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The Dynamic Integrated Climate-Economy model, referred to as the DICE model or Dice model, is a neoclassical integrated assessment model developed by 2018 Nobel Laureate William Nordhaus that integrates in the neoclassical economics, carbon cycle, climate science, and estimated impacts allowing the weighing of subjectively guessed costs and subjectively guessed benefits of taking steps to slow climate change. Nordhaus also developed the RICE model (Regional Integrated Climate-Economy model), a variant of the DICE model that was updated and developed alongside the DICE model.[1][2][3][4] Researchers who collaborated with Nordhaus to develop the model include David Popp, Zili Yang, and Joseph Boyer.[2]

The DICE model is one of the three main integrated assessment models used by the United States Environmental Protection Agency, and it provides estimates intermediate between the other two models.[4][5]

History

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Precursors

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According to a summary of the DICE and RICE models prepared by Stephen Newbold,[1] the earliest precursor to DICE was a linear programming model of energy supply and demand in two 1977 papers of William Nordhaus.[6][7] Although dynamic (in that it considered the changing levels of supply of fuel based on supply and demand and the consequence impact on carbon dioxide emissions) the model did not attempt to measure the economic impact of climate change.[1] A 1991 paper by Nordhaus developed a steady-state model of both the economy and climate, coming quite close to the DICE model.[1][8]

The model

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The model appears to have first been proposed by economist William Nordhaus in a discussion paper for the Cowles Foundation in February 1992.[9] He also wrote a brief note outlining the main ideas in an article for Science in November 1992.[10] A subsequent revised model was published in Resource and Energy Economics in 1993.[11][12]

Nordhaus published an improved version of the model in the October 1994 book Managing the Global Commons: The Economics of Climate Change,[13] with the first chapter as well as an appendix containing a computer program both freely available online.[14][15] Marian Radetzki reviewed the book for The Energy Journal.[16]

In 1996, Nordhaus and Zili Yang published an article titled A regional dynamic general-equilibrium model of alternative climate-change strategies at The American Economic Review, established the RICE (Regional Integrated model of Climate and the Economy) model.[17]

In 1998, Nordhaus published a revised version of the DICE model in multiple papers, one of which was coauthored with Joseph Boyer in order to understand the effects of the proposed Kyoto Protocol.[18][19]

In 1999, Nordhaus published computer programs and spreadsheets implementing a revised version of the DICE model as well as a variant called the RICE model (RICE stands for Regional Integrated Climate-Economics, signifying that the modeling of economics and climate are being done only for a particular region rather than the whole world).[20][21]

In 2000, Nordhaus and Boyer co-authored a book published by MIT Press titled Warming the World: Economic Models of Global Warming with a detailed description of the updated DICE and RICE models.[22]

In 2001, Nordhaus published revised spreadsheets for the RICE model.[23]

In November 2006, Nordhaus published a new version of the DICE model with updated data, and used it to review the Stern Review.[2][24][25]

In 2010, updated RICE and DICE models were published, and the new RICE model was explained by Nordhaus in an article for the Proceedings of the National Academy of Sciences (US).[26][27]

In 2013, the book The Climate Casino by Nordhaus, with updated discussion of the DICE and RICE models and the broader policy implications, was published by Yale University Press.[28] A background on the latest version of the models as used in the book was published on Nordhaus' website.[29][30]

2020 rework

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In 2020, modelers from the Potsdam Institute for Climate Impact Research (PIK) reported a rerun of the DICE model using updated climate and economic information and found that the economically optimal climate goal was now less than 2.0 °C of global warming — and not the 3.5 °C that Nordhaus had originally calculated.[31][32] The PIK team employed current understandings of the climate system and more modern social discount rates.[33] This new result therefore broadly supports the Paris Agreement goal of holding global warming to "well below 2.0 °C". Their revised AMPL code and data are available under open licenses.[34]

Assumptions and outcomes

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According to the original formulation of DICE, staying below the 2 °C as agreed by the Paris agreement would cost more in mitigation investments than would be saved in damage from climate change. A 2020 paper by Glanemann, Willner and Levermann, which used an updated damage function, revised this conclusion, showing that a warming of around 2 °C would be "optimal", depending on the climate sensitivity to greenhouse gases.[35]

The DICE model is an example of a neoclassical energy-economy-environment model. The central assumption of this type of model is that market externalities create costs not captured in the price system and that government must intervene to assure that these costs are included in the supply price of the good creating the externality. Innovation is assumed to be exogenous; as such, the model is a pre-ITC model (it does not yet include Induced Technological Change).[36] An extension of the model (DICE-PACE) that does include induced technological change, has strongly different outcomes: the optimal path would be to invest strongly early on in mitigation technology.[37] In contrast to non-equilibrium models, investment in low carbon technology is assumed to crowd-out investments in other parts of the economy, leading to a loss of GDP.[36]

Reception

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Academic reception

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A number of variants of the DICE model have been published by researchers working separately from Nordhaus.[38][39] The model has been criticised by Steve Keen for a priori assuming that 87% of the economy will be unaffected by climate change, misrepresenting contributions from natural scientists on tipping points, and selecting a high discount rate.[40]

Reception in the public policy world

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The DICE and RICE models have received considerable attention from others studying the economic impact of climate change. It is one of the models used by the Environmental Protection Agency for estimating the social cost of carbon.[4][5] Stephen Newbold of the Environmental Protection Agency in the United States reviewed the models in 2010.[1]

The Basque Centre for Climate Change, in an October 2009 review of integrated assessment models for climate change, discussed the DICE model in detail.[41]

A report from The Heritage Foundation, a conservative and climate change denying think tank in the United States, called the DICE model "flawed beyond use for policymaking" on account of its extreme sensitivity to initial assumptions.[5] Similar criticisms, including criticisms of the specific choice of discount rates chosen in the model, have been made by others.[42][43] Many of these criticisms were addressed in the 2020 rework listed above.

See also

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References

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Revisions and contributorsEdit on WikipediaRead on Wikipedia

DICE model

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from Grokipedia
The Dynamic Integrated Climate-Economy (DICE) model is a neoclassical integrated assessment model (IAM) developed by economist William D. Nordhaus that couples a global economic growth framework with simplified representations of atmospheric carbon cycles and climate responses to evaluate the economic costs and benefits of greenhouse gas emissions and abatement policies. First introduced in 1992, the model optimizes intertemporal welfare by balancing consumption, investment, and emissions reductions, producing estimates of the social cost of carbon (SCC)—the marginal damage from an additional ton of CO2—as well as optimal carbon prices and emission trajectories. Nordhaus's draws from Ramsey optimal growth theory, incorporating a driven by capital, labor, and inputs; a damage function that reduces output based on temperature increases; and backstop technologies for low-carbon substitution, all solved via dynamic programming to maximize discounted utility over centuries-long horizons. The model's projections have informed U.S. regulatory analyses, including SCC values used by agencies like the EPA, with recent iterations like DICE-2023 yielding an SCC of approximately $85 per ton of CO2 in 2020 dollars under updated damage and climate sensitivity assumptions. Nordhaus received the 2018 in partly for pioneering such IAMs, recognizing their role in quantifying trade-offs between short-term and long-term climate stabilization. Despite its influence, has faced substantial scrutiny for its damage function, which relies on quadratic approximations of empirical data that critics argue systematically understate catastrophic risks, non-market damages, and tipping points like thaw or collapse, leading to lower SCC estimates compared to surveys of damage projections. Other debates center on its high pure rate of in (implying future generations' welfare is undervalued) and assumptions of smooth technological progress enabling cheap , which empirical backtests and alternative suggest may overestimate optimal emissions pathways and laxity. These limitations highlight DICE's roots in equilibrium neoclassical assumptions rather than empirical validation of fat-tailed or sectoral disaggregation, though proponents defend its transparency and consistency in revealing that aggressive near-term abatement yields net welfare losses under baseline parameters.

