Elyès Jouini

Clotilde Napp

Cognitive biases and the representative agent

Keywords: G11, D81, D84, D87, D03, H43 behavioral agent, hyperbolic discounting, probability weighting function, representative agent, neuro…nance, ambiguity aversion

est destinée au dépôt

Introduction

In this paper, we analyze a model of Pareto optimal allocations with von Neuman Morgenstern utility maximizing agents and exponential discounting. Agents are heterogeneous, in the sense Corresponding author: jouini@ceremade.dauphine.fr, +33144054226 1 Electronic copy available at: http://ssrn.com/abstract=1517780 that they might di¤er in their beliefs and in their time preference rates. At the aggregate level, the social welfare function of this economy is characterized by a social/representative belief and a social/representative time preference rate. We examine these social characteristics and we show that we retrieve, at the aggregate level, behavioral properties that have been proved to be true at the individual level in recent literature. The group acts as a behavioral agent and these behavioral properties at the aggregate level are generated by heterogeneity alone.

We start by introducing natural notions of optimism and pessimism and we assume that beliefs are heterogeneous enough in order to allow for optimistic as well as pessimistic agents in the initial set of von Neuman Morgenstern utility maximizing agents. In such a setting, we obtain that the representative agent can neither be everywhere optimistic nor everywhere pessimistic; she is optimistic for the good states of the world and pessimistic for the bad states of the world. As in the SP/A Theory of [START_REF] Lopes | Between hope and fear: the psychology of risk[END_REF], the representative agent behaves as if she had fear (need for security) for very bad events and hope (desire for potential) for very good events. The representative agent puts more weight on extreme events. We show that the distribution of outcomes from the representative agent point of view is portfolio dominated by the objective distribution. This means that heterogeneity generates doubt at the aggregate level. This e¤ect is reinforced when agents are more risk tolerant or when there is more heterogeneity among agents.

The representative agent distorts the objective distribution of aggregate endowment. We analyze this distortion and we show that the distortion function (de…ned as the transformation of the objective decumulative distribution function into the decumulative distribution function of the representative agent) is inverse S-shaped as is the probability weighting function in Cumulative Prospect Theory. We show that we are able to …t relatively well standard probability weighting functions of the Cumulative Prospect Theory literature [START_REF] Tversky | Advances in Prospect Theory: Cumulative Representation of Uncertainty[END_REF][START_REF] Tversky | Ambiguity Aversion and Comparative Ignorance[END_REF], Prelec, 1998, among others). We analyze how the distribution of individual characteristics in the group governs the shape of the resulting representative agent distortion function. Following [START_REF] Gonzalez | On the shape of the probability weighting function[END_REF] terminology, we show that attractiveness at the aggregate level is directly related to the average level of optimism while discriminability is related to beliefs heterogeneity.

As far as discount rates are concerned, we obtain that the aggregate time preference rate is decreasing, which is consistent with "hyperbolic discounting". It converges to the time preference rate of the most patient individual. These properties are similar to those obtained in a deterministic setting by [START_REF] Gollier | Aggregation of heterogeneous time preferences[END_REF] and [START_REF] Lengwiler | Heterogeneous patience and the term structure of real interest rates[END_REF] but are derived here in a stochastic setting that takes into account beliefs heterogeneity.

As a main application of these results we obtain that a behavioral individual (i.e. an individual whose preferences are governed by hyperbolic discounting and who distorts the distributions through a probability weighting function) behaves as would a group of standard heterogeneous vNM individuals with exponential discounting. Our results can then be related to Neuroeconomics. As underlined by [START_REF] Cohen | The Vulcanization of the Human Brain: a Neural Perspective on Interactions beetween Cognition and Emotion[END_REF], …ndings from neuroscience "provide support for a view of the brain as a confederation of systems and behavior as the outcome of an interaction among these, (...) when disagreements arise, behavior re ‡ects the outcome of a competition among systems". We show that a model of the brain as a central planner who, for the evaluation of a given prospect, maximizes the social welfare of a collection of competing doers (mental processes with heterogeneous time preference rates and noisy beliefs about the prospect under consideration) leads to hyperbolic discounting and to probability weighting functions that have the same shape as in the Cumulative Prospect Theory (CPT). Furthermore, such a model of individual behavior accomodates with a distinction between risk and uncertainty and exhibits, for some values of the parameters, ambiguity aversion. In such a framework, ambiguity aversion, inverse S-shaped probability weighting functions and hyperbolic discounting result from doers heterogeneity.

Note that we don't pretend to retrieve all features of CPT on the aggregate belief nor all features of the time preference rate as in e.g. [START_REF] Loewenstein | Anomalies in intertemporal choices: evidence and an interpretation[END_REF]. We only retrieve one of the three main features of CPT, namely the inverse S-Shaped probability distribution weighting function (the other two being the presence of a reference point and the presence of loss aversion). This comes from the fact that we have introduced heterogeneity on the beliefs only, hence the behavioral property that we retrieve deals with the belief only. We also only obtain the "hyperbolic"property of the time preference rate and not other behavioral properties such as the di¤erent (discounting) treatment of gains and losses.

The paper is organised as follows. Section 2 presents the model. Section 3 analyses the properties of the belief of the representative agent, while Section 4 analyses the properties of the time preference rate of the representative agent. Section 5 provides possible applications.

The Setting

We consider an economy with a single consumption good and with agents who have the same utility function but heterogeneous beliefs. Aggregate endowment in the consumption good is described by a random variable e de…ned on a probability space ( ; F; P ) : We let I denote the set of heterogeneous agents. We assume that the common utility function is CRRA with derivative given by u 0 (x) = x 1 . Each agent has a subjective belief Q i and wants to maximize her von Neumann Morgenstern utility for consumption of the form

U i (c) = E Q i [u (c)] :
We let M i denote the density of Q i with respect to the probability P , hence agent i's utility for consumption can equivalently be written in the form

U i (c) = E M i u (c) :
In such an economy, we consider the aggregate utility function U de…ned as the solution of the following maximization program

U (e ) max P i2I y i =e X i2I i E M i u(y i )
where ( i ) are given positive weights. The aggregate utility function corresponds to the value of the social welfare function at the Pareto optimum when agent i is granted a weight i by a social planner. The index i may also represent a group of agents with common beliefs M i and y i then represents the total consumption of the group and i the sum of the weights granted by the social planner to the individuals in the group. When the social planner grants the same weight to all the agents in the economy, the weight i represents the proportion of agents that have the same belief M i : From a social planner point of view, the aggregate utility function corresponds to the highest social utility level among all possible endowment distributions across agents.

The number of agents can be …nite or in…nite. In the case of an in…nite number of agents, sums are replaced by integrals. We obtain the following representation result.

Proposition 1 Representative Agent

The aggregate utility for consumption is given by

U (e ) = E [M u(e )] with M = X i2I i M i ! 1
(1)

for i = i :
The representative agent belief is then given by M =

P i2I i M i 1 .
This means that, at the Pareto optimum, the aggregate utility is given by the utility of a representative agent endowed with an average belief (and the same utility function as each of the agents). In particular, if all the agents share the same belief, then the representative agent will share this common belief. If we think of e as a given prospect for the group I of agents, the aggregate utility U (e ) corresponds to the social welfare associated with the optimal allocation of e across the members of the group and is given by the utility of the representative agent.

