Rational expectation can preclude trades (Mathematical Economics : Game Theory)

Rational

expectation

can

preclude

trades

*

Takashi Matsuhisa1 _{and Ryuichiro Ishikawa}2

1 _Department

_of

_{Liberal Arts andSciences. Ibaraki}$Nation\dot{a}l$ College

of

_Technology

866 Nakane, Hitachinaka-shi, Ibaraki 312-8508, Japan

E-mail:[email protected]

2 _{Graduate School}

of

Economics, Hitotsubashi University

Naka 2-1, Kunitachi-shi, Tokyo 186-8601 Japan

E-rnail: [email protected]

Abstract. In apure exchange economy under uncertainty the traders are

willing to trade of the amounts of state-contingent commodities and they

know their expectations. Common-knowledge aboutthese conditionsamong

all traders canpreclude trade if the initial endowments allocation is a$\mathrm{r}\mathrm{a}\mathrm{t}\mathrm{i}\triangleright$

nal expectations equilibrium, evenwhen the traders have the non-partition

structureofinformationwithout thecommonprior aslumption. In the proof

itplays essential roletoextend thenotion ofarationalexpectations

equilib-rium andto characterize$\mathrm{e}\mathrm{x}$-ante Paretooptimal endowments as the

equilib-rium. From the epistemic point of view it is emphasized that the partition

structure of information for the traders plays no roles in the notrade theo

$\mathrm{r}\mathrm{e}\mathrm{m}$.

Keywords: Pure exchange economy with knowledge, Rational

expecta-tions equilibrium, No trade theorem, Ex-ante Pareto optimal,

Common-knowledge,

1. Introduction

One of the purposes of this paper is to introduce apure exchange economy under

generalized information structure and to extend the notion of rational expectations

equilibria for the economy. The anotherpurpose isto characterize

an

$\mathrm{e}\mathrm{x}$-ante Pareto

optimalendowments allocationasarationalexpectations equilibrium,and to extend

the no trade theorem of Milgrom and Stokey (1982) in the economy under

gener-alized information structure: It is assumed that (a) the traders are willing to trade

of theamounts of state-contingent commodities, and that (b) they have rationality

such that theyknow their expectedutilities. We shallshow that common-knowledge

about (a) and (b)

can

preclude trade provided that the traders have

-the reflexive and transitive information;

-the subjective priors that are not common for them; and

-the strictly monotone preferences.

In theirpaperMilgrom and Stokey (1982) show the

no

trade theorem

as

follows:1

Let us consider apure exchange economy with traders in uncertain environment.

Let $\Omega$ _{$=\Theta\cross X$} and the state of $\Omega$ consists of apair

$(\theta,x)$ where 0ranging

over

.

_{The paper is an extended abstract and the final form will be published elsewhere.}

1 _{See Fudenberg and Tirole (1991),}

Chapter 14, Subsection(14.3.3), pp. 550-553

数理解析研究所講究録 1264 巻 2002 年 227-236

the contingencies

on

whichcommodities

are

defined. Theset $\Theta$ is interpreted

as

the

set of payoff-relevant events; endowments and utility functions may depend

on 0.

The set $X$ is interpreted

as

consisting ofpayoff-irrelevant events; these events do

not affect endowments ortaste directly. It isassumed here that the contingent

com-moditiesare$\mathrm{e}\mathrm{x}$-anteParetO-0ptimallyallocated, and the tradersreceive information

about the state of$\Omega$ representable byinformation partition, and it is assumed that

the traders’ beliefs

are

acommon

prior distribution;

we

call it the

common

prior assumption. Now, atrading

process

takes place where traders try to maximize their

expected utilities. We

assume

that in any equilibrium of this

process

traders’

in-tended trades

are

both jointlyfeasible and

common

knowledge amongthem. In this

set-up Milgrom and Stokey show that iftraders

are

strictlyrisk-averse, equilibrium

trade is null.

Theserious limitationsofthe analysis in apureexchangeeconomy under

uncer-tainty such

as

MilgromandStokey’s

are

its

use

oftheinformation partitionstructure by which the tradersreceiveinformation andofthe

common

prior assumption. From

the epistemic pointofviewtheinformationpartition structure represents the trades’ knowledge: Precisely, the structure is equivalent to the standard model

of

knowledge that includes the ‘factivity’ of knowledg$\dot{\mathrm{e}}\mathrm{T}$ (what is known is true) and the

‘intr0-spection’ properties Axioms 4and 5(that

we

know what

we

do and do not know).