Overview

Core Structure and Purpose

The DICE (Dynamic Integrated Climate-Economy) model is an integrated assessment model (IAM) that quantifies the interactions between human economic activity and the climate system to evaluate the costs and benefits of climate policies, such as carbon pricing. Developed by economist , it optimizes global welfare by balancing the marginal costs of emissions abatement against the marginal damages from , typically yielding estimates for the (SCC) around $80 per ton of CO2 in recent calibrations. The model's purpose centers on providing a framework for cost-benefit analysis of mitigation, assuming a representative global agent maximizes intertemporal utility subject to resource and environmental constraints, thereby informing policy on efficient emissions paths. At its core, DICE structures the economy as a neoclassical optimal growth model akin to the Ramsey framework, where output is produced via a Cobb-Douglas function combining labor, capital, and total factor productivity, with gross output reduced by a climate damage term and net output further diminished by abatement expenditures. Capital accumulates through savings from net output, while labor grows exogenously based on population projections; emissions arise as a fraction of gross output, modulated by abatement efforts that incur convex costs rising with the share of emissions reduced. The climate module simplifies geophysical processes: atmospheric CO2 stocks evolve from net emissions (fossil fuels minus abatement plus land sinks), driving radiative forcing that influences global mean temperature via a two-equation energy balance model incorporating ocean heat uptake. Damages are modeled as a quadratic function of temperature change, aggregating empirical estimates of sectoral losses (e.g., agriculture, sea-level rise) into a percentage reduction in global output, calibrated from meta-analyses of damage studies. This integrated structure enables dynamic programming solutions over discrete time periods (typically 10-year steps from onward), discounting future utilities at a rate combining pure and elasticity of , often around 1.5-4% annually depending on parameterization. Unlike purely economic or climate models, endogenously links emissions to growth decisions and feedbacks from damages to , highlighting trade-offs where early abatement slows short-term growth but averts long-term losses, though critics note its reliance on extrapolated damage functions beyond observed warming levels. The model operates as a single-region global aggregator, abstracting from regional heterogeneity or geopolitical factors to focus on aggregate efficiency.

Integration of Climate and Economic Modules

The DICE model couples its economic and subsystems through dynamic, recursive interactions that link anthropogenic emissions to climate responses and subsequent economic impacts. The core economic module employs a neoclassical Ramsey optimal growth framework, where gross output is produced via a Cobb-Douglas incorporating capital, labor, and , but adjusted downward by fractions representing abatement costs (Λ) and climate damages (Ω). Emissions of CO₂ and non-CO₂ greenhouse gases arise endogenously from economic output, scaled by carbon intensity (σ) and modulated by an emissions control rate (μ), with abatement costs modeled as a of μ to represent expenditures. These emissions serve as direct inputs to the climate module, which simulates the global carbon cycle using a four-reservoir diffusion energy balance model (DFAIR), tracking flows from emissions into atmospheric CO₂ concentrations (MAT), upper ocean, deep ocean, and pre-industrial levels. Radiative forcing (F) is then derived primarily from logarithmic increases in MAT relative to pre-industrial levels, augmented by exogenous forcings and abatement of non-CO₂ gases, leading to temperature dynamics governed by a two-layer energy balance model calibrated to an equilibrium climate sensitivity of 3.0°C and transient climate response of 1.8°C. This unidirectional flow from economy to climate establishes the causal pathway by which growth-driven emissions accumulate and alter global temperatures over multi-decade horizons. The feedback from to occurs via the function, where Ω is specified as a quadratic in anomaly (T_AT), empirically estimated to capture rising marginal impacts such as sea-level rise, , and productivity losses, reducing net output and thus future and emissions. In the optimization, the model solves for time paths of μ and consumption that maximize welfare, subject to these linkages, yielding endogenous responses where elevated damages incentivize higher abatement to curb emissions and mitigate feedbacks. This integrated structure enables to evaluate trade-offs between short-term abatement costs and long-term avoidance, though the simplified representations—such as global aggregation and quadratic forms—have been critiqued for understating nonlinear risks.

Historical Development

Precursors in Climate-Economic Modeling

Early efforts to integrate economic activity with climate dynamics emerged in the 1970s, primarily through partial equilibrium models linking energy use, emissions, and atmospheric concentrations. developed one of the initial frameworks in his 1975 International Institute for Applied Systems Analysis (IIASA) working paper, "Can We Control ?", which analyzed combustion as the primary source of anthropogenic CO2 buildup and evaluated policy options like fuel switching and carbon taxes to limit emissions growth. This model projected CO2 levels under business-as-usual scenarios reaching 600-700 ppm by the mid-21st century, assuming exponential energy demand growth tied to GDP expansion at 3-4% annually, but treated economic output as exogenous rather than endogenously responding to climate feedbacks. Building on this, Nordhaus's 1977 paper, "Economic Growth and Climate: The Case of ," published in the , extended the analysis to incorporate neoclassical growth theory, estimating that unchecked CO2 accumulation could reduce global welfare by altering long-term temperature and patterns, with damages potentially equivalent to 1-2% of GDP under moderate warming scenarios. These models emphasized causal chains from economic expansion to emissions via but omitted comprehensive damage functions or abatement costs, relying instead on simplified climate response functions derived from contemporaneous geophysical estimates, such as those from the 1975 report on climate variability. Nordhaus's 1978 updates further refined energy substitution elasticities, introducing backstop technologies like with high initial costs declining over time at 2-3% annually, foreshadowing endogenous technological progress in later frameworks. These precursors laid foundational linkages between macroeconomic variables and subsystems but remained sector-specific and static compared to dynamic general equilibrium approaches. Broader global modeling efforts, such as the Limits to Growth study by Meadows et al., incorporated pollution stocks as constraints on growth using but lacked explicit climate-economy feedbacks or emissions tracing to specific greenhouse gases like CO2. By the late , as geophysical understanding advanced—e.g., via the 1985 Villach Conference consensus on —models began evolving toward fuller integration, setting the stage for Nordhaus's by addressing endogeneity in , damages, and within a unified intertemporal optimization. This progression highlighted the need for causal realism in representing trade-offs, avoiding overreliance on exogenous assumptions that understated adaptation potentials or overstated abrupt tipping risks without empirical calibration.

