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Inferring the rate of pure time preference under uncertainty
Ecological Economics 74 (2012) 27–33
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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
a r t i c l e
i n f o
Article history: Received 31 August 2011 Received in revised form 7 November 2011 Accepted 11 November 2011 Available online 20 December 2011 Keywords: Time preference Discounting Ramsey Rule Uncertainty
a b s t r a c t This paper studies how to infer the rate of pure time preference (ρ) from the Ramsey Rule when multiple asset returns exist due to uncertainty. Using a Generalized Uncertainty Ramsey Rule derived from a model that separates intertemporal substitution and risk aversion, we find that the U.S. historical data on consumption growth and asset returns imply that (i) for the reciprocal of the elasticity of intertemporal substitution less than or equal to one, ρ lies within ± 1% from zero for a plausible range of the coefficient of relative risk aversion; and (ii) for the larger reciprocal of the elasticity of intertemporal substitution, ρ tends to be negative. These results contradict the widely-held belief in the environmental economics literature that the inferred ρ must be significantly larger than zero and suggest that it is appropriate to use ρ = 0 as a benchmark for 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 future instantaneous utility – plays a critical role in long-horizon economic analyses. Greenhouse gas (GHG) abatement policy modeling in the area of climate change is an example. Cline (1992) assumed ρ = 0 and found that a 40% reduction in GHG emissions is optimal. In contrast, Nordhaus (1994) assumed ρ = 3% and found that only modest emissions abatement can be justified on economic grounds. According to Nordhaus (1994, 1999, 2008), although the DICE model (Dynamic Integrated model of Climate and the Economy) had been through many versions, its main sensitivity is to the rate of pure time preference. More recently, Stern (2007) essentially let ρ = 0, but his rejection of discounting future utility has drawn criticism. 1 There has been some controversy over how the value of ρ should be 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 with revealed intertemporal choices. 2 Specifically, the latter school
⁎ Tel.: + 1 979 845 7723; fax: + 1 979 845 6636. E-mail address: [email protected]. 1 ρ is arbitrarily set at 0.1% in the Stern Review to represent the small risk of the extinction of the human race. For reviews of the Stern Review with a focus on discounting, see Nordhaus (2007); Weitzman (2007a). For a review of Stern's critics, see Quiggin (2008). 2 A criticism of the revealed preferences approach is that observed intertemporal choices in the marketplace are based on individual time preferences, which may not be the same as those of the social planner (see Azar and Sterner, 1996; Howarth, 1996). In this paper, we abstract from any divergence between individual and social time preferences by assuming a representative individual (the social decision-maker). 0921-8009/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2011.11.007
believes that ρ should be determined according to the following equilibrium condition for the Ramsey growth model (the Deterministic Ramsey Rule):
α
1 þ r ¼ ð1 þ ρÞð1 þ g Þ ;
ð1Þ
where r is the rate of return on capital, α is the (absolute value of the) elasticity of the marginal 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 value of r and g (which are observable in principle) and have a reasonable estimate of α. The issue of inferring ρ is different from, though related to, the issue of choosing a discount rate. In terms of the Deterministic Ramsey Rule, r represents the discount rate that should be used for future consumption-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 the continuous-time counterpart of Eq. (1) for the discrete-time model. In this paper we work with a discrete-time model. Note that r = ρ + αg can also be understood as a linearization of Eq. (1) with respect to g, which makes it an approximate formula suitable for use in discrete-time models when g is sufficiently small. 4 Howarth (2003) focused on the discount rate choice in the environment of uncertainty with an eye on climate change. See Zeckhauser and Viscusi (2008) for a survey of recent 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 accompanied by corresponding choices of α so that the same r is reached according 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 parameter in the equation. The value of g is relatively well understood. For example, Mehra and Prescott (1985) estimated the average annual growth rate of real per-capita consumption to be 1.8% based on the annual U.S. data from 1889 to 1978. However, determining the values of r and α is less straightforward. First, a question arises as to which r should be put in the left-hand side of Eq. (1) because of the coexistence of multiple asset returns. Different values of r would generate different values of ρ given α and g. For example, suppose α = 1 (corresponding to the logarithmic instantaneous utility function) and g = 1.8%. Then according to Eq. (1), r = 7.0% (corresponding to the mean equity return) implies ρ = 5%, but r = 0.8% (corresponding to the risk-free rate) implies ρ = − 1%. 6 Second, estimates of α are very diverse, and its appropriate value depends on the context in which it is applied. Specifically, the reciprocal of α, namely, the elasticity of intertemporal substitution, is the parameter of focus in the macro literature on life-cycle consumption. Letting α be one (i.e., the logarithmic utility) is the preferred choice in macroeconomic models, which seems to be due to the evidence of relatively strong intertemporal substitution as well as convenience considerations. 7 On the other hand, α itself, as the coefficient of relative risk aversion, is the parameter of focus in the finance literature on asset returns and risks. Data in this context imply a value of α much larger than one. 8 The literature on climate change economics appears to view α from the perspective of intertemporal substitution by adopting a value of α between one and two. For example, α = 1 in Nordhaus (1994) and Stern (2007), α = 1.5 in Cline (1992) and α = 2 in Nordhaus (2008). However, larger values of α consistent with financial data should also be relevant when capital market uncertainty is explicitly considered. 9 These conceptual difficulties surrounding using the Deterministic Ramsey Rule to infer ρ suggest that one should incorporate uncertainty and multiple asset returns into the analysis, and separate the two preference parameters traditionally represented by a single parameter α. This paper investigates the issue of how to infer the rate of pure time preference in a saving-portfolio model, adopting a utility specification 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 implications of different combinations of time discount rates and consumption elasticities.” On the other hand, the implications of different combinations of ρ and α given r for near-term decisions are relatively less important. 6 The data on asset returns are from Mehra and Prescott (1985). That the Deterministic Ramsey Rule (Eq. (1)) is inapplicable in a world of uncertainty with multiple asset classes 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 elasticity of substitution in demand between produced goods and environmental services is low and the produced goods sector grows faster than the environmental services sector. 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 utility function for consumption and that for the value function for wealth, and found that the former could be 1.25 to 10 times the latter. In this paper, the utility function is for consumption. 9 More recently, models that separate risk aversion from intertemporal substitution were incorporated in environmental studies (Kousky et al., 2011; Traeger, 2009). For example, Crost and Traeger (2010) incorporated a utility specification that disentangles the two attitudes into a DICE type model.
intertemporal substitution. Based on the Generalized Uncertainty Ramsey Rule derived from this setup, we find that inferring ρ in a way consistent with the U.S. historical data on consumption growth and asset returns yields the following main numerical findings: (i) for αu (the reciprocal of the elasticity of intertemporal substitution) equal to or less than one, the value of ρ lies within ± 1% from zero regardless 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 environmental economics literature that the inferred ρ must be significantly larger than zero, and suggest that it is appropriate to use ρ = 0 as a benchmark for long-term economic analysis. 2. The Generalized Uncertainty Ramsey Rule Following Gollier (2002), we adopt a simple version of the utility specification that distinguishes between risk aversion and intertemporal substitutability of consumption due to Epstein and Zin (1991) and Kreps and Porteus (1978).10 The preferences are jointly represented by two utility functions: u(⋅) captures the preferences for intertemporal substitution and v(⋅) captures the preferences for risk. Both functions are assumed to exhibit constant elasticity of marginal utility with respect to consumption, with the constant elasticity (absolute value) being αu (which is also the reciprocal of the elasticity of intertemporal substitution) and αv (which is also the coefficient of relative 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 special limiting cases, u(c) = ln c when αu = 1 and v(c) = ln c when αv = 1. The representative individual's overall intertemporal utility is given by uðc0 Þ þ
uðmÞ ; 1þρ
ð2Þ
where c0 is consumption at t = 0, and m is the certainty equivalent consumption at t = 1, which is determined by vðmÞ ¼ Evðc~1 Þ;
ð3Þ
where c~1 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 are only two assets: a risky asset (stocks) and a riskless asset (risk-free bonds).11 The former asset has a random rate of return ĩSand the latter has a certain rate of return iB. In the standard saving-portfolio problem, a representative consumer chooses savings in period 0,s0, and the fraction of savings allocated to the risky asset, γ, to maximize the overall utility Eq. (2), subject to Eq. (3) and c0 ¼ W 0 −s0 h i c~1 ¼ s0 1 þ iB þ γ ~i S −iB ;
ð4Þ
where W0 is the initial wealth. From the first-order conditions of the individual's optimization problem, we obtain the following Generalized Uncertainty Ramsey Rule (see Appendix A for a proof of the proposition).
