Community composition and the provision of local public goods

Journal

of Public

Economics

44 (1991) 217-237.

North-Holland

and the Community composition provision of local public goods A normative

analysis

Robert

and

University of Maryland. Received

March

College Park, MD 20742, USA

1990, revised version received July 1990

There is considerable evidence suggesting that the composition of the community - that is, the characteristics of the residents themselves - plays a central role in determining levels of important public outputs such as education and public safety. This paper explores the normative implications of this evidence. We show that optimal community composition involves a trade-off between the gains from homogeneity in demands among residents and the gains from heterogeneity in the production of those goods. The analysis establishes a role for equalizing intergovernmental grants on efliciency grounds: such grants can provide the needed incentives for sustaining heterogeneous communities.

1. Introduction The cornerstone of the theory of local finance is the Tiebout model under which mobile consumers seek out a community of residence that best satisfies their preferences for local public goods. In the limiting case described by Tiebout, this sorting process leads to an efficient outcome, one in which the residents in each community have identical demands for the public good. The Tiebout outcome depends critically upon a number of quite strong assumptions: costless mobility and the ‘exogeneity’ of income, to name only two. These assumptions have been analyzed extensively in the literature. A seemingly more innocuous assumption in the model is the presumption of identical production functions and input prices - with the implication that all communities have the same cost function for the provision of local public services. The assumption of identical cost functions is commonly made in the analysis of competitive firms in the private sector - indeed, it is one to which we hardly give a second thought. *Oates is also a University Fellow at Resources for the Future. We are grateful to the National Science Foundation for support of this research. We also thank Jan Brueckner, Richard Arnott, David Wildasin, and two anonymous referees for their helpful comments and suggestions on earlier drafts. 0047-2727/91/$03SO

0

1991-Elsevier

Science Publishers

B.V. (North-Holland)

218

R.M. Schwab and WE. Oates, Community composition

This assumption, however, is much more suspect in the context of local public goods. It is our contention that cost functions for the most important local services are unlikely to be identical; our claim, in short, is that in its application to certain key local public goods, the ‘production function’ envisioned in the basic Tiebout model is incomplete and misleading [Oates (1977, 1981)]. There is considerable evidence suggesting that the composition of the cotimunity - that is, the characteristics of the residents themselves plays a central role in determining levels of public outputs. Many studies indicate, for example, that the level of attainment in a school system or the level of safety in a neighborhood depends not so much on the instructional staff or frequency of police patrols as on the characteristics of the residents of the jurisdiction. Our objective in this paper is to explore the normative implications of this contention.’ We set forth a model in which the production function for local public goods contains as an explicit argument the composition of the community. By focusing the analysis on the ‘price’ of final outputs, we are able to derive a set of intelligible first-order conditions that characterize an optimal allocation of individuals among communities. We find that, in general, efficient community composition no longer involves homogeneous communities. The gains from interactions in the production of local public goods must be played off against the gains from having populations that are homogeneous in demand. In this general framework, the Tiebout outcome emerges as a special, limiting case in which community composition has no effect on the level of public output.’ Further consideration of these conditions for optimal community composition suggests that, unlike the Tiebout world, decentralized choice will not necessarily lead to an efficient outcome. In the model, individual location decisions encompass a form of externality, since an individual’s presence in a community will affect the cost per head of providing local services. Efficiency requires that communities be allowed to discriminate among residents in the levying of local taxes; typically, this implies that local communities would ‘There has, incidentally, been some work developing the positive implications of this matter. Hamilton (1983) has argued convincingly that the failure to account for community characteristics in the production of local services has led to significant biases in the estimation of the income elasticity of demand and excessively large estimates of the magnitude of the flypaper effect. Schwab and Zampelli (1987), in an empirical study of public safety, provide some evidence in support of Hamilton’s contention. ‘There is another class of mobility models that gives rise to heterogeneous jurisdictions. These ‘regional’ models require that individuals work and reside in the same jurisduction. The source of heterogeneity in these models is the need for both ‘skilled’ and ‘unskilled’ workers in the production of private goods. There thus emerges a trade-off between the needed mix of residents for private production and the gains from homogeneity in the consumption of local public goods [Berglas (1976), McGuire (1988)]. Our model does not require individuals to live and work in the same community. The gain from heterogeneity in our model stems from interactions in the production of the public, not the private, good. Interestingly, Brueckner (1989) has shown recently that the two problems have strong formal similarities.