Initial Formulation and Early Iterations (1990s)

The Dynamic Integrated Climate-Economy (DICE) model was initially formulated by in 1992 as an intertemporal general-equilibrium framework combining neoclassical theory with simplified representations of the . The model, detailed in Nordhaus's Cowles Foundation Discussion Paper 1009 and a contemporaneous Science article, optimized global emissions paths to balance economic costs of abatement against climate damages, using a single representative agent for the world economy. Key components included a Ramsey-style growth module with , labor from population projections, and ; emissions linked to gross output via a carbon intensity parameter; a three-box model for atmospheric, upper ocean, and deep ocean concentrations; and a two-equation climate module for and temperature dynamics based on equilibrium responses calibrated to historical data. Damages were modeled as a quadratic function reducing net output, with an estimated 1.5% global GDP loss for a 3°C warming, derived from rudimentary sector-specific studies due to the absence of comprehensive empirical aggregates at the time. In baseline simulations without policy intervention, the 1992 version (DICE-1992 or DICE.1) projected cumulative industrial CO₂ emissions reaching 89.1 Gt by 2100, atmospheric concentrations stabilizing near 700 ppm, and a rise of 3.2°C above pre-industrial levels, reflecting mid- emissions data and a stagnationist assumption that implied slowing growth after 2025. Optimal trajectories, solved via dynamic programming, recommended gradual emissions reductions starting in the , achieving a rising to about $5–10 per ton by century's end (in 1989 dollars), with total welfare gains from mitigation estimated at small fractions of GDP. The model's discount rate incorporated pure and elasticity of , yielding a 3% annual rate, while abatement costs followed a logarithmic form calibrated to estimates for substitution. Limitations included reliance on outdated data for economic baselines, simplistic damage aggregation without micro-foundations, and no explicit calculation of the , which awaited later versions. Early iterations in the mid-1990s refined the 1992 structure with updated inputs but preserved the core neoclassical and reduced-form approach. The version, published in Nordhaus's book Managing the Global Commons, incorporated revised population and GDP projections from sources like the World Bank, slightly lowering baseline emissions while raising projected 2100 warming to around 3°C under no-policy scenarios due to adjusted sensitivities. estimates remained basic, drawing on expanded but still limited sectoral data (e.g., , sea-level rise), yielding similar quadratic impacts without probabilistic tail risks. These updates emphasized the model's tractability for , such as evaluating carbon taxes versus quantity controls, but Nordhaus noted persistent uncertainties in functions, which were "put together based on very rudimentary estimates" amid scarce empirical evidence. By the late , DICE had influenced related models like (regional disaggregation) but underwent no major structural overhauls, focusing instead on data recalibrations to track emerging IPCC assessments.

Evolution Through the 2000s and 2010s

In the , the DICE model underwent refinements to incorporate emerging empirical data on , damages, and dynamics, with notable updates in versions such as DICE-2007 and DICE-2008. These iterations addressed limitations in earlier formulations by shifting valuations from market exchange rates to for more accurate global output comparisons and revising growth assumptions to eliminate stagnationist biases observed in retrospective analyses. functions were adjusted upward, reflecting new studies on impacts, while estimates aligned with updated IPCC assessments, leading to projected temperature increases of approximately 3.2°C by 2100 under baseline emissions. The (SCC) rose modestly in these models, influenced by responses to external critiques like the 2006 , which highlighted discounting and underestimation; Nordhaus countered by emphasizing empirical calibration over prescriptive rates, resulting in stable long-term discount rates around 3.5%. By the 2010s, further iterations like DICE-2010, DICE-2013R, and DICE-2016R introduced structural enhancements and parameter recalibrations based on expanded datasets. The DICE-2013R version integrated regional variations via linkages to the model and updated abatement cost functions using data from the Modeling Uncertainty Project, while baseline global output projections for 2100 were revised upward by 35% to $816 trillion (in 2010 dollars) due to stronger historical growth evidence. In DICE-2016R, damage functions were significantly revised using 26 independent studies, increasing estimated losses to 2.1% of income at 3°C warming and 8.5% at 6°C, correcting prior errors in meta-analyses like Tol's survey; this adjustment raised the SCC to $31 per ton of CO₂ in 2015 dollars, a near-sixfold increase from early estimates. modules saw improvements, including a more precise extending to 4,000 years and equilibrium of 3.1°C, alongside faster decarbonization rates of -1.5% annually; abatement costs were recalibrated slightly higher than in 2013R. These changes, driven by peer-reviewed evidence rather than policy advocacy, elevated overall damage projections by 191% across the model's history while maintaining neoclassical growth foundations.

Recent Updates Including DICE-2023

The DICE-2023 model, developed by and Lint Barrage, incorporates significant revisions to reflect updated empirical data on climate damages, carbon cycles, and abatement technologies, as detailed in their analysis published in April 2023. A primary structural change is the integration of the DFAIR module, an adaptation of the (Finite Amplitude Impulse-Response) framework, which replaces prior linear approximations in the with models accounting for saturation effects in ocean and land uptake, drawing from Joos et al. (2013) and Millar et al. (2017). This enhances accuracy for large emission pulses, projecting atmospheric retention at approximately 70% for a 5,000 GtC impulse. Parameter updates include a revised damage function estimating 3.1% global GDP loss at 3°C warming, up from 1.2% in DICE-2016, informed by syntheses from Piontek et al. (2021), IPCC AR6 assessments, and Dietz et al. (2021) on tipping points, with additional judgmental adjustments for underrepresented impacts. parameters align with IPCC AR6, setting equilibrium at 3.0°C and transient response at 1.8°C. The pure rate of is lowered to 1.0% annually from 1.5%, and the elasticity of of consumption raised to 1.5 from 1.45, reducing the average discount rate to 3.9% over 2020–2100. Abatement costs feature a backstop price of $515/tCO₂ in 2050, declining at 1% annually until then and 0.1% thereafter, enabling net-zero emissions at 2.7% of output by 2100. These modifications yield a cost-benefit optimal atmospheric concentration stabilizing at 2.6–2.7°C warming by 2100, lower than in prior iterations, with carbon prices rising to $90/tCO₂ by 2040 and $148/tCO₂ by 2060. The increases to $50–66/tCO₂ in 2020 under baseline and optimal scenarios, compared to $18/tCO₂ in a 1992 rerun, reflecting heightened estimates. Achieving a 2°C target becomes more feasible, requiring 99% emissions control by 2100 at a offset by avoided , potentially increasing global by $107–120 relative to baseline. Emissions coverage expands to non-industrial CO₂ and non-CO₂ gases, boosting abatable fractions by 35% by 2050. No further major updates beyond DICE-2023 have been documented as of late 2024.

