Ecological Economics 74 (2012) 27–33

Contents lists available at SciVerse ScienceDirect

Ecological Economics

j ourna l homepage: www.e lsev ie r .com/ locate /eco lecon

Methods

Inferring the rate of pure time preference under uncertainty

Liqun Liu ⁎Private Enterprise Research Center, 4231 TAMU, Texas A&M University, College Station, TX 77843-4231, USA

⁎ Tel.: +1 979 845 7723; fax: +1 979 845 6636.E-mail address: lliu@tamu.edu.

1 ρ is arbitrarily set at 0.1% in the Stern Review to reptinction of the human race. For reviews of the Stern Reving, see Nordhaus (2007); Weitzman (2007a). For aQuiggin (2008).

2 A criticism of the revealed preferences approach ichoices in the marketplace are based on individual timbe the same as those of the social planner (see Azar1996). In this paper, we abstract from any divergencetime preferences by assuming a representative individu

0921-8009/$ – see front matter © 2011 Elsevier B.V. Alldoi:10.1016/j.ecolecon.2011.11.007

a b s t r a c t

a r t i c l e i n f o

Article history:Received 31 August 2011Received in revised form 7 November 2011Accepted 11 November 2011Available online 20 December 2011

Keywords:Time preferenceDiscountingRamsey RuleUncertainty

This paper studies how to infer the rate of pure time preference (ρ) from the Ramsey Rule when multipleasset returns exist due to uncertainty. Using a Generalized Uncertainty Ramsey Rule derived from a modelthat separates intertemporal substitution and risk aversion, we find that the U.S. historical data on consump-tion growth and asset returns imply that (i) for the reciprocal of the elasticity of intertemporal substitutionless than or equal to one, ρ lies within ±1% from zero for a plausible range of the coefficient of relativerisk aversion; and (ii) for the larger reciprocal of the elasticity of intertemporal substitution, ρ tends to benegative. These results contradict the widely-held belief in the environmental economics literature that theinferred ρmust be significantly larger than zero and suggest that it is appropriate to use ρ=0 as a benchmarkfor economic analysis of environmental policies.

© 2011 Elsevier B.V. All rights reserved.

1. Introduction

The rate of pure time preference (ρ) – the discount rate for futureinstantaneous utility – plays a critical role in long-horizon economicanalyses. Greenhouse gas (GHG) abatement policy modeling in thearea of climate change is an example. Cline (1992) assumed ρ=0and found that a 40% reduction in GHG emissions is optimal. In con-trast, Nordhaus (1994) assumed ρ=3% and found that only modestemissions abatement can be justified on economic grounds.According to Nordhaus (1994, 1999, 2008), although the DICEmodel (Dynamic Integrated model of Climate and the Economy)had been through many versions, its main sensitivity is to the rateof pure time preference. More recently, Stern (2007) essentially letρ=0, but his rejection of discounting future utility has drawncriticism.1

There has been some controversy over how the value of ρ shouldbe determined. Some economists, Frank Ramsey in particular(Ramsey, 1928), believe that ρ should be zero on moral grounds.Other economists argue that the value of ρ must be consistent withrevealed intertemporal choices.2 Specifically, the latter school

resent the small risk of the ex-iew with a focus on discount-review of Stern's critics, see

s that observed intertemporale preferences, which may notand Sterner, 1996; Howarth,between individual and socialal (the social decision-maker).

rights reserved.

believes that ρ should be determined according to the following equi-librium condition for the Ramsey growth model (the DeterministicRamsey Rule):

1þ r ¼ 1þ ρð Þ 1þ gð Þα ; ð1Þ

where r is the rate of return on capital, α is the (absolute value of the)elasticity of themarginal utility with respect to consumption (i.e., α=−u″(c)c/u′(c)), and g is the consumption growth rate.3 Presumably,the value of ρ can be readily inferred from (1) if we know the valueof r and g (which are observable in principle) and have a reasonableestimate of α.

The issue of inferring ρ is different from, though related to, theissue of choosing a discount rate. In terms of the Deterministic RamseyRule, r represents the discount rate that should be used for futureconsumption-equivalent benefits or costs. 4 According to Nordhaus(1994, 1999, 2008), different choices of ρ have real and quantitatively

3 The Deterministic Ramsey Rule is better known as r=ρ+αg, which is thecontinuous-time counterpart of Eq. (1) for the discrete-time model. In this paper wework with a discrete-time model. Note that r=ρ+αg can also be understood as a lin-earization of Eq. (1) with respect to g, which makes it an approximate formula suitablefor use in discrete-time models when g is sufficiently small.

4 Howarth (2003) focused on the discount rate choice in the environment of uncer-tainty with an eye on climate change. See Zeckhauser and Viscusi (2008) for a survey ofrecent literature on discount rate choice in the context of climate change.

28 L. Liu / Ecological Economics 74 (2012) 27–33

significant consequences, especially in the long-run, even accompa-nied by corresponding choices of α so that the same r is reachedaccording to Eq. (1).5 Therefore, ρ by itself is a value worth inferring.

To infer ρ from Eq. (1), we need a value for every other parameterin the equation. The value of g is relatively well understood. For ex-ample, Mehra and Prescott (1985) estimated the average annualgrowth rate of real per-capita consumption to be 1.8% based on theannual U.S. data from 1889 to 1978. However, determining the valuesof r and α is less straightforward.

