The Market Selection Hypothesis
Lawrence Blume and David Easley
Department of Economics
Cornell University
1 June 1999
The authors thank Alvaro Sandroni for stimulating conversation, and The National Science Foundation
for research support under grant SES 9808690. Research support for Blume from the John D. and
Catherine T. MacArthur Foundation is gratefully acknowledged.
1
1
Introduction
It is conventional to assume that traders in asset markets have rational expectations about asset
returns, and choose savings rates and portfolios as if they maximize expected utility using these
beliefs. As the hypothesis that traders are expected utility maximizers places few restrictions
on behavior in the absence of the rational expectations hypothesis, much attention has been
focused on the validity of assuming rationality (correctness) of expectations. Although traders
would certainly prefer to hold accurate rather than inaccurate beliefs, an explanation of how
traders come to correctly forecast endogenous equilibrium rates of return is lacking.
Various approaches have been devised to provide the missing foundation for the rational
expectations hypothesis. One approach posits that correct beliefs can be learned. In other
words, rational expectations are stable steady states of learning dynamics; see [2] and [5] for
more discussion. The learning approach is not completely satisfactory. Achieving rational expectations through learning requires too much prior knowledge, to the point where the requirements
for learning essentially assume the conclusion. An alternative approach based on evolutionary
forces operating through wealth dynamics has arisen in recent years. This approach actually has
a long tradition. Both Alchian [1] and Friedman [7] argued in the early 1950’s that evolutionary
forces would eventually result in behavior consistent with correct maximization. Although their
argument is plausible, until recently there was no careful analysis of the market dynamics that
would supposedly select for expected utility maximizers, and, within the class of expected utility
maximizers, select for those with rational expectations.
In [3] we analyzed an economy with repeated markets for risky, one period assets. We
showed that if savings rates are equal across traders, then wealth dynamics do not necessarily
lead to traders acting as if they maximize expected utility using rational expectations. Expected
utility maximization may fail to emerge because the market selects for traders whose investment
portfolios generate higher expected growth rates of their share of wealth. Although our primary
focus was not on the link between portfolio rules and expectations, we showed that a trader with
logarithmic utility and correct expectations maximizes the expected growth rate of wealth share
and so dominates the market. However, traders with correct expectations and non-logarithmic
utility need not maximize the expected growth rate of wealth share and so can be driven out of
the market even by traders with incorrect expectations.
Our analysis addressed the possible emergence of expected utility maximization. The
question of belief selection among expected utility maximizers was examined by Sandroni [9],
who analyzed an economy with infinitely lived risky assets. He showed that if markets are
dynamically complete, and some assumptions are made on returns, then when savings rates are
endogenous, and all traders are expected utility maximizers with a common discount factor, only
traders with rational expectations survive. This occurs because when markets are complete,
traders can place bets on any disagreement about the probability of states and traders with
correct expectations will win the bets. So in Sandroni’s world the market selects for those
traders whose expectations are correct. This analysis differs from [3] in that savings behaviors are
2
endogenous, and the exogenous parameters are the primitives involved in describing preferences,
such as payoff functions and discount factors.
In this paper we explore more completely when selection occurs, when it does not occur
and why. We return to the asset structure of [3], so first we show that with our assets, market
completeness and some ancillary assumptions imply selection for rational expectations. Second,
we show that dynamic completeness of markets is necessary to guarantee selection for rational
expectations. In economies with incomplete markets, a trader who is overly optimistic about the
return on some asset in some state can choose to save enough to more than overcome the poor
asset allocation decision that his incorrect expectations create. This result is even more striking
than it seems, because when traders’ beliefs are heterogeneous, some market incompleteness
is inevitable. For with heterogeneous beliefs, market completeness implies that traders can
bet on any differences in beliefs. This amounts to opening a new set of markets every time a
new trader with different beliefs enters the economy. In the context of the model, the relevant
state space contains the union of the supports of each player’s beliefs. Thus adding a new
trader can require expanding the state space, and therefore adding new markets. We conclude
that, within the evolutionary framework, the conditions required to ensure market selection for
rational expectations are too strong to be useful. In general there are no market impediments
to long-run heterogeneous beliefs.
2
The Model
Our model imagines a group of traders repeatedly buying shares in exogenously supplied assets.
The assets deliver units of a single consumption good, corn. Traders are infinitely-lived, and all
assets have a life of one period. That is, each asset available today promises delivery of corn
tomorrow. Asset returns are uncertain, and will be modeled as functions of some stochastic
process of states. An equilibrium is a stochastic process of prices for assets in terms of corn with
the property that the asset markets clear when traders are optimizing given their beliefs about
the future evolution of the economy.
