IDEAS home Printed from https://ideas.repec.org/p/kob/dpaper/dp2012-27.html
   My bibliography  Save this paper

Existence, Uniqueness and Stability of Stationary Distributions: An Extension of the Hopenhayn-Prescott Theorem

Author

Listed:
  • Takashi Kamihigashi

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

  • John Stachurski

    (Research School of Economics, Australian National University, Australia)

Abstract

This paper strengthens the Hopenhayn-Prescott stability theorem for monotone economies by extending it to a significantly larger class of models. We provid general conditions for existence, uniqueness and stability of stationary distributions. The conditions in our main result are both necessary and sufficient for global stability of monotone economies that satisfy a weak mixing condition introduced in the paper. Through our analysis we develop new insights on the nature and causes of stability and instability.

Suggested Citation

  • Takashi Kamihigashi & John Stachurski, 2012. "Existence, Uniqueness and Stability of Stationary Distributions: An Extension of the Hopenhayn-Prescott Theorem," Discussion Paper Series DP2012-27, Research Institute for Economics & Business Administration, Kobe University.
    • Handle: RePEc:kob:dpaper:dp2012-27
    as

    Download full text from publisher

    File URL: https://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/DP2012-27.pdf
    File Function: First version, 2012
    Download Restriction: no
    ---><---