10 See (Gollier 2001, chapter 20) for a detailed description of the simple version of Kreps–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.
L. Liu / Ecological Economics 74 (2012) 27–33
Proposition 1. (Generalized Uncertainty Ramsey Rule). Intertemporal utility maximization implies,
1 þ iB ¼ 1 þ iB ¼
α −α u n h io v 1−α v α v −1 ð1 þ ρÞ E ð1 þ g~ Þ E½ð1 þ g~ Þ−α v
f or
α v ≠1
f or
αv ¼ 1
ðα −1ÞE½ lnð1þg~ Þ
ð1 þ ρÞe u E ð1 þ g~ Þ−1
From the Deterministic Ramsey Rule (Eq. (1)) to the Generalized Uncertainty Ramsey Rule (Eq. (5)), the rate of return r, which is ambiguous when multiple asset returns exist in the capital market, is replaced n h ioααv −α u v −1 1−α v with iB, the risk-free rate. On the other hand, E ð1 þ g~ Þ = h i −α −1 E ð1 þ g~ Þ v , or eðα u −1ÞE½ lnð1þg~ Þ =E ð1 þ g~ Þ when αv =1, replaces (1+g)α on the right-hand side to incorporate the uncertainty in consumption growth and to distinguish between αu and αv. To infer ρ from the Generalized Uncertainty Ramsey Rule, we need estimates of iB, αu and αv, as well as the distribution of 1 þ g~ . Leaving a detailed discussion of the parameter values to the next section, we present here a more operational formula for the Generalized Uncertainty Ramsey Rule when 1 þ g~ is assumed to follow a lognormal distribution (see Appendix B for a proof of the proposition). Proposition 2. (Generalized Uncertainty Ramsey Rule with lognormally distributed consumption growth). When 1 þ g~ follows a lognormal distribution, the Generalized Uncertainty Ramsey Rule becomes ð5′Þ
where μ and σ2 are the mean and variance of the normal distribution lnð1 þ g~ Þ, respectively.12 As a special case of Eqs. (5) and (5′), we have by imposing αu = αv ≡ α that 1 þ iB ¼
1þρ E½ð1 þ g~ Þ−α
ð6Þ
and 1 2 2 lnð1 þ iB Þ ¼ lnð1 þ ρÞ þ αμ− α σ ; 2
ð6′Þ
where Eq. (6) will be referred to as the Simple Uncertainty Ramsey Rule, and Eq. (6′) is its more operational form when 1 þ g~ is assumed to follow a lognormal distribution. 13 Bradford (2003) and Howarth (2003) used Eq. (6) as a constraint that must be satisfied by parameters ρ, iB, α and those depicting consumption growth. The Simple Uncertainty Ramsey Rule explicitly incorporates uncertainty and hence represents an improvement on the well-known Deterministic Ramsey Rule (Eq. (1)). However, αu = αv is too strong an assumption that is not supported by empirical evidence. 14 Further, as explained in Weitzman (2007a), the
12
Unlike Eq. (5), Eq. (5′) is a uniform formula for all αv > 0. See Mehra (2003) for a direct derivation of Eq. (6′). For example, Barsky et al. (1997) found no significant relationship, either statistically or economically, between the two attitudes. In addition, Atkinson et al. (2009) found that correlations among the three preference parameters that respectively govern risk aversion, intertemporal substitution and inequality aversion are weak. 13 14
Table 1 Parameter values for asset returns and consumption growth (The U.S. economy, 1889–1978). iB
i¯S
Eð1 þ g~ Þ
varð1 þ g~ Þ
μ≡E½ lnð1 þ g~ Þ
σ 2 ≡ var½ lnð1 þ g~ Þ
0.8%
7.0%
1.018
0.00130
0.0172
0.00125
ð5Þ
where 1 þ g~ ≡ c~1 =c0 :
1 2 lnð1 þ iB Þ ¼ lnð1 þ ρÞ þ α u μ þ σ ðα u −α v −α u α v Þ; 2
29
conventional model of investment under uncertainty that does not separate αu and αv, based on which Eq. (6) is derived, cannot organize data on asset returns and consumption growth, as a manifestation of the equity premium puzzle and the closely-related risk-free rate puzzle. 15 An implication of Weitzman's critique of the Simple Uncertainty Ramsey Rule is that ρ should be more appropriately inferred with the Generalized Uncertainty Ramsey Rule Eq. (5) or Eq. (5′) derived from a saving-portfolio model that separates the attitude towards time and that towards risk. 16
3. Numerical Results In this section, we use the Generalized Uncertainty Ramsey Rule Eq. (5) or Eq. (5′) to infer the rate of pure time preference ρ. To do so, we need estimates of the risk-free rate iB, the reciprocal of the elasticity of intertemporal substitution αu, the coefficient of relative risk aversion αv, as well as the distribution of the gross consumption growth rate 1 þ g~ . The data on asset returns and consumption growth are from the U.S. economy for the 1889–1978 period. Table 1 lists the relevant parameter values produced in Mehra and Prescott (1985) for that period. The values of Eð1 þ g~ Þ and varð1 þ g~ Þ are based on their estimates of the mean and standard deviation of the annual per-capita consumption growth rate g~ , which are 1.8% and 3.6%, respectively. If 1 þ g~ follows a lognormal distribution, then lnð1 þ g~ Þ follows a normal distribution. The mean (μ) and variance (σ 2) of the normal distribution can be calculated from Eð1 þ g~ Þ and varð1 þ g~ Þ using Fact 2 in Appendix C. In contrast, there is little empirical consensus about the values of αu and αv. Mankiw et al's. (1985) estimates of αu center around 0.3, under the assumption that consumption and leisure are separable in the utility function. Ogaki and Reinhart's (1998) estimates are around 0.35. Attanasio and Weber (1989) obtained an estimate of 0.51, and Hansen 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 consider a range of αu from 0.5 to 5. 17
15
See Mehra and Prescott (1985) and Weil (1989). Quiggin (2008) also claimed, in general, that failure to account for the equity premium puzzle can lead to inconsistent policy recommendations. Separating intertemporal substitution and risk aversion in utility specification has been proposed to solve the asset pricing puzzles (see Epstein and Zin, 1991; Weil, 1989). For other approaches to solving these puzzles, see Kocherlakota (1996) and Mehra (2003) for two comprehensive surveys. 17 Note that αu cannot be empirically estimated independently of the value of ρ, because 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 ρ between − 0.74% and 0.37%; Barsky et al. (1997) suggested a negative ρ without pinning down its value. In contrast, this paper studies an issue often raised in environmental economics regarding the value of ρ that is implied by a Ramsey Rule given acceptable estimates of α (or αu and αv) . 16
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L. Liu / Ecological Economics 74 (2012) 27–33
Table 2 Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1 þ g~ following a lognormal distribution, iB = 0.8%). αu 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
αv
0.5
1
2
3
4
5
6
7
8
9
10
0.0% − 0.9% − 1.8% − 2.6% − 3.5% − 4.3% − 5.2% − 6.0% − 6.8% − 7.6%
0.0% − 0.9% − 1.7% − 2.5% − 3.4% − 4.2% − 5.0% − 5.8% − 6.6% − 7.4%
0.1% − 0.7% − 1.6% − 2.4% − 3.2% − 4.0% − 4.8% − 5.5% − 6.3% − 7.1%
0.2% − 0.6% − 1.4% − 2.2% − 3.0% − 3.7% − 4.5% − 5.3% − 6.0% − 6.8%
0.3% − 0.5% − 1.2% − 2.0% − 2.7% − 3.5% − 4.2% − 5.0% − 5.7% − 6.4%
0.4% − 0.4% − 1.1% − 1.8% − 2.5% − 3.2% − 4.0% − 4.7% − 5.4% − 6.1%
0.5% − 0.2% − 0.9% − 1.6% − 2.3% − 3.0% − 3.7% − 4.4% − 5.0% − 5.7%
0.6% − 0.1% − 0.8% − 1.4% − 2.1% − 2.8% − 3.4% − 4.1% − 4.7% − 5.3%
0.7% 0.0% − 0.6% − 1.3% − 1.9% − 2.5% − 3.1% − 3.8% − 4.4% − 5.0%
0.8% 0.1% − 0.5% − 1.1% − 1.7% − 2.3% − 2.9% − 3.5% − 4.0% − 4.7%
0.8% 0.3% − 0.3% − 0.9% − 1.5% − 2.0% − 2.6% − 3.2% − 3.7% −4.3%
