Background on the DICE Model.

The analysis begins with a discussion of the DICE-2016R model, which is a revised version of the DICE-2013R model (1, 3). It is the latest version of a series of models of the economics of global warming developed in collaboration with colleagues at Yale University. The first version of the global dynamic model was in ref. 4. The discussion explains the major modules of the model, and describes the major revisions since the 2013 version. (The current version of the DICE-2016R is available at www.econ.yale.edu/∼nordhaus/homepage/DICEmodels09302016.htm.)

The DICE model views climate change in the framework of economic growth theory. In a standard neoclassical optimal growth model known as the Ramsey model, society invests in capital goods, thereby reducing consumption today, to increase consumption in the future. The DICE model modifies the Ramsey model to include climate investments, which are analogous to capital investments in the standard model. The model contains all elements from economics through climate change to damages in a form that attempts to represent simplified best practice in each area.

Equations of the DICE-2016R Model.

Most of the analytical background is similar to that in the 2013R model, and, for details, readers are referred to ref. 3. Major revisions are discussed as the equations are described.

The model optimizes a social welfare function, W, which is the discounted sum of the population-weighted utility of per capita consumption. The notation here is that V is the instantaneous social welfare function, U is the utility function, c(t) is per capita consumption, and L(t) is population. The discount factor on welfare is R(t) = (1+ρ)^−t, where ρ is the pure rate of social time preference or generational discount rate on welfare.W=∑t=1TmaxV[c(t), L(t)]R(t)=∑t=1TmaxU[c(t)]L(t)R(t).[1]The utility function has a constant elasticity with respect to per capita consumption of the form U(c)=c1−α/(1−α). The parameter α is interpreted as generational inequality aversion.

Net output is gross output reduced by damages and mitigation costs,Q(t)=Ω(t)[1−Λ(t)]Y(t).[2]In this specification, Q(t) is output net of damages and abatement, and Y(t) is gross output, which is a Cobb−Douglas function of capital, labor, and technology. Total output is divided between total consumption and total gross investment. Labor is proportional to population, whereas capital accumulates according to an optimized savings rate.

The current version develops global output in greater detail than earlier versions. The global output concept is purchasing power parity (PPP) as used by the International Monetary Fund (IMF). The growth concept is the weighted growth rate of real gross domestic product (GDP) of different countries, where the weights are the country shares of world nominal GDP using current international dollars. We constructed our own version of world output, and this corresponds closely to the IMF estimate of the growth of real output in constant international (PPP) dollars. The earlier model used the World Bank growth figures, but the growth rates by region could not be replicated.

The present version substantially revised both the historical growth estimates and the projections of per capita output growth. Future growth is based largely on a survey of experts conducted by Peter Christensen and colleagues at Yale University. Growth in global per capita output over the 1980–2015 period was 2.2% per year. Growth in global per capita output from 2015 to 2050 is projected at 2.1% per year, whereas that to 2100 is projected at 1.9% per year. The revisions are updated to incorporate the latest output, population, and emissions data and projections. Population data and projections through 2100 are from the United Nations. CO₂ emissions are from the Carbon Dioxide Information Analysis Center and are updated using various sources. Non-CO₂ radiative forcings for 2010 and projections to 2100 are from projections prepared for the IPCC Fifth Assessment.

The additional variables in the production function are Ω(t) and Λ(t), which represent the damage function and the abatement cost function, respectively. The abatement cost function, Λ(t) in Eq. 2, was recalibrated to the abatement cost functions of other IAMs as represented in the Modeling Uncertainty Project (MUP) study (5). The result was a slightly more costly abatement function than earlier estimates.

The damage function is defined as Ω(t)=D(t)/[1+D(t)], whereD(t)=φ1TAT(t)+φ2[TAT(t)]2.[3]Eq. 3 describes the economic impacts or damages of climate change, which is a key component in calculating the SCC. The DICE-2016R model takes globally averaged temperature change (T_AT) as a sufficient statistic for damages. Eq. 3 assumes that damages can be reasonably well approximated by a quadratic function of temperature change.

The damage function was revised in the 2016 version to reflect new findings. The 2013 version relied on estimates of monetized damages from ref. 6. It turns out that that survey contained several numerical errors (7). The current version continues to rely on existing damage studies, but these were collected by Andrew Moffat and the author and independently verified (see Supporting Information for details). Including all factors, the final estimate is that the damages are 2.1% of global income at a 3 °C warming, and 8.5% of income at a 6 °C warming.

