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A nonsmooth, nonconvex model of optimal growth

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  • Kamihigashi, Takashi
  • Roy, Santanu

Abstract

This paper analyzes the nature of economic dynamics in a one-sector optimal growth model in which the technology is generally nonconvex, nondifferentiable, and discontinuous. The model also allows for irreversible investment and unbounded growth. We provide sufficient conditions for boundedness, extinction (convergence to zero), survival (boundedness away from zero), and unbounded growth. These conditions reveal that boundedness and survival are symmetrical phenomena, so are extinction and unbounded growth. Since many of the conditions are only local, it is possible that extinction occurs from small capital stocks, while unbounded growth occurs from large capital stocks. Despite such nonclassical results and nonclassical features such as nonconvexity and discontinuity, the model behaves much like a classical one as the discount factor approaches unity. In particular, we show that in most cases, if the discount factor is close to one, any optimal path from a given initial capital stock converges to a small neighborhood of what we define as the golden rule capital stock. If this stock is not finite, i.e., if sustainable consumption is maximized atinfinity, then as the discount factor approaches one, unbounded growth at least almost occurs.

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Bibliographic Info

Article provided by Elsevier in its journal Journal of Economic Theory.

Volume (Year): 132 (2007)
Issue (Month): 1 (January)
Pages: 435-460

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Handle: RePEc:eee:jetheo:v:132:y:2007:i:1:p:435-460

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Web page: http://www.elsevier.com/locate/inca/622869

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References

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  1. Kamihigashi, Takashi, 2003. "Necessity of transversality conditions for stochastic problems," Journal of Economic Theory, Elsevier, vol. 109(1), pages 140-149, March.
  2. Dutta, Prajit K & Mitra, Tapan, 1989. "On Continuity of the Utility Function in Intertemporal Allocation Models: An Example," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 30(3), pages 527-36, August.
  3. Dechert, W. Davis & Nishimura, Kazuo, 1983. "A complete characterization of optimal growth paths in an aggregated model with a non-concave production function," Journal of Economic Theory, Elsevier, vol. 31(2), pages 332-354, December.
  4. Kamihigashi, Takashi & Roy, Santanu, 2007. "A nonsmooth, nonconvex model of optimal growth," Journal of Economic Theory, Elsevier, vol. 132(1), pages 435-460, January.
  5. Larry E. Jones & Rodolfo E. Manuelli, 1994. "The Sources of Growth," GE, Growth, Math methods 9410002, EconWPA, revised 05 Mar 1999.
  6. Yano, Makoto, 1984. "Competitive Equilibria on Turnpikes in a McKenzie Economy, I: A Neighborhood Turnpike Theorem," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 25(3), pages 695-717, October.
  7. Olson, Lars J. & Roy, Santanu, 1996. "On Conservation of Renewable Resources with Stock-Dependent Return and Nonconcave Production," Journal of Economic Theory, Elsevier, vol. 70(1), pages 133-157, July.
  8. J. Dolmas, 2010. "Endogenous Growth with Multisector Ramsey Models," Levine's Working Paper Archive 1383, David K. Levine.
  9. Joshi, Sumit, 1997. "Turnpike Theorems in Nonconvex Nonstationary Environments," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 38(1), pages 225-48, February.
  10. Guerrero-Luchtenberg, C.L., 2000. "A uniform neighborhood turnpike theorem and applications," Journal of Mathematical Economics, Elsevier, vol. 34(3), pages 329-357, November.
  11. Montrucchio, Luigi, 1995. "A New Turnpike Theorem for Discounted Programs," Economic Theory, Springer, vol. 5(3), pages 371-82, May.
  12. Montrucchio, Luigi, 1994. "The neighbourhood turnpike property for continuous-time optimal growth models," Ricerche Economiche, Elsevier, vol. 48(3), pages 213-224, September.
  13. Takashi Kamihigashi, 2000. "Indivisible labor implies chaos," Economic Theory, Springer, vol. 15(3), pages 585-598.
  14. Majumdar, Mukul & Mitra, Tapan, 1982. "Intertemporal allocation with a non-convex technology: The aggregative framework," Journal of Economic Theory, Elsevier, vol. 27(1), pages 101-136, June.
  15. Dolmas, Jim, 1996. "Endogenous Growth in Multisector Ramsey Models," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 37(2), pages 403-21, May.
  16. Kaganovich, Michael, 1998. "Sustained endogenous growth with decreasing returns and heterogeneous capital," Journal of Economic Dynamics and Control, Elsevier, vol. 22(10), pages 1575-1603, August.
  17. Amir, Rabah & Mirman, Leonard J & Perkins, William R, 1991. "One-Sector Nonclassical Optimal Growth: Optimality Conditions and Comparative Dynamics," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 32(3), pages 625-44, August.
  18. Majumdar, Mukul & Mitra, Tapan, 1983. "Dynamic Optimization with a Non-Convex Technology: The Case of a Linear Objective Function," Review of Economic Studies, Wiley Blackwell, vol. 50(1), pages 143-51, January.
  19. Jones, Larry E & Manuelli, Rodolfo E, 1990. "A Convex Model of Equilibrium Growth: Theory and Policy Implications," Journal of Political Economy, University of Chicago Press, vol. 98(5), pages 1008-38, October.
  20. Azariadis, Costas & Drazen, Allan, 1990. "Threshold Externalities in Economic Development," The Quarterly Journal of Economics, MIT Press, vol. 105(2), pages 501-26, May.
  21. de Hek, Paul & Roy, Santanu, 2001. "On Sustained Growth under Uncertainty," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 42(3), pages 801-13, August.
  22. Takashi Kamihigashi & Santanu Roy, 2004. "Dynamic Optimization with a Nonsmooth, Nonconvex Technology: The Case of a Linear Objective Function," Discussion Paper Series 161, Research Institute for Economics & Business Administration, Kobe University.
  23. McKenzie, Lionel W., 2005. "Optimal economic growth, turnpike theorems and comparative dynamics," Handbook of Mathematical Economics, in: K. J. Arrow & M.D. Intriligator (ed.), Handbook of Mathematical Economics, edition 2, volume 3, chapter 26, pages 1281-1355 Elsevier.
  24. McKenzie, Lionel W., 1982. "A primal route to the Turnpike and Liapounov stability," Journal of Economic Theory, Elsevier, vol. 27(1), pages 194-209, June.
  25. Alexandre Scheinkman, Jose, 1976. "On optimal steady states of n-sector growth models when utility is discounted," Journal of Economic Theory, Elsevier, vol. 12(1), pages 11-30, February.
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Citations

