数理解析研究所講究録 1215 巻 2001 年 1-13

1

On a Problem of Proving the Existence of an
Equilibrium in a Large Economy without Free
Disposal: A problem of a purely finitely additive
measure arising from the Fatou’s lemmain several dimensions
Akira Yamazaki
Graduate Faculty of Economics
Hitotsubashi University
Abstract
The purpose of our paper, however, is to show that the assumption of the ffee
disposability nor the desirability of the commodities is not needed to prove the existence of an equilibrium in alarge economy with acontinuum of economic agents
provided that negative pricqs are allowed and that the preference distribution among
the agents satisfy amild requirement that “if there are unboundedly desirable commodities, they must be unanimously regarded as such by almost all members of theeconomy. ”

1Introduction
It is well known in the literature that the free disposability or the desirability of thecommodities is needed to ensure the existence of an equilibrium in alarge economy withacontinuum of economic agents. This is astark contrast to the case of economies withafinite number of economic agents, where an equilibrium can be shown to exist without
assuming the free disposability nor the desirability of all the commodities provided thatnegative prices are allowed. This difference originates in the fact that feasible allocations
to individuals are bounded by the totally available resources in case of finite economies
whereas in case of economies with an infinite number of agents what each individual canfeasibly consume need not be bounded by the average of the totally available resources.
The purpose of our paper, however, is to show that the assumption of the free disposability nor the desirability of all the commodities is not needed to prove the existence ofan equilibrium even in alarge economy with acontinuum of economic agents provided
that negative prices are allowed and that the preference distribution among the agentssatisfy amild requirement that (‘if there are unboundedly desirable commodities, theymust be unanimously regarded as such by almost all members of the economy.”
In the literature there have been two types of proofs showing the existence of an equilibrium in alarge economy. One is by Aumann [2] and Schmeidler [16] while the othe

2is by Hildenbrand [8]. In both of these types, they first show the existence of equilibria where individual consumption sets are effectively bounded. Then, by considering asequence of equilibria in an economy with bounded consumption sets where bounds areallowed to increase indefinitely, an equilibrium is shown to exist by appealing to strictly
positive limit prices in case of Aumann [2] and Schmeidler [16], and to the Fatous’s lemmain several dimensions in case of Hildenbrand [8].
The strictly positive prices at the limit establish bounds for budget sets along theconvergent subsequence eventually, implying that “bounded partial equilibria” along thesubsequence eventually become equilibria.
However, the very reason that equilibria (without free disposal of commodities) existed
is due to the fact that people did not want to discard any commodities as all the commodities are (unboundedly) desirable. Thus, strictly speaking, from theoretical point ofview, if one assumes the desirability of all the commodities, one cannot answer the question of whether the market price mechanism can coordinate supply and demand when thedisposal activity of commodities is costly. The point here is that one needs to establishthat market prices can indeed coordinate market forces of supply and demand even if
unwanted commodities cannot be discarded freely in alarge economy.
Once the assumption that all the commodities are desirable is dropped, we are in the
set-up of Hildenbrand [8]. Adifficulty arises in applying the Fatou’s lemma in several
dimensions. This lemma is applied to obtain at alimit point an equilibrium with freedisposal. In this step one cannot hope to obtain an equilibrium with exact feasibility
unless the result of the Fatou’s lemma in several dimensions is strengthened.
Before we give astatement of the main result of this paper, we would like to take time
to explain in more detail the nature of the difficulty in providing an equilibrium existence
result in alarge economy.

2AProblem of aPurely Finitely Additive Measure
Arising from the Fatou’s Lemma in Several Dimen$\mathrm{s}$

2.1

ome
Two typical types of existence proofs

Let us briefly explain some basic features of existing existence proofs in large economies
with ameasure space of economic population so that difficulties in providing an existence proof of an equilibrium without assuming the free disposability of commodities nor
assuming all the commodities to be unboundedly desirable.
Let us consider alarge economy :with an atomless measure
$(A,
A,
\nu)$
space of economic population given bywith $\nu(A)=1$ . is ameasurable map,$\mathcal{E}$$(A, A, \nu)arrow \mathcal{P}\cross \mathbb{R}^{\ell}$$\mathcal{E}$and$\mathcal{E}(a)=(\succ_{a}, e(a))$

