DICE Simplified

Abstract

We analyze Nordhaus’ DICE model and show that the temperature and CO₂ equations are needlessly complicated and can be simplified without loss of essence. In addition, we argue that the damage function can be altered in such a way that it lends itself to experiments involving extreme risk. We conclude that, within the philosophy of the DICE model, significant simplifications can be made which make the model more transparent, more robust, and easier to apply.

“Entia non sunt multiplicanda praeter necessitatem” William of Ockham, 1285(?)–1347

1 Introduction

This paper provides a critical assessment of the 2016 DICE (Dynamic Integrated model of Climate and the Economy) model originally developed by Nordhaus [17], but since then continuously updated and altered (see [22]). Integrated assessment models (IAMs) are being used extensively for the analysis of climate change policy and DICE has played an important part in projecting greenhouse gas emissions and temperature under various social and economic scenarios.

The assumptions and functional forms used in DICE have been under considerable scrutiny and criticism. Among others, [8, 9, 23, 24] argue that IAMs should be made simpler and more transparent, and an editorial in Nature Climate Change [6] supports this view. DICE is complex because it employs an equation for radiative forcing, two equations (a two-box model) for the climate system, and a three-reservoir model for the carbon cycle. Some of the other IAMs (for example, the FUND and PAGE models) are somewhat less complex, as they employ a single-equation climate model (see [3]).

The purpose of this paper is to show that the temperature and carbon dioxide (CO₂) equations in DICE are needlessly complicated and can be much simplified. The technical reason why this simplification is possible lies in the fact that the matrix connecting the two dynamic equations describing temperature in the DICE model has one eigenvalue close to 1, and that the matrix connecting the three equations describing CO₂ has one exact and one approximate unit eigenvalue. We derive the equivalent equations after differencing out the auxiliary variables and we provide simplifications.

In response to criticism that IAMs give the impression of being “black boxes” [6], one approach is to propose significant changes to IAMs such as DICE, leading to simpler IAMs with an analytical formula for the optimal carbon price; see [5, 10, 25, 29,30,31]. Among studies employing closed-form IAMs, the specifications of the climate system and the carbon cycle vary. Golosov et al. [10] specify a two-and-a-half-box carbon system consisting of a permanent component (about 20% of carbon) and a transient component. The climate system is omitted. Under assumptions such as logarithmic utility, Cobb-Douglas technology, constant saving rate, and full depreciation of capital, they derive the optimal-tax formula analytically. Rezai and Van der Ploeg [25] employ a similar two-box carbon system. They allow for a lag between temperature and atmospheric carbon in addition to more general functional forms than those assumed by [10], and they derive a simple rule for the optimal carbon price.

The omission of the climate system in [10] is justified by recent findings in climate science that the climate response to a CO₂ emission is nearly instantaneous and remains almost constant over time; see [26, 27]. Based on the same findings, Dietz and Venmans [5] assume that the global mean temperature is linearly proportional to cumulative CO₂ emissions, in contrast to the large thermal inertia of the climate system assumed in DICE and [16]. These differences may lead to different optimal transition paths of temperature. Thus, the optimal carbon price follows Hotelling’s rule in [5], whereas it grows more slowly in [16].

Our approach is to stay as close as possible to Nordhaus’ DICE model, but to simplify it using Ockham’s razor quoted above (“more things should not be used than are necessary”). A complex model is not necessarily better than a simple model. Leaving things out is arguably more difficult and more important than putting things in. Many statisticians believe that a more complex model will reduce the bias and increase the variance, but this is only half true. A more complex model does indeed increase the variance, but it does not necessarily reduce the bias (see [4]). Hence, simplicity matters given the large uncertainties in the exogenous variables (such as population and technical knowledge) and the parameters.

We shall not propose a completely different climate system nor do we examine climate sensitivity in the DICE model. Our purpose is more modest. We shall show that the temperature and CO₂ equations in the DICE model are needlessly complicated and that they can be much simplified. We also argue that the specification of the damage function can be altered in such a way that it lends itself to experiments involving extreme risk. Finally, we briefly discuss the assumption in DICE that the abatement fraction for CO₂ is allowed to become larger than 1, which implies that emissions can become negative.