R.M. Schwab and LYE. Oates, Community composition

219

have to set higher taxes on low-income families who drive up the cost of providing public goods in the community. Since local taxes with such a ‘super-regressive’ pattern of incidence will understandably prove objectionable on equity grounds, the issue naturally arises of alternative policy instruments that can address this source of distortions in local finance. We find that a second-best outcome can indeed be achieved by a more conventional and favorably received fiscal instrument: equalizing intergovernmental grants. In short, the analysis establishes a case on efficiency grounds for the introduction of such a grant system. By making more generous grant payments to local governments with relatively large numbers of low-income households, a higher level government (like a metropolitan area government) can overcome the opposition of communities to the kinds of heterogeneity that are required for efficient outputs of local public goods. We also find that equalizing grants appear to solve an existence problem: they create a local-finance equilibrium by effectively offsetting an incentive for continued ‘pursuit’ of higher-income households. Such a system of grants may, in this way, eliminate (or at least reduce) the powerful incentives for the adoption by local governments of various sorts of exclusionary devices. In the next section of the paper, we set forth the basic model, derive and discuss the conditions for Pareto efficiency, and examine efficiency under decentralized choice. In section 3 we examine the role of higher levels of government in the problem and develop the case for a system of equalizing intergovernmental grants. Section 4 includes a summary and conclusions. At the outset, we need to place our analysis in the context of two recent, provocative papers that explore community composition in the presence of peer-group effects: Brueckner and Lee (1989) and de Bartoleme (1990). At first glance, these two papers appear to reach exactly the opposite conclusion: Brueckner and Lee find that decentralized choice produces an equilibrium that is socially optimal, while de Bartoleme stresses that decentralized decisons result in an inefficient configuration of local communities. The contradiction, however, is only apparent; these opposing findings result from a basic difference in the orientation of the two papers. Brueckner and Lee employ a club theory framework within which ‘competitive developers’ are in a position to charge differential prices to prospective residents and thereby effect ‘intraclub transfers’ that are needed to sustain an equilibrium. De Bartoleme, in contrast, works within the context of local governments that are constrained to treat all residents alike. As we show in this paper, the ability to treat groups of people differently has quite fundamental implications for the efficiency properties of a decentralized equilibrium.3 ‘De Bartoleme also imposes some further restrictions on the interaction between peer groups in the production of local public goods and on the structure of the communities that give his equilibria somewhat different properties from those in this paper. Finally, we must cite the

220

R.M. Schwab and WE. Oates, Community composition

2. Basic model

Our conceptual framework draws on the distinction from Bradford, Malt and Oates (1969) between outputs directly produced by the public sector (termed ‘D-output’) and final output which is of primary concern to the individual consumer (called ‘C-output’). It is C-output that enters as an argument in individual utility functions. Consider, for example, public safety. By combining police officers with other police inputs, a local government produces a vector of D-output, including as elements such activities as police patrols, traffic control at busy intersections, etc. A resident of the jurisdiction, however, is primarily interested in the level of public safety (C-output) which depends not only on D-output but also on the characteristics of the community. Poor communities, for instance, burdened with high unemployment and individuals with a high propensity to commit crimes, will exhibit lower levels of safety for any given vector of D-outputs than will more affluent communities. Likewise, the level of educational attainment in a school district depends not only on the teaching staff, library facilities, and other inputs, but in fundamental ways on the characteristics of the pupils themselves. Community characteristics (by which we shall mean the characteristics of the residents of the community) can influence the production of local public goods in two ways. First, a household’s own characteristics may influence the level of C-output that it realizes from any given set of public inputs. Children from more prosperous and highly educated families will, on average, tend to perform relatively well in school. If this were the only way in which community characteristics entered the production function for local public goods, then efficiency considerations would still call for communities that are homogeneous in the demand for purchased inputs. These communities would obviously not be homogeneous in the levels of consumption of final output C-outputs would differ among households - but they would be homogeneous in their demands for D-outputs. Second, the characteristics of other households may influence the C-output consumed by a given resident. The characteristics of neighbors, for example, will influence the level of public safety in an area; similarly, peer-group effects are well documented as important contributors to a student’s achievement [Summers and Wolfe (1977), Henderson, Mieszkowski and Sauvageau (1978)]. In a general specification, then, the production function for C-output of a particular household should include both the characteristics of the household insightful paper by Arnott and Rowse (1987) which explores the implications of peer-group effects for the design of an effkient educational system. Their careful analysis reveals that whether mixing of pupils or ‘perfect streaming’ is socially optimal depends in fundamental ways on the curvature properties of the educational production function.

