Existence and uniqueness of equilibria with increasing returns

Journal

of Mathematical

EXISTENCE

Economics

17 (1988)

149-178.

North-Holland

AND UNIQUENESS OF EQUILIBRIA INCREASING RETURNS

WITH

Kazuya KAMIYA* Osaka University, Osaka 560, Japan Submitted

August

1986, accepted

January

1988

It is well known that firms’ profit maximizing behavior is inconsistent with the existence of equilibria in an economy with increasing returns to scale (or non-convex) technologies. The main purpose of this paper is to show that if firms adopt price setting behaviors then, under certain assumptions, there exists an equilibrium even in an economy with non-convex technologies. In addition, the method of the proof allows us to deduce a uniqueness condition.

1. Introduction

The existence of a competitive equilibrium is proved assuming, in addition to the assumptions on consumers, that firms have convex production sets and that firms are price taking profit maximizers. However, the presence of fixed costs or increasing returns to scale gives rise to non-convex technologies. The main purpose of this paper is to show that if firms adopt price setting behavior, i.e., if they follow pricing rules which are mappings from the boundaries of their production sets to the price simplex, then, under certain assumptions, there exists an equilibrium even in economies with non-convex technologies. Here, the conditions for a (price setting) equilibrium are (i) all firms set the same prices, (ii) consumers maximize their utilities subject to their budget constraints, and (iii) all markets clear. Of course, in general, this is not a competitive equilibrium in that firms need not be maximizing their profits in equilibria, but firms with convex technologies following marginal cost pricing will be maximizing their profits. As for price setting equilibria, the existence of marginal cost pricing equilibria was first studied by Beato (1982) and Mantel (1979); they proved the existence of equilibria in economies with one non-convex technology. Their results were extended by Beato and Mas-Cole11 (1985), Bonnisseau and Cornet (1987), Brown and Heal (1982), Brown et al. (1986), and Cornet *This grateful Herbert Support

paper is based on chapter 2 of my dissertation submitted to Yale University. I am very to Professors Donald Brown, Bernard Cornet, Atsushi Higuchi, Andreu Mas-Colell, Scarf, and anonymous referees of this journal for their helpful comments and advice. from CORE is also gratefully acknowledged.

0304-4068/88/$3.50

0

1988, Elsevier Science Publishers

B.V. (North-Holland)

150

K. Kamiya, Existence and uniqueness of equilibria

(1982). Among them, Bonnisseau and Cornet (1987) proved the most general existence theorem, i.e., they proved the existence of equilibria in an economy with several non-convex firms whose production sets have smooth boundaries. On the other hand, other pricing rules, e.g., average cost pricing; have been studied by Brown and Heal (1983), Dierker et al. (1985), MacKinnon (1979) and Reichert (1986). However, their attention was restricted to special models. Brown and Heal’s (1983) model is a two sector model with average cost pricing firms and Reichert’s (1986) model is a generalized Leontief model with average cost pricing firms. Dierker et al. (1985) proved the existence of equilibria in an economy with competitive firms and non-competitive firms which follow special pricing rules. However, they assumed that the input requirement sets are convex and each price setting firm produces goods which are not produced by the other price setting firms. MacKinnon’s (1979) model is similar to Dierker et al.% model. We will present a general existence theorem. Our model has several nonconvex firms which follow pricing rules, i.e., mappings (correspondences) from the boundaries of the production sets to the price simplex, and guarantees the profit per scale of production to be asymptotically nonnegative. (See A.2 in section 2.) Thus both average cost pricing and mark-up pricing are included in our model. Bonnisseau and Cornet (1988) proved a related result: (i) they showed the existence of equilibria in which demands equal supplies while, in our paper, supplies are larger than or equal to demands, i.e., we prove the existence of free disposal equilibria, (ii) an important class of production sets, e.g., 5 E {(yj, , yjz) E lR2Jyjl 50, yj2 ~(yj1)2}, is included in their model while such production sets are not covered by our model, and (iii) our pricing rule allows for a certain class of unbounded losses while losses must be bounded in their model. It is worth noting that Bonnisseau (1988) showed that Dierker et al.‘s (1985) existence theorem and our theorem result in Bonnisseau and Cornet’s (1988) theorem by changing production sets and pricing rules. Finally, we note that Vohra (1988) presented a simple proof of the existence of equilibria when firms follow lossfree pricing rules. In order to derive a uniqueness condition, Dierker (1972) introduced topological degree into general equilibrium analysis of an exchange economy. However, even in an economy with convex technologies, there are few contributions on uniqueness, e.g., Kehoe (1980, 1983). In an economy with a non-convex technology, only Brown and Heal (1982) studied this problem; they derived a uniqueness condition in an economy with one non-convex marginal cost pricing firm. We present a general uniqueness theorem in an economy with several non-convex firms which follow general pricing rules. This paper is organized as follows. In section 2, we present the model and the existence theorem. Section 3 is devoted to the proof of the existence theorem and to the derivation of the topological property of our model. In

K. Kamiya, Existence and uniqueness of equilibria

151

section 4, this topological property is used to derive a uniqueness condition. In section 5, we discuss pricing rules which are worthy of attention, i.e., marginal (cost) pricing, average cost pricing, and mark-up pricing 2. The model and existence theorem We consider an economy with 1 goods, m consumers, and n firms. Let L, M, and N be the sets {1,. . . , l}, { 1,. . . , m), and {1,. . . , n}, respectively. The jth firm has a production set 5~ R’ and a pricing rule (a correspondence)’ 4j: a 5 -+ S, i.e., each firm sets prices for all production vectors on the boundary of its production set. We denote by WE R’ the total initial endowment. The ith consumer has a consumption set Xi CR’, a utility function u,:X, + R, and a revenue function r,:S x ny= 1 a? + R; each consumer maximizes his utility subject to his budget constraint. We denote by XiE(Xil,..., xil)~Xi, yj=(yji ,..,, yjr)el$ and p=(pl ,..., PJES, a consumption vector of the ith consumer, a production plan of the jth firm, and a price vector, respectively. We use the following assumptions. A.1. (i) For all ie M, Xi is a closed convex subset of R’, containing 0. (ii) For all iE M, ui:Xi + R’ is continuous and, for all xieXi, the set 1s convex and, for all E>O, there exists xi E Xi n {XiEXilUi(Xi)~~i(Xi)} B,(xJ such that u~(xJ > Ui(Xi).