Uncontrolled industrial CO₂ emissions are given by a level of carbon intensity, σ(t), times gross output. Total CO₂ emissions, E(t), are equal to uncontrolled emissions reduced by the emissions reduction rate, μ(t), plus exogenous land-use emissions.E(t)=σ(t)[1−μ(t)]Y(t)+ELand(t).[4]The model has been revised to incorporate a more rapid decline in the CO₂−output ratio to reflect the last decade’s observations. The decade through 2010 showed relatively slow decarbonization, with the global CO₂/GDP ratio changing at −0.8% per year. However, the most recent data indicate a sharp downward tilt, with the global CO₂/GDP ratio changing at −2.1% per year over the 2000–2015 period (preliminary data). Whether this is structural or the result of climate policies is unclear at this point. For the DICE model, we assume that the rate of decarbonization going forward is −1.5% per year (using the IMF output concept).

The geophysical equations link greenhouse gas emissions to the carbon cycle, radiative forcings, and climate change. Eq. 5 represents the equations of the carbon cycle for three reservoirs.Mj(t)=ϕ0 jE(t)+∑i=13ϕi jMi(t−1).[5]The three reservoirs are j = AT, UP, and LO, which are the atmosphere, the upper oceans and biosphere, and the lower oceans, respectively. The parameters ϕ i j represent the flow parameters between reservoirs per period. All emissions flow into the atmosphere. The 2016 version incorporates new research on the carbon cycle. Earlier versions of the DICE model were calibrated to fit the short-run carbon cycle (primarily the first 100 y). Because we plan to use the model for long-run estimates, such as the impacts on the melting of large ice sheets, it was decided to change the calibration to fit the atmospheric retention of CO₂ for periods up to 4,000 y. Based on ref. 8, the 2016 version of the three-box model does a much better job of simulating the long-run behavior of larger models with full ocean chemistry. This change has a major impact on the estimate of the SCC (Table 4 below).

The relationship between GHG accumulations and increased radiative forcing is shown in Eq. 6.F(t)=η{log2[MAT(t)/MAT(1,750)]}+FEX(t).[6]F(t) is the change in total radiative forcings from anthropogenic sources such as CO₂. F_EX(t) is exogenous forcings, and the first term is the forcings due to atmospheric concentrations of CO₂.

Forcings lead to warming according to a simplified two-level global climate model,TAT(t)=TAT(t−1)+ξ1{F(t)−ξ2TAT(t−1)−ξ3[TAT(t−1)−TLO(t−1)]}[7]TLO(t)=TLO(t−1)+ξ4[TAT(t−1)−TLO(t−1)].[8]In these equations, T_AT(t) is the global mean surface temperature and T_LO(t) is the mean temperature of the deep oceans.

The climate module has been revised to reflect recent Earth system models. We have set the equilibrium climate sensitivity (ECS) using the analysis in ref. 9. Ref. 9 uses a Bayesian approach, with a prior based on previous studies and a likelihood based on observational or modeled data. The reasons for using this approach are provided in ref. 5. The final estimate is mean warming of 3.1 °C for an equilibrium CO₂ doubling. We adjust the transient climate sensitivity (TCS) (sometimes called the transient climate response) to correspond to models with an ECS of 3.1 °C, which produces a TCS of 1.7 °C.

The treatment of discounting is identical to that in DICE-2013R. We always distinguish between the welfare discount rate (ρ) and the goods discount rate (r). The welfare discount rate applies to the well-being of different generations, whereas the goods discount rate applies to the return on capital investments. The former is not observed, whereas the latter is observed in markets. When the term “discount rate” is used without a modifier, this will always refer to the discount rate on goods.

The economic assumption behind the DICE model is that the goods discount rate should reflect actual economic outcomes; this implies that the assumptions about model parameters should generate savings rates and rates of return on capital that are consistent with observations. With the current calibration, the discount rate (or, equivalently, the real return on investment) averages 4¼% per year over the period to 2100. The discount rate is the global average of a lower figure for the United States and a higher figure for other countries and is consistent with estimates in other studies that use US data. (This specification is sometimes called the “descriptive approach” to discounting. The alternative approach, used in ref. 10 and elsewhere, is called the “prescriptive discount rate.” Under this second approach, the discount rate is assumed on a normative basis and determined largely independently of actual market returns on investments.)

Definition of the SCC.