2.1
Notation and Basics
Formally, we assume that time is discrete, beginning at date 1. Furthermore, the state available
at each date comes from a finite set {1, . . . , S}. The set of all sequences of states is Σ, and Σ
together with its product σ-field is the measurable space on which everything will be built. Let
p denote the “true” probability meaasure on Σ. For many of the examples we construct, the
stochastic process of states will be assumed to be i.i.d. In this case p = (p1 , . . . , pS ) will refer
to the single-period probability distribution on states. In the next few paragraphs we introduce
a number of random variables of the form xt (σ). All such random variables are assumed to be
date-t measurable; that is, their value depends only on the realization of states through date t.
3
For a given path σ, σt is the state at date t and σ t = (σ1 , . . . , σt ) is the partial history through
date t of the evolution of states.
There are A assets. The number Rta (σ) is the amount of corn asset a, which is traded at
a
date t − 1, pays off at date t in realization σ. At date t − 1 the price of this asset is qt−1
(σ).
∞
A price sequence is a collection of random variables q = {qt }t=1 , where qt is the price vector
(qt1 , . . . , qtA ). At each date t, 1 unit of each assetPis available to the economy. Thus the total
supply of corn at date t along path σ is Rt (σ) = a Rta (σ).
The economy contains I infinitely lived traders. Each trader has a discount factor βi < 1
and a single-period payoff function ui : R → R defined on units of corn consumption. A consumption plan ci for trader i is a collection of random variables {cit }∞
t=1 . For
P a given consumption
plan ci and path σ, the utility of the consumption plan is v i (ci )(σ) = t βit−1 ui cit (σ) . Each
payoff function ui is C 1 , strictly increasing, strictly concave and satisfies an Inada condition at
0
the origin: limc→0 ui (c) = +∞. Each player i also has beliefs pi , which is a probability measure
on Σ. In the i.i.d. case, pi will refer to a probability measure on {1, . . . , S}. In every case, E i
refers to the expectations operator for beliefs pi .
At the beginning of each date t on path σ, all the assets pay off. Total corn wealth is
Rt (σ), and trader i owns share rti (σ). (The wealth shares sum to 1.) Thus trader i’s date t
wealth on path σ is wti (σ) = rti (σ)Rt (σ). At date 1 traders have an aggregate endowment R1 of
corn, and trader i’s share is r1i > 0. These date 1 parameters are exogenously given.
2.2
Demand and Equilibrium
Each trader i chooses a savings plan and a portfolio plan to maximize the expected value of
v i (ci )(σ) given his beliefs pi . A savings plan δ i for trader i is a collection of [0, 1]-valued random
variables {δti }∞
t=1 , which record the fraction of current wealth which is to be invested in assets
for tomorrow. The remaining fraction is eaten. A portfolio plan αi for trader i is a collection of
i
i
i
random variables {αti }∞
t=1 such that each αt = (α1t , . . . , αAt ) is a vector representing the shares
of savings going to each asset a. The coefficients of this vector sum to 1, but it may have
negative terms which correspond to short sales. Thus trader i’s optimization problem is to solve
the problem
P∞ t−1
i
i
ui ct (σ)
P :
maxci ,δi ,αi E
t=1 βi
1 − δti (σ) wti (σ)
s.t. cit (σ) =
A
(1)
i
X
(σ)δti (σ)wti (σ) a
αat
i
Rt+1 (σ)
wt+1 (σ) =
qta (σ)
a=1
w1i (σ)
=
r1i R1
A sufficient condition for the existence of optimal plans is that feasible consumption not grow
to fast. See [8, Theorem 16.12, pg. 116]. Although this dynamic programming problem is not
4
stationary, it nonetheless is solvable by backward induction. Let P (σ t ) denote the dynamic
programming problem beginning at time t after partial history σ t .
Equilibrium in this model is given by the condition that asset markets clear. That is, for
all paths σ, dates t and assets a,
I
X
αi (σ)δ i (σ)wi (σ)
t
i=1
t
t
qta (σ)
=1
This condition can be rewritten in terms of wealth shares.
I
X
αi (σ)δ i (σ)ri (σ)
t
i=1
t
t
a
qt (σ)
Rt (σ) = 1
(2)
Total wealth is exogenously given, and so (2) shows that prices are determined by the optimal
savings and portfolio plans of each agent, and the path of wealth shares.
3
Evolution
The evolution of wealth shares determines both the fate of individual traders and the path
of market aggregates such as equilibrium prices. Here we work out the implications for both
individual selection and the long-run behavior of market price.
