Can Preferences for Catastrophe Avoidance Reconcile ... - CiteSeerX
Can Preferences for Catastrophe Avoidance Reconcile Social Discounting with Intergenerational Equity?∗ Antoine Bommier† CNRS and University of Toulouse (GREMAQ)
St´ephane Zuber‡ University of Toulouse (GREMAQ)
Abstract A social welfare function treating all generations equally is derived from a set of axioms that, contrary to the Utilitarian approach, allows for preferences for Catastrophe Avoidance. Implications for the case where there is a risk of world extinction are studied. We show that substantial time discounting can arise from the planner’s taste for Catastrophe Avoidance, even if the probability of the world ending is infinitesimally small. Keywords: Time Discounting, Intergenerational Equity, Social Welfare Function, Catastrophe Avoidance. JEL Classification numbers: D63, D69, D99.
∗
We wish to thank Marc Fleurbaey and Thibault Gajdos for many valuable comments. Universit´e des Sciences Sociales - Manufacture des Tabacs, 21 all´ees de Brienne, 31000 Toulouse, France. E-mail: [email protected]. ‡ Universit´e des Sciences Sociales - Manufacture des Tabacs, 21 all´ees de Brienne, 31000 Toulouse, France. E-mail: [email protected]. †
1
Introduction
Many major policy decisions involve making trade-offs between the welfare of current and future generations. Policy guidance has then to rely on a given social objective. The most common approach involves assuming that the social planner aims at maximizing a social welfare function:
SW =
∞ X
γ t Ut
t=0
where t indicates time, Ut is the aggregate utility of cohort t and γ t is the social discount factor (the discount rate being 1 − γ). Such an approach has been extensively criticized by economists and philosophers. The main point is that generations are not treated equally, since they are assigned a weight that depends on when they were born. This is generally considered to be unfair, a point of view most clearly expressed by philosopher Henry Sidgwick who argued that “[...] the time at which a man exists cannot affect the value of his happiness from a universal point of view” (Sidgwick, 1907, p. 414). To our knowledge, there is only one argument that has been suggested to provide an ethical ground for using such a welfare function. Initially developed by Dasgupta and Heal (1974 & 1979), it involves assuming that between any two dates there exists a positive probability 1 − γ that the world ends. An equitable Utilitarian objective leads then to the above social welfare function. Still, this argument is problematic when we turn to quantitative aspects. Social discount rates are usually taken between 1% and 5 % per year. But, for most people it would seem excessively pessimistic to assume a yearly probability of the word ending of 1% or 5%. With a 5% probability we would have more than a 50% chance of disappearing in the next 14 years. Even with a probability of the world ending of 1% per year, there would be the less than a 50% chance of seeing the world lasting more than 69 years. It is hard to believe that this should reflect 1
the beliefs of a reasonable social planner. The aim of this paper is to provide a theoretical foundation for a social welfare function displaying time discounting, but which is ethically acceptable, in the sense that it treats all generations equally. Our argument, just as that of Dasgupta and Heal, relies on the idea that at any time there is a positive probability that the world ends. However, the novelty is that we suggest different foundations that make it possible to distinguish the social discount rate from the probability of the world ending. Indeed we are able to break down social discount into two parts: one representing the risk of the extinction of the world, and the other related to aversion to correlated risks. It is thus possible to believe that the instantaneous probability of the world ending is very low, but that accounting for such a risk leads to introduce a discount rate that significantly differs from zero. Theoretically speaking, our work relies on an axiomatic construction of the planner’s preferences that largely resembles the one suggested by Harsanyi (1955). It extends Harsanyi’s approach by considering a weaker version of the Pareto axiom, that allows us to consider preferences for “Catastrophe Avoidance”, a notion that was initially suggested by Keeney (1980) and discussed further in Fishburn (1984). The remainder of the paper is organized as follows. In Section 2, we introduce the notation. Section 3 will expose the axiomatic construction of the planner’s preferences. Preferences for Catastrophe Avoidance are discussed in Section 4. In Section 5, we explore the consequences when there is a positive (but tiny) instantaneous probability of the world ending and see that it actually yields a rate of discount that is possibly far from zero. Last, in Section 6, we show how the additive representation can be recovered as a limit case of the more general representation that we derive.
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2
Evaluation of intertemporal welfare
We consider a society composed of consecutive generations. Time is discrete and infinite, and a period is denoted by t ∈ N (with N the set of natural numbers). In each period, there are N (potential) individuals. We use the adjective “potential” to emphasize the fact that these individuals may never exist (for example because the end of the world occurs before the time they were supposed to be born). Generations do not overlap: people can live only one period. Therefore, n o the set of all individuals is I = (i, t) such that i ∈ {1, · · · , N }, t ∈ N . An individual is completely identified by a pair (i, t). The situation of an individual is described by a unidimensional outcome xit . This is an element of the set of possible outcomes X = K ∪{d}. K is a non-empty compact subset of the positive real line, R+ . It can be seen as representing a commodity space. d indicates the non-existence status. We wish to consider uncertain prospects. Let P be a set of Borel probability measures on X. For simplicity, individuals are assumed to be identical and selfish (they have the same self-regarding preferences). Each individual’s preferences are represented by a binary relation ºI defined on P . We denote by ÂI the strict preference relation and by ∼I the indifference relation. Like Harsanyi (1955) we restrict the study to the case where individual preferences admit an expected utility representation. Stated differently, there exists a Bernoulli utility function u(.) such that: Z
Z
∀P, Pˆ ∈ P : P ºI Pˆ ⇔
u(x)dPˆ (x)
u(x)dP (x) ≥ X
X
By normalization, we can assume without loss of generality that u(d) = 0. Let us now describe social outcomes and preferences. The literature on the evaluation of infinite utility streams makes no assumption as to the asymptotic properties of feasible utility streams, so that impatience has to be assumed in order to assure the convergence of the social evaluation criterion (Koopmans,
3
1960; Diamond, 1965; and, for recent developments, Fleurbaey and Michel, 2003). In order to avoid such a shortcoming, we are led to put some structure on possible intertemporal outcomes: we assume that the society will end in a finite time. The termination date is however unknown. Q To be more specific, if we denote (i,t)∈I X the Cartesian product of individual outcome spaces, the social outcome space is ( χ=
x∈
Y
) X such that ∃T : ∀i ∈ {1, · · · , N }, ∀t ≥ T , xit = d
(i,t)∈I
Uncertainty at the social level can be described by a probability measure on χ. Let Q be a set of Borel probability measures on χ. The social planner’s preferences are defined on Q, and are denoted by ºS . Again, we follow Harsanyi in assuming that the social planner’s preferences admit an expected utility representation. In other words, there exists a utility function U (.) defined on χ such that: Z
Z
b ∈ Q : Q ºS Q b⇔ ∀Q, Q
b U (x)dQ(x)
U (x)dQ(x) ≥ χ
χ
For any subset of the set of individuals, I ⊂ I, and for any Q ∈ Q, we denote QI the marginal measure of Q on the outcomes of individuals in I . In particular, we denote Q(i,t) the individual marginal measure on individual’s (i, t) outcomes. We will assume that P is such that Q(i,t) ∈ P for any Q ∈ Q and any (i, t) ∈ I. Last we introduce a subset of Q that will be of particular importance. Let Q (i,t)∈I
P be the set of product measures of measures in P for all individuals. We
define the set of “independent measures”, R, by:
R=Q∩
Y
P
(i,t)∈I
This is the set of aggregate measures, which are also product measures. An element of R thus describes a societal risk composed of independent individual 4
risks. An experiment where each individual would be asked to flip a coin could be represented by an element of R. The case of a social planner who flips a coin to determine all individual outcomes cannot be represented by an element of R. This is also the case for any aggregate risk, as with the risk of having good weather tomorrow, or that associated with the world ending.
3
Axioms of social choice and the representation theorem
In this section, we state the assumptions made on the planner’s preferences and then provide a representation result. Our first axiom is a restricted Pareto axiom: Axiom 1 Restricted Pareto (RP):
ˆ ∈ R, if ∀(i, t) ∈ I, Q(i,t) ºI Q ˆ (i,t) then Q ºS Q ˆ ∀Q, Q
ˆ (j,t0 ) then Q ÂS Q ˆ If, furthermore, ∃(j, t0 ) / Q(j,t0 ) ÂI Q The above axiom is called Restricted Pareto, since we apply Pareto’s principle only to product measures. This axiom is weaker than the standard strong Pareto axiom, which would be obtained by replacing R by Q in the above definition. The reason for using a restricted version of the Pareto axiom, instead of the standard one, is that we want the planner’s preferences to reflect individual preferences when independent risks are concerned, but possibly to deviate from individual preferences when collective risks come into play. This possibility was first considered by Keeney (1980) and Fishburn (1984). More recently we also find it in Manski and Tetenov (2005). Such a restricted axiom is necessary if we want to allow for social judgments on how individual risks are combined. 5
This seems reasonable if we consider that the social planner should not only care for individuals’ happiness but also implement some coordination in individuals’ behaviors in order to avoid undesirable social outcomes, as with a major social catastrophe. Fishburn argued that social risk diversification would be particularly “appealing if the fate of the human race were at stake”(Fishburn, 1984, p. 904). This is precisely the issue that is discussed in the present paper. The second major axiom we use is an independence axiom. It states that the choice between two socially risky prospects does not depend on the situation of individuals who receive the same non random endowment in both situations. To express the axiom, we need to introduce further notation. First, we denote δx the one-point measure on x ∈ X, that is the measure such that δx (x) = 1 and δx (y) = 0, ∀y ∈ X \ {x}. Then, for any two probability measures Q, Q0 ∈ Q, we denote n o C(Q, Q0 ) the set C(Q, Q0 ) = (i, t) ∈ I such that ∃x ∈ X : Q(i,t) = Q0(i,t) = δx . This is the set of individuals receiving the same non-random outcome in both Q and Q0 . Our independence axiom is then as follows: Axiom 2 Independence of Unconcerned Individuals (IUI): Let Q, Q0 , ˆ and Q ˆ 0 be four probability measures in Q such that C(Q, Q0 ) = C(Q, ˆ Q ˆ 0 ) = C. Q ˆ I\C and Q0 ˆ0 Assume furthermore that QI\C = Q I\C = QI\C . Social preferences ˆ and Q ˆ 0 , the following property satisfy the (IUI) axiom if, for any such Q, Q0 , Q holds: ˆ ºS Q ˆ0 Q º S Q0 ⇔ Q Last, we would like the social planner to treat all generations impartially. In the literature, the concept of intergenerational equity has been represented using permutations of individual outcomes1 . In our framework, this gives the following axiom: 1 For some discussion of permutations as expressing impartiality, and the presentation of different permutation conditions, see Fleurbaey and Michel (2003).