In Section 2, we present the Nordhaus DICE 2016R model. In Sections 3 and 4, we discuss and simplify the DICE equations for temperature and CO₂ concentration. In Section 5, we provide an alternative to the DICE damage function. Section 6 presents and optimizes the S-DICE (simplified DICE) model and concludes.

2 Nordhaus’ DICE 2016R Model

The following equations are the equations from the beta version of DICE-2016R [20, 21], a version with the identification DICE-2016R-091916ap.gms. A number of equations are redundant and have been deleted. A new variable ω_t has been introduced, some equations have been combined, and the equations have been reordered; see [14] for the details. Still, this is precisely the same model as Nordhaus’ 2016R model.

Everybody works. In period t, the labor force L_t together with the capital stock K_t generates GDP Y_t through a Cobb-Douglas production function:

$$ Y_{t}=A_{t}K_{t}^{\gamma}L_{t}^{1-\gamma} \qquad (0<\gamma<1), $$

(1)

where A_t represents technological efficiency and γ is the elasticity of capital. Capital is accumulated through

$$ K_{t+1}=(1-\delta)K_{t}+I_{t} \qquad (0<\delta<1), $$

(2)

where I_t denotes investment and δ is the depreciation rate of capital.

CO₂ emissions consist of industrial and non-industrial (“land-use”) emissions. We denote the latter type by ${E_{t}^{0}}$ and consider it to be exogenous to our model. Total CO₂ emissions E_t are then given by:

$$ E_{t}=\sigma_{t}(1-\mu_{t})Y_{t} + {E_{t}^{0}}, $$

(3)

where σ_t denotes the emissions-to-output ratio for CO₂ and μ_t is the abatement fraction for CO₂. The associated CO₂ concentration increase M_t in the atmosphere (GtC from 1750) accumulates through:

$$ \begin{array}{@{}rcl@{}} M_{t+1} &=& (1-b_{0})M_{t} + b_{1}X_{1,t} + E_{t}, \end{array} $$

(4a)

$$ \begin{array}{@{}rcl@{}} X_{1,t+1} &=& b_{0}M_{t} + (1-b_{1}-b_{3})X_{1,t} + b_{2}X_{2,t}, \end{array} $$

(4b)

$$ \begin{array}{@{}rcl@{}} X_{2,t+1} &=& b_{3}X_{1,t} + (1-b_{2})X_{2,t}, \end{array} $$

(4c)

where X_1,t and X_2,t are auxiliary variables representing CO₂ concentration increases in shallow and lower oceans, respectively, also measured in GtC from 1750.

Temperature increase H_t (degrees Celsius from 1900) develops according to:

$$ \begin{array}{@{}rcl@{}} H_{t+1} &=& (1-a_{0}) H_{t} + a_{1} \log(M_{t+1}) + a_{2} Z_{t} + F_{t+1}, \end{array} $$

(5a)

$$ \begin{array}{@{}rcl@{}} Z_{t+1} &=& (1-a_{3})Z_{t} + a_{3}H_{t}, \end{array} $$

(5b)

where Z_t is an auxiliary variable representing the temperature increase of the lower oceans, also measured in degrees Celsius from 1900, and F_t+ 1 is exogenous radiative forcing.

In each period t, the fraction of GDP not spent on abatement or “damage” is either consumed (C_t) or invested (I_t) along the budget constraint:

$$ \left( 1 - \omega_{t} - \xi {H_{t}^{2}}\right) Y_{t} = C_{t} + I_{t}. $$

(6)

A fraction ω_t of Y_t is spent on abatement, and we specify the abatement cost fraction as:

$$ \omega_{t}=\psi_{t}\mu_{t}^{\theta} \qquad (\theta>1). $$

(7)

When μ_t increases then so does ω_t, and a larger fraction of GDP will be spent on abatement.

Damage is represented by a fraction $\xi {H_{t}^{2}}$ of Y_t and it depends only on temperature. The optimal temperature is H_t = 0, the temperature in 1900. Deviations from the optimal temperature cause damage. For very high and very low temperatures, the fraction becomes large, but (given the value of ξ) it will still be a fraction between 0 and 1, unless in truly catastrophic cases.

