1

Do Markets Favor Agents Able to Do Markets Favor Agents Able to Make Accurate Predictions?Make Accurate Predictions?

Alvaro Sandroni

Reporter:Lena HuangLena Huang

2

IntroductionA Model of ReinvestmentEndogenous Investment and Savings –ExamplesA Model of Endogenous Investment and Savings Basic ConceptsPredictions and SurvivalConvergence to Rational ExpectationsConclusion

3

Introduction (1)Introduction (1) A long-standing theory in economics is that agents

who do not predict as accurately as others are driven out of the market, and it underlies the efficient-markets hypothesis and the use of rational expectations equilibrium as a solution concept because it implies that asset prices will eventually reflect the beliefs of agents making accurate predictions.

However, under certain conditions the agents who have accumulated more wealth are also those who have made the worst prediction.Blume and Easley (1992) is an example.

4

Introduction (2)Introduction (2) Blume and Easley (1992) show that if agents’ have

the same savings rule, those who maximize the expected logarithm of next period’s outcomes will eventually hold all wealth. However, if no agent adopts this rule then the most prosperous are not necessarily those who make the most accurate predictions.

Agents with incorrect beliefs, but equally averse to risk, may choose an investment rule closer to the MEL rule, and so eventually accumulates more wealth than the agent with correct beliefs.

5

Introduction (3)Introduction (3) The recent literature casts serious doubt on the

theory that agents with incorrect beliefs will be driven out of the market by those with correct beliefs.This paper seeks to resurrect this intuitive theory.

The main difference is that,in the recent literature, savings are exogenously fixed and agents’ choices are solely restricted to investment decisions.In this model, I assume that agents make savings and investments decisions that fully maximize expected discounted utility.

6

Introduction (4)Introduction (4) In this paper, I show that if markets are

dynamically complete then, among agents who have the same intertemporal discount factor (but not necessarily the same degree of risk-aversion), the most prosperous will be those making accurate predictions, and convergence to rational expectations obtains.

This holds even though agents with identical beliefs, but different utility functions (i.e., diverse preference over risk), may choose different savings and portfolios, and therefore, the relative wealth of agents with different preferences over risk is a random variable.

7

Introduction (5)Introduction (5) The intuition is that if agents’ choices are

restricted to investment decisions, then they may optimally choose to allocate small amounts of wealth to events they believe likely to occur.

However, if agents can maximize over both savings and investment decisions, then I will show that although they may not maximize wealth accumulation, they will still allocate relatively more wealth to future events they believe more likely to occur;therefore,agents who eventually make accurate predictions survive in the market.

8

Introduction (6)Introduction (6) I examine who survives in the market without the

assumptions that agents have identical discount factors and some make accurate predictions.

I show that any agent who has strictly smaller entropy than another agent is driven out of the market.

This result is particularly appealing because agents’entropy depends only on exogenous parameters.Therefore, it is possible to compute it without solving for equilibrium.

9

A Model of Reinvestment (1) A Model of Reinvestment (1)

This model (following Blume and Easley(1992)) show that if savings are exogenously fixed, then agents with correct expectations may accumulate less wealth than agents with incorrect expectations.

The argument is completed by observing that agents with incorrect beliefs may choose an investment strategy closer to the MEL rule (which maximizes the growth rate of wealth) than what agents with correct beliefs choose.

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A Model of Reinvestment (2)A Model of Reinvestment (2)

- N+ : N {0}∪ -The set of states of nature is given by T≡{1, …,L}, L N. -Tt, t N {∞}, be the t-Cartesian product of ∪ T. -For every finite history , a cylinder with base

on st is the set C(st) = of all finite histories whose t initial elements coincide with st.

- :the σ-algebra consisting of all finite unions of cylinders with base on Tt.

- 0 ≡ ; hence,∪ is the smallest σ-algebra that contains 0 .

NtTs tt ,

,...)}({ tssTs

Some notations~

tS~

S~ tNt S~

S~S~

11

A Model of Reinvestment (3)A Model of Reinvestment (3) The true stochastic process of states of nature is

given by an arbitrary probability measure P defined on (T∞, ) .

Given a t-history , let be the posterior probabilities of P, defined by :

Let EP and be the expectation operators associated with P and , respectively.

