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Local public goods as indivisible commodities
Regional 7 ~
and Urban Eoanam~ 17 (19S"~ ] 9 ! ~ -
N~4q~
/.DCAL PUBLIC GOODS AS iNDIVISIBI~ C O M M O D ~ Ra~iv V O H R A *
R~,a~ved December 1995, fiaal vetsloa m a ~
~q~rfl ~gi~
A sharp~ yemen o~ ~--rr~:soa's (19~)J rcmtt on ~ ¢ ~ d ,~ ~ ~ Ti~ equilibrium is shown to be trm~ m ~ , - ~t_~a urea.kerr a ~ z ~ i 0 ~ r ~ | ~ i i v . ~ ~ h o ~ ~u~ l h ~ ~is~s a _q~md-bes¢ policy regarding the b,-,~lc of pebllc ~ ~ L,:~__.!~ o t k n ~_a and Rashid (i982) to e~oaomies _withproducfioa, tn th~ ~¢¢~,-..~.~ ,~ ~ ma~ ~ _.d~.__uat!~y~:~ that the a ~ ' r , ~ ~m.,~-b'd-m~~ 0 ~ does ~ ~ ia the ~ t ~ u : _~ ~ sum~a~_~ t.~q:L.~.~ Of trn~s'-~'~cmare ~ t ~ , ~ . e .
1. Intretlaefioe
As Bewley (I981) has shown, a rigorous f o ~ o a or a ~otio~ .r,f equilibrium which captures all of the e~e~tial idt~s of Tieboul (1956) d o ~ not lead to an equih'bdum con~p-~ which .Compar~ favorably with ~ q n l ~ * five equilibrium. He points out thal~ in general, Tie.bout equilibrium 'does a ~ have the nice properties of genei~al oampetitive equilibrium'; eq~tibti'e~._._ ~ma_y not exist or may not be Pareto o p t i m a l . A n d t h e a ~ o m p t i o l ~ w h i c h h a v e tO be made to avoid these diffioAties, 1 in _trarfi~dar the assmnption that there be as m a n y c o m m u n i t i e s as t h e r e a r e t y p e s o f g o a s u i a e r s , r o b T i e b o u t ' s
nGfion of almost all its significance. One might, thetefore~ coadude that Tiebout equilibrium cannot be given th~ same kind of rigorous basis as competitive equilibrium, in this paper we show that under oertaln conditions, Tiebout equilibrium does compare favorably with a Iess classk:~tl form of competitive equilibrium, namely one in which some ¢ommodites are indivisible. It should be clear from Bewley's work that the positive results about Tiebout equilibrium in the literature do not do justice to all the essential *This paper is based on Ch. 6 of the author's Phi). dissertation at Johns Hopldns University. Bruce Hamilton and Ati Khan have had a sig~cant inlluenee on this paper and the author is 8fateful to them for encouragement and many helpful ~ o n s . Than~s are also d ~ to Vernon Henderson, Peter Newman and Nicholas "tannelis for valuable comments. This research has been supported in part by NSF Grant No. SES-8410229. ISee Theorem 9.12 in Bew!ey (1981). 0t66-0462/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North.HoUaad)
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I~ Vohra, l.~,cal public goods as indivisible commoditie~
aspects of Tiebout (1956). Different authors have formalized the notion of Tiebout equilibrium in quite diverse ways - stressing some aspects of the Tiebout model while abstracting from others, in this respect the present paper is no exception. We consider a model of local public goods in which there is a fixed number of communities. We do not model land and a consumer's choice of a community depends only on the public services being provided and taxes being levied in each community. Each local government finances its expenditures through lump-sum taxes on its ~esidents wi~out discriminating by name or on the basis of taste. We assume that the publ/¢ goods are pure pubfic services so that their cost is proportional to the size of the commurfity. In the context of a large economy this is not a restrictive assumption as long as the cost of public goods eventually becomes proportional to the size of the comn~unity since the economy can then be rcp!/ea~d in order to satisfy this assumption. We first establish that~ for any giv._~ mix and level of per capita services in each community, tbel~ exists an appco~mate Tiebout equilibrium for an economy with a suffidenfly lar_~ number of consumers. The approximation is of the same n~t-~h-¢ as that inYb!ved ha showing the existence of competitive equilibrium when some ¢ommod/6es are indivisible. Our other as~.-zmptions are weaker than those, for examp~ of Bewley (I981, Theorem 9.12). A restrictive asp~t of this model is the assumption that the ~ of per capita public services in each community is give~ exogenously. Th/s ~t.m!p~ fion was also made by Ellickson (1979), W~__~n (!980) a ~ S c c ~ . . ~ (198!). Our main result can, in fs¢~ be ,~'_~-'gz~l as an ex~e,~t+~, of corresponding result of FJfickson (1979}. Given this ~ ~ model should be viewed as shon-run "F~ou! model H ~ , ~e ~k~ s.~-,w that if each community choof~ its bundle of p ~ b ~ sert~:~ from a ~ p e ~ set, there exists an appro~nate Tkbout equ~-b6um whl¢~ is uxxmcl-b~ .~ la Mantel (I975). It is only iu th~ ~ tha~ we cau ~ per ¢2p~t public scrvioes to h:, e~dogcnouS. A ~¢~Rcd a p p t o a ~ to m a k / ~ ~he of local public goods endogenous is that of Richter (1992~i~ ~enerAli~e~ subsequently by Greenberg (1983), in wh__-_mh_each_ local govecamen~s chooses. such a bundle op_fim_al!yfor its residenLs. However+ there is z~ g,a:srs~e t~_~. the equilibrium is Pareto o _l~m_'__~1 for the economy as a ~_h¢~ [~ Weghoff 0977) each oommuaity decides UlXm its level of the lmre ¢mb~_L through majori~ voting. A quit~ d ~ t ~ fto~ al~ o[ ~ ~ iks.~ of Weeders (1980) in which not only is the m/~ ~ra~ ~ ~ ~ ¢~g~ services in a community cndog~ous but so i~ tL~ number of c o m m ~ A cooperative solution concept is analy~d and _ . ~ ~ ~ ~ including one concenfing the ~[afio__n~_~p between the n w m ~ o f ~ of dh~erent types and the tedmotogy, ~ _~._'~o~ d an t core is ~ m ~ . However, the taxes are not required to s a t i s b / ~ ~ ~ i t ~ mp, a t t k ~ +
~. Vohr~ lg~cal public goods as indi~i~ble commodities
193
they may depend on consumers' preferences.2 A related model has recently been introduced by Greenbc~g and Weber (1986) in which the total taxes of a community are either spread evenly among the residents or in proportion to income. This has the effect that the equilibrium need not be Pareto optima. It is worth mentioning that in the case of pure public services, equilibrium in their model leads to the formation of as many communities as there are types of consumers. In the remaining part of this scetion we shall indicate the essential difficulties involved in general equilibrium theory with indivisibilities, point out the inadequacies of previous attempts to consider local public goods as indivisible commodities, and describe the notion of approximation we shall USe.
The idea of treating a finite number of communities as indivi-,ible commodities goes back to Schweizer, Varaiya and Hartwick (1976). However, their resuR on the existence of a compensated equilibrium was based on the assumption that the number of traders is 'large' - a notion which was never pre.cisely defined.3 The. work of E|lickson (1979) can be seen as the first rigorous attempt in which toca! public goods are treated as indivisible commodities and the existence of an approximate Tiebout equi~brium is shown. Unfortunately, an argument in Ellickson's existence proof seems to be open to question. Moreover~ his notion of approximate equilibrium is quite weak in relation to the recent advances that have been made in the theory of indivisibilities. There.are two major difficulties encou,tered in proving the existeno: of aft equilibrium with m~visible commodities [s¢¢ Mas~Co!ell (1977)]. Fh'sfly, ihe demand correspondence may fail to be upper hemi-c.ontinuous. A small change in prices may induce a utility-maxim~_ng consumer to ~ttmp" fco/n one community tO another. This is tiroome's (1972) "problem of the edge'~ Secondly, the demand correspondence may not be convex valued. It is worthwhile to emphasize that even though, in an economy with a continuum of traders, the problem of non-convexity can be eliminated through aggregation, the same is not true of the continuity problem unless endowments_ are assumed to be dispersed as in Mas-ColeH (1977). For a fini~ economy with indivisibilities both the lack of convexity and continuity are inherent and~ unless rather strict assttmplions ~'e made, the equilibrium notion has to be modified to an approximate one~ Eltickson conside~ a --orion of equilibrium in which the approximation can be ascribed solely to the non-convexity problem. The 'problem of the 2Both Westboff (1977) and Wo~ca-s (I980) assm~ thltt there g~~ ~ publi~ go~__, ~Although lhey consider a finite ¢wnomy, t i ~ do no~ n:quir¢ the m~mbc¢ of residers ia a
communityto be an integer.~hus the results.., are a good approzimatioaonly if the neanbe¢ of-households is "lat~"" [Schweizer, Varaiya arai Hartwick (1976, footnote 9}],
194
edge' is dealt with by assuming that the consumer is not willing to spend his entire income on public goods and by assuming that all private goods~p~c~ ~re strictly positive. Recognizing that upper-hemicontinuity of the demand correspondence depends on all private goods' pric~s being strictly positive he nevertheless uses Debreu's (1959) theorem to prove the existence of ~uilibrium in the convexified economy. This procedure seems to be invalid since Debreu's theorem treats prices as belonging to the closed simplex which clearly permits prices of some private goods to be zero. Instead of attempting to prove a "quasi equilibrium' version of Ellickson's result, however, we consider an alternative notion of approximation which does not assume away 'the problem of the edge'.' t We work with the approTAmate demand corr~ pondence of Khan and Yamazaki (1981) which is upper hemicontinuous eve~ when th~ consumption set is non-c~nvex~ A consumption bundle is said to belong to the approximate demand correspondence if it belongs to the budget set and there does not e~$t •anether bundle in the interior of the budget set which is preferred to iL If utility functions exist thi.~ is nothing but the usual definitiou of a-demand correspondence assc~ated with a compen~ted equilibrium. Although we do not require utility functions to exist we shall refer to an equilibrium associated with the approximate demand corresponden~ ~ a c~mpens~ed equilibrium. 5 In economies with non-convex p r d ~ it ~ u~tmlly be shown that equilibria exist which are approximate in the sen~ that feasibility is violated by an amount which is independent cf t~e size of the ¢¢o~omy~ Eili~sc~ (1979) follows Staff's (1969) a p p r o ~ of s h o ~ that if O~e n o n ~ z ~ are uniformly bounded an approximate Tiebot~t e q ~ b 6 e m exists. Recemly, Anderson, Khan and ~ d (1982) have shown ~ ~ x i ~ of ~.-~rordmate equilibrium with bounds independent of ~ As t_..~ .__~_.k i~ Anderson, Khan and R:~h;d (1982) indicate~ for an e ~ ~ y ~ indivisibilities, an approxh~m~ c o m ~ ! e ~ _eq_.t~'~um ( ~ t h bo~dls independent oi p r d ~ ) can be shown to e~d~t by ~ t ~ the ~pproxi~ m~te demand correspondence of Khan and Y a m a n ~ (198l~o I¢ ~ ~_¢ apply their notion of approximation to a Ttebo~t equ~hbritm~ how~c~, we shall also need to extend their result to e~:vnomies with p ~ o ~ A t~hnical diff}c~ty ~ t e d with ~howi~ the ~ c u c e o f . ~ ~ppro~imate equilibrium of this kind i~ that ti~ eqtt~briem ~ n ~ce~ no~ belong to the attainable set. Even ttm~gh a ~ sets w~ty be ~ the
standard ~o¢:dure of consi~'ing ~u~es~y ~ *Under the assumpczn floatthe ~ ~~ public goodsour notionof an appro~m,~ ~ versionofE]]icksoa'~notio~
eq~ibrkua is identical~ a qa.~q~'~k~z~
~ ~
~
~ ~ .