As in [20, 21], one period is 5 years. Period 1 refers to the time interval 2015–2019, period 2 to 2020–2024, and so on. Stock variables are measured at the beginning of the period; for example, K₁ denotes capital in the year 2015. We choose the exogenous variables such that L_t > 0, A_t > 0, ${E_{t}^{0}}>0$, σ_t > 0, and 0 < ψ_t < 1. The policy variables must satisfy

$$ C_{t}\geq0,\quad I_{t}\geq0,\quad \mu_{t}\geq0. $$

(8)

Nordhaus [21, 22] allows negative-emission technologies by setting an upper bound on μ_t of 1.2 (rather than 1.0) from period 30 onwards (year 2160), which implies that emissions can become negative by Eq. 3, and in fact this upper bound is reached in the DICE output from period 46 onwards (year 2240). The idea of negative emissions is controversial. Anderson and Peters [2] state that negative-emission technologies are unjust and a high-stakes gamble, while a recent editorial in Nature ([7]) discusses the enormous effort required to carry out such technologies—an effort which would lead to a deterioration of the environment.

Given a utility function U, we define welfare in period t as

$$ W_{t}=L_{t} U(C_{t}/L_{t}). $$

(9)

The policy maker has a finite horizon and maximizes total discounted welfare:

$$ W=\sum\limits_{t=1}^{T}\frac{W_{t}}{(1+\rho)^{t}} \qquad (0<\rho<1), $$

(10)

where ρ denotes the discount rate and T = 100 (500 years). Letting x denote per capita consumption, the utility function U(x) is assumed to be defined and strictly concave for all x > 0. There are many such functions, but a popular choice is

$$ U(x)=\frac{x^{1-\alpha}-1}{1-\alpha} \qquad (\alpha>0), $$

(11)

where α denotes the elasticity of marginal utility of consumption. This is the so-called power function. Many authors, including Nordhaus, select this function. In earlier versions of the DICE model, Nordhaus [18] chooses α = 2 in which case U(x) = 1 − 1/x. Also popular is α = 1 in which case $U(x)=\log (x)$; see [15, 28]. In the 2016 version of the DICE model, α = 1.45.

3 Temperature

The DICE model thus consists of the seven Eqs. 1–7. Four of these, Eqs. 1–3 and 7, are not controversial. In the next three sections, we shall discuss the CO₂ equation (4c), the temperature equation (5a), and the budget constraint (6).

We start with the temperature equations in Eq. 5a, which we now write in matrix form as

$$ x_{t+1}=A x_{t} + a_{t+1}, $$

(12)

where

$$ A= \begin{pmatrix} 1-a_{0} & a_{2} \\ a_{3} & 1-a_{3} \end{pmatrix} $$

and

$$ x_{t}= \begin{pmatrix} H_{t} \\ Z_{t} \end{pmatrix}, \qquad a_{t}= \begin{pmatrix} a_{1} \log(M_{t}) + F_{t} \\ 0 \end{pmatrix}. $$

The matrix A has two eigenvalues given by

$$ 1 - \frac{1-\eta_{1}}{2}\pm \frac{1}{2}\sqrt{(1-\eta_{1})^{2} - 4\eta_{2}}, $$

where η₁ = 1 − a₀ − a₃ = 0.8468 and η₂ = (a₀ − a₂)a₃ = 0.0030, so that the eigenvalues are 0.9771 and 0.8697, respectively. The largest eigenvalue is thus close to 1 and it would be equal to 1 if (and only if) η₂ = 0.

We can “difference out” the auxiliary variable Z and this gives

$$ \begin{array}{@{}rcl@{}} H_{t+1} &=&(1+\eta_{1})H_{t}-(\eta_{1}+\eta_{2})H_{t-1} \\ &&+ \eta_{3}\log(M_{t+1})-(\eta_{3}-\eta_{4})\log(M_{t})+ \eta_{0t}, \end{array} $$

(13)

where η₃ = a₁ = 0.5338, η₄ = a₁a₃ = 0.0133, and η_0t = F_t+ 1 − (1 − a₃)F_t. Equation 13 does not contain Z but, compared to Eq. 5a, it contains an additional lag in both H and $\log (M)$. Note that Eq. 13 is not invariant to scaling in M.

