Tiebout without politics

Regional

Science and Urban

Tiebout

Economics

without

21 (1991) 469489.

North-Holland

politics*

David Pines Tel Aviv University, Received October

Ramat Aviv 69978, Tel Aviv, Israel 1989, fmal version

received August

1990

This paper re-examines the existence of a Tiebout-type equilibrium with price-taking developers. It is shown that, under Tiebout’s assumptions, including perfect replicability of communities, such an equilibrium exists and is efficient. But if communities cannot be replicated, equilibrium with profit-maximizing developers may not exist, unless strong assumptions regarding the preference and the production technology are adopted. Since the assumption of perfect replicability is in itself unrealistic in view of the uneven local amenities’ distribution in space, equilibrium with profit-maximizing developers may not exist in the real world. Consequently, some form of political process is indispensable for efficient inter-community population distribution.

1. Introduction

In his seminal paper, Tiebout (1956) suggested that, with costless migration, jurisdictional competition guarantees an efficient allocation of resources. This holds for both the distribution of population among communities and the provision of local public goods within communities. However, the model suggested by Tiebout was not fully specified; in particular, he did not spell out either the motivation of the local governments or the type of filled in the gap in the resulting equilibrium. Later, other researchers Tiebout’s model and then evaluated his assertions. They showed that for some specifications of Tiebout-type models, equilibrium may not exist and, even if it does exist, it may be inefficient. Most of the examples for the market failure in these Tiebout-type models are associated with the integer problem [Stiglitz (1977), Scotchmer (1983, Bewley (1981)], with imperfect competition [Epple and Zelenitz (1981), Scotchmer (198541, or with distortive taxation [Helpman (1988), Epple and Zelenitz (1981)]. One of the interesting specifications discussed intensively in the literature is when the local governments are price-taking (or, equivalently, utility taking) profit maximizing developers. According to this specification the developer, *The first version of this paper was completed while I was on leave at the University of Western Ontario. I am indebted to R. Arnott, E. Helpman, 0. Hochman, Y. Papageorgiou, S. Scotchmer, K. Stahl, J.D. Wilson, D. Wildasin, I. Zilcha, and two anonymous referees for helpful comments and suggestions. 016~462/91/$03.50

0

1991-Elsevier

Science Publishers

B.V. All rights reserved

470

D. Pines, Tiebout without politics

seeking to maximize the net revenue from land, finances the supply of the local public good from the land rent revenue. [See, for example, Scotchmer (1985, 1986) and Wildasin (1986, 1987).] Following Henderson (1985) this specification is referred to hereafter as ‘Tiebout without politics’. In reply to the question ‘Does profit-maximizing behavior lead to efficient is charoutcomes?, Wildasin (1986, 1987) asserts ‘. developer’s equilibrium acterized by both expenditure efficiency.. . and local efficiency. Thus given the utility-taking assumption, profit-maximizing entrepreneurs price congestion externalities efliciently, so that jurisdictions contain the correct number and types of households, and they solve the Tiebout-Samuelson problem of efficient public good provision’. A similar conclusion is derived by Henderson (1985), who, responding to Epple and Zelenitz (1981) argues that with active profit-maximizing developers, and without any political process, a Tiebouttype model produces efficient (second-best) solutions. As a matter of fact, the problem posed by Wildasin (1986) can be divided into two distinct sub issues: (a) Is Tiebout without politics viable, that is, does an equilibrium of a Tiebout-type model specified with profit maximizing developers exist?; and (b) Is the allocation of such equilibrium efficient?. It is clear that the second question is vacuous if the answer to the first question is negative, or is positive but requires restrictive assumptions. Wildasin (1986, 1987) and Schotchmer (1986) are explicitly concerned by the second issue only. The existence issue in a Tiebout-type model with price-taking prolitmaximizing developers is explicitly discussed in Wooders (1980) Bewley (1981) and Scotchmer (1985). Wooders (1980), like Tiebout, requires ‘ . . . “large” choice set of jurisdictions’, an assumption referred to hereafter as ‘perfect replicability’.’ Bewley’s analysis is confined to the case of publiclyprovided private goods financed by residence-based income taxes. His proof cannot be extended to the more general cases, including the case of a pure public good discussed by Stiglitz (1977). Restricting her analysis to specific preferences, Scotchmer (1985) suggests quite strong sufficient conditions for the viability of Tiebout without politics. Since strong sufficient conditions do not define the domain over which this Tiebout-type equilibrium exists, the problem of Tiebout without politics viability, especially under imperfect replicability, still remains basically unresolved. In this paper I investigate the issue of existence, showing that whenever communities cannot be perfectly replicated, the necessary conditions for existence of Tiebout without politics are quite strong. In the case of club theory [see Pines and Berglas (1980, 1981) and Scotchmer and Wooders (1987)], perfect replicability is a natural assumption ‘Wooders (1980) is mainly concerned with the integer problem involved in rigorously Tiebout’s conjectures. She proves these conjectures by assuming that the formation communities is costly and the economy is sufficiently large.