The estimates of αv are similarly divergent. Metrick (1995) found near risk-neutrality (αv = 0) but Beetsma and Schotman (2001) estimated αv to be around 7 in a game show environment. Kaplow (2005) estimated it to be around 0.5 based on labor market responses. Chetty (2006) established a relationship between αv and labor supply elasticities and found existing estimates of labor supply elasticities imply a mean value of αv at 0.71. Using asset return data, Epstein and Zin (1991) put αv at 1 but Attanasio and Weber (1989) put it at 30. Barsky et al. (1997) obtained a mean estimate of αvat 4 using survey data. In addition, resolving the asset return puzzle found in Mehra and Prescott (1985) generally requires a value 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 1 þ g~ follows a lognormal distribution) and by letting iB, μ and σ 2 take their respective 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 considered range of αv. Second, for larger αu, ρ tends to lie in the negative zone. These results indicate that the rate of pure time preference inferred from the Generalized Uncertainty Ramsey Rule should not be significantly larger than zero and could easily be negative, contradicting the widely-held belief in the environmental economics literature that data on market interest rates and consumption growth would imply a value of ρ that is significantly larger than the one chosen on moral grounds. Further, since αu = 1 is by far the default choice in environmental policy analyses, the first main finding above suggests that it is appropriate to use ρ = 0 as a benchmark for these long-run economic analyses. There is also a third finding that deserves mentioning. According to the Deterministic Ramsey Rule (Eq. (1)), a larger α implies a smaller ρ. In contrast, Table 2 (which is based on the Generalized Uncertainty Ramsey Rule) indicates that ρ must increase in αv and decrease in αu, highlighting the differential effects on ρ from parameters αu and αv. These findings are obtained with specific parameter values on consumption growth and the risk-free rate (see Table 1), and with the assumption that the gross consumption growth rate 1 þ g~ follows the lognormal distribution. In the following, we investigate how sensitive these findings are to the chosen risk-free rate and the assumption of the lognormally distributed consumption growth. In our first sensitivity analysis, we increase iB from 0.8% to 1.5% while maintaining the lognormal distribution. We (only) consider a larger value of iB for a sensitivity analysis because a smaller iB would further push the value of ρ into the negative zone, strengthening our findings in a way. Another reason for considering a larger iB is that while the 0.8% risk-free rate is based on historical average of short-term Treasury bond interest rates, the real government long-
term borrowing rates fluctuated around 3% in recent years. 18 The results of this sensitivity analysis are reported in Table 3 in Appendix D. Comparing Table 3 with Table 2, the inferred ρ is almost uniformly higher by 1.5%–0.8% = 0.7%. However, the main findings discussed above for Table 2 have barely changed. We can still say: for αu less than or equal to 1.5, ρ lies within ± 1.5% from zero for the considered range of values of αv; for larger αu, ρ tends to lie in the negative zone. The next step is to see how sensitive our numerical findings are to the assumption of the lognormally distributed 1 þ g~ . We consider two alternative distributions of 1 þ g~ , both of which are consistent with Mehra and Prescott's (1985) estimates that the mean and the standard deviation of per capita consumption growth rate are respectively 1.8% and 3.6% per year. The alternative distributions of 1 þ g~ and the corresponding sensitivity analysis results are presented in Appendix E. As shown in Table 4 and Table 5, alternative distributions of 1 þ 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 and the distribution of 1 þ g~ , our conclusions remain the same: ρ = 0 is a reasonable benchmark for long-run policy analysis, and it is unlikely that the ρ inferred from the Ramsey Rule would be significantly larger than the one chosen on moral grounds. 4. Conclusion The economics of climate change has recently focused attention on the importance of the rate of pure time preference (ρ) in making policy decisions with long-run consequences. While ρ = 0 is often imposed on moral grounds in climate change studies, it is widely believed in the environmental economics literature that the value of ρ inferred from the Ramsey Rule based on observed market interest rates and consumption growth rates must be significantly larger than zero. This belief is behind the criticism of the Stern Review that centers on Stern's using the extreme combination of ρ = 0 and α = 1, where α is the elasticity of marginal utility of consumption. This paper investigates how ρ should be inferred in an uncertain world with multiple rates of return and a separation between the (reciprocal) of elasticity of intertemporal substitution (αu) and the coefficient of relative risk aversion (αv). One main numerical finding is 18 For example, reflecting various long-term government borrowing rates, the federal government used a real interest rate of 2.75% in 2009 and a real interest rate of 2.6% in 2010 to determine the pension liabilities to federal employees and veterans, and Social Security Trustees have assumed a 2.9% real interest rate in recent years for Social Security trust funds (Social Security Trustees, 2008–2011). Note that these long-term borrowing rates are before personal income taxes, and the corresponding after-tax longterm government borrowing rates should be lower. For the purpose of this paper, iB should be understood as the after-tax risk-free bond rate. iB = 1.5% can be looked upon as an average of short-term and long-term after-tax Treasury bond interest rates.
L. Liu / Ecological Economics 74 (2012) 27–33
that the combination of ρ = 0 and αu ≤ 1 is very much consistent with the Generalized Uncertainty Ramsey Rule for all the values of αv within a plausible range. Another main numerical finding is that the value of ρ tends to fall in the negative zone for larger values of αu, suggesting that a value of ρ significantly larger than zero (say 3%) is extremely unlikely. Moreover, according to the Deterministic Ramsey Rule (Eq. (1)), a larger α implies a smaller ρ. In contrast, to be consistent with the Generalized Uncertainty Ramsey Rule, ρ must increase in αv and decrease in αu, highlighting the differential effects on ρ from parameters αu and αv.
31
Substituting Eq. (A4) into Eq. (A3), the latter becomes
1 þ iB ¼
α −α u n h io v α v −1 ð1 þ ρÞ E ð1 þ g~ Þ1−α v E½ð1 þ g~ Þ−α v
f or
α v ≠1
f or
αv ¼ 1
~
1 þ iB ¼
ð1 þ ρÞeðα u −1ÞE½ lnð1þg Þ E ð1 þ g~ Þ−1
Acknowledgments
which is the Generalized Uncertainty Ramsey Rule (Eq. 5).
I would like to thank the late David Bradford for raising and discussing the question of inferring the rate of pure time preference in an uncertain world, and Richard Howarth, Charles Mason, Jack Meyer, Andrew Rettenmaier and two referees for helpful comments and suggestions on earlier drafts. Editorial assistance from Courtney Collins and Jeremy Nighohossian are greatly appreciated.
Appendix B. Proof of Proposition 2 Taking log of Eq. (5) and applying Fact 1 (see below), when αv ≠ 1,
h i α v −α u 1−α v −α − lnE ð1 þ g~ Þ v lnE ð1 þ g~ Þ α v −1 1 2 ¼ lnð1 þ ρÞ þ α u μ þ σ ðα u −α v −α u α v Þ; 2
lnð1 þ iB Þ ¼ lnð1 þ ρÞ þ
Appendix A. Proof of Proposition 1 The first order conditions of utility maximization are
which is Eq. (5′). 0
−u ðc0 Þ þ dm ¼ 0; dγ
1 dm ′ ⋅u ðmÞ ¼0 1þρ ds0
When αv = 1, on the other hand, ðA1Þ
h i lnð1 þ iB Þ ¼ lnð1 þ ρÞ þ ðα u −1ÞE½ lnð1 þ g~ Þ− lnE ð1 þ g~ Þ−1 1 2 ¼ lnð1 þ ρÞ þ α u μ− σ ; 2 which is also represented by Eq. (5′).
where, from Eqs. (3) and (4),
Appendix C. Some Facts about the Lognormal Distribution n h io ′ ~ dm E v ðc~1 Þ 1 þ iB þ γ i S −iB ¼ ds0 v′ ðmÞ h i ′ ~ dm s0 E v ðc~1 Þ i S −iB ¼ : dγ v′ ðmÞ
ðA2Þ
variance of lnX~ as μ and σ 2, respectively. 1 2 2 Fact 1. E X~ λ ¼ eλμþ2λ σ . This well-known fact about the lognormal distribution can be ~λ readily shown by noting that X~ λ ¼ e lnðX Þ and ln X~ λ follows the 2 2 normal distribution with mean λμ and σ . variance λ 2 ~ ~ Fact 2. μ and σ are related to E X and var X through
From Eqs. (A1) and (A2), we have
1 þ iB ¼
Suppose that a random variable X~ follows a lognormal distribution. Then, by definition, lnX~ follows a normal distribution. Denote the mean and variance of X~ as E X~ and var X~ , and the mean and
α v −α u
ð 1 þ ρÞ c ⋅ 0 ; E½ð1 þ g~ Þ−α v mα v −α u
ðA3Þ
h i 1 2 μ ¼ ln E X~ − σ 2 9 8 > < = var X~ > σ 2 ¼ ln 1 þ h i2 > > : ; E X~
where 1 þ g~ ¼ c~1 =c0 . h i 1 2 Proof. First, E X~ ¼ eμþ2σ from Fact 1. So μ ¼ ln E X~ − 12 σ 2 . Sec h i2 h i2 2 2 ¼ e2μþ2σ −e2μþσ ¼ E X~ ond, var X~ ¼ E X~ 2 − E X~ 2 eσ −1 .