A.2. (i) For all jE N, yj is closed, OE 5, and T- R’+c Yj.2 (ii) For all je N, $j:aYj + S is an upper-hemicontinuous correspondence with non-empty, closed, convex values and satisfies

$5

4jCYq)YJ_ . . -= hm mf{i$)pP$,(y$}tO ((Y& q<

for all sequences

{yg}(y;~aq,llyq))~ + co as 4 + co).

A.3. (i) Y - cj”= 1 I; is closed. (ii) (coA(Y))n(-coA(Y))={O}, w h ere A(Y) is the asymptotic and co A(Y) is the convex hull of A(Y).’

cone of Y

‘S is the unit simplex, i.e., S={p~R’I~,=,p,= 1, p,,zO for all h}. For a set BcR’, dB, int B, and cl B denote its boundary, its interior, and its closure, respectively. Moreover, riS={pE R’l Lb= 1p,, = 1, ph > 0 for all h} and J’S =S\ri S. For x=(x,,), y = (y,,) in R’, the notation x g y means x,,~y,, for all heL and xy denotes Ci=t x,y,. ~~~~~~=rnax,,~.x~~and 11x1=(xx)~” denote the max norm and the Euclidean norm of x. respectively, and, for E> 0, B,(x) = I y E IW’I 1(x- y/l CC}. *We can use Yj#O instead of OE Yj in the proof. However, in this paper, we assume OE I; for the simplicity of the proof. 3(co A( Y)) n (-co A(Y)) = (0) is a stronger assumption than A(Y) n (-A(Y)) = (0) which is commonly assumed in general equilibrium theory. Note that if Y is convex then co A(Y) = A( Y).

152

K. Kamiya, Existence and uniqueness of equilibria

A.4. (i) For all ieM, r,:S x ny=, 8q + R is continuous. (ii) CY=I ri(p, Y) = p(Cj”= 1 yj + w), where Y E (~1,. . . , Y,). (iii) For all ieM and (p, y) ES x n;= 1

p(C3=

1

_Vj

+

W)

>

0

imply

A..?. (i) For all (p, y) E S x

r&4

n,?=

Y)

1 83,

>

~35,

p

E

n;=

1

4j(Yj)

and

0.

p E n;=

1 ~j(yj)

implies

p(cj”,

I yj + w) >

0.

A.2(ii), the most important assumption in our model, says that pricing rules guarantee the profit per scale of production to be asymptotically nonnegative when the scale of production becomes infinity. Our pricing rule is general enough to include not only pricing rules which guarantee nonnegative profit, e.g., average cost pricing, but also pricing rules which allow for relatively small losses. That is (1) if {4j(yj)yjlyje aq) is bounded from below then A.2(ii) holds, and (2) a certain class of (~j, I;-) satisfies A.2(ii) even IS if {4j(Yj)YjlYjEaql . not bounded from below. In section 5, we discuss pricing rules which satisfy A.2(ii). Below we present conditions on a private ownership economy to satisfy A.4. We recall that a private ownership economy is an economy for which ri(p, y) =poi +c;= 1 Bijpyj for all i E M, where wi is the ith consumer’s initial endowment (CL1 wi=m) and Bij is the ith consumer’s share holding in the jth firm (~~= 1 t9,,= 1 for all je N and eijzO for all i E M and j E N). First, if (eij) and (oi) satisfy a fixed structure of revenue, i.e., Oij=ai>O (cr= 1 ai= 1) and oi=aio for all iE M and jE N then ri(p,y) =a,(pw+~~= 1 pyj) holds for all in M and A.4 is satisfied. Next, suppose the government uses lump-sum taxation and follows the rule si = si(po +C;= 1pyj) -((Pw~ + C’j= 1 Oijpyj), where si is the lump-sum tax for the ith consumer, and CT= I pi = 1 and /Ii 20 for all i E M, i.e., the government keeps the ith consumer’s share in the total revenue. Then ri(p, y) = Bi(pW + c;= 1 pyj) for all i E M, hence A.4 is satisfied. A.5, the so-called survival assumption, says that if all firms set the same prices then the total revenue is positive. It is worth noting that if 4j(Yj)YjzO holds for all yj E 85 and all j E N, and Oi E R’, + holds for all i E M then A.4 and A.5 are satisfied in private ownership economies. Definition 1. An (m + n + I)-tuple ((xi*),(yj*>,p*) E nK 1 Xi x ny= 1 5 x S said to be an equilibrium if the following conditions hold.

(i) For all i E M, X: E arg max {Ui(Xi)1Xi E Xi and p*xi 5 ri(p*, y*)}. (ii) For all jEN, y;Eal;. and p*E4j(Yj*)* (iii) C~=ix:-Cj”=ryj*o
is

K. Kamiya, Existence and uniqueness of equilibria

Theorem 1. Proof:

Under A.l-AS,

153

there exists an equilibrium.

See section 3.

Remark 1. In our model, the pricing rule 4j depends only on yj. However, even if 4j depends on (p, y) ES x ny= i aYj, we can prove the existence of equilibria by changing A.2(ii) as follows. A.2. (ii’) For all Jo N, +j:S x fly,= i aI;, -+ S is an upper hemi-continuous correspondence with non-empty, closed, convex values and satisfies

for all sequences{(p4,y4)}((P4,yq) E S x l-I;,= i a?., llyj”llm-+ cc as q + 00). The proof is almost the same as in the proof of Theorem 1, i.e., we can prove the existence of equilibria by substituting 4j(p, y) into ~j(yj) in the proof of Theorem 1. 3. Proofs 3.1. Preliminary lemmas

First we prove that the attainable designate the attainable set by As

((Xi),(yj))E fi Xix i=l

consumption

fJ,q ~ xi~ ~

j=l

i=l

j=

sets are bounded.

We

yj+w 1

and the projections of A on Xi by Xi for all iE M. Lemma 1. Proof.

Under A. 1, A.2, and A.3, zi is bounded for all i E M.