proving of new

D. Pines, Tiebout without politics

471

and, consequently, the issue of existence is not disturbing. However, in view of the uneven distribution in space of localized amenities, this assumption is highly unrealistic where local governments are concerned. Though the number of communities per se can be increased, the new communities cannot have exactly the same locational characteristics as the old ones. With such imperfect replicability, we show in the paper that the viability of Tiebout without politics is questionable. Though we provide much weaker sufficient conditions for existence of equilibrium than Scotchmer (1985), we show that, in the case of imperfect replicability, the necessary conditions for existence are still quite strong. The assertions of Henderson (1985), Scotchmer (1986, and Wildasin (1986, 1987) may, therefore, be vacuous. Section 2 presents the basic model; we specify agents and their objective functions, define equilibrium, and characterize configurations of surplus functions which are and are not consistent with equilibrium. In section 3, we relate the above configurations to the underlying structure of the economy, specified by land scarcity, preference, and technology. We suggest sufficient conditions for the existence and nonexistence of equilibrium. In section 4 we show that our rsults can be extended to the more elaborate models in the literature. Section 5 compares our results with those of the literature, and section 6 summarizes the paper and provides a concluding comment. 2. The basic model Consider a fully decentralized price-taking economy. Households maximize their utility, and developers maximize some net profit function; both take some price system as given. No political entity cares about the common welfare or can impose income or lump sum taxes. There are M identical islands, where M is a very large number. Four commodities are defined for each island: land, labor, private good, and public good. The technology of the economy is that of Stiglitz (1977). The private good is produced on any given island i by land, Li, and labor, Ni, according to a linear homogeneous production function, F(L,,N), with positive Iirstorder and negative second-order derivatives. Without loss of generality, let L,=l. so that F(Li,Ni)=j(Ni). W e assume that f(Ni) is bounded. The private good is used either for direct consumption or for producing the public good. We assume a fixed production coefficient which, without loss of generality, is 1. There are N households with identical preferences and endowments. The welfare of a representative household which lives on island i depends on its consumption of the private good, Zi, and the provision of the public good, Gi, according to a strictly quasi-concave function u(Z,, G,), where both goods are normal. Each household is endowed with l/N of the land of each of the M islands and with one unit of labor per period of time. As a landlord, the

D. Pines, Tiebout without politics

472

household is entitled to an income of Zl; this represents l/N of the total profit derived from the land of the M islands. As a supplier of labor, the household determines the wage rate it earns by choosing the environment in which it works, represented by the public good. In other words, being confronted with a wage schedule, w(G), the household chooses G and thus ascertains its income from work, w(G). Accordingly, the household’s disposable income is w(G) + II, which is spent on the private good, Zi. Hence, for a household living on island i, choosing G,, we have Zi = w( Gi) + Z7 and the chosen

(1)

Gi satisfies

u(w(G,)fn,

G,)=max

u(w(G)+n,

G).

(2)

G

Each island is developed by one profit-maximizing producer. Confronted with the wage schedule w(G), the developer of any given island i chooses Gi and Ni to maximize the net profit from the production of the private and the public goods, ni, that is,

17,s f( Ni) - NiW(Gi) -

=max

G

the market

.

(3)

I

for the private

z Cf(NJ -NiZi and clearing

N.G

f(N)-min[Nw(G)+G]

N

Clearing

Gi = max f(N) - Nw( G) - G

good implies

- Gil= 0,

the labor market

(4)

implies

zNi=N. With the above equilibrium as

specification

Definition I (Dl). Equilibrium and a function w(G) satisfying

(5) of the economy,

is a set of parameters (1)+5).

we define

a wage-taking

((Zp, Gp, NY, Z7p)z r; Z7’}

D. Pines, Tiebout without politics

473

Though the economy defined above is finite, the price-taking behavior is justified by assuming that the number of occupied islands is very large. It follows from (Dl) that, if equilibirum exists, then there must exist some U”, such that for an inhabited island

u(Zp,G,O)= u(w(G;) + ZZ’,G,O)= U”,

(64

and for an empty island

u(Z;, G;) = u(w(0) + Suppose

that equilibrium

e( I/N:,

Uo) E

no,0) 5

U”.

exists, yielding

1.

some U” and NY. Then,

min (Zi + G,/N”) s.t. u(Zi, Gi) = Uo Z,, G,

is the minimum expenditure on Zi and l/N: and the utility is U”. With this notation we can prove Lemma

(6b)

If equilibrium

Gi when

the relative

(7)

price of Gi is

exists, then, for NY, Zp, Go and U”,

NpZF + Gp = NFe( l/NO, U’). Proof.

Suppose

not. Then there must exist some 2 and G such that

(9

2 + c/No < Z;

-I-

G"IN"

and (ii)

u(Z, G) = u”.

But from (2) and (ii), (iii)

U” = u(w(G;) + III’, G;) 2 u(w(@ + Z7’, c).

Hence, (ii) and (iii) imply

D. Pines, Tiebout without politics

414

and, therefore,

by (1) (i), and (iv),

N;w(Gy) + G; = Np(Zp - Z7’) + G: > NY2 + G - NyIZ” _2N;,(G) which violates

(3).

+ G,

0

Lemma 1 implies that, given the distribution of the population among the islands, the (wage-taking) equilibrium is efficient - that is, the resources required to provide each island’s population with utility level U” is minimized. The first-order condition for this minimization is &/u;,=

l/N:,

(8)

or N$&u;~

= 1.

