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On optimal extinction in the matchbox two-sector model

Author

Listed:
  • Liuchun Deng

    (Yale-NUS College)

  • Minako Fujio

    (Yokohama National University)

  • M. Ali Khan

    (The Johns Hopkins University)

Abstract

We provide a complete characterization of optimal extinction in a two-sector model of economic growth through three results, surprising in both their simplicity and intricacy. (i) When the discount factor is below a threshold identified by the well-known $$\delta $$ δ -normality condition for the existence of a stationary optimal stock, the economy’s capital becomes extinct in the long run. (ii) This extinction may be staggered if and only if the investment-good sector is capital-intensive. (iii) We uncover a sequence of thresholds of the discount factor, identified by a family of rational functions, that represent bifurcations for optimal postponements on the path to extinction. We also report various special cases of the model having to do with unsustainable technologies and equal capital-intensities that showcase long-term optimal growth, all of topical interest and all neglected in the antecedent literature.

Suggested Citation

  • 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.
    • Handle: RePEc:spr:joecth:v:76:y:2023:i:2:d:10.1007_s00199-022-01462-0
    DOI: 10.1007/s00199-022-01462-0
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    as
    1. , & ,, 2014. "Stochastic stability in monotone economies," Theoretical Economics, Econometric Society, vol. 9(2), May.
    2. Rose-Anne Dana & Cuong Le Van & Tapan Mitra & Kazuo Nishimura, 2006. "Handbook on optimal growth (volume 1)," Post-Print halshs-00101345, HAL.
    3. Kamihigashi, Takashi & Roy, Santanu, 2007. "A nonsmooth, nonconvex model of optimal growth," Journal of Economic Theory, Elsevier, vol. 132(1), pages 435-460, January.
    4. Paul A. Samuelson & Robert M. Solow, 1956. "A Complete Capital Model Involving Heterogeneous Capital Goods," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 70(4), pages 537-562.
    5. Galor, Oded, 1992. "A Two-Sector Overlapping-Generations Model: A Global Characterization of the Dynamical System," Econometrica, Econometric Society, vol. 60(6), pages 1351-1386, November.
    6. Grandmont, Jean-Michel, 1985. "On Endogenous Competitive Business Cycles," Econometrica, Econometric Society, vol. 53(5), pages 995-1045, September.
    7. Kazuo Nishimura & Makoto Yano, 2012. "Non-linear Dynamics and Chaos in Optimal Growth: An Example," Springer Books, in: John Stachurski & Alain Venditti & Makoto Yano (ed.), Nonlinear Dynamics in Equilibrium Models, edition 127, chapter 0, pages 127-150, Springer.
    8. Khan, M. Ali & Mitra, Tapan, 2005. "On topological chaos in the Robinson-Solow-Srinivasan model," Economics Letters, Elsevier, vol. 88(1), pages 127-133, July.
    9. Tapan Mitra & Gerhard Sorger, 2014. "Extinction in common property resource models: an analytically tractable example," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 57(1), pages 41-57, September.
    10. David Cass, 1965. "Optimum Growth in an Aggregative Model of Capital Accumulation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 32(3), pages 233-240.
    11. Tapan Mitra & Santanu Roy, 2022. "Propensity to consume and the optimality of Ramsey–Euler policies," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 73(1), pages 55-89, February.
    12. Rose-Anne Dana & Cuong Le Van & Tapan Mitra & Kazuo Nishimura (ed.), 2006. "Handbook on Optimal Growth 1," Springer Books, Springer, number 978-3-540-32310-5, December.
    13. Minako Fujio, 2005. "The Leontief Two-sector Model and Undiscounted Optimal Growth with Irreversible Investment: The Case of Labor-intensive Consumption Goods," Journal of Economics, Springer, vol. 86(2), pages 145-159, November.
    14. Takashi Kamihigashi, 2006. "Almost sure convergence to zero in stochastic growth models," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 29(1), pages 231-237, September.
    15. Deng Liuchun & Fujio Minako & Khan M. Ali, 2019. "Optimal growth in the Robinson-Shinkai-Leontief model: the case of capital-intensive consumption goods," Studies in Nonlinear Dynamics & Econometrics, De Gruyter, vol. 23(4), pages 1-18, September.
    16. Morishima, Michio, 1969. "Theory of Economic Growth," OUP Catalogue, Oxford University Press, number 9780198281641.
    17. Tjalling C. Koopmans, 1967. "Economic Growth at a Maximal Rate," International Economic Association Series, in: E. Malinvaud & M. O. L. Bacharach (ed.), Activity Analysis in the Theory of Growth and Planning, chapter 0, pages 3-42, Palgrave Macmillan.