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Cited by:
  1. Olivier Bruno & Cuong Van & Benoît Masquin, 2009. "When does a developing country use new technologies?," Economic Theory, Springer, vol. 40(2), pages 275-300, August.
  2. Ken-Ichi Akao & Takashi Kamihigashi & Kazuo Nishimura, 2011. "Monotonicity and Continuity of the Critical Capital Stock in the Dechert-Nishimura Model," Discussion Paper Series DP2011-20, Research Institute for Economics & Business Administration, Kobe University, revised Sep 2011.
  3. Nguyen Manh Hung & Cuong Le Van & Philippe Michel, 2009. "Non-convex Aggregate Technology and Optimal Economic Growth," Université Paris1 Panthéon-Sorbonne (Post-Print and Working Papers) halshs-00267100, HAL.
  4. Takashi Kamihigashi & Santanu Roy, 2003. "A Nonsmooth, Nonconvex Model of Optimal Growth," Discussion Paper Series 158, Research Institute for Economics & Business Administration, Kobe University.
  5. Takashi Kamihigashi, 2012. "Elementary Results on Solutions to the Bellman Equation of Dynamic Programming: Existence, Uniqueness, and Convergence," Discussion Paper Series DP2012-31, Research Institute for Economics & Business Administration, Kobe University.
  6. Takashi Kamihigashi, 2005. "Stochastic Optimal Growth with Bounded or Unbounded Utility and with Bounded or Unbounded Shocks," Discussion Paper Series 176, Research Institute for Economics & Business Administration, Kobe University.
  7. repec:hal:journl:halshs-00267100 is not listed on IDEAS
  8. repec:hal:journl:halshs-00197539 is not listed on IDEAS
  9. Serena Brianzoni & Cristiana Mammana & Elisabetta Michetti, 2012. "Local and Global Dynamics in a Discrete Time Growth Model with Nonconcave Production Function," Working Papers 70-2012, Macerata University, Department of Finance and Economic Sciences, revised Dec 2012.

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