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Interregional spillovers in the presence of perfect and imperfect household mobility
ELSEVIER
Journal
of Public
Economics
55 (1994)
167-184
Interregional spillovers in the presence of perfect and imperfect household mobility Dietmar
Well&h
Vniversitiit Dortmund, WiSo-Fakultiit, Lehrstuhl VWL II, 44221 Dortmund, Germany Received
August
1992, final version
received
August
1993
Abstract
When households are immobile across regions, the decentralized provision of public goods that generate interregional benefit spillovers is inefficiently low. We analyze this problem in an environment of interregional household migration and obtain the following results: if households are perfectly mobile across regions, then the decentralized provision of public goods generating spillovers is socially efficient. There is no need for central government intervention. If households are attached to particular regions for cultural or nationalistic reasons, however, at least one of two regions undersupplies the public good. Central government intervention can enhance efficiency. Key words: Household
JEL classification:
mobility;
Interregional
spillovers
H70, H77
1. Introduction Traditional economic reasoning suggests that the decentralized provision of public goods generating spillovers into neighboring regions is a typical field for central government intervention. Oates (1972), for example, argues that the provision of such goods is inefficiently low since regional governments ignore the well-being of non-residents. Therefore, a central government must provide a Pigovian subsidy to encourage the supply of public goods by regional governments, or the central government itself should provide these public goods to achieve the fiscal equivalence. Typically, 0047-2727/94/$07.00 0 1994 Elsevier SSDZ 0047-2727(94)05001-X
Science
B.V. All rights
reserved
168
D. Wellisch I Journal of Public Economics
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previous models in this area ignore any household mobility across regions.’ Another strand in the literature analyzes the decentralized supply of local public goods without spillovers in the presence of interregional household mobility (see, for example, Flatters et al., 1974; Stiglitz, 1977; and Boadway, 1982). Recent studies have concluded that a Nash equilibrium of competing regional governments is not only characterized by an optimal supply of public goods (Boadway, 1982), but also a socially efficient population distribution obtains, provided regions have some instrument for making interregional transfers (Myers, 1990). More recently, Mansoorian and Myers (1993) demonstrated that this conclusion remains true if households are attached to particular regions and are therefore imperfectly mobile. It is our purpose to integrate both models, i.e. to analyze the decentralized provision to public goods causing spillovers in an environment of household mobility, since both phenomena occur simultaneously. Revealing preferences by voting with one’s feet presumably characterizes typical federal economies like the United States or Germany. There are also many examples of public services provided by lower-level governments in these countries that generate spillovers across jurisdictions. Sewage treatment by an upstream city thereby reducing the need for purification by downstream cities is a prominent one. These phenomena also occur within the EU. One important public good that generates spillovers into surrounding member states is environmental quality. Germany’s efforts in recent years to clean the water of the river Rhine provides an example. The inhabitants of the downstream country, Holland, also benefit from this effort. The increasing degree of household mobility within the EU is made possible by the fact that the EU countries are legally committed to a common labor market. According to the Treaty of Rome (Articles 48 and 51), EU member states may not exclude citizens of other member states from employment and social rights and other benefits available to their own residents. Citizens of any EU country are legally entitled to work in any other member state and must be treated identically to native citizens with respect to taxation, transfers, access to education, and all other social benefits. While this is a rather de jure description of current circumstances, it is nonetheless true that the EU member states are drawing closer to a common labor market, and this trend will continue in the future. Hence there are practical reasons for integrating both approaches. One
these authors 1 Exceptions are Pauly (1970), Boskin (1973), and Gordon (1983). However, conclude that regional governments provide public goods generating spillovers inefficiently since they ignore the well-being of non-residents.
D. Wellisch I Journal of Public Economics
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169
should keep in mind, however, that the degree of household mobility is likely to differ in various federations. It is probably much higher in the United States or in Germany than within the EU or Canada since the latter federations consist of culturally diverse regions. There are different languages and the social attachments of households to particular regions are much stronger than in the United States or in Germany. In such federations, it is reasonable to assume that households have a preference for a particular region for cultural or nationalistic reasons. Therefore, we will analyze the decentralized provision of public goods generating spillovers in an environment of a mobile homogeneous population (referred to as perfect household mobility) which applies to such countries as the United States or Germany and in an environment of a mobile heterogeneous population (imperfect household mobility) which applies, for example, to the EU. In the latter case, the heterogeneity of the population will be characterized by different psychic attachments to particular regions within the federation (see DePalma and Papageorgiou, 1988, and Mansoorian and Myers, 1993). This difference in household mobility will prove to be important since our conclusions differ in both cases: decentralized provision of public goods generating spillovers in an environment of perfect household mobility is socially efficient since regions agree upon their objectives. The reason for agreeing is that the migration equilibrium, which is an important constraint for decentralized decision-making, is characterized in this case by equal utilities of households across regions. In the case of imperfect mobility, regions disagree about their objectives since the migration equilibrium can no longer be characterized by equal utilities in each region. At least one (of two) regions cannot achieve its desired resource distribution within the federation, and this region has no incentives to provide a socially efficient level of public goods generating spillovers. The decentralized Nash equilibrium is inefficient. This result underlines the importance of the degree of household mobility for the efficiency of decentralized decision-making if there are spillovers. The plan of the paper is as follows. Section 2 describes the model structure and derives the social optimum. Section 3 analyzes the decentralized decision-making of Nash-competing regional governments. It demonstrates that this Nash equilibrium is socially efficient in the perfect mobility case, but no longer if households are imperfectly mobile. Section 4 analyzes whether a central government can enhance efficiency by implementing an interregional transfer that deviates from the transfer performed by regions in the imperfect mobility case. Section 5 provides some conclusions and, finally, the appendix demonstrates that a region which would like to make a negative interregional transfer necessarily underprovides the public good and that there exist cases in which a deviation from the interregional transfer chosen by regions can enhance efficiency.
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2. Social optimum
We consider a federation which consists of two regions, indexed by i = 1, 2. In each region one group of mobile households resides. Households are heterogeneous, if at all, only with respect to their attachment to a region. We assume one household of each type, denoted by 12. IZ varies between 0 and N, the entire population of the federation. Furthermore, we assume that the utility function of households is additively separable with respect to attachment to a region (as in DePalma and Papageorgiou, 1988, and Mansoorian and Myers, 1993).2 Hence, the utility function of household II, V(n), can be characterized as V(n) =
u’(x~,G~,G~)+~(N-~), 1 ~~62, G,, G,) + an ,
ifthehouseholdliveinl, if the household lives in 2 .
(1)
ui(xi, G,, Gj) is a strictly quasi-concave sub-utility function, where x, is the consumption of the private good and Gi denotes the consumption of the public good provided by region i. The public good generates spillovers into the neighboring region. All residents in i have the same sub-utility ui(.). The parameter II measures the psychic benefit a household derives from living in region 2 and the parameter (N - n) the benefit from living in region 1. Households with relatively small 11’s are at home in region 1, while households with relatively large 11’s are at home in 2. The parameter a (a 2 0) expresses the degree of heterogeneity in tastes for a region (see DePalma and Papageorgiou, 1988, p. 41). For a = 0, there is no attachment to a region. Households are perfectly mobile across regions. The case of imperfect mobility is indicated by a >O. Neither the social planner nor regional governments can affect the psychic benefit a household derives from a particular region. Households are free to choose their region of residence. Since they differ in their attachment to a region, the migration equilibrium must be characterized by the marginal household, identified by n,, being indifferent between locating in either region. Households with II less than n, locate in regidn 1 and households with it greater than IZ~locate in region 2:
u’(x,,G~,G~)+~(N-~,)=u~(~,,G~,G~)+u~~, u~(x~,G~,G~)+u(N-~)>u*(x,,G~, u~(x~,G~,G~)+u(N-~)
G,)+un,
Vn
(2)
V’n>n,.
’ We employ this simplification for illustrative purposes only. None of our conclusions would change if we alternatively assumed that the utility function V(n) is weakly separable between attachment to a region and the sub-utility ui(.).
