# A 4-Stated DICE: Quantitatively Addressing Uncertainty Effects in Climate Change

- 977 Downloads
- 33 Citations

## Abstract

We introduce a version of the DICE-2007 model designed for uncertainty analysis. DICE is a wide-spread deterministic integrated assessment model of climate change. Climate change, long-term economic development, and their interactions are highly uncertain. The quantitative analysis of optimal mitigation policy under uncertainty requires a recursive dynamic programming implementation of integrated assessment models. Such implementations are subject to the curse of dimensionality. Every increase in the dimension of the state space is paid for by a combination of (exponentially) increasing processor time, lower quality of the value or policy function approximations, and reductions of the uncertainty domain. The paper promotes a state-reduced, recursive dynamic programming implementation of the DICE-2007 model. We achieve the reduction by simplifying the carbon cycle and the temperature delay equations. We compare our model’s performance and that of the DICE model to the scientific AOGCM models emulated by MAGICC 6.0 and find that our simplified model performs equally well as the original DICE model. Our implementation solves the infinite planning horizon problem in an arbitrary time step. The paper is the first to carefully analyze the quality of the value function approximation using two different types of basis functions and systematically varying the dimension of the basis. We present the closed form, continuous time approximation to the exogenous (discretely and inductively defined) processes in DICE, and we present a numerically more efficient re-normalized Bellman equation that, in addition, can disentangle risk attitude from the propensity to smooth consumption over time.

## Keywords

Climate change Uncertainty Integrated assessment DICE Dynamic programming Risk aversion Intertemporal substitution MAGICC Basis Recursive utility## JEL Classification

Q54 Q00 D90 C63## Notes

### Acknowledgments

I thank Larry Karp, Benjamin Crost, Svenn Jensen, Derek Lemoine, David Kelly, Tony Smith, Klaus Keller, Robert Nicholas, Inez Fung, Ujjayant Chakravorty, the referees, and the editor, Jared Carbone. Partial funding by the Giannini Foundation and the National Science Foundation under Grant No. GEO-1240507 on Sustainable Climate Risk Management (SCRiM) is gratefully acknowledged.

