Discussion Paper

No. 2009-45 | October 16, 2009
Statistical Theories of Income and Wealth Distribution


The distributions of income and wealth in countries across the world are found to possess some robust and stable features independent of the specific economic, social and political conditions of the countries. We discuss a few physics-inspired multi-agent dynamic models along with their microeconomic counterparts, that can produce the statistical features of the distributions observed in reality. A number of exact analytical methods and solutions are also provided.

JEL Classification:

C00, C31, D51


  • Downloads: 3373


Cite As

Anindya S. Chakrabarti and Bikas K. Chakrabarti (2009). Statistical Theories of Income and Wealth Distribution. Economics Discussion Papers, No 2009-45, Kiel Institute for the World Economy. http://www.economics-ejournal.org/economics/discussionpapers/2009-45

Comments and Questions

Enrico Scalas - Related discrete models
November 05, 2009 - 14:56

I read with interest this paper reviewing several probabilistic models for the distribution of wealth. The models presented in this paper are continuous-space, discrete-time Markov chains. The classical book by Meyn and Tweedie [1] provides a nice introduction to the theory of continuous-space Markov chains.

The reader of this ...[more]

... paper might find a series of two papers on finite Markov chains of some interest, which are strictly related to the models presented here [2-4]. In [2], we analyze the prototypical model in which n coins divided into g agents are exchanged in the following way:

- two agents are selected at random, the first being the loser (selected among the agents with one coin at least), the second being the winner (selected among all the agents);

- the loser gives one coin to the winner.

This simple stochastic game is an irreducible and aperiodic finite Markov chain whose average wealth distribution (the average number of coins per agent) converges to the exponential distribution in the appropriate thermodynamic and continuous limit.

In [4], a model is considered for taxation and redistribution where n coins "jump" among n agents. This is equivalent to the Ehrenfest -Brillouin model discussed in [5], but only with unary moves. The invariant measure of the Markov chain is now the multivariate Polya distribution which has many interesting continuous limits, such as the Beta and Gamma distributions.

Coming back to the present paper, equation (6) is not clear. If $m_i + m_j$
represents the sum of two random variables, on the left-hand side one should find a convolution and not a product, the right equation being perhaps
P(m_i)P(m_j) = P(m_i,m_j)
where $P(m_i,m_j)$ is the joint probability density function for the two variables and $m_i$, $m_j$ are _independent_ random variables.
The authors should better explain what they mean here.

There is also a large body of rigorous results on similar models obtained by mathematicians. References [6-8] contain a summary for physicists which should also be accessible to economists who are not specialist on stochastic processes. In this literature, the emphasis is on the analysis of Boltzmann's like equations.

This whole body of literature has been criticized in various papers written by economists, the main problem being that there is no conservation of money in a real economy. Steve Keen also wrote a specific paper on this point [9] as a follow-up to a short paper on "worrying trends in Econophysics" [10]. Indeed, in recent times, the "inflation" of monetary masses goes even beyond what is expected from the standard theory of fractional banking [11] (see the plots in the Wikipedia article).
For what I can see, in the present version of the paper, a deep discussion of these issues is missing.

Finally, income, wealth and money are different quantities. Money is discussed above. As for income and wealth, one should always keep in mind that income is a flux, whereas wealth is a stock. Mixing them up is like mixing position and velocity in a Physics paper. In my opinion, the kinetic models dealt with in this paper can be considered as describing wealth exchanges and they are not describing income at all. In any case, also a discussion on this point is necessary when comparing empirical distributions with theoretical results.


[1] S.P. Meyn and R.L. Tweedie (1993), Markov chains and stochastic stability. Springer-Verlag.

[2] E. Scalas, U. Garibaldi, S. Donadio
Statistical equilibrium in simple exchange games I - Methods of solution and application to the Bennati-Dragulescu-Yakovenko (BDY) game
Eur. Phys. J. B 53 2 (2006) 267-272
DOI: 10.1140/epjb/e2006-00355-x

[3] E. Scalas, U. Garibaldi, S. Donadio
Erratum: Statistical equilibrium in simple exchange games I
Eur. Phys. J. B 60 2 (2007) 271-272
DOI: 10.1140/epjb/e2007-00345-6

[4] U. Garibaldi, E. Scalas, P. Viarengo
Statistical equilibrium in simple exchange games II. The redistribution game
Eur. Phys. J. B 60 2 (2007) 241-246
DOI: 10.1140/epjb/e2007-00338-5

[5] Enrico Scalas and Ubaldo Garibaldi (2009). A Dynamic Probabilistic Version of the Aoki–Yoshikawa Sectoral Productivity Model. Economics: The Open-Access, Open-Assessment E-Journal, Vol. 3, 2009-15. http://www.economics-ejournal.org/economics/journalarticles/2009-15

[6] B. Duering, G. Toscani, Hydrodynamics from kinetic models of conservative economies, Physica A: Statistical Mechanics and its Applications, 384 (2007) 493-506

[7] B. Duering, D. Matthes, G.Toscani, Kinetic Equations modelling Wealth Redistribution: A comparison of Approaches, Phys. Rev. E, 78, (2008) 056103

[8] G. Toscani, Wealth redistribution in conservative linear kinetic models with taxation, Europhysics Letters 88 (1) (2009) 10007.

[9] http://www.debtdeflation.com/blogs/wp-content/uploads/2007/09/KeenNonConservationMoney.pdf

[10] Mauro Gallegati, Steve Keen, Thomas Lux, Paul Ormerod, (2006). “Worrying trends in econophysics”, Physica A 370, pp. 1–6.

[11] http://en.wikipedia.org/wiki/Fractional-reserve_banking

Anindya S. Chakrabarti - Reply to Reader Comment
November 12, 2009 - 11:18

See attached file

Anindya S. Chakrabarti - Revised Version
November 12, 2009 - 11:22

See attached file

Anonymous - Referee Report 1
November 30, 2009 - 15:06

Generally speaking the paper is re-stating that the steady state of nature is reflected by the normal distribution. The synergy with the interraction between economic agents is though making things interesting. Although: (i) the main reference sources are the Physical journals on economics systems (e.g. the World Bank variety of ...[more]

... data is not mentioned, not to mention my paper in Journal of Economic Forecasting presented at Essex Conf in 2005) and (ii) there is little difference between persons and companies behaviour description (e.g. the saving propensity variable is not thoroughly explained); the model is providing for enough behaviourial variety to be usefull. As a sugestion: it would be nice to see behaviour near the steady state (as much as such model allows) that could describe domains of diferent behaviour.

Anonymous - Referee Report 2
December 04, 2009 - 09:12

See attached file

Anindya S. Chakrabarti - Reply to Referee Report 1 and 2
December 07, 2009 - 09:42

See attached file

Anindya S. Chakrabarti - Revised Version
December 07, 2009 - 09:43

See attached file