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171
Hence, n1 is also the number of households residing in region 1. If households are perfectly mobile (a = 0), (2) reduces to the standard equal utility migration equilibrium (see, for example, Wildasin, 1986). Each household is endowed with one unit of homogeneous labor which is supplied in the region of residence. Furthermore, we assume a concave production function for the private good in region i, f’(T,, n,), where Ti is the quantity of the immobile production factor in i, say land. The private good in i can alternatively be used as a public good, Gi. Hence, MRT = 1; thus, Gi denotes the costs of providing Gi units of the public good in region i. The social planner is constrained by free locational choices of households and therefore faces (2) as a constraint. Thus, we are only interested in the efficient allocation that is compatible with free mobility of households. Besides the migration equilibrium, the social planner is also constrained by the feasibility restriction for the entire federation: f’(T,,n,)+f2(T2,N-n,)-n,x,-(IV-n,)x,-G,-G,=O.
(3)
Entire production must cover the consumption of private and public goods by all households living in the individual regions. The population constraint has already been inserted into (3). An efficient allocation is defined as a subset of feasible allocations [satisfying (2) and (3)] at which it is impossible to raise the utility of one household in the federation without reducing the utility of another household. For our problem, this means that the set of efficient allocations is obtained if it is impossible to increase ul(xl, G,, G2) without reducing u*(x,, G,, G,) and vice versa. Equivalently, we can characterize all possible efficient allocations by maximizing a linear combination of u1 and u*, or Su’(x,, G,, G2) + (1 - S)u*(x,, G,, G,) for 6 E [0,11. Although these subutilities ignore locational tastes, any locational changes that accompany a change in u1 and u* must further raise total utilities. This is a revealed preference argument: if a change in location did not increase utility, it would not be made. Thus, if 6~’ + (1 - 6)u2 is not maximized, the allocation cannot be Pareto efficient.3 Therefore, the set of efficient allocations can be found by solving the following problem for all 6 : 4 3 I thank Jay Wilson for suggesting this explanation to me. 4 Maximizing the function social welfare I,“‘[u’(X,,G1,Gz)+a(N-n)]dn+ ],“, [u2(x2, G,, G,) + an] dn subject to (2) and (3) yields the same condition for an optimal provision of public goods generating spillovers, but it does not indicate an efficient allocation since it determines a unique population distribution for a given optimal allocation {G,, G2}. However, there is a range of efficient population distributions for which a redistribution of households across regions increases the utilities of residents in one region only at the expense of the utilities of residents in the other region.
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D. Wellisch I Journal of Public Economics
maximize
6u’(x,, G,, G,) + (1
-+*(x2,
55 (1994) 167-184
G,,
G,)
(x~xz.G,,Gz.n,)
subject to (2) and (3) .
(4)
When there is perfect mobility (a = 0) for any 6 the problem collapses to the standard efficiency problem consistent with perfect mobility (see Wildasin, 1986). Maximizing u1 in the case of attachment to a region does not mean maximizing U* since both utilities differ with respect to the planner’s choice variable II 1, as the migration equilibrium (2) indicates. Therefore, the efficient allocation consists of a range of allocations corresponding to all 6 E [0,11. Defining A, and A, as the Lagrange multipliers associated with, respectively, the migration equilibrium constraint (2) and the feasibility constraint (3), we achieve the following first-order conditions (with instruments being optimized shown in parentheses): (x1): [6 +
A&; -
(G,): [6 + A&;,
h,n,
= 0
(54
)
+ [(l - S) - A&& - A, = 0,
[( 1 - S) - A,]uZ - A,n, = 0 ,
(x,):
(G2): [S + A&&
@I>:-24
+ [(l - 6) - A,]&
(5b)
(54
- A, = 0 ,
(54
+ A,[(ff,-xl> -
where the subscripts indicate partial derivatives. For example, L& = &l aGi. Inserting both (5a) and (5~) into (5b) and (5d), respectively, yields as a necessary condition for the socially efficient provision of public goods: I
UG
n,++ni+=l UX
U'G
UX
fori,j=1,2,i#j.
(6)
Expression (6) is the familiar Samuelson Rule in the case of public good spillovers (see, for example, Oates, 1972, and Boadway and Wildasin, 1984). The sum of the MRS of the federation’s entire population between the consumption of the private and the public good must be equal to the marginal costs (the MRT) of providing the public good (here unity). The Samuelson Rule (6) holds both for the perfect and the imperfect mobility
sa$e.~auaEiwq~ ~00% 3yqnd aql asueuy 01 saxel peaq paseq-amaprsal spa~~o:, ~uau~u_~a~o%p~~o$la~ ayL ‘t u! sluaplsal 01 Quo sanme TV ‘amaH .s!seq el!der, lad Ienba UI? uo uor8a.I $eq$ 30 swap!sal Icq 61~0 pauMo s! t uo$al u! pue~ )ey~ awnsse aM .:J!u - (!u “~)~j= !u 01 spuodsallo3 ? u! jual pm1 IIZJOJaql ‘amaH slyold ou wea 01 paumssc an SU.I~~ .yJ ‘Impold ~s.I~S~~I SJ! 01 pmba a&M e loqel Aed (sfseq auo-ovauo e uo poo8 3yqnd c se pasn poo8 aleayd ag 8urmpold ‘T uo$?al u! SI.I.I.ILJ.suo$ial aq os[e UKI qcq~) qloq 103 suog~puo3 agl!ladtuoD aumssc am ‘2uglas pazqc.waDap mo UI um!Jq!l!nba
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D. Wellisch I Journal of Public Economics
174
55 (1994) 167-184
spillovers, Gi, and a non-negative interregional transfer, Z, 2 0.5 Using these assumptions, the feasibility constraint for region i becomes fi(T,,ni)-Gi-(Zij-Zji)-nixi=
fori, j=1,2,i#j.
(8)
Regional governments will be assumed to maximize the utility of a representative resident, ui(xi, G, , Gj). As the psychic benefit each household derives from residing in region i is a parameter, the regional government in effect maximizes the utility of each household in the region. We assume a decentralized equilibrium of the Nash-Cournot type. More precisely, in choosing {Z,, Gi} , the regional government in i takes {Zji, Gj} as given.6 Regional authorities cannot directly control the migration decisions of households. This means that the migration equilibrium (2) is an important constraint in their maximization problem. Inserting (8) for xi into (2) shows that the migration equilibrium condition, (2), determines ni as an implicit function of the regional control variables: n, = h(ZLj, Zji, Gi, Gj) .
(9)
Region i’s problem is to maximize ui (Zi,%C,)
fi(Ti, ni) -
Gi - (Zij - Z,i) 9 Gi>
Gj,)
(10)
2
Iii
where n, depends on Zil and Gi. The first-order conditions are
au’ -=
I
an.
--2+u:,&~O,Z,r0 azij ‘I fori,j=1,2,i#j, I
and
ad
Z. -= 11az,
0 (11)
5 Instead of assuming that ex post residence confers land ownership, we could alternatively suppose that households are endowed with TLIN units of land in each region. Regional governments could finance public goods by levying residence-based head taxes and sourcebased taxes on land rents. The land rent tax then serves as an indirect interregional transfer instrument. Using this alternative approach and playing the Nash game with the land rent tax and the public good yields the same results as playing the Nash game with Z,, and G, and using the land ownership assumption in our main text. ’ There are, however, alternative specifications of the Nash game. With governments playing a Nash game in head taxes and interregional transfers, public good levels would adjust to balance the governments’ budgets. If households were perfectly mobile or the public good did not generate interregional spillovers, the equilibria of this alternative Nash game would not differ from the ones analyzed in the basic text. If households are imperfectly mobile and public goods generate spillovers, we found that the equilibrium using this game as well as the equilibrium in the main text are inefficient. However, both equilibria differ since we are in a second-best environment. A similar point has been made by Wildasin (1988) in a model of capital tax competition.
D. Wellisch I Journal of Public Economics
ad
-= dGi
I
ani
-?+ui, ,
I
+&x=0
I
55 (1994) 167-184
fori, j=l,2,i#j,
175
(12)
where U’ = (u:lni)(fl, -xi). We assume an interior solution for Gi. Implic:‘; differentiation of (9) yields the perceived migration responses:’ I
2+2
an,=
’
azi,
D
Ui,
an.