Key Assumptions

Economic Growth and Discounting Framework

The economic growth module of the DICE model is grounded in neoclassical optimal growth theory, utilizing a discrete-time Ramsey framework that endogenously determines savings, , and consumption paths while accounting for damages and abatement expenditures. Gross output QtQ_t is produced according to a Cobb-Douglas : Qt=AtKtγLt1−γQ_t = A_t K_t^\gamma L_t^{1-\gamma}, where AtA_t represents , KtK_t is the capital stock, LtL_t is labor input, and γ\gamma (approximately 0.3) is the capital's share of income; this gross output is then reduced by a damage term Ωt\Omega_t from and abatement costs Λt\Lambda_t. Net output Yt=Qt(1−Ωt)(1−Λt)Y_t = Q_t (1 - \Omega_t)(1 - \Lambda_t) is allocated between consumption CtC_t and ItI_t, with capital accumulating via Kt+1=(1−δ)Kt+ItK_{t+1} = (1 - \delta) K_t + I_t, where δ\delta (around 0.1) is the depreciation rate. Labor LtL_t follows exogenous population projections from the , while AtA_t grows exogenously at a rate that asymptotes to a long-run value, incorporating economy-wide technological progress and carbon-saving innovations that reduce emissions intensity over time. Discounting in DICE occurs through maximization of a social welfare function aggregating period utility across time, weighted by population and a discount factor: W=∑tLtct1−ϕ1−ϕΠtW = \sum_t L_t \frac{c_t^{1-\phi}}{1-\phi} \Pi_t, where ctc_t is per capita consumption, ϕ\phi is the elasticity of marginal utility of consumption (calibrated to values around 2 in prior iterations, reflecting inequality aversion and risk), and Πt\Pi_t incorporates the pure rate of time preference ρ\rho (typically 1.5% in historical DICE versions) along with adjustments for consumption growth. The effective consumption discount rate follows the Ramsey rule, approximately rt=ρ+ϕgtr_t = \rho + \phi g_t, where gtg_t is expected per capita consumption growth (around 2% in baseline projections), yielding near-term real rates of return on capital around 4.5% in DICE-2023, calibrated to observed market interest rates, risk premia, and equity returns. Recent updates introduce time-varying elements to ρt∗=ρ−12ϕ2σC2t+βCLIMπ\rho_t^* = \rho - \frac{1}{2} \phi^2 \sigma_C^2 t + \beta_{CLIM} \pi, accounting for consumption growth uncertainty (σC≈1%\sigma_C \approx 1\% per year) and climate-related risk premia (βCLIM=0.5\beta_{CLIM} = 0.5), with an overall risk-free rate of 2% and equity risk premium of 5%, ensuring the framework aligns with empirical asset pricing data rather than ad hoc low-discount assumptions. This setup implies declining discount rates over time due to uncertainty adjustments and potential productivity slowdowns, influencing the optimal timing of emissions reductions by balancing intergenerational equity against opportunity costs of capital; for instance, higher ϕ\phi amplifies the growth-augmented discounting term, reducing the present value of distant damages relative to low-ρ\rho alternatives like those in the Stern Review, which Nordhaus critiques for understating empirical time preferences. Parameter choices prioritize consistency with long-run economic data, such as historical GDP growth (revised upward in later versions using purchasing power parity, projecting 2100 per capita GDP at $73,367 in 2017-equivalent terms) and observed returns, over prescriptive ethical priors.

Climate System Representation

The in the DICE model is depicted via a reduced-form geophysical module that connects anthropogenic emissions to atmospheric concentrations, , and surface temperature anomalies, calibrated to empirical data and more complex simulations rather than solving full general circulation models. This module comprises three primary components: a representation for CO₂ partitioning, a logarithmic function linking concentrations to imbalance, and a two-box balance model for temperature evolution incorporating ocean heat uptake. In earlier iterations, such as DICE-2016, the relied on a three-reservoir linear box-diffusion scheme assuming constant fractional transfers between atmosphere, upper ocean, and deep ocean layers, with parameters yielding an airborne fraction of approximately 45-50% over centuries. DICE-2023 introduces substantial refinements to enhance realism, particularly in the , by adopting the DFAIR (Dynamic Finite-Amplitude Impulse-Response) framework with four s to model CO₂ dynamics. This update replaces prior linear assumptions with saturation-dependent , where efficiency declines under high cumulative emissions; for instance, the fraction of emissions remaining airborne rises from roughly 30% for a 100 GtC pulse to 70% at 5,000 GtC total, reflecting of weakening and s. The equations are given by ΔRi(t)=ξiE(t)−Ri(t)α(t)τi\Delta R_i(t) = \xi_i E(t) - \frac{R_i(t) \alpha(t)}{\tau_i}, where Ri(t)R_i(t) denotes carbon in ii, ξi\xi_i the initial partitioning fraction, α(t)\alpha(t) the time-varying saturation factor tied to cumulative emissions Cacc(t)C_{acc}(t), τi\tau_i the turnover time, and E(t)E(t) gross CO₂ emissions; atmospheric concentration MAT(t)MAT(t) sums the reservoirs relative to preindustrial levels. Non-CO₂ greenhouse gases are handled separately via exogenous forcing adjustments, with total forcing F(t)=FCO2×2log⁡2(MAT(t)MAT(1765))+FABATE(t)+FEX(t)F(t) = F_{CO_2 \times 2} \log_2 \left( \frac{MAT(t)}{MAT(1765)} \right) + F_{ABATE}(t) + F_{EX}(t), where FCO2×2≈3.93F_{CO_2 \times 2} \approx 3.93 W/m² calibrates the CO₂ doubling effect. Temperature dynamics employ a two-layer linear response model distinguishing upper ocean/atmosphere (Box 1) from deep (Box 2), with global mean surface anomaly TATM(t)=TBOX1(t)+TBOX2(t)T_{ATM}(t) = T_{BOX1}(t) + T_{BOX2}(t). Evolution follows TBOXj(t)=TBOXj(t−1)e−dj+teq(F(t)−TBOXj(t−1))(1−e−dj)T_{BOXj}(t) = T_{BOXj}(t-1) e^{-d_j} + t_{eq} (F(t) - T_{BOXj}(t-1)) (1 - e^{-d_j}) for j=1,2j=1,2, where decay rates d1=0.324d_1 = 0.324 and d2=0.440d_2 = 0.440 (implying adjustment times τ1≈4.07\tau_1 \approx 4.07 years and τ2≈236\tau_2 \approx 236 years) and equilibrium parameter teqt_{eq} ensure calibration to IPCC AR6 medians of equilibrium (ECS) at 3.0°C and transient climate response (TCR) at 1.8°C per CO₂ doubling. heat uptake is implicit in the differential adjustment rates, capturing lagged deep- warming without explicit ; this simplification assumes constant and neglects tipping points or feedbacks, prioritizing tractability for economic optimization over detailed process representation. These elements yield temperature projections aligned with CMIP6 ensemble means under comparable forcings, though the model's neutrality on responses may overestimate near-term atmospheric retention.