s ~ot
.to ~ ~ e~ ~ ~ ~ ~ ~ ~ qu~ _.~p~__'~.__m~_.
R. Vohra, l.~cal public goods as indivisible commodities
195
applied since that does not ensure that an approximate equilibrium allocation lies in the interior of the appropriate set. For an exchange economy considered in Anderson, Khan and Rashid (1982), the consumption sets were effectively truncated since prices were construc*~,cd such that they ~l had a positive lower bound. This approach evidently does not suffice for an economy with production. We shall truncate production sc2s taking into account the affect of such a truncation on the deviation of aggregate excess demand from zero. To apply this procedure we need to consider an economy in which the number of consumers is sufficiently large. The two different kinds of local governments we consider are passive governments, as in Hamilton (1975) and entrepreneurial governments, as in Ellickson (1979). Passive local governments treat as given the number of residents in their respective jurisdictions and minimiTe the costs of providing the requisite level of public ~rvi~s. "~a¢ torn! publ_;c expenditure in each community is spread equally among all the residents. An entrepreucuriai local government behaves like a profit maximizing producer and determines the aggregate level of public goods that it wishes to supply. This also constitutes a decision about how many residents to provide for. Notice that these governments are not entrept~neurial in the sense of Bewley (1981)since their public goods bundles are fixed; they choose only their leeds of services. If there are increasing returns in the constmaption of public goods, Le., ff the cost, per resident, of providing local pubiic g o ~ declines "as the number of residents increases, then such profit maximifiag behavior cannot be consistent with taking prices as given~ At best, ¢ ~ hope to show the existence of an approximate equilibrium in which not only are some consumers not maximizing utility bat there are also some producers not magimizing profits Isle Theorem 2 in Eili_ckso_n (1979~]. In any case, such an equilibrium would hs,~ the undesirab!e property ~ govefw ments" budgets might not be baIanced. 6 H ow¢~_, E!!ickson also e ~ w _ : ~ that, eventually, all economies of scale are egha~ted and the average cost of providing public goods to each resident b e c o ~ constanL He then shows (in his Theorem 4) that under ~ t replications of the economy, economies of scale are exhausted in equilibriun~ This seems to be a more interes~g result since ia this case, the gov~-nments ~ budgets are ~ o c ~ L "~ We ~ i show that in an economy in w h i ~ aIi scate ¢conom/cs arc exhanst¢~ an approximate Tiebout equilibrium exists under much weaker conditions then those postulated in Eilickson (I979). In parf~L.h._.r, the mean e x ~ d~_a_~d is shown to be bounded i ~ t i y of prd~enocs.
*there is no way of earning that a la~duc~ who_/~not maximim~¢mSt~.iS n¢~ making loses. ~Bewley(1981)alsoimfav:~Ih¢balancedbudg~,ocn~tlo~ Tim is I t i s ~ 2.4.
t96
R~ Vohra, Local public goods as indivLffble commodities
2. The medd We consider an economy with c consumers, m+ l communities and u private producers. The first m communities are interpreted as those which pro~de ~ m ¢ public goods while the (m+ 1)th community provides no pubfic goods, thus all-3wing a consumer the possibility of not consuming any public goods. Each of the first m communities can provide h different types of public goods. Let Q~, R km be the choice set of community i with respect to the per capita public services it provides to its residents. While only h coordinates of Qt can be non-zero, we define Qf as belonging of R ~" rather than Rs so as to distinguish between the same kind of public good in different communities. Any b-.mdle qi~Qi will alternatively be referred to as an admissible policy of the ith- local government. Note that no more than h elements of q~ can be noN-zero. Since a consumer can reside only in one corardunity, the choice of local public goods is a choice of a particular community. Given a vector of admissible policies, his consumption of local public goods is restricted to lie in a finite set of bundles. We refer to a consumption of consumer t by ~(t)---(xp(t),x~(t))~X(O--Xp(t)xX~t), where Xp(t) is his consumption of private goods and x,(O that of public goods. Xp=R~+ is the consumption set corresponding to private goods X~(t)~ {Or.... .q=,O} is the set of public goods consumption possibilities offered by b e market. We do not assume that Lhe actual consumption set e o n e s ~ n d ing to public goods is non-co-vex, i.e~, it may b~ p~y~ically possible for a consumer to consume a convex combination of points in X.(t). What makes our model inh~-r~ntly one of indivisible commodities is the fact that the choice set of public goods made available by the m communities is nonconvex. Although X(t) depends on q~qt x---×q=, for the sake of notational convenience, we shall not always make ~hat explicit. Note, h~wv,~er, tttet the consumption set correspondence X(t): ] ' L ~ x {0}--,R r-k= is contit~ttotts. Th,'. u private producers have convex, production sets YI~Rt and bel~ve as pl~fit maximizers. The extended production set 1?~ is defined as ~'~= {(f,0)~Rt+m[y~e YJ-}. The production set e r a local government p r o d u t ~ is I a ~ R t_ ×R+~. For a given vector of admirable policies q, we denote the ifix local government's production set by Y~=R~_x R ~ and its production ve~:tor I . t~ by f---(Yj,,Y;I, where -I?i. are the inputs of private commodities mad y~ the output of pubfic goods; ~---nt~, where ni¢N+ - the ~et of non-negative integers - is the number of residents in the ith community. Clearly, Oven any admissible, policy ¢, I~ is non-convvx because # must necessary be integervalued. Y* could also be non-convex because of inmmsing returns to scale but we shall follow EUickson in assuming that eventually, all economies of scale with respect to the number of residents are exhausted and each government produces under conditions of constant average ¢oSL Since we are interested in large economies, without loss of ge~rality, we can assume that
R v.~2a,,g Lcc.cd t~ub!~_ goo6s ae- mdielsible o0mmod~[es
.197
each govemmc-nt produces under conditions of constant average cost at all levels of outpuL This follows from Ell/ckson's result (his Theorem 4) that for a large enough rcpfication, all economies of scal:~ are exhausted in equilibrium~ We can now formally state our assumptions:
A.I. For each consumer t, the preference relation >-t~Ri++~r~ is irreflexiv¢, transitive and continuous.
A.2. For each consumer t, e(t)~Xp{t) is his endowment and 0i(t) his share in the profits of the flh firm. A.3. For each private producer, j, j ~ I ..... u, the production set YJ~R t satisfies: (i) YJ r~ R +J- {0}, (ii) yi is convex
A.4. For each local government/producer i, i--1,..., m, the production set F~mRtx Rh+ m is con :ex, and for any given vectors of admissible policies q, Yi satisfies: (i) Oe Y~, (ii) if yE~con (yt) and y~=n~q for any nt~N+, then f ~ Y~. (iii) i f y ~ y i then ay~e Y: for any ~ e N + .
/LS. (~2--i ?~+Y.~'=~con(~) is clo~l. Assumption A.4(ii) captures the idea that non-convexity of ~ arises only from the fact that the number of residents must be integer valued. This corr~ponds to the assumption of integer convexity in Frank (I969, p. 34). Assumption A.4 imptics that con (Y~) is a convex cone gith vertex zero. A local public goods economy is de~ed by {(X(t),>~,t~t)),(Y~),(]~)~(L~)}.
The budget set for consumer t is defined as
~ , O= {~(O s X(O l p" x(t) ~_p," e(t) + x(p,t)}, where p--(pp,p~) refers to private and public goods" p r k ~ and
~,t)-=j~ @(t)roaxpr" YJ+l ~ maxpY jNOtice that, given AA, the second teJ'ul in ~(l:, 0 will be 0 in ¢qu/t/br/um.
198
R. Vohra, Local public goods as indivisible commodities
The compensated demand set for consumer t is defined as d~p~t~ = {x(t) ~ B(p, t) ]~t)>- t x(t)=~p- ~ t ) - p ~ . e(t) + ~t)}.
The supply set for a .~rivate producer j is defined as nJ(P)={Y~ PIP')g>--P'E for all C e ~").