In the same way as the representative agent belief M is an average of the individual beliefs M i , we show in the next corollary that the density f M of e for the representative agent is an average of the densities f i of e for the di¤erent agents. For this purpose, let us assume that for all i 2 I; the distribution of e for agent i admits a density1 (with respect to the Lebesgue measure on the real line), denoted by f i : We also assume that the distribution of e under the probability P admits a density and we denote it by f: Since we don't have E [M ] = 1 (except in the speci…c logarithmic utility setting) we need the following technical de…nition. We say that the distribution of a random variable X admits a "density f X for the representative agent" if for all function h, we have Corollary 2 The distribution of e admits the following density for the representative agent

E [M h (X)] = R h (x) f X (x) dx: Moreover,
f M = X i2I i f i ! 1=
which is a power average of the initial densities. In particular, for = 1; the distribution of e for the representative agent is a mixture of the individual subjective distributions.

As an immediate consequence of Corollary 2, we get that for any measurable real-valued function '; the distribution of ' (e ) admits the density f M;' = P i2I i f i;' 1= for the representative agent where f i;' denotes the density of the distribution of ' (e ) for agent i: This implies in particular that in the case = 1; if each agent anticipates a normal distribution on log e , then the distribution of log e is a mixture of normal distributions.

3 Behavioral properties of the group

Qualitative properties

The next two simple examples illustrate the qualitative properties of the endowment distribution from the representative agent point of view. The proofs can be found in the Appendix.

Example 1. Let us assume that all utility functions are logarithmic ( = 1): We have

E Q [e ] = X i2I i E Q i [e ] ;
which means that the mean at the aggregate level is given by an arithmetic average of the individual means. The variance is given by

V ar Q [e ] = X i2I i V ar Q i [e ] + V ar i E Q i [e ] ;
where

V ar i E Q i [e ] P i2I i E Q i [e ] 2 P i2I i E Q i [e ]
2 measures beliefs (on the mean) heterogeneity. This means that the variance at the aggregate level is given not only by an arithmetic average of the individual variances, but also by an additional term related to beliefs dispersion. The variance is "increased" at the aggregate level and this increase is proportional to the level of beliefs heterogeneity: beliefs heterogeneity generates "doubt".

Example 2. Let us assume that the objective distribution of aggregate endowment is lognormal with e P ln N ( ; 2 ) and that we have two equally weighted groups of agents, both with lognormal subjective distributions for aggregate endowment, e Q i ln N ( i ; 2 ) for i = 1; 2:

The distribution of log e for the representative agent is not Gaussian and when agents'beliefs are heterogeneous enough (j 1 2 j > 2 p ), the distribution of log e is bimodal (see Figure 1).

When = 1 + 2 2 ; the distribution of log e for the representative agent is Portfolio Dominated 2 2 Let us recall that a distribution f dominates a distribution g in the sense of Portfolio Dominance (f P D g) if we have R u 0 (x)(x a)f (x)dx = 0 =) R u 0 (x)(x a)g(x)dx = 0 for any real number a and any non-decreasing concave function u: This concept has been introduced in the context of portfolio problems by [START_REF] Landsberger | Mean Preserving Portfolio Dominance[END_REF] and further studied by [START_REF] Gollier | A Note on Portfolio Dominance[END_REF]. It characterizes the changes in the distribution of the by the objective distribution. Hence, aggregate endowment e is considered as more risky by the representative agent than it actually is. In particular, we have V ar

Q [log e ] > V ar P [log e ].
This last property still holds for general ( i ). Figures 1 and 2 illustrate these conclusions in di¤erent settings. Note that Figure 1 is similar to Figure 8.2 in [START_REF] Shefrin | A Behavioral Approach to Asset Pricing[END_REF]. For > 0 and associated representative agent probability measures Q and Q 0 ; we have dQ dQ 0 = h ; 0 (e ) where h ; 0 is symmetric with respect to 1 + 2 2 , decreasing before 1 + 2 2 and increasing after 1 + 2 2 .

A higher level of risk tolerance induces then a portfolio dominated shift in the representative agent's distribution. In particular, V ar Q [log e ] increases with the level or risk tolerance :

The interpretation is the following. When there is heterogeneity, each agent consumes a larger proportion of aggregate endowment in states of the world that she considers more likely. This leads to heterogeneous allocations and generates variance at the aggregate level. However, this e¤ect is counterbalanced by risk aversion. Consequently, the higher the level of risk tolerance, the more heterogeneous the members of the group are in their optimal allocations. Figure 2 illustrates this result.

For normal distributions N ( i ; 2 ), there is a natural order on the set of possible densities induced by the natural order on the means ( i ). Agents with a larger (resp. smaller) i can be referred to as more optimistic (resp. pessimistic). We generalize these notions of relative pessimism/optimism in the following de…nition. If we assume that P is the objective probability, then we are also able to introduce absolute notions of pessimism/optimism.

De…nition 1 For i; j 2 I; agent i is said to be more optimistic than agent j and we denote it by f i < opt f j if and only if f i f j is nondecreasing. The optimism relation < opt is an order on the set (f i ) i2I : If P is the objective probability, then agent i is said to be optimistic (resp. pessimistic)

if f i f is nondecreasing (resp. nonincreasing).
De…nition 1 can be rephrased in terms of Monotone Likelihood Ratio Dominance (MLR) 3 returns of the risky asset that lead to an increase in demand for the risky asset irrespective of the risk-free rate.

It is then related to the degree of riskiness. See also [START_REF] Jouini | Discounting and Divergence of Opinion[END_REF]. 3 This concept is widely used in the statistical literature and was …rst introduced in the context of portfolio 8 : agent i is more optimistic than agent j if the distribution of e for agent i (i.e., under Q i )

dominates the distribution of e for agent j (i.e., under Q j ) in the sense of the MLR. For a given agent i; if we let g i denote the transformation of the objective decumulative distribution function F into the agent's subjective decumulative distribution function F i , i.e. such that A MLR dominated shift for a given distribution reduces the mean and if f i < opt f j then

F i = g i F; it
we have

E Q i [e ] E Q j [e ]
. This last condition characterizes the MLR dominance when we restrict our attention to a family of lognormal distributions with the same variance parameter and we then retrieve that agent i is more optimistic than agent j if and only if i > j : In that case, optimistic (resp. pessimistic) agents are then characterized by i > (resp. i < ) as in [START_REF] Shefrin | A Behavioral Approach to Asset Pricing[END_REF].

Proposition 3 We suppose that there are at least one optimistic agent denoted by f opt and one pessimistic agent denoted by f pess in the set I of agents: We also assume that lim

+1 fopt f = lim 1 fpess f = +1 and lim 1 fopt f = lim +1 fpess f = 0:
1. The representative agent can neither be optimistic, nor pessimistic, i.e. f M f is non monotone.

