Voting with Ballots and Feet: Existence of Equilibrium in a Local Public Good Economy*

Transcription

1 ournal of economic theory 68, (1996) article no Voting with Ballots and Feet: Existence of Equilibrium in a Local Public Good Economy* Hideo Konishi Department of Economics, Southern Methodist University, Dallas, Texas Received August 5, 1993; revised January 24, 1995 This paper proves a general existence theorem for equilibrium in a local public good economy with free mobility by extending Greenberg and Shitovitz's (J. Econ. Theory 46, 1988, ) approach. Each urisdiction's collective choice rule is the d-maority voting rule proposed by Greenberg (Econometrica 47, 1979, ). Spillover effects of local public goods, externalities due to the population distribution, and snob effects are all allowed. We need transitivity and completeness of preferences; these assumptions are not used by Greenberg and Shitovitz because they do not permit consumers to move. Free mobility of consumers causes disconnections in consumption sets, a problem addressed here. Journal of Economic Literature Classification Numbers: C62, D72, H72, R Academic Press, Inc. 1. Introduction Since a pioneering work by Tiebout [51], many articles on local public good economies have been written in the past few decades. Tiebout [51] argues that with free mobility, consumers choose their most preferable urisdictions as their place of residence (voting with their feet), and this voting leads to, a market-type competition of urisdictions for voters and generates the optimal allocation of resources and population. However, the assumptions he states are informal and ambiguous; for instance, each urisdiction's obective function or collective choice rule is not specified. * I am most grateful to my advisor Marcus Berliant for his continuous guidance and encouragement. Conversations with Ryoichi Nagahisa and Tom Nechyba, and comments by Karl Dunz, Norman Schofield, and Myrna H. Wooders were invaluable. Encouragement and insightful comments by the referees were quite helpful. I am grateful to David Austen-Smith, Andrew Austin, Jeff Banks, John H. Boyd III, Lionel McKenzie, William Thomson, and John Yinger for their comments, and Steve Ching, Jiping Guo, Kohtaro Hitomi, Wan-hsiang Pan, Jim Peck, Fernando Perera-Tallo, Suzanne Scotchmer, Shige Serizawa, Tomoichi Shinotsuka, and Shlomo Weber for helpful discussions. I also thank conference participants at the Regional Science Association International Meeting in Houston, the Econometric Society Winter Meeting in Boston, the Social Choice and Welfare Meeting in Rochester, and the Center of Political Economy Meeting in St. Louis Copyright 1996 by Academic Press, Inc. All rights of reproduction in any form reserved. 480

2 VOTING WITH BALLOTS AND FEET 481 Following Tiebout's idea, many subsequent papers have looked for the urisdiction's behavior which assures the validity of the first welfare theorem. 1 In contrast, this paper adopts a positive approach to investigate a local public good economy. We use maority voting as the urisdictions' collective choice mechanism and prove the existence of a market equilibrium. Maority voting is one of the most plausible candidates to describe a urisdiction's decision mechanism in the actual economy, and many empirical papers in this field adopt this rule (see, e.g., Rubinfeld [38]). It therefore appears important to give a theoretical foundation to this kind of applied work. By adopting maority voting as a collective choice mechanism, the model contains two types of voting processes: voting with feet a la Tiebout and voting by ballot. This type of existence problem has been analyzed by Westhoff [52], Rose-Ackerman [36], Epple et al. [14, 15], and Dunz [1012]. Although there are some variations, the structure of the problem takes the following form: Each consumer chooses her living place by taking the taxpublic good bundles as given. The local public good supply is financed by tax revenue collected in each urisdiction. Maority voting by the residents determines the level of local public good provision. Westhoff [52] proves the existence of maority voting equilibria with an arbitrary finite number of urisdictions in a two-good economy without land. He assumes that there is a continuum of consumers who are sorted by the marginal rates of substitution (which we call ``sorting assumption'') and there is only one private good (composite good) and one local public good in each urisdiction. Each consumer's wealth (endowment) is given in terms of private good and is therefore fixed. The local public good is financed by a wealth tax. Westhoff's [52] model is not satisfactory for two reasons. First, since there is no land and no congestion in his model, it is difficult to explain why consumers scatter over urisdictions. Consumers can enoy a much higher public good supply by living together. Therefore, it is not clear how important it is to show the existence of equilibria for an 1 Several articles investigate the validity of the first welfare theorem in local public good economies. Given the number of urisdictions fixed, Richter [35] and Greenberg [20] use public competitive governments (see Foley [16]) and show that a public competitive equilibrium exists but is not Pareto optimal (see also Richter [34] and Greenberg [18]). Ellickson [13] and Bewley [5] prove the existence of an equilibrium and the first welfare theorem under very restrictive assumptions. Sonstelie and Portney [48], Scotchmer [45], and Wildasin [53] assume that both urisdiction managers and consumers know a (hedonic) price function which relates public good provision levels to land prices. They prove a first welfare theorem with this price function (a continuum of prices). Bewley [5] and Starrett [50] summarize the difficulties in obtaining efficient equilibria in local public good economies. With endogenous formation of coalitions Wooders [54, 55] proves core convergence theorems in replica economies using complete price systems (including participation fees).