with $e(a)\geqq 0,0<[ed\nu$ $<\infty$ and apreference$\mathrm{r}\mathrm{e}1\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\succ_{a}$on3$X_{a}\equiv \mathbb{R}_{+}^{\ell}$which is continuous, $i.e$ . open in$X_{a}\cross X_{a}$, and negatively transitive, $i.e$ . $z\neq_{a}x$

if $z\mu_{a}y$ and $y\neq_{a}x$ . In an integral we will often omit the symbol

and write$d\nu$$\int f$is locally nonsatiated if for any $x\in X_{a}$ and. Apreference
instead offor any neighborhood of there is $z\in U$ such that $x\succ_{a}x$ . Apreference relation
andis said to be monotone or desirable if for any
defined on
and any $t>0$ one has
one has $x+v\succ_{a}x$ . Commodity is desirable if for any
is the vector with 1in the -th place. If aU the commodities$x+tu_{j}\succ_{a}x$ whereare desirable, then preference relation is monotone.
. in . It
is an integrable function such that
An allocation :, and exactly feasible if $\int f=\int e$ .
is feasible if$\int fd\nu$$\mathrm{r}\mathrm{e}1\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\succ_{a}$$U$$\succ_{a}$$x$$v\in \mathbb{R}_{+}^{\ell}\backslash \{0\}$$x\in \mathbb{R}_{+}^{\ell}$$\mathbb{R}_{+}^{\ell}$$x\in \mathbb{R}_{+}^{\ell}$

$j$

$j$

$u_{j}\in \mathbb{R}^{\ell}$$f$$f(a)\in X_{a}\mathrm{a}.\mathrm{e}$$Aarrow \mathbb{R}^{\ell}$$A$$\int f\leqq\int e$such that $p\neq 0$ .
Aprice vector is amember ofAn equilibrium for is apair $(p, f)$ consisting of aprice vector
such that
allocation :$\mathbb{R}^{\ell}$$p$$\mathcal{E}$$f$$p\in \mathbb{R}^{\ell}\backslash \{0\}$, and an$Aarrow \mathbb{R}^{\ell}$

1. $f(a)\in B(a,p)$ and. $a\in A$ , where
and, for each $y\in X_{a},$ $\succ_{a}(y)\equiv\{z\in X_{a}|z\succ_{a}y\}$ .

$B(a,p)\cap\succ_{a}(/(\mathrm{a}))=\emptyset \mathrm{a}.\mathrm{e}$$B(a,p)\equiv\{z\in X_{a}|p\cdot z\leqq \mathrm{p}- \mathrm{e}(\mathrm{a})\}$

2. $\int f=\int e$
In the above definition, if one has $\int f\leqq\int e$ , but $\int f\neq\int e$ , then the pair $(p, f)$ is calledan equilibrium with free disposal.

Define, for each positive integer $k=1,2$ ,$B^{k}(a,p)$$\equiv$, -bounded budget sets by$k$$\ldots$$\{z\in X_{a}|p\cdot z\leqq p\cdot e(a)\}\cap\{x\in \mathbb{R}_{+}^{\ell}|x\leqq k(1+\sum_{i=1}^{\ell}e^{i}(a))u\}$

(2.1)

, 1).
where is the vector (1,, and an allocation is ak-bounded$(p,f)$
consisting of aprice vector
Apairpartial equilibrium for if it is defined with respect to $B^{k}(a,p)$ instead of $B(a,p)$ .
There are two types of existing fundamental results on the existence of equilibrium inthe literature. One is by Aumann [2] and by Schmeidler [16], and the other by Hildenbrand
[8].
$u$

$\ldots$$f$$p\in \mathbb{R}^{\ell}\backslash \{0\}$$\mathcal{E}$

THEOREM by Aumann and Schmeidler (Aumann [2] and Schmeidler [16]): Given. $a\in A,$
an economy , assume,
satisfies the desirability of all commodities. Then,.
an equilibrium $(p, f)$ exists with$\mathcal{E}$$\mathrm{a}.\mathrm{e}$$\succ_{a}$$p\in \mathbb{R}_{++}^{\ell}$THEOREM by Hildenbrand (Hildenbrand [8]):.
$(p, f)$ with free disposal exists where$p\in \mathbb{R}_{+}^{\ell}\backslash \{0\}$Given an economy , an equilibrium$\mathcal{E}$4
We lke to comment in the next subsection on some basic features of the proofs by theabove papers in order to understand the nature of the problem at hand.

