Transcript Do Markets Favor Agents Able to Make Accurate Predictions?

Do Markets Favor Agents Able to
Make Accurate Predictions?
Alvaro Sandroni
Reporter:Lena Huang
1

Introduction
 A Model of Reinvestment
 Endogenous Investment and Savings –Examples
 A Model of Endogenous Investment and Savings
 Basic Concepts
 Predictions and Survival
 Convergence to Rational Expectations
 Conclusion
2
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.
3
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.

4
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.
5
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.
6
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.

7
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.
8
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.
9
A Model of Reinvestment (2)
Some notations~
- 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 st  T t , t  N , a cylinder with

base on st is the set C(st) ={s  T s  ( st ,...)} of all
finite histories whose t initial elements coincide with st.
~
- S t :the σ-algebra consisting of all finite unions of
cylinders with base on Tt.
~
~
~
0
- S ≡∪ tN S;t hence, Sis the smallest σ-algebra that
10
contains S~ 0 .
A Model of Reinvestment (3)

The true stochastic process of states of nature is
given by an arbitrary probability measure P
~) .
defined on (T∞, S
 Given a t-history st  T t , let Ps be the posterior
probabilities of P, defined by :
t
P ( As )

s
T
Where
is
the
set
of
all
paths
A
t
Ps ( A) 
s
P (C ( st )) such that s=(st , ~s ), ~s  A .
t
t
t

~
Let
E ( St )( s ) be the expectation
operators associated with P and Ps , respectively.
EP and
P
t
11
A Model of Reinvestment (4)

A decision maker has initial wealth w0=1.At
period t, she chooses a portfolio
at  a1,t ,...,aM ,t 
M
among M assets such that  am ,t  1.
m 1
 An investment strategy is a sequence of portfolios
a  (at ,t  N  ) ,and gross returns of assets are given
~
by the positive St -measure functions rt  r1,t ,...rM ,t 
 If consumption is a fixed portion,1-δ,of wealth,
then agents who invest under strategy a
t
t
accumulate wealth w(a)t    ak 1rk .
k 1
 The entropy of a portfolio at is E P logat rt 1  S~t 
12
A Model of Reinvestment (5)


Proposition 1: Let α1 and α2 be investment strategies such
2
1
that
are
uniformly
bounded away from

and
r
t rt 
1
t t 1
zero and infinity.If there is
such
,
  0 that, for all
~
~
P
1
P
2
t

N







E
log

r
S

E
log

r
S
P-a.s., then

t t 1
t.
t t 1
t  
2
w( )t
t
 0

1
w( )t


Proof :Let
,
.By
~
2
1
yk  log
 k 1rk  k 1rk zk  yk .By
 E Pthe
yk law
S k 1 of
the law of iterated
expectations,
large numbers for uncorrelated random
E P zk  variables,P-a.s.,
0



1
1
~
P
P
lim  z k  E  z k   0  lim  yk  E yk S k 1  0
t  t 1 k t
t  t 1 k t
By assumption, lim sup 1  E P ( yk S~k 1 )  0. Hence,
t 1k t
2
2




w

w

1
t
t

y






log



0
lim sup  yk  0 1k t k t
1
1
w t
w t
t 1k t
t 
t 
13
A Model of Reinvestment (6)

The MEL rule is the myopic investment strategy
a* defined by at*  arg max E P {log( at rt 1 ) S~t }.
 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.
14
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
15
general conditions.
Endogenous Investment and Savings
-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.
16
Example 1 of Endogenous Investment
and Savings (1)

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.
17
Example 1 of Endogenous Investment
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
i
are given by cti & k miLet
be
w
.
t agent i’s wealth,
,t
wti   p1,t  1k1i,t 1  p2,t k 2i ,t 1
wti  p1,t k1i,t 1   p2,t  2 k 2i ,t 1

(State l )
(State h)
Agents’ maximizing problem is :

1
2

Max E  (0.5) t u i (cti )
u
(
c
)

2
c
,
u
(c)  log(c)

t 0

s.t.
cti  p1,t k1i,t  p2,t k 2i ,t  wti
pi
ct1  ct2  et , k11,t  k12,t  1, k 21,t  k 22,t  1
18
Example 1 of Endogenous Investment
and Savings (3)

0.5t
From two agents’ f.o.c, 0.5   n 0.5   t n
1
 2 (A)
1

ct
ct2
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,
t
0.5t
0.5   n 0.5   t n

0.50.5

2

  (B)
 t

 0.5   0.5    
By (A) and (B),
1
t
ct2

 ct2 t
 0

c
t
 0

( ct1  2)
(C)
(D)
19
Example 1 of Endogenous Investment
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.
 1 2 
2 2
  0.5 log