2. The representative agent overestimates the weight of the "good states of the world" (high problems by [START_REF] Landsberger | Demand for risky assets: a portfolio analysis[END_REF]. More precisely, [START_REF] Landsberger | Demand for risky assets: a portfolio analysis[END_REF] showed that in the standard portfolio problem a MLR shift in the distribution of returns of the risky asset leads to an increase in demand for the risky asset for all agents with nondecreasing utilities. 4 More precisely, in an expected utility framework [START_REF] Abel | An exploration of the e¤ects of pessimism and doubt on asset returns[END_REF] de…nes pessimism by the condition Fi F (First Stochastic Dominance) that corresponds to the condition gi Id introduced by [START_REF] Chateauneuf | Risk seeking with diminishing marginal utility in a nonexpected utility model[END_REF] in a RDEU setting.

values of e ) as well as the weight of "bad states of the world" (low values of e ), i.e. It appears from this proposition that as long as there are optimistic as well as pessimistic agents in the set I of agents, the representative agent behaves like the individual agents considered in the behavioral economics and/or psychology literature. Indeed, she puts more weight on small probability events with large consequences as in the Cumulative Prospect Theory of [START_REF] Tversky | Advances in Prospect Theory: Cumulative Representation of Uncertainty[END_REF]. She has fear (need for security) for very bad events and hope (desire for potential) for very good events as in the SP/A Theory of [START_REF] Lopes | Between hope and fear: the psychology of risk[END_REF]. Everything works then as if the representative agent distorted the objective distribution of e . In the next section, we analyze more precisely how this distortion operates.

f M (x) f (x) for x x and f M (x) f (x)

The distortion function

Let us …rst recall that in the context of Cumulative Prospect Theory, the probability weighting function is de…ned as the function that transforms the decumulative objective distribution function into the decumulative subjective distribution function. This probability weighting function operates on any prospect. In our framework, we have seen that the representative agent distorts the objective distribution of e : In the present section, we will describe this distortion in terms of a transformation of the decumulative distribution function of e and we will see that this distortion function has the same shape as the probability weighting function of the Cumulative Prospect Theory. We will then use the typology introduced by [START_REF] Gonzalez | On the shape of the probability weighting function[END_REF] on the probability weighting functions to describe our distortion function and to relate its characteristics to the individual beliefs of the agents. In particular, we analyze how shifts in the distribution of individual beliefs impact the distortion function.

With the same notations as in Section 2, we denote by g the distortion function that transforms the objective decumulative distribution function

R 1 x f (s)ds into the decumulative distri- bution function of the group R 1 x f M (s)ds; i.e. such that g( R 1 x f (s)ds) = R 1 x f M (s)ds:
The next proposition assumes that there are at least one optimistic and one pessimistic agent in the set I and characterizes the shape of g under this assumption.

Proposition 4 1. In the lognormal setting with log e Q i N i ; 2 for i = 1; :::; N and if the set I is made of both optimistic and pessimistic agents then the function g is inverse S-shaped: concave then convex.

2. In the general setting, if there are at least one optimistic agent f opt and one pessimistic agent f pess with lim +1

fopt f = lim 1 fpess f = +1 and lim 1 fopt f = lim +1 fpess f
= 0 and if g is continuously twice di¤ erentiable on [0; 1], then g is concave for small probabilities, and convex for high probabilities.

The function g has then the same shape as the probability weighting function of the Cumulative Prospect Theory. This is in particular illustrated in Figure 3. A variety of methods have been used to determine the shape of the probability weighting function. [START_REF] Tversky | Advances in Prospect Theory: Cumulative Representation of Uncertainty[END_REF], Fox and [START_REF] Tversky | Ambiguity Aversion and Comparative Ignorance[END_REF] and [START_REF] Prelec | The Probability Weighting Function[END_REF] among others specify parametric forms

(respectively ! (p) = p [p +(1 p) ] 1= , ! (p) = p [ p +(1 p) ] 1= and ! (p) = exp ( log p )) and
estimate them through standard techniques. Figure 4 permits to show that with a well chosen distribution of agents'characteristics we obtain a distortion function that perfectly …ts Prelec (1998)'s function. [START_REF] Gonzalez | On the shape of the probability weighting function[END_REF]Gonzalez (1996, 1998) and [START_REF] Abdellaoui | Parameter-free Elicitation of Utilities and Probability Weighting Functions[END_REF] avoid the potential problems of parametric estimation and directly derive from experimental studies the shape of the probability weighting function at the aggregate or individual level. The results of all these studies are (mostly) consistent with an inverse S-shaped weighting function, concave for small probabilities, and convex for moderate and high probabilities.

In the lognormal setting, if we denote by i the quantity i = i ; it is interesting to remark that the distortion function g only depends on the i s and on the relative proportions i s and is independent of and . In other words, the distortion function only depends on how much the agents deviate from the objective mean in terms of standard deviation.

Discriminability, attractiveness and individual agents'beliefs

Let us analyse how the main features for the shape of the representative agent's distortion function g; for a given aggregate endowment e ; are related to the individual characteristics of the agents in I. [START_REF] Gonzalez | On the shape of the probability weighting function[END_REF] exhibit two main features for the shape of the probability weighting function in the context of CPT: diminishing sensitivity and attractiveness.

Attractiveness characterizes the absolute level of the function. Indeed, an inverse S-shaped function can be completely below the identity line, can cross the identity line at some point or can be completely above the identity line. Betting on the chance domain is more attractive when the graph of the probability weighting function graph is more "elevated". The de…nition of attractiveness is expressed in terms of First Stochastic Dominance (FSD). The probability weighting function g 1 is more attractive than the probability weighting function g 2 when the subjective density f 1 dominates the subjective density f 2 in the sense of the FSD. In our setting, we will say that a (representative agent's) distortion function g 1 associated with a set I 1 of agents is more attractive than a (representative agent's) distortion function g 2 associated with a set I 2 of agents if f M I 1 dominates f M I 2 in the sense of the FSD. Attractiveness of the distortion function 12

is related to the level of optimism of the representative agent. In particular, since FSD is weaker than MLR, a more optimistic representative agent is associated with a more attractive distortion function.

Let ( i ) and ( 0 i ) denote two possible distributions of agents' density functions. If the set (f i ) i2I of agents' density functions is totally ordered with respect to the FSD order, we will say that the distribution ( 0 i ) dominates the distribution ( i ) in the sense of the FSD if for any increasing family (f i ) ; we have

P 0 i f i < F SD P i f i :
In other words, the distribution ( 0 i ) puts more weight on more attractive distributions. If the set (f i ) i2I of agents'density functions is totally ordered with respect to the optimism order < opt ; we will say that the distribution ( 0 i )

dominates the distribution ( i ) in the sense of the MLR if whenever f i < opt f j we have

0 i i 0 j j :
In other words the ratio between the two densities ( 0 i ) and ( i ) increases with agents'optimism and, in particular, the distribution ( 0 i ) puts more weight on more optimistic agents.

In the next proposition we analyze the impact of shifts in the distribution of agents characteristics on the attractiveness of the distortion function and on the level of optimism of the representative agent.

Proposition 5 1. For log-utility functions and in the case of lognormal distributions log e Q i N i ; 2 for i = 1; :::; N; with the same variance parameter 2 , a FSD shift in the distribution of the means ( i ) increases attractiveness of the representative agent's distortion function.

2. For log-utility functions and general distributions, if the set (f i ) i2I of agents'density functions is totally ordered with respect to the FSD order then a FSD shift in the distribution of agents' density functions increases attractiveness of the representative agent's distortion function.