Basic features of existing existence proofs

2,2

Let us give abrief description of each of the proofs by Schmeidler and Hildenbrand below.
2.2.1

Proof by Schmeidler

Step SHI:$p_{k}\in \mathbb{R}_{+}^{\ell}\backslash \{0\}$Given an economy , a -bounded partial equilibrium
for each integer $k>1$ .
$k$$\mathcal{E}$

Step S2: One takes aconvergent subsequence
all commodities implies.

$p_{k}arrow p$$(p_{k}, f_{k})$exists withand show that the desirability of$p\in \mathbb{R}_{++}^{\ell}$

Step S3: The last step is to show that the sequence of -bounded partial equilibria
eventually becomes an equilibrium along the subsequence.
$k$$(p_{k}, f_{k})$2.2.2Proof by Hildenbrand

Step SHI: Given an economy , a -bounded partial equilibrium
disposal exists withfor each integer $k>1$ .
$\mathcal{E}$$k$$(p_{k}, f_{k})$with free$p_{k}\in \mathbb{R}_{+}^{\ell}\backslash \{0\}$Given asequence of -bounded partial equilibria with free disposal,,, it is shown that the Fatou Lemma in several dimensions implies theexistence of an integrable function :such that is alimit point of.
$a\in A$ ancl.
Step H2:$k$$(p_{k}, f_{k})$$fs$$p_{k}\in \mathbb{R}_{+}^{\ell}\backslash \{0\}$$f$$Aarrow \mathbb{R}^{\ell}$$f$$f_{k}.(a)\mathrm{a}.\mathrm{e}$$\int f\leqq\int e$

Step H3: It is shown that apair $(p, f)$ with , alimit point of the sequenceequilibrium with free disposal.
$p$

2.2.3

, is anFatou’s Lemma in Several Dimensions

Fatou’s Lemma in Several Dimensions: Let . :$f_{k}$

grable and$\lim_{k}\int f_{k}$$(A, A, \nu)arrow \mathbb{R}_{+}^{\ell}$

1. $f(a)$ is a limit point of the sequence$[f$, $k=1,2$ ,exists. Then, there exists an integrable function

that

2.

$\{p_{k}\}$$\leqq \mathrm{h}.\mathrm{m}_{k}[f_{k}$.

$\{f_{k}(a)\}$, $a.e$ . in

$A$

;

$f$

:

$\ldots$$f$

be inte-$Aarrow \mathbb{R}_{+}^{\ell}$

such

5
Aproof of this lemma first appeared is by Schmeidler [17]. To understand amathematical difficulty involved in applying the lemma to the sequence of -bounded partial
equilibrium to obtain an equilibrium at alimit as in the proof in 2.2.2, we like to show
next the steps of the proof by Hildenbrand and Mertens [10].
$\mathrm{A}\ovalbox{\tt\small REJECT}$

Proof by Hidenbrand and Mertens:
Define $\mu_{k}(E)=\int_{E}f_{k}d\nu$ for $E\in A$ and each $k=1$ , 2,

Step 1:$\ldots$. Then,$\mu_{k}\in \mathrm{b}\mathrm{a}^{\ell}$, theis bounded, by-fold product of bounded additive measures on $(A, A)$ . Sincehas-compact. Thus,is relative
the Theorem of Alaoglu.
-accumulation point$\{\mu_{k}(A)\}_{k}$$\ell$$\{\mu_{k}(A)\}_{k}$$\sigma^{\ell}(ba, L_{\infty})$$\{\mu_{1}, \mu_{2}, \ldots\}$$\mu\in \mathrm{b}\mathrm{a}^{\ell}$$\sigma^{\ell}(ba, L_{\infty})$

Step 2: By the Theorem of Yoshida-Hewitt
away that it can be written as,$\mu_{p}$can be decomposed into two parts in such,is purely finitely additive,$\mu_{c}$$\mu=\mu_{c}+\mu_{\mathrm{p}}\mu_{c}\in \mathrm{c}\mathrm{a}^{\ell}’$$\mu$$\mu_{p}\geqq 0$(2.2)

is the Mold product of countably additive measures on $(A, A)$ .
where
Take aRadon-Nikodym derivative of with respect to . Then, one has$\mathrm{c}\mathrm{a}^{\ell}$$\ell$$g$$\nu$$\mu_{c}$$\int g=\mu_{c}(A)\leqq\mu(A)=\lim_{k}\int f_{k}$.

