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【論文】

商学論集 第59巻第3号 1991年3月

The Existence of Lindah1 Equilibria Reconsidered*

ToshihiroSato

Abstract:The purpose of this paper is to present an alternative proof to the existence of Lindahl equilibria. It is carried out in a direct fashion:that is,as in the case of existence proof of cornpetitiveequilibria in economies with private goods only,we make the quantities of public and private goods as choice variables and the prices as parameters in each agent's choice problem, Such an approach enables us to prove the existence of Lindahl equilibria without any assumptions other than those imposedordinarily.The techniques may be applied separately te ether fields of research_

Jottrna1of Economic Literature Classification Nos Keymords: Public good, Lindah1 equilibrium_

021,022

1 Introduction

The purpose of this paper is to present an alternative proof to the existence theorem of Lindah1 equilibria. Since the publication of Samuelson[1954],a significant volume of literature has been devoted to this task,and the existence is now a well established fact. Amongst it,three mathematical techniques employed are prominent for proving existence. The first approach is to apply the proof of competitive equilibria in economies with private goods only,where the consumption bundle of public goods consumed by each individual agent is taken tobe a sepa- rate one of private goods(see,for example,Foley[1970]), The second approach which is often called a duality approach bases on the idea that the original consumer choice problem among the consumption bundles with respect to given prices can be reformulated as the choice problem among the prices with respect to prefixed consumption bundles via indirect utility functions defined on price space(see Mi11eron[1972]and Ruys [1972]). The third approach is that of

率This research was supported in part by the Japanese Ministry of Education through Grant・In-Aid fo「 Scientific Research,Grant No.02730003.

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Roberts ([1973]and[1974]),who treats the personalized prices of the public goods and the quantities of private goods as choice variables of each individual agent,with the quantities of public goods and prices of private goods being parameters in the individual choices. That is, he took the quantities which are free to vary between agents as choice variables while those common fo al]agents being taken as parameters. For further discussions,refer to Roberts [1974]which gives a good survey about this problem_

The technique developed here constitutes the fourth approach to this problem. The basic idea itself may not be new but is familiar to those engaged in the field of general equilibrium theory. But it is the first attempt to apply the method of Arrow and Hahn[1971]to economies with public goods,thereby establishing the existence of Lindahl equilibria,while they invented it in order to prove the existence of compensated equilibria in economies with private goods only_ Such an approach may be called an indirect method in that it involves the(ordinal)uti1_ itv distributions,especial]y Pareto efficient ones,as a requisite device_ But it should still be called a direct method in that the prices are taken as parameters and the quantities of the public and private goods as choice variables in each agent's choice problem as in the case of demand correspondences in the private goods economies. The most important feature of our method is that the existence of Lindahi equilibria can be proved under weaker conditions than Roberts's. Tobe more concrete,we can show the convex-valued property of our price corre_ spondence without any additive assumption a la Roberts(see Section3.2).

It is noteworthy that our method will be applied separately in other field of research.See for example,Sate[1991].

2 The Model and an Existence Theorem

We consider an economy with m public goods and?private goods whose amounts are denoted by,r = (?l,?2,_.,?m)andy = (yl,y2,_ ,yf),respectively.N is a finite index set representing the set of agents (consumers)and n is its cardinality. Agents will be distinguished by superscripts.

All agents have the same consumption set R?+1,the non-negativeorthant of (m十f )- dimensional Euclidian space. Thus we can characterize each agent zby his/her utility tunc_ tionu definedon the identical consumption set and by the initial endowment of private goods (o'?一二R Throughout this paper we will impose the following assumptions on agents'charac_ teristics:for each i ∈N

A.l A.2 A.3

u'(・)is continuous;u'(・)is quasi-concave;

(:r ,y')> (、r,y)? u'(r ',y')≧u'(?,l/)w ith strict inequality ji lt > y ;

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A.4 A.5

ui(0)= 0 at > 0.

Sate,T.:Existence of LindahlEquilibria

Here the convention of vector inequality signs (> ,> ,≧)is followed。 Note that A.41S a no「- malization assumption and is not crucial for the results of this paper.A11of these assumptions are standard,so they may not need further explanations.However,it should be pa「tiCula「ly noted that these assumptions are weaker than those employed by Roberts ([1973]and[1974]).