3.1
Selection across Traders’ Characteristics
Let 1st (σ) denote the indicator function for state s at date t; that is, it equals 1 if σt = s and 0
otherwise. The evolution of trader i’s wealth share is given by the equation
i
rt+1
(σ)
s
s X
A
a
i
Y
(σ) 1t (σ)
αat
(σ)δti (σ)wti (σ) Rt+1
=
qta (σ)
Rt+1 (σ)
s=1 a=1
s
s X
A
a
i
Y
Rt+1
(σ) 1t (σ)
αat
(σ)δti (σ)rti (σ)
=
Rt (σ)
a
q
(σ)
Rt+1 (σ)
t
s=1 a=1
(3)
i
In the first equation, δti wti is the amount of wealth trader i saves. Fraction αat
is invested in
asset a, and so the fraction represents the number of shares of asset a purchased by trader i.
Multiplying this by Rta gives the wealth next period that trader i accrues from his holdings of
asset a. Summing over all the assets gives the gross return on his investment, and dividing by
Rt turns that into a share. Using the indicator function as an exponent and multiplying over
all states picks out the right state realization for the path σ.
5
The equilibrium condition (2) determines the equilibrium price q a of asset a as a function
of the savings, wealth shares and portfolio decisions of all traders. Substituting from (2) into
(3) gives
i
rt+1
(σ)
s X
A
Y
=
s=1
a=1
a
i
(σ)
αat
(σ)δti (σ)rti (σ) Rt+1
P j
j
j
R (σ)
j αat (σ)δt (σ)rt (σ) t+1
1st (σ)
(4)
This equation identifies trader i’s share tomorrow on path σ as the average of his share of
investment in each asset, weighted by the share of total wealth tomorrow that asset generated.
Our interest is in the long-run fate of traders. Traders can vanish if their wealth share
goes to 0, or survive if it does not. Formally,
Definition 3.1. Let S ⊂ Σ denote a measurable set of paths. Trader i vanishes on S if for
p-almost all σ ∈ S, limt rti (σ) = 0. If trader i does not vanish, he survives.
The criteria for survival worked out in [3] apply in this model too. These criteria are
based on the notion of relative entropy.
Definition 3.2. The relative entropy
P of a probability measure ρ on S with respect to the probability measure q on S is Iq (ρ) = s qs log(qs /ρs ).
The relative entropy is a way of measuring a distance from ρ to q. Clearly this function
is minimized at ρ = q, where it takes the value 0.
In [3] we directly analyzed the limit behavior of (3) using appropriate strong laws of
large numbers. A weakness of our approach there is the difficulty in working backwards from
behaviors to primitives such as preferences. Here we characterize the long run distribution
of wealth directly in terms of preference primitives, in a manner similar to [9]. Our methods
directly extend the Euler equation techniques of [4] to the stochastic environment of the model
developed here. The main result in this section is a characterization of who vanishes and who
survives based on dynamically complete markets with bounded asset returns. All assets pay
off in the numeraire good, and so the necessary and sufficient condition for dynamic market
completeness is that each A × S matrix of asset returns have rank S. Let Rt (σ) denote the
asset returns matrix at date t on path σ. This is the matrix whose as’th element is Rta (σ) where
σt = s. This matrix-valued random variable is measurable with respect to state observations
through date t. The following Theorem is the basic result which enables the demonstration
of various propositions about market selection for traders with heterogeneous preferences and
beliefs in dynamically complete markets.
This Theorem compares two traders, i and j. Let
Zt (σ) = log pj (σ τ +1 |σ τ )/pi (σ τ +1 |σ τ ) . Notice that the conditional expectation E(Zt |Ft−1 ) with
respect to the probability distribution p is a difference of relative entropies with respect to
p( · |σ t−1 ).
Theorem 3.1. Consider 2 agents i and j. Suppose that the following conditions hold:
6
i. For all t and σ the asset returns matrix Rt (σ) has rank S.
ii. Every trader’s utility function is bounded from below.
iii. There are B > R > 0 such that for all assets a and dates t, B > Rta (σ) > R;
P −2
iv.
Var(Zt |Ft−1 ) < ∞;
tt
P
v. lim supt t−1 tτ =1 E{Zτ |Ft−1 } > log(βj /βi ).
Then rtj (σ) → 0 p-almost surely.
Theorem 3.1 concerns the asymptotic behavior of traders consumption and wealth paths
in economies with complete markets. The role of the variance condition iv is to justify the use of
a law of large numbers. It is easy to state conditions implying that iv holds almost surely. For
instance, fix > 0 and suppose that almost surely, each trader believes at every date that each
state tomorrow will happen with probability at least . The weight of the Theorem is carried by
condition v, which is a condition on the distances of beliefs of the two traders from the truth.