First, a question arises as to which r should be put in the left-handside of Eq. (1) because of the coexistence of multiple asset returns.Different values of r would generate different values of ρ given αand g. For example, suppose α=1 (corresponding to the logarithmicinstantaneous utility function) and g=1.8%. Then according to Eq.(1), r=7.0% (corresponding to the mean equity return) impliesρ=5%, but r=0.8% (corresponding to the risk-free rate) impliesρ=−1%.6

Second, estimates of α are very diverse, and its appropriate valuedepends on the context in which it is applied. Specifically, the recip-rocal of α, namely, the elasticity of intertemporal substitution, is theparameter of focus in the macro literature on life-cycle consumption.Letting α be one (i.e., the logarithmic utility) is the preferred choice inmacroeconomic models, which seems to be due to the evidence ofrelatively strong intertemporal substitution as well as convenienceconsiderations.7 On the other hand, α itself, as the coefficient of rela-tive risk aversion, is the parameter of focus in the finance literature onasset returns and risks. Data in this context imply a value of α muchlarger than one.8 The literature on climate change economics appearsto view α from the perspective of intertemporal substitution byadopting a value of α between one and two. For example, α=1 inNordhaus (1994) and Stern (2007), α=1.5 in Cline (1992) andα=2 in Nordhaus (2008). However, larger values of α consistentwith financial data should also be relevant when capital marketuncertainty is explicitly considered.9

These conceptual difficulties surrounding using the DeterministicRamsey Rule to infer ρ suggest that one should incorporate uncertain-ty and multiple asset returns into the analysis, and separate the twopreference parameters traditionally represented by a single parameterα.

This paper investigates the issue of how to infer the rate of puretime preference in a saving-portfolio model, adopting a utility specifi-cation due to Epstein and Zin (1991) and Kreps and Porteus (1978)that separates the consumer's attitude towards risk from that towards

5 As Nordhaus (2008, pp61–62) put it, “There are important long-term implicationsof different combinations of time discount rates and consumption elasticities.” On theother hand, the implications of different combinations of ρ and α given r for near-termdecisions are relatively less important.

6 The data on asset returns are from Mehra and Prescott (1985). That the Determin-istic Ramsey Rule (Eq. (1)) is inapplicable in a world of uncertainty with multiple assetclasses has been pointed out in Ackerman et al. (2009), Cochrane (2005), Howarth(2003), Weitzman (2007a) and Weitzman (2009). In addition, Hoel and Sterner(2007) showed that the Ramsey-type discount rate must be modified if the elasticityof substitution in demand between produced goods and environmental services islow and the produced goods sector grows faster than the environmental servicessector.

7 For example, see Browning et al. (1999) and Gomme and Rupert (2007).8 See Kocherlakota (1996), Mehra and Prescott (1985), and Weil (1989). Meyer and

Meyer (2005) distinguished between the relative risk aversion measure for the utilityfunction for consumption and that for the value function for wealth, and found that theformer could be 1.25 to 10 times the latter. In this paper, the utility function is forconsumption.

9 More recently, models that separate risk aversion from intertemporal substitutionwere incorporated in environmental studies (Kousky et al., 2011; Traeger, 2009). Forexample, Crost and Traeger (2010) incorporated a utility specification that disentan-gles the two attitudes into a DICE type model.

intertemporal substitution. Based on the Generalized UncertaintyRamsey Rule derived from this setup, we find that inferring ρ in away consistent with the U.S. historical data on consumption growthand asset returns yields the following main numerical findings:(i) forαu (the reciprocal of the elasticity of intertemporal substitution)equal to or less than one, the value of ρ lies within ±1% from zero re-gardless of the relative risk aversion within a plausible range; and(ii) for larger αu, the value of ρ tends to lie in the negative zone.These results contradict the widely-held belief in the environmentaleconomics literature that the inferred ρ must be significantly largerthan zero, and suggest that it is appropriate to use ρ=0 as a bench-mark for long-term economic analysis.

2. The Generalized Uncertainty Ramsey Rule

Following Gollier (2002), we adopt a simple version of the utilityspecification that distinguishes between risk aversion and intertem-poral substitutability of consumption due to Epstein and Zin (1991)and Kreps and Porteus (1978).10 The preferences are jointlyrepresented by two utility functions: u(⋅) captures the preferences forintertemporal substitution and v(⋅) captures the preferences for risk.Both functions are assumed to exhibit constant elasticity of marginalutility with respect to consumption, with the constant elasticity (abso-lute value) being αu (which is also the reciprocal of the elasticity ofintertemporal substitution) and αv (which is also the coefficient of rel-ative risk aversion), respectively. That is, u(c)=(c1−αu−1)/(1−αu)for αu>0 and v(c)=(c1−αv−1)/(1−αv) for αv>0. Note that, as spe-cial limiting cases, u(c)=ln cwhen αu=1 and v(c)=ln cwhen αv=1.

The representative individual's overall intertemporal utility isgiven by

u c0ð Þ þ u mð Þ1þ ρ

; ð2Þ

where c0 is consumption at t=0, and m is the certainty equivalentconsumption at t=1, which is determined by

v mð Þ ¼ Ev ~c1ð Þ; ð3Þ

where ~c1 is the random consumption at t=1.For simplicity, we work with a two-period framework in which the

two periods are indexed by t=0 or t=1 and assume that there areonly two assets: a risky asset (stocks) and a riskless asset (risk-freebonds).11 The former asset has a random rate of return ĩSand the latterhas a certain rate of return iB. In the standard saving-portfolio problem,a representative consumer chooses savings in period 0,s0, and thefraction of savings allocated to the risky asset, γ, to maximize the over-all utility Eq. (2), subject to Eq. (3) and

c0 ¼ W0−s0

~c1 ¼ s0 1þ iB þ γ ~iS−iB� �h i

;ð4Þ

where W0 is the initial wealth.From the first-order conditions of the individual's optimization

problem, we obtain the following Generalized Uncertainty RamseyRule (see Appendix A for a proof of the proposition).