From Eq. (3),
1 h i 1−α v 1−α ~ v m ¼ E c1 ~
m ¼ eEð lnc 1 Þ
f or
α v ≠1
f or
αv ¼ 1
ðA4Þ
Therefore, σ 2 ¼ ln 1 þ
varðX~ Þ
2 ½EðX~ Þ
:.
32
L. Liu / Ecological Economics 74 (2012) 27–33
Appendix D. Sensitivity Analysis with Respect to iB Table 3 Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1 þ g~ following a lognormal distribution, iB = 1.5%). αu
αv
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
1
2
3
4
5
6
7
8
9
10
0.6% − 0.2% − 1.1% − 2.0% − 2.8% − 3.7% − 4.5% − 5.3% − 6.2% − 7.0%
0.7% − 0.2% − 1.0% − 1.9% − 2.7% − 3.5% − 4.4% − 5.2% − 6.0% − 6.8%
0.8% 0.0% − 0.9% − 1.7% − 2.5% − 3.3% − 4.1% − 4.9% − 5.7% − 6.5%
0.9% 0.1% − 0.7% − 1.5% − 2.3% − 3.1% −3.8% − 4.6% − 5.4% − 6.1%
1.0% 0.2% − 0.6% − 1.3% − 2.1% − 2.8% − 3.6% − 4.3% − 5.0% − 5.8%
1.1% 0.3% − 0.4% − 1.1% − 1.9% − 2.6% − 3.3% − 4.0% − 4.7% − 5.4%
1.2% 0.5% − 0.2% − 0.9% − 1.6% − 2.3% − 3.0% − 3.7% − 4.4% − 5.0%
1.3% 0.6% − 0.1% − 0.8% − 1.4% − 2.1% − 2.7% − 3.4% − 4.0% − 4.7%
1.4% 0.7% 0.1% − 0.6% − 1.2% − 1.8% − 2.5% − 3.1% − 3.7% − 4.3%
1.5% 0.8% 0.2% − 0.4% − 1.0% − 1.6% − 2.2% − 2.8% − 3.4% − 4.0%
1.5% 1.0% 0.4% − 0.2% − 0.8% − 1.3% − 1.9% − 2.5% − 3.0% − 3.6%
Table 4 Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1 þ g~ following distribution (E1), iB = 0.8%). αu
αv
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
1
2
3
4
5
6
7
8
9
10
0.0% − 0.9% − 1.8% − 2.6% − 3.5% − 4.3% − 5.2% − 6.0% − 6.8% − 7.6%
0.0% − 0.9% − 1.7% − 2.5% − 3.4% − 4.2% − 5.0% − 5.8% − 6.7% − 7.5%
0.1% − 0.7% − 1.6% − 2.4% − 3.2% − 4.0% − 4.8% − 5.6% − 6.3% − 7.1%
0.2% − 0.6% − 1.4% − 2.2% − 3.0% − 3.7% − 4.5% − 5.3% − 6.0% − 6.8%
0.3% − 0.5% − 1.2% − 2.0% − 2.7% − 3.5% − 4.2% − 5.0% − 5.7% − 6.4%
0.4% − 0.4% − 1.1% − 1.8% − 2.5% − 3.2% − 4.0% − 4.7% − 5.4% − 6.0%
0.5% − 0.2% − 0.9% − 1.6% − 2.3% − 3.0% − 3.7% − 4.4% − 5.0% − 5.7%
0.6% − 0.1% − 0.8% − 1.4% − 2.1% − 2.8% − 3.4% − 4.1% − 4.7% − 5.3%
0.7% 0.0% − 0.6% − 1.3% − 1.9% − 2.5% − 3.1% − 3.8% − 4.4% − 5.0%
0.7% 0.1% − 0.5% − 1.1% − 1.7% − 2.3% − 2.9% − 3.5% − 4.0% − 4.6%
0.8% 0.3% − 0.3% − 0.9% − 1.5% − 2.0% − 2.6% − 3.2% − 3.7% −4.3%
Table 5 Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1 þ g~ following distribution (E2), iB = 0.8%). αu
αv
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
1
2
3
4
5
6
7
8
9
10
0.0% − 0.9% − 1.8% − 2.6% − 3.5% − 4.3% − 5.2% − 6.0% − 6.8% − 7.6%
0.0% − 0.9% − 1.7% − 2.6% − 3.4% − 4.2% − 5.0% − 5.8% − 6.7% − 7.5%
0.1% − 0.7% − 1.6% − 2.4% − 3.2% − 4.0% − 4.8% − 5.6% − 6.3% − 7.1%
0.2% − 0.6% − 1.4% − 2.2% − 3.0% − 3.7% − 4.5% − 5.3% − 6.0% − 6.8%
0.3% − 0.5% − 1.2% − 2.0% − 2.8% − 3.5% − 4.2% − 5.0% − 5.7% − 6.4%
0.4% − 0.4% − 1.1% − 1.8% − 2.5% − 3.3% − 4.0% − 4.7% − 5.4% − 6.1%
0.5% − 0.2% − 0.9% − 1.6% − 2.3% − 3.0% − 3.7% − 4.4% − 5.0% − 5.7%
0.6% − 0.1% − 0.8% − 1.5% − 2.1% − 2.8% − 3.4% − 4.1% − 4.7% − 5.4%
0.6% 0.0% − 0.6% − 1.3% − 1.9% − 2.6% − 3.2% − 3.8% − 4.4% − 5.0%
0.7% 0.1% − 0.5% − 1.1% − 1.7% − 2.3% − 2.9% − 3.5% − 4.1% − 4.7%
0.8% 0.2% − 0.4% − 0.9% − 1.5% − 2.1% − 2.7% − 3.2% − 3.8% −4.4%
Appendix E. Sensitivity Analysis with Respect to the Distribution of 1 þ g~ Specifically, we consider the following two alternative distributions of 1 þ g~ : 8 < 1:018 þ 0:051 1 þ g~ ¼ 1:018 : 1:018−0:051
with probability 1=4 with probability 1=2 with probability 1=4
ðE1Þ References
and 1 þ g~ ¼
1:018 þ 0:036 1:018−0:036
with probability 1=2 with probability 1=2
These alternative distributions are chosen for their simplicity, not for realism, because we need to use the more demanding Eq. (5) rather than the simpler Eq. (5′) to infer ρ when 1 þ g~ is not lognormal. However, they are sufficient to suggest, as shown in Table 4 and Table 5 below, that alternative distributions of 1 þ g~ would barely alter the inferred value of ρ.
ðE2Þ
The difference between the two alternative distributions lies in the thickness of tails, with (E1) having thicker tails than Eq. (E2). 19 19 Barro (2006), Rietz (1988) and Weitzman (2007b) suggested that the super-thin tails of 1 þ g~ implicit in studies finding the equity premium puzzle may be the heart of the puzzle, and showed that the puzzle can be solved by modifying probability distributions to admit rare but disastrous events.
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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 Monetary Economics 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 equity premium puzzle, and habit formation utility. Journal of Monetary Economics 52, 1497–1515. Nordhaus, W.D., 1994. Managing the Global Commons: The Economics of Climate Change. 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 climate change. Journal of Economic Literature 45, 686–702. Nordhaus, W.D., 2008. A Question of Balance: Weighing the Options on Global Warming Policies. Yale University Press, New Haven and London. Ogaki, M., Reinhart, C.M., 1998. Measuring intertemporal substitution: the role of durable goods. Journal of Political Economy 106, 1078–1098. Quiggin, J., 2008. Stern and his critics on discounting and climate change: an editorial essay. 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 Trust Funds. Washington, DC. Stern, N., 2007. The Economics of Climate Change: The Stern Review. Cambridge University Press, 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 of Monetary Economics 24, 401–421. Weitzman, M., 2007a. A review of the Stern review on the economics of climate change. Journal of Economic Literature 45, 703–724. Weitzman, M., 2007b. Subjective expectations and asset-return puzzles. The American Economic Review 97, 1102–1130. Weitzman, M., 2009. On modeling and interpreting the economics of catastrophic climate change. The Review of Economics and Statistics 91, 1–19. Zeckhauser, R.J., Viscusi, W.K., 2008. Discounting dilemmas: editors' introduction. Journal of Risk and Uncertainty. 37, 95–106.