(9)

Version (8) of the necessary condition says that the implicit price of the public good is l/N, [see Stiglitz (1977)]; version (9) is the Samuelson rule for efficient provision of public goods. Substituting (1) into (3) it follows from Lemma 1 that, if equilibrium exists, then I7: =max n(Ni; U”, no), N,

(10)

rc(Ni; U”, no) -S(N,;

(11)

where U”) + NiZ7’,

and S(N;; UO)sf(Ni)-N;e(l/Ni,

(12)

UO).

the resources produced and the S(N,; U”) is the difference between resources consumed on the island.’ rr(Ni; U”, Ii’“) is the difference between the resources available and the resources consumed there. (The resources available include, in addition to the resources produced on the island, the net claims of the island’s population on profits derived elsewhere.) Hence, (lO)(12) imply that in equilibrium (Dl), if it exists, the difference between the resources available and the resources used on each island is maximized. Assume that Ni is continuous and S(N,; U”) is differentiable. Then, (10) and (11) imply

‘The surplus in this paper is equivalent to the maximum Hochman (1981) and Hochman and Pines (1982).

level of the surplus

as defined

in

D. Pines, Tiebout without politics

s’( No;

415

U”) = - Z7,

(13)

and, therefore, Z7o=S(NF;

U”)- N$'(Nf;

U’).

Using (lo), (1 l), and the definition ll”=~17~/N=~[S(N;; M

(14)

of II, we obtain

U”)+N;17’]/N=xS(N;;

M

U’)/N+L”,

(15)

M

and, therefore, xS(N:; M

U’)=O.

(16)

(lo), (1 l), (13), (14) and (16) imply three conditions: Condition 1 (Cl).

S’(Ny; U’)lO.

Condition 2 (C2).

I+, S( NF; U”) = 0.

Condition 3 (C3).

S(N,; U”) - N,S’(Ny;

U”) 5 S(Ny; U”) - NpS’(Ny;

U”).

(Cl) follows from non-negativity of the profits, (13), and the lirst equality of (15). (C2) is (16) and (C3) follows from (IO), (11) (13) and (14). Condition (C3) is further discussed below. The right-hand side of (C3) is the intercept of the tangent line to the surplus function at NY (referred to below as ‘the tangent line’). The left-hand side of (C3) is the intercept of a line, which is parallel to the tangent line [i.e., its slope is S’(Ny; U’)], and which, for the given Ni, passes through the point S(N,; U’). Diagrammatically, (C3) implies that the tangent line does not intersect the surplus function, or, in other words, the surplus function is nowhere strictly above the tangent line. Condition (C3) says that NY is an equilibrium value of Ni, only if Nj’ globally maximizes the profit function, rc( .), where Ii’ is equal to -S’(Np; U’). In other words, NF is the value of Ni which globally maximizes n(N,; U”, - S’(N;; U”)). Observe that Ni and U which satisfy (Cl)+C3) and (5) are their equilibrium values. Therefore, (Cl)-(C3) and (5) are necessary and sufficient conditions for equilibrium (Dl). The above three conditions are used below to verify whether any given function S(Ni; U”) and an associated parameter NF can simultaneously be consistent with equilibrium (Dl). The surplus function portrayed in panel (i) of fig. 1 satisfies the three

R.S.U.E.--

F

D. Pines, Tiebout without politics

476

niN,$J,O)

(i)

m

Ni

x (-1 n(&+J,-tng ~1, S(*l r(b++J-tngp) t ‘Ikl..

n=-tng

p

(iii)

n(N,;U,-tnga) x(N,;lJ,

-tngr) (iv)

Fig. 1. Maximum

profit levels according

to alternative

configurations

of the surplus

function.

conditions when NY = N 1. To see this, observe that, in this case, S’(N,; U”) = S(N,; U”) =O. Therefore, (Cl) and (C2) are satisfied. In addition, (C3) is reduced to S(N,; U’)sS(NF; U”) for all N,; it is satisfied with strict inequality for all N, satisfying Ni # N 1, as reflected in this panel. The surplus function portrayed in panel (ii) with NF=N, is also consistent with the three conditions. To see this, observe that, in this case, S’(N,; U”) < 0, thus satisfying condition (Cl). In addition, since S(N,; U’)=O, condition (C2) is satisfied. Finally, the surplus function is completely below the tangent line to S(N,; U")at N,, satisfying (C3). The configuration portrayed in panel (iii) of fig. 1 with NY = N,, N4 is, once again, consistent with the three conditions. To see this, notice first that the slopes of the surplus function at both N, and N, are negative, thus satisfying (Cl). Second, since S( N,; U”) > 0 and S(N,; U”) < 0, (C2) can be

471

D. Pines, Tiebout without politics

satisfied if the M cities are appropriately distributed between those with N, and those with N, households. Third, the surplus function is wholly below the tangent line at N3 and N,, which satisfies (C3). Finally, turning to pane1 (iv) of fig. 1, we observe that a multiple maxima [as in pane1 (iii)] is impossible. Hence, if equilibrium exists at all, (C2) requires that NF= N, with S(N,; U’)=O. However, since the tangent line at N, intersects the surplus function, the left-hand side of (C3) exceeds the right-hand side at some values of Ni (Ns is an example). We conclude that (C3) is violated and, therefore, the configuration of S(N,; U’), as portrayed in panel (iv), is inconsistent with the existence of equilibrium. The surplus function in each pane1 differs from the others by its position relative to the horizontal axis and by its shape. In the next section, we analyze the determinants of position and shape. Then we show that the configurations portrayed in panels (ii)+iv) of fig. 1 reflect land scarcity, and the configuration portrayed in pane1 (iv) is not an outcome of pathological preference or technology. We then conclude that, in the case of land scarcity, strong assumptions are required to guarantee the existence of equilibrium.

