Kinetic exchange models of markets
Encyclopedia
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
.
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
that the tail (typically 5 to 10 percent of agents in any country) of the income
/wealth
distribution indeed follows a power law
. The rest (bulk) of the population (i.e., the low-income population) follow 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.
, there is an analogy with statistical mechanics
, where many particles interact. This similarity was noted long back by Meghnad Saha
and B. N. Srivastava
(see Fig. 6 in Sinha & Chakrabarti, 2009, in "Further Reading", reproducing the page in the book giving this analogy) and also by Benoit Mandelbrot
. J. Angle was the first to propose an elementary version of the stochastic exchange model. In the context of kinetic theory of gases, such an exchange model was first investigated by A. Dragulescu, V. Yakovenko. The main modelling effort has been put to introduce the concepts of savings, and taxation 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 and any monetary transaction therefore, represents a redistribution of money from one agent to another. Millions of such conservative transactions lead to a steady state distribution of money (gamma function
-like in the Chakraborti-Chakrabarti model with uniform savings [6],
and a gamma-like bulk distribution ending with a Pareto tail in the Chatterjee-Chakrabarti-Manna model with distributed savings [7]) and the economy converges to it. The distributions derived thus have close resemblance with those found in case of income/wealth distributions. Though this theory has been originally derived from the entropy maximization principle of statistical mechanics
, it has recently been shown that the same could be derived from the utility maximization principle as well, following a standard exchange-model with Cobb-Douglas
utility function.
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.
Statistical physics
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...
of energy distribution, which try to explain the robust and universal features of income/wealth distributions.
Understanding the distributions
Distribution of wealth
The distribution of wealth is a comparison of the wealth of various members or groups in a society. It differs from the distribution of income in that it looks at the distribution of ownership of the assets in a society, rather than the current income of members of that society.-Definition of...
of income
Income
Income is the consumption and savings opportunity gained by an entity within a specified time frame, which is generally expressed in monetary terms. However, for households and individuals, "income is the sum of all the wages, salaries, profits, interests payments, rents and other forms of earnings...
and wealth
Wealth
Wealth is the abundance of valuable resources or material possessions. The word wealth is derived from the old English wela, which is from an Indo-European word stem...
in an economy
Economy
An economy consists of the economic system of a country or other area; the labor, capital and land resources; and the manufacturing, trade, distribution, and consumption of goods and services of that area...
has been a classic problem in economics
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
for more than a hundred years. Today it is one of the main branches of Econophysics
Econophysics
Econophysics is an 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 and nonlinear dynamics...
.
Data and Basic tools
In 1897, Vilfredo ParetoVilfredo Pareto
Vilfredo Federico Damaso Pareto , born Wilfried Fritz Pareto, was an Italian engineer, sociologist, economist, political scientist and philosopher. He made several important contributions to economics, particularly in the study of income distribution and in the analysis of individuals' choices....
first found a universal feature in the distribution of wealth
Distribution of wealth
The distribution of wealth is a comparison of the wealth of various members or groups in a society. It differs from the distribution of income in that it looks at the distribution of ownership of the assets in a society, rather than the current income of members of that society.-Definition of...
. 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 income
Income
Income is the consumption and savings opportunity gained by an entity within a specified time frame, which is generally expressed in monetary terms. However, for households and individuals, "income is the sum of all the wages, salaries, profits, interests payments, rents and other forms of earnings...
/wealth
Wealth
Wealth is the abundance of valuable resources or material possessions. The word wealth is derived from the old English wela, which is from an Indo-European word stem...
distribution indeed follows a power law
Power law
A power law is a special kind of mathematical relationship between two quantities. When the frequency of an event varies as a power of some attribute of that event , the frequency is said to follow a power law. For instance, the number of cities having a certain population size is found to vary...
. The rest (bulk) of the population (i.e., the low-income population) follow a different distribution which is debated to be either Gibbs
Gibbs sampling
In statistics and in statistical physics, Gibbs sampling or a Gibbs sampler is an algorithm to generate a sequence of samples from the joint probability distribution of two or more random variables...
or log-normal.
Basic tools used in this type of modelling are probabilistic
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
and statistical
Statistics
Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....
methods mostly taken from the kinetic theory
Kinetic theory
The kinetic theory of gases describes a gas as a large number of small particles , all of which are in constant, random motion. The rapidly moving particles constantly collide with each other and with the walls of the container...
of statistical physics
Statistical physics
Statistical physics is the branch of physics that uses methods of probability theory and statistics, and particularly the mathematical tools for dealing with large populations and approximations, in solving physical problems. It can describe a wide variety of fields with an inherently stochastic...
. 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 agentsAgent (economics)
In economics, an agent is an actor and decision maker in a model. Typically, every agent makes decisions by solving a well or ill defined optimization/choice problem. The term agent can also be seen as equivalent to player in game theory....
, there is an analogy with statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, where many particles interact. This similarity was noted long back by Meghnad Saha
Meghnad Saha
Meghnad Saha FRS was an Indian astrophysicist best known for his development of the Saha equation, used to describe chemical and physical conditions in stars.-Early life:...
and B. N. Srivastava
(see Fig. 6 in Sinha & Chakrabarti, 2009, in "Further Reading", reproducing the page in the book giving this analogy) and also by Benoit Mandelbrot
Benoît Mandelbrot
Benoît B. Mandelbrot was a French American mathematician. Born in Poland, he moved to France with his family when he was a child...
