Supply-constrained equilibria in economies with indexed prices

JOURNAL

OF ECONOMIC

THEORY

43, 203-222 (1987)

Supply-Constrained with

Equilibria in Economies Indexed Prices

CLAUS WEDDEPOHL Faculty University

of Actuarial of Amslerdam,

Science and Econometrics. Amsterdam, The Nerherlands

Received January 2, 1986; revised November 18, 1986

In an exchange economy prices are rigid and the prices are restricted by index functions. The indexes can be of a general type and can be different for each commodity. Indexed prices can appear in other indexes. Also considered are restrictions in terms of a general price index. Existence of a Drize equilibrium, and particularly a supply-constrained Drize equilibrium, is proved. The system of indexes must be solvable in terms of unconstrained prices and for the unconstrained commodities a monotonicity condition must hold. Journal of Economic Literature Classification Number: 021. 1“ 1987 Acadermc Press, Inc

1. INTRODUCTION We study Dreze equilibria in an exchange economy where prices are restricted by index functions. In Dreze [3] prices are restricted in relation to a non-constrained numeraire commodity, and both supply and demand constraints were permitted. Van der Laan [S] introduced supplyconstrained equilibria, also called unemployment equilibria, and he proved existence of an equilibrium where at least one commodity remains unconstrained, which however cannot be chosen a priori. In a model with prices partially free and partially linked to free prices (the linkages represent both upper and lower bounds of linked prices), Kurz [4] showed that rationing of some commodities with free prices cannot be excluded, since otherwise equilibrium may not exist. Dehez and Dreze [2] studied an economy with production, with prices restricted above and below in terms of index functions on all non-numeraire commodities. An equilibrium with no rationing of the commodities of which the prices are not at their lower bound could only be proved to exist with the prices of “index makers” * Comments of an unknown referee that led to a substantial improvement of the paper are gratefully acknowledged. All remaining errors are mine. 203 OO22-0531/87 $3.00 CopyrIght (~8 1987 by Academic Press, Inc. All rights of reproduction in any form reserved

204

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WEDDEPOHL

fixed. Van der Laan [6] studied an exchange economy where some prices are restricted by a price index in terms of “index makers,” the index makers being prices that can vary freely. Also, the price of at least one unconstrained numeraire commodity was free. In equilibrium all index makers may be constrained, if at least one indexed price attains its upper bound. Below we generalize the latter results in the following respects: (1) A much wider set of price restrictions by index functions is permitted; particularly prices may be indexed above and below by different indexes or linked by a single index, as in [4]. Indexes differ for different goods and may contain other indexed prices, but not in a circular way. An obvious example of a chained index is a wage linked to another wage, which is indexed by commodity prices. (2) If prices are restricted above and below in terms of a linear general price index of all prices, as studied in [Z, 61, then constrained prices appear in the index, which means circularity: a price is also related to itself. Then at least one price must be unconstrained and the upper and lower bounds must fulfill an extra condition, in order to ensure that a sufficiently large set of prices remains that satisfies all restrictions. (3) It is shown that if a certain monotonicity assumption holds, not only commodities with free prices that do not appear in the indexes, but also part of the index makers, can a priori be chosen unconstrained. The remaining index makers are constrained if some indexed commodity attains its upper bound. In [S] an equilibrium was defined in terms of an arbitrary rationing scheme, but existence was proved for uniform rationing. In many studies this line of argument is followed. Reference [4] is an exception: quantity constraints are related to initial resources. The definition of an equilibrium in Benassy [ 1 ] contains a rationing scheme defined as a function of effective demand. In [7] an alternative rationing system was considered in an economy with production. Below we introduce rationing in terms of constraint functions that permit a great variety of (non-manipulable) rationing systems including uniform rationing, rationing by priority, and the system applied in [4]. This method also simplifies proofs. Actually it contains part of the argument usually applied within proofs. In Section 2 the constraint functions are introduced, and in Section 3 we consider the index functions. Both non-circular indexes and upper and lower bounds relative to a general price index are discussed. In Section 4 equilibria are defined and existence is proved. Parts of the proofs are given in Appendixes A and B.

SUPPLY-CONSTRAINED

2. THE MODEL

20.5

EQUILIBRIA

OF AN EXCHANGE ECONOMY EXTENDED RATIONING SYSTEM

BY A

Consider an exchange economy with set of consumers A = ( 1,2, .... 2) and set of commodities N= ( 1, 2, .... fl>. Each consumer a EA has a consumption set X” c l@‘, preferences 2” on X” ( >, denoting the derived strict preference) and initial resources wQE RN. A price is a vector p E ‘wfl For Zc N, p, denotes the prices of commodities in I: p, = (P~)~~,E R’, I denoting the number of elements in I. x5, zL;, etc., are similarly defined. If net trades z;=xpw; are rationed, there exist numbers ~950 and dpz0, such that c:Iz’tId;, I- I-

iEN,

aEA

and xi E O;n C, WY,

ASSUMPTION

(ii) (iii) (iv) such that

c; (r;) = 2;

if

v,zO

d:(r;)=$

if

r, 5 0.

