

90

W.F. Richter, D, Wellisch / Journal of Public Economics 60 (1996) 73-93

were no rent outflow. In the absence of this intervention, jurisdictions will undersupply local public factors and they will tax mobile households and mobile firms inefficiently high if doing so restricts the rent outflow. However, the Samuelson Rule for providing local public goods is met in this second-best world. The potential rent outflow is not the only reason for inefficient local behavior. In many countries, local governments have no local head tax at their disposal. Although mobile households cause crowding costs in the supply of public goods when they enter a jurisdiction, no marginal cost pricing is applied. In this case, jurisdictions undersupply local public goods to displace mobile residents. They also tax mobile firms inefficiently high and undersupply local public factors if doing so decreases the marginal product of mobile labor and therefore displaces mobile households.

Appendix

A.1. First-order conditions and migration responses We first demonstrate how the first-order conditions (34)-(37) are obtained. Since the derivations show strong parallels, we state them explicitly only in the case of (34). (35)-(37) follow analogously. After substituting (30) for x A into (29), we differentiate UA(x A, z) with respect to r A, making use of the envelope theorem:

OU A 1 OS _ Cn 0n13 d'r B UAx _ 0713 0713 Om(F

na

Om nm -Or- B+

0n13 0"r13

z a - - + n

a

~.)

+ o----¢ ( -m

LOm n13 d F n ) dFt - - - 0r13 + m d,r B - ~-5d,T- B = O. F t -m2

(A.1)

By (18)-(23), f - (n13/m)F~ - ~i"= lFt + r F, and by (28), dF~/dr a = 1. Inserting these expressions into (A.1) yields the first-order condition (34). We next prove that the matrix

(on0m) 0r13 ' 0n13 0r F '

has full rank.

0r13 Om Or v

(A.2)

W.F. Richter, D. Wellisch / Journal o f Public Economics 60 (1996) 73-93

91

This makes it necessary to derive the migration responses OnB/O* and ~m/O * for * E {r s, rF}. We extend the derivations by including * ~ {z, g} since all derivations are needed in the following. Differentiation of (26) and (27) yields /drB\ (V~a; P,~;

V,,;;)(dna)=(-V,.; Pm \ d m ) \-P,~;

-V~; -P,F;

-Vz; -Pz;

\ dg/ (A.3)

where

F,,,

V,,. = Uax --m- , P,,. =

Vm = -

uB{ L._~

L n~ mz Ft,, - - ~ F,,,,

V.B = -Uax ,

na

)

X \ m z Ft,~+ - - ~ F,,,, , L2 n"L P" = m 3 Fu + 2--'-T- Ft"

V.v = O,

Vz = U ~ ,

(n_~!~ F,,,, ,

B

V~ = UxF,,g, B

P.. = 0 ,

P.F = --1,

Pz=0,

L n Pg=Fg---~Ftg---~F,,g. (A.4)

Denoting the Jacobian matrix of (A.3) by A, we have L2 IZl = uB-----g[F,,,,Fu- F2nt] . m

(A.5)

The determinant ]A] is positive due to the strict concavity of F in l and n. Let

a =- m l a l

"

Using Cramer's Rule yields: Onn = a[lZFu + 2lnFtn + nZF,,,,] < 0 ~,r B Om O,.rB

- a[IFl,, + nF,,,,l

OnB Om O.rV = a[lF., + nF,,.] - 0r a ,

(A.6) (A.7)

(A.8)

92

W.F. Richter, D. WeUisch I Journal o f Public Economics 60 (1996) 73-93

Om 0T F =

aF.. < 0,

(A.9)

On B

(A.10)

Og - -alF"g[lFu + nFt"] - a[Fg - lF~g]llF., + nF..] , Om

(A.11)

Og = -a[F,,.(Fg - IFtg ) + lFt.F.g ] ,

On B

U B

B

Oz = -S:-.-~,B z a[l 2 F. + 2nlFt. + nZF..] = Ux Om Oz

U ~ a[nF,, + tFt, ] = ~,a Uz

U ~ 0n B > U~ OTB

U a Om Ua~ Or B .

0

'

(A. 12)

(A. 13)

Inserting the migration responses (A.6)-(A.9) into the matrix (A.2) yields:

deJ

~On a

Om

\OT F ,

oTF

= am > 0.

(A.14)

Hence, (A.2) is of full rank. A . 2 . Rent outflow

Here we prove Proposition 3. Setting S -= 0 and rewriting (34) and (35) in matrix form yields: / dFt\ Or B '

Or B "~

T B _ Cn

(A.15) On" 0T F '

Om;/\T~_H,,j

\ dC /

0T F

Using Cramer's Rule and inserting (A.6)-(A.9), we get B dF~ F~. r - C n = 6 ~ n a =6~ m ' F

dFt

T - H , , = 8 -d--~m= - 6

(A.16)

lFtt + nFm

m

(A.17)

Inserting S-= 0, (A.12), and (A.13) into (36) and exploiting (34) yields

W.F. Richter, D. Wellisch / Journal of Public Economics 60 (1996) 73-93

U A q'-nB

- -

x

~

=

C z .

93

(A.18)

Ux

By inserting (A.10) and (A.11) into (37), it finally follows that ,,F, -/4 8 = a

(A.19)

References

Bergstrom, T.C. and R.P. Goodman, 1973, Private demands for public goods, American Economic Review 63, 280-296. Borcherding, T.E. and R.T. Deacon, 1972, The demand for the services of non-federal governments, American Economic Review 62, 613-633. Brueckner, J.K., 1983, Property value maximization and public sector efficiency, Journal of Urban Economics 14, 1-16. Hercowitz, Z. and D. Pines, 1991, Migration with fiscal externalities, Journal of Public Economics 46, 163-180. Hoyt, W.H., 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. Krelove, R., 1992, Efficient tax exporting, Canadian Journal of Economics 25, 145-155. Krelove, R., 1993, The persistence and inefficiency of property tax finance of local public expenditures, Journal of Public Economics 51,415-435. Richter, W.F., 1994, The efficient allocation of local factors in Tiebout's tradition, Regional Science and Urban Economics 24, 323-340. Richter, W.F. and D. Wellisch, 1993, AIIokative Theorie eines interregionalen Finanzausgleichs bei unvollst~indiger Landrentenabsorption, Finanzarchiv 50, 433-457. Stiglitz, J.E., 1977, The theory of local public goods, in: M. Feldstein and R.P. Inman, eds., The economics of public services (Macmillan, London) 274-333. Wellisch, D., 1995, Locational choices of firms and decentralized environmental policy with various instruments, Journal of Urban Economics 37, 167-184. Wildasin, D.E., 1983, The welfare effects of intergovernmental grants in an economy with independent jurisdictions, Journal of Urban Economics 13, 147-164. Wildasin, D.E., 1986, Urban public finance (Harwood, New York). Wildasin, D.E., 1987, Theoretical analysis of local public economics, in: E.S. Mills, ed., Handbook of regional and urban economics, Vol. 2 (North-Holland, Amsterdam) 1131-1178. Wildasin, D.E., 1991, Income redistribution in a common labor market, American Economic Review 81,757-774.