R.M. Schwab and WE. Oates, Community composition

221

and the characteristics of the other households in the community. It is this interaction among residents in the production of local public goods that enriches and complicates the problem of determining the optimal composition of the community. We shall now set forth a simple model of local finance that captures these dual effects of residents on the production of a local public good. There are two types of individuals in our model, A and B. We consider a ‘metropolitan area’ that consists of a number of communities and that has an exogenously determined population of u A’s and /I B’s. Both types of individuals receive utility from a single private good X and a publicly provided good 2. X is purchased in private markets at a price of $1 per unit. 2 is produced within the community, and the technology for producing Z plays an important role in the model. 2 in our model is equivalent to C-output in the Bradford, Malt and Oates (1969) framework. In that spirit, we assume that the level of 2 an individual consumes depends on three factors: the quantities of publicly purchased inputs employed by the community, community characteristics, and the characteristics of the individual (i.e. whether the person is an A or a B). For example, we might think of 2 as educational achievement, where a child’s test scores are a function of the number of teachers and textbooks the local schools provide, the characteristics of the other children in the classroom, and the child’s family background. We make three assumptions about the technology for producing 2. First, we assume constant returns to scale: for a given population size and composition, a doubling of per capita consumption of 2 would require a doubling of purchased inputs. Second, in order to abstract from the question of optimal community size and to allow us to focus on optimal community composition, we assume that 2 is fully subject to congestion. Holding per capita consumption and community composition constant, a doubling of the population thus requires a doubling of purchased inputs. Third, following the formulation suggested in Oates (1977), we assume that we can capture the role of community characteristics by including the fraction of the community’s population that are A’s in the production function for 2; without loss of generality, we assume that an increase in this proportion leads to an increase (or more formally, does not lead to a decrease) in the quantity of 2 consumed by all residents. Throughout the paper, we use subscripts to index individuals and superscripts to index communities. Using this notation, our three assumptions on the production of the public good imply the existence of a unit cost function for Z in community i for a type j person of Pj(e’), where 19’is defined as 8’-

,ui

CY.‘+B”

222

R.M. Schwab and WE. Oates, Community composition Table Community

Community Community Community

1 2 3

1

composition

A’s

B’s

e

0 u-a3 c?

8-P 0 fi3

0 1 a3/(a3 + f13)

Pi(@) is independent of the quantity of 2 produced and the number of people in the community, and aPi/% SO. That is, if an increase in 13were to cause output to rise holding purchased inputs constant, then an increase in 6’ holding output constant at 1 unit causes costs to fall. Consumers have exogenous incomes Y, and Y, that are independent of the community in which they live.4 The production of the local public good in community i is financed through a set of head taxes T:. An individual’s consumption of the private good (Xi) equals income less local taxes ( Yj- T:). We allow for three communities in our model.5 As shown in table 1, only B’s live in community 1, only A’s live in community 2, and both A’s and B’s live in community 3. A key issue in the paper is the composition of this heterogeneous community 3. Throughout the paper, we let a3 and fl” represent the number of A’s and B’s who live in the heterogeneous community. Then by definition, (a-a3) A’s live in community 2 and (/?-fi3) B’s live in community 1. 2.1. Efficiency

in the basic model

We can think of the efficiency problem as one of choosing the T’s, the Z’s, fi” in order to maximize the welfare of the A’s who live in community 3 subject to several constraints. First, we require the A’s in 2 to reach the same level of utility as the A’s in 3. Second, the B’s living in communities 1 and 3 must reach a specified level of utility.(j Third, aggregate taxes must equal aggregate public spending. The fourth constraint focuses on the equal provision of public goods a3, and

4This assumption is consistent with our goal of constructing a model where consumers are not required to live in the community where they work. ‘Given our assumption that the public good is fully subject to congestion, community size is indeterminate. It would perhaps be better to say that we have incorporated three types of communities in our model, rather than three communities. 6This requirement, that identical people in different communities reach the same level of utility, can be explained in several ways. First (and admittedly this is a weak argument), it is a common assumption in the literature. Second, as Wildasin (1987) argues, only equilibria with equal utility across communities are sustainable if free mobility is possible. Since we want to compare the efficiency properties of alternative equilibrium outcomes, it makes sense to restrict our attention at the outset to the class of outcomes which rules out identical people reaching different levels of utility. The analysis is, in this sense, second-best in character.

R.M. Schwab and WE. Oates, Community composition

223

within the heterogeneous community. As Shoup (1969, ch. 5; 1989) has emphasized, equality can have at least two different meanings. We might interpret equality as a prohibition against unequal spending and therefore unequal D-output within a community; all students must be given the same number of textbooks, all neighborhoods must be provided within the same number of police officers. In our framework, an equal spending restriction would require PiZi = P$?$.Alternatively, equality might be interpreted in terms of equal consumption (equal C-output); all students must reach the same level of achievement, the crime rate must be reduced to the same level in all neighborhoods. An equal output restriction would require Zi = Z& We have considered both cases. We present here the analysis for the equal spending restriction. The results are not altered in any substantive way if this condition is replaced by an equal output restriction.’ Finally, we require non-negative numbers of both types of people in all three communities, We can state the formal efficiency problem as max U,(Z:,

Y, - Ti)

(2)

subject to

U,(Zi,

Y*-

Ti) - U,(Zi, Y, - Ti) 10,

(34

U,(Z;, Y,- T;)-U; 20, U,(Z;,

Y,-

(3b)

T;)-U,*zO,

(34

p3z3-p;z;=o 7 A A

0sb3sp.

(3g)

Let wj be the marginal willingness to pay for Z by a type j person who lives in community i; w thus equals the marginal rate of substitution between Z and X. The solution of this efficiency problem requires in part that

wi=P'B9 ‘We will be happy

(44

to provide

upon request

the analysis

for this alternative

case.