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Axiom 3 Anonymity (A): Denote by Π the set of permutation on X N ×N .
∀x ∈ χ, ∀π ∈ Π such that π(x) ∈ χ :
δx ∼S δπ(x) Notice that the axiom holds on sure prospects. This is enough to obtain our result. Its counterpart in terms of expected utility would be stronger but is more difficult to write. We prefer this weaker axiom which is simpler and sufficient for our purpose. Remark also that the anonymity axiom rules out any pure time preference of the social planner. We are then able to state a representation result. Proposition 1 Assume that individual and social preferences admit an expected utility representation. Then social preferences satisfy (RP), (IUI) and (A), if and only if they can be represented by the Bernoulli utility function:
U (x) =
Y ¡ ¢ 1 1 − εu(xit ) × 1 − ε
(1)
i∈[1,N ] t∈N
with ε 6= 0 such that εu(xit ) < 1 for all xit ∈ X, or by:
U (x) =
X
u(xit ).
(2)
i∈[1,N ] t∈N
Proof. See Appendix 1. Note that the above representation of the social planner’s preferences is contingent on the normalization assumption that has been made on u (it has been 7
assumed that u(d) = 0). In particular, since u(d) = 0, the infinite product and sum that appear in (1) and (2) are well defined for any x ∈ χ. The additive welfare function (equation 2) corresponds to the limit of the multiplicative representation (equation 1) when ε tends to zero. We will therefore consider the multiplicative representation as the general one, remembering that the additive one is obtained by taking ε = 0. Remark that only the additive case would be obtained if we were to replace (RP) by the standard strong Pareto axiom2 . Note lastly that, u(.) being bounded from above, the case ε > 0 is possible. This representation of preferences is similar to the multiplicative utility function discussed in Keeney and Raiffa (1993), and is derived through a very similar set of axioms. The paternity of such a result should thus be attributed to Keeney and Raiffa, even if their proof had to be adapted to account for the fact that we do not consider a finite set of objectives (there is no upper limit on the number of individuals that may exist). Our contribution is to explore the consequence of such representation when there is an exogenous risk of the world ending. However, before going in that direction, we explain in the next section how the parameter ε that enters equation (1) is related to Catastrophe Avoidance.
4
Catastrophe avoidance
Keeney (1980), who discusses the social evaluation of fatality risks, defines Catastrophe Avoidance as follows: social preferences are said to exhibit a preference for Catastrophe Avoidance if the probability π1 of having n1 fatalities is preferred to a probability π2 of having n2 fatalities when n1 < n2 and π1 n1 = π2 n2 . Stated otherwise, under the assumption of a preference for Catastrophe Avoidance, for a given number of expected fatalities, the social planner prefers the case of an accident that kills few people to a less likely accident that kills more people. In our setting, we are not concerned by fatality risks, but by the mere existence 2
Proof available upon request.
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of individuals. Still, the transposition of Keeney’s definition to our framework is trivial. The planner’s utility exhibits preferences for Catastrophe Avoidance if and only if, in the case where all existing individuals are provided with the same positive utility level, the planner prefers the lottery where the number of individuals that will ever exist is M with probability (1 − π1 ) and M − n1 with probability π1 , to the lottery where the number of individuals is M with probability (1 − π2 ) and M − n2 with probability π2 , when π1 n1 = π2 n2 and n1 < n 2 . Preferences for Catastrophe Avoidance therefore occur if for all u > 0 and all n1 , n2, π1 , π2 such that π1 n1 = π2 n2 and n1 < n2 we have:
(1 − π1 )(1 − εu)M + π1 (1 − εu)M −n1 > (1 − π2 )(1 − εu)M + π2 (1 − εu)M −n2
Simplifying the above inequality by (1 − εu)M this leads to: £ ¤ £ ¤ π1 (1 − εu)−n1 − 1 > π2 (1 − εu)−n2 − 1
Using π1 n1 = π2 n2 it yields: (1 − εu)−n1 − 1 (1 − εu)−n2 − 1 > n1 n2 −n
Thus preferences for Catastrophe Avoidance exist if and only if f (n) = (1−εu)n −1 h i −n (1−εu)n −1 is a decreasing function. But f 0 (n) = (1−εu) log(1 − εu) + which is n n negative for all n > 0 if and only ε > 0. Assuming preference for Catastrophe Avoidance is therefore equivalent to taking ε > 0. Indeed, as shown by the second theorem of Keeney (1980), preference for Catastrophe Avoidance is equivalent to risk aversion with respect to the number of existing individuals (when all existing individuals are provided with the same utility level). To measure the strength of the preference for Catastrophe Avoid-
9
ance, we can therefore use an Arrow-Pratt coefficient of risk aversion with respect to the number of existing individuals. More precisely, assume that all individuals that ever exist are provided with the same amount of commodity x , which yields the same utility level u(x). For any M denote
U(M, x) = U (x, · · · , x, d, · · · ) | {z } M times
the social utility when the number of individuals that will ever exist is M . We can define the Index of Catastrophe Avoidance as the following Arrow-Pratt coefficient: ICA(x) = −
∂2U ∂M 2 ∂U ∂M
It is a matter of simple calculation to compute that, when the social welfare function takes the form shown in (1), we have U(M, x) = 1² − 1² (1 − εu(x))M and ICA(x) = − log(1 − εu(x)). In particular the Index of Catastrophe Avoidance equals zero when ε = 0 . This means that Utilitarian preferences (which fulfill the unrestricted Pareto axiom) cannot exhibit preferences for Catastrophe Avoidance. It is obvious that the Index of Catastrophe Avoidance is positive if and only if ε > 0 and that when ε is small, then ICA(x) ≈ εu(x). We will now see that Catastrophe Avoidance plays a key role when looking at time discounting arising from the risk of the world ending.
5
The risk of the end of the society and the social discount rate
For the sake of simplicity we consider the case where uncertainty only bears on the timing of the world’s disappearance. The planner’s problem involves ranking infinitely long consumption plans, knowing that for a reason that is independent of his/her behavior, the world will stop existing at a finite date (consumption 10
becoming then impossible). This problem is similar to the one considered by Dasgupta and Heal (1979). To simplify matters further, we assume that at each period there is a probability p that the end of the world occurs and a probability (1 − p) that the world survives. Moreover, all the people of generation t are assumed to receive the same xt (xit = xt , ∀i ∈ I and ∀t ∈ N). Consider an infinitely long consumption plan x = (xt )t∈N . For any T ≥ 0 there is a probability p(1 − p)T that the world will last exactly T periods. In such a ³ ¢N ´ QT ¡ 1 case the consumption plan x yields a social utility ε × 1 − t=1 1 − εu(xt ) if T > 0 and zero if T = 0. The expected utility associated with x is therefore: ∞
1X W (x) = p(1 − p)T ε T =1
Ã
T Y ¡ ¢N 1− 1 − εu(xt )
! .
