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Discounting an uncertain future
Journal of Public Economics 85 (2002) 149–166 www.elsevier.com / locate / econbase
Discounting an uncertain future Christian Gollier Universite´ de Toulouse and Institut Universitaire de France, GREMAQ , Place Anatole France, 31042 Toulouse Cedex, France Received 24 March 1999; received in revised form 16 July 2000; accepted 30 October 2000
Abstract The objective of this paper is to determine the socially optimal discount rate for public investment projects that entail costs and benefits in the very long run. We suppose that there is an exogenous process for the growth of consumption per capita, which is stochastic. We first evaluate the determinants of the discount rate for a specific horizon when the representative agent has a recursive utility. We then explore the influence of the time horizon in the expected utility model. Under various conditions on preferences, as positive prudence, decreasing relative risk aversion or decreasing absolute risk aversion, we prove that (1) the fact that growth is uncertain reduces the efficient discount rate at any horizon, and that (2) this discount rate should be smaller for more distant futures. We characterize the asymptotic value of the discount rate. 2002 Elsevier Science B.V. All rights reserved. Keywords: Discounting; Uncertain growth; Log-supermodularity; Prudence; Kreps–Porteus preference JEL classification: D81; D91; Q25; Q28
1. Introduction Much of life is made of investments. Costly actions are taken today in the prospect of future benefits. In the presence of efficient financial markets, the investment decision process is based on the classical concept of the Net Present Value (NPV). The argument sustaining this decision rule is based on arbitrage. E-mail address: [email protected] (C. Gollier). 0047-2727 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0047-2727( 01 )00079-2
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Instead of undertaking the planned investment, one could invest in the financial markets. As a consequence, the return of the planned investment should yield at least the risk free rate. For standard investment projects, this rule is equivalent to having a positive NPV. If financial markets are frictionless, the use of the observed risk free rate to discount public investment projects leads to a socially efficient level of investment. The analysis is less easy to perform when benefits and costs of the set of current potential actions are expected to last in the long run. The carbon dioxide that one emits today will not be recycled for a couple of centuries, yielding long term costs like global warming. Some nuclear wastes like plutonium have half-life in the tens of thousands years. Obviously, financial markets are not very helpful to provide a guideline for investing in technologies that prevent this kind of long-lasting risks to occur. Liquid financial instruments with such large durations do not exist. For the sake of comparison, US Treasury Bonds have time horizons that do not exceed 30 years. We must thus rely on the use of an economic model to value the distant future. ¨ The name of Bohm-Bawerk and Fisher are intimately related to these questions. The first reason offered by these authors is purely psychological. Namely, agents may have a pure preference for the present, i.e. they are impatient. The second reason to discount the future is related to a wealth effect. We expect that the quantity of available consumption goods will increase over time. After all, in the western world at least, we experienced an uninterrupted growth during the two last centuries. Given decreasing marginal utility of consumption, an investment which gives one unit of the consumption good in the future in exchange for one unit of the consumption good in the present should not be acceptable. Investing for the future in a growing economy will increase consumption inequalities over time. Since agents have preferences for the smoothing of consumption over time, this investment should be implemented only if its rate of return is large enough to compensate for this negative impact on welfare. The larger the growth rate of the economy, the larger the socially efficient discount rate. A problem arises with this wealth effect if the growth rate is not known with certainty. Estimating the growth rate for the coming year is already a difficult task. No doubt, any estimation of growth for the next century / millennium is subject to potentially enormous errors. The history of the western world before the industrial revolution is full of important economic slumps, as the one due to the invasion of the Roman Empire, or the one due to the Black Death during the middle ages. The recent debate on the notion of a sustainable growth is an illustration of the degree of uncertainty we face to think about the future of society. Some will argue that the effects of the improvements in information technology have yet to be realized, and the world faces a period of more rapid growth. On the contrary, those who emphasize the effects of natural resource scarcity will see lower growth rates in the future. Some even suggest a negative growth of the GNP per head in the future, due to the deterioration of the environment, population growth and decreasing returns to scale. They claim that the wealth effect goes the other
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direction, so that everything should be made to improve the future. This uncertainty at least casts some doubt on the relevance of the wealth effect to justify the use of a large discount rate. In this paper, we provide an analysis of the effect of the uncertainty on growth on the socially optimal discount factor. Instrumental to this analysis is the concept of prudence that has been formalized by Kimball (1990). An agent is prudent if his willingness to save increases in the face of an increase in his future income risk. Technically, an agent is prudent if the third derivative of his utility function is positive. As shown in this paper, prudence justifies taking a discount rate that is less than the one that would have been obtained by assuming a certain growth. The magnitude of the effect depends upon the degree of prudence and the degree of uncertainty on growth. This analysis is provided in Section 2. Some numerical simulations are performed in Section 3. In Section 4, we examine the relationship between the time horizon and the socially optimal discount rate. In order to counterbalance the exponential effect of discounting, several authors suggested to take a smaller discount rate per period to discount more distant cash flows. Harvey (1994) discusses the reasonableness of such a strategy. Our aim is to determine whether this strategy is socially efficient. An intuitive argument would rely on the increased risk of longer horizons due to the accumulation of period to period growth risks. The longer the horizon is, the larger is the uncertainty on future wealth, the smaller should the discount rate be. However, there is a potentially counterbalancing wealth effect: the expected level of consumption is increasing with time. Thus, there is an argument to take care less of more distant horizons, thereby selecting a larger discount rate for such horizons. Which of the precautionary effect or of the wealth effect dominates the other? The main result of the paper is to show that the precautionary effect dominates the wealth effect when relative risk aversion is decreasing and when there is no risk of recession. In such a case, it is socially efficient to reduce the discount rate per year for more distant horizons. This section is related to the literature on the term structure of interest rate. A reinterpretation of our work is indeed that, if the growth rate of consumption is stationary over time, the yield curve is increasing or decreasing depending upon whether relative risk aversion is increasing or decreasing. Section 5 is devoted to the analysis of the asymptotic value of the discount rate per period. In Section 6, we compare our results to the existing literature and, in particular, to a recent paper by Martin Weitzman (1998) who also proves that the discount rate should be decreasing with time horizon. We show that the two approaches are complementary.
2. The socially optimal discount rate In this section, we consider a model with one period and two dates, t 5 0,1. We consider the case where cost–benefit analysis can be applied. That is, we assume that all costs and benefits can be measured in one dimension, i.e. money. People
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consume c at date t 5 0, and c˜ 1 at date t 5 1. Seen from t 5 0, c˜ 1 is uncertain. Thus, in our model, we have some form of dynamic uncertainty. In order to disentangle the attitude towards time and the attitude towards risk, we assume that the planner has Kreps–Porteus–Selden preferences. Kreps and Porteus (1978) and Selden (1979) proposed two equivalent preference functionals that allow for disentangling these two concepts. The simplest way of presenting this family of preference functionals is to first define the certainty equivalent consumption m at date t 5 1 as is standard under expected utility: v(m) 5 Ev(c˜ 1 )
(1)
Then the social planner evaluates the intertemporal welfare as: u(c) 1 b u(m)
(2)
We assume that u and v are increasing and weakly concave in their argument. In words, the planner first computes the certainty equivalent consumption of the future generation by using its aversion to future risks characterized by the concavity of v. Then the planner aggregates consumptions of the two generations in a nonlinear way. The concavity of u characterizes the agent’s aversion to consumption fluctuations over time. Parameter b is the discount factor of utility, representing the pure preference for the present (‘pure discounting’). The intertemporal time-separable expected utility model is obtained by taking u ; v, which yields a classical objective function v(c) 1 b Ev(c˜ 1 ). Another particular case is when u is linear. In that case, the social welfare function is a weighted sum of the certainty equivalent consumption, i.e. certainty equivalent consumptions at different dates are perfect substitutes. Kimball and Weil (1992) examine the saving decision of an agent having Kreps–Porteus–Selden preferences. This economy has an exogenous consumption process. It is similar to a tree economy a` la Lucas: each agent is endowed with a tree which generates a number of fruits c and c˜ 1 , respectively, at date t 5 0 and t 5 1. There is no possibility to plant new trees. The only possibility is to invest in some prevention effort to improve future individual crops. Consider a marginal project which costs a small x (Dc 5 2 x) at date t 5 0 and which brings a sure benefit (1 1 r) x at date t 5 1.1 This project can be to reduce the pollution of the air, or the preservation of animal species. Should society invest in this project? This sure benefit increases the certainty equivalent consumption at date t 5 1 by:
1 In many instances, the future benefits are to reduce risks borne by future generations. When the benefits are uncertain, but independent of the growth rate of the economy, the same model can be used with 1 1 r denoting the certainty equivalent benefit at date 1. Notice that the presence of the background uncertainty on the future wealth level affects this certainty equivalent. Under risk vulnerability (see Gollier and Pratt (1996)), a condition on preferences that is satisfied by most familiar utility functions, the presence of an uncertain future wealth increases the certainty equivalent benefit of a risk-reducing investment. This provides another argument to select a smaller discount rate.