The producers are assumed to be aggregated and represented by a social production Set Y : an jnput_output combination(.r c)? R?xRf belongs toY if the amounts of public goods r js capable of being produced by using the private goods by the amounts - c. We Still make the following assumptions on Y :

B.l y is a closed and convex cone with vertex 0;

B.2 (1rreversibimy) Y ∩(- y)= {0};

B.3 (Free Dist)osability) -R「 1⊂Y ;B.4 ∃(?,?) ∈Y such that :f1> 0.

In what follows,we often describe Y as

Y = {(r c)? r1> 0, brr 十b?c < 0}U (- R?+つ,

where bx= (br1,br2,_ ,brm)and b?= (b?, by2,_ ,by1)are the coeff icients of production with br > 0and by> 0,and the product of any two vectors denotes their inner product.

An at]ocatjon is an (rt十1)-tuple of vectors(:r,1f1,_ If「')=- (fr it)who「ey'「eP「eSentS the amounts of private goods assigned to agent i. An allocation(.:t:・,If)is feasible if it is nonnegative and satisfies

( r , ∑ ,111_ ∑ j a)i ) ? Y .

Let z be the set of an feasible allocations. Obviously,Z is a non-empty and compact Subset of f1+m+nf An allocation is Pareto efficient if it is feasible and there is no other feasible allocation(:?,y')such that u'(;t:',If ')> u'(:r,? )for all i with strict inequality for Some t.

Alternatjvely,we can define the feasibility and Pareto efficiency with respect to utility distributions Let u -_ (u,,u2;_ ;un)be a utility distribution where ut is the utility level assigned to agent i. A utility distribution u then is said to be Pareto efficient(resp.feaSible)if there is some Pareto efficient (resp.feasjble)allocation which achieves u. We denote by V the set of all pareto efficient utility distributions which is easily shown to be a non-empty and compact subset oi Rn.

A price system is an(vt十1)-tuple of vectors(1)',P2,_ ,1)n,q)_ (P,q) ? Rmn+f Who「e p is the personalized prices of public goods for agent i and q is the prices of private goods In what fo11owsf we normalize the prices sothat qf二 St. Here,given a positive intege「k,Sk denotes the(k_1)-dimensional unit simplex.

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With these preliminaries we are now ready te state the formal definition of Lindahi equilibria and the result of this paper.

DEFINmON. A Lindahlequilibrium is the pair of feasible allocation(:r,y)and price system (P,q)such that

i ) f or each i ? N, if u i(?',If '、)> ui (,r ,y )then p'lr f十qy > p';r 十qy = qω' ;

i i ) ∑1p t;r 十q (∑jil l 一∑t(n i) > ∑11)f ir ' 十q c f or a ll (? ', c ) ? y .

THEOREM. Under AssumPtionsA.1to A.5 and B。l to B.4,a Lindah1 equMbrium ensts

These definition and the existence theorem are now familiar ones What is different in this paper from others is the method of proving the existence theorem,which is summarized in the next section.

3 Proof of the Theorem

The proof of our theorem proceeds as the Arrow and Hahn's[1971]proof of the existence for compensated equilibria in economies with private goods only. The existence of public goods; however,makes it rather long,self may be convenient to divide it into four steps.

3.1 u-feasible Allocations

Given a utility distribution u = (u,,u2,_ ,un),consider two kinds of upper contour sets such as the following:

X (u i) = {(.「,y) ? R 「 l ? t,t '(.:1' y ) ≧ u };

X '(z・t ,?) _ {? ? R? l (?,?l) ? X '(uj )} for .r ? R? .

It is intuitively clear that both X'(・)and .X'(・,.r)for a fixed .r are continuous with respect to the utility level.

