Kinetic exchange models are multi-agent dynamic models inspired by the

statistical physics Statistical physics is a branch of physics that evolved from a foundation of statistical mechanics, which uses methods of probability theory and statistics, and particularly the Mathematics, mathematical tools for dealing with large populations ...

energy distribution Electric power distribution is the final stage in the delivery of electric power; it carries electricity from the transmission system to individual consumers. Distribution substations connect to the transmission system and lower the transmissio ...

, which try to explain the robust and universal features of income/wealth distributions. Understanding the distributions of

income Income is the consumption and saving opportunity gained by an entity within a specified timeframe, which is generally expressed in monetary terms. Income is difficult to define conceptually and the definition may be different across fields. For ...

and

wealth Wealth is the abundance of Value (economics), valuable financial assets or property, physical possessions which can be converted into a form that can be used for financial transaction, transactions. This includes the core meaning as held in the ...

in an

economy An economy is an area of the production, distribution and trade, as well as consumption of goods and services. In general, it is defined as a social domain that emphasize the practices, discourses, and material expressions associated with the ...

has been a classic problem in

economics Economics () is the social science that studies the Production (economics), production, distribution (economics), distribution, and Consumption (economics), consumption of goods and services. Economics focuses on the behaviour and intera ...

for more than a hundred years. Today it is one of the main branches of

econophysics Econophysics is a Heterodox economics, heterodox interdisciplinary research field, applying theories and methods originally developed by physicists in order to solve problems in economics, usually those including uncertainty or stochastic processes ...

Data and basic tools

In 1897,

Vilfredo Pareto Vilfredo Federico Damaso Pareto ( , , , ; born Wilfried Fritz Pareto; 15 July 1848 – 19 August 1923) was an Italian polymath (civil engineer, sociologist, economist, political scientist, and philosopher). He made several important contribut ...

first found a universal feature in the

distribution of wealth The distribution of wealth is a comparison of the wealth of various members or groups in a society. It shows one aspect of economic inequality or economic heterogeneity. The distribution of wealth differs from the income distribution in that ...

. 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 that the tail (typically 5 to 10 percent of agents in any country) of the

distribution indeed follows a

power law In statistics, a power law is a Function (mathematics), functional relationship between two quantities, where a Relative change and difference, relative change in one quantity results in a proportional relative change in the other quantity, inde ...

. 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 In probability theory, a log-normal (or lognormal) distribution is a continuous probability distribution of a random variable whose logarithm is normally distributed. Thus, if the random variable is log-normally distributed, then has a normal ...

. Basic tools used in this type of modelling are

probabilistic Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

and

statistical Statistics (from German: ''Statistik'', "description of a state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of data. In applying statistics to a scientific, industria ...

methods mostly taken from the

kinetic theory Kinetic (Ancient Greek: κίνησις “kinesis”, movement or to move) may refer to: * Kinetic theory, describing a gas as particles in random motion * Kinetic energy, the energy of an object that it possesses due to its motion Art and ente ...

Monte Carlo simulations Monte Carlo methods, or Monte Carlo experiments, are a broad class of computational algorithms that rely on repeated random sampling to obtain numerical results. The underlying concept is to use randomness to solve problems that might be determi ...

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 In physics, statistical mechanics is a mathematical framework that applies statistical methods and probability theory to large assemblies of microscopic entities. It does not assume or postulate any natural laws, but explains the macroscopic be ...

, where many particles interact. This similarity was noted by

Meghnad Saha Meghnad Saha (6 October 1893 – 16 February 1956) was an Indian astrophysicist who developed the Saha ionization equation, used to describe chemical and physical conditions in stars. His work allowed astronomers to accurately relate the sp ...

and B. N. Srivastava in 1931 and thirty years later by

Benoit Mandelbrot Benoit B. Mandelbrot (20 November 1924 – 14 October 2010) was a Polish-born French-American mathematician and polymath with broad interests in the practical sciences, especially regarding what he labeled as "the art of roughness" of phy ...

