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The Dynamics of Pareto Distributed Wealth in a Small Open Economy

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  • Matthias Birkner
  • Niklas Scheuer
  • Klaus Wälde

Abstract

We study a small open economy displaying Pareto-distributed wealth resulting from random death. The government runs a distribution scheme on inheritance. We present the mathematical background that allows to study the dynamics of means. We end up with ordinary differential equations for the mean of age and of individual and government wealth. We also study distributional dynamics analytically. Starting from any distribution of age and wealth, the aggregate distribution converges, both on a transition path towards a steady state and on a transition path towards balanced growth, to an exponential distribution of age and a Pareto-distribution of wealth. The findings are illustrated for different distribution schemes.

Suggested Citation

  • Matthias Birkner & Niklas Scheuer & Klaus Wälde, 2021. "The Dynamics of Pareto Distributed Wealth in a Small Open Economy," CESifo Working Paper Series 9058, CESifo.
    • Handle: RePEc:ces:ceswps:_9058
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    1. Benjamin Moll & Lukasz Rachel & Pascual Restrepo, 2022. "Uneven Growth: Automation's Impact on Income and Wealth Inequality," Econometrica, Econometric Society, vol. 90(6), pages 2645-2683, November.
    2. 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.
    3. Raoul Bouccekine & Fabien Prieur & Weihua Ruan & Benteng Zou, 2021. "Uncertainty-driven symmetry-breaking and stochastic stability in a generic differential game of lobbying," DEM Discussion Paper Series 21-10, Department of Economics at the University of Luxembourg.
    4. Jess Benhabib & Alberto Bisin & Shenghao Zhu, 2011. "The Distribution of Wealth and Fiscal Policy in Economies With Finitely Lived Agents," Econometrica, Econometric Society, vol. 79(1), pages 123-157, January.
    5. Reiichiro Kawai, 2009. "Sensitivity Analysis And Density Estimation For The Hobson-Rogers Stochastic Volatility Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(03), pages 283-295.
    6. Markus K. Brunnermeier & Yuliy Sannikov, 2014. "A Macroeconomic Model with a Financial Sector," American Economic Review, American Economic Association, vol. 104(2), pages 379-421, February.
    7. Oleg Itskhoki & Benjamin Moll, 2019. "Optimal Development Policies With Financial Frictions," Econometrica, Econometric Society, vol. 87(1), pages 139-173, January.
    8. 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.
    9. Zhiguo He & Arvind Krishnamurthy, 2013. "Intermediary Asset Pricing," American Economic Review, American Economic Association, vol. 103(2), pages 732-770, April.
    10. Greg Kaplan & Benjamin Moll & Giovanni L. Violante, 2018. "Monetary Policy According to HANK," American Economic Review, American Economic Association, vol. 108(3), pages 697-743, March.
    11. Aghion, Philippe & Howitt, Peter, 1992. "A Model of Growth through Creative Destruction," Econometrica, Econometric Society, vol. 60(2), pages 323-351, March.
    12. Bossmann, Martin & Kleiber, Christian & Walde, Klaus, 2007. "Bequests, taxation and the distribution of wealth in a general equilibrium model," Journal of Public Economics, Elsevier, vol. 91(7-8), pages 1247-1271, August.
    13. Sennewald, Ken, 2007. "Controlled stochastic differential equations under Poisson uncertainty and with unbounded utility," Journal of Economic Dynamics and Control, Elsevier, vol. 31(4), pages 1106-1131, April.
    14. Marc J. Melitz, 2003. "The Impact of Trade on Intra-Industry Reallocations and Aggregate Industry Productivity," Econometrica, Econometric Society, vol. 71(6), pages 1695-1725, November.
    15. Barro, Robert J, 1974. "Are Government Bonds Net Wealth?," Journal of Political Economy, University of Chicago Press, vol. 82(6), pages 1095-1117, Nov.-Dec..
    16. Stephen J. Turnovsky, 1997. "International Macroeconomic Dynamics," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262201119, December.
    17. 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.
    18. d'Albis, Hippolyte, 2007. "Demographic structure and capital accumulation," Journal of Economic Theory, Elsevier, vol. 132(1), pages 411-434, January.
    19. Charles I. Jones, 1999. "Growth: With or Without Scale Effects?," American Economic Review, American Economic Association, vol. 89(2), pages 139-144, May.
    20. Benhabib, Jess & Bisin, Alberto & Zhu, Shenghao, 2016. "The Distribution Of Wealth In The Blanchard–Yaari Model," Macroeconomic Dynamics, Cambridge University Press, vol. 20(2), pages 466-481, March.
