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Dynamic optimization with a nonsmooth, nonconvex technology: The case of a linear objective function

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  • Takashi Kamihigashi

    (Research Institute for Economics & Business Administration (RIEB), Kobe University, Japan)

  • Santanu Roy

    (Department of Economics, Southern Methodist University, USA)

Abstract

This paper studies a one-sector optimal growth model with linear utility in which the production function is only required to be increasing and upper semicontinuous. The model also allows for a general form of irreversible investment. We show that every optimal capital path is strictly monotone until it reaches a steady state; further, it either converges to zero, or reaches a positive steady state in finite time and possibly jumps among different steady states afterwards. We establish conditions for extinction (convergence to zero), survival (boundedness away from zero), and the existence of a critical capital stock below which extinction is possible and above which survival is ensured. These conditions generalize those known for the case of S-shaped production functions. We also show that as the discount factor approaches one, optimal paths converge to a small neighborhood of the capital stock that maximizes sustainable consumption.

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File URL: http://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/dp175.pdf
File Function: First version, 2005
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Bibliographic Info

Paper provided by Research Institute for Economics & Business Administration, Kobe University in its series Discussion Paper Series with number 175.

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Length: 21 pages
Date of creation: Aug 2005
Date of revision:
Handle: RePEc:kob:dpaper:175

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Keywords: Nonconvex; nonsmooth; and discontinuous technology; Extinction; Survival; Turnpike; Linear utility;

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References

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  1. Spence, A Michael & Starrett, David, 1975. "Most Rapid Approach Paths in Accumulation Problems," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 16(2), pages 388-403, June.
  2. Skiba, A K, 1978. "Optimal Growth with a Convex-Concave Production Function," Econometrica, Econometric Society, vol. 46(3), pages 527-39, May.
  3. Azariadis, Costas & Drazen, Allan, 1990. "Threshold Externalities in Economic Development," The Quarterly Journal of Economics, MIT Press, vol. 105(2), pages 501-26, May.
  4. Kamihigashi, Takashi, 1999. "Chaotic dynamics in quasi-static systems: theory and applications1," Journal of Mathematical Economics, Elsevier, vol. 31(2), pages 183-214, March.
  5. 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.
  6. 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.
  7. 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.
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Citations

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Cited by:
  1. Kamihigashi, Takashi & Roy, Santanu, 2007. "A nonsmooth, nonconvex model of optimal growth," Journal of Economic Theory, Elsevier, vol. 132(1), pages 435-460, January.
  2. 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.
  3. N. M. Hung & Cuong Le Van & P. Michel, 2008. "Non-convex Aggregate Technology and Optimal Economic Growth," Working Papers 05, Development and Policies Research Center (DEPOCEN), Vietnam.
  4. Takashi Kamihigashi & Taiji Furusawa, 2006. "Immediately Reactive Equilibria in Infinitely Repeated Games with Additively Separable Continuous Payoffs," Discussion Paper Series 199, Research Institute for Economics & Business Administration, Kobe University.
  5. Nguyen Manh Hung & Cuong Le Van & Philippe Michel, 2009. "Non-convex Aggregate Technology and Optimal Economic Growth," Post-Print halshs-00267100, HAL.
  6. 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.
  7. Vassili Kolokoltsov & Wei Yang, 2012. "Turnpike Theorems for Markov Games," Dynamic Games and Applications, Springer, vol. 2(3), pages 294-312, September.
  8. La Grandville, O. de, 2014. "Optimal growth theory: Challenging problems and suggested answers," Economic Modelling, Elsevier, vol. 36(C), pages 608-611.
  9. Takashi Kamihigashi & Taiji Furusawa, 2007. "Global Dynamics in Infinitely Repeated Games with Additively Separable Continuous Payoffs," Discussion Paper Series 210, Research Institute for Economics & Business Administration, Kobe University.

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