Asymptotic analysis of the model for distribution of high-tax payers. (English) Zbl 1151.91660

Yamamoto, Hiroshi; Ohtsuki, Toshiya; Fujihara, Akihiro; Tanimoto, Satoshi; Yamamoto, Keizo; Miyazima, Sasuke

Japan J. Ind. Appl. Math. 24, No. 2, 211-218 (2007).

Summary: The \(z\)-transform technique is used to investigate the model for distribution of high-tax payers, which is proposed by two of the authors and others. Our analysis shows an asymptotic power-law of this model with the exponent \(-5/2\) when a total “mass” has a certain critical value. Below the critical value, the system exhibits an ordinary critical behavior, and scaling relations hold. Above the threshold, numerical simulations show that a power-law distribution coexists with a huge “monopolized” member. It is argued that these behaviors are observed universally in conserved aggregation processes, by analizing an extended model.

MSC:

91B64	Macroeconomic theory (monetary models, models of taxation)
62P20	Applications of statistics to economics
62E20	Asymptotic distribution theory in statistics
91B82	Statistical methods; economic indices and measures

Keywords:

power-law distribution; high income model; asymptotic analysis

References:

[1]	G.D. Champernowne, A model of income distribution. Economic Journal,63 (1953), 318–351. · doi:10.2307/2227127
[2]	P.L. Krapivsky and S. Render, Transitional aggregation kinetics in dry and damp environments. Phys. Rev. E,54 (1996), 3553–3561. · doi:10.1103/PhysRevE.54.3553
[3]	M.H. Lee, On the validity of the coagulation equation and the nature of Runaway Growth. Icarus,143 (2000), 74–86. · doi:10.1006/icar.1999.6239
[4]	F. Leyvraz, Scaling theory and exactly solved models in the kinetics of irreversible aggregation. Phys. Rep.,383 (2003), 95–212. · doi:10.1016/S0370-1573(03)00241-2
[5]	S.N. Majumdar, S. Krishnamurthy, and M. Barma, Nonequilibrium phase transitions in models of aggregation, adsorption, and dissociation. Phys. Rev. Lett.,81 (1998), 3691–3694. · doi:10.1103/PhysRevLett.81.3691
[6]	S. Miyazima, Y. Lee, T. Nagamine and H. Miyajima, Power-law distribution of family names in Japanese societies. Physica A,278 (2000), 282–288. · doi:10.1016/S0378-4371(99)00546-4
[7]	M.E.J. Newman, Power laws, Pareto distributions and Zipf’s law. Contemporary Physics,46 (2005), 323–351. · doi:10.1080/00107510500052444
[8]	V. Pareto, Le Cours d’Economie Politique. Macmillan, London, 1897.
[9]	J. Schmelzer, G. Röpke and R. Mahnke, Aggregation Phenomena in Complex Systems. Wiley-Vch, 1999. · Zbl 0989.00003
[10]	A.C. Silva and V.M. Yakovenko, Temporal evolution of the ’thermal’ and ’superthermal’ income classes in the USA during 1983–2001. Europhysics Letters,69 (2005), 304–310. · doi:10.1209/epl/i2004-10330-3
[11]	H. Takayasu, I. Nishikawa and H. Tasaki, Power-law mass distribution of aggregation systems with injection. Phys. Rev. A,37 (1988), 3110–3117. · doi:10.1103/PhysRevA.37.3110
[12]	K. Yamamoto, S. Miyazima, R.K. Koshal, M. Koshal and Y. Yamada, A model for distribution of high-tax payers. Japan J. Indust. Appl. Math.,20 (2003), 147–154. · Zbl 1074.91588 · doi:10.1007/BF03170424
[13]	K. Yamamoto, S. Miyazima, R.K. Koshal, Y. Yamada, A model for distribution of high-tax payers. Physica A,314 (2002), 743–748. · Zbl 1001.91086 · doi:10.1016/S0378-4371(02)01171-8
[14]	K. Yamamoto, S. Miyazima, H. Yamamoto, T. Ohtsuki and A. Fujihara, The power-law exponent and the competition rule of the high income model. Practical Fruits of Econophysics, Springer, 2006, 349–353. · Zbl 1154.91584
[15]	G.K. Zipf, Selected studies of the principle of relative frequency in language. Harvard University Press, Cambridge, MA, 1932.