The variable ri E [ - 1, l] is called a rationing index. It represents the state of the market: if ri < 0 ( > 0) there is no demand (supply) rationing of commodity 1, and at ri = 0 there is no rationing (see Fig. l), since Cp can never be binding contraints and $ cannot be binding constraints in equilibrium. The constraint functions are not unique: given ci(ri) and di(ri) and a continuous and strictly increasing function g: [: - I, l] + [ - 1, I], with g( - 1) = - 1, g( 1) = 1, g(0) = 0, (Ci(g(S), d,(g(s))) specify the same rationing system. From this it follows that arbitrary values - 1 > s > 0 and 0 < t < 1 can be chosen such that Ci(S)< 0

and

d,(t) > 0.

206

CLAUS

WEDDEPOHL

FIGURE

Taking s = ( - (J-

1

1)/J and t = (J- 1 )/J, we obtain the following

PROPOSITION 2.1. For Jc N, there exist constraint di(ri), for i E J, such that

-(I-l)s;Cris(J-1)*3EJ:ci(ri)<0

functions

and

ci(ri)

and

di( ri) s 0.

For an n-tuple of rationing indices that satisfy the hypothesis of the proposition, it is ensured that not all J-commodities are rationed in such a way that all buying or all selling is forbidden. This will rule out “trivial” equilibria. Examples

of Constraint

Uniform

rationing.

Functions

For yk=max {w;},bk>~w; a A ‘k

a= -y,min{l,

d;=6,min(l,

l+r) l-r}.

by market shares (see [‘i’]). For Yk and 8k as above and given LX;2 1, & 2 1 (where the potential market share under supply equals a;Ea a:)

Rationing

numbers rationing

c; =max(

-Yk, -(l+r)a~yk}=-ykmin{a~(l+r),l}

d;:=6,min{&(l-r),

l}.

SUPPLY-CONSTRAINED

“Kurz” rationing,

with rie [ - 1, 01, supply rationing c;= -%$(l

only (see [4]):

-Y).

Rationing by priority. Consumers are rationed &=6,m(l

207

EQUILIBRIA

in the order 1, 2, .... m:

-r)

$=6,max{O,m-1-mr} .......................... i;‘i6,max{O,m-s+ 1 -mrJ

and

d;=min{6,,

&}

. . . .. . .. . . . . . . . . . . .. . . .. . . . . . .

$J=6,max{O, (similarly

1 -mr}

for c;). 3. PRICE RESTRICTIONS BY INDEX FUNCTIONS

DEFINITION 3.1. An index function is a mapping for A>O, pEUP: cp(Q)=;icp(p).

cp: Iwy + R, such that,

Note that it is implied that ~(0) = 0. The index cp depends on pi, if there exists PE Ry, such that (P(P~,~, pi) # (P(P~,~, 0). If JC N is the set of prices on which cpdepends we shall write (no confusion being possible) cp(PJ) = ‘p(PJ? 0).

(3.1)

We consider price restrictions in terms of index functions. We decompose the set of commodities N into the subsets Z, L, and K, where (i) for i EZ, pi is unrestricted; (ii) for i E L, pi is only restricted below; and (iii) for i EK, pi is restricted both above and below (however, the lower bound of K-prices might be identically zero). Thus we have index functions cp, and tii such that Pi 2

Vi(P)

9,(P) 5Pi s $,(P)

for

iEL

(3.2)

for

iEK.

(3.3)

If cp,(p)=$,(p) for all p, then pi is fully determined. We then have a “linked” price as studied in [4]. The price pi’is said to be indexed by jE N, if (pi or ll/i depends on pj. The price pi is said to be indirectly indexed by pi if there exists a sequence . . I = zl, z2, .... i, =j, such that pi, is indexed by pi,+, , for t = 1, .... n - 1. The system of indexes is said to be non-circular if no commodity is directly or indirectly indexed by itself.

208

CLAUSWEDDEPOHL

All ‘pi and I,+, are continuous; (R2) For MEL: (pi(p)>0 ifp>O);’ For iEK: O~~p,(p)g$~(p), allp and rjJp)>O ifp>O. ASSUMPTION

R.