10 See (Gollier 2001, chapter 20) for a detailed description of the simple version ofKreps–Porteus–Epstein–Zin utility specification and several applications.11 It should be pointed out, in light of the recent near-default of the U.S. government,that the concept of a “risk-free bond” is an idealization.

15 See Mehra and Prescott (1985) and Weil (1989).16 Quiggin (2008) also claimed, in general, that failure to account for the equity pre-mium puzzle can lead to inconsistent policy recommendations. Separating intertem-

Table 1Parameter values for asset returns and consumption growth (The U.S. economy,1889–1978).

iB iS E 1þ ~gð Þ var 1þ ~gð Þ μ≡E ln 1þ ~gð Þ½ � σ2≡ var ln 1þ ~gð Þ½ �0.8% 7.0% 1.018 0.00130 0.0172 0.00125

29L. Liu / Ecological Economics 74 (2012) 27–33

Proposition 1. (Generalized Uncertainty Ramsey Rule). Intertemporalutility maximization implies,

1þ iB ¼1þ ρð Þ E 1þ ~gð Þ1−αv

h in oαv−αu

αv−1

E 1þ ~gð Þ−αv½ � f or αv≠1

1þ iB ¼ 1þ ρð Þe αu−1ð ÞE ln 1þ~gð Þ½ �

E 1þ ~gð Þ−1� � f or αv ¼ 1

where 1þ ~g≡~c1=c0:

ð5Þ

From the Deterministic Ramsey Rule (Eq. (1)) to the GeneralizedUncertainty Ramsey Rule (Eq. (5)), the rate of return r, which is ambig-uouswhenmultiple asset returns exist in the capitalmarket, is replaced

with iB, the risk-free rate. On the other hand, E 1þ ~gð Þ1−αvh in oαv−αu

αv−1=

E 1þ ~gð Þ−αv� �

, or e αu−1ð ÞE ln 1þ~gð Þ½ �=E 1þ ~gð Þ−1h i

when αv=1, replaces(1+g)α on the right-hand side to incorporate the uncertainty in con-sumption growth and to distinguish between αu and αv.

To infer ρ from the Generalized Uncertainty Ramsey Rule, we needestimates of iB, αu and αv, as well as the distribution of 1þ ~g . Leaving adetailed discussion of the parameter values to the next section, we pre-sent here a more operational formula for the Generalized UncertaintyRamsey Rule when 1þ ~g is assumed to follow a lognormal distribution(see Appendix B for a proof of the proposition).

Proposition 2. (Generalized Uncertainty Ramsey Rule with lognor-mally distributed consumption growth). When 1þ ~g follows a lognor-mal distribution, the Generalized Uncertainty Ramsey Rule becomes

ln 1þ iBð Þ ¼ ln 1þ ρð Þ þ αuμ þ 12σ2 αu−αv−αuαvð Þ; ð5′Þ

where μ and σ2 are the mean and variance of the normal distributionln 1þ ~gð Þ, respectively.12

As a special case of Eqs. (5) and (5′), we have by imposingαu=αv≡α that

1þ iB ¼ 1þ ρE 1þ ~gð Þ−α½ � ð6Þ

and

ln 1þ iBð Þ ¼ ln 1þ ρð Þ þ αμ−12α2σ2

; ð6′Þ

where Eq. (6) will be referred to as the Simple Uncertainty RamseyRule, and Eq. (6′) is its more operational form when 1þ ~g is assumedto follow a lognormal distribution.13

Bradford (2003) and Howarth (2003) used Eq. (6) as a constraintthat must be satisfied by parameters ρ, iB, α and those depictingconsumption growth. The Simple Uncertainty Ramsey Rule explicitlyincorporates uncertainty and hence represents an improvement onthe well-known Deterministic Ramsey Rule (Eq. (1)). However,αu=αv is too strong an assumption that is not supported by empiricalevidence.14 Further, as explained in Weitzman (2007a), the

12 Unlike Eq. (5), Eq. (5′) is a uniform formula for all αv>0.13 See Mehra (2003) for a direct derivation of Eq. (6′).14 For example, Barsky et al. (1997) found no significant relationship, either statisti-cally or economically, between the two attitudes. In addition, Atkinson et al. (2009)found that correlations among the three preference parameters that respectively gov-ern risk aversion, intertemporal substitution and inequality aversion are weak.

conventional model of investment under uncertainty that does notseparate αu and αv, based on which Eq. (6) is derived, cannot organizedata on asset returns and consumption growth, as a manifestation ofthe equity premium puzzle and the closely-related risk-free rate puz-zle.15 An implication of Weitzman's critique of the Simple UncertaintyRamsey Rule is that ρ should be more appropriately inferred with theGeneralized Uncertainty Ramsey Rule Eq. (5) or Eq. (5′) derived froma saving-portfolio model that separates the attitude towards time andthat towards risk.16

3. Numerical Results

In this section, we use the Generalized Uncertainty Ramsey RuleEq. (5) or Eq. (5′) to infer the rate of pure time preference ρ. To doso, we need estimates of the risk-free rate iB, the reciprocal of theelasticity of intertemporal substitution αu, the coefficient of relativerisk aversion αv, as well as the distribution of the gross consumptiongrowth rate 1þ ~g .