3 482 HIDEO KONISHI arbitrary number of urisdictions in his model. 2 Second, since there are no prices in the model, we do not know whether his result is widely applicable (e.g., when there are multiple private goods). Rose-Ackerman [36] adds divisible land to Westhoff's [52] model, and shows that there might not exist an equilibrium if a property (land) tax is used to finance the local public good. She assumes that there is a finite number of urisdictions, and each urisdiction has a fixed amount of perfectly divisible homogeneous land. The price of land is determined in the market, but land rents go to an absentee landlord. Under these assumptions, she shows that a voting equilibrium might not exist even when utility functions are specified as a CobbDouglas type in each urisdiction. The problem is that it is difficult to obtain single-peaked preferences over tax rates if the tax tool employed is a land tax, since under property tax we have nonconvexity in extended budget sets (which are budget sets including public goods consumption). 3 The relative curvatures of the surfaces of extended budget sets and indifference surfaces are important to exclude a cyclic voting outcome. Epple et al. [14, 15] employ a generalized version of Rose-Ackerman's model and prove the existence of a market equilibrium using a relative curvature assumptionindifference curves of indirect utility functions are concave. However, even CobbDouglas preferences do not necessarily satisfy this assumption as Rose-Ackerman [36] points out. Therefore, as an assumption employed in proving an existence theorem, it looks too strong. Dunz [1012] considers a model a la Rose-Ackerman but with indivisible land (house) and proves the existence of equilibrium by assuming that a wealth tax is employed to finance the local public good. Since housing supply is fixed, the model is intrinsically an assignment problem. His model is an extension of classical general equilibrium models in the sense that land ownership and a price system are incorporated. Since there is no absentee landlord, consumers' wealth levels vary with house prices. He uses a wealth tax as a policy tool instead of a property tax. Instead of assuming the sorting condition for consumers' preferences, he assumes that each agent's most preferred tax rate is independent of the total wealth level 2 To be fair to Westhoff [52], we have to make one remark. His intention is not only to prove the existence of equilibria, but also to show the existence of a stratification equilibrium, in which similar types of consumers tend to live together. The sorting condition is used in proving the existence of a stratification equilibrium. 3 Roberts [37] proves the existence of a voting market equilibrium in a distortionary income tax economy without using single-peaked preference property. The trick is that he assumes fixed prices (linear production functions). If there is a commodity whose price can vary such as land, then we cannot apply his argument. Although we can prove that there exists voting outcome for each price, we cannot prove the voting outcome mapping is convexvaluedit is not even connected-valued.

4 VOTING WITH BALLOTS AND FEET 483 in the urisdiction (an independence assumption). 4 Although the clarifies that the sorting and the fixed income assumptions are not necessary, the indivisible land assumption supposes that population in each urisdiction is fixed, and here is no way to construct a large population urisdiction. 5 Nechyba [32] introduces a central government into Dunz's economy, and assumes that the central government provides a national public good financed by a wealth tax, and urisdictions provide local public goods financed by land taxes. Public goods provision levels are determined by a nationwide vote, and local public goods levels are determined by within-urisdiction votes. He proves an existence theorem. 6 In economies without mobility of consumers, much more general results have been obtained by extending the standard general equilibrium model. In a one-urisdiction economy with one public good, Denzau and Parks [9] prove a general existence result of maority voting equilibria. Greenberg and Shitovitz [21] further generalize Denzau and Parks's [9] result. First, by using Greenberg's [19] d-maority voting rule, they can treat the many local public goods case, and they drop transitivity and completeness of consumers' preferences. Greenberg [19] shows that a necessary and sufficient condition for an equilibrium to exist in general is that the maximal set of winning coalitions in a community consists of all coalitions that contain more than d(d+1) of its residents given appropriate convexity of preferences, where the agenda space is a d-dimensional Euclidean space. Obviously, this rule degenerates to the simple maority voting rule when d=1. Thus, Greenberg and Shitovitz [21] is strictly more general than Denzau and Parks [9]. 7 Second, Greenberg and Shitovitz [21] introduce many urisdictions to ustify each local government's price taking behavior. They also allow spillover effects of local public goods. However, they assume that consumers cannot move across urisdictions, which technically makes their model a one-urisdiction economy. 4 His independence assumption is rather strong. If the public good production function is linear, then the CobbDouglas utility function is the only candidate satisfying this assumption. However, we can drop this assumption by modifying his fixed point mapping slightly using the population mapping in this paper (see also Nechyba [32]). 5 An important point made by Dunz [12] is that there is, in general, no stratification equilibrium if land ownership is included in the model. Nechyba [32] derives some stratification results in Dunz's economy under restrictive assumptions. 6 This might look contradictory with Rose-Ackerman's [36] result. The difference is that in Nechyba [32], houses are indivisible goods. Hence, consumers cannot select the sizes of houses. This is the reason why there is no nonconvexity in extended budget constraints. Similarly, if there are zoning restrictions (c.f. Hamilton [24]), then we could recover convexity in extended budget sets. 7 Slutsky [49] handles a many public good economy with voting using a restricted maority voting rule. Thus, Slutsky's method is an alternative to d-maority voting rule.