Hurwicz and Reiter (1973, Corollary A’-

((XJ,(Yj))E fi Xix i=l

fi j=l

5

1) showed that

5 Xi= i

i=l

J’j+O

j=l

is bounded under the conditions that (i) A(xr= 1 Xi) n A( Y)= {0}, (ii) A(Cyzi Xi) n(-A(Cr=“=, Xi))=(O), and (iii) A(Y) E( -A( Y)) = (0). (i) follows from Xic[w’+ for all iEM, q-Iw’+cYj for all HEN, and (coA(Y))n

154

K. Kamiya, Existence and uniqueness of equilibria

(-coA(Y))={O}. (ii) fo 11ows from Xi c I@+ for all in M. (iii) follows from (coA(Y))n(-coA(Y))={O}. Suppose Xi is not bounded for some ie M. Then there exists a sequence such that x~ES~ for all in M, y‘j~ Yj for all jeN, {(X4,...,XZ,Yf,...,Y~)~ +c.c as q-00, and ‘j$lx~~~;=,y‘j+o. Let u~=~jnzlyq+wx?llm m i=l XpE[W'+ and j’j = ~4, uj. Then, by A.2(i), jj E Yi and cr’ 1 xf = yl + k the boundedness of ~~=zyj+o holds for all q. This contradicts A’. Q.E.D. For all iE M, we choose a compact set Ai which includes Xi in its interior. We denote Xi= Ai n Xi. For all i E M, let mi(p, y) c arg max {Ui(Xi)( Xi~~i and pxi s rip, y)]. For all iE M, let the ith consumer’s demand be di(P,

Y)

=argmax{Pxil

xi E mi(P,

Y)}

if

ri(P,

Y) >

0,

=(xi:qpxi~ri(p,y)) =

{Ol

if

ri(p, y) =O,

if

ri(p, y) < 0.

and

Let the excess demand be f(p, y) = CT= 1di(P, y) - cj”= 1Yj - CO. Lemma 2 (Walras’ law). Under A.4 and A.5, for (p, y) ES x ny= 1 al;-, p E n;= 1 +j(yj) implies pz SOfor all zEf(p, y). Proof: From A.4 and AS, p E (7;= 1 ~j(yj) implies ri(p, y) >O for all ie N. Then, by the well-known argument [see, for example, ch. 5 in Debreu (1959)], pz50 holds for all ZE f( p,y). Q.E.D. Lemma 3. Under A.1 and A.4, di:S x mzl q -+ I?%’is an upper hemicontinuous correspondence with non-empty, convex, compact values for all iEM. Proof.

An immediate consequence of Lemma 2 in Debreu (1962).

3.2. Function case In order to simplify the proof, we assume that f and 4;s are functions in

K. Kamiya, Existence and uniqueness of equilibria

155

this section. In 3.3, we prove the existence of equilibria when f and ~j’s are correspondences. Lemma 4. Under A.1, A.2, and A.3, (a) int(coA(Y))“#@ and (b) for p” lint (coA( Y))O n S, there exists E>O such that llyjllrn>E and yj~ 85 for some j E N imply p’f(p, y) >0.4 Proof:

(i) By A.3(ii) and a separation argument, int (co A( Y))” #8. (ii) Below we prove (b). Since A(YJcA(~,“,, 5) then (A(Y (A(x;= 1 ?))o = (A( Y))o holds. On the other hand, A(Y) ccoA( Y) holds so that (A( Y))”I>(co A( Y))O holds. Hence (co A( Y))”c (A( 5))” holds so that p” E int(A( 5))” holds for all j E N. (iii) Below, we show that {yj~a~lpoyj20} is bounded for all jEN. Suppose the contrary. Then, for some jE N, there exists a sequence (yjf) such that y:~ay, p”y,‘20, and limq,,llyj411m= +co. Hence l&,,,pOyq/llyiglIa,~O holds. On the other hand, we can choose a subsequence {q’} such that lim,. _ myj”‘/lJy~‘(l mE A(T). Hence km,, _ i. p”y~‘/lly~‘lIm< 0 holds because p’~int (A(q))*. This is a contradiction. Hence aj-sup{p”yjIyjE8~$ is finite for all jgN. We set asmax{a, ,..., a,}. (iv) We show that, for all jEN, there exists kj>O such that ~~~85 and IlYjllm>kj imply p”yj < -pow - (n - 1)a. Suppose the contrary. Then, for someje N and all k >O, there exists yi E a? such that lly~llrn> k and p”y; 2 -p”-w(n1)a. Hence

holds. On the other hand, by choosing a subsequence {k’}, lim,., myj”‘/ IIyr’llmEA( 5) holds so that lim,. _ mp”y!‘/lly3’((, ~0. This is a contradiction. (v) We.set R=max{k,,..., L,,}. If (lyj,jl,zE holds for some j’EN then, from (iii) and (iv),

f di- i yj-0 i=l

2

-PO i

j=l

yj-p”W

j=l

=

-poyj*-po

1

yj-poo

j#j’

&Fora cone Cc KS’,we all YEC}.

denote by

Co the negative

polar

cone of C, i.e., Co = {x E R’lxy40

for

K. Kamiya,Existence and uniqueness of equilibria

156

2 -p”yj,-(n-

l)cc-p”o

s-0

holds. Note that the first inequality follows p” E int ((A( Y))o) n S c I@+ + and dik 0 for all i E M.

from the Q.E.D.

facts

that

We choose an arbitrary K > k and 8,(q) E {z E EK( T) 1 denote d ZE EK( Yj), ZS z and .T,, < zh for all h E L such that z,, > 0}, where E,(Y$)=)r(~+{Ke})

n R’,

and e=(l,...,l). Lemma 5. Under A.3, if rj- R’+ c Yj and OE Yj then fiK( Yj) is homeomorphic to the unit simplex S, i.e., there exists a continuous mapping vl:S --+EK( 5) which is bijective with a continuous inverse. Moreover, we can choose vy:S + 8,(q) such that

yj+Ke

($-'(yj+Ke)=

cfi=l (Yjh+W Proof.

See

(v~)-~(JJ~+

lemma, proof. Note

Lemma 2 in Brown et al. (1984). Note that although Ke)=(yj+ Ke)/cfiE1 (y,+K) was not stated explicitly in the it is easily derived from the construction of vy in the Q.E.D. that

determined uniquely by (v:)-‘(Yj+ Ke)= Furthermore, vf(S’) E {z E I@+(zr = *.. = zI> holds, l/t). Let ~Y=vT(~~)- Ke then Y~EC?%holds and yy does

vr

is

(yj+ Ke)/CL= 1 (yjh + K).

where ho-(l/1,..., not depend on K. Lemma 6.

Under A.3, if $-

l >

R”+c Yj and 0 E Yj then, for 6j”E ZS,

1 ‘---,

1

~~yj~~rn=~fi=~(Yj~+K)=2~~~Yj~~rn

where

Proof. SinceIJYjllm>,K, C:,=1(~jh+K)~Cf,=1(I~jhl+K)~21)lyjll,.

