Equilibrium among local jurisdictions: toward an integrated treatment of voting and residential choice

Journal

of Public

Economics

EQUILIBRIUM INTEGRATED

24 (1984) 28lL308.

North-Holland

AMONG LOCAL JURISDICTIONS: TREATMENT OF VOTING AND CHOICE Dennis Carnegie-Mellon

TOWARD AN RESIDENTIAL

EPPLE

University, Pittsburgh, PA 15213, USA

Radu FILIMON Tulane University, New Orleans, LA 70118, USA

Thomas Carnegie-Mellon Received January

ROMER*

University, Pittsburgh, PA 15213, USA 1983, revised version received

March

1984

In models of local public goods with mobile consumers, existence of equilibrium is problematic. Difficulties with existence of equilibrium that arise in models with discrete locations and in models with voting are compoinded when both features are introduced into the same model. We present conditions under which equilibrium exists in a model where freely mobile households choose community of residence and amount of housing consumption, and vote on the level of public goods provision. These conditions involve restrictions on preferences and the technology of public goods supply. At least some of these conditions appear consistent with empirical observations. We discuss the implications of the conditions, and their role in assuring existence of equilibrium. A series of computational examples provide illustrations of the way these conditions interact, and the diffkulties that must be confronted if they are to be relaxed.

1. Introduction The study of the political economy of local jurisdictions has evolved along two relatively distinct paths.’ Along one, the focus has been investigation of interjurisdictional population mobility as a mechanism for inducing local jurisdictions to offer efficient tax-public good bundles. Along the other, the focus has been characterization of the tax-public good bundles that emerge from intrajurisdictional collective choice processes. Attempts to study in a single model the implications of intrajurisdictional choice mechanisms and *This research was supported by NSF grants DAR79-17576 and SES83-10383. We are grateful to Vernon Henderson, as well as to participants at a session of the 1982 Winter Meetings of the Econometric Society and in seminars at Washington University, Caltech, University of Maryland, and University of Virginia. Two anonymous referees also provided valuable comments. ‘See Epple and Zelenitz (1981) and references cited there. 0047-2727/84/$3X@

0

1984, Elsevier Science Publishers

B.V. (North-Holland)

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D. Epple et al., Equilibrium

among local jurisdictions

mobility of households among jurisdictions have encountered a fundamental difficulty ~ the absence of a satisfactory characterization of equilibrium in such a model. What has emerged from these attempts are relatively pessimistic conclusions about the prospects for identifying conditions that insure equilibrium, and a variety of examples of models in which no equilibrium exists. Rose-Ackerman (1979) examines a model in which each jurisdiction has a housing market, and voting by residents determines the quantity of a publicly provided good and a property tax rate for that jurisdiction. Households consume housing, the publicly provided good, and a numeraire good. They choose jurisdiction of residence, vote, and allocate private expenditure between housing and the numeraire good to maximize their utility. Rose-Ackerman argues, and we agree, that this structure embodies the important elements that a model of local jurisdictions should possess. Her investigation of the properties of the model, buttressed by an example, leads her to conclude that failure of existence of equilibrium is likely to be a pervasive problem in this model. Stiglitz (1977) also presents examples in which no equilibrium exists. In contrast to these authors, Westhoff (1977) establishes existence in a model with multiple local jurisdictions. Westhoffs model is simpler than Rose-Ackerman’s in an important respect. It contains only two goods, a publicly provided good and a numeraire commodity; there is no housing market. While Westhoff offers a model in which an equilibrium exists, an investigation of the properties of his model leads him to be pessimistic about the prospects for developing a model in which a unique, stable equilibrium exists. In his model, he finds that a stable equilibrium exists only when there are multiple equilibria; a unique equilibrium will always be unstable [Westhoff (1979)J It is perhaps not surprising that conditions for existence and uniqueness of equilibrium are difficult to establish in models with multiple local jurisdictions. Locational equilibrium may not exist in models with less than perfect divisibility of locations even if there is no problem of public good supply [Koopmans and Beckmann (1957)]. Yet indivisibility of locations is inherent in a model with a finite number of local jurisdictions. Similarly, existence and stability of voting equilibrium is problematic except under relatively restrictive conditions. It is natural to expect that problems of existence and stability of equilibrium would be compounded when both voting and locational choice are introduced into a single model. Hence, one should not expect to be able to establish existence and stability of equilibrium in models combining voting and locational choice without adding restrictions on tastes and technology beyond those customarily employed in economic models.’ ‘Recognizing the difficulties associated with characterizing political choice model with multiple jurisdictions, Richter (1982) studies a model in which

mechanisms in a properties of the

D. Epple et al., Equilibrium among local jurisdictions

283

This conclusion does not lead us to abandon the hope of establishing existence and stability of equilibrium in an empirically relevant model of local jurisdictions. On the contrary, we will argue in this paper that there is reason to believe that conditions for existence of a stable equilibrium can be established in such a model. Our case will be based on two types of evidence. The first is our previous work establishing existence of equilibrium in a model embodying what we believe to be empirically relevant structural features of a model of local jurisdictions. The second is a set of numerical results in which an equilibrium not only exists but is unique and stable. In addition to arguing that greater attention to existence and stability of local equilibrium is likely to bear fruit, we will identify and discuss key problems on which further research is needed. Roughly, these problems fall into two broad areas. One is the problem of the intimate links between intrajurisdictional equilibrium and mobility among jurisdictions. The other centers on assumptions about the technology of local public goods production. It has generally been assumed that the major problem of existence of a multiple-community equilibrium is mobility of population across communities. Put simply, the argument is that equilibrium will fail because the poor seek to enter wealthy communities in order to consume the high level of public goods provided there, the wealthy move elsewhere to avoid being taxed to provide a high level of public services shared by the poor, and a game of ‘musical suburbs’ results. 3 Our examples show that this can indeed happen, and they suggest conditions under which it is most likely to occur. Unfortunately, conditions that rule out this problem also impose restrictions on the set of assumptions that can be made in proving existence of equilibrium within a given community. In our discussion, we attempt to distinguish between conditions required for just one part of the problem (internal equilibrium or mobility equilibrium) and conditions that arise due to the linkages across pieces of the puzzle. Assumptions about the technology of public goods production play a crucial role in determining the existence and properties of an equilibrium. Conditions on the technology of public good supply that rule out the problem of ‘the poor chasing the rich’ are conditions that, in our examples, frequently give rise to multiple equilibria. Somewhat surprisingly, the problem of the poor chasing the rich is less troublesome than the case in which the wealthy prefer the equilibrium in the poor community and the poor

outcomes of intrajurisdictional political choice processes are assumed but the processes are not specified. With this strategy, he is able to establish existence of equilibrium in a model with a relatively general characterization of private good markets. However, his approach does not permit study of the role of political choice mechanisms, a central concern in the study of the political economy of local governments, 3This potential source of community instability is discussed by Mills and Oates (1975, p. 4).