By utilizing the last mapping,we can formalize the set of tt-possjblea11ocatjons as

'「 (u)= U{{.rl x ILX'(u,,.r)1? ? .叫 ,

and the set of tt-feastblea11ocattonsas

? (u) = '「 (u)∩Z

Obviously,.:「(')constitutes a continuous and convex_valued correspondence with respect to u And then ??u) becomes a compact-and convex-valued,and is upper hemi continuous

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correspondence in u The fact that j:(tt)≠a for any u ?V may be intuitively Clea「.The「e- fore,we have showr,

LEMMA 1 :r (u) ts non emptyfor any u ? V. MoreolJer,i ;(・)Is comlMt- and convex-tlalued

artd i.s upper heml contimtous m u.

3.2 Price Correspondence

Second,we tum our attention to the price systems associated with Pareto efficient utMty distributions.

First of all,choose rn real numbers{rhl sothat

十00> rh> b,h/∑kb?k, h= 1,2,_ ,m,

and define two cubes Qand Qas

Q_ 11h[0,r h], and ,0 = Qn.

Then we can state the following result which is well-known but its boundary property with respect to public goods prices is not mentioned yet anywhere.

pRoposmoN If u = (u,u2,_.,ur,)isa Pareto eがcierltutility distnbutiort there enst an allocation (?,y)= (1f11f',1f2,_ ,fn) ? .i (u)and a price system (P,(1)-_ (Pi,P2,_ ,tl'') ∈? X S?

sttch that,for all i ? N

i) u (.r ,1/)= u,;

jj) if u '(1r',?')> u'(?,? ), then P .r '十qy'> P':r 十qy';

jjj) ∑,p'1r:'十q c d" ∑t tl':r 十q (∑,lf i 一∑,col、) = 0 f or a l l (.r ', c) ? y .

Proof. We already know the existence of an allocation(?,?)? '「(u)and a price system(p,e)f K「- with the properties (i),(ii)and(iii)(see,for example,Foley[1970J). Here,.since the relations in(ii)and(iii)are all homogeneous of degree zero with respect to the prices,we can normalize (p q) se as to satisfy q ?Se.0n the other hand,the profit maximi- zation property(iii)implies that

ヨ,l> 0 such that ∑,p・' ≦λb,, and q = λb?

In view of the fact that q ∈S' we have λ= 1/∑kbyh and hence

p i d"_ ∑tp t ≦ b x/∑h b?h,

Thus,1)?Q holds trivially for ali i . Therefore,we can conclude that(1)q)∈OX So as Was to be proved. 11

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Now let P(・)bea correspondencefrom V intoQx S'defined as

P(?)= j (- f Q X S t

From the Proposition it follows that

LEMMA 2 P(u)≠fi t(or any u ? V

第59巻第3 号

-i (- ) ? ? (ti); (r,f f) If)satisfies }

(i),(ii)and(iii)in the Proposition

We also see that P(u)is compact,partly because of the c1osedness of i「(u)and y and partly because of the boundedness of Ox St :

LEMMA 3 .P(,) 1,s a compact-va11ued correspondence

Next,we want to show that ii(・)is convex-valued.This,however,is not always an easy matter to prove. In order to ensure the convex-valuedness of this price correspondence,for example,Roberts had to make an additional assumption which in effect is equivalent to assuming the convexity itself (see Roberts[1973,Assumption N.3,pp.361-362]). But when we define the price correspondence as ours and when we restrict ourselves to Pareto efficient utility distributions,we do not need an assumption a la Roberts. Indeed,we can show the following property.

LEMMA 4. Let ti be a Pareto eがicient tttlmy distribution and take any(p'q')and(f q2)in ii (u). Also let (1r]11') and (:fl2 1f2) In .f (tt) be the a11ocations correspmding to (p'q')and

(か2,q2),resPecti11,ely. Then,for any a ? [0,1], (tｻ(a),q(a),1r:(a),y(a))satisftes the three condj_ tions(i),(ii)and (iii)m thePropostt1on,tohere

(j )(a ) q (a ),:1l(a ),y (a ・)) - a (? l ,q 1,? 1,y 1) 十(1 _ a )(1)2,q2, r 2 y 2)

Proof Since ,0 and St are both convex,1)(a)? ? and q(a,)∈S'hold trivially. Similarly,(:r(a),y(a))? 、f (u),which in turn implies the validity of (i)with respect to this (?(a) If (a))because of the Pareto efficiencyoi u and the Assumption A.2.