K. Kamiya, Existence and uniqueness of equilibria

157

Hence

On the other hand, if [[yj[[,=Iyj,,l= -yjh then -yj,=K holds. Since 86 n ( - R’+ +)= 8 follows from rj- IX’,c Yj and 0 E rj, there exists h’ E L such that yj,* 2 0. Hence yjh, + K 2 K = - yj,, = yj m holds. If I(yjllm= lyjh(= yjh, then IIyj(lm holds. Hence ckzl (yj,+K)z IIyj IIoDalways holds. Hence =yjh

Q.E.D. Let #(67) z 4j(vr(SjK) - Ke) = 4j(yF). Lemma 7. Under A.2 and A.3, let p” lint (co A( Y))” n S then there exists EzE>O (E was defined in Lemma 4) such that

holdsfor all K>z, Proof

jEN,

DIETS,

and tE[O,l].

Suppose the contrary. Then, for all k>f;,

holds for some K > k, j E N, t E [0, I] and SF E 8’s. Hence (1-t)(s;-s0)(s~-P)+t($f(6~)-p0)(6;-P)=0

(1)

holds. The first term of the above equation is non-negative because

(s:-su)(+so)~‘>o. l(l- 1)

(2)

On the other hand,

=

(4@f, - PO)

yr+Ke

yy+Ke

C:,=l(~~+K)-Cf,=,(yiq,+K)

>

158

K. Kamiya, Existence and uniqueness of equilibria

(3) holds because 1; = 1 $$Sy) = ch = 1pz = 1 and K$je = Kp’e = K hold. By Lemma 6 and A.2(ii), if &I,_, o. $j”(SjK)yy20 then

(4) holds and if l&

_ m #(6y)yr

5 0 then

(5) holds. Note that lim K+ m IIyj”llm= cc because 6jK~a’S. Next, we show

(6) Suppose the contrary then

holds. On the other hand lim K_ coch= 1(yyh+ K) = + co holds. Hence we can choose sequences {Kq} and {yjKq)such that lim,, o. Kq= + 00,

However, since

p” E

int (A( yj>)o then

This is a contradiction.

K. Kamiya,

Existence

and uniqueness

of equilibria

159

Since $x(8:) is bounded and lim, _,mI:= 1(yjoh+ K) = + 03 holds then (7)

(8) From (3) (4), (5), (6), (7), and (8), there exists kI >O such that (CJK(s~)-pO)(+P)>O

(9)

holds for all K > k,, SF E d’s, and j E N. From (2) and (9),

holds (1).

for all Q.E.D.

K >k,,

SUE 8’S,

TV [0, I],

and

jfz N.

This

contradicts

We choose an arbitrary K >k: Let djr6r, c$~-$;, and vi- vr for all jE N. Since CL= 1p,, = 1 and ck= 1dj, = 1 for all j E N hold, we can use T x T” as the domain instead of S x S”, where T E {x E R’; 1 ~~~~‘,x,,~ l}. Let $jz(~jl,...,~jl-l)

for all

jEN,

F-(P,,...,Pr-11,

Jj(sj) E (bj(Zj, 1- CiZ\ 6,)

for all

j E N,

and

l-l

_f(jb%t- @,I-

c

h=l

PhyV1

&,l-

1 6lh)-Ke,..s,v.(&,

11:

Let H”:(T x T”) x [0, l] + [W(“+‘)(‘-r)be

H;;h(d> 8,l) = Ph- ph” for

~EL\(I}

and te[O, 11,

l-x

anh)-Ke).

160

K. Kamiya,

for

Existence

and uniqueness

h~L\{l},jeN,

of equilibria

and CE[O, 11.

We denote by deg(+, D) the degree of a mapping I/KT x T” + IR(‘-~)(“+‘) at 0 in an open set D; for the definition of degree, see, for example, Ortega and Rheinboldt (1970). Note that deg(H”(. ,O), D) = 1 holds for all open sets Dc(T x T”) such that cl Dcint (T x T”) and (fi”,zo) E D, where H”( ., t) is the restriction of the mapping H” to T x T” x {t}. Lemma 8. Under A.2 and A.3, there exists an open set D’c T x T” such that cl D”c int (T x T”) and I?‘(@,8, t) # 0 for all (pj& t) E (T x T”\D”) x [O, 11. Moreover, deg (H”( * , l), D”)= 1. Proof. If fi E dT then H”,(& 8, t) #O. If aj~ aT then, by Lemma 7, there exists h E L such that H‘j,,(fi,5, t) # 0 because

holds. Hence there exists an open set Dac T x T” such that cl D”c int (T x T”) and Ha@, 8, t) #O for all (Aa, t) E(T x T”\D”) x [0, 11. Hence, by the homotopy invariance theorem, deg (H”( . , l), D”)= deg (H”( . , 0), D”)= 1. For the homotopy invariance theorem, see Theorem 6.2.2 in Ortega and Rheinboldt Q.E.D. (1970). We use the following auxiliary assumption. A.6. There is no equilibrium on the boundary of the price simplex, i.e., if z 50 for some z E J(fi, 8) and (fi, 1 -Et:\ ph) E ~j(~j) for all j E N hold then (p”,8) 4 c?T x T”.5 Let Hb:T x T” x [O,l] -P EZW(n+lf(‘-ll be

f%,(A&t) = (1- t)(ph-P,“)+ m,( - .6,(&8)) H$@, $7t) = $jhCsj)- Ph for Lemma 9.

for

h E L\{ I} and t E CO,11,

h E L\{ I}, j E N, and t E [0, 11.

Under A.l, A.2, A.3, A.4, AS, and A.6, there exists an open set

5This assumption is written correspondence case.

in terms

of correspondences

because

it will be used in the

K. Kamiya, Existence and uniqueness of equilibria

161

D*c T x T” such that cl D* tint (T x T”) and H*(p”,8, t) #O for all (p”,8, t) E (T x T”\D*) x [O, 11. Moreover, deg (H*( . , 0), D*) = deg (ZY*( . , l), Db). Proof: (i) If a price vector is on the boundary of the domain and r# (0, l} then it cannot be a solution to H* =O, i.e., (p”,8, t) E(~T x T”) x (0,l) implies H*(i, 8, t) #O. Otherwise ~j(~j) =(fi, cL1ii p,,) -p holds for all j E N then, by A.4, A.5, and Lemma 2, pf(fi,$) j0 (Walras’ law) holds. Moreover,

holds for all h E L\(Z). Hence Q4P,-PP)

holds so that

(1-

Q(Pl_ PP) + tpd - .m

4) 2 0.

