1 † ‡§ † ‡ § Elyès Jouini and Clotilde Napp July 17, 2008 How to aggregate experts discount rates: an equilibrium approach The nancial support of the GIP-ANR ("Croyances" project) and of the Risk Foun- dation (Groupama chair) is gratefully acknowledged. Université Paris-Dauphine, Ceremade, F-75016 Paris, France, jouini@cere- made.dauphine.fr, phone: + 33 1 44 05 42 26 CNRS, UMR7088, F-75016 Paris, France Université Paris-Dauphine, DRM, F-75016 Paris, France, [email protected], phone: + 33 1 44 05 46 42
Transcript
1
�
† ‡§
�
†
‡§
Elyès Jouini and Clotilde Napp
July 17, 2008
How to aggregate experts discount rates:an equilibrium approach
The �nancial support of the GIP-ANR ("Croyances" project) and of the Risk Foun-dation (Groupama chair) is gratefully acknowledged.
How to aggregate experts discount rates:an equilibrium approach
We address the problem of a social planner who, as in Weitzman (2001),gathers data on experts� discount rates and wants to infer the socially effi-cient consumption discount rate. We propose an �equilibrium approach� andwe analyse the expression and the properties of the resulting �equilibriumdiscount rate�. We compare our expression for the discount rate with thedifferent expressions that have been previously proposed in the literature.We analyse the impact of shifts in the distributions of experts discountrates. Finally, we apply our approach to Weitzman (2001)�s data to pro-pose discount rates for public sector Cost-Bene�t Analysis, in particularfor the long term.Key-words: consumption discount rate; equilibrium discount rate; expertsdiscount rate; hyperbolic discounting; cost-bene�t analysis; gamma dis-counting; divergence of opinion;
2
1
�
1
= 0 = +� R � ��.
12
2
12
2
1 Introduction
= + (1 + )
= = 0 = 0= (1 + )
utilityconsumption
t,
R�
R � �� � � � ��
� � �� � �
The original Ramsey equation (Ramsey, 1928) was derived in a deterministic setting( ) and is given by The extended Ramsey equation corresponds to adirect generalization in a stochastic setting. For the sake of completeness, we rederive itin the Appendix.
The appropriate social discount rate to apply in public sector cost-bene�tanalysis is a contentious issue. This is especially true for long term projects,for which �nancial markets cannot provide any guideline. As Weitzman(2001, p.261) states �There does not now exist, nor has ever existed, any-thing remotely resembling a consensus, even -or, perhaps one should sayespecially- among the �experts� on this subject�.
In this paper we address the problem of a social planner, who gathersdata on experts� discount rates and wants to infer the socially efficientdiscount rate. More precisely, as in Weitzman (2001), the social plannerhas consulted a group of experts about the discount rate to apply for costsor bene�ts occurring at a given date each expert has proposed a discountrate and the problem of the social planner is to aggregate these proposeddiscount rates into a socially efficient discount rate.
As underlined by e.g. Nordhaus (2007) or Weitzman (2007), there isan important distinction between the social discount rate and the
social discount rate. The former refers to a pure time pref-erence rate that discounts utility. It re�ects the level of impatience or, forlong time horizon projects, the relative weights of different people or gen-erations. The latter is the rate used to discount future consumption; it isdetermined by the time preference rate, but also by the anticipations aboutthe future of the economy. The (extended) Ramsey equation illustratesthe distinction and the relation between the utility discount rate and theconsumption discount rate. Letting denote the consumption discountrate, and the utility discount rate, Ramsey formula gives the relation
, where is the growth rate of the economy andis the elasticity of marginal utility. Apart in the speci�c settings of a
stationary economy ( ) or a risk neutral investor ( ) or whenthe wealth and precautionary savings effect cancel out ( ),the two rates differ. In this paper, the rates proposed by the experts, aswell as the socially efficient discount rate to be inferred, are consumptiondiscount rates, since they are to be applied to cost-bene�t analysis. Thisis also the case in Weitzman (2001); indeed, as Weitzman makes it clear in
3
2
2
( )
i
i
i ii
i R�
� , �R
Moreover, the fact that the given rates are on average equal to 4% con�rms that theexperts actually gave their discount rate for consumption.
his questionnaire : �What I am here after is the relevant interest rate fordiscounting real-dollar changes in future goods and services �as opposed tothe rate of pure time preference on utility� .
Weitzman (1998, 2001) deal with this problem by adopting a certaintyequivalent approach. In this certainty equivalent approach, the social dis-count factor is taken to be the (arithmetic) average of the discount factorsproposed by the experts. Weitzman (1998) derives properties of the cer-tainty equivalent discount rate and in particular, its convergence towardsthe lowest individual discount rate, whatever the distribution of experts�discount rates. Weitzman (2001) applies this approach to a speci�c exam-ple. Starting from the results of a survey based on the opinions of 2,160economists about the consumption discount rate, he estimates the distrib-ution of experts� discount rates and derives an explicit expression for thecertainty equivalent discount rate in this case (Gamma discounting). Gol-lier (2004) underlines that the approach of Weitzman (1998, 2001) amountsto ranking the projects according to their expected net present value. Byadopting the criterion that projects should be ranked according to theirexpected future value, Gollier (2004) reaches opposite conclusions and con-cludes that �both criteria are arbitrary as they do not rely on realisticpreferences of human beings towards risk and time� suggesting that anequilibrium analysis is maybe the cost to be paid to make policy recom-mendations that have an economic sense.
We propose an approach to aggregate experts� discount rates into a con-sensus discount rate, which relies on an equilibrium analysis. Our approachis the following. We consider that each expert in the panel has consultedan equilibrium model, calibrating it with her own tastes and beliefs para-meters, in order to propose her own discount rate. For instance, expert
has applied the Ramsey formula and proposed a discount rate thatcorresponds to her own pure time preference rate (or as previously un-derlined, to her own conception of intergenerational equity) and her ownbeliefs about the future growth of the economy. The divergence inthe proposed individual discount rates stems then from divergence inindividual tastes and beliefs. We consider that experts� tastes and beliefsare representative in the sense that each of them represents the tastes andbeliefs of a portion of the population. It is then natural to adopt as the
4
�1
′
�
3
4
3
4
t
t
�
� �� �
6
t
t t t
tA
A
c�
( ) ( ) ( )
( )( )
( )
00 ln
= 1
1 2 3
1 1 1 2 2 2 3 3 3
1 1 1
2 2 2
3 3 3
1
1
R , R R� , � , � � , � , � � , � , � ,
� , � , �� , � , �
� , � , � .