Letting Δ be the (backward) difference operator defined by Δx_t+ 1 = x_t+ 1 − x_t, we can write Eq. 13 alternatively as

$$ \varDelta H_{t+1} = \eta_{1} \varDelta H_{t} + \eta_{3}\varDelta\log(M_{t+1}) + \eta^{*}_{0t}, $$

where $\eta ^{*}_{0t}=-\eta _{2} H_{t-1} + \eta _{4}\log (M_{t}) + \eta _{0t}$. This equation in first differences can in turn be integrated to

$$ H_{t+1} = \eta_{0} + \eta_{1} H_{t} + \eta_{3} \log(M_{t+1}) + \sum\limits_{j=1}^{t}\eta^{*}_{0j}, $$

(14)

where η₀ = − 3.3291 is an integration constant. Notice that H_t+ 1 in Eq. 14 depends on H_t but that the effect of H_t− 1 (through $\eta ^{*}_{0t}$) is negligible, which is another way of saying that the largest eigenvalue of the matrix A in Eq. 12 is close to 1. Both are caused by the fact that η₂ is small. We emphasize that Eqs. 12, 13, and 14 are equivalent descriptions of the DICE temperature equations. No approximation has yet taken place.

Given the DICE parameter values, in particular the fact that η₂ and η₄ are small, the partial sums ${\sum }_{j}\eta ^{*}_{0j}$ are well approximated by a linear trend with slope 0.025. This implies that if we run a regression on the equation:

$$ H_{t+1} = \eta^{*}_{0} + \eta^{*}_{1} H_{t} + \eta^{*}_{2} \log(M_{t+1}) + \eta^{*}_{3} t $$

(15)

we will get a good fit. If we leave out the linear trend, then the estimate of $\eta ^{*}_{1}$ increases somewhat and the estimate of $\eta ^{*}_{2}$ decreases somewhat.

More precisely, we obtain the results in Table 1, where we note that in all regressions, figures, and numerical experiments that follow, H_t (and similarly M_t and other variables) takes the optimal values as obtained from the General Algebraic Modeling System (GAMS) routine which optimizes welfare (10) in the DICE system.

Table 1 Simplified temperature equations

Under (a), we report the estimated coefficients and standard errors from a regression of H_t+ 1 on a constant, H_t, $\log (M_{t+1})$, and a time trend, as in Eq. 15. The fit is very good. In particular, letting e denote the vector of residuals and $\hat {H}_{t}$ the predicted value of H_t from the regression, and defining the regression variance s² and the relative deviations Q_t as

$$ s^{2}=e'e/(n-k), \qquad Q_{t}=100(\hat{H}_{t}-H_{t})/H_{t}, $$

(16)

we find that s = 0.0032 and $\max \limits _{t} |Q_{t}|=0.43$ with n = 99 and k = 4. This shows that for a temperature increase of, say, 3 °C the maximum error will be 0.013°.

If we leave out the linear trend, we obtain (b) which is almost as good, except that the error in the first few periods is somewhat higher. This is illustrated in Figs. 1 and 2. In Fig. 1, the time paths of temperature in DICE and the two models (a) and (b) are indistinguishable, reaching a maximum of 7.2 in 2270. The relative deviations Q_t are graphed in Fig. 2. They are all below 0.5% except the first four periods in model (b). Even though the estimated coefficient on the time trend is “significant,” it is not important, and the fit is essentially the same.

Fig. 1

Temperature—Time path for DICE 2016R and two simplified models. Model (a) is based on Eq. 15. Model (b) is based on Eq. 15 without a time trend

Fig. 2

Temperature—Deviations (%) of two simplified models relative to DICE 2016R. Model (a) is based on Eq. 15. Model (b) is based on Eq. 15 without a time trend

Summarizing, the simplified equation (15) provides a good approximation because (a) the coefficients in Eq. 13 correspond approximately to a first-order difference equation; and (b) the omitted variable is essentially constant. The second approximation (without trend) is almost as good as the approximation in Eq. 15, and suffices for practical applications.