(Rl)

for

Define CQ (k E K) and qr (1E L) by (3.4)

PI = V,(P) + q/ Pk

=

t1

-

ak)

qk(P)

+

(3.5)

ak$k(P)

and denote by qL E [w” and ~1~E [w” the vectors. Clearly the restrictions (3.2) and (3.3) are equivalent to qr. 10 and c(kE [0, 11, respectively; p, is at its lower bound if q, = 0. The price restrictions are expressed in terms of p,, qL, and CI~: pk is at its lower (Upper) bound if ak = 0 (ak = 1). If q,(p) = tik(p) then the value of ak is arbitrary. The variable elk will however be retained in this case, since it plays a role in existence proofs below. We can try to solve the system (3.4), (3.5) such that the constrained prices are expressed only in the “free” variables p,, qL, and ~1~. We obtain reduced indexes Bi(p,, qL, ~1~). Consider the following condition: CONDITION

S. There exists a continuous

function

8: rWy + rWy with

p = Qp,, qL, CI~) such that (3.2) and (3.3) hold if qL 2 0 and tlk E [0, l] for i E K. Furthermore

if p,>O,qi>O if p,>O, clj>O

@i(PIYClLIaK)‘o WP,,

for for

iEL ieK

(Sii)

qL9a,) = ~(~PP,,lq,, a&

For any non-circular

system of indexes Condition

PROPOSITION 3.2. Under Assumptions Rl cularity, Condition S is satisfied.

Pi)

S holds.

and R2 and under non-cir-

Proof. Let N(i) c N be the set of commodities such that cpi(p)=cpi(pNci,) and Il/i(p)=$i(pN(ij); hence N(i) is the set of prices on

which ‘pi and tii depend. Define J, = {in N~N(~)cz} .I,= {iENIN(i)cZuJ,} . . . .. . . . .. . . .. . . .. . . .. . . .. . . . . . . . . . . . Jk= {iENJN(i)cZuJ,u ... uJk_,} if ZuJ,u ... uJ,_,#N. ’ For

x, ye R”, x>

y means

xi>

y, for all i=

1, 2, . ... n.

if for

ZuJl#N ksE+R

SUPPLY-CONSTRAINED

209

EQUILIBRIA

3.3. I# 0, J, # $3.

LEMMA

Proof of the Lemma. (i) Suppose I= 0. Take any in N and construct a sequence il, i2, .... iR+ i as follows: for t = 1, 2, .... in, i,, i E N(i,), which is not empty, since Z= 0, so this sequence exists. For s > 0, i, is (indirectly) no ie N can occur in this sequence indexed by i, +s. By non-circularity twice, but since the sequence contains m+ 1 elements, that is impossible, a contradiction. Hence Z # 0. (ii) Suppose J, = 0. Starting with any i, EL u K we can construct a sequence ii, i2, .... iL + R+, , with i, + , E N(i,)\Z. This sequence exists since by hypothesis N(i,)\Z# 0. This gives a contradiction as above. Hence J, # 0.

Given J, # 0, we have either J1 u I = N or J2 # 0, which is proved in the same way as Lemma 3.3. The procedure of constructing Jk ends in at most E + R steps. 0 is obtained by successive substitution, for i E I:

Q,(P,Y qL> aA ‘Pi

foriEJ,nL:

ei(P,>

qL9

UK)

foriEJ,nK:

~;(PIY

qL3

a~)=ei(P19

foriEJ*nL:

~,(p,~ qLy ad = mh

= ~i(P,?

4i)

= cpi(Pl)

“I)=~~CP~(PI)+

+ 4i (1

-a,)

tii(Pl)

qJ,, qiy a,,) = (P~(P,, qJ,, aJ,) + qil

and so on. The remaining properties of Considtion S are easily seen to hold by inspection. Continuity follows from the continuity of the index functions and from the continuity of (3.4) and (3.5). If circularities occur, Condition S need not hold. EXAMPLE.

Let R= 3 and vi(p) = $,(p) for i= 2,3, with Z= { 1). Let P2 = (P2(Pl, P3

=

(P3(PlTP2)

P3)

=Pl

+P3

=Pl

+p2.

It is not possible to find @,(p,). In fact only price vectors pi =O, p2 =p3 satisfy the constraints; hence the constraints also restrict the “free” price pl. Similarly, with only lower bounds,

give 2p, + q2 + q3 = 0, hence p1 = q2 = q3 = 0. This example shows that for Condition S to hold under circularity, additional assumptions on the system of restrictions may be necessary.

