Dynamics of Income Distribution — A Diffusion Analysis

Author(s)
Fariba Hashemi

ABSTRACT

The study is motivated by the observation that the distribution of income across countries varies as a function of time. It would not be unreasonable to assume that there exists a statistical equilibrium distribution of income with a certain mean and variance, towards which the ensemble of countries considered tend to converge, and there is a speed of adjustment towards this said equilibrium. In order to quantify this process, the evolution through time of income around its trend is modeled using a classic stochastic differential equation. The model describes the diffusion of shocks across space, via an income adjustment process with noise. The dynamics rely on two opposing flows: (i) a factor equalization process, and (ii) a counteracting diffusion process. It is hypothesized that these flows follow simple evolutionary laws that can be described with five parameters — parameters that can be estimated from historical data with some accuracy. The dynamic behavior of the model is analytically derived. Both the extent and speed of adjustment of income are analyzed. An empirical application of the proposed model to the evolution of the distribution of income for 25 countries in the European Union tests the validity of the proposed method and suggests that diffusion may be a preferable technique for the analysis of income dynamics.

The study is motivated by the observation that the distribution of income across countries varies as a function of time. It would not be unreasonable to assume that there exists a statistical equilibrium distribution of income with a certain mean and variance, towards which the ensemble of countries considered tend to converge, and there is a speed of adjustment towards this said equilibrium. In order to quantify this process, the evolution through time of income around its trend is modeled using a classic stochastic differential equation. The model describes the diffusion of shocks across space, via an income adjustment process with noise. The dynamics rely on two opposing flows: (i) a factor equalization process, and (ii) a counteracting diffusion process. It is hypothesized that these flows follow simple evolutionary laws that can be described with five parameters — parameters that can be estimated from historical data with some accuracy. The dynamic behavior of the model is analytically derived. Both the extent and speed of adjustment of income are analyzed. An empirical application of the proposed model to the evolution of the distribution of income for 25 countries in the European Union tests the validity of the proposed method and suggests that diffusion may be a preferable technique for the analysis of income dynamics.

Cite this paper

F. Hashemi, "Dynamics of Income Distribution — A Diffusion Analysis,"*Theoretical Economics Letters*, Vol. 1 No. 2, 2011, pp. 33-37. doi: 10.4236/tel.2011.12008.

F. Hashemi, "Dynamics of Income Distribution — A Diffusion Analysis,"

References

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[13] D. Fudenberg and D. Levine, “Learning and Equilibrium,” Annual Review of Economics, Vol. 1, 2009, pp. 385-419. doi:10.1146/annurev.economics.050708.142930

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[16] R. Taylor, “Predation, Population and Community Biol-ogy Series,” Chapmann and Hall, Kolberg, 1984.

[17] M, -O. Hongler, R. Filliger and P. Blanchard, “Soluble Models for Dynamics Driven by a Super-Diffusive Noise,” Physical A, Vol. 30, No. 2, 2006, pp. 301-315. doi:10.1016/j.physa.2006.02.036

[18] M, -O. Hongler, H. Soner and L. Streit, “Stochastic Control for a Class of Random Evolution Models,” Applied Mathematics and Optimization, Vol. 49, No. 2, 2004, pp. 113-121. doi:10.1007/BF02638147

[19] O. Besson and G. de Montmollin, “Space-Time Integrated Least Squares: a Time-Marching Approach,” International Journal for Numerical Mathods in Fluids, Vol. 44, 2004, pp. 525-543. doi:10.1002/fld.655

[20] M. Friedman, “Do Old Fallacies Ever Die?” Journal of Economic Literature, Vol. 30, No. 4, 1992, pp. 2129-2132.

[21] D. Quah, “Empirics for Economic Growth and Convergence,” The Economics of Structural Change, Elgar Reference Collection, Vol. 3, 2003, pp. 174-196.

[22] L. Kevin, H. Pesaran and R. Smith, “Growth Empirics: a Panel Data Approach a Comment,” Quarterly Journal of Economics, Vol. 113, No. 1, 1998, pp. 319-323. doi:10.1162/003355398555504

[23] P. Krugman, “The Self-Organizing Economy,” Blackwell Publishers, England, 1996.

[24] R. Axtel, “Zip Distribution of US Firm Sizes,” Science, Vol. 293, No. 5536, 2001, pp. 1818-1820. doi:10.1126/science.1062081

[25] F. Hashemi, “A Dynamic Model of Size Distribution of Firms Applied to U.S. Biotechnology and Trucking Industries,” Small Business Economics, Vol. 21, No. 1, 2003, pp. 27-36. doi:10.1023/A:1024433203253

[26] F. Hashemi, “An Evolutionary Model of the Size Distribution of Firms,” Journal of Evolutionary Economics, Vol. 10, No. 2, 2000, pp. 507-521.