For example, if βi = βj , condition v states that the relative entropy of i’s beliefs with respect
to the truth is smaller than that of j’s beliefs. The remaining two conditions are necessary to
move from the conclusion of small consumption to small wealth share. Condition iii bounds
gross returns. This condition implies that rates of return are bounded from below by B/R.
Theorem 3.1 compares the experiences of different traders to determine who survives and
who vanishes. The following Corollaries are obvious consequences of the Theorem.
Corollary 3.1. Suppose that all traders have a common discount factor and at least one trader
has correct expectations, and that conditions i, ii, iii and iv are satisfied. Then for any trader
j, on the set of paths such that
lim sup t
t
−1
t
X
E(log p(σ t+1 |σ t )/pj (σ t+1 |σ t ) < 0
τ =1
rtj → 0 p-almost surely.
Corollary 3.2. Suppose that all traders have the same beliefs, but different discount factors. If
conditions i, ii and iii are satisfied, then only those traders with the highest discount factor in
the population survive, and all others vanish p-almost surely.
The main point of these Corollaries (and of Theorem 3.1) is that so long as the Inada
condition is satisfied and the Euler equations work, payoff functions have nothing to do with
survival. The market selects across discount factors and beliefs, but not across payoff functions.
Unfortunately the hypotheses of the Theorem exclude important utility functions such
as the log function, which can in fact be handled by distinct arguments, and CRRA utility with
7
relative risk aversion coefficients exceeding 1. The difficulty with unbounded payoff functions
comes in ensuring that the Euler conditions used in the proof are in fact necessary conditions
for an optimal path.
The proof of 3.1 is an extension of the Euler equation arguments of [4], applied to the
optimization problem 1 under the assumption of dynamically complete markets. It is not a
best posible theorem. For instance, suppose that trader i has correct beliefs, such that all
conditional probabilities are uniformly bounded away from 0. Suppose that trader j’s beliefs
have the probability of one state rapidly converging to 0. Then trader j will be investing ever
less in this state, which keeps recurring at some fixed rate, so his wealth should go to 0 almost
surely. But the variance condition iv fails, and so the Strong Law argument of the proof will
not work.
Proof of Theorem 3.1. Since markets are complete, we can construct portfolios, one for each
state s at date t, such that the payoff
holding the entire supply of this portfolio, which is
P to
s
a
taken to be 1 share, is Wt+1 (σ) = a Rt+1 (σ t , s) in state s at date t, and 0 in all other states.
These portfolios can be priced from the equilibrium asset prices. Let πts denote the price of the
portfolio that pays off in state s at date t.
In equilibrium each consumer is optimizing, and the solution to (1) induces a solution to
the optimization problem in which shares of these portfolios are traded at the prices π. In this
solution savings are the same, and portfolio allocations are such that the returns to savings at
each date-event pair are identical in the two problems.
At each date agents’ consumptions are bounded above by A · B, and so ui (A · B)/(1 − βi )
is an upper bound on the value of the problem. Since this bound is finite, and since utility
is bounded from below, Euler equations are necessary first-order conditions that optimal plans
must satisfy.
W σt+1 (σ)
u0i cit (σ) = βi pi (σ t+1 |σ t )u0i cit+1 (σ) σt t+1
πt (σ)
Iterating this equation through time along the path,
u0i cit+1 (σ) = βi−t
t+1
2
πtσ (σ) · · · π1σ (σ)
1
0 i
u
c
(σ)
σ
i
1
σ
t+1
2
pi (σ t+1 |σ t ) · · · pi (σ 2 |σ1 ) Wt (σ) · · · W1 (σ)
(5)
Now define the random variable
t
u0i cit (σ)
u0i ci1 (σ)
βj 1 X
1
1
log 0 j = log
+
Zτ (σ) + log 0 j t
βi
t τ =1
t
uj ct (σ)
uj c1 (σ)
(6)
A
[6, Corollary 4.5b, pg. 108] shows that on the set of paths V satisfying condition (iv),
PSLLN
−1
almost surely. The usual Kronecker lemma argument now implies that on
t t Zt converges
P
this set t−1 τ =1t Xτ − E(Xτ |Ft−1 ) converges to 0 almost surely. It follows that on the set V ,
!
t
u0i cit (σ)
1
βj
1X
pj (σ τ +1 |σ τ )
lim log 0 j − log
−
E log i τ +1 τ |Ft−1
→ 0 p a.s.
t→∞ t
βi
t τ =1
p (σ |σ )
uj ct (σ)
8
Let W denote the set of states
0 satisfying
(v), that the term in large brackets has a negative
j
0 i
lim sup . On V ∪ W , ui ct (σ) /uj ct (σ) converges to 0 p-almost surely. This can happen either
because cit becomes large or cjt becomes small. Because aggregate corn stocks are bounded
uniformly from above, cjt must be converging to 0.