S~t

t Ts tsP

))(()(

)(t

ss sCP

APAP t

t Where is the set of all paths

such that s=(st , ), .tsA Tst

s~ s~ A

))(~( sSE tP

tsP

12

A Model of Reinvestment (4)A Model of Reinvestment (4) A decision maker has initial wealth w0=1.At

period t, she chooses a portfolio among M assets such that .

An investment strategy is a sequence of portfolios ,and gross returns of assets are given

by the positive -measure functions If consumption is a fixed portion,1-δ,of wealth,

then agents who invest under strategy a accumulate wealth .

The entropy of a portfolio at is

tMtt aaa ,,1 ,...,1

1,

M

mtma

)( , Ntaa t

tS~ tMtt rrr ,,1 ,...

k

t

kk

tt raaw

11)(

tttP SraE ~log 1

13

A Model of Reinvestment (5)A Model of Reinvestment (5) Proposition 1: Let α1 and α2 be investment strategies such

that and are uniformly bounded away from zero and infinity.If there is such that, for all , P-a.s., then .

Proof :Let , .By the law of iterated expectations, .By the law of large numbers for uncorrelated random variables,P-a.s.,

11

tt r 12

tt r0

Nt tttP

tttP SrESrE ~log~log 1

21

1

0)()(

1

2

tt

t

ww

kkkkk rry 11

21log 1

~ kk

Pkk SyEyz

0kP zE

tkkk

Pkttk

kP

ktSyEy

tzEz

t 11

10~1lim01lim

By assumption, . Hence,0)~(11

1suplim

kk

tk

P

tSyE

t

tk

kt

yt 1

01suplim

0log 1

2

1

2

1

t

t

t

tt

tkk w

wwwy

14

A Model of Reinvestment (6)A Model of Reinvestment (6) The MEL rule is the myopic investment strategy

a* defined by . A corollary of Proposition 1 is that agents who

adopt the MEL rule eventually obtain greater wealth than the others.

Blume and Easley (1992) extended this result to an equilibrium model, and show that only agents investing under MEL rule survive. But if no one have a log utility function, agents with the highest entropy (hence, accumulate most wealth and survive) are not necessarily those making correct predictions.

}~){log(maxarg 1*

tttP

t SraEa

15

A Model of Reinvestment (7)A Model of Reinvestment (7) The key observation is that agents having correct

beliefs and maximizing expected utility may not optimally invest under the MEL rule and maximize wealth accumulation, because of their preference over risk.

Agents with incorrect beliefs, but equally averse to risk, may choose an investment rule closer to the MEL rule, and hold portfolios with higher entropy, which eventually result in more wealth.

Under the assumption of savings being exogenously fixed, similar examples results can be constructed in many frameworks under fairly general conditions.

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Endogenous Investment and SavingsEndogenous Investment and Savings-Examples -Examples

In contrast with the above results, we will show that if savings are endogenous then agents making inaccurate predictions will necessarily be driven out of the market.

This section illustrate some of these ideas in simple examples of dynamically complete market economies.

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Example 1 of Endogenous Investment Example 1 of Endogenous Investment and Savings (1)and Savings (1) Assumptions~Assumptions~

-Two long-lived agents, 1 and 2,two long-lived trees, 1 and 2, two states of nature, h and l, and one consumption good c.

-Tree 1 gives 0 units of consumption in state h and 1 unit in state l. Tree 2 gives 2 units of consumption in state h and 0 unit in state l .

-The probability of state h is 0.5 in every period. -Agent 1 has correct beliefs.Agent2 believes that

the probability of state h is 0.5+ε.Let Pi be the probability measure associated with the agent i’s beliefs.

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Example 1 of Endogenous Investment Example 1 of Endogenous Investment and Savings (2)and Savings (2)

At period t, the price of tree m is given by pm,t, agent i’s consumption and share holdings of tree m are given by Let be agent i’s wealth,

Agents’ maximizing problem is :

.& ,i

tmit kc i

tw

itt

itt

it

itt

itt

it

kpkpw

kpkpw

1,2,21,1,1

1,2,21,1,1

2

1

(State l )

(State h)

1,1,

..