284

D. Epple et nl., Equilibrium

among local jurisdictions

prefer the equilibrium in the wealthy community - everyone wants to move. This case, too, can be ruled out by restrictions on the technology of public good supply. The crucial role of the technology of public goods supply suggests this area as a second important focus of further research. This paper draws on our earlier work [Epple, Filimon and Romer (1983)] in which we establish existence of equilibrium in a model similar to that discussed in this paper. Here our purposes are to attempt to explain the role of various assumptions employed in the proof, to indicate the difficulties that arise when those assumptions are relaxed, and to point to some potentially promising avenues for future research on uniqueness as well as existence. Unlike the previous model, the model in this paper does not require housing supply functions in communities to be perfectly inelastic. In addition, an ad valorem rather than a specific tax on housing is assumed in this paper. The strategy used to prove existence of equilibrium in the earlier model is easily modified to apply to the model in this paper. This being the case, we outline rather than detail the proof of existence in this paper. To make this paper relatively self-contained, we summarize at various points arguments that appear in our earlier paper. The plan of the remainder of this paper is as follows. In section 2 we introduce the model and state some necessary conditions for existence of equilibrium. The problem of intracommunity equilibrium is discussed in section 3. In section 4 we discuss intercommunity equilibrium. In section 5 we introduce an example and compute equilibria for a variety of parameter settings. We present cases with unique equilibria (both stable and unstable), with multiple equilibria, and with no equilibrium. Section 6 contains our conclusions. 2. The model and some necessary conditions for equilibrium The economy of our model consists of a continuum of consumers. There are three goods: a local public (i.e. collectively provided) good, x; a local housing good, h; and a ‘composite’ private good, b. There are Tcommunities (where T is an exogenously given positive integer). An individual lives in one community only, and consumes housing and the public good only in the community where he lives. Moving from one place to another, however, is costless. In each community, the provision of the public good is financed by taxing residents on the amount of housing they consume. The tax rate and amount of the public good supplied in the community are determined by a vote of residents of the community. Our concern is with establishing conditions for existence of equilibrium in this multicommunity model.4 We define an inter-community equilibrium of our 4More precisely, we are interested in finding equilibria other than the ‘trivial’ equilibrium of T communities identical in every respect. Such trivial equilibria will typically exist under conditions much weaker than those we use in this paper. The study of these equilibria is not very interesting.

285

D. Epple et al., Equilibriumamong localjurisdictions

model to be an allocation of individuals to communities, of goods to individuals such that:

communities

(1) Each individual lives in one and only one community. (2) Each community has positive population. (3) No individual can increase his utility by moving munity. (4) Each community is in internal equilibrium. We (1) (2) (3) (4) utility.

and,

within

to another

com-

define an internal equilibrium to be an allocation such that: The aggregate demand for housing equals the supply of housing. The community budget is balanced. There is political equilibrium. No individual can, by altering his consumption bundle, increase

his

We assume that all individuals have the same preferences, which can be represented by a utility function U(x,h, b). We take U to be increasing, strictly quasi-concave, and twice continuously differentiable in all its arguments. Individuals differ in their exogenously given income Y, whose value is normalized to lie in the interval [ 1,2]. In this exposition, we will assume that y is uniform on this interval - although this is not a critical restriction. It proves to be convenient to represent consumer preferences in (x,P) space. Therefore, we let V(x, p, y) = max U(x, h, b) (h, b)

s.t. yzph+b

(1)

be the indirect utility function obtained when an individual with income y optimally chooses h and b subject to given values of x and the price of housing, p. We let h(p, y) denote the housing demand function that results from the solution of the above maximization problem. As this notation suggests, we assume that individual housing demand does not depend on the amount of the locally provided good. For a given consumer, the slope of an ‘indirect’ indifference curve in (x, p) space is given by

M(x,P,Y)=g

=_ =

v v

-

K(X?P, Y) yA%P?Y)

u, Cx, h(P, Y), Y - PWP, = h(P> Y)

ub

Lx,

h(p,

A

y -

41

ph(p,

Y)]

. (2)

A key assumption of our model concerns how the slope of an indirect indifference curve changes as income changes. We assume, following Ellickson (1970) and Westhoff (1977), that indifference curves in the (x,p) plane

D. Epple et al., Equilibrium

286

become

steeper

as income

~M(--G P>Y)> ay

among local jurisdictions

changes:

()

(3)

.

This powerful assumption permits for intercommunity equilibrium:

us to establish

three necessary

conditions

(Cl) Stratification. Each community is formed of individuals with incomes in a single interval. The set of community incomes partitions the interval [ 1,2]. This stratification condition, analogous to that of Ellickson and Westhoff, implies that communities can be ordered by the incomes of consumers. For each pair of ‘adjacent’ communities in this partition, there will be a ‘border’ consumer who has the highest income in one community and the lowest income in the other. This condition is illustrated in fig. 1 for the case of two communities. The indifference curve of the border individual is denoted by that individual’s income, yr. Indifference curves of two other individuals, whose incomes are y’ and y”, are also shown in fig. 1. The assumption that indifference curves become steeper as income increases implies y”> y’ > y’. Since utility increases downward to the right, it is clear that y” prefers the community with (x2, p2) , while y’ prefers the community offering (x’, p’). P’ Y”

Fig.

I

287

D. Epple et al., Equilibrium among local jurisdictions

The following condition, assumption of a continuum (C2) Boundary indifference. communities is indifferent

illustrated by y’ in fig. 1, is implied of consumers. The ‘border’ consumer between between the two communities.

That the following condition must conditions is obvious from fig. 1.

be satisfied

(C3) Ascending bundles. If y’ is the highest income the highest income in community j, then in pi <$ iffyicy’.

given

by the

any two ‘adjacent’

the two preceding

in community i and yj is equilibrium xi
These three necessary conditions follow from our assumptions about preferences and our specification of the consumer’s decision problem. They do not depend on assumptions about the technology of producing the public good. In defining the indirect utility function, we assumed that consumers take x and p as fixed. We also assumed that individuals are free to consume as much of the housing and numeraire commodity as they wish at the specified values of x and p. The three necessary conditions hold for any mechanism for allocating the public good that satisfies these assumptions about the consumer’s choice problem.’ These three conditions simplify the search for existence conditions because they restrict the set of allocations that need to be considered as potential candidates for equilibrium allocations. The problem of establishing conditions for existence of internal equilibrium is logically prior to the problem of establishing conditions for existence of a multicommunity equilibrium. Therefore, we take up the problem of internal equilibrium in the next section. Intercommunity equilibrium will then be considered in section 4. 3. Internal equilibrium To characterize the internal equilibrium, it is necessary to specify how individuals behave in their role as voters. In particular, we must specify what predictions voters make about changes in the price of housing, the tax base, and the supply of the public good that will result from changes in the tax rate. Given these assumptions about voter behavior, we must prove that a political equilibrium exists. Finally, to establish internal equilibrium, it is necessary to prove that there exists an allocation for which the political equilibrium and housing market equilibrium are mutually consistent. We assume that voters take community population, the net-of-tax housing price, and the community tax base as given. Voters are assumed to know the ‘These minimum

conditions might be violated if, for example, level of housing consumption on residents.