The data on asset returns and consumption growth are fromthe U.S. economy for the 1889–1978 period. Table 1 lists the rele-vant parameter values produced in Mehra and Prescott (1985) forthat period. The values of E 1þ ~gð Þ and var 1þ ~gð Þ are based ontheir estimates of the mean and standard deviation of the annualper-capita consumption growth rate ~g , which are 1.8% and 3.6%,respectively. If 1þ ~g follows a lognormal distribution, thenln 1þ ~gð Þ follows a normal distribution. The mean (μ) and variance(σ2) of the normal distribution can be calculated from E 1þ ~gð Þand var 1þ ~gð Þ using Fact 2 in Appendix C.

In contrast, there is little empirical consensus about the values ofαu and αv. Mankiw et al's. (1985) estimates of αu center around 0.3,under the assumption that consumption and leisure are separable inthe utility function. Ogaki and Reinhart's (1998) estimates are around0.35. Attanasio and Weber (1989) obtained an estimate of 0.51, andHansen and Singleton (1983) had it around 1. On the other hand,Epstein and Zin (1991) estimated αu to be between 1.25 and 5,Barsky et al. (1997) estimated it to be around 5, whereas Hall's(1988) estimates are over 10. Based on these estimates, we considera range of αu from 0.5 to 5.17

poral substitution and risk aversion in utility specification has been proposed tosolve the asset pricing puzzles (see Epstein and Zin, 1991; Weil, 1989). For other ap-proaches to solving these puzzles, see Kocherlakota (1996) and Mehra (2003) fortwo comprehensive surveys.17 Note that αu cannot be empirically estimated independently of the value of ρ, be-cause both parameters are about the time dimension of the preferences. For example,Mankiw et al. (1985) also estimated ρ to be around 0.3%; Attanasio and Weber (1989)put their central estimate of ρ at 2.8%; Epstein and Zin (1991) had their estimates of ρbetween −0.74% and 0.37%; Barsky et al. (1997) suggested a negative ρ without pin-ning down its value. In contrast, this paper studies an issue often raised in environmen-tal economics regarding the value of ρ that is implied by a Ramsey Rule givenacceptable estimates of α (or αu and αv) .

Table 2Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1þ ~g following a lognormal distribution, iB=0.8%).

αvαu 0.5 1 2 3 4 5 6 7 8 9 10

0.5 0.0% 0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.7% 0.8% 0.8%1 −0.9% −0.9% −0.7% −0.6% −0.5% −0.4% −0.2% −0.1% 0.0% 0.1% 0.3%1.5 −1.8% −1.7% −1.6% −1.4% −1.2% −1.1% −0.9% −0.8% −0.6% −0.5% −0.3%2 −2.6% −2.5% −2.4% −2.2% −2.0% −1.8% −1.6% −1.4% −1.3% −1.1% −0.9%2.5 −3.5% −3.4% −3.2% −3.0% −2.7% −2.5% −2.3% −2.1% −1.9% −1.7% −1.5%3 −4.3% −4.2% −4.0% −3.7% −3.5% −3.2% −3.0% −2.8% −2.5% −2.3% −2.0%3.5 −5.2% −5.0% −4.8% −4.5% −4.2% −4.0% −3.7% −3.4% −3.1% −2.9% −2.6%4 −6.0% −5.8% −5.5% −5.3% −5.0% −4.7% −4.4% −4.1% −3.8% −3.5% −3.2%4.5 −6.8% −6.6% −6.3% −6.0% −5.7% −5.4% −5.0% −4.7% −4.4% −4.0% −3.7%5 −7.6% −7.4% −7.1% −6.8% −6.4% −6.1% −5.7% −5.3% −5.0% −4.7% −4.3%

18 For example, reflecting various long-term government borrowing rates, the federalgovernment used a real interest rate of 2.75% in 2009 and a real interest rate of 2.6% in2010 to determine the pension liabilities to federal employees and veterans, and SocialSecurity Trustees have assumed a 2.9% real interest rate in recent years for Social Secu-rity trust funds (Social Security Trustees, 2008–2011). Note that these long-term bor-rowing rates are before personal income taxes, and the corresponding after-tax long-term government borrowing rates should be lower. For the purpose of this paper, iBshould be understood as the after-tax risk-free bond rate. iB=1.5% can be looked uponas an average of short-term and long-term after-tax Treasury bond interest rates.

30 L. Liu / Ecological Economics 74 (2012) 27–33

The estimates of αv are similarly divergent. Metrick (1995) foundnear risk-neutrality (αv=0) but Beetsma and Schotman (2001) esti-mated αv to be around 7 in a game show environment. Kaplow(2005) estimated it to be around 0.5 based on labor market re-sponses. Chetty (2006) established a relationship between αv andlabor supply elasticities and found existing estimates of labor sup-ply elasticities imply a mean value of αv at 0.71. Using asset returndata, Epstein and Zin (1991) put αv at 1 but Attanasio and Weber(1989) put it at 30. Barsky et al. (1997) obtained a mean estimateof αvat 4 using survey data. In addition, resolving the asset returnpuzzle found in Mehra and Prescott (1985) generally requires avalue of αv greater than 10. In the end, we consider a range of αv

from 0.5 to 10.Table 2 presents the central numerical results of the paper, in

which ρ is inferred using Eq. (5′) (i.e., assuming that 1þ ~g follows alognormal distribution) and by letting iB, μ and σ2 take their respec-tive values according to Table 1.