It follows that$\int g\leqq\lim_{k}\int f_{k}=\int e$.

However, at this stage one cannot say that $g(a)$ is alimit point of $\{f_{k}(a)\}$ ,order to achieve this one needs one more step.

,, , with
One can show that there are, , each in the set $\{f_{k}(a), f_{k+1}(a), \ldots\}$ satisfying, $i=0$ ,Step 3:$y_{k}^{i}$$\delta_{k}^{i}\geqq 0,i=0$$\ell$$\ldots$$\ell$$\ldots$$g(a)$$=$$\lim_{n}\sum_{i=0}^{\ell}\delta_{k_{\mathfrak{n}}}^{i}y_{k_{n}}^{i}$$=$$\sum_{i=0}^{\ell}\lim_{n}\delta_{k_{n}}^{i}y_{k_{n}}^{i}\geqq$$\delta^{\mathrm{i}}$$\lim_{n,i_{-}^{-}0}y_{k_{n}}^{i}$$. \sum_{\delta^{*}>0}^{\ell},$

where$\delta^{i}=\lim_{n}\delta_{k_{n}}^{i}.$.

$\mathrm{a}.\mathrm{e}$. in $A$ . In$\sum_{i=0}^{\ell}\delta_{k}^{i}=1$

, and

6
2,3

Aproblem of apurely finitely additive measure arising fromthe Fatou’s lemma in several dimensions

Aproblem in proving the existence of an equilibrium in alarge economy may clearly beunderstood by looking at the steps of typical proofs provided in above subsections 2.2.1,
2.2.2, and 2.2.3. The first proof by Aumann [2] and asubsequent proof by Schmeidler [16]as illustrated by the steps in 2.2.1 relies on the assumption that all the commodities are(unboundedly) desirable. This assumption was essential in establishing Step S2 where itis shown that there is asubsequence of -bounded partial equilibria whose prices converge
to strictly positive prices. The strictly positive prices at the limit establish bounds forbudget sets along the convergent subsequence eventually, implying that -bounded partial
equilibria along the subsequence eventually become equilibria.
However, the very reason that equilibria (without free disposal of commodities) existed
is due to the fact that people did not want to discard any commodities as all the commodities are (unboundedly) desirable. Thus, strictly speaking, ffom theoretical point ofview, if one assumes the desirability of all the commodities, one cannot answer the question of whether the market price mechanism can coordinate supply and demand when thedisposal activity of commodities is costly. The point here is that one needs to establishthat market prices can indeed coordinate market forces of supply and demand even if
unwanted commodities cannot be discarded freely in alarge economy.
Once the assumption that all the commodities are desirable is dropped, we are in the
set-up of Hildenbrand [8] except that his model is that of aproduction economy. In thatframework one could establish the existence of a -bounded partial equilbrium without
free disposal because budget sets are bounded. Adifficulty arises in the last step where
he applies the Fatou’s lemma in several dimensions. His method of proof is to apply
the Fatou’s lemma in several dimensions to asequence of -bounded partial equilibria toobtain at alimit point an equilibrium with free disposal. In this step one cannot hopeto obtain an equilibrium with exact feasibility unless the result of the Fatou’s lemma inseveral dimensions is strengthened. More precisely, in the statement of the lemma inthe subsection 2.2.3, the inequality in the second condition needs to be strengthened toequality.
In following the steps of the proof by Hildenbrand and Mertens [10], one sees that there
appear to be two sources of this inequality. One is in Step 2and apurely finitely additive
part of the weak limit of the sequence of bounded measures generated by the sequence ofallocations associated with -bounded partial equilibria gives rise to this inequality. The
second source is in Step 3where one ignores the terms with coefficients that go to zerothat in turn imply that corresponding’s might be unbounded. However it may appear
that these two sources are independent, it all boils down to anon-vanishing purely finitely
additive part in Step 2of the proof of the Fatou’s lemma. Aproblem caused by it is thatit could correspond to circumstances where asequence of groups of agents with declining
weights down to null are assigned the commodity vectors
which are unboundedalong the sequence of -bounded partial equilibria. This type of phenomenon will not
$k$

$k$

$k$

$k$

$k$

$y_{k_{n}}^{\dot{l}}$$f_{k}(a)$$k$7axise when all the commodities are assumed to be (unboundedly) desirable since strictly
positive prices bound budget sets of agents. Nevertheless, on an intuitive basis it shouldbe also clear that if everyone thinks commodities are not unboundedly desirable, thisphenomenon cannot occur. We shall give aformal statement of this intuition in the next
section.

