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When the long run matters

The joint effect of carbon decay and discounting

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Roughly 20 percent of current CO2 emissions will likely remain in the atmosphere for thousands of years (Solomon et al. 2008). Despite this, climate damages attributable to current emissions that occur beyond 150 years or so have almost no effect on the current optimal carbon tax in typical integrated assessment models. The source of this strong result is conventional economic discounting. The current paper builds on recent work by Gerlagh and Liski (2013) and Iverson (J Environ Econ Manag 66:598–608, 2013a, b) to demonstrate this fact in a simple way and to show that it is not robust to plausible changes in the calibration approach for discounting parameters. Specifically, when time preference rates decline, a possibility supported by a wide variety of studies from psychology and economics, long run consumption impacts are potentially very important and so are long run features of the carbon cycle. The paper follows (Gerlagh and Liski 2013) in showing that this remains true even when the discounting parameters are calibrated to match historical interests rates, thus avoiding the main economic critique of the Stern Review (Stern 2007; Nordhaus 2008; Weitzman Rev Econ Stat 91:1–19 2009). The effects are quantified using a formula for the optimal carbon tax from Iverson (J Environ Econ Manag 66:598–608, 2013a, b), which we use to decompose the current optimal tax into the cumulative contribution from consumption impacts at different horizons.

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  1. Gerlagh and Liski (2013) (hereafter “GL”) refer to this insensitivity to long-term outcomes under conventional discounting assumptions as a climate-policy “puzzle”. Their paper is discussed below.

  2. This implies a subgame perfect equilibrium.

  3. The employed decomposition is not possible in typical, quantitatively-interesting integrated assessment models (e.g. Nordhaus 2008; Tol 1997; Hope 2008) because the optimal tax in those models is inferred numerically from the shadow value on the carbon stock constraint at the numerical optimum.

  4. The formula is presented in the Appendix.

  5. This approach is consistent with Utilitarian ethics when the consumption good is public, so the same amount of the aggregate good is consumed by all agents. Gollier and Zeckhauser (2005) consider the problem in which consumption shares are endogenous while income is exogenous, and Heal and Millner (2013) consider the problem in which consumption shares and income are both endogenous.

  6. This is seen by computing \({r(t) = -\frac {\dot {D}(t)}{D(t)} = \frac {p_{S} \cdot e^{-r_{S} t} }{p_{S} \cdot e^{-r_{S} t} + (1-p_{S}) \cdot e^{-r_{N} t}} r_{S} + \frac {(1-p_{S}) \cdot e^{-r_{N} t}}{p_{S} \cdot e^{-r_{S} t} + (1-p_{S}) \cdot e^{-r_{N} t}} r_{N}}\), where \(\dot {D}(t)\) is the time derivative of D(t). Weitzman (1998) proves the same limiting result for an arbitrary finite probability distribution. Iverson (2013b) shows that an analogous result holds when decision makers are unable to assign probabilities to potential discount rates and employ the non-Bayesian decision criterion Minimax Regret.

  7. The qualitative results of the paper are unchanged if we instead adopt the more conventional calibration procedure above.

  8. Climate sensitivity in this equation is set to 3.0 degrees Centigrade.

  9. In contrast, Golosov et al. (2014) fix it at 0.2 based on summary evidence from Solomon et al. (2008).

  10. The remaining calibration assumptions follow (Iverson 2013a), which with slight differences to account for model differences, are the same as in Golosov et al. (2014). Total factor productivity evolves as in Nordhaus (2013).

  11. In the notation of GL, we take a = (ϕ L , (1 − ϕ L )ϕ 0), η = (0, ϕ), and π = γ GHKT = 2.379×10−5.

  12. Our calibration is hybrid between GHKT and GL. With the full climate module calibration from GL, including their higher damage parameter, the tax under Nordhaus discounting is $ 27 per ton carbon.