Then, for h E Z(p) = (h E L [Ph= 0}, - (1 - t)pi 2 0 holds. This is a contradiction. (ii) If (fi, $0) E dT x T” x (0) then obviously EZ*($,~$0) = 0 cannot hold. (iii) For a sequence {(i4, zq’“, tq)}((jY,sq”,tq) E T x T” x [0,1) and (p”q,~q,tq)+(~,&l)~~TxT”x(l} as q + co), Hb(fiq, sq, t”) = 0 cannot hold for all q. Otherwise, for h E I(p), lim, _ co fh(fiq, 8) 5 0 holds because 14PR(- Jt($

sq)) 2 - ( 1 - t’) (Pi -Pi)

2 0

holds for sufficiently large q, while, for h $ I(p), lim, _ m3h(fiq,sq) SO holds.6 By choosing a subsequence {q’} of {q}, 3&J) 50 holds, where (p”,8)= lim,, _ m(bq’,Jq’). This contradicts A.6, since ~j(~j) =(p”, 1 -cLZir ph) holds for all HEN. (iv) If (fi,& t) E T x i?T” x [0, l] then H*(p, 8, t) =0 cannot hold. Suppose the contrary then

(1- 4(P/l-

Pho)+ lPh( - _ufi, ai) = 0

for all h E L\(l). Hence, by the same argument as in (i),

holds. Hence 61f h=l

then we must use the same argument

as in (i).

162

K. Kamiya,

(1-t)

(

1-E

>

Existence

and uniqueness

of equilibria

+t(-j;(fi,8))~0

holds for all h E L. Hence

(10) Since p-1

and

’

p’=argmin

1 h=l

(PF

__

Ph

’

(l-i1 @p(l-il @f)=o. On the other hand, by Lemma 4, rf= r pi( - f&j, $)) CO holds. Hence, if O
holds. This contradicts (1). If t=O then p=p”, hence Ozp3(/7,8) = p’f(fi, 8) > 0 holds by Lemma 4. This is also a contradiction. (v) By (i), (ii), (iii), (iv), and A.6, we can find an open set Dbc T x T” such that cl D* c int (T x T”) and Hb(p”,& t) #O for all (p”,a, t) E (T x T”\Db) x [0, 11. the homotopy invariance theorem, deg (Hb( *, 0), Db) = Hence, by Q.E.D. deg (Hb( . , l), D*). Proposition equilibrium.

1.

Under

A.1,

A.2,

A.3,

A.4,

A.5,

and

A.6,

there

exists

an

Proof. First we show that deg (H”( . , l), DC)= deg Hb(. , 0), DC), where DC= D” u Db. Obviously deg (H”( *, l), DC)= deg (H”( . , l), D”) and deg (H*(. ,O), DC)= deg(Hb( . , 0), Db). Let

H’(jX 8, t) = (1 - t)H”(fi,

8,1) + tH”(fi,

a, 0).

We can easily check that the solutions to Zf’@,& t) = 0 are exactly those of H”@, 8,l) =0 and Hb(b, &O) =O. Hence there is no solution to H’(p’,8, t) on

K. Kamiya, Existence and uniqueness

163

of equifibria

aD x [0, l] and, by the homotopy invariance theorem, deg(H”( . , l), DC)= deg(Hb( . ,O), DC). Hence, by Lemmas 8 and 9, deg(Hb( . , I), DC)= 1. Hence there exists a solution (fi*, $*) ED” to Hb(. , 1) = 0, i.e., p$3$*, $*) = 0 for all L\(1) and $j(@)=p*z(fi*, 1 -xiZ\ pf) for all Jo N. Hence, by A.4, A.5 and Lemma 2, p3(p”*,9*)50. Since p”*EintT, _&(~*,~*)=O for all ~EL\{Z} and ./G*, 6*) 2 0. Let yj*= Vj(~~’ 1 -ck= 1 d$,) - Ke for all Jo N and x: =di(p*, y*) for all iEM, where y*z(y:,..., y,*). Below we prove that (p*,(xF),(yT)) is an equilibrium. First, (ii) of Definition 1 is obviously satisfied, i.e., ~*=4~(yj*) for all jEN. Next, we show (iii) of Definition 1. First, we prove p*&(p*, y*) =ri(p*, y*) for all i E M. By A.4, A.5 and ~j(yj*) =p* for all j E N, p*di(p*, y*) 5 ri(p*, y*) holds for all iE N. If p*di(p*, y*) ui($) and p*Xi < ri(p*, y*). This is a contradiction and p*di(p*, y*) = ri(p*, y*) holds for all i E M. Hence, by A.4, p*f(p*, y*) = 0. Hence, by fi* E int T and 3$*, $*) = 0 for all h E L\(1), f(p*, y*) = 0 holds. Finally, we prove (i) of Definition 1. Suppose there exists XisXi such that Ui(Xi)> ui(xT) and p*Xi 5 ri(p*, y*). Then, by the continuity of Ui, ri(p*, y*) > 0, OE Xi, and the convexity of Xi,. there exists iiEXi in a neighborhood of ii such that u,(,i-,)> ui(xT) and p*$ < ri(p*, y*). Let x, = (1 - t)xF + tfi. Then U&X,)2 uj(xT) and p*x, < ri(p*, y*) holds for all t E [IO,I], and, since xi‘ E int Xi, x,gXi holds for sufficiently small t. Hence, by A.l(ii), there exists XOE$ such that ui(xo) > ui(xF) and p*x” < ri(p*, y*). This is a contradiction. Q.E.D. We are now ready to prove Theorem 1. Proof of Theorem 1 (function case).

If A.6 holds then, by Proposition 1, there exists an equilibrium. If A.6 does not hold then there exists (@*,$*)E dTxT”such thatp,*3h(fi*,$*)=Oforall hEL\{I} andp*=@*,l-C::\p,*)= ~j(~~) for all js N. Then, by almost the same argument as in Proposition 1, (p*,(xT),(yT)) is an equilibrium, where yj*= ~~($7, 1 -CL:\ 6%) for all jE N and XT=di(p*, y*) for all iE M. Q.E.D. 3.3. Correspondence case

In this section, we prove Theorem 1 without assuming that f and ~j’S are functions. Following Cellina and Lasota (1969), we define the degree of correspondences and compute the degree of our model.’ First, we introduce a definition. For sets B, C~IR(‘-~)(~+‘) and a vector c~lF!(‘-~‘(“‘+~),we let ‘Cellina and Lasota (1969) defined the degree of fixed point mappings However, the same argument is applicable to our case, i.e., the case of general

(correspondences). correspondences.

