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Existence, Uniqueness and Stability of Invariant Distributions in Continuous-Time Stochastic Models

Author

Listed:
  • Christian Bayer

    (Department of Mathematics, University of Vienna, Austria)

  • Klaus Waelde

    (Department of Economics, Johannes Gutenberg-Universitaet Mainz, Germany)

Abstract

We study a dynamic stochastic general equilibrium model in continuous time. Related work has proven that optimal consumption in this model is a smooth function of state variables. This allows us to describe the evolution of optimal state variables (wealth and labour market status) by stochastic differential equations. We derive conditions under which an invariant distribution for state variables exists and is unique. We also provide conditions such that initial distributions converge to the long-run distribution.

Suggested Citation

  • Christian Bayer & Klaus Waelde, 2011. "Existence, Uniqueness and Stability of Invariant Distributions in Continuous-Time Stochastic Models," Working Papers 1111, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz, revised 21 Jul 2011.
    • Handle: RePEc:jgu:wpaper:1111
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    as
    1. Matsuyama, Kiminori, 2001. "Growing through Cycles in an Infinitely Lived Agent Economy," Journal of Economic Theory, Elsevier, vol. 100(2), pages 220-234, October.
    2. Robert Shimer & Lones Smith, 2000. "Assortative Matching and Search," Econometrica, Econometric Society, vol. 68(2), pages 343-370, March.
    3. Fwu-Ranq Chang & A. G. Malliaris, 2005. "Asymptotic Growth under Uncertainty: Existence and Uniqueness," World Scientific Book Chapters, in: Economic Uncertainty, Instabilities And Asset Bubbles Selected Essays, chapter 1, pages 3-8, World Scientific Publishing Co. Pte. Ltd..
    4. Pierre Cahuc & Fabien Postel-Vinay & Jean-Marc Robin, 2006. "Wage Bargaining with On-the-Job Search: Theory and Evidence," Econometrica, Econometric Society, vol. 74(2), pages 323-364, March.
    5. Lo, Andrew W., 1988. "Maximum Likelihood Estimation of Generalized Itô Processes with Discretely Sampled Data," Econometric Theory, Cambridge University Press, vol. 4(2), pages 231-247, August.
    6. Gong, Liutang & Zou, Heng-fu, 2002. "Direct preferences for wealth, the risk premium puzzle, growth, and policy effectiveness," Journal of Economic Dynamics and Control, Elsevier, vol. 26(2), pages 247-270, February.
    7. Leo Kaas & Philipp Kircher, 2015. "Efficient Firm Dynamics in a Frictional Labor Market," American Economic Review, American Economic Association, vol. 105(10), pages 3030-3060, October.
    8. Lars Peter Hansen & José A. Scheinkman, 2009. "Long-Term Risk: An Operator Approach," Econometrica, Econometric Society, vol. 77(1), pages 177-234, January.
    9. repec:hal:spmain:info:hdl:2441/c8dmi8nm4pdjkuc9g8grh35j2 is not listed on IDEAS
    10. Fernando Alvarez & Robert Shimer, 2011. "Search and Rest Unemployment," Econometrica, Econometric Society, vol. 79(1), pages 75-122, January.
    11. Fabien Postel-Vinay & Jean-Marc Robin, 2002. "Equilibrium Wage Dispersion with Worker and Employer Heterogeneity," Econometrica, Econometric Society, vol. 70(6), pages 2295-2350, November.
    12. Turnovsky, Stephen J, 1993. "Macroeconomic Policies, Growth, and Welfare in a Stochastic Economy," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 34(4), pages 953-981, November.
    13. Fabien Postel-Vinay & Jean-Marc Robin, 2002. "Equilibrium Wage Dispersion with Worker and Employer Heterogeneity," Post-Print hal-03458567, HAL.
    14. Futia, Carl A, 1982. "Invariant Distributions and the Limiting Behavior of Markovian Economic Models," Econometrica, Econometric Society, vol. 50(2), pages 377-408, March.
    15. Patrick Francois & Huw Lloyd-Ellis, 2003. "Animal Spirits Through Creative Destruction," American Economic Review, American Economic Association, vol. 93(3), pages 530-550, June.
    16. Jessica A. Wachter, 2013. "Can Time-Varying Risk of Rare Disasters Explain Aggregate Stock Market Volatility?," Journal of Finance, American Finance Association, vol. 68(3), pages 987-1035, June.
    17. Lentz, Rasmus, 2010. "Sorting by search intensity," Journal of Economic Theory, Elsevier, vol. 145(4), pages 1436-1452, July.
    18. Robert Shimer, 2005. "The Cyclical Behavior of Equilibrium Unemployment and Vacancies," American Economic Review, American Economic Association, vol. 95(1), pages 25-49, March.
    19. Bental, Benjamin & Peled, Dan, 1996. "The Accumulation of Wealth and the Cyclical Generation of New Technologies: A Search Theoretic Approach," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 37(3), pages 687-718, August.
    20. Kenneth Burdett & Carlos Carrillo‐Tudela & Melvyn G. Coles, 2011. "Human Capital Accumulation And Labor Market Equilibrium," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 52(3), pages 657-677, August.