210

CLAUSWEDDEPOHL

Below we study the case of restrictions in terms of a general price index, where also prices appearing in the index are restricted, which leads to circularity. A restriction of the sum of lower and upper bounds is the additional assumption needed. In [2,6] such an index was also introduced, but for existence only the case was considered where the index makers were completely fixed and completely free, respectively, thus restoring non-circularity. Let a4=cP,+cP,+cPk I L

(3.6) K

be a general price index (without loss of generality we may assume that all prices are given the same weight). For i E L u K lower bounds pi > 0 are given such that Vi(P)

(3.7)

=_Pin(P)

and for k E K upper bounds jik Qk

are given also, with

$k(P) =F,n(p);

(3.8)

hence (3.4) and (3.5) become

Pl=pinm+cl/> Pk

=

[(

1 -

ak)

pk

+

akPkl

n(p)

1EL

(3.9)

keK.

(3.10)

Note that the above indexes satisfy Assumptions R. Define s(aK)=&h+c

PROPOSITION 3.4.

(3.11)

[(l-ak)pk+akPkl. L

K

Given the system of indexes defined by (3.6), (3.7), and

(3.8). zf ti)

cLpI+

CKpk

<

1

then Condition S holds and tii) tiii)

e,(p~y

qL?

aK)=pl(l

-s(aK))-‘c~pi+

CLqjl

+q!for lELT

ek(~l,~L,aK)=[(1-ak)~k+ak~kl(1-~(aK))-’CIPi+CLq,l

for k EK.

Prooj The derivation of 13~is given in Appendix B. Clearly 9 is continuous. The first part of Condition S holds if and only if (1 - S(a,)) > 0, for all aK, that is, if S(a,) < 1 for aK= (1, 1, .... l), because I)~ @k. (Si) and (Sii) are obvious.

SUPPLY-CONSTRAINED

4.

EQUILIBRIA

211

EQUILIBRIA

Let

denote the exchange economy as defined in Sections 2 and 3, where the constraint functions satisfy Assumptions C and the index functions satisfy Assumptions R. On the consumers we assume ASSUMPTION

A.

For all a E A,

(Al ) X” is closed and convex; X” c rWT; if x E X” and x’ 2 X, then x’ E A-“; (A2) u$’ is in the interior of x”; (A3) kU is complete, transitive, convex, and continuous; (A4) for Y, X” E X”, if xU 1 X” and .uy > X7, then x’ >” .U” (desirability of I-commodities). We study Dreze equilibria in this model. First, a Dreze equilibrium with rationing of both supply and demand is defined (compare 33). Following Van der Laan [S, 61, Kurz [4], and Dehez and Dreze [2], we then define supply-constrained equilibria (where only supply is rationed ). DEFINITION 4.1. A Dreze equilibrium in E is a net trade allocation {z”} (QEA), a‘price PE IWN, and a vector of rationing indexes r E 1w’, such that

(i) d;(h)}; (ii) (iii) (iv) (v) (vi)

-’ + wU is best from {(i” + wU)E X”Ipi” 5 0, V’i: c;(ri) 5 2; 5 * C, 27 5 0; qi(p)spi for itsL~~K;p~S$~(p) for iEK; riO=pi=t+b,(p) for ieK; rj=O for FEZ, TIE [-l,O] for MEL, r,e C-1, l] for iEK.

By (iv) there can be supply rationing only if a price pi (in L u K) is at its minimum, and by (v) demand rationing can occur only if pi (i E K) is at its maximum. By (vi), Z-commodities are not rationed and L-commodities are only supply rationed. (Recall that d;(O) = @.) Assumption C ensures that at most one side of the market is constrained. In fact it is known that under suitable assumptions a non-constrained commodity can be chosen a priori (as in [3]). Note that the set of equilibria may be different for different choices of the unconstrained good.

212

CLAUS

WEDDEPOHL

The model studied in Dreze [3] fits into the present model with I= {0}, the numeraire commodity and all price restrictions expressed in pO. If L = fa and K= 52/ the Dreze equilibrium of Definition 4.1 is a Walras equilibrium. THEOREM 4.2. Under Assumptions A, C, and R and Condition S, and provided that I # 0, a DrPze equilibrium exists.

A sketch of a proof of this theorem is given in Appendix A, using the proof of Theorem 4.4. Theorem 4.2 has not much to add to known theorems in [3], except that it allows for a wide set of price indexes as constraints. If only suppfy rationing is permitted, rationing of only L- and K-commodities (K# 0) may not be sufficient for achieving equilibrium. If there remains excess demand for some K-commodity at its maximum price, and excess supply of all Z-commodities, a decrease of prices of Z-commodities may not reduce supply of the K-commodity, since K-prices follow the price decrease through the index, and hence relative prices do not change. In order to remove excess demand on K-commodities and to prevent the indexes from becoming too low, supply rationing must also be made possible on some commodities with unrestricted prices. This implies that such commodities are supply rationed, although their price is not at its minimum, which is zero. From the set 1, a non-empty set F is split off. Thus the set N is decomposed into the four sets: A4

no supply restrictions, free prices (possibly empty); supply rationing, free prices (non-empty); supply rationing, prices restricted below (possibly empty); supply rationing, prices restricted below and above (non-empty).