## Supplementary material

## References

- Alex L. Marten SCN (2013) Temporal resolution and DICE. Nat Clim Change 3(6):526Google Scholar
- Bansal R, Yaron A (2004) Risks for the long run: a potential resolution of asset pricing puzzles. J Financ 59(4):1481–1509CrossRefGoogle Scholar
- Bansal R, Kiku D, Yaron A (2010) Long run risks, the macroeconomy, and asset prices. Am Econ Rev Pap Proc 100:542–546CrossRefGoogle Scholar
- Cai Y, Judd KL, Lontzek TS (2012a) Continuous-time methods for integrated assessment models. NBER Working Papers 18365, National Bureau of Economic Research IncGoogle Scholar
- Cai Y, Judd KL, Lontzek TS (2012b) DSICE: a dynamic stochastic integrated model of climate and economy. Working Paper 12-02, RDCEPGoogle Scholar
- Cai Y, Judd KL, Lontzek TS (2012c) Open science is necessary. Nat Clim Change 2:299Google Scholar
- Campbell JY (1996) Understanding risk and return. J Political Econ 104(2):298–345CrossRefGoogle Scholar
- Crost B, Traeger CP (2014) Optimal CO2 mitigation under damage risk valuation. Nat Clim Change. doi: 10.1038/nclimate2249
- Crost B, Traeger CP (2013) Optimal climate policy: uncertainty versus Monte-Carlo. Econ Lett 120:552–558Google Scholar
- Epstein LG, Zin SE (1989) Substitution, risk aversion, and the temporal behavior of consumption and asset returns: a theoretical framework. Econometrica 57(4):937–969CrossRefGoogle Scholar
- Fischer C, Springborn M (2011) Emissions targets and the real business cycle: intensity targets versus caps or taxes. J Environ Econ Manag 62(3):352–366Google Scholar
- Heutel G (2012) How should environmental policy respond to business cycles? Optimal policy under persistent productivity shocks. Rev Econ Dyn 15:244–264Google Scholar
- Hoel M, Karp L (2001) Taxes and quotas for a stock pollutant with multiplicative uncertainty. J Public Econ 82:91–114CrossRefGoogle Scholar
- Hoel M, Karp L (2002) Taxes versus quotas for a stock pollutant. Res Energy Econ 24:367–384CrossRefGoogle Scholar
- Interagency Working Group on Social Cost of Carbon, U. S. G. (2010) Technical support document: social cost of carbon for regulatory impact analysis under executive order 12866, Department of EnergyGoogle Scholar
- Interagency Working Group on Social Cost of Carbon, U. S. G. (2013) Technical support document: technical update of the social cost of carbon for regulatory impact analysis under executive order 12866, Department of EnergyGoogle Scholar
- IPCC (2000) Emissions scenarios. Cambridge University Press, CambridgeGoogle Scholar
- IPCC (2007) Contribution of working group I to the fourth assessment report of the Intergovernmental Panel on Climate Change, 2007. Cambridge University Press, CambridgeGoogle Scholar
- IPCC (2013) Climate Change 2013: The Physical Science Basis, Intergovernmental Panel on Climate Change. Preliminary Draft, GenevaGoogle Scholar
- Jensen S, Traeger CP (2013) Optimally climate sensitive policy under uncertainty and learning, Working paper, University of California, BerkeleyGoogle Scholar
- Jensen S, Traeger CP (2014) Growth uncertainty in the integrated assessment of climate change. Eur Econ Rev 69:104–125Google Scholar
- Karp L, Zhang J (2006) Regulation with anticipated learning about environmental damages. J Environ Econ Manag 51:259–279Google Scholar
- Keller K, Bolker BM, Bradford DF (2004) Uncertain climate thresholds and optimal economic growth. J Environ Econ Manag 48:723–741Google Scholar
- Kelly DL (2005) Price and quantity regulation in general equilibrium. J Econ Theory 125(1):36–60CrossRefGoogle Scholar
- Kelly DL, Kolstad CD (1999) Bayesian learning, growth, and pollution. J Econ Dyn Control 23:491–518CrossRefGoogle Scholar
- Kelly DL, Kolstad CD (2001) Solving infinite horizon growth models with an environmental sector. Comput Econ 18:217–231CrossRefGoogle Scholar
- Kelly DL, Tan Z (2013) Learning and climate feedbacks: optimal climate insurance and fat tails, University of Miami Working paperGoogle Scholar
- Leach AJ (2007) The climate change learning curve. J Econ Dyn Control 31:1728–1752CrossRefGoogle Scholar
- Lemoine D, Traeger C (2014) Watch your step: optimal policy in a tipping climate. Am Econ J Econ Policy 6:1–31Google Scholar
- Meinshausen M, Raper S, Wigley T (2011) Emulating coupled atmosphere-ocean and carbon cycle models with a simpler model, magicc6—part 1: model description and calibration. Atmos Chem Phys 11:1417–1456Google Scholar
- Miranda M, Fackler PL (eds) (2002) Applied computational economics and finance. Massachusetts Institute of Technology, CambridgeGoogle Scholar
- Moss R, Babiker M, Brinkman S, Calvo E, Carter T, Edmonds J, Elgizouli I, Emori S, Erda L, Hibbard K, Jones R, Kainuma M, Kelleher J, Lamarque JF, Manning M, Matthews B, Meehl J, Meyer L, Mitchell J, Nakicenovic N, O’Neill B, Pichs R, Riahi K, Rose S, Runci P, Stouffer R, vanVuuren D, Weyant J, Wilbanks T, vanYpersele JP, Zurek M (2007) Towards new scenarios for analysis of emissions, climate change, impacts, and response strategies. Intergovernmental Panel on Climate Change, GenevaGoogle Scholar
- Nakamura E, Steinsson J, Barro R, Ursua J (2010) Crises and recoveries in an empirical model of consumption disasters. Am Econ J Macroecon 5(3):35–74Google Scholar
- Nordhaus W (2008) A question of balance: economic modeling of global warming. Yale University Press, New Haven (online preprint: a question of balance: weighing the options on global warming policies)Google Scholar
- Nordhaus W, Boyer J (2000) Warming the world. MIT Press, CambridgeGoogle Scholar
- Traeger CP (2009) Recent developments in the intertemporal modeling of uncertainty. ARRE 1:261–85Google Scholar
- Traeger CP (2010) Intertemporal risk aversion. CUDARE Working Paper No. 1102Google Scholar
- Traeger CP (2012) Why uncertainty matters—discounting under intertemporal risk aversion and ambiguity, CESifo Working Paper No. 3727Google Scholar
- Vissing-Jørgensen A, Attanasio OP (2003) Stock-market participation, intertemporal substitution, and risk-aversion. Am Econ Rev 93(2):383–391CrossRefGoogle Scholar
- von Neumann J, Morgenstern O (1944) Theory of games and economic behaviour. Princeton University Press, PrincetonGoogle Scholar
- Weil P (1990) Nonexpected utility in macroeconomics. Q J Econ 105(1):29–42CrossRefGoogle Scholar