3Statement of the Main Theorem
In order to obtain an equilibrium existence result without free disposal we shall introduce
an assumption which requires unanimous perception among economic agents as to which
commodities, if they exist, are unboundedly desirable.
be
Let $J=\{1, \ldots, \ell\}$ be the set of indices of all commodities, and$\kappa$ofnumber sufficiently large. Define asubset$a\inA$
commodities for agent , that is, for each
$J_{\succ_{a}}^{+}$$J$$> \sum_{i=1}^{\ell}\int e^{:}$$\mathrm{a}$consiting of “unboundedly desirable$a$$J_{\succ_{a}}^{+}=\{j\in J| (\forall x\in X_{a} :x^{j}>\kappa)(\exists t>0)x+tu_{j}\succ_{a}x\}$.

According to this definition, commodity is an unboundedly desirable commodity for, if, whenever aconsumption of commodity exceeds acertain
agent , $i.e$ .
amount, regardless of how much the agent already consumes that amount there always isafurther increase of that commodity consumption that will be preferred by the agent.
If all the commodities are desirable, then preference relation is monotone. Adesirable
commodity is unboundedly desirable but not vice versa. In this paper we do not requiremonotonicity of preferences. In fact, it is not necessary that even one desirable commodity exists. Instead, what we require is that if, in fact, for one agent acommodity is
unboundedly desirable, then this perception must be unanimously held by all the agents.
$j$

$a$

$j$

$j\in J_{\succ_{a}}^{+}$

Assumption [Unanimous Perception of Unboundedly Desirable Commodities (UP. $a\in A$
of , possibly empty, such that
UDC)]: There exists asubset$J^{+}$$J$$\mathrm{a}.\mathrm{e}$

, and

1.

$J_{\succ_{a}}^{+}=J^{+}$

2.

$(\forall x\in X_{a})x\neq_{a}x^{\kappa}(J\backslash J^{+})$$(x^{\kappa}(J \backslash J^{+}))^{i}=\min\{x^{i}, \kappa\}$

where, for any
, and
for

$x\in \mathbb{R}_{+}^{\ell}$$i\in J\backslash J^{+}$,$x^{\kappa}(J\backslash J^{+})$

is defined by
for $i\in J^{+}$ .

$(x^{\kappa}(J\backslash J^{+}))^{i}=x^{i}$

The above assumption of Unanimous Perception of Unboundedly Desirable Commodities (UPUDC) says that almost every agent in the economy unanimously agrees on which
commodities, if any, are unboundedly desirable. The assumption is stated in two parts
to express that either almost everyone wants acommodity unboundedly or there is aunanimously perceived limit as to how much each agent wants that commodity. Mathematically speaking, the second requirement can be weakened. That is, instead of requirin$\mathrm{g}$8the unanimous perception of almit g of the commodities that are not unboundedly desirable, this lmit can depend on each agent a as long as
as afunction from A into
\yen is integrable.
We now give astatement of our main theorem.
$\ovalbox{\tt\small REJECT}(\ovalbox{\tt\small REJECT})$

Theorem 1Given an economy , an equilibrium $(p, f)$ exists with
providedthat the preference distribution of satisfies the assumption of Unanimous Perception of
Unboundedly Desirable Commodities.
$\mathcal{E}$$p\in \mathbb{R}^{\ell}\backslash \{0\}$$\mathcal{E}$Our main theorem confirms our intuition that market prices can indeed coordinate
market forces of supply and demand even if unwanted commodities cannot be discarded
ffeely in alarge economy if everyone agrees on as to which commodities are desirable
without any bounds. Amathematical difficulty of apurely finitely additive measure arising ffom the Fatou’s lemma is avoided because the unanimous perception of agents as towhich commodities are unboundedly desirable induces either strictly positive prices forunboundedly desirable commodities that in turn lmit the consumption of those commodities by agents, or agents themselves want not to consume commodities without limits.