The supply set for a local government/prodttcer i, given a vector of admissible policies q, is defined as
The con~exified supply set for a local government producer ~ given a vector of admissible policies q, is defined as con O/'(p)) ---{y~contY~)] p" y~>-p'~', for all ~ f c o n (Y~)}.
A compensated, approximate Tiebout equ[I~brium for an economy ~ h entrepreneurial local governments, giveu a vector of admi~ible policies q, is a (c+u+m+l) tuple {(x(t)), (f), 0/), p} such that (i) for each consumer t,x(t) ¢d(t,,t), (ii) for each private producerj, y / z t / ~ ) , (iii) for each local government/producer/,y~e~ffp), (iv) (l/c) ~[,+-__t~ max [z~,O] =6, where
and 6, referred to as ~ ¢ error, is a finite, computab~ bou~d sada f~h-~
A compensate& approMmate Tiebout equilibrium fc,r an ~ , i~ passive local g ~ given a vector of ad:n~itge poIicie~ q, i~ d ~ in the same way as above except that c~ndition ('th~) is c-_~angedto (iii)~ for each local government/ptodu~nr/t
R. Vohra, Local public goods a~ indivisible commodities
199
When feasibility is satisfied exactly, in the sense that
l:l
we
i=l
also have
Clearly, under Assumption AA, an equilibrium for an economy ~ t h entrepreneurial governments is also an equilibrium for an economy M | h passive gove~,nents. ~(Q;6) is the equilibrium cort~pondenc~ which associates with each q~Q, a set of compensated, approximate Tiebout equilibria, with error 6. We denote by g=(Q;6) the coordinates of ~ f ( ~ eorrcsponding to consw';ota'~ consumption bundles, q ~ Q is said to be a second-best policy ~ h r e s ~ .to ff there is an x ¢ ~ ( q ; , ~ such that ~z~¢Q and £ ~ ( ~ 6 ) with £.'>.~ for ~ L & R ~
In this section we present the main result o~' this p~per.
Theorem L I f Assumptions A.I-A.~ are ~atisfie~ for any ~ of ~fvfts~bl¢ policies there exists a compensated, approximate Tie.bo~ ecagI[brium for large economy with ~trepreneurfal local g o z ~ m ~ s . The term "large economy' is taken to ~ that the number of o ~ c, is large enough. An actual value for the error, ,~, is_ a l ~ compu . ~ in t ~ proof of Theorem 1 and show~ to depend onty on the e n d o ~ t ~ .the technology, the distribution of profits and the number .of commodities, T~his result can also be interpreted as stating that if there ere no ~large~ t t ~ k ~ ('large' both in terms of endowments and share of profits), dg~- ia ~ there exists a comp~saled Tiebout eqt[ilibfium. I1 s~e~s rec.somtbie to exp¢~ :-~t this resuk would also be true for an ¢~c~n~n~y ~ t b a. co~tis,uam of consumers. That this is indeed so czm be .deduced from Votu'a 09g4o Theorvm I), which shows the exist¢l~ of a compcasated Tiebout eq _~.'.._ibrium (mth~ t h ~ that of a comtg-asated, appro]rim~te Tiebout _¢q_ui_.h_'.briam) in an economy with passive local gove,nmcnts.~ The reset_ iu Vda_..r~{I_9~t) eSinc~ the n~ult in Vchm (I~,Q ~ trot depe~_ _-on.~b¢ ute.r¢ d ~ l F ~ t t i ~ ~ 1~ ¢c/nsmnption of p u b ~ goods it a~d~s to the ~ iu which the avetal~ rest -of goveramcnt~s out~ut is ~ t
equil~L=i=mfor ~ a t ~ z r l a ]
so that an ¢qu/fila'ium for p=ssiv¢ g ~ , ~
is ~ls¢ an
200
R. Vohro.l.oc~!publicgoods m indivisiblecommodities
shows that if we consider an economy with a continuum of consumers not only can we ¢~eal with cases in which economies of scale in consuming public goods are not exhausted but also el~rn~nate the approximation involved in the feasibility condition. Rcstorafio~ of the feasibility conditiott also makes the welfare implications of our equilibrium notion more attractive. It is clear that a compensated Tiebout equilibrium for entrepreneurial governments is approximately Pareto optimal, given the particular vector of admissible policies, or locafionally efficient in the sense of Wildasin (1980); if the aggregate endowment is reduced by an arbitrarily small amount it is not po~_~.'b!e to make any non-negligible set of consumers better off without making the rest worse off. However, it is worthwhile to stress the fact that while the approximation involved in the feasibility condition can be eliminated by consid¢6ng a continuum of consumers the same is not true for the approximation involved in the demand correspondence of each consumer, unless further assumptions are made, one must be satisfied with a compensated equilibrium. Theorem I is also of some interest for economies without public goods but with xg(t) interpreted as the tth consumer's demand for indivisible commodities. However, as it is stated, Theorem I is valid only for economies in which each producer has a single Output and con(Y ~) is a cone. While the latter assumption is not used in any of our proofs, Lemma 1 depends critically on the assumption that each producer has a single outpuL For the multiproduc-t case we would have to assume that every f ¢ c o n ( Y i) can ~ written as a convex combination of vectors including one which had each output as the highest integer less than or equal to the cor~sponding output in y~. With such a 'bounded non-convexity' assumption 9 replacing the assumption of integer convexity, Theorem 1 can be interpreted as stating the existence of a compensated, approximate equilibrium in an economy with indivisible cornmodifies. It can be seen as an extension of Anderson, Khan and Rashid~s (1982) result to economies with production but with bounded aonconvexities in production and no indivisible inputs. Theorem Z ! f in addition to Assumptions A . 1 - A J it is assumed that Q is compact, then there exists for every q~Q, an approximate equilibrium, as in Theorem I, with the additional property that the error i5" is independent of q~ Moreover, there exists a q*~Q such that q* is a second-best policy wi~h respect to ~*. This result shows that we can find an error independently of q and, then, given the set of admi~bIe policies Q a~,~d the equilibrium correspondence 9The assumption we use here is similar ~ough trot identi~ to the he,reded nma-couvegity assumption in Start (t969).
R, Vohra,Local public goods as indivisible commodities
201
~'(Q;6*) find a policy which is second-best. Of course, second-best optimaJity is relative to the equilibrium concept being considered. There raay, for instance, be another poiicy which is unanimously preferred if a different kind of tax scheme is adopted. 4. Proofs We first prove existence of equilibrium for an e ~ n o m y in which the production sets of local governments are replaced by their convex hurls. Such an economy will be referred to as a convexified economy.
Auxiliary Theorem. If Assumptions A.I-A.5 are satisfied, for any vector of admissible 'policies there exists a compensated, approximate Tiebout equilibrium for a convexified, large economy with entrem'eneurial local governments. Proof Consider the attainable set defined independently of q as
j=|
-
"=
0)}
. ) , - - 0 t
Since all the conditions of 5.4(2) of Debren (1959) are satisfied, A is compact, t° For vector x~R', let Hxil=~Y~_-t[x'l. The convention we use for ordering vectors is >>, > , __>. Define g,={max~gJ]]g~A] and e.=max, i]e(0l[. Given compactness of A, g= clearly exists and is a non-decreeing function of aggregate endowment;
where C,(cem)=G(cl'e,=) and I is the unit vector in R *÷~'. Since aggregate production is subject to non-increasing returns g=<~cG(e~.). Note that C-(-) is independent of q. Throughout the remainder of this proof we shall assume that a vector of admissible policies q is given and consider Yi, i = l , . . . , m to be convex hulls of the corresponding sets. Let K be a cube in R z+h= oentered a 0 with length to(~= I Y'~+~'= ~ yt) n RC+÷~- {0}. by A.3(i) and because Y~ ~ R L x RX.*~, ~ t y follows again, from the fsA~t that Y~ = RL x R ~ wh/c,h also ~npl/es that publ/v goods are not , ~ I as inputs.
202
R. Vohra, Local public goods as i~ulivisible commodities
2 {cG(e,~)+o0, where ~ is some positive number. The truncated production sets are defined as ~f= yt n K, F J= ~ n K. Prices are chosen, as in Anderson, Khan and Rashid (1982) from
The truncated consumption set for consumer t is
where ~mfmax~(t) and ~(t)~is the maximum profit that t can receive $~en that production plans belong to ~'~ and Y~ and prices belong to ~. Clearly, ~t, is well defined. For each consumer t, the compensated demand c.orrespondence d(p,t) is non~-mp~ and u p ~ r hemi-continuons Esee Khan and Yamazaki (1981)]. The aggregate, compensated demand ~trespondent~.~/~. S ' ~ , where 2=~_#2(t)is defined by o{p)---Eg=zd(p,Oand it too is aonempty and upper hemi-continuous.It is easy to check that D(p)e _~.~_.(t)~The supply correspondences~(a) and V~tp)definedover the truncated sets Y and P~ are clearlynon-empty, convex-valued and upper hemi-continuous.There* fore, the aggregate supply correspondence ~/: $ 4 ~, where ~'=(~
~'i+ ~ ~ )
and
tl(p)--~q~p)+ ~ ' ~ ) ,
also satisfiestheseproperties. Define ~con(~') × IY--*Wby
for all/~eS'},
where x denotes ,~.~=~x(t) and y deeotes
Consider ~b:S'x eon(R')x F-+S'x con(A~)x F
i¢, Vola'#.,Localpublicgoodsas ~,div~bIe commodities
203
where 4 ~ , x, y) = ¢(x, y) x con (o(p)) × n(p).
Since ~ is non-empty and convex valued, we can apply Kakutani's Theorem to obtain a fixed point (p, x, y) a 0(P, x, y). By the Shapley-Folkman Theorem we can write x'(t) +
Xm t= i
Z
x(t),
l=|+/~m+ I
where x'(t) ~con(d(p,t)) and x(t)Ed(p, O. Thus, for l+hm comumet~ there is a possible discrepanvy between x'(t), the assignment corresponding to the fixed point and x(t), that belonging to the actual, compenxated demand set d(p,t). Since X ( t ) = R ~ ~ and con(X(t))=Rt++~, (x(t)-x'(t))<~x(t). Let x~.t), k=- 1,..., l+hm denote the kth element of x(t). Then, l+ltm
l+/tm
,=~ m = f ( x ( 0 - x ' ( 0 ) , 0 ] ' ~
t~x~0.