)log()(,2)()()5.0(

2,2

1,2

2,1

1,1

21

,2,2,1,1

21

0

ttttttt

it

itt

itt

it

t

it

itpi

kkkkecc

wkpkpcts

ccuccucuEMax

Total dividends at t

19

Example 1 of Endogenous Investment Example 1 of Endogenous Investment and Savings (3)and Savings (3)

From two agents’ f.o.c,

where n is # of times that state h occurs until t. (A) shows that the ratio of probabilities times MU

of consumption is constant. By the law of large numbers, if t is large, n should

be close to t/2.Therefore,

By (A) and (B),

2

1

1

2

5.05.05.0

t

tntn

t

cc

(A)

t

t

ntn

t 2

5.05.05.05.0

5.05.05.0

(B)

01

2

tt

t

cc

(C)

)2(0 12 ttt cc (D)

20

Example 1 of Endogenous Investment Example 1 of Endogenous Investment and Savings (4)and Savings (4)

Equation (D) shows that agent 2 allocates vanishing consumption in some paths believing they will not occur, and share prices converge to REE in which agent 1 is alone in the economy.

However,in both states, agent 1’s portfolio are further away from MEL rule than agent 2’s portfolio.This is possible because of the difference in savings behavior.

Agent 2’s compounded saving ratio(the product of agent 2’s savings ratio in all period) is eventually an arbitrarily small fraction of agent 1’s.This fraction will eventually be small enough to compensate for difference in portfolio selection, making the relative wealth of agent 2 small.

The expected log of agent 1’s saving ratio:

is greater than the expected log of agent 2’s saving ratio (log(0.5))

2622log5.0

2321log5.0

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Assume thatε=0, i.e., the beliefs of both agent are identical.

There will be no speculative trade and agent may achieve an efficient allocation by trading just once.

Agent 1 always holds 2/3 of tree 1 and ¾ of tree 2; Agent 2 always holds 1/3 of tree 1 and ¼ of tree 2, and consuming dividends given by these shares.

Agents’ wealth and consumption depend only on the current state.

Both agents survive, although they do not choose similar savings and portfolios.

In both cases, share prices are different, but eventually close to a REE.

tttt

t eccc

c 21

2

1

1

2

,

Example 2 of Endogenous Investment & SavingsExample 2 of Endogenous Investment & Savings

22

A Model of Endogenous Investment & SavingsA Model of Endogenous Investment & Savings-The general case-The general case

Assumptions of the model : -I long-lived agents, M long-lived trees, L states of

nature, h and l, and one consumption good c. -Agents are born with shares of the trees and receive

no other endowments. -Dividends: ; e=(et , )

-Share price: ; p=(pt , )

-Agent i’s consumption: -Share holding: ;*Hence, agent i’s wealth is given by

Ii

imk1 1, )1(

tMtt eee ,,1 ,... Nt tMtt ppp ,,1 ,... Nt

Ntccc it

iit ,,

itM

it

it kkk ,,1 ,... Ntkk i

ti ,

ittt

it kepw 1

23

A Model of Endogenous Investment A Model of Endogenous Investment And Savings (1)And Savings (1)

The model of dynamically complete markets: -Let P & Pi represent the true probability measure &

agent i’s beliefs respectively. -Let Ht be the -measurable function defined by

-The markets are dynamically complete.That is L=M, and the rank of Ht(s) is L.So agents may transfer wealth across states of nature by trading the existing assets.

tS~

LseLsp

sespsH

tt

tt

t

11

11

:11

24

A Model of Endogenous Investment A Model of Endogenous Investment And Savings (2)And Savings (2)

Agents’ maximizing problem is

Where ui is a strictly increasing, strictly concave, continuously differentiable utility function that satisfies the Inada conditions.

)...1(1,

,0,0,..

~

1,

1,

1

1

0

Mmkec

Nrcwkepkpcts

ScuEMax

I

i

itm

M

mtm

I

i

it

irt

irt

irtrtrt

irtrt

irt

tr

irt

iriPi

25

Basic Concepts Basic Concepts

SurvivalThe Accuracy of Agents’ PredictionsEntropy

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Survival Survival Def 1Def 1:Agent i is driven out of the market on

a path if agent i’s wealth, , converges to zero as t goes to infinity. Agent i survives on a path if he is not driven out of the market on s.

I focus on accumulation of wealth as main criteria to define survival because only agents with positive wealth influence prices.