4 CO₂ Concentration

Next, we consider the CO₂ equations in Eq. 4c, which we also write in matrix form as:

$$ x_{t+1}=A x_{t} + a_{t}, $$

(17)

where now

$$ A= \begin{pmatrix} 1-b_{0} & b_{1} & 0 \\ b_{0} & 1-b_{1}-b_{3} & b_{2} \\ 0 & b_{3} & 1-b_{2} \end{pmatrix} $$

and

$$ x_{t}= \begin{pmatrix} M_{t} \\ X_{1,t} \\ X_{2,t} \end{pmatrix}, \qquad a_{t}= \begin{pmatrix} E_{t} \\ 0 \\ 0 \end{pmatrix}. $$

One of the three eigenvalues of A equals 1, and the two remaining eigenvalues are given by

$$ 1 - \frac{1-\phi_{1}}{2}\pm \frac{1}{2}\sqrt{(1-\phi_{1})^{2} - 4\phi_{2}}, $$

where ϕ₁ = 1 − b₀ − b₁ − b₂ − b₃ = 0.675535 and ϕ₂ = b₀b₂ + b₀b₃ + b₁b₂ = 0.001303, so that the two remaining eigenvalues take the values 0.995933 and 0.679602, respectively. The largest eigenvalue is thus equal to 1 and the next eigenvalue is close to 1; it would be equal to 1 if (and only if) ϕ₂ = 0. This suggests that we should difference not once (as in the previous section) but twice, and this is precisely what we shall do.

As in the previous section, we can “difference out” the auxiliary variables X₁ and X₂, and this gives:

$$ \begin{array}{@{}rcl@{}} M_{t+1} &=& (\phi_{1}+2)M_{t} - (1+2\phi_{1}+\phi_{2})M_{t-1} \\ &&+ (\phi_{1}+\phi_{2})M_{t-2}+E_{t}^{*}, \end{array} $$

(18)

where

$$ E_{t}^{*}=E_{t} - (1+\lambda_{1})E_{t-1} + (\lambda_{1}+\lambda_{2}) E_{t-2}. $$

This equation does not contain X₁ and X₂ but it contains two additional lags in both M and E. We can write Eq. 18 alternatively as

$$ \begin{array}{@{}rcl@{}} \varDelta M_{t+1} &=(\phi_{1}+1)\varDelta M_{t}-(\phi_{1}+\phi_{2})\varDelta M_{t-1} + E_{t}^{*}, \end{array} $$

where we notice that there is no remainder term in M_t− 2 because the largest eigenvalue of A equals 1 exactly given the DICE parameters. This leads to

$$ M_{t+1} =\phi_{0} + (\phi_{1}+1)M_{t}-(\phi_{1}+\phi_{2})M_{t-1}+ E_{t}^{**}, $$

(19)

where $E_{t}^{**}={\sum }_{j=1}^{t} E_{j}^{*}$ and ϕ₀ = 0.8761 is an integration constant. This, in turn, can be written as

$$ \varDelta M_{t+1} = \phi_{0} + \phi_{1} \varDelta M_{t} -\phi_{2} M_{t-1} + E_{t}^{**}, $$

so that

$$ M_{t+1} = \phi_{00} + \phi_{0} t + \phi_{1} M_{t} - \phi_{2}\sum\limits_{j=1}^{t-1}M_{j} + \sum\limits_{j=1}^{t}w_{tj}E_{j}, $$

(20)

where ϕ₀₀ = 263.2837 is an integration constant and

$$ \begin{array}{@{}rcl@{}} \sum\limits_{j=1}^{t}w_{tj}E_{j} &=&\sum\limits_{j=1}^{t}E_{j}^{**} =\sum\limits_{j=1}^{t}\sum\limits_{i=1}^{j} E_{i}^{*} =\sum\limits_{j=1}^{t}(t-j+1)E_{j}^{*}\\ &=&E_{t} + (1 - \lambda_{1})\sum\limits_{j=1}^{t-1}E_{j} +\lambda_{2}\sum\limits_{j=1}^{t-2}(t - j - 1)E_{j}, \end{array} $$

so that w_tt = 1 and

$$ w_{tj}=1-\lambda_{1} + (t-j-1)\lambda_{2} \qquad (j=1,\dots,t-1). $$

The DICE weights w_tj are thus slightly increasing rather than decreasing, which is a little awkward. Notice that Eqs. 17–20 are equivalent descriptions of the DICE CO₂ equations. No approximation has yet taken place.