3. For general CARA utility functions and general distributions, if the set (f i ) i2I of agents' density functions is totally ordered with respect to the optimism order < opt then a MLR dominated shift in the distribution of agents'density functions increases attractiveness of the representative agent's distortion function and the level of pessimism of the representative agent.

When all agents have logarithmic utility functions, attractiveness at the representative agent level increases with the weight granted to the more attractive density functions. Since FSD is weaker than MLR, attractiveness at the representative agent level increases with the weight granted to the more optimistic agents. This is illustrated by Figure 5. As shown in Proposition 5, this last property can be extended to power utility functions if we replace FSD shifts on the distribution of agents'density functions by MLR shifts.

Diminishing sensitivity corresponds to the fact that people become less sensitive to changes in probability as they move away from a reference point. In the probability domain, the two endpoints 0 (certainly will not happen) and 1 (certainly will happen) serve as reference points and under this principle, increments near the endpoints of probability loom larger than increments near the middle of the scale. This concept is related to the concept of discriminability in psychophysics literature and can be illustrated by two extreme cases: a function that approaches a step function and a function that is almost linear.

In our setting we say that a representative agent's distortion function g 1 associated with a set I 1 of agents exhibits more discriminability than a representative agent's distortion function g 2 associated with a set I 2 of agents if there exists x 2 [0; 1] such that g 1 g 2 for x x and g 1 g 2 for x x : In the next proposition we show that the level of discriminability of the representative agent's distortion function is closely related to the level of disagreement among agents.

Let us consider as above a family of agents with lognormal distributions ln N ( i ; 2 ). We denote by ( i ) the support of the distribution of the mean parameter and by ( i ) the associated weights. Recall that a mean preserving spread is de…ned as a modi…cation of the distribution set ( i ) on a set of three locations 1 < 2 < 3 with associated increments 1 0; 2 0 14 and 3 0 such that P 3 i=1 i = 0 and P 3 i=1 i i = 0: A mean preserving spread will be said symmetric if 1 = 3 :

Proposition 6 For log-utility functions and in the case of lognormal distributions log e

Q i N ( i ;
2 ), a symmetric mean-preserving spread on the distribution of the means ( i ) decreases discriminability of the representative agent's distortion function.

Intuitively, this proposition means that when the level of disagreement among agents increases, then the representative agent focuses more on the endpoints of the probability domain and is less sensitive to probability variations in the middle of the scale. Figure 6 illustrates this result. It shows, in the setting with two agents, that discriminability decreases with the level of disagreement. When both agents agree on the objective distribution, the probability weighting function is linear. When the agents disagree, one of them overestimating the average payo¤ by twice the standard deviation and the other underestimating it by twice the standard deviation, we obtain a function that approaches a step function.

The Setting with Heterogeneous Time Preference Rates

In this section, we extend our framework in order to take into account the impact of time and of heterogeneous time preference rates across the agents. Aggregate endowment at a given date

t
P i2I y i t =e t X i2I i E M i t D i t u(y i t )
where ( i ) are given positive weights. Each agent is then characterized by a belief M i t ; a discount factor D i t and a weight i :

We will say that the characteristics M i t ; D i t ; i i2I are independent if for almost all states of the world !; M i t (!) ; D i t and i are independent 5 as random variables on I: This property will be, in particular, satis…ed when I can be written in the form I = J K L and when there exist characteristics M j t j2J

; D k t k2K and ` `2L such that for i = (j; k; `) we have

M i t ; D i t ; i = M j t ; D k t ; ` :
Roughly speaking, this property means that there is no speci…c correlation between beliefs and time preferences and that the weights granted by the social planner to the individuals in the economy are independent of their time and belief characteristics.

This condition is, in particular, satis…ed when beliefs and time preferences are independent and when the agents are uniformly weighted in the social welfare function. This is also the case when the agents'weights are given by their relative wealth and when wealth, beliefs and time preferences are independent.

Assuming uniform weights is quite reasonable since there is no particular reason for the social planner to favor one agent with respect to another agent. The independence of beliefs and time preference rates is more disputable. They may be positively as well as negatively correlated, the independence condition may then be analyzed as a central scenario.

We easily obtain the following analog of Proposition 1 in the framework with heterogeneous time preference rates.

Proposition 7 If the characteristics M i t ; D i t ; i i2I are independent, then the aggregate utility for consumption is given by

U (e t ) = E [M t D t u(e t )]
5 More precisely, for any real valued (measurable) functions f; g; h de…ned on the real line, we have

1 jIj X i2I f M i t g D i t h ( i) = 1 jIj X i2I f M i t ! 1 jIj X i2I g D i t ! 1 jIj X i2I h ( i) ! a.e. with M t = 1 jIj X i2I M i t ! 1 and D t = 1 jIj X i2I D i t ! 1 :
The representative agent belief is then given by M t = 1 jIj P i2I M i t 1 and the representative agent time discount factor is given by D

t = 1 jIj P i2I D i t 1 :
This means that all the properties established in the previous section on the belief of the representative agent remain valid.

The properties of the representative agent time preference rate are easy to obtain. Note that the properties of a "consensus" time preference rate when there is heterogeneity on the individual time preference rates (and not on the beliefs) have already been studied in varying contexts. Indeed, the problem of the aggregation of the utility discount rates has been studied by [START_REF] Reinschmidt | Aggregate Social Discount Rate derived from individual discount rates[END_REF] through a certainty equivalent approach, by [START_REF] Gollier | Aggregation of heterogeneous time preferences[END_REF] and [START_REF] Nocetti | Properties of the social discount rate in a Benthamite framework with heterogeneous degrees of impatience[END_REF] through a Benthamite/Pareto optimal approach, and by [START_REF] Lengwiler | Heterogeneous patience and the term structure of real interest rates[END_REF] through an equilibrium approach. All these papers adopt a deterministic setting with no divergence on the beliefs of the agents. On the contrary our aim here is to derive the properties at the aggregate level simultaneously on the beliefs and on the time preference rate (and in a quite general stochastic setting).

We know that the representative agent time discount factor is given by D

(t) = 1 t log " 1 N N X i=1 exp ( i t) # 1= ; D m (t) = N X i=1 exp ( i t) P N i=1 exp ( i t) i :
2. The representative agent time preference rates are lower than the average of the time preference rates, i.e.

D m (t) 1 N N X i=1 i = D m (0) and D a (t) 1 N N X i=1 i = D a (0)
with strict inequalities when i 6 = j for some (i; j) in I.

"Behavioral Properties" :

The representative agent time preference rates are decreasing with time. Moreover, the asymptotic discount rates are given by the lowest time preference rate, i.e.

lim t!+1 D a (t) = lim t!+1 D m (t) = inf i ( i ) :
The representative agent behaves for t large enough like the most patient agent.

These formulas permit explicit computations for speci…c distributions of the individual time preference rates. For instance, if we assume a Gamma6 distribution ( ; ) for the i s we

obtain D m (t) = m 2 m + v 2 t
where m and v 2 respectively denote the mean and the variance of the considered distribution.

It is immediate on this simple example that the marginal discount rate decreases with time and is hyperbolic as in Weitzman (1998Weitzman ( , 2001)). Furthermore, the speed of the decrease increases with the level of heterogeneity v 2 as well as with the level of risk tolerance.