Now need to show that rtj (σ) converges to 0. Suppose not. Then we can find a state s
and a subsequence of dates with the following properties: wtjk (σ) ≥ 2, ≤ cjtk (σ) → 0, and
αtjsk (σ) ≥ 1/S, where αtjs is the fraction of savings invested by j at date t in the portfolio that
pays off in state s. Let σst denote the partial history of length t + 1 which agrees with σ through
date t and ends with state s at date t+1. Let vj (w, σ t ) denote the value function for the dynamic
program beginning at node σ t in the date-event tree with wealth w. It is necessary to index
the value function by the entire partial history because the dynamic program is not in general
stationary. Standard arguments show that despite the non-stationarity, the value function is
concave, and the marginal value of wealth equals the marginal utility of consumption in the
current period. Consequently the Euler equation implies
u0j (cjtk σ tk ) = βrs (σ tk )v 0 wtjk (σ tk ), σstk pj (σstk |σ tk )
Since trader j invests at least /A in portfolio s, and since by assumption it pays off at most B,
we have the following inequality:
B
u0j (cjtk σ tk ) ≤ β v 0 wtjk (σ tk ), σstk
A
Since consumption is converging to 0, the left hand side of the inequality converges to +∞,
and therefore so does the right hand side. Furthermore, wtjk (σ tk ) ≥ R/BA (taking account of
the lower bound on the rate of return of the portfolio given by the assumed upper and lower
bounds on gross
returns). Since the value functions are increasing in wealth, it follows that
v 0 R/BA, σstk diverges. But this cannot happen. The value function at any node is bounded
from below by v(0) = u(0)/(1−β). Then, using the subgradient inequality for concave functions,
R 0
v R/BA, σstk ↓ −∞
v(0) = v 0, σstk ≤ v R/BA, σstk −
BA
which is a contradiction.
3.2
Long-Run Asset Pricing
PUT THE ASSET-PRICING FORMULA HERE.
4
Selection
Theorem 3.1 shows that the market selects for rational expectations under some assumptions.
The most important of these assumptions are:
9
1. Traders’ disagreement about the future is limited to differing probabilities on a common
state space. In particular, all traders know the returns on asset holdings and equilibrium
prices in each future state. Disagreement about realizations of these variables can be
modeled as disagreement about the probability distribution on an enriched state space,
but this then calls into question the next important assumption.
2. Markets are dynamically complete. This assumption allows traders to place bets on all
possible disagreements and it is crucial for the analysis. It is an important limitation
particularly in light of the first assumption.
3. Returns on assets are uniformly bounded from above and away from zero. This prohibits,
for example, the case of assets with dividends that grow at a non-zero, constant rate.
The following three example economies show that each of these assumptions is necessary for selection for rational expectations. In each of these economies there is one trader with logarithmic
utility and correct expectations and one trader with CRRA utility and incorrect expectations.
We show that the logarithmic trader vanishes and that equilibrium prices do not converge to
their rational expectations equilibrium values. We endow the trader with correct expectations
with logarithmic utility for two reasons. First, it is easy to compute the optimal policy for this
utility function. Second, and more important, we know from our previous work [1] that if all
traders save at the same rate a trader with logarithmic utility and correct expectations will survive and any trader whose portfolio is not asymptotically equivalent to the log traders portfolio
will vanish. Thus our examples give selection for rational expectations the best possible chance
of success.
ADD BELIEFS EXAMPLE HERE
Example I:
This first example illustrates the effects of traders disagreeing about what will happen in the
exogenously given states.
There is one state and one asset with a gross return of 1 at each date. There are two
traders. Trader 1 has logarithmic utility, u1 (c1t ) = log c1t , and trader 2’s utility function exhibits
constant relative risk aversion (CRRA), u2 (c2t ) = γ −1 (c2t )γ , for γ < 1, γ 6= 0. Both traders have
discount factor β = 1/2. Trader 1 knows the return process and trader 2 believes at each date
t that the gross return on the asset will be
e
Rt+1
9 + 6(2/3)t−1
.
=
4 + 4(2/3)t−1
(7)
The return trader 2 expects is greater than one at each date and converges to 9/4 as t → ∞.
Actutal total wealth at each date is 1 and traders begin with equal shares of this wealth.