Damage and Abatement Cost Functions

In the DICE model, the damage function quantifies climate-induced economic losses as a fraction of gross output, assuming damages arise primarily from and scale proportionally with global levels. This function takes a : the damage fraction Ωt=a1⋅Tt+a2⋅Tt21+a1⋅Tt+a2⋅Tt2\Omega_t = \frac{a_1 \cdot T_t + a_2 \cdot T_t^2}{1 + a_1 \cdot T_t + a_2 \cdot T_t^2}, where TtT_t denotes the global mean surface relative to 1900 levels, and parameters a1a_1 and a2a_2 (typically small positive values, such as a1≈0a_1 \approx 0 and a2≈0.0027a_2 \approx 0.0027 in earlier calibrations) are derived from meta-analyses of sector-specific impact studies, including , sea-level rise, and . Net output is then Ytnet=Yt⋅(1−Ωt)Y_t^{net} = Y_t \cdot (1 - \Omega_t), implying that damages reduce without altering capital or labor inputs directly. This specification, retained in DICE-2023 with minor parameter tweaks from updated reviews, presumes smooth, convex damage escalation but excludes abrupt tipping elements like thaw or collapse, drawing parameters from peer-reviewed syntheses that aggregate tangible impacts while imputing intangibles conservatively. The abatement cost function captures the resource costs of emissions reductions, modeled as a convex in the control rate μt\mu_t (the share of baseline emissions abated in period tt). It is formulated as Θt=θ1⋅Yt⋅μtθ2\Theta_t = \theta_1 \cdot Y_t \cdot \mu_t^{\theta_2}, where Θt\Theta_t is the abatement expenditure as a share of gross output YtY_t, θ1>0\theta_1 > 0 scales baseline costs (often around 2-3 for CO2), and θ2>1\theta_2 > 1 (typically 2.8-3.0) enforces rising marginal costs due to technological and behavioral frictions. Parameters are calibrated from bottom-up assessments and econometric studies of historical efforts, assuming costs fall over time via exogenous backstop technologies (e.g., advanced renewables at declining prices). In DICE-2023, the function extends to non-CO2 gases with analogous convex forms, but retains proportionality to output and no endogenous beyond baseline trends. Combined, these functions yield net output Ytnet=Yt⋅(1−Ωt)⋅(1−Θt/Yt)Y_t^{net} = Y_t \cdot (1 - \Omega_t) \cdot (1 - \Theta_t / Y_t), balancing marginal abatement costs against avoided damages in optimization.

Model Mechanics and Outputs

Simulation Dynamics

The DICE model simulates the coupled evolution of the global economy and climate system over discrete time periods, with recent versions such as DICE-2023 employing five-year time steps from the base year (typically 2015 or 2020) extending to a terminal period around 2600 to avoid end-of-world boundary effects. This discretization approximates continuous-time processes, allowing numerical solution of the model's intertemporal optimization problem, where a representative social planner maximizes discounted global utility subject to economic production, emissions, carbon cycle, radiative forcing, and temperature dynamics. The solution method typically involves recursive dynamic programming or nonlinear programming solvers, computing backward from the terminal condition to derive optimal controls (e.g., savings rates and abatement efforts) and forward-simulating state trajectories. Key state variables include the capital stock KtK_t, which evolves via net investment after ; atmospheric carbon concentration MATtM_{AT_t}; carbon in upper and lower boxes MUPtM_{UP_t} and MLOtM_{LO_t}; and temperatures in the atmosphere TATtT_{AT_t} and upper TUPtT_{UP_t}. Economic dynamics follow a Ramsey-style growth model, with output Yt=AtKtα(Ltvt)1−αΩtY_t = A_t K_t^\alpha (L_t v_t)^{1-\alpha} \Omega_t, where AtA_t is , α\alpha the capital share (around 0.3), LtL_t labor, vtv_t labor-augmenting technical progress, and Ωt=1/(1+D(TATt))\Omega_t = 1 / (1 + D(T_{AT_t})) the damage factor from quadratic temperature damages D(T)=a1T2/(1+a2T2)D(T) = a_1 T^2 / (1 + a_2 T^2). Gross output is allocated to consumption CtC_t, ItI_t, and abatement costs, yielding capital accumulation Kt+1=(1−δ)Kt+ItK_{t+1} = (1 - \delta) K_t + I_t with δ≈0.06\delta \approx 0.06 per five years. Emissions arise as Et=σtYt(1−μt)E_t = \sigma_t Y_t (1 - \mu_t), where σt\sigma_t is the carbon intensity (declining exogenously) and μt\mu_t the abatement rate, incurring costs θ(μt)Yt\theta(\mu_t) Y_t calibrated to quadratic forms matching empirical marginal abatement curves. These feed into a three-box carbon cycle with transition equations: atmospheric uptake to oceans via ϕ12(MATt−ζ1MUPt)\phi_{12} (M_{AT_t} - \zeta_1 M_{UP_t}) and ϕ23(MUPt−ζ2MLOt)\phi_{23} (M_{UP_t} - \zeta_2 M_{LO_t}), plus exogenous decay, updating stocks as MATt+1=MATt+Et(1−δCO2)−ϕ12(⋅)+ϕ23(⋅)M_{AT_{t+1}} = M_{AT_t} + E_t (1 - \delta_{CO2}) - \phi_{12} (\cdot) + \phi_{23} (\cdot), where ϕ\phi parameters reflect ocean mixing rates from empirical calibrations. Radiative forcing Ft=ηln⁡(MATt/M0)+FexF_t = \eta \ln(M_{AT_t}/M_0) + F_{ex} drives temperature dynamics via energy balance: TATt+1=TATt+λ(Ft−σTATt−γ(TATt−TUPt))T_{AT_{t+1}} = T_{AT_t} + \lambda (F_t - \sigma T_{AT_t} - \gamma (T_{AT_t} - T_{UP_t})), with ocean heat uptake λ\lambda, feedback σ\sigma, and diffusion γ\gamma, similarly for TUPt+1T_{UP_{t+1}}. In simulation, optimal μt\mu_t and investment rise gradually to balance marginal abatement costs against , yielding trajectories where emissions peak under calibrated parameters, global temperature stabilizes around 2.5–3°C above pre-industrial levels, and GDP growth slows modestly due to (e.g., 0.5% annual loss at 3°C). Sensitivity to time step length is low for five- versus ten-year discretizations in earlier versions, as continuous-time approximations confirm stability, though finer steps increase computational demands without altering core dynamics significantly. Utility is isoelastic in per-capita consumption, discounted at a pure rate ρ≈0.015\rho \approx 0.015 plus elasticity-weighted growth, aggregating over calibrated to UN projections.

Optimal Policy Trajectories

The optimal policy trajectories in the DICE model emerge from solving a centralized optimal control problem, where a social planner maximizes the present value of global utility over time by choosing abatement rates that balance marginal abatement costs against marginal climate damages. This intertemporal optimization incorporates the model's neoclassical growth framework, with emissions reductions implemented via a shadow price on carbon equivalent to the social cost of carbon (SCC). The resulting paths feature gradual intensification of policy effort, as immediate aggressive abatement would impose high short-term economic costs while deferring action risks irreversible climate feedbacks. In baseline calibrations, optimal gross emissions from economic output peak around the 2030s-2040s before declining toward net-zero by 2100, reflecting technological learning and backstop substitutions in the abatement cost function. The corresponding trajectory rises monotonically: for instance, in DICE-2023, the optimal SCC starts at approximately $45 per metric ton of CO₂ in , escalating to $111 per ton by 2050, driven by accumulating atmospheric concentrations and escalating valuations. This pricing incentivizes a transition where abatement rates increase from under 20% initially to over 50% by century's end, stabilizing below levels implying 3°C equilibrium warming. These trajectories assume perfect commitment and global coordination, yielding welfare gains over business-as-usual scenarios through avoided damages exceeding abatement expenditures in terms. Sensitivity analyses show that higher damage elasticities or lower rates accelerate emissions declines and elevate peak carbon prices, potentially doubling the 2100 abatement level. Empirical implementation often proxies this via uniform carbon taxes, though outputs underscore the inefficiency of uniform targets like net-zero by 2050 absent corresponding price signals.