Two main findings can be readily seen from Table 2. First, for αu

equal to 0.5 or 1, ρ lies within ±1% from zero for the consideredrange of αv. Second, for larger αu, ρ tends to lie in the negative zone.These results indicate that the rate of pure time preference inferredfrom the Generalized Uncertainty Ramsey Rule should not be signifi-cantly larger than zero and could easily be negative, contradicting thewidely-held belief in the environmental economics literature thatdata on market interest rates and consumption growth would implya value of ρ that is significantly larger than the one chosen on moralgrounds. Further, since αu=1 is by far the default choice in environ-mental policy analyses, the first main finding above suggests that it isappropriate to use ρ=0 as a benchmark for these long-run economicanalyses.

There is also a third finding that deserves mentioning. Accordingto the Deterministic Ramsey Rule (Eq. (1)), a larger α implies a smal-ler ρ. In contrast, Table 2 (which is based on the Generalized Uncer-tainty Ramsey Rule) indicates that ρ must increase in αv anddecrease in αu, highlighting the differential effects on ρ from param-eters αu and αv.

These findings are obtained with specific parameter values onconsumption growth and the risk-free rate (see Table 1), and withthe assumption that the gross consumption growth rate 1þ ~g fol-lows the lognormal distribution. In the following, we investigatehow sensitive these findings are to the chosen risk-free rate andthe assumption of the lognormally distributed consumptiongrowth.

In our first sensitivity analysis, we increase iB from 0.8% to 1.5%while maintaining the lognormal distribution. We (only) consider alarger value of iB for a sensitivity analysis because a smaller iB wouldfurther push the value of ρ into the negative zone, strengtheningour findings in a way. Another reason for considering a larger iB isthat while the 0.8% risk-free rate is based on historical average ofshort-term Treasury bond interest rates, the real government long-

term borrowing rates fluctuated around 3% in recent years.18 The re-sults of this sensitivity analysis are reported in Table 3 in Appendix D.Comparing Table 3 with Table 2, the inferred ρ is almost uniformlyhigher by 1.5%–0.8%=0.7%. However, the main findings discussedabove for Table 2 have barely changed. We can still say: for αu lessthan or equal to 1.5, ρ lies within ±1.5% from zero for theconsidered range of values of αv; for larger αu, ρ tends to lie in thenegative zone.

The next step is to see how sensitive our numerical findings are tothe assumption of the lognormally distributed1þ ~g . We consider twoalternative distributions of 1þ ~g , both of which are consistent withMehra and Prescott's (1985) estimates that the mean and the stan-dard deviation of per capita consumption growth rate are respective-ly 1.8% and 3.6% per year. The alternative distributions of 1þ ~g andthe corresponding sensitivity analysis results are presented inAppendix E. As shown in Table 4 and Table 5, alternative distributionsof 1þ ~g would barely alter the inferred value of ρ quantitatively(compared with Table 2).

After these sensitivity analyses with respect to the value of iB andthe distribution of 1þ ~g , our conclusions remain the same: ρ=0 is areasonable benchmark for long-run policy analysis, and it is unlikelythat the ρ inferred from the Ramsey Rule would be significantly largerthan the one chosen on moral grounds.

4. Conclusion

The economics of climate change has recently focused attentionon the importance of the rate of pure time preference (ρ) in makingpolicy decisions with long-run consequences. While ρ=0 is oftenimposed on moral grounds in climate change studies, it is widely be-lieved in the environmental economics literature that the value of ρinferred from the Ramsey Rule based on observed market interestrates and consumption growth rates must be significantly largerthan zero. This belief is behind the criticism of the SternReview that centers on Stern's using the extreme combination ofρ=0 and α=1, where α is the elasticity of marginal utility ofconsumption.

This paper investigates how ρ should be inferred in an uncertainworld with multiple rates of return and a separation between the(reciprocal) of elasticity of intertemporal substitution (αu) and the co-efficient of relative risk aversion (αv). One main numerical finding is

31L. Liu / Ecological Economics 74 (2012) 27–33

that the combination of ρ=0 and αu≤1 is very much consistent withthe Generalized Uncertainty Ramsey Rule for all the values of αv withina plausible range. Another main numerical finding is that the value of ρtends to fall in the negative zone for larger values of αu, suggestingthat a value of ρ significantly larger than zero (say 3%) is extremelyunlikely.

Moreover, according to the Deterministic Ramsey Rule (Eq. (1)), alarger α implies a smaller ρ. In contrast, to be consistent with theGeneralized Uncertainty Ramsey Rule, ρ must increase in αv and de-crease in αu, highlighting the differential effects on ρ from parametersαu and αv.

Acknowledgments

I would like to thank the late David Bradford for raising and dis-cussing the question of inferring the rate of pure time preference inan uncertain world, and Richard Howarth, Charles Mason, JackMeyer, Andrew Rettenmaier and two referees for helpful commentsand suggestions on earlier drafts. Editorial assistance from CourtneyCollins and Jeremy Nighohossian are greatly appreciated.