4ProofDefine$K(a)$$=$$\{x\in \mathbb{R}_{+}^{\ell}|x\leqq k(1+\sum_{\dot{l}=1}^{\ell}e^{:}(a))u\}$$K=$$\{x\in \mathbb{R}_{+}^{\ell}|x\leqq k(1+\sum_{\dot{\iota}=1}^{\ell}\int e^{i})u\}$

,

,
(4.3)

$P=$
$D_{<}^{k}(a,p)$$\{p\in \mathbb{R}^{\ell}|\sum_{=1}^{\ell}|p^{\dot{l}}|=1\}$,$=\{x\in B(a,p)|(\forall z\in B_{<}(a,p))z\neq_{a}x\}\cap K(a)$.

The first step is to follow the standard proofs as in [2], [8], [16], and show the existenceof an (quasi-)equilibrium in -bounded economies. The only difference from these proofs
is that negative prices are allowed so that prices can coordinate to achieve exact equality
between demands and supplies even if some of the commodities are unwanted, $i.e$ . “bads”,
as in the case of the proofs of existence of an equilibrium (without free disposal nor
monotonicity) in economies with afinite number of agents (see [12], [13],[7], [3], [15]).
For this purpose we shall define the correspondences : $Karrow P$ , : $Parrow K$ , and$\varphi:P\cross Karrow P\cross K$ by
$k$

$\pi$

$\varphi$9$\pi(x)$$=$$\{p\in P|(\forall q\in P)p\cdot(x-\int e)\geqq q\cdot$$\varphi(p)$$=$$\{x\in \mathbb{R}_{+}^{\ell}|$( an integrable function $f$ :$\exists$and$\Psi(p,x)$$(x- \int e)\}$ ,$=\pi(x)\cross\varphi(p)$$Aarrow \mathbb{R}_{+}^{\ell}$) $x= \int f$(4.4)$f(a)\in D_{<}^{k}(a,p)\mathrm{a}.\mathrm{e}$. $a\in A\}$ ,.

The correspondence is well defined and satisfies the conditions of Kakutanf fixed
point theorem (see, $e.g.$ , [2] pp.8-10, [16] pp.581-582, or [18] pp.550-551 ). So, let $(p,x)\in$$\Psi(p, x)$ .
such that $x= \int f$ , andSince $x\in\varphi(p)$ , there is an alocation :$\Psi$$f$$\mathrm{f}(\mathrm{a})\in D_{<}^{k}(a,p)\mathrm{a}.\mathrm{e}$.

$a\in A$ .

$Aarrow \mathbb{R}_{+}^{\ell}$

We shall show that the allocation$f$is exactly feasible.

Suppose we had $\int f-\int e\neq 0$ . Since $x= \int f\in\varphi(p)$ , it follows that$p\cdot f(a)\leqq p\cdot e(a)$$p \in\pi(\int f)$$\mathrm{a}.\mathrm{e}$.

$a\in A$ ,. Thus, we have $p \cdot(\int f-\int e)\leqq 0$ . On the other hand, since, we would have,$p \cdot(\int f-\int e)\geqq\frac{1}{\sum_{i=1}^{\ell}|\int f^{i}-\int e^{i}|}(\int f-\int e)\cdot(\int f-\int e)>0$

which is acontradiction. Therefore, must be exactly feasible.
Thus, we have established that for each integer $k\geqq 1$ there exists$f$$(p_{k}, f_{k})$satisfying

1. $p_{k}\in P$ ,2.

$f(a)\in D_{<}^{k}(a,p_{k})\mathrm{a}.\mathrm{e}$3.

$\int f_{k^{\sim}}=\int e$. $a\in A$ , and.

. We canis compact, there is aconvergent subsequence of the sequence
Since
assume without loss of generality that the sequence itself is convergent so that $p_{k}arrow p$ .
eventually becomes an equilibrium.
We shall prove that the sequenceFirst let$P$$\{p_{k}\}$$(p_{k}, f_{k})$$W_{p}= \{a\in A|p\cdot e(a)>\inf p\cdot X_{a}\}$If$(\exists j\in J)p^{\mathrm{J}}<0$.