community

zoning

laws

imposed

a

288

D. Epple et al., Equilibrium

among local jurisdictions

community budget constraint and the identity relating net-of-tax and grossof-tax prices of housing. Using these assumptions, the voter predicts the gross-of-tax price of housing and the supply of public good that will result from any proposed tax rate. To make this more precise, let t be an ad valorem tax rate on housing, P,, be the net-of-tax housing price, H be the aggregate amount of housing consumed, and N be community population. Let the technology of public good supply be represented by the cost function c(x, N) which specifies the total cost of supplying x units of the public good to each of the N members of the community. The community budget constraint is then: tp,,H = c(x, N).

(4)

The gross-of-tax housing price is related rate by the following identity:

to the net-of-tax

price and the tax

(5) Using

this identity

P=Ph+-

and the community

4x2 N) H

budget

constraint,

we have:

(6)

At the time of voting, the voter takes phr N, and H as given. Hence, given values for these variables, (6) characterizes the voter’s perceived tradeoff between the gross-of-tax price of housing and the supply of the public good. We assume that the provision of the public good in the community is determined by an idealized form of majority rule. This process requires that the amount of the local public good that is chosen be preferred to any other available alternative by at least half the citizens. In the preceding paragraph we specified the way a voter predicts the value of p that will result from a given value of x. Thus, when voting, each resident selects from the available (x,p) pairs the alternative that maximizes his utility. We have already stated that when voting, residents take the aggregate amount of housing in the community as fixed. It is also necessary to specify what the voter assumes about changes in his own housing consumption that he will make in response to changes in x and p. One might assume that, when voting, each voter anticipates adjusting his own housing consumption. This implies that the voter evaluates alternative (x,p) pairs by taking into account the housing choice that would be optimal for him at each (x,p) pair along the community budget constraint (6). This approach allows the voter to consider adjustments about which he is likely to be knowledgeable namely his own potential housing decisions. While such a view of voter

D. Epple et al., Equilibrium among local jurisdictions

289

behavior assumed

has some appeal, it involves a certain inconsistency. Voters are to consider adjustments in their own housing consumption, yet treat aggregate housing demand in the jurisdiction as fixed; that is, they do not expect others to adjust. Alternatively, we could assume that citizens regard voting and housing decisions as quite distinct. Because of the atemporal nature of our model, we cannot formally treat home buying and voting as sequential decisions. It may, however, be useful to think of a citizen as someone who has taken up residence in a jurisdiction and made a housing decision there. Given this decision, a voter may well evaluate alternative tax rates under the assumption that his own (and others’) consumption of housing is fixed, and will not change in response to a change in the tax rate. Fortunately, we do not have to choose between these alternative characterizations of voter behavior. It can be shown that any internal equilibrium in which voters anticipate varying their own housing consumption will also be an internal equilibrium if voters do not anticipate varying their own housing consumption.6 Henceforth, we make the former assumption. With two additional assumptions, the existence of a unique political equilibrium for any finite values of P,,, H, and N can be established. The first assumption is that voter indifference curves in the (x,p) plane are concave. The second assumption is that the function c(x, N) is weakly convex in x for any N. This assumption rules out increasing returns to scale in the production of x. Under the conditions specified above, it can be shown that a unique political equilibrium exists for any finite ph, H, and N. For given P,,, H, and N, voters have single-peaked preferences in x. Our assumption (3) implies that a voter’s most-preferred level of x on the community budget constraint increases with y. The (x,p) pair chosen in political equilibrium will be the pair on the community budget constraint that maximizes the utility of the median income resident. Let y” denote the income of the median income resident. This individual’s utility-maximizing choice of (x,p) is determined by the point of tangency between his indirect indifference curve and the community budget constraint:

The existence of political equilibrium for arbitrary p,, and H does not guarantee existence of internal equilibrium. For internal equilibrium, the taxspending decisions of voters must be consistent with the actual housing decisions that result from individual adjustments to the gross-of-tax housing price implied by the political equilibrium. In other words, the aggregate tax 6This is demonstrated

in Epple, Filimon

and Romer (1983).

290

D. Epple et al., Equilibrium

among local jurisdictions

base at the politically selected tax rate must be exactly the tax base that yields the revenue required to finance the level of public spending determined by the political process. Therefore, for internal equilibrium, it is necessary that conditions (6)+8) be satisfied simultaneously, where condition (8) specifies that aggregate community housing demand equal community housing supply:

H,(P)= Hs(P/J. Aggregate

housing

demand

(8) is defined

as follows:

H,(P)= j NP,Y) dy, 2

where y and j are respectively commt&ty. Recall from (Cl)

the poorest and wealthiest that in an intercommunity

individuals in the equilibrium, the

incomes of residents will be continuous on [y, j]. We assume a fixed amount of land in the community and a constant returns production function for converting land and numeraire inputs into housing. These conditions and competitive behavior by land owners and housing housing supply function. In general, producers generate the aggregate Hi (P,,) 2 0. We assume that land is owned by and rents accrue to ‘absentee landlords’ who reside outside the model. For the study of internal equilibrium to be of interest, an equilibrium must be viable. That is, there must be some values of p,,, p, and x such that eqs. (6) and (8) can be simultaneously satisfied. Were this not the case, there would be no allocation that would simultaneously satisfy the community budget constraint and the housing market equilibrium conditions. Henceforth, we will assume that this basic feasibility condition is satisfied; the community is viable. Note that if c(O,N) =O, then this condition is trivially satisfied. However, in section 4 we wish to consider cost functions for which there is a so that ~(0, N) >O. For such cost fixed cost of community formation, functions, the viability conditions will not be satisifed for communities with arbitrarily small population. Difficulties that can arise in establishing internal equilibrium are depicted in fig. 2 where, for ease of illustration, we suppose that c(x,N)=c,x, with c1 a positive constant. In fig. 2(a) an internal equilibrium is illustrated. Voters take ah and fi as given when they vote. They select 2 with the expectation that price 8 will then prevail in the housing market and that fi,, and c? will not change. In the case shown in fig. 2(a), these expectations are confirmed. Heuristically, the following situation must be ruled out. Given a housing market equilibrium as depicted in lig. 2(b), voters choose an 2 and expect $, l?, and Lj,, to be the housing market equilibrium. However, at j less housing