Appendix A. Proof of Proposition 1

The first order conditions of utility maximization are

−u0 c0ð Þ þ 11þ ρ

⋅u′ mð Þ dmds0

¼ 0

dmdγ

¼ 0;ðA1Þ

where, from Eqs. (3) and (4),

dmds0

¼E v′ ~c1ð Þ 1þ iB þ γ ~iS−iB

� �h in ov′ mð Þ

dmdγ

¼s0E v′ ~c1ð Þ ~iS−iB

� �h iv′ mð Þ :

ðA2Þ

From Eqs. (A1) and (A2), we have

1þ iB ¼ 1þ ρð ÞE 1þ ~gð Þ−αv½ � ⋅

c0αv−αu

mαv−αu; ðA3Þ

where 1þ ~g ¼ ~c1=c0.

From Eq. (3),

m ¼ E ~c11−αv

� �h i 11−αv f or αv≠1

m ¼ eE ln~c1ð Þ f or αv ¼ 1

ðA4Þ

Substituting Eq. (A4) into Eq. (A3), the latter becomes

1þ iB ¼1þ ρð Þ E 1þ ~gð Þ1−αv

h in oαv−αu

αv−1

E 1þ ~gð Þ−αv½ � f or αv≠1

1þ iB ¼ 1þ ρð Þe αu−1ð ÞE ln 1þ~gð Þ½ �

E 1þ ~gð Þ−1� � f or αv ¼ 1

which is the Generalized Uncertainty Ramsey Rule (Eq. 5).

Appendix B. Proof of Proposition 2

Taking log of Eq. (5) and applying Fact 1 (see below), when αv≠1,

ln 1þ iBð Þ ¼ ln 1þ ρð Þ þ αv−αu

αv−1lnE 1þ ~gð Þ1−αv

h i− lnE 1þ ~gð Þ−αv

� �

¼ ln 1þ ρð Þ þ αuμ þ 12σ2 αu−αv−αuαvð Þ;

which is Eq. (5′).

When αv=1, on the other hand,

ln 1þ iBð Þ ¼ ln 1þ ρð Þ þ αu−1ð ÞE ln 1þ ~gð Þ½ �− lnE 1þ ~gð Þ−1h i

¼ ln 1þ ρð Þ þ αuμ−12σ2

;

which is also represented by Eq. (5′).

Appendix C. Some Facts about the Lognormal Distribution

Suppose that a random variable ~X follows a lognormal distribu-tion. Then, by definition, ln~X follows a normal distribution. Denote

the mean and variance of ~X as E ~X� �

and var ~X� �

, and the mean and

variance of ln~X as μ and σ2, respectively.

Fact 1. E ~Xλ� �

¼ eλμþ12λ

2σ2 .

This well-known fact about the lognormal distribution can be

readily shown by noting that ~Xλ ¼ e ln ~X λð Þ and ln ~Xλ� �

follows the

normal distribution with mean λμ and variance λ2σ2.

Fact 2. μ and σ2 are related to E ~X� �

and var ~X� �

through

μ ¼ ln E ~X� �h i

−12σ2

σ2 ¼ ln 1þvar ~X

� �

E ~X� �h i2

8><>:

9>=>;

Proof. First,E ~X� �

¼ eμþ12σ

2from Fact 1. Soμ ¼ ln E ~X

� �h i− 1

2σ2. Sec-

ond, var ~X� �

¼ E ~X2� �

− E ~X� �h i2

¼ e2μþ2σ2 −e2μþσ2 ¼ E ~X� �h i2

eσ2−1

� �.

Therefore, σ2 ¼ ln 1þ var ~Xð ÞE ~Xð Þ½ �2

� �:.

32 L. Liu / Ecological Economics 74 (2012) 27–33

Appendix D. Sensitivity Analysis with Respect to iB

Table 3Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1þ ~g following a lognormal distribution, iB=1.5%).

αvαu 0.5 1 2 3 4 5 6 7 8 9 10

0.5 0.6% 0.7% 0.8% 0.9% 1.0% 1.1% 1.2% 1.3% 1.4% 1.5% 1.5%1 −0.2% −0.2% 0.0% 0.1% 0.2% 0.3% 0.5% 0.6% 0.7% 0.8% 1.0%1.5 −1.1% −1.0% −0.9% −0.7% −0.6% −0.4% −0.2% −0.1% 0.1% 0.2% 0.4%2 −2.0% −1.9% −1.7% −1.5% −1.3% −1.1% −0.9% −0.8% −0.6% −0.4% −0.2%2.5 −2.8% −2.7% −2.5% −2.3% −2.1% −1.9% −1.6% −1.4% −1.2% −1.0% −0.8%3 −3.7% −3.5% −3.3% −3.1% −2.8% −2.6% −2.3% −2.1% −1.8% −1.6% −1.3%3.5 −4.5% −4.4% −4.1% −3.8% −3.6% −3.3% −3.0% −2.7% −2.5% −2.2% −1.9%4 −5.3% −5.2% −4.9% −4.6% −4.3% −4.0% −3.7% −3.4% −3.1% −2.8% −2.5%4.5 −6.2% −6.0% −5.7% −5.4% −5.0% −4.7% −4.4% −4.0% −3.7% −3.4% −3.0%5 −7.0% −6.8% −6.5% −6.1% −5.8% −5.4% −5.0% −4.7% −4.3% −4.0% −3.6%

Table 4Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1þ ~g following distribution (E1), iB=0.8%).