The postulate 5is indeed

so

strong that describe the hyper-rationality oftraders,

and thusit is particularlyobjectionable. Alsois the common-knowledge assumption because the common-knowledge operator is defined by an infinite recursion of the knowledge operators. The recent idea of ‘bounded rationality’ suggests dropping such assumptions since real people

are

not complete

reasoners.

The

common

prior assumption also

seems

to be problematic.

Thisraises thequestion towhat extent results

as

the

no

tradetheoremdepend

on

both common-knowledgeand the information partition structure (ortheequivalent

postulates of knowledge.) The

answer

is that results strengthen the Milgrom and

Stokey ’s theorem

can

be obtained in twoways: First, Tanaka (2000) investigates the theorem on the information partition by iterated elimination reasoning instead

of common-knowledge. Secondly, in this paper

we

drop the hypothesis that the initial endowments

are

$\mathrm{e}\mathrm{x}$-anteParetooptimal and

we

extendthe

no

trade theorem

to thereflexive and transitive information structurewithout the traders being risk-aversion and having the

common

prior assumption. We show the results

as

follows: In apure exchange economy under reflexive and transitive information structure,

the traders are assumed to have their subjective priors not

common

and to have

strictly monotone preferences. Then

Main Theorem 1. Any price system

for

which the initial endowments allocation

is a rational expectations equilibrium allocation

can

preclude trade

_if

all the traders

commonly know that they

are

willing to trade

_of

the amounts

_of

state-contingent

commodities and

_if

they knowtheirexpectations everywherewith respect to the price.

To prove it we extend the notion ofrational expectations equilibrium for

econ-omy under uncertainty to that ofeconomy under reflexive and transitive

informa-tion structure, and

we

establish the the existence theorem for the equilibrium: The

traders are further assumed to be strictlyrisk-averse

Main Theorem 2. There eists a rational expectations equilibrium allocation

rel-ative to aprice with respect to which the traders know their expectations everywhere.

Moreover,

_we

_{show ageneralized version of fundamental theorem of welfare}

economics, apart of which plays essential role in proving Main Theorem 1:

Main Theorem 3. The initial endowments allocation is $ex$-ante Pareto optimal

if

and only

_if

itis a rational expectations equilibr$\tau rium$allocation relative to a price with

respect to which the traders are rational everywhere about their expectations. Thispaper organizes asfollows: In

Section

2wefirst recallageneralized

informa-tionstructure;_the$RT$-informationstructure,and theknowledge operatormodel

cor-responding to it. Secondly we introduce the economy under $RT$-infomation

struc-ture, called an economy with knowledge, which is ageneralization of

an

economy

under uncertainty. In Section 3we extend the notion of rational expectations equi-librium for economy under uncertainty to that ofeconomy with knowledge, and

we

establish the fundamental theorem ofwelfareeconomics and the existence theorem for the equilibrium. Main Theorem 1is proved as aconsequence of apart of the fundamental theorem. At the end of this section we give the existence theorem for the rational expectations equilibrium. InSection4we state the generalized notrade theorem ofMilgrom and Stokey. In Section 5weremark that the X-information

structure plays

an

essential role in the no trade theorem.

2. The Model

Let $\Omega$ be anon-empty

finite

set called astate space, $N=\{1,2, \cdots,n\}$ aset of

finitely many traders, and let $2^{\Omega}$ denote the field ofall subsets of$\Omega$

.

Each member

of$2^{\Omega}$ is called an

event and each element of $\Omega$ called astate. 2.1. Information and $\mathrm{K}\mathrm{n}\mathrm{o}\mathrm{w}1\mathrm{e}\mathrm{d}\mathrm{g}\mathrm{e}^{9}\sim$

An

_information

structure

_{

$P$

{

_$)ieN$ is aclass of mappings $P_{i}$ of $\Omega$ into $2^{\Omega}$

.

It is said

to be

_refleive

ifthe following property is true:

Ref $\omega$ $\in P_{i}(\omega)$ for every $\omega$ $\in\Omega$,

and it is said to be transitive if the following property is true: Trn $\xi\in P_{i}(\omega)$ implies $P_{i}(\xi)\subseteqq P_{i}(\omega)$ for all $\xi,\omega\in\Omega$.

