IDEAS home Printed from
   My bibliography  Save this paper

Substitutability and the social cost of carbon in a solvable growth model with irreversible climate change


  • Quaas, Martin F.
  • Bröcker, Johannes


We develop an overlapping generations endogenous growth model with stocks of produced capital, human capital, a non-renewable resource, and irreversibly accumulated greenhouse gases in deterministic and stochastic versions. The model allows for analyzing different elasticities of substitution. We present a full analytical solution and characterization of the transition dynamics. We show that, as a rule of thumb, the social cost of carbon grow at a rate equal to the economy's growth rate divided by the elasticity of substitution. We analytically study sensitivity of the social cost of carbon with respect to key parameters: the intergenerational discount rate, the elasticity of substitution, and climate uncertainty. We show that the social cost of carbon explode at a finite level of log-normally distributed climate uncertainty. We illustrate results in a calibrated version of the model.

Suggested Citation

  • Quaas, Martin F. & Bröcker, Johannes, 2016. "Substitutability and the social cost of carbon in a solvable growth model with irreversible climate change," Economics Working Papers 2016-09, Christian-Albrechts-University of Kiel, Department of Economics.
  • Handle: RePEc:zbw:cauewp:201609

    Download full text from publisher

    File URL:
    Download Restriction: no

    References listed on IDEAS

    1. Christian Traeger, 2015. "Closed-Form Integrated Assessment and Uncertainty," CESifo Working Paper Series 5464, CESifo Group Munich.
    2. Daron Acemoglu & Philippe Aghion & Leonardo Bursztyn & David Hemous, 2012. "The Environment and Directed Technical Change," American Economic Review, American Economic Association, vol. 102(1), pages 131-166, February.
    3. Sjak Smulders & Michael Toman & Cees Withagen, 2014. "Growth theory and ‘green growth’," Oxford Review of Economic Policy, Oxford University Press, vol. 30(3), pages 423-446.
    4. Joseph E. Stiglitz, 1974. "Growth with Exhaustible Natural Resources: The Competitive Economy," Review of Economic Studies, Oxford University Press, vol. 41(5), pages 139-152.
    5. Robert S. Pindyck, 2013. "Climate Change Policy: What Do the Models Tell Us?," Journal of Economic Literature, American Economic Association, vol. 51(3), pages 860-872, September.
    6. Yongyang Cai & Kenneth L. Judd & Thomas S. Lontzek, 2015. "The Social Cost of Carbon with Economic and Climate Risks," Papers 1504.06909,, revised Apr 2015.
    7. Llavador, Humberto & Roemer, John E. & Silvestre, Joaquim, 2011. "“A dynamic analysis of human welfare in a warming planet”," Journal of Public Economics, Elsevier, vol. 95(11), pages 1607-1620.
    8. Martin L. Weitzman, 2009. "On Modeling and Interpreting the Economics of Catastrophic Climate Change," The Review of Economics and Statistics, MIT Press, vol. 91(1), pages 1-19, February.
    9. Lucas Bretschger & Christos Karydas, 2014. "Optimum Growth and Carbon Policies with Lags in the Climate System," OxCarre Working Papers 144, Oxford Centre for the Analysis of Resource Rich Economies, University of Oxford.
    10. R. M. Solow, 1974. "Intergenerational Equity and Exhaustible Resources," Review of Economic Studies, Oxford University Press, vol. 41(5), pages 29-45.
    11. Joseph Stiglitz, 1974. "Growth with Exhaustible Natural Resources: Efficient and Optimal Growth Paths," Review of Economic Studies, Oxford University Press, vol. 41(5), pages 123-137.
    12. Kenneth Stollery, 1998. "Constant Utility Paths and Irreversible Global Warming," Canadian Journal of Economics, Canadian Economics Association, vol. 31(3), pages 730-742, August.
    13. Mikhail Golosov & John Hassler & Per Krusell & Aleh Tsyvinski, 2014. "Optimal Taxes on Fossil Fuel in General Equilibrium," Econometrica, Econometric Society, vol. 82(1), pages 41-88, January.
    14. Robert S. Pindyck & Neng Wang, 2013. "The Economic and Policy Consequences of Catastrophes," American Economic Journal: Economic Policy, American Economic Association, vol. 5(4), pages 306-339, November.