3. Replicability, equilibria

land scarcity, and the existence,

uniqueness and efficiency of

We define the concept of optimal community size, use it to define land scarcity, and show that land scarcity leads to one of the configurations represented in panels (ii)-( Then we investigate under what conditions each of the configurations portrayed in panels (ii)emerges when land is scarce.

3.1. Optimal community Let V(N,) Hence,

size

be the maximum

utility

attainable

on an

V(Ni) E max u(Z,, Gi)

autarkic

island

i.

(17)

Zi. G,

such that NiZi + Gi = f( Ni).

(18)

Eqs. (17) and (18) allow us to define Definition 2 (D2). 1/(Ni) for any Ni.

N* is said to be an optimal

community

size if V(N*)z

D. Pines, Tiebout without politics

478

We will always follow Tiebout

in assuming

Assumption 1 (Al). There exists (numbers), N*, satisfying (D2).

(exist)

a finite

(finite)

positive

number

Observe that (Al) is a standard assumption in the literature [e.g., see Stiglitz (1977), Wooders (1980) Berglas and Pines (1981) and Scotchmer (1985)]. In examining the issues of existence, uniqueness, and efficiency of equilibria, we first distinguish between two alternative regimes: (i) Islands (ii) Islands Later regime.

are perfectly replicable (land is free). are non-replicable, and land is scarce.

we will

3.2. Pecfectly The concept can be adapted Definition infinite.

replicahle

imperfect

replicability,

3 (D3).

4 (D4).

The

which

is an

intermediate

islands (land is ,free)

of perfect replicability in Tiebout’s to our model as follows.

For our purpose, Definition

discuss

islands

however,

are said

we require

sense and in club

to be perfectly

a less-demanding

replicable

theory

if M is

condition:

Land is said to be free if M > N/min N*.

We start with a few observations regarding the existence, uniqueness and effkiency of equilibrium when land is free (D4). We differentiate (12) with respect to U”, and use the envelope theorem to obtain aS(Ni; LJ/iXJ) = - Ni ae( l/N,; U)jC?U = -Ni[au(Zi;

G,)/az,] - l


(19)

since Zi is normal. Observe that the allocation: NF=N* in N/N* islands, where N* is any maximand of V(N,), NY =0 in the remainder, and U” = V(N*) is feasible. Also, it is evident from the definition of V(Ni) and S(N,; U) that S( Ni; V(N,)) = 0. Therefore, by (A 1) and (19), S(N,; V(N*)) 5 0 for Ni # N*, while S(N*; V(N*)) =O. It follows that with U” = V(N*), the surplus function

D. Pines, Tiebout without politics

479

can be represented by panel (i) of fig. 1, which satisfies conditions (Cl)+C3). Hence, equilibrium with Ni = N, and with U = V(N*) exists. Next, notice that there is no other U”# V(N*) consistent with (Cl)-(C3). If U”> V(N*), it follows from eq. (19) and S(N*; V(N*))=O that the surplus function is negative for all Ni. Therefore, (C2) cannot be satisfied. If U” < V(N*), the surplus function assumes positive values for some range of Ni. This follows from (19) and S(N*; V(N*))=O. Since ZZ is non-negative, (11) implies that ni must be positive, and, therefore, all of the islands are developed. Thus, if every island is inhabited with more than N/M households, then (5) is violated, and the allocation is infeasible. We prove in Lemma 2 that, indeed, if U”< V(N*), every island inhabits more than N/M households. Lemma

2.

lf

U”< V(N*),

then the profit maximizing

Ni, NY, is larger

than

NIM. Proof.

We proceed

(i) We begin Differentiating

in steps.

by showing that d27(( ‘)/aNiaU <0 and (11) and (12) with respect to Ni we have

t%(N,; U, Il)(aNi=S’(N,; =

a2rc( .)/dNi8n>0.

U) + ll

f’(NJ-c( l/N,,

U) + e,( l/Ni, U)/Ni + II

=f’(Ni) -e(l/Ni, u) +g(l/‘Ni, U)/Ni+17 =

f’(Ni)-Z(l,lNi,

(20)

U) + IZ,

where e, is the partial derivative of e( .) with respect to its first argument, z(l/N,, U) is the compensated demand for the private good, and g(l/N,, Uj is the implicit compensated demand for the public good. Then, differentiate (20) with respect to U and Zl to get

a2n(.)/a~, au = a%( .)/a~, au = - az( .)/au CO,

(21)

and a2rt( .)/aNi aZl= a2S( .)/aNi where the inequality

an + 1 =

1 > 0,

in (21) follows from the normality

(22) of Zi.

(ii) We have already shown that, for U”< V(N*), l7 must be positive, while, for U”= V(N*), IZ vanishes. This consideration and (i) above imply that, for all Ni, Tt’(Ni; U”, 17)>d(Ni; V(N*), 0). It must be the case that the profit maximizing level of Ni is larger than any N*.

D. Pines, Tiebout without politics

480

(iii) Since land is free, then, by (D4) and (ii) above we have NY >min which completes

N* > NJM, the proof of Lemma

(23) 2.