291

D. Epple et al., Equilibrium among local jurisdictions

HI

h 0 Fig. 2(a)

H,(P)/

)

X

X

t

--_-

H

H

0 Fig. 2(b)

x x

is consumed, say 8, and the net-of-tax price is 8,. The new community budget constraint then has a lower intercept and higher slope than the previous one. Voters=choose i and expect j to prevail in the housing market, but at fi less than H units of housing are consumed. To establish internal equilibrium, it is necessary to show that the process depicted in fig. 2(b) does not result in an infinite gross-of-tax housing price. Our strategy for proving existence of equilibrium is as follows. Let H = H,(p) in (6) and (7). Solve eq. (8) to obtain an expression for ph as a function of p. Substitute the result into (6). Then (6) becomes an expression in x and p for a community of fixed population. Solve this expression for x

292

D. Epple et al., Equilibrium among local jurisdictions

in terms of p and denote the resulting function, x(p). Substituting x(p) into the left-hand side of (7) we obtain an expression in p which we denote 4(p):

4(P)= WX(P)>P, $9. Similarly, expression

substituting x(p) into the right-hand in p which we denote 8(p):

e4 = cx(X(P)? WHd(P).

(9) side of (7) we obtain

another

(10)

If some fairly restrictive conditions on preferences are satisfied, we can show that there exists a finite p for which &p)=B(p), and hence that an internal equilibrium exists. 7 These conditions guarantee that B’(p) 20 and d’(p) 5 0 for all p, and that lim 4(p) = co P’P:

and lim 4(p) = 0. P-m

‘The additional

conditions

aM(x, P%Y)
=’

aM(x3 P. Y)
=’

aM aM z+-
are: (4

(ii)

(iii)

lim M(x, p, y) = co, x-0 lim M(x, p, y) = 0, P-m

(iv)

wfdi4,

(vi)

ap =

o.

(4

Conditions (i))(v) impose further restrictions on the slopes of indirect indifference curves in the (x,p) plane. Conditions (i)+ii) require that the slope of an indifference curve be non-increasing in x and p and, at any point, decreasing in at least one of these variables. Condition (iv) requires that the slope of an indirect indifference curve become infinite as x goes to zero, while (v) requires that the slope go to zero as p goes to infinity. Finally, (vi) requires that the price elasticity of housing demand not exceed one in absolute value.

D. Epple et al., Equilibrium among local jurisdictions

As a consequence, equilibrium exists.’

293

4 and 8 cross once as in tig. 3(a), and an internal 8(p)

.5 -

I

0

I

#

5

6

6

8

-P

(a) A

+,e

20lo! 0

,Y? 2

4

P

(bl Fig. 3

4. Intercommunity

equilibrium

In section 2 we identified three conditions necessary for equilibrium. Those three conditions and the requirement that each community be in internal ‘The conditions of footnote 7 establish not only existence but also uniqueness of internal equilibrium. As we point out in footnote 12, uniqueness of internal equilibrium plays a role in the proof of intercommunity equilibrium. Existence of internal equilibrium can be shown under weaker conditions than those in footnote 7. In particular, a referee has pointed out that ‘The assumption that housing demand be inelastic, which is not innocuous, is not required for internal equilibrium.’ When c(O,N)=O, the condition H,(p)>O,Vp, can be substituted for (vi) in footnote 7. [As we shall note later, ~(0, N) =0 causes problems in proving intercommunity equilibrium.] If c(O,N)>O and we omit (vi), then we need some condition that assures viability for arbitrary p, so that continuity of 4(p) and B(p) is preserved.

294

D. Epple et al., Equilibrium among local jurisdictions

equilibrium are sufficient for equilibrium. From our definition of equilibrium, it can readily be verified that an allocation satisfying these conditions also satisfies all the criteria for an allocation to be an equilibrium. Therefore, in this section we discuss restrictions on tastes and technology in our model that will assure the existence of an allocation satisfying the sufficient conditions. We discuss the two-community case, but the results generalize to an arbitrary number of communities.’ To the extent possible, we wish to distinguish conditions required for internal equilibrium from those required for intercommunity equilibrium. Consequently, in this section we assume that viable communities are in internal equilibrium. We retain the assumptions on tastes specified in section 2. In addition, we keep the assumptions employed in proving political equilibrium in section 3; namely, the assumptions on voter behavior, the assumption that c(x, N) is weakly convex in x, and the assumption that indirect indifference curves are concave in the (x,p) plane.” Let y’ be the border individual between communities 1 and 2 in an allocation satisfying (Cl). A choice of y’ determines the population in each community. Thus, for each choice of y’ that defines two viable communities, the internal equilibria in both communities can be determined. Let (x’(y’),p’(y’)) denote the (x,p) pair in community i that arises in internal equilibrium when y’ is the border individual. Let V’(yl) = V(x’(y’), p’(y’), y’) be the utility achieved by border individual y’ in community i, and let F(y’) be the difference in the utility levels y’ can obtain in the two communities: F(y’) = V'(y')- V2(y’).

(11)

Condition (C2) will be satisfied for values of y’ for which F(y’)=O. One strategy for attempting to prove intercommunity equilibrium is to establish conditions to assure that there exists a y’ such that F(y’)=O. An allocation for which F(y’) =0 would, by construction, satisfy (Cl) and (C2) and define two communities in internal equilibrium. A difficulty in pursuing this strategy is that such an allocation need not satisfy (C3). The problem is illustrated in fig. 4. The border individual is indifferent between communities 1 and 2, so that (C2) is satisfied. Suppose, further, that (Cl) is also satisfied, so that all individuals in community 2 have incomes greater than y’, while all individuals in community 1 have incomes less than y’. Finally, suppose that each of the communities is viable and in internal equilibrium. It follows ‘Under conditions assumed in this section, existence of a two-community equilibrium assures existence of a one-community equilibrium. Indeed, existence of equilibrium with T communities implies existence of equilibrium with T’ communities, for any 7” < 7: i°Conditions (i)+;i) of footnote 7 play no role in our discussion of intercommunity eauilibrium. Should conditions for internal equilibrium weaker than (i)qvi) but still cdnsistent with intercommunity equilibrium 2 be-found, the entire discussion of section 4 would be valid under those weaker conditions.