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≠m ] ≠x
U
x 50
Ev9(c˜ 1 ) 5 ]]] (1 1 r) v9(m)
153
(3)
The social welfare function (2) is increased by this marginal investment if: ≠m 2 u9(c) 1 b u9(m) ] u x 50 $ 0 ≠x or, equivalently, if: u9(c) v9(m) r $ d 5 def ]] ]]] 2 1 b u9(m) Ev9(c˜ 1 )
(4)
where d is the socially efficient discount rate. Notice that d would be the equilibrium risk-free rate in this exchange economy. The socially optimal d depends upon social values ( b ), together with the attitude towards risk and the elasticity of intertemporal substitution, as is apparent in Eq. (4). We want to evaluate the effect of the uncertain growth on the socially optimal discount rate. To do this, we compare d to d c , the socially optimal discount rate in an economy with a sure consumption level equaling Ec˜ 1 at date t 5 1. Without uncertainty, risk aversion does not matter, and the certainty equivalent is just m 5 Ec˜ 1 . u9(c) d c []]] 2 1 b u9(Ec˜ 1 ))
(5)
The socially optimal discount rate d is smaller than d c if and only if: u9(c) v9(m) u9(c) d # d c ⇔]] ]]] # ]]] u9(m) Ev9(c˜ 1 ) u9(Ec˜ 1 )
(6)
We first examine the classical case of time-separable expected utility preferences with u ; v. In this case, we can rewrite condition (6) as:
d # d c ⇔Ev9(c˜ 1 ) $ v9(Ec˜ 1 )
(7)
The latter condition holds for any c and for any distribution of c˜ 1 if and only if v9 is convex. The convexity of marginal utility is a well-known condition since ` and Modigliani (1972). It is a necessary and sufficient Leland (1968) and Dreze condition for an increase in future risk to increase (precautionary) savings. Kimball (1990) coined the term ‘prudent’ to characterize individuals who behave in this way. Proposition 1. Suppose that u ; v: preferences are time separable. Then, the uncertainty affecting the growth rate of the economy should induce Society to select a smaller (resp. greater) discount rate if agents living in the future are prudent (resp. imprudent).
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A prudent society should care more about the future when it is more uncertain. This is induced by reducing the discount rate, which implies that more investment favoring the future become profitable, viz. have a positive NPV. Matters are more complex under Kreps–Porteus preferences. We obtained the following result: Proposition 2. Under Kreps–Porteus preferences, the uncertainty affecting the growth rate of the economy should induce society to select a smaller discount rate if agents are decreasingly absolute risk-averse. Moreover, this condition is necessary if function u is linear. Proof. We have to show that condition (6) holds under decreasing absolute risk aversion. Observe first that m is less than Ec˜ 1 , by risk aversion. It implies that: u9(m) $ u9(Ec˜ 1 )
(8)
It remains to prove that v9(m) is less than Ev9(c˜ 1 ) under decreasing absolute risk aversion. Since m is defined as v(m) 5 Ev(c˜ 1 ), this is the case if and only if 2 v9 is more concave than v. But this is precisely the definition of decreasing absolute risk aversion: 2 v-(.) /v0(c) $ 2 v0(c) /v9(c) for all c. When u is linear, condition (8) is an equality, and d # d c if and only if v exhibits decreasing absolute risk aversion. h If we compare this Proposition with Proposition 1, we see that we end up here with a stronger condition to guarantee that growth uncertainty affects the discount rate negatively. Indeed, decreasing absolute risk aversion of v is more demanding than prudence. This is not a surprise, since Kreps–Porteus preferences are a more flexible preferences form than the time-separable expected utility model used in Proposition 1. This is the weakest sufficient condition that one can get without restricting the time aggregator u. The necessity of decreasing absolute risk aversion can be explained as follows. In the extreme case where u is linear, the efficient discount rate in excess of the rate of pure preference for the present is equal to the inverse of the sensitivity of the certainty equivalent to a change in wealth. As suggested by the intuition, the risk on future consumption increases this sensitivity under decreasing absolute risk aversion: without risk, it is equal to unity, whereas it is larger than unity when c˜ 1 is risky, under DARA. Indeed, one must take into account of the fact that the risk premium is decreasing with wealth in that case.
3. Numerical estimation of the efficient discount rate for the short term We can estimate the socially efficient discount rate in an uncertain environment by assuming that the support of growth rate of consumption is in a small neighborhood of zero. This is compatible only with a relatively short time horizon,
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i.e. with b → 1. In the next section, we will determine whether socially efficient discount rates for longer time horizons should be smaller than this benchmark. ˜ with y˜ small and m 2 5 (Ey˜ )2 < Ey˜ 2 . s 2 . The Thus, suppose that c˜ 1 5 c 1 y, Arrow–Pratt approximation tells us that: m . c 1 m 2 0.5s 2 A v
(9)
where A f is the absolute risk aversion of function f evaluated at c. Using second-order Taylor approximations, we get in turn that: u9(c) ]] . 1 1 m A u 2 0.5s 2 A u A v u9(m)
(10)
v9(m) ]]] 5 1 1 0.5s 2 A 9v Ev9(c˜ 1 )
(11)
and
where A v9 is the derivative of A v evaluated at c. This second equality represents the sensitivity of the certainty equivalent to changes in wealth. It is negative when absolute risk aversion is decreasing. We finally obtain that:
F G
1 d . ] 2 1 1fEc˜ 1 2 cg A u 1 0.5 Var(c˜ 1 )(A v9 2 A u A v ) b
(12)
We recognize in this equation the three determinants of the discount rate: the first bracketed term in the left-hand side of condition (12) is the rate of pure preference for the present. The second term in the sum is the pure wealth effect. It is positive when (1) the representative agent is averse to consumption fluctuations over time, and (2) the expected growth of consumption is positive. A larger expected growth of the economy should induce agents to care less about the future, by increasing the discount rate. The third term in the sum measures the effect of uncertainty on the discount rate. In accordance to Proposition 2, this term is unambiguously negative when the absolute risk aversion of v is decreasing. Notice that the necessity of A 9v # 0 can be derived directly from (12) by considering temporal preferences with A u 5 0. Notice also that, A 9v 5 A v (A v 2 Pv ), where Pv 5 2 v-(c) / v0(c) is absolute prudence. Thus, in the special case of time separable preferences (u ; v), condition (12) becomes:
F G
1 d . ] 2 1 1fEc˜ 1 2 cg A 2 0.5 Var(c˜ 1 )AP b
(13)
This condition is similar to what was obtained by Hansen and Singleton (1983). It confirms that positive prudence is necessary for the precautionary effect to reduce the socially efficient discount rate in the classical time-separable utility model (Proposition 1). Let us alternatively assume that u9(c) 5 c 2gu and v9(c) 5 c 2gv , for some gu $
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Table 1 The socially efficient discount rate per year for short horizons
gu 5 1 gu 5 5 gu 5 10
gv 5 0
gv 5 1
gv 5 5
gv 5 10
1.8% 9.0% 18.0%
1.7% 8.6% 17.3%
1.1% 7.1% 14.4%
0.5% 5.1% 10.9%
and gv $ 0. Let also g˜ 5 (c˜ 1 /c) 2 1 denote the growth rate of consumption. Under this specification, condition (12) can be rewritten as:
F G
1 d . ] 2 1 1 gu Eg˜ 2 0.5gv (1 1 gu ) Var(g˜ ) b
(14)
Kocherlakota (1996), using United States annual data over the period 1889–1978, estimated the mean and the standard deviation of the growth of consumption per capita to, respectively, 1.8% per year and 3.6% per year. If we use these values of the parameters together with b 5 1, approximation (14) gives us socially efficient discount rates as documented in Table 1. Barsky et al. (1997) estimated gu and gv based on survey responses to hypothetical situations dealing either with consumption smoothing under certainty (for gu ) or with static risky choices (for gv ). They estimated gv , the degree of relative risk aversion, to around 4, which is compatible to the range of what is considered as reasonable by economists. The measurement of the aversion to consumption fluctuation over time, gu , is more problematic. Barsky et al. (1997) obtained an estimation around gu 5 5, whereas Epstein and Zin (1991) found a value ranging from 1.25 to 5. Based on these estimates of individual preferences, we recommend using a discount rate for the short term in a range between 1.5 and 7%, with a preference for the mid-point around 4.5%.
4. Compound growth rates When considering environmental risks, we are not only interested in discounting benefits and costs occurring in the next few years, but also for much longer time horizons. Obviously, the uncertainty prevailing on the level of consumption per head prevailing t periods from now is an increasing function of t.2 In this section, we address the question of how this increased uncertainty affects the optimal discount rate per period. In fact, we examine whether the yield curve should be 2
As indicated by Samuelson (1963), compounding risks borne at different points in time, even if they are independent, is not a diversification device.
