IDEAS home Printed from https://ideas.repec.org/p/jgu/wpaper/1004.html
   My bibliography  Save this paper

Matching and Saving in Continuous Time: Theory

Author

Listed:
  • Christian Bayer

    (Institute of Mathematics, Technische Universität Berlin, Germany)

  • Klaus Wälde

    (Chair in Macroeconomics, Johannes Gutenberg-Universität Mainz, Germany)

Abstract

We analyse optimal saving of risk-averse households when labour income stochastically jumps between two states. The generalized Keynes-Ramsey rule includes a precautionary savings term. A phase diagram analysis il- lustrates consumption and wealth dynamics within and between states. There is an endogenous lower and upper limit for wealth. We derive the Fokker-Planck equations for the densities of individual wealth and em- ployment status. These equations also characterize the aggregate distrib- ution of wealth and allow us to describe general equilibrium. An optimal consumption path exists and distributions converge to a unique limiting distribution.

Suggested Citation

  • Christian Bayer & Klaus Wälde, 2010. "Matching and Saving in Continuous Time: Theory," Working Papers 1004, Gutenberg School of Management and Economics, Johannes Gutenberg-Universität Mainz, revised 13 Jan 2010.
    • Handle: RePEc:jgu:wpaper:1004
    as

    Download full text from publisher

    File URL: https://download.uni-mainz.de/RePEc/pdf/Discussion_Paper_1004.pdf
    File Function: First version, 2010
    Download Restriction: no
    ---><---

    Other versions of this item:

    References listed on IDEAS

    as
    1. 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.
    2. Andrey Launov & Klaus Wälde, 2013. "Estimating Incentive And Welfare Effects Of Nonstationary Unemployment Benefits," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 54(4), pages 1159-1198, November.
    3. John Hassler & José V. Rodríguez Mora & Kjetil Storesletten & Fabrizio Zilibotti, 2005. "A Positive Theory Of Geographic Mobility And Social Insurance," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 46(1), pages 263-303, February.
    4. Mortensen, Dale & Pissarides, Christopher, 2011. "Job Creation and Job Destruction in the Theory of Unemployment," Ekonomicheskaya Politika / Economic Policy, Russian Presidential Academy of National Economy and Public Administration, vol. 1, pages 1-19.
    5. Giuseppe Bertola & Luigi Guiso & Luigi Pistaferri, 2005. "Uncertainty and Consumer Durables Adjustment," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 72(4), pages 973-1007.
    6. Rasmus Lentz & Torben Tranas, 2005. "Job Search and Savings: Wealth Effects and Duration Dependence," Journal of Labor Economics, University of Chicago Press, vol. 23(3), pages 467-490, July.
    7. Daron Acemoglu & Robert Shimer, 1999. "Efficient Unemployment Insurance," Journal of Political Economy, University of Chicago Press, vol. 107(5), pages 893-928, October.
    8. repec:hal:wpspec:info:hdl:2441/8623 is not listed on IDEAS
    9. Jeremy Lise, 2006. "On-the-Job Search and Precautionary Savings: Theory and Empirics of Earnings and Wealth Inequality," 2006 Meeting Papers 137, Society for Economic Dynamics.
    10. Rasmus Lentz, 2009. "Optimal Unemployment Insurance in an Estimated Job Search Model with Savings," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 12(1), pages 37-57, January.
    11. repec:hal:spmain:info:hdl:2441/c8dmi8nm4pdjkuc9g8grh35j2 is not listed on IDEAS
    12. Christian BAYER & Klaus WALDE, 2010. "Matching and Saving in Continuous Time: Proofs," LIDAM Discussion Papers IRES 2010014, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
    13. 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.
    14. Per Krusell & Toshihiko Mukoyama & Ayşegül Şahin, 2010. "Labour-Market Matching with Precautionary Savings and Aggregate Fluctuations," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 77(4), pages 1477-1507.
    15. Toche, Patrick, 2005. "A tractable model of precautionary saving in continuous time," Economics Letters, Elsevier, vol. 87(2), pages 267-272, May.
    16. Huggett, Mark & Ospina, Sandra, 2001. "Aggregate precautionary savings: when is the third derivative irrelevant?," Journal of Monetary Economics, Elsevier, vol. 48(2), pages 373-396, October.
    17. Turnovsky, Stephen J. & Smith, William T., 2006. "Equilibrium consumption and precautionary savings in a stochastically growing economy," Journal of Economic Dynamics and Control, Elsevier, vol. 30(2), pages 243-278, February.
    18. Yongsung Chang & Sun-Bin Kim & Mark Bils, 2007. "Comparative Advantage in Cyclical Unemployment," 2007 Meeting Papers 508, Society for Economic Dynamics.
    19. Fabien Postel-Vinay & Jean-Marc Robin, 2002. "Equilibrium Wage Dispersion with Worker and Employer Heterogeneity," Post-Print hal-03458567, HAL.
    20. S. Rao Aiyagari, 1994. "Uninsured Idiosyncratic Risk and Aggregate Saving," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 109(3), pages 659-684.
    21. Christopher J. Flinn, 2006. "Minimum Wage Effects on Labor Market Outcomes under Search, Matching, and Endogenous Contact Rates," Econometrica, Econometric Society, vol. 74(4), pages 1013-1062, July.
    22. Gerard J. van den Berg, 1990. "Nonstationarity in Job Search Theory," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 57(2), pages 255-277.
    23. Krusell, Per & Mukoyama, Toshihiko & Rogerson, Richard & Sahin, Aysegül, 2008. "Aggregate implications of indivisible labor, incomplete markets, and labor market frictions," Journal of Monetary Economics, Elsevier, vol. 55(5), pages 961-979, July.
    24. Robert Shimer, 2005. "The Cyclical Behavior of Equilibrium Unemployment and Vacancies," American Economic Review, American Economic Association, vol. 95(1), pages 25-49, March.
    25. Guo, Xin & Miao, Jianjun & Morellec, Erwan, 2005. "Irreversible investment with regime shifts," Journal of Economic Theory, Elsevier, vol. 122(1), pages 37-59, May.
    26. Robert Shimer & Iván Werning, 2007. "Reservation Wages and Unemployment Insurance," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 122(3), pages 1145-1185.
    27. Hassler, John & Rodríguez Mora, José V., 2008. "Unemployment insurance design: Inducing moving and retraining," European Economic Review, Elsevier, vol. 52(5), pages 757-791, July.
    28. Makoto Nakajima, 2006. "Business Cycles in the Equilibrium Model of Labor Search and Self-Insurance," Computing in Economics and Finance 2006 426, Society for Computational Economics.
    29. Mark Bils & Yongsung Chang & Sun-Bin Kim, 2008. "Heterogeneity and Cyclical Unemployment," RCER Working Papers 543, University of Rochester - Center for Economic Research (RCER).
    30. Burdett, Kenneth & Mortensen, Dale T, 1998. "Wage Differentials, Employer Size, and Unemployment," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 39(2), pages 257-273, May.
    31. repec:hal:spmain:info:hdl:2441/8623 is not listed on IDEAS
    32. Merz, Monika, 1995. "Search in the labor market and the real business cycle," Journal of Monetary Economics, Elsevier, vol. 36(2), pages 269-300, November.
    33. Walde, Klaus, 1999. "Optimal Saving under Poisson Uncertainty," Journal of Economic Theory, Elsevier, vol. 87(1), pages 194-217, July.
    34. Pieter A. Gautier & Coen N. Teulings & Aico Van Vuuren, 2010. "On-the-Job Search, Mismatch and Efficiency ," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 77(1), pages 245-272.
    35. Merton, Robert C, 1969. "Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case," The Review of Economics and Statistics, MIT Press, vol. 51(3), pages 247-257, August.
    36. Ait-Sahalia, Yacine, 2004. "Disentangling diffusion from jumps," Journal of Financial Economics, Elsevier, vol. 74(3), pages 487-528, December.
    37. 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.
    38. Andolfatto, David, 1996. "Business Cycles and Labor-Market Search," American Economic Review, American Economic Association, vol. 86(1), pages 112-132, March.
    39. Lucas, Robert Jr. & Prescott, Edward C., 1974. "Equilibrium search and unemployment," Journal of Economic Theory, Elsevier, vol. 7(2), pages 188-209, February.
    40. Hassler, John & Rodriguez Mora, Jose V., 1999. "Employment turnover and the public allocation of unemployment insurance," Journal of Public Economics, Elsevier, vol. 73(1), pages 55-83, July.
    41. Giuseppe Moscarini, 2005. "Job Matching and the Wage Distribution," Econometrica, Econometric Society, vol. 73(2), pages 481-516, March.
    42. repec:hal:spmain:info:hdl:2441/dc0ckec3fcb29ms985085gkbp is not listed on IDEAS
    43. 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.
    Full references (including those not matched with items on IDEAS)

    Citations

    Citations are extracted by the CitEc Project, subscribe to its RSS feed for this item.
    as


    Cited by:

    1. Khieu, Hoang & Wälde, Klaus, 2023. "Capital income risk and the dynamics of the wealth distribution," Economic Modelling, Elsevier, vol. 122(C).
    2. Flórez, Luz A., 2017. "Informal sector under saving: A positive analysis of labour market policies," Labour Economics, Elsevier, vol. 44(C), pages 13-26.
    3. Donsimoni Jean Roch & Glawion René & Plachter Bodo & Wälde Klaus, 2020. "Projecting the spread of COVID-19 for Germany," German Economic Review, De Gruyter, vol. 21(2), pages 181-216, June.
    4. Bayer, Christian & Rendall, Alan D. & Wälde, Klaus, 2019. "The invariant distribution of wealth and employment status in a small open economy with precautionary savings," Journal of Mathematical Economics, Elsevier, vol. 85(C), pages 17-37.
    5. Carlos Miguel Silva & Ana Paula Ribeiro, 2011. "The Impacts of Structural Changes in the Labor Market: a Comparative Statics Analysis Using Heterogeneous-agent Framework," CEF.UP Working Papers 1104, Universidade do Porto, Faculdade de Economia do Porto.
    6. Jeremy Lise, 2013. "On-the-Job Search and Precautionary Savings," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 80(3), pages 1086-1113.