αvαu 0.5 1 2 3 4 5 6 7 8 9 10

0.5 0.0% 0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.7% 0.7% 0.8%1 −0.9% −0.9% −0.7% −0.6% −0.5% −0.4% −0.2% −0.1% 0.0% 0.1% 0.3%1.5 −1.8% −1.7% −1.6% −1.4% −1.2% −1.1% −0.9% −0.8% −0.6% −0.5% −0.3%2 −2.6% −2.5% −2.4% −2.2% −2.0% −1.8% −1.6% −1.4% −1.3% −1.1% −0.9%2.5 −3.5% −3.4% −3.2% −3.0% −2.7% −2.5% −2.3% −2.1% −1.9% −1.7% −1.5%3 −4.3% −4.2% −4.0% −3.7% −3.5% −3.2% −3.0% −2.8% −2.5% −2.3% −2.0%3.5 −5.2% −5.0% −4.8% −4.5% −4.2% −4.0% −3.7% −3.4% −3.1% −2.9% −2.6%4 −6.0% −5.8% −5.6% −5.3% −5.0% −4.7% −4.4% −4.1% −3.8% −3.5% −3.2%4.5 −6.8% −6.7% −6.3% −6.0% −5.7% −5.4% −5.0% −4.7% −4.4% −4.0% −3.7%5 −7.6% −7.5% −7.1% −6.8% −6.4% −6.0% −5.7% −5.3% −5.0% −4.6% −4.3%

Table 5Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1þ ~g following distribution (E2), iB=0.8%).

αvαu 0.5 1 2 3 4 5 6 7 8 9 10

0.5 0.0% 0.0% 0.1% 0.2% 0.3% 0.4% 0.5% 0.6% 0.6% 0.7% 0.8%1 −0.9% −0.9% −0.7% −0.6% −0.5% −0.4% −0.2% −0.1% 0.0% 0.1% 0.2%1.5 −1.8% −1.7% −1.6% −1.4% −1.2% −1.1% −0.9% −0.8% −0.6% −0.5% −0.4%2 −2.6% −2.6% −2.4% −2.2% −2.0% −1.8% −1.6% −1.5% −1.3% −1.1% −0.9%2.5 −3.5% −3.4% −3.2% −3.0% −2.8% −2.5% −2.3% −2.1% −1.9% −1.7% −1.5%3 −4.3% −4.2% −4.0% −3.7% −3.5% −3.3% −3.0% −2.8% −2.6% −2.3% −2.1%3.5 −5.2% −5.0% −4.8% −4.5% −4.2% −4.0% −3.7% −3.4% −3.2% −2.9% −2.7%4 −6.0% −5.8% −5.6% −5.3% −5.0% −4.7% −4.4% −4.1% −3.8% −3.5% −3.2%4.5 −6.8% −6.7% −6.3% −6.0% −5.7% −5.4% −5.0% −4.7% −4.4% −4.1% −3.8%5 −7.6% −7.5% −7.1% −6.8% −6.4% −6.1% −5.7% −5.4% −5.0% −4.7% −4.4%

Appendix E. Sensitivity Analysis with Respect to the Distributionof 1þ ~g

Specifically, we consider the following two alternative distribu-tions of 1þ ~g:

1þ ~g ¼1:018þ 0:051 with probability 1=41:018 with probability 1=2

1:018−0:051 with probability 1=4

8<: ðE1Þ

and

1þ ~g ¼ 1:018þ 0:036 with probability 1=21:018−0:036 with probability 1=2

�ðE2Þ

The difference between the two alternative distributions lies inthe thickness of tails, with (E1) having thicker tails than Eq. (E2).19

19 Barro (2006), Rietz (1988) and Weitzman (2007b) suggested that the super-thintails of 1þ ~g implicit in studies finding the equity premium puzzle may be the heartof the puzzle, and showed that the puzzle can be solved by modifying probability dis-tributions to admit rare but disastrous events.

These alternative distributions are chosen for their simplicity, notfor realism, because we need to use the more demanding Eq. (5) rath-er than the simpler Eq. (5′) to infer ρ when 1þ ~g is not lognormal.However, they are sufficient to suggest, as shown in Table 4 andTable 5 below, that alternative distributions of 1þ ~g would barelyalter the inferred value of ρ.

References

Ackerman, F., DeCanio, S.J., Howarth, R.B., Sheeran, K., 2009. Limitations of integratedassessment models of climate change. Climatic Change 95, 297–315.

Atkinson, G., Dietz, S., Helgeson, J., Hepburn, C., Soelen, H., 2009. Siblings, not triplets:social preferences for risk, inequality and time in discounting climate change.Economics. doi:10.5018/economics-ejournal.ja.2009-26.

Attanasio, O.P., Weber, G., 1989. Intertemporal substitution, risk aversion and the Eulerequation for consumption. The Economic Journal 99, 59–73.

Azar, C., Sterner, T., 1996. Discounting and distributional considerations in the contextof global warming. Ecological Economics 19 (2), 169–184.

Barro, R.J., 2006. Rare disasters and asset markets in the twentieth century. QuarterlyJournal of Economics 121, 823–866.

Barsky, R.B., Juster, F.T., Kimball, M.S., Shapiro, M.D., 1997. Preference parameters andbehavioral heterogeneity: an experimental approach in the health and retirementstudy. Quarterly Journal of Economics 112, 537–579.

33L. Liu / Ecological Economics 74 (2012) 27–33

Beetsma, R., Schotman, P., 2001. Measuring risk attitudes in a natural experiment: datafrom the television game show lingo. The Economic Journal 111, 821–848.

Bradford, D.F., 2003. Are we doing our duty towards future generations? unpublishedmanuscript (dated October 25, 2003).

