Heterogeneous Beliefs, Asset Prices

and Wealth Dynamics

Viktor Tsyrennikov

April 2010

Abstract

This paper studies asset prices and wealth dynamics in the environ-ment with heterogeneous beliefs and endogenously incomplete markets.Agents can trade the full set of Arrow securities. But the markets areincomplete because agents are allowed to renege on their contract obli-gations.

Unlike in previous studies on limited commitment participationconstraints always bind in equilibrium. This happens because differ-ences in opinions lead to a highly volatile wealth distribution. Unlikein the previous studies on survival in the economies with completemarkets all the agents always survive because they can repudiate theirdebts. This allows us to study stationary distribution of asset returnsand wealth dynamics with heterogeneous beliefs.

Our setting can also accommodate learning and our results applyto a setting with any meaningful borrowing limit.

1 Model

Time is discrete and indexed by t ∈ Z+. There is one perishable good in eachdate. The aggregate state of the economy is fully summarized by a stochasticprocess st. It is a first-order Markov process with a finite set of statesS = 1, ..., S and a transition matrix Π = πij where πij = prob(zt+1 =j|zt = i). We let st to denote the history of the state (s0, s1, ..., st). Agentsbeliefs about the state of the economy differ. We denote the transitionprobability matrix of agent i by Πi.

Agents are infinitely lived. Agent i receives income yi(st) in period t andstate st. Every agent manages his income to achieve the highest utility from

1

consumption:

U(ci) =∞∑

t=0

∑

st

βtπit(s

t)u(cit(st)), β ∈ (0, 1), (1)

subject to a sequence of budget constraints:

cit(st) +

∑

st+1

Qt(st+1)ait+1(s

t+1) = yi(st) + ait(st), ∀t. (2)

Agents can choose not to fulfill their commitments. That is there islimited contract enforcement. After an agent chooses to renege on his obli-gations he is banned from the financial market forever. An agent livingin financial autarky has no more choices to make and simply consumes hisincome every period. We denote the life-time utility of agent i living inautarky by V i

aut(st):

V iaut(st) = u(yi(st)) + β

∑

st+1

πi(st+1|st)Viaut(st+1). (3)

Note that this value is a function of a state st rather than a history st.This is a consequence of our assumption that the state of the economy is afirst-order Markov process and that income is a function of the current stateonly.

Information and Learning

In our setting the assumption of common knowledge is unnecessary andunreasonable. We imagine an economy in which individual agents tradebased on their beliefs and their expectations of future prices.

We assume that there is no learning. This is justified on the groundsthat our model is necessarily simplified. So, even the agents were allowedto learn from past observations and prices their beliefs may not converge letalone converge to the true measure.

The second justification is that we aim to obtain a stationary distributionof asset prices. With learning the effect of heterogenous beliefs would be onlytransient. For example, Cogley & Sargent (2009) study the transition pathwhere agents are still learning the true distribution of the state.

It is possible to force beliefs to fluctuate forever. For example, Veronesi(2000) studies an economy with a hidden Markov state. An ever-changingunobserved regime prevents the agents from learning the truth even in thelimit. However, such an economy is a special case of a model with ex-ogenously specified beliefs. The reason is that the evolution of beliefs isexogenous to the model.

2

2 Special Cases

This model includes as special cases two important classes of models. Thefirst class of models features limited participation. Agents are allowed toexit their market arrangements at a cost of a penalty, usually perpetual lossof access to the market. These models are used to study incomplete risk-sharing. The second class of models has complete markets and heterogeneousbeliefs. Agents with the same information take different asset positionsbecause of their beliefs. This type of models is used to study survival ofagents with with incorrect beliefs. We briefly discuss each type of modelsnext.