At,
t R A
r
� ,�
Jouini et al. (2008) deals with the determination of the equilibrium discount rate inan economy in which agents have heterogeneous beliefs and heterogeneous time preferencerates.
The case with more general power utility functions would be much more difficult tohandle.
socially efficient discount rate the equilibrium discount rate in the economymade of agents with the heterogeneous beliefs and tastes of the experts. Forexample, if the panel is made of three experts, the �rst expert proposingthe discount rate the second and the third corresponding re-spectively to characteristics , and and ifwe assume that the three experts are equally representative, then we takeas the socially efficient discount rate the equilibrium discount rate in aneconomy made of one third of agents with characteristics , onethird of agents with characteristics and one third of agents withcharacteristics This means that we have transformed the prob-lem of aggregating data on heterogeneous discount rates into the problemof aggregating data on heterogeneous beliefs and tastes.
We can then apply the techniques of Jouini et al. (2008) in order toobtain the expression of the socially efficient discount rate. If we letdenote the price at date of a zero coupon bond maturing at date theaverage discount rate between date and date is given by
and the marginal discount rate is given by . We obtain thatboth rates are weighted averages of the individual discount rates. They areboth decreasing with time and converging to the lowest individual expertdiscount rate. These properties hold for both constant and decreasing puretime preference rates.
In the present paper we consider logarithmic utility functions. Thereare essentially two reasons for such a restriction. The �rst reason is ana-lytical tractability. Indeed, as underlined by Rubinstein (1975), �log utilityfunctions are singular in their capacity to cope with heterogeneous beliefswhile not imposing unreasonable restrictions on tastes�. This choice en-ables us to obtain simple formula, while considering reasonable levels ofrisk aversion . The second reason is the central role of logarithmic utilityfunctions. Jouini et al. (2008) shows that the log-utility setting is central inthe analysis of beliefs heterogeneity: some biases are induced when we dealwith power utility function with these biases being in oppositedirections depending on the position of with respect to 1. This is thenan additional argument in favor of the log-utility setting. Note �nally that
5
�
5
5
= 1
The problem of aggregating individual utility discount rates has been studied by,among others, Reinschmidt (2002) through a certainty equivalent approach, Nocetti etal. (2008) through a Benthamite approach, Gollier-Zeckhauser (2005) through a Paretooptimality approach and Lengwiler (2005) through an equilibrium approach.
the choice of is also made in the Stern Review.We compare our expression for the socially efficient discount rate with
other possible functions of the individual discount rates previously consid-ered in the literature. In particular, our formula are different from Weitz-man (1998). The equilibrium discount rate coincides with Weitzman (1998,2001) certainty equivalent discount rate when all experts have the same puretime preference rate. In a more general setting, the discount rates of themore impatient experts are granted a higher weight. A possible interpreta-tion is as follows. When considering its intertemporal rate of substitution,the group must weigh more the agents with a higher shadow price of theintertemporal budget constraint, i.e. the more impatient members of thegroup. The overweighting of impatient experts discount rates implies thatwhen tastes and beliefs are independent, the equilibrium discount rate ishigher than the certainty equivalent discount rate for all horizons.
We determine the explicit expression of the socially efficient discountrate for speci�c distributions of the experts� discount rates. We considerGaussian as well as Gamma distributions. We calibrate the model witha Gamma distribution on Weitzman (2001)�s data. Our results suggestusing the following approximation of within-period marginal discount ratesfor long term public projects: Immediate Future about 5 per cent; NearFuture about 4 percent; Medium Future about 3 percent; Distant Futureabout 1.5 per cent and Far-Distant Future about 0 per cent. Except forthe Far-Distant Future, these rates are slightly higher than those obtainedby Weitzman (2001) .
Finally, we determine which concepts of stochastic dominance on thedistributions of experts/individual discount rates lead to a clear impact onthe equilibrium discount rate. We analyze the impact of standard shifts, like�rst or second stochastic dominance shifts as well as monotone likelihoodratio dominance shifts. The impact of these shifts are different depending onwhich discount rate (average or marginal) we consider. Roughly speaking,more pessimism, more patience, more doubt as well as more heterogeneityin individual discount rates reduce the equilibrium discount rate.
Note that our approach also permits to aggregate utility discount rates(pure time discount rates). It suffices to consider the speci�c case where
6
� 2
2
( )
= +
0
( )
i
i
ii i i
i i i
i
n R.i
tt
Ri
R � � �
� > , � �
R
2 Equilibrium discount rate
utilityconsumption
equilibrium
there is no beliefs heterogeneity. We obtain that the equilibrium utilitydiscount rate is a weighted average of the individual ones. Our formulasare analogous to those of Lengwiler (2005). They coincide with those ofNocetti et al. (2008) and Gollier and Zeckhauser (2005) only for speci�cchoices of Pareto weights. They differ from those in Reinschmidt (2002), inthe same way as our formulas for the consumption discount rate differ fromthose in Weitzman (1998). We emphasize that, while these papers aim ataggregating individual discount rates, the aim of the present paperis to aggregate individual discount rates and to do it throughan approach.
All proofs are in the Appendix.
Let us consider experts, who propose different discount rates forcost-bene�t analysis of public projects as in Weitzman (2001)
We assume that the discount rate proposed by expert for costs orbene�ts occurring at date comes from a general equilibrium model withlog utility and lognormal aggregate consumption at date . According tothe extended Ramsey equation, the consumption discount rate proposedby expert is given by
where and are respectively the pure time preference rate, themean and the variance (by unit of time) of the distribution of the growthrate of aggregate consumption that the expert uses in order to calibratethe model. The divergence on the discount rates results then fromdivergence on these parameters.
Assuming that experts differ in their expectation about the growth rateis fairly natural. Indeed, the expected growth rate re�ects the opinion aboutthe future. It suffices to look at experts forecasts to realise that there isno consensus about the future of the economy. Indeed, forecasting for thecoming year is already a difficult task. It is natural that forecasts for thenext century/millennium are subject to potentially enormous divergence.It is doubtful that agents or economists currently have a complete under-standing of the determinants of long term economic evolutions. It is alsonatural to assume that experts differ in their pure time preference rate sinceit may re�ect their point of view about intergenerational equity as well as
7
6
6
i i� � .