D. Epple et al., Equilibrium among local jurisdictions

295

WX

I

Fig. 4

that the indifference curve of the median income resident of each community will be tangent to the budget constraint of the community, as depicted in fig. 4. Fig. 4 thus presents an allocation in which each community is in internal equilibrium and conditions (Cl) and (C2) are satisfied. However, (C3) is not satisfied. Everyone in community 2 prefers the bundle available in community 1, and everyone in community 1 prefers the bundle in community 2. In the allocation shown in fig. 4, everyone wants to move. Conditions on tastes and technology imposed thus far are not sufficient to rule out this problem, as we demonstrate by examples in section 5. An allocation for which IQ’) = 0 need not be an equilibrium. By imposing further restrictions on the technology of public good supply and the housing demand function, it is possible to rule out the problem that arises in fig. 4. Suppose the cost function for supplying the publicly provided good is c(x,N)=c,+c,xN,

(14

where c0 20 and c1 > 0. Substituting this cost function into eq. (6), we obtain the following equation for the budget constraint in community i:

D. Epple et al., Equilibrium among local jurisdictions

296

xiNi pi=p:+s+c-

Hi

.

(13)

For the case depicted in fig. 4, the slope of the budget constraint in community 2 must exceed the slope of the budget constraint of community 1. This follows from the requirement of political equilibrium that the slope of the median income voter’s indifference curve must be tangent to the community budget constraint. From (13), this slope condition implies c,N2/H2 > c,N’/H’. This implies that HI/N’ > H2/N2; i.e. housing consumption per capita is greater in the low- than in the high-income community. If we assume housing demand to be non-decreasing in income, then this can happen only if the price of housing is lower in community 1 than in community 2. But the case depicted in fig. 4 requires that the price of housing be higher in the low-income community. Thus, with the cost function in (12), the problem case of fig. 4 cannot occur. Fig. 5 illustrates one of the cases that can occur. Intercommunity equilibria in which the budget line for community 1 cuts the budget line for community 2 from below are also possible. With the cost function in (12) an allocation satisfying the requirement F(y’) =0 in (11) will satisfy all of the sufficient conditions for an equi1ibrium.i’ The final problem is to establish conditions to guarantee that there is a y’ such that F(y’) =O. To this end, we introduce the following additional restrictions on preferences. Bundles that provide positive consumption of all goods strictly dominate bundles in which any good is not consumed; i.e. for any x>O, h>O, b>O and X~O,6~0,&~0:

U(x, h, b) > U(X, 0,b), and U(x, h, b) > U(X, h, 0). “The problem illustrated in tig. 4 also brings out a difftculty that relates to our discussion of internal equilibrium in section 3. We could have established existence of internal equilibrium more directly by assuming that voters correctly predict how p, ph, H, and x vary with t. In this way, housing market adjustments would enter voters’ calculus directly, and political equilibrium would necessarily be consistent with housing market equilibrium. We would still need to impose restrictions on preferences and aggregate housing demand to guarantee that such equilibria exist and are well behaved. While far from innocuous, these restrictions are somewhat weaker than those used in section 3. When it comes to intercommunity equilibrium, however, assumptions that ensure existence of internal equilibrium when voters fully take into account housing market adjustments are not enough to rule out the case shown in tig. 4. Indeed, with this alternative assumption of voter behavior, restricting c(x, N) to the form given in (12) is not sufficient to rule out the problem of fig. 4. Thus, although getting internal equilibrium may become ‘easier’, intercommunity equilibrium requires stronger restrictions.

D. Epple et al., Equilibrium among local jurisdictions

291

““+p:, H’

3 +p; Ha ,X

0 Fig. 5

In addition, we further restrict the technology of public good supply in (12) by requiring c,, > 0. With cO>O, there will be a y’ sufficiently small, say yr, such that the lowincome community cannot raise sufficient tax revenue to pay the fixed cost of community formation and provide a positive quantity of the public good. Under these conditions, given the assumptions on preferences introduced immediately above, the border individual would prefer the high-income community; hence F(y,) ~0. Similarly, there is a value of y’ sufficiently large, say y,, such that the high-income community cannot raise sufficient taxes to pay the fixed cost of community formation and provide a positive amount of the public good. Under these conditions, the border individual will prefer the low-income community; F(y,) > 0. F(y’) is continuous in y’ if internal equilibrium is continuous in y’. Continuity and the above conditions imply that there will be at least one value of y1 such that F(JJ’)=O.‘~ Indeed, there may be more than one such value. Panels (a) and (b) in fig. 6 illustrate cases in which there are one and three equilibria, respectively. ‘*The conditions discussed in the previous section (and presented existence of unique internal equilibrium for any viable community. valued and continuous in y’.

in footnote 7) guarantee the Given this, F(y’) is single-

D. Epple et al., Equilibrium among local jurisdictions

298

F&1

(a) F(y’)

(b) Fig. 6

Fig. 6 also illustrates an important point about the stability of equilibrium. An equilibrium in which F(y’) crosses the horizontal axis from below will be unstable. For example, in fig. 6(a), consider an allocation in which the border individual has less income, say j’, than the equilibrium value of y’. Then F(j’) < 0 and j1 will prefer the high-income community. If j1 were to move in accord with her preferences, the income of the new border individual would be lower still. Thus, those individuals wishing to move would not move in the direction that would tend toward the equilibrium. By a similar argument, it can be shown that if the border individual had income greater than the equilibrium level in fig. 6(a), that individual would prefer the lowincome community - a movement away from the equilibrium. Allocations in which the function F(y’) crosses the horizontal axis from above are stable. Of the three equilibria in fig. 6(b), the intermediate value of y’ is stable, and the others are unstable. Fig. 6 illustrates an unresolved problem regarding stability of equilibdiscussed in this section, rium.’ 3 Given the strategy of proof of equilibrium there will be a stable equilibrium only if there are multiple equilibria. A unique equilibrium will be unstable. Within the framework we have presented, there are two possible tacks we might follow in addressing this problem. One is to ignore unstable equilibria as irrelevant and to attempt to 13Westhoff (1979) raised the issues of stability and multiple equilibria in a model without a housing market. In discussions with us, Westhoff suggested - correctly - that his approach to analyzing these problems would prove useful in our model.

D. Epple et al., Equilibrium among local jurisdictions

299

establish conditions under which there will be one stable equilibrium, i.e. to attempt to establish conditions under which the function F(y’) conforms to the pattern in fig. 6(b). The second alternative is to attempt to prove that F(y’) has a unique zero without imposing restrictions that make small communities unviable. The objective would then be to find conditions such that, as y’ increases from its lower to its upper bound, F(y’) is positive, crosses the horizontal axis once, and remains negative thereafter. In the next section, we introduce an example and investigate existence and stability of equilibrium for a variety of parameter values. These results suggest the types of conditions that are likely to be required to establish existence of a unique, stable equilibrium if either of the strategies discussed in the preceding paragraph is pursued. The example also illustrates the roles of various assumptions and suggests problems that will have to be confronted if those assumptions are to be relaxed. 5. A two-community The utility

function

example in our example

U(x, h, b) = [CYX i’P+(l

is:

-cY) [w(h,b)]“P]P,

(14)

where w(h, b) =

[/?I++ (1 -/I)!?“] ljS.