10
In this economy there is no asset decision; traders simply choose savings rates at each
date. For trader i the optimal savings rate rate at date t is characterized by the Euler equation
i i
i
Rt+1
0
i
i
0 wt δt i
ie
ui (wt (1 − δt )) = βui
Rt+1 (1 − δt+1 )
,
(8)
1
qt
qt1
i
ie
where Rt+1
is the return trader i expects at date t + 1 and δt+1
is the savings rate that trader i
plans at date t to employ at date t + 1. Using the traders’ utility funcations (8) simplifies to
(1 −
δti )γ−1
γ−1 i γ
Rt+1
i
ie
= β δt (1 − δt+1 )
qt1
(9)
From (9) we see that a trader’s savings plan depends only on his expectations about rates of
Ri
return ( qt+1
1 ) and his preference parameter γ.
t
The solution of Equation (9) for the logarithmic trader (γ = 0) is straightforward. He
saves at rate β = 1/2 at each date. The solution for the CRRA trader depends on the expected
return process (7) and the price process. Consider the price process
qt =
3 + 2(2/3)t−1
.
4 + 4(2/3)t−1
(10)
If trader 2 anticipates that prices will follow this process then Equation (8) implies that his
savings rate will be 3/4 at each date. At date 1 this yields aggregate savings, and thus aggregate
investment in the asset, of 1/2(1/2 + 3/4) = 5/8 which is the value of q11 given by Equation (10).
Then at date 2, trader 2 will have share 3/5 of the one unit gross return on the asset. When he
saves 3/4 of this wealth and trader 1 saves 1/2 of his wealth the market clearing price will be
q2 from Equation (10). Continuing in this way we see that Equation (10) gives an equilibrium
price process.
Because trader 2 always saves at a higher rate than does trader 1 his wealth share, and
his actual wealth, converges to 1. So trader 1 vanishes. The equilibrium evolution of wealth
share for trader 2 is given by
−1
rt2 = 1 + (2/3)t−1 .
In this economy the price of the asset converges to 3/4, the savings rate of the irrational trader 2.
It is easy to check that if trader 2 had correctly anticpated the gross returns on the asset, then
in equilibrium both traders would have saved at rate 1/2 at each date, there would have been no
evolution of wealth shares, and the price of the asset would have been its rational expectations
equilibrium value of 1/2 at each date. Thus in this economy a trader with incorrect beliefs drives
out a trader with rational expectations, because his overly optimistic beliefs cause him to save
at a higher rate. In consequence market behavior does not converge to a rational expectations
equilibrium.
11
At first glance this example is hard to compare with the models of Theorem 3.1 because there
seems to be no room in the models of the Theorem for the kind of disagreement we have posited
here. But in fact we can recapture the conventional understanding of states (that all agents
know what happens in each state) by expanding the state space and appropriately modifying
beliefs.
Example II:
Another way of describing the economy in Example I is to expand the state space so that at each
date there is a state in which trader 1’s beliefs hold, and a state in which trader 2’s beliefs hold.
Each player believes that only “his” state will happen, and trader 1 is correct in those beliefs.
Formally, suppose that there are two states, S = {s1 , s2 }, which are i.i.d. with probability
p = (1, 0). Trader 1’s beliefs are correct, p1 = p, and trader 2’s beliefs are p2 = (0, 1). There is
one asset with gross returns matrix
Rt (σ) = (1, Rte ),
for all σ. The analysis of this economy is exactly as in I. Trader 2, who has incorrect beliefs,
survives and trader 1, who has correct beliefs, vanishes.
Notice that in this economy, markets are incomplete. There are always two possible
states tomorrow, but only one asset. One unsatisfactory feature of this example is that traders
disagree so completely. If markets were complete, equilibrium would fail to exist because of their
extreme disagreement. Each trader would want to infinitely short the state he believes to be
impossible. Furthermore, condition iv fails. But this orthogonality of beliefs does not drive the
example. The next Example shows that the economy can easily be modified so that the traders
have overlapping beliefs.
Example III:
Consider again Example II, but now suppose that the beliefs of trader 2 are p2 = (1 − , ) for
> 0. Change the return on the asset in state 2 to
2
t−1 1/2
e
−1 9 + 6(2/3)
−1
Rt+1 = 4 + 4(2/3)t−1
The return
2 expects to receive in state 2 is greater than one at each date and converges
√ trader
2
to (1 + 5/2) as t → ∞. Actutal total wealth at each date is 1 as state 1 always occurs.
The logarithmic utility trader continues to save 1/2 of his wealth at each date regardless
of his expectations. Calculation shows that the CRRA utility trader continues to save 3/4 of his
wealth at each date. Thus as before the prices in Equation (10) clear the market at each date.