    References listed on IDEAS

    as
    1. Gong, Liutang & Zhao, Xiaojun & Yang, Yunhong & Hengfu, Zou, 2010. "Stochastic growth with social-status concern: The existence of a unique stable distribution," Journal of Mathematical Economics, Elsevier, vol. 46(4), pages 505-518, July.
    2. Kamihigashi, Takashi & Stachurski, John, 2012. "An order-theoretic mixing condition for monotone Markov chains," Statistics & Probability Letters, Elsevier, vol. 82(2), pages 262-267.
    3. Marcet, Albert & Obiols-Homs, Francesc & Weil, Philippe, 2007. "Incomplete markets, labor supply and capital accumulation," Journal of Monetary Economics, Elsevier, vol. 54(8), pages 2621-2635, November.
    4. Samaniego, Roberto M., 2008. "Can technical change exacerbate the effects of labor market sclerosis," Journal of Economic Dynamics and Control, Elsevier, vol. 32(2), pages 497-528, February.
    5. Diaz, Antonia & Pijoan-Mas, Josep & Rios-Rull, Jose-Victor, 2003. "Precautionary savings and wealth distribution under habit formation preferences," Journal of Monetary Economics, Elsevier, vol. 50(6), pages 1257-1291, September.
    6. Antunes, Antonio R. & Cavalcanti, Tiago V. de V., 2007. "Start up costs, limited enforcement, and the hidden economy," European Economic Review, Elsevier, vol. 51(1), pages 203-224, January.
    7. Sokbae Lee & Oliver Linton & Yoon-Jae Whang, 2009. "Testing for Stochastic Monotonicity," Econometrica, Econometric Society, vol. 77(2), pages 585-602, March.
    8. Chatterjee, Partha & Shukayev, Malik, 2012. "A stochastic dynamic model of trade and growth: Convergence and diversification," Journal of Economic Dynamics and Control, Elsevier, vol. 36(3), pages 416-432.
    9. Hopenhayn, Hugo A & Prescott, Edward C, 1992. "Stochastic Monotonicity and Stationary Distributions for Dynamic Economies," Econometrica, Econometric Society, vol. 60(6), pages 1387-1406, November.
    10. repec:hal:spmain:info:hdl:2441/8623 is not listed on IDEAS
    11. Takashi Kamihigashi & John Stachurski, 2011. "Existence, Stability and Computation of Stationary Distributions: An Extension of the Hopenhayn-Prescott Theorem," Discussion Paper Series DP2011-32, Research Institute for Economics & Business Administration, Kobe University.
    12. Manjira Datta & Leonard Mirman & Olivier Morand & Kevin Reffett, 2002. "Monotone Methods for Markovian Equilibrium in Dynamic Economies," Annals of Operations Research, Springer, vol. 114(1), pages 117-144, August.
    13. Rabah Amir, 2002. "Complementarity and Diagonal Dominance in Discounted Stochastic Games," Annals of Operations Research, Springer, vol. 114(1), pages 39-56, August.
    14. Leonard Mirman & Olivier Morand & Kevin Reffett, "undated". "A Qualitative Theory of Markovian Equilibrium in Infinite Horizon Economies with Capital," Working Papers 2133376, Department of Economics, W. P. Carey School of Business, Arizona State University.
    15. Antunes, António & Cavalcanti, Tiago & Villamil, Anne, 2008. "Computing general equilibrium models with occupational choice and financial frictions," Journal of Mathematical Economics, Elsevier, vol. 44(7-8), pages 553-568, July.
    16. repec:hal:wpspec:info:hdl:2441/8623 is not listed on IDEAS
    17. Razin, Assaf & Yahav, Joseph A, 1979. "On Stochastic Models of Economic Growth," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 20(3), pages 599-604, October.
    18. Owen, Ann L. & Weil, David N., 1998. "Intergenerational earnings mobility, inequality and growth," Journal of Monetary Economics, Elsevier, vol. 41(1), pages 71-104, February.
    19. Cabrales, Antonio & Hopenhayn, Hugo A., 1997. "Labor-market flexibility and aggregate employment volatility," Carnegie-Rochester Conference Series on Public Policy, Elsevier, vol. 46(1), pages 189-228, June.
    20. Zhang, Yuzhe, 2007. "Stochastic optimal growth with a non-compact state space," Journal of Mathematical Economics, Elsevier, vol. 43(2), pages 115-129, February.
    21. Huw Lloyd-Ellis & Dan Bernhardt, 2000. "Enterprise, Inequality and Economic Development," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 67(1), pages 147-168.
    22. Mirman, Leonard J. & Morand, Olivier F. & Reffett, Kevin L., 2008. "A qualitative approach to Markovian equilibrium in infinite horizon economies with capital," Journal of Economic Theory, Elsevier, vol. 139(1), pages 75-98, March.