12
2 2
2
ln (( ) )
+
i
i
i i i
ii i i
�� �� �� �
� N � �
�� � �
N
w i
� i
t
� � t, � t it
R � � � i
i
This is the case for instance if aggregate consumption is a geometric Brownian motionwith drift and volatility We will in Section 4 consider a more general setting withgeneral distributions for aggregate consumption. The formulas are then less easy to handle.
one�s level of impatience. The important debate among economists (andalso among philosophers) on the notion of intergenerational equity is anillustration of this possible divergence. Some will argue that intergenera-tional choices should be treated as intertemporal individual choices leadingto weigh more present welfare. Others will argue that fundamental ethicsrequire intergenerational neutrality and that the only ethical basis for plac-ing less value on the welfare of future generations is the uncertainty aboutwhether or not the world will exist and whether or not these generationswill be present.
The problem now is to determine how to aggregate these experts� rec-ommended discount rates into a consensus discount rate. We consider thatthe panel experts� tastes and beliefs re�ect those of the population. Weshall then consider a complete markets economy with heterogeneous agentsendowed with the beliefs and tastes chosen by the experts and we shalladopt the equilibrium discount rate in this economy as our consensus dis-count rate. There is no speci�c reason to discriminate between the experts,hence we suppose that each expert is representative of the same proportionof the agents in the economy and that all the agents have the same initialendowment.
To summarise, we have
groups of agents,
relative size of group ,
pure time preference rate of the agents in group ,
the time at which a cost or bene�t is incurred, relative to thepresent time,
group �s anticipated distribution of aggregateconsumption at date ,
log utility functions,
, the individual discount rate for agents in group ,i.e. the equilibrium discount rate that would prevail if the economywas made of group agents only.
8
∞
′
7
7
t
t
it
∫
∑∑
Rt.
� �
� �
� � �
1
10
=1 =1
0 ln
=
1ln exp
= inf
(1)(1)
i
t
t t t
t tA
A
t tts
i i
t
t
N
i
i iNj j j
i
ii
t
w
A tt. R A
t
A . rt
R r ds.i
r R .
R
Rt
w �
w �R t,
tR R .
R
2.1 Expression of the equilibrium discount rate andproperties
In a more general setting, the individual discount rates are not given by the Ramseyformula and may then be dependent upon
The weights model then the distribution of agents�c haracteristics inthe economy, which also correspond by construction to the distribution ofexperts characteristics.
We denote by the equilibrium discount factor for horizon , i.e. the priceat date of $1 at date We denote by the discount rate forhorizon , i.e. the rate which if applied constantly for all intervening yearswould yield the discount factor We denote by the marginaldiscount rate for horizon , i.e. the rate of change of the discount factor.We have Marginal and average rates of discount coincidewhen the discount rate is constant. In particular, for all , the individualmarginal discount rate coincides with the individual discount rateHowever, the distinction between the two notions of discount rates canbecome important when the discount rate is time dependent (Groom et al.,2005).
We easily get, as in Jouini et al. (2008, Proposition 5.1), that theaverage discount rate is given by
(1)
that it decreases with and that it converges to the lowest proposed rateNote that our setting is slightly different from the setting in
Jouini et al. (2008). Indeed, in Jouini et al. (2008), aggregate consumptionfollows a speci�c diffusion process. However, the proofs remain essentiallythe same. For the sake of completeness, we provide the proof of Equation
at the beginning of the Appendix. Moreover, we will show in Section4 that the formula for in Equation remains valid in a very generalArrow Debreu setting, with continuous or discrete time, a �nite number or acontinuum of agents, and general distributions for aggregate consumption .
Analogously, we easily obtain the following results on the marginal dis-count rate.
9
2
8
8 ��
i
i
∞∞
∑∑
∑∑ � �
� ��
� ��
�
=1 =1
=1 =1
2
2
Proposition 1
exp ( )
exp ( )
= =
exp ( )
exp+
=inf =
=
)
t
N
i
i ii
Nj j j
j
i
i i
t
N
i
i i i
Nj j j j
i
t
ii
i
i
i
i
rw � r t
w � r tr .
� �, � �
rw � � t
w � � t� � � .
r t,r
r R .
� �i
��
�
Note that optimism/pessimism in the form of a higher/lower or overcon�-dence/doubt in the form of a lower/higher have no impact on the relative weights,but this might be due to the speci�c logarithmic utility function.
1. The equilibrium marginal discount rate is given by
2. In the case of homogeneous beliefs ( ), the equilibriummarginal discount rate is given by
3. The discount rate decrease with and the asymptotic equilibriumdiscount rate is given by the lowest individual discount rate, i.e.
(2)
(3)
As in the certainty equivalent approach of Weitzman (1998), the con-sensus discount factors obtained through our equilibrium approach are av-erages of the individual discount factors proposed by the experts. However,except in the case of homogeneous pure time preference rates, i.e.for all , our expressions for the rates are different from those of Weitzman(1998). There is a bias towards the more impatient agents in the consensusequilibrium discount rates. A possible interpretation is as follows. Whenconsidering its intertemporal rate of substitution, the group must weighmore the agents with a higher shadow price of the intertemporal budgetconstraint, i.e. the more impatient members of the group .
As far as asymptotic properties are concerned, we obtain, as in Jouiniet al. (2008) that the relevant rate in the long run is given by the lowestindividual discount rate. This rate corresponds to the discount rate of themost patient agent (lowest when there is no beliefs heterogeneity, or tothe most pessimistic agent (lowest ) when there is no pure time preferencerate heterogeneity and all the agents have the same volatility parameter orto the least con�dent agent (highest ) when there is no pure time prefer-ence rate heterogeneity and all the agents have the same drift parameter.Moreover, the equilibrium approach leads to decreasing discount rates, notonly utility discount rates, but also consumption discount rates. As in the
10
=1
��
∑ ∑
∑∑
∑∑
∑
i i iN
j j jj
��
� �
� �
� �
� � �
Proposition 2
(3)exp (3)
exp exp( )
1ln exp
=1exp( )
exp( )
=1 =1 =1 =1
2
=1
� � t
� �
�w � � t � � t
�
w �
Rw �
w �R r
w �
w �r .