F L K

It is not interesting to consider K= 0, since in that case no demand rationing is needed for a Drbze equilibrium satisfying Definition 4.1, and existence follows for Theorem 4.2. DEFINITION 4.3. A supply-constrained Dreze equilibrium in the economy E, for given F c Z, is a net trade allocation { zLI}, a price p E RN, and rationing indexes ri such that

za+w’I

is best from

{(i’+

wa) E XalpiaS

dy(rj)il)

(ii) (iii) (iv)

CA zU5 0; (Pi(p)SpiforiELuK;piSIl/,(p), ri
for icK;

0, Vi: cy(ri) 5 2; 5

SUPPLY-CONSTRAINED

(v)

3feF: r/
(vi)

ri=O,ifiEM;riE[-l,O]ifiEFuLuK.

213

EQUILIBRIA

and rk=O;

Condition (i)-(iv) are as in Definition 4.1. By condition (v) supply rationing of non-indexed commodities in F can occur only if the price of at least one K-commodity is at its maximum. This condition seems quite natural, because the rationing of F has been introduced precisely because of the lack of upward price flexibility of K-commodities. However, van der Laan [6] does not include condition (v) in his definition of an unemployment equilibrium; he proves existence of an equilibrium where it holds. Moreover (v) ensures that the equilibrium is non-trivial even if M = @. Condition (vi) defines the set of feasible rationings and it differs from the condition in Definition 4.1, in that rationing of F-commodities is permitted and demand rationing is excluded, since r, 5 0. It will be shown that a supply-constrained Dreze equilibrium exists if the following assumption holds. E. There exists a subset Fc I such that

ASSUMPTION

(i) (ii)

for ak>O;

p,#O~VkEK:e,(p,,q,,a,)>O,

VUEA: x’~,?’

and x”,>.u$=>x”>,x”.

A set satisfying Assumption E is chosen as the set Fin Definition 4.3. This assumption requires that the upper bound of the price of every K-commodity be positive, if at least one F-commodity has a positive price, and that the F-commodities be desired (in combination, not necessarily separately). Clearly the latter condition ensures that pF cannot be zero, because otherwise excess demand would be positive. Condition (i) may hold because pk depends directly on pF, but the dependence may also be indirect via pK or pL. Note that E(i) is satisfied for the system (3.6), (3.7), (3.8), if I# 0, and we can choose any non-empty Fc Z for which E(ii) holds. If Assumption E holds for the set Z, then choose F = Z, M = @. But if Assumption E holds for Z and there also exists a desirable subset G of Z (for which E(ii) holds) we may also choose F= G, M= Z/G. Particularly, if some fg Z is desired separately (Y 2 .Y, x; > .Y; 3 xU >” X”) and E(i) holds for Z then we may choose F= {f), M= Zj(f}. Note however that, if F satisfies Assumption E, it is in general neither true that every subset F’ c F satisfies Assumption E since (ii) might fail) nor true that any superset F’ 3 F satisfies Assumption E (since (i) might fail). Clearly iE Z, where p, does not occur in any index, cannot be in F. EXAMPLE.

If

7= 3

and

the

upper

bounds

of

pk

satisfy

214

CLAUS

$,Jp,, p2, p3) = a,$, + bk & but F= { 1) does.

WEDDEPOHL

then F= ( 1,2, 3) does not satisfy E(i),

THEOREM 4.4. Under Assumptions R, Al, A2, A3, C, and E and Condition S, a supply-constrained Drtze equilibrium exists.

Proof.

X” - { w” 1 is a’s set of possible net trades. Define the compact set

Z” by

with C; and $ satisfying Assumption C(iv). As noted in Section 2, CTand @ are never binding constraints if x 2: 5 0 (implying that equilibria in Z” are also equilibria in X” - { ~1~) ). The price simplex S is defined by s=

1

(pn,,pF,q[,)EIW/ii+C+t

/ ;p,,+;p,+;q/=

11

(4.2)

and (4.3)

T= [0, 11”

is the set of possible weights c(~, as defined in (3.5): q, is defined by (3.4). rF, rL, and rK denote the vectors of rationing indexes of the three types of commodities. (rM plays no role since it is identically zero.) For all F-commodities a single index r0 is defined, hence rF= (r,,, rO, .... ro). The set of possible rationing schemes is (rg,rL,rg)E[-l,0]1+17+R

r,+Cr,+Cr,?