Since, when U”< V(N*), all the islands are inhabited than N/M, eq. (5) is violated. We can, therefore, state

by more households

Proposition 1. If (Al) is satisfied und the islands are perfectly replicable uccording to (03). or land is ,free uccording to (D4), then there exists a unique equilibrium

according

A few comments (a) If N* is unique,

to (Dl).

are in order: then,

there exists a unique

equilibrium

population

size,

0

:)‘The case covered by Propositon 1 is the one discussed by Tiebout (1956), with the gap in his specification filled by assuming that the communities are run by profit-maximizing developers. This is also the case elaborated in club theory. (c) The zero profit implied in this case, i.e., f‘( Ni) - f’(N,)N,G,=O, is sometimes referred to as the Henry George rule, that is, the expenditure on the public good is perfectly capitalized in land value [see Stiglitz (1977) Berglas and Pines (1981), and Wildasin (1986)].

3.3. Non-replicable

islunds (land is scurce)

In this subsection, we assume that the number land is scarce in the following sense: Definition

5 (D5).

of islands

is fixed and that

Land is said to be scarce if M < N/max N*

If equilibrium exists when land is scarce according to (D5), then, by (Al), U” -C V(Nf). Therefore, by (19) it must be that S(N,; U”) > 0 for some range of Ni. We can have either the case represented by panels (ii) and (iii), which are consistent with the existence of equilibrium, or that represented by panel (iv), which is not. It follows from the discussion of the case represented in panel (ii) that: Proposition 2. If land is scurce according to (DS) and, for NY = N/M and U”= V(N/M), condition (C3) is satisfied, then equilibrium (DI), with No= N/M

and U” = V(N/M),

exists.

D. Pines, Tiebout without politics

481

In particular, Corollary I. If the surplus function with unique NY exists.

is globally

concave,

then equilibrium

(Dl)

The condition for global concavity of the surplus function, in terms of the underlying utility and production functions, can be derived by differentiating (20) to obtain: S”(Ni; Cr) =f”(Ni)

-gi(

l/N,, U)/N?

= Cf’(Ni)INJ(VS, =

-SL/O),

Cf’(NJS~I(N~I~)l { ~0 - Cf(NJ - Nif’(NJlIGi}

5 0.

(24)

where of substitution along the isoquant corresponding CJ =elasticity Y) =price elasticity of the compensated demand for G, SG = public good expenditure-wage ratio, i.e. Gi( l/Ni)/f ‘, S, = the relative share of land in the production costs.

to f(Ni),

The first term in parentheses on the right-hand side of the second equation in (24) is related to the benefit of cost-sharing; the second is related to the loss from the decline in the marginal product of labor resulting from the increase in the population. If the first term exceeds the second, the negative effect of the increased population is increasingly mitigated by the positive effect; this is reflected in the positive second derivative of S( .). A few comments are in order: (a) A global concavity of the surplus function is precisely a global concavity of V(Ni). Therefore, (24) also guarantees that any extreme value of V(N,) is a global maximum. Stiglitz (1977), however, suggests a slightly different condition, ays 1, to which our condition collapses when the Henry George rule, [f(Ni) -f’(N,)N,]/G,= 1, is satisfied. 3 Applied globally, Stiglitz’ condition does not require concavity of the increasing section of S( .) and V( .), but it does so regarding the decreasing section. Furthermore, it imposes on the decreasing section even more stringent constraint than just concavity. (b) The conditions specified in Proposition 2, Corollary 1, and in Stiglitz (1977) are sufficient, not necessary. Thus, equilibria can prevail, even when the conditions specified in Proposition 2 and, a fortiori, in Corollary 1 and Stiglitz, are violated. To see this, recall from the analysis in the preceding section that the surplus function portrayed in panel (iii), which violates (C3) 3This relationship

between

the two conditions

was pointed

out to me by J.D. Wilson.

482

D. Pines, Tiebout without politics

for NY = N/M, can still be consistent with (Cl)-(C3). If the distribution of city size according to N, and N, happens to be consistent with (5), then equilibrium characterized by panel (iii) can prevaiL4 (c) As reflected in panel (iii) of fig. 1, the surplus corresponding to N, is positive and that which corresponds to N, is negative. Of course, this is possible only if resources are transferred from the small to the large community. This transfer is possible because a community with N, households has a larger share of profits than a community with N, households (N,Z7 vs. N,17).5 (d) Both in panels (ii) and (iii) the value of land exceeds the expenditure on the public good - that is, the expenditure on the public good is overcapitalized. [This follows from Lemma 1, (12), the elaboration in (20), and the negative sign of the surplus function, which, together, yield f(Ni)Nif’(Ni)>Gi]. But the d$ference between the expenditures on the public good in the two communities is fully capitalized.6 (e) It follows from Scotchmer (1986) that the competitive allocation guaranteed by the conditions of Proposition 2 is efficient. The surplus function portrayed in panel (iv) is convex for sufficiently large Ni, and its derivative converges to zero as Ni tends to infinity. It is shown in the preceding section that in this case conditions (Cl)-(C3) cannot be satisfied simultaneously, so that no equilibrium exists. This configuration in the following represents a more general case, which is described proposition. Proposition 3. If land is scarce according to (D.5) and, for any U” < V(N*) and NY satisfying condition (Cl), condition (C3) is violated, then no equilibrium according to (Dl) exists.