Given our interpretation, an trader $i$ for whom $P_{i}(\omega)\subseteqq E$ knows, in the state $\omega$, that somestate in the event $E$ has occurred. In this case we say that at the state $\omega$ the trader $i$ knows E. $i’ \mathrm{s}$ knowledge operator $K_{i}$ on $2^{\Omega}$ i

$\mathrm{s}$defined by

$K_{i}E=\{\omega\in\Omega|P\dot{.}(\omega)\subseteqq E\}$. (1)

The set $P_{i}(\omega)$ will be interpreted as the set of all the states of nature that $i$ knows

to be possible at $\omega$, and $K\{E$ will be interpreted

as

the set ofstates of nature for

which $i$ knows $E$ to be possible. We will therefore call $P_{i}i$’s possibility operator on

$\Omega$ and also will call

P.

$\cdot(\omega)$ $i’ \mathrm{s}$ possibility setat $\omega$.

It is noted that $i’ \mathrm{s}$ knowledge operator satisfies the following properties: For

every $E$,$F$ of$2^{\Omega}$,

2 $\mathrm{S}$ Bacharach (1985), Binmore (1992)

$\mathrm{N}$ $K_{i}\Omega=-- \mathit{0}$ and $K_{}\emptyset=\emptyset$ ;

$\mathrm{K}$ $Ki\{E\cap F$) $=K_{i}E\cap K_{i}F$; $\mathrm{T}$

$\mathrm{K}\mathrm{i}\{\mathrm{E}$) $\subset E$. for every $E\in 2^{\Omega}$

.

4 $\mathrm{K}\mathrm{i}\{\mathrm{E}$)

$\overline{\overline{\subseteqq}}K_{\dot{*}}(K_{i}(E))$ for every $E\in 2^{\Omega}$

.

It is also noted that the possibility operator$P_{}$ is uniquely determinedbythe knowl-edge operator $K_{i}$ such

as

$P \dot{.}(\omega)=\bigcap_{\omega\in K:E}E$

.

The mutual knowledge operator $K_{E}$ on $2^{\Omega}$ i

$\mathrm{s}$ defined by $K_{E}F= \bigcap_{:\in N}K\dot{.}F$

.

The event $K_{E}F$ isinterpreted

as

that ‘all traders know F.’ The common-knowledge operator $K_{C}$ is defined by the infinite recursion of knowledge operators:

$K_{C}E:=\cap\ldots\cap\dot{.}K_{1}\dot{.}K_{\dot{*}_{2}}\cdots K_{k}\dot{.}E^{3}k=1,2,\{:_{1},i_{2\prime\cdots\prime k}\}\subset N^{\cdot}$

The cornrnunal possibility operator is the mapping$M$ : $\Omegaarrow 2^{\Omega}$ definedby _$M(\omega)=$

$\mathrm{r}\mathrm{u}_{\epsilon}K_{C}E$ $E$

.

All traders commonly know $E$ at $\omega$ if$\omega$ $\in KcE$;which is equivalent

to that $M(\omega)\subseteqq E$.

2.2. Economy with knowledge

Apure exchange economyunderuncertainty is atuple$\langle N, \Omega, (e:):\in N, (U_{}):\in N, (h.):\in N\rangle$

consisting of the following structure and interpretations: There

are

$l$ commodities in each state of the statespace $\Omega$ , and it is assumed that $\Omega$ is

finite

and that the

consumption set of trader $i$ is $\mathrm{R}_{-\{-}^{l}$;

$-N=\{1,2, \cdots,n\}$ is the set of$n$traders;

$-e$

:

: $\Omegaarrow \mathrm{R}_{[perp]}^{l}$ is $i’ \mathrm{s}$ endoumenb,

$-U\dot{.}$ : $\mathrm{R}^{\underline{\iota_{\mathfrak{l}}}}\cross\Omega$$arrow \mathrm{R}$ is $i’ \mathrm{s}$ utility function;

$-\mu$

:is

asubjective prior

on

$\Omega$ for $i$

.

For simplicity it is assumed that $(\Omega,\mu.)$ is afinite probability space with _$\mu$.

full

$support^{4}\mathrm{f}\mathrm{o}\mathrm{r}$every i _{$\in N$}

.

Definition 1. An economy $wid\iota$ knowledge $\mathcal{E}^{K}$ is astructure

$\langle \mathcal{E}, (P_{i}):\in N\rangle$, in which

$\mathcal{E}$ is apure exchange economy under uncertainty with astatespace $\Omega$ finite and

with $(P_{i})$ areflexive and transitive information structure

on

$\Omega$

.

