A Physicist’s attempt to model wealth distributions in Economic systems Anirban Chakraborti Brookhaven National Laboratory, Department of Physics, Upton,

A Physicist’s attempt to model wealth distributions in

Economic systems

Anirban Chakraborti

Brookhaven National Laboratory,

Department of Physics,

Upton, New York 11973, USA.

SINP, Kolkata, 2005.

Acknowledgements

Collaborators:

Rui Carvalho, UCL, London, UKBikas K. Chakrabarti, SINP, Kolkata, IndiaGiulia Iori, CU, London, UKKimmo Kaski, LCE, HUT, FinlandMarco Patriarca, UM, Marburg, Germany Srutarshi Pradhan, NTNU, Trondheim, Norway.

The work at Brookhaven National Laboratory was carried out under Contract No. DE-AC02-98CH10886, Division of Material Science, U. S. Department of Energy.

Econophysics of Wealth distributions

Conference title

What is Econophysics?

What is Economics?

– Economics is defined as the “Social science that analyzes and describes the consequences of choices made concerning scarce productive resources.”

– “Economics is the study of how individuals and societies choose to employ those resources: what goods and services will be produced, how they will be produced, and how they will be distributed among the members of society.”

What is Economics?

Economics deals with the real life around us: market place, environment, family life!

What is Physics?

– Physics is the “Science that deals with the structure of matter and the interactions between the fundamental constituents of the observable universe.”

– In particular, statistical mechanics is the “Branch of physics that combines the principles and procedures of statistics with the laws of both classical and quantum mechanics...and aims to predict and explain the measurable properties of macroscopic (bulk) systems on the basis of the properties and behavior of their microscopic constituents.”

What is Statistical Physics?

An assembly of methods for analyzing the physical properties of matter in bulk, in terms of the dynamical behavior of the microscopic constituents

It includes the tricks of extracting the average propertiesof a macroscopic system from the microscopic dynamics of the system

What is Econophysics?

It is the interdisciplinary field consisting of variousconceptual approaches (originating from the physicalsciences) of economic problems.

Its aim is to study the global behavior of economic systems with the help of concepts such as stochastic dynamics, disordered systems, correlation effects, self-similarity and scaling, without going into the detailed microscopic equation of the economic system.

Differences in the approaches

There are prominent differences in the methods and approaches of physicists and economists, right from defining quantities to identifying problems!

e.g. definition of metre:The metre is the length of the path travelled by light in

vacuum during a time interval of 1/299 792 458 of a second.

e.g. definition of second:The duration of 9,192,631,770 periods of the radiation corresponding

to the transition between the two hyperfine levels of the ground state of

the caesium-133 atom, at rest at a temperature of 0 K.

What is Wealth?

Definition of Wealth

Wealth usually refers to money and property. It is the abundance of objects of value and also the state of having accumulated these objects. The use of the word itself assumes some socially-accepted means of identifying objects, land, or money as "belonging to" someone, i.e. a broadly accepted notion of property and a means of protection of that property that can be invoked with minimal (or, ideally, no) effort and expense on the part of the owner. Concepts of wealth vary among societies.

Anthropological view of Wealth

A rudimentary notion of wealth

Great Apes seem to have notions of "turf" and control of food-gathering ranges, but it is questionable whether they understand this as a form of wealth. They acquire and use limited tools but these objects typically do not change, are simple to re-create, and therefore are unlikely to be seen as objects of wealth. Gorillas seem to have the capacity to recognize and protect pets and children, but this seems less an idea of wealth than of family.

Anthropological view of Wealth

The interpersonal concept of wealth

Early hominids seem to have started with incipient ideas of wealth, similar to that of the great apes. But as tools, clothing, and other mobile infrastructural capital became important to survival, ideas such as the inheritance of wealth, political positions, leadership, and ability to control group movements (to perhaps reinforce such power) emerged. Neanderthal societies had elaborate funerary rites and cave painting which implies at least a notion of shared assets that could be spent for social purposes, or preserved for social purposes. Wealth may have been collective.