I=
8Gi
KM
i
’
fori, j=l,2,i#j,
(13)
ui,+ui, I
D
’ fori, j=1,2,i#j,
(14)
where D = u:,, + uLn - a. Stability of the migration equilibrium requires D to be negative (see Boadway, 1982).’ Then, according to (13), the population size in region i decreases with increased interregional transfers from i to j. Our basic conclusion in this section will be that a region has incentives to provide the public good according to the Samuelson Rule if it achieves its desired interregional resource distribution. Hence, it is important to know whether regions achieve their desired resource distribution. We begin by analyzing this issue. Inserting the migration response (13) into (11) and rearranging, the first-order condition for choosing Z, becomes
Bringing the first-order
conditions for both regions together
yields
(16)
‘The migration responses are perceived rather than actual to achieve consistency with the regional government’s conjecture that the policy actions of the other government are given. ’ In the perfect mobility case, instability arises for an underpopulated federation (see Stiglitz, 1977, and Boadway and Flatters, 1982). If a socially efficient population distribution can be achieved in this case, stability means that the equalized marginal net benefit of mobile households to both regions, fi --x1, must be negative. Attachment to a region improves the conditions for a stable interior migration equilibrium. Given the population distribution and the consumption bundles, the denominator D is smaller in the case of attachment to a region (a > 0). Therefore, ties to particular regions can preclude complete depopulation of regions that would otherwise be possible (see Boadway and Flatters, 1982).
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D. Wellisch I Journal of Public Economics
55 (1994) 167-184
A comparison of (16) and (7) indicates that decentralized decisions of regional governments achieve the first-order condition for a socially efficient population distribution, no matter if households are perfectly mobile (a = 0) or are attached to a particular region (a > 0). Proposition 1. The decentralized Nash equilibrium is characterized by the first-order condition for a socially efjicient population distribution across regions. If region i’s first-order condition with respect to 2,. holds as a strict inequality, then we say that this region is transfer-constrained since it would like to perform a negative interregional transfer, but is restricted from doing 2, 2 0. We can state Proposition 1 so by the non-negativity constraint, without knowing whether regions are transfer-constrained. There is a range of efficient allocations. If both regions were transfer-constrained, then no region would achieve its desired interregional resource distribution. Ignoring inefficiencies associated with the public good supply, we would have an efficient allocation for which the welfare weight 6 satisfies 0 < S < 1. If region l(2) were not transfer-constrained, then we would have an efficient allocation with 6 = 1 (6 = 0). In the imperfect mobility case, we cannot exclude that both regions are transfer-constrained. Take, for example, the case of two identical regions. Then, the terms between the two inequalities of (16) sum to zero in a symmetric equilibrium. The two inequalities are strict implying that both regions are transfer-constrained. At least one region must be transferconstrained. To see this, assume that region i is not transfer-constrained, Inserting this into the first-order condition (fi--r,)-(f:-x,)=2anj/u~. for region j, (15), it follows that -2anjlu’, <2anilu:. This is necessarily a transfer-constrained corner solution. If one region is not transfer-constrained, then the equilibrium is at the boundary of the set of efficient allocations between both regions (if we further ignore inefficiencies associated with the provision of public goods). In this case, the non-transfer-constrained region performs an interregional transfer that solves the welfare maximization problem given by (4) for the case where its welfare weight is equal to 1. This equilibrium is efficient, but is suboptimal under any social welfare function where all utilities have a positive weight. This could provide a strong incentive for a central government to force the non-transfer-constrained region to increase its interregional transfer. The resulting interregional redistribution of resources is still efficient, but is no longer voluntarily performed by any region. According to its welfare weight 6, a central government can choose any efficient distribution of resources by forcing one region to make the supporting interregional transfer.
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177
In the perfect mobility case (a = 0), both regions are not transfer-constrained [see (16)] and achieve their desired interregional resource distribution. We can summarize these findings by Proposition 2. Zf households are imperfectly mobile across regions, then at least one region must be transfer-constrained. Zf households are perfectly mobile, then no region is transfer-constrained.
In the perfect mobility case, regional governments face the equal utility migration equilibrium as a constraint for their decisions. Maximizing the utility of residents in effect means maximizing the utilities of non-residents too. Hence, both governments have the same objectives and therefore agree upon the distribution of resources across regions that maximizes the common utility level of all households in the federation. Agreeing upon interregional resource distribution means agreeing upon the net interregional transfer. That is why both regions are not transfer-constrained in this case. For imperfectly mobile households, the migration equilibrium (2) indicates that maximizing the utility of residents by a regional government no longer means maximizing the utilities of non-residents. Both regions disagree about the population distribution ni. They no longer have the same objectives as in the perfect mobility case. Since the population size is determined by the net interregional transfer [see (13)], both regions disagree over this transfer. Consequently, at least one region would like to make a negative interregional transfer, but is bound by the non-negativity constraint. This region is not able to influence the resources of the federation like a social planner (or a central government). We emphasize this conclusion because it has important implications for the regional provision of public goods generating spillovers, to which we turn now. Inserting the migration response (14) into (12) and rearranging yields I
UGi nii+nj+
UX
U’G ux
(
fb-q
1 =l
f;-Xj-3 X
fori,j=1,2,i#j.
(17)
Comparing (17) with (15), it is clear that the Samuelson Rule can only be achieved if a region is not transfer-constrained. In this case, the second term on the LHS of (17) corresponds to the MRS of non-residents for providing an additional unit of the public good. Applying Proposition 1 and Proposition 2 to this result, we can state our basic conclusion: Proposition
3. Zf households
are imperfectly
mobile
and regions provide
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D. Wellisch / Journal of Public Economics
55 (1994) 167-184
public goods generating interregional spillovers, then the decentralized Nash equilibrium is inejjicient. If households are perfectly mobile, however, the Nash equilibrium is socially efficient.
Only with a perfectly mobile homogeneous population do regions have incentives for complete policy coordination. In considering the equal utility migration equilibrium as a constraint, both regions agree upon their objectives and this dominates inefficiencies associated with public good spillovers. When households are attached to their home region, at least one region is transfer-constrained and this region has no incentives to provide public goods according to the Samuelson Rule if there are spillovers. We demonstrate in Section A.1 of the appendix that a transfer-constrained region necessarily undersupplies the public good generating spillovers relative to the Samuelson criterion. Since at least one region fails to provide an efficient level of the public good, it is of limited guidance for a central government that the Nash equilibrium is characterized by the first-order condition for a socially efficient population distribution (even if the central government were to accept all efficient allocations). This is a second-best argument. Since at least one region fails to provide public goods according to the Samuelson Rule, a central government might be able to enhance efficiency by implementing an interregional transfer that deviates from the transfer chosen by regions, if doing so improves the efficiency of public good provision. We explore in the next section whether such cases exist.
4. Interregional
transfer
by a central government
We now assume that public good supplies are left in the hands of regional governments and a central government controls the net interregional transfer, S,, = Z,, - Z,,.9 Regions take S,, as given when deciding on the supply of public goods. In a Nash equilibrium (playing {Gi}) they provide public goods according to (17). The first-order conditions (17) for both regions determine Gi as an implicit function of S,,: 9 The issue of the appropriate governmental level to perform an optimal interregional transfer is addressed in Hartwick (1980), Boadway and Flatters (1982), Myers (1990) and Mansoorian and Myers (1993). We extend this discussion to the case of imperfect mobility and spillovers in the public good. We should mention, however, that we are still in a second-best economy. Clearly, the best response of a central government would be to leave the interregional transfer and the public good provision in the hands of regional governments and to encourage the public good supply of a transfer-constrained region by a Pigovian subsidy. A socially efficient outcome would then be possible.
D. Wellisch I Journal of Public Economics
G,=I(S,*)
55 (1994) 167-184
fori=1,2.
179
(18)
The central government cannot interfere with locational choices of households and has to take (2) as a constraint. Inserting (8) for xi into (2) determines ni as an implicit function of the central and regional control variables: ni = m(S,,, Gi, G,) .