� b � �
� b
Rt
w R t.
i iNi
w � � t
w � � t i
i
i i i i i
i
i i
t
N
i
i iNj j j
it
N
i
i iNj j j
i
i i i i
i i
t
N
i
ii
2.2 Comparison with other formula for the sociallyefficient discount rate
1. The equilibrium discount rate is lower than the puretime preference weighted arithmetic average of the individual discountrates, i.e.
and
2. If the tastes and beliefs characteristics and are in-dependent or if they are comonotonic, i.e. individuals with highertastes characteristics have higher beliefs characteristics , then theequilibrium discount rate is higher than an average of the individualdiscount rates, i.e.
certainty approach of Weitzman (1998), this leads to use lower discountrates for long term projects in a cost-bene�t analysis.
In the case of homogeneous beliefs, Equation involves the covariancebetween and as in Lengwiler (2005). Equation also gives usthe expression for the consensus utility discount rate .
Although of the same nature, this expression is slightly different from theone obtained through the Pareto optimality of Gollier and Zeckhauser(2005) or the Benthamite approach of Nocetti et al. (2008). Indeed,our weights in the weighted averages of the are given by the quantities
whereas they are given by in Gollier and Zeck-hauser (2005) or Nocetti et al. (2008), where the are Pareto weightschosen by the social planner. Notice that this means that our equilibriumapproach and the Pareto/Benthamite approach would lead to the samesocial utility discount rate if the Pareto weights were proportional to .
The following proposition clari�es the relation between our socially efficientdiscount rate and the different expressions that have been provided in theliterature.
(4)
(5)
11
9
9
2
∑∑
2
=1 =1
2
2
22
� �
� ��
N�
N �
i i i i
t
N
i
ii
Nj j
j
i
i
i
i i
ii
i
i i ii
tv
3.1 Gaussian and Gamma distributions
See Section 4 for a proof in a more general setting.
exp
exp
(4)
(5) (6)
( ) (1)
( ) ( )
(1)(2) ( )
= ( )= ( )( + ) = +
3 Speci�c Distributions and Dominance Prop-erties
� b � �
rw r t
w r tr .
�
R .R r
R r .
� , RR r
� �, m, vb � � . R
� m, v R � m t.
3. If the tastes and beliefs characteristics and are inde-pendent, then the equilibrium marginal discount rate is higher than anaverage of the individual marginal discount rates, i.e.
(6)
Equation means that, as expected, the discount rate to use is lowerthan the simple arithmetic average (with the same weights) of the individualdiscount rates. Moreover, Equations and imply that, when tastesand beliefs characteristics are independent, our discount rates are higherthan those of Weitzman (2001). This is intuitive since our weights are givenby the pure time preference rates hence higher weights are granted tohigher individual discount rates.
Let us now determine the equilibrium discount rate for speci�c distributionsof the experts discount rates Equation as well as Proposition 1provide the expression of the discount rates and as a function of theexperts discount rates or
We whall consider continuous sets of experts. It is easy to show that theexpression of the discount rates remains the same in the setting with acontinuum of agents . The problem is that according to Equations and
, we need to make extra assumptions on the joint distribution ofin order to determine the discount rates and .
Consider �rst the case with homogeneous pure time preference ratesand with a normal distribution on the beliefs parameters
The discount rates then follow a normal distributionand we easily obtain that Reinschmidt
(2002) obtains a similar formula for the consensus utility discount ratewhen the individual utility discount rates follow a normal distribution.
12
��
2
�
� �
10
11
10
11
�( )1 2
�� �
� �
�
��
� ��
��
( )
exp( ) =
Proposition 3
�, �
x �x . m v m
.
1 1
1
2 2
2
1
1
2
2
21
21
121
22
2 2
2 2
2
2
1 1 2 2
+1+ +
+1+ +
+
+ +
121 2
22
1 2 1 21
++
1+
222
121
2
=( ) ( )
= ln ln = + = +
( ) ( )( ) ( )
= ( ) = + = + ln 1 +
= = +
( )( )
( )
( )( )
( )
( )
i i i i
i i
t�t
�� t
�t
�� t t
�� t
�� t
m v
m tv
m
m tv
i i
it
Wt t
t�
tm vm tv
Wt � t
Wt
Wt
i
i t
t
i
i
t
i i
i
it
� b � �� � , � b � , � ,
R r
m , v m , v� b .
� � R �, � � � � , R R
r r r R
m, vR .
m vb r
Rm �
v �r t
�,R r
m Rv
R rt
We focus on the case of independent distributions as a central case; explicit formulasmay also be derived in the case of a given correlation between pure time preference ratesand beliefs.
Recall that the density function of a gamma distribution is given byIts mean and its variance are respectively given by
and
If pure time preference rates and beliefs areindependently distributed with and then
1. and
where and respectively denote the mean and varianceof and
2. If then with
and where and respectively denotethe marginal discount rate and the discount rate obtained through thecertainty equivalent approach of Weitzman and where denotethe mean and variance of
Suppose now that utility discount rates and beliefs are independentlyand gamma distributed. We obtain the following result.
A decrease in the mean or an increase in the variance of theindividual beliefs decreases the marginal discount rate (hence thediscount rate ). The same result occurs with a decrease in the mean
of the individual pure time preference rates . An increase in thevariance of the individual pure time preference rates decreases themarginal discount rate for large enough.
When beliefs and tastes are independent and follow gamma distribu-tions with the same parameter the distribution of the individual dis-count rates or is a sufficient statistics for the equilibrium discountrate. As in Weitzman (2001), experts discount rates then follow a Gammadistribution. As shown in the previous section, our equilibrium discountrates are higher than Weitzman (2001)�s discount rates but converge to thesame value. A decrease in the mean of the individual discount ratesdecreases the marginal discount rate and an increase in the variance ofthe individual discount rates decreases the marginal discount ratefor large enough.