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decreasing, increasing or flat. We do this by considering the special case of expected utility with a time separable utility function. Empirical evidence suggests that the discount rate used by individuals to value the future is a decreasing function of the time horizon. Lowenstein and Thaler (1989) for example obtained discount rates ranging around 15% for the short run (less than 5 years), and then dropping to a level as low as 2% for the long run (100 years). Can this be due to the increase of uncertainty about one’s future wealth, rather than to a deviation of the standard model with exponential discounting of utilities? 3 In accordance with this empirical evidence, our intuition is that the interaction between intertemporal growth risks is an aggravating factor. A longer time horizon should induce a smaller discount rate per period. Because the uncertainty affecting future consumption is increasing with the time horizon, the precautionary argument applies in favor of a reduction of the discount rate per year for more distant futures under prudence (Proposition 1). But, on the other side, more distant generations will enjoy a larger level of consumption, in expectation. This wealth effect goes the opposite direction than the precautionary effect. Thus, the global effect of time horizon on the efficient discount rate per year is a priori ambiguous. Let us consider a two-period model with three dates t 5 0, 1, 2. The growth rate of consumption from t 5 0 to t 5 1 is g˜ 1 , whereas the growth rate from t 5 1 to t 5 2 is g˜ 2 . We assume that random variables g˜ 1 and g˜ 2 are independently and ˜ We assume that the representative identically distributed. They are distributed as g. agent has a time-separable utility function v. Therefore, the socially optimal rate d1 to discount costs and benefits occurring at date t 5 1 is as in Section 2 with u ; v: v9(c) 1 1 d1 5 ]]]]] b Ev9(c(1 1 g˜ 1 ))
(15)
The growth rate over the two periods is (1 1 g˜ 1 )(1 1 g˜ 2 ) 2 1. The socially optimal discount rate per period d2 for a cash flow occurring at t 5 2 should be such that: v9(c) (1 1 d2 )2 5 ]]]]]]] 2 b Ev9(c(1 1 g˜ 1 )(1 1 g˜ 2 ))
(16)
Usually, d2 is not equal to d1 , except in the case of the isoelastic utility function as we show now. Suppose that v9(c) 5 c 2 f , for some constant relative risk aversion f . 0. Conditions (15) and (16) yield: (1 1 d1 )2
5 [ b E(1 1 g˜ 1 )2 f ] 22 22 2 5 b [E(1 1 g˜ 1 ) f E(1 1 g2 )2 f ] 21 5b
3
22
[E((1 1 g˜ 1 )(1 1 g2 ))
2 f 21
]
5 (1 1 d2 )
(17) 2
Laibson (1997) discusses the effect of non-exponential discounting of utilities on individual strategic behaviors.
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This implies that d2 5 d1 . The wealth effect of a longer time horizon just compensates the precautionary effect when relative risk aversion is constant. More generally, using conditions (16) and (18), d2 is less than d1 if and only if: v9(c) Ev9(c(1 1 g˜ 1 )(1 1 g˜ 2 )) $ Ev9(c(1 1 g˜ 1 )) Ev9(c(1 1 g˜ 2 ))
(18)
In the following Proposition, we present a simple sufficient condition to satisfy this inequality. Proposition 3. Suppose that the growth rate of consumption is nonnegative almost surely, and that v is three time differentiable. The long term discount rate is smaller (resp. larger) than the short term one if relative risk aversion is decreasing (resp. increasing). Proof. Let function h from R 21 to R be defined as h(x 1 , x 2 ) 5 v9(cx 1 x 2 ). Suppose that this function be log supermodular.4 This means that: h(min(x 1 , x 91 ), min(x 2 , x 92 )) h(max(x 1 , x 19 ), max(x 2 , x 29 )) $ h(x 1 , x 2 ) h(x 19 , x 29 ) (19) 2 1
for all (x 1 , x 2 ) and (x 19 , x 29 ) in R . Taking x 1 5 x 29 5 1, the log supermodularity of h implies that: h(1, 1) h(x 91 , x 2 ) $ h(1, x 2 ) h(x 19 , 1)
(20)
for all x 91 and x 2 that are both larger than 1. This inequality is equivalent to: v9(c) v9(c(1 1 g1 )(1 1 g2 )) $ v9(c(1 1 g1 )) v9(c(1 1 g2 ))
(21)
for all g1 5 x 91 2 1 and g2 5 x 2 2 1 that are both positive. Suppose now that the growth rate g˜ t per period is positive almost surely. Then, the log supermodularity of h is sufficient to guarantee that condition (21) holds almost everywhere. Taking the expectation of this inequality directly yields inequality (18), which implies in turn that d2 is less than d1 . If the utility function is three time differentiable, the log supermodularity of h also means that the cross derivative of log h is positive. It is easily seen that this is equivalent to require that relative risk aversion is decreasing (DRRA). h Decreasing relative risk aversion is compatible with the well-documented observation that the relative share of wealth invested in risky assets is an increasing function of wealth. Kessler and Wolff (1991) for example show that the portfolios of US households with low wealth contains a disproportionately large share of risk free assets. Measuring by wealth, over 80% of the lowest quintile’s portfolio was 4
For an analysis of the usefulness of this concept in the economics of uncertainty, see Jewitt (1988), Athey (1998) and Gollier (2000b).
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in liquid assets, whereas the highest quintile held less than 15% in such assets. Guiso et al. (1996), using cross-section of Italian households, observed portfolio compositions which are also compatible with decreasing relative risk aversion. Ogaki and Zhang (1999) tested whether relative risk aversion is decreasing or increasing from various consumption data in developing countries. They obtained strong evidences that relative risk aversion is decreasing. Thus one should select a smaller rate to discount far distant benefits than the rate to discount benefits realized in the near future. Growth risks are mutually aggravating. To illustrate, let us consider again the scenario in which the growth rate of consumption per capita is either 0 or 75% with, respectively, probability 0.36 and 0.63 in each period of 20 years. These probabilistic scenarios yield the same two first moments than those of the historical data for France over the period 1870–1990, as documented by Maddison (1991). We consider the case of power utility functions: v9(z) 5 (z 2 k)2g, z $ k
(22)
for some g $ 0, and k which can be interpreted as some minimum subsistence level if it is positive. For those functions, prudence is positive, and relative risk aversion equals: 2 cv0(c) gc f (c) 5 ]]] 5 ]] c2k v9(c) When parameter k is positive, relative risk aversion is decreasing from plus infinity at c 5 k to g as c tends to infinity. We hereafter examine the special case where g 5 1. We also normalize consumption at date t 5 0 to unity. We also take b 5 1. We computed the socially optimal discount rate for horizons up to 45 periods, using different values of the parameter k. The results are drawn in Fig. 1. As an illustration, consider the case k 5 0.9 for which relative risk aversion equals 10 at current consumption and goes to 1 as consumption goes to infinity. The discount rate per year that should be used to value benefits arising within the next twenty years equals 4.14%. For benefits arising in 800 years from today, it should be reduced to less than 2%. An important requirement of the above Proposition is that the growth rate of consumption per capita be nonnegative almost surely. When there is a risk of recession in an economy with a positive expected growth, it is not true anymore that DRRA is sufficient for the yield curve to be strictly decreasing. Let us illustrate this point by the following example, using One-Switch utility functions introduced by Bell (1988). Take v9(z) 5 a 1 z 2b with a . 0 and b . 0. It yields 2 zv0(z) /v9(z) 5 bfaz 2b 1 1g 21 , which is decreasing in z. In addition, take a 5 b 5 1 and g˜ t | (250%, 1 / 3; 1 100%, 2 / 3). In such a situation, straightforward computations generate d1 5 d2 5 0: the yield curve is flat in spite of DRRA! This result does not contradict the above Proposition, since g˜ t is not positive with probability 1. Thus, extending the analysis to economies with a risk of a recession
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Fig. 1. The socially efficient discount rate per year as a function of the time horizon. The upper curve is for k 5 0.9, whereas the horizontal line is for k 5 0.
requires restricting the set of DRRA utility functions to guarantee that the socially optimal discount rate be decreasing with time horizon. Gollier (2000a) examines this question in more details.
5. The asymptotic value of the discount rate It is interesting to determine the asymptotic value of the per-period discount rate when time horizon recedes to infinity. The rate dt to discount the cash flows at date t is implicitly defined by the following equation: v9(c) (1 1 dt )t 5 ]]]]]] t b t Ev9 c (1 1 g˜ i )
SP
i 51
D
(23)
where g˜ i is the growth rate per head between date i 2 1 and i. Suppose first that, at each period i, there is a positive probability that a recession ( gi , 0) occurs. Suppose moreover that there exists a positive k such that lim z →k v9(z) 5 1 `. In that case, the denominator in the right-hand side of Eq. (23) tends to infinity for large t. This is possible only if dt becomes negative. We conclude that the risk of repeated recessions over long periods eventually induces the selection of a nonpositive rate to discount far-distant futures. The low probability of such events is overweighted by the large marginal utility of wealth in these bad states. We now turn to the more interesting case where there is no risk of recession (Prfg˜ # 0g 5 0). Remember that f (z) 5 2 zv0(z) /v9(z) denote the relative risk aversion function, with f` 5 lim z →` f (z). The following Proposition shows that
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convergence of the socially efficient discount rate as the horizon becomes distant occurs if f` exists. ˜ g˜ 1 , g˜ 2 , . . . are i.i.d. variables whose support is in Proposition 4. Suppose that g, R 1 . Suppose also that f` 5 lim z →` 2 zv0(z) /v9(z) exists. It implies that the socially efficient discount rate dt defined by Eq. (23) tends to d` for very far-distant futures, with:
d` 5f b E(1 1 g˜ )2 f `g 21 2 1
(24)
Proof. See the Appendix.5 This Proposition states that the discount rate for far-distant futures tends to the short-term interest rate in an economy with agents having a constant relative risk aversion equaling f` . For example, it implies that all curves in Fig. 1 tend to a discount rate of 1.62%, independent of the value of k.
6. Relation with the literature The distribution of the growth rate of consumption has been treated here as a given primitive. We could have specified a full equilibrium model as Cox et al. (1985a,b). But what matters for the result is the partial equilibrium reduced form ˜ however it comes about. Moreover, it is doubtful that economists dispersion of g, currently have a complete understanding of the determinants of (very) long term consumption growth, as impacted by technological progresses, political choices and wars, in top of the accumulation of capital. Dybvig et al. (1996) and Weitzman (1998) propose a very different argument for why the far-distant future should be discounted at a lower rate. They took the equilibrium short-term interest rate as the primitive, rather than the process of consumption growth. To keep their argument simple, consider an economy where the per-annum short-term risk-free interest rate r 5 R 2 1 will remain the same for ever, but is unknown as of today. It will be revealed within a very short period of time. Let again dT denote the socially efficient discount rate for horizon T, before R˜ is revealed. Notice that after R˜ is revealed, the yield curve is completely flat at level R˜ 2 1. Dybvig et al. (1996) show that the socially efficient discount rate for the very long run must be equal to r min 5 R min 2 1, the lowest possible future rate with a positive probability of occurrence. They use a simple arbitrage argument. Namely,
5
I am grateful to Jean-Charles Rochet for suggesting this proof.