    21. Gene M. Grossman & Elhanan Helpman, 2005. "Outsourcing in a Global Economy," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 72(1), pages 135-159.
    22. Ernst Eberlein & Kathrin Glau, 2014. "Variational Solutions of the Pricing PIDEs for European Options in Lévy Models," Applied Mathematical Finance, Taylor & Francis Journals, vol. 21(5), pages 417-450, November.
    23. Jess Benhabib & Alberto Bisin & Mi Luo, 2019. "Wealth Distribution and Social Mobility in the US: A Quantitative Approach," American Economic Review, American Economic Association, vol. 109(5), pages 1623-1647, May.
    24. Emmanuel Saez & Gabriel Zucman, 2016. "Editor's Choice Wealth Inequality in the United States since 1913: Evidence from Capitalized Income Tax Data," The Quarterly Journal of Economics, President and Fellows of Harvard College, vol. 131(2), pages 519-578.
    25. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    26. Gene M. Grossman & Elhanan Helpman, 1991. "Quality Ladders in the Theory of Growth," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 58(1), pages 43-61.
    27. Narayana R. Kocherlakota, 2010. "The New Dynamic Public Finance," Economics Books, Princeton University Press, edition 1, number 9222.
    28. Wälde, Klaus, 2011. "Production technologies in stochastic continuous time models," Journal of Economic Dynamics and Control, Elsevier, vol. 35(4), pages 616-622, April.
    29. Reed, William J., 2001. "The Pareto, Zipf and other power laws," Economics Letters, Elsevier, vol. 74(1), pages 15-19, December.
    30. 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).
    31. Matthias Birkner & Niklas Scheuer & Klaus Wälde, 2023. "The dynamics of Pareto distributed wealth in a small open economy," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 76(2), pages 607-644, August.
    32. Benavie, Arthur & Grinols, Earl & Turnovsky, Stephen J., 1996. "Adjustment costs and investment in a stochastic endogenous growth model," Journal of Monetary Economics, Elsevier, vol. 38(1), pages 77-100, August.
    33. Jess Benhabib & Alberto Bisin, 2006. "The distribution of wealth and redistributive policies," Levine's Working Paper Archive 122247000000001162, David K. Levine.
    34. Blanchard, Olivier J, 1985. "Debt, Deficits, and Finite Horizons," Journal of Political Economy, University of Chicago Press, vol. 93(2), pages 223-247, April.
    35. Weil, Philippe, 1989. "Overlapping families of infinitely-lived agents," Journal of Public Economics, Elsevier, vol. 38(2), pages 183-198, March.
    36. Alexis Akira Toda & Kieran Walsh, 2015. "The Double Power Law in Consumption and Implications for Testing Euler Equations," Journal of Political Economy, University of Chicago Press, vol. 123(5), pages 1177-1200.
    37. Philippe Aghion & Christopher Harris & Peter Howitt & John Vickers, 2001. "Competition, Imitation and Growth with Step-by-Step Innovation," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 68(3), pages 467-492.
    38. Walde, Klaus, 1999. "Optimal Saving under Poisson Uncertainty," Journal of Economic Theory, Elsevier, vol. 87(1), pages 194-217, July.
    39. Piketty, Thomas & Zucman, Gabriel, 2014. "Wealth and Inheritance in the Long Run," CEPR Discussion Papers 10072, C.E.P.R. Discussion Papers.
    40. Mortensen, Dale T, 1982. "Property Rights and Efficiency in Mating, Racing, and Related Games," American Economic Review, American Economic Association, vol. 72(5), pages 968-979, December.
    41. 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.
    42. Diamond, Peter A, 1982. "Aggregate Demand Management in Search Equilibrium," Journal of Political Economy, University of Chicago Press, vol. 90(5), pages 881-894, October.
    43. Toda, Alexis Akira, 2012. "The double power law in income distribution: Explanations and evidence," Journal of Economic Behavior & Organization, Elsevier, vol. 84(1), pages 364-381.
    44. Kasa, Kenneth & Lei, Xiaowen, 2018. "Risk, uncertainty, and the dynamics of inequality," Journal of Monetary Economics, Elsevier, vol. 94(C), pages 60-78.
    45. Brendan K. Beare & Alexis Akira Toda, 2022. "Determination of Pareto Exponents in Economic Models Driven by Markov Multiplicative Processes," Econometrica, Econometric Society, vol. 90(4), pages 1811-1833, July.
    46. Stephen J. Turnovsky, 2000. "Methods of Macroeconomic Dynamics, 2nd Edition," MIT Press Books, The MIT Press, edition 2, volume 1, number 0262201232, December.