3

2
6

2




The expected log of agent 1’s saving ratio: 0.5 log
is greater than the expected log of agent 2’s saving ratio (log(0.5))

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 20
wealth of agent 2 small.
Example 2 of Endogenous Investment & Savings
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.
ct2
1 1
2

,
c

c
t
t  et
2
1

ct
 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.
21
A Model of Endogenous Investment & Savings
-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. (iI1 k mi , 1  1)
-Dividends: et  e1,t ,...eM ,t ; e=(et , t  N )
-Share price: pt   p1,t ,... pM ,t ; p=(pt , t  N )
-Agent i’s consumption: cti , c i  cti , t  N  
-Share holding: kti  k1i,t ,...k Mi ,t  ; k i  kti , t  N  
*Hence, agent i’s wealth is given by wti   pt  et kti1
22
A Model of Endogenous Investment
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 S~t-measurable function defined by
 pt 1 s 1  et 1 s 1 


H t s   
:

 p s L   e s L 
t 1
 t 1

-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.
23
A Model of Endogenous Investment
And Savings (2)

Agents’ maximizing problem is

~
i r i
i





Max E    u ct r St 
 r 0

s.t. ctir  pt r ktir   pt r  et r  ktir 1 , wtir  0, ctir  0, r  N 
Pi
I
M
 c   em,t
i 1

i
t
m 1
I
,  k mi ,t  1
i 1
(m  1...M )
Where ui is a strictly increasing, strictly concave,
continuously differentiable utility function that
satisfies the Inada conditions.
24
Basic Concepts

Survival
 The Accuracy of Agents’ Predictions
 Entropy
25
Survival

Def 1:Agent i is driven out of the market on
i

a path s T if agent i’s wealth, wt s ,
converges to zero as t goes to infinity. Agent
i survives on a path
ifs T
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.
26
The Accuracy of Agents’ Predictions (1)
~
 The difference between two probability measures Q & Q
~
~
 A  Q  A
Q

Q

Q
sup
in the sup-norm, is given by
~
AS
~
 The difference between Q & Q , in the dl-metric, is
~
~


 A .
d
Q
,
Q

Q
A

Q
given by l
max
~


ASl
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 lperiods.
27
The Accuracy of Agents’ Predictions (2)

Def 2: Agent i eventually makes accurate
predictions on a path s T , s=(st,…), if Psi  Ps t
 0

 Def 3:Agent i eventually makes accurate next
period predictions on a path s T , s=(st,…), if
d1 Psi , Ps  t
 0 .

 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.
t
t
t
t
28
The Accuracy of Agents’ Predictions (3)

Def 4: :Agent i eventually makes inaccurate next
period predictions on a path s T , s=(st,…), if
there isε>0 such that d1 Psi , Ps   
t  N .
 Clearly, an agent who does not eventually make
accurate next period predictions need not always
make inaccurate next period predictions.
 Def 5:Some agents eventually make accurate (next
period) predictions if, P-a.s., in every path s T  ,
there exists at least one agent who eventually
makes accurate (next period) predictions on s.
(Not necessarily the same agents on different paths)
t
t
29
Entropy (1)
Let dQt ( s)  QC st , then the probability of the
dQt s 
states of nature at period t, Qt, is defined as Qt s  
dQt 1 s 
i
 In particular, P & P are the true probabilities and
t
t
agent i’s beliefs over states of nature at period t,
given past data, respectively.
 Def 6 :The entropy of agent i’s beliefs at period
i
t,  ti, is given by i


~
P
P
t 1
 St  Write on the
 t  E  log
  Pt 1   blackboard
  ti  0and it is zero iff agent i’s belief and the true
probabilities over states of nature in the next
period are identical.
30

Entropy (2)

Def 7:The entropy of agent i, is given by:
1
  log(  )  lim   ki
t  t 1 k t
i

i
Hence, the entropy of an agent does not depend on
the characteristic of the other agents.
 Def 8:The ratio of beliefs and true probabilities
over states of nature in next period uniformly
bounded away from zero and infinity if there
i
P
exists u>0 and U<∞ such that u  t s   U
Pt s 
31
Predictions 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
32
Main Results-Proposition 2 (1)

Proposition 2:Assume that all agents have the same
intertemporal discount factor and some agents eventually
make accurate predictions.Then in every equilibrium, Pa.s.:
1.Any agent who does not eventually make accurate
predictions on a path s T  is driven out of the market on
the path s.
2.Any agent who eventually makes accurate predictions on

a path s T 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.
33
Main Results -Proposition 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.
34
Main Results -Proposition 2 (3)