Anthropological view of Wealth

Wealth as the accumulation of non-necessities

Humans back to and including the Cro-Magnons seem to have had clearly defined rulers and status hierarchies. Archaeological findings in Russia have revealed elaborate funeral clothing on a pair of children buried there over 35,000 years ago. This indicates a considerable accumulation of wealth by some individuals or families. The high artisan skill also suggest the capacity to direct specialized labor to tasks that are not of any utility to the group's survival.

Anthropological view of Wealth

Wealth as control of arable land

Irrigation and urbanization, especially in ancient Sumer and later Egypt, are thought to have triggered a shift that unified the ideas of wealth and control of land and agriculture. To feed a large stable population, it was possible and necessary to achieve universal cultivation and city-state protection. The notion of the state and the notion of war are said to have emerged at this time. Tribal cultures were formalized into what we would call feudal systems, and many rights and obligations were assumed by the monarchy and related aristocracy.

Anthropological view of Wealth

Wealth as control of arable landProtection of infrastructural capital built up over generations became critical: city walls, irrigation systems, sewage systems, aqueducts, buildings, all impossible to replace within a single generation, and thus a matter of social survival to maintain. The social capital of entire societies was often defined in terms of its relation to infrastructural capital (e.g. castles or forts or an allied monastery, cathedral or temple), and natural capital, (i.e. the land that supplied locally grown food). Agricultural economics continues these traditions in the analyses of modern agricultural policy and related ideas of wealth, e.g. the ark of taste model of agricultural wealth.

Anthropological view of Wealth

The capitalist notion of wealth

Industrialization emphasized the role of technology. Many jobs were automated. Machines replaced some workers while other workers became more specialized. Labour specialization became critical to economic success. However, physical capital, as it came to be known, consisting of both the natural capital (raw materials from nature) and the infrastructural capital (facilitating technology), became the focus of the analysis of wealth. Adam Smith saw wealth creation as the combination of materials, labour, land, and technology in such a way as to capture a profit. The theories of David Ricardo, John Locke, John Stuart Mill, and later, Karl Marx, in the 18th century and 19th century built on these views of wealth that we now call classical economics and Marxist economics.

Creation of Wealth

The creation of wealth

Wealth is created through several means.• Natural resources can be harvested and sold to

those who want them. • Material can be changed into something more

valuable through proper application of labor and equipment.

• Better methods also create additional wealth by allowing faster creation of wealth.

• Ideas create additional wealth by allowing it to be created faster or with new methods.

Limits to Wealth creation

The limits to wealth creation

There is a debate in economic literature, usually referred to as the limits to growth debate in which the ecological impact of growth and wealth creation is considered. Many of the wealth creating activities mentioned above (cutting down trees, hunting, farming) have an impact on the environment around us. Sometimes the impact is positive (for example, hunting when herd populations are high) and sometimes the impact is negative (for example, hunting when herd populations are low).

Distribution of Wealth

The distribution of wealthFrom the era of the tribal society to the modern era, all societies have had means of moderating the acquisition and use of wealth.

Wealth inequality varies drastically between countries. It may be expressed as a Gini coefficient, which measures the concentration of wealth.

Gini Coefficient

The Gini coefficient is a measure of inequality developed by the Italian statistician Corrado Gini and published in his 1912 paper "Variabilità e mutabilità".

It is usually used to measure income inequality, but can be used to measure any form of uneven distribution.

The Gini coefficient is a number between 0 and 1, where 0 corresponds with perfect equality (where everyone has the same income) and 1 corresponds with perfect inequality (where one person has all the income, and everyone else has zero income).

Gini Coefficient

2004 Gini coefficients in selected countries (UNHDR 2004)

Denmark: 0.247Japan: 0.249Sweden: 0.250Germany: 0.283India: 0.325France: 0.327Australia: 0.354UK: 0.360USA: 0.408China: 0.447Russia: 0.456Mexico: 0.546Chile: 0.571

It is an interesting fact that while the most developed European nations tend to have values between 0.24 and 0.36, the United States has been above 0.4 for several decades, showing the United States has greater inequality. This is an approach to quantify the perceived differences in welfare and compensation policies and philosophies.