Solving Eqs. 1−8 by optimizing the social welfare function (W) yields a path of all variables. We then define the SCC at time t asSCC(t)≡∂W∂E(t)/∂W∂C(t)≡∂C(t)/∂E(t).[9]Looking at the middle term, the numerator is the marginal welfare impact of emissions at time t, and the denominator is the marginal welfare impact of a unit of aggregate consumption in period t; this gives the third term of Eq. 9, in which the SCC equals the economic impact of a unit of emissions in terms of t-period consumption as a numéraire. In actual calculations, we take a discrete approximation to Eq. 9. Note that the SCC is time-indexed because the marginal damage of emissions changes over time.

We have estimated the SCC in the DICE-2016R model for several alternative scenarios. These scenarios reflect differing assumptions about policy and discounting. The units are 2010 US international dollars (that is, in PPP) per metric ton of CO₂ and are expressed in terms of consumption in the given year.

SCC for Standard DICE Model Parameters.

The central cases for the SCC are shown in the first two rows of Table 1. The first row shows the estimate for the standard DICE model with baseline or current climate policy. The SCC figure here is $31.2 per ton of CO₂ for emissions in 2015, with the value rising at 3% per year in real terms through 2050. The SCC along an optimized path, shown in row 2, is slightly lower than the baseline path. The difference between the two cases is small because marginal damages change relatively little between the optimal and baseline case.

Alternative Estimates.

Table 1 shows alternative estimates. We show a calculation for constraining temperature to a 2½ °C limit in two cases: one with a hard cap of 2½ °C, and the second where that cap is for an average of 100 y rather than a single period. The average would be a more sensible objective if damages are a function of average rather than peak temperature. (The hard cap is infeasible for a maximum of 2 °C and would only be feasible if technologies were available that allow substantial negative emissions by around 2050.) The SCCs for the two limit cases are $184 and $107 per ton of CO₂ in 2015 for the two cases of maximum and average limit.

It is well known that the discount rate has an important impact on the SCC. A closer look shows that the key variable is the “growth-corrected discount rate,” which is the difference between the discount rate on goods and the rate of growth of output (see Supporting Information and ref. 2). The estimates of the SCC with different discount rates on goods are shown at the bottom of Table 1. Table 1 also shows the SCC for the discounting procedure proposed in ref. 10. The relationship between the growth-corrected discount rate and the SCC is shown in Fig. 3.

• Download figure
• Open in new tab
• Download powerpoint

Fig. 3.

Social cost of carbon and growth-corrected discount rate in DICE model. The growth-corrected discount rate equals the discount rate on goods minus the growth rate of consumption. The solid line shows the central role of the growth-corrected discount rate on goods in determining the SCC in the DICE model. The square is the SCC from the full DICE model, and the triangle uses the assumptions of The Stern Review (10). A further discussion and derivation of the growth-corrected discount rate is given in Supporting Information.

Regional SCCs.

A few IAMs disaggregate the global SCC into the regional SCCs. These regional estimates represent the marginal damages of emissions for a particular country or region, that is, the SCC when only the damages to that particular region are included in the calculation. These estimates are important for understanding the impacts on individual regions as well as the problem of noncooperative behavior. (In noncooperative behavior, national efforts will be determined by the national SCCs rather than the global SCC and will therefore be much lower; see ref. 12.) Table 2 shows four different sets of estimates of the regional composition of the SCC. The first three are from the three models used by the Interagency Working Group on Social Cost of Carbon (IAWG), and the fourth shows an estimate based on the discounted value of GDP of the regions. One point is clear: The regional estimates are poorly understood, often varying by a factor of 2 across the three models. Moreover, regional damage estimates are highly correlated with output shares (R = 0.71).

The dollar estimates of regional SCCs shown in the first numerical column of Table 2 allocate the global SCC based on the output shares. This estimate is used partially because the regional damage estimates are both incomplete and poorly understood. Additionally, regional output shares are well defined and easy to replicate and, in most cases, fall within the estimates of the different models. A key message here is that there is little agreement on the distribution of the SCC by region—except for the important point that each country’s SCC is well below the global total.

Comparison with the IAWG.

The US government has relied on the work of the IAWG to develop estimates of the SCC (see ref. 13 for different versions). The IAWG concept is conceptually comparable to the baseline in the first row of Table 1. The IAWG combines estimates from three models and multiple scenarios. Table 3 compares the latest round of estimates of the IAWG with estimates from the DICE-2013R and DICE-2016R models for the baseline model and different discount rates. The preferred SCC estimate of the most recent DICE model is about one-fifth lower than the IAWG’s preferred SCC. At comparable discount rates, the DICE model estimate would be roughly twice that of the IAWG.