Ts swit

Ts

27

The Accuracy of Agents’ Predictions (1)The Accuracy of Agents’ Predictions (1) The difference between two probability measures

in the sup-norm, is given by

The difference between , in the dl-metric, is given by .

QQ &~

AQAQQQSA

~~ sup~

QQ &~

AQAQQQdlSA

l~~, max~

Two probability measure are close, in the sup-norm, if they assign “similar” probability to all events.

Two probability measure are close, in the dl-metric, if they assign “similar” probability to events within l-periods.

28

The Accuracy of Agents’ Predictions (2)The Accuracy of Agents’ Predictions (2)

Def 2Def 2: Agent i eventually makes accurate predictions on a path , s=(st,…), if

Def 3Def 3:Agent i eventually makes accurate next period predictions on a path , s=(st,…), if

. Def of ‘merge’:Agent i’s beliefs (weakly) merge

with the truth if, P-a.s., agent i eventually makes accurate (next period or l-period) predictions. Merging implies weak merging, but not conversely.

Ts 0 tsi

s ttPP

Ts 0,1 ts

is tt

PPd

29

The Accuracy of Agents’ Predictions (3)The Accuracy of Agents’ Predictions (3)

Def 4Def 4: :Agent i eventually makes inaccurate next period predictions on a path , s=(st,…), if there isε>0 such that

Clearly, an agent who does not eventually make accurate next period predictions need not always make inaccurate next period predictions.

Def 5Def 5:Some agents eventually make accurate (next period) predictions if, P-a.s., in every path , there exists at least one agent who eventually makes accurate (next period) predictions on s.

(Not necessarily the same agents on different paths)

Ts .,1 NtPPd

tt si

s

Ts

30

Entropy (1)Entropy (1) Let , then the probability of the

states of nature at period t, Qt, is defined as In particular, are the true probabilities and

agent i’s beliefs over states of nature at period t, given past data, respectively.

Def 6 Def 6 :The entropy of agent i’s beliefs at period t, , is given by

and it is zero iff agent i’s belief and the true probabilities over states of nature in the next period are identical.

tt sCQsdQ )(

sdQsdQsQ

t

tt

1

i

tt PP &

it

t

t

itPi

t SPPE ~log

1

1

0it

Write on the blackboard

31

Entropy (2)Entropy (2) Def 7Def 7:The entropy of agent i, is given by:

Hence, the entropy of an agent does not depend on the characteristic of the other agents.

Def 8Def 8:The ratio of beliefs and true probabilities over states of nature in next period uniformly bounded away from zero and infinity if there exists u>0 and U<∞ such that

tk

ik

t

ii

t 1

1)log( lim

UsPsPu

t

it

i.e.Pt ε≧ ＞ 0 Pt

i ε≧ ＞ 0

32

Predictions and SurvivalPredictions and Survival Main Results -proposition 2 -proposition 3 -proposition 4 -proposition 5 Proofs and Intuition -Basic Results -Results in Probability Theory -Proof of Proposition 2-5

33

Main ResultsMain Results--Proposition 2Proposition 2 (1) (1) Proposition 2:Assume that all agents have the same

intertemporal discount factor and some agents eventually make accurate predictions.Then in every equilibrium, P-a.s.:

1.Any agent who does not eventually make accurate predictions on a path is driven out of the market on the path s.

2.Any agent who eventually makes accurate predictions on a path survives on s.

Agents with diverse preferences over risk survive with probability 1 if their beliefs merge with the truth.This holds although the relative wealth of agents is a random variable because they may choose different savings and portfolios.

Ts

Ts

34

Main Results Main Results --Proposition 2Proposition 2 (2) (2) Proposition 2 is surprisingly strong. For example,

agent 1’s belief merge with the truth, but agent 2’s weakly merge with the truth.The differences in belief have vanishingly small impact on agents’ savings and investment decisions. However, by proposition 2, agent 2 is driven out of the market with probability 1.

A corollary of proposition 2: Assume that all agents have the same intertemporal discount factor. In every equilibrium, agent i survives Pi-a.s.

Consider the case P=Pi.Then agent i’s beliefs are exactly correct.By proposition 2, he survives Pi-a.s.

35

Main Results Main Results --Proposition 2Proposition 2 (3) (3)Intuition Intuition -Agents who maximize expected discounted

utility functions allocate relatively more wealth to paths they believe more plausible than to paths they believe less plausible.