The next proposition provides comparative statics results for shifts in the distribution f of the individual time preference rates:

Applications

We have seen that starting from a standard model with optimistic as well as pessimistic vNM and exponential discounting agents, we obtain, at the representative agent level, properties such as an inverse S-shaped distortion function and hyperbolic discounting, that are in line with recent empirical and experimental results. A possible interpretation of such a result is to consider that each individual subject to experiments behaves as a group of individuals at the equilibrium. This provides us with a possible representation of the brain as an organization with a social planner and heterogeneous doers. Some doers are overoptimistic while others are overpessimistic. Similarly, some doers are impatient while others are more patient. Such an approach is in the same spirit as [START_REF] Brocas | The brain as a hierarchical organization[END_REF] where the authors divide the brain into two systems: an impulsive/myopic one and a cognitive/forward-looking one. However, while [START_REF] Brocas | The brain as a hierarchical organization[END_REF] model mainly relies on information asymmetries and principalagent models, our model relies on decentralization and optimal allocation approaches. Such a decomposition of the brain into di¤erent systems that are possibly in con ‡ict are based on recent neuroscience and psychology evidences (see [START_REF] Cohen | The Vulcanization of the Human Brain: a Neural Perspective on Interactions beetween Cognition and Emotion[END_REF] related to intrapersonal tensions:

temporal horizon con ‡icts, information con ‡icts (that may lead to information asymmetries as in [START_REF] Brocas | The brain as a hierarchical organization[END_REF], but also to information diversity and beliefs heterogeneity as in our model) or utility evaluation con ‡icts. In economics, such decompositions have been …rst

considered by [START_REF] Thaler | An Economic Theory of Self-Control[END_REF] and [START_REF] Shefrin | The Behavioral Life-Cycle Hypothesis[END_REF].

Noisy Beliefs and Cumulative Prospect Theory

In this section we show that an individual who evaluates lotteries through the social welfare function associated with a collection of agents (neural systems or brain areas), each of them with speci…c noisy beliefs, is a CPT agent in the sense that she distorts the distribution of the lotteries through an inverse S-shaped weighting function (commpon to all lotteries).

We start by considereing normal distributions. Let us consider an individual who when facing a lottery whose payo¤ x is described by a normal distribution N ( ; 2 ) passes this information for evaluation to separate systems. Each system i has a subjective belief Q i under which x has a normal distribution N + i ; 2 : The parameter i is …xed independently of x and characterizes the system i. It might result from noise in the information transmission. In that case there is no speci…c reason for the average perceived signal to be biased and we should have

P i = 0.
It might also result from a specialization of the di¤erent systems, some systems being optimistic, i.e., i > 0 and others being pessimistic, i.e., i < 0: We assume that the individual acts like a central planner looking for a Pareto optimal decomposition of the payo¤s from the lottery among the systems and evaluates the lottery through the social welfare function, i.e., U (x) = max

P x i =x X i i E Q i u(x i ) (2)
where the parameters i are the weights granted to the systems by the central planner or the distribution of the i s.

Proposition 10 Consider an individual who evaluates any lottery x in the space X of lotteries with normal payo¤ s through U (x):

1. The individual is a CPT agent over the space X in the sense that there exists a probability weighting function g such that, for all lotteries x in X with density f x we have

U (x) = R f x; (s)u (s) ds where g ( R 1 t f x (s)ds) = R 1 t f x; (s)ds:
2. If there exist at least one optimistic and one pessimistic system, then g is inverse Sshaped.

3. A MLR shift on the distribution of the i s increases attractiveness of the probability weighting function g .

This means that a Pareto optimal decomposition leads to an overall (representative agent) evaluation that corresponds to the valuation that would be provided by a behavioral agent.

The level of discriminability would then be directly related to the level of noise as illustrated in Proposition 6 in the case of log utility functions. The level of attractiveness would be associated to the level of systematic bias (if any) as a direct corollary of Proposition 5.

The behavior of the individual and the de…nition of the social welfare function U can be naturally generalized to any lottery whose payo¤ is a function of a normal distribution. Indeed, consider a lottery whose payo¤ is of the form v = '(x) where x is normally distributed as above and where ' is a Borelian function. We may de…ne U (v) by

U (v) = max P v i =v X i i E Q i u(v i )
where the Q i s and the i s are the same as for x.

The following result extends the result of Proposition 10 to general lotteries. It relies on the fact that any random variable is distributed as a function of a given normally distributed variable.

Proposition 11 Consider an individual who evaluates through U any lottery v = '(x) where

x is normally distributed and whose preferences over the set of all possible lotteries only depend on the distribution of the lottery under consideration. The individual is a CPT agent over the space of all possible lotteries in the sense that her preferences can be represented by the utility function U extended to the space of all possible lotteries and there exists a probability weighting function g such that, for all lottery v with density f v we have

U (v) = R f v; (s)u (s) ds where g ( R 1 t f v (s)ds) = R 1 t f v; (s)ds:
This corollary provides then the following possible interpretation of CPT: the result of a possibly noisy transmission of the objective distribution to separate (specialized) systems (neural systems or brain areas), the overall evaluation resulting from a social welfare function applied to these systems. The construction of the Q i s in the general case is very similar to their construction in the normal case. The resulting global behavior might then be associated intuitively with a possible behavior of the systems that consists in describing any random variable in terms of Gaussian distributions. For instance, a random variable that takes values 0 and 1 with probability 1/2 may be described as a random variable that takes value 1 when a given Gaussian variable N (0; 1) is positive and that takes value 0 when the Gaussian variable is negative. The process i will then transform this binomial distribution into a binomial distribution that is equal to 1 when a Gaussian variable N ( i ; 1) is positive -or equivalently when a Gaussian variable N (0; 1) is smaller than i -and is equal to 0 when the Gaussian variable N ( i ; 1) is negative.

Heterogeneous Beliefs and Ambiguity aversion

In this section, we show that an individual who evaluates ambiguous lotteries through the social welfare function associated with a collection of agents, each of them being endowed with a plausible probability, may be ambiguity averse.

Let us consider an individual facing a lottery whose payo¤ x has an ambiguous distribution.

The ambiguity of the lottery is characterized by a set of plausible distributions with a set of associated probabilities x = (Q x 1 ; :::; Q x n ). We assume that the agent passes the information to di¤erent systems (neural systems or brain areas), each endowed with one of the plausible distributions, in order to evaluate this lottery. We assume that the individual acts like a central planner looking for a Pareto optimal decomposition of the gains/losses from the lottery among the di¤erent processes and evaluates the lottery through the social welfare function, i.e., U (x) = max

P x i =x X i2I E Q x i u(x i ) :
We assume that the common utility function is given by u(x) = x 1 1 1 1

with < 1:

Consider …rst the classical Ellsberg 3-color urn problem. An urn contains 30 red balls and 60 black and yellow balls in unspeci…ed proportion. A lottery A that pays $100 if a red ball is drawn is evaluated at U (A) = 1 3 u (100) : Since the probability for one black ball to be drawn is unknown, and if all possible probabilities of the form i 90 for i = 0; :::; 60; are considered as equally plausible, a lottery B that pays $100 if a black ball is drawn is evaluated Such an individual prefers lottery D to lottery C. As in Ellsberg paradox, Savage's independence axiom is violated, even though each process is an expected utility maximizer and satis…es the independence axiom. The individual exhibits ambiguity aversion. In order to generalize this result let us recall the following de…nitions adapted to our setting from Schmeidler7 (1989).

