# Kinetic exchange models of markets

Kinetic exchange models are multi-agent dynamic models inspired by the statistical physics of energy distribution, which try to explain the robust and universal features of income/wealth distributions.

Understanding the distributions of income and wealth in an economy has been a classic problem in economics for more than a hundred years. Today it is one of the main branches of econophysics.

## Data and basic tools

In 1897, Vilfredo Pareto first found a universal feature in the distribution of wealth. After that, with some notable exceptions, this field had been dormant for many decades, although accurate data had been accumulated over this period. Considerable investigations with the real data during the last fifteen years (1995–2010) revealed[1] that the tail (typically 5 to 10 percent of agents in any country) of the income/wealth distribution indeed follows a power law. However, the majority of the population (i.e., the low-income population) follows a different distribution which is debated to be either Gibbs or log-normal.

Basic tools used in this type of modelling are probabilistic and statistical methods mostly taken from the kinetic theory of statistical physics. Monte Carlo simulations often come handy in solving these models.

## Overview of the models

Since the distributions of income/wealth are the results of the interaction among many heterogeneous agents, there is an analogy with statistical mechanics, where many particles interact. This similarity was noted by Meghnad Saha and B. N. Srivastava in 1931[2] and thirty years later by Benoit Mandelbrot.[3] In 1986, an elementary version of the stochastic exchange model was first proposed by J. Angle.[4]

In the context of kinetic theory of gases, such an exchange model was first investigated by A. Dragulescu and V. Yakovenko.[5][6] The main modelling effort has been put to introduce the concepts of savings,[7][8] and taxation[9] in the setting of an ideal gas-like system. Basically, it assumes that in the short-run, an economy remains conserved in terms of income/wealth; therefore law of conservation for income/wealth can be applied. Millions of such conservative transactions lead to a steady state distribution of money (gamma function-like in the Chakraborti-Chakrabarti model with uniform savings,[7] and a gamma-like bulk distribution ending with a Pareto tail[10] in the Chatterjee-Chakrabarti-Manna model with distributed savings[8]) and the distribution converges to it. The distributions derived thus have close resemblance with those found in empirical cases of income/wealth distributions.

Though this theory had been originally derived from the entropy maximization principle of statistical mechanics, it had been shown by A. S. Chakrabarti and B. K. Chakrabarti [11] that the same could be derived from the utility maximization principle as well, following a standard exchange-model with Cobb-Douglas utility function. Recently it has been shown [12] that an extension of the Cobb-Douglas utility function (in the above-mentioned Chakrabarti-Chakrabarti formulation) by adding a production savings factor leads to the desired feature of growth of the economy in conformity with some earlier phenomenologically established growth laws in the economics literature. The exact distributions produced by this class of kinetic models are known only in certain limits and extensive investigations have been made on the mathematical structures of this class of models.[13][14] The general forms have not been derived so far.

## Criticisms

This class of models has attracted criticisms from many dimensions.[15] It has been debated for long whether the distributions derived from these models are representing the income distributions or wealth distributions. The law of conservation for income/wealth has also been a subject of criticism.

## References

1. ^ Chatterjee, A.; Yarlagadda, S.; Chakrabarti, B.K. (2005). Econophysics of Wealth Distributions. Springer-Verlag (Milan).
2. ^ Saha, M.; Srivastava, B.N. (1931). A Treatise on Heat. Indian Press (Allahabad). p. 105. (the page is reproduced in Fig. 6 in Sitabhra Sinha, Bikas K Chakrabarti, Towards a physics of economics, Physics News 39(2) 33-46, April 2009)
3. ^ Mandelbrot, B.B. (1960). "The Pareto-Levy law and the distribution of income". International Economic Review. 1 (2): 79–106. doi:10.2307/2525289. JSTOR 2525289.
4. ^ Angle, J. (1986). "The surplus theory of social stratification and the size distribution of personal wealth". Social Forces. 65 (2): 293–326. doi:10.2307/2578675. JSTOR 2578675.
5. ^ Dragulescu, A.; Yakovenko, V. (2000). "The statistical mechanics of money". European Physical Journal B. 17 (4): 723–729. arXiv:cond-mat/0001432. Bibcode:2000EPJB...17..723D. doi:10.1007/s100510070114. S2CID 16158313.
6. ^ Garibaldi, U.; Scalas, E.; Viarenga, P. (2007). "Statistical equilibrium in exchange games". European Physical Journal B. 60 (2): 241–246. Bibcode:2007EPJB...60..241G. doi:10.1140/epjb/e2007-00338-5. S2CID 119517302.
7. ^ a b Chakraborti, A.; Chakrabarti, B.K. (2000). "Statistical mechanics of money: how savings propensity affects its distribution". European Physical Journal B. 17 (1): 167–170. arXiv:cond-mat/0004256. Bibcode:2000EPJB...17..167C. doi:10.1007/s100510070173. S2CID 5138071.
8. ^ a b Chatterjee, A.; Chakrabarti, B.K.; Manna, K.S.S. (2004). "Pareto law in a kinetic model of market with random saving propensity". Physica A. 335 (1–2): 155–163. arXiv:cond-mat/0301289. Bibcode:2004PhyA..335..155C. doi:10.1016/j.physa.2003.11.014. S2CID 120904131.
9. ^ Guala, S. (2009). "Taxes in a simple wealth distribution model by inelastically scattering particles". Interdisciplinary Description of Complex Systems. 7 (1): 1–7. arXiv:0807.4484. Bibcode:2008arXiv0807.4484G.
10. ^ Chakraborti, A.; Patriarca, M. (2009). "Variational Principle for the Pareto Power Law". Physical Review Letters. 103 (22): 228701. arXiv:cond-mat/0605325. Bibcode:2009PhRvL.103v8701C. doi:10.1103/PhysRevLett.103.228701. PMID 20366128. S2CID 909820.
11. ^ A. S. Chakrabarti; B. K. Chakrabarti (2009). "Microeconomics of the ideal gas like market models". Physica A. 388 (19): 4151–4158. arXiv:0905.3972. Bibcode:2009PhyA..388.4151C. doi:10.1016/j.physa.2009.06.038. S2CID 14908064.
12. ^ D. S. Quevedoa; C. Jos ́e Quimbay (2020). "Non-conservative kinetic model of wealth exchange with saving of production". European Physical Journal B. 93: 186.
13. ^ During, B.; Matthes, D.; Toscani, G. (2008). "Kinetic equations modelling wealth distributions: a comparison of approaches" (PDF). Physical Review E. 78 (5): 056103. Bibcode:2008PhRvE..78e6103D. doi:10.1103/physreve.78.056103. PMID 19113186.
14. ^ Cordier, S.; Pareschi, L.; Toscani, G. (2005). "On a kinetic model for a simple market economy". Journal of Statistical Physics. 120 (1–2): 253–277. arXiv:math/0412429. Bibcode:2005JSP...120..253C. doi:10.1007/s10955-005-5456-0. S2CID 10218909.
15. ^ Mauro Gallegati, Steve Keen, Thomas Lux and Paul Ormerod (2006). "Worrying Trends in Econophysics". Physica A. 371 (1): 1–6. Bibcode:2006PhyA..370....1G. doi:10.1016/j.physa.2006.04.029.CS1 maint: multiple names: authors list (link)