In particular, Corollary 2. there exists

If land is scarce

according

to (D5)

4Equilibria of the type portrayed in panel (iii) prevails function, the following condition is met:

and, for

if for U”, which

any U” < V(N*),

generates

this surplus

N/M=(S,N,-SS,N,)/(S,-SA where Si is the surplus in a city with Ni inhabitants. This condition guarantees that the population can be distributed between the two city types such that both (5) and (C2) are simultaneously satisfied. [There are NS,/(S,N,-S,NJ cites with N, individuals and the rest with N, individuals]. %ee further elaboration on this issue in section 5. 6Let the subscripts 3 and 4 refer to the islands with N, and N, households, respectively. [See panel (iii) of lig. 1.1 Then, f‘(N,)-N,w(G,)-Gj=rc(N,;U,II=rr(N,;U,17)= N,w(G,)]-[f(N,)N,w(G,)]. The terms in the f‘(NJ - N,w(G,)G, or G4-G3=[/(NIbrackets represent land values.

D. Pines, Tiebout without politics

some iTi(

lim,,,

such that, for

~11 Ni> Ai(

483

(u) S(Ni; U”) is CO~LWX and (b) to (Dl) exists.

m S’(N,; U”) =O, then no equilibrium according

It must be emphasized that the sufficient conditions for market failure, described in Proposition 3, are not outcomes of pathological underlying utility or production functions. For example, standard functions can imply the conditions (a) and (b) of Corollary 2. ’ Furthermore, Proposition 3 is relevant not only to the extreme case of land scarcity (D5), but also to a lessrestrictive case of imperfect replicability discussed below. 3.4. Imperfect

replicubility

In this case, the quality of the islands varies. Some islands are larger, and therefore more productive; some have better amenities than others. Although the total number of the islands is unbounded, the superior islands are scarce in the sense that their number is not sufficient to accommodate all of the population in optimal community size. By a superior island, we mean Definition

6 (D6).

Island

i is superior

to island

j if Si(N; U) > S,(N; V) for

any given N and U. The scarcity of the superior islands implies that, in equilibrium (if it exists at all) part of the population lives on inferior islands. This requires that the surplus function of any superior island i be positive for some range of Ni. Otherwise, it follows from (D6) and (19) that the surplus functions of the inferior islands must be strictly negative for all Ni. Hence, the surplus of all the islands is not positive, and the surplus of some is negative - which ‘S assumes example,

the shape

portrayed

forNi
in (iv) (including

S is zero) if, for

1

and u(Z,, GJ = ZiGi. With this specification S(N,; U)=ln

the case where maximum

we have for Ni 2 1:

N;-2fi,/i?,

s’=l/Ni-~/fi=(~)(l/fi-fi)$O S”=(fi&-2)/2N,230

as N,zl/U, as

NiP4/U,

and

lim S’(.)= lim [l/Ni-$/fi]=O. NPrn N,-r, In calculating ‘s’ we can, of course, follow eq. (24). In this example ‘I= l/2, &=a, c = In N, - l)/ln Ni and S,=(ln Nr- l)/ln Ni. Therefore, S” =( f ‘(NJ/NJ(& -SJu) = ( /- UNi-2)/(2Nf), as above. With the above specification, N*=e*, where e is the basis of natural logarithm and Y(N*)= l/e’. Hence, if N/M >e ',V(N/M)-Cl/e’. Therefore, by eq. (12), we have S(e’; V(N/M))>S(e*; l/e2)=Ine2-2Je2Jl/e2=2-2=0. It follows that for some range of Ni, S(N,; V(N/M)) is positive. This, and the shape described above, yields the configuration of panel (iv) of fig. 1.

484

D. Pines, Tiebout without politics

violates condition (C2). It follows, therefore, that the surplus function of the superior islands must be represented by one of the panels (ii)of fig. 1. But then Proposition 3 can apply, such that no equilibrium exists. We conclude that the case of imperfect replicability reaises the same difficulty as that of land scarcity.

3.5. Comparison It may be useful to clarify the relationships among the various sufficient conditions discussed in this section and to summarize our results. Denote by [Al], [IS], [2], [3], [Cl], and [C2], the sets defined by (Al), the sufficient conditions of Stiglitz, Proposition 2, Proposition 3, Corollary 1, and Corollary 2, respectively. Then, the following relationships hold: (a) [Cl]u[S]

c[2],

i.e., the union

of [Cl]

(b) CC21 = C31, (c) [2] u [3] c [Al],

i.e., the union

of [2] and [3] is a subset

Sufficient (a) [Al]

conditions

for existence

and perfect replicability

and [S] is a subset

of equilibrium

of [2], of [Al].

(Dl) are:

or

(b) C21. Sufficient

conditions

for market

failure

[no

equilibrium

(Dl)

exists]

are

c31.

4. Extensions We did not lose any generality by using the simple specification of Stiglitz (1977), rather than the more complicated and cumbersome specifications often used in the literature. To show this, we introduce housing and congestion and re-examine the results of the preceding sections. Let the utility of an inhabitant of island i be a function u(.Z,, Hi, AJ of a composite good, Zi, housing, Hi, and the rate of utilization of a public facility, Ai. The land is now allocated not only to the production of the composite good, Li, but also to housing, 1 -Li. The production function of the composite good is f(&, Ni) and that of housing is h( 1 -L,, Xi), where Xi is the composite good input of housing. Both functions are linear homogeneous, with positive firstorder and negative second-order partial derivatives. The cost of providing a public facility with capacity Gi and utilization level N,A, (in terms of the composite good) is c(Gi, NJ,), where the partial derivative with respect to NiAi is positive. With this specification, the resource constraint for housing is

D. Pines, Tiebout without politics

NiHi-h(l-Li,Xi)~O and for the composite

(i=l,...,M),

485

(25)

good

F C(N,Zi+Xi) +c(Gi,

NiAi)-j”(Li,

NJ] ~0.