This utility $unction, with O
kY)=Y

[ 1-s 1 >

where

By substituting

these

demand

functions

into

the utility

function

we obtain

300

D. Epple et al., Equilibrium among local jurisdictions

the indirect

utility

function:

W4 P, Y) = tax “p+(l

-cI) [ti(p,y)ll’p-JP,

where

Given our assumption that income is uniformly distributed on [ 1,2], aggregate housing demands in communities 1 and 2 for arbitrarily chosen ‘border’ y’ are:14 H;

=

C(Y')'-

11

’

&(P')

,,=C4-W121 d

&dP2)

.

Viability of community 1 requires that there exist a tax rate such that it is feasible to supply a positive amount of the publicly provided good. Let y, be the value of y’ at which the community could just pay cO with the maximum tax revenue that could be raised in the community. Viability of community 1 will then require y’ >y,. Tax revenue in community 1 is: t’p,‘H; =-

t’p’ fp= t'C(Y')'-

1lP'

2(1+ t’k(P’)

1 +t’

(15)

.

The first equality follows from (5) and the second from use of the housing demand function for community 1. The market clearing price of housing can be written as a function of t’ by using (5) to eliminate ph from (8) and solving the resulting expression for p ‘. Denote the resulting function p’(t’). By substituting p’(t’) into (15) and differentiating the resulting expression with respect to t’, we can verify that tax revenue increases as t’ increases. Hence, y, is defined by the following equality:

lim t’C(yd’- llP’(t’) fl+oD 2(1 +t’)g(pl(t’)) It is easily

verified

that

14Hi and H,’ satisfy condition

=co.

p’(tl)--+a

as t’-+cO.

(vi) of footnote

7 if 6 20.

Thus,

taking

the above

limit

D. Epple et al., Equilibrium among local jurisdictions

301

gives: 112

2%

1’)

if 6=0,

7-l

y,=

(2c, - 1)“2

It can be similarly where:

if 6 < 0.

be shown

that viability

of community

2 requires

y’
if 6=0, Yh =

if 6 < 0. Our viability assumption requires that there be a yr for which both communities are viable, i.e. yI
where K is the non-land

input

and Lis land. This can be rewritten

as:

where k is the ratio of non-land to land inputs. The competitive protitmaximizing choice of k is given by the following relationship, in which P, is the price of non-land inputs:

1 llr

k=

$(1-p) [

k

Substituting this expression housing supply function:

H

.

back into the production

function,

we obtain

the

1

PhPPL) (l--p)‘B

=L

s [

pk

Since the price of non-land inputs is exogenously given and uniform across jurisdictions, it is convenient to choose units of K such that P, = 1 -,u. Also, let 0=(1 -p)/p. Then the housing supply function is: H, = Lpft.

D. Epple et al., Equilibrium among local jurisdictions

302

We begin by choosing a set of parameter values that may be used as a benchmark for investigating the effects of perturbing the parameters. In our derivation of the housing supply function, p is the share of expenditure on housing that accrues to land. This share is on the order of 20-25 percent [Mills (1972, p. 80)]. Hence, a long-run housing supply elasticity of 3 or 4 is probably the right order of magnitude. We let the benchmark price elasticity of housing demand be - 1, which implies 6 =O. Housing expenditure net of tax as a proportion of consumer income is approximately 25-30 percent. Property tax rates are on the order Therefore, we assume the gross-of-tax share of exof 2&30 percent.15 penditure on housing to be one-third (/?=0.33). With a property tax rate of 0.25, the net-of-tax share of expenditure on housing will then be roughly 25 percent of income. Communities are each assumed to have 0.01 unit of land: L=O.Ol. Other parameter values were chosen to yield equilibrium property tax rates of about 25 percent. A variety of parameter values yield equilibrium tax rates of this magnitude. We chose the following values as a convenient benchmark: 6=0,

p=o.33,

p= -0.2,

co=o,

ci =0.125,

9 = 3.0

The only one of these benchmark values that does not satisfy the restrictions imposed in section 4 is cO. However, as we show below, equilibria with quite desirable properties are obtained when c0 equals zero. As we indicated in section 4, an allocation with F(y’)=O will satisfy all conditions for an equilibrium using cost function (12). Thus, the allocations computed in these examples with cc, = 0 are, indeed, equilibria. We will also illustrate the effects of choosing various positive values for cO. In tables l-3 we illustrate the effects of varying the parameters about the benchmark values. The most interesting finding in tables 1 and 2 is that in all cases there is a unique, stable equilibrium. That is to say, the function F is positive for small values of yl, negative for large values of y’, and zero at only one value of y’. In tables 1 and 2, cc, is equal to zero. Beyond these examples, we have not established existence of equilibrium for the case in which c0 is zero, and we have not established stability or uniqueness of equilibrium for any case. Hence, these results are both interesting and encouraging, since they suggest that a unique, stable equilibrium exists for a wide range of parameter values. Examining the tables in more detail, we find considerable regularity in the effects of varying individual parameters. In table 1 we vary the the housing ‘5Property tax rates are normally specified as a fraction of property value rather than a fraction of annual implicit rental. The latter is appropriate in the context of our model. Observed property tax rates of 2 or 3 percent of property value translate to taxes on implicit rental on the order of 2C-30 percent.

1o-2

1o-2 lo-’ 1o-2

tom7 1o-2

1O-2

LZ 0.47 0.69 0.64 0.62

1.51

0.64

0.73 1.99 l.OtMOl 0.49 1.OOOOO20.61

1.56 1.38 1.10 1.02

Variables x1 y’

0.88

0.94 0.73 0.91

0.64 0.96 0.94 0.92

x2

2.52

3.09 2.99 1.34

8.67 1.60 1.33 1.29

p’

2.71

3.42 3.09 1.39

9.17 1.72 1.39 1.35

p2

0.24

0.22 0.23 0.30

0.16 0.28 0.30 0.30

t’

“Other benchmark parameter values: a=0.2, a=O.33, p= -0.2, c,=O, c, =0.125, 6=0. Feasible y’: 1 i y’ < 2.

3.0

3.0 3.0 50.0

Land areas, L’, L2, and housing supply elasticity, 0

Benchmark values”

0.5 10.0 50.0 100.0

Housing supply elasticity, 0

10-l

Parameters 0 L’

Effects of Varying:

Table 1

0.027 0.094 0.025 0.005

H’

0.24

0.083

0.22 0.16 0.22 14x lo-’ 0.296 44x lo-’

0.16 0.28 0.30 0.30

tZ

0.105

22 x lo-’ 0.16 0.35

0.028 0.200 0.332 0.363

HZ

9

S : OS 6 B 2. 3. P i; =1

2 c. =: 9 5

h 2 6 r: “F

?