R.M. Schwab and WE. Oates, Community composition

224

w:=Pi,

(W (W

Eqs. (4a) and (4b) state that in the homogeneous communities the public good should be provided on the same basis as any private good: price must equal marginal willingness to pay. (4~) is the analog to the Samuelson condition in this model. The term (Pi/Pi) appears on the left-hand side of (4~) because we have required equal spending, rather than equal output, per capita.’ It is helpful to begin the discussion of the remaining necessary conditions for optimality by defining

v*= [(T: - T:) -

(P:z:

- P:z:)]

-- [(aP:/aa3)a3(w~/P~)z:

v, = [( 7-i - 7-A)- (P;z; -

+(aP~/aa3)B3(w~/P~)Z~],

(54

P;z;)]

where, for example, dPi/aa3 = (aP~/af33)(at13/aa3) SO and aPi/af13 = (aP~/a6”)(ae”/aa”)~O. Eq. (5a) measures the net gain to society from the movement of an additional type-A person from community 2 to community 3; eq. (5b) measures this gain for the migration of a type-B individual from 1 to 3. The first set of terms within brackets reflects the marginal gain from the perspective of the migrant, while the latter set of terms within brackets indicates the externality (the welfare effect of the migrant operating through his impact on the cost of providing local services in community 3). Efficiency requires that one of the following sets of conditions be satisfied: (i) V, = 0 and Oga3 ga; (ii) V,O and a3 =a. Similarly, (i) V,=O and 0 5 /I3 5 fl; (ii) V, < 0 and p3 = 0; or (iii) V, > 0 and b3 = p.’ 2.2. Homogeneous communities In this subsection

we examine

the conditions

under

which the solution

of

*The more familiar version of the Samuelson condition emerges if we focus on the purchased input. Marginal willingness to pay for the purchased input equals the marginal product of the input times marginal willingness to pay for Z. Pi and P: equal the price of the purchased input divided by the marginal product of the purchased input. Eq. (4~) thus implies that the sum of the marginal willingness to pay for the purchased input must equal the cost of providing an additional unit of the input to each person in the community. ?See Arnott and Rowse (1987) for a careful and illuminating treatment of the ways in which optimal community composition (or, in their case, class composition within schools) depends on the properties and parameter values of the production function.

R.M. Schwab and WE. Oates, Community composition

225

the efficiency problem we have posed would require only homogeneous communities. Consider a solution where the A’s live in both communities 2 and 3 and all of the B’s live in 1. The last term on the right-hand side of (5b) vanishes in this case since, by assumption, B’s do not live in community 3. Any B who were to move to 3 would have to reach the same level of utility as the B’s who remain in 1. It follows that (Ti- Tk) must equal the value to the consumer of the difference in Z provided in the two communities so that we can write (7;~T;)-(P;Z;-P;Z;)=T(w,(Z)-P;)dZ-(P;--P;)Z:. Z:, Therefore, if homogeneous into (5b) implies that

communities

are efficient, then substituting

(6) (6)

(5b’) must be less than or equal to zero.” The first term on the right-hand side of eq. (5b’) is the difference between a B’s willingness to pay for the incremental (Zs-Zk) units of public goods he receives if he moves from community 1 to community 3, and the cost of providing those goods. wg evaluated at Zi must be equal to PA and therefore, as long as the marginal willingness to pay curve is downward sloping, this term must be negative. Thus, the first term on the right-hand side of (5b’) is the deadweight loss from heterogeneity; focusing only on the demand for public goods, efficiency requires homogeneous communities. The second term is the difference in the cost of providing public goods to a B in the two communities. It must be negative since the cost of public goods in community 3, where 0= 1, is lower than the cost in community 1, where 8=0. The final term represents the cost imposed on the existing residents in 3 when a B moves moves into their community; it must be positive since by assumption raising 0 raises the cost of public goods. Putting these three terms together, we come to the following conclusion: homogeneous communities are efticient if, were a B to move to community 3, the sum of the deadweight loss from distorting consumers’ choices of local public goods and the cost imposed on current residents is greater than the difference in the cost of public goods in the two communities. Or, put slightly differently, heterogeneous communities will be the efficiency-dominating “If only A’s live in community 3, then efficiency requires that the level of the public good be provided which equates their marginal willingness to pay and price.. w:/Pi thus equals 1 in this special case and therefore the last term in (Sb’) is simpler than the corresponding term in (5b).