    21. William A. Brock & Leonard J. Mirman, 2001. "Optimal Economic Growth And Uncertainty: The Discounted Case," Chapters, in: W. D. Dechert (ed.), Growth Theory, Nonlinear Dynamics and Economic Modelling, chapter 1, pages 3-37, Edward Elgar Publishing.
    22. Olaf POSCH & Klaus WALDE, 2009. "On the non-causal link between volatility and growth," LIDAM Discussion Papers IRES 2009025, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    23. Magill, Michael J. P., 1977. "A local analysis of N-sector capital accumulation under uncertainty," Journal of Economic Theory, Elsevier, vol. 15(1), pages 211-219, June.
    24. Julien Prat, 2007. "The Impact of Disembodied Technological Progress on Unemployment," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 10(1), pages 106-125, January.
    25. Walde, Klaus, 2002. "The economic determinants of technology shocks in a real business cycle model," Journal of Economic Dynamics and Control, Elsevier, vol. 27(1), pages 1-28, November.
    26. Xavier Gabaix, 1999. "Zipf's Law for Cities: An Explanation," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 114(3), pages 739-767.
    27. Evans, Lynne & Kenc, Turalay, 2003. "Welfare Cost Of Monetary And Fiscal Policy Shocks," Macroeconomic Dynamics, Cambridge University Press, vol. 7(2), pages 212-238, April.
    28. 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.
    29. Robert C. Merton, 1975. "An Asymptotic Theory of Growth Under Uncertainty," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 42(3), pages 375-393.
    30. Posch, Olaf, 2009. "Structural estimation of jump-diffusion processes in macroeconomics," Journal of Econometrics, Elsevier, vol. 153(2), pages 196-210, December.
    31. Grinols, Earl L & Turnovsky, Stephen J, 1998. "Consequences of Debt Policy in a Stochastically Growing Monetary Economy," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(2), pages 495-521, May.
    32. 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.
    33. Kiminori Matsuyama, 1999. "Growing Through Cycles," Econometrica, Econometric Society, vol. 67(2), pages 335-348, March.
    34. Fabien Postel-Vinay & Jean-Marc Robin, 2002. "Equilibrium wage dispersion with worker and employer heterogeneity," Post-Print hal-03587660, HAL.
    35. Gokan, Yoichi, 2002. "Alternative government financing and stochastic endogenous growth," Journal of Economic Dynamics and Control, Elsevier, vol. 26(4), pages 681-706, April.
    36. Bismut, Jean-Michel, 1975. "Growth and optimal intertemporal allocation of risks," Journal of Economic Theory, Elsevier, vol. 10(2), pages 239-257, April.
    37. Gertler, Mark & Grinols, Earl, 1982. "Monetary randomness and investment," Journal of Monetary Economics, Elsevier, vol. 10(2), pages 239-258.
    38. Scott, Louis O., 1987. "Option Pricing when the Variance Changes Randomly: Theory, Estimation, and an Application," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 22(4), pages 419-438, December.
    39. Pissarides, Christopher A, 1985. "Short-run Equilibrium Dynamics of Unemployment Vacancies, and Real Wages," American Economic Review, American Economic Association, vol. 75(4), pages 676-690, September.
    40. Giuseppe Moscarini & Fabien Postel-Vinay, 2009. "The Timing of Labor Market Expansions: New Facts and a New Hypothesis," NBER Chapters, in: NBER Macroeconomics Annual 2008, Volume 23, pages 1-51, National Bureau of Economic Research, Inc.
    41. Bandi, Federico M. & Nguyen, Thong H., 2003. "On the functional estimation of jump-diffusion models," Journal of Econometrics, Elsevier, vol. 116(1-2), pages 293-328.
    42. Brock, William A & Magill, Michael J P, 1979. "Dynamics under Uncertainty," Econometrica, Econometric Society, vol. 47(4), pages 843-868, July.
    43. 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.
    44. Steger, Thomas M., 2005. "Stochastic growth under Wiener and Poisson uncertainty," Economics Letters, Elsevier, vol. 86(3), pages 311-316, March.
    45. Giuseppe Moscarini, 2005. "Job Matching and the Wage Distribution," Econometrica, Econometric Society, vol. 73(2), pages 481-516, March.
    46. repec:hal:spmain:info:hdl:2441/dc0ckec3fcb29ms985085gkbp is not listed on IDEAS
    47. Huggett, Mark, 1993. "The risk-free rate in heterogeneous-agent incomplete-insurance economies," Journal of Economic Dynamics and Control, Elsevier, vol. 17(5-6), pages 953-969.
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    More about this item

    Keywords

    uncertainty in continuous time; Poisson process; existence; uniqueness; stability;
    All these keywords.

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • D91 - Microeconomics - - Micro-Based Behavioral Economics - - - Role and Effects of Psychological, Emotional, Social, and Cognitive Factors on Decision Making
    • E24 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Employment; Unemployment; Wages; Intergenerational Income Distribution; Aggregate Human Capital; Aggregate Labor Productivity
    • J63 - Labor and Demographic Economics - - Mobility, Unemployment, Vacancies, and Immigrant Workers - - - Turnover; Vacancies; Layoffs

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