Wedenote by $\mathcal{F}\dot{.}$ the field generated by $\{P\dot{.}(\omega)|\omega\in\Omega\}$ and by $\mathcal{F}$ the join ofall $\mathcal{F}\dot{.}(i\in N)$;i.e. $\mathcal{F}=\bigvee_{:\in N}\mathcal{F}_{i}$. It is noted that the atoms $\{A_{i}(\omega)|\omega\in\Omega\}$ of$\mathcal{F}_{i}$ is

the partition induced from $P\ldots$ We denote by $\{A(\omega)|\omega \in\Omega\}$ the set of all atoms

$A(\omega)$ containing$\omega$ of thefield $\mathcal{F}=\vee:\in N\mathcal{F}\dot{.}$

.

By an allocation

we mean

aprofile $a=(a:)$ of$\mathcal{F}.\cdot$-measurable functions_$a$

:from

$\Omega$ into $\mathbb{R}_{\mathrm{A}}^{l}$ such that for every $\omega\in\Omega$,

$\sum_{i\in N}a:(\omega)\leqq\dot{.}\sum_{\in N}e:(\omega)$.

3 _{That is, when}

$\omega$occursthen for all $k$ and for $\mathrm{a}\mathbb{I}$traders $i\iota$,i2,$\ldots.i_{k}$, it is true that $‘ i_{1}$

knowsthat[i2 knows that [.

. .

$\mathrm{i}\mathrm{k}-\mathrm{i}$ knows that $[i_{k}$ knows$X]]\ldots$ ].’ Thisis the iterated

notionof common-knowledge.

4 _{I.e., $\mu.(\omega)\neq>0$ for every} $\omega$$\in\Omega$

.

We denote by $A$ the set of all allocations and denote by $A_{i}$ the set of all the

$\mathrm{z}’ \mathrm{t}\mathrm{h}$ components: $A$ $=\cross_{i\in N}A_{i}$

.

Atrade $t$ $=(t_{i})_{i\in N}$ is aprofile of

$\mathcal{F}_{i^{-}}\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{a}\mathrm{b}1\mathrm{e}$

functions$t_{i}$ ffom$\Omega$ into

$\mathbb{R}^{l}$. It is saidtobe

feasible

iffor all$i\in N$ and for all$\omega$ $\in\Omega$,

$e_{i}(\omega)+t:(\omega)\geqq 0$; and

$\sum_{i\in N}t_{i}(\omega)\leqq 0$

.

We shall oftenrefer to the following conditions: For every $i\in N$,

A-l The function $e_{i}(\cdot)$ is $\mathcal{F}_{i}$-measurable with$\sum_{i\in N}e_{i}(\omega)>\neq 0$ for all

$\omega\in\Omega$

.

A-2 For each$x\in \mathbb{R}_{+}^{l}$, the function $U\dot{.}(x, \cdot)$ is $\mathcal{F}_{i}$-measurable.

A-2 For each$\omega$ $\in\Omega$, the function $U_{i}(\cdot,\omega)$ is strictly monotone on

$\mathrm{R}_{-\mathrm{I}-}^{l}$

.

A-4 For each $\omega\in\Omega$, the function $U_{i}(\cdot,\omega)$ is continuous, strictly quasi-concave

and $non- saturated^{5}\mathrm{o}\mathrm{n}\mathrm{R}_{+}^{l}$

.

Here it is noted that $\mathrm{A}-4$ implies to A-3.

2.3. Pareto optimality and Acceptability

We set by $\mathrm{E}_{i}[U\dot{.}(a_{i})]$ the $ex$-ante expectation defined by

$\mathrm{E}.\cdot[U_{i}(a_{i})]:=\sum_{\omega\in\Omega}U_{i}(a:(\omega),\omega)\mu\dot{.}(\omega)$

for each $a_{i}\in A_{i}$

.

The endowments (ei)\^i $N$

are

said to be $ex$-ante ParetO-Optimal if there is no

allocation $(a_{i})_{i\in N}$ such that for all $i\in N$,

$\mathrm{E}:[U\dot{.}(a\dot{.})]\geqq \mathrm{E}_{i}[U_{i}(e:)]$;

and that for

some

$j\in N$,

$\mathrm{E}_{j}[U_{j}(a_{j})]$

a

$\mathrm{E}_{j}[U\wedge e_{j})]$.

Let $\mathrm{E}_{i}[U_{i}(a_{i})|P_{i}](\omega)$ denotethe interirnexpectation defined by $\mathrm{E}_{:}[U\dot{.}(a\dot{.})|P_{\dot{*}}](\omega):=\sum_{\xi\in\Omega}U_{i}(a_{i}(\xi),\xi)\mu_{i}(\xi|P\dot{.}(\omega))$ .