-Thus, agents who eventually make accurate predictions allocate large amounts of wealth to paths that have, in fact, high probability and, hence, survive.

36

Main Results Main Results --Proposition 3Proposition 3 (1) (1) In the following proposition, I relax the

assumptions that all agents have the same discount factor and that some agents eventually make accurate predictions.

Proposition 3:Assume that the ratio of beliefs and true probabilities over states of nature in the next period is uniformly bounded away from zero and infinity. In every equilibrium, P-a.s., if the entropy of agent i is strictly smaller than the entropy of agent j on a path , , then agent i is driven out of the market on s.(remind entropy)

Ts )()( ss ji

37

Main Results Main Results --Proposition 3Proposition 3 (2) (2) Agents’ entropy is a function of exogenous

parameters, so there is no need to solve for equilibrium to compute it.Moreover, the entropy of an agent does not depend on preferences over risk, dividends in each state of nature, and beliefs and discount factors of the other agents.

Agents whose entropy is not smaller than any others’ do not necessarily survive.For example, agent 1 has correct beliefs; agent 2’s beliefs weakly merge with the truth.The average entropy of agent 2’s beliefs is zero.So, their entropy is identical.However, by proposition 2, agent 2 is driven out of the market.

38

Main Results Main Results --Proposition 4Proposition 4 Because that the entropy of an agent who always makes

inaccurate next period predictions is strictly smaller than that of an agent who makes accurate next period predictions, if they have the same discount factor.(Show in next section)Proposition 4 follows from this observation and Proposition 3.

Proposition 4:Assume that the ratio of beliefs and true probabilities over states of nature in the next period is uniformly bounded away from 0 and ∞; that all agents have the same intertemporal discount factor;and that some agents eventually make accurate next period predictions.In every equilibrium, P-a.s., if agent i always makes inaccurate next period predictions on a path ,then agent i is driven out of the market on s. (However)

Ts

39

Main Results Main Results --Proposition 5Proposition 5 Proposition 5:Under the same assumptions of

proposition 4, in every equilibrium, P-a.s., if there exists andε>0 such that

on a path , s=(st,…), then agent i is driven out of the market on s.

An open question is whether proposition 3~5 are true without the assumption that the the ratio of beliefs and true probabilities over states of nature in the next period is uniformly bounded away from 0 and infinity.

Nl NtPPdtt s

isl ,

TsStronger than proposition 4

40

Proofs and IntuitionProofs and Intuition-Basic Results-Basic Results (1) (1) Agents’ f.o.c of the maximization problem imply

that,

Lemma 1:In every equilibrium, for any path and agent ,

Lemma 2:Fix an agent and a path In every equilibrium, if there exists an agent such that then agent i is driven out of the market on s. Moreover, if for all agents

there exists ε>0 such that then agent i survives on s.

jj

ii

jt

jjt

tj

it

iit

ti

cucu

cudPcudP

0'

0'

'

'

( * )

Ts Ii ,...,1 00 t

itt

it swiffsc

Ii ,...,1 Ts Ij ,...1

0 tjt

tj

it

ti

sdPsdP

sdPsdP

jt

tj

it

ti

41

Proofs and IntuitionProofs and Intuition-Basic Results-Basic Results (2) (2) Lemma 2 implies that, almost surely, if agent i

believes that a path s is much less likely to occur than agent j does, and they have the same discount factor, then agent i allocates much less wealth on s than agent j does and, hence, is driven out of the market on s.

Lemma 2 can be used to determine who survives in simple examples. Here is an example in which an agent whose beliefs weakly merge with the truth is driven out of the market although no other agent eventually makes accurate next period predictions.

42

Proofs and IntuitionProofs and Intuition-The example (1)-The example (1) The true probability of state a is 1, and agent 1

believes that a will occur next period with probability , so agent 1 eventually makes accurate next period predictions.At period 0, agent 1 believes that state a will always occur until period t with probability .

Let t(k), , be the smallest natural number such that .At periods t(k), agent 2 believes that state a will occur next period with probability 0.5.In all other periods, agent 2 believes that state a will occur next period with probability 1.Agent 2 does not eventually make accurate next period predictions because, infinitely often, agent 2 believes that state a will occur next period with probability 0.5.