De…nition 2 An ambiguous lottery x; (Q x i ) i2I is de…ned by a random variable x, whose values (x (!)) !2 represent the set of possible outcomes and by a set (Q x i ) i2I of plausible probabilities over .

De…nition 3 If x; Q 1;x i i2I
and x; Q 2;x i i2I

are two ambiguous lotteries associated to the same random variable x, the even mixture of these ambiguous lotteries is the ambiguous

lottery x; 1 2 Q 1;x i + 1 2 Q 2;x i i2I :
De…nition 4 An agent is called ambiguity averse if she prefers the even mixture of two ambiguous lotteries that she values equally to either of the two lotteries.

Proposition 12 When < 1; an individual who evaluates ambiguous lotteries through the social welfare function associated to a collection of agents each of them being endowed with one of the plausible probabilities is ambiguity averse.

We have considered as equally plausible all probabilities associated to a given ambiguous lottery. Our results can be generalized to the setting where the individual, based on his subjective information, associates a probability distribution (q i ) i2I over the set (Q x i ) i2I where q i 0 is the subjective probability that Q x i be the probability associated to the true distribution of the lottery payo¤s. In such a setting, the q i s might be integrated in the individual overall utility function as the weights in the social welfare function, i.e.

U (x) = max

P x i =x X i2I q i E Q x i u(x i ) :

Dynamic setting

We have until now only considered a two dates setting even though we have considered di¤erent possible time horizons for the second date. It is useful to understand how our model might be embedded in a dynamic setting.

Consider as in …nancial models a di¤usion setting: We denote by W a Brownian motion and we assume that e follows the following stochastic di¤erential equation with constant parameters

de t = + 1
2 2 e t dt + e t dW t . The distribution of e t is then, for all t; lognormal of the form log e t N ( t; 2 t):

Let us …rst assume as in Section 5.1 that agents' deviation from the objective mean is constant in terms of standard deviation, i.e., that the subjective distributions are of the form log N ( i t; 2 t) with i t = t + i p t: Following Proposition 10, there exists a probability weighting function g that distorts the objective distribution into the distribution of the group.

This function is independent of t and the behavior of the group is then consistent across time.

Let us now consider the case where beliefs heterogeneity results from ambiguity. Di¤erent probabilities are considered as plausible and each agent is endowed with one of them. In our di¤usion setting, let us consider the simplest case where each probability is de…ned by its density dM i = i M i dW where the i s are constant. Following Girsanov Theorem, the distribution of log e t from agent i point of view is given by N ( i t ; 2 t) with i t = ( 1 2

2 )t + i t: As seen in Example 1, the ex-ante variance of log e t from the representative agent point of view is given by

V ar M [log e t ]) = V ar [log e t ] + V ar i i t = 2 t + 2 t 2 V ar i i :
The long term ex-ante variance from the representative agent point of view, is then higher than the objective variance and their ratio converges to in…nity.

It is interesting to note that, even though the di¤erent agents do not learn and continue to believe in a constant drift i = (

1 2
2 ) + i ; the group as a whole learns and the drift anticipated by the group converges to the objective drift as long as the objective probability lies in the initial set of plausible probabilities, i.e. as long as we have i = 0 for some i. This point can be easily checked in the log utility setting where the representative agent probability is

given by M = 1 jIj P M i and satis…es dM = M M dW with M = P i M i P M i : By Girsanov Theorem, the date-t drift anticipated by the representative agent is then given by i = ( 1 2

2 ) + M which converges almost surely to the objective drift ( 1 2

2 ).

Proof of Proposition 1

At the Pareto optimum, we have i M i u 0 (y i ) = q for some random variable q: It follows that

y i = q i M i hence e = X i2I q i M i = q X i2I 1 i M i and y i = e i M i P i2I [ i M i ] :
We have then

X i2I i E M i u(y i ) = X i2I i E 2 4 M i i M i 1 P i2I [ i M i ] 1 1 u(e ) 3 5 = E 2 4 P i2I i M i P i2I [ i M i ] 1 1 u(e ) 3 5 = E 2 4 " X i2I i M i # 1=
u(e )

Proof of Corollary 2

We have

E [M h (e )] = E 2 4 X i2I i M i ! 1= h (e ) 3 5 = E 2 4 X i2I i f i f (e ) ! 1=
h (e )

3 5 = E " P i2I i f i (e ) 1= f (e ) h (e ) # = Z P i2I i f i (x) 1= f (x) h (x) f (x) dx = Z X i2I i f i (x) ! 1= h (x) dx hence f M = P i2I i (f i ) 1= :
Proofs for Example 2

1. Proof that the distribution of log e is bimodal for j 1 2 j > 2 = p and unimodal for j 1 2 j 2 = p : We have

f log = 1 2 f log 1 + 1 2 f log 2 = 1 2 p 2 exp (x 1 ) 2 2 2 ! + 1 2 p 2 exp (x 2 ) 2 2 2 ! :
This function has either two maxima that are symmetric with respect to 1 + 2 2 or only one maximum at 1 + 2 2 : In the …rst case 1 + 2 2 would be a local minimum. It su¢ ces then to analyse the sign of the second derivative of f log at 1 + 2 2 : We obtain that the distribution is bimodal for j 1 2 j > 2 = p and unimodal for j 1 2 j 2 = p :

2. Proof that for = 1 + 2 2 the distribution of log e is portfolio dominated by the objective distribution. The ratio between the density of log e under Q and the density of log e under P is given by

f M log f log (x) = 1 2 exp 2 (x ) ( 1 ) + 2 2 1 2 2 + 1 2 exp 2x( 2 ) + 2 2 2
2 2 1 which is clearly symmetric with respect to , decreasing before and increasing after :

f M f = lim 1 f M f = 1 and f M f can not be monotone.
2. This is immediate according to lim 1

f M f = lim 1 f M f = 1:
f i f max opt 1=
is bounded away from 0 and 1 in the neighborhood of 1 and we have f M 1 f max opt : The result at the neighborhood of 1 is obtained similarly.

Proof of Proposition 4

1. Let g be given by

R 1 u f M (x)dx = g R 1 u f (x)dx : We have f M (x) = g 0 R 1 u f (x)dx f (u) and g 0 R 1 u f (x)dx = f M f (u) : We also have f (u)g 00 R 1 u f (x)dx = f M f 0 (u) which
gives that the concavity of g is governed by the sign of

f M f 0 : Remark that f M f 0 is
negative in a neighborhood of 1 and then that g 00 is positive and g is convex in a neighborhood of 1. Similarly, we have that f M f 0 is positive in a neighborhood of 1 and then that g 00 is negative and g is concave in a neighborhood of 1 : Finally, The function g is then inverse S-shaped: concave then convex.

Since g

0 R 1 u f (x)dx = f M f (u), we have g 0 (0) = f M f (1) = 1: If g 00 (0
) is well de…ned, we have g 00 (0) < 0 and hence g 00 (x) < 0 in a neighborhood of 0: The probability weighting function is then concave for small probabilities. The result in the neighborhood of 1 is obtained similarly.

