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

Author

Listed:
  • 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.

Suggested Citation

  • Takashi Kamihigashi & Santanu Roy, 2005. "Dynamic optimization with a nonsmooth, nonconvex technology: The case of a linear objective function," Discussion Paper Series 175, Research Institute for Economics & Business Administration, Kobe University.
    • Handle: RePEc:kob:dpaper:175
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    File URL: https://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/dp175.pdf
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    Citations

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    Cited by:

    1. Stefano Bosi & Thai Ha-Hui, 2023. "A multidimensional, nonconvex model of optimal growth," Documents de recherche 23-07, Centre d'Études des Politiques Économiques (EPEE), Université d'Evry Val d'Essonne.
    2. repec:hal:cesptp:hal-03261262 is not listed on IDEAS
    3. Ha-Huy, Thai & Tran, Nhat Thien, 2020. "A simple characterisation for sustained growth," Journal of Mathematical Economics, Elsevier, vol. 91(C), pages 141-147.
    4. Dam, My & Ha-Huy, Thai & Le Van, Cuong & Nguyen, Thi Tuyet Mai, 2020. "Economic dynamics with renewable resources and pollution," Mathematical Social Sciences, Elsevier, vol. 108(C), pages 14-26.
    5. N. Hung & C. Le Van & P. Michel, 2009. "Non-convex aggregate technology and optimal economic growth," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 40(3), pages 457-471, September.
    6. Kamihigashi, Takashi & Roy, Santanu, 2007. "A nonsmooth, nonconvex model of optimal growth," Journal of Economic Theory, Elsevier, vol. 132(1), pages 435-460, January.
    7. repec:hal:pseptp:hal-03261262 is not listed on IDEAS
    8. repec:hal:journl:hal-03261262 is not listed on IDEAS
    9. Olivier Morand & Kevin Reffett & Suchismita Tarafdar, 2018. "Generalized Envelope Theorems: Applications to Dynamic Programming," Journal of Optimization Theory and Applications, Springer, vol. 176(3), pages 650-687, March.
    10. Somdeb Lahiri, 2025. "A Deterministic and Linear Model of Dynamic Optimization," Papers 2502.17012, arXiv.org, revised Jun 2025.
    11. Ken-Ichi Akao & Hitoshi Ishii & Takashi Kamihigashi & Kazuo Nishimura, 2019. "Existence of an optimal path in a continuous-time nonconcave Ramsey model," RIEEM Discussion Paper Series 1905, Research Institute for Environmental Economics and Management, Waseda University.
    12. Takashi Kamihigashi, 2014. "Elementary results on solutions to the bellman equation of dynamic programming: existence, uniqueness, and convergence," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 56(2), pages 251-273, June.
    13. Ha-Huy, Thai & Tran, Nhat-Thien, 2019. "A simple characterization for sustained growth," MPRA Paper 94576, University Library of Munich, Germany.
    14. Vassili Kolokoltsov & Wei Yang, 2012. "Turnpike Theorems for Markov Games," Dynamic Games and Applications, Springer, vol. 2(3), pages 294-312, September.
    15. Ali Khan, M. & Zhang, Zhixiang, 2023. "The random two-sector RSS model: On discounted optimal growth without Ramsey-Euler conditions," Journal of Economic Dynamics and Control, Elsevier, vol. 146(C).
    16. Liuchun Deng & Minako Fujio & M. Ali Khan, 2023. "On optimal extinction in the matchbox two-sector model," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 76(2), pages 445-494, August.
    17. 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 Sep 2015.
    18. La Grandville, O. de, 2014. "Optimal growth theory: Challenging problems and suggested answers," Economic Modelling, Elsevier, vol. 36(C), pages 608-611.
    19. 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.
    20. 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.
    21. Michetti, Elisabetta, 2015. "Complex attractors and basins in a growth model with nonconcave production function and logistic population growth rate," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 108(C), pages 215-232.
    22. Akao, Ken-Ichi & Kamihigashi, Takashi & Nishimura, Kazuo, 2025. "Critical capital stock in a continuous-time growth model with a convex-concave production function," Journal of Mathematical Economics, Elsevier, vol. 119(C).
    23. Bosi, Stefano & Ha-Huy, Thai, 2023. "A multidimensional, nonconvex model of optimal growth," Journal of Mathematical Economics, Elsevier, vol. 109(C).
    24. Liuchun Deng & Minako Fujio & M. Ali Khan, 2022. "On Sustainability and Survivability in the Matchbox Two-Sector Model: A Complete Characterization of Optimal Extinction," Papers 2202.02209, arXiv.org.

    More about this item

    Keywords

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    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General
    • O41 - Economic Development, Innovation, Technological Change, and Growth - - Economic Growth and Aggregate Productivity - - - One, Two, and Multisector Growth Models
    • Q20 - Agricultural and Natural Resource Economics; Environmental and Ecological Economics - - Renewable Resources and Conservation - - - General

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