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suppose by contradiction that d` is larger than r min . Then buying a claim on the delivery of R Tmin at date T that will be liquidate right after the realization of R˜ is an arbitrage when T tends to infinity. Indeed, buying the claim costs R Tmin /(1 1 dT )T , which tends to zero as T tends to infinity, since 1 1 d` is larger than R min by assumption. On the contrary, liquidating the position after the realization of R˜ T yields a cash flow equaling y T (R) 5 R min /R T . As T tends to infinity, this tends to zero, except for R 5 R min , where y T (R min ) 5 1. Thus, there is a positive probability that this costless strategy yields a positive cash flow. This cannot be an equilibrium. Thus, it must be that d` # r min . Thus the discount rate for far distant time horizons must be equal to the lowest possible realization of the future interest rate. However, this result does not tell us the relationship between the discount rate and time horizon. Weitzman (1998) answered this question in this framework by assuming that agents are risk neutral. Suppose that an agent wants to get a sure benefit of 1 dollar ˜ he will have to at date T. If he saves the money right after the observation of R, 2T ˜ save R to get this benefit. He could alternatively decide to invest in a long coupon bond with maturity at T, whose rate is dT by definition. To attain the objective, he will have to invest (1 1 dT )2T . Because the agent is risk neutral, there 2T would be an arbitrage if (1 1 dT )2T would not be equal to the expectation of R˜ . Thus, they must be equal. This yields: 2T 21 / T 1 1 dT 5fER˜ g
(25)
˜ it is less than its arithmetic for all T . 0. Because dT is an harmonic mean of R, mean, and it tends to its lowest possible rate R min for large T. In words, the yield curve must be decreasing in such an environment. It corresponds to the idea that a risk-neutral agent with an objective of a fixed future benefit (and thus with a random contribution) likes the randomization of the per-period interest rate. Because the net present value of a cash flow is a convex function of the interest rate, what he will be able to borrow on average will be larger. This raises his willingness to invest in the project. Notice, however, that the argument would be reversed for a risk-neutral agent who wants fixed contributions, and thus a random benefit. The model presented in this paper differs on what is considered as exogenous. When the consumption growth is taken as the primitive and when the process of growth is i.i.d. over time, we showed that the effect of time horizon on the discount rate can be made unambiguous only by restricting the set of preferences of the representative agent. Without surprise, other conditions are necessary when other economic variables are taken as exogenous. One can reconcile these results only by considering a general equilibrium model where the sources of uncertainty are completely specified, as in Cox et al. (1985a,b). This means endogenizing the
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consumption growth g˜ in our model, or endogenizing the interest rates in Dybvig et al. (1996) and Weitzman (1998).
7. Conclusion When public investment projects entail costs and benefits in the very long run, a question arises about the selection of the relevant discount rate to use for the cost–benefit analysis. Indeed, financial markets do not provide any guideline in this case. The main argument for using a positive discount rate is the fact that the GNP per head is expected to grow over time. Therefore, projects whose costs today are as large as benefits in the future are clearly not desirable, since we do not see why current generations should sacrifice part of their consumption today for the benefit of future generations who will already be better off. But growth is an uncertain phenomenon. The recognition of this fact should induce society to take this argument with caution / prudence. By how much has been the question we tried to answer in this paper. We showed that the answer mainly depends upon the time horizon under scrutiny, the degree of relative prudence and the degree of resistance to intertemporal substitution. For commonly accepted levels of these indexes, the effect of uncertainty on the socially optimal discount rate may vary from 2 to 8%. Another important message is that the discount rate to be used for long-lasting investments should be a decreasing function of their duration. This is due to the negative effect of accumulating the per period growth risk in the long run. The French Commissariat au Plan recommends to use a constant 8% per year as the discount rate for all public investments, and most developed countries use a rate between 5 and 8%. From our simulations, we feel that this range of rates is too large with respect that what would be socially efficient, given our current expectation on growth, and the uncertainty that prevails on it. We recommend using the risk free rate that is observable on financial markets for short time horizons. A discount rate not larger than 5% should be used in the medium run (between 50 and 100 years), whereas a decreasing rate down to around 1.5% would be relevant for flows of benefits and costs occurring in the very long run (more than 200 years).
Acknowledgements I am grateful to two anonymous referees, David Bradford, Claude Henry, Jean-Pierre Florens, Jean-Jacques Laffont and especially to Jean-Charles Rochet for their helpful comments. I also want to thank participants of seminars at
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Princeton, Paris, Louvain, Toulouse, and ESEM. Financial support of the European Commission is gratefully acknowledged.
Appendix A. Proof of Proposition 4 In order to simplify the notation, we assume that b 5 1. By definition of function f, we have: z
3E
f (s) v9(z) 5 K exp 2 ]] ds s
4
0
(A.1)
It implies that: ce x
3E
v9(ce x) ]] 5 exp 2 v9(c)
x
4 3E
f (s) ]] ds 5 exp 2 f (ce u) du s
c
0
4
(A.2)
Take any scalar ´ . 0. Define: x
E
q2 (x) 5 ff (ce u) 2 f` 2 ´g du
(A.3)
0
and x
E
q1 (x) 5 ff (ce u) 2 f` 1 ´g du
(A.4)
0
Function q2 (x) tends to minus infinity when x tends to infinity. It implies that q2 has a maximum in R 1 that is denoted M. It implies in turn that the right-hand side of condition (A.2) is bounded below by expf 2 (f` 1 ´) x 2 Mg. Similarly, q1 has a minimum m in R 1 , implying that the right-hand side of condition (A.2) is bounded above by expf 2 (f` 2 ´) x 2 mg. To sum up, we know that for any ´ . 0, there exists two scalars A 5 e 2M and B 5 e 2m such that, for any x [ R 1 : v9(ce x) A expf 2 (f` 1 ´) xg # ]] # B expf 2 (f` 2 ´)xg v9(c)
(A.5)
Observe now that condition (23) can be rewritten as: x˜
v9sce td (1 1 dt )2t 5 E ]] v9(c)
(A.6)
where x˜ t 5 o it 51 h˜ i and h˜ i 5 log(1 1 g˜ i ), whose support is in R 1 . Combining conditions (A.5) and (A.6) yields:
C. Gollier / Journal of Public Economics 85 (2002) 149 – 166 t t 2t AfE expf 2 (f` 1 ´) h˜gg # (1 1 dt ) # BfE expf 2 (f` 2 ´)h˜gg
165
(A.7)
or, equivalently: 1 2 ] log B 2 logfE expf 2 (f` 2 ´) h˜gg # log(1 1 dt ) t
(A.8)
1 # 2 ] log A 2 logfE expf 2 (f` 1 ´) h˜gg t
(A.9)
Taking the limit when t tends to infinity yields: 2 logfE expf 2 (f` 2 ´) h˜gg #lim log(1 1 dt ) t →`
# 2 logfE expf 2 (f` 1 ´) h˜gg
(A.10)
Since this condition holds for any ´ . 0, it implies that: lim log(1 1 dt ) 5 2 logfE expf 2 f` h˜gg
t→`
(A.11)
or, equivalently: lim(1 1 dt ) 5 Ef(1 1 g˜ )2 f `g 21
t→`
This concludes the proof.
(A.12)
h
References Arrow, K.J., 1994. Discounting climate change: Planning for an uncertain future. University of Toulouse, IDEI lecture. Athey, S., 1998. Comparative statics under uncertainty: single crossing properties and log-supermodularity. MIT, Cambridge, Mimeo. Barsky, R.B., Juster, F.T., Kimball, M.S., Shapiro, M., 1997. Preference parameters and behavioral heterogeneity: An experimental approach in the health and retirement study. Quarterly Journal of Economics 62, 537–579. Bell, D.E., 1988. One-switch utility functions and a measure of risk. Management Science 34, 1416–1424. Cox, J., Ingersoll, J., Ross, S., 1985a. A theory of the term structure of interest rates. Econometrica 53, 385–403. Cox, J., Ingersoll, J., Ross, S., 1985b. An intertemporal general equilibrium model of asset prices. Econometrica 53, 363–384. ` Dreze, J.H., Modigliani, F., 1972. Consumption decisions under uncertainty. Journal of Economic Theory 5, 308–335. Dybvig, P.H., Ingersoll, J.E., Ross, S.A., 1996. Long forward and zero-coupon rates can never fall. Journal of Business 69, 1–25. Epstein, L.G., Zin, S., 1991. Substitution, risk aversion and the temporal behavior of consumption and asset returns: An empirical framework. Journal of Political Economy 99, 263–286. Gollier, C., Pratt, J.W., 1996. Risk vulnerability and the tempering effect of background risk. Econometrica 64, 1109–1123.