    15. Martin L. Weitzman, 2007. "A Review of the Stern Review on the Economics of Climate Change," Journal of Economic Literature, American Economic Association, vol. 45(3), pages 703-724, September.
    16. Thomas Sterner & U. Martin Persson, 2008. "An Even Sterner Review: Introducing Relative Prices into the Discounting Debate," Review of Environmental Economics and Policy, Association of Environmental and Resource Economists, vol. 2(1), pages 61-76, Winter.
    17. Christian Gollier, 2012. "Pricing the Planet's Future: The Economics of Discounting in an Uncertain World," Economics Books, Princeton University Press, edition 1, number 9894, June.
    18. Nordhaus, William D, 1993. "Optimal Greenhouse-Gas Reductions and Tax Policy in the "Dice" Model," American Economic Review, American Economic Association, vol. 83(2), pages 313-317, May.
    19. Moritz Drupp & Mark Freeman & Ben Groom & Frikk Nesje, 2015. "Discounting disentangled," GRI Working Papers 172, Grantham Research Institute on Climate Change and the Environment.
    20. Brock, William A. & Mirman, Leonard J., 1972. "Optimal economic growth and uncertainty: The discounted case," Journal of Economic Theory, Elsevier, vol. 4(3), pages 479-513, June.
    21. Geir Asheim, 2013. "A Distributional Argument for Supply-Side Climate Policies," Environmental & Resource Economics, Springer;European Association of Environmental and Resource Economists, vol. 56(2), pages 239-254, October.
    22. Per Krusell & Lee E. Ohanian & JosÈ-Victor RÌos-Rull & Giovanni L. Violante, 2000. "Capital-Skill Complementarity and Inequality: A Macroeconomic Analysis," Econometrica, Econometric Society, vol. 68(5), pages 1029-1054, September.
    23. Gollier, Christian, 2015. "Discounting, inequality and economic convergence," Journal of Environmental Economics and Management, Elsevier, vol. 69(C), pages 53-61.
    24. Heckman, James J, 1976. "A Life-Cycle Model of Earnings, Learning, and Consumption," Journal of Political Economy, University of Chicago Press, vol. 84(4), pages 11-44, August.
    25. Partha Dasgupta & Geoffrey Heal, 1974. "The Optimal Depletion of Exhaustible Resources," Review of Economic Studies, Oxford University Press, vol. 41(5), pages 3-28.
    26. Schneider, Maik T. & Traeger, Christian P. & Winkler, Ralph, 2012. "Trading off generations: Equity, discounting, and climate change," European Economic Review, Elsevier, vol. 56(8), pages 1621-1644.
    27. Frederick Ploeg & Cees Withagen, 2014. "Growth, Renewables, And The Optimal Carbon Tax," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 55, pages 283-311, February.
    28. Tahvonen, Olli & Kuuluvainen, Jari, 1991. "Optimal growth with renewable resources and pollution," European Economic Review, Elsevier, vol. 35(2-3), pages 650-661, April.
    Full references (including those not matched with items on IDEAS)


    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.

    Cited by:

    1. Hiroaki Sakamoto & Masako Ikefuji & Jan R. Magnus, 2017. "Adaptation for mitigation," Discussion papers e-16-014, Graduate School of Economics , Kyoto University.
    2. Gerlagh, Reyer & Jaimes, Richard & Motavasseli, Ali, 2017. "Global demographic change and climate policies," Discussion Paper 2017-035, Tilburg University, Center for Economic Research.

    More about this item


    overlapping generations; substitutes vs. complements; stochastic resource dynamics; optimum growth; climate policy;

    JEL classification:

    • Q54 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - Environmental Economics - - - Climate; Natural Disasters and their Management; Global Warming
    • O44 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - Environment and Growth
    • Q32 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - Nonrenewable Resources and Conservation - - - Exhaustible Resources and Economic Development

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:zbw:cauewp:201609. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (ZBW - German National Library of Economics). General contact details of provider: .

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.