1/1exp t

01

1exp1

0

1

t

t

kt k

sdPNk

kkt skdP 5.01

)(

43

Proofs and IntuitionProofs and Intuition-The example (2)-The example (2) At period t, t(k)<t≦t(k+1), .Hence,

By Lemma 2, agent 1 is driven out of the market. If there were another agent, agent 3, who

eventually makes accurate predictions, then By definition, .Thus, by

Lemma 2, agent 2 is driven out of the market. ＊ In the example above, the true distribution is

deterministic.The results of the next section deal with arbitrary stochastic processes.

kt sdP 5.02

001

5.0 2

11)(

2

1

tt

tkk

kt

t

t

sdPsdP

ksdP

sdPsdP

03lim

sdPtt

05.0limlim 2

k

kt

tsdP

44

Proofs and Intuition Proofs and Intuition -Results in Probability Theory-Results in Probability Theory (1) (1)

Lemma 3:For every agent , P-a.s., .Moreover, P-a.s., agent i eventually makes accurate predictions on a path iff . Lemma 4:Assume that the ratio of beliefs and true

probabilities over states of nature in the next period is uniformly bounded away from 0 and ∞. Agent i eventually makes accurate next period predictions on a path iff . Agent i eventually makes accurate next period predictions on a path s iff there existsδ>0 such that .

Ii ,...,10lim

t

it

t dPdP

Ts 0lim

sdPsdP

t

it

t

Ts 0 tit

sit

45

Proofs and Intuition Proofs and Intuition -Results in Probability Theory-Results in Probability Theory (2) (2)

Lemma 4 is not true without assumption that the ratio of beliefs and true probabilities over states of nature in the next period is uniformly bounded away from 0 and∞.

For example, there are two states of nature a & b. At period t, agent i believes that state a will occur

next period with probability .The true probability is 1/t.Agent i weakly merge with the truth.However,

tt /exp

111

)exp(1log11)exp(log1

t

it

t

tt

tt

t

46

Proofs and Intuition –Proofs and Intuition –Proof of Proposition 2-5Proof of Proposition 2-5

Proof of Proposition 2Proof of Proposition 3Proof of Proposition 4Proof of Proposition 5

47

Proof of Proposition 2Proof of Proposition 2 If there is an agent j who eventually makes

accurate predictions on s, by Lemma 3, .If agent i does not

eventually make accurate predictions, by Lemma 3 .Thus, By Lemma 2 agent i is driven out of the market. If agent i eventually makes accurate predictions,

by Lemma 3, .Moreover for for all agents j, on s.Hence, on s for all j.

By Lemma 2, agent i survives on s.

sdPsdP jttt /lim

0/ tti

t sdPsdP 0/ tj

ti

t sdPsdP

0/lim sdPsdP ti

tt

0/lim sdPsdP jttt

0/lim sdPsdP jt

itt

48

Proof of Proposition 3 (1)Proof of Proposition 3 (1) Let , and let . By the law of iterated

expectations, .By the law of large number for uncorrelated random variables,

If , by definition, . Hence,

NkIxPPy kx

kxk ,,...,1,/log

1~

kxk

Pxk

xk SyEyz

0xk

P zE

0))(~()(10)()(11

11

ttk

kxk

Pxkt

tk

xk

Pxk sSyEsy

tzEsz

t

ss ji

tk

kik

P

t

i sSyEt 1

1~1log lim

tk

kj

kP

t

j sSyEt 1

1~1log lim

0expexploglog

0loglog1(**)

1

1

1 1

1 1suplim

ttk

jt

tjtk

it

ti

ttk tk

jijt

it

tk tk

jijt

it

t

sysytsysy

tsysyt

49

Proof of Proposition 3 (2)Proof of Proposition 3 (2)

By definition,

By Lemma 2, agent i is driven out of the market on s.

0

expexp

1

1

tjt

it

tj

ti

jt

it

tkj

t

tkit

sdPsdP

sdPsdP

sysy

50

Proof of Proposition 4Proof of Proposition 4 Assume that agent i always makes inaccurate next

period predictions on a path s, that agent j eventually makes accurate next period predictions, and they have the same discount factorβ, by Lemma 4, and

logsj

logsi

tk

kj

kP

tsSyE

t 11

~1suplim≦ δ ＜ 0

By proposition 3, agent i is driven out of the market.