But from the budget constraint and deletion of p. we have t+/tm /t-~l
Thus,
Lt z = Z" x ( t ) f=x
-
-
\$~t
/
t=t
(t. \r=l
~),
.
j
Then, for any ~6~Y,
From the properties of the fo~c~lpoint we b o w that the ~ d the right-hand side of (2) is uoa-p~tive. ~ g to 0), ~ rewrite (2)
~ on we.
R. Vohra~Local public goods as indivisible coramodiffes
L e t / ~ . 1 ifzk>0 and I / x / c if~__<0. The.,.,,
~.~= ~ ~ + ~
~ ~
k~K L
~4)
k~g 2
l+hm)lz~
where K ~ = { k ~ ( ! ..... t+~)l z~>0) and K 2 :- {ke(1,..., Combining (3) and (4) we have ~+tm 1 max(z~,O)=<(l+hm)/c(e=+n~)---7 k=l
~ zt.
% / G /ceK 2
The last term can be rewritten as 1
1
~
- ~
~'
and by adding (1/x/'c~.~= l x(t) to the right-hand side we got 1
¢
Combining (5), (6) and recalling that I[yl]
l.(a÷a/x/c)] which
implies that
From the definition of G it follows tkat for all producers i and j.
Convexity of the production sets implies
X/ c /
(5)
R. Vohra,Local publicgoods as indivisible commodities
205
We now show that yt belongs to the interior of ~ and y~ belongs to the interior of ~7 for all i and j. Let G(a) + 1
which is well defined and positive as long as ~ i / x / ~ ) < I. G is non-increasing i n c and G(0)--0. Moreover, there exists a c large enough such that O(1/x/~)< 1. This follows from the continuity of g,(-) at 0, which in turn is a consequence of the continuity of *.he correspondence A(~.~e(t)) at 0. H With defined as above it is easy to see that= for sufficiently large c., ~ a ) + • G(1/x/~)<~. Thus, from (8), Given that y~ and yl belong to the interior of their respective truncated production sets, for all i and j, it follov~ from usual arguments, that they magimiT~ profits not only over ~ and Y~ but also over Y~ and 1~ Thus, conditions (ii) and (iii) of equilibrium are satisfiet~ Condition (i) is el_early satisfied. If no consumer is large either in terms of endowment ~r profits. i.e., if ~ and ~,~, are constants G(a)=G(x/~ ) so that ~./c----~l/,Jc)and ;,t follows from (7)' that I + tJm
Ic k~x=max (zt, 0) -
2;, and
x/e l.
(9)
cj
\,¢'c7
,,10)
so that condition (iv) i~ also satisfied for a convexilied e~onomy. l_emma 1. Under Assumption A.4, for every/, = 1 .... , n g ~ff~{con(Yi)}./Ir ~ and p . 3 / > p . ~ , for all j~con(Y/), then there exists ~/~ yl and satisfying (a) p.$~>_p.j/ for all y~Y~, (b) y~Yp* Proof. Let y~=n~q ~. Since fe{con(Y~)}/y ~, n ¢ is not an integer. By Caratheodory's Theorem, 3/ can be expressed as a convex combination of 1+2 vectors in ]n. Moreover, at least one of these vectors must have pubiic "C-d.) is clearly upper hemi-continuous at a $i,o~ O~G(]~_~e(t)I] for all e(t) add G(O)={O}, also lower h~'fi-continuous.
C_d-) is
R. Vohva,Local public goods as in&'zg~ibiecommodities
206
goods ou~pm- tcvzt , . __, less than
We ~ - +he~r,-,~ ~-ita
..i
y~----alyU+~t2y2~+ .-- +a~+2y~+2~, where 1+2
e~>O foralli,
~=J = l ,
~ ( 0 , t1 and
Yit,
Let N be the highest integer less than nt. Clearly, N = p n f + ( 1 - / ~ ~i for some
pe[O,1). Let . ¢ = p / + ( l - / ~ y ' . sin= ~ = g q ' we know from Asmmptioa A.4(ii) that .¢e Ia. Since j / ~ profits over co.(Y~ ,rod is a convex combination of y~, .... yJ+2~ we must have p.y~=p.yU . . . . . p . y ! ~ . Thus, p-~=pp.y[+(l _~p.yU=p.yX and conditions(a) and (b) are s a ~
Since p~S' and p.~p.~=maxp-con(Y~, the~ does not ex~ another j ~ e ~ n ( Y ~) such that ~>j~. But y ~ > ~ which implies that y ~ < ~ . This completes the Ftoof of Lemma 1.
Proof of T~orem 1. Acoording to tbe Awr~i~ry Theorem there exists {(s(t)),iyt),(~,p} satisfying ~nditions 03-{iv) except that for all t o ~ go~-t~ ment/produce~ y~econ(Yi). By I_~mmR I, for each i, i = 1..... u there exists ¢e P such that p-¢>p"¢ for allj~e ka,~+q~>y~ Thus
Z m~[e.0]
I l+l~ -
C k=l
1 ~+~
r(
1 z+i=
I-(
~
/
.
\
-
f.
:
~.)
I
l.~i I
Combining (I0) we have, 1 l+/tm
-c ,~= max[~,O]<,~,
where
.=~{(,+~)(e. + ~.)+,.÷ ~e.)+;÷~}.
(,,,
Thus, {x(t),(y~,OC),p} is a compensated, al~roximate Ttebout equilibrium. Proof of Theorem 2. Given compactness of Q we can find a 6* suck that an approximate equilibrium with respect ot 6* exists for any qeQ. DeFine q = - -
IL Vohra, Loc~lFublic goods as indivisible commodities
207
max,.,ellqr[]_~.nd-6*=O&/~)'{(i+~n)(e~+~,)+e,+C~e~)+~/c÷qj~}. By Theorem ! and the definition of& as in (11), for any qeQ, ~'(q;6*) # ~. We shah now show that ~'(Q;6*) is compact. Since aggregate excess demand is bounded by 6"c.I for any qeQ, ~f(Q;6*) is bounded. To show that ~f(.;J*) has a closed graph, consider (q~,x~,f,f)-~(q,x,y,~) such that (x~,f , f ) e ¢(q~; 6*) for all v. Ce~tainly,
-'~ 1 l+~'max [zt,0] =<6,. C
ffi
Also, yJ maximizes profits with respect to p for all private producers. Suppose there exists a local government i for which there exists ~ ~ such that p. ~ > p- f or, p-(6- q) > p-(n- q). Then, there exists a o large enough such that f.(~.q)>p°.(n~.q), a contradi~don~ It now remains to be shown that x(t)Ed(p, t) for all t. Suppose not. Then there exists a t and ~ ( t ) s ~ h that ~O>-~x(t) and p-~t)-t, there exists a v.large enough such that ~(t)~X(t,q~ X~t)~:~t} f-~(t)- over elements o f e~Q;,6*) such that ~>-x iff~(t)~x(t) for all t. Clearly, ~- inherits irrellexivity, continuity and transitivity from the individual preference relations, By standard argument [see, for .example, Mantel (1975, p. t93)], there exists a maximal element i~ 8(Q;6") with respect to >.-; i-¢., there exist 6*, q" and x*f~f~(q~,6 *) such that ~txEd'ffQ;6*) such that x(t)~-t:~°(t)for all t. 5. ConelesJan Bewley (1981) demonstrated, quite convincingly, that there seems to be no reasonable way of providing the Ticbottt hypothesis with the ~_m_e Hurl of rigorous basis as the classical model of competitive equilibrium. He showed that the only situation in which an exact equt~brium could be shown to exist and be Pareto optimal required the assumption that there be as many communities as there were different types of consumers and that the average cost of providing public goods be constant with respect to community size. We have stressed the fact that, in an economy with a finite number of communitieS, the opportunities for consuming local public goods ~ restricted to a finite number of bundles; the choice for pubfic goods is essentially a choi~ for indivisible ¢omm.odities. Our result shows that even if no assumptions are made on the number of commtttdti~ vis-a-vis the number of different types of consumers, under fairly general conditions, aB approximate, competitive equilibrium exists in which consumers shop for local public goods the same way as they do for indivisible private goods.
R,S.U.E.~ B
208
K Vohra, Loca~ public goods as indivisible commodities
T h r o u g h o u t this p a p e r we have considered a m o d e l in which economies of scale in providing public g o o d s have been exhausted. This restriction o n the nature o f congestion can be removed if we consider an e c o n o m y w i t h a c o n t i n u u m of consumers. In that setting even the a p p r o x i m a t i o n with respect to m e a n excess d e m a n d c a n be eliminated. I t wouid be of interest to s t u d y the effect of increasing the n u m b e r of communities on the e c o n o m y ' s welfare. A d r a w b a c k o f Elliekson's w o r k a n d of o u r s is that, in each c o m m u n i t y , the b u n d l e of public g o o d s to be provided t o a c o n s u m e r is fixed. However, as we have shown, this a s s u m p t i o n is n o t as restrictive as it seems since we can find a second-best equilibrhtm with respect t o g o v e r n m e n t policies regarding the per capita mix a n d level of p u b l i c goods.