Theories of Wealth distribution

Supply-side theory:

It is a form of time-deferred philanthropy. The theory is that newly created wealth eventually "trickles down" to all strata of society. The argument goes that although wealth is created primarily by the wealthy, they will tend to reinvest their wealth, and this process will create even more wealth. As the economy grows, it is said that more and more people will share in the newly created wealth.

Theories of Wealth distribution

Keynesian theory:Government redistributions and expenditures have a multiplier effect that stimulates the economy and creates wealth. Supply-siders claim that wealth is created primarily by investment (supply), whereas Keynesians claim that wealth is driven by expenditure (demand).

Today most economists agree that growth can be stimulated by either the supply or demand side, and some of them argue that these are really two sides of the same coin, in the sense that you seldom get one without the other.

Wealth Condensation

Wealth condensation is a theoretical process by which, in certain conditions, newly-created wealth tends to become concentrated in the possession of already-wealthy individuals or entities. According to this theory, those who already hold wealth have the means to invest in new sources of creating wealth or to otherwise leverage the accumulation of wealth, thus are the beneficiaries of the new wealth.

Some advocates believe the theory of wealth condensation applies to democratic countries with free market economies, which they claim exemplify the old phrase "The rich get richer and the poor get poorer."

Pareto’s law

Vilfredo Pareto

• Vilfredo Pareto (July 15, 1848 - August 19, 1923) made several important contributions to economics, sociology and moral philosophy, especially in the study of income distribution and in the analysis of an individual’s choices.

Vilfredo Pareto

• He introduced the concept of Pareto efficiency and helped develop the field of microeconomics with ideas such as indifference curves. Pareto’s social policies were put on paper in his work, Mind and Society, which is sometimes criticized as being elitist.

• He is well known for the observation that 20% of the population owned 80% of the property in Italy, later generalised (by Joseph M. Juran and others) into the Pareto principle, and generalised further to the concept of a Pareto distribution.

• Pareto's first work, Cours d'economie politique (1896-97), included this famous 'law' of income distribution.

Pareto distribution

• The Pareto distribution is a power law probability distribution found in a large number of real-world situations:

If X is a random variable with a Pareto distribution, then the probability distribution of X is characterized by the statement

where x is any number greater than xmin, which is the

(necessarily positive) minimum possible value of X, and k is a positive parameter.

Pareto distribution & Zipf’s law

• Pareto distributions are continuous probability distributions. "Zipf's law", also sometimes called the "zeta distribution", may be thought of as a discrete counterpart of the Pareto distribution.

• Originally the term Zipf's law meant the observation of the Harvard linguist George Kingsley Zipf that the frequency of use of the nth most-frequently used word in any natural language is approximately inversely proportional to n.

Pareto, Zipf & Power laws

The phrase "The r th largest city has n inhabitants" is equivalent to saying "r cities have n or more inhabitants". Whereas for Zipf, r is on the x-axis and n is on the y-axis, for Pareto, r is on the y-axis and n is on the x-axis. Simply inverting the axes, we get that if the rank exponent is b, i.e. in Zipf,

n ~ r-b,   (n = income, r = rank of person with income n)then the Pareto exponent is 1/b so that

r ~ n-1/b  ,  (n = income, r = number of people whose income is n or higher). Of course, since the power-law distribution is a direct derivative of Pareto's Law, its exponent is given by (1+1/b). This also implies that any process generating an exact Zipf rank distribution must have a strictly power-law probability density function.

What is Money?

Money

Money is an agreement within a community, to use something as a medium of exchange, which acts as an intermediary market good. It can be traded and exchanged for other goods. The agreement can either be explicit or implicit, freely chosen, or coerced.

Characteristics of Money

1. Medium of exchange

When an object is in demand primarily for its use inexchange -- for its ability to be used in trade to exchange for other things -- then it has this property.

This characteristic allows money to be a standard of deferred payment, i.e., a tool for the payment of debt.