51

Proof of Proposition 5 (1)Proof of Proposition 5 (1) Let , by def,

Let By the law of large numbers for uncorrelated random variables,

kkkli

ki

kikl dPdPPdPdPP 1,1, ,

lr rk

irk

kl

ikl

PP

PP

1,

, loglog

k

lr

irk

Pk

kl

iklPi

kl SESPP

Es ~~log1,

,,

∵ By the law of iterated

expectations,

rk

irki

rk PPlog

lri

rki

kly 1,

01

0~1

1,

1

1,,

tlk

ikl

lr

irk

tlk

ki

klPi

kl

sst

sSyEsyt

52

Proof of Proposition 5 (2)Proof of Proposition 5 (2) Assume that agent j eventually makes accurate

next period predictions, and . By Lemma 4, .Thus,

Therefore,

By the same argument given in proposition 4, agent i is driven out of the market.

0, kk s

isl PPd

0,0 , iklk

jk s

01suplim1

11

1,

1 1

1 11

suplim

suplimsuplim

tk

ikl

tk lr

irk

t

lr tk

irk

ttk

ik

t

st

st

st

st

l

loglog ij

tk

ik

ts

tl

1

1suplim

53

Convergence to Rational Expectations (1)Convergence to Rational Expectations (1) Def 9Def 9:A rational expectations equilibrium is a

probability measure , share prices , dividends , and initial shares of the

trees , such that maximizes

Def 10Def 10:Given a set of agents , an -

rational expectations equilibrium in which all agents have no share of the trees.

P Ntpp t ,ˆˆ Ntee t ,ˆˆ

ik 1 Ntkc it

it ,ˆ,ˆ

1ˆ,ˆˆ

0,0,ˆˆˆ..

1,

1,

1

1

0

ˆ

I

i

itm

M

mtm

I

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54

Convergence to Rational Expectations (2)Convergence to Rational Expectations (2)

Def 11Def 11:Let be a norm on a finite-dimensional Euclidean space.Given two arrays

, let The metric measures the distance between

arrays of measurable functions.Share prices, dividends, consumption goods, and share holdings may be represented by arrays.

Def 12Def 12:Given a t-history ,and array , let be the induced array of

measurable functions given by

2

),( Njzz j

),ˆ(ˆ& Njzz j

2

;1ˆˆ, sup szszzzd jj

Tsljl

ldjS~

tt Ts ),( Njzz j

),( , Njzz jss ttjS~

Tssszsz tjtjst

~,~,~,

55

Convergence to Rational Expectations (3)Convergence to Rational Expectations (3) Def 13Def 13: A sequence of induced arrays weakly

converges to z on s (denoted ) if .Analogously, a sequence of probability

measure weakly converges to Q if Def 14Def 14:The economy weakly converges to an -

rational expectations equilibrium on s, if there is an -rational expectations equilibrium

such that , and

tszzz w

st 0, tsl zzd

t

Nl tsQ 0, tsl QQd

t

I

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eepp ws

ws tt

ˆ,ˆ .ˆ,ˆ PPIi ist

56

Convergence to Rational Expectations (4)Convergence to Rational Expectations (4) Proposition 6:Assume that all agents have the same

intertemporal discount factor and some eventually make accurate predictions.Then, in every equilibrium, P-a.s., in every path the economy weakly converges to an -rational expectations equilibrium.Moreover, the set is the nonempty set of agents who eventually make accurate predictions on s.

The intuition is that agents who are driven out of the market ultimately do not influence prices, and by proposition 2, all surviving agents must be making accurate predictions.Hence, prices are eventually determined by agents with almost correct beliefs.

Ts sI

sI

57

Convergence to Rational Expectations (5)Convergence to Rational Expectations (5) Using sup-norm, instead of the , we defined

convergence to rational expectations. An open question is whether, under the

assumptions of proposition 2, convergence to rational expectation also obtains.

Weakly convergence to rational expectation also obtains under assumptions of proposition 5, i.e., all agents have the same intertemporal discount factor;and that some agents eventually make accurate next period predictions.

ll dd &

58

Conclusion Conclusion

If markets are dynamically complete and agents have the same discount factor, then all agents who eventually make accurate predictions survive. All agents who do not eventually make accurate predictions are driven out of the market.Hence, share prices converge to share prices of a rational expectations equilibrium.