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Ecological Economics journal homepage: www.elsevier.com/locate/ecolecon
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
a r t i c l e
i n f o
Article history: Received 31 August 2011 Received in revised form 7 November 2011 Accepted 11 November 2011 Available online 20 December 2011 Keywords: Time preference Discounting Ramsey Rule Uncertainty
a b s t r a c t This paper studies how to infer the rate of pure time preference (ρ) from the Ramsey Rule when multiple asset returns exist due to uncertainty. Using a Generalized Uncertainty Ramsey Rule derived from a model that separates intertemporal substitution and risk aversion, we find that the U.S. historical data on consumption growth and asset returns imply that (i) for the reciprocal of the elasticity of intertemporal substitution less than or equal to one, ρ lies within ± 1% from zero for a plausible range of the coefficient of relative risk aversion; and (ii) for the larger reciprocal of the elasticity of intertemporal substitution, ρ tends to be negative. These results contradict the widely-held belief in the environmental economics literature that the inferred ρ must be significantly larger than zero and suggest that it is appropriate to use ρ = 0 as a benchmark for 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 future instantaneous utility – plays a critical role in long-horizon economic analyses. Greenhouse gas (GHG) abatement policy modeling in the area of climate change is an example. Cline (1992) assumed ρ = 0 and found that a 40% reduction in GHG emissions is optimal. In contrast, Nordhaus (1994) assumed ρ = 3% and found that only modest emissions abatement can be justified on economic grounds. According to Nordhaus (1994, 1999, 2008), although the DICE model (Dynamic Integrated model of Climate and the Economy) had been through many versions, its main sensitivity is to the rate of pure time preference. More recently, Stern (2007) essentially let ρ = 0, but his rejection of discounting future utility has drawn criticism. 1 There has been some controversy over how the value of ρ should be 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 with revealed intertemporal choices. 2 Specifically, the latter school
⁎ Tel.: + 1 979 845 7723; fax: + 1 979 845 6636. E-mail address: [email protected]. 1 ρ is arbitrarily set at 0.1% in the Stern Review to represent the small risk of the extinction of the human race. For reviews of the Stern Review with a focus on discounting, see Nordhaus (2007); Weitzman (2007a). For a review of Stern's critics, see Quiggin (2008). 2 A criticism of the revealed preferences approach is that observed intertemporal choices in the marketplace are based on individual time preferences, which may not be the same as those of the social planner (see Azar and Sterner, 1996; Howarth, 1996). In this paper, we abstract from any divergence between individual and social time preferences by assuming a representative individual (the social decision-maker). 0921-8009/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2011.11.007
believes that ρ should be determined according to the following equilibrium condition for the Ramsey growth model (the Deterministic Ramsey Rule):
α
1 þ r ¼ ð1 þ ρÞð1 þ g Þ ;
ð1Þ
where r is the rate of return on capital, α is the (absolute value of the) elasticity of the marginal 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 value of r and g (which are observable in principle) and have a reasonable estimate of α. The issue of inferring ρ is different from, though related to, the issue of choosing a discount rate. In terms of the Deterministic Ramsey Rule, r represents the discount rate that should be used for future consumption-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 the continuous-time counterpart of Eq. (1) for the discrete-time model. In this paper we work with a discrete-time model. Note that r = ρ + αg can also be understood as a linearization of Eq. (1) with respect to g, which makes it an approximate formula suitable for use in discrete-time models when g is sufficiently small. 4 Howarth (2003) focused on the discount rate choice in the environment of uncertainty with an eye on climate change. See Zeckhauser and Viscusi (2008) for a survey of recent 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 accompanied by corresponding choices of α so that the same r is reached according 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 parameter in the equation. The value of g is relatively well understood. For example, Mehra and Prescott (1985) estimated the average annual growth rate of real per-capita consumption to be 1.8% based on the annual U.S. data from 1889 to 1978. However, determining the values of r and α is less straightforward. First, a question arises as to which r should be put in the left-hand side of Eq. (1) because of the coexistence of multiple asset returns. Different values of r would generate different values of ρ given α and g. For example, suppose α = 1 (corresponding to the logarithmic instantaneous utility function) and g = 1.8%. Then according to Eq. (1), r = 7.0% (corresponding to the mean equity return) implies ρ = 5%, but r = 0.8% (corresponding to the risk-free rate) implies ρ = − 1%. 6 Second, estimates of α are very diverse, and its appropriate value depends on the context in which it is applied. Specifically, the reciprocal of α, namely, the elasticity of intertemporal substitution, is the parameter of focus in the macro literature on life-cycle consumption. Letting α be one (i.e., the logarithmic utility) is the preferred choice in macroeconomic models, which seems to be due to the evidence of relatively strong intertemporal substitution as well as convenience considerations. 7 On the other hand, α itself, as the coefficient of relative risk aversion, is the parameter of focus in the finance literature on asset returns and risks. Data in this context imply a value of α much larger than one. 8 The literature on climate change economics appears to view α from the perspective of intertemporal substitution by adopting a value of α between one and two. For example, α = 1 in Nordhaus (1994) and Stern (2007), α = 1.5 in Cline (1992) and α = 2 in Nordhaus (2008). However, larger values of α consistent with financial data should also be relevant when capital market uncertainty is explicitly considered. 9 These conceptual difficulties surrounding using the Deterministic Ramsey Rule to infer ρ suggest that one should incorporate uncertainty and multiple asset returns into the analysis, and separate the two preference parameters traditionally represented by a single parameter α. This paper investigates the issue of how to infer the rate of pure time preference in a saving-portfolio model, adopting a utility specification 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 implications of different combinations of time discount rates and consumption elasticities.” On the other hand, the implications of different combinations of ρ and α given r for near-term decisions are relatively less important. 6 The data on asset returns are from Mehra and Prescott (1985). That the Deterministic Ramsey Rule (Eq. (1)) is inapplicable in a world of uncertainty with multiple asset classes 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 elasticity of substitution in demand between produced goods and environmental services is low and the produced goods sector grows faster than the environmental services sector. 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 utility function for consumption and that for the value function for wealth, and found that the former could be 1.25 to 10 times the latter. In this paper, the utility function is for consumption. 9 More recently, models that separate risk aversion from intertemporal substitution were incorporated in environmental studies (Kousky et al., 2011; Traeger, 2009). For example, Crost and Traeger (2010) incorporated a utility specification that disentangles the two attitudes into a DICE type model.
intertemporal substitution. Based on the Generalized Uncertainty Ramsey Rule derived from this setup, we find that inferring ρ in a way consistent with the U.S. historical data on consumption growth and asset returns yields the following main numerical findings: (i) for αu (the reciprocal of the elasticity of intertemporal substitution) equal to or less than one, the value of ρ lies within ± 1% from zero regardless 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 environmental economics literature that the inferred ρ must be significantly larger than zero, and suggest that it is appropriate to use ρ = 0 as a benchmark for long-term economic analysis. 2. The Generalized Uncertainty Ramsey Rule Following Gollier (2002), we adopt a simple version of the utility specification that distinguishes between risk aversion and intertemporal substitutability of consumption due to Epstein and Zin (1991) and Kreps and Porteus (1978).10 The preferences are jointly represented by two utility functions: u(⋅) captures the preferences for intertemporal substitution and v(⋅) captures the preferences for risk. Both functions are assumed to exhibit constant elasticity of marginal utility with respect to consumption, with the constant elasticity (absolute value) being αu (which is also the reciprocal of the elasticity of intertemporal substitution) and αv (which is also the coefficient of relative 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 special limiting cases, u(c) = ln c when αu = 1 and v(c) = ln c when αv = 1. The representative individual's overall intertemporal utility is given by uðc0 Þ þ
uðmÞ ; 1þρ
ð2Þ
where c0 is consumption at t = 0, and m is the certainty equivalent consumption at t = 1, which is determined by vðmÞ ¼ Evðc~1 Þ;
ð3Þ
where c~1 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 are only two assets: a risky asset (stocks) and a riskless asset (risk-free bonds).11 The former asset has a random rate of return ĩSand the latter has a certain rate of return iB. In the standard saving-portfolio problem, a representative consumer chooses savings in period 0,s0, and the fraction of savings allocated to the risky asset, γ, to maximize the overall utility Eq. (2), subject to Eq. (3) and c0 ¼ W 0 −s0 h i c~1 ¼ s0 1 þ iB þ γ ~i S −iB ;
ð4Þ
where W0 is the initial wealth. From the first-order conditions of the individual's optimization problem, we obtain the following Generalized Uncertainty Ramsey Rule (see Appendix A for a proof of the proposition).