(19)
Since we want to analyze whether an interregional transfer that deviates from the one chosen by regional governments can enhance efficiency, the central government must solve, for all 6 E [0,11, the problem maximize SU’ (SlZ)
(f1(L4-G,-%
(1 - 6)U2
+
,G
G
1,
n1
f’(7’2, (
N
-
nl)
-
G,
+
2
>
Sl,
,G,,G,),
N-n,
where G, and G, depend on S,, and n1 depends on G,, G,, and S,,. The first-order condition becomes [using (12)]:
(21) The expression which is multiplied by 6 characterizes the change in u1, and the term which is multiplied by (1 - 6) is the first-order impact on u2. From (14) and (12), we have ub,(anjlaGj) - z& = 2a(anjlaGj). Inserting this and the migration response an,laS,, set out in (13) into (21) and allowing 6 to vary from 0 to 1, we obtain _- 2an, 4
<-
2an, 4
n2 l+Du,‘aGl
--an, dG,
dS,,
(22)
where for 6 = 1 the first relation holds as an equality, and for 6 = 0 the second relation holds as an equality. According to (14), the migration responses anJaG,, i = 1, 2, do not generally vanish. The responses dG,/dS,,, i = 1, 2, can be derived from the first-order conditions (17) taking the migration responses into account. Even when using second-order conditions for the regional choice of Gj, i = 1, 2, and stability conditions of the Nash equilibrium, (17), I was unable to derive
180
D. Wellisch I Journal of Public Economics
55 (1994) 167-184
clear-cut results for the signs of dG,/dS,,, i = 1, 2, in the general case. All that one can say is that they do not vanish in general. D is negative by stability of the migration equilibrium. Hence, the interval defined by (22) does not coincide with the one set out in (16) where the regions choose the transfer. It may well be the case that these two intervals intersect, suggesting that there may exist cases in which the transfer chosen by regional governments is efficient. Although it is impossible to derive general results, we can demonstrate, by considering a simple example, that there exist cases in which a deviation from the transfer chosen by regions will enhance efficiency. Suppose that region 1 is not transfer-constrained and that G, does not generate spillovers. In this situation, region 2 must be transfer-constrained and its provision of G,, causing interregional spillovers, is inefficiently low relative to the Samuelson criterion (see Section A.1 in the appendix). Let us further assume that a marginal transfer to region 2 increases its supply of G,, dG,/dS,, > 0. This will be the case if the utility function is additively separable with respect to all arguments.” To see that a marginal transfer improves efficiency, consider first the change in U* set out in (21). Following (14) and (17), an,laG, = 0 if 2 = 0. According to (21), the change in u2 reduces to (utln,) - u~,(&z,/ Q* c?S,~). This expression must be positive since we consider a situation in which region 2 is transfer-constrained [see (ll)]. Therefore, u2 increases in response to a higher transfer. Now, consider the change in u1 set out in (21). Taking into account that region 1 is not transfer-constrained, -(~‘ln,) + ui,(dn, /as,,) = 0, and using (12) and (14), the entire effect on U’ becomes -2a(an,/aG,)(dG,/ dS,,). We demonstrate in Section A.2 of the appendix that an,laG, is negative. Hence, the effect on u1 is positive if dG,/dS,, > 0. Therefore, if the interregional transfer improves the efficiency of public good provision by a transfer-constrained region, there are cases in which a deviation from the transfer chosen by regions can be recommended for efficiency reasons. Such cases do not exist if there is no attachment (a = 0) or if public goods do not generate spillovers. If there is no attachment (a = 0), (22) indicates that S,, is equal to the net interregional transfer chosen by regions in a decentralized Nash equilibrium [see (16)]. Both regions supply public goods generating spillovers according to the Samuelson Rule. Hence, we are in a first-best economy and the central government should not violate the first-best allocation of labor. Even if we have attachment, but public goods do not generate spillovers (the case analyzed by Mansoorian and Myers, 1993), a central government cannot increase efficiency by deviating from the interregional transfer chosen by regions. To see this, note that, according to “A
proof is available from the author upon request.
D. Wellisch I Journal of Public Economics
55 (1994) 167-184
181
(17), both regions achieve the Samuelson condition of public good supply (uij = 0). According to (14), the migration response ani/aGi vanishes at an equilibrium. Thus, (22) reduces to (16). The decentralized Nash equilibrium is socially efficient and a central government has no incentive to change this allocation for efficiency reasons.
5. Conclusions
This paper analyzed the regional provision of public goods that generate interregional spillovers. We modeled alternative degrees of household mobility and derived that the decentralized Nash equilibrium is socially efficient when households are perfectly mobile. If households are attached to their home region, at least one region undersupplies the public good relative to the Samuelson Rule. The decentralized Nash equilibrium is inefficient. Hence, the result of Manssorian and Myers (1993) that the efficiency of decentralized decisions can be extended to an environment of imperfect household mobility breaks down if there are spillovers in the provision of public goods. Although regions achieve an ‘efficient’ population distribution by choosing interregional transfers, there exist cases in which a central government can enhance efficiency by deviating from this transfer. The conclusions derived in this paper also apply to other policy considerations. For instance, one can demonstrate regional environmental policy to be socially efficient in the perfect mobility case even if there are negative external effects harming neighboring jurisdictions. Regions conducting environmental policy take into account their damage to non-residents due to their migration responses (see Wellisch, 1992). However, if households are attached to a particular region, this only holds for a region that is not transfer-constrained. Furthermore, an extension to the multi-region case for perfect mobility is straightforward and is not found to alter any of the basic conclusions (see Wellisch, 1991). An extension of the heterogeneous population case in this direction is rather difficult and is a task for future research.
Acknowledgements
I am ments. helpful TAPES Gordon
indebted to David Wildasin for encouragement and helpful comI also thank Gordon Myers, Wolfram Richter, and Uwe Walz for discussion and correspondence. This paper was presented at the 1992 conference in Munich. I would like to thank the organizers, Roger and Hans-Werner Sinn, for the invitation and Harris Schlesinger
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D. Wellisch I Journal of Public Economics
~(!$+~)-~a-~<~ x
I
or
183
55 (1994) 167-184
O
(A-5)
x
which is a contradiction. Therefore, if region i is transfer-constrained, undersupplies the public good. Q.E.D. A .2. Central government
it
intervention
Here we prove that, in a situation in which region 1 is not transferconstrained and the provision of G, does not generate spillovers, a central government can enhance efficiency by increasing the transfer if dGZ/dSi2 is positive. Since u2 increases and the first-order impact on u1 is -2a(&z,l =,)(dG/d%) (see Section 4), we only have to demonstrate that an,laG, is negative. Inserting (12) into (14), we get
G4.6) or, using the definition of D: 1
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64.7)
’
Following (A.7), we must demonstrate that uin - 2a is negative. If region 1 is not transfer-constrained, (15) yields fi - x2 =f,‘, -x1 + (2an,lut). Inserting this into the condition for a stable migration equilibrium yields (f:-X,)($+-$)-2a+3<0
or
fi-Xi
(A-8)
Therefore, ui, - 2a = (u~ln,)(ffi -x1) - 2a must be negative and, according to (A.73, an, / aG, must also be negative. Q.E.D.
References Boadway, R.W., 1982, On the method of taxation and the provision of local public goods: Comment, American Economic Review 72, 846-851. Boadway, R.W. and F.R. Flatters, 1982, Efficiency and equalization payments in a federal system of governments: A synthesis and extension of recent results, Canadian Journal of Economics 15, 613-633. Boadway, R.W. and D.E. Wildasin, 1984, Public sector economics, 2nd edn. (Little Brown, Boston).
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D. Wellisch / Journal of Public Economics 55 (1994) 167-184
Boskin, M.J., 1973, Local government tax and product competition and the optimal provision of public goods, Journal of Political Economy 81, 203-210. DePalma, A. and Y.Y. Papageorgiou, 1988, Heterogeneity in tastes and urban structures, Regional Science and Urban Economics 18, 37-56. Flatters, F.R., V.J. Henderson and P. Mieszkowski, 1974, Public goods, efficiency and regional equalization, Journal of Public Economics 3, 99-112. Gordon, R.H., 1983, An optimal taxation approach to fiscal federalism, Quarterly Journal of Economics 98, 567-586. Hartwick, J.M., 1980, Henry George rule, optimal population, and interregional equity, Canadian Journal of Economics 13, 695-700. Mansoorian, A. and G.M. Myers, 1993, Attachment to home and efficient purchases of population in a fiscal externality economy, Journal of Public Economics 52, 117-132. Myers, G.M., 1990, Optimality, free mobility, and the regional authority in a federation, Journal of Public Economics 43, 107-121. Oates, W.E., 1972, Fiscal federalism (Hartcourt Brace Jovanovich, New York). Pauly, M.V., 1970, Optimality, public goods, and local governments: A general theoretical analysis, Journal of Political Economy 78, 572-584. 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) 247-333. Wellisch, D., 1991, On the decentralized provision of public goods with spillovers in the presence of interregional migration, Ttibinger Diskussionsbeitrlge No. 19, Tiibinger University. Wellisch, D., 1992, Dezentrale Umweltpolitik, Mobilitat von Kapital, Haushalten und Firmen und grenziiberschreitende Umweltschlden, Zeitschrift fiir Umweltpolitik 15, 433-458. Wildasin, D.E., 1986, Urban public finance (Harwood, New York). Wilasdin, D.E., Nash equilibrium in Models of Fiscal competition, Journal of Public Economics 35, 229-240.