K. Kamiya, Existence and uniqueness

164

d*(B,

C)E

sup

of equilibria

d(b, C).

bsB

Note that d*(B, C) is not necessarily equal to d*(C, B). Definition 2. Let D c R(‘-“(m+l). If @:D + [WC’l)(m+l) is an upper hemicontinuous correspondence with non-empty, convex, closed values and G(D) is totally bounded then @ is said to be a compact vector field. Definition 3.

Let D c R” - 1’(m+I’ and let @:D -+ iR”- ‘No+‘) be a correspondence. A sequence of correspondences !P:D + R(l-l)(m+l) is said to converge to @ (denoted by lim,, m Qiq= @) if lim d*( G4, G) = 0, 4-+“,

where G4 and G are the graphs of (Pq and @. Definition 4. Let DC RCl- ‘jcrn+” be an open bounded set and &cl D + I@‘- “Cm+” be a compact vector field and O$ @(aD). Let $‘:D + ~TXJ(~-~‘(‘“+~’

be a sequence of single valued vector fields such that lim, _ mtjq = @. We define the topological degree of @ as follows. deg (CD,D) = lim deg ($4, D). q-00

Cellina and Lasota (1969) showed that the above definition of the degree of correspondences is well defined, i.e., (i) given any such @, there exists a sequence of single-valued vector fields $q converging to @, (ii) for q sufficiently large, 0 4 J/,(aD), so that deg($,, D) is defined, and (iii) lim, -+a, deg(II/,, D) exists and does not depend on the choice of the sequence. They also proved the following theorem. Theorem 2. (i) Let Qo, Q1 be homotopic avoiding zero, i.e., there is a family @,(t E [0, 11) of compact vector fields which depend continuously on t (Qtp+ @, in the sense of Definition 3, when tq + t) such that 0 4 @,(aD) for t E [0, 11. Then deg(%,D)=deg(@,,D). (ii) Zfdeg(@, D) #O, then there exists Z~EC~D such that OE @(zJ.

By almost the same argument as in the function case, we can prove the following lemma.

K. Kamiya, Existence and uniqueness of equilibria

165

Lemma 10. Under A.1, A.2, A.3, A.4, A.5, and A.6 there exists an open set D c( T x T”) such that 0 4 H’(p”, 8, t), 0 $ Hb(d, 8, t), and 0 $ H’(fi, 8, t) hold for all (A 8, t) E aD x [O, 11, where H”, Hb, and H’ are defined in section 3.2.

We can now give the proof of Theorem 1. Proof of Theorem

1 (correspondence case). and the above lemma, we can show that

If A.6 holds then, by Theorem 2(i)

1 = deg (H”( . , 0),D) = deg (H”( . , l), D) = deg (H'( . , 0),D) = deg (H’( . , l), D) = deg ( Hb( . , 0), D) = deg (Hb( . , l), D).

Hence, by Theorem 2(ii), there exists (fi*, b*) E cl D such that 0 E Hb@*, 8*, 1). By the same argument as in the function case, we can show the existence of equilibria. If A.6 does not hold then, by using the same argument as in the function Q.E.D. case, there exists an equilibrium. 4. A uniqueness theorem In this section, we present a uniqueness section 3. We use the following assumptions.

theorem

using the results in

A. Z For all j E N, $j:al;. -+ S is a C’ function. f:S x m= 1 iJlj -+ IL!’ is C’ for (p, y) E ri S x ny= 1 a 6 such that p(~~, 1yj + w) > 0. A.8.

(i) For all Jo N, y ={& E R’lgj(yj)~O} with gj : R’ + R continuously differentiable and Vgj(yj) # 0 if gj(y .)= O.* (ii) For all HEN, if (jr,y)~E={(p,y)~Sx fi:=i aYj/f(p,y)=O and 4j(y_J=P for all j E N} then agj(yi>/ayjl # 0. Let Cj(Sj) E(Vjl(Fj, 1-C~~~

6,) ,...,vjl_1(~j,l_Cfi=l18jh))foralljEN,

&, Y) = Cd,CCYy, + Ke), . . . , t; ‘(y, + Ke))

E

(3, F),

“RP, Y) = (fi(P, Y), . . .? f; - 104 Y)), 8Under boundary.

J.Math-C

A.2(i) and A.3(ii), A.S(i) is equivalent

to saying

that

q is a C’ submanifold

with

166

K. Kamiya, Existence and uniqueness

$j(Yj)

Proposition 2.

*3 4jI-I(Yj))

E(+jl(Yj)7..

ofequilibria

for all jeN, and

Under A.I-A. 7 and A.@$, if det (Cc-‘) #O for all (fi, 8) E .!? z b) =O and ~j(~j) = p for all j E N) then

((p, 8) E T x T”( f(fi,

1 signdetD(G<-‘)=l (b,J)E~ holds, where D(Gt-‘) is the Jacobian matrix of G<-’ determinant of D(G<- ‘).

and det D(Gg-‘)

is the

Proof: We showed that Vj is a homeomorphism for all jEN. Let n be rr(z)=z/llzII, where ZE I@+.Let rc’ be the restriction of K to E,(Yj) then n’ is a C’-function in int S. [See Brown and Heal (1982).] Since n’(Yj

vJ’l(yj+Ke)=&

Ke) 1 7C6(yj+Ke) +

holds then vj and v,:’ are C’ functions for all je N. Hence Cc-’ is a C’-function for (p, y) E ri S x g;= 1 ri (8x( rj) - {Ke)) such that p(cj”, 1 yi + o) > 0, where ri (EK( 5) - (I(e)) = (EK( 5) n R’, +)- (Ke} for all j E N. By the proof of Proposition 1 and a well-known property of degree, 1=

deg (Hb(& 8, l), DC)

=,,F $gn(h x

... XP,-~ xdetD(Gc-‘)(fi,$))

9 E

=

,,,Lsign det

D(G<- ‘)(j$).

Q.E.D. By Proposition 2 and the well-known argument by Dierker (1972), det D(Gg-‘)(@,8) >O for all (6, b) EE implies the uniqueness of equilibrium. However, Gt- ‘(p”,6) includes artificial variables and functions, d and cj4 Below we present a uniqueness condition expressed only by f(p, y) and 4j(Yj)‘s.