(3)
t=1
Since the social planner aims at maximizing expected utility, W (x) is a utility function representing the planner’s preferences over consumption plans. A consumption plan x will be preferred to an alternative plan x0 if and only if W (x) > W (x0 ) Now, let us examine the implicit social discount rate associated with such preferences. In a conventional fashion, we define the rate of time discounting by looking at how the marginal utility of consumption changes, when consumption is (locally) constant. Formally, for any x such that xt = xt+1 the rate of time discounting at time t is defined by:
ρ(x, t) = 1 −
∂W ∂xt+1 | ∂W xt =xt+1 ∂xt
(4)
When preferences are given by the utility function W (x) shown in equation (1) we have the following result: Proposition 2 Along a constant allocation path giving x to each living individ-
11
ual, the social rate of discount is
ρ(x) = p + (1 − p)(1 − [1 − εu(x)]N )
Proof. See Appendix 2. Note that when ε = 0, that is in the standard Utilitarian case, we find that the rate of discount equals the instantaneous probability of the world ending, which is consistent with the results of Dasgupta and Heal. From Proposition 2 we also have the immediate consequences: Corollary 1 Along constant allocation paths : 1) The rate of discount is greater than the hazard rate of the world disappearing whenever social preferences exhibit preferences for Catastrophe Avoidance (i.e. ρ > p whenever εu(x) > 0) 2) When the probability of the world ending is infinitesimally small the rate of discount approximately equals 1 − [1 − εu(x)]N . These two points deserve some comment. First we see that preference for Catastrophe Avoidance makes the social planner discount the future more heavily. The reason is that the planner wants to avoid the worst catastrophe in which: (1) only few individuals ever come to life; (2) those few individuals sacrifice most of their resources for the sake of future generations that will actually never exist. Secondly, even if the instantaneous probability of the world ending is very small, the rate of discount may be quite large. This may seem counterintuitive, as one might expect that when p is infinitesimally small, the planner would not exhibit time preferences, just as in the case where there is no uncertainty. There is however a simple intuition that explains why the rate of discount does not tend to zero when p tends to zero. The point is that when p tends to zero the expected 12
number of individuals that will ever exist (which equals
N ) p
tends to infinity. As a
consequence, the smaller p, the greater the loss when the world ends. Thus, when p tends to zero, the probability of a catastrophe occurring does tend to zero, but the magnitude of the catastrophe tends to infinity. Both factors compensate and lead the social planner to use a non-negligible social discount even if p is very small. Lastly, we may want to look at the size of the discount rate when p is infinitesimally small. At the limit p → 0, the discount rate is 1 − [1 − εu(x)]N . When ε is small, 1 − [1 − εu(x)]N ≈ N εu(x), and the discount rate is therefore approximately the product of the Index of Catastrophe Avoidance by the population size3 . Thus even if the planner has a very weak inclination for Catastrophe Avoidance, the social discount rate may be non-negligible if the population size is large enough.
6
Recovering the additive social welfare function
An indisputable drawback of our multiplicative representation is that it is no longer additive. For most applications, this is a source of substantial increase in complexity. Still, following the idea suggested in Bommier (2006), additivity can be recovered by considering the limit case where the difference in welfare between existing or not existing is assumed to be much greater than the difference in welfare between having a low or a high level of consumption. More precisely, assume that the xt (when different from d) remain in a bounded domain [xmin , xmax ] and 3
This simply reflects that the larger the population, the greater the catastrophe in case of the world termination.
13
that the function u is such that:
u(xt ) = 0 when xt = d u(xt ) = 1 + λv(xt ) when xt 6= d
where λ is very small and v is a bounded function over [xmin , xmax ]. In such a case the difference in welfare between existence and non-existence is approximately equal to 1, while the difference in welfare between having xmin or xmax equals λ (v(xmax ) − v(xmin )), which is assumed to be much smaller than 1. Consider now the limit case where λ is infinitesimally small. Note that in such a case, the Index of Catastrophe Avoidance is independent of x and equals − log(1 − ε). We now give the following result: Proposition 3 In the limit where λ → 0 the social planner’s preferences are represented by the social welfare function
Wε (x) =
∞ X
γ t N v(xt )
t=0
where γ = (1 − p)(1 − ε)N . Proof. See Appendix 3. We therefore obtain the standard additive representation, with a discount ¤ £ rate that equals p + (1 − p) 1 − (1 − ε)N . Thus, we end up with a formulation that is the same as that of Dasgupta and Heal, though with the fundamental difference that the rate of discount is now augmented by a factor that depends on the Index of Catastrophe Avoidance and the population size. Thus, there is no contradiction between assuming that the probability that the world disappears is very low, and that the rate of discount is significantly greater than zero. Nor is there any inequitable bias in favor of present generations. 14
Despite its simplicity, the additive approximation may be controversial, for we have to assume that the difference in welfare between existence and non-existence is much larger than the difference in welfare between possible lives. This is of course disputable. Blackorby, Bossert and Donaldson (1995) argue for example there are some states of extreme poverty that are, from the planner’s point of view, worse than non-existence. If one is reluctant to use the additive approximation, there is no other solution than to rely on the non-additive formula given by (3). However, as it emerges from Appendix 2 (see equation 9), W (x) can be written as:
³ ´Y 1X W (x) = (1 − p)T +1 1 − [1 − εu(xT +1 )]N [1 − εu(xτ )]N ε T =0 τ =1 +∞
T
or, alternatively: +∞ T ³ X ´ 1−pX W (x) = uε (xT +1 ) exp − vε,p (xτ ) ε T =0 τ =1
with uε (xt ) = 1 − εu(xt ) and vε,p (xt ) = −N log(1 − εu(xt )) − log(1 − p). We recognize here recursive preferences, as originally introduced by Uzawa (1968), in continuous time, and Epstein (1983), in discrete time. As is well known, and as is found in Proposition 2, those preferences are characterized by endogenous time discounting. Thus preference for Catastrophe Avoidance permits to reconcile intergenerational equity with endogenous time discounting. The simplifying assumption of exogenous time discounting can be seen as a limit case that corresponds to the additive approximation detailed above. Whether or not this limit case may be considered as relevant depends on how wide we think the welfare gap is between existence and non-existence.
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7
Conclusion
We have extended the standard Utilitarian aggregation of preferences `a la Harsanyi to account for a possible planner’s taste for Catastrophe Avoidance. This was done by replacing the usual Pareto axiom by a weaker axiom. Basically, this axiom stipulates that there is no divergence between the social planner’s and the individuals’ preferences as long as uncorrelated risks are considered, but that some divergence may occur when correlated risks are at play. The axiom is reasonable if we consider that individual judgements have to be followed only when individual consequences are at play. This allows planners to express their own views on collective consequences. Preferences for Catastrophe Avoidance were found to play a key role when accounting for the probability that the world may end. We showed that an equitable social planner who has no pure time preference, but preferences for Catastrophe Avoidance, discounts the welfare of future generation with a rate that is greater than the instantaneous probability of the world coming to an end. More importantly this rate does not vanish when the instantaneous probability of the world ending tends towards zero. In other words, substantial time discounting does not necessarily reflect the planner’s lack of equity, or the planner’s belief that the world will soon end. There is a third source of social discounting that results from the combination of preference for Catastrophe Avoidance with the belief that there is indeed a positive (but, possibly very small) probability that the world will end. The end of the world is a very stylized representation of an event with durable consequences. Its key characteristics, for our analysis of time discounting, is that it durably and negatively impacts individuals’ utilities and marginal utilities4 . In fact, it can be shown that, when the planner exhibits preferences for Catastrophe Avoidance, the planner’s discount rate increases with the likelihood of an event having these characteristics occurring. Natural extensions of this paper therefore 4
When the world ends, the utilities and marginal utilities of future individuals are irreversibly set to zero.
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involve considering less caricatured risks, such as the possibility of an ecological catastrophe, and see how they may affect time discounting.
Appendices Appendix 1: Proof of Proposition 1 It is straightforward to check that if the planner’s preferences are represented by the multiplicative or additive utility function shown in (1) and (2) they satisfy (RP), (IUI) and (A). We will therefore focus on showing that (RP), (IUI) and (A) imply these particular specifications. For simplicity, we will slightly change the notation along this proof. Instead of identifying individuals by a pair (i, t), with t and i ∈ [1, N ], we will use a single index k = i + N t. This way, xk denotes in fact xit where t is the integer part of
k N
and i = k − N t. An element of x ∈ χ is then just a sequence x =
(x1 , x2 , · · · , xk , · · · ) such that xk = d for all k greater than a finite number kx . The Bernoulli utility function that represents the planner’s preferences is:
U (x1 , x2 , · · · , xk , · · · )
Consider a given c0 ∈ K. From (RP):
(c0 , d, d, · · · ) ºS (d, d, d, · · · ) ⇔ c0 ºI d
Thus, by normalization of U it can be assumed that:
U (d, d, d, · · · ) = u(d) = 0 U (c0 , d, d, · · · ) = u(c0 )
17
(5) (6)
ˆ and Q ˆ 0 , four probability measures in Q such that Q(1) = Consider now Q, Q0 , Q ˆ (1) = Q ˆ 0 = δx1 , QI\(1) = Q ˆ I\(1) and Q0 ˆ0 Q0(1) = δd , Q (1) I\(1) = QI\(1) . ˆ ºS Q ˆ 0 . Thus: According to the (IUI) axiom, Q ºS Q0 ⇔ Q Z
Z U (d, x2 , · · · )dQI\(1) (x2 , · · · )
≥
χ
χ
⇐⇒
Z U (x1 , x2 , · · · )dQI\(1) (x2 , · · · )
Z
≥
χ
U (d, x2 , · · · )dQ0I\(1) (x2 , · · · )
χ
U (x1 , x2 , · · · )dQ0I\(1) (x2 , · · · )
That implies that Vd (x2 , x3 , · · · ) = U (d, x2 , x3 , · · · ) and Vx1 (x2 , x3 , · · · · · · ) = U (x1 , x2 , x3 , · · · ) are two Bernoulli utility functions representing the same preference orderings on the probability measures on χ. The function Vx1 must therefore be obtained from Vd by a positive affine transformation. In other words, for any x1 ∈ X there exits v(x1 ) and w(x1 ) > 0 such that
U (x1 , x2 , x3 , · · · ) = v(x1 ) + w(x1 ) × U (d, x2 , x3 , · · · )
The normalization conditions (5) and (6) imply that:
v(d) = 0 and v(c0 ) = u(c0 )
(7)
Now remember that xk = d when k is larger than kx and that by the (A) axiom, U (d, x2 , x3 · · · xkx , d, d, · · · ) = U (x2 , x3 , · · · , xkx , d, d, d · · · ), so that, for any x ∈ X: U (x1 , x2 , x3 · · · ) = v(x1 ) + w(x1 )U (x2 , x3 , · · · ) Iterating once we obtain
U (x1 , x2 , x3 , · · · , xkx , d, d, d · · · ) = v(x1 ) + w(x1 )v(x2 ) +w(x1 )w(x2 )U (x3 , · · · , xkx , d, d, d · · · )
18
By repeating the iteration kx times we get:
U (x1 , x2 , · · · , xkx , d, d, d · · · ) =
kx X
v(xk )
k=1
k−1 Y
w(xl ) +
l=1
kx Y
w(xl )U (xκ+1 , xκ+2 , ...)