Definition 2. Let $\mathcal{E}^{K}$ b

$\mathrm{e}$ an economy with knowledge and

$t$ _{$=(t:)_{i\in N}$} afeasible

trade. We say that $t\dot{.}$ is acceptable for $i$ at state$\omega$ provided that $\mathrm{E}_{i}[U_{i}(t_{i}+e_{i})|P_{i}](\omega)\geqq \mathrm{E}_{i}[U_{i}(e:)|P_{i}](\omega)$

.

Denote by $A\varphi(t_{i})$ the set of all the states in which $t_{i}$ is acceptable for $i$, and by

Act(t)$)$ the intersection $\bigcap_{i\in N}A\varphi(ti)$.

3. Rational Expectations Equilibrium

In this section we extend the notion of rational expectations equilibrium for an economy under uncertainty to that for an economy with knowledge. We show the fundamental theorem of welfare economics concerning about the relationship

be-tween $\mathrm{e}\mathrm{x}$-ante Pareto optimal allocations and rational expectations equilibria.

5 _{I.e.; For any}$x\in \mathrm{R}_{+}^{l}$ thereexists an $x’\in \mathrm{R}_{+}^{\mathrm{t}}$ such that U.$\cdot$

$(x’, \omega)$ $\neq>U.\cdot(x,\omega)$.

3.1.

Price system and rational expectations equilibrium

Let $\mathcal{E}^{K}=\langle N, \Omega, (e:):\in N, (U.\cdot):\in N, (\mu:):\in N, (P.\cdot):\in N\rangle$ be apure exchange economy

with knowledge. Aprice system is

anon-zero

function$p:\Omegaarrow \mathrm{R}_{+}^{l}$

.

We denote by $\sigma(p)$ the set ofall atoms ofthe smallest field that $p$ is measurable, and by $\sigma(p)(\omega)$

the component containing $\omega$

.

The budget set of atrader

:at

astate $\omega$ for aprice

system$p$ is defined by

$B_{:}(\omega,p)=\{a\in \mathrm{R}_{+}^{l}|p(\omega)\cdot a\leqq p(\omega)\cdot e:(\omega)\}$

.

Let $\sigma(p)\cap P\dot{.}$ : $\Omega$ $arrow 2^{\Omega}$ be defined by _{$(\sigma(p)\cap P_{})(\omega):=\sigma(p)(\omega)\cap P_{}(\omega)$};it is

plainly observed that $\sigma(p)\cap P_{}$ is areflexive and transitive information structure of trader $i$

.

We denote by $\sigma(p)\vee \mathcal{F}4$ the field generated by $(\sigma(p)\cap P_{})$ and denote by

$A_{i}(p)(\omega)=\sigma(p)\cap A:(\omega)$ the atomcontaining $\omega$

.

Definition 3. Arational expectations equilibrium for

an

economy $\mathcal{E}^{K}$ with

knowl-edgeis apair $(p,x)$, in which $p$ is aprice system and $x$ $=(x:):\in N$ is

an

allocation

satisfying the followingconditions:

$\mathrm{R}\mathrm{E}1$ For every $i\in Nx$

:is

$\sigma(p)\vee \mathcal{F}.\cdot$-measurable.

$\mathrm{R}\mathrm{E}2$ For every $i\in N$ and for every $\omega\in\Omega$, _{$x:(\omega)\in B_{:}(\omega,p)$}

.

RE 3 For all$i\in N$, if$y_{i}$ : $\Omegaarrow \mathrm{R}_{+}^{l}$is $\sigma(p)\vee \mathcal{F}.\cdot$-measurable with$y:(\omega)\in B_{:}(\omega,p)$

for all $\omega$ $\in\Omega$, then

$\mathrm{E}_{:}[U.\cdot(X:)|\sigma(p)\cap P_{}](\omega)\geqq \mathrm{E}_{:}[U_{}(y_{})|\sigma(p)\cap P\dot{.}](\omega)$

pointwise

on

$\Omega$

.

The profile

x

$=(x:):\in N$ is called arational expectations equilibrium allocation.

We denote by $R.(p)$ the event that $i$ is rational about his expectation; i.e.,

$R.(p)=\{\omega\in\Omega |(\sigma(p)\cap P.\cdot)(\omega)\subseteqq[\mathrm{R}.[U_{}(\cdot)|\sigma(p)\cap P_{}](\omega)]\}$

and denote by $R(p)$ the event that all traders

are

rational: i.e., $R(p)= \bigcap_{i\in N}R.(p)$.

Definition 4. Atrader $i$ is said to be rational about his expectation with respect

to price system$p$ at$\omega$ if\mbox{\boldmath $\omega$}\in R%(p). And all traders

are

rational everywhere about

their expectations if$R(p)=\Omega$

.