    47. Tor Jakob Klette & Samuel Kortum, 2004. "Innovating Firms and Aggregate Innovation," Journal of Political Economy, University of Chicago Press, vol. 112(5), pages 986-1018, October.
    48. Alessandro Citanna, 2007. "The dynamics of wealth distribution with asymmetric incentives and endogenous matching," Post-Print halshs-00170561, HAL.
    49. Shenghao Zhu, 2019. "A Becker–Tomes model with investment risk," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 67(4), pages 951-981, June.
    50. Joonbae Lee & Hanna Wang, 2021. "Ranking and Search Effort in Matching," Working Papers 1242, Barcelona School of Economics.
    51. Charles I. Jones, 2015. "Pareto and Piketty: The Macroeconomics of Top Income and Wealth Inequality," Journal of Economic Perspectives, American Economic Association, vol. 29(1), pages 29-46, Winter.
    52. Eric Smith & Duncan Foley & Benjamin Good, 2013. "Unhedgeable shocks and statistical economic equilibrium," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 52(1), pages 187-235, January.
    53. Menahem E. Yaari, 1965. "Uncertain Lifetime, Life Insurance, and the Theory of the Consumer," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 32(2), pages 137-150.
    54. Klaus Wälde, 2005. "Endogenous Growth Cycles," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 46(3), pages 867-894, August.
    55. Gross, Till & Klein, Paul, 2022. "Optimal tax policy and endogenous growth through innovation," Journal of Public Economics, Elsevier, vol. 209(C).
    56. Alessandro Citanna, 2007. "The dynamics of wealth distribution with asymmetric incentives and endogenous matching," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 33(2), pages 243-261, November.
    57. Reed, William J., 2003. "The Pareto law of incomes—an explanation and an extension," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 319(C), pages 469-486.
    58. Cox, John C. & Ross, Stephen A., 1976. "The valuation of options for alternative stochastic processes," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 145-166.
    59. Toda, Alexis Akira, 2014. "Incomplete market dynamics and cross-sectional distributions," Journal of Economic Theory, Elsevier, vol. 154(C), pages 310-348.
    60. Aoki, Shuhei & Nirei, Makoto, 2013. "Pareto Distributions and the Evolution of Top Incomes in the U.S," MPRA Paper 47967, University Library of Munich, Germany.
    61. 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.
    62. 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).
    63. Hyun Park, 2009. "Ramsey fiscal policy and endogenous growth," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 39(3), pages 377-398, June.
    64. Ben-Gad, Michael, 2003. "Fiscal policy and indeterminacy in models of endogenous growth," Journal of Economic Theory, Elsevier, vol. 108(2), pages 322-344, February.
    65. Kaymak, Barış & Poschke, Markus, 2016. "The evolution of wealth inequality over half a century: The role of taxes, transfers and technology," Journal of Monetary Economics, Elsevier, vol. 77(C), pages 1-25.
    66. Masanao Aoki, 1995. "Economic Fluctuations With Interactive Agents: Dynamic And Stochastic Externalities," The Japanese Economic Review, Japanese Economic Association, vol. 46(2), pages 148-165, June.
    67. Galo Nuno & Benjamin Moll, 2018. "Social Optima in Economies with Heterogeneous Agents," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 28, pages 150-180, April.
    68. Sebastian Di Tella, 2017. "Uncertainty Shocks and Balance Sheet Recessions," Journal of Political Economy, University of Chicago Press, vol. 125(6), pages 2038-2081.
    69. Lucas, Robert Jr., 1988. "On the mechanics of economic development," Journal of Monetary Economics, Elsevier, vol. 22(1), pages 3-42, July.
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    1. Matthias Birkner & Niklas Scheuer & Klaus Wälde, 2023. "The dynamics of Pareto distributed wealth in a small open economy," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 76(2), pages 607-644, August.

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    More about this item

    Keywords

    analytical dynamics of mean and distribution; age and wealth; government budget; stochastic differential equation; Kolmogorov backward equation;
    All these keywords.

    JEL classification:

    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis
    • D31 - Microeconomics - - Distribution - - - Personal Income and Wealth Distribution
    • E21 - Macroeconomics and Monetary Economics - - Consumption, Saving, Production, Employment, and Investment - - - Consumption; Saving; Wealth

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