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.
35
Main Results -Proposition 3 (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 s T ,  i (s)   j (s) , then agent i
is driven out of the market on s.(remind entropy)
36
Main Results -Proposition 3 (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.
37

Main Results -Proposition 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 s T ,then
agent i is driven out of the market on s. (However)
38
Main Results -Proposition 5

Proposition 5:Under the same assumptions of
proposition 4, in every equilibrium, P-a.s., if there
exists l  N andε>0 such that dl Psi , Ps    t  N
on a path s T , s=(st,…), then agent i is driven
out of the market on s. Stronger than proposition 4
 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.
t
t
39
Proofs and Intuition-Basic Results (1)

Agents’ f.o.c of the maximization problem imply
that,  i t dPt i  u i ' cti  u i ' c0i 

  dP  u  c  u  c 
j
t
j
t

j
'
j
t
j
'
j
0
(*)
Lemma 1:In every equilibrium, for any path s T 
i
and agent i  1,..., I , cti s  t


0
iff
w
 0
t s  t



Lemma 2:Fix an agent i  1,..., I and a path s T
In every equilibrium,
if there exists an agentj  1,... I 
t

 i  dPt i s 
 0 then agent i is driven
such that j t j t

  dPt s 
out of the market on s. Moreover, if for all agents
i t

 dPt i s   then agent

there exists ε>0 such that
j t
j


s 

dP
t
i survives on s.
40
Proofs and Intuition-Basic Results (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.
41
Proofs and Intuition-The example (1)

The true probability of state a is 1, and agent 1
believes that a will occur next period with
probability exp  1/ t  1 , so agent 1 eventually
makes accurate next period predictions.At period 0,
agent 1 believes that state a will always
occur until
t 1  1

period t with probability dPt1 s   exp  
 0.
 t

 k 0 k  1 
 Let t(k), k  N, be the smallest natural number such
k
that kdPt1( k ) s   0.5 .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.
42
Proofs and Intuition-The example (2)

k
At period t, t(k)<t≦t(k+1), dPt 2 s   0.5.Hence,
dPt s 

2
dPt s 
1
dPt1( k ) s 
0.5
k
dPt1 s 
1
 k
 0 
t
 0


2
k
dPt s 
By Lemma 2, agent 1 is driven out of the market.
 If there were another agent, agent 3, who
eventually makes accurate predictions, then lim dPt 3 s   0
t 
k
2
By definition, lim dPt s   lim 0.5  0.Thus, by
t 
k 
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.
43
Proofs and Intuition
-Results in Probability Theory (1)
Lemma 3:For
every agent i  1,..., I , P-a.s.,
i
dPt
  lim
 0 .Moreover, P-a.s., agent i
t  dPt

eventually makes
accurate
predictions
on
a
path
s
T
dPt i s 
 0.
iff   lim
t  dPt s 
 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
 0 . Agent i eventually makes
a path s T iff  ti t

accurate next period predictions on a path s iff there
existsδ>0 such that  ti s   .

44
Proofs and Intuition
-Results in Probability Theory (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 exp  t  / t .The true
probability is 1/t.Agent i weakly merge with the
exp( t )
truth.However, 1
1
1


 ti  log exp( t )  1   log
t
 t
t
1
1
t
t
 1

45
Proofs and Intuition –
Proof of Proposition 2-5

Proof of Proposition 2
 Proof of Proposition 3
 Proof of Proposition 4
 Proof of Proposition 5
46
Proof of Proposition 2

If there is an agent j who eventually makes
accurate predictions on s, by Lemma 3,
limt  dPt s  / dPt j s    .If agent i does not
eventually make accurate predictions, by Lemma 3
i
j



dP
s
/
dP
 0
dPt i s  / dPt s  t
 0.Thus,
t
t s  t


By Lemma 2 agent i is driven out of the market.
 If agent i eventually makes accurate predictions,
by Lemma 3, limt  dPt i s  / dPt s   0 .Moreover
for for all agents j, limt  dPt s  / dPt j s   0 on
i
j



dP
s
/
dP
s.Hence, limt  t
t s   0 on s for all j.
47
By Lemma 2, agent i survives on s.
Proof of Proposition 3 (1)

Let ykx  logPkx / Pk , x  1,..., I , k  N , and let
x
x
P
x ~
z k  yk  E yk S k 1 . By the law of iterated
expectations, E P zkx   0.By the law of large
number for uncorrelated random variables,


1
1
x
P
x
x
P
x ~
 0   yk ( s)  E ( yk S k 1 )( s) t
 0
 zk ( s)  E ( zk ) t


t 1k t
t 1k t


1
P
i ~
 If  s    s , by definition,log   lim  E yk S k 1 s 
t  t 1 k t
1
j
P
j ~
 log   lim  E yk S k 1 s . Hence,
t  t 1 k t
i
i
j