3.2. Fundamental Theorem in Welfare Economics

We establish ageneralizedversion ofthe fundamentaltheorem ofwelfareeconomics

for initial endowments in the economy with knowledge (Propositions 2and 3), and Proposition 1below is also akey to proving Main Theorem 1:

Proposition 1. Let $\mathcal{E}^{K}b$ an economy with knowledge satisfying the conditions

A-l, A-2 and A-3. Then the initial endowments allocation $e=(e:):\in N$ is ex-ante

Pareto optimal

_if

it is arational expectations equilibrium allocation relative to some

price system $p$ with respect to which all traders are rational everywhere about their

expectations

Thenext proposition states that theconversein Proposition 1is alsovalid under the additional assumption that the traders are strictly risk-averse for traders: Proposition 2. Let$\mathcal{E}^{K}$ be an economy ettith knowledge satisfying the conditions

A-1, A-2 and

_A-4.

_If

the initial endowments allocation $e=(e_{i})_{i\in N}$ is ex-ante-Pareto optimal then it is a rational expectations equilibrium allocation relative to

some

price system$p$ with respect to which all traders are rational every where about their

expectations.

Proof.

For each $\omega\in\Omega$

we

denote by $G(\omega)$ the set of all vectors $\sum_{i\in N}e_{i}(\omega)$

-$\sum_{i\in N}y_{i}$ such that $y_{i}\in \mathrm{R}_{\mathrm{T}}^{l}|$ and $U_{i}(y_{i},\omega)\geqq U_{i}(e_{i}(\omega),\omega)$ for all $i\in N$.

First, in view of the conditions A-l, A-2 and A-4

we

note that that $G(\omega)$ is

convex

and closed in$\mathbb{R}_{+}^{l}$

.

We

can

establish the propositioninobserving the following

three points: First

Claim 1: For each $\omega\in\Omega$ there exists $p^{*}(\omega)\in \mathbb{R}_{+}^{l}$ such that $p^{*}(\omega)\cdot$$v\leqq 0$ for all $v\in G(\omega)$.

Secondly, let $p$ be the price system defined

as

follows: For each $\omega\in\Omega$ and for

all $\xi\in A(\omega)$, $p(\xi):=p^{*}(\omega)$. We

can

show

Claim 2: The pair $(p, (e:)_{i\in N})$ is arational expectations equilibrium for $\mathcal{E}^{K}$

.

Finally, it is observed that all traders are rational with respect to the price$p$

.

$\square$

3.3. Main Theorem 3

We now state Main Theorem 3explicitlyas follows:

Theorem 1. Let $\mathcal{E}^{K}$ _$be$ an economy $with$ knowledge satisfying the conditions A-l,

A-2 and

_A-4.

The initial endowments allocation is $ex$-ante Pareto optimal

if

and

only

_if

it is a rational expectations equilibrium allocation relative to $a$ $p\tau\dot{\mathrm{v}}ce$ with

respect to which the traders are rational everywhere about their expectations.

Proof.

Follows immediately from Propositions 1and 2.

The following remark has been already proved in the proof of Proposition 1:

Remark 1. Let $\mathcal{E}^{K}$ be apure exchange economy with knowledge satisfying the

conditions A-l, A-2 and A-3. Ifthe allocation of initial endowments $e=(e_{i})_{\dot{*}\in N}$

is arational expectations equilibrium allocation relative tosomepricesystem$p$with

respect to which all traders

are

rational every where about their expectations then the pair $(p(\omega), (e:(\omega))_{i\in N})$ constitutes an $ex$-post competitive equilibrium for the

pure exchange economy $\mathcal{E}^{K}(\omega)$ with complete information for each $\omega\in\Omega$.

3.4. Existence Theorem

It will well end this section in giving the explicit statement of Main Theorem 2:

The existence theorem of rational expectations equilibrium for an economy with knowledge.

