Hostname: page-component-848d4c4894-4hhp2 Total loading time: 0 Render date: 2024-05-29T05:48:36.245Z Has data issue: false hasContentIssue false


Published online by Cambridge University Press:  13 July 2021

Brendan K. Beare*
University of Sydney
Won-Ki Seo
University of Sydney
Alexis Akira Toda
University of California San Diego
Address correspondence to Bredan K. Beare, School of Economics, University of Sydney, Sydney, Australia; e-mail:


This article concerns the tail probabilities of a light-tailed Markov-modulated Lévy process stopped at a state-dependent Poisson rate. The tails are shown to decay exponentially at rates given by the unique positive and negative roots of the spectral abscissa of a certain matrix-valued function. We illustrate the use of our results with an application to the stationary distribution of wealth in a simple economic model in which agents with constant absolute risk aversion are subject to random mortality and income fluctuation.

© The Author(s), 2021. Published by Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)



Achdou, Y., Han, J., Lasry, J.M., Lions, P.L., & Moll, B. (2021) Income and wealth distribution in macroeconomics: A continuous-time approach. To appear in the Review of Economic Studies.CrossRefGoogle Scholar
Albornoz, F., Fanelli, S. & Hallak, J. C. (2016) Survival in export markets. Journal of International Economics 102, 262281.CrossRefGoogle Scholar
Aoki, S. & Nirei, M. (2017) Zipf’s law, Pareto’s law, and the evolution of top incomes in the United States. American Economic Journal: Macroeconomics 9, 3671.Google Scholar
Arkolakis, C. (2016) A unified theory of firm selection and growth. Quarterly Journal of Economics 131, 89155.CrossRefGoogle Scholar
Asmussen, S. (2003) Applied Probability and Queues, 2nd Edition. Springer.Google Scholar
Beare, B. K. & Seo, W. K. (2020) Representation of I(1) and I(2) autoregressive Hilbertian processes. Econometric Theory 36, 773802.CrossRefGoogle Scholar
Beare, B. K. & Toda, A. A. (2020) On the emergence of a power law in the distribution of COVID-19 cases. Physica D: Nonlinear Phenomena 412, 132649.CrossRefGoogle Scholar
Beare, B.K. & Toda, A.A. (2021) Determination of Pareto exponents in economic models driven by Markov multiplicative processes. Preprint. Scholar
Benhabib, J., Bisin, A. & Zhu, S. (2016) The distribution of wealth in the Blanchard-Yaari model. Macroeconomic Dynamics 20, 466481.CrossRefGoogle Scholar
Cao, D. & Luo, W. (2017) Persistent heterogeneous returns and top end wealth inequality. Review of Economic Dynamics 26, 301326.CrossRefGoogle Scholar
Champernowne, D. G. (1953) A model of income distribution. Economic Journal 63, 318351.CrossRefGoogle Scholar
Chen, T. & Francis, B. (1995) Optimal Sampled-Data Control Systems. Springer.CrossRefGoogle Scholar
Deutsch, E. (1975) The spectral abscissa of partitioned matrices. Journal of Mathematical Analysis and Applications 50, 6673.CrossRefGoogle Scholar
Fleming, W. H. & Soner, H. M. (2006) Controlled Markov Processes and Viscosity Solutions. Springer.Google Scholar
Franchi, M. & Paruolo, P. (2020) Cointegration in functional autoregressive processes. Econometric Theory 36, 803839.CrossRefGoogle Scholar
Gabaix, X., Lasry, J. M., Lions, P. L. & Moll, B. (2016) The dynamics of inequality. Econometrica 84, 20712111.CrossRefGoogle Scholar
Gouin-Bonenfant, É. & Toda, A.A. (2020) Pareto extrapolation: An analytical framework for studying tail inequality. Preprint. Scholar
Hartman, P. (1982) Ordinary Differential Equations, 2nd Edition. Birkhauser.Google Scholar