In order to prove the properties (ii)and(iii),we need the following relations:for each z,

( 1 ) 1)I t :r 1 十 q i l f l i = f) i ' :1:・2 十 11 t l f ' ;

(2 ) f〕l21、r 2 十q 2 1/2' = tl2' r 1 十q 2 1f l'

The first equality may be shown as follows_ Since (r2,y2') ? X'(u')and u'= u'(?',If ''), we have

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(3 ) tit ,? 2 十(11y 2'≧ p l i :r 1十(111/ l ' f or a l i i .

Take the summation of these inequalities with respect te l to Obtain

(4 ) ∑ , p i ':r 2 十q 1∑ ,y -1 > ∑ ,j)i i :r 1 十q ∑ ,lf l '.

Here since(? ,∑,lf2,_∑,to')f Y by definition,the validity of property (ii)with 「eSPeCt to (p i,q')Implies that 0 、>∑,1)'ilr2十q'∑,y-'_∑,al',and hence that the left hand side of (4)iS no greater than∑,(o' Similarly;since(0,0)? Y, property (ii) also implies that the right hand side of (4)is no jess than∑,at'_ Therefore (3)must hold with equalities fo「 ali i ・ w ith this (1)is proved. (2)will be proved in the same way.

We see from(1)and(2)that

(5 ) (p hi , q h)(? (a ) If (a )) = a (p h',q h) (:r 1,1f l i ) 十( 1 _ a )(1)hi , q h)( r 2, 112' )

= (1)h,qり(:1:hyh) for ali i ? N and h= 1,2.

provided with (5),we can easily prove that the pair of (P(a),q(a))and (:「(a'),y(a)) satisfies the property (jjj). Indeed choose any(.:r,c)E y arbitrarily,then We have

(∑,p'(a) q,(a,))(:r c)

≦ (∑ j t)l t:r 1 十q 1∑ j y l ' _ q 1∑j ωi ) 十 (∑ ,p21:z2 十q 2 ∑ 1f 2' - q 2∑ ,(il l )

= 0

the inequality and equality resulting from the fact that the pair of (?h,qh)and(1「h,yh)SatiS・ fies (iii), h= 1,2. Moreover,

(∑ip '(a ),q(a ))(r(a ),∑,・lf '(a )一∑,(,) )

= a ∑ j (j )11,q 1)(:r (a ),y i (a )) 十(1 - a )∑ ,(1)2 ',(12) (;r (a ),11 (a )) - q (a )∑ t (0 1

= a (∑ I t)1':f 、1十q l ∑ ,111' _ q 1∑ ,a ') 十( 1 _ a )(∑ I f )2':I ,2 十If ∑ If 21 - q ∑ at )

= 0,

which together with above relation proves the property (iii)with respect to the Pal「 Of (p(a),q(a・))and (?(a),y (a)).

The property (jj)may be proved in a similar way. For any t, choose (?,y)Such that ui(.r,If)> uj,then

p (a )x:十q (a )? = a (p l i ? 十q 11f )十( 1 - a )(1)2 tit 十(121f )

> a (j ?1':rl 十q11f t')十(1_ a)(p2i r 2十q2y2') (by (iii))

= p (a).r(a)十q(a)y(a), (by (5))

which implies that the property (ii)holds true. ll

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COROLLARY. P(・) isa conυex-tJahted correspondence on V .

The following Lemma asserts the upper hemi continuity of P(.)

LEMMA 5 i i(・) is upper hemi continuous(m V

第59巻第3号

P「cot Since the range of -P(・)is the compact setQx St,we are able to prove the Lemma by means of sequences.