(**)
1
lim sup   yti s    ytj s   t log  i  log  j   0
t 
t  1k t
1k t

1k t
  yti s    ytj s   t log  i  log  j
1k t

      exp 
  exp 
i t
t 
i
y
1k t t s 
j t
1k t
yt  s 
j
48t
 0

Proof of Proposition 3 (2)

By definition,
exp 1k t yti s  dPt i s 

j
exp 1k t yt s  dPt j s 

 

 
i t
j t

dPt i s 
t
 0

j
dPt s 
By Lemma 2, agent i is driven out of the
market on s.
49
Proof 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,  j s   log  and
1
 i s   log   lim sup  E P ykj S~k 1 s 
≦δ＜0
t 1k t
t 
By proposition 3, agent i is driven out of
the market.
50
Proof of Proposition 5 (1)
i
i
i
P

dP
dP
 Let l ,k
k 1
k , Pl ,k  dPk 1 dPk , by def,
 Pl i,k 
 Pki r 


log
log


 P   1
P
r l


l
,
k
k

r



i

 Pl ,k  ~ 
i
P
 S k   E P    ki  r S~k 
 l ,k s   E  log
 
 1r l

P
l
,
k

 



i
k r
 Pki r
 log 
 Pk  r



Let yli,k  1r l  ki r By the law of large numbers for
uncorrelated random variables,


 
1
~
i
P
i


y
s

E
y
S
 0
 l ,k
l ,k
k s  t

t 1k l
1
     ki r s    li,k s  t
 0


t 1k l  1r l
51
Proof of Proposition 5 (2)

Assume that agent j eventually makes accurate
i

d
P
next period predictions, and l s , Ps     0 .
i
 By Lemma 4,  kj s  k
.Thus,


0
,

l ,k    0

k
k
1
1
i



s

 k
 lim sup   ki r s 
t 1k t
t 1k t
1 r l
t 
t 
1
1
 lim sup    ki r s   lim sup   li,k s   0
t 1k t 1r l
t 1k t
t 
1
 Therefore,  j  log    i  log   l lim sup  ki s 
t 1k t
t 
l lim sup

By the same argument given in proposition 4,
agent i is driven out of the market.
52
Convergence to Rational Expectations (1)

Def 9:A rational expectations equilibrium is a
probability measure Pˆ , share prices pˆ   pˆ t , t  N  
, dividends eˆ  eˆt , t  N   , and initial shares of the
trees kˆi 1 , such that cˆti , kˆti , t  N   maximizes

t

E   i  u i cti 

t 0

ˆ t kti   p
ˆ t  eˆt kti1 , wti  0, cti  0
cti  p
ˆ
P
s.t.
I
M
I
m 1
i 1
 cˆ   eˆm ,t ,  kˆmi ,t  1
i 1

i
t
Def 10:Given a set of agents Iˆ  1,...I , an Iˆrational expectations equilibrium in which all i  Iˆ
agents have no share of the trees.
53
Convergence to Rational Expectations (2)

Def 11:Let
be a norm on a finite-dimensional
2
Euclidean space.Given two arrays z  ( z j , j  N  )


& zˆ  ( zˆ j , j  N  ), let d l  z, zˆ    sup z j s   zˆ j s  2 
 1 j l ;sT

The metric d l measures the distance between
arrays of S~j measurable functions.Share prices,
dividends, consumption goods, and share holdings
may be represented by arrays.
t
 Def 12:Given a t-history st  T ,and array z  ( z j , j  N  )
~
, let z s  ( z s , j , j  N  ) be the induced array of S j 
measurable functions given by zs , j ~
s   zt  j st , ~
s , ~
s T 
t
t
t
54
Convergence to Rational Expectations (3)

Def 13: A sequence of induced arrays z s weakly
w
converges to z on s (denoted z s 
 0
z) if d l z s , z  t

l  N .Analogously, a sequence of probability
measure Qs weakly converges to Q if d l Qs , Q  t
 0

 Def 14:The economy weakly converges to an Iˆrational expectations equilibrium on s, if there is
ˆ, p
an Iˆ-rational expectations equilibrium P
ˆ , eˆ, kˆi 1 
w
w
i
ˆ
ˆ
ˆ
ˆ
p


p
,
e


e
such that s
,
and

i

I
,
P
s
s  P.
t
t
t
t
t
t
t
t
55
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 s T  the
economy weakly converges to an Iˆs -rational
expectations equilibrium.Moreover, the set Iˆs 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. 56
Convergence to Rational Expectations (5)

Using sup-norm, instead of the d l & d l, 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.
57
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.
58