, then $W_{p}=A$ . If not, $p\geqq 0$ and thus $\int e>0$ iimmpplliieess $p \cdot\int e>0$ , which

in turn implies $\nu(W_{p})>0$ . Define, then, an integer$b$and the set $B$ by

10

b

B

\yen
$>$

$\mathrm{p}(\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT}\ovalbox{\tt\small REJECT} \mathrm{p})\ovalbox{\tt\small REJECT} 7$$\mathrm{Y}^{\ovalbox{\tt\small REJECT}_{\ovalbox{\tt\small REJECT} \mathrm{n}\mathrm{d}}}${z$\ovalbox{\tt\small REJECT}$|r$\mathrm{c}_{\ovalbox{\tt\small REJECT}}\mathrm{R}\ovalbox{\tt\small REJECT}.$$\ovalbox{\tt\small REJECT}$

(4.5)

h}.

of strictly positive measure having the property that$f_{k}(a)\in B$ for infinitely many
(4.6),
. $a\in U$.

Lemma 1There is a subset$U\subset W_{p}$

$k$

Proof
Thus,

If not, for$\mathrm{a}.\mathrm{e}$$\sum_{i=1}^{\ell}f_{k}^{i}(a)>b$.

$a\in W_{p}$

,

$a.e$

$(\exists k_{0})k>k_{0}\Rightarrow f_{k}(a)\not\in B$. It follows that

. So, for such$\lim\inf_{k}\sum_{i=1}^{\ell}f_{k}^{\dot{l}}(a)\geqq b$, $(\exists j)f_{k}^{j}(a)>b$ .
and this implies$k$.

$\int_{W_{p}}\mathrm{h}.\mathrm{m}_{k}\inf\sum_{\dot{*}=1}^{\ell}f_{k}^{\dot{l}}(a)\geqq b\nu(W_{p})$Then, by Fatou’s lemma we obtain$\int_{W_{p}}\lim_{k}\inf \mathrm{I}$$f_{k}\dot{.}(a)$$\leqq$$\lim_{k}\inf\int_{W_{p}}\dot{.}\sum_{=1}^{\ell}f_{k}\dot{.}(a)$$\leqq$$\lim_{k}\inf\int\dot{.}\sum_{=1}^{\ell}f_{k}\dot{.}(a)$

$=$

(4.7)

.

$\lim_{k}\inf\int\sum_{\dot{*}=1}^{\ell}e^{\dot{l}}(a)=\int\sum_{=1}^{\ell}e^{:}(a)$Therefore, we must have$b \nu(W_{p})\leqq\int.\cdot\sum_{=1}^{\ell}e^{:}(a)<b\nu(W_{p})$acontradiction. This establishes the above claim.

,$\bullet$Next, we prove the following lemma.

Lemma 2For every$j\in J_{f}^{+}$

we have $p^{j}>0$ .

has a
Proof By lemma 1, the compactness of implies that, for . $a\in U$ ,$f_{k}(a)arrow
y$
Then,
$y\in
B$
.
. Taking asubsequence if necessary, one can assume
limit point$B$$\{f_{k}(a)\}$$\mathrm{a}.\mathrm{e}$$p \cdot y=\lim_{k}p_{k}\cdot f_{k}(a)=\lim_{k}p_{k}\cdot e(a)=p\cdot e(a)$

.

11
,
. $a\in A$ , by thefor $j\in J^{+}$ . Then, sinceNow, suppose we had
assumption of unanimous perception of imboimdedly desirable commodities,$p^{j}\leqq 0$$j\in J^{+}=J_{\succ_{t}}^{+}$$(\exists z\in \mathbb{R}_{+}^{\ell})z^{j}>\kappa$,$z^{i}=y\dot{.}$for $i\neq j$ , and$z\succ_{a}y$.

$y=p\cdot$ $e(a)$ . $a\in W_{p}$ implies that there is
For this , we havethat $p\cdot z_{0}<p\cdot e(a)$ . So, define for $n=1,2$ ,$p\cdot z\leqq p\cdot$$z$$\mathrm{a}.\mathrm{e}$$z_{0}\in X_{a}=\mathbb{R}_{+}^{\ell}$such$\ldots$$z_{n}= \frac{1}{n}z_{0}+\frac{n-1}{n}z$.