-(L+K)

It is assumed without loss of generality that the functions c,(r,) and d,(r,) are specified in such a way that riz

-(E+R)/(t+R+

l)*Va:

cp(ri)
(4.5)

which is possible by Proposition 2.1. The restriction on R ensures that it is impossible for all commodities to be rationed in such a way that for some agent CT= 0, for all i. Hence also for M = 0 the “trivial equilibrium,” with ri = - 1, cy = 0 for all iE N and all a E A, is excluded. Agent a’s budget restriction is {z” E Z”lpz 2 0, Vi: cy(r,) 5 z; 5 dy(r;)}.

(4.6)

SUPPLY-CONSTRAINED

215

EQUILIBRIA

Since the price indexes satisfy Assumption R, 8,(p,-, qL, a,) can be substituted in (4.6) for pi (ie L u K), giving a budget correspondence (4.7)

Given the continuity of 0; by Condition S, this correspondence is upper hemi-continuous in each YE Y (by Assumption A2 it is nowhere empty) and lower hemi-continuous in each ~1, satisfying 3ig N:

[p, > 0 and cg(r,) < 01,

(4.8 1

which ensures that B”( 4’) has a non-empty interior (see [3, p. 3041). Our restrictions do not a priori rule out the possibility that for some a, Vi: p, = 0 or cp = 0. Therefore we define a “restricted budget correspondence,” where (4.8) always holds. Define i,: Y -+ [ - I, 0] by i,,( .v) = max { - max G(~,I’() j J,(J)=max{ -1 +q,,r,),, i,(JF)=max(-l+a,,r,)

IEL

(4.9 1

kEK

and define the budget correspondence B” by B”(y) = B”(p,,, P/.., ql., a,, J,,(a,, rd, T*,.(qLl r,-), jK(aK, rK)). (4.10) Hence the budget correspondence is computed after an adjustment of the rationing scheme as defined in (4.9) which by Lemma 4.5 ensures that (4.8) holds. It will be shown that in an equilibrium (z, IX), i(y) is always equal to r, hence B(J) = B( J’). LEMMA

4.5.

For I

as dejined

in (4.9), condition

(4.8) is sati$ed

(1) If pn, # 0, then (4.8) holds for some m, since r,,, = 0. (2) If pM=O, pF=O, qL#O, then for some /EL, q,Ll/L and r^,z -1 +q,L -(i;-- l)/(l+L+R)Z -(e+R)/(l +L+R), and (4.8) holds for f by condition (4.5.). (3) Let p,.> 0, fE F. By Assumption E, $,(p) >O, for all k~ K. Let maxk ak = ~1~.Then either ProoJ

(a) ioL -crnL -(E+R)/(l +E+R), and (4.8) holds forf, or (b) -xG _I -(L+R)/(l+L+K), hence tn 2 -1 +a, 2 -1/(1+L+R)>-(E+R)/(1+~+R)and(4.8)holdsfork,sinceccx>0, hence pk > 0 by Assumption E.

216

CLAUS

WEDDEPOHL

Since (4.8) holds for all YE Y, B” is lower hemi-continuous, hence continuous, and it is convex compact valued. Under Assumptions (Al ), (A2), and (A3), the net trade correspondence z”(y): Y + Z”, with zU(y) = (z” EZ~IV’Z~EB(~):~“+M~~~~Z~+W~)

(4.11)

is convex, compact, non-empty valued, and upper hemi-continuous, by the maximum theorem, and so is the excess demand correspondence z(y): Y-,Z=CZ”, with

We define a correspondence G: Z x Y ---fZ x Y. First note that if Vc R” is compact convex and x E R”, then the correspondence u(x)= {UE VIVEE V:ux~fix) is upper hemi-continuous by the maximum theorem. In order to simplify notation we denote the above correspondence by u(x) = {u E I/( u is a maximizer

of fix}.

The correspondence G is the Cartesian product G = X,4= I G, of the following four correspondences: G,(GY)=z(Y)

(Gl)

G,(z, y) = {(FM, bF, gL) E S\(J?,,,, bF, dL) is a maximizer of diblZM+PFzF+qL(ZL+rL)) G3(z, y) = {OiKe T(oi, is a maximizer

(G2)

of E,(z, + rK)}

G,(z, y) = { FE Rl(r^,, r^,, FK) is a maximizer

(G3)

of

fo( 1 - maxk elk - maxk zk) + 7,z, + ?,z,).

(G4)

Since Z x Y is compact and all G, are upper hemi-continuous correspondences, G has a fixed point (z, y) E G(z, y). It remains to prove that this fixed point is an equilibrium. LEMMA

4.6.

A fixed

point

(z, y) E G(z, y)

is

a supply-constrained

equilibrium. ProofI

Conditions

We prove that conditions (i)-(vi) of Definition (iii) and (vi) hold by the construction of G.

4.3 are satisfied.

