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Sustained Positive Consumption in a Model of Stochastic Growth: The Role of Risk Aversion

  • Mitra, Tapan

    (Cornell University)

  • Roy, Santanu

    (Southern Methodist Univesrity)

An intriguing problem in stochastic growth theory is as follows: even when the return on investment is arbitrarily high near zero and discounting is arbitrarily mild, long run capital and consumption may be arbitrarily close to zero with probability one. In a convex one-sector model of optimal stochastic growth with i.i.d. shocks, we relate this phenomenon to risk aversion near zero. For a Cobb-Douglas production function with multiplicative uniformly distributed shock, the phenomenon occurs with high discounting if, and only if, risk aversion diverges to infinity sufficiently fast as consumption goes to zero. We specify utility functions for which the phenomenon occurs even when discounting is arbitrarily mild. For the general version of the model, we outline sufficient conditions under which capital and consumption are bounded away from zero almost surely, as well as conditions under which growth occurs almost surely near zero; the latter ensures a uniform positive lower bound on long run consumption (independent of initial capital). These conditions require the expected marginal productivity at zero to be above the discount rate by a factor that depends on the degree of risk aversion near zero.

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Paper provided by Cornell University, Center for Analytic Economics in its series Working Papers with number 10-03.

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Date of creation: Jul 2010
Date of revision:
Handle: RePEc:ecl:corcae:10-03
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  1. Olson, Lars J. & Roy, Santanu, 2000. "Dynamic Efficiency of Conservation of Renewable Resources under Uncertainty," Journal of Economic Theory, Elsevier, vol. 95(2), pages 186-214, December.
  2. Danyang Xie, 1999. "Power Risk Aversion Utility Functions," CEMA Working Papers 22, China Economics and Management Academy, Central University of Finance and Economics, revised Oct 2000.
  3. Mendelssohn, Roy & Sobel, Matthew J., 1980. "Capital accumulation and the optimization of renewable resource models," Journal of Economic Theory, Elsevier, vol. 23(2), pages 243-260, October.
  4. Takashi Kamihigashi, 2006. "Almost sure convergence to zero in stochastic growth models," Economic Theory, Springer, vol. 29(1), pages 231-237, September.
  5. Chatterjee, Partha & Shukayev, Malik, 2008. "Note on positive lower bound of capital in the stochastic growth model," Journal of Economic Dynamics and Control, Elsevier, vol. 32(7), pages 2137-2147, July.
  6. Mirman, Leonard J. & Zilcha, Itzhak, 1977. "Characterizing optimal policies in a one-sector model of economic growth under uncertainty," Journal of Economic Theory, Elsevier, vol. 14(2), pages 389-401, April.
  7. Mitra, Tapan & Roy, Santanu, 2007. "On the possibility of extinction in a class of Markov processes in economics," Journal of Mathematical Economics, Elsevier, vol. 43(7-8), pages 842-854, September.
  8. Mirman, Leonard J. & Zilcha, Itzhak, 1975. "On optimal growth under uncertainty," Journal of Economic Theory, Elsevier, vol. 11(3), pages 329-339, December.
  9. Boylan, Edward S., 1979. "On the avoidance of extinction in one-sector growth models," Journal of Economic Theory, Elsevier, vol. 20(2), pages 276-279, April.
  10. Mitra, Tapan & Roy, Santanu, 2003. "Optimal Exploitation of Renewable Resources under Uncertainty and the Extinction of Species," Working Papers 03-10, Cornell University, Center for Analytic Economics.
  11. 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.
  12. Mirman, Leonard J & Zilcha, Itzhak, 1976. "Unbounded Shadow Prices for Optimal Stochastic Growth Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 17(1), pages 121-32, February.
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