5 484 HIDEO KONISHI In this paper, we will prove a general equilibrium existence theorem in a local public good economy with free mobility by extending Greenberg and Shitovitz's [21] model. Land is assumed to be divisible, and a proportional wealth tax is used to finance the local public goods. 8 A consumer chooses her residence from a finite number of urisdictions by watching their policy packages (tax rates and local public good provisions) and the population distributions. An allocation is called a voting market equilibrium in a local public good economy, if no consumer wants to unilaterally change her demand bundle and her residential place, 9 and if the policy package in each urisdiction is consistent with the voting outcome by its residents and its budget constraint. Free mobility adds the consumers' location choice problem to the model. Without mobility, convexity of consumption sets and preferences might be a plausible assumption, but with mobility, such convexity is an unreasonable one. We think free mobility of consumers is essential in Tiebout-type local public good economies since it allows, for instance, population agglomeration and capitalization. After introducing free mobility, we can no longer use the same method as the one that Greenberg and Shitovitz [21] use, since their approach depends crucially on convexity of preferences and consumption sets. In the proof of existence of equilibrium, we need to know each consumer's demand bundle to get a voting outcome in each urisdiction. So, to apply Kakutani's fixed point theorem, we need either convex-valued individual demand correspondences (the classical way to prove the existence of equilibrium) or (semi) convexity of strictly better sets (see Shafer and Sonnenschein [47]). Obviously, both requirements fail, since the consumers' location choice problem leads to nonconvexity of consumption sets across locations even though consumption sets in each location are convex. Therefore, we introduce a dummy consumer for each type to each urisdiction, and we hypothetically fix her location. Consumers' location choice problems are solved separately. Then, we can find a dummy consumer's demand bundles by the recovery of convexity. However, we need more assumptions to prove that the fixed point is an equilibrium, i.e., transitivity and completeness of preferences. Moreover, as a result of nonconvex consumption sets, we introduce a continuum of consumers to avoid integer problems (see Bewley [5]). This 8 Sato [39] independently provided an existence proof of a voting market equilibrium in a simplified version of our economy by using a different method. 9 In this paper we do not allow for a consumer's coalitional deviation to form a new urisdiction. All the deviations are unilateral (Nash-type behavior). For various types of research on equilibrium concepts which allow for coalitional deviations, see Bewley [5], Greenberg and Weber [22, 23], Ichiishi [27], and Wooders [54, 55].

6 VOTING WITH BALLOTS AND FEET 485 causes one conceptual problem. Since consumers can move freely, we might have empty urisdictions (urisdictions where only a zero measure of consumers are living) in the equilibrium. How are the policy packages in empty urisdictions determined? Since only a zero measure of consumers are living there, the voting problems in such urisdictions become trivial and we might have only unreasonable equilibria. 10 Hence, in the theorem, we will prove the existence of a voting market equilibrium without empty urisdictions. A continuum of consumers requires a few more subtle arguments in the proof. We use wealth taxes (instead of property taxes) as policy tools to obtain convex extended budget sets. This convexity is crucial in applying Greenberg's [19] method to prove the existence of a voting equilibrium. Since we assume perfect divisibility in commodities (including land), population size in each urisdiction can vary freely, different from Dunz [10] and Nechyba [32]. We adopt Greenberg [19] and Greenberg and Shitovitz's [21] d-maority voting rule as each urisdiction's collective choice rule. 11 We will allow spillover effects of local public goods, externalities from other residents, and snob effects as in Greenberg [20]. Our specification of externalities treats differential crowding (externalities from population distribution: see Wooders [55]) which contains nondifferentiated crowding as a special case. This type of generalization is helpful to see what determines a consumer's location choices. Since our focus is on consumer behavior, we will not exhaust all the possibilities to weaken the set of assumptions to assure the existence of an equilibrium (for generalizations, see Section 5). Section 2 describes the model and the consumer choice problem. At the end of the section we state our main theorem. Section 3 explains the assumptions employed. Section 4 proves the existence theorem. Section 5 is devoted to concluding remarks. 10 In an empty urisdiction, we cannot find any admissible winning coalition. Thus, in the formal sense, any feasible policy package is admitted as a voting outcome. However, since only a zero measure of consumers are living in the urisdiction, no local public good will be provided. As a result, some empty urisdiction might charge a very high tax rate without providing any local public good. Surely, consumers do not want to move in such a urisdiction. If this is the case, even if we can prove the existence of an equilibrium, the equilibrium might not make sense. The author thanks one of the referees for pointing out this issue. We will come back to this point in Section We can easily generalize Greenberg and Shitovitz's [21] result by replacing d-maority voting rule by a nonanonymous voting rule using Schofield's [41, 42] results (see Konishi [28]). However, in our economy we believe that the ``anonymity'' property of the voting rule is especially important, since the population distribution over urisdictions is endogenous because of free mobility of consumers.

7 486 HIDEO KONISHI 2. The Model After setting up the model, we will discuss each agent's problem and define a voting market equilibrium. The description of a political economy is as follows. (a) Private and Public Goods. There are J urisdictions in the economy. The urisdiction set [1,..., J] is also denoted by J. There are I private goods (commodities) in the whole economy, and the set of private goods is also denoted by I. These private goods can include freely traded commodities and location-specific commodities (say, land and labor at a certain location). Freely traded commodities are transported among urisdictions without costs, while location-specific commodities are transported with costs or prohibited to be transported among urisdictions. We can partition location-specific commodities into two groups, although we will not distinguish them explicitly in the analysis. The first group is composed of commodities which are mobile with transportation costs, such as labor, and the second is composed of commodities which are immobile, such as land. Even if commodities in different urisdictions are physically the same, they are treated as different commodities unless they are freely traded commodities. For example, coffee in Rochester and coffee in Dallas are different commodities. Transportation costs of commodities can be described by transportation technology sets (production sets) which transform commodities in a urisdiction to those in another urisdiction (see Schweizer et al. [44]). The second group of commodities is not transportable because the transportation technology sets do not permit the transportation of them among urisdictions. The private commodity set I is partitioned into I f and (I ) # J, where I f and I denote the set of freely traded commodities and the set of location-specific commodities at, respectively. Land in urisdiction, denoted by l # I, is homogeneous and perfectly divisible. Jurisdiction provides Q local public goods. Thus there are Q# # J Q local public goods in the economy. Sets of local public goods are also represented by Q and Q. (b) Consumers. There are A types of consumers. The type set is denoted by A, and a # A is its representative element. For each type a we have a continuum of consumers. Consumers can move over urisdictions, and they have to choose one of them to live in. Each consumer is endowed with private goods, but not with local public goods. We denote type a consumer's (private good) endowment by a function e a : J R I (see + Greenberg [20]). That is, if a type a consumer decides to live in the th urisdiction, then her endowment vector is e a ( )#R I +. This device is required, since the endowment of the first group of location-specific commodities (especially labor) can change depending on the place of