Since ϕ₂ = 0.0013 and λ₂ = 0.0003 are close to 0, Eq. 20 will be well approximated by

$$ M_{t+1} \approx \phi_{00} + \phi_{0} t + \phi_{1} M_{t} + E_{t} + (1-\lambda_{1})\sum\limits_{j=1}^{t-1}E_{j}. $$

In fact, we will run regressions on the equation

$$ M_{t+1} = \phi^{*}_{0} + \phi^{*}_{1} M_{t} + \phi^{*}_{2} E_{t} + \phi^{*}_{3} t $$

(21)

and simplifications thereof.

This leads to the results in Table 2. Under (a), we regress M_t+ 1 on all four variables; under (b), we delete the trend; under (c), we also delete the constant term; and under (d), we restrict the coefficient of E_t to be 1. The last model is the simplest and mirrors capital accumulation in Eq. 2. The fit is very good in all cases, as can be seen from the values of s and ${\max \limits } |Q_{t}|$, and also from Figs. 3 and 4.

Table 2 Simplified CO₂ equations

Fig. 3

CO₂ concentration—Time path for DICE 2016R and four simplified models. Model (a) is based on Eq. 21. Model (b) is based on Eq. 21 without a time trend. Model (c) is based on Eq. 21 without both a time trend and a constant term. Model (d) additionally restricts the coefficient of E_t to be 1

Fig. 4

CO₂ concentration—Deviations (%) of four simplified models relative to DICE 2016R. Model (a) is based on Eq. 21. Model (b) is based on Eq. 21 without a time trend. Model (c) is based on Eq. 21 without both a time trend and a constant term. Model (d) additionally restricts the coefficient of E_t to be 1

In Fig. 3, the time paths of CO₂ of DICE and the four models (a)–(d) are indistinguishable, reaching a maximum of 2707 in 2230. The relative deviations Q_t are graphed in Fig. 4. In models (a) and (b), the relative deviations are all below 0.4% in absolute value. In model (c), the relative deviations are all below 0.6% in absolute value, except in the first three periods. In model (d), the relative deviations are larger than 0.6% up to period 36 (year 2190) and smaller than 0.6% afterwards, with a maximum of 1.7% in period 12 (year 2070) where M_t = 1402 (the DICE output) and $\hat {M}_{t}=1425$ (the predicted value of M_t from the regression).

The simplified equation (21) thus provides an excellent approximation to the DICE results because the coefficients in Eq. 18 correspond approximately to a second-order difference equation. Model (b) is possibly the preferred approximation although the simplest model (d) will suffice for most practical applications.

5 Damage and Abatement

The damage-abatement function in DICE specifies two fractions, ω_t (abatement) and $\xi {H_{t}^{2}}$ (damage), of Y_t which reduce Y_t so that less money is available for investment and consumption along the budget constraint. In DICE, this fraction is specified as:

$$ 1 - \omega_{t} - \xi {H_{t}^{2}}. $$

For very high and very low temperatures, the fraction becomes large, but (given the value of ξ) it will still be a fraction between 0 and 1, unless in truly catastrophic cases when H_t > 20.58, that is, when the temperature in period t is more than 20 °C higher than in 1900. Of course, other forms of the damage function are possible; see [1, 19, 28, 32]. Howard and Sterner [11] emphasize the importance of the damage function in accurately estimating coefficient and standard error bias.

In Fig. 5, the graph labeled DICE 2016R contains the time path of this fraction. Models (a) and (b) use an alternative specification, namely

$$ \frac{1 - \omega_{t}}{1+ \xi {H_{t}^{2}}}, $$

based on the fact that the difference

$$ \left( 1-\omega_{t}-\xi {H_{t}^{2}}\right) - \frac{1-\omega_{t}}{1+\xi {H_{t}^{2}}} = \frac{-(\omega_{t}+\xi {H_{t}^{2}})(\xi {H_{t}^{2}})}{1+\xi {H_{t}^{2}}} $$

is small, about 1% in relative terms.