References ~d,.',rson, R.M., M. All Khan and S. R~hid, 1982, Approximate equilibria with bounds . independent of preferences, Review of Economic Studies 447 473-475. Bewley+T., 1981, A critique of Tiebout's theory of local eXl~'nditm-es,F_zonometrica49, 713-740. Broome~ L, 1972, Existen~ of equilibrium in economies with indivisa'blecommoditim, 5ournal of Economic Theory 5, 224--2f@. Debreu, G~ 1959,Theory of value (Wiley, New York). Eitickeon, 13, 1979, Competitive equilibrium with local public goods, Journal of Econtmfi¢ Theory 21, 46-61. Frank, C.R~ 1969, Produ~on theory and indivisible commodities (Princeton Univet~ty Pre~ Prim:~on, NJ). Grec:nberg, J~ 1983, Local p.blin g~:~s with mobility:. Existence and optimality of a geoetal eqt~ilibrium.Journal of Economic Theory 30, I7-33. Greenberg, L and S. Weber, I986, Strong Tiebout equi~brium under rm~rk~.ed domain, Journal of Economic Theory 38, 10I-IIZ Hamilton, B.W., 1975, Zoning and property taxation iv. a system of local g o v ~ t s , Urban Studies 12, 205-21L Khan, M. All and A. y~ma~ld: 1981, On the cores of =.mtomies with imii~i~L: ¢mnmodifies mad a continuum of traders, Journal of Economic Theory 24, 2 1 8 - 9 . Mant~ 1hR., 1975, General eqeih'brium and o p t ~ l taxe~ Journal of Mathematical Economics 2, 187-200. Mvs~-.olell. A, 1977, Indivisible commodid~ and general equilibrium theory, J ~ of Economic Theory 16, 443-456. Richwr, D.K~ 1982, Weakly democratic ~3ular tax equifibriain a local public goods economy with perfectconsumer mobility,Journal of Economic Theory 27, 137-162. %hweiz~, U~ P. Vamiya and L Hartwick, 1976, General equilibrium and location theory, ~ournal of Urban Economics 3, 285-303. Scotchmer, S., 1981, Hedonic prices, crowding atul optimal ~ of populatioa, ~ o n paper no. 864 (Harvard Iastitute of Economic Remar~ Cambridge, MA)+ Starr, R., 1969, Quasi-equilibrium in markets with n o n - c o n ~ + ~ E~oom~k:a 37, 25--38. TicbouL C, 1956, "A pun: theory of local ~ Journal of Political Economy 64, 416..424. Vohra, P~ 1984, Local public .go~L~ and average cost pri~iDg, Joenml of Mathematical Economies 13, 51-67. Westhoff, F . 1977, Existence of equilibria with a Iocal public good, Journal of F.conomi¢ Theory 14, 84-112. Wiidasin, D~ 1980, Locatiottal efficiency in a federal system, R~omd ~ and Urban E~nomi~ 10, 453-471. Woodem, M., 1980, The Tiebout hypothesis: Near optimality in local pubtic good economies, Econometrica 48, 1467-1485.
and Urban Eoanam~ 17 (19S"~ ] 9 ! ~ -
N~4q~
/.DCAL PUBLIC GOODS AS iNDIVISIBI~ C O M M O D ~ Ra~iv V O H R A *
R~,a~ved December 1995, fiaal vetsloa m a ~
~q~rfl ~gi~
A sharp~ yemen o~ ~--rr~:soa's (19~)J rcmtt on ~ ¢ ~ d ,~ ~ ~ Ti~ equilibrium is shown to be trm~ m ~ , - ~t_~a urea.kerr a ~ z ~ i 0 ~ r ~ | ~ i i v . ~ ~ h o ~ ~u~ l h ~ ~is~s a _q~md-bes¢ policy regarding the b,-,~lc of pebllc ~ ~ L,:~__.!~ o t k n ~_a and Rashid (i982) to e~oaomies _withproducfioa, tn th~ ~¢¢~,-..~.~ ,~ ~ ma~ ~ _.d~.__uat!~y~:~ that the a ~ ' r , ~ ~m.,~-b'd-m~~ 0 ~ does ~ ~ ia the ~ t ~ u : _~ ~ sum~a~_~ t.~q:L.~.~ Of trn~s'-~'~cmare ~ t ~ , ~ . e .
1. Intretlaefioe
As Bewley (I981) has shown, a rigorous f o ~ o a or a ~otio~ .r,f equilibrium which captures all of the e~e~tial idt~s of Tieboul (1956) d o ~ not lead to an equih'bdum con~p-~ which .Compar~ favorably with ~ q n l ~ * five equilibrium. He points out thal~ in general, Tie.bout equilibrium 'does a ~ have the nice properties of genei~al oampetitive equilibrium'; eq~tibti'e~._._ ~ma_y not exist or may not be Pareto o p t i m a l . A n d t h e a ~ o m p t i o l ~ w h i c h h a v e tO be made to avoid these diffioAties, 1 in _trarfi~dar the assmnption that there be as m a n y c o m m u n i t i e s as t h e r e a r e t y p e s o f g o a s u i a e r s , r o b T i e b o u t ' s
nGfion of almost all its significance. One might, thetefore~ coadude that Tiebout equilibrium cannot be given th~ same kind of rigorous basis as competitive equilibrium, in this paper we show that under oertaln conditions, Tiebout equilibrium does compare favorably with a Iess classk:~tl form of competitive equilibrium, namely one in which some ¢ommodites are indivisible. It should be clear from Bewley's work that the positive results about Tiebout equilibrium in the literature do not do justice to all the essential *This paper is based on Ch. 6 of the author's Phi). dissertation at Johns Hopldns University. Bruce Hamilton and Ati Khan have had a sig~cant inlluenee on this paper and the author is 8fateful to them for encouragement and many helpful ~ o n s . Than~s are also d ~ to Vernon Henderson, Peter Newman and Nicholas "tannelis for valuable comments. This research has been supported in part by NSF Grant No. SES-8410229. ISee Theorem 9.12 in Bew!ey (1981). 0t66-0462/87/$3.50 © 1987, Elsevier Science Publishers B.V. (North.HoUaad)
192
I~ Vohra, l.~,cal public goods as indivisible commoditie~
aspects of Tiebout (1956). Different authors have formalized the notion of Tiebout equilibrium in quite diverse ways - stressing some aspects of the Tiebout model while abstracting from others, in this respect the present paper is no exception. We consider a model of local public goods in which there is a fixed number of communities. We do not model land and a consumer's choice of a community depends only on the public services being provided and taxes being levied in each community. Each local government finances its expenditures through lump-sum taxes on its ~esidents wi~out discriminating by name or on the basis of taste. We assume that the publ/¢ goods are pure pubfic services so that their cost is proportional to the size of the commurfity. In the context of a large economy this is not a restrictive assumption as long as the cost of public goods eventually becomes proportional to the size of the comn~unity since the economy can then be rcp!/ea~d in order to satisfy this assumption. We first establish that~ for any giv._~ mix and level of per capita services in each community, tbel~ exists an appco~mate Tiebout equilibrium for an economy with a suffidenfly lar_~ number of consumers. The approximation is of the same n~t-~h-¢ as that inYb!ved ha showing the existence of competitive equilibrium when some ¢ommod/6es are indivisible. Our other as~.-zmptions are weaker than those, for examp~ of Bewley (I981, Theorem 9.12). A restrictive asp~t of this model is the assumption that the ~ of per capita public services in each community is give~ exogenously. Th/s ~t.m!p~ fion was also made by Ellickson (1979), W~__~n (!980) a ~ S c c ~ . . ~ (198!). Our main result can, in fs¢~ be ,~'_~-'gz~l as an ex~e,~t+~, of corresponding result of FJfickson (1979}. Given this ~ ~ model should be viewed as shon-run "F~ou! model H ~ , ~e ~k~ s.~-,w that if each community choof~ its bundle of p ~ b ~ sert~:~ from a ~ p e ~ set, there exists an appro~nate Tkbout equ~-b6um whl¢~ is uxxmcl-b~ .~ la Mantel (I975). It is only iu th~ ~ tha~ we cau ~ per ¢2p~t public scrvioes to h:, e~dogcnouS. A ~¢~Rcd a p p t o a ~ to m a k / ~ ~he of local public goods endogenous is that of Richter (1992~i~ ~enerAli~e~ subsequently by Greenberg (1983), in wh__-_mh_each_ local govecamen~s chooses. such a bundle op_fim_al!yfor its residenLs. However+ there is z~ g,a:srs~e t~_~. the equilibrium is Pareto o _l~m_'__~1 for the economy as a ~_h¢~ [~ Weghoff 0977) each oommuaity decides UlXm its level of the lmre ¢mb~_L through majori~ voting. A quit~ d ~ t ~ fto~ al~ o[ ~ ~ iks.~ of Weeders (1980) in which not only is the m/~ ~ra~ ~ ~ ~ ¢~g~ services in a community cndog~ous but so i~ tL~ number of c o m m ~ A cooperative solution concept is analy~d and _ . ~ ~ ~ ~ including one concenfing the ~[afio__n~_~p between the n w m ~ o f ~ of dh~erent types and the tedmotogy, ~ _~._'~o~ d an t core is ~ m ~ . However, the taxes are not required to s a t i s b / ~ ~ ~ i t ~ mp, a t t k ~ +
~. Vohr~ lg~cal public goods as indi~i~ble commodities
193
they may depend on consumers' preferences.2 A related model has recently been introduced by Greenbc~g and Weber (1986) in which the total taxes of a community are either spread evenly among the residents or in proportion to income. This has the effect that the equilibrium need not be Pareto optima. It is worth mentioning that in the case of pure public services, equilibrium in their model leads to the formation of as many communities as there are types of consumers. In the remaining part of this scetion we shall indicate the essential difficulties involved in general equilibrium theory with indivisibilities, point out the inadequacies of previous attempts to consider local public goods as indivisible commodities, and describe the notion of approximation we shall USe.