l=1
Using (5) and (6) we eventually obtain:
U (x1 , x2 , · · · , xkx , d, d, d · · · ) =
∞ X
v(xk )
k=1
k−1 Y
w(xl )
l=1
Now, remark that the (A) axiom imposes that U (x, y, d, · · · ) = U (y, x, d, · · · ) so that for any x, y ∈ X we must have:
v(x) + w(x)v(y) = v(y) + w(y)v(x)
(8)
or equivalently (whenever x 6= d and y 6= d): 1 − w(x) 1 − w(y) = v(x) v(y) The ratio
1−w(x) v(x)
is therefore constant for any x ∈ K. Let us denote it by ε.
Applying (8) to x = d it follows that w(d) = 1. Therefore, we have w(x) = 1 − εv(x) for all x ∈ X. Since w(x) > 0 we must have εv(x) < 1. First remark that if ε = 0, we obtain the additive Bernoulli function:
U (x1 , x2 · · · , · · · ) =
+∞ X k=1
19
v(xk )
When ε 6= 0 we compute:
1 − εU (x1 , x2 , · · · ) = 1 − = 1+ = 1+
kx X
εv(xk )
k=1 kx X
k−1 Y
(1 − εv(xl ))
l=1
(1 − εv(xk ))
k=1 kx Y k X
k−1 Y
kx k−1 X Y
l=1
k=1 l=1
(1 − εv(xl )) −
(1 − εv(xl )) −
k=1 l=1
=
kx Y
kx k−1 X Y
(1 − εv(xl ))
(1 − εv(xl ))
k=1 l=1
(1 − εv(xl )) =
l=1
+∞ Y
(1 − εv(xl ))
l=1
We eventually get the multiplicative specification: 1 U (x1 , x2 , · · · ) = ε
à 1−
+∞ Y
! (1 − εv(xl ))
l=1
It only remains to check that the functions v(.) and u(.) are identical. Since, from (7), v(d) = u(d) and v(c0 ) = u(c0 ), we only need to show that v(.) is a Bernoulli utility function that represents individuals’ preferences. Consider the ˆ such that Q(k) = Q ˆ (k) = δx , ∀k 6= 1. Uncertainty thus only measures Q and Q k bears on individual 1. From the (RP) axiom:
ˆ ⇔ Q(1) ºI Q ˆ (1) Q ºS Q ˆ if and only if But Q ºS Q
R X
U (x1 , x2 , · · · )dQ(1) (x1 ) ≥
R X
ˆ (1) (x1 ). U (x1 , x2 , · · · )dQ
Since Z
Ã+∞ ! µ ¶ Z Y¡ ¢ 1 1 U (x1 , x2 , · · · )dQ(1) (x1 ) = − − v(x1 )dQ(1) (x1 ) 1 − εv(xk ) × ε ε X X k=2
ˆ if and only if we obtain that Q ºS Q
R
v(x1 )dQ(1) (x1 ) ≥ X
20
R X
ˆ (1) (x1 ). It v(x1 )dQ
follows that: Z Q(1)
Z
ˆ (1) ⇔ º Q I
ˆ (,1) (x1 ) v(x1 )dQ
v(x1 )dQ(1) (x1 ) ≥ X
X
which implies that v(.) is indeed a Bernoulli utility function that represents individual preferences.
Appendix 2: Proof of Proposition 2 For any x and T > 0 denote 1 U (x, T ) = × ε
Ã
T Y
1−
! [1 − εu(xτ )]N
τ =1
and, for T = 0, U (x, 0) = 0 We have, for any T ≥ 0: T ´ Y 1 ³ N U (x, T + 1) − U (x, T ) = × 1 − [1 − εu(xT +1 )] × [1 − εu(xτ )]N , ε τ =1
Now write W (x) = = =
P+∞ T =0
p(1 − p)T U (x, T ) =
P+∞
T =0 (1
− p)T U (x, T ) −
P+∞
T T =0 (1 − p) U (x, T ) −
P+∞ £ T =0
P+∞
T =0 (1
P+∞
T =0 (1
¤ (1 − p)T − (1 − p)(1 − p)T U (x, T )
− p)T +1 U (x, T ) − p)T +1 U (x, T + 1)
P+∞
− p)T +1 [U (x, T + 1) − U (x, T )] ³ ´ P+∞ N N QT 1 T +1 = ε T =0 (1 − p) 1 − [1 − εu(xT +1 )] τ =1 [1 − εu(xτ )] +
T =0 (1
(9)
21
Now, we can easily compute the partial derivative of W (x) with respect to xt : t Y ∂W (x) −1 0 t = N u (xt ) (1 − εu(xt )) (1 − p) [1 − εu(xτ )]N ∂xt τ =1 0
−N u (xt ) [1 − εu(xt )]
−1
+∞ X
T +1
(1 − p)
³
N
1 − [1 − εu(xT +1 )]
T ´Y
[1 − εu(xτ )]N
τ =1
T =t
Along a constant allocation path, xt = x, ∀t, denoting u(x) and u0 (x) by u and u0 , the above expression reduces to: ∂W (x) = N u0 (1 − p)t (1 − εu)N t−1 ∂xt +∞ ³ ´ X −N u0 (1 − p)T +1 1 − [1 − εu]N [1 − εu]N T −1 T =t
Ã
= N u0 (1 − p)t (1 − εu)N t−1
+∞ ³ ´ X N T −t+1 1− (1 − p) 1 − [1 − εu] [1 − εu]N (T −t)
à = N u0 (1 − p)t (1 − εu)N t−1
1−
T =t +∞ X
³
´
(1 − p)τ +1 1 − [1 − εu]N [1 − εu]N τ
τ =0
We eventually obtain that:
ρ(x) = 1 −
∂W (x) ∂xt+1 ∂W (x) ∂xt
= 1 − (1 − p)(1 − εu)N = p + (1 − p)[1 − (1 − εu)N ]
Appendix 3: Proof of Proposition 3 The planner’s preferences over consumption plans is represented by: +∞
1X W (x) = p(1 − p)T ε T =1
Ã
22
1−
T Y τ =1
! [1 − εu(xτ )]N
!
!
Substitute u(x) = 1 + λv(x) in the above formula and write that:
W (x) ' W (x)|λ=0 + λ
∂W |λ=0 ∂λ
to obtain
W (x) '
+∞ ³ ´ 1X p(1 − p)T 1 − [1 − ε]N T ε T =1 Ã T ! +∞ X X N (T −1) T +λ p(1 − p) [1 − ε] N v(xt ) t=1
T =1
The first term is a constant and does not affect preferences. Switching the summation signs, the second term equals
λ
+∞ X t=1
N v(xt )
+∞ X
p(1 − p)T [1 − ε]N (T −1)
T =t
+∞ +∞ X X £ ¤T = λp [1 − ε]−N N v(xt ) (1 − p)(1 − ε)N t=1
T =t
−N
=
λp [1 − ε] 1 − (1 − p)(1 − ε)N
where γ = (1 − p)(1 − ε)N . The term
+∞ X
γ t N v(xt )
t=1
λp[1−ε]−N 1−(1−p)(1−ε)N
is a positive multiplicative
factor which does not affect preferences. We therefore see that the planner’s preferences can be represented by +∞ X
γ t N v(xt ).
t=1
23
References Bommier, A. (2006), “Uncertain Lifetime and Intertemporal Choice: Risk Aversion as a Rationale for Time Discounting”, forthcoming in the International Economic Review. Blackorby, C., W. Bossert and D. Donaldson (1995), “Intertemporal Population Ethics: Critical-Level Utilitarian Principles ”, Econometrica, Vol. 63, 1303-1320.