. J. Angle was the first to propose an elementary version of the stochastic exchange model. In the context of kinetic theory of gases, such an exchange model was first investigated by A. Dragulescu, V. Yakovenko. The main modelling effort has been put to introduce the concepts of savings, and taxation in the setting of an ideal gas
Ideal gas
An ideal gas is a theoretical gas composed of a set of randomly-moving, non-interacting point particles. The ideal gas concept is useful because it obeys the ideal gas law, a simplified equation of state, and is amenable to analysis under statistical mechanics.At normal conditions such as...
-like system. Basically, it assumes that in the short-run, an economy remains conserved in terms of income/wealth and any monetary transaction therefore, represents a redistribution of money from one agent to another. Millions of such conservative transactions lead to a steady state distribution of money (gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
-like in the Chakraborti-Chakrabarti model with uniform savings [6],
and a gamma-like bulk distribution ending with a Pareto tail in the Chatterjee-Chakrabarti-Manna model with distributed savings [7]) and the economy converges to it. The distributions derived thus have close resemblance with those found in case of income/wealth distributions. Though this theory has been originally derived from the entropy maximization principle of statistical mechanics
Statistical mechanics
Statistical mechanics or statistical thermodynamicsThe terms statistical mechanics and statistical thermodynamics are used interchangeably...
, it has recently been shown that the same could be derived from the utility maximization principle as well, following a standard exchange-model with Cobb-Douglas
Cobb-Douglas
In economics, the Cobb–Douglas functional form of production functions is widely used to represent the relationship of an output to inputs. Similar functions were originally used by Knut Wicksell , while the Cobb-Douglas form was developed and tested against statistical evidence by Charles Cobb and...
utility function.
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 Law of conservation for income/wealth has also been a subject of criticism.See also
- Economic inequalityEconomic inequalityEconomic inequality comprises all disparities in the distribution of economic assets and income. The term typically refers to inequality among individuals and groups within a society, but can also refer to inequality among countries. The issue of economic inequality is related to the ideas of...
- EconophysicsEconophysicsEconophysics is an 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 and nonlinear dynamics...
- ThermoeconomicsThermoeconomicsThermoeconomics, also referred to as biophysical economics, is a school of heterodox economics that applies the laws of thermodynamics to economic theory. The term "thermoeconomics" was coined in 1962 by American engineer Myron Tribus, and developed by the statistician and economist Nicholas...
- Wealth condensation
Further reading
- Brian Hayes, Follow the money, American ScientistAmerican ScientistAmerican Scientist is the bimonthly science and technology magazine published since 1913 by Sigma Xi. Each issue includes four to five feature articles written by scientists and engineers. These authors review research in all fields of science...
, 90:400-405 (Sept.-Oct.,2002) - Jenny Hogan, There's only one rule for rich, New ScientistNew ScientistNew Scientist is a weekly non-peer-reviewed English-language international science magazine, which since 1996 has also run a website, covering recent developments in science and technology for a general audience. Founded in 1956, it is published by Reed Business Information Ltd, a subsidiary of...
, 6-7 (12 March 2005) - Peter Markowich, Applied Partial Differential Equations, Springer-Verlag (Berlin, 2007)
- Arnab Chatterjee, Bikas K ChakrabartiBikas K ChakrabartiBikas Kanta Chakrabarti is an Indian physicist.He is a professor of Physics at Saha Institute of Nuclear Physics, Kolkataand Visiting Professor of Economics, Indian Statistical Institute, Kolkata....
, Kinetic exchange models for income and wealth distribution, European Physical Journal B, 60:135-149(2007) - Victor Yakovenko, J. B. Rosser, Colloquium: statistical mechanics of money, wealth and income, Reviews of Modern PhysicsReviews of Modern PhysicsThe Reviews of Modern Physics is a journal of the American Physical Society. The journal started in paper form. All volumes are also online by subscription.Issue 1, Volume 1 consisted of the review by...
81:1703-1725 (2009) - Thomas Lux, F. Westerhoff, Economics crisis, Nature PhysicsNature PhysicsNature Physics, is a monthly, peer reviewed, scientific journal published by the Nature Publishing Group. It was first published in October 2005 . The Chief Editor is Alison Wright, who is a full-time professional editor employed by this journal...
, 5:2 (2009) - Anindya S. Chakrabarti, Bikas K ChakrabartiBikas K ChakrabartiBikas Kanta Chakrabarti is an Indian physicist.He is a professor of Physics at Saha Institute of Nuclear Physics, Kolkataand Visiting Professor of Economics, Indian Statistical Institute, Kolkata....
, Statistical theories of income and wealth distribution, Economics E-journal, 4 (2010) (http://www.economics-ejournal.org/economics/journalarticles/2010-4) - Sitabhra Sinha, Bikas K ChakrabartiBikas K ChakrabartiBikas Kanta Chakrabarti is an Indian physicist.He is a professor of Physics at Saha Institute of Nuclear Physics, Kolkataand Visiting Professor of Economics, Indian Statistical Institute, Kolkata....
, Towards a physics of economics, Physics News 39(2) 33-46 (April 2009).