be such that
. For each let$p_{k}\cdot z_{n}<p_{k}\cdot e(a)$ for $k\geqq k_{1}(n)$ .
$z\succ_{a}y$such that,
integerthereisan
As we havegreater, there is an integer. Since $f_{k}(a)arrow y$ , for eachfor all$k\geqq
k_{2}(n)$. This contradicts the fact thatforsuch that we havethan$p^{j}>0$$a\inA$Therefore,
have
for every $j\in J^{+}$ .
.
we must.
Then, $z_{n}arrow z$ , and for all$n$,$p\cdot$$z_{n}<p\cdot e(a)$$n$$k_{1}(n)$$n_{0}$$z_{n}\succ_{a}y$$n\geqq n_{0}$$n\geqq n_{0}$$k_{2}(n)$$z_{n}\succ_{a}f_{k}(a)$$k_{1}(n)$$\bullet$$fk(\mathit{0})\in D_{<}^{k}(a,p_{k})\mathrm{a}.\mathrm{e}$It follows from lemma 2that there is apositive integer.
for all $k>k_{0}$ for some
have
$j\in
J^{+}$$z\in
B(a,p_{k})$Then,.
we haveandNow, let$k_{0}$

such that for ffi $j\in J^{+}$ we$\delta>0$$p_{k}^{j}\geqq\delta$$\delta z^{j}\leqq p_{k}^{?}.z^{j}$$\leqq$$i \in J\sum_{+}^{\ell}p_{k}^{i}z^{i}$$\leqq$$p_{k} \cdot e(a)-\sum_{+i\not\in J}^{\ell}p_{k}^{i}z^{i}$$\leqq$$\sum_{i=1}^{\ell}|p_{k}^{i}|e^{i}(a)+\sum_{+i\not\in J}^{\ell}|p_{k}^{i}|z^{i}$$\leqq$$\sum_{i=1}^{\ell}e^{:}(a)+\sum_{+i\not\in J}^{\ell}z^{i}$$\leqq$$\sum_{i=1}^{\ell}e^{i}(a)+(\ell-\# J^{+})\kappa$

(4.8)

.

Thus, one obtains$z^{j} \leqq\frac{1}{\delta}(\sum_{i=1}^{\ell}e^{i}(a)+(\ell-\# J^{+})\kappa)$

Define an integer$k_{1}$

.

(4.9)

by

$k_{1}= \max\{k_{0}$,$\frac{\kappa}{1+\sum_{i}e^{i}(a)}$,$\frac{\sum_{i}e^{i}(a)+(\ell-\# J^{+})\kappa)}{\delta(1+\sum_{i}e^{i}(a))}\}$.

12Then, for any k.
when zE
we havefor each jE$K(a)$, then it must be that
Thus, if we have zf. So, define
for some id,“partiallybytruncated” vector of z
and athe subset J of
$B(ap_{k)}\rangle\rangle$$>k_{\mathit{1}}$$z_{\ovalbox{\tt\small REJECT}}\ovalbox{\tt\small REJECT}$$J^{+}$$z^{\ovalbox{\tt\small REJECT}}>\mathrm{x}$$J^{\ovalbox{\tt\small REJECT} s_{\ovalbox{\tt\small REJECT}}}J^{+}$$J^{+}$$k(1+\ovalbox{\tt\small REJECT} \mathrm{i}:_{\ovalbox{\tt\small REJECT}}e^{\ovalbox{\tt\small REJECT}}(a))$$\mathrm{z}^{\mathrm{K}}$$J^{-}$$=$$\{i\in J\backslash J^{+}|z^{\dot{l}}>\kappa\}$$(z^{\kappa})^{:}$

$=$

$\{$

if
if

$z^{i}$$\kappa$,$i\in J\backslash J^{-}$(4.10)$i\in J^{-}$

and $f_{k}(a)\in D_{<}^{k}(a,p_{k})$ ,Since we havefor any $k>k_{1}$ . It follows
ffom the assumption of the unanimous perception of unboundedly desirable commodities. Hence, by the negative transitivity ofthat we havethatfor any $k>k_{1}$ . This establishes that$z^{\hslash}\in K(a)$$z^{\kappa}\neq_{a}f_{k}(a)$$z\neq_{a}z^{\kappa}$$\mathrm{p}\mathrm{r}\mathrm{e}\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{e}\mathrm{n}\mathrm{c}\mathrm{e}\succ_{a}$$f_{k}(a)\in D^{k}(a,p_{k})$,$\mathrm{a}.\mathrm{e}$. $a\in A$$z\neq_{a}f_{k}(a)$for $k>k_{1}$ .

Therefore, apair $(p_{k}, f_{k}(a))$ is an equilibrium for each $k>k_{1}$ .

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