Dasgupta, P., and G. Heal (1974), “The Optimal Depletion of Exhaustible Resources”, Review of Economic Studies, Vol. 41, 3-28. Dasgupta, P., and G. Heal (1979), Economic Theory and Exhaustible Resources, Cambridge: Cambridge University Press. Diamond, P.A. (1965), “The Evaluation of Infinite Utility Streams”, Econometrica, Vol. 33, 170-177. Epstein, L.G. (1983), “Stationary Cardinal Utility and Optimal Growth Under Uncertainty”. Journal of Economic Theory, 31: 133-52. Fishburn, P.C. (1984), “Equity Axioms for Public Risks”, Operations Research, Vol. 32, 901-908. Fleurbaey, M., and P. Michel (2003), “Intertemporal equity and the extension of the Ramsey criterion”, Journal of Mathematical Economics, Vol. 39, 777-802. Harsanyi, J.C. (1955), “Cardinal welfare, individualistic ethics, and interpersonal comparisons of utility”, Journal of Political Economics, Vol. 63, 309-321. Keeney, R.L. (1980), “Equity and Public Risk”, Operations Research, Vol. 28, 527-534. Keeney, R.L., and H. Raiffa (1993), Decisions with Multiple Objectives: Preferences and Value Tradeoffs, Cambridge and New York: Cambridge University Press. 24
Koopmans, T.C. (1960), “Stationary Ordinal Utility and Impatience”, Econometrica, Vol. 28, No. 2, 287-309. Manski, C.F., and A. Tetenov (2005), Admissible treatment rules for a risk-averse planner with experimental data on an innovation, mimeo. Sidgwick, H. (1907), The Methods of Ethics, seventh edition (first edition: 1891), London: MacMillan. Uzawa, H. (1968), “Time Preference, The Consumption Function and Optimum Asset Holdings”. In J.N. Wolfe (ed.), Value Capital and Growth: Papers in Honour of Sir John Hicks, Edinburgh: University of Edinburgh Press.
25
St´ephane Zuber‡ University of Toulouse (GREMAQ)
Abstract A social welfare function treating all generations equally is derived from a set of axioms that, contrary to the Utilitarian approach, allows for preferences for Catastrophe Avoidance. Implications for the case where there is a risk of world extinction are studied. We show that substantial time discounting can arise from the planner’s taste for Catastrophe Avoidance, even if the probability of the world ending is infinitesimally small. Keywords: Time Discounting, Intergenerational Equity, Social Welfare Function, Catastrophe Avoidance. JEL Classification numbers: D63, D69, D99.
∗
We wish to thank Marc Fleurbaey and Thibault Gajdos for many valuable comments. Universit´e des Sciences Sociales - Manufacture des Tabacs, 21 all´ees de Brienne, 31000 Toulouse, France. E-mail: [email protected]. ‡ Universit´e des Sciences Sociales - Manufacture des Tabacs, 21 all´ees de Brienne, 31000 Toulouse, France. E-mail: [email protected]. †
1
Introduction
Many major policy decisions involve making trade-offs between the welfare of current and future generations. Policy guidance has then to rely on a given social objective. The most common approach involves assuming that the social planner aims at maximizing a social welfare function:
SW =
∞ X
γ t Ut
t=0
where t indicates time, Ut is the aggregate utility of cohort t and γ t is the social discount factor (the discount rate being 1 − γ). Such an approach has been extensively criticized by economists and philosophers. The main point is that generations are not treated equally, since they are assigned a weight that depends on when they were born. This is generally considered to be unfair, a point of view most clearly expressed by philosopher Henry Sidgwick who argued that “[...] the time at which a man exists cannot affect the value of his happiness from a universal point of view” (Sidgwick, 1907, p. 414). To our knowledge, there is only one argument that has been suggested to provide an ethical ground for using such a welfare function. Initially developed by Dasgupta and Heal (1974 & 1979), it involves assuming that between any two dates there exists a positive probability 1 − γ that the world ends. An equitable Utilitarian objective leads then to the above social welfare function. Still, this argument is problematic when we turn to quantitative aspects. Social discount rates are usually taken between 1% and 5 % per year. But, for most people it would seem excessively pessimistic to assume a yearly probability of the word ending of 1% or 5%. With a 5% probability we would have more than a 50% chance of disappearing in the next 14 years. Even with a probability of the world ending of 1% per year, there would be the less than a 50% chance of seeing the world lasting more than 69 years. It is hard to believe that this should reflect 1
the beliefs of a reasonable social planner. The aim of this paper is to provide a theoretical foundation for a social welfare function displaying time discounting, but which is ethically acceptable, in the sense that it treats all generations equally. Our argument, just as that of Dasgupta and Heal, relies on the idea that at any time there is a positive probability that the world ends. However, the novelty is that we suggest different foundations that make it possible to distinguish the social discount rate from the probability of the world ending. Indeed we are able to break down social discount into two parts: one representing the risk of the extinction of the world, and the other related to aversion to correlated risks. It is thus possible to believe that the instantaneous probability of the world ending is very low, but that accounting for such a risk leads to introduce a discount rate that significantly differs from zero. Theoretically speaking, our work relies on an axiomatic construction of the planner’s preferences that largely resembles the one suggested by Harsanyi (1955). It extends Harsanyi’s approach by considering a weaker version of the Pareto axiom, that allows us to consider preferences for “Catastrophe Avoidance”, a notion that was initially suggested by Keeney (1980) and discussed further in Fishburn (1984). The remainder of the paper is organized as follows. In Section 2, we introduce the notation. Section 3 will expose the axiomatic construction of the planner’s preferences. Preferences for Catastrophe Avoidance are discussed in Section 4. In Section 5, we explore the consequences when there is a positive (but tiny) instantaneous probability of the world ending and see that it actually yields a rate of discount that is possibly far from zero. Last, in Section 6, we show how the additive representation can be recovered as a limit case of the more general representation that we derive.
2
2
Evaluation of intertemporal welfare
We consider a society composed of consecutive generations. Time is discrete and infinite, and a period is denoted by t ∈ N (with N the set of natural numbers). In each period, there are N (potential) individuals. We use the adjective “potential” to emphasize the fact that these individuals may never exist (for example because the end of the world occurs before the time they were supposed to be born). Generations do not overlap: people can live only one period. Therefore, n o the set of all individuals is I = (i, t) such that i ∈ {1, · · · , N }, t ∈ N . An individual is completely identified by a pair (i, t). The situation of an individual is described by a unidimensional outcome xit . This is an element of the set of possible outcomes X = K ∪{d}. K is a non-empty compact subset of the positive real line, R+ . It can be seen as representing a commodity space. d indicates the non-existence status. We wish to consider uncertain prospects. Let P be a set of Borel probability measures on X. For simplicity, individuals are assumed to be identical and selfish (they have the same self-regarding preferences). Each individual’s preferences are represented by a binary relation ºI defined on P . We denote by ÂI the strict preference relation and by ∼I the indifference relation. Like Harsanyi (1955) we restrict the study to the case where individual preferences admit an expected utility representation. Stated differently, there exists a Bernoulli utility function u(.) such that: Z
Z
∀P, Pˆ ∈ P : P ºI Pˆ ⇔
u(x)dPˆ (x)
u(x)dP (x) ≥ X
X
By normalization, we can assume without loss of generality that u(d) = 0. Let us now describe social outcomes and preferences. The literature on the evaluation of infinite utility streams makes no assumption as to the asymptotic properties of feasible utility streams, so that impatience has to be assumed in order to assure the convergence of the social evaluation criterion (Koopmans,
3
1960; Diamond, 1965; and, for recent developments, Fleurbaey and Michel, 2003). In order to avoid such a shortcoming, we are led to put some structure on possible intertemporal outcomes: we assume that the society will end in a finite time. The termination date is however unknown. Q To be more specific, if we denote (i,t)∈I X the Cartesian product of individual outcome spaces, the social outcome space is ( χ=
x∈
Y
) X such that ∃T : ∀i ∈ {1, · · · , N }, ∀t ≥ T , xit = d
(i,t)∈I
Uncertainty at the social level can be described by a probability measure on χ. Let Q be a set of Borel probability measures on χ. The social planner’s preferences are defined on Q, and are denoted by ºS . Again, we follow Harsanyi in assuming that the social planner’s preferences admit an expected utility representation. In other words, there exists a utility function U (.) defined on χ such that: Z
Z
b ∈ Q : Q ºS Q b⇔ ∀Q, Q
b U (x)dQ(x)
U (x)dQ(x) ≥ χ
χ
For any subset of the set of individuals, I ⊂ I, and for any Q ∈ Q, we denote QI the marginal measure of Q on the outcomes of individuals in I . In particular, we denote Q(i,t) the individual marginal measure on individual’s (i, t) outcomes. We will assume that P is such that Q(i,t) ∈ P for any Q ∈ Q and any (i, t) ∈ I. Last we introduce a subset of Q that will be of particular importance. Let Q (i,t)∈I
P be the set of product measures of measures in P for all individuals. We
define the set of “independent measures”, R, by:
R=Q∩
Y
P
(i,t)∈I
This is the set of aggregate measures, which are also product measures. An element of R thus describes a societal risk composed of independent individual 4
risks. An experiment where each individual would be asked to flip a coin could be represented by an element of R. The case of a social planner who flips a coin to determine all individual outcomes cannot be represented by an element of R. This is also the case for any aggregate risk, as with the risk of having good weather tomorrow, or that associated with the world ending.