We now calibrate this model with two independent gamma distributionsand on Weitzman (2001)�s data. We impose
that and where and respectively denotethe mean and the variance of the individual discount rates computed onWeitzman(2001)�s sample. We further impose that (same ratiobetween mean and standard deviation for both distributions), which leadsto and for some positive Note that for , we
get which corresponds to the calibrationin Weitzman (2001). We have then a family of stastistical models thatcontains Weitzman (2001)�s statistical model and we maximize the log-likelihood with respect to the parameter to choose the best calibration.We obtain henceand . Tosummarise, the best calibration corresponds to a gamma distribution onthe individual pure time preference rates with an average time preferencerate among experts equal to and a median equal to and agamma distribution on the individual beliefs with an average belief para-meter (about the growth of the economy) equal to and a medianequal to More precisely, the belief parameter can be in-terpreted as a risk adjusted growth rate. The values we obtain are thenreasonable values for both an average pure time preference rate and an av-erage risk-adjusted growth rate. Stern report considers values for the puretime preference rate (utility discount rate) between 0.1 and 1.5 and valuesfor the growth rate ranging from 0 per cent to 6 per cent. Arrow (1995)states that the pure time preference rate should be about 1% and surveyingthe evidence, the HM Treasury�s Green Book (2003) suggests a long rungrowth rate of 2.1 per cent.
Figure 1 represents the log-likelihood as a function of Figure 2 rep-resents the distribution of the individual discount rates for the parameter
that maximizes the log-likelihood ( ) as well as the empiricaldistribution and Weitzman (2001)�s distribution. Figure 3 represents thecorresponding marginal discount rate curve and compares it to the discountrate curve of Weitzman (2001). Table 1 presents the corresponding recom-mended sliding-scale discount rates: Immediate Future about 5 per cent;Near Future about 4 percent; Medium Future about 3 percent; DistantFuture about 1.5 per cent and Far-Distant Future about 0 per cent.
14
Y
X
� �
Proposition 4
X Y X Yff
ii
t
ii
t
it
( ) ( ) ( ) ( )
( )
( )
( )
4
3.3 Dominance properties
r
YX, X Y �
f x f y f y f x y x
� , RR
� ,r r
R R
1. If all the agents have the same time preference ratethen a FSD (resp. SSD) shift in the distribution of increases
the discount rate for all horizons.
2. If all the agents have the same time preference then a MLR shiftin the distribution of the increases the marginal discount ratefor all horizons.
3. If all the agents have the same beliefs, then a MLR shift in the dis-tribution of the increases the discount rate for all horizons.
In the speci�c setting of Gamma distributions considered in Proposition 3,we have seen the impact of an increase in the mean and in the variance ofthe distribution of individual discount rates on the socially efficient discountrates. The impact was the same for the average and for the marginaldiscount rates. We now analyze in a more general setting which shifts onthe distribution of individual discount rates have a clear impact on thesocially efficient discount rates. We consider First and Second StochasticDominance shifts, as de�ned in e.g. Rothschild and Stiglitz (1970). In orderto obtain a clear impact on the socially efficient marginal discount rate ,we also consider Monotone Likelihood Ratio dominance (MLR) shifts. Thisconcept of dominance has been studied by Landsberger and Meilijson (1990)and is de�ned as follows: a random variable dominates a random variable
if and have densities with respect to some dominating measuresuch that for all (roughly speaking, theratio is nondecreasing).
Proposition makes it clear which concepts of dominance (correspond-ing to the notions of pessimism, doubt, patience, or heterogeneity) oneshould consider in order to have a clear impact on the discount rates.Roughly speaking, it means that a country where experts are more pes-simistic and/or exhibit more doubt about future growth and/or have lowerpure time preference rates (more patient or more altruistic with respectto future generations) should apply a lower discount rate for cost-bene�tanalysis. More heterogeneity in experts beliefs about future growth ratesalso leads to lower discount rates.
15
12
12
4 Extensions
12
16
13
110
810
110
1 2 3
3 2 1
.
.
. , . , . , . . , .
, .
.
w w w .w < w < w .
[1 5 2 5] [2 5 3 5] [3 5 4 5]
[1; 3] [2; 4] [3; 4]
4.1 General distributions for aggregate consumption
In this paragraph, for simplicity, we refer to groups and their size instead of expertsand the size of the population that they represent.
More precisely, suppose that in one population, say (A), we have threeequally large groups with discount rates of 2%, 3% and 4%. In a sec-ond population (B), there are also three groups with the same anticipatedgrowth rates but their proportion in the population is , and Popula-tion (B) is more pessimistic than population (A) (in the sense of the FSD)and the discount rate to apply is lower for (B). In a third population (C),there are three groups with anticipated growth rates 1%, 3% and 5% andtheir proportion in the population is , and Populations (A) and(C) have the same average level of pessimism but population (C) is moreheterogeneous (in the sense of the SSD) than population (A) and the dis-count rate to apply is lower for (C). Let us assume now that experts provideforecasts with a 95% con�dence interval. Let us assume that these intervalsin population (A) are given by and while in afourth population (D) also with three equally large groups, these intervalsare given by and There is more doubt in population (D)and the discount rate to apply is then lower for (D). The MLR (monotonelikelihood ratio) dominance is stronger than the FSD dominance. Let usconsider two populations (E) and (F) In population (E), there are threeequally large groups of experts with pure time preference rates respectivelyequal to 0.5%, 1% and 1.5%. In population (F) there are also three groupswith the same pure time preference rates but with proportions in the pop-ulation respectively equal to , and The population (E) is morepatient (in the sense of the MLR) if In this case, the discountrate to apply for cost-bene�t analysis is lower for population (E).
In this section, we examine two possible extensions: more general subjectiveand objective distributions for aggregate consumption and time dependentpure time preference rates.
We �rst show that Proposition 1 remains valid in a very general completemarkets Arrow-Debreu setting. Time can be continuous or discrete. We
16
,
13
13
(1)
∫∫
� �
∫
�
�
� � �
��
�
1
010
10
2
2
Proposition 5
Proposition 6
it
i i
i
iit
ti i
j j
it
it
i iit
iit t
ti i i
it i i i
i i
4.2 Time dependent pure time preference rates
[0 1]
[0 1]( )
( )
1ln
( )exp ( )
(1)
( ( )) = ( ) += ( ) +
( ( )) ( )
, , �
iQ t
i i� , w .
, , �� ,
w t Q
Rt
w �
w � d� jR t d� i
Ri
w � .R t.
� t R � s ds � �r � t � �
� t , w t
In particular, if aggregate consumption is lognormal, then the individual discountrates are time independent, and the expression for the socially efficient discount rate isthe same as in Equation replacing sums by integrals.