(26)

As before, we can define V( Ni) =

max

u(zi,

Hi,

(27)

Ai)

Z,,Hi.A,,G,,X,,Li

s.t. eq. (25) and (26). We can also define E( Ni, L,; U)

min

E

NiZi + Xi +

C(

Gi, NiAi)

(28)

Z,,Hi.Ai,Gi.X,

such that g( I- Li, Xi) 2 NiHi,

dzi,

Ai) 2

Hi,

S(Ni; U)

E

u,

max f( Li, Ni) - E( Ni, L,; U), L,

(29) (30) (31)

and 71(Ni; U, n) ~ Ni17 + S(Ni; U),

(32)

which is maximized in a price-taking equilibrium. With these definitions, the reduced form of the extended model is similar to that of the basic model. Therefore, the analysis associated with fig. 1 applies to the extended model as well as do Proposition 1 and (with the appropriate adjustments) Propositions 2 and 3. 5. Comparison

with other models

In this section we compare our findings with those derived in related recent works. Bewley (1981), for example, discusses a case of publicly provided private goods, financed by head taxes levied by profit maximizing local governments. He shows that a competitive allocation, where all the goods are privately provided, is isomorphic to a Tiebout-type equilibrium, where communities are segregated by types and some of the goods are

486

D. Pines, Tiebout

without politics

publicly provided by profit maximizing local governments, being financed by head taxes. He uses the standard existence of competitive equilibrium theorem and the above isomorphism to prove existence of Tiebout-type equiilibrium with profit-seeking local governments who use head taxes to finance the provision of the publicly-provided private goods. Unfortunately, this exercise is confined to publicly-provided private goods and cannot be extended to more general cases of pure or impure public goods. Scotchmer (1985) discusses a Symmetrical Nash Equilibrium of a fixed number of clubs (communities) with positive profits (SNE) and a Two-Stage Symmetrical Nash Equilibrium of a variable number of clubs (communities) with zero profits (TSSNE). With replications of the economy, Scotchmer’s SNE (TSSNE) converges to price-taking equilibrium when land is scarce (free). Confining the analysis to specific utility function (transferable utility) Scotchmer provides (strong) sgficient conditions for the existence of SNE and, therefore, for a price-taking equilibrium. (These conditions are extended somewhat in Proposition 2.) Being restrictive sufficient conditions, they are useless in proving our main assertion: when land is scarce, strong assumptions are required to guarantee the existence of price-taking equilibrium. Proposition 3 of this paper and the discussion thereof provide sufficient conditions for non-existence of price-taking equilibrium when land is scarce. Since these conditions do not involve pathological utility or production function, we conclude that the market can fail even in standard cases. This conclusion, of course, does not follow from Scotchmer’s analysis.* While Bewley (1981) and Scotchmer explicitly distinguish between the issues of existence and efticienccy of a Tiebout-type model with profitmaximizing local governments and analyze them separately, both Henderson (1985) and Wildasin (1986, 1987) investigate only the efficiency issue. Yet both make some disturbing statements which can be interpreted as asserting that equilibrium is always viable even under imperfect replicability. According to Henderson (1985), ‘Tiebout without politics produces efficient solutions’ and that ‘. . . fixing community numbers and sizes need not be critical’, in that ‘The no-politics solution is generally consistent with a characterization where equilibrium is unique.. .‘.’ Wildasin (1986) states ‘Now it is clear that the developer’s profit maximizing solution can be sustained as a

“In presenting Scotchmer’s specifications I ignored the integer issue with which she is concerned. For example, the zero profits which I ascribe to TSSNE is represented in Scotchmer’s specification by non-positive profits for k and negative profits for k+ 1, where k - an integer number - is the equilibrium number of lirms (communities). ‘While Bewley (1981), Wildasin (1986, 1987) and our specification allow the use of a head tax, Henderson (1985), disallows its use and, instead, assumes that the publicly-provided private good is partially financed by a property tax.

D. Pines, Tiebout without politics

487

equilibrium with profit-maximizing private producers.. .‘. If these statements mean that equilibrium exists even under imperfect replicability assumption, they are, of course, in a sharp contrast to Proposition 3 and the discussion thereof. Albeit, their model differs from ours in several respects. But, as explained below, these differences cannot reconcile the conflicting conclusions. First, their model includes housing and congestion, while our model does not. As shown in the preceding section, however, this is not consequential. Second, the case represented in panel (iv) of fig. 1 appears to indicate that it is the household’s share of the profits, II, which upsets the equilibrium. This ‘disturbing’ element is absent in Henderson (1985) and Wildasin (1986, 1987), where the developers represent absentee landlords - rather than households as in our model. In their case, therefore, one tends to conclude that equilibrium exists even when land is scarce. The developer, on behalf of the absentee landlords, maximizes S(N,; U”) rather than n(Ni; U”, no) = S(N,; U”)+Nini. Therefore, it follows from (20) that the first-order condition for profit maximization is f’(N,)-z(l/N,; U)=O, where z( .) is, as before, the compensated demand for the private good. Now, suppose that N/M >N*. Then, U is uniquely determined by f’(N/M) = z( M/N; U).