3
Axioms of social choice and the representation theorem
In this section, we state the assumptions made on the planner’s preferences and then provide a representation result. Our first axiom is a restricted Pareto axiom: Axiom 1 Restricted Pareto (RP):
ˆ ∈ R, if ∀(i, t) ∈ I, Q(i,t) ºI Q ˆ (i,t) then Q ºS Q ˆ ∀Q, Q
ˆ (j,t0 ) then Q ÂS Q ˆ If, furthermore, ∃(j, t0 ) / Q(j,t0 ) ÂI Q The above axiom is called Restricted Pareto, since we apply Pareto’s principle only to product measures. This axiom is weaker than the standard strong Pareto axiom, which would be obtained by replacing R by Q in the above definition. The reason for using a restricted version of the Pareto axiom, instead of the standard one, is that we want the planner’s preferences to reflect individual preferences when independent risks are concerned, but possibly to deviate from individual preferences when collective risks come into play. This possibility was first considered by Keeney (1980) and Fishburn (1984). More recently we also find it in Manski and Tetenov (2005). Such a restricted axiom is necessary if we want to allow for social judgments on how individual risks are combined. 5
This seems reasonable if we consider that the social planner should not only care for individuals’ happiness but also implement some coordination in individuals’ behaviors in order to avoid undesirable social outcomes, as with a major social catastrophe. Fishburn argued that social risk diversification would be particularly “appealing if the fate of the human race were at stake”(Fishburn, 1984, p. 904). This is precisely the issue that is discussed in the present paper. The second major axiom we use is an independence axiom. It states that the choice between two socially risky prospects does not depend on the situation of individuals who receive the same non random endowment in both situations. To express the axiom, we need to introduce further notation. First, we denote δx the one-point measure on x ∈ X, that is the measure such that δx (x) = 1 and δx (y) = 0, ∀y ∈ X \ {x}. Then, for any two probability measures Q, Q0 ∈ Q, we denote n o C(Q, Q0 ) the set C(Q, Q0 ) = (i, t) ∈ I such that ∃x ∈ X : Q(i,t) = Q0(i,t) = δx . This is the set of individuals receiving the same non-random outcome in both Q and Q0 . Our independence axiom is then as follows: Axiom 2 Independence of Unconcerned Individuals (IUI): Let Q, Q0 , ˆ and Q ˆ 0 be four probability measures in Q such that C(Q, Q0 ) = C(Q, ˆ Q ˆ 0 ) = C. Q ˆ I\C and Q0 ˆ0 Assume furthermore that QI\C = Q I\C = QI\C . Social preferences ˆ and Q ˆ 0 , the following property satisfy the (IUI) axiom if, for any such Q, Q0 , Q holds: ˆ ºS Q ˆ0 Q º S Q0 ⇔ Q Last, we would like the social planner to treat all generations impartially. In the literature, the concept of intergenerational equity has been represented using permutations of individual outcomes1 . In our framework, this gives the following axiom: 1 For some discussion of permutations as expressing impartiality, and the presentation of different permutation conditions, see Fleurbaey and Michel (2003).
6
Axiom 3 Anonymity (A): Denote by Π the set of permutation on X N ×N .
∀x ∈ χ, ∀π ∈ Π such that π(x) ∈ χ :
δx ∼S δπ(x) Notice that the axiom holds on sure prospects. This is enough to obtain our result. Its counterpart in terms of expected utility would be stronger but is more difficult to write. We prefer this weaker axiom which is simpler and sufficient for our purpose. Remark also that the anonymity axiom rules out any pure time preference of the social planner. We are then able to state a representation result. Proposition 1 Assume that individual and social preferences admit an expected utility representation. Then social preferences satisfy (RP), (IUI) and (A), if and only if they can be represented by the Bernoulli utility function:
U (x) =
Y ¡ ¢ 1 1 − εu(xit ) × 1 − ε
(1)
i∈[1,N ] t∈N
with ε 6= 0 such that εu(xit ) < 1 for all xit ∈ X, or by:
U (x) =
X
u(xit ).
(2)
i∈[1,N ] t∈N
Proof. See Appendix 1. Note that the above representation of the social planner’s preferences is contingent on the normalization assumption that has been made on u (it has been 7
assumed that u(d) = 0). In particular, since u(d) = 0, the infinite product and sum that appear in (1) and (2) are well defined for any x ∈ χ. The additive welfare function (equation 2) corresponds to the limit of the multiplicative representation (equation 1) when ε tends to zero. We will therefore consider the multiplicative representation as the general one, remembering that the additive one is obtained by taking ε = 0. Remark that only the additive case would be obtained if we were to replace (RP) by the standard strong Pareto axiom2 . Note lastly that, u(.) being bounded from above, the case ε > 0 is possible. This representation of preferences is similar to the multiplicative utility function discussed in Keeney and Raiffa (1993), and is derived through a very similar set of axioms. The paternity of such a result should thus be attributed to Keeney and Raiffa, even if their proof had to be adapted to account for the fact that we do not consider a finite set of objectives (there is no upper limit on the number of individuals that may exist). Our contribution is to explore the consequence of such representation when there is an exogenous risk of the world ending. However, before going in that direction, we explain in the next section how the parameter ε that enters equation (1) is related to Catastrophe Avoidance.
4
Catastrophe avoidance
Keeney (1980), who discusses the social evaluation of fatality risks, defines Catastrophe Avoidance as follows: social preferences are said to exhibit a preference for Catastrophe Avoidance if the probability π1 of having n1 fatalities is preferred to a probability π2 of having n2 fatalities when n1 < n2 and π1 n1 = π2 n2 . Stated otherwise, under the assumption of a preference for Catastrophe Avoidance, for a given number of expected fatalities, the social planner prefers the case of an accident that kills few people to a less likely accident that kills more people. In our setting, we are not concerned by fatality risks, but by the mere existence 2
Proof available upon request.
8
of individuals. Still, the transposition of Keeney’s definition to our framework is trivial. The planner’s utility exhibits preferences for Catastrophe Avoidance if and only if, in the case where all existing individuals are provided with the same positive utility level, the planner prefers the lottery where the number of individuals that will ever exist is M with probability (1 − π1 ) and M − n1 with probability π1 , to the lottery where the number of individuals is M with probability (1 − π2 ) and M − n2 with probability π2 , when π1 n1 = π2 n2 and n1 < n 2 . Preferences for Catastrophe Avoidance therefore occur if for all u > 0 and all n1 , n2, π1 , π2 such that π1 n1 = π2 n2 and n1 < n2 we have:
(1 − π1 )(1 − εu)M + π1 (1 − εu)M −n1 > (1 − π2 )(1 − εu)M + π2 (1 − εu)M −n2
Simplifying the above inequality by (1 − εu)M this leads to: £ ¤ £ ¤ π1 (1 − εu)−n1 − 1 > π2 (1 − εu)−n2 − 1
Using π1 n1 = π2 n2 it yields: (1 − εu)−n1 − 1 (1 − εu)−n2 − 1 > n1 n2 −n
Thus preferences for Catastrophe Avoidance exist if and only if f (n) = (1−εu)n −1 h i −n (1−εu)n −1 is a decreasing function. But f 0 (n) = (1−εu) log(1 − εu) + which is n n negative for all n > 0 if and only ε > 0. Assuming preference for Catastrophe Avoidance is therefore equivalent to taking ε > 0. Indeed, as shown by the second theorem of Keeney (1980), preference for Catastrophe Avoidance is equivalent to risk aversion with respect to the number of existing individuals (when all existing individuals are provided with the same utility level). To measure the strength of the preference for Catastrophe Avoid-
9
ance, we can therefore use an Arrow-Pratt coefficient of risk aversion with respect to the number of existing individuals. More precisely, assume that all individuals that ever exist are provided with the same amount of commodity x , which yields the same utility level u(x). For any M denote
U(M, x) = U (x, · · · , x, d, · · · ) | {z } M times
the social utility when the number of individuals that will ever exist is M . We can define the Index of Catastrophe Avoidance as the following Arrow-Pratt coefficient: ICA(x) = −
∂2U ∂M 2 ∂U ∂M
It is a matter of simple calculation to compute that, when the social welfare function takes the form shown in (1), we have U(M, x) = 1² − 1² (1 − εu(x))M and ICA(x) = − log(1 − εu(x)). In particular the Index of Catastrophe Avoidance equals zero when ε = 0 . This means that Utilitarian preferences (which fulfill the unrestricted Pareto axiom) cannot exhibit preferences for Catastrophe Avoidance. It is obvious that the Index of Catastrophe Avoidance is positive if and only if ε > 0 and that when ε is small, then ICA(x) ≈ εu(x). We will now see that Catastrophe Avoidance plays a key role when looking at time discounting arising from the risk of the world ending.
5
The risk of the end of the society and the social discount rate
For the sake of simplicity we consider the case where uncertainty only bears on the timing of the world’s disappearance. The planner’s problem involves ranking infinitely long consumption plans, knowing that for a reason that is independent of his/her behavior, the world will stop existing at a finite date (consumption 10
becoming then impossible). This problem is similar to the one considered by Dasgupta and Heal (1979). To simplify matters further, we assume that at each period there is a probability p that the end of the world occurs and a probability (1 − p) that the world survives. Moreover, all the people of generation t are assumed to receive the same xt (xit = xt , ∀i ∈ I and ∀t ∈ N). Consider an infinitely long consumption plan x = (xt )t∈N . For any T ≥ 0 there is a probability p(1 − p)T that the world will last exactly T periods. In such a ³ ¢N ´ QT ¡ 1 case the consumption plan x yields a social utility ε × 1 − t=1 1 − εu(xt ) if T > 0 and zero if T = 0. The expected utility associated with x is therefore: ∞
1X W (x) = p(1 − p)T ε T =1
Ã
T Y ¡ ¢N 1− 1 − εu(xt )
! .