8 VOTING WITH BALLOTS AND FEET 487 residence. 12 For commodities such as land, the endowment mapping is a constant mapping since a consumer's location choice does not affect her land endowment at all; i.e., e a l ( $)=e a l ( ") for any, $, ". Even if I decide to live in Dallas instead of Rochester, my land endowment in Rochester will not be transformed to be land endowment in Dallas. Each type a # A is composed of her preferences (defined below), endowment, and share holdings (defined below). (c) Consumption set. A type a consumer's consumption set at urisdiction is denoted by 0 a /RI+Q + _J, i.e., private commodity consumption, local public good consumption, and residential choice. We partition 0 a into three parts: 0 a #X a _G_J, where X a /RI + with (private good consumption set) and G=R Q + (public good consumption set). Depending on types of commodities, we partition X a : X a #X a f _Xa 1 _}}}_Xa, where J X a f and X a $ are freely traded commodity consumption set and location $-specific commodity consumption set, respectively. We let X a $ =[0] for any a # A and for any ${. This implies that if a consumer chooses urisdiction as her residential place, then she cannot consume commodities which are specific to other urisdictions. 13 Let G =R Q + (public good consumption set in th urisdiction). Representative elements of X a, f X$, a G are x f, x $, and g, respectively. Let 0 a # # J 0 a /R I+Q + _J. Note that by including G in 0 a we can capture the case where there are spillover effects from local public goods. Let X=R I + be potential commodity consumption set. (d) Price vector. The price vector is defined over private and public goods. Price vectors of private and public goods are denoted by r=(r f, r 1,..., r J )#R I and s=(s + 1,..., s J )#R Q +, respectively. Let p#(r, s). We use the following simplex normalization: 2#[p#R I+Q : + i#i r i + q#q s q =1]. (e) Population distribution. There is a continuum of consumers for each type a # A, and the measure of the type a population is denoted by m a >0. Therefore, the total population in the economy is described by m # (m 1,..., m A ). The set of possible population distributions over urisdictions given total population is denoted by M#M 1 _M 2 _}}}_M A, where M a #[m a # R J : + # J m a=m a ]. (f) Firms. To concentrate on consumer choice, we greatly stylize the producer side. With more effort and notation our result can be generalized 12 In an economy with many locations, it usually makes more sense to use trading sets (McKenzie [30]), instead of consumption sets. However, since we have wealth tax in our economy, we employ consumption sets together with endowment functions. 13 We can easily change our model to allow for the case where a consumer's residential place and her working place are different (see Section 5).

9 488 HIDEO KONISHI in a straightforward fashion. There are H firms in the economy, and each of them is located in a certain urisdiction. The firm set is represented by H. The firms' location choice problem can be avoided by simply decomposing a freely mobile firm into J firms. Each firm h # H has a production set Y h /R I+Q, such that 0 # Y h. The aggregated production set is defined by Y# h # H Y h. The aggregated profit income function is denoted by?: 2 R + _ [] such that?( p)=sup y # Y p } y. Let 2$#[p#2:?(p)<]. Type a consumer's profit income function : a : 2$ R + is such that a # A m a : a ( p)=?(p) for all p # 2$. We assume : a : 2$ R + to be continuous [17]. The aggregated supply relation ': 2$ Y is such that '( p)#[y#y:\y$#yp}yp}y$], where dot ( } ) denotes an inner product operation. This might be empty-valued for some p # 2$. (g) Preferences. Type a (a # A) consumers' weak preference relation is represented by a binary relation R a /(0 a _M_2)_(0 a _M_2). Strict preference is defined in the usual way: P a /(0 a _M_2)_(0 a _M_2). This general specification allows for spillover effects of local public goods, the possibilities of differentiated crowdings (Wooders [55]: congestions or type conflicts in each urisdiction such as race and religion), and snob effects (some consumers may not like to live with richer or poorer people). We adopt the convention of denoting the weak (strong) preferred set correspondence by R a : 0 a _M_2 0 a _M_2 (P a : 0 a _M_2 0 a _M_2). To state assumptions on preferences in our theorem, we need to define some restrictions of R a. For example, denote the preference relation R a when (, m, p)#j_m_2 is fixed by R a [, m, p]/(x a_g)_ (X a _G). This is the preference relation on private and public good consumption when location choice, population distribution, and price vector are given. In this way, we denote restricted weak (strict) preference relations by R a [}] (P a [ } ]). The weak (strong) preferred set correspondence when some of the elements are held fixed is also denoted R a [}] (P a [}]). Now, we can denote a consumer's type (a # A) by a list (R a, e a, : a ). (h) Jurisdictions. Each urisdiction imposes a proportional wealth tax t on its residents independent of types of consumers, and provides Q local public goods. 14 A set of possible wealth tax rates is denoted T /[0, 1]. We denote a product of sets of possible tax rates by T (=> # J T ). A local public good provision level in urisdiction is denoted by a vector g # G. 14 This is the proportional wealth tax is proposed by Foley [16]. Denzau and Parks [9] and Greenberg and Shitovitz [21] use a general linear tax scheme which treats different types of consumers differently. It is easy to generalize our result by employing their tax scheme. However, we think that the ``anonymity'' property of the tax is particularly important in our economy, since consumers can move freely and the population distribution is not determined ex ante.