Let us consider a model with a measured space ( )of log-utility agents that have pure time preference rates wealth shares
and date probability measures . We assume that all these proba-bilities are equivalent, i.e. the agents agree on the events of zero probability.The equilibrium discount rate is then given by
where is the equilibrium discount rate that would prevail if the economywas made of agent only.
If agents have time-dependent positive pure time prefer-ence rates wealth shares and date distributions for aggregate
allow for a �nite number or a continuum of agents. For this purpose, theset of agents is represented by a measured space ( ). Furthermorewe do not need to assume speci�c individual distributions for aggregateconsumption. It suffices to assume that agent has a probability measure
that represents the distribution of date aggregate consumption fromagent point of view. As in previous sections, agent has a pure timepreference rate a share of total wealth and a log-utility
(7)
In such a general setting, the equilibrium discount rate is still a weightedaverage of the individual discount rates, and as previously, the weights areproportional to The only difference with the setting of Equationis the fact that the individual discount rates may depend upon
It is also easy to adapt our approach to the case with time-dependent puretime preference rates . We then have and
. We obtain the following expression for the sociallyefficient discount rate.
17
∞ �
→∞ →∞ →∞
5 Conclusion
N �
� � �
���
� � �
�
∑∑
� �
∑� ∫ �
∑ � ∫ �
∫ �∫ � ∫ � �
� �
12
2 2
=1 =1
=1
0
=1 0
10
2 20 0
1
2
ln (( ) )
1ln exp
exp
exp
= ( ) + = ( )+ = exp ( )
( ( ))
lim = lim = inf inf = inf + lim ( )
� � t, � t
Rt
w �
w �R t
rw � r ds
w � r dsr
R � s ds � � , r � t � � � � s ds dt .
� t tR r t
R r r � � � t .
i i i
t
N
i
i iNi j j
it
t
N
i
i it is
Ni j j
t is
it
it t
ti i i
it i i i i
ti
i
t t
tt
tt
i t
it
ii i t i
consumption , the equilibrium discount rates are givenby
for andIf the pure time preference rates are decreasing with , then the
discount rates and are also decreasing with and we have
(8)
(9)
In this paper, we propose an equilibrium approach to aggregate the discountrates proposed by experts into a consensus discount rate. We emphasizethat our approach enables to deal with consumption discount rates and notonly with utility discount rates (pure time preference rates).
We start with the recognition that divergence among experts on whatthe discount rate should be is rooted in fundamental differences of opinionabout inter-generational equity as well as about future growth of the quan-tity of available consumption. This enables us to translate the problem ofaggregating experts discount rates into a problem of aggregating heteroge-neous beliefs and time preference rates. We can then use the techniques ofJouini et al. (2008) in order to obtain the expression of the socially efficientdiscount rate. The equilibrium discount rate is a weighted average of theexperts proposed discount rates, in which more impatient experts are moreheavily weighted; the equilibrium discount rate is decreasing and convergesto the lowest expert discount rate, which does not necessarily correspondto the discount rate of the most �patient� expert.
We show that the equilibrium discount rate is higher than Weitzman(1998)�s certainty equivalent discount rate for all horizon. More divergenceof opinion about future growth rates among experts (in the form of secondstochastic dominated shifts) leads to lower discount rates for all horizons.
18
References
More doubt (larger con�dence intervals) as well as more pessimism (in theform of �rst stochastic or monotone likelihood ratio dominated shifts) inexperts� forecasts also leads to lower discount rates.
We calibrate the model on Weitzman (2001)�s data. We show that thevery wide spread of opinion on discount rates makes the effective equilib-rium discount rate decline signi�cantly over time from 5 per cent per annumfor Immediate Future to 0 per cent per annum for Far-Distant Future.
[1] Arrow, K.J., 1995. Inter-generational equity and the rateof discount in long-term social investment. Paper at IEAWorld Congress (December), available at http://www-econ.stanford.edu/faculty/workp/swp97005.pdf.
[2] Groom, B., Hepburn, C., Koundouri, P., and D. Pearce, 2005. Declin-ing Discount Rates : the Long and the Short of it. Environmental andResource Economics, 445-493.
[3] Gollier, C., 2004. Maximizing the expected net future value as an al-ternative strategy to gamma discounting. Finance Research Letters, 1,85-89.
[4] Gollier, C., and R. Zeckhauser, 2005. Aggregation of heterogeneoustime preferences. Journal of Political Economy, 113, 4, 878-898.
[5] HM Treasury, 2003. The Green Book � Appraisal and Evaluation inCentral Government. HM Treasury, London.
[6] Jouini, E., Marin, J.-M., and C. Napp, 2008. Discount-ing and divergence of opinion. Working Paper, available athttp://papers.ssrn.com/sol3/papers.cfm?abstract_id=915380#PaperDownload
[7] Landsberger, M., and I. Meilijson, 1990. Demand for risky �nancialassets: A portfolio analysis. Journal of Economic Theory, 50, 204-213.
[8] Lengwiler, Y., 2005. Heterogeneous patience and the term structure ofreal interest rates. American Economic Review, 95, 890-896.
19
′
�[ ]
�
� �
t
t t
t�
t
Appendix
t q
q �t u e
Rt
E q �t
E e .
= exp( ) ( )
=1
ln [ ] =1
ln
Derivation of the extended Ramsey equation
[9] Nocetti, D., Jouini, E., and C. Napp, 2008. Properties of the SocialDiscount Rate in a Benthamite Framework with Heterogeneous De-grees of Impatience. Management Science, To appear.
[10] Nordhaus, W., 2007. A Review of the Stern Review on the Economicsof Climate Change. Journal of Economic Literature, 45(3), 686-702.
[11] Ramsey, F., 1928. A mathematical theory of savings. Economic Jour-nal, 38, 543-559.
[13] Rothschild, M. and J.E. Stiglitz, 1970. Increasing risk: I. A de�nition.Journal of Economic Theory, 2, 225-243
[14] Rubinstein, M., 1975. The strong case for the generalized LogarithmicUtility Model as the Premier Model of Financial Markets. The Journalof Finance, 31, 2, 551-571.
[15] Weitzman, M., 1998. Why the Far-Distant Future Should Be Dis-counted at its Lowest Possible Rate. Journal of Environmental Eco-nomics and Management, 36, 201-208.