(3)
t=1
Since the social planner aims at maximizing expected utility, W (x) is a utility function representing the planner’s preferences over consumption plans. A consumption plan x will be preferred to an alternative plan x0 if and only if W (x) > W (x0 ) Now, let us examine the implicit social discount rate associated with such preferences. In a conventional fashion, we define the rate of time discounting by looking at how the marginal utility of consumption changes, when consumption is (locally) constant. Formally, for any x such that xt = xt+1 the rate of time discounting at time t is defined by:
ρ(x, t) = 1 −
∂W ∂xt+1 | ∂W xt =xt+1 ∂xt
(4)
When preferences are given by the utility function W (x) shown in equation (1) we have the following result: Proposition 2 Along a constant allocation path giving x to each living individ-
11
ual, the social rate of discount is
ρ(x) = p + (1 − p)(1 − [1 − εu(x)]N )
Proof. See Appendix 2. Note that when ε = 0, that is in the standard Utilitarian case, we find that the rate of discount equals the instantaneous probability of the world ending, which is consistent with the results of Dasgupta and Heal. From Proposition 2 we also have the immediate consequences: Corollary 1 Along constant allocation paths : 1) The rate of discount is greater than the hazard rate of the world disappearing whenever social preferences exhibit preferences for Catastrophe Avoidance (i.e. ρ > p whenever εu(x) > 0) 2) When the probability of the world ending is infinitesimally small the rate of discount approximately equals 1 − [1 − εu(x)]N . These two points deserve some comment. First we see that preference for Catastrophe Avoidance makes the social planner discount the future more heavily. The reason is that the planner wants to avoid the worst catastrophe in which: (1) only few individuals ever come to life; (2) those few individuals sacrifice most of their resources for the sake of future generations that will actually never exist. Secondly, even if the instantaneous probability of the world ending is very small, the rate of discount may be quite large. This may seem counterintuitive, as one might expect that when p is infinitesimally small, the planner would not exhibit time preferences, just as in the case where there is no uncertainty. There is however a simple intuition that explains why the rate of discount does not tend to zero when p tends to zero. The point is that when p tends to zero the expected 12
number of individuals that will ever exist (which equals
N ) p
tends to infinity. As a
consequence, the smaller p, the greater the loss when the world ends. Thus, when p tends to zero, the probability of a catastrophe occurring does tend to zero, but the magnitude of the catastrophe tends to infinity. Both factors compensate and lead the social planner to use a non-negligible social discount even if p is very small. Lastly, we may want to look at the size of the discount rate when p is infinitesimally small. At the limit p → 0, the discount rate is 1 − [1 − εu(x)]N . When ε is small, 1 − [1 − εu(x)]N ≈ N εu(x), and the discount rate is therefore approximately the product of the Index of Catastrophe Avoidance by the population size3 . Thus even if the planner has a very weak inclination for Catastrophe Avoidance, the social discount rate may be non-negligible if the population size is large enough.
6
Recovering the additive social welfare function
An indisputable drawback of our multiplicative representation is that it is no longer additive. For most applications, this is a source of substantial increase in complexity. Still, following the idea suggested in Bommier (2006), additivity can be recovered by considering the limit case where the difference in welfare between existing or not existing is assumed to be much greater than the difference in welfare between having a low or a high level of consumption. More precisely, assume that the xt (when different from d) remain in a bounded domain [xmin , xmax ] and 3
This simply reflects that the larger the population, the greater the catastrophe in case of the world termination.
13
that the function u is such that:
u(xt ) = 0 when xt = d u(xt ) = 1 + λv(xt ) when xt 6= d
where λ is very small and v is a bounded function over [xmin , xmax ]. In such a case the difference in welfare between existence and non-existence is approximately equal to 1, while the difference in welfare between having xmin or xmax equals λ (v(xmax ) − v(xmin )), which is assumed to be much smaller than 1. Consider now the limit case where λ is infinitesimally small. Note that in such a case, the Index of Catastrophe Avoidance is independent of x and equals − log(1 − ε). We now give the following result: Proposition 3 In the limit where λ → 0 the social planner’s preferences are represented by the social welfare function
Wε (x) =
∞ X
γ t N v(xt )
t=0
where γ = (1 − p)(1 − ε)N . Proof. See Appendix 3. We therefore obtain the standard additive representation, with a discount ¤ £ rate that equals p + (1 − p) 1 − (1 − ε)N . Thus, we end up with a formulation that is the same as that of Dasgupta and Heal, though with the fundamental difference that the rate of discount is now augmented by a factor that depends on the Index of Catastrophe Avoidance and the population size. Thus, there is no contradiction between assuming that the probability that the world disappears is very low, and that the rate of discount is significantly greater than zero. Nor is there any inequitable bias in favor of present generations. 14
Despite its simplicity, the additive approximation may be controversial, for we have to assume that the difference in welfare between existence and non-existence is much larger than the difference in welfare between possible lives. This is of course disputable. Blackorby, Bossert and Donaldson (1995) argue for example there are some states of extreme poverty that are, from the planner’s point of view, worse than non-existence. If one is reluctant to use the additive approximation, there is no other solution than to rely on the non-additive formula given by (3). However, as it emerges from Appendix 2 (see equation 9), W (x) can be written as:
³ ´Y 1X W (x) = (1 − p)T +1 1 − [1 − εu(xT +1 )]N [1 − εu(xτ )]N ε T =0 τ =1 +∞
T
or, alternatively: +∞ T ³ X ´ 1−pX W (x) = uε (xT +1 ) exp − vε,p (xτ ) ε T =0 τ =1
with uε (xt ) = 1 − εu(xt ) and vε,p (xt ) = −N log(1 − εu(xt )) − log(1 − p). We recognize here recursive preferences, as originally introduced by Uzawa (1968), in continuous time, and Epstein (1983), in discrete time. As is well known, and as is found in Proposition 2, those preferences are characterized by endogenous time discounting. Thus preference for Catastrophe Avoidance permits to reconcile intergenerational equity with endogenous time discounting. The simplifying assumption of exogenous time discounting can be seen as a limit case that corresponds to the additive approximation detailed above. Whether or not this limit case may be considered as relevant depends on how wide we think the welfare gap is between existence and non-existence.
15
7
Conclusion
We have extended the standard Utilitarian aggregation of preferences `a la Harsanyi to account for a possible planner’s taste for Catastrophe Avoidance. This was done by replacing the usual Pareto axiom by a weaker axiom. Basically, this axiom stipulates that there is no divergence between the social planner’s and the individuals’ preferences as long as uncorrelated risks are considered, but that some divergence may occur when correlated risks are at play. The axiom is reasonable if we consider that individual judgements have to be followed only when individual consequences are at play. This allows planners to express their own views on collective consequences. Preferences for Catastrophe Avoidance were found to play a key role when accounting for the probability that the world may end. We showed that an equitable social planner who has no pure time preference, but preferences for Catastrophe Avoidance, discounts the welfare of future generation with a rate that is greater than the instantaneous probability of the world coming to an end. More importantly this rate does not vanish when the instantaneous probability of the world ending tends towards zero. In other words, substantial time discounting does not necessarily reflect the planner’s lack of equity, or the planner’s belief that the world will soon end. There is a third source of social discounting that results from the combination of preference for Catastrophe Avoidance with the belief that there is indeed a positive (but, possibly very small) probability that the world will end. The end of the world is a very stylized representation of an event with durable consequences. Its key characteristics, for our analysis of time discounting, is that it durably and negatively impacts individuals’ utilities and marginal utilities4 . In fact, it can be shown that, when the planner exhibits preferences for Catastrophe Avoidance, the planner’s discount rate increases with the likelihood of an event having these characteristics occurring. Natural extensions of this paper therefore 4
When the world ends, the utilities and marginal utilities of future individuals are irreversibly set to zero.
16
involve considering less caricatured risks, such as the possibility of an ecological catastrophe, and see how they may affect time discounting.