226

R.M. Schwab and WE. Oates, Community composition

outcome when the cost-savings to B’s from mixing with A’s outweigh the sum of the increased production costs to A’s and the deadweight losses from distorted consumption levels for both groups. If the cost gains from mixing are negative or if they are small relative to the losses from uniform consumption among those with different demands, then homogeneous communities will be efficient. It is not difficult to see that the Tiebout model emerges as a special case of our model. In the Tiebout model, the price of public goods is independent of community composition. Given Tiebout’s assumptions, the last two terms on the right-hand side of (5b’) vanish; V, is thus always negative, and the homogeneous communities solution satisfies the efficiency conditions in our model. ’ 1

2.3. The gains from heterogeneity and corner solutions We will say that there are gains from heterogeneity if there exists a solution to the efficiency problem we have posed where both A’s and B’s live in the heterogeneous community, and there does not exist a second solution where all communities are homogeneous. That is, heterogeneity is valuable if society could not do just as well by leaving community 3 empty. We now show that in solutions where all three communities are occupied there are no gains from heterogeneity; if heterogeneity is valuable, then alternatively all of the A’s live in 3, all of the B’s live in 3, or both. The proof is as follows. Suppose we have a solution where people live in all three communities, i.e., 0 < ~1~
homoof (Sb) (ii) of the vanish which in the

R.M. Schwab and WE. Oates, Community

composition

227

as good as the solution where all three communities are occupied; there are, therefore, no gains from heterogeneity in this case. If heterogeneity is valuable, then either community 1 is empty, community 2 is empty, or both. In the remainder of the paper we will focus on a solution where all of the A’s live in community 3 (~~=a) and the B’s live in communities 1 and 3 (O-CD”
Y, - ri),

(7)

subject to

‘sA diagrammatic description of the preceding analysis is available upon request. ‘“This type of model is sometimes referred to as a ‘utility-taking’ model. As Atkinson and Stigma (1980) argue, this is the natural analog to a price-taking model in a model without a local public sector. Note that since this equal ut;lity constraint has been built into the model, community 3 does not have to erect any barriers .o entry (such as restrictive zoning provisions) to stop a greater than optimal amount of migration from community 1. The chosen level of taxes and spending and free migration leads to the optimal number of B’s in the heterogeneous community.

JPE.

D

228

R.M. Schwab and WE. Oates, Community

u3T; p3z3 A

+ /I3 T; - tx3P;Z: A

- /3”P;Z;

composition

= 0,

(W

- P3Z3 = 0. B B

The solution to this problem Samuelson condition) and that (T,: - P;Z;)

(84

requires

that eq. (4~) hold (the analog

- [(cYP;/dp3)a3(w;/P:)Z;

+(dP;/dp3)/33(~;/P;)Z;]

to the

=O. (9)

Eq. (9) has the following interpretation. The benefit to the residents of community 3 from allowing an additional B to enter the community is that marginal person’s contribution to the local treasury. The costs to the community include the required expenditure to provide public goods to the marginal person, PzZ& and the monetized value of the increase in costs that result from the externality, (dP$dp3)~3(~~/P~)Z~+(dP$dg3)~3(w~/P~)Z& Eq. (9) states that the community should continue to allow B’s to enter to the point where marginal benefits equal marginal costs. It is not difficult to see that this solution must satisfy the necessary conditions for efficiency. We know that in community 1 utility will be maximized and the budget will be balanced. Thus, (4a) will hold and PAZ: - Ti will equal zero. But if PAZ;- TL equals zero, then when (9) holds, V, as defined in (5b) must equal zero. All of the first-order conditions in the efficiency problem are therefore satisfied under the decentralized solution. The intuition behind this result is straightforward. When a B leaves community 1, he imposes no harm on those who remain since the lost tax revenue in 1 is exactly offset by the reduction in the cost of providing public goods. Those already in 3 are neither harmed nor benefited by the new entrant since, at the margin, the fiscal surplus just equals the increase in costs. Therefore, in equilibrium there will be no gain to society from any further migration so that the outcome is consistent with efficiency. In a median-voter framework we find that the equilibrium outcome does not violate the necessary conditions for efficiency. Interestingly, Brueckner and Lee (1989) reach a similar result for a quite different institutional setting. Their characterization of the equilibrium outcome under a system of competitive developers also satisfies the conditions for Pareto efficiency. Developers in their model achieve this outcome by making ‘intraclub transfers’ between the A’s and B’s (which amounts to charging the A’s and B’s different prices for consuming the collective good). 2.5. Equilibrium Before

leaving

when the B’s are in the majority in both communities

this topic,

we wish

to consider

briefly

the nature

of an

R.M. Schwab and WE. Oates, Community composition

229

equilibrium where the group that constitutes a majority in the mixed community, say the B’s, also live in a homogeneous community. Suppose that the B’s (now, by assumption, a majority in the mixed community 3) attempt to exploit the minority group of As. They will be unable to do so because of the (potential) competition from the B’s in community 1. The A’s are, in a sense, a desirable commodity since their presence lowers the cost of providing local public services. Competition between the two sets of B’s in communities 1 and 3 will thus, in the limit, defeat completely any attempts to exploit the A’sI The result, more formally, will be the same as in the previous section: the utility of the A’s will effectively be maximized given that the utility of the B’s is held at the level in the homogeneous community. Thus, we have again that the objective function in (7) is effectively maximized subject to the constraints in (8a) through (8~) so that the outcome continues to satisfy the necessary conditions for Pareto optimality.