Journal
of Public
Economics
55 (1994)
167-184
Interregional spillovers in the presence of perfect and imperfect household mobility Dietmar
Well&h
Vniversitiit Dortmund, WiSo-Fakultiit, Lehrstuhl VWL II, 44221 Dortmund, Germany Received
August
1992, final version
received
August
1993
Abstract
When households are immobile across regions, the decentralized provision of public goods that generate interregional benefit spillovers is inefficiently low. We analyze this problem in an environment of interregional household migration and obtain the following results: if households are perfectly mobile across regions, then the decentralized provision of public goods generating spillovers is socially efficient. There is no need for central government intervention. If households are attached to particular regions for cultural or nationalistic reasons, however, at least one of two regions undersupplies the public good. Central government intervention can enhance efficiency. Key words: Household
JEL classification:
mobility;
Interregional
spillovers
H70, H77
1. Introduction Traditional economic reasoning suggests that the decentralized provision of public goods generating spillovers into neighboring regions is a typical field for central government intervention. Oates (1972), for example, argues that the provision of such goods is inefficiently low since regional governments ignore the well-being of non-residents. Therefore, a central government must provide a Pigovian subsidy to encourage the supply of public goods by regional governments, or the central government itself should provide these public goods to achieve the fiscal equivalence. Typically, 0047-2727/94/$07.00 0 1994 Elsevier SSDZ 0047-2727(94)05001-X
Science
B.V. All rights
reserved
168
D. Wellisch I Journal of Public Economics
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previous models in this area ignore any household mobility across regions.’ Another strand in the literature analyzes the decentralized supply of local public goods without spillovers in the presence of interregional household mobility (see, for example, Flatters et al., 1974; Stiglitz, 1977; and Boadway, 1982). Recent studies have concluded that a Nash equilibrium of competing regional governments is not only characterized by an optimal supply of public goods (Boadway, 1982), but also a socially efficient population distribution obtains, provided regions have some instrument for making interregional transfers (Myers, 1990). More recently, Mansoorian and Myers (1993) demonstrated that this conclusion remains true if households are attached to particular regions and are therefore imperfectly mobile. It is our purpose to integrate both models, i.e. to analyze the decentralized provision to public goods causing spillovers in an environment of household mobility, since both phenomena occur simultaneously. Revealing preferences by voting with one’s feet presumably characterizes typical federal economies like the United States or Germany. There are also many examples of public services provided by lower-level governments in these countries that generate spillovers across jurisdictions. Sewage treatment by an upstream city thereby reducing the need for purification by downstream cities is a prominent one. These phenomena also occur within the EU. One important public good that generates spillovers into surrounding member states is environmental quality. Germany’s efforts in recent years to clean the water of the river Rhine provides an example. The inhabitants of the downstream country, Holland, also benefit from this effort. The increasing degree of household mobility within the EU is made possible by the fact that the EU countries are legally committed to a common labor market. According to the Treaty of Rome (Articles 48 and 51), EU member states may not exclude citizens of other member states from employment and social rights and other benefits available to their own residents. Citizens of any EU country are legally entitled to work in any other member state and must be treated identically to native citizens with respect to taxation, transfers, access to education, and all other social benefits. While this is a rather de jure description of current circumstances, it is nonetheless true that the EU member states are drawing closer to a common labor market, and this trend will continue in the future. Hence there are practical reasons for integrating both approaches. One
these authors 1 Exceptions are Pauly (1970), Boskin (1973), and Gordon (1983). However, conclude that regional governments provide public goods generating spillovers inefficiently since they ignore the well-being of non-residents.
D. Wellisch I Journal of Public Economics
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169
should keep in mind, however, that the degree of household mobility is likely to differ in various federations. It is probably much higher in the United States or in Germany than within the EU or Canada since the latter federations consist of culturally diverse regions. There are different languages and the social attachments of households to particular regions are much stronger than in the United States or in Germany. In such federations, it is reasonable to assume that households have a preference for a particular region for cultural or nationalistic reasons. Therefore, we will analyze the decentralized provision of public goods generating spillovers in an environment of a mobile homogeneous population (referred to as perfect household mobility) which applies to such countries as the United States or Germany and in an environment of a mobile heterogeneous population (imperfect household mobility) which applies, for example, to the EU. In the latter case, the heterogeneity of the population will be characterized by different psychic attachments to particular regions within the federation (see DePalma and Papageorgiou, 1988, and Mansoorian and Myers, 1993). This difference in household mobility will prove to be important since our conclusions differ in both cases: decentralized provision of public goods generating spillovers in an environment of perfect household mobility is socially efficient since regions agree upon their objectives. The reason for agreeing is that the migration equilibrium, which is an important constraint for decentralized decision-making, is characterized in this case by equal utilities of households across regions. In the case of imperfect mobility, regions disagree about their objectives since the migration equilibrium can no longer be characterized by equal utilities in each region. At least one (of two) regions cannot achieve its desired resource distribution within the federation, and this region has no incentives to provide a socially efficient level of public goods generating spillovers. The decentralized Nash equilibrium is inefficient. This result underlines the importance of the degree of household mobility for the efficiency of decentralized decision-making if there are spillovers. The plan of the paper is as follows. Section 2 describes the model structure and derives the social optimum. Section 3 analyzes the decentralized decision-making of Nash-competing regional governments. It demonstrates that this Nash equilibrium is socially efficient in the perfect mobility case, but no longer if households are imperfectly mobile. Section 4 analyzes whether a central government can enhance efficiency by implementing an interregional transfer that deviates from the transfer performed by regions in the imperfect mobility case. Section 5 provides some conclusions and, finally, the appendix demonstrates that a region which would like to make a negative interregional transfer necessarily underprovides the public good and that there exist cases in which a deviation from the interregional transfer chosen by regions can enhance efficiency.
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I Journal of Public Economics
55 (1994) 167-184
2. Social optimum
We consider a federation which consists of two regions, indexed by i = 1, 2. In each region one group of mobile households resides. Households are heterogeneous, if at all, only with respect to their attachment to a region. We assume one household of each type, denoted by 12. IZ varies between 0 and N, the entire population of the federation. Furthermore, we assume that the utility function of households is additively separable with respect to attachment to a region (as in DePalma and Papageorgiou, 1988, and Mansoorian and Myers, 1993).2 Hence, the utility function of household II, V(n), can be characterized as V(n) =
u’(x~,G~,G~)+~(N-~), 1 ~~62, G,, G,) + an ,
ifthehouseholdliveinl, if the household lives in 2 .
(1)
ui(xi, G,, Gj) is a strictly quasi-concave sub-utility function, where x, is the consumption of the private good and Gi denotes the consumption of the public good provided by region i. The public good generates spillovers into the neighboring region. All residents in i have the same sub-utility ui(.). The parameter II measures the psychic benefit a household derives from living in region 2 and the parameter (N - n) the benefit from living in region 1. Households with relatively small 11’s are at home in region 1, while households with relatively large 11’s are at home in 2. The parameter a (a 2 0) expresses the degree of heterogeneity in tastes for a region (see DePalma and Papageorgiou, 1988, p. 41). For a = 0, there is no attachment to a region. Households are perfectly mobile across regions. The case of imperfect mobility is indicated by a >O. Neither the social planner nor regional governments can affect the psychic benefit a household derives from a particular region. Households are free to choose their region of residence. Since they differ in their attachment to a region, the migration equilibrium must be characterized by the marginal household, identified by n,, being indifferent between locating in either region. Households with II less than n, locate in regidn 1 and households with it greater than IZ~locate in region 2:
u’(x,,G~,G~)+~(N-~,)=u~(~,,G~,G~)+u~~, u~(x~,G~,G~)+u(N-~)>u*(x,,G~, u~(x~,G~,G~)+u(N-~)
G,)+un,
Vn
(2)
V’n>n,.