Fig. 5

Damage-abatement functions. Model (a) uses the DICE value of ξ. Model (b) uses the optimal value of ξ

The alternative specification is of interest because we may wish to randomize ξ, as in [14]; see also [12, 13]. This is difficult under the DICE specification because $1 - \omega _{t} - \xi {H_{t}^{2}}$ could become negative (under extreme circumstances), while the alternative specification is always positive provided ξ > 0.

In model (a), we use the same value for ξ as in the DICE model, while in model (b) we use an “optimal” value ξ^∗ = 0.00265 which brings the lines closer together; in fact, DICE and model (b) are indistinguishable in the figure. The value ξ^∗ is obtained by minimizing the sum of squares:

$$ \sum\limits_{t=1}^{T}\left( 1 - \omega_{t} - \xi {H_{t}^{2}} -\frac{1 - \omega_{t}}{1+ \xi^{*} {H_{t}^{2}}}\right)^{2}. $$

with respect to ξ^∗. In summary, for a suitable choice of ξ, we obtain an alternative for the DICE damage function which lends itself better to studying situations of uncertainty or catastrophe.

6 The S-DICE Model

We summarize our proposed S-DICE (simplified DICE) model with the relevant parameters. The S-DICE model is the DICE model, but with the temperature equation replaced by the new (simplest) temperature equation:

$$ H_{t+1} = \eta^{*}_{0} + \eta^{*}_{1} H_{t} + \eta^{*}_{2} \log(M_{t+1}) $$

(22)

with

$$ \eta^{*}_{0}=-2.8672, \quad \eta^{*}_{1}=0.8954, \quad \eta^{*}_{2}=0.4622, $$

and the CO₂ equation replaced by

$$ M_{t+1} = \phi^{*}_{1} M_{t} + E_{t}, \qquad \phi^{*}_{1}=0.9942. $$

(23)

For the damage-abatement equation, we propose:

$$ \frac{1 - \omega_{t}}{1+ \xi^{*} {H_{t}^{2}}}, \quad \xi^{*}=0.00265, $$

(24)

instead of the DICE specification

$$ 1 - \omega_{t} - \xi {H_{t}^{2}}, \quad \xi=0.00236. $$

In addition, one may wish to set μ ≤ 1.0 instead of the upper bound μ ≤ 1.2 as used in DICE. Thus, there are four differences between DICE and S-DICE.

So far, we have worked within the framework of the optimized DICE model (to be precise, the baseline version with ifopt = 0), so that variables such as H_t and M_t take their optimal values as obtained from the DICE GAMS routine, as emphasized in Section 3.

To judge how the S-DICE model compares to DICE, we should optimize S-DICE itself. In Table 3, we compare three scenarios: DICE and two scenarios of S-DICE, namely (a) where we use the new temperature equation (22) and the CO₂ equation (23), but not the new damage-abatement equation, while also keeping the 1.2 bound for μ as in DICE; and (b) where we use the full S-DICE model with all four changes. We present these results for the short term (up to 40 years) and the longer term (100–200 years).

Table 3 DICE versus S-DICE (a) and (b)

The most important conclusion from Table 3 is that the difference between DICE and version (a) of S-DICE is small. This is true for the short and “longer” term (up to 200 years) and remains true for the very long term (up to 500 years), as the three figures below will demonstrate. Hence, the idea that simplifications of DICE tend to have poor long-term properties is not true, at least when we consider simplifications of the temperature and CO₂ equations, which is the main topic of the current paper. In fact, the comparisons we report here give an upper bound on the approximation error of S-DICE to DICE: we could have further reduced the approximation error by calibrating the S-DICE parameters so that optimal S-DICE is as close as possible to optimal DICE.

When we add two further changes, namely the damage-abatement function and the upper bound on μ, then we obtain version (b) of S-DICE. Here, the CO₂ concentration and hence the temperature is much lower than in DICE and S-DICE(a) because the new damage-abatement function of scenario (b) leads to a rapid increase in μ. In order to better understand the reason behind this difference, we study below not only versions (a) and (b) but also the two intermediate versions where only one of the additional changes is implemented.