Proof of Proposition 5

1. Let us consider a distribution of the means that is described by a density function h . The associated representative agent cumulative distribution function is given by

1 p 2 2 R dh ( ) R x 1 exp (s ) 2 2 2 ds: Since the function ! R x 1 exp (s ) 2 2 2 ds is decreas- ing a FSD shift of h decreases the value of R dh ( ) R x 1 exp (s ) 2
2 2 ds and leads then to a FSD dominating distribution function for the representative agent.

2. Let us consider a distribution ( 0 i ) and a FSD dominated shift ( i ). We want to prove that P 0 i F i P i F i : For a given x; letting x i denote the quantity F i (x), it su¢ ces to prove that P 0 i x i P i x i for a nondecreasing family (x i ) i2I which is true since ( 0 i ) dominates ( i ) in the sense of the FSD.

3. Let us consider a distribution ( 0 i ) and a MLR dominated shift ( i ). It su¢ ces to prove that (

P 0 i f i ) 1 ( P i f i )
1 is increasing or that

P 0 i G i P i G i is increasing with G i = f i . Without any loss
of generality, we may assume that all the considered functions are di¤erentiable and let us consider the derivative of

P 0 i G i P i G i P 0 i G i P i G i 0 = ( P 0 i G 0 i ) ( P i G i ) ( P 0 i G i ) ( P i G 0 i ) ( P i G i ) 2 = P f i f j i j 0 i i 0 j j G 0 i G j G i G 0 j ( P i G i ) 2 : Remark that for f i f j we have G i G j and then G 0 i G j G i G 0 j 0: Furthermore, for
f i f j we also have 0 i i 0 j j 0 which leads to the conclusion.

Proof of Proposition 6

It is immediate that 1 ; 2 ; and 3 can be written in the form 2 h; 2 ; 2 + h for some h > 0: For the distribution of individual characteristics ( i ) ; the representative agent distribution function is given by

1 p 2 2 P i i R x 1 exp (s i ) 2
2 2 ds: The symmetric mean preserving spread induces a modi…cation of this distribution that is positively proportional to

1 p 2 2 ( R x 1 exp (s 2 +h) 2 2 2 2 exp (s 2 ) 2 2 2 + exp (s 2 h) 2 2 2
)ds: Simple computations permit to show that this modi…cation is positive for x 2 and negative for x 2 : A symmetric mean preserving spread leads then to a distribution function that is above (resp. below) the original distribution function below a given threshold. We have then an increase of the level of discriminability.

Proof of Proposition 7

Replacing M i by M i D i in the proof of Proposition 1, we easily get that

U (e t ) = " X i2I i M i t D i t # 1= : Now, if the characteristics i ; M i t ; D i t are independent, then " X i2I i M i t D i t # 1= = " 1 jIj X i2I M i t !# 1= " 1 jIj X i2I D i t !# 1= and
X i2I i E M i t D i t u(y i t ) = E 2 4 1 jIj X i2I M i t ! 1= 1 jIj X i2I D i t ! 1=
u(e t ) 3 5 :

Proof of Proposition 8

We prove the proposition for D m since it is easy to check that all the derived properties are 1. Immediate.

The representative agent time preference rate

D m (t) = P N i=1 exp( i t) P N i=1 exp( i t)
i is an average of the i s with weights that decrease with i : Such an average is smaller than the equally weighted average.

Denote

i = exp ( i t). We have d D m (t) dt = P i2I i 2 i P i2I i P i2I i i P i2I i 2 which is nega- tive. We have D m (t) = inf + P N i6 =inf exp( ( i inf )t) i 1+ P N i6 =inf exp( ( i inf )t)
and exp ( ( i inf )t) i ! 1 0 we have then D m (t) ! inf :

Proof of Proposition 9 34

The proof of 1. is inspired from [START_REF] Jouini | Discounting and Divergence of Opinion[END_REF] and the proof of 2. is inspired from [START_REF] Nocetti | Properties of the social discount rate in a Benthamite framework with heterogeneous degrees of impatience[END_REF].

1. We have

D a (t) 1 t ln E [exp ( t)]
where E is the expectation operator associated with the distribution of ( i ). For a given

(t) = E [ exp ( t)] E [exp ( t)] :
where E is the expectation operator associated with the distribution of ( i ) :

Let us now consider P 1 and P 2 ; two distributions such that P 2 M LR P 1 : By de…nition, the density = dP 2 dP 1 is nondecreasing in (in other words i ! i and i ! i are comonotonic). We have then,

E P 2 [ exp( t)] E P 2 [exp( t)] = E P 1 [ exp( t)] E P 1 [ exp( t)] = E Q [ ] E Q [ ]
where Q is de…ned by a density with respect to P 1 equal (up to a constant) to exp ( t) : Since is nondecreasing in , we have

E Q [ ] E Q [ ] E Q [ ] ;
hence The function g is completely de…ned on [0; 1] and by a simple change of variables, we have

E P 2 [
g 1 p 2 R 1 t exp (x ) 2 2 2 ds = R 1 t h 1 n P N i=1 1 p 2 exp (x i ) 2 2 2 i 1=
ds for all t and we then have R 1 t f x; (s)ds = g R 1 t f x (s)ds :

2. The function g is the same as in Proposition 4.

3. Direct application of Proposition 5.

Proof of Proposition 11

Let v such that v = '(x) where x is normally distributed. By de…nition, we have Figure 2: In this …gure we represent the consensus belief for 3 di¤erent levels of risk aversion. We assume that a proportion of 47% of the agents believe that log e N (0; 1) and the remaining 53% believe that log e N (2:5; 1): The upper curve corresponds to = 2; the lower curve to = 0:8 and the middle curve to = 1: An increase of increases the distance between the peaks and their size.

U (v) = sup P v i E Q i [u(v i )].

  in order to analyse the relative weights of the di¤erent states of the world from the representative agent point of view, we introduce the probability measure Q de…ned by dQ dP M E[M ] .

  for x x where x and x are given real numbers. 3. If one of the agents denoted by f max opt is more optimistic than all the other agents and if one of the agents denoted by f max pess is more pessimistic than all the other agents, then the representative agent behaves like the most pessimistic individual for low values of e and behaves like the most optimistic individual for high values of e ; i.e. f M +1 f max opt and nondecreasing (resp. nonincreasing). In Proposition 3, we slightly reinforce these conditions by further assuming that the values of fopt f (resp. fpess f )range from zero to in…nity. Notice that these conditions are satis…ed in the case of lognormal distributions.

  FSD (resp. SSD) dominated shift in the distribution f of individual time preference rates decreases the representative agent average time preference rate D a : 2. A MLR (resp. PD) dominated shift in the distribution f of individual time preference rates decreases the representative agent marginal time preference rate D m : Second Stochastic Dominance as well as Portfolio Dominance are related to a notion of risk or of dispersion while First Stochastic Dominance and Monotone Likelihood Ratio Dominance are related to notions of shifts from low values to high values. Roughly speaking, Proposition 9 introduces the right concepts of dispersion and shifts and shows that more dispersion in agents' time preference rates as well as shifts to lower values of individual time preference rates decrease the representative agent's time preference rate.

u

  prefers lottery A to lottery B. Now, the same individual evaluates the lottery C that pays $100 if a red or a yellow ball is drawn at U (C) = (100) . The lottery D that pays $100 if a black or a yellow ball is drawn is evaluated at U (D) = 2 3 u (100).

  e ) = E [M u(e )] with M = P i2I i M i 1= .