304

D. Epple et al., Equilibrium

among local jurisdictions

supply elasticity and the relative amounts of land in the two communities. An increase in the housing supply elasticity decreases the population of the low-income community, decreases equilibrium gross-of-tax housing prices in both communities, and increases equilibrium tax rates in both communities. The level of public good supplied in both communities rises to a peak and then declines as the housing supply elasticity increases from 0.3 to 100. The values of y’ in table 1 reveal that the poor community is very small for large values of 0. In fact, if we continue to increase 0, we find that 8 can be made large enough that no equilibrium exists. The failure of existence can be intuitively interpreted as arising because ‘the poor chase the rich’. That is to say, the individual yi at the ‘border’ of the two communities will always prefer the wealthy community to the poor community for every value of y’. Put differently, the wealthiest member of the poor community will always wish to move to the wealthy community. While an equilibrium can fail to exist when c0 equals zero, it is interesting to note that, in our example, this occurs only for high values of the housing supply elasticity. This suggests that, when there is no fixed cost of community formation, equilibrium is likely to fail to exist when the stock of housing can be increased with relatively little increase in unit housing prices net of tax. It might be thought that a similar outcome would occur if land area in one community were set very high relative to land in the other community. However, we have found that even with land areas differing by as much as a factor of lOi’, a unique two-community equilibrium exists. In table 2 the per capita unit cost of the local public good, the substitution parameters p and 6, and the intensity parameter CI are varied. In all cases in Table 2 Parameters

Effects of varying:

Cl

Per capita unit cost of local public good, ~1

0.05 0.15 0.2 0.5

Substitution parameter,

0

Intensity parameter,

a

Substitution parameter,

6

Benchmark

values”

Variables y’ x’

t’

t2

H’

HZ

0.10 0.29 0.39 1.26

0.10 0.28 0.38 1.14

0.097 0.080 0.072 0.042

0.109 0.104 0.101 0.080

0.0

1.57 0.54 0.75 2.53 2.55 0.19 0.19 0.096 1.53 0.61 0.84 2.52 2.66 0.22 0.22 0.088 1.47 0.70 0.96 2.51 2.80 0.28 0.27 0.076

0.098 0.108 0.109

0.1 0.3 0.4

0.0

1.52 0.57 0.78 2.48 2.64 0.21 0.20 0.086 1.50 0.70 0.96 2.55 2.76 0.27 0.26 0.081 1.50 0.75 1.03 2.58 2.80 0.29 0.29 0.079

0.106 0.104 0.104

-0.2

0.2

0.3 -0.5

1.45 0.69 0.95 2.54 2.89 0.56 0.58 0.043 1.53 0.59 0.80 2.61 2.74 0.14 0.14 0.119

0.061 0.140

-0.2

0.2

0.0

1.51 0.64 0.88 2.52 2.71 0.24 0.24 0.083

0.105

a

6

-0.2

0.2

0.0

1.54 1.50 1.48 1.39

0.125

-0.05 -0.15 -0.33

0.2

0.125

-0.2

0.125 0.125

“Other benchmark parameter Feasible yl: 1~ y’ < 2.

p

x2

p’

p2

0.78 1.07 2.35 2.44 0.62 0.85 2.58 2.80 0.58 0.79 2.70 2.98 0.44 0.60 3.64 4.28

values: /I = 0.33, c,, =O, 0 = 3.0, L’ = L2 = 0.01.

305

D. Epple et al., Equilibrium among local jurisdictions

table 2 a unique, stable equilibrium exists. Note in table 2 that the condition 6 50, which is required to satisfy the conditions in footnote 7, is not necessary for existence of equilibrium. However, if 6 is increased sufficiently, a two-community equilibrium will fail to exist because internal equilibria will not exist. This is illustrated in figs. 3(a) and 3(b), where the functions 4(p) and 8(p) defined in (9) and (10) are plotted for two values of 6. These functions were computed with parameters other than 6 set at their benchmark values and all individuals located in a single community. Fig. 3(a) demonstrates that an internal equilibrium exists for 6 = -0.1, which satisfies the conditions in footnote 7, while fig. 3(b) demonstrates that no internal equilibrium exists for 6 = 0.42. The fixed cost of community formation is varied in table 3. Multiple equilibria occur at small positive values of cO. These multiple equilibria appear to be an artifact of imposing a fixed cost of community formation. In each case in which multiple equilibria occur, one occurs near the lower bound of feasible values of y’, one occurs near the upper bound of feasible values of y’, and one occurs at an intermediate value. Only the intermediate Table 3 Effects of varying the fixed cost of the local public good, c,,.

(1) Parameters

; P CO cl 0 6

(2) (a)

(3) (b)

(cl

0.33 0.2 0.33 0.2 0.33 0.2 0.33 0.2 -0.2 -0.2 -0.2 -0.2 0 0.01 0.01 0.01 0.125 0.125 0.125 0.125 3.0 3.0 3.0 3.0 0 0 0 0

(a)

(4) (b)

(5)

(cl

0.2 0.2 0.2 0.2 0.2 0.33 0.33 0.33 0.33 0.33 -0.2 -0.2 -0.2 -0.2 -0.2 0.025 0.025 0.025 0.05 0.075 0.125 0.125 0.125 0.125 0.125 3.0 3.0 3.0 3.0 3.0 0 0 0 0 0

Endogenous variables

0.50 0.74 3.01 3.12 2.51 0.26 0.006 0.153

1.49 0.63 0.87 2.61 2.81 0.32 0.29 0.078 0.104

1.97 0.12 0.93 3.12 3.47 0.26 3.41 0.153 0.005

1.18 0.53 0.77 3.01 3.15 1.33 0.32 0.021 0.137

1.46 0.61 0.85 2.78 2.98 0.48 0.36 0.067 0.104

1.92 0.70 0.92 3.17 3.49 0.31 2.10 0.140 0.014

1.80 0.67 0.88 3.28 3.59 0.46 1.37 0.113 0.035

1.67 0.62 0.84 3.56 3.85 0.75 1.23 0.083 0.052

1.03 1.98

1.03 1.98

1.03 1.98

1.07 1.96

1.07 1.96

1.07 1.96

1.14 1.92

1.20 1.88

Y1

1.51

1.06

;: PI PZ t’ t2

0.64 0.88 2.52 2.71 0.24 0.24 0.083 0.105

Feasible y1 : y, < y' < y, YI Yh

1 2

“Benchmark parameter values.