10 VOTING WITH BALLOTS AND FEET 489 Thus each urisdiction collects taxes and buys local public goods with the tax revenue. Each urisdiction's budget is always balanced: that is, s } g =t a # A m a [r } ea ( )+: a (p)]. By this requirement we can determine the tax rates from any given public good provision levels, population distribution, and price vector. (i) Economy. An economy E is a list (J, (0 a,r a,m a,e a,: a ) a#a,t,y). Consumers choose their locations and demand bundles given a price vector, population distribution, and each urisdiction's offer (tax rate and local public good provision level). Residents in a urisdiction vote over public good provision levels, taking the resulting wealth tax rate into account. The consumers' decision problems and the urisdictions' collective choice processes are described below. () Consumers choice problem. A type a consumer chooses her most preferred demand bundle and residence. Her restricted budget correspondence when her residence is in # J, B a : T _2$ X a, is such that B a (t, p)=[x # X a : r } x(1&t )(r } e a ( )+: a (p))]. Let type a consumer's budget correspondence, B a : T_2$ X a _J, be such that B a (t, p)# # J (B a (t, p)_[]). Therefore, a type a consumer's choice problem is summarized as the following correspondence: h a : T_M_2$_G X a _J such that h a (t, m, p, g)=[(x, )#B a (t,p): \(x$, $) # B a (t, p), (x$, $) P a [m, p, g](x, )]. Her location choice is summarized by a correspondence, pro J h a : T_M_2$_G J, which is the latter part of h a (where pro (}) ( } ) denotes a proection operation). That is, if # pro J h a (t, m, p, g), then a type a consumer chooses to live in urisdiction. (k) Consumers' demand decision problem when their locations is hyperthetically fixed. Although the consumer's consumption decision is done simultaneously with her location choice, it is convenient for us to consider the case where her choice is done sequentially to prove the existence theorem. Note that this is not needed to define a voting market equilibrium, but will be used later in our proof. Define h a : T _M_2$_ G X a such that h a (t, m, p, g)=[x # B a (t, p): \x$#b a (t,p), x$ P a [m, p, g, ](x)]. That is, h a denotes type a consumer's restricted demand correspondence and shows her demand when her residence is restricted to # J. (l) Equal treatment consumption plans. Before we describe our voting mechanism, we first define equal treatment consumption plans. Since there is a continuum of consumers in our economy, even if a pair of consumers belong to the same type, they do not necessarily choose the same urisdiction andor consumption bundle. A consumer's voting behavior depends on her private good consumption bundle. Hence, if consumers belonging to the same type are choosing a continuum of different consumption bundles

11 490 HIDEO KONISHI in the same urisdictions, then there is a continuum of types of voters. 15 Nevertheless, Greenberg's [19] voting equilibrium existence theorem is applicable only to a society with a finite number of voters (or types of voters). 16 To overcome this point, we will prove the existence of an equal treatment voting market equilibrium instead of that of a voting market equilibrium itself. Hence, we prove the existence of a stronger notion of equilibrium. An equal treatment consumption plan is a consumption plan such that if a pair of consumers belong to the same type and are living in the same urisdiction, then both of them choose the same consumption bundle. Of course, we do not require that the same type of consumers must live in the same urisdiction. Actually an ``equal treatment'' simplifies the definition of a consumption plan quite a bit (c.f. Hildenbrand [26]). Formally, an equal treatment consumption plan is a list ((m a ) a # A#J, (x a ) a#a#j)#m_(> a # A > # J (X a )), where xa # X a. (m) The voting mechanism. We define the voting decision problem over tax rates when a consumer is assigned to live in a urisdiction. To show the existence of a (Nash type) equilibrium in the sense that there is no consumer who deviates, we can separate the consumer's voting decision problem from demand and location choice problems. Consider each consumer's voting decision given her location and demand bundle. Consumers are price takers, and consumers vote over pairs of public good provision vectors taking the resulting wealth tax rates into account (via her urisdiction's budget constraint). 17 Let { : M_2$_G T be a tax mapping at urisdiction such that { (m, p, g )=[t # T : s } g =t a # A m a [r } ea ( )+: a (p)]]. Let G : M_2 G be the feasible public good provision correspondence such that G (m, p)=[g #G : s }g t a#a m a [:a (p)+r }e a ()] for some t # T ]. This denotes a set of feasible public good provision in. The voting mechanism employed in the th urisdiction is a Q -maority voting rule used in Greenberg [19]. One agenda g is 15 If a consumer has strictly convex preferences within each urisdiction, she has a unique optimal consumption plan in any given urisdiction, so this problem does not occur. This is the reason why Dunz [1012] assumes strictly convex preferences in his papers. 16 If there is an infinite number of types of voters, we have a technical difficulty in applying Greenberg's [19] theorem (an infinite intersection of open sets might not be open). This is the reason why we assume that there is a finite number of types of consumers in the economy. Caplin and Nalebuff [6, 7] provide some existence results in a continuum of voter case. However, it seems to be difficult to apply their results to our model. Schofield and Tovey [43] relates Greenberg's [19] result to Caplin and Nalebuff's [6] in the limit in the Euclidean preference case. 17 In our paper and in Slutsky [49] and Denzau and Parks [9], it is assumed that consumers are price takers for all goods including local public goods. On the other hand, Greenberg and Shitovitz [21] assume that consumers take private good prices as given, and vote over the efficient production set that is owned by the local urisdiction. We can adopt Greenberg and Shitovitz's [21] type of voting behavior with a little more work.