  13. Specifically, we report a revised version of Fig. 4, which includes the information presented in Fig. 3.

  14. The corresponding carbon taxes are different from those reported in Table 1, where the long run emission fraction is 0.2. With ϕ L = 0, the corresponding carbon taxes for the discounting scenarios in the table are, respectively, ($ 38, $ 67, $ 96, $ 126, $ 157, $ 189, and $ 327).

  15. The analytic formula for the optimal tax is unchanged when there are an arbitrary number of energy sectors of each type.

  16. This strong assumption is partly offset by the fact that the period length in the calibrated model is a decade. It is relaxed later on.

  17. The period length in the model is a decade.


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The authors thank William Brock, Reyer Gerlagh, and three anonymous referees for helpful comments.

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Correspondence to Terrence Iverson.

Appendix: Technical appendix

Appendix: Technical appendix

1.1 A.1 Model

The model is a multi-sector neoclassical growth model that includes a final-goods sector, one fossil energy sector (“coal”) and one clean energy sector (“wind”)Footnote 15. Net output in the final-goods sector is determined by

$$Y_{t} = \omega(S_{t}) A_{0,t}K_{t}^{\alpha}N_{0,t}^{1-\alpha-v}E_{t}^{\nu}, $$

where ω(S t ) is a multiplicative damage function described below and

$$E_{t} = (\kappa_{1} E_{1,t}^{\rho} + \kappa_{2} E_{2,t}^{\rho})^{1/\rho}. $$

Here, E 1, t is energy output from the “coal” (or composite fossil) sector and E 2, t is energy output from the “wind” (or composite low carbon energy) sector. Both are measured in units of carbon. The parameter ρ < 1 pins down the elasticity of substitution between coal and wind, while the κ parameters determine the relative energy-efficiency of each energy source. Output in each energy sector i is linear in labor:

$$E_{i,t}=A_{i,t}N_{i,t}. $$

Labor is mobile across sectors with

$$N_{0,t}+N_{1,t}+ N_{2,t}=N_{t}, $$

where N t is the total workforce in the economy at date t. Both N t and A i, t are exogenous. The aggregate resource constraint for final consumption goods assumes 100 % depreciation of physical capital each periodFootnote 16. Thus,

$$Y_{t} = C_{t} + K_{t+1}. $$

In addition, climate change arises because CO2 is produced as a byproduct of burning fossil fuels. The stock of atmosphere carbon accumulates according to

$$S_{t} - \bar{S} = \sum\limits_{j=0}^{t+H} (1-d_{j})E^{f}_{\tau-j}, $$


$${E^{f}_{s}} = \sum\limits_{i=1}^{n}E_{i,s} $$

is total carbon emissions (from fossil sectors) in period s. In the summation above, H is the number of periods between period 0 in the model and the start of the industrial revolution. Thus, the concentration of carbon in the atmosphere is an arbitrary linear function of all prior anthropogenic emissions. 1 − d k is the fraction of a unit of emissions that remains in the atmosphere k periods after it is emitted, and it is given in the paper by the decay formula in Eq. 2.

Cumulative emissions lead to climate damages that impact economic output through a multiplicative “exponential–linear” damage function:

$$ \omega(S_{t}) = \exp\left(- \gamma_{\tau} (S_{t}-\bar{S})\right). $$

γ τ , an elasticity, is the percent output loss associated with an extra unit of atmospheric carbon in τ. It is stochastic.

The key innovation in the model is to to allow for non-constant time preference. Specifically, utility is discounted with a sequence of potentially non-constant discount factors, {β j }. For each j, 0 < β j < 1. The typical case in which the TPR declines implies {β j } increasing. For each time horizon k periods ahead, \(R_{k} = {\prod }_{j=1}^{k} \beta _{j}\) is the cumulative discount factor. Using this, the the representative household in generation t is evaluates future consumption streams using the following “social welfare function”:

$$\mathbb{E}_{t} \sum\limits_{\tau=t}^{T} R_{\tau-t} \ln (C_{\tau}). $$

Utility is logarithmic.