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Gollier, C., 2000a. Time horizon and the discount rate. University of Toulouse, Mimeo. Gollier, C., 2000b. The Economics of Risk and Time. MIT Press, Cambridge, MA, forthcoming. Guiso, L., Jappelli, T., Terlizzese, D., 1996. Income risk, borrowing constraints, and portfolio choice. American Economic Review 86, 158–172. Hansen, L., Singleton, K., 1983. Stochastic consumption, risk aversion and the temporal behavior of assets returns. Journal of Political Economy 91, 249–268. Harvey, C.M., 1994. The reasonableness of non-constant discounting. Journal of Public Economics 53, 31–51. Jewitt, I., 1988. Risk and risk aversion in the two risky asset portfolio problem. University of Bristol, Mimeo. Kessler, D., Wolff, E.N., 1991. A comparative analysis of household wealth patterns in France and in the United States. Review of Income and Wealth 37, 249–266. Kimball, M.S., 1990. Precautionary saving in the small and in the large. Econometrica 58, 53–73. Kimball, M.S., Weil, P., 1992. Precautionary saving and consumption smoothing across time and possibilities. NBER Working Paper 3976. Kocherlakota, N.R., 1996. The equity premium: It’s still a puzzle. Journal of Economic Literature 34, 42–71. Kreps, D.M., Porteus, E.L., 1978. Temporal resolution of uncertainty and dynamic choice theory. Econometrica 46, 185–200. Laibson, D.I., 1997. Golden eggs and hyperbolic discounting. Quarterly Journal of Economics 62, 443–479. Leland, H.E., 1968. Saving and uncertainty: The precautionary demand for saving. Quarterly Journal of Economics 33, 465–473. Lowenstein, G., Thaler, R., 1989. Intertemporal choice. Journal of Economic Perspectives 3, 181–193. Maddison, A., 1991. Phases of Economic Development. Oxford Economic Press, Oxford. Ogaki, M., Zhang, Q., 1999. Decreasing relative risk aversion and tests of risk sharing. Ohio State University, Mimeo. Samuelson, P.A., 1963. Risk and uncertainty: The fallacy of the law of large numbers. Scientia 98, 108–113. Selden, L., 1979. An OCE analysis of the effect of uncertainty on saving under risk independence. Review of Economic Studies, 73–82. Weitzman, M.L., 1998. Why the far-distant future should be discounted at its lowest possible rate. Journal of Environmental Economics and Management 36, 201–208.
Discounting an uncertain future Christian Gollier Universite´ de Toulouse and Institut Universitaire de France, GREMAQ , Place Anatole France, 31042 Toulouse Cedex, France Received 24 March 1999; received in revised form 16 July 2000; accepted 30 October 2000
Abstract The objective of this paper is to determine the socially optimal discount rate for public investment projects that entail costs and benefits in the very long run. We suppose that there is an exogenous process for the growth of consumption per capita, which is stochastic. We first evaluate the determinants of the discount rate for a specific horizon when the representative agent has a recursive utility. We then explore the influence of the time horizon in the expected utility model. Under various conditions on preferences, as positive prudence, decreasing relative risk aversion or decreasing absolute risk aversion, we prove that (1) the fact that growth is uncertain reduces the efficient discount rate at any horizon, and that (2) this discount rate should be smaller for more distant futures. We characterize the asymptotic value of the discount rate. 2002 Elsevier Science B.V. All rights reserved. Keywords: Discounting; Uncertain growth; Log-supermodularity; Prudence; Kreps–Porteus preference JEL classification: D81; D91; Q25; Q28
1. Introduction Much of life is made of investments. Costly actions are taken today in the prospect of future benefits. In the presence of efficient financial markets, the investment decision process is based on the classical concept of the Net Present Value (NPV). The argument sustaining this decision rule is based on arbitrage. E-mail address: [email protected] (C. Gollier). 0047-2727 / 02 / $ – see front matter 2002 Elsevier Science B.V. All rights reserved. PII: S0047-2727( 01 )00079-2
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Instead of undertaking the planned investment, one could invest in the financial markets. As a consequence, the return of the planned investment should yield at least the risk free rate. For standard investment projects, this rule is equivalent to having a positive NPV. If financial markets are frictionless, the use of the observed risk free rate to discount public investment projects leads to a socially efficient level of investment. The analysis is less easy to perform when benefits and costs of the set of current potential actions are expected to last in the long run. The carbon dioxide that one emits today will not be recycled for a couple of centuries, yielding long term costs like global warming. Some nuclear wastes like plutonium have half-life in the tens of thousands years. Obviously, financial markets are not very helpful to provide a guideline for investing in technologies that prevent this kind of long-lasting risks to occur. Liquid financial instruments with such large durations do not exist. For the sake of comparison, US Treasury Bonds have time horizons that do not exceed 30 years. We must thus rely on the use of an economic model to value the distant future. ¨ The name of Bohm-Bawerk and Fisher are intimately related to these questions. The first reason offered by these authors is purely psychological. Namely, agents may have a pure preference for the present, i.e. they are impatient. The second reason to discount the future is related to a wealth effect. We expect that the quantity of available consumption goods will increase over time. After all, in the western world at least, we experienced an uninterrupted growth during the two last centuries. Given decreasing marginal utility of consumption, an investment which gives one unit of the consumption good in the future in exchange for one unit of the consumption good in the present should not be acceptable. Investing for the future in a growing economy will increase consumption inequalities over time. Since agents have preferences for the smoothing of consumption over time, this investment should be implemented only if its rate of return is large enough to compensate for this negative impact on welfare. The larger the growth rate of the economy, the larger the socially efficient discount rate. A problem arises with this wealth effect if the growth rate is not known with certainty. Estimating the growth rate for the coming year is already a difficult task. No doubt, any estimation of growth for the next century / millennium is subject to potentially enormous errors. The history of the western world before the industrial revolution is full of important economic slumps, as the one due to the invasion of the Roman Empire, or the one due to the Black Death during the middle ages. The recent debate on the notion of a sustainable growth is an illustration of the degree of uncertainty we face to think about the future of society. Some will argue that the effects of the improvements in information technology have yet to be realized, and the world faces a period of more rapid growth. On the contrary, those who emphasize the effects of natural resource scarcity will see lower growth rates in the future. Some even suggest a negative growth of the GNP per head in the future, due to the deterioration of the environment, population growth and decreasing returns to scale. They claim that the wealth effect goes the other
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direction, so that everything should be made to improve the future. This uncertainty at least casts some doubt on the relevance of the wealth effect to justify the use of a large discount rate. In this paper, we provide an analysis of the effect of the uncertainty on growth on the socially optimal discount factor. Instrumental to this analysis is the concept of prudence that has been formalized by Kimball (1990). An agent is prudent if his willingness to save increases in the face of an increase in his future income risk. Technically, an agent is prudent if the third derivative of his utility function is positive. As shown in this paper, prudence justifies taking a discount rate that is less than the one that would have been obtained by assuming a certain growth. The magnitude of the effect depends upon the degree of prudence and the degree of uncertainty on growth. This analysis is provided in Section 2. Some numerical simulations are performed in Section 3. In Section 4, we examine the relationship between the time horizon and the socially optimal discount rate. In order to counterbalance the exponential effect of discounting, several authors suggested to take a smaller discount rate per period to discount more distant cash flows. Harvey (1994) discusses the reasonableness of such a strategy. Our aim is to determine whether this strategy is socially efficient. An intuitive argument would rely on the increased risk of longer horizons due to the accumulation of period to period growth risks. The longer the horizon is, the larger is the uncertainty on future wealth, the smaller should the discount rate be. However, there is a potentially counterbalancing wealth effect: the expected level of consumption is increasing with time. Thus, there is an argument to take care less of more distant horizons, thereby selecting a larger discount rate for such horizons. Which of the precautionary effect or of the wealth effect dominates the other? The main result of the paper is to show that the precautionary effect dominates the wealth effect when relative risk aversion is decreasing and when there is no risk of recession. In such a case, it is socially efficient to reduce the discount rate per year for more distant horizons. This section is related to the literature on the term structure of interest rate. A reinterpretation of our work is indeed that, if the growth rate of consumption is stationary over time, the yield curve is increasing or decreasing depending upon whether relative risk aversion is increasing or decreasing. Section 5 is devoted to the analysis of the asymptotic value of the discount rate per period. In Section 6, we compare our results to the existing literature and, in particular, to a recent paper by Martin Weitzman (1998) who also proves that the discount rate should be decreasing with time horizon. We show that the two approaches are complementary.
2. The socially optimal discount rate In this section, we consider a model with one period and two dates, t 5 0,1. We consider the case where cost–benefit analysis can be applied. That is, we assume that all costs and benefits can be measured in one dimension, i.e. money. People
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consume c at date t 5 0, and c˜ 1 at date t 5 1. Seen from t 5 0, c˜ 1 is uncertain. Thus, in our model, we have some form of dynamic uncertainty. In order to disentangle the attitude towards time and the attitude towards risk, we assume that the planner has Kreps–Porteus–Selden preferences. Kreps and Porteus (1978) and Selden (1979) proposed two equivalent preference functionals that allow for disentangling these two concepts. The simplest way of presenting this family of preference functionals is to first define the certainty equivalent consumption m at date t 5 1 as is standard under expected utility: v(m) 5 Ev(c˜ 1 )
(1)
Then the social planner evaluates the intertemporal welfare as: u(c) 1 b u(m)
(2)
We assume that u and v are increasing and weakly concave in their argument. In words, the planner first computes the certainty equivalent consumption of the future generation by using its aversion to future risks characterized by the concavity of v. Then the planner aggregates consumptions of the two generations in a nonlinear way. The concavity of u characterizes the agent’s aversion to consumption fluctuations over time. Parameter b is the discount factor of utility, representing the pure preference for the present (‘pure discounting’). The intertemporal time-separable expected utility model is obtained by taking u ; v, which yields a classical objective function v(c) 1 b Ev(c˜ 1 ). Another particular case is when u is linear. In that case, the social welfare function is a weighted sum of the certainty equivalent consumption, i.e. certainty equivalent consumptions at different dates are perfect substitutes. Kimball and Weil (1992) examine the saving decision of an agent having Kreps–Porteus–Selden preferences. This economy has an exogenous consumption process. It is similar to a tree economy a` la Lucas: each agent is endowed with a tree which generates a number of fruits c and c˜ 1 , respectively, at date t 5 0 and t 5 1. There is no possibility to plant new trees. The only possibility is to invest in some prevention effort to improve future individual crops. Consider a marginal project which costs a small x (Dc 5 2 x) at date t 5 0 and which brings a sure benefit (1 1 r) x at date t 5 1.1 This project can be to reduce the pollution of the air, or the preservation of animal species. Should society invest in this project? This sure benefit increases the certainty equivalent consumption at date t 5 1 by:
1 In many instances, the future benefits are to reduce risks borne by future generations. When the benefits are uncertain, but independent of the growth rate of the economy, the same model can be used with 1 1 r denoting the certainty equivalent benefit at date 1. Notice that the presence of the background uncertainty on the future wealth level affects this certainty equivalent. Under risk vulnerability (see Gollier and Pratt (1996)), a condition on preferences that is satisfied by most familiar utility functions, the presence of an uncertain future wealth increases the certainty equivalent benefit of a risk-reducing investment. This provides another argument to select a smaller discount rate.