[16] Weitzman, M., 2001. Gamma Discounting. The American EconomicReview, 91, 1, 260-271.
[17] Weitzman, M., 2007. A Review of the Stern Review on the Economicsof Climate Change. Journal of Economic Literature, 45(3), 703-724.
At the equilibrium, the date state price density is given by
and the discount rate is given by
20
1
=1
� � � �
�
��
��
�
�
�
�
�
�
� �
∞∞ ∞
� �
� �
∞ ∞
�
i i
i iN
j j j
i
i i
t
i i
i
� � [ ]
∑ ∑
∑
∫∫ ∫
∑
[ ]
∑ ∑
∫ [ ] ∫
∑
12
2 2 2 12
2
2
=1 =1
0
0 0
=1
1
=1 =1
1
0 0
=1
Proof of Equation 1 and of Proposition 1
�t
tNi i
R t Ni i
r ti
w �
w �t
it
i ff t
iit
it
Ptit i
Pt t
ii
it i
tt
i
ii
itt
it
t
N
i ii
it
t
r t R ti
it e
t t
N
i i
r tN
i i
R t
�
Ptit
i ii
Pt t
i
i i
Nj j j
ln ( (1 + ) )
= +1
2(1 + )
= exp = exp =
( )
exp( ) [ log ][ ] [ ]
1exp( )
1=
( )
1exp( )
1=
=1
exp( )1
exp = exp = exp( )
= [ ] =1
exp =1
exp
=1
= [ ]
1=
e
� � � t � � t. E c t �� � � �
R � �� � � � .
A , . q t
P y i. M ei
� t E M y dtE q y dt w E q e dt.
�� t M
yq i
�
�� t M
qy
q�
� t Me.
E � t M ,
A E q� �
.
.
E q y dt� �
w E q e dt
�
� w
� w.
The random variable follows a log normal distribution with parametersand We then have =
and
We �rst prove that withLet us denote by the date state-price density (with respect
to ) and by the consumption of group We let denote thedensity of agent �s belief with respect to the true probability. Each groupmaximizes its aggregate utility under its budgetconstraint This leads to the followingEuler conditions
for all
for some positive Lagrange multipliersWe have then
and summing all these equations leads to
Now, in our setting, hence
It remains to determine the equilibrium weights From the Euler andbudget conditions we have
which leads to
21
� �
�
��
��
��
→∞
=1
=1
2
=1
2
=1
exp exp
=1
Proof of Proposition 2
∑ ∑∑∑
∑ ∑
∑∑
� � ∑∑
∑∑
∑∑
∑
∑∑
�
�
� � �
� 6→
� �
�
� � �
�
� ��
��
�� � �
�
i ii
N
j j jj
t
N
i i ii i
N
i i ii
i ii
N
j j jj
t � �
�
�
i iN
j j j
� � �
�
=1
exp
exp( )
exp
exp( )=1
exp
exp( )
2
exp2 2
=1 =1 =1 =1
=1 =1
=1 =1
=1 =1
, .
r r r ,
r .
Pw � r t . E r E r .
r t.r
r r j r rr r r .
Rt
A
A R t.
w �
w �R t
w �
w �R t.
Rw �
w �R .
rw � r t
w � r tr
E rt r
E rt
P . r t r ,
E rt r E r E rt .
r E rw �
w �r .
(1) (2) (3)( )
=( ( ) )
( )
( )( )
exp ( ) = [ ] [ ] 0
inf =(2)
1ln
exp
exp exp
exp ( )
exp ( )
=[exp ( ) ]
[exp ( )]
exp ( )
[exp ( ) ] [ ] [exp ( )]
[ ] =
tNi
w � r t
w � r t
i
drdt
w � r t r
w � r t
Ni
w � r t
w � r t
i
�
i ii dr
dtP P
t
iii j i
jt t
i
t t
ti
N
i
i iNj j j
iN
i
i iNj j j
i
t
N
i
i iNj j j
i
t
N
i
i ii
Nj j j
j
i
P
P
�w �
w �
i i
P i P P
tP
N
i
i iNj j j
i
We easily deduce Equations as well as Equation
As far as the monotony of is concerned, we have
then Let us con-
sider the probability measure whose weights are proportional to byWe have The marginal
discount rate decreases then withAs far as the asymptotic behavior of the marginal discount rate is
concerned, let and let be such that , then the relativeweight of in Equation converges to zero and
1. According to Proposition 1, we have
where is an arithmetic average of the Since the arithmeticaverage is larger than the geometric average, we have
2. Let us denote by the probability measure with weights . Sinceand are independent, we have
Now, since and are anticomonotonic, we have
which gives
3. We have
where denotes the probability measure whose weights are proportionalto We have then
If pure time preference rates and beliefs are independentand are distributed as follows and then
23
∫ ∫∫
∫ ∫∫
) )
∞ ∞ � �
∞ �
∞ ∞ �∞
��
��
Proof of Proposition 4
�
w
w
w
w
w
wPw
Pw
Pw
Pw
Q
Q
w
w
w
w
� � ��
� ��
� � �
→ �
�
��
�→
→
�
�
�� �
� ��
2 1 1 2
1
1
1
2 1 2
1
1 2
2
1
2
2
1
1
exp
exp
exp exp exp
2
2exp
1
1
2
2
1
1
0 01
11
2
�( ) 01
1
2
2
0 1 0 21
0 1
1
1
1+2
2
1 2 2 1
[ exp ]
[exp( )]
[ exp ]
[ exp( )]
[ ]
[ ] exp1
t
� � � �
��
�
� � �
�
� �
i
tP
w
i
Pt
i
t
P
P
w w w MLR wdPdP
i
i E r rt
E rt
E �r rt
E � rt
E �r
E � w
Q Q Q
P
PQ
P
P
A�
�
�
�
� � b t � � � b � b d�db
�� � � d�,
�
�
� � t � d� b � t b db
� � � d�,
�
� t
�
� t.
R r
�
Rt
E Rt
P t,R Rt
RE Rt R .
�
rE r rt
E rt.
P P , P P .
� r i �
i r
Q P
rt . � r
E �r E � E r ,
E r rt
E rtE r ,
E r rt
E rt.