Consider a sequence {“レ1レ= 1,2,_.}⊂ Vet utility distributions converging towarduo ? 1/ and a sequence{(t,?,q') 1レ= 1 2,_ }⊂? x St satisfying (t,''qレ、) ? ii (uレ); レ= 12,_.,and converging toward(j)o,qo)? QxSf. By definition,we can find a sequence{(x;, yつ? .i (u')1レ= 1;2,_ }⊂Z of allocations such that the pair of (p?,(1')and(r''yつsatisfies the three conditions in the Proposition,レ= 1,2,_ _Since Z is compact,the last sequence has a limit point:let(:reye) be a limit point of this sequence and take its subsequence converging toward this limit point。 Without loss of generality,we can regard {(1:?,y )}as such a subsequence;and take{uレ}and{? ?,qつ}as the corresponding subsequences。We want to show that(Po?qo)? ii(uo),or that the pair of (Po,qo)and(ro,yo)satisfies the three conditions in the Proposition with respect to uo.

The validity of condition (i)is trivial because u'(・)'s are continuous by A.1.In order to prove (ii) suppose that(pc,(Io),(1:・e ye)and uOviolate it。This supposition

guarantees the existence of some(1rf,If')E 、j (uo)such that

(6 ) ti e l :I:o 十q Oy o1 > p Oi :f1 十q Oy i ' f o r s o m e z

But,since X'(・)is continuous (especial]v iewer hemi continuous)and since(?'If if)? x i(uOt), We Can Construct a sequence{(1r' If ?')1レ= 1,2,_ }satisfying (.rレf,?- )∈X,(u?t)for ali i andレ,and converging toward(r,y)_ (6)then implies that

f)''.「 '十fr y ?' > P? :r: '十q レy レ'' f o r レ la r g e en oug h ,

which contradicts the supposition that the condition (ii)is valid with respect to(j)',qつand (r ,y')_ With this(ii)is proved。

Finally,observe that

(∑ 1tl 11 q 1つ (? ,C ) ≦ 0 = (∑ ・t'P i t , ｫ1つ (;C?, ∑ tit リt 一∑ j ω1)

for all レand(:r c )? Y. By letting レ→十00;we have

(∑ ,f?0'? q 0) (? , C ) ≦ 0 = (∑ ,ti c',q 0) (.「 0, ∑ ty 0 1一 ∑ ,te l)

for any (:r c)?Y _ Thus(f)o,q?)and(ro,yo)satisfy (iii). With this our proof is complete 11

The tot]owing Lemma will be needed in the last subsection.

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LEMMA 6 Choose any u ? V and (?,q)f i)(tt). Then for any(、fr y)? i (ze),the Pal「

of (?,q)artd (:r ?)n tt? os the three condi'ttons (i),(ii)and (iii)in the ProPoSttiOn.

proof since u ?V,any(?,?)? j (u)satisfies the condition(i)trivially. Now,(P,q) ?i i(tj)implies the existence of some (?',?)?f (u) for which(i),(ii)and(iii)hold true. Note that

t,'lrl 十q 1? = p'.r 十q 11 ' f o r a l i i

by the same reason as in the cases of (1)and(2). From these equalities as well as f「Om the fact that the condition(jj)holds with respect to(j)q)and (?',y「),it follows that,fo「 ali i,

t)ill 十qlf > p :z1十qlf i for any (i ,?)? int .X t(u,),

which warrants the validity of (ii).On the other hand,for any(i ,c)?Y,

(∑,p ' q ) (ii, c) く 0 = (∑j1) ,q ) (:「 ∑ilfi 一∑tOつ

which together with(?;∑,yi)∈Y establishes (m). 11

3.3 Pareto Efficient Utility Distributions

concerning the set of Pareto efficient utility distributions we know a well established 「elation as the following:

LEMMA7 There1.s a homeomorphism f f rom V znto Sn st,tch that u,= c it and Only if vj = 0ωheneuer f (u) = t1.

As such a homeomorphism f we consider

f (u) = u/∑,;tt.,・, u ? V .

By the Assumptions A.3,A.4 and A.5,we have∑,u,> 0 for all u ?V,sothatf is well-defined onv.For further discussions,see Arrow and Hahn[1971,pp.111-113]. In What follows,We use the notation g _ f -':S「'→V.

As a last preliminary,we will construct a correspondence F from(?XSつXZ intOSnaS follows:given (」:,q)?Qx Stand(?,?)∈Z, define

F (?, q, .r? ?) - S nn {v l t1 = c it s'(j ), (I f , y) < 0},

where

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商 学 論 集

s ' (P , q :r , 11 ) - q (ii ' - f) :r - q y ', I E N

Thenwecanshow thefo11owing Lemma.