The idea of treating a finite number of communities as indivi-,ible commodities goes back to Schweizer, Varaiya and Hartwick (1976). However, their resuR on the existence of a compensated equilibrium was based on the assumption that the number of traders is 'large' - a notion which was never pre.cisely defined.3 The. work of E|lickson (1979) can be seen as the first rigorous attempt in which toca! public goods are treated as indivisible commodities and the existence of an approximate Tiebout equi~brium is shown. Unfortunately, an argument in Ellickson's existence proof seems to be open to question. Moreover~ his notion of approximate equilibrium is quite weak in relation to the recent advances that have been made in the theory of indivisibilities. There.are two major difficulties encou,tered in proving the existeno: of aft equilibrium with m~visible commodities [s¢¢ Mas~Co!ell (1977)]. Fh'sfly, ihe demand correspondence may fail to be upper hemi-c.ontinuous. A small change in prices may induce a utility-maxim~_ng consumer to ~ttmp" fco/n one community tO another. This is tiroome's (1972) "problem of the edge'~ Secondly, the demand correspondence may not be convex valued. It is worthwhile to emphasize that even though, in an economy with a continuum of traders, the problem of non-convexity can be eliminated through aggregation, the same is not true of the continuity problem unless endowments_ are assumed to be dispersed as in Mas-ColeH (1977). For a fini~ economy with indivisibilities both the lack of convexity and continuity are inherent and~ unless rather strict assttmplions ~'e made, the equilibrium notion has to be modified to an approximate one~ Eltickson conside~ a --orion of equilibrium in which the approximation can be ascribed solely to the non-convexity problem. The 'problem of the 2Both Westboff (1977) and Wo~ca-s (I980) assm~ thltt there g~~ ~ publi~ go~__, ~Although lhey consider a finite ¢wnomy, t i ~ do no~ n:quir¢ the m~mbc¢ of residers ia a
communityto be an integer.~hus the results.., are a good approzimatioaonly if the neanbe¢ of-households is "lat~"" [Schweizer, Varaiya arai Hartwick (1976, footnote 9}],
194
edge' is dealt with by assuming that the consumer is not willing to spend his entire income on public goods and by assuming that all private goods~p~c~ ~re strictly positive. Recognizing that upper-hemicontinuity of the demand correspondence depends on all private goods' pric~s being strictly positive he nevertheless uses Debreu's (1959) theorem to prove the existence of ~uilibrium in the convexified economy. This procedure seems to be invalid since Debreu's theorem treats prices as belonging to the closed simplex which clearly permits prices of some private goods to be zero. Instead of attempting to prove a "quasi equilibrium' version of Ellickson's result, however, we consider an alternative notion of approximation which does not assume away 'the problem of the edge'.' t We work with the approTAmate demand corr~ pondence of Khan and Yamazaki (1981) which is upper hemicontinuous eve~ when th~ consumption set is non-c~nvex~ A consumption bundle is said to belong to the approximate demand correspondence if it belongs to the budget set and there does not e~$t •anether bundle in the interior of the budget set which is preferred to iL If utility functions exist thi.~ is nothing but the usual definitiou of a-demand correspondence assc~ated with a compen~ted equilibrium. Although we do not require utility functions to exist we shall refer to an equilibrium associated with the approximate demand corresponden~ ~ a c~mpens~ed equilibrium. 5 In economies with non-convex p r d ~ it ~ u~tmlly be shown that equilibria exist which are approximate in the sen~ that feasibility is violated by an amount which is independent cf t~e size of the ¢¢o~omy~ Eili~sc~ (1979) follows Staff's (1969) a p p r o ~ of s h o ~ that if O~e n o n ~ z ~ are uniformly bounded an approximate Tiebot~t e q ~ b 6 e m exists. Recemly, Anderson, Khan and ~ d (1982) have shown ~ ~ x i ~ of ~.-~rordmate equilibrium with bounds independent of ~ As t_..~ .__~_.k i~ Anderson, Khan and R:~h;d (1982) indicate~ for an e ~ ~ y ~ indivisibilities, an approxh~m~ c o m ~ ! e ~ _eq_.t~'~um ( ~ t h bo~dls independent oi p r d ~ ) can be shown to e~d~t by ~ t ~ the ~pproxi~ m~te demand correspondence of Khan and Y a m a n ~ (198l~o I¢ ~ ~_¢ apply their notion of approximation to a Ttebo~t equ~hbritm~ how~c~, we shall also need to extend their result to e~:vnomies with p ~ o ~ A t~hnical diff}c~ty ~ t e d with ~howi~ the ~ c u c e o f . ~ ~ppro~imate equilibrium of this kind i~ that ti~ eqtt~briem ~ n ~ce~ no~ belong to the attainable set. Even ttm~gh a ~ sets w~ty be ~ the
standard ~o¢:dure of consi~'ing ~u~es~y ~ *Under the assumpczn floatthe ~ ~~ public goodsour notionof an appro~m,~ ~ versionofE]]icksoa'~notio~
eq~ibrkua is identical~ a qa.~q~'~k~z~
~ ~
~
~ ~ .
s ~ot
.to ~ ~ e~ ~ ~ ~ ~ ~ ~ qu~ _.~p~__'~.__m~_.
R. Vohra, l.~cal public goods as indivisible commodities
195
applied since that does not ensure that an approximate equilibrium allocation lies in the interior of the appropriate set. For an exchange economy considered in Anderson, Khan and Rashid (1982), the consumption sets were effectively truncated since prices were construc*~,cd such that they ~l had a positive lower bound. This approach evidently does not suffice for an economy with production. We shall truncate production sc2s taking into account the affect of such a truncation on the deviation of aggregate excess demand from zero. To apply this procedure we need to consider an economy in which the number of consumers is sufficiently large. The two different kinds of local governments we consider are passive governments, as in Hamilton (1975) and entrepreneurial governments, as in Ellickson (1979). Passive local governments treat as given the number of residents in their respective jurisdictions and minimiTe the costs of providing the requisite level of public ~rvi~s. "~a¢ torn! publ_;c expenditure in each community is spread equally among all the residents. An entrepreucuriai local government behaves like a profit maximizing producer and determines the aggregate level of public goods that it wishes to supply. This also constitutes a decision about how many residents to provide for. Notice that these governments are not entrept~neurial in the sense of Bewley (1981)since their public goods bundles are fixed; they choose only their leeds of services. If there are increasing returns in the constmaption of public goods, Le., ff the cost, per resident, of providing local pubiic g o ~ declines "as the number of residents increases, then such profit maximifiag behavior cannot be consistent with taking prices as given~ At best, ¢ ~ hope to show the existence of an approximate equilibrium in which not only are some consumers not maximizing utility bat there are also some producers not magimizing profits Isle Theorem 2 in Eili_ckso_n (1979~]. In any case, such an equilibrium would hs,~ the undesirab!e property ~ govefw ments" budgets might not be baIanced. 6 H ow¢~_, E!!ickson also e ~ w _ : ~ that, eventually, all economies of scale are egha~ted and the average cost of providing public goods to each resident b e c o ~ constanL He then shows (in his Theorem 4) that under ~ t replications of the economy, economies of scale are exhausted in equilibriun~ This seems to be a more interes~g result since ia this case, the gov~-nments ~ budgets are ~ o c ~ L "~ We ~ i show that in an economy in w h i ~ aIi scate ¢conom/cs arc exhanst¢~ an approximate Tiebout equilibrium exists under much weaker conditions then those postulated in Eilickson (I979). In parf~L.h._.r, the mean e x ~ d~_a_~d is shown to be bounded i ~ t i y of prd~enocs.
*there is no way of earning that a la~duc~ who_/~not maximim~¢mSt~.iS n¢~ making loses. ~Bewley(1981)alsoimfav:~Ih¢balancedbudg~,ocn~tlo~ Tim is I t i s ~ 2.4.
t96
R~ Vohra, Local public goods as indivLffble commodities
2. The medd We consider an economy with c consumers, m+ l communities and u private producers. The first m communities are interpreted as those which pro~de ~ m ¢ public goods while the (m+ 1)th community provides no pubfic goods, thus all-3wing a consumer the possibility of not consuming any public goods. Each of the first m communities can provide h different types of public goods. Let Q~, R km be the choice set of community i with respect to the per capita public services it provides to its residents. While only h coordinates of Qt can be non-zero, we define Qf as belonging of R ~" rather than Rs so as to distinguish between the same kind of public good in different communities. Any b-.mdle qi~Qi will alternatively be referred to as an admissible policy of the ith- local government. Note that no more than h elements of q~ can be noN-zero. Since a consumer can reside only in one corardunity, the choice of local public goods is a choice of a particular community. Given a vector of admissible policies, his consumption of local public goods is restricted to lie in a finite set of bundles. We refer to a consumption of consumer t by ~(t)---(xp(t),x~(t))~X(O--Xp(t)xX~t), where Xp(t) is his consumption of private goods and x,(O that of public goods. Xp=R~+ is the consumption set corresponding to private goods X~(t)~ {Or.... .q=,O} is the set of public goods consumption possibilities offered by b e market. We do not assume that Lhe actual consumption set e o n e s ~ n d ing to public goods is non-co-vex, i.e~, it may b~ p~y~ically possible for a consumer to consume a convex combination of points in X.(t). What makes our model inh~-r~ntly one of indivisible commodities is the fact that the choice set of public goods made available by the m communities is nonconvex. Although X(t) depends on q~qt x---×q=, for the sake of notational convenience, we shall not always make ~hat explicit. Note, h~wv,~er, tttet the consumption set correspondence X(t): ] ' L ~ x {0}--,R r-k= is contit~ttotts. Th,'. u private producers have convex, production sets YI~Rt and bel~ve as pl~fit maximizers. The extended production set 1?~ is defined as ~'~= {(f,0)~Rt+m[y~e YJ-}. The production set e r a local government p r o d u t ~ is I a ~ R t_ ×R+~. For a given vector of admirable policies q, we denote the ifix local government's production set by Y~=R~_x R ~ and its production ve~:tor I . t~ by f---(Yj,,Y;I, where -I?i. are the inputs of private commodities mad y~ the output of pubfic goods; ~---nt~, where ni¢N+ - the ~et of non-negative integers - is the number of residents in the ith community. Clearly, Oven any admissible, policy ¢, I~ is non-convvx because # must necessary be integervalued. Y* could also be non-convex because of inmmsing returns to scale but we shall follow EUickson in assuming that eventually, all economies of scale with respect to the number of residents are exhausted and each government produces under conditions of constant average ¢oSL Since we are interested in large economies, without loss of ge~rality, we can assume that
R v.~2a,,g Lcc.cd t~ub!~_ goo6s ae- mdielsible o0mmod~[es
.197
each govemmc-nt produces under conditions of constant average cost at all levels of outpuL This follows from Ell/ckson's result (his Theorem 4) that for a large enough rcpfication, all economies of scal:~ are exhausted in equilibrium~ We can now formally state our assumptions:
A.I. For each consumer t, the preference relation >-t~Ri++~r~ is irreflexiv¢, transitive and continuous.