First, we change the parameterization of a neighborhood of an equilibrium. Let (p, y) E E. We take an e-neighborhood of yj, i.e., U,(yj) - B,(yj) A aI;. Let

K. Kamiya, Existence and uniqueness of equilibria

ij(Yjl,...,Yjt)~(Yjl,...,Yjt-1)~~j

for

161

.iEN.

By A.8(ii) and the implicit function theorem, for j E N, rj: U,(Yj) + RI- ’ is one to one and c,: ’ is C’ for sufficiently small E> 0. Hence i=(C 0,.

. .) in): U,(p)

X fi

U,(Yj)

+

R(‘-l)(m+

‘)

j=l

is one to one and [- ’ is C’ for sufficiently small a>O, where U,(p) z B,(p) n S and co(p) = fi. Since G~-‘=G~-‘~[-’ holds then, by the chain rule, D(G[-‘)= D(G<-‘)D(S[-‘) holds. Hence sign det D(G[-‘) =sign(det D(Gt- ‘) det D(li-‘)). Lemma 11. Proof:

Under A.2 and A.8, det D(<[- ‘) > 0 holds.

See appendix.

By Lemma 11, sign det D(G[- ‘) = sign det D(Gc- ‘) holds. Then, by Proposition 2, the following theorem holds. Theorem 3. Under A.&A.8, if det D(GY-')(p, jj) >O holds for all (p”,y) such that {-‘(jj, j) E E then the equilibrium is unique.

Let J(P, Y)= (fI(P? Y),. . .Tf; - l(P? YN and for all

j E N.

Then

... where D,f is the Jacobian

matrix of 3 with respect to z=j,jjl,.

. . ,y, and

K. Kamiya, Existence and uniqueness of equilibria

168

D,,~i is the Jacobian matrix of ii with respect to Fi for all j E N. Note that D(b[- ‘) does not include ~j(~j). -

5. Pricing rules

In this section, we discuss pricing rules which are worthy of attention, i.e., marginal (cost) pricing, average cost pricing, and mark-up pricing. 5.1. Marginal (cost) pricing

Following Cornet (1982), we define marginal (cost) pricing as follows. Definition 5.

MCj:al;. --t S is called marginal pricing if

where Nr,(yj) is the normal cone to y at yj in the sense of Clarke (1973, i.e., for yjELJI;., N,,(yj)Z{PER’Ipu~O

for all UE T,,(yj)},

where

E (0 E R’I for all sequences tqlO,’ yj”+ yj E cl Yj, there exists a sequence uq -+ u such that, for all 4, yj”+ tquqECI q}. Ty,(Yj)

Proposition 3. Under A.2(i), MCj:aYj + S is an upper hemicontinuous correspondence with non-empty conuex values. Proof.

See Rockafellar (1979).

not satisfy A.2(ii), i.e., Marginal pricing does necessarily lim,,, MCj(y,4)yi’/lly~ll,
Bonnisseau and Cornet (1988) showed that the star-shapedness of 5 leads to the boundedness (from below) of {MC,(y,)y,j y,cz a%} and A.2(ii) holds in this case. Note that it is easy to find an example such that {MCj(yj)yjlyjE aq} is not bounded from below but A.2(ii) is satisfied. 9f 10 denotes a sequence of positive numbers converging to zero.

K. Kamiya, Existence and uniqueness of equilibria

169

o,

Input

Fig. 1

5.2. Average cost pricing The definition of average cost pricing is as follows. Definition 6

ACj(yj):a~

+ S is called

average

cost pricing

if ACj(Yj)E

{PEslPYj=O)* To guarantee the non-emptiness of ACj(yj) we assume 5 n R*+= (0). Proposition 4. A.2(ii). Proof

Under A-2(i) and rj n l@+= {0}, average cost pricing satisfies

Self-explanatory.

We consider an economy d =((Xi, ui, r$‘= i, (5, #j)y= 1, CO)in which 41 is average cost pricing. Suppose there exists an equilibrium in the economy without firm 1, i.e., 6” = ((Xi, ni, ri)y=r, (5, dj)3=2,0)). Then, setting Yi =O, the equilibrium of 8’ is also that of 8, since 41(O) = AC1(0) = S. Hence firm 1 may not produce in equilibria of 6. Below we give a condition which implies that, roughly speaking, y, #O in equilibria. Lemma 12. Suppose A.2(i), a? n (-Ilk?+) = {0}, and 5 n R’+ = (0). Then ACj:a~ * S is a lower hemi-continuous correspondence with non-empty, closed convex values for yj E fYq\(O}. Proof: First, P E ACj(Yj).

(i) Suppose

we

prove

first there

lower

hemi-continuity.

exists h, EL

such that

Let

Yj~aI;.\{O}

and

yjhr #O and p,,, >O. If

K. Kamiya, Existence and uniqueness of equilibria

170

Yjh,>O then, by PYj=O, there exists h,EL such that hz#hl, yjhl0. If yjh, O, and ph2>0. Without loss of generality, we can assume that h, = 1, h, =2, p1 >O, and yj, >O. For a sequence {yj’}(yl E 85, y‘j --) Yj as q -P 00). We set for

Pf’P*

h4(1,2),

if pq E S is guaranteed. Note that, for sufhciently large q, we pick an arbitrary pq E ACj(y3). Obviously, pqyq =O,

S holds. If lim, ~ a, pq =p,

pq E

pq $ S,

and

pqEs.

(ii) Suppose now there is no h E L such that yih#O and p,,>O. Let W(yj)={hE LJyjh=O). Let (~1) b e a sequence such that yJ E aI;. and lim q+myq=yj. If &,.Wtyj)p,,y~O and we set P%, =

1-

1

uqph,

uq=y$,,

where

~1 =uqph

If xhsW(yj)

PhY:

>O

he W(yj),

for

pf =p,,=O

phyyh

I(

hsW(Yj)

for

-h.&)

+

ylh 1

J

,

>

and

h$W(yj) u {h,}.

then, by 3 n IA?+= {0}, we can pick h, EL such that Yjh,CO

and set pff,= 1-

1

vqph,

where

vq= -y$#,

hG;ty,JP&h -

Y;h,

hs W(Yj)

pf=v’p, p#=p,,=O

for for

9 >

hE W(yj), h$W(yj)

and u {h,}.