12 VOTING WITH BALLOTS AND FEET 491 defeated by another g $ if and only if strictly more than Q (Q +1) of residents prefer g$tog. An agenda g # G (m, p) isaq -maority voting outcome, if g is not defeated by any other agenda in G (m, p). If we have only a one-dimensional agenda space, this rule degenerates to the simple (``absolute'' in Denzau and Parks [9]) maority voting rule. Formally, a Q -maority voting outcome given an equal treatment consumption allocation is defined as follows. We will define P, collective preferences generated by a Q -maority voting, in the following way. Let F a : X a _G_M_2 G be such that F a (xa, g, m, p)=[g $#G (m,p): _x a $#B ak ({ (m, p, g $), p) s.t. (x a $, g $) # P a [g &, m, p, ](x a, g )]. That is, g $#F a (xa,g,m,p) implies that a type a consumer prefers g $tog. Thus, she votes for g $ and makes an obection to g. Let W : M > a # A M a be such that W (m)=[m $#> a#a M a :m a $ma for all a # A and a # A m a $ a # A m a >Q (Q +1)]; i.e., W (m) is the set of winning coalitions (in terms of measure) in. Let A: > a # A M a A be such that A(m$)=[a # A: m a $>0]. Let W : M 2 A be such that W (m)= m $#W (m)a(m $), where 2 A is the power set of A. That is, W (m) is a collection of sets of types which can form winning coalitions in. Let P :(> a#a X a )_G_M_2 G be such that P (x, g, m, p)= W # W (m) a # w F a (xa, g, m, p); i.e., P (x, g, m, p) represents public good provision levels which are strictly preferred within the set of feasible public good vectors. An agenda g is a Q -maority voting outcome if and only if P (x, g, m, p)=<. Let us define an equal treatment voting market equilibrium. From the reason that we explained in Section l, we require that there is no empty urisdiction in the equilibrium. (n) Equal treatment voting market equilibrium without empty urisdictions. An equal treatment voting market equilibrium without empty urisdictions in an economy E is a list (t, m, p, g, y, (x a ) a#a#j), where (a) t # T for all # J, (b) m a # M a for all a # A, (c) p # 2, such that (1) \a # A \ # J, ifm a >0 then # pro J h a (t, m, p, g), 18 (2) \a # A \ # J, ifm a >0 then (xa, )#ha (t,m,p,g), (3) y # '(p), 18 Although condition (1) is implied by (2), we require (1) to stress that a consumer's location choice is consistent.

13 492 HIDEO KONISHI (4) a # A # J m a xa&y X& a # A # J m a ea ( )=0, (5) g&y G =0, (6) \ # J, s } g =t a # A m a [r } ea ( )+: a (p)], (7) \ # J, P (x, g, m, p)=<, (8) \ # J, a # A m a>0, where y=(y X, y G ): y X and y G denote private and public production vector. (o) The existence theorem. Our main result takes the following form. Theorem. An economy E has a voting market equilibrium without empty urisdictions under the following assumptions: (a1) I f 1, (c1) For all a # A and # J, X a is convex, bounded below, and 0#X a, (f1) Y is closed, convex, 0#Y,and Y & R I+Q + is bounded, (f2) R I+Q & /Y, (f3) For all p=(0, s)#2,?(p)=, (f4) For all # J and p, p$#2 such that (r &l, s)=k(r$ &l, s$) for some k # R ++,?( p)=k?( p$), (g1) For all a # A, R a is closed, (g2) For all a # A, R a is reflexive, transitive, and complete, (g3) For all a # A, # J, g & # G &, m # M, and p # 2, R a [ g &,, m, p]: X a_g X a_g is convex-valued, (g4) For all a # A, # J, g # G, m # M, p # 2, and x # X a, for any x$ f >x f,(x$ f,x &f )#P a [g,,m,p](x). 19 (g5) There exists a # A such that for all g # G, m # M, p # 2, and, $#J, for any x # X a and x$#x a, $ there exists x f # X a $f such that (x f, x$ &f, $) # P a [g, m, p](x, ), (g6) There exists a # A such that for all g # G, m # M, p # 2, and, $#J, for any x # X a and x$#x a $, there exists x l $ >0 such that (x l, x$ &l $, $) # P a [g, m, p](x, ), (1) For all # J, T #[0, t ] satisfies 0<t <1, (2) For all a # A and # J, e a (i)>0 for all i # I i f _ I, where g & # G & => ${ G $, and (x$ f, x &f )=(x" i ) i # I such that x" i =x$ i if i # I f, and x" i =x i if i I f. 19 We use the convention of vector inequalities:, >, and r.

14 VOTING WITH BALLOTS AND FEET Remarks on Assumptions Here, we will discuss the important assumptions assuring the existence of an equal treatment voting market equilibrium without empty urisdictions. Although convexity of preferences (g3) is not needed in a private market economy with an infinite number of consumers, we need convexity of preferences to get (d-)maority voting equilibria (see Greenberg [19]). 20 In a voting problem, we have no convexifying effect of a continuum of consumers, unlike a market demand correspondence (see Aumann [1], Schmeidler [40], and Hildenbrand [25]). The reason is that voting does not imply the summation of individual demands. Next we will discuss assumptions on preferences over all. To do that, let us review the previous work on existence theorems. Shafer and Sonnenschein [47] prove a quite general existence theorem for market equilibrium without transitivity and completeness. Greenberg [19] applies their result to a voting equilibrium problem in an ingenious way and proves the existence of a voting equilibrium under the same set of assumptions as theirs. Greenberg and Shitovitz [21] synthesize these two papers to prove an existence theorem for a voting market equilibrium in a local public good economy without mobility. As a result, they are able to prove the theorem without transitivity and completeness. In our existence theorem we need transitivity and completeness since we allow free mobility of consumers. As long as consumers are fixed in a urisdiction, we can assume that each consumer's consumption set is convex. However, once free mobility is allowed, obviously the consumption set becomes disconnected. Since the Shafer and Sonnenschein [47] and Greenberg [19] proofs depend crucially upon convexity of consumption sets and preferences, we cannot apply their method to our model directly. At first we need acyclicity of preferences to get a stable population allocation. This, however, is not enough. In contrast with standard market equilibrium existence theorems, we have to know not only market demand but also each consumer's individual demand bundle to apply Greenberg's [19] argument. In our economy, each consumer's consumption set is disconnected, which implies that we cannot expect continuity of each individual demand correspondence. Therefore, we will use dummy consumers to get continuity; i.e., although consumers are able to move, we hypothetically fix each type of consumer to each urisdiction to get 20 Strictly speaking, Shafer and Sonnenschein [47] (appears below) and Greenberg [19] only require semi-convex-valuedness of strict preference correspondences instead of convexvaluedness. However, if we apply these two theorems to a public good economy with voting, semi-convex-valuedness is not enough to prove the existence of equilibrium because a consumer's voting behavior depends on her private good consumption (see Greenberg and Shitovitz [21] and Konishi [28]). We need convex-valuedness of preference correspondences.