:

  Moreover, since the distributions of log e under Q and under P are both symmetric with respect to , we haveE Q [log e ] = E P [log e ] = : These properties give V ar Q [log e ] > V ar P [log e ] (see[START_REF] Jouini | Discounting and Divergence of Opinion[END_REF].3. Proof that for general (i ) ; V ar Q [log e ] > V ar P [log e ] : For general ( i ), f M log is symmetric with respect to 1 + 2 2 which gives E Q [log e ] = 1 + 22 : Furthermore, we may apply the same reasoning as in 2. to compare the distribution of log e under Q with the distribution whose density is given by 1 We then haveV ar Q [log e ] > 2 = V ar P [log e ] :4. Proof that a higher level of risk tolerance induces a Portfolio Dominated shift in the representative agent distribution. For two di¤erent values and 0 of the risk tolerance parameter; it su¢ ces to consider f M log 0 f M log and to apply the same reasoning as in 2. function is such that lim 1

3.:

  It su¢ ces to remark that f M = f max opt max opt + P i=1;:::;N i6 =opt f i f max opt 1= If f i f max opt is nonincreasing for all i then max opt + P i=1;:::;N i6 =opt

  where the decreasing exponentials have a negative weight and the increasing exponentials have a positive weight. The function f M f 0 is then increasing.

Figure 1 :

 1 Figure1: In this …gure, we have represented in black the consensus belief in a log-utility agents setting. A proportion of 47% of the agents believe that log e N (0; 1) and the remaining 53% believe that log e N (2:5; 1): The beliefs of these two categories of agents are represented in grey.

Figure 3 :

 3 Figure 3: In this …gure we represent in black the representative agent probability weighting function in a model with two logarithmic utility agents. One of them overestimates the objective mean by one standard deviation and the other one underestimates it by one standard deviation. We also represent in grey the individual probability weighting functions (the concave one corresponds to the optimistic agent).

Figure 4 :Figure 5 :

 45 Figure 4: In this …gure we represent Prelec's function exp( ( ln p) ) with = 0:73 that corresponds to a standard speci…cation. We also represent the probability weighting function corresponding to a model with two log-utility agents. The …rst one underestimates the objective average by 120% of the standard deviation and has a weight of 30%. The second one overestimates the objective average by 60% of the standard deviation and has a weight of 70%.

Figure 6 :

 6 Figure6: The probability weighting function for di¤erent levels of divergence of belief. Both agents agree on a normal distribution but one of them overestimates the objective mean by times the standard deviation while the other one underestimates it by times the standard deviation. The value of ranges from 0 to 2. The discriminability decreases with (in other words the curvature increases with ).

  is described by a random variable e t : Agents have di¤erent time preference rates ( i ) and di¤erent subjective beliefs Q i : We let M i t denote the density at date t of Q i with respect to the objective probability P and D i

t exp ( i t) the discount factor of agent i between date 0 and date t. As previously, we consider the aggregate utility function U de…ned as the solution of the following maximization program U (e t ) = max

  Let us state the following properties of the average and marginal time preference rates.

	t	R t 0 m (s) ds:
	Proposition 8 Properties of the representative agent time preference rate
	1. The representative agent average and marginal time preference rates are given by
	D a
						1
					i2I	1 jIj D i t
	where D i
	D m (t)	D 0 t D t	and D a (t)	1 t	log D t :
	The average discount rate corresponds to the rate which, if applied constantly for all in-

t = P t exp ( i t) : We introduce the representative agent marginal time preference rate m as well as the representative agent average time preference rate a ; respectively de…ned by tervening years, would yield the discount factor D t; whereas the marginal discount rate is the rate of change of the discount factor. It is easy to recover the average discount rate from the marginal discount rate since a (t) = 1

:

  Let us assume now that P 2 P D P 1 and let us consider D;P 2 the associated representative agent time preference rates: We have D;P 2 Proof of Proposition 10 1. Let x 2 X with x N ; 2 : We havex Q i N + i ; 2 : From Proposition 1, there exists Q such that U (x) = E Q [x]and the density of x under Q is given by f x; (s) = 1= where f x; i is the density of x underQ i : We then have U (x) = R f x; (s)u (s) ds: It su¢ ces to prove that R 1 t f x; (s)ds is a function of R 1 t f x (s)dsthat does not depend on x; i.e. that does not depend upon and : Let g be the function

	and then E P 2 h	u 0 ( )(		D;P 2 m	i )	= 0 with u( ) = exp(	m t): By de…nition, this implies (t) = E P 2 [ exp( t)] E P 2 [exp( t)]
	E P 1 h	u 0 ( )(	D;P 2 m	) i	0 hence D;P 2 m		D;P 1 m
	h i de…ned by g 1 n P N i=1 (f x; i ) 1 p 2 R 1 t exp x 2 2 ds =	R 1 t	h	1 n	P N i=1	1 p 2 exp	(x i ) 2 2	i 1=	ds for all t:
							exp ( E P 2 [exp (	t)] t)]	E Q [ ] ;
							E P 1 [ exp ( E P 1 [exp (	t)] t)]	:

m

(t) and D;P 1 m (t)

  We denote by f i v the density of v with respect to Q We have then the result over the set of transformations of normal distributions.Let us consider now a random variable v and a normally distributed random variable x: We know that v has the same distribution asF 1 v [F x (x)] where F 1 v (p) is de…ned by F 1 v (p) = inf ft : F v (t) pg : If the individual has preferences that only depend on the distribution, it su¢ ces to set U (v) = U ('(x)) with ' = F 1

	i v = ' 0 f i x ' and since	R 1 t f x; (s)ds =	R 1 t	P	f i x	1	(s)ds = g	R 1 t f x (s)ds ;
	a simple change of variable leads to	R 1 t f v; (s)ds = g	R 1 t f

i : By Proposition 1 and Corollary 2, we have

U (v) = R f v; (s)u (s) ds with f v; = P f i v 1 : We clearly have f v; = ' 0 f x; ' and f v (s)ds : v F

x which is perfectly de…ned. We then have

U (x) = R 1 t f '(x); (s)u(s)ds with R 1 t f '(x); (s)ds = g R 1 t f '(x) (s)ds and since v is distributed like '(x); we have R 1 t f '(x); (s)ds = g R 1 t f v

(

s)ds ; and the result follows.

Proof of Proposition 12

In other words, the distribution of e under Qi is absolutely continuous with respect to the Lebesgue measure.

As mentioned in Section 2, sums should be replaced by integrals when dealing with continuous distributions. The density function of a gamma distribution ( ; ) is given by ( ) x 1 exp( x): Its mean m and its variance v 2 are respectively given by m = and v 2 = 2 :

See also[START_REF] Ghirardato | De…ning ambiguity and ambiguity attitude[END_REF].

Let us consider x; Q 1;x i i2I and x; Q 2;x i i2I two lotteries that our agent values equally.

For all i in I; let us denote by M 1 i and M 2 i the densities respectively associated to Q 1;x i and Q 2;x i :

We