Horn, R.A. & Johnson, C.R. (2013) Matrix Analysis, 2nd Edition. Cambridge University Press.Google Scholar
Howland, J. S. (1971) Simple poles of operator-valued functions. Journal of Mathematical Analysis and Applications 36, 1221.CrossRefGoogle Scholar
Jones, C. I. & Kim, J. (2018) A Schumpeterian model of top income inequality. Journal of Political Economy 126, 17851826.CrossRefGoogle Scholar
Kasa, K. & Lei, X. (2018) Risk, uncertainty, and the dynamics of inequality. Journal of Monetary Economics 94, 6078.CrossRefGoogle Scholar
Lukacs, E. (1970) Characteristic Functions, 2nd Edition. Griffin.Google Scholar
Nakagawa, K. (2007) Application of Tauberian theorem to the exponential decay of the tail probability of a random variable. IEEE Transactions on Information Theory 53, 32393249.CrossRefGoogle Scholar
Norris, J. R. (1997) Markov Chains. Cambridge University Press.CrossRefGoogle Scholar
Nussbaum, R. D. (1986) Convexity and log convexity for the spectral radius. Linear Algebra and its Applications 73, 59122.CrossRefGoogle Scholar
Reed, W. J. (2001) The Pareto, Zipf and other power laws. Economics Letters 74, 1519.CrossRefGoogle Scholar
Rutherford, R. S. G. (1955) Income distributions: A new model. Econometrica 23, 277294.CrossRefGoogle Scholar
Sargan, J. D. (1957) The distribution of wealth. Econometrica 25, 568590.CrossRefGoogle Scholar
Sato, K. (1999) Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press.Google Scholar
Schumacher, J.M. (1986) Residue formulas for meromorphic matrices. In Byrnes, C.I. & Lindquist, A. (eds.), Computational and Combinatorial Methods in Systems Theory, pp. 97111. North-Holland.Google Scholar
Schumacher, J.M. (1991) System-theoretic trends in econometrics. In Antoulas, A.C. (ed.), Mathematical System Theory: The Influence of R.E. Kalman, pp. 559577. Springer.CrossRefGoogle Scholar
Simon, H. A. (1955) On a class of skew distribution functions. Biometrika 42, 425440.CrossRefGoogle Scholar
Simon, H. A. & Bonini, C. P. (1958) The size distribution of business firms. American Economic Review 48, 607617.Google Scholar
Smith, H. L. (1995) Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems. American Mathematical Society.Google Scholar
Stachurski, J. & Toda, A. A. (2019) An impossibility theorem for wealth in heterogeneous-agent models with limited heterogeneity. Journal of Economic Theory 182, 124.CrossRefGoogle Scholar
Steinberg, S. (1968) Meromorphic families of compact operators. Archive for Rational Mechanics and Analysis 31, 372379.CrossRefGoogle Scholar
Toda, A. A. (2014) Incomplete market dynamics and cross-sectional distributions. Journal of Economic Theory 154, 310348.CrossRefGoogle Scholar
Toda, A. A. (2017) Huggett economies with multiple stationary equilibria. Journal of Economic Dynamics and Control 84, 7790.CrossRefGoogle Scholar
Toda, A. A. & Walsh, K. (2015) The double power law in consumption and implications for testing Euler equations. Journal of Political Economy 123, 11771200.CrossRefGoogle Scholar
Wang, N. (2003) Caballero meets Bewley: The permanent-income hypothesis in general equilibrium. American Economic Review 93, 927936.CrossRefGoogle Scholar
Widder, D. V. (1941) The Laplace Transform. Princeton University Press.Google Scholar
Wold, H. O. A. & Whittle, P. (1957) A model explaining the Pareto distribution of wealth. Econometrica 25, 591595.CrossRefGoogle Scholar
Yaari, M. E. (1965) Uncertain lifetime, life insurance, and the theory of the consumer. Review of Economic Studies 32, 137150.CrossRefGoogle Scholar
Zhang, Z. (2013) Variational, Topological, and Partial Order Methods with Their Applications. Springer.CrossRefGoogle Scholar