2.1 Heterogeneous beliefs, full commitment

Sandroni (2000) shows that agents with more accurate forecasts survive inthe limit. Those with less accurate beliefs will be driven out of the market.Blume & Easley (2006) show that this result extends to all the economiesin which the allocation is Pareto optimal and to the environments withlearning. Unfortunately, in the limit only the agent with the more accuratebeliefs survives.1 So, even though these economies have the potential todeliver interesting asset market predictions they are non-stationary. Blume& Easley (2009) write:

Analysis of [infinite horizon, stochastic, general equilibrium economieswith heterogeneous agents] are beginning to appear in both fi-nance and macroeconomics in response to the inability of repre-sentative agent models to fit asset pricing and macro data. ...[But] we do not expect them to stand up to market selection.

A stationary equilibrium requires a constraint on the agents’ borrowingwhich would prevent them from loosing all their wealth in the limit. Impos-ing an arbitrary exogenous borrowing limit is not attractive. This issue issolved by adding an endogenous borrowing/participation constraint.

2.2 Homogenous beliefs, no commitment

Kocherlakota (1996) studies the efficient outcome in the model with twosided lack of commitment.2 Jermann & Alvarez (2000) show that the equi-

1If there are several agents with equally accurate beliefs then the log ratio of marginalutilities is a mean-zero random walk.

2Two-sided lack of commitment is fundamentally different from one-sided lack of com-mitment, e.g. Eaton-Gersovitz. In the latter an agent is allowed to renege on his promises

3

librium with two sided lack of commitment can be decentralized by imposingon agents endogenous borrowing limits. The authors then study the impli-cations for asset prices. Jermann & Alvarez (2002) calibrate the model tothe U.S. household data and demonstrate that the model can generate thehigh equity premium and the low risk-free rate close to those observed in thedata. To achieve this the authors must assume a very low discount factor.Without the latter the borrowing constraints are unrestrictive.

In the equilibrium with limited commitment default option is never ex-ercised. But the default constraint may or may not be binding. In the lattercase, the economy’s characteristics match those of the complete marketseconomy. In the case of homogeneous beliefs it easy to characterize a set ofparameters for which the constraint is binding. Figure 1 plots the defaultregion for u(c) = −1/c and an iid state st. Consistently with the resultsin Jermann & Alvarez (2000) the endogenous borrowing constraint is activewhen agents discount future heavily (β is low) or when agents face a signif-icant amount of idiosyncratic risk (e is high). In both cases the value of theautarky increases relative to the arrangement with full risk-sharing. Notethat for the degree of volatility in the individual data, σ(log(yit)) ≈ 0.25 tohave the constraint bind in equilibrium one needs to assume an unconven-tionally low3 discount factor.

Now let’s look at the same question from another side. For β = 0.96the maximum volatility of income for which participation constraints bindis only ≈ 0.04. For a typical quarterly discount factor β = 0.99 participationconstraint is never binding.

3 Recursive Formulation

Consider a decentralized environment in which agents face endogenous bor-rowing constraints. Let ω = (ω1, ..., ωI) be a distribution of wealth in theeconomy. The state vector is (ω, s). Agents rationally expect that wealthdistribution in state s′ tomorrow will be Ω(ω, s, s′).

Let Q(ω, s, s′) be the price of a security paying one unit of good in states′ tomorrow when the wealth distribution is ω and the current state is s.Let V i(ω, a, s) be the optimal life-time utility of an agent i who has assetsa and when the economy’s state is (ω, s). The value function must satisfy

to the principal/planner who is assumed to be risk-neutral.3For a macroeconomic study.

4

Figure 1: Default region

0.00

0.10

0.20

0.30

0.40

0.50

0.60

0.50 0.55 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 1.00

Vo

latilit

y o

f in

co

me

, σ(

ln(y

t))

Discount factor, β

Default region

No default region

the following Bellman equation:

V i(a, ω, s) = maxc,a′(s′)

[

u(c) + β∑

s′

πi(s′|s)V i(Ω(ω, s, s′), a′(s′), s′)]

(4)

subject to the budget constraint

c+∑

s′

Q(ω, s, s′)a′(s′) = yi(s) + a, ∀t (5)

and a borrowing limit

a′(s′) > −Bi(Ω(ω, s, s′), s′), ∀s′ ∈ S. (6)

An endogenous borrowing limit Bi(ω, s, s′) is implicitly defined by the fol-lowing relation:

V i(−Bi,Ω(ω, s, s′), s′) = V iaut(s

′), ∀s′. (7)

It is easy to show that V i is increasing in a.4 Then the borrowing constraintis equivalent to:

V i(a′(s′),Ω(ω, s, s′), s′) > V iaut(s

′), ∀s′. (8)

We will refer to this constraint as participation constraint.

4This follows from the monotonicity of the budget set.

5

3.1 Competitive equilibrium

A recursive competitive equilibrium is a list of functionsQ(ω, s, s′), Bi(ω, s)i∈I ,ci(a, ω, s), ai ′(a, ω, s, s′)i∈I such that:

a) given the price system Q and borrowing limit Bi policy functions(ci, ai′) solve agent i’s optimization problem;

b) good market clears:

I∑

i=1

ci(ω, s) =

I∑

i=1

yi(s); (9)

c) financial markets clear:

I∑

i=1

ai′(ω, s, s′) = 0, ∀s′ ∈ S; (10)

d) the wealth evolution is consistent with individual decisions:

Ω(ω, s, s′) = (a1′(ω, s, s′), ..., aI ′(ω, s, s′)), ∀s′ ∈ (11)

3.2 Binding borrowing limits

With homogenous information borrowing limits may remain slack in equi-librium. For example, when the agents are infinitely patient (β = 1) or theagents’ income very volatile std(yi(s)) ≫ 0 they will never choose to default.The reason for this is a declining attractiveness of the financial autarky. Inthe environment with heterogeneous beliefs borrowing limits always bind.This is a consequence of the survival result in Blume and Easley (2006).If the borrowing limit did not bind consumption of an individual with thelowest survival index must converge to zero. This means that at least in thelimit the value of staying in the financial market is lower than the value offinancial autarky. The latter is not possible with limited participation. Weformalize this result with a proposition.

Proposition 1. Suppose u(0) = −∞. If there exists i ∈ 1, .., I such thatΠi 6= Π then there must exist an agent for whom borrowing limits are bindingin equilibrium.

6

Proof. First, note because endowments are bounded away from zero theautarky value is finite for each agent and state.

Next, without loss of generality, suppose that agent 1 and 2 have differentbeliefs. If agent 1’s beliefs are less accurate as measured by the relativeentropy then lim supt→∞ c1t = 0 by theorem 2 of Blume & Easley (2006).Because u is continuous, this implies that lim supt→∞ U1

t = u(0)/(1 − β) <V 1aut(1), where U i

t denotes the continuation utility of agent i starting fromdate t.

If agent 1 and agent 2 hold equally accurate beliefs then lim inft→∞ c1t = 0(see Blume & Easley (2009), p.11). This implies that for any ε > 0, c1t < εinfinitely often. Choose ε such that U(ε) ≡ u(ε)+βu(maxs[e(s)])/(1−β) =0.5mins[V

1aut(s)]. Then U1

t < U(ε) < V 1aut(s),∀s infinitely often. So, the

participation constraint must be binding as V 1aut(1) > −∞.

3.3 Equivalence of Recursive and Sequential Formulations

[TBW]

3.4 Asset prices

Since we have a full set of Arrow securities traded in equilibrium we canprice any asset. Let pd(ω, s) be the ex-dividend price of an asset payingthe dividend d(s) when the economy’s state is (ω, s). This price can berepresented recursively as follows:

pd(ω, s) =∑

s′

Q(ω, s, s′)(p(Ω(ω, s, s′), s′) + d(s′)). (12)

Realized return on the asset is:

R(ω, s, s′) =p(Ω(ω, s, s′), s′) + d(s′)

p(ω, s). (13)

“Stock market index” is the asset paying the aggregate endowment. Longterm bond pays one unit of good in each period and state. We denote theprices of these two assets by ps(ω, s) and pb(ω, s) respectively.

3.5 Solution

We are not able to solve for the RCE analytically. We will compute themodel solution numerically for the case with two agents, I = 2. In this casethe state vector contains only one continuous state variable: financial wealth

7

of the first agent, ω1. Financial wealth of the second agent can be readilyobtained via the market clearing condition, ω2 = 1−ω1. This simplificationmakes the problem very tractable.

Because in equilibrium ω = a1 = −a2 we can state the system of equi-librium conditions in terms of auxiliary value functions:

v1(ω, s) = V 1(ω, ω, s),

v2(ω, s) = V 2(−ω, ω, s).

Then the solution to the model is fully characterized by the following systemof equations:

Q(ω, s, s′) = βπi(s′|s)u′(ci(Ω(ω, s, s′), s′)) + µi(ω, s, s′)

u′(ci(ω, s)), i ∈ I, s′ ∈ S, (14a)

y1(s) + ω = c1(ω, s) +∑

s′

Q(ω, s, s′)a1′(ω, s, s′), (14b)

y2(s)− ω = c2(ω, s) +∑

s′

Q(ω, s, s′)a2′(ω, s, s′), (14c)

Ω(ω, s, s′) = a1′(ω, s, s′), ∀s′ ∈ S, (14d)

and

vi(ω, s) = u(ci(ω, s)) + β∑

s′

πi(s′|s)vi(ω′, s′), i ∈ I, (14e)

vi(ω′, s′) > V iaut(s

′), 0 = µi(ω, s, s′)(vi(ω′, s′)− V iaut(s

′)), i ∈ I, s′ ∈ S. (14f)

The first four equations are enough to characterize an equilibrium with-out borrowing constraints. The last two equations allow us to keep track ofthe endogenous borrowing limits.

To compute the RCE we iteratively solve the system of first-order condi-tions (14). The difficulty lies in computing the borrowing limits as they aredefined via value functions. So, in addition to solving a system of equilibriumconditions we also compute the value functions on each iteration.

We solve the following system of equations:

Φ(c, a′, ω, s|ρc, ρa) = 0,

Ω(ω, s, s′) = a1′(s′).

The solution to this system consists of two functions:

c = ρc(ω, s), a′(s′) = ρa(ω, s, s′).

We use this solution to update the initial policy functions. The solution isprojected on the space of cubic splines and we iterate until supω,s |ρ(ω, s)−ρ(ω, s)| < 10−6.

8

4 Stylized Example

We assume that state st takes S = 2 values: two income states. In state 1incomes are (1+ e, 1− e) and in state 2 they are (1− e, 1+ e) with e ∈ [0, 1).The initial state is s0 = 1; so, agent 1 has an advantage as he receives highincome in the first period.

Beliefs can be specified in many different ways. But the task is consid-erably simpler with only 2 states. Let the true transition probability matrixbe:

Π =

[

ρ 1− ρ1− ρ ρ

]

. (15a)

Beliefs of agent 1 are:

Π =

[

ρ1 1− ρ1

1− ρ ρ

]

. (15b)

Beliefs of agent 2 are:

Π =

[

ρ 1− ρ1− ρ2 ρ2

]

. (15c)

That is agent i is wrong about the conditional distribution in state i butholds correct beliefs in the other state. We choose ρi to be in the α% the-oretical confidence interval for the data of length 100 (100 years of annualdata). That is given a sample of 100 observations the two agents wouldagree that with a confidence level α% their beliefs are statistically indistin-guishable. When ρi > ρ agent i is an optimist; otherwise he is a pessimist.The following lemma helps us to compute the confidence interval.

Lemma 1. Let the true process and the beliefs of the agents be described asin (15a–15c). Then limiting distribution of the maximum likelihood estimateof ρi from a sample of length N is Beta(ρN/2, (1 − ρ)N/2).

The parameters of the beta distribution are the expected number oftransitions from state i to state i and from state i to state −i respectively.The validity of the lemma is demonstrated by a numerical example withρ = 0.5, an i.i.d. environment, and N = 100. Numerical Monte-Carlosimulations would be unavoidable in more complicated environments. Wegenerate a sample of length 100 using the true data generating process andestimate ρ. Then we repeat the estimation procedure 1,000,000 times andcompute the distribution of ρ. The empirical distribution and the analyticdistribution of the estimates are plotted in figure 2. Note the volatility of

9

Figure 2: Empirical and theoretical distributions of ρ

0

1

2

3

4

5

6

7

0.2 0.3 0.4 0.5 0.6 0.7 0.8

Pro

ba

bili

ty d

en

sity

Estimate of ρ

the estimated parameter. The theoretic 50% confidence interval for ρ is[0.452, 0.548]. This allows for a lot of freedom when specifying the beliefs ofthe two agents. For this example we set

ρ1 = ρ2 = 0.548.

Equilibrium consumption and wealth dynamics. Under the complete mar-kets when the two agents hold correct beliefs consumption of both agentsis constant and equals c = 1 + (e,−e)β/(1 + β − 2βρ). Had we endowedthe agents with heterogeneous beliefs wealth of one of the agents would con-verge to the natural borrowing limit.5 That is a consequence of the survivalresult in Blume & Easley (2006). With the presence of borrowing limits6

both agents remain in the market because they can walk away from an ac-cumulated debt. Figure 3 plots time path of the agent 1’s wealth under thethree scenarios for a random but fixed sequence of shocks. The less volatileseries in each panel denote the wealth series under correct beliefs. Threefacts emerge from the figure. First, under homogenous beliefs wealth is verystable. (In fact, wealth distribution is strongly stationary – that is it is onlya function of the exogenous state.) Second, under complete markets butheterogeneous beliefs wealth is non-stationary. Third, under endogenouslyincomplete markets and heterogeneous beliefs wealth is volatile but station-ary. With incomplete markets agents will never be allowed to accumulate“too much” debt because they can repudiate their obligations. This also

5Equivalent statement is that consumption of an agent converges to zero.6An exogenous borrowing limit may have a similar effect. When an agent’s borrowing

is limited he or she in general will always have income to repay the debt and have apositive consumption.

10

Figure 3: Wealth dynamics under complete and incomplete markets

-10

-5

0

5

10

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

We

alth

of a

ge

nt 1

Period

A) Complete markets

-10

-5

0

5

10

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Wealth o

f agent

1

Period

B) Endogenously incomplete markets

limits the positions that market participants can take and hence makes itimpossible to loose all your wealth. The natural borrowing limit in this en-vironment equals 9.98, while in the incomplete markets model the most thatthe agent can borrow/owe is 4.78. Note also the frequently with which agentshit the borrowing constraints. under the current parametrization, with ho-mogeneous beliefs the endogenous borrowing constrains are non-binding.

Figure 4 plots asset returns. The red dashed line denotes the return inthe economy with correct beliefs which equals 1/β. As expected, the averagestock market price in the economy with endogenous borrowing constraintsis higher than in the economy with the natural borrowing limits. However,the difference is only 0.5%. Asset returns in the two economies are also veryclose. However, with more restrictive borrowing limits assets returns are16.33 times more volatile. Note also a characteristic volatility clustering ofstock returns.

Interestingly, that heterogeneous beliefs per se does not have much of an

11

Figure 4: Asset returns under complete and incomplete markets

1.07

1.08

1.09

1.10

1.11

1.12

1.13

1.14

1.15

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Sto

ck m

ark

et re

turn

Period

A) Complete markets

1/β

1.07

1.08

1.09

1.10

1.11

1.12

1.13

1.14

1.15

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

Sto

ck m

ark

et

retu

rn

Period

B) Endogenously incomplete markets

1/β

effect on asset prices. The role of heterogeneous beliefs is to amplify wealthdynamics. The latter, in turn, never disappears because the endogenousborrowing constraints keep the agent trading in the market. When agent 1looses wealth prices on the Arrow securities change favorably and so he orshe can increase wealth. Similar forces exist even with natural borrowinglimit but they are too large to let the agent accumulate new wealth quickly.

Finally we look at the trading volume in the financial market. Tradingvolume is defined as the change in the wealth of the first agent. Figure 5plots the trading volume in the two markets environments. It can be seenthat trading is much larger in the economy with the endogenous borrowinglimit. Interestingly, in the latter economy trade periodically halts. Thishappens when one of the agents hits the endogenous borrowing limit and isprepared to exist the financial market. Note also that the trading volume isthe highest when wealth is equally distributed in the economy.

12

Figure 5: Asset returns under complete and incomplete markets

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0 50 100 150 200 250 300 350 400 450 500

Period

A) Trading volume

-1.00

0.00

1.00

2.00

3.00

4.00

5.00

0 50 100 150 200 250 300 350 400 450 500

Period

B) Wealth dynamics

13

4.1 Default

There is no explicit default in the model.7 But we can interpret states inwhich the borrowing constraint is binding as default. This can be doneas when the borrowing limit binds any debt is simply postponed. Thisresembles repayment renegotiation. But in strict sense of the word it is adefault.

More interesting is the pattern of binding borrowing constraints. Usu-ally in models with participation constraints agents want to default in highincome states. But this may or may not be true here. Suppose that theeconomy that we analyzed above is actually i.i.d.. Each state is equallyprobable. Suppose that agent 1 puts probability 0.999... on state 1, whilethe other agent has correct beliefs. In this example agent 1 will never defaultin state 1 (when he has high income) because that is when he gains a lotof wealth! So, his value of staying in the market goes up by more than thevalue of autarky.

5 Concluding Remarks

The economy that we analyze and the results that apply are in fact truefor any borrowing limits that are strictly tighter than the natural borrowinglimits. The endogenous borrowing limit has a virtue as it is endogenous.

6 Appendix

6.1 Example with Aggregate Uncertainty

We assume that state st takes S = 2 values: two income states. In state 1 incomesare (1, 1) and in state 2 they are (0.9, 0.9). The initial state is s0 = 1. The truetransition matrix is:

Π =

[

0.980 0.0100.500 0.500

]

. (16a)

Beliefs of agent 1 are:

Π =

[

0.985 0.0150.500 0.500

]

. (16b)

Beliefs of agent 2 are:

Π =

[

0.980 0.0100.400 0.600

]

. (16c)

7We work on an extension that allows default.

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Figure 6: Wealth dynamics with aggregate uncertainty

-1.00

-0.50

0.00

0.50

1.00

1.50

2.00

2.50

0 50 100 150 200 250 300 350 400 450 500

Period

This case is interesting because we can interpret agent 1 as an optimistic bank-ing sector and agent 2 as a pessimistic general public. Here if state 1 realizes thenagent 1 accumulates wealth. Since the probability of state 1 under the stationarydistribution is 0.9756 agent 1 is steadily accumulating wealth for protracted peri-ods of time. But whenever state 2 realizes agent 1 loses a large amount of wealth.Average fraction of the period income lost is 65.6% and the maximal share lost is146.2%! Figure 6 plots wealth a typical path for an economy starting with ω = 0.

6.2 Asset Prices with Heterogeneous Beliefs

Consider the case with CRRA preferences. With natural borrowing limits the CEallocation is Pareto optimal. So, there exists a set of Pareto weights such that theCE allocation is:

ci(st) =ki(st)

∑

j kj(st)

e(st), (17)

where k(st) = [θiπi(st)]1/γ .

Let qs(st) and qb(st) denote the price of a stock paying aggregate endowment

15

and the price of a risk-free bond respectively. Then:

qs(st) =∑

τ>t

∑

sτ |st

βτ−tπi(sτ |st)[ ci(st)

ci(sτ )

]1/γ

e(sτ )

=∑

τ>t

∑

sτ |st

βτ−t[

∑

j kj(sτ )

∑

j kj(st)

]γ

[e(sτ )]1−γ [e(st)]γ , (18a)

qb(st) =∑

st+1

βπi(st+1|st)[ ci(st)

ci(st+1)

]1/γ

= β∑

st+1

[

∑

j kj(st+1)

∑

j kj(st)

]γ[

e(st)

e(st+1)

]γ

. (18b)

Note that the term (∑

j kj(sτ )/

∑

j kj(st))γ plays the role of conditional probability

of history sτ |st. If every agent held the same belief then this term would becomeπ(sτ |st). The task is to understand the behavior of this term.

π(sτ |st) ≡

[

∑

j kj(sτ )

∑

j kj(st)

]γ

=

[

∑

j(θjπj(sτ ))1/γ

∑

j(θjπj(st))1/γ

]γ

. (19)

We derive a second-order Taylor approximation to the above formula. The approx-imation “point” is the economy in which every agent holds an average belief definedbelow:

π(st|s0) = (1/I)I

∑

j=1

πj(st|s0).

We assume further that all individuals are ex-ante identical. Then the competitiveequilibrium allocation can be obtained by setting θj = θ, ∀j. Then:

π(sτ |st) ≈ π(sτ |st)(

1− (γ + 1/γ − 2)∑

j

(∆πj(st))2)

+O([maxj

|πj − π|]3),

where ∆πj(sτ ) = πj(sτ ) − π(sτ ). So, belief heterogeneity has only second ordereffect on asset prices. Moreover the second order effect is proportional to γ+1/γ−2.With logarithmic preferences, γ = 1, even the second-order effect vanishes. In fact,with logarithmic preferences we can compute the stock prices analytically:

qs(st) = e(st)/(1− β).

So, equity prices and returns are the same as in the economy with homogeneousbeliefs:

Return on equity after history st = β−1e(st)/e(st−1).

Yet, the price of the bond depends on individual beliefs. This example demonstratesthat asset price volatility is not an automatic consequence of belief heterogeneity.Note also that wealth is still more volatile that in the homogeneous beliefs economy.So, volatile wealth is not sufficient for volatile asset prices in such an economy.

16

References

Blume, L. E. & Easley, D. (2006), ‘If you’re so smart, why aren’t you rich?belief selection in complete and incomplete markets’, Econometrica74(4), 929–966.

Blume, L. E. & Easley, D. (2009), ‘Heterogeneity, selection and wealth dy-namics’, Cornell University manuscript .

Cogley, T. & Sargent, T. J. (2009), ‘Diverse beliefs, survival, and the marketprice of risk’, NYU manuscript .

Jermann, U. & Alvarez, F. (2000), ‘Efficiency, equilibrium, and asset pricingwith risk of default’, Econometrica pp. 775–797.

Jermann, U. & Alvarez, F. (2002), ‘Efficiency, equilibrium, and asset pricingwith risk of default’, Econometrica pp. 775–797.

Kocherlakota, N. (1996), ‘Implications of efficient risk sharing without com-mitment’, Review of Economic Studies pp. 595–609.

Sandroni, A. (2000), ‘Do markets favor agents able to make accurate pre-dictions?’, Econometrica 68(6), 1303–1341.

Veronesi, P. (2000), ‘How does information quality affect stock returns?’,Journal of Finance 55(2), XXX.

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