A.2. For each consumer t, e(t)~Xp{t) is his endowment and 0i(t) his share in the profits of the flh firm. A.3. For each private producer, j, j ~ I ..... u, the production set YJ~R t satisfies: (i) YJ r~ R +J- {0}, (ii) yi is convex
A.4. For each local government/producer i, i--1,..., m, the production set F~mRtx Rh+ m is con :ex, and for any given vectors of admissible policies q, Yi satisfies: (i) Oe Y~, (ii) if yE~con (yt) and y~=n~q for any nt~N+, then f ~ Y~. (iii) i f y ~ y i then ay~e Y: for any ~ e N + .
/LS. (~2--i ?~+Y.~'=~con(~) is clo~l. Assumption A.4(ii) captures the idea that non-convexity of ~ arises only from the fact that the number of residents must be integer valued. This corr~ponds to the assumption of integer convexity in Frank (I969, p. 34). Assumption A.4 imptics that con (Y~) is a convex cone gith vertex zero. A local public goods economy is de~ed by {(X(t),>~,t~t)),(Y~),(]~)~(L~)}.
The budget set for consumer t is defined as
~ , O= {~(O s X(O l p" x(t) ~_p," e(t) + x(p,t)}, where p--(pp,p~) refers to private and public goods" p r k ~ and
~,t)-=j~ @(t)roaxpr" YJ+l ~ maxpY jNOtice that, given AA, the second teJ'ul in ~(l:, 0 will be 0 in ¢qu/t/br/um.
198
R. Vohra, Local public goods as indivisible commodities
The compensated demand set for consumer t is defined as d~p~t~ = {x(t) ~ B(p, t) ]~t)>- t x(t)=~p- ~ t ) - p ~ . e(t) + ~t)}.
The supply set for a .~rivate producer j is defined as nJ(P)={Y~ PIP')g>--P'E for all C e ~").
The supply set for a local government/prodttcer i, given a vector of admissible policies q, is defined as
The con~exified supply set for a local government producer ~ given a vector of admissible policies q, is defined as con O/'(p)) ---{y~contY~)] p" y~>-p'~', for all ~ f c o n (Y~)}.
A compensated, approximate Tiebout equ[I~brium for an economy ~ h entrepreneurial local governments, giveu a vector of admi~ible policies q, is a (c+u+m+l) tuple {(x(t)), (f), 0/), p} such that (i) for each consumer t,x(t) ¢d(t,,t), (ii) for each private producerj, y / z t / ~ ) , (iii) for each local government/producer/,y~e~ffp), (iv) (l/c) ~[,+-__t~ max [z~,O] =6, where
and 6, referred to as ~ ¢ error, is a finite, computab~ bou~d sada f~h-~
A compensate& approMmate Tiebout equilibrium fc,r an ~ , i~ passive local g ~ given a vector of ad:n~itge poIicie~ q, i~ d ~ in the same way as above except that c~ndition ('th~) is c-_~angedto (iii)~ for each local government/ptodu~nr/t
R. Vohra, Local public goods a~ indivisible commodities
199
When feasibility is satisfied exactly, in the sense that
l:l
we
i=l
also have
Clearly, under Assumption AA, an equilibrium for an economy ~ t h entrepreneurial governments is also an equilibrium for an economy M | h passive gove~,nents. ~(Q;6) is the equilibrium cort~pondenc~ which associates with each q~Q, a set of compensated, approximate Tiebout equilibria, with error 6. We denote by g=(Q;6) the coordinates of ~ f ( ~ eorrcsponding to consw';ota'~ consumption bundles, q ~ Q is said to be a second-best policy ~ h r e s ~ .to ff there is an x ¢ ~ ( q ; , ~ such that ~z~¢Q and £ ~ ( ~ 6 ) with £.'>.~ for ~ L & R ~
In this section we present the main result o~' this p~per.
Theorem L I f Assumptions A.I-A.~ are ~atisfie~ for any ~ of ~fvfts~bl¢ policies there exists a compensated, approximate Tie.bo~ ecagI[brium for large economy with ~trepreneurfal local g o z ~ m ~ s . The term "large economy' is taken to ~ that the number of o ~ c, is large enough. An actual value for the error, ,~, is_ a l ~ compu . ~ in t ~ proof of Theorem 1 and show~ to depend onty on the e n d o ~ t ~ .the technology, the distribution of profits and the number .of commodities, T~his result can also be interpreted as stating that if there ere no ~large~ t t ~ k ~ ('large' both in terms of endowments and share of profits), dg~- ia ~ there exists a comp~saled Tiebout eqt[ilibfium. I1 s~e~s rec.somtbie to exp¢~ :-~t this resuk would also be true for an ¢~c~n~n~y ~ t b a. co~tis,uam of consumers. That this is indeed so czm be .deduced from Votu'a 09g4o Theorvm I), which shows the exist¢l~ of a compcasated Tiebout eq _~.'.._ibrium (mth~ t h ~ that of a comtg-asated, appro]rim~te Tiebout _¢q_ui_.h_'.briam) in an economy with passive local gove,nmcnts.~ The reset_ iu Vda_..r~{I_9~t) eSinc~ the n~ult in Vchm (I~,Q ~ trot depe~_ _-on.~b¢ ute.r¢ d ~ l F ~ t t i ~ ~ 1~ ¢c/nsmnption of p u b ~ goods it a~d~s to the ~ iu which the avetal~ rest -of goveramcnt~s out~ut is ~ t
equil~L=i=mfor ~ a t ~ z r l a ]
so that an ¢qu/fila'ium for p=ssiv¢ g ~ , ~
is ~ls¢ an
200
R. Vohro.l.oc~!publicgoods m indivisiblecommodities
shows that if we consider an economy with a continuum of consumers not only can we ¢~eal with cases in which economies of scale in consuming public goods are not exhausted but also el~rn~nate the approximation involved in the feasibility condition. Rcstorafio~ of the feasibility conditiott also makes the welfare implications of our equilibrium notion more attractive. It is clear that a compensated Tiebout equilibrium for entrepreneurial governments is approximately Pareto optimal, given the particular vector of admissible policies, or locafionally efficient in the sense of Wildasin (1980); if the aggregate endowment is reduced by an arbitrarily small amount it is not po~_~.'b!e to make any non-negligible set of consumers better off without making the rest worse off. However, it is worthwhile to stress the fact that while the approximation involved in the feasibility condition can be eliminated by consid¢6ng a continuum of consumers the same is not true for the approximation involved in the demand correspondence of each consumer, unless further assumptions are made, one must be satisfied with a compensated equilibrium. Theorem I is also of some interest for economies without public goods but with xg(t) interpreted as the tth consumer's demand for indivisible commodities. However, as it is stated, Theorem I is valid only for economies in which each producer has a single Output and con(Y ~) is a cone. While the latter assumption is not used in any of our proofs, Lemma 1 depends critically on the assumption that each producer has a single outpuL For the multiproduc-t case we would have to assume that every f ¢ c o n ( Y i) can ~ written as a convex combination of vectors including one which had each output as the highest integer less than or equal to the cor~sponding output in y~. With such a 'bounded non-convexity' assumption 9 replacing the assumption of integer convexity, Theorem 1 can be interpreted as stating the existence of a compensated, approximate equilibrium in an economy with indivisible cornmodifies. It can be seen as an extension of Anderson, Khan and Rashid~s (1982) result to economies with production but with bounded aonconvexities in production and no indivisible inputs. Theorem Z ! f in addition to Assumptions A . 1 - A J it is assumed that Q is compact, then there exists for every q~Q, an approximate equilibrium, as in Theorem I, with the additional property that the error i5" is independent of q~ Moreover, there exists a q*~Q such that q* is a second-best policy wi~h respect to ~*. This result shows that we can find an error independently of q and, then, given the set of admi~bIe policies Q a~,~d the equilibrium correspondence 9The assumption we use here is similar ~ough trot identi~ to the he,reded nma-couvegity assumption in Start (t969).
R, Vohra,Local public goods as indivisible commodities
201
~'(Q;6*) find a policy which is second-best. Of course, second-best optimaJity is relative to the equilibrium concept being considered. There raay, for instance, be another poiicy which is unanimously preferred if a different kind of tax scheme is adopted. 4. Proofs We first prove existence of equilibrium for an e ~ n o m y in which the production sets of local governments are replaced by their convex hurls. Such an economy will be referred to as a convexified economy.
Auxiliary Theorem. If Assumptions A.I-A.5 are satisfied, for any vector of admissible 'policies there exists a compensated, approximate Tiebout equilibrium for a convexified, large economy with entrem'eneurial local governments. Proof Consider the attainable set defined independently of q as
j=|
-
"=
0)}
. ) , - - 0 t
Since all the conditions of 5.4(2) of Debren (1959) are satisfied, A is compact, t° For vector x~R', let Hxil=~Y~_-t[x'l. The convention we use for ordering vectors is >>, > , __>. Define g,={max~gJ]]g~A] and e.=max, i]e(0l[. Given compactness of A, g= clearly exists and is a non-decreeing function of aggregate endowment;
where C,(cem)=G(cl'e,=) and I is the unit vector in R *÷~'. Since aggregate production is subject to non-increasing returns g=<~cG(e~.). Note that C-(-) is independent of q. Throughout the remainder of this proof we shall assume that a vector of admissible policies q is given and consider Yi, i = l , . . . , m to be convex hulls of the corresponding sets. Let K be a cube in R z+h= oentered a 0 with length to(~= I Y'~+~'= ~ yt) n RC+÷~- {0}. by A.3(i) and because Y~ ~ R L x RX.*~, ~ t y follows again, from the fsA~t that Y~ = RL x R ~ wh/c,h also ~npl/es that publ/v goods are not , ~ I as inputs.