15 494 HIDEO KONISHI individual demand bundles. Population distribution is derived from consumers' location choice separately. 21 Then, the problem is whether or not each consumer's restricted demand bundles when she is fixed in her most favorite urisdiction coincide with her unrestricted demand bundles (the demand bundles chosen when she can move). If these two coincide with each other (consistency of demand), we can obtain equal treatment consumption plans that are consistent to consumers' choice problem even if we separate consumers' demand decision and their location choice (see () and (k) in Section 2). To satisfy this consistency of demand, we need transitivity and completeness of preferences (see Lemma 2 and its remark below). Without the consistency, a fixed point allocation using restricted demand may not satisfy the equilibrium condition (2). Some consumers might want to move out to some other urisdictions to choose a better consumption bundle than the assigned one. 22 Now we move to (c1), ( 1), and ( 2). These ensure the existence of cheaper points for each type in each urisdiction. 23 Assumption ( 1) requires that a 1000 wealth tax is prohibited. Allowing for a 1000 wealth tax, Denzau and Parks [9] and Greenberg and Shitovitz [21] prove the existence of a minimum expenditure equilibrium, which is equivalent to a quasi-equilibrium under convexity of consumption sets. (For these equilibrium concepts, see Hildenbrand [25].) Even in our economy, we can still prove the existence of a minimum expenditure equilibrium without ( 1). However, although if consumption sets are convex then a minimum expenditure equilibrium is equivalent to a market equilibrium under monotonicity of preferences or irreducibility, this equivalence does not hold if consumption sets are nonconvex. We discuss alternative assumptions of ( 1) in Section 5. Finally, we make some remarks on technical assumptions. Assumption (a1) requires that there exists at least one commodity that is freely tradable, that is, there is a commodity which can be transported without costs. Although this assumption is not very desirable from the spirits of Schweizer, et al. [43], it is unavoidable to keep the set of assumptions 21 Population of a type of consumer in a urisdiction is positive only if the urisdiction is one of the most preferable urisdictions for her given the price vector, policy packages, and population distributions over urisdictions (see the equilibrium condition (1)). The population mapping has nice propertiesit is nonempty and convex-valued and has a closed graph (see Fact 3 below). 22 If we do not use a voting rule to determine the public good provision vector, (that is, if we simply need aggregated demand instead of individual demands of consumers), acyclicity of preferences is enough to prove an existence theorem as long as budget correspondence is continuous (see Schmeidler [40]). 23 Assumptions (c1) and (1) cannot be replaced by a dispersed wealth distribution type of assumption (see Mas-Colell [29] and Yamazaki [56]), because we need a finite number of types of consumers (see footnote 16).

16 VOTING WITH BALLOTS AND FEET 495 simple. Assumptions (a1), (g4), and (g5) are used to assure that there exists at least one commodity which has a positive price and is tradable by any consumer in any urisdiction. Otherwise, in some urisdiction, the prices of all the relevant commodities for residents might vanish. Then their budget sets at become the whole consumption sets at, and we lose the lower hemicontinuity of budget correspondences (see also Greenberg [20], (2), p. 24). Assumption (g4) is a monotonicity of preferences in freely traded commodities. Assumption (g5) says that there exists a type of consumer who strongly prefers freely traded commodities (a freely traded commodity lover). Although it might look unusual, it is also important in proving r f r0. (Mas-Colell's [29] assumption (2) plays the same role in an indivisible commodity economy.) Assumption (f 3) is used to exclude the case where r=0. Since public goods are produced by private goods, it seems reasonable to assume that the firms can earn infinite amount of profits under r=0 (see also Denzau and Parks [9], (c3) p. 253). Assumptions (f4) and (g6) are made only to exclude the possibility of having empty urisdictions. Assumption (f 4) requires that land is not used in production. Assumption (g6) says that there exists a type of consumer who strongly prefers land (a land lover). 24 Although we can prove the existence of equal treatment voting market equilibria without (f4) and (g6), all the possible equilibria might contain empty urisdictions with some unreasonable tax rates (see footnote 10 and Section 5). We can replace (f3), (f4), (g4), (g5), and (g6) by weaker but more technical assumptions. 4. The Existence Proof Here, we prove the existence of voting market equilibria in our economy E. Although our model is close to that of Greenberg and Shitovitz [21], there are some important differences. First, since consumers can move freely, we have disconnected consumption sets. Second, to avoid integer problems we need a continuum of consumers with each consumer having zero measure (see Bewley [5]). In this case, the usual truncation method developed in a finite economy does not work. Since we have a continuum of consumers, even if we have boundedness of the aggregated (or average) attainable consumption set, some small portion of consumers can consume a very large amount. Third, we need to show that the prices of freely traded commodities are positive in order to ensure the continuity of the budget set in each urisdiction. Fourth, we want each urisdiction to have positive measure of consumers as we explained in the Introduction. 24 Although these (especially, (f4)) are not preferable assumptions, we adopt them to keep the assumption set simple.