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≠m ] ≠x
U
x 50
Ev9(c˜ 1 ) 5 ]]] (1 1 r) v9(m)
153
(3)
The social welfare function (2) is increased by this marginal investment if: ≠m 2 u9(c) 1 b u9(m) ] u x 50 $ 0 ≠x or, equivalently, if: u9(c) v9(m) r $ d 5 def ]] ]]] 2 1 b u9(m) Ev9(c˜ 1 )
(4)
where d is the socially efficient discount rate. Notice that d would be the equilibrium risk-free rate in this exchange economy. The socially optimal d depends upon social values ( b ), together with the attitude towards risk and the elasticity of intertemporal substitution, as is apparent in Eq. (4). We want to evaluate the effect of the uncertain growth on the socially optimal discount rate. To do this, we compare d to d c , the socially optimal discount rate in an economy with a sure consumption level equaling Ec˜ 1 at date t 5 1. Without uncertainty, risk aversion does not matter, and the certainty equivalent is just m 5 Ec˜ 1 . u9(c) d c []]] 2 1 b u9(Ec˜ 1 ))
(5)
The socially optimal discount rate d is smaller than d c if and only if: u9(c) v9(m) u9(c) d # d c ⇔]] ]]] # ]]] u9(m) Ev9(c˜ 1 ) u9(Ec˜ 1 )
(6)
We first examine the classical case of time-separable expected utility preferences with u ; v. In this case, we can rewrite condition (6) as:
d # d c ⇔Ev9(c˜ 1 ) $ v9(Ec˜ 1 )
(7)
The latter condition holds for any c and for any distribution of c˜ 1 if and only if v9 is convex. The convexity of marginal utility is a well-known condition since ` and Modigliani (1972). It is a necessary and sufficient Leland (1968) and Dreze condition for an increase in future risk to increase (precautionary) savings. Kimball (1990) coined the term ‘prudent’ to characterize individuals who behave in this way. Proposition 1. Suppose that u ; v: preferences are time separable. Then, the uncertainty affecting the growth rate of the economy should induce Society to select a smaller (resp. greater) discount rate if agents living in the future are prudent (resp. imprudent).
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A prudent society should care more about the future when it is more uncertain. This is induced by reducing the discount rate, which implies that more investment favoring the future become profitable, viz. have a positive NPV. Matters are more complex under Kreps–Porteus preferences. We obtained the following result: Proposition 2. Under Kreps–Porteus preferences, the uncertainty affecting the growth rate of the economy should induce society to select a smaller discount rate if agents are decreasingly absolute risk-averse. Moreover, this condition is necessary if function u is linear. Proof. We have to show that condition (6) holds under decreasing absolute risk aversion. Observe first that m is less than Ec˜ 1 , by risk aversion. It implies that: u9(m) $ u9(Ec˜ 1 )
(8)
It remains to prove that v9(m) is less than Ev9(c˜ 1 ) under decreasing absolute risk aversion. Since m is defined as v(m) 5 Ev(c˜ 1 ), this is the case if and only if 2 v9 is more concave than v. But this is precisely the definition of decreasing absolute risk aversion: 2 v-(.) /v0(c) $ 2 v0(c) /v9(c) for all c. When u is linear, condition (8) is an equality, and d # d c if and only if v exhibits decreasing absolute risk aversion. h If we compare this Proposition with Proposition 1, we see that we end up here with a stronger condition to guarantee that growth uncertainty affects the discount rate negatively. Indeed, decreasing absolute risk aversion of v is more demanding than prudence. This is not a surprise, since Kreps–Porteus preferences are a more flexible preferences form than the time-separable expected utility model used in Proposition 1. This is the weakest sufficient condition that one can get without restricting the time aggregator u. The necessity of decreasing absolute risk aversion can be explained as follows. In the extreme case where u is linear, the efficient discount rate in excess of the rate of pure preference for the present is equal to the inverse of the sensitivity of the certainty equivalent to a change in wealth. As suggested by the intuition, the risk on future consumption increases this sensitivity under decreasing absolute risk aversion: without risk, it is equal to unity, whereas it is larger than unity when c˜ 1 is risky, under DARA. Indeed, one must take into account of the fact that the risk premium is decreasing with wealth in that case.
3. Numerical estimation of the efficient discount rate for the short term We can estimate the socially efficient discount rate in an uncertain environment by assuming that the support of growth rate of consumption is in a small neighborhood of zero. This is compatible only with a relatively short time horizon,
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i.e. with b → 1. In the next section, we will determine whether socially efficient discount rates for longer time horizons should be smaller than this benchmark. ˜ with y˜ small and m 2 5 (Ey˜ )2 < Ey˜ 2 . s 2 . The Thus, suppose that c˜ 1 5 c 1 y, Arrow–Pratt approximation tells us that: m . c 1 m 2 0.5s 2 A v
(9)
where A f is the absolute risk aversion of function f evaluated at c. Using second-order Taylor approximations, we get in turn that: u9(c) ]] . 1 1 m A u 2 0.5s 2 A u A v u9(m)
(10)
v9(m) ]]] 5 1 1 0.5s 2 A 9v Ev9(c˜ 1 )
(11)
and
where A v9 is the derivative of A v evaluated at c. This second equality represents the sensitivity of the certainty equivalent to changes in wealth. It is negative when absolute risk aversion is decreasing. We finally obtain that:
F G
1 d . ] 2 1 1fEc˜ 1 2 cg A u 1 0.5 Var(c˜ 1 )(A v9 2 A u A v ) b
(12)
We recognize in this equation the three determinants of the discount rate: the first bracketed term in the left-hand side of condition (12) is the rate of pure preference for the present. The second term in the sum is the pure wealth effect. It is positive when (1) the representative agent is averse to consumption fluctuations over time, and (2) the expected growth of consumption is positive. A larger expected growth of the economy should induce agents to care less about the future, by increasing the discount rate. The third term in the sum measures the effect of uncertainty on the discount rate. In accordance to Proposition 2, this term is unambiguously negative when the absolute risk aversion of v is decreasing. Notice that the necessity of A 9v # 0 can be derived directly from (12) by considering temporal preferences with A u 5 0. Notice also that, A 9v 5 A v (A v 2 Pv ), where Pv 5 2 v-(c) / v0(c) is absolute prudence. Thus, in the special case of time separable preferences (u ; v), condition (12) becomes:
F G
1 d . ] 2 1 1fEc˜ 1 2 cg A 2 0.5 Var(c˜ 1 )AP b
(13)
This condition is similar to what was obtained by Hansen and Singleton (1983). It confirms that positive prudence is necessary for the precautionary effect to reduce the socially efficient discount rate in the classical time-separable utility model (Proposition 1). Let us alternatively assume that u9(c) 5 c 2gu and v9(c) 5 c 2gv , for some gu $
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Table 1 The socially efficient discount rate per year for short horizons
gu 5 1 gu 5 5 gu 5 10
gv 5 0
gv 5 1
gv 5 5
gv 5 10
1.8% 9.0% 18.0%
1.7% 8.6% 17.3%
1.1% 7.1% 14.4%
0.5% 5.1% 10.9%
and gv $ 0. Let also g˜ 5 (c˜ 1 /c) 2 1 denote the growth rate of consumption. Under this specification, condition (12) can be rewritten as:
F G
1 d . ] 2 1 1 gu Eg˜ 2 0.5gv (1 1 gu ) Var(g˜ ) b
(14)
Kocherlakota (1996), using United States annual data over the period 1889–1978, estimated the mean and the standard deviation of the growth of consumption per capita to, respectively, 1.8% per year and 3.6% per year. If we use these values of the parameters together with b 5 1, approximation (14) gives us socially efficient discount rates as documented in Table 1. Barsky et al. (1997) estimated gu and gv based on survey responses to hypothetical situations dealing either with consumption smoothing under certainty (for gu ) or with static risky choices (for gv ). They estimated gv , the degree of relative risk aversion, to around 4, which is compatible to the range of what is considered as reasonable by economists. The measurement of the aversion to consumption fluctuation over time, gu , is more problematic. Barsky et al. (1997) obtained an estimation around gu 5 5, whereas Epstein and Zin (1991) found a value ranging from 1.25 to 5. Based on these estimates of individual preferences, we recommend using a discount rate for the short term in a range between 1.5 and 7%, with a preference for the mid-point around 4.5%.