Trader 2 saves at a higher rate than does trader 1, so trader 1 vanishes. This occurs regardless
of trader 1’s expectations. So even if trader 1 has correct expectations he vanishes.
12
In this economy traders disagree about the probability distribution on S, but because
markets are incomplete, their ability to bet on this disagreement is limited. This disagreement
about states leads to trader 2 being overly optimistic and therefore saving at a higher rate than
does trader 1. The state space in this economy was constructed in order to incorporate differing
expectations about returns on assets. It is apparent from (9) that it is only expectations over
rates of return that matter. So a similar construction could be used to incorporate differing
expectations about future asset prices.
An economy has dynamically complete markets only if the set of assets is rich enough
to allow wealth transfers from any partial history to any other partial history. Example III
illustrates the fact the number of markets this requires is controlled by the amount of subjective
disagreement among traders. Consequently, we find the assumptions of dynamically complete
markets and heterogeneous beliefs to be implausible when taken together.
In the preceding examples the source of adverse selection effects came from the effect of
differing beliefs and market incompleteness on savings behavior. Blume and Easley [3] forced
agents to have identical savings behavior and studied the effects of selection on portfolio choices.
The next example shows how, even when investors have identical savings behavior, portfolio
effects can cause incorrect beliefs to survive and even prosper.
Example IV:
Consider an economy with two assets and 3 states. Asset 1 pays off 1 unit of corn in state 1
and 0 in the other two states. Asset 2 pays off 0 in state 1, but 1 unit in each of states 2 and
3. There are two traders with log payoff functions and common discount factor β. The state
probabilities and beliefs are described in the following table:
states
s1
s2
s3
truth
1/2 1/2 − trader 1
1/2 1/2 − trader 2
1/2
1/2 − The parameter > 0 is very small. Trader 1 has rational expectations, while trader 2 does not.
Both traders will save at the same rate β. Savings must be allocated between assets 1 and 2. At
any asset price at every date, asset 2 will have the same distribution of returns for each trader.
Consequently they will hold the same portfolio at each date. and so the distribution of wealth
never changes despite the configuration of traders’ beliefs.
To push this point farther, consider the following configuration of beliefs where δ > 0 is
13
very small:
states
truth
trader 1
trader 2
s1
s2
s3
1/2
1/2 − (1 − δ)/2 (1/2 − )(1 + δ) (1 + δ)
1/2
1/2 − Trader 1 has slightly incorrect beliefs while trader 2 has grossly incorrect beliefs. But trader 2’s
beliefs lead him to make the same decisions that a trader knowing the truth would make, while
trader 1 will do something else. Consequently trader 1 will vanish and trader 2 will dominate
the market.
The next example addresses the boundedness assumptions on asset returns. It demonstrates that if these do not hold even with complete markets, market selection for rational
expectations may fail.
Example V:
Now consider an economy in which markets are dynamically complete but returns are either
unbounded above or are not bounded away from zero. There are two states and two assets.
Asset a paysoff Rta (σ) at date t if σt = a and nothing in the other state. States are i.i.d. and
the probability of state 1 is p1 = 1 − for 1/2 ≥ > 0. Trader 1 has logarithmic utility and
correct beliefs. Trader 2 has CRRA utility, u2 (c2t ) = γ −1 (c2t )γ , for γ < 1, γ 6= 0, and beliefs
p2 = (1/2, 1/2). The traders have common discount factor β. Initially the traders have equal
wealth shares of total wealth 1.
It is apparent from the Euler equation that the logarthmic utility trader saves fraction β
of his wealth and invests fraction 1 − of his savings in asset 1 at each date regardless of current
or future prices. The optimal policy for the CRRA utility trader is more complex. Suppose that
prices are such that the rate of return on each asset, in the state in which it paysoff, is x at each
date. That is, for each asset a
a
Rt+1
(σ)
=x
a
qt (σ)
at each date on all paths σ. We will construct returns and prices so that this conjecture is
satisfied in equilibrium. With this constant rate of return the Euler equation implies that the
1
optimal policy for the CRRA utility trader is to save fraction (βxγ ) 1−γ and invest 1/2 of his
savings in asset 1 at each date.
With the policies above the equilibrium pricing equations are
1
qt1 (σ) = Rts (σ) (βxγ ) 1−γ rt2 (σ)1/2 + β(1 − )rt1 (σ)
1
qt2 (σ) = Rts (σ) (βxγ ) 1−γ rt2 (σ)1/2 + βrt1 (σ)
(11)
(12)
14
where Rts (σ) is the gross return on the asset that paysoff at date t in state σ.