. In 1986, an elementary version of the stochastic exchange model was first proposed by J. Angle. In the context of kinetic theory of gases, such an exchange model was first investigated by A. Dragulescu and V. Yakovenko. The main modelling effort has been put to introduce the concepts of

savings Wealth is the abundance of valuable financial assets or physical possessions which can be converted into a form that can be used for transactions. This includes the core meaning as held in the originating Old English word , which is from an I ...

, and

taxation A tax is a compulsory financial charge or some other type of levy imposed on a taxpayer (an individual or legal person, legal entity) by a governmental organization in order to fund government spending and various public expenditures (regiona ...

in the setting of an

ideal gas An ideal gas is a theoretical gas composed of many randomly moving point particles that are not subject to interparticle interactions. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is a ...

-like system. Basically, it assumes that in the short-run, an economy remains conserved in terms of income/wealth; therefore ''

law of conservation In physics, a conservation law states that a particular measurable property of an isolated physical system does not change as the system evolves over time. Exact conservation laws include conservation of energy, conservation of linear momentum, co ...

'' for income/wealth can be applied. Millions of such conservative transactions lead to a steady state distribution of money (

gamma function In mathematics, the gamma function (represented by , the capital letter gamma from the Greek alphabet) is one commonly used extension of the factorial function to complex numbers. The gamma function is defined for all complex numbers except ...

-like in the ''Chakraborti-Chakrabarti'' model with uniform savings, and a gamma-like bulk distribution ending with a

Pareto tail In statistics and business, a long tail of some distributions of numbers is the portion of the distribution having many occurrences far from the "head" or central part of the distribution. The distribution could involve popularities, random nu ...

in the ''Chatterjee-Chakrabarti-Manna'' model with distributed savings) and the distribution converges to it. The distributions derived thus have close resemblance with those found in

empirical Empirical evidence for a proposition is evidence, i.e. what supports or counters this proposition, that is constituted by or accessible to sense experience or experimental procedure. Empirical evidence is of central importance to the sciences and ...

cases of income/wealth distributions. Though this theory had been originally derived from the

entropy maximization The principle of maximum entropy states that the probability distribution which best represents the current state of knowledge about a system is the one with largest entropy, in the context of precisely stated prior data (such as a proposition ...

principle of

, it had been shown by A. S. Chakrabarti and B. K. Chakrabarti that the same could be derived from the

utility maximization Utility maximization was first developed by utilitarian philosophers Jeremy Bentham and John Stuart Mill. In microeconomics, the utility maximization problem is the problem consumers face: "How should I spend my money in order to maximize my uti ...

principle as well, following a standard exchange-model with Cobb-Douglas

utility function As a topic of economics, utility is used to model worth or value. Its usage has evolved significantly over time. The term was introduced initially as a measure of pleasure or happiness as part of the theory of utilitarianism by moral philosopher ...

. Recently it has been shown 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. The general forms have not been derived so far.

Criticisms

This class of models has attracted criticisms from many dimensions. It has been debated for long whether the distributions derived from these models are representing the income distributions or wealth distributions. The ''

'' for income/wealth has also been a subject of criticism.

References

(Oxford 2013)
* Kishore Chandra Dash, "Story of Econophysics
Cambridge Scholars Press (UK, 2019)
* Marcelo Byrro Ribeiro, ''Income Distribution Dynamics of Economic Systems: An Econophysical Approach''
Cambridge University Press (Cambridge, UK, 2020)
* Giuseppe Toscani, Parongama Sen and Soumyajyoti Biswas (Eds), "Kinetic exchange models of societies and economies
Philosophical Transactions of the Royal Society A 380: 20210170
(Special Issue, May 2022) {{Physics-footer Applied and interdisciplinary physics Distribution of wealth Schools of economic thought Statistical mechanics Interdisciplinary subfields of economics

Data and basic tools

Overview of the models

Criticisms

See also

References

Further reading