=� ( ) � ( )
exp ( ( + ) ) exp ( ) exp ( )
exp ( )
=� ( )
( exp ( ( + )) ) ( exp ( ( + )) )
exp ( )
=+ +
1ln [exp ]
exp( )
[exp ]
=[ exp ( )]
[exp ( )]
=
= =
exp( )
[ ] [ ] [ ]
[ exp ]
[exp ( )][ ]
[ exp ]
[exp ]
The results on the expression of and are then easily derived.
1. Let us assume that all the agents have the same , we have then
where is de�ned as in the proof of Proposition . For a given thefunction is decreasing (and convex) and, by de�nition, aFSD (resp. SSD) shift in the distribution of decreases the value of
and increases2. We still assume that all the agents have the same , we have then
Let us consider and two distributions such that Byde�nition, the density is nondecreasing in (in other words
and are comonotonic). We have then,
where is de�ned by a density with respect to equal (up toa constant) to Since is nondecreasing in , we have
hence
24
� � �
∞
∞ ∞
∫ [ ]
∫ ∫
∑
2
2
1
1
2
1
2
2
1
1
� � �
�
�
� � �
� � �
� �
�
��
�
�
�
Proof of Proposition 5
w
w
Pw
Pw
Pw
Pw
Q�
Q�w
w
� � �
w
w
�
w
w
it
it
1 2 2 1
[ exp ]
[ ]
[ exp ]
[ ]
[ exp ]
[ ]
1
01
0 0
=1
1ln
[ exp ]
[ ]
= = =
exp
[ exp ] [ ] [exp ]
[ exp ]
[ ][exp ]
[ exp ]
[ ]
exp( )
[ ] [ ]
1exp( )
1=
1exp( )
1=
=1
exp( )1
t
P
P
w w w MLR w
E � �t
E �
E �� �t
E ��
E � �t
E �
dPdP
� w
Q Q Q
P
PQ
P
P
it
it
it t
it
iQ
yP
tit i
Pt t
ii
it i
tt
ii
itt
it
t
N
i ii
it
t
Rt
E � �t
E �.
P P , P P .
�
Q P�. � r �t
E � �t E � E �t ,
E � �t
E �E �t ,
E � �t
E �.
M Q PQ . q t
P y
i. � t E dt
E q y dt w E q e dt.
�� t M
yq .
�� t M
qy
q�
� t Me.
i
3. If we now assume that all the agents have the same belief, we have
Let us consider and two distributions such that We
have then, where and
where is de�ned by a density with respect to equal (up to a constant)to Since is nondecreasing in and then nonincreasing in , wehave
hence
Let denote the density of with respect to a given probabilityequivalent to all the probability measures Let us denote by the datestate-price density (with respect to ) and by the consumption of group
Each group maximizes its aggregate utility under
its budget constraint This leads to thefollowing Euler condition
We have then
and summing all these equations leads to
Now in an economy made of group only, we would have
25
=1
�
�
�
�
∞ ∞
∞ ∞
i
i
i iN
j j j
Proof of Proposition 6
[ ]
∑
∑
∑
∫ [ ] ∫
∑
∑ ∑
∑∑
∫
∫ [ ] ∫ ∫
0 =11
=1
=1
1
0 0
=1
=1
=1 =1
0
0 0 0
exp( )1
=
=1
ln
= 1 = 1
=1
=1
=1
= [ ]
1=
= = [ ]
=
(7)
=1
exp( ( ) )1
=1
exp( ( ) )
� t Me
q
rt
E q .
q
q�q
A�A .
.
E q y dt� �
w E q e dt
�
� w
� w
q q A E q ,
A� w
� wA ,
q�
� s ds My
E q y dt�
� s ds dt
iit
t
it
it
it
Ni �
t
N
i i
it
t
N
i i
it
�
Ptit
i ii
Pt t
i
i i
Nj j j
tNi
� w
� w
it t t
t
N
i
i i
Nj j j
it
ti
t
iit i
t
Ptit
i
t
i
and
If all prices are expressed in terms of today�s consumption units, we haveand which leads to
hence
It remains to determine the equilibrium weights From the Euler condi-tions and budget conditions we have
which leads to
and . Since we have
hence Equation .It is easy to see that the formulas in the proof
of Proposition 5 above have to be adapted as follows
26
�∞ �
∑∑
�∫ ∫ �=1 =1
0 0
1
=
= exp( ( ) )
t
N
i
i i
Nj j j
it
it
i
A� w
� wA
� � s ds dt .
The same steps as above lead to
with
27
Figure 1: We calibrate a model with two independent gamma distributions (tastes andbeliefs) on Weitzman (2001)�s data. We assume that the two distributions are homothetic(the �rst one is obtained from the second one through a change of variable x! �x where �is a given parameter) and we calibrate the model in order to �t the mean and the variance ofthe empirical distribution. We have then a family of stastical models that contains Weitzman(2001)�s statistical model (it corresponds to � = 1) and we maximize the log-likelihood withrespect to the parameter � to choose the best calibration. We obtain � = 0:4116.
28
Figure 2: This �gure represents the distribution of the individual discount rates for the value� = 0:4116 that maximizes the log-likelihood (upper curve) as well as the empirical distribu-tion and Weitzman (2001)�s distribution (lower curve). Our distribution corresponds to thesum of two independent gamma distributions with parameters (�1; �1) and (�2; �2) given by(�1; �1; �2; �2) = (1:04; 89:45; 1:04; 36:82) : These parameters correspond to mean and vari-ance levels given by (m1; v
Weitzman�s distribution corresponds to a gamma distribution with parameters (1:78; 44:44):All represented distributions have the same mean and variance levels (m; v2) = (4�10�2; 9�10�4):
29
5003752501250
0.05
0.04
0.03
0.02
0.01
t
r
t
r
Figure 3: This �gure represents the marginal discount rate curve rt =PNi=1
wi�i exp(�rit)PNj=1 wj�j exp(�rjt)
ri = �1+1�1+t
+ �2�2+t
obtained through our calibration (upper curve)
and compares it to the discount rate curve rt = ��+t
of Weitzman (2001) (lowercurve). The intermediate curve represents, with our calibration, the unweighted averagePN
i=1
wi exp(�rit)PNj=1 wj exp(�rjt)
ri = �1�1+t
+ �2�2+t
: It is clear that the di¤erence between our discount
rate curve and Weitzman (2001)�s curve mainly results from the fact that, contrarily to thecertainty equivalent approach, more impatient experts are more heavily weighted in theequilibrium approach.