17 496 HIDEO KONISHI To deal with the first point above, we need to assume transitivity and completeness of preferences to ensure the restricted demand in the most preferable urisdiction to be the unrestricted demand, as we have seen before. For the second point, we have to use a convergence argument. First we compactify each consumer's consumption set by an arbitrary hypercube and derive a voting market equilibrium, and then by making the hypercube larger and larger we get an equilibrium sequence, of which the accumulation point is an equilibrium of the original economy. This method is often used to measure theoretical general equilibrium models due to nonconvex preferences (see Hildenbrand [26]). Although we have convex preferences within each urisdiction, we must use this type of argument since consumption sets are not connected. For the third point, we truncate the price set and apply a convergence argument making use of (g4) (see Hildenbrand [26]). For the fourth point, we truncate the population distribution set and use a convergence argument once again. However, once the population distribution set is truncated, we need to force some portion of consumers to choose unfavorable urisdictions. Then we decrease the portion to zero as a limit, and show that the measure of residents in each urisdiction does not go to zero. This procedure is similar to the one employed in proving the existence of a perfect equilibrium in game theory (see Selten [46]). Let us start to prove the existence of an equal treatment voting market equilibrium. First, we will define the attainable sets of the economy. Let X a =m a X a. Define the attainable set of E as follows: a state a ( y, m, x, g)#y_m_[> a # A > # J (X )]_G is attainable if y X a # A a # J [x &ma ea ( )] and y G g. Then the attainable production set is defined by Y #[y# Y:( y, m, x, g) is attainable for some (m, x, g)]. Similarly, the attainable aggregated consumption set of type a is defined by a X #[x a a # X :(y,m,x,g) is attainable for some ( y, m, x &(a, ), g)]. The following shows the boundedness of the attainable production and aggregated consumption sets (standard: see Appendix). Lemma 1. The attainable production set Y and the attainable aggregated a consumption set of each type in each urisdiction X are bounded. As we explained in Section 3, we need ``consistency of demand'' due to disconnected consumption sets. The following lemma proves the consistency under transitivity and completeness of preferences. Lemma 2. Suppose that # pro J h a (t, m, p, g). If R a [m, p, g] is reflexive, transitive, and complete, then (x, )#h a (t,m,p,g) if and only if x # h a (t, m, p, g). Proof. Let (x, )#h a (t,m,p,g). Then there is no (x$, $) # B a (t, p) such that (x$, $) # P a [m, p, g](x, ). Therefore, there is no z # B a (t, p) such that

18 VOTING WITH BALLOTS AND FEET 497 z # P a [m, p, g, ](x), and x # h a (t, m, p, g). Conversely, let x # h a (t, m, p, g). Suppose that (x, ) h a (t, m, p, g). Then, there exists (x$, $) # B a (t, p) such that (x$, $) # P a [m, p, g](x, ) and ${. By assumption there exists (z, )#h a (t,m,p,g). By completeness and transitivity, (z, )#P a [m,p,g] (x,). Hence, z # P a [m, p, g, ](x), a contradiction. K Remark. This property is called full rationality in decision theory. Without transitivity and completeness, the latter part of the lemma does not hold. Notice that we did not use any other properties besides reflexivity, transitivity, and completeness in the proof. Although the domain of preferences is restricted in our economy, we need transitivity and completeness to get consistency. 25 By this lemma, we can replace the equilibrium condition (2) by the following: (2$) \a # A \ # J, ifm a >0 then xa # h a (t, m, p, g). Now, we truncate the economy in the following way. First we truncate the production set and the public good consumption set. Let D=D X _ D G /R I _R Q be a closed hypercube such that Y /int(d), where int( } ) denotes a topological interior of ( } ). Let Y /Y & D. This is the truncated production set which is compact and convex. Since an equilibrium allocation must be attainable, we can confine our attention to compact sets which contain attainable sets in the interior of them. Let G =D G & R Q.We + will replace Y, G, and G by Y, G, and G, respectively. Because of the modification of the production set, the supply correspondence and profit income correspondences must be replaced. Let 2"#[p# 2:'( p)&y {<], which denotes the set of attainable prices in the original economy. Then, 2" is nonempty and closed (see Lemma 2 in Gale and Mas-Colell [17]). Let 2 #co(2"), where co( } ) denotes a convex-hull operation of ( } ). The set 2 is closed and convex, and 2 /2$ since 2$ is convex. Let '~ : 2 Y be such that for any p # 2, y # '~ ( p ), p } yp } y$ for any y$#y. Notice that '~ satisfies the following: if p # 2", then '~ ( p )='(p)&y, and '~ ( p ){< for any p # 2. Next,?~ : 2 R + is similarly defined. Let :~ a : 2 R + be a continuous function such that :~ a ( p )=: a (p) for p # 2". By this modification, we can define the consumers' profit incomes for all p # 2, and as long as prices are supporting an attainable production, these artificial consumers' profit incomes coincide with their actual profit incomes. We truncate the population distribution set in the following manner: M k #[m # M: m am a kj for all a # A, # J], where k is a positive integer (remember that J represents either the set or the number of urisdictions in the economy). Using the same k, we truncate each consumer's consumption set with a hypercube C(k) which is defined below. Let C/R I be a closed 25 We have counterexamples for the statement of Lemma 2 when either completeness is dropped or transitivity is weakened to quasi-transitivity. These are available on request.