Analytical tractability is a consequence of five key assumptions: log utility, Cobb Douglass production (in capital), “exponential-linear” damages, 100 percent depreciation of physical capital each period, and the exclusion of capital as a factor of production in the various energy sectors.

1.2 A.2 Equilibrium concept

Non-constant time preference introduces the potential for time inconsistency. This is true in particular if the problem is solved in a way that (explicitly or implicitly) assumes a commitment device. Unfortunately, standard optimal control methods do precisely this. To avoid this problem, we follow Iverson (2013a) in solving for the subgame perfect equilibrium in a finite horizon version of the model. This means that we assume that the policy variables will be chosen each period by the living generation who controls policy for one period onlyFootnote 17. At the same time, the planner in each period correctly anticipates the way in which future policy makers will set policy (in response to the stock of physical capital and the stock of atmospheric carbon that they inherit from the past). This assumption implies an equilibrium problem that must be solved using backward induction. The corresponding equilibrium is unique. For more details, see Iverson (2013a).

1.3 A.3 Optimal carbon taxes

The optimal carbon tax is the tax on the carbon content of fossil fuels in the model economy that would induce firms to choose the abatement path that corresponds to the subgame perfect equilibrium to the planner’s problem without commitment. This is the best one can do on cost-benefit grounds in a world in which commitment is infeasible.

The adopted notation follows Iverson (2013a). He defines \(R_{l,m}^{(t)}\) as the price at t + l of a unit of utility received at t + m, viewed by an agent in t. For ml,

$$ R_{l,m}^{(t)}=\left\{\begin{array}{cl} {\prod}_{j=l+1}^{m} \beta_{j}^{(t)} & m>l\\ 1 & m=l. \end{array}\right. $$

With this notation, the optimal tax in fossil sector i can be written

$$ \tau_{i,t} = \mathbb{E}_{t} \left[\sum\limits_{k=0}^{\infty} R_{k}^{(t)} \gamma_{t+k} (1-d_{k}) \cdot {\Gamma}_{k}^{(t)}\right] Y_{t}, $$


$${\Gamma}_{k}^{(t)} = \frac{{\sum}_{m=0}^{\infty} \alpha^{m} R_{k,k+m}^{(t)}}{{\sum}_{n=0}^{\infty} \alpha^{n} R_{n}^{(t)}}. $$

The optimal tax on clean-energy firms is zero. Here \(\mathbb {E}_{t} \gamma _{t+k}\) is the expected damage elasticity parameter in future period t + k given that we are standing today in period t. α is the elasticity of final goods production with respect to physical capital. Y t is Gross World Product in t.

1.4 A.4 Results with temperature inertia

Temperature inertia is introduced by considering a two-box version of the GL climate module. Maintaining the carbon cycle and climate damage calibration assumptions from GHKT implies a = (0.2, 0.3144), η = (0, 0.0228), and \(\mathbb {E}_{t} \gamma _{t+k} = 2.379e-5\). Meanwhile, temperature inertia is calibrated so peak temperature occurs 70 years after an emission pulse, as in GL and (Nordhaus 2008). This implies 𝜖 = 0.5.

Comparing Fig. 5 to Fig. 4 shows the sense in which the results in the text are conservative. For any given value of ϕ L , the impact of increasing the weight on Stern is more pronounced with temperature inertia (Fig. 5) then it is without temperature inertia (Fig. 4). The difference is especially large when the weight on Stern is small.

Fig. 5
figure 5

Percent of current optimal carbon tax attributable to consumption impacts beyond 200 years (in percent). Horizontal axis depicts alternative discounting assumptions. Vertical axis depicts the fraction of emissions that remain in the very long run (ϕ L )

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Iverson, T., Denning, S. & Zahran, S. When the long run matters. Climatic Change 129, 57–72 (2015).

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