    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. Wälde, Klaus & Bayer, Christian, 2011. "Describing the Dynamics of Distribution in Search and Matching Models by Fokker-Planck Equations," VfS Annual Conference 2011 (Frankfurt, Main): The Order of the World Economy - Lessons from the Crisis 48736, Verein für Socialpolitik / German Economic Association.
    2. Andrey Launov & Klaus Wälde, 2013. "Estimating Incentive And Welfare Effects Of Nonstationary Unemployment Benefits," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 54(4), pages 1159-1198, November.
    3. Chaumont, Gaston & Shi, Shouyong, 2022. "Wealth accumulation, on-the-job search and inequality," Journal of Monetary Economics, Elsevier, vol. 128(C), pages 51-71.
    4. Andreas Hornstein & Per Krusell & Giovanni L. Violante, 2011. "Frictional Wage Dispersion in Search Models: A Quantitative Assessment," American Economic Review, American Economic Association, vol. 101(7), pages 2873-2898, December.
    5. Tom Krebs & Martin Scheffel, 2013. "Macroeconomic Evaluation of Labor Market Reform in Germany," IMF Economic Review, Palgrave Macmillan;International Monetary Fund, vol. 61(4), pages 664-701, December.
    6. Per Krusell & Toshihiko Mukoyama & Ayşegül Şahin, 2010. "Labour-Market Matching with Precautionary Savings and Aggregate Fluctuations," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 77(4), pages 1477-1507.
    7. Violante, Giovanni & Hornstein, Andreas, 2006. "Frictional Wage Dispersion in Search Models: A Quantitative Approach," CEPR Discussion Papers 5935, C.E.P.R. Discussion Papers.
    8. Rogerson, Richard & Shimer, Robert, 2011. "Search in Macroeconomic Models of the Labor Market," Handbook of Labor Economics, in: O. Ashenfelter & D. Card (ed.), Handbook of Labor Economics, edition 1, volume 4, chapter 7, pages Pages: 61, Elsevier.
    9. Leena Rudanko, 2008. "Aggregate and Idiosyncratic Risk in a Frictional Labor Market," Boston University - Department of Economics - Working Papers Series wp2008-009, Boston University - Department of Economics.
    10. Mukoyama, Toshihiko, 2013. "Understanding the welfare effects of unemployment insurance policy in general equilibrium," Journal of Macroeconomics, Elsevier, vol. 38(PB), pages 347-368.
    11. Edouard Challe & Xavier Ragot, 2016. "Precautionary Saving Over the Business Cycle," Economic Journal, Royal Economic Society, vol. 126(590), pages 135-164, February.
    12. Jan Eeckhout & Alireza Sepahsalari, 2024. "The Effect of Wealth on Worker Productivity," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 91(3), pages 1584-1633.
    13. Richard Rogerson & Robert Shimer & Randall Wright, 2004. "Search-Theoretic Models of the Labor Market-A Survey," NBER Working Papers 10655, National Bureau of Economic Research, Inc.
    14. Yashiv, Eran, 2007. "Labor search and matching in macroeconomics," European Economic Review, Elsevier, vol. 51(8), pages 1859-1895, November.
    15. Lyon, Spencer G. & Waugh, Michael E., 2018. "Redistributing the gains from trade through progressive taxation," Journal of International Economics, Elsevier, vol. 115(C), pages 185-202.
    16. Simon Jäger & Benjamin Schoefer & Samuel Young & Josef Zweimüller, 2020. "Wages and the Value of Nonemployment," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 135(4), pages 1905-1963.
    17. Andrey Launov & Irene Schumm & Klaus Walde, 2008. "Estimating insurance and incentive effects of labour market reforms," CDMA Conference Paper Series 0813, Centre for Dynamic Macroeconomic Analysis.
    18. Ortego-Marti, Victor, 2016. "Unemployment history and frictional wage dispersion," Journal of Monetary Economics, Elsevier, vol. 78(C), pages 5-22.
    19. Balleer, Almut & Duernecker, Georg & Forstner, Susanne & Goensch, Johannes, 2023. "Wage Bargaining and Labor Market Policy with Biased Expectations," CEPR Discussion Papers 18019, C.E.P.R. Discussion Papers.
    20. Makoto Nakajima, 2012. "Business Cycles In The Equilibrium Model Of Labor Market Search And Self‐Insurance," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 53(2), pages 399-432, May.

    More about this item

    Keywords

    matching model; optimal saving; incomplete markets; Poisson uncertainty; Fokker-Planck equations; general equilibrium;
    All these keywords.

    JEL classification:

    • 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
    • J64 - Labor and Demographic Economics - - Mobility, Unemployment, Vacancies, and Immigrant Workers - - - Unemployment: Models, Duration, Incidence, and Job Search

    NEP fields

    This paper has been announced in the following NEP Reports:

    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:jgu:wpaper:1004. 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: Research Unit IPP (email available below). General contact details of provider: https://edirc.repec.org/data/vlmaide.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.