Theorem 2. Suppose a pure exchange economy with knowledge

_satisfies

the

condi-tions A-l, A-2 and

_{A-4. If}

the initial endowments allocation $e=(e:)_{i\in N}$

satisfies

the additional condition that$e_{i}(\omega)\neq>0$

for

all$\omega\in\Omega$ and

for

each$i\in N$ then there

eists$.a$ rational expectations equilibrium

for

the economy such that all traders

are

rational about their expectations with respect to the price

4. No Trade Theorem

In this section

we

shall givetwoextensions ofthe

no

trade theorem ofMilgromand Stokey (1982): First

we

give the below theorem that directly extends the

no

trade theorem to an economy with knowledge, and secondlywe give Main Theorem 1. 4.1. Theorem of Milgrom and Stodcy

Theorem 3. Let $\mathcal{E}^{K}$ be an economy with knowledge satisfying the conditions A-l,

A-2 and A-3, and let $t$ $=(t:):\in N$ be a

feasible

trade. Suppose that the initial

en-doeryments _allocation $(e:):\in N$ is $ex$-ante ParetO-Optimal. Then the traders can never

agree to any non null trade at each state where they commonly know both the

ac-ceptable trade $t=(t:)$ and rationality

_of

their expectations; that is, $t(\omega)=0$ at

every $\omega\in K_{C}(Ad(t) \cap R)$

.

Proof.

Follows from the key lemma below.

Lemma 1. Let $\mathcal{E}^{K}$, t _{$=(t:)_{\dot{*}\in N}$} and _{$(e:):\in N$} be the same as in Theorem 3.

If

$\omega\in K_{C}(A\varphi(t:)\cap \mathrm{R}\mathrm{i})$

for

each i $\in N$ then the equality is true:

$\mathrm{E}_{:}[U\dot{.}(t^{*}\dot{.}+\mathrm{q}.)|P_{}](\omega)=\mathrm{E}:[U.\cdot(\mathrm{q}.)|P.\cdot](\omega)$, (2)

where the trade $t^{*}=(t^{*}.\cdot):\in N$ is

defined

by $t^{*}.\cdot(\xi):=\{$

$t_{:}(\xi)$ $\dot{l}f\xi\in M(\omega)$

,

0if

not (3)

4.2. Rational expectations equilibrium and No trade theorem

It is interesting to consider what

can

be said if

we

drop the hypothesis that the endowments

are

$\mathrm{e}\mathrm{x}$-anteParetooptimal in Theorem

3.

Is the

no

trade theoremstill

true ifthe endowments allocation is rational expectations equilibrium aUocations?

We shall give

an

affirmativeanswer.To state it explicitly

we

introduce the knowledge operator $K^{(p)}\dot{.}$ on $2^{\Omega}$ induced from the information structure

$\sigma(p)\cap P_{}$ ; which is

defined by

$K^{(p)}.\cdot(E)=\{\omega\in\Omega|(\sigma(p)\cap P_{})(\omega)\subseteqq E\}$,

and let $K_{C}^{(p)}$ be thecommon-knowledge operator defined by theinfinite recursion of the operators $\{K^{(\mathrm{p})}\dot{.}\}:\in N\cdot 6$ We

can now

explicitly state Main Theorem 1asfollows: Theorem 4. Let $\mathcal{E}^{K}$ be

an

economy with knowledge satisfying the conditions A-l,

A-2 andA-3,

_If

$e=(e:):\in N$ is arational expectations _{equilibrium allocation relative}

to some price system$p$ withrespectto which all traders

are

rationaleverywhere about

their expectations, then the traders

can

never agree to any

non

null trade at each

state where they commonly know both the acceptable

_feasible

trade $t=(t:):\in Nj$ that is, $t(\omega)=\mathrm{O}$ at every$\omega\in K_{C}^{(\mathrm{p})}$(Act(t)).

6 _{That is, $K_{C}^{(p)}E:= \bigcap_{k=1,2},\ldots\bigcap_{\{:::\}}1,2,\ldots,k\subset NK_{i_{1}}^{(p)}K_{_{2}}^{(\mathrm{p})}\cdots K_{_{k}}^{(p)}E$}

_.

Proof.

Consider

now

the economic with knowledge

$\mathcal{E}^{K(p)}=\langle N, \Omega, (e_{i})_{i\in N}, (U_{i})_{i\in N}, (\mu_{i})_{i\in N}, (\sigma(p)\cap P_{i})_{i\in N}\rangle$ .

By the similar argument in the proof of Theorem 3it

can

be plainly observed that $t(\omega)=0$ at every cv $\in K_{C}^{(p)}$(Act(t ) if $e$ is $\mathrm{e}\mathrm{x}$-ante Pareto optimal, and

$\mathrm{t}\mathrm{h}\mathrm{u}\mathrm{s}\square$

Theorem4follows from Proposition 1.

5. Concluding Remarks

Our

real

concern

is to what extent the

no

trade theorem of Milgrom and Stokey

(1982) depends on the information partition and on the hypothesis that the initial endowments

are

$\mathrm{e}\mathrm{x}$-ante Pareto optimal. As

we

have observed, the reflexivity

and

transitivity of information structure

can

preclude trade if the traders commonly know that theyarewillingtotradeofthe amounts ofstate-contingent commodities.