SUPPLY-CONSTRAINED

217

EQUILIBRIA

(ii) The properties (a)-(k) are subsequently proved. to(y)=

-1 *zizo

forall

i,(y)=

-1 *z,zo

iEF (4

ieLvK

(with i defined in (4.9)). Vu: c;( - 1) = 0 5 z;, hence Z, = C $‘z 0. -1

z,
=,
(I’EL)

and

l-maxcc,-max~,
cc,=0

(iE K)

lb)

-1.

At least one of the components of (G4) is nonnegative: if max zk < 0 then for each ke K, by (G4), rk 5 0 and by (G3), CY~ = 0, hence l-maxcc,-maxi, > 0. By condition (4.5), L + R rationing indexes can be equal to - 1. Hence (b) follows from (G4). Zk 2 0

(k E K).

(cl

Suppose zk < 0; by (b), rk = - 1, and c1/,= 0; since i, = - 1 + elk = rk = - I, (c) follows from (a). (d) If :k > 0, then rk = 0 (G4) and elk = 1 (G3). ro= -1 = -maxcc,=r^,, hence ~~20 (a). q,=O=>z,~O

By (b), this implies

(IE L).

(e)

Suppose 2, < 0. By (b), r,= - 1, and i, = - 1 + q, = r, hence (e) folows from (a). PlblzM +psp Suppose

(f) does not

hold.

+ qL(zL + rL 15 0. From

pz 20

and

(0 (c) it follows

that

pMzM +pFzF+pLzL 5 0. Subtracting pMzM +pFzf+ qL(zL + I.~) > 0 gives pLzL - qL(zL + r,) -C0. For I EL, we have either (i) q, = 0, implying Z, = 0 (e); this gives p,z, - q,(z, + r,) 2 0; or (ii) q, > 0; this implies (z, + rl) > 0, for by the hypothesis, pizi>O for some ie Mu F, or qi(z,+ ri) >O for some Jo L; since (pM, pF, qL) is a maximizer (G2), z, + r( 2 0 would imply q, = 0. Since (z,+r,j>O, z,ZO (if z,
M2/43/2-2

z, so.

(8)

218

CLAUS

WEDDEPOHL

By (f) and since (P,,,,, pF, qJ is a maximizer (G2), we obtain zis 0 for and (z,+r,)sO for IEL; z,> 0 is impossible, since it would imply by (G4), Y,= 0, and (zl + r,) > 0. ieMuF,

PF#O.

By desirability of F (Assumption contradicts (f). PM~M

l

(h)

E, pF= 0 would imply

-PFZF+PLZL

+PKZK=O

follows from pz _I 0 and (g) and the desirability Zk 5 0,

kEK.

zF> 0, which

ci)

of F. (k)

Suppose zK > 0. By (d), & = 1 and by (h) and Assumption E, pk > 0, hence By (d) and (g), cf;=O, hence p+, = 0. By (c), pKzK 2 0. It follows from (j) that there exists m E M with pmz, < 0 or 1~ L with p,z, < 0, which by (e) implies qlzr < 0 and by (a), qr(z, + Y,) < 0. But since (pM, pF, qJ is a maximizer (G2), with zF = 0, pmz, < 0 and q,(z, + r,) < 0 are impossible. By (g) and (k), (ii) is satisfied.

pkzk>O.

(iv) p, = q,(p). Pk

=

By (G2) and with Z, = 0, Y,< 0 implies q,= 0, which implies Similarly from (G3) it follows for rk
q,(P).

(v) If y0 ~0, then 1 - max ak 50, hence max ak = 1 and then for some k, pk = (I/k(p) and by (G3), rk = 0. (i) It is to be shown that B(y) = 8(~‘), which is equivalent to i(y) = r. For I E L, we have by (G2): q,= 0 or r, = 0. This implies r^,= max( - 1 + q,, r,) = r, (see (4.9)). Similarly, by (G3) for k E K: ak = 0 or rk=O, implying ik=rk. And by (G4): r,=O or (1 -max ak)=O, hence to(y) = ro. This completes the proof of Theorem 4.4.