2.6. Equity

and efficiency

A further examination of (9) suggests that this favorable conclusion as to efficiency should be tempered by some important concerns about equity and the political feasibility of such solutions. Eq. (9) states that in equilibrium, T&PzZi must be greater than zero, since (aP:/ag3)a3(w:/P~)Z:+ (8P~/8/13)fi3(w~/P~)Z~ is greater than zero. The balanced budget and equal spending requirements then imply T3B > P3Z3 BB

= P3Z3 AA

> T3A’

(10)

That is, the B’s living in community 3 must pay per capita taxes that exceed the cost of the public goods they consume and the per capita taxes paid by the As. We have framed the problem in very general terms by simply talking about two groups, A’s and B’s. But clearly in many important problems the B’s have lower incomes than the A’s; increasing the number of low-income families in a community often raises the cost of providing a given level of education or establishing a given level of public safety. The result that decentralized decision-making achieves efficiency thus turns crucially on the assumption that local governments have the ability to levy higher taxes on lowincome families.

This is not an altogether convincing result. Local governments typically do not have wide latitude in setting tax policy, and on equity grounds we might 15As a referee noted, A’s can escape exploitation to the extent that they realize the gains offered by the next highest bidder. At worst, they will realize the level of utility available in a homogeneous community.

230

R.M.

Schwab and WE. Oates, Community

composition

well applaud the limitations that are often imposed. An interesting set of questions then arises. How should we characterize equilibrium if the choice of local tax instruments is limited? Will decentralized choice achieve this second-best optimum? What sorts of policies by a metropolitan government might be appropriate? We turn to these issues in the next section.

3. The efficiency

case for equalizing

grants

Our goal in this section to examine the major results set out above when the choice of tax instruments is more limited. We focus on the simple case where all individuals in a community must be charged the same taxes. The analysis of other cases (e.g. the taxes paid by the A’s must be some multiple of the taxes paid by the B’s) proceeds along similar lines. 3.1. Efficiency in the second-best model Efficiency in this second-best mode1 requires the maximization subject to the constraints in eqs. (3a) through (3g) and the equal constraint

of (2) taxation

T; = T;.

(11)

To compare the results under equal taxation with those presented above, we continue to focus on solutions where all of the A’s and some of the B’s live in community 3 (~1~=o! and OSP” 58.) Let the common tax on all individuals in community 3 be T 3. The solution to the social efficiency problem requires in part (T;-

T;) -(P;Z;-

PAZ;) -yA(aP~/ag3)Z~-yyg(aP~/alj3)Zi:

=O,

(12)

where yA and ~a are the weights to be applied to the change in cost of providing public goods (i3Pi/8p3)Zi and (IYP~/cY~~)Z~.In the appendix we show that these weights have two characteristics that are helpful in what follows. First, yA and yB must both be positive. Second, in the special case where the Samuelson condition in (4~) holds even though we are now imposing equal taxation in communiy 3, then yA equals cr3(w:/P:), yB equals p3(wi/P& and the left-hand side of (12) reduces to the optimality condition in the first-best problem in (5b). In short, the logic of the rule for the optima1 allocation of the B’s is unchanged, although the weights to be used to evaluate the change in the cost of providing public goods are now different since we are forced to use equal taxation. It is not difficult to show that a decentralized system cannot achieve this second-best outcome. In particular, with equal taxation the A’s would never

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231

agree to live in a community with the B’s. The proof is straightforward. Suppose, as in section 2, the A’s in 3 are in the majority and must choose public policy so as to maximize their utility subject to the constraints in (8a) through (8~) and the equal taxation constraint in (11). Suppose the solution to this problem requires fl” >O. Then consider an alternative solution with p” = 0 and 22 unchanged. Since ~9equals 1 in this alternative, Pi has been reduced. Therefore, Ti is lower, consumption of the private good is higher, and consumption of the public good is unchanged under this alternative. If equal taxation is required, the A’s would neoer choose to live in a heterogeneous community. This result is not surprising. The presence of B’s raises the cost of providing public goods. In the model we presented in the previous section, A’s are willing to live in the same community with B’s despite the latter’s impact on cost because the B’s pay higher taxes. That is not possible under equal taxation and equal spending - so the A’s incentive for integration vanishes. Of course, in a setting of free migration among local communities, the A’s may not be in a position to exclude the B’s. With equal taxes per capita, it is likely to be in the interest of a B-household to locate in a community populated by A’s, where local public goods are available at a lower cost. i6 This raises the troublesome issue of the existence of an equilibrium. The problem here is that the presence of an equal-tax constraint (or one that at least prohibits differentially high taxation of B’s) produces what looks like a recipe for a game of musical chairs. The A’s will try to move to a community to get away from the B’s, but the B’s will have every incentive to remain in hot pursuit. The problem is much like that which arises in the presence of local income taxation: as Wheaton (1975) has shown, there may exist no equilibrium, because everyone will have a preference for a high-income community in which the tax base (and hence the tax rate to finance any given level of public outputs) will be relatively low. In our model, the incentive is not the larger tax base; it is the lower costs of provision available in an A-type community. But the implications for location decisions are much the same. This existence problem, incidentally, may well provide part of the rationale for observed zoning ordinances. Since differential taxes are not available to regulate community composition, residents may employ other exclusionary devices in an attempt to control the characteristics of residents. As Mills and Oates (1975) have argued, large-lot zoning ordinances and other such measures may to some extent represent a kind of ‘public-goods zoning’ by 161t is possible that in terms of the tastes of a B individual, the level of local services available in the A-dominated community will be suffkiently inferior to that in community 1 to more than offset the advantages stemming from the lower cost per unit in the former.