’ We employ this simplification for illustrative purposes only. None of our conclusions would change if we alternatively assumed that the utility function V(n) is weakly separable between attachment to a region and the sub-utility ui(.).
D. Wellisch I Journal of Public Economics
55 (1994) 167-184
171
Hence, n1 is also the number of households residing in region 1. If households are perfectly mobile (a = 0), (2) reduces to the standard equal utility migration equilibrium (see, for example, Wildasin, 1986). Each household is endowed with one unit of homogeneous labor which is supplied in the region of residence. Furthermore, we assume a concave production function for the private good in region i, f’(T,, n,), where Ti is the quantity of the immobile production factor in i, say land. The private good in i can alternatively be used as a public good, Gi. Hence, MRT = 1; thus, Gi denotes the costs of providing Gi units of the public good in region i. The social planner is constrained by free locational choices of households and therefore faces (2) as a constraint. Thus, we are only interested in the efficient allocation that is compatible with free mobility of households. Besides the migration equilibrium, the social planner is also constrained by the feasibility restriction for the entire federation: f’(T,,n,)+f2(T2,N-n,)-n,x,-(IV-n,)x,-G,-G,=O.
(3)
Entire production must cover the consumption of private and public goods by all households living in the individual regions. The population constraint has already been inserted into (3). An efficient allocation is defined as a subset of feasible allocations [satisfying (2) and (3)] at which it is impossible to raise the utility of one household in the federation without reducing the utility of another household. For our problem, this means that the set of efficient allocations is obtained if it is impossible to increase ul(xl, G,, G2) without reducing u*(x,, G,, G,) and vice versa. Equivalently, we can characterize all possible efficient allocations by maximizing a linear combination of u1 and u*, or Su’(x,, G,, G2) + (1 - S)u*(x,, G,, G,) for 6 E [0,11. Although these subutilities ignore locational tastes, any locational changes that accompany a change in u1 and u* must further raise total utilities. This is a revealed preference argument: if a change in location did not increase utility, it would not be made. Thus, if 6~’ + (1 - 6)u2 is not maximized, the allocation cannot be Pareto efficient.3 Therefore, the set of efficient allocations can be found by solving the following problem for all 6 : 4 3 I thank Jay Wilson for suggesting this explanation to me. 4 Maximizing the function social welfare I,“‘[u’(X,,G1,Gz)+a(N-n)]dn+ ],“, [u2(x2, G,, G,) + an] dn subject to (2) and (3) yields the same condition for an optimal provision of public goods generating spillovers, but it does not indicate an efficient allocation since it determines a unique population distribution for a given optimal allocation {G,, G2}. However, there is a range of efficient population distributions for which a redistribution of households across regions increases the utilities of residents in one region only at the expense of the utilities of residents in the other region.
172
D. Wellisch I Journal of Public Economics
maximize
6u’(x,, G,, G,) + (1
-+*(x2,
55 (1994) 167-184
G,,
G,)
(x~xz.G,,Gz.n,)
subject to (2) and (3) .
(4)
When there is perfect mobility (a = 0) for any 6 the problem collapses to the standard efficiency problem consistent with perfect mobility (see Wildasin, 1986). Maximizing u1 in the case of attachment to a region does not mean maximizing U* since both utilities differ with respect to the planner’s choice variable II 1, as the migration equilibrium (2) indicates. Therefore, the efficient allocation consists of a range of allocations corresponding to all 6 E [0,11. Defining A, and A, as the Lagrange multipliers associated with, respectively, the migration equilibrium constraint (2) and the feasibility constraint (3), we achieve the following first-order conditions (with instruments being optimized shown in parentheses): (x1): [6 +
A&; -
(G,): [6 + A&;,
h,n,
= 0
(54
)
+ [(l - S) - A&& - A, = 0,
[( 1 - S) - A,]uZ - A,n, = 0 ,
(x,):
(G2): [S + A&&
@I>:-24
+ [(l - 6) - A,]&
(5b)
(54
- A, = 0 ,
(54
+ A,[(ff,-xl> -
where the subscripts indicate partial derivatives. For example, L& = &l aGi. Inserting both (5a) and (5~) into (5b) and (5d), respectively, yields as a necessary condition for the socially efficient provision of public goods: I
UG
n,++ni+=l UX
U'G
UX
fori,j=1,2,i#j.
(6)
Expression (6) is the familiar Samuelson Rule in the case of public good spillovers (see, for example, Oates, 1972, and Boadway and Wildasin, 1984). The sum of the MRS of the federation’s entire population between the consumption of the private and the public good must be equal to the marginal costs (the MRT) of providing the public good (here unity). The Samuelson Rule (6) holds both for the perfect and the imperfect mobility
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D. Wellisch I Journal of Public Economics
174
55 (1994) 167-184
spillovers, Gi, and a non-negative interregional transfer, Z, 2 0.5 Using these assumptions, the feasibility constraint for region i becomes fi(T,,ni)-Gi-(Zij-Zji)-nixi=
fori, j=1,2,i#j.
(8)
Regional governments will be assumed to maximize the utility of a representative resident, ui(xi, G, , Gj). As the psychic benefit each household derives from residing in region i is a parameter, the regional government in effect maximizes the utility of each household in the region. We assume a decentralized equilibrium of the Nash-Cournot type. More precisely, in choosing {Z,, Gi} , the regional government in i takes {Zji, Gj} as given.6 Regional authorities cannot directly control the migration decisions of households. This means that the migration equilibrium (2) is an important constraint in their maximization problem. Inserting (8) for xi into (2) shows that the migration equilibrium condition, (2), determines ni as an implicit function of the regional control variables: n, = h(ZLj, Zji, Gi, Gj) .
(9)
Region i’s problem is to maximize ui (Zi,%C,)
fi(Ti, ni) -
Gi - (Zij - Z,i) 9 Gi>
Gj,)
(10)
2
Iii
where n, depends on Zil and Gi. The first-order conditions are
au’ -=
I
an.
--2+u:,&~O,Z,r0 azij ‘I fori,j=1,2,i#j, I
and
ad
Z. -= 11az,
0 (11)
5 Instead of assuming that ex post residence confers land ownership, we could alternatively suppose that households are endowed with TLIN units of land in each region. Regional governments could finance public goods by levying residence-based head taxes and sourcebased taxes on land rents. The land rent tax then serves as an indirect interregional transfer instrument. Using this alternative approach and playing the Nash game with the land rent tax and the public good yields the same results as playing the Nash game with Z,, and G, and using the land ownership assumption in our main text. ’ There are, however, alternative specifications of the Nash game. With governments playing a Nash game in head taxes and interregional transfers, public good levels would adjust to balance the governments’ budgets. If households were perfectly mobile or the public good did not generate interregional spillovers, the equilibria of this alternative Nash game would not differ from the ones analyzed in the basic text. If households are imperfectly mobile and public goods generate spillovers, we found that the equilibrium using this game as well as the equilibrium in the main text are inefficient. However, both equilibria differ since we are in a second-best environment. A similar point has been made by Wildasin (1988) in a model of capital tax competition.
D. Wellisch I Journal of Public Economics
ad
-= dGi
I
ani
-?+ui, ,
I
+&x=0
I
55 (1994) 167-184
fori, j=l,2,i#j,
175
(12)
where U’ = (u:lni)(fl, -xi). We assume an interior solution for Gi. Implic:‘; differentiation of (9) yields the perceived migration responses:’ I
2+2
an,=
’
azi,
D
Ui,
an.