5. CONCLUSION

By Theorem 4.4 existence of a supply-constrained Drbze equilibrium according to Definition 4.3. was proved, for systems of index functions that permit one to derive the reduced indexes in terms of free variables that satisfy Condition S and Assumption E. This permits as special cases noncircular indexes satisfying Assumption R and price constraints in terms of a general price index, for which (i) of Proposition 3.4 holds. In both cases

SUPPLY-CONSTRAINED

219

EQUILIBRIA

some prices occurring in the indexes must be unconstrained. In an equilibrium where some price attains its upper bound, some commodities with free price may have to be supply-constrained. A priori, all commodities that are not indexed and are not index makers are unconstrained, e.g., money and monetary commodities. Also, any complement in the set of index makers (F-commodities) that satisfy Assumption E can a priori be left unconstrained. All F-commodities may be a simultaneously constrained if at least one index attains its upper bound. Prices of F-commodities must be kept up in order to keep the upper index high, which requires excess supply of F-commodities to be removed by supply constraints (which also removes part of the excess demand of K-commodities). The reason why existence can be proved seems to be the following: if the price of an indexed commodity reaches its upper bound the model works as if the index is inverted; an upper restriction on the price of an indexed commodity is translated into a simultaneous lower restriction on (a part of) the prices that appear in the index and those commodities are accordingly supply-constrained. However, we have considered only the case where F-commodities are supply-constrained. Equilibrium might also be attained in some cases by supply constraints on L- or K-commodities of which the prices are not at their minimum.

APPENDIX

A: SKETCH OF THE PROOF OF THEOREM 4.2.

The theorem can be proved similarly to Theorem 4.4. A price simplex is defined by (4.3) with the provision that p,,, and pF need not be distinguished. A set of possible rationing schemes is, instead of (4.4),

R'=(rL,rK(rL~[O,

l]r,r&-1,

+l]“j

and no sum constraint is necessary, since Fu A4# 0. For ensuring continuity, only the additional restriction needed;

with

y = (pM, pF, qL,

UK,

IL,

on rL of (4.9) is

rK),

i,(y)=max{-l+q,,r,J

(ZGL).

The excess demand correspondence z(y) is upper hemi-continuous (compare (4.12)). The correspondence G is now defined by (Gl ), (G2), (G3) and by G&, .Y)= {in N( F,i,)

is a maximizer

of ?,z, + ?,z,}.

((34')

220

CLAUS WEDDEPOHL

To prove that a fixed point of G is an equilibrium we can use a simplified version of the proof of Lemma 4.6: since there is no sum constraint on r, it follows directly that z,=o

(c’)

qr=O*z,~O

and from monotonicity (h), we obtain

(e’)

(A4), (pF, p,+,) # 0. Taking

into account (f) and

z -I 0. The rest of the proof is similar, including rk>o*Q=

APPENDIX

1 *pk=

that from (G2) and (G4’), Y&I)

(k E K).

B: PROOF OF (ii) AND (iii) OF PROPOSITION 3.4.

by eL, e K, e, vectors with all components equal to unity and with - Denote respectively, and e = (e,, eK). Define v = ( vL, vK) with

L, K, and 7 components,

VL’PL ‘K=

tUk)kcK

and

vk=%#k+(1--k)Pk

then (3.9) and (3.10) can be rewritten in matrix notation (vectors are row vectors if they have the supertix ‘; otherwise, column vectors):

031) For h=L+Klet

h I

Then, for S = e’v,

SUPPLY-CONSTRAINED

221

EQUILIBRIA

where S equals S(a,) as defined in (3.11) u,+l-S

M-‘=(l’(l

u, VI v2+1 -s v, 02 . . . . . . . . . . . . . . . . . . ..*.......

-s))

vh

Vh

vh

...

VI

1’: . . . . . . .Y: ...

v,+l-s

. I

since (with ve’ve’ = Sve’) (I-

ve’)(Z+ (l/( 1 - S)) ve’) =Z-ve’+(l/(l-S))ue’-(l/(1-S))ue’ue’ =Z+(-l+l/(l-S)-S/(1-S))ve’=Z.

The diagonal elements of AC’ elements of row i are vi/( 1 - S). From (Bl) we obtain

are 1 + vi/(1 -S)

and all off-diagonal

U-32)

and taking into account that

and

we obtain (ii) and (iii) of Proposition

3.4.

REFERENCES

1. J. P. BENASSY, Neo-Keynesian Stud. 42 (I 975),

disequilibrium

in a monetary economy, Rev. Econ.

503-523.

2. P. DEHEZ AND J. DRUZE, On supply-constrained equilibria, J. Econ. Theory 33 (1984). 172-182. 3. J. H. DOZE, Existence of exchange equilibrium under price rigidities, ht. Econ. Rev. 16(1975),

301-320.

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WEDDEPOHL

4. M. KURZ, Unemployment equilibrium in an economy with linked prices, J. Econ. Theory 26 (1982), 10&123. 5. G. VAN DER LAAN. Equilibrium under rigid prices with compensation for the consumers. ht. Econ. Rev. 21 (1980). 63-74. 6. G. VAN DER LAAN, Supply constrained fixed price equilibria in monetary economies. J. Math. Econ. 13 (1984), i71-187. 7. C. WEDDEPOHL, Fixed price equilibria in a multifirm model, J. Econ. Theory 29 (1983), 95-108.