4. Compound growth rates When considering environmental risks, we are not only interested in discounting benefits and costs occurring in the next few years, but also for much longer time horizons. Obviously, the uncertainty prevailing on the level of consumption per head prevailing t periods from now is an increasing function of t.2 In this section, we address the question of how this increased uncertainty affects the optimal discount rate per period. In fact, we examine whether the yield curve should be 2
As indicated by Samuelson (1963), compounding risks borne at different points in time, even if they are independent, is not a diversification device.
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decreasing, increasing or flat. We do this by considering the special case of expected utility with a time separable utility function. Empirical evidence suggests that the discount rate used by individuals to value the future is a decreasing function of the time horizon. Lowenstein and Thaler (1989) for example obtained discount rates ranging around 15% for the short run (less than 5 years), and then dropping to a level as low as 2% for the long run (100 years). Can this be due to the increase of uncertainty about one’s future wealth, rather than to a deviation of the standard model with exponential discounting of utilities? 3 In accordance with this empirical evidence, our intuition is that the interaction between intertemporal growth risks is an aggravating factor. A longer time horizon should induce a smaller discount rate per period. Because the uncertainty affecting future consumption is increasing with the time horizon, the precautionary argument applies in favor of a reduction of the discount rate per year for more distant futures under prudence (Proposition 1). But, on the other side, more distant generations will enjoy a larger level of consumption, in expectation. This wealth effect goes the opposite direction than the precautionary effect. Thus, the global effect of time horizon on the efficient discount rate per year is a priori ambiguous. Let us consider a two-period model with three dates t 5 0, 1, 2. The growth rate of consumption from t 5 0 to t 5 1 is g˜ 1 , whereas the growth rate from t 5 1 to t 5 2 is g˜ 2 . We assume that random variables g˜ 1 and g˜ 2 are independently and ˜ We assume that the representative identically distributed. They are distributed as g. agent has a time-separable utility function v. Therefore, the socially optimal rate d1 to discount costs and benefits occurring at date t 5 1 is as in Section 2 with u ; v: v9(c) 1 1 d1 5 ]]]]] b Ev9(c(1 1 g˜ 1 ))
(15)
The growth rate over the two periods is (1 1 g˜ 1 )(1 1 g˜ 2 ) 2 1. The socially optimal discount rate per period d2 for a cash flow occurring at t 5 2 should be such that: v9(c) (1 1 d2 )2 5 ]]]]]]] 2 b Ev9(c(1 1 g˜ 1 )(1 1 g˜ 2 ))
(16)
Usually, d2 is not equal to d1 , except in the case of the isoelastic utility function as we show now. Suppose that v9(c) 5 c 2 f , for some constant relative risk aversion f . 0. Conditions (15) and (16) yield: (1 1 d1 )2
5 [ b E(1 1 g˜ 1 )2 f ] 22 22 2 5 b [E(1 1 g˜ 1 ) f E(1 1 g2 )2 f ] 21 5b
3
22
[E((1 1 g˜ 1 )(1 1 g2 ))
2 f 21
]
5 (1 1 d2 )
(17) 2
Laibson (1997) discusses the effect of non-exponential discounting of utilities on individual strategic behaviors.
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This implies that d2 5 d1 . The wealth effect of a longer time horizon just compensates the precautionary effect when relative risk aversion is constant. More generally, using conditions (16) and (18), d2 is less than d1 if and only if: v9(c) Ev9(c(1 1 g˜ 1 )(1 1 g˜ 2 )) $ Ev9(c(1 1 g˜ 1 )) Ev9(c(1 1 g˜ 2 ))
(18)
In the following Proposition, we present a simple sufficient condition to satisfy this inequality. Proposition 3. Suppose that the growth rate of consumption is nonnegative almost surely, and that v is three time differentiable. The long term discount rate is smaller (resp. larger) than the short term one if relative risk aversion is decreasing (resp. increasing). Proof. Let function h from R 21 to R be defined as h(x 1 , x 2 ) 5 v9(cx 1 x 2 ). Suppose that this function be log supermodular.4 This means that: h(min(x 1 , x 91 ), min(x 2 , x 92 )) h(max(x 1 , x 19 ), max(x 2 , x 29 )) $ h(x 1 , x 2 ) h(x 19 , x 29 ) (19) 2 1
for all (x 1 , x 2 ) and (x 19 , x 29 ) in R . Taking x 1 5 x 29 5 1, the log supermodularity of h implies that: h(1, 1) h(x 91 , x 2 ) $ h(1, x 2 ) h(x 19 , 1)
(20)
for all x 91 and x 2 that are both larger than 1. This inequality is equivalent to: v9(c) v9(c(1 1 g1 )(1 1 g2 )) $ v9(c(1 1 g1 )) v9(c(1 1 g2 ))
(21)
for all g1 5 x 91 2 1 and g2 5 x 2 2 1 that are both positive. Suppose now that the growth rate g˜ t per period is positive almost surely. Then, the log supermodularity of h is sufficient to guarantee that condition (21) holds almost everywhere. Taking the expectation of this inequality directly yields inequality (18), which implies in turn that d2 is less than d1 . If the utility function is three time differentiable, the log supermodularity of h also means that the cross derivative of log h is positive. It is easily seen that this is equivalent to require that relative risk aversion is decreasing (DRRA). h Decreasing relative risk aversion is compatible with the well-documented observation that the relative share of wealth invested in risky assets is an increasing function of wealth. Kessler and Wolff (1991) for example show that the portfolios of US households with low wealth contains a disproportionately large share of risk free assets. Measuring by wealth, over 80% of the lowest quintile’s portfolio was 4
For an analysis of the usefulness of this concept in the economics of uncertainty, see Jewitt (1988), Athey (1998) and Gollier (2000b).
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in liquid assets, whereas the highest quintile held less than 15% in such assets. Guiso et al. (1996), using cross-section of Italian households, observed portfolio compositions which are also compatible with decreasing relative risk aversion. Ogaki and Zhang (1999) tested whether relative risk aversion is decreasing or increasing from various consumption data in developing countries. They obtained strong evidences that relative risk aversion is decreasing. Thus one should select a smaller rate to discount far distant benefits than the rate to discount benefits realized in the near future. Growth risks are mutually aggravating. To illustrate, let us consider again the scenario in which the growth rate of consumption per capita is either 0 or 75% with, respectively, probability 0.36 and 0.63 in each period of 20 years. These probabilistic scenarios yield the same two first moments than those of the historical data for France over the period 1870–1990, as documented by Maddison (1991). We consider the case of power utility functions: v9(z) 5 (z 2 k)2g, z $ k
(22)
for some g $ 0, and k which can be interpreted as some minimum subsistence level if it is positive. For those functions, prudence is positive, and relative risk aversion equals: 2 cv0(c) gc f (c) 5 ]]] 5 ]] c2k v9(c) When parameter k is positive, relative risk aversion is decreasing from plus infinity at c 5 k to g as c tends to infinity. We hereafter examine the special case where g 5 1. We also normalize consumption at date t 5 0 to unity. We also take b 5 1. We computed the socially optimal discount rate for horizons up to 45 periods, using different values of the parameter k. The results are drawn in Fig. 1. As an illustration, consider the case k 5 0.9 for which relative risk aversion equals 10 at current consumption and goes to 1 as consumption goes to infinity. The discount rate per year that should be used to value benefits arising within the next twenty years equals 4.14%. For benefits arising in 800 years from today, it should be reduced to less than 2%. An important requirement of the above Proposition is that the growth rate of consumption per capita be nonnegative almost surely. When there is a risk of recession in an economy with a positive expected growth, it is not true anymore that DRRA is sufficient for the yield curve to be strictly decreasing. Let us illustrate this point by the following example, using One-Switch utility functions introduced by Bell (1988). Take v9(z) 5 a 1 z 2b with a . 0 and b . 0. It yields 2 zv0(z) /v9(z) 5 bfaz 2b 1 1g 21 , which is decreasing in z. In addition, take a 5 b 5 1 and g˜ t | (250%, 1 / 3; 1 100%, 2 / 3). In such a situation, straightforward computations generate d1 5 d2 5 0: the yield curve is flat in spite of DRRA! This result does not contradict the above Proposition, since g˜ t is not positive with probability 1. Thus, extending the analysis to economies with a risk of a recession
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Fig. 1. The socially efficient discount rate per year as a function of the time horizon. The upper curve is for k 5 0.9, whereas the horizontal line is for k 5 0.
requires restricting the set of DRRA utility functions to guarantee that the socially optimal discount rate be decreasing with time horizon. Gollier (2000a) examines this question in more details.