[1] R. Barro, “Economic Growth in a Cross Section of Countries,” Quarterly Journal of Economics, Vol. 106, No. 2, 1991, pp. 407-43. doi:10.2307/2937943

[2] N. Mankiw, D. Romer and D. Weil, “A Contribution to the Empirics of Economic Growth,” Quarterly Journal of Economics, Vol. 107, No. 2, 1992, pp. 407-437. doi:10.2307/2118477

[3] D. Quah, “Empirics for Eco-nomic Growth and Convergence,” European Economic Review, 1996.

[4] R. Lucas, “Trade and the Diffusion of the Industrial Revolution,” American Economic Journal: Macroeconomics, Vol. 1, No. 1, 2009, pp. 1-25. doi:10.1257/mac.1.1.1

[5] O. Blanchard and L. Katz, “Regional Evolutions,” Brookings Papers on Economic Activity, No. 1, 1992, pp. 1-61.

[6] R. Barro and X. Sala-i-Martin, “Technological Discussion,” Convergence and Growth Eco-nomic Growth, Vol. 2, 1997, pp. 126.

[7] R. Gibrat, “Les Inegalite Economiques,” Sirey, Paris, 1931.

[8] V. Yakovenko and B. Rosser, “Statistical Mechanics of Money, Wealth, and Income,” Reviews of Morden Physics, Vol. 81, No. 4, 2009, pp.1703-1725.

[9] A. Toda, “Income Dynamics with a Stationary Double Pareto Distribution,” Physical Review, Vol. 83, 2011, pp. 1-4.

[10] J. Hirshleifer, “Evolutionary Models in Economics and Law: Cooperation versus Conflict Strategies,” Economic Approaches to Law series, Vol. 3, 2007, pp. 189-248.

[11] D. Levine, “Is Behavioral Economics Doomed? The Ordinary versus the Extraordinary,” Max Weber Lecture, 2009.

[12] J. Sachs and A.Warner, “Economic Re-form and the Process of Global Integration,” Brookings Papers on Economic Activity, 1995, pp. 1-95.

[13] D. Fudenberg and D. Levine, “Learning and Equilibrium,” Annual Review of Economics, Vol. 1, 2009, pp. 385-419. doi:10.1146/annurev.economics.050708.142930

[14] A.Okubo, “Diffusion and Ecological Problems: Mathematical Models Perspectives,” Springer Verlag, New York, Vol. 14, 1980.

[15] L. Ricciardi, “Diffusion Processes and Related Topics in Biomathematics,” Springer-Verlag, New York, 1977.

[16] R. Taylor, “Predation, Population and Community Biol-ogy Series,” Chapmann and Hall, Kolberg, 1984.

[17] M, -O. Hongler, R. Filliger and P. Blanchard, “Soluble Models for Dynamics Driven by a Super-Diffusive Noise,” Physical A, Vol. 30, No. 2, 2006, pp. 301-315. doi:10.1016/j.physa.2006.02.036

[18] M, -O. Hongler, H. Soner and L. Streit, “Stochastic Control for a Class of Random Evolution Models,” Applied Mathematics and Optimization, Vol. 49, No. 2, 2004, pp. 113-121. doi:10.1007/BF02638147

[19] O. Besson and G. de Montmollin, “Space-Time Integrated Least Squares: a Time-Marching Approach,” International Journal for Numerical Mathods in Fluids, Vol. 44, 2004, pp. 525-543. doi:10.1002/fld.655

[20] M. Friedman, “Do Old Fallacies Ever Die?” Journal of Economic Literature, Vol. 30, No. 4, 1992, pp. 2129-2132.

[21] D. Quah, “Empirics for Economic Growth and Convergence,” The Economics of Structural Change, Elgar Reference Collection, Vol. 3, 2003, pp. 174-196.

[22] L. Kevin, H. Pesaran and R. Smith, “Growth Empirics: a Panel Data Approach a Comment,” Quarterly Journal of Economics, Vol. 113, No. 1, 1998, pp. 319-323. doi:10.1162/003355398555504

[23] P. Krugman, “The Self-Organizing Economy,” Blackwell Publishers, England, 1996.

[24] R. Axtel, “Zip Distribution of US Firm Sizes,” Science, Vol. 293, No. 5536, 2001, pp. 1818-1820. doi:10.1126/science.1062081

[25] F. Hashemi, “A Dynamic Model of Size Distribution of Firms Applied to U.S. Biotechnology and Trucking Industries,” Small Business Economics, Vol. 21, No. 1, 2003, pp. 27-36. doi:10.1023/A:1024433203253

[26] F. Hashemi, “An Evolutionary Model of the Size Distribution of Firms,” Journal of Evolutionary Economics, Vol. 10, No. 2, 2000, pp. 507-521.