IDEAS home Printed from
   My bibliography  Save this paper

The Invariant Distribution of Wealth and Employment Status in a Small Open Economy with Precautionary Savings


  • Christian Bayer
  • Alan D. Rendall
  • Klaus Wälde


We study optimal savings in continuous time with exogenous transitions between employment and unemployment as the only source of uncertainty in a small open economy. We prove the existence of an optimal consumption path. We exploit that the dynamics of consumption and wealth between jumps can be expressed as a Fuchsian system. We derive conditions under which an invariant joint distribution for the state variables, i.e., wealth and labour market status, exists and is unique. We also provide conditions under which the distribution of these variables converges to the invariant distribution. Our analysis relies on the notion of T-processes and applies results on the stability of Markovian processes from Meyn and Tweedie (1993a, b,c).

Suggested Citation

  • Christian Bayer & Alan D. Rendall & Klaus Wälde, 2018. "The Invariant Distribution of Wealth and Employment Status in a Small Open Economy with Precautionary Savings," CESifo Working Paper Series 7397, CESifo.
  • Handle: RePEc:ces:ceswps:_7397

    Download full text from publisher

    File URL:
    Download Restriction: no

    Other versions of this item:

    References listed on IDEAS

    1. Yves Achdou & Jiequn Han & Jean-Michel Lasry & Pierre-Louis Lions & Benjamin Moll, 2017. "Income and Wealth Distribution in Macroeconomics: A Continuous-Time Approach," NBER Working Papers 23732, National Bureau of Economic Research, Inc.
    2. Shuhei Aoki & Makoto Nirei, 2016. "Pareto Distribution of Income in Neoclassical Growth Models," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 20, pages 25-42, April.
    3. Robert M. Anderson & Roberto C. Raimondo, 2008. "Equilibrium in Continuous-Time Financial Markets: Endogenously Dynamically Complete Markets," Econometrica, Econometric Society, vol. 76(4), pages 841-907, July.
    4. SeHyoun Ahn & Greg Kaplan & Benjamin Moll & Thomas Winberry & Christian Wolf, 2018. "When Inequality Matters for Macro and Macro Matters for Inequality," NBER Macroeconomics Annual, University of Chicago Press, vol. 32(1), pages 1-75.
    5. S. Rao Aiyagari, 1994. "Uninsured Idiosyncratic Risk and Aggregate Saving," The Quarterly Journal of Economics, Oxford University Press, vol. 109(3), pages 659-684.
    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. Jesus Fernandez-Villaverde & Samuel Hurtado & Galo Nuno, 2019. "Financial Frictions and the Wealth Distribution," PIER Working Paper Archive 19-015, Penn Institute for Economic Research, Department of Economics, University of Pennsylvania.
    2. Toda, Alexis Akira, 2019. "Wealth distribution with random discount factors," Journal of Monetary Economics, Elsevier, vol. 104(C), pages 101-113.
    3. Khieu, Hoang & Wälde, Klaus, 2018. "Capital Income Risk and the Dynamics of the Wealth Distribution," IZA Discussion Papers 11840, Institute of Labor Economics (IZA).
    4. Karsten O. Chipeniuk, 2020. "Optimal Grid Selection for the Numerical Solution of Dynamic Stochastic Optimization Problems," Computational Economics, Springer;Society for Computational Economics, vol. 56(4), pages 883-928, December.
    5. Per Krusell & Anthony Smith & Joachim Hubmer, 2015. "The historical evolution of the wealth distribution: A quantitative-theoretic investigation," 2015 Meeting Papers 1406, Society for Economic Dynamics.
    6. Boppart, Timo & Krusell, Per & Mitman, Kurt, 2018. "Exploiting MIT shocks in heterogeneous-agent economies: the impulse response as a numerical derivative," Journal of Economic Dynamics and Control, Elsevier, vol. 89(C), pages 68-92.
    7. Pietro Dindo & Andrea Modena & Loriana Pelizzon, 2019. "Risk Pooling, Leverage, and the Business Cycle," CESifo Working Paper Series 7772, CESifo.
    8. Dan Cao & Wenlan Luo, 2017. "Persistent Heterogeneous Returns and Top End Wealth Inequality," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 26, pages 301-326, October.
    9. Giusto, Andrea & Ä°ÅŸcan, Talan B., 2019. "Market Power And The Aggregate Saving Rate," Macroeconomic Dynamics, Cambridge University Press, vol. 23(6), pages 2269-2297, September.
    10. Ma, Qingyin & Stachurski, John & Toda, Alexis Akira, 2020. "The income fluctuation problem and the evolution of wealth," Journal of Economic Theory, Elsevier, vol. 187(C).
    11. Cao, Dan, 2020. "Recursive equilibrium in Krusell and Smith (1998)," Journal of Economic Theory, Elsevier, vol. 186(C).
    12. Joachim Hubmer & Per Krusell & Anthony A. Smith Jr., 2020. "Sources of US Wealth Inequality: Past, Present, and Future," NBER Chapters, in: NBER Macroeconomics Annual 2020, volume 35, National Bureau of Economic Research, Inc.
    13. Papp, Tamás K. & Reiter, Michael, 2020. "Estimating linearized heterogeneous agent models using panel data," Journal of Economic Dynamics and Control, Elsevier, vol. 115(C).
    14. Lei, Xiaowen, 2019. "Information and Inequality," Journal of Economic Theory, Elsevier, vol. 184(C).
    15. Feld, Lars P. & Schmidt, Christoph M. & Schnabel, Isabel & Truger, Achim & Wieland, Volker, 2019. "Den Strukturwandel meistern. Jahresgutachten 2019/20," Annual Economic Reports / Jahresgutachten, German Council of Economic Experts / Sachverständigenrat zur Begutachtung der gesamtwirtschaftlichen Entwicklung, volume 127, number 201920, September.
    16. Mariacristina De Nardi & Giulio Fella, 2017. "Saving and Wealth Inequality," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 26, pages 280-300, October.
    17. Gouin-Bonenfant, Emilien & Toda, Alexis Akira, 2018. "Pareto Extrapolation: Bridging Theoretical and Quantitative Models of Wealth Inequality," University of California at San Diego, Economics Working Paper Series qt90n2h2bb, Department of Economics, UC San Diego.
    18. Lehrer, Ehud & Light, Bar, 2018. "The effect of interest rates on consumption in an income fluctuation problem," Journal of Economic Dynamics and Control, Elsevier, vol. 94(C), pages 63-71.
    19. Łukasz Rachel & Lawrence H. Summers, 2019. "On Secular Stagnation in the Industrialized World," NBER Working Papers 26198, National Bureau of Economic Research, Inc.
    20. Toda, Alexis Akira, 2017. "Huggett economies with multiple stationary equilibria," Journal of Economic Dynamics and Control, Elsevier, vol. 84(C), pages 77-90.

    More about this item


    uncertainty in continuous time; counting process; existence; uniqueness; stability;

    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
    • J63 - Labor and Demographic Economics - - Mobility, Unemployment, Vacancies, and Immigrant Workers - - - Turnover; Vacancies; Layoffs

    NEP fields

    This paper has been announced in the following NEP Reports:


    Access and download statistics


    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:ces:ceswps:_7397. See general information about how to correct material in RePEc.

    For technical questions regarding this item, or to correct its authors, title, abstract, bibliographic or download information, contact: (Klaus Wohlrabe). General contact details of provider: .

    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 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.

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

    IDEAS is a RePEc service hosted by the Research Division of the Federal Reserve Bank of St. Louis . RePEc uses bibliographic data supplied by the respective publishers.