If Chs W(yj)PhYh 4 -- 0, we set pq = p. Obviously, pqyj = 0, lim, + mpq = p, and pq E S

holds. The non-empty, self-explanatory. Q.E.D.

closed,

convex,

valuedness

of

AC,

is

171

K. Kamiya, Existence and uniqueness of equilibria

We define a modified average cost pricing as follows. ACj(Yj) = ACj(Yj)

=N,,(yj) I? S Lemma 13.

Suppose A.2(i),

for

yjsaq\{O}

for

yj=O.

a? n (- W+)=

and

{o),I$ n Rt+= (O}, NYj(O) n PS= fj

and 85 is smooth at the origin. Then AC, is a lower hemi-continuous correspondence with non-empty, closed, convex values for all yj E a?.. Proof: It is sufficient to prove the lower hemi-continuity of AC, at OE I$ By smoothness of 85 at OE~Y$, ACj(0)= {p}, i.e., a singleton. Let {y;> be a sequence such that y;~aq\{O} and yg--+Oas q --) co. Let pq=argmin,EA~jo ) then there exists a subsequence {q’} of {q} suck q’+ myj”‘/llyj”‘llm = b. Since, by the smoothness of 85 at 0, b is a tangent vector to aYj at 0 and pb=O holds. On the other hand, by NYj(0) n 8’s = 4, b $( - Rf+) and b $ R’, hold. Hence we can define 6‘7’as pq in (i) in the proof of Lemma 12 by substituting b into yj and y$/)lyq’ll, into yg. Hence fiq’yj4’/llyj4’//m =O, ~“‘ES, and lim,, _ mj”‘=p hold. On Ltnt;;ic;tiind, ll~‘,$l~ llpq’-pll so that lim,., a,pq =p holds. This is a . . . Lemma 14 [Michael (1956, Theorem 3.2”)]. Let DC R’-’ be compact and @:D + R’-’ be a lower hemi-continuous correspondence with non-empty, closed, convex values. Then there is a continuous function g:D -+ R’-’ such that g(z)E@(z) for all ZED.

Let

for a;.E i7 (For the definition of Vj, see section 3.) BY Lemma is a lower hemi-continuous correspondence with non-empy, values. Hence we can apply Lemma 14 to A”c. so that continuous function gj: T + T such that gj(~j) ~kj($j) for define I-1

Zj(Yj)

s 2jlvi,'(_Yj+Ke),

. . V,Tll(yj+Ke)), .9

1-

C

h=l

~j~

EACj(yj).

13, ACj: T + T closed, convex there exists a all ~jE 7: We

K. Kamiya, Existence and uniquenessof equilibria

172

If, in the existence proof, we use gj instead of ACj then the possibility of y,=O in equilibria is, roughly speaking, zero because sj(O) is a singleton. 5.3. Mark-up

pricing

First, we discuss the conditions under which mark-up pricing is well defined. Suppose there are two goods in the economy and the output vector is (2,6) and the input vector is (-3, -2). Then the production vector is (2,6)+(-3, -2)=(-1,4). If p=(l, 1) then the mark-up rate is (2+6-3-2)/ (3 +2) = 3/5. On the other hand, the production vector (- 1,4) can be attained by (0,4) and (- l,O), and the mark-up rate is (4- 1)/l = 3 if p=(l, 1). Therefore, it is impossible to define mark-up pricing only by the production vector (- 1,4). The simplest way to overcome this difficulty is to assume that inputs and outputs are different goods, e.g., we do not use oil to produce oil. Another way is to introduce time, e.g., oil in the first period (input) and oil in the second period (output) are different goods. Note that, in this case, the number of periods must be finite in order to guarantee the finiteness of the number of goods. Hence, we assume that inputs and outputs are different goods. Moreover, if y,sO then firm j cannot make profit no matter what the pricing rule is. Thus, in this case, we use marginal (cost) pricing. Mark-up pricing is defined as follows. Definition

Suppose 7. Let O(Yj)- {h E L) yjh> 0) and Z(yj)~{h~LIyj,,0 if MUj(yj)=

PtlYjtl _~hsI(yjbPhyjh=l

1I pES

ChEO(yj)

+a

1

when yjEM1~{yjEal;IO(yj)#~}, =iV,,(yj)nS

when

yjEM,E{yjEdI;.IO(yj)=O,

and there exists E>O such that O(y:) =8 for all y;~ au;. n B,(yj)}, =CO {PESI

there exist sequences {yj}(yfi4~M, u M1, y! + yj as 4 + co) and {pq)(pq~MUj(y$pq+p as q+ co)}, when yjo aq.\M,\M,.

If yj~ Ml, MUj(yj) is defined because there exists h,, h, E L such that Yjh,> 0 and Yjh*
K. Kamiya, Existence and uniqueness of equilibria

Proposition 5. Proof:

Under .4.2(i) and I$ n I?+= (01, MUiaYj~

173

S satisfies A.Yii).

Self-explanatory.

Appendix: Proof of Lemma 11 WtIi;‘) 0

{ 0

detD(t[-‘)=det I

0 ..* *I

... ...

0

... %X1)

0

= jfi det D(tjii ‘)

d :I

Iz-1.

holds. Hence if det D(cj:jrJ:‘) > 0 holds for all j E N then det D(t<- ‘) > 0 holds. Below we prove that det D(O. By the definitions of tj and cj,

holds, where Z~xz:(yj,,+K), Hence

~j~(yjl,...,yjl_l),

and e”=(l,...,l)~R’-l.

sign det D(sj[,~ ‘) =sign det [[Z+ c,T‘(~j) + Km2(‘-‘)

1

-(l+z)(_Vjl+K)

Z+l,;l+K-(l+E)(Yjl+K)

X

-

.*a

JI (1+?& )

.

(Yjz+K)

...

...

...

s sign det [[Z + cJy‘(Jj) + K] - 2(‘- I) X Bj] holds so that

.

.

J

.. .

-(l+E)(Yji+K)

174

K. Kamiya, Existence and uniqueness ofequilibrin

sign det D(ej[,~ ‘) = sign det Bj holds. Note that ac,;‘/dyj,,, the marginal rate of transformation, is nonpositive for all hEL\(I) and jEN. We show (see fig. A.l) that det Bj>O Q.E.D. holds. Consequently, det D(Tj[,: ‘) > 0 holds.

K. Kamiya, Existence and uniqueness of equilibria

& _+ 1% LJ

. ..

.. r= ..

. .. n

. . 9 3

2

+J$ +

‘; e

z+ 3

/

2

s

b

w

2

176

K. Kamiya, Existence and uniqueness of equilibria

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177

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