Combining these two equation systems provides the evolution of gross returns necessary
to verify the conjectured constant rate of return on assets.
1
1
Rt+1
(σ) = xRts (σ) (βxγ ) 1−γ rt2 (σ) + β(1 − )rt1 (σ)
(13)
1
2
Rt+1
(σ) = xRts (σ) (βxγ ) 1−γ rt2 (σ) + βrt1 (σ) .
(14)
Finally, wealth share evolution is

1

(βxγ ) 1−γ


rt2 (σ) if σt = 1

1

2
2
γ

1−γ
 (βx ) )rt (σ) + β(1 − )(1 − rt (σ))
2
rt+1 (σ) =

1


(βxγ ) 1−γ



rt2 (σ)
if σt = 2
1

2
2
γ
1−γ
(βx ) rt (σ) + β(1 − rt (σ))
(15)
Now for any specification of β, γ and x Equations (11) through (15) describe the evolution of
the economy.
We first consider a specification that yields unbounded returns on the assets. Suppose
that β = 1/4, γ = 1/2 and x = 10. This yields an optimal savings rate for trader 2 of 5/8.
Trader 2 invests one-half of this, or 5/16, in each asset. Trader 1’s optimal savings rate is
β = 1/4. So no matter how trader 1 divides his savings between assets (that is no matter what
) he purchases less of each asset than does trader 2. So trader 1’s wealth share converges to 0; he
vanishes. In this economy relative asset prices converge to one as trader 2 invests one-half of his
savings in each asset. These limit prices are independent of the actual probability distribution
on states and are not rational expectations equilibrium prices for the economy consisting of only
the CRRA trader unless = 1/2.
We next consider a specification in which the gross returns of assets converge to 0. For
1−γ
any 0 < β < 1 and 0 6= γ < 1 let x = β −1 3 γ . As before, trader 1 saves at rate β and invests
fraction 1 − of his savings in asset 1. Calculation shows that the savings rate of trader 2,
1
(βxγ ) 1−γ , times the share that he invests in each asset, 1/2, is greater than the savings rate β,
of trader 1. So trader 2 will purchase more of each asset than will trader 1 and thus trader 1
will vanish. In this economy the growth rate of the return on asset 1 at date t is
1
1
3γ
1−γ
Rt+1
(σ)
2
γ
≤
3
1
+
(1/2)r
(σ)
≤
t
Rt1 (σ)
2
If γ < 0, then regardless of the wealth share evolution, gross returns on asset 1 are converging to
0. A similar result holds for asset 2. Thus in this economy the trader with correct expectations
vanishes and the relative prices of the assets converge to one for all possible state probabilities.
15
In each of the economies above selection for traders with incorrect beliefs is driven by
savings rates. In these economies discount factors, beliefs and preferences interact to determine
savings rates and portfolios. When traders have identical savings rates, markets select for
portfolio rules that maximize the conditional expected growth rate of wealth share (as in [3]).
In general, this does not lead to selection for correct beliefs, but it does if some trader has
logarithmic utility. But when savings rates are endogenous, even logarithmic utility traders
with correct beliefs can be driven out of the market by expected utility maximizing traders with
incorrect beliefs. This occurs because traders with incorrect beliefs may choose to save so much
that the savings rate effect overcomes the consequences of less fit portfolios.
References
[1] Armen Alchian. Uncertainty, evolution and economic theory. Journal of Political Economy,
58:211–221, 1950.
[2] David Easley Blume, Lawrence and Margaret Bray. Introduction to the stability of rational
expectations. Journal of Economic Theory, 26(2):313–17, August 1982.
[3] Lawrence Blume and David Easley. Evolution and market behavior. Journal of Economic
Theory, 58(1):9–40, October 1992.
[4] Lawrence Blume and David Easley. Optimality and natural selection in markets. unpublished, Cornell University, December 1997.
[5] Lawrence Blume and David Easley. Rational expectations and rational learning. In Mukul
Majumdar, editor, Organizations with Incomplete Information: Essays in Economic Analysis, pages 61–109. Cambridge University Press, Cambridge, UK, 1998.
[6] David A. Freedman. Tail probabilities for martingales. Annals of Probability, 3(1):100–118,
1975.
[7] Milton Friedman. Essays in Positive Economics. University of Chicago Press, Chicago, 1953.
[8] Karl Hinderer. Foundations of Non-Stationary Dynamic Programming with Discrete Time
Parameter. Lecture Notes in Operations Research and Mathematical Systems. SpringerVerlag, Berlin, 1970.
[9] Alvaro Sandroni. Do markets favor agents able to make accurate predictions. unpublished,
Northwestern University, March 1999.