10 See (Gollier 2001, chapter 20) for a detailed description of the simple version of Kreps–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.
L. Liu / Ecological Economics 74 (2012) 27–33
Proposition 1. (Generalized Uncertainty Ramsey Rule). Intertemporal utility maximization implies,
1 þ iB ¼ 1 þ iB ¼
α −α u n h io v 1−α v α v −1 ð1 þ ρÞ E ð1 þ g~ Þ E½ð1 þ g~ Þ−α v
f or
α v ≠1
f or
αv ¼ 1
ðα −1ÞE½ lnð1þg~ Þ
ð1 þ ρÞe u E ð1 þ g~ Þ−1
From the Deterministic Ramsey Rule (Eq. (1)) to the Generalized Uncertainty Ramsey Rule (Eq. (5)), the rate of return r, which is ambiguous when multiple asset returns exist in the capital market, is replaced n h ioααv −α u v −1 1−α v with iB, the risk-free rate. On the other hand, E ð1 þ g~ Þ = h i −α −1 E ð1 þ g~ Þ v , or eðα u −1ÞE½ lnð1þg~ Þ =E ð1 þ g~ Þ when αv =1, replaces (1+g)α on the right-hand side to incorporate the uncertainty in consumption growth and to distinguish between αu and αv. To infer ρ from the Generalized Uncertainty Ramsey Rule, we need estimates of iB, αu and αv, as well as the distribution of 1 þ g~ . Leaving a detailed discussion of the parameter values to the next section, we present here a more operational formula for the Generalized Uncertainty Ramsey Rule when 1 þ g~ is assumed to follow a lognormal distribution (see Appendix B for a proof of the proposition). Proposition 2. (Generalized Uncertainty Ramsey Rule with lognormally distributed consumption growth). When 1 þ g~ follows a lognormal distribution, the Generalized Uncertainty Ramsey Rule becomes ð5′Þ
where μ and σ2 are the mean and variance of the normal distribution lnð1 þ g~ Þ, respectively.12 As a special case of Eqs. (5) and (5′), we have by imposing αu = αv ≡ α that 1 þ iB ¼
1þρ E½ð1 þ g~ Þ−α
ð6Þ
and 1 2 2 lnð1 þ iB Þ ¼ lnð1 þ ρÞ þ αμ− α σ ; 2
ð6′Þ
where Eq. (6) will be referred to as the Simple Uncertainty Ramsey Rule, and Eq. (6′) is its more operational form when 1 þ g~ is assumed to follow a lognormal distribution. 13 Bradford (2003) and Howarth (2003) used Eq. (6) as a constraint that must be satisfied by parameters ρ, iB, α and those depicting consumption growth. The Simple Uncertainty Ramsey Rule explicitly incorporates uncertainty and hence represents an improvement on the well-known Deterministic Ramsey Rule (Eq. (1)). However, αu = αv is too strong an assumption that is not supported by empirical evidence. 14 Further, as explained in Weitzman (2007a), the
12
Unlike Eq. (5), Eq. (5′) is a uniform formula for all αv > 0. See Mehra (2003) for a direct derivation of Eq. (6′). For example, Barsky et al. (1997) found no significant relationship, either statistically or economically, between the two attitudes. In addition, Atkinson et al. (2009) found that correlations among the three preference parameters that respectively govern risk aversion, intertemporal substitution and inequality aversion are weak. 13 14
Table 1 Parameter values for asset returns and consumption growth (The U.S. economy, 1889–1978). iB
i¯S
Eð1 þ g~ Þ
varð1 þ g~ Þ
μ≡E½ lnð1 þ g~ Þ
σ 2 ≡ var½ lnð1 þ g~ Þ
0.8%
7.0%
1.018
0.00130
0.0172
0.00125
ð5Þ
where 1 þ g~ ≡ c~1 =c0 :
1 2 lnð1 þ iB Þ ¼ lnð1 þ ρÞ þ α u μ þ σ ðα u −α v −α u α v Þ; 2
29
conventional model of investment under uncertainty that does not separate αu and αv, based on which Eq. (6) is derived, cannot organize data on asset returns and consumption growth, as a manifestation of the equity premium puzzle and the closely-related risk-free rate puzzle. 15 An implication of Weitzman's critique of the Simple Uncertainty Ramsey Rule is that ρ should be more appropriately inferred with the Generalized Uncertainty Ramsey Rule Eq. (5) or Eq. (5′) derived from a saving-portfolio model that separates the attitude towards time and that towards risk. 16
3. Numerical Results In this section, we use the Generalized Uncertainty Ramsey Rule Eq. (5) or Eq. (5′) to infer the rate of pure time preference ρ. To do so, we need estimates of the risk-free rate iB, the reciprocal of the elasticity of intertemporal substitution αu, the coefficient of relative risk aversion αv, as well as the distribution of the gross consumption growth rate 1 þ g~ . The data on asset returns and consumption growth are from the U.S. economy for the 1889–1978 period. Table 1 lists the relevant parameter values produced in Mehra and Prescott (1985) for that period. The values of Eð1 þ g~ Þ and varð1 þ g~ Þ are based on their estimates of the mean and standard deviation of the annual per-capita consumption growth rate g~ , which are 1.8% and 3.6%, respectively. If 1 þ g~ follows a lognormal distribution, then lnð1 þ g~ Þ follows a normal distribution. The mean (μ) and variance (σ 2) of the normal distribution can be calculated from Eð1 þ g~ Þ and varð1 þ g~ Þ using Fact 2 in Appendix C. In contrast, there is little empirical consensus about the values of αu and αv. Mankiw et al's. (1985) estimates of αu center around 0.3, under the assumption that consumption and leisure are separable in the utility function. Ogaki and Reinhart's (1998) estimates are around 0.35. Attanasio and Weber (1989) obtained an estimate of 0.51, and Hansen 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 consider a range of αu from 0.5 to 5. 17
15
See Mehra and Prescott (1985) and Weil (1989). Quiggin (2008) also claimed, in general, that failure to account for the equity premium puzzle can lead to inconsistent policy recommendations. Separating intertemporal substitution and risk aversion in utility specification has been proposed to solve the asset pricing puzzles (see Epstein and Zin, 1991; Weil, 1989). For other approaches to solving these puzzles, see Kocherlakota (1996) and Mehra (2003) for two comprehensive surveys. 17 Note that αu cannot be empirically estimated independently of the value of ρ, because 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 ρ between − 0.74% and 0.37%; Barsky et al. (1997) suggested a negative ρ without pinning down its value. In contrast, this paper studies an issue often raised in environmental economics regarding the value of ρ that is implied by a Ramsey Rule given acceptable estimates of α (or αu and αv) . 16
30
L. Liu / Ecological Economics 74 (2012) 27–33
Table 2 Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1 þ g~ following a lognormal distribution, iB = 0.8%). αu 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
αv
0.5
1
2
3
4
5
6
7
8
9
10
0.0% − 0.9% − 1.8% − 2.6% − 3.5% − 4.3% − 5.2% − 6.0% − 6.8% − 7.6%
0.0% − 0.9% − 1.7% − 2.5% − 3.4% − 4.2% − 5.0% − 5.8% − 6.6% − 7.4%
0.1% − 0.7% − 1.6% − 2.4% − 3.2% − 4.0% − 4.8% − 5.5% − 6.3% − 7.1%
0.2% − 0.6% − 1.4% − 2.2% − 3.0% − 3.7% − 4.5% − 5.3% − 6.0% − 6.8%
0.3% − 0.5% − 1.2% − 2.0% − 2.7% − 3.5% − 4.2% − 5.0% − 5.7% − 6.4%
0.4% − 0.4% − 1.1% − 1.8% − 2.5% − 3.2% − 4.0% − 4.7% − 5.4% − 6.1%
0.5% − 0.2% − 0.9% − 1.6% − 2.3% − 3.0% − 3.7% − 4.4% − 5.0% − 5.7%
0.6% − 0.1% − 0.8% − 1.4% − 2.1% − 2.8% − 3.4% − 4.1% − 4.7% − 5.3%
0.7% 0.0% − 0.6% − 1.3% − 1.9% − 2.5% − 3.1% − 3.8% − 4.4% − 5.0%
0.8% 0.1% − 0.5% − 1.1% − 1.7% − 2.3% − 2.9% − 3.5% − 4.0% − 4.7%
0.8% 0.3% − 0.3% − 0.9% − 1.5% − 2.0% − 2.6% − 3.2% − 3.7% −4.3%