2. Unit of account

When the value of a good is frequently used to measure or compare the value of other goods or where its value is used to denominate debts then it is functioning as a unit of account.

Characteristics of Money

3. Store of valueWhen an object is purchased primarily to store value for future trade then it is being used as a store of value. Most non-perishable goods have this quality.

Many goods or tokens have some of the characteristics outlined above. However no good or token is money unless it can satisfy all three criteria.

History of Money

Before money

Prior to the introduction of money, barter was the only way to exchange goods. Bartering has several problems, most notably timing constraints. If you wish to trade pigs for wheat, you can only do this when the pigs and wheat are both available at the same time and place - and without proper storage that may be a very brief time. With a trade standard like gold, you can sell your pigs at the "best time" and take the gold coins. You can then use that gold to buy wheat when the harvest comes in. Thus the use of money makes all commodities become more liquid.

Money

Commodity money

Precious metals have been a common form of money, such as gold.

The first instances of money were objects which were useful for their intrinsic value. This was known as commodity money and included any commonly-available commodity that has intrinsic value; historical examples include pigs, rare seashells, whale's teeth, and cattle. In medieval Iraq, bread was used as an early form of currency. In India, cows were considered as a measure.

Money

Even in the industrialised world, in the absence of other types of money, people have occasionally used commodities such as tobacco as money. This last happened on a wide scale after World War II when cigarettes became used unofficially in Europe, in parallel with other currencies, for a short time.

Fluctuations in the value of commodity money can be strongly influenced by supply and demand whether current or predicted (if a local gold mine is about to run out of ore, the relative market value of gold may go up in anticipation of a shortage).

Money

Representative moneyAn example of representative money, this 1896 note could be exchanged for five US Dollars worth of silver.

The system of commodity money in many instances evolved into a system of representative money. In this system, the material that constitutes the money itself had very little intrinsic value, but none the less such money achieves significant market value through being scarce as an artefact.

Money

Paper currency and non-precious coinage was backed by a government or bank's promise to redeem it for a given weight of precious metal, such as silver.

The dominant coins and bills used within a particular country or trade region is called a currency.

This is the origin of the term "British Pound" for instance; it was a unit of money backed by a Tower pound of sterling silver - hence the currency Pound Sterling.

For much of the nineteenth and twentieth centuries, many currencies were based on representative money through the use of the gold standard.

Money

Fiat moneyAn example of fiat money is the new, international currency, the Euro.

Fiat money refers to money that is not backed by reserves of another commodity. The money itself is given value by government fiat (Latin for "let it be done") or decree, enforcing legal tender laws, previously known as "forced tender", whereby debtors are legally relieved of the debt if they (offer to) pay it off in the government's money. By law the refusal of "legal tender" money in favor of some other form of payment is illegal, and has at times in history invoked the death penalty.

Money

Governments through history have often switched to forms of fiat money in times of need such as war, sometimes by suspending the service they provided of exchanging their money for gold, and other times by simply printing the money that they needed. When governments produce money more rapidly than economic growth, the money supply overtakes economic value. Therefore, the excess money eventually dilutes the market value of all money issued. This is called inflation.

In 1971 the US finally switched to fiat money indefinitely. At this point in time many of the economically developed countries' currencies were fixed to the US dollar (Bretton Woods Conference), and so this single step meant that much of the western world's currencies became fiat money based.

Money

Credit money

Credit money often exists in parallel with other money such as fiat money or commodity money, and from the user's point of view is indistinguishable from it. Most of the western world's money is credit money derived from national fiat money currencies.

During the Crusades in Europe, precious goods would be entrusted to the Catholic Church's Knights Templar, who effectively created a system of modern credit accounts. Over time this system grew into the credit money that we know today, where banks create money by approving loans - although the risk and reserve policies of each national central bank sets a limit on this, requiring banks to keep reserves of fiat money to back their deposits.

Credit & Money

Credit is often loosely referred to as money. However credit only satisfies items one and three of the above "Essential Characteristics of Money" criteria. Credit completely fails criteