Whom should we believe? Aggregation of heterogeneous beliefs

Abstract

We examine the collective risk attitude of a group with heterogeneous beliefs. We prove that the wealth-dependent probability distribution used by the representative agent is biased in favor of the beliefs of the more risk tolerant consumers. Moreover, increasing disagreement on the state probability raises the state probability of the representative agent. It implies that when most disagreements are concentrated in the tails of the distribution, the perceived collective risk is magnified. This can help to solve the equity premium puzzle. We show that the trade volume and the equity premium are positively correlated.

Appendix: The case of ISHARA preferences

In this appendix, we first prove Proposition 2.

Proof of Proposition 2 Fully differentiating Eq. 8 with respect to z and using property (6) yields that $\partial R/\partial z$ evaluated at (z,P) has the same sign that

$$\frac{\partial T^{u}}{\partial c}(c(z,P,\theta ),\theta )-\sum_{\theta ^{\prime }=1}^{N}\frac{T^{u}(c(z,P,\theta ^{\prime }),\theta ^{\prime })}{ T^{v}(z,P)}\frac{\partial T^{u}}{\partial c}(c(z,P,\theta ^{\prime }),\theta ^{\prime }),$$

(21)

For ISHARA preferences, $\partial T^{u}/\partial c$ is a constant, which implies that the above expression is uniformly equal to zero, implying that R is independent of the per capita wealth in the group. Reciprocally, R independent of z implies that

$$ \frac{\partial T^{u}}{\partial c}(c(z,P,\theta ),\theta )=\sum_{\theta ^{\prime }=1}^{N}\frac{T^{u}(c(z,P,\theta ^{\prime }),\theta ^{\prime })}{ T^{v}(z,P)}\frac{\partial T^{u}}{\partial c}(c(z,P,\theta ^{\prime }),\theta ^{\prime }) $$

for all θ and P. This can be possible only if $\partial T^{u}/\partial c$ is independent of c and θ, which means that the group has ISHARA preferences. □

We now derive an analytical solution in the ISHARA case. It is easy to check that the set of utility functions that satisfies the ISHARA property must be parameterized as follows:

$$ u(c,\theta )=\kappa \left( \frac{c-a(\theta )}{\gamma }\right) ^{1-\gamma } $$

(22)

These utility functions are defined over the consumption domain such that γ ^− 1( c − a(θ)) > 0. In this particular case, the first-order condition to state-dependent the Pareto program (3) implies that

$$ c(z,P,\theta )-a(\theta )=k\left[ \lambda (\theta )p(\theta )\right] ^{1/\gamma }. $$

Since T ^u(c,θ) = (c − a(θ))/γ, property (8) can be rewritten in the ISHARA case as

$$ R(z,P,\theta )=\frac{\left[ \lambda (\theta )p(\theta )\right] ^{1/\gamma }}{ NE\left[ \lambda (\widetilde{\theta })p(\widetilde{\theta })\right] ^{1/\gamma }}, $$

(23)

where $Ef(\widetilde{\theta })=N^{-1}\sum_{\theta =1}^{N}f(\theta ).$ The definition of R applied to the ISHARA case implies that

$$ R(z,P,\theta )=\frac{p(\theta )p_{\theta }^{v}(P)}{p^{v}(P)}, $$

(24)

where $p_{\theta }^{v}=\partial p^{v}/\partial p(\theta ).$ Combining Eqs. 23 and 24 yields

$$ \frac{p_{\theta }^{v}(P)}{p^{v}(P)}=\frac{\lambda (\theta )^{1/\gamma }p(\theta )^{-1+1/\gamma }}{NE\left[ \lambda (\widetilde{\theta })p( \widetilde{\theta })\right] ^{1/\gamma }} \label{C3} $$

(25)

for θ = 1,...,N. The solution to this system of partial differential equations has the following form:

$$ p^{v}(P)=C\quad \left[ E\left[ \lambda (\widetilde{\theta })p(\widetilde{ \theta })\right] ^{1/\gamma }\right] ^{\gamma }, \label{pvC} $$

(26)

where C is a constant. In order for p ^v to be a probability distribution, we need to select the particular solution with

$$ p^{v}(P(s))=\frac{\left[ E_{\widetilde{\theta }}\left[ \lambda (\widetilde{ \theta })p(s,\widetilde{\theta })\right] ^{1/\gamma }\right] ^{\gamma }}{ \sum_{t=1}^{S}\left[ E_{\widetilde{\theta }}\left[ \lambda (\widetilde{ \theta })p(t,\widetilde{\theta })\right] ^{1/\gamma }\right] ^{\gamma }}. \label{ISHsol} $$

(27)

Calvet, Grandmont and Lemaire (2001) and Jouini and Napp (2004) obtained the same solution. Jouini and Napp (2004) and Chapman and Polkovnichenko (2006) derived this result in the special case of CRRA (a = 0).

Three special cases are worthy to examine.

• Consider first the case with γ tending to zero. This corresponds to risk-neutral preferences above a minimum level of subsistence. Under this specification, condition (27) is rewritten as

  $$\begin{array}{lll} p^{v}(P(s))&=p^{n}(P(s))\nonumber\\ &=_{def}\frac{\max_{\theta \in \Theta }\lambda (\theta )p(s,\theta )}{\sum_{t=1}^{S}\max_{\theta \in \Theta }\lambda (\theta )p(t,\theta )}\text{ \ for all }s\text{. (risk-neutral case)} \end{array}$$

  (28)

  With risk-neutral preferences, the efficient allocation produces a flip-flop strategy where the cake in state s is entirely consumed by the agent with the largest Pareto-weighted probability associated to that state. It implies that the group will use a state probability p ⁿ proportional to it to determine its attitude toward risk ex ante.

• In the case of logarithmic preferences (a = 0, γ = 1), the denominator in Eq. 27 equals $E\lambda (\widetilde{\theta })$ since

  $$ \sum_{t=1}^{S}E_{\widetilde{\theta }}\lambda (\widetilde{\theta })p(t, \widetilde{\theta })=E_{\widetilde{\theta }}\left[ \lambda (\widetilde{ \theta })\sum_{t=1}^{S}p(t,\widetilde{\theta })\right] =E\lambda (\widetilde{ \theta })=1. $$

  It implies that

  $$ p^{v}(P(s))=p^{\ln }(P(s))=_{def}E\lambda (\widetilde{\theta })p(s, \widetilde{\theta })\text{ \ for all }s\text{. (logarithmic case)} $$

  (29)

  With these Bernoullian preferences, the efficient probability that should be associated to any state s is just the weighted mean p ^ln (s) of the individual subjective probabilities of that state s. This is the limit case $T_{c}^{u}\equiv 1$ of the result presented in Proposition 5.

• In the CARA case, we assume that $u(c,t,\theta )=-\exp (-c/t(\theta ))$ , which is equivalent to γ tending to + ∞ , and a(θ)/γ tending to − t(θ). Equation 8 implies in that case that

  $$ p^{v}(P)=p^{CARA}(P)=K\prod_{\theta =1}^{N}p(\theta )^{\frac{t(\theta )}{ \sum_{\theta ^{\prime }=1}^{N}t(\theta ^{\prime })}}, $$

  (30)

  where K is a normalizing constant. Aggregation rule (30) and (29) are due to Rubinstein (1974). This aggregation rule is particularly easy to use when all individual beliefs are normally distributed. Suppose that agent θ, θ = 1,...,N, believes that states are normally distributed with mean μ(θ) and variance σ ²(θ). An easy consequence of Eq. 30, first observed by Lintner (1969), is that the collective beliefs p ^v are also normally distributed with mean

  $$ \mu ^{v}=\frac{\sum_{\theta =1}^{N}\frac{t(\theta )\mu (\theta )}{\sigma ^{2}(\theta )}}{\sum_{\theta =1}^{N}\frac{t(\theta )}{\sigma ^{2}(\theta )}}, $$

  (31)

  and variance

  $$ \sigma ^{v}=\left[ \frac{\sum_{\theta =1}^{N}\frac{t(\theta )}{\sigma ^{2}(\theta )}}{\sum_{\theta =1}^{N}t(\theta )}\right] ^{-0.5}. $$

  (32)