I=
8Gi
KM
i
’
fori, j=l,2,i#j,
(13)
ui,+ui, I
D
’ fori, j=1,2,i#j,
(14)
where D = u:,, + uLn - a. Stability of the migration equilibrium requires D to be negative (see Boadway, 1982).’ Then, according to (13), the population size in region i decreases with increased interregional transfers from i to j. Our basic conclusion in this section will be that a region has incentives to provide the public good according to the Samuelson Rule if it achieves its desired interregional resource distribution. Hence, it is important to know whether regions achieve their desired resource distribution. We begin by analyzing this issue. Inserting the migration response (13) into (11) and rearranging, the first-order condition for choosing Z, becomes
Bringing the first-order
conditions for both regions together
yields
(16)
‘The migration responses are perceived rather than actual to achieve consistency with the regional government’s conjecture that the policy actions of the other government are given. ’ In the perfect mobility case, instability arises for an underpopulated federation (see Stiglitz, 1977, and Boadway and Flatters, 1982). If a socially efficient population distribution can be achieved in this case, stability means that the equalized marginal net benefit of mobile households to both regions, fi --x1, must be negative. Attachment to a region improves the conditions for a stable interior migration equilibrium. Given the population distribution and the consumption bundles, the denominator D is smaller in the case of attachment to a region (a > 0). Therefore, ties to particular regions can preclude complete depopulation of regions that would otherwise be possible (see Boadway and Flatters, 1982).
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D. Wellisch I Journal of Public Economics
55 (1994) 167-184
A comparison of (16) and (7) indicates that decentralized decisions of regional governments achieve the first-order condition for a socially efficient population distribution, no matter if households are perfectly mobile (a = 0) or are attached to a particular region (a > 0). Proposition 1. The decentralized Nash equilibrium is characterized by the first-order condition for a socially efjicient population distribution across regions. If region i’s first-order condition with respect to 2,. holds as a strict inequality, then we say that this region is transfer-constrained since it would like to perform a negative interregional transfer, but is restricted from doing 2, 2 0. We can state Proposition 1 so by the non-negativity constraint, without knowing whether regions are transfer-constrained. There is a range of efficient allocations. If both regions were transfer-constrained, then no region would achieve its desired interregional resource distribution. Ignoring inefficiencies associated with the public good supply, we would have an efficient allocation for which the welfare weight 6 satisfies 0 < S < 1. If region l(2) were not transfer-constrained, then we would have an efficient allocation with 6 = 1 (6 = 0). In the imperfect mobility case, we cannot exclude that both regions are transfer-constrained. Take, for example, the case of two identical regions. Then, the terms between the two inequalities of (16) sum to zero in a symmetric equilibrium. The two inequalities are strict implying that both regions are transfer-constrained. At least one region must be transferconstrained. To see this, assume that region i is not transfer-constrained, Inserting this into the first-order condition (fi--r,)-(f:-x,)=2anj/u~. for region j, (15), it follows that -2anjlu’, <2anilu:. This is necessarily a transfer-constrained corner solution. If one region is not transfer-constrained, then the equilibrium is at the boundary of the set of efficient allocations between both regions (if we further ignore inefficiencies associated with the provision of public goods). In this case, the non-transfer-constrained region performs an interregional transfer that solves the welfare maximization problem given by (4) for the case where its welfare weight is equal to 1. This equilibrium is efficient, but is suboptimal under any social welfare function where all utilities have a positive weight. This could provide a strong incentive for a central government to force the non-transfer-constrained region to increase its interregional transfer. The resulting interregional redistribution of resources is still efficient, but is no longer voluntarily performed by any region. According to its welfare weight 6, a central government can choose any efficient distribution of resources by forcing one region to make the supporting interregional transfer.
D. Wellisch I Journal of Public Economics 55 (1994) 167-184
177
In the perfect mobility case (a = 0), both regions are not transfer-constrained [see (16)] and achieve their desired interregional resource distribution. We can summarize these findings by Proposition 2. Zf households are imperfectly mobile across regions, then at least one region must be transfer-constrained. Zf households are perfectly mobile, then no region is transfer-constrained.
In the perfect mobility case, regional governments face the equal utility migration equilibrium as a constraint for their decisions. Maximizing the utility of residents in effect means maximizing the utilities of non-residents too. Hence, both governments have the same objectives and therefore agree upon the distribution of resources across regions that maximizes the common utility level of all households in the federation. Agreeing upon interregional resource distribution means agreeing upon the net interregional transfer. That is why both regions are not transfer-constrained in this case. For imperfectly mobile households, the migration equilibrium (2) indicates that maximizing the utility of residents by a regional government no longer means maximizing the utilities of non-residents. Both regions disagree about the population distribution ni. They no longer have the same objectives as in the perfect mobility case. Since the population size is determined by the net interregional transfer [see (13)], both regions disagree over this transfer. Consequently, at least one region would like to make a negative interregional transfer, but is bound by the non-negativity constraint. This region is not able to influence the resources of the federation like a social planner (or a central government). We emphasize this conclusion because it has important implications for the regional provision of public goods generating spillovers, to which we turn now. Inserting the migration response (14) into (12) and rearranging yields I
UGi nii+nj+
UX
U’G ux
(
fb-q
1 =l
f;-Xj-3 X
fori,j=1,2,i#j.
(17)
Comparing (17) with (15), it is clear that the Samuelson Rule can only be achieved if a region is not transfer-constrained. In this case, the second term on the LHS of (17) corresponds to the MRS of non-residents for providing an additional unit of the public good. Applying Proposition 1 and Proposition 2 to this result, we can state our basic conclusion: Proposition
3. Zf households
are imperfectly
mobile
and regions provide
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D. Wellisch / Journal of Public Economics
55 (1994) 167-184
public goods generating interregional spillovers, then the decentralized Nash equilibrium is inejjicient. If households are perfectly mobile, however, the Nash equilibrium is socially efficient.
Only with a perfectly mobile homogeneous population do regions have incentives for complete policy coordination. In considering the equal utility migration equilibrium as a constraint, both regions agree upon their objectives and this dominates inefficiencies associated with public good spillovers. When households are attached to their home region, at least one region is transfer-constrained and this region has no incentives to provide public goods according to the Samuelson Rule if there are spillovers. We demonstrate in Section A.1 of the appendix that a transfer-constrained region necessarily undersupplies the public good generating spillovers relative to the Samuelson criterion. Since at least one region fails to provide an efficient level of the public good, it is of limited guidance for a central government that the Nash equilibrium is characterized by the first-order condition for a socially efficient population distribution (even if the central government were to accept all efficient allocations). This is a second-best argument. Since at least one region fails to provide public goods according to the Samuelson Rule, a central government might be able to enhance efficiency by implementing an interregional transfer that deviates from the transfer chosen by regions, if doing so improves the efficiency of public good provision. We explore in the next section whether such cases exist.
4. Interregional
transfer
by a central government
We now assume that public good supplies are left in the hands of regional governments and a central government controls the net interregional transfer, S,, = Z,, - Z,,.9 Regions take S,, as given when deciding on the supply of public goods. In a Nash equilibrium (playing {Gi}) they provide public goods according to (17). The first-order conditions (17) for both regions determine Gi as an implicit function of S,,: 9 The issue of the appropriate governmental level to perform an optimal interregional transfer is addressed in Hartwick (1980), Boadway and Flatters (1982), Myers (1990) and Mansoorian and Myers (1993). We extend this discussion to the case of imperfect mobility and spillovers in the public good. We should mention, however, that we are still in a second-best economy. Clearly, the best response of a central government would be to leave the interregional transfer and the public good provision in the hands of regional governments and to encourage the public good supply of a transfer-constrained region by a Pigovian subsidy. A socially efficient outcome would then be possible.
D. Wellisch I Journal of Public Economics
G,=I(S,*)
55 (1994) 167-184
fori=1,2.
179
(18)
The central government cannot interfere with locational choices of households and has to take (2) as a constraint. Inserting (8) for xi into (2) determines ni as an implicit function of the central and regional control variables: ni = m(S,,, Gi, G,) .