(33)

It follows that the first-order condition is always satisfied. But a closer look reveals that the difticulty pointed out in this paper is also relevant when the land belongs to absentee landlords. We cannot be sure that the secondorder condition, Y-CO, is satisfied for Ni=N/M and U, which solves eq. (33). In other words, we cannot guarantee that for U which solves eq. (33), N/M maximizes S(N,; U); N/M can be a minimum, or an inflexion point. Furthermore, even when S” ~0, N/M can be a local maximum. Only by assuming that S(N,; U) has a n shape for every U, can we be sure that NjM is a unique maximand of S( .) for U which solves (33). If land is scarce, therefore, the existence of equilibrium requires strong assumptions regarding S( .), even in the case of absentee landlords. In contrast to our conclusion regarding the case represented in panel (iii) of fig. 1, Helpman (1978) shows that, if the optimal allocation requires multiple sizes of population, it cannot be supported by a competitive price system. The reason is that Helpman’s model is not free of politics; the local public good in a given community is financed by a local head tax, More specifically, the net return from developing the islands, which is distributed to all the households as their share in the profits, is cM [f(Ni) - Niw(Gi)]/N while in our specification it is ,& [f(Ni) -Nw(Gi)Gil/N. The implied difference between the two tax systems is that Helpman’s is a local head tax,

488

D. Pines, Tiebout without politics

while ours is a nation-wide head tax.” In Helpman’s case, an optimal allocation which implies different Ni for different i cannot be sustained without some non-market transfer of resources - which, in turn, requires some political process. In our case this allocation is sustainable, since the appropriate transfers are built into the redistribution of profits.

6. Summary

and concluding comments

We have shown that, as in the club theory, if communities can be perfectly replicated, equilibrium with price-taking profit-maximizing developers exists and is efficient. The provision of the local public good can then be left to the market. But if communities cannot be perfectly replicated and land is scarce, then equilibrium with developers may not exist. Since we based our main analysis on a very simple specification, one could suspect that our conclusions apply only to a very restricted case. But we demonstrated that the concept of the surplus function is general, and can be applied to much more elaborate specifications. Once its configuration assumes the shape portrayed in panel (iv) of fig. 1 (or, more generally, once the conditions of Proposition 3 are satisfied), equilibrium fails to exist. Since the above configuration is not necessarily an outcome of pathological preferences or technologies, we need strong assumptions to guarantee the existence of equilibrium when land is scarce. Furthermore, land scarcity is a realistic assumption; even when the number of communities is unlimited, sites cannot be perfectly replicated. Hence, costless migration and wage-taking profit-maximizing developers cannot guarantee a viable and efficient allocation. We conclude that Tiebout, with the unrealistic assumption of perfect replicability, can work without politics. But in the real world, the efficient provision of public goods needs politics - even on a local level. “‘The Wildasin

distortive effect of a local (1986, 1987).

head

tax,

when

city

size is not

optimal,

is discussed

in

References Berglas, E. and D. Pines, 1980, Clubs as a case of competitive industry with goods of variable quality, Economics Letters 5, 363-366. Berglas, E. and D. Pines, 1981, Clubs, local public goods, and transportation models: A synthesis, Journal of Public Economics 15, 141-162. Bewley, T.F., 1981, A critique of Tiebout’s theory of local public expenditures, Econometrica 49, no. 3,713-739. Epple, D. and A. Zelenitz, 1981, The implications of competition among jurisdictions: Does Tiebout need politics?, Journal of Political Economy 89, 1197-1217. Helpman, E., 1978, On optimal community formation, Economics Letters 1, 289-293. Henderson, J.V., 1985, The Tiebout model: Bring back the entrepreneurs, Journal of Political Economy 93, 248-264. Hochman, O., 1981, Land rents, optimal taxation and local fiscal independence in an economy with local public good, Journal of Public Economics 15, 59-85.

D. Pines, Tiebout without politics

489

Hochman, 0. and D. Pines, 1982, Cost of adjustment and spatial pattern of a growing open city, Econometrica 50, no. 6, 1371-1391. Scotchmer, S., 1985, Profit maximizing clubs, Journal of Public Economics 27, 2545. Scotchmer, S., 1986, Local public goods in an equilibrium, Regional Science and Urban Economics 16.4633481. Scotchmer, S. and M. Wooders, 1987, Competitive equilibrium and the case in club economies with anonymous crowding, Journal of Public Economics 34, 1599174. Stiglitz, J.E., 1977, The theory of local public good, in: M. Feldstein and R.P. Inman, eds., The economics of public services (Macmillan, London) 274333. Tiebout, CM., 1956, A pure theory of local expenditure, Journal of Political Economy 64, 416424. Wildasin, D.E., 1986, Urban public finance (Harwood Academic Publishers, New York). Wildasin, D.E., 1987, Theoretical analysis of local public economics, in: ES. Mills, ed., Handbook of regional and urban economcs, vol. 2 (North-Holland, New York). Wooders, M., 1980, The Tiebout hypothesis: Near optimality in local public good economies, Econometrica 48, 1467-2485.