LEMMA 8.on OxS?xZ

第59巻第3号

F(') ts non empty, compact-and con11ex-oalued and is upper hemi contlnuotts

For the proof of this Lemma as well as for the meanings of s''s and F,see Arrow and Hahn [1971,.pp_114-115].

3.4 Proof of the Theorem

We are now ready to prove the Theorem. Consider a correspondence from Snx(0xSつxz into itself defir1ed as

F(f),q :r y)x i i [g (v)]x f [g(,,)]

Obviously,the domain of this correspondence is a non-empty,compact and convex subset of the Euclidian space of dimension n十(n十1)(m十?)_ On the other hand,the Lemmas and Corollary,except for Lemma 6,thus far developed ensure that this correspondence is non_ empty,compact-and convex-valued,and is upper hemi continuous.Thus we can apply the Kakutani;s Fixed Point Theorem which warrants the existence of some(v:let,*q?、:r*,y*)? Snx(Qx Sりx Z such that

(7) 11'' ? F (?? q *,:r*,y *),

(8) (1);q*) ? P(u*),

(9) (:?* y本) ? .f (u*),

where u*-- g(f ). In what follows,we will show that this fixed point constitutes a Ljndah1 equilibrium.

(8)and (9)as well as Lemma 6 imply that (j)*,q,*,:r?,If?)satisfies the three conditions (i);(ii)and(iii)in the Proposition,which in turn assures the validity of Definition (i)and(ii)excepting for the equalities p *:r*十q*y'*= q*al'or s'(p*,q*,r*,y*)= 0 for ali i

The last inequalities may be proved as follows. Since the condition (iii)in the Proposj_ tion holds with respect to(p?q???11*),we see

(10 ) ∑ ,S '(j )'?,q *,:「?, If ? ) = 0 .

NOw Suppose that st(1?*,q?,1;f;*,y本)< 0 for some 1. Then 1,t* = 0by the definition of F(.) and the Lemma 7_Since(0,(,)')? Xi(0)by the Assumptions A.3 and A.5,the first part of (jj) imPlieS tl'*:r?? q*y'* ≦q*(,1'or s'(f q? :f1本If?)> 0, which is a contradiction. Therefore

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si(p*,q*,?*,?*)≧0 must hold for all i . These inequalities and(10)together imply that s,(p*,q*,?*;11?) = 0,∀1? N . We can thus conclude that (P? q*,?? If?) is a Lindah1

equilibrium.

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Arrow,K.J.and Hahn,F.H.[1971],General Com? tiffυeAm1??,North-Holland,Amsterdam.

Debreu,G. [1959], Theory of V;alue,Yale University Press New Haven.

Debreu,G.[1962],“New Concepts and Techniques for Equilibrium Analysis,''International EconomicRetneu・ 3, 257-273.

Foley;D.[1967],“Resource Allocation and the Public Sector'Yale EconomtcEssays7, 43-98.

Foley,D.[1970],“Lindah「s Solution and the Core of an Economy with Public Goods,”Econometrica36, 66-72.

Mjlleron,J._C_[1972],“Theory of Value with Public Goods:A Survey Article,''Journa1of EconOmtCTheory 5, 419-477.

Roberts,D_J_[1973],“Existence of LindahlEquilibrium with a Measure Space of Consumers,”Journalof EcononucTheory 6, 355-381.

Roberts,D_J.[1974],''The Lindahl Solution for Economies with Public Goods,''Jonrna1of Public Eco-nomics 3, 23-42_

Ruys;p.H M.[1972],''On the Existence of an Equilibrium for an Economy with Public Goods Only''ZeltschriftfiirNationa1okonomie32, 189-202.

sate,T [1991],“Ordinal Values for Economies with Public Goods: Definition and Some Properties,''mimeo Fukushima University.

samuelson,P A.[1954],“The Pure Theory of Public Expenditure''Revieu1of Economtcs and Statistics36, 387-289.

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