202
R. Vohra, Local public goods as i~ulivisible commodities
2 {cG(e,~)+o0, where ~ is some positive number. The truncated production sets are defined as ~f= yt n K, F J= ~ n K. Prices are chosen, as in Anderson, Khan and Rashid (1982) from
The truncated consumption set for consumer t is
where ~mfmax~(t) and ~(t)~is the maximum profit that t can receive $~en that production plans belong to ~'~ and Y~ and prices belong to ~. Clearly, ~t, is well defined. For each consumer t, the compensated demand c.orrespondence d(p,t) is non~-mp~ and u p ~ r hemi-continuons Esee Khan and Yamazaki (1981)]. The aggregate, compensated demand ~trespondent~.~/~. S ' ~ , where 2=~_#2(t)is defined by o{p)---Eg=zd(p,Oand it too is aonempty and upper hemi-continuous.It is easy to check that D(p)e _~.~_.(t)~The supply correspondences~(a) and V~tp)definedover the truncated sets Y and P~ are clearlynon-empty, convex-valued and upper hemi-continuous.There* fore, the aggregate supply correspondence ~/: $ 4 ~, where ~'=(~
~'i+ ~ ~ )
and
tl(p)--~q~p)+ ~ ' ~ ) ,
also satisfiestheseproperties. Define ~con(~') × IY--*Wby
for all/~eS'},
where x denotes ,~.~=~x(t) and y deeotes
Consider ~b:S'x eon(R')x F-+S'x con(A~)x F
i¢, Vola'#.,Localpublicgoodsas ~,div~bIe commodities
203
where 4 ~ , x, y) = ¢(x, y) x con (o(p)) × n(p).
Since ~ is non-empty and convex valued, we can apply Kakutani's Theorem to obtain a fixed point (p, x, y) a 0(P, x, y). By the Shapley-Folkman Theorem we can write x'(t) +
Xm t= i
Z
x(t),
l=|+/~m+ I
where x'(t) ~con(d(p,t)) and x(t)Ed(p, O. Thus, for l+hm comumet~ there is a possible discrepanvy between x'(t), the assignment corresponding to the fixed point and x(t), that belonging to the actual, compenxated demand set d(p,t). Since X ( t ) = R ~ ~ and con(X(t))=Rt++~, (x(t)-x'(t))<~x(t). Let x~.t), k=- 1,..., l+hm denote the kth element of x(t). Then, l+ltm
l+/tm
,=~ m = f ( x ( 0 - x ' ( 0 ) , 0 ] ' ~
t~x~0.
But from the budget constraint and deletion of p. we have t+/tm /t-~l
Thus,
Lt z = Z" x ( t ) f=x
-
-
\$~t
/
t=t
(t. \r=l
~),
.
j
Then, for any ~6~Y,
From the properties of the fo~c~lpoint we b o w that the ~ d the right-hand side of (2) is uoa-p~tive. ~ g to 0), ~ rewrite (2)
~ on we.
R. Vohra~Local public goods as indivisible coramodiffes
L e t / ~ . 1 ifzk>0 and I / x / c if~__<0. The.,.,,
~.~= ~ ~ + ~
~ ~
k~K L
~4)
k~g 2
l+hm)lz~
where K ~ = { k ~ ( ! ..... t+~)l z~>0) and K 2 :- {ke(1,..., Combining (3) and (4) we have ~+tm 1 max(z~,O)=<(l+hm)/c(e=+n~)---7 k=l
~ zt.
% / G /ceK 2
The last term can be rewritten as 1
1
~
- ~
~'
and by adding (1/x/'c~.~= l x(t) to the right-hand side we got 1
¢
Combining (5), (6) and recalling that I[yl]
l.(a÷a/x/c)] which
implies that
From the definition of G it follows tkat for all producers i and j.
Convexity of the production sets implies
X/ c /
(5)
R. Vohra,Local publicgoods as indivisible commodities
205
We now show that yt belongs to the interior of ~ and y~ belongs to the interior of ~7 for all i and j. Let G(a) + 1
which is well defined and positive as long as ~ i / x / ~ ) < I. G is non-increasing i n c and G(0)--0. Moreover, there exists a c large enough such that O(1/x/~)< 1. This follows from the continuity of g,(-) at 0, which in turn is a consequence of the continuity of *.he correspondence A(~.~e(t)) at 0. H With defined as above it is easy to see that= for sufficiently large c., ~ a ) + • G(1/x/~)<~. Thus, from (8), Given that y~ and yl belong to the interior of their respective truncated production sets, for all i and j, it follov~ from usual arguments, that they magimiT~ profits not only over ~ and Y~ but also over Y~ and 1~ Thus, conditions (ii) and (iii) of equilibrium are satisfiet~ Condition (i) is el_early satisfied. If no consumer is large either in terms of endowment ~r profits. i.e., if ~ and ~,~, are constants G(a)=G(x/~ ) so that ~./c----~l/,Jc)and ;,t follows from (7)' that I + tJm
Ic k~x=max (zt, 0) -
2;, and
x/e l.
(9)
cj
\,¢'c7
,,10)
so that condition (iv) i~ also satisfied for a convexilied e~onomy. l_emma 1. Under Assumption A.4, for every/, = 1 .... , n g ~ff~{con(Yi)}./Ir ~ and p . 3 / > p . ~ , for all j~con(Y/), then there exists ~/~ yl and satisfying (a) p.$~>_p.j/ for all y~Y~, (b) y~
C_d-) is
R. Vohva,Local public goods as in&'zg~ibiecommodities
206
goods ou~pm- tcvzt , . __, less than
We ~ - +he~r,-,~ ~-ita
..i
y~----alyU+~t2y2~+ .-- +a~+2y~+2~, where 1+2
e~>O foralli,
~=J = l ,
~ ( 0 , t1 and
Yit,
Let N be the highest integer less than nt. Clearly, N = p n f + ( 1 - / ~ ~i for some
pe[O,1). Let . ¢ = p / + ( l - / ~ y ' . sin= ~ = g q ' we know from Asmmptioa A.4(ii) that .¢e Ia. Since j / ~ profits over co.(Y~ ,rod is a convex combination of y~, .... yJ+2~ we must have p.y~=p.yU . . . . . p . y ! ~ . Thus, p-~=pp.y[+(l _~p.yU=p.yX and conditions(a) and (b) are s a ~
Since p~S' and p.~p.~=maxp-con(Y~, the~ does not ex~ another j ~ e ~ n ( Y ~) such that ~>j~. But y ~ > ~ which implies that y ~ < ~ . This completes the Ftoof of Lemma 1.
Proof of T~orem 1. Acoording to tbe Awr~i~ry Theorem there exists {(s(t)),iyt),(~,p} satisfying ~nditions 03-{iv) except that for all t o ~ go~-t~ ment/produce~ y~econ(Yi). By I_~mmR I, for each i, i = 1..... u there exists ¢e P such that p-¢>p"¢ for allj~e ka,~+q~>y~ Thus
Z m~[e.0]
I l+l~ -
C k=l
1 ~+~
r(
1 z+i=
I-(
~
/
.
\
-
f.
:
~.)
I
l.~i I
Combining (I0) we have, 1 l+/tm
-c ,~= max[~,O]<,~,
where
.=~{(,+~)(e. + ~.)+,.÷ ~e.)+;÷~}.
(,,,
Thus, {x(t),(y~,OC),p} is a compensated, al~roximate Ttebout equilibrium. Proof of Theorem 2. Given compactness of Q we can find a 6* suck that an approximate equilibrium with respect ot 6* exists for any qeQ. DeFine q = - -
IL Vohra, Loc~lFublic goods as indivisible commodities
207
max,.,ellqr[]_~.nd-6*=O&/~)'{(i+~n)(e~+~,)+e,+C~e~)+~/c÷qj~}. By Theorem ! and the definition of& as in (11), for any qeQ, ~'(q;6*) # ~. We shah now show that ~'(Q;6*) is compact. Since aggregate excess demand is bounded by 6"c.I for any qeQ, ~f(Q;6*) is bounded. To show that ~f(.;J*) has a closed graph, consider (q~,x~,f,f)-~(q,x,y,~) such that (x~,f , f ) e ¢(q~; 6*) for all v. Ce~tainly,
-'~ 1 l+~'max [zt,0] =<6,. C
ffi
Also, yJ maximizes profits with respect to p for all private producers. Suppose there exists a local government i for which there exists ~ ~ such that p. ~ > p- f or, p-(6- q) > p-(n- q). Then, there exists a o large enough such that f.(~.q)>p°.(n~.q), a contradi~don~ It now remains to be shown that x(t)Ed(p, t) for all t. Suppose not. Then there exists a t and ~ ( t ) s ~ h that ~O>-~x(t) and p-~t)
R,S.U.E.~ B
208
K Vohra, Loca~ public goods as indivisible commodities
T h r o u g h o u t this p a p e r we have considered a m o d e l in which economies of scale in providing public g o o d s have been exhausted. This restriction o n the nature o f congestion can be removed if we consider an e c o n o m y w i t h a c o n t i n u u m of consumers. In that setting even the a p p r o x i m a t i o n with respect to m e a n excess d e m a n d c a n be eliminated. I t wouid be of interest to s t u d y the effect of increasing the n u m b e r of communities on the e c o n o m y ' s welfare. A d r a w b a c k o f Elliekson's w o r k a n d of o u r s is that, in each c o m m u n i t y , the b u n d l e of public g o o d s to be provided t o a c o n s u m e r is fixed. However, as we have shown, this a s s u m p t i o n is n o t as restrictive as it seems since we can find a second-best equilibrhtm with respect t o g o v e r n m e n t policies regarding the per capita mix a n d level of p u b l i c goods.
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