Social Cost of Carbon Estimates

The (SCC) in the DICE model quantifies the present discounted value of incremental global economic from emitting one additional metric ton of CO₂, computed along the cost-benefit optimal emissions trajectory where marginal abatement costs equal marginal . This metric rises over time within each model run due to accumulating atmospheric concentrations, rising temperatures, and escalating damage vulnerabilities. SCC estimates have trended upward across model versions, driven by empirical updates to damage functions (e.g., higher loss fractions per degree of warming), refined carbon cycle dynamics (e.g., reduced ocean and land efficacy), and economic inputs like growth.
Model VersionEmission YearSCC ($/tCO₂)Currency/Base YearKey Update Factors
DICE-1992~19904.541989 USDInitial low damage specifications; retrospective calculation.
DICE-2013R201518.62005 USDBaseline reference path; 3% annual real growth projected.
DICE-2016R201531.22010 Intl. $ (PPP)Revised growth, , and quadratic ; 3% annual rise to 2050.
DICE-2023 (optimal)2020502019 USDDoubled vs. 2016 (3.1% output loss at 3°C); updated sinks, non-CO₂ GHGs; time-varying discount ~4.5% near-term.
DICE-2023 also computes a higher baseline SCC of $66 per for 2020 emissions (absent policy), highlighting the model's sensitivity to abatement assumptions, as optimal controls internalize benefits from avoided future damages. Sensitivity analyses in DICE-2023 show SCC varying sharply with constant discount rates: $87 at 3%, $176 at 2%, and $485 at 1%. By 2050, the optimal SCC reaches $125 per , underscoring long-term escalation from sustained warming.

Empirical Foundations and Validation

Data Inputs and Parameter Estimation

The DICE model's parameters are calibrated using a combination of historical observations, projections from international agencies, and syntheses of peer-reviewed studies on , emissions, climate dynamics, and impacts. Economic inputs draw from World Bank and IMF data on in terms, with baseline emissions following the Shared Socioeconomic Pathway 2 (SSP2) scenario from the International Institute for Applied Systems Analysis (IIASA). Population projections are sourced from estimates, capping at approximately 10.8 billion by the late , while growth is informed by econometric analyses such as those in Christensen et al. (2018) and Müller et al. (2022). Climate module parameters, including equilibrium (ECS) of 3.0°C and transient climate response (TCR) of 1.8°C, are updated from (IPCC) Sixth Assessment Report (AR6) syntheses of paleoclimate, observational, and process-based modeling evidence. The carbon cycle incorporates the diffusive finite-amplitude impulse-response (DFAIR) function calibrated to historical atmospheric CO2 concentrations and emissions data, with nonlinear retention rates reflecting updated process models from Millar et al. (2017). and initial atmospheric conditions are aligned with observed 2020 levels from global monitoring networks. Damage function parameters employ a quadratic specification in global mean temperature, with the coefficient ψ₂ set at 0.003467 to yield about 3.1% global GDP loss at 3°C warming, derived from meta-analyses of sector-specific impact studies (e.g., Piontek et al., 2021; IPCC AR6) augmented by probabilistic assessments of tipping points from Dietz et al. (2021) and judgmental adjustments for underrepresented risks. This represents a doubling from DICE-2016 estimates, reflecting accumulated from econometric panels on weather-related damages and sea-level rise. Abatement cost parameters, including a backstop price of $515 per metric ton CO2 in 2050, are calibrated from multi-model intercomparisons like the Energy, Greenhouse Gas, and Global Economic Model Intercomparison (ENGAGE) exercise by Riahi et al. (2021), which aggregates engineering cost data and deployment scenarios. Parameter evolution across DICE versions incorporates iterative updates to match improving datasets; for instance, damage estimates rose 60% from 1.3% to 2.1% GDP loss at 3°C between 1992 and 2017, driven by refined IMF GDP series, expanded historical emissions records, and IPCC-derived projections showing higher warming paths (from 3.2°C to 4.3°C under business-as-usual). prioritizes consistency with observed trends rather than formal statistical estimation, with sensitivity analyses addressing uncertainties in growth (standard deviation 1% per year for consumption) and response.

Comparisons with Observed Climate and Economic Data

The DICE model's climate module, a simplified energy balance representation, is calibrated to reproduce observed historical trends in atmospheric CO2 concentrations and global mean surface temperatures. Parameters such as equilibrium climate sensitivity (3.0°C in DICE-2023) and transient climate response (1.8°C) are drawn from IPCC AR6 assessments, which synthesize paleoclimate proxies, instrumental records, and climate model ensembles spanning 1850–2020. Initial conditions for 2020 include CO2 at 416.2 ppm and a 1.25°C warming anomaly from pre-industrial levels (circa 1765), closely aligning with direct measurements from Mauna Loa observatory (NOAA) and global temperature datasets like HadCRUT5, which report approximately 1.1–1.2°C warming over the same baseline. The updated FAIR carbon cycle submodule in DICE-2023 simulates airborne fractions and ocean uptake to match historical emissions-to-concentration pathways, with simulations confirming close tracking of observed levels from anthropogenic sources estimated at 43.2 GtCO2 in 2020. For economic variables, parameters are estimated via regression on historical data, including global GDP in PPP terms ($118.3 trillion for 2019 from World Bank/IMF aggregates), population growth, and carbon emissions intensities derived from energy statistics (e.g., CDIAC and EDGAR databases covering 1950–2020). growth and capital shares are fitted to Solow-model residuals from post-1950 observations, enabling the baseline scenario to replicate observed decoupling of emissions from GDP via historical efficiency gains. Emissions control rates are initialized at 5% for 2020, consistent with actual abatement efforts implied by carbon prices around $6/tCO2. However, these calibrations reflect in-sample fitting rather than predictive validation. Hindcasting exercises reveal limitations in the model's out-of-sample performance, particularly for economic dynamics. In a test using DICE's neoclassical Ramsey growth module on data (1870–2010 from Piketty-Zucman estimates), training on 1870–1920 data to forecast (TFP) and GDP from 1920 onward yielded systematic underpredictions, with 30-year horizons missing realized growth rates and 50-year projections failing pre-WWII but partially capturing postwar accelerations within widened confidence intervals. This underscores the challenge of extrapolating stylized exogenous TFP trends to periods of unanticipated technological shifts, as the model assumes steady-state convergence absent historical shocks like wars or interventions. Climate-economy linkages, including , lack direct historical benchmarking due to sparse pre-2100 damage data; quadratic damage functions are extrapolated from cross-sectional estimates (e.g., country-level GDP impacts from events), but no rigorous hindcast exists for cumulative effects, with critics noting potential underestimation if nonlinear risks materialized earlier than projected.