Browning, M., Hansen, L.P., Heckman, J.J., 1999. Micro data and general equilibriummodels, In: Taylor, J.B., Woodford, M. (Eds.), Handbook of Macroeconomics,vol. 1A. Elsevier Sci, Amsterdam.

Chetty, R., 2006. A newmethod of estimating risk aversion. American Economic Review96, 1821–1834.

Cline, W.R., 1992. The Economics of Global Warming. Institute for InternationalEconomics, Washington, DC.

Cochrane, J.H., 2005. Asset Pricing, revised edition. Princeton University Press,Princeton.

Crost, B., Traeger, C., 2010. Risk and aversion in the integrated assessment of climatechange. CUDARE Working Paper 1104. Berkeley, CA, Department of Agricultural& Resources Economics, UC Berkeley.

Epstein, L.G., Zin, S.E., 1991. Substitution, risk aversion, and the temporal behavior ofconsumption growth and asset returns: an empirical analysis. Journal of PoliticalEconomy 99, 263–286.

Gollier, C., 2001. The Economics of Risk and Time. TheMIT Press, CambridgeMassachusettsand London England.

Gollier, C., 2002. Discounting an uncertain future. Journal of Public Economics 85,149–166.

Gomme, P., Rupert, P., 2007. Theory, measurement and calibration of macroeconomicmodels. Journal of Monetary Economics 54, 460–497.

Hall, R.E., 1988. Intertemporal substitution in consumption. Journal of Political Economy96, 339–357.

Hansen, L.P., Singleton, K.J., 1983. Stochastic consumption, risk aversion, and thetemporal behavior of asset returns. Journal of Political Economy 91, 249–265.

Hoel, M., Sterner, T., 2007. Discounting and relative prices. Climatic Change 84,265–280.

Howarth, R.B., 1996. Climate change and overlapping generations. ContemporaryEconomic Policy 14 (4), 100–111.

Howarth, R.B., 2003. Discounting and uncertainty in climate change policy analysis.Land Economics 79, 369–381.

Kaplow, L., 2005. The value of a statistical life and the coefficient of relative riskaversion. Journal of Risk and Uncertainty 31, 23–34.

Kocherlakota, N.R., 1996. The equity premium: it's still a puzzle. Journal of EconomicLiterature 34, 42–71.

Kousky, C., Kopp, R.E., Cooke, R.M., 2011. Risk premia and the social cost of carbon: areview. Economics Discussion Paper, No. 2011–19.

Kreps, D.M., Porteus, E.L., 1978. Temporal resolution of uncertainty and dynamic choicetheory. Econometrica 46, 185–200.

Mankiw, N.G., Rotemberg, J.J., Summers, L.H., 1985. Intertemporal substitution inmacroeconomics. Quarterly Journal of Economics 99, 225–251.

Mehra, R., 2003. The equity premium: why is it a puzzle? Financial Analysts Journal 59,54–69.

Mehra, R., Prescott, E.C., 1985. The equity premium: a puzzle. Journal of MonetaryEconomics 15, 145–161.

Metrick, A., 1995. A natural experiment in ‘Jeopardy!’ American Economic Review. 85,240–53.

Meyer, D., Meyer, J., 2005. Risk preferences in multi-period consumption models, the eq-uity premium puzzle, and habit formation utility. Journal of Monetary Economics 52,1497–1515.

Nordhaus, W.D., 1994. Managing the Global Commons: The Economics of ClimateChange. The MIT Press, Cambridge, Massachusetts.

Nordhaus, W.D., 1999. Discounting and public policies that affect the distant future.In: Portney, P.R., Weyant, J.P. (Eds.), Discounting and Intergenerational Equity.Resources for the Future, Washington, DC.

Nordhaus, W.D., 2007. A review of the Stern review on the economics of climatechange. Journal of Economic Literature 45, 686–702.

Nordhaus, W.D., 2008. A Question of Balance: Weighing the Options on GlobalWarming Policies. Yale University Press, New Haven and London.

Ogaki, M., Reinhart, C.M., 1998. Measuring intertemporal substitution: the role of durablegoods. Journal of Political Economy 106, 1078–1098.

Quiggin, J., 2008. Stern and his critics on discounting and climate change: an editorialessay. Climatic Change 89, 195–205.

Ramsey, F., 1928. A mathematical theory of savings. The Economic Journal 38, 543–559.Rietz, T.A., 1988. The equity risk premium: a solution. Journal of Monetary Economics

22, 117–131.Social Security Trustees, 2008–2011. The Annual Report of the Board of Trustees of the

Federal Old-Age and Survivors Insurance and Federal Disability Insurance TrustFunds. Washington, DC.

Stern, N., 2007. The Economics of Climate Change: The Stern Review. Cambridge UniversityPress, Cambridge.

Traeger, C., 2009. Recent developments in the intertemporal modeling of uncertainty.Annual Review of Resource Economics 1 (1), 261–286.

Weil, P., 1989. The equity premium puzzle and the risk-free rate puzzle. Journal ofMonetary Economics 24, 401–421.

Weitzman, M., 2007a. A review of the Stern review on the economics of climatechange. Journal of Economic Literature 45, 703–724.

Weitzman, M., 2007b. Subjective expectations and asset-return puzzles. The AmericanEconomic Review 97, 1102–1130.

Weitzman, M., 2009. On modeling and interpreting the economics of catastrophicclimate change. The Review of Economics and Statistics 91, 1–19.

Zeckhauser, R.J., Viscusi, W.K., 2008. Discounting dilemmas: editors' introduction. Journalof Risk and Uncertainty. 37, 95–106.