Both the information partition and the strictly risk-aversion for the traders of the

amounts ofcommodities play no roles in the no trade theorem.

Could we prove the theorem under the generalized information structure jet-tisoning the reflexivity

or

the $\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{i}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}^{7}$ The following two examples show that

the reflexivity Ref and the transitivity Trn of the information structure (or the equivalent postulates Axioms 4and T) do play

an

essential role.

Example 1. Let $\mathcal{E}^{K}=\langle N, \Omega, (e_{i}):\in N, (U_{i})_{i\in N},\mu, (P_{i})_{i\in N}\rangle$ the economy with

knowl-edge in which $N$,$\Omega$,

$e:$,$U\dot{.}$ are the

same

in Section ??, and

$- \mu(\omega)=\frac{1}{2}$ for each$\omega\in\Omega$;

$-P_{i}$ is defined by

$P_{1}(\omega):=\{\omega_{2}\}$ and $P_{9,\sim},(\omega):=\{\omega_{1}\}$

for eachci $\in\Omega$.

It is plainly observed the two points: First that both $P_{i}$ $(i=1, 2)$

are

not reflexive but transitive, and second that the endowments $(e_{i})_{i=1},\underline’$

are

both $\mathrm{e}\mathrm{x}$-ante Pareto

optimal. Let $t=(t:):=1,2$ be the feasible

non-zero

tradedefined by

$t_{1}(\omega):=\{$ -2 if

$\omega$ $=\omega_{1}$

0if $\omega$ $=\omega_{\sim}$’

and $t_{2}(\omega):=\{$ 2if

$\omega$ $=\omega_{1}$

0if $\omega$ $=\omega_{-}’$

.

Then itcanbe verified that Act(t) $=R=\Omega$and thus $Kc(Act(t)\cap R)=\Omega$

.

$\mathrm{H}\mathrm{o}\mathrm{w}\mathrm{e}\mathrm{v}\mathrm{e}\mathrm{r}\square$

the trade $t$ is not nullat $\omega_{1}\in K\mathrm{c}\{A\mathrm{c}\mathrm{t}(t)\cap R)$.

Example 2. Let $\mathcal{E}^{K}=\langle N, \Omega, (e:)_{i\in N}, (U\dot{.})_{i\in N}, \mu, (P_{i})_{i\in N}\rangle$the economy with

knowl-edge in which $N$,$e_{i}$ are the

same

in Section ??, and

$-\Omega$$=\{\omega_{1},\omega_{\sim}"\omega_{3}\}$

$-\mu(\omega)=31$ for each$\omega\in\Omega$;

$-U_{i}$ : $\mathrm{R}_{+}^{l}\cross\Omegaarrow \mathrm{R}$is defined by

Ui$(x,\omega)=(x+1)^{\mathrm{o}}\sim$ and $U_{-},(x,\omega)=\sqrt{x+3}$;

$-P_{i}$ is defined by

$P_{1}(\omega):=\{$

$\{\omega_{1}\}$ if$\omega$ $=\omega_{1}$

$\{\omega_{2},\omega_{3}\}$ if$\omega$ $=\omega_{\sim}’ \mathrm{o}\mathrm{r}\omega_{3}$

$P_{2}(\omega):=\{$

$\{\omega_{1},\omega_{3}\}$ if$\omega$ $=\omega_{1}$

$\{\omega_{2},\omega_{3}\}$ if$\omega=\omega_{2}$

or

$\omega_{3}$

.

It is plainly observed that $P_{\underline{9}}$

are

reflexive and not transitive. Let $t$ $=(t:):=1,2$

be the feasible

non-zero

trade defined by $t_{1}(\omega):=\{$ 1if

$\omega$ $=\omega_{1}$

or

$\omega_{2}$

-1.5 if$\omega$ $=\omega_{3}$

$t_{2}(\omega):=\{$ -1 if

$\omega=\omega_{1}$

or

$\omega_{-}$’

1.5 if$\omega=\omega_{3}$

.

Then it folows that Act(t) $=R=\Omega$ and $K_{C}$ Act(t) $\cap R)=\Omega$

.

However the trade

$t$ is not null at any_{$\omega\in K_{C}(Act(t)\cap R)$}

.

Cl

Nevertheless, common-knowledge of the acceptance of feasible trades

seems a

rather strong assumption. Could not

we

get away withless, saywithmutual

knowl-edge? The

answer

is

no

again: For the counter example see Fudenberg and Tirole

(1991, p.552).

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