Appendices Appendix 1: Proof of Proposition 1 It is straightforward to check that if the planner’s preferences are represented by the multiplicative or additive utility function shown in (1) and (2) they satisfy (RP), (IUI) and (A). We will therefore focus on showing that (RP), (IUI) and (A) imply these particular specifications. For simplicity, we will slightly change the notation along this proof. Instead of identifying individuals by a pair (i, t), with t and i ∈ [1, N ], we will use a single index k = i + N t. This way, xk denotes in fact xit where t is the integer part of
k N
and i = k − N t. An element of x ∈ χ is then just a sequence x =
(x1 , x2 , · · · , xk , · · · ) such that xk = d for all k greater than a finite number kx . The Bernoulli utility function that represents the planner’s preferences is:
U (x1 , x2 , · · · , xk , · · · )
Consider a given c0 ∈ K. From (RP):
(c0 , d, d, · · · ) ºS (d, d, d, · · · ) ⇔ c0 ºI d
Thus, by normalization of U it can be assumed that:
U (d, d, d, · · · ) = u(d) = 0 U (c0 , d, d, · · · ) = u(c0 )
17
(5) (6)
ˆ and Q ˆ 0 , four probability measures in Q such that Q(1) = Consider now Q, Q0 , Q ˆ (1) = Q ˆ 0 = δx1 , QI\(1) = Q ˆ I\(1) and Q0 ˆ0 Q0(1) = δd , Q (1) I\(1) = QI\(1) . ˆ ºS Q ˆ 0 . Thus: According to the (IUI) axiom, Q ºS Q0 ⇔ Q Z
Z U (d, x2 , · · · )dQI\(1) (x2 , · · · )
≥
χ
χ
⇐⇒
Z U (x1 , x2 , · · · )dQI\(1) (x2 , · · · )
Z
≥
χ
U (d, x2 , · · · )dQ0I\(1) (x2 , · · · )
χ
U (x1 , x2 , · · · )dQ0I\(1) (x2 , · · · )
That implies that Vd (x2 , x3 , · · · ) = U (d, x2 , x3 , · · · ) and Vx1 (x2 , x3 , · · · · · · ) = U (x1 , x2 , x3 , · · · ) are two Bernoulli utility functions representing the same preference orderings on the probability measures on χ. The function Vx1 must therefore be obtained from Vd by a positive affine transformation. In other words, for any x1 ∈ X there exits v(x1 ) and w(x1 ) > 0 such that
U (x1 , x2 , x3 , · · · ) = v(x1 ) + w(x1 ) × U (d, x2 , x3 , · · · )
The normalization conditions (5) and (6) imply that:
v(d) = 0 and v(c0 ) = u(c0 )
(7)
Now remember that xk = d when k is larger than kx and that by the (A) axiom, U (d, x2 , x3 · · · xkx , d, d, · · · ) = U (x2 , x3 , · · · , xkx , d, d, d · · · ), so that, for any x ∈ X: U (x1 , x2 , x3 · · · ) = v(x1 ) + w(x1 )U (x2 , x3 , · · · ) Iterating once we obtain
U (x1 , x2 , x3 , · · · , xkx , d, d, d · · · ) = v(x1 ) + w(x1 )v(x2 ) +w(x1 )w(x2 )U (x3 , · · · , xkx , d, d, d · · · )
18
By repeating the iteration kx times we get:
U (x1 , x2 , · · · , xkx , d, d, d · · · ) =
kx X
v(xk )
k=1
k−1 Y
w(xl ) +
l=1
kx Y
w(xl )U (xκ+1 , xκ+2 , ...)
l=1
Using (5) and (6) we eventually obtain:
U (x1 , x2 , · · · , xkx , d, d, d · · · ) =
∞ X
v(xk )
k=1
k−1 Y
w(xl )
l=1
Now, remark that the (A) axiom imposes that U (x, y, d, · · · ) = U (y, x, d, · · · ) so that for any x, y ∈ X we must have:
v(x) + w(x)v(y) = v(y) + w(y)v(x)
(8)
or equivalently (whenever x 6= d and y 6= d): 1 − w(x) 1 − w(y) = v(x) v(y) The ratio
1−w(x) v(x)
is therefore constant for any x ∈ K. Let us denote it by ε.
Applying (8) to x = d it follows that w(d) = 1. Therefore, we have w(x) = 1 − εv(x) for all x ∈ X. Since w(x) > 0 we must have εv(x) < 1. First remark that if ε = 0, we obtain the additive Bernoulli function:
U (x1 , x2 · · · , · · · ) =
+∞ X k=1
19
v(xk )
When ε 6= 0 we compute:
1 − εU (x1 , x2 , · · · ) = 1 − = 1+ = 1+
kx X
εv(xk )
k=1 kx X
k−1 Y
(1 − εv(xl ))
l=1
(1 − εv(xk ))
k=1 kx Y k X
k−1 Y
kx k−1 X Y
l=1
k=1 l=1
(1 − εv(xl )) −
(1 − εv(xl )) −
k=1 l=1
=
kx Y
kx k−1 X Y
(1 − εv(xl ))
(1 − εv(xl ))
k=1 l=1
(1 − εv(xl )) =
l=1
+∞ Y
(1 − εv(xl ))
l=1
We eventually get the multiplicative specification: 1 U (x1 , x2 , · · · ) = ε
à 1−
+∞ Y
! (1 − εv(xl ))
l=1
It only remains to check that the functions v(.) and u(.) are identical. Since, from (7), v(d) = u(d) and v(c0 ) = u(c0 ), we only need to show that v(.) is a Bernoulli utility function that represents individuals’ preferences. Consider the ˆ such that Q(k) = Q ˆ (k) = δx , ∀k 6= 1. Uncertainty thus only measures Q and Q k bears on individual 1. From the (RP) axiom:
ˆ ⇔ Q(1) ºI Q ˆ (1) Q ºS Q ˆ if and only if But Q ºS Q
R X
U (x1 , x2 , · · · )dQ(1) (x1 ) ≥
R X
ˆ (1) (x1 ). U (x1 , x2 , · · · )dQ
Since Z
Ã+∞ ! µ ¶ Z Y¡ ¢ 1 1 U (x1 , x2 , · · · )dQ(1) (x1 ) = − − v(x1 )dQ(1) (x1 ) 1 − εv(xk ) × ε ε X X k=2
ˆ if and only if we obtain that Q ºS Q
R
v(x1 )dQ(1) (x1 ) ≥ X
20
R X
ˆ (1) (x1 ). It v(x1 )dQ
follows that: Z Q(1)
Z
ˆ (1) ⇔ º Q I
ˆ (,1) (x1 ) v(x1 )dQ
v(x1 )dQ(1) (x1 ) ≥ X
X
which implies that v(.) is indeed a Bernoulli utility function that represents individual preferences.
Appendix 2: Proof of Proposition 2 For any x and T > 0 denote 1 U (x, T ) = × ε
Ã
T Y
1−
! [1 − εu(xτ )]N
τ =1
and, for T = 0, U (x, 0) = 0 We have, for any T ≥ 0: T ´ Y 1 ³ N U (x, T + 1) − U (x, T ) = × 1 − [1 − εu(xT +1 )] × [1 − εu(xτ )]N , ε τ =1
Now write W (x) = = =
P+∞ T =0
p(1 − p)T U (x, T ) =
P+∞
T =0 (1
− p)T U (x, T ) −
P+∞
T T =0 (1 − p) U (x, T ) −
P+∞ £ T =0
P+∞
T =0 (1
P+∞
T =0 (1
¤ (1 − p)T − (1 − p)(1 − p)T U (x, T )
− p)T +1 U (x, T ) − p)T +1 U (x, T + 1)
P+∞
− p)T +1 [U (x, T + 1) − U (x, T )] ³ ´ P+∞ N N QT 1 T +1 = ε T =0 (1 − p) 1 − [1 − εu(xT +1 )] τ =1 [1 − εu(xτ )] +
T =0 (1
(9)
21
Now, we can easily compute the partial derivative of W (x) with respect to xt : t Y ∂W (x) −1 0 t = N u (xt ) (1 − εu(xt )) (1 − p) [1 − εu(xτ )]N ∂xt τ =1 0
−N u (xt ) [1 − εu(xt )]
−1
+∞ X
T +1
(1 − p)
³
N
1 − [1 − εu(xT +1 )]
T ´Y
[1 − εu(xτ )]N
τ =1
T =t
Along a constant allocation path, xt = x, ∀t, denoting u(x) and u0 (x) by u and u0 , the above expression reduces to: ∂W (x) = N u0 (1 − p)t (1 − εu)N t−1 ∂xt +∞ ³ ´ X −N u0 (1 − p)T +1 1 − [1 − εu]N [1 − εu]N T −1 T =t
Ã
= N u0 (1 − p)t (1 − εu)N t−1
+∞ ³ ´ X N T −t+1 1− (1 − p) 1 − [1 − εu] [1 − εu]N (T −t)
à = N u0 (1 − p)t (1 − εu)N t−1
1−
T =t +∞ X
³
´
(1 − p)τ +1 1 − [1 − εu]N [1 − εu]N τ
τ =0
We eventually obtain that:
ρ(x) = 1 −
∂W (x) ∂xt+1 ∂W (x) ∂xt
= 1 − (1 − p)(1 − εu)N = p + (1 − p)[1 − (1 − εu)N ]
Appendix 3: Proof of Proposition 3 The planner’s preferences over consumption plans is represented by: +∞
1X W (x) = p(1 − p)T ε T =1
Ã
22
1−
T Y τ =1
! [1 − εu(xτ )]N
!
!
Substitute u(x) = 1 + λv(x) in the above formula and write that:
W (x) ' W (x)|λ=0 + λ
∂W |λ=0 ∂λ
to obtain
W (x) '
+∞ ³ ´ 1X p(1 − p)T 1 − [1 − ε]N T ε T =1 Ã T ! +∞ X X N (T −1) T +λ p(1 − p) [1 − ε] N v(xt ) t=1
T =1
The first term is a constant and does not affect preferences. Switching the summation signs, the second term equals
λ
+∞ X t=1
N v(xt )
+∞ X
p(1 − p)T [1 − ε]N (T −1)
T =t
+∞ +∞ X X £ ¤T = λp [1 − ε]−N N v(xt ) (1 − p)(1 − ε)N t=1
T =t
−N
=
λp [1 − ε] 1 − (1 − p)(1 − ε)N
where γ = (1 − p)(1 − ε)N . The term
+∞ X
γ t N v(xt )
t=1
λp[1−ε]−N 1−(1−p)(1−ε)N
is a positive multiplicative
factor which does not affect preferences. We therefore see that the planner’s preferences can be represented by +∞ X
γ t N v(xt ).
t=1
23
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