5. The asymptotic value of the discount rate It is interesting to determine the asymptotic value of the per-period discount rate when time horizon recedes to infinity. The rate dt to discount the cash flows at date t is implicitly defined by the following equation: v9(c) (1 1 dt )t 5 ]]]]]] t b t Ev9 c (1 1 g˜ i )
SP
i 51
D
(23)
where g˜ i is the growth rate per head between date i 2 1 and i. Suppose first that, at each period i, there is a positive probability that a recession ( gi , 0) occurs. Suppose moreover that there exists a positive k such that lim z →k v9(z) 5 1 `. In that case, the denominator in the right-hand side of Eq. (23) tends to infinity for large t. This is possible only if dt becomes negative. We conclude that the risk of repeated recessions over long periods eventually induces the selection of a nonpositive rate to discount far-distant futures. The low probability of such events is overweighted by the large marginal utility of wealth in these bad states. We now turn to the more interesting case where there is no risk of recession (Prfg˜ # 0g 5 0). Remember that f (z) 5 2 zv0(z) /v9(z) denote the relative risk aversion function, with f` 5 lim z →` f (z). The following Proposition shows that
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convergence of the socially efficient discount rate as the horizon becomes distant occurs if f` exists. ˜ g˜ 1 , g˜ 2 , . . . are i.i.d. variables whose support is in Proposition 4. Suppose that g, R 1 . Suppose also that f` 5 lim z →` 2 zv0(z) /v9(z) exists. It implies that the socially efficient discount rate dt defined by Eq. (23) tends to d` for very far-distant futures, with:
d` 5f b E(1 1 g˜ )2 f `g 21 2 1
(24)
Proof. See the Appendix.5 This Proposition states that the discount rate for far-distant futures tends to the short-term interest rate in an economy with agents having a constant relative risk aversion equaling f` . For example, it implies that all curves in Fig. 1 tend to a discount rate of 1.62%, independent of the value of k.
6. Relation with the literature The distribution of the growth rate of consumption has been treated here as a given primitive. We could have specified a full equilibrium model as Cox et al. (1985a,b). But what matters for the result is the partial equilibrium reduced form ˜ however it comes about. Moreover, it is doubtful that economists dispersion of g, currently have a complete understanding of the determinants of (very) long term consumption growth, as impacted by technological progresses, political choices and wars, in top of the accumulation of capital. Dybvig et al. (1996) and Weitzman (1998) propose a very different argument for why the far-distant future should be discounted at a lower rate. They took the equilibrium short-term interest rate as the primitive, rather than the process of consumption growth. To keep their argument simple, consider an economy where the per-annum short-term risk-free interest rate r 5 R 2 1 will remain the same for ever, but is unknown as of today. It will be revealed within a very short period of time. Let again dT denote the socially efficient discount rate for horizon T, before R˜ is revealed. Notice that after R˜ is revealed, the yield curve is completely flat at level R˜ 2 1. Dybvig et al. (1996) show that the socially efficient discount rate for the very long run must be equal to r min 5 R min 2 1, the lowest possible future rate with a positive probability of occurrence. They use a simple arbitrage argument. Namely,
5
I am grateful to Jean-Charles Rochet for suggesting this proof.
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suppose by contradiction that d` is larger than r min . Then buying a claim on the delivery of R Tmin at date T that will be liquidate right after the realization of R˜ is an arbitrage when T tends to infinity. Indeed, buying the claim costs R Tmin /(1 1 dT )T , which tends to zero as T tends to infinity, since 1 1 d` is larger than R min by assumption. On the contrary, liquidating the position after the realization of R˜ T yields a cash flow equaling y T (R) 5 R min /R T . As T tends to infinity, this tends to zero, except for R 5 R min , where y T (R min ) 5 1. Thus, there is a positive probability that this costless strategy yields a positive cash flow. This cannot be an equilibrium. Thus, it must be that d` # r min . Thus the discount rate for far distant time horizons must be equal to the lowest possible realization of the future interest rate. However, this result does not tell us the relationship between the discount rate and time horizon. Weitzman (1998) answered this question in this framework by assuming that agents are risk neutral. Suppose that an agent wants to get a sure benefit of 1 dollar ˜ he will have to at date T. If he saves the money right after the observation of R, 2T ˜ save R to get this benefit. He could alternatively decide to invest in a long coupon bond with maturity at T, whose rate is dT by definition. To attain the objective, he will have to invest (1 1 dT )2T . Because the agent is risk neutral, there 2T would be an arbitrage if (1 1 dT )2T would not be equal to the expectation of R˜ . Thus, they must be equal. This yields: 2T 21 / T 1 1 dT 5fER˜ g
(25)
˜ it is less than its arithmetic for all T . 0. Because dT is an harmonic mean of R, mean, and it tends to its lowest possible rate R min for large T. In words, the yield curve must be decreasing in such an environment. It corresponds to the idea that a risk-neutral agent with an objective of a fixed future benefit (and thus with a random contribution) likes the randomization of the per-period interest rate. Because the net present value of a cash flow is a convex function of the interest rate, what he will be able to borrow on average will be larger. This raises his willingness to invest in the project. Notice, however, that the argument would be reversed for a risk-neutral agent who wants fixed contributions, and thus a random benefit. The model presented in this paper differs on what is considered as exogenous. When the consumption growth is taken as the primitive and when the process of growth is i.i.d. over time, we showed that the effect of time horizon on the discount rate can be made unambiguous only by restricting the set of preferences of the representative agent. Without surprise, other conditions are necessary when other economic variables are taken as exogenous. One can reconcile these results only by considering a general equilibrium model where the sources of uncertainty are completely specified, as in Cox et al. (1985a,b). This means endogenizing the
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consumption growth g˜ in our model, or endogenizing the interest rates in Dybvig et al. (1996) and Weitzman (1998).
7. Conclusion When public investment projects entail costs and benefits in the very long run, a question arises about the selection of the relevant discount rate to use for the cost–benefit analysis. Indeed, financial markets do not provide any guideline in this case. The main argument for using a positive discount rate is the fact that the GNP per head is expected to grow over time. Therefore, projects whose costs today are as large as benefits in the future are clearly not desirable, since we do not see why current generations should sacrifice part of their consumption today for the benefit of future generations who will already be better off. But growth is an uncertain phenomenon. The recognition of this fact should induce society to take this argument with caution / prudence. By how much has been the question we tried to answer in this paper. We showed that the answer mainly depends upon the time horizon under scrutiny, the degree of relative prudence and the degree of resistance to intertemporal substitution. For commonly accepted levels of these indexes, the effect of uncertainty on the socially optimal discount rate may vary from 2 to 8%. Another important message is that the discount rate to be used for long-lasting investments should be a decreasing function of their duration. This is due to the negative effect of accumulating the per period growth risk in the long run. The French Commissariat au Plan recommends to use a constant 8% per year as the discount rate for all public investments, and most developed countries use a rate between 5 and 8%. From our simulations, we feel that this range of rates is too large with respect that what would be socially efficient, given our current expectation on growth, and the uncertainty that prevails on it. We recommend using the risk free rate that is observable on financial markets for short time horizons. A discount rate not larger than 5% should be used in the medium run (between 50 and 100 years), whereas a decreasing rate down to around 1.5% would be relevant for flows of benefits and costs occurring in the very long run (more than 200 years).
Acknowledgements I am grateful to two anonymous referees, David Bradford, Claude Henry, Jean-Pierre Florens, Jean-Jacques Laffont and especially to Jean-Charles Rochet for their helpful comments. I also want to thank participants of seminars at
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Princeton, Paris, Louvain, Toulouse, and ESEM. Financial support of the European Commission is gratefully acknowledged.
Appendix A. Proof of Proposition 4 In order to simplify the notation, we assume that b 5 1. By definition of function f, we have: z
3E
f (s) v9(z) 5 K exp 2 ]] ds s
4
0
(A.1)
It implies that: ce x
3E
v9(ce x) ]] 5 exp 2 v9(c)
x
4 3E
f (s) ]] ds 5 exp 2 f (ce u) du s
c
0
4
(A.2)
Take any scalar ´ . 0. Define: x
E
q2 (x) 5 ff (ce u) 2 f` 2 ´g du
(A.3)
0
and x
E
q1 (x) 5 ff (ce u) 2 f` 1 ´g du
(A.4)
0
Function q2 (x) tends to minus infinity when x tends to infinity. It implies that q2 has a maximum in R 1 that is denoted M. It implies in turn that the right-hand side of condition (A.2) is bounded below by expf 2 (f` 1 ´) x 2 Mg. Similarly, q1 has a minimum m in R 1 , implying that the right-hand side of condition (A.2) is bounded above by expf 2 (f` 2 ´) x 2 mg. To sum up, we know that for any ´ . 0, there exists two scalars A 5 e 2M and B 5 e 2m such that, for any x [ R 1 : v9(ce x) A expf 2 (f` 1 ´) xg # ]] # B expf 2 (f` 2 ´)xg v9(c)
(A.5)
Observe now that condition (23) can be rewritten as: x˜
v9sce td (1 1 dt )2t 5 E ]] v9(c)
(A.6)
where x˜ t 5 o it 51 h˜ i and h˜ i 5 log(1 1 g˜ i ), whose support is in R 1 . Combining conditions (A.5) and (A.6) yields:
C. Gollier / Journal of Public Economics 85 (2002) 149 – 166 t t 2t AfE expf 2 (f` 1 ´) h˜gg # (1 1 dt ) # BfE expf 2 (f` 2 ´)h˜gg
165
(A.7)
or, equivalently: 1 2 ] log B 2 logfE expf 2 (f` 2 ´) h˜gg # log(1 1 dt ) t
(A.8)
1 # 2 ] log A 2 logfE expf 2 (f` 1 ´) h˜gg t
(A.9)
Taking the limit when t tends to infinity yields: 2 logfE expf 2 (f` 2 ´) h˜gg #lim log(1 1 dt ) t →`
# 2 logfE expf 2 (f` 1 ´) h˜gg
(A.10)
Since this condition holds for any ´ . 0, it implies that: lim log(1 1 dt ) 5 2 logfE expf 2 f` h˜gg
t→`
(A.11)
or, equivalently: lim(1 1 dt ) 5 Ef(1 1 g˜ )2 f `g 21
t→`
This concludes the proof.
(A.12)
h
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