J.P.E.-

E

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which ‘existing residents try to control the characteristics of newcomers that result in higher levels of particularly important local services’ (p. 8). At any rate, it does seem clear that the second-best outcome in the presence of an equal-tax constraint, if it involves some heterogeneity in community composition, is unlikely to emerge from decentralized fiscal behavior. 3.2. Equalizing intergovernmental

grants

We might then ask: What policy could a higher level of government (say, a metropolitan government) follow in order to reach our second-best outcome? Suppose the metropolitan government were to offer a grant to community 3 of fi3s3 and a grant of (b-fi3)s’ to community 1; such a system would effectively reward communities for accepting the B’s, where the awards are s1 and s3. Suppose for simplicity that these grants are financed by setting a head tax of tM on everyone in the metropolitan area.” Now that we have introduced grants, we have to distinguish between taxes paid to one’s community and total taxes paid. Let t’ be the taxes paid to the community government; then in this notation, T’= t’+ t”. In the presence of the grant system, the A’s who live in 3 will seek to maximize (7) subject to the minimum utility constraint for the B’s living in 3 in (8a), the equal spending constraint in (8c), an equal tax constraint, and the public budget constraint: (a +

83)t3- dJ;z: - p3p;z; + p3s3= 0.

(13)

The solution to this problem requires

s3= P;z; - t3 + y&P:/ag”)z;

+ yB(aP;/ap)z;.

(14)

Efficiency requires the metropolitan government to set s3 so that (14) holds at the optimal fi3. This has an interesting interpretation. In the absence of the grant, a B entering community 3 imposes a cost on the citizens of 3 equal to the cost of the public goods he consumes plus the external damages from rising costs, and offers a benefit equal to the taxes he pays. Under a constitution that requires equal taxes and equal expenditures, costs will always exceed benefits and as a consequence heterogeneous communities will not come into being if the A’s have exclusionary powers. But if the intergovenmental grant is set equal to those net costs, then B’s will continue “This requirement of uniform taxation is not particularly restrictive since the metropolitan government can always undo the effects of equal taxation by varying the grant to community 1. It would be outside the spirit of this second-best model to allow the metropolitan government to tax the A’s and B’s living in 3 differently.

R.M.

to enter community 3 until at the margin their contribution to revenue, which includes taxes paid and the increase in the grant from the metropolitan government, equals the cost they impose on the current residents. The characterization of the grant to community 1, sl, is also of interest. If the community utility maximization condition in (14) and the efficiency condition in (12) both hold, then it must be the case that s3=T3-t3-Tl+P;Z;

=p’z’-t’ B B =s

1

(15)

)

where the second line in (15) follows from the definitions of T3 and T’ and the fourth line reflects a balanced public budget requirement in community 2 that states that local taxes plus grants must equal the cost of public goods. Efficiency thus requires the metropolitan government to set the same subsidy in both communities given that the head tax set by the metropolitan government is uniform throughout the region. On first glance, the result in eqs. (15) may seem puzzling. It is clear why a grant is required to community 3 - to induce the A’s to overcome their aversion to B’s and to set fiscal parameters to create the right incentives for the entry of B’s into community 3. It is not so clear, however, why the very same subsidy per capita must be paid to community 1, which is wholly composed of B’s. This result stems from the equal-utility constraint on B’s The grant to community 3, s3, provides the needed incentive to A’s in 3 but we need, in addition, an incentive to keep the requisite number of B’s in community 1. The intergovernmental grant to community 1, sl, serves precisely this function: it offsets what would otherwise be an excessive tendency for B’s to move to community 3. The discussion makes clear a further and interesting property of this system of equalizing grants. We were troubled earlier by the likely nonexistence of an equilibrium in a setting of free mobility with a requirement of equal taxation within each community. The grant system appears to resolve this problem: It creates the requisite incentive for A’s to accept B’s into community 3 and for B’s to restrict their movement to 3 to the efficient level. l8 This has a potentially important implication for local policy. We noted “There conditions

are admittedly some complicated here that we have not addressed.