    18. Mitra, Tapan & Roy, Santanu, 2012. "Sustained positive consumption in a model of stochastic growth: The role of risk aversion," Journal of Economic Theory, Elsevier, vol. 147(2), pages 850-880.
    19. Takashi Kamihigashi & Santanu Roy, 2006. "Dynamic optimization with a nonsmooth, nonconvex technology: the case of a linear objective function," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 29(2), pages 325-340, October.
    20. Stachurski, John, 2002. "Stochastic Optimal Growth with Unbounded Shock," Journal of Economic Theory, Elsevier, vol. 106(1), pages 40-65, September.
    21. J. v. Neumann, 1945. "A Model of General Economic Equilibrium," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 13(1), pages 1-9.
    22. Tomohiro HIRANO & Joseph E. Stiglitz, 2021. "The Wobbly Economy; Global Dynamics with Phase Transitions and State Transitions," CIGS Working Paper Series 21-008E, The Canon Institute for Global Studies.
    23. Ali Khan, M. & Mitra, Tapan, 1986. "On the existence of a stationary optimal stock for a multi-sector economy: A primal approach," Journal of Economic Theory, Elsevier, vol. 40(2), pages 319-328, December.
    24. Burmeister, Edwin, 1974. "Synthesizing the Neo-Austrian and Alternative Approaches to Capital Theory: A Survey," Journal of Economic Literature, American Economic Association, vol. 12(2), pages 413-456, June.
    25. Benhabib, Jess & Day, Richard H., 1982. "A characterization of erratic dynamics in, the overlapping generations model," Journal of Economic Dynamics and Control, Elsevier, vol. 4(1), pages 37-55, November.
    26. Spear, Stephen E. & Young, Warren, 2014. "Optimum Savings And Optimal Growth: The Cass–Malinvaud–Koopmans Nexus," Macroeconomic Dynamics, Cambridge University Press, vol. 18(1), pages 215-243, January.
    27. Liuchun Deng & Minako Fujio & M. Ali Khan, 2021. "Eventual periodicity in the two-sector RSL model: equilibrium vis-à-vis optimum growth," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 72(2), pages 615-639, September.
    28. Tomohiro Hirano & Joseph E. Stiglitz, 2022. "The Wobbly Economy: Global Dynamics with Phase and State Transitions," NBER Working Papers 29806, National Bureau of Economic Research, Inc.
    29. Cropper, M. L., 1988. "A note on the extinction of renewable resources," Journal of Environmental Economics and Management, Elsevier, vol. 15(1), pages 64-70, March.
    30. Kamihigashi, Takashi, 2007. "Stochastic optimal growth with bounded or unbounded utility and with bounded or unbounded shocks," Journal of Mathematical Economics, Elsevier, vol. 43(3-4), pages 477-500, April.
    31. Fujio, Minako & Lei, Yan & Deng, Liuchun & Khan, M. Ali, 2021. "The miniature two-sector model of optimal growth: The neglected case of a capital-intensive investment-good sector," Journal of Economic Behavior & Organization, Elsevier, vol. 186(C), pages 662-671.
    32. Tapan Mitra & Santanu Roy, 2006. "Optimal exploitation of renewable resources under uncertainty and the extinction of species," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 28(1), pages 1-23, May.
    33. Fujio, Minako, 2008. "Undiscounted optimal growth in a Leontief two-sector model with circulating capital: The case of a capital-intensive consumption good," Journal of Economic Behavior & Organization, Elsevier, vol. 66(2), pages 420-436, May.
    34. Spear, Stephen & Young, Warren, 2015. "Two-Sector Growth, Optimal Growth, And The Turnpike: Amalgamation And Metamorphosis," Macroeconomic Dynamics, Cambridge University Press, vol. 19(2), pages 394-424, March.
    35. Kazuo Nishimura & John Stachurski, 2012. "Stability of Stochastic Optimal Growth Models: A New Approach," Springer Books, in: John Stachurski & Alain Venditti & Makoto Yano (ed.), Nonlinear Dynamics in Equilibrium Models, edition 127, chapter 0, pages 289-307, Springer.
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    More about this item

    Keywords

    Extinction; Capital intensity; Two-sector; $$delta $$ δ -Normality; Bifurcation;
    All these keywords.

    JEL classification:

    • C60 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - General
    • D90 - Microeconomics - - Micro-Based Behavioral Economics - - - General
    • O21 - Economic Development, Innovation, Technological Change, and Growth - - Development Planning and Policy - - - Planning Models; Planning Policy

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