    23. Hidalgo Cabrillana, Ana, 2009. "Endogenous capital market imperfections, human capital, and intergenerational mobility," Journal of Development Economics, Elsevier, vol. 90(2), pages 285-298, November.
    24. Kiminori Matsuyama, 2000. "Endogenous Inequality," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 67(4), pages 743-759.
    25. Josep Pijoan-Mas, 2006. "Precautionary Savings or Working Longer Hours?," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 9(2), pages 326-352, April.
    26. Kiminori Matsuyama, 2004. "Financial Market Globalization, Symmetry-Breaking, and Endogenous Inequality of Nations," Econometrica, Econometric Society, vol. 72(3), pages 853-884, May.
    27. Buly A. Cardak, 2004. "Ability, Education, and Income Inequality," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 6(2), pages 239-276, May.
    28. Samaniego, Roberto M., 2006. "Industrial subsidies and technology adoption in general equilibrium," Journal of Economic Dynamics and Control, Elsevier, vol. 30(9-10), pages 1589-1614.
    29. de Hek, Paul A, 1999. "On Endogenous Growth under Uncertainty," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 40(3), pages 727-744, August.
    30. Balbus, Łukasz & Reffett, Kevin & Woźny, Łukasz, 2012. "Stationary Markovian equilibrium in altruistic stochastic OLG models with limited commitment," Journal of Mathematical Economics, Elsevier, vol. 48(2), pages 115-132.
    31. Hopenhayn, Hugo & Rogerson, Richard, 1993. "Job Turnover and Policy Evaluation: A General Equilibrium Analysis," Journal of Political Economy, University of Chicago Press, vol. 101(5), pages 915-938, October.
    32. Couch, K.A. & Morand, O.F., 2005. "Inequality, mobility, and the transmission of ability," Journal of Macroeconomics, Elsevier, vol. 27(2), pages 365-377, June.
    33. Gilles Joseph & Thomas Weitzenblum, 2003. "Optimal Unemployment Insurance: Transitional Dynamics vs. Steady State," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 6(4), pages 869-884, October.
    34. 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.
    35. Philippe Aghion & Patrick Bolton, 1997. "A Theory of Trickle-Down Growth and Development," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 64(2), pages 151-172.
    36. 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.
    37. Thomas Piketty, 1997. "The Dynamics of the Wealth Distribution and the Interest Rate with Credit Rationing," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 64(2), pages 173-189.
    38. Ranjan, Priya, 2001. "Dynamic evolution of income distribution and credit-constrained human capital investment in open economies," Journal of International Economics, Elsevier, vol. 55(2), pages 329-358, December.
    39. Diaz, Antonia & Pijoan-Mas, Josep & Rios-Rull, Jose-Victor, 2003. "Precautionary savings and wealth distribution under habit formation preferences," Journal of Monetary Economics, Elsevier, vol. 50(6), pages 1257-1291, September.
    40. Das, Satya P., 2006. "Trade, skill acquisition and distribution," Journal of Development Economics, Elsevier, vol. 81(1), pages 118-141, October.
    41. Hopenhayn, Hugo A, 1992. "Entry, Exit, and Firm Dynamics in Long Run Equilibrium," Econometrica, Econometric Society, vol. 60(5), pages 1127-1150, September.
    42. Morand, Olivier F. & Reffett, Kevin L., 2007. "Stationary Markovian equilibrium in overlapping generation models with stochastic nonclassical production and Markov shocks," Journal of Mathematical Economics, Elsevier, vol. 43(3-4), pages 501-522, April.
    43. Olivier F. Morand & Kevin L. Reffett, 2005. "Stationary Markovian Equilibrium in Overlapping Generation Models with Stochastic Nonclassical Production," Working papers 2005-52, University of Connecticut, Department of Economics.
    44. 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.
    45. H. Lloyd-Ellis, 1995. "Occupational Choice and the Growth-Inequality Relationship," Working Papers lloydell-95-02, University of Toronto, Department of Economics.
    46. Olson, Lars J., 1989. "Stochastic growth with irreversible investment," Journal of Economic Theory, Elsevier, vol. 47(1), pages 101-129, February.
    47. Bhattacharya, Rabi & Majumdar, Mukul, 2001. "On a Class of Stable Random Dynamical Systems: Theory and Applications," Journal of Economic Theory, Elsevier, vol. 96(1-2), pages 208-229, January.
    Full references (including those not matched with items on IDEAS)