issues

concerning

stability

and

second

order

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R.M. Schwab and WE. Oates, Community composition

earlier the incentives for ‘public-goods’ zoning: the use of local exclusionary devices to keep out households with adverse effects on the cost of providing local services. A system of equalizing intergovernmental grants that effectively rewards communities for the presence of such households will work to offset the incentives for public-goods zoning - it will eliminate the gains from exclusionary zoning measures. 4. Summary and further reflections The central argument of this paper flows from the premise that interaction among individuals with differing characteristics is likely to have significant cost implications for the provision of local public goods. Where such interactions produce cost-savings that are sufficiently pronounced, the Tiebout solution with communities that are homogeneous in demand for local services will no longer represent an efficient outcome. Such cost-savings require that the gains from heterogeneity in the provision of public outputs must be traded off against the losses from less homogeneous groups of consumer-residents. The implications for decentralized local choice are important. Efficient heterogeneous outcomes can be sustained only if communities have the capacity for differential taxation of a specific form: the levying of higher taxes on those whose presence raises the cost per capita of supplying local services. This result suggests that an efficient local tax system is likely to be ‘superregressive’ with higher tax bills required for lower-income households. The understandable opposition to such a tax system creates serious problems for local finance. In a setting of free mobility, more equitable tax systems create incentives for the poor to ‘chase’ the rich - and for the latter, in turn, to try to escape to other communities. There may exist no equilibrium in the system. This problem may well explain (to some extent at least) the observed propensity for high-income communities to erect barriers in the form of exclusionary zoning ordinances to keep out lower-income households. Such segregated outcomes, while perhaps introducing stability, are likely to be inefficient in the presence of significant interactions in the production of local public outputs. We have found in this paper that these inefficiencies can be addressed to some degree through a system of equalizing intergovernmental grants. An appropriately designed set of grants that effectively rewards communities for the presence of cost-increasing households can overcome the aversion that is the source of the inefficiency and sustain a second-best outcome. In so doing, it also appears that such grants can introduce an equilibrium into the system, removing the incentive for a continuing game of musical chairs. An intriguing aspect of this result is that it introduces explicitly an efficiency argument in support of a system of equalizing grants. The case for

R.M. Schwab and WE. Oates, Community composition

235

fiscal equalization in the public finance literature and policy arena has been made primarily on equity grounds [Bradbury et al. (1984)]. In terms of horizontal equity, for example, the claim is that equalizing grants are needed to lessen differentials among jurisdictions in fiscal capacity and fiscal need [e.g. King (1984, ch. 4)] so that individuals in a fiscally weak jurisdiction are not disadvantaged relative to their counterparts elsewhere. Such grants will also tend, on average, to redistribute income from higher- to lower-income households. The argument in this paper suggests that such grants may also have an efficiency-enhancing role. At the same time, we must not claim too much. As the equations make clear, the determination of the needed set of grants is not straightforward. It involves the measurement of the effects of differing group composition on the cost of producing local services. Moreover, most local governments provide a menu of services. For some of them, cost-savings from heterogeneity are likely to be important (although in differing ways), while for others such savings will be of little moment. The construction of the optimal set of equalizing grants in such a setting is obviously no easy task. Even so, the analysis does extend the case for such grants. A system of equalizing intergovernmental grants may not only enhance equity in the local public sector; it may also help the local sector to function more efficiently. There are a number of ways of extending this research. In particular, we suspect that it may prove important to incorporate capitalization in an extended version of the model.” We have not explored this issue formally, but we would conjecture that under some circumstances capitalization and free mobility can sustain an efficient outcome even if governments are constrained to levy equal taxes on all citizens. Suppose, for example, that different types of people prefer different types of housing and that communities therefore include several housing submarkets. Then in the spirit of the Brueckner and Lee (1989) model, we might expect a profit-maximizing land developer to choose an efficient community composition; differentials in housing prices might in that case play the same role as differentials in tax burdens. This capitalization argument also suggests a potentially interesting line of empirical research. One way to estimate the role of community composition in the production of local public goods would be to examine land markets. If heterogeneity is valuable, we would expect that value to be reflected in the prices different people are willing to pay for housing. Appendix

The purpose of the appendix is to characterize 19We thank

a referee for raising

this point.

the weights to be used to

236

evaluate problem

R.M. Schwab and W.E. Oates, Community composition

the changes in the cost of providing public goods in the second-best analyzed in section 3. Formally, these weights are given by

and

64.2) It can be shown that as in the first-best problem, it must be true in the second-best problem that either (i) wgP:, or (ii) wi> Pi and wi< Pi. In either case, it is then straightforward to see that both yA and yB must be positive. It is also easy to see that if the first-best condition in (4~) holds, then these weights are the same as the weights in the first-best problem: yA equals (w:/Pi) and yB equals (wi/Pi).

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