The estimates of αv are similarly divergent. Metrick (1995) found near risk-neutrality (αv = 0) but Beetsma and Schotman (2001) estimated αv to be around 7 in a game show environment. Kaplow (2005) estimated it to be around 0.5 based on labor market responses. Chetty (2006) established a relationship between αv and labor supply elasticities and found existing estimates of labor supply elasticities imply a mean value of αv at 0.71. Using asset return data, Epstein and Zin (1991) put αv at 1 but Attanasio and Weber (1989) put it at 30. Barsky et al. (1997) obtained a mean estimate of αvat 4 using survey data. In addition, resolving the asset return puzzle found in Mehra and Prescott (1985) generally requires a value 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 1 þ g~ follows a lognormal distribution) and by letting iB, μ and σ 2 take their respective 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 considered range of αv. Second, for larger αu, ρ tends to lie in the negative zone. These results indicate that the rate of pure time preference inferred from the Generalized Uncertainty Ramsey Rule should not be significantly larger than zero and could easily be negative, contradicting the widely-held belief in the environmental economics literature that data on market interest rates and consumption growth would imply a value of ρ that is significantly larger than the one chosen on moral grounds. Further, since αu = 1 is by far the default choice in environmental policy analyses, the first main finding above suggests that it is appropriate to use ρ = 0 as a benchmark for these long-run economic analyses. There is also a third finding that deserves mentioning. According to the Deterministic Ramsey Rule (Eq. (1)), a larger α implies a smaller ρ. In contrast, Table 2 (which is based on the Generalized Uncertainty Ramsey Rule) indicates that ρ must increase in αv and decrease in αu, highlighting the differential effects on ρ from parameters αu and αv. These findings are obtained with specific parameter values on consumption growth and the risk-free rate (see Table 1), and with the assumption that the gross consumption growth rate 1 þ g~ follows the lognormal distribution. In the following, we investigate how sensitive these findings are to the chosen risk-free rate and the assumption of the lognormally distributed consumption growth. In our first sensitivity analysis, we increase iB from 0.8% to 1.5% while maintaining the lognormal distribution. We (only) consider a larger value of iB for a sensitivity analysis because a smaller iB would further push the value of ρ into the negative zone, strengthening our findings in a way. Another reason for considering a larger iB is that while the 0.8% risk-free rate is based on historical average of short-term Treasury bond interest rates, the real government long-
term borrowing rates fluctuated around 3% in recent years. 18 The results of this sensitivity analysis are reported in Table 3 in Appendix D. Comparing Table 3 with Table 2, the inferred ρ is almost uniformly higher by 1.5%–0.8% = 0.7%. However, the main findings discussed above for Table 2 have barely changed. We can still say: for αu less than or equal to 1.5, ρ lies within ± 1.5% from zero for the considered range of values of αv; for larger αu, ρ tends to lie in the negative zone. The next step is to see how sensitive our numerical findings are to the assumption of the lognormally distributed 1 þ g~ . We consider two alternative distributions of 1 þ g~ , both of which are consistent with Mehra and Prescott's (1985) estimates that the mean and the standard deviation of per capita consumption growth rate are respectively 1.8% and 3.6% per year. The alternative distributions of 1 þ g~ and the corresponding sensitivity analysis results are presented in Appendix E. As shown in Table 4 and Table 5, alternative distributions of 1 þ 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 and the distribution of 1 þ g~ , our conclusions remain the same: ρ = 0 is a reasonable benchmark for long-run policy analysis, and it is unlikely that the ρ inferred from the Ramsey Rule would be significantly larger than the one chosen on moral grounds. 4. Conclusion The economics of climate change has recently focused attention on the importance of the rate of pure time preference (ρ) in making policy decisions with long-run consequences. While ρ = 0 is often imposed on moral grounds in climate change studies, it is widely believed in the environmental economics literature that the value of ρ inferred from the Ramsey Rule based on observed market interest rates and consumption growth rates must be significantly larger than zero. This belief is behind the criticism of the Stern Review that centers on Stern's using the extreme combination of ρ = 0 and α = 1, where α is the elasticity of marginal utility of consumption. This paper investigates how ρ should be inferred in an uncertain world with multiple rates of return and a separation between the (reciprocal) of elasticity of intertemporal substitution (αu) and the coefficient of relative risk aversion (αv). One main numerical finding is 18 For example, reflecting various long-term government borrowing rates, the federal government used a real interest rate of 2.75% in 2009 and a real interest rate of 2.6% in 2010 to determine the pension liabilities to federal employees and veterans, and Social Security Trustees have assumed a 2.9% real interest rate in recent years for Social Security trust funds (Social Security Trustees, 2008–2011). Note that these long-term borrowing rates are before personal income taxes, and the corresponding after-tax longterm government borrowing rates should be lower. For the purpose of this paper, iB should be understood as the after-tax risk-free bond rate. iB = 1.5% can be looked upon as an average of short-term and long-term after-tax Treasury bond interest rates.
L. Liu / Ecological Economics 74 (2012) 27–33
that the combination of ρ = 0 and αu ≤ 1 is very much consistent with the Generalized Uncertainty Ramsey Rule for all the values of αv within a plausible range. Another main numerical finding is that the value of ρ tends to fall in the negative zone for larger values of αu, suggesting that a value of ρ significantly larger than zero (say 3%) is extremely unlikely. Moreover, according to the Deterministic Ramsey Rule (Eq. (1)), a larger α implies a smaller ρ. In contrast, to be consistent with the Generalized Uncertainty Ramsey Rule, ρ must increase in αv and decrease in αu, highlighting the differential effects on ρ from parameters αu and αv.
31
Substituting Eq. (A4) into Eq. (A3), the latter becomes
1 þ iB ¼
α −α u n h io v α v −1 ð1 þ ρÞ E ð1 þ g~ Þ1−α v E½ð1 þ g~ Þ−α v
f or
α v ≠1
f or
αv ¼ 1
~
1 þ iB ¼
ð1 þ ρÞeðα u −1ÞE½ lnð1þg Þ E ð1 þ g~ Þ−1
Acknowledgments
which is the Generalized Uncertainty Ramsey Rule (Eq. 5).
I would like to thank the late David Bradford for raising and discussing the question of inferring the rate of pure time preference in an uncertain world, and Richard Howarth, Charles Mason, Jack Meyer, Andrew Rettenmaier and two referees for helpful comments and suggestions on earlier drafts. Editorial assistance from Courtney Collins and Jeremy Nighohossian are greatly appreciated.
Appendix B. Proof of Proposition 2 Taking log of Eq. (5) and applying Fact 1 (see below), when αv ≠ 1,
h i α v −α u 1−α v −α − lnE ð1 þ g~ Þ v lnE ð1 þ g~ Þ α v −1 1 2 ¼ lnð1 þ ρÞ þ α u μ þ σ ðα u −α v −α u α v Þ; 2
lnð1 þ iB Þ ¼ lnð1 þ ρÞ þ
Appendix A. Proof of Proposition 1 The first order conditions of utility maximization are
which is Eq. (5′). 0
−u ðc0 Þ þ dm ¼ 0; dγ
1 dm ′ ⋅u ðmÞ ¼0 1þρ ds0
When αv = 1, on the other hand, ðA1Þ
h i lnð1 þ iB Þ ¼ lnð1 þ ρÞ þ ðα u −1ÞE½ lnð1 þ g~ Þ− lnE ð1 þ g~ Þ−1 1 2 ¼ lnð1 þ ρÞ þ α u μ− σ ; 2 which is also represented by Eq. (5′).