Sensitivity to Parametric Uncertainty

The DICE model's projections, including optimal carbon prices and the (SCC), demonstrate substantial sensitivity to uncertainties in key parameters such as (TFP) growth, (ECS), and components of the discount rate. Global sensitivity analyses, including those employing Sobol's method on DICE-2007, identify TFP growth as the dominant driver of variance in outputs, with perturbations in this parameter explaining a large share of in SCC estimates due to its compounding effects on future economic damages relative to abatement costs. Discounting parameters, particularly the pure rate of (ρ) and the elasticity of of consumption (α, often denoted as η in related formulations), further amplify this sensitivity; variations in ρ from 0.1% to 3% can shift SCC values by factors of 5–10 or more, as lower ρ values elevate the of distant damages while higher α increases the concavity of and thus the weight on inequality. Nordhaus's own one-at-a-time sensitivity experiments in earlier iterations, varying up to nine parameters including these, illustrate how such changes alter optimal policy trajectories, with carbon prices ranging from near-zero to over $100 per ton CO₂ depending on the combination. Climate response parameters like ECS and transient climate response (TCR) introduce additional variability, with DICE-2023 calibrations adjusting to central estimates from IPCC AR6 (ECS around 3°C) but acknowledging that the full uncertainty range (1.5–4.5°C historically, narrowed in recent assessments) propagates to wide bands in SCC, compounded by interactions with growth and damage functions; Nordhaus estimates this leads to SCC uncertainty spanning an or more when integrating TFP, ECS, and quadratic damage coefficients. Damage function parameters, including the quadratic exponent and abatement cost elasticities, also prove influential in Monte Carlo simulations; for example, raising the damage exponent from DICE's default (around -0.0027 per °C² in 2016R) to reflect higher-order losses can double SCC under baseline growth assumptions, highlighting how parametric choices in empirically calibrated but uncertain functions drive divergent policy implications. These sensitivities underscore the model's reliance on probabilistic distributions for inputs like population and TFP, where expected-value calibrations (as in standard DICE runs) may understate risks if distributions feature fat tails, though Nordhaus emphasizes using best-guess means for central projections to avoid over-weighting extremes without empirical warrant.

Criticisms and Controversies

Underestimation of Climate Risks and Tipping Points

Critics contend that the model's quadratic damage function, which posits damages as a continuous and reversible function of temperature rise, fails to capture the abrupt and potentially irreversible shifts posed by tipping points, thereby underestimating long-term economic risks. This formulation assumes smooth marginal increases in harm without discontinuities, excluding phenomena such as the rapid disintegration of polar ice sheets or the release of from thawing , which could amplify warming through feedback loops. Peer-reviewed analyses incorporating tipping elements into frameworks akin to illustrate how such omissions skew policy recommendations toward less stringent emissions controls. For instance, Cai et al. (2015) integrated multiple tipping points— including Amazon dieback and North Atlantic circulation slowdown—into a cost-benefit model, finding that these elements raise the optimal by factors of 2 to 10 and necessitate near-immediate deep cuts in emissions to avert high-risk scenarios. Similarly, Dietz et al. (2021) elicited expert probabilities for nine tipping points, estimating a 25% chance of substantial economic losses from events like collapse or boreal shift by 2100 under moderate warming, with tail risks implying damages far exceeding DICE's baseline projections of 2-3% global GDP loss per degree Celsius. Subsequent DICE revisions, such as the 2023 version, incorporate limited adjustments for tipping risks, adding a 1% output reduction at 3°C warming derived from Dietz et al.'s findings, alongside a judgmental 0.5-1% for unmodeled impacts. However, Lemoine (2010) argues that ambiguity surrounding tipping thresholds—where low-probability, high-impact events defy standard expected-value calculations—further biases DICE toward understating uncertainties, as the model relies on deterministic representations rather than robust decision rules under deep uncertainty. These critiques, often from interdisciplinary teams blending climate science and economics, highlight that DICE's exclusion of cascading tipping interactions—such as interconnected ice sheet and ocean circulation failures—may propagate underestimation into social cost of carbon estimates, which remain below $50 per ton CO2 in recent iterations despite evidence of higher tail risks.

Debates on Discount Rates and Intergenerational Equity

The discount rate in the model, derived from the as the sum of the pure rate of (ρ) and the product of the elasticity of of consumption (η) and expected per capita consumption growth (g), determines the relative weighting of future versus present utilities in optimal climate policy calculations. In early versions of , Nordhaus calibrated ρ at 1.5%, with η around 1.5 and g approximately 2%, yielding a near-term consumption discount rate of about 4%. Later iterations, such as and , retained a positive ρ (starting at 1.5% and declining gradually to 0.1% over centuries to reflect declining uncertainty), emphasizing empirical alignment with observed market rates and savings behavior rather than purely ethical prescriptions. A central debate arose from the 2006 , which employed a near-zero ρ (0.1%) and lower overall discount rate of 1.4%, arguing that ethical considerations demand minimal impatience toward unaffected by present choices, as future welfare at positive rates implies a willingness to forgo consumption for the living at the expense of the yet unborn. Nordhaus countered that such a low ρ is ethically arbitrary and empirically unsupported, as it derives from a "dictatorial" ethical stance ignoring observed , where positive reflects risks like or productivity differences; applying Stern's parameters to yields socially suboptimal policies with excessive near-term abatement costs exceeding 10% of GDP by 2030, while Nordhaus's calibration balances costs and benefits more efficiently. This disagreement halved versus doubled estimates of the (SCC) in comparable models, with Stern's approach implying immediate global emissions cuts of 25% or more. Intergenerational equity critiques of DICE focus on how positive discount rates systematically devalue distant future damages, potentially justifying insufficient mitigation and perpetuating inequity for later cohorts facing amplified climate risks; for instance, extensions to DICE show that standard parameters allow cumulative damages to impose up to 10-20% GDP losses on post-2100 generations while current ones bear minimal abatement burdens. Proponents of lower or declining rates, including some ethical frameworks, advocate ρ approaching zero to equalize per-generation welfare impacts, as in proposals limiting climate costs to 3% of GDP per cohort through adjusted mitigation paths. Nordhaus responds that equity requires not zero discounting but efficient resource allocation, as undiscounted futures overprioritize uncertain long-term harms over verifiable present needs, and empirical evidence from asset markets supports ρ > 0 to avoid overinvestment in abatement at the cost of poverty reduction today. Sensitivity analyses confirm that varying ρ from 0% to 3% swings optimal carbon prices from $100+ per ton to under $10, underscoring the parameter's leverage but also the need for robust justification beyond ethics alone.