We denote by μDMH the scenario where all four changes have been implemented: μ for the 1.0 upper bound on μ, D for the damage-abatement function, M for the CO₂ concentration equation, and H for the temperature equation. A 0 indicates that this change has not been implemented. Hence, DICE is 0000, model (a) is 00MH and model (b) is μDMH. To these three scenarios, we now add the intermediate scenarios μ0MH and 0DMH.

In Fig. 6, we present the time paths of consumption of the five scenarios. Consumption (and, similarly, capital accumulation) is not much affected by the different scenarios. The simplified model 00MH with the original damage-abatement function and constraint on μ yields similar time paths to DICE, and even the full S-DICE model μDMH remains close to the DICE results, even in the long run. The consumption paths with a positive emissions constraint are lower than the paths without the constraint. This is because the resource allocation to abatement is larger when negative emissions are not allowed.

Fig. 6

Consumption—DICE and S-DICE compared. Refer to the main text for the meaning of scenario acronyms such as μDMH

In the full S-DICE scenario, the new damage-abatement function causes a rapid reduction of CO₂ in the short term. In fact, μ reaches its maximum (μ = 1) in 2100 when CO₂ concentration, and hence temperature, reaches its highest value. After 2100, CO₂ concentration and temperature gradually decline. Although μ reaches its maximum, consumption becomes larger than under the original damage- abatement function, due to smaller damage.

In Fig. 7, we present the time paths of temperature. With the old damage-abatement function, the time paths are close. Hence, the different behavior is not caused by the constraint on μ but by the form of the damage-abatement fraction. If we include the new damage-abatement fraction but keep the constraint on μ at 1.2 (0DMH), so that we allow negative emissions, then the difference with DICE is much larger than if we include all four changes (μDMH). When the modified damage-abatement function and the constraint of positive emissions are both used, then the positivity constraint on temperature increase imposed by Nordhaus’ GAMS code is binding from the year 2400 onwards, which affects emissions and causes a higher abatement rate and a lower level of CO₂ emissions in the short term. The figure for CO₂ concentration is similar to temperature.

Fig. 7

Temperature—DICE and S-DICE compared. Refer to the main text for the meaning of scenario acronyms such as μDMH

Finally, we present the social cost of carbon (SCC), which attempts to answer the question how much we should be willing to pay to avert future climate damages. More precisely, the SCC tries to add up all the quantifiable costs and benefits of emitting one additional tonne of CO₂. This value can then be used to weigh the benefits of reduced warming against the costs of cutting emissions. In Fig. 8, we present the SCC for the same five scenarios as before. We see that when negative emissions are allowed the social cost of carbon is lower than when negative emissions are not allowed. With the original damage-abatement function and the simplified temperature and CO₂ equations, the SCC paths are similar to DICE, but the new damage-abatement function leads to a higher SCC in the short run so as to reduce CO₂ rapidly. Thus, when negative emissions are not allowed, the social cost of carbon becomes much higher.

Fig. 8

Social cost of carbon—DICE and S-DICE compared. Refer to the main text for the meaning of scenario acronyms such as μDMH

From the tables and figures, we draw the following general conclusions. First, replacing the temperature and CO₂ equations by two simpler, more transparent, more robust, and easier to interpret equations does not much affect the DICE output, not even in the long run. Second, the optimized results are sensitive to the form of the damage-abatement function, especially if we allow for negative emissions. Third, the full S-DICE model leads to optimized values that are different from DICE, but within the bounds of reason. After all, there is no reason to believe that the DICE model is the truth. If S-DICE deviates from DICE then this does not necessarily mean that S-DICE is further from the truth than DICE or is less useful as a policy instrument. As noted in the “Introduction,” there is value in simplicity. Models should be, as Einstein put it, “as simple as possible but not simpler.”

Appendix

In this Appendix, we present two tables which together contain all variable and parameter definitions required to compute the optimum in DICE (the Nordhaus model) and S-DICE (our simplified version of DICE): the variables employed in S-DICE and DICE and their relationship, and the initial values of the six state variables of DICE (Table 4); and the parameters employed in S-DICE and DICE and their relationship (Table 5).

Table 4 Variables in S-DICE and DICE

Table 5 Parameters in S-DICE and DICE