306

D. Epple

et al., Equilibrium

among local jurisdictions

equilibrium of the three is stable. At relatively large values of cO, we find a unique equilibrium. As discussed in section 4 and illustrated in fig. 4, we have not been able to establish conditions sufficient to guarantee existence of equilibrium when the publicly provided good is a pure public good, e.g. with a cost function c(x, N) = c,+c,x. We can, however, investigate whether an equilibrium exists for our example. Table 4 illustrates some types of outcomes that can occur in the pure public good case. Recall from section 4 that when we do not use cost functions of the form given in eq. (12), equilibrium can fail to exist because the ‘ascending bundles’ condition, (C3), may not be satisfied. If it is satisfied when F(y’)=O, then the allocation is an equilibrium. The unique equilibrium in column (1) was obtained with the benchmark parameter values, The results in columns (2a)-(2c) were obtained with ci =0.05 and all other parameters set at the benchmark values. Columns (2a) and (2b) are equilibria, but (2~) is not. Note that in column (c), x1 >x* and p1 >p2; the ‘ascending bundles’ condition is violated. Hence, while the individual with income y’= 1.86 is indifferent between the two communities, all other individuals wish to move. The results in table 4 are encouraging in that they demonstrate that there are conditions under which an equilibrium will exist when the locally Table 4 Locally

provided

pure public good, c(.u, N) = c,x

(2) Parameters

(1)”

(a)

(b)

(c)

;

0.33 0.2

0.33 0.2

0.33 0.2

0.33 0.2

I’ (‘0

-0.2 0

;;’ ii

-0.2 0

-0.2 0

-0.2 0

0.125 3.0 0

0.05 3.0 0

0.05 3.0 0

0.05 3.0 0

1.61 0.59 0.74 2.90

1.12 0.45 0.89 2.48

1.54 0.69 0.92 2.47 2.62 0.18 0.21 0.092 0.102

1.86 0.82 0.8 1 2.732 2.73 0.11 0.84 0.149 0.032

Endogenous variables

!‘I x’ x2 p: ;I-

tZ H’

HZ Feasible

0.40 3.20 2.80 1.05 0.65 0.11 0.090 0.018 0.073 0.161 y’ : 1 < 4” < 2 for all cases

“Benchmark

parameter

values

D. Epple et al., Equilibrium among local jurisdictions

provided good is a pure public good. the problem to be solved in establishing an equilibrium exists.

307

At the same time, the results exhibit general conditions under which such

6. Conclusion We have discussed two broad areas in which further work on equilibrium among local jurisdictions is needed. One concerns the relaxation of the relatively strong assumptions that our proof of existence of equilibrium has relied on so far. The other concerns identification of conditions under which a unique, stable equilibrium can be shown to exist. In our discussion of existence results, we have highlighted the roles of key restrictive assumptions. We have also distinguished between conditions used in establishing internal equilibrium and those employed in establishing intercommunity equilibrium. Our numerical examples illustrate several points. They demonstrate that a unique, stable equilibrium exists for a variety of parameter values. Hence, they suggest that a search for conditions sufficient to guarantee uniqueness and stability may be fruitful. The results also indicate conditions under which such an equilibrium can fail to exist; namely, conditions in which housing supply is relatively elastic. This finding suggests that restrictions on the housing supply function may play an important role in establishing existence of a unique, stable equilibrium. The numerical results also demonstrate that equilibrium exists for conditions on the technology of public good supply more general than those for which we have thus far proved existence. For both the formal analysis of existence and the interpretation of numerical results, the sufficient conditions we enumerated at the beginning of section 4 play a critical role. For the formal analysis of existence, they provide a structure for evaluating whether a specified set of restrictions on tastes and technology will generate an equilibrium allocation. For evaluating numerical results, they provide a set of criteria for determing whether a numerically computed outcome is an equilibrium. This is particularly useful for studying models in which assumptions regarding tastes and technology do not satisfy conditions known to generate an equilibrium allocation. Our sufficient conditions provide a useful point of departure for further study of existence, uniqueness, and stability of equilibrium. Our results to date rest on restrictions stronger than those customarily imposed on tastes and technology in economic models. We have suggested areas in which future research may lead to relaxation of some of these restrictions. However, as we indicated in the Introduction, we do not expect that it will be possible to eliminate such restrictions completely. Moreover, implementation of our suggestions for research on uniqueness and stability may result in imposing restrictions beyond those discussed in this paper.

308

D. Epple et al., Equilibrium

among local jurisdictions

Clearly, restrictions on preferences and technology cannot be accepted merely because they give rise to a model that has equilibria with desirable properties. By the same token, a model cannot be rejected simply because it embodies more stringent restrictions than the usual concavity and convexity assumptions. However, where such restrictions are imposed, it is desirable to test their validity directly where possible. Assumptions about the price elasticity of housing demand and about the form of the cost function for locally provided public goods can be evaluated using results from existing empirical studies. For these assumptions, the bulk of available evidence is consistent with the restrictions used in this paper.16 16This evidence is discussed in Epple, indifference curves in the (x,p) plane method for using available evidence to in condition (3). While the evidence is for this assumption in existing empirical

Fihmon and Romer (1983). Restrictions on the slopes of are harder to evaluate. Bucovetsky (1981) provides a test the validity of the fundamental assumption embodied not entirely unambiguous, he finds considerable support results.

References Bucovetsky, S., 1981, Optimal jurisdictional fragmentation and mobility, Journal of Public Economics 16, 171-19i. Ellickson. B.C.. 1970. Metronolitan residential location and the local public sector, Ph.D. Dissertation: MIT.’ I Ellickson, B.C., 1971, Jurisdictional fragmentation and residential choice, American Economic Review Papers and Proceedings 61, 334339. Epple, D., R. Filimon and T. Romer, 1983, Housing, voting, and moving: Equilibrium in a model of local public goods with multiple jurisdictions, in: J.V. Henderson, ed., Research in urban economics, vol. iI1 (JAI Press, Greenwich, Connecticut) 59-90. Eoole. D. and A. Zelenitz. 1981. The roles of iurisdictional competition and of collective choice L‘institutions in the market for local public goods, American- Economic Review Papers and Proceedings 71, 87-92. Koopmans, T.C. and M. Beckmann, 1957, Assignment problems and the location of economic activities, Econometrica 25, 53-76. Mills, E.S., 1972, Urban economics (Scott, Foresman, Glenview, Illinois). Mills. ES. and W.E. Oates, eds., 1975, Fiscal zoning and land use controls (D.C. Heath, Lexington, Mass.). Richter, D.K., 1982, Weakly democratic regular tax equilibria in a local public goods economy with nerfect consumer mobility, Journal of Economic Theory 27, 1377162. Rose-Ackerman, S., 1979, Marketmodels of local government: Exit, voting, and the land market, Journal of Urban Economics 6, 319-337. Stiglitz, J.E., 1977, The theory of local public goods, in: M.S. Feldstein and R.P. Inman, eds., The economics of public services (Macmillan, London) 274333. Westhoff, F., 1977, Existence of equilibrium in economies with a local public good, Journal of Economic Theory 14, 84112. Westhoff, F., 1979, Policy inferences from community choice models: A caution, Journal of Urban Economics 6, 5355549.