Overreliance on Quadratic Damage Specifications

The DICE model's damage function specifies economic losses from climate change as a quadratic function of global mean temperature increase relative to pre-industrial levels, typically expressed as a fraction of global output reduced by terms proportional to temperature anomaly TT and T2T^2. This form, calibrated primarily on empirical estimates from observed variations in temperature and output at low warming levels (up to approximately 2.5°C), implies damages of around 2-3% of global GDP at 3°C warming, with losses scaling smoothly and symmetrically without abrupt thresholds. Critics argue this specification relies on extrapolation from limited historical data, underestimating damages at higher temperatures where empirical evidence from sectoral studies—such as agriculture, sea-level rise, and extreme weather—suggests accelerating, non-linear impacts. A core limitation is the quadratic form's inability to incorporate tipping points or discontinuous risks, such as thaw, collapse, or Amazon dieback, which could amplify beyond proportional scaling; even the model's developers acknowledge that this functional form does not capture such thresholds at 1.5-2°C warming. Pindyck has highlighted the arbitrary nature of such damage functions in integrated assessment models like , noting their weak empirical foundation outside narrow temperature ranges and failure to reflect potential catastrophic outcomes, as they treat as reversible fractions of output rather than existential threats to or growth. Meta-analyses of damage estimates, drawing from hundreds of peer-reviewed studies on sector-specific impacts, consistently yield higher loss projections than DICE's quadratic calibration; for instance, one such estimates non-catastrophic at 3.2-9.2% of GDP for 3°C warming when including growth effects, implying a 3- to 4-fold underestimation of the social cost of carbon in DICE-2013R. This overreliance persists despite alternatives proposed in the , such as cubic or probabilistic functions that better fit emerging evidence of convex damages, yet updates through 2023 have retained the quadratic structure for tractability, potentially biasing optimal policy toward gradual over aggressive action. Empirical critiques further point to the function's assumption of proportionate scaling with income growth, which ignores differential vulnerabilities in developing economies or long-term limits, leading to systematically low damage projections under scenarios of sustained high emissions. While the facilitates analytical solutions and aligns with some aggregate historical correlations, its dominance in has drawn charges of model conservatism, as it precludes even under extreme warming (e.g., >6°C), contrasting with paleoclimate records and biophysical models indicating potential irreversible declines.

Reception and Influence

Academic Debates and Extensions

Scholars have extended the DICE model by incorporating stochastic elements to address parametric uncertainty and climate variability, as in the Dynamic Stochastic Integrated Climate and Economy (DSICE) model developed by Cai et al. in 2013, which adds productivity shocks and tipping point risks to the deterministic framework of DICE-2012, yielding higher social cost of carbon (SCC) estimates under uncertainty. Another extension is the Resilience Integrated Model of Climate and Economy (RIMCE), proposed in 2023, which builds on DICE by integrating resilience metrics and social-economic perspectives to better capture adaptive capacities beyond standard growth models. Academic debates center on the dynamic realism of DICE's representation of economic-climate feedbacks, with critics arguing that its stylized Ramsey growth core and simplified modules overlook nonlinearities and path dependencies observed in empirical , potentially understating long-term risks. Validation efforts, such as hindcasting experiments comparing DICE projections to historical and GDP from 1970–2010, reveal mixed performance, with the model accurately hindcasting temperature trends but overestimating economic damages relative to observed outcomes, prompting calls for refined techniques. Extensions have also regionalized DICE into the Regional Integrated Climate and Economy (RICE) model since 1993, allowing heterogeneous damage functions and policy responses across regions, which Nordhaus used to analyze burden-sharing in climate agreements. Debates persist on the physics embedded in DICE, where simplifications of carbon cycles and radiative forcing—drawn from multi-model ensemble means—have been critiqued for insufficient resolution of ocean heat uptake and aerosol interactions, as evidenced by comparisons with comprehensive Earth system models showing divergences in transient climate response. Further scholarly work has integrated DICE-like structures into broader frameworks, such as FeliX 2.0 (2024), which extends damage modules to include and non-market sectors while retaining DICE's core optimization, to assess with carbon capture scenarios. Critics like Pindyck (2017) contend that such , including , lack robust empirical foundations for damage elasticities, rendering SCC outputs sensitive to ad hoc parameters rather than validated , though proponents counter with iterative updates like DICE-2023, which incorporate post-2020 data to refine projections. These debates underscore ongoing refinements, with model ensembles increasingly used to quantify ranges in policy analyses.

Adoption in Public Policy and Government Assessments

The DICE model has been prominently adopted in federal government assessments for estimating the (SCC), a metric used to quantify the economic damages from emitting one additional ton of CO₂. In 2010, an interagency working group comprising the , , and Department of Energy, among others, incorporated DICE alongside the FUND and PAGE models to generate SCC values for regulatory impact analyses, yielding estimates ranging from $21 per ton in 2010 (in 2007 dollars) under a 3% discount rate to higher values under alternative assumptions. This approach informed rules such as vehicle standards and power plant emissions regulations. Subsequent updates by the U.S. Environmental Protection Agency (EPA) have continued to rely on for SCC calculations, including in its 2016 technical update, which employed the model's three-box representation to project damages through 2100 under varying emissions scenarios. The model's integration into these assessments stems from its neoclassical framework linking , emissions, dynamics, and quadratic damage functions, providing a baseline for benefit-cost evaluations of policies. By 2023, remained one of the primary integrated assessment models (IAMs) referenced in EPA reports on costs, though alongside others to capture ranges. Beyond the U.S., DICE's influence appears more indirect in international assessments, with its parameters and methodology informing discussions on optimal carbon pricing in forums like the IPCC's III, where collectively underpin analyses for pathways. However, explicit adoption in non-U.S. government evaluations is less documented, as agencies such as the European Commission's have favored ensemble approaches over single-model reliance. Critics note that DICE's use in has sparked over its conservative estimates, yet its tractability has sustained its role in official valuations despite alternatives.

Contrasts with Other Integrated Assessment Models

The DICE model differs from other prominent integrated assessment models (), such as PAGE and FUND, primarily in its neoclassical economic framework, which integrates a global aggregate with simplified dynamics, contrasting with the probabilistic and regionally disaggregated approaches of PAGE and FUND. While DICE employs a deterministic structure in its baseline formulation—optimizing intertemporal utility via Ramsey-style growth theory—PAGE and FUND incorporate simulations to propagate uncertainties in parameters like and economic damages, enabling probabilistic distributions of outcomes such as the (SCC). These structural choices yield SCC estimates ranging from approximately $5–18 per ton CO2 in FUND, $18–30 in DICE, to $71 in PAGE, with variations attributable to differences in damage specifications and equity weighting rather than solely projections. In terms of climate representation, DICE features a more mechanistic and temperature module, including heat exchange between atmosphere, ocean surface, and deep ocean layers, which allows for transient climate response dynamics not fully captured in the simpler single-equation used by FUND and PAGE. FUND, by contrast, disaggregates impacts into 16 regions and 14 economic sectors (e.g., , ), permitting net benefits in cooler regions or certain damages like reduced cold-related mortality, whereas applies a uniform quadratic function to global output, scaling losses as a fraction of GDP (e.g., 0.2% at 2.7°C warming in early versions). PAGE emphasizes regional heterogeneity and equity weights, often producing higher SCC values due to greater assumed in developing regions. These disparities in modeling explain 60–95% of SCC inconsistencies across the models when coupled with unified emulators. FUND's focus on distributional effects and —evident in its calibration to observed sector and inclusion of —positions it as more responsive to heterogeneity, unlike DICE's centralized global optimum that assumes perfect foresight and markets. PAGE, developed for analysis, integrates expert elicitations for parameters, diverging from DICE's reliance on econometric estimates from historical . Later DICE iterations (e.g., DICE-2016R) address parametric via perturbation analysis, but retain less emphasis on structural compared to PAGE's full probabilistic framework. Overall, DICE's parsimony facilitates analytical tractability and long-run projections, but critics argue it underrepresents tail risks and regional inequities relative to its counterparts.

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