(19)
Since we want to analyze whether an interregional transfer that deviates from the one chosen by regional governments can enhance efficiency, the central government must solve, for all 6 E [0,11, the problem maximize SU’ (SlZ)
(f1(L4-G,-%
(1 - 6)U2
+
,G
G
1,
n1
f’(7’2, (
N
-
nl)
-
G,
+
2
>
Sl,
,G,,G,),
N-n,
where G, and G, depend on S,, and n1 depends on G,, G,, and S,,. The first-order condition becomes [using (12)]:
(21) The expression which is multiplied by 6 characterizes the change in u1, and the term which is multiplied by (1 - 6) is the first-order impact on u2. From (14) and (12), we have ub,(anjlaGj) - z& = 2a(anjlaGj). Inserting this and the migration response an,laS,, set out in (13) into (21) and allowing 6 to vary from 0 to 1, we obtain _- 2an, 4
<-
2an, 4
n2 l+Du,‘aGl
--an, dG,
dS,,
(22)
where for 6 = 1 the first relation holds as an equality, and for 6 = 0 the second relation holds as an equality. According to (14), the migration responses anJaG,, i = 1, 2, do not generally vanish. The responses dG,/dS,,, i = 1, 2, can be derived from the first-order conditions (17) taking the migration responses into account. Even when using second-order conditions for the regional choice of Gj, i = 1, 2, and stability conditions of the Nash equilibrium, (17), I was unable to derive
180
D. Wellisch I Journal of Public Economics
55 (1994) 167-184
clear-cut results for the signs of dG,/dS,,, i = 1, 2, in the general case. All that one can say is that they do not vanish in general. D is negative by stability of the migration equilibrium. Hence, the interval defined by (22) does not coincide with the one set out in (16) where the regions choose the transfer. It may well be the case that these two intervals intersect, suggesting that there may exist cases in which the transfer chosen by regional governments is efficient. Although it is impossible to derive general results, we can demonstrate, by considering a simple example, that there exist cases in which a deviation from the transfer chosen by regions will enhance efficiency. Suppose that region 1 is not transfer-constrained and that G, does not generate spillovers. In this situation, region 2 must be transfer-constrained and its provision of G,, causing interregional spillovers, is inefficiently low relative to the Samuelson criterion (see Section A.1 in the appendix). Let us further assume that a marginal transfer to region 2 increases its supply of G,, dG,/dS,, > 0. This will be the case if the utility function is additively separable with respect to all arguments.” To see that a marginal transfer improves efficiency, consider first the change in U* set out in (21). Following (14) and (17), an,laG, = 0 if 2 = 0. According to (21), the change in u2 reduces to (utln,) - u~,(&z,/ Q* c?S,~). This expression must be positive since we consider a situation in which region 2 is transfer-constrained [see (ll)]. Therefore, u2 increases in response to a higher transfer. Now, consider the change in u1 set out in (21). Taking into account that region 1 is not transfer-constrained, -(~‘ln,) + ui,(dn, /as,,) = 0, and using (12) and (14), the entire effect on U’ becomes -2a(an,/aG,)(dG,/ dS,,). We demonstrate in Section A.2 of the appendix that an,laG, is negative. Hence, the effect on u1 is positive if dG,/dS,, > 0. Therefore, if the interregional transfer improves the efficiency of public good provision by a transfer-constrained region, there are cases in which a deviation from the transfer chosen by regions can be recommended for efficiency reasons. Such cases do not exist if there is no attachment (a = 0) or if public goods do not generate spillovers. If there is no attachment (a = 0), (22) indicates that S,, is equal to the net interregional transfer chosen by regions in a decentralized Nash equilibrium [see (16)]. Both regions supply public goods generating spillovers according to the Samuelson Rule. Hence, we are in a first-best economy and the central government should not violate the first-best allocation of labor. Even if we have attachment, but public goods do not generate spillovers (the case analyzed by Mansoorian and Myers, 1993), a central government cannot increase efficiency by deviating from the interregional transfer chosen by regions. To see this, note that, according to “A
proof is available from the author upon request.
D. Wellisch I Journal of Public Economics
55 (1994) 167-184
181
(17), both regions achieve the Samuelson condition of public good supply (uij = 0). According to (14), the migration response ani/aGi vanishes at an equilibrium. Thus, (22) reduces to (16). The decentralized Nash equilibrium is socially efficient and a central government has no incentive to change this allocation for efficiency reasons.
5. Conclusions
This paper analyzed the regional provision of public goods that generate interregional spillovers. We modeled alternative degrees of household mobility and derived that the decentralized Nash equilibrium is socially efficient when households are perfectly mobile. If households are attached to their home region, at least one region undersupplies the public good relative to the Samuelson Rule. The decentralized Nash equilibrium is inefficient. Hence, the result of Manssorian and Myers (1993) that the efficiency of decentralized decisions can be extended to an environment of imperfect household mobility breaks down if there are spillovers in the provision of public goods. Although regions achieve an ‘efficient’ population distribution by choosing interregional transfers, there exist cases in which a central government can enhance efficiency by deviating from this transfer. The conclusions derived in this paper also apply to other policy considerations. For instance, one can demonstrate regional environmental policy to be socially efficient in the perfect mobility case even if there are negative external effects harming neighboring jurisdictions. Regions conducting environmental policy take into account their damage to non-residents due to their migration responses (see Wellisch, 1992). However, if households are attached to a particular region, this only holds for a region that is not transfer-constrained. Furthermore, an extension to the multi-region case for perfect mobility is straightforward and is not found to alter any of the basic conclusions (see Wellisch, 1991). An extension of the heterogeneous population case in this direction is rather difficult and is a task for future research.
Acknowledgements
I am ments. helpful TAPES Gordon
indebted to David Wildasin for encouragement and helpful comI also thank Gordon Myers, Wolfram Richter, and Uwe Walz for discussion and correspondence. This paper was presented at the 1992 conference in Munich. I would like to thank the organizers, Roger and Hans-Werner Sinn, for the invitation and Harris Schlesinger
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D. Wellisch I Journal of Public Economics
~(!$+~)-~a-~<~ x
I
or
183
55 (1994) 167-184
O
(A-5)
x
which is a contradiction. Therefore, if region i is transfer-constrained, undersupplies the public good. Q.E.D. A .2. Central government
it
intervention
Here we prove that, in a situation in which region 1 is not transferconstrained and the provision of G, does not generate spillovers, a central government can enhance efficiency by increasing the transfer if dGZ/dSi2 is positive. Since u2 increases and the first-order impact on u1 is -2a(&z,l =,)(dG/d%) (see Section 4), we only have to demonstrate that an,laG, is negative. Inserting (12) into (14), we get
G4.6) or, using the definition of D: 1
an,= % aG2
ui,-2a
64.7)
’
Following (A.7), we must demonstrate that uin - 2a is negative. If region 1 is not transfer-constrained, (15) yields fi - x2 =f,‘, -x1 + (2an,lut). Inserting this into the condition for a stable migration equilibrium yields (f:-X,)($+-$)-2a+3<0
or
fi-Xi
(A-8)
Therefore, ui, - 2a = (u~ln,)(ffi -x1) - 2a must be negative and, according to (A.73, an, / aG, must also be negative. Q.E.D.
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Boskin, M.J., 1973, Local government tax and product competition and the optimal provision of public goods, Journal of Political Economy 81, 203-210. DePalma, A. and Y.Y. Papageorgiou, 1988, Heterogeneity in tastes and urban structures, Regional Science and Urban Economics 18, 37-56. Flatters, F.R., V.J. Henderson and P. Mieszkowski, 1974, Public goods, efficiency and regional equalization, Journal of Public Economics 3, 99-112. Gordon, R.H., 1983, An optimal taxation approach to fiscal federalism, Quarterly Journal of Economics 98, 567-586. Hartwick, J.M., 1980, Henry George rule, optimal population, and interregional equity, Canadian Journal of Economics 13, 695-700. Mansoorian, A. and G.M. Myers, 1993, Attachment to home and efficient purchases of population in a fiscal externality economy, Journal of Public Economics 52, 117-132. Myers, G.M., 1990, Optimality, free mobility, and the regional authority in a federation, Journal of Public Economics 43, 107-121. Oates, W.E., 1972, Fiscal federalism (Hartcourt Brace Jovanovich, New York). Pauly, M.V., 1970, Optimality, public goods, and local governments: A general theoretical analysis, Journal of Political Economy 78, 572-584. 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) 247-333. Wellisch, D., 1991, On the decentralized provision of public goods with spillovers in the presence of interregional migration, Ttibinger Diskussionsbeitrlge No. 19, Tiibinger University. Wellisch, D., 1992, Dezentrale Umweltpolitik, Mobilitat von Kapital, Haushalten und Firmen und grenziiberschreitende Umweltschlden, Zeitschrift fiir Umweltpolitik 15, 433-458. Wildasin, D.E., 1986, Urban public finance (Harwood, New York). Wilasdin, D.E., Nash equilibrium in Models of Fiscal competition, Journal of Public Economics 35, 229-240.