    Most related items

    These are the items that most often cite the same works as this one and are cited by the same works as this one.
    1. Takashi Kamihigashi & John Stachurski, 2011. "Stability of Stationary Distributions in Monotone Economies," ANU Working Papers in Economics and Econometrics 2011-561, Australian National University, College of Business and Economics, School of Economics.
    2. Takashi Kamihigashi & John Stachurski, 2011. "Existence, Stability and Computation of Stationary Distributions: An Extension of the Hopenhayn-Prescott Theorem," Discussion Paper Series DP2011-32, Research Institute for Economics & Business Administration, Kobe University.
    3. , & ,, 2014. "Stochastic stability in monotone economies," Theoretical Economics, Econometric Society, vol. 9(2), May.
    4. Kamihigashi, Takashi & Stachurski, John, 2016. "Seeking ergodicity in dynamic economies," Journal of Economic Theory, Elsevier, vol. 163(C), pages 900-924.
    5. Takashi Kamihigashi & John Stachurski, 2014. "Interlinkage between Real Exchange rate and Current Account Behaviors: Evidence from India," Working Papers 2014-86, Department of Research, Ipag Business School.
    6. Takashi Kamihigashiw & John Stachurski, 2014. "Seeking Ergodicity in Dynamic Economies," Working Papers 2014-402, Department of Research, Ipag Business School.
    7. repec:ipg:wpaper:2014-086 is not listed on IDEAS
    8. Takashi Kamihigashi & John Stachurski, 2014. "Stability Analysis for Random Dynamical Systems in Economics," Discussion Paper Series DP2014-35, Research Institute for Economics & Business Administration, Kobe University.
    9. John Stachurski, 2009. "Economic Dynamics: Theory and Computation," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262012774, December.
    10. Azariadis, Costas & Stachurski, John, 2005. "Poverty Traps," Handbook of Economic Growth, in: Philippe Aghion & Steven Durlauf (ed.), Handbook of Economic Growth, edition 1, volume 1, chapter 5, Elsevier.
    11. Tapan Mitra & Santanu Roy, 2023. "Stochastic growth, conservation of capital and convergence to a positive steady state," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 76(1), pages 311-351, July.
    12. Cai, Yiyong & Kamihigashi, Takashi & Stachurski, John, 2014. "Stochastic optimal growth with risky labor supply," Journal of Mathematical Economics, Elsevier, vol. 50(C), pages 167-176.
    13. Lars J. Olson & Santanu Roy, 2006. "Theory of Stochastic Optimal Economic Growth," Springer Books, in: Rose-Anne Dana & Cuong Le Van & Tapan Mitra & Kazuo Nishimura (ed.), Handbook on Optimal Growth 1, chapter 11, pages 297-335, Springer.
    14. Jaime McGovern & Olivier Morand & Kevin Reffett, 2013. "Computing minimal state space recursive equilibrium in OLG models with stochastic production," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 54(3), pages 623-674, November.
    15. Josep Pijoan-Mas, 2006. "Precautionary Savings or Working Longer Hours?," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 9(2), pages 326-352, April.
    16. Masako Ikefuji & Ryo Horii, 2007. "Wealth Heterogeneity and Escape from the Poverty–Environment Trap," Journal of Public Economic Theory, Association for Public Economic Theory, vol. 9(6), pages 1041-1068, December.
    17. Balbus, Łukasz & Reffett, Kevin & Woźny, Łukasz, 2013. "A constructive geometrical approach to the uniqueness of Markov stationary equilibrium in stochastic games of intergenerational altruism," Journal of Economic Dynamics and Control, Elsevier, vol. 37(5), pages 1019-1039.
    18. Kevin Reffett & Olivier Morand, 2008. "Isotone recursive methods for Stationary Markov Equilibra in OLG models with stochastic nonclassical production," 2008 Meeting Papers 470, Society for Economic Dynamics.
    19. Mirman, Leonard J. & Morand, Olivier F. & Reffett, Kevin L., 2008. "A qualitative approach to Markovian equilibrium in infinite horizon economies with capital," Journal of Economic Theory, Elsevier, vol. 139(1), pages 75-98, March.
    20. José Wynne, 2005. "Wealth as a Determinant of Comparative Advantage," American Economic Review, American Economic Association, vol. 95(1), pages 226-254, March.
    21. Leonard J. Mirman & Kevin Reffett & John Stachurski, 2005. "Some stability results for Markovian economic semigroups," International Journal of Economic Theory, The International Society for Economic Theory, vol. 1(1), pages 57-72, March.

    More about this item

    Keywords

    Stability; Simulation; Stationary equilibria;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques

    Statistics

    Access and download statistics

    Corrections

    All material on this site has been provided by the respective publishers and authors. You can help correct errors and omissions. When requesting a correction, please mention this item's handle: RePEc:kob:dpaper:dp2012-27. See general information about how to correct material in RePEc.

    If you have authored this item and are not yet registered with RePEc, we encourage you to do it here. This allows to link your profile to this item. It also allows you to accept potential citations to this item that we are uncertain about.

    If CitEc recognized a bibliographic reference but did not link an item in RePEc to it, you can help with this form .

    If you know of missing items citing this one, you can help us creating those links by adding the relevant references in the same way as above, for each refering item. If you are a registered author of this item, you may also want to check the "citations" tab in your RePEc Author Service profile, as there may be some citations waiting for confirmation.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: Office of Promoting Research Collaboration, Research Institute for Economics & Business Administration, Kobe University (email available below). General contact details of provider: https://edirc.repec.org/data/rikobjp.html .

    Please note that corrections may take a couple of weeks to filter through the various RePEc services.

    IDEAS is a RePEc service. RePEc uses bibliographic data supplied by the respective publishers.