where, from Eqs. (3) and (4),
Appendix C. Some Facts about the Lognormal Distribution n h io ′ ~ dm E v ðc~1 Þ 1 þ iB þ γ i S −iB ¼ ds0 v′ ðmÞ h i ′ ~ dm s0 E v ðc~1 Þ i S −iB ¼ : dγ v′ ðmÞ
ðA2Þ
variance of lnX~ as μ and σ 2, respectively. 1 2 2 Fact 1. E X~ λ ¼ eλμþ2λ σ . This well-known fact about the lognormal distribution can be ~λ readily shown by noting that X~ λ ¼ e lnðX Þ and ln X~ λ follows the 2 2 normal distribution with mean λμ and σ . variance λ 2 ~ ~ Fact 2. μ and σ are related to E X and var X through
From Eqs. (A1) and (A2), we have
1 þ iB ¼
Suppose that a random variable X~ follows a lognormal distribution. Then, by definition, lnX~ follows a normal distribution. Denote the mean and variance of X~ as E X~ and var X~ , and the mean and
α v −α u
ð 1 þ ρÞ c ⋅ 0 ; E½ð1 þ g~ Þ−α v mα v −α u
ðA3Þ
h i 1 2 μ ¼ ln E X~ − σ 2 9 8 > < = var X~ > σ 2 ¼ ln 1 þ h i2 > > : ; E X~
where 1 þ g~ ¼ c~1 =c0 . h i 1 2 Proof. First, E X~ ¼ eμþ2σ from Fact 1. So μ ¼ ln E X~ − 12 σ 2 . Sec h i2 h i2 2 2 ¼ e2μþ2σ −e2μþσ ¼ E X~ ond, var X~ ¼ E X~ 2 − E X~ 2 eσ −1 .
From Eq. (3),
1 h i 1−α v 1−α ~ v m ¼ E c1 ~
m ¼ eEð lnc 1 Þ
f or
α v ≠1
f or
αv ¼ 1
ðA4Þ
Therefore, σ 2 ¼ ln 1 þ
varðX~ Þ
2 ½EðX~ Þ
:.
32
L. Liu / Ecological Economics 74 (2012) 27–33
Appendix D. Sensitivity Analysis with Respect to iB Table 3 Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1 þ g~ following a lognormal distribution, iB = 1.5%). αu
αv
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
1
2
3
4
5
6
7
8
9
10
0.6% − 0.2% − 1.1% − 2.0% − 2.8% − 3.7% − 4.5% − 5.3% − 6.2% − 7.0%
0.7% − 0.2% − 1.0% − 1.9% − 2.7% − 3.5% − 4.4% − 5.2% − 6.0% − 6.8%
0.8% 0.0% − 0.9% − 1.7% − 2.5% − 3.3% − 4.1% − 4.9% − 5.7% − 6.5%
0.9% 0.1% − 0.7% − 1.5% − 2.3% − 3.1% −3.8% − 4.6% − 5.4% − 6.1%
1.0% 0.2% − 0.6% − 1.3% − 2.1% − 2.8% − 3.6% − 4.3% − 5.0% − 5.8%
1.1% 0.3% − 0.4% − 1.1% − 1.9% − 2.6% − 3.3% − 4.0% − 4.7% − 5.4%
1.2% 0.5% − 0.2% − 0.9% − 1.6% − 2.3% − 3.0% − 3.7% − 4.4% − 5.0%
1.3% 0.6% − 0.1% − 0.8% − 1.4% − 2.1% − 2.7% − 3.4% − 4.0% − 4.7%
1.4% 0.7% 0.1% − 0.6% − 1.2% − 1.8% − 2.5% − 3.1% − 3.7% − 4.3%
1.5% 0.8% 0.2% − 0.4% − 1.0% − 1.6% − 2.2% − 2.8% − 3.4% − 4.0%
1.5% 1.0% 0.4% − 0.2% − 0.8% − 1.3% − 1.9% − 2.5% − 3.0% − 3.6%
Table 4 Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1 þ g~ following distribution (E1), iB = 0.8%). αu
αv
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
1
2
3
4
5
6
7
8
9
10
0.0% − 0.9% − 1.8% − 2.6% − 3.5% − 4.3% − 5.2% − 6.0% − 6.8% − 7.6%
0.0% − 0.9% − 1.7% − 2.5% − 3.4% − 4.2% − 5.0% − 5.8% − 6.7% − 7.5%
0.1% − 0.7% − 1.6% − 2.4% − 3.2% − 4.0% − 4.8% − 5.6% − 6.3% − 7.1%
0.2% − 0.6% − 1.4% − 2.2% − 3.0% − 3.7% − 4.5% − 5.3% − 6.0% − 6.8%
0.3% − 0.5% − 1.2% − 2.0% − 2.7% − 3.5% − 4.2% − 5.0% − 5.7% − 6.4%
0.4% − 0.4% − 1.1% − 1.8% − 2.5% − 3.2% − 4.0% − 4.7% − 5.4% − 6.0%
0.5% − 0.2% − 0.9% − 1.6% − 2.3% − 3.0% − 3.7% − 4.4% − 5.0% − 5.7%
0.6% − 0.1% − 0.8% − 1.4% − 2.1% − 2.8% − 3.4% − 4.1% − 4.7% − 5.3%
0.7% 0.0% − 0.6% − 1.3% − 1.9% − 2.5% − 3.1% − 3.8% − 4.4% − 5.0%
0.7% 0.1% − 0.5% − 1.1% − 1.7% − 2.3% − 2.9% − 3.5% − 4.0% − 4.6%
0.8% 0.3% − 0.3% − 0.9% − 1.5% − 2.0% − 2.6% − 3.2% − 3.7% −4.3%
Table 5 Inferring ρ based on the Generalized Uncertainty Ramsey Rule (1 þ g~ following distribution (E2), iB = 0.8%). αu
αv
0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
0.5
1
2
3
4
5
6
7
8
9
10
0.0% − 0.9% − 1.8% − 2.6% − 3.5% − 4.3% − 5.2% − 6.0% − 6.8% − 7.6%
0.0% − 0.9% − 1.7% − 2.6% − 3.4% − 4.2% − 5.0% − 5.8% − 6.7% − 7.5%
0.1% − 0.7% − 1.6% − 2.4% − 3.2% − 4.0% − 4.8% − 5.6% − 6.3% − 7.1%
0.2% − 0.6% − 1.4% − 2.2% − 3.0% − 3.7% − 4.5% − 5.3% − 6.0% − 6.8%
0.3% − 0.5% − 1.2% − 2.0% − 2.8% − 3.5% − 4.2% − 5.0% − 5.7% − 6.4%
0.4% − 0.4% − 1.1% − 1.8% − 2.5% − 3.3% − 4.0% − 4.7% − 5.4% − 6.1%
0.5% − 0.2% − 0.9% − 1.6% − 2.3% − 3.0% − 3.7% − 4.4% − 5.0% − 5.7%
0.6% − 0.1% − 0.8% − 1.5% − 2.1% − 2.8% − 3.4% − 4.1% − 4.7% − 5.4%
0.6% 0.0% − 0.6% − 1.3% − 1.9% − 2.6% − 3.2% − 3.8% − 4.4% − 5.0%
0.7% 0.1% − 0.5% − 1.1% − 1.7% − 2.3% − 2.9% − 3.5% − 4.1% − 4.7%
0.8% 0.2% − 0.4% − 0.9% − 1.5% − 2.1% − 2.7% − 3.2% − 3.8% −4.4%
Appendix E. Sensitivity Analysis with Respect to the Distribution of 1 þ g~ Specifically, we consider the following two alternative distributions of 1 þ g~ : 8 < 1:018 þ 0:051 1 þ g~ ¼ 1:018 : 1:018−0:051
with probability 1=4 with probability 1=2 with probability 1=4
ðE1Þ References
and 1 þ g~ ¼
1:018 þ 0:036 1:018−0:036
with probability 1=2 with probability 1=2
These alternative distributions are chosen for their simplicity, not for realism, because we need to use the more demanding Eq. (5) rather than the simpler Eq. (5′) to infer ρ when 1 þ g~ is not lognormal. However, they are sufficient to suggest, as shown in Table 4 and Table 5 below, that alternative distributions of 1 þ g~ would barely alter the inferred value of ρ.
ðE2Þ
The difference between the two alternative distributions lies in the thickness of tails, with (E1) having thicker tails than Eq. (E2). 19 19 Barro (2006), Rietz (1988) and Weitzman (2007b) suggested that the super-thin tails of 1 þ g~ implicit in studies finding the equity premium puzzle may be the heart of the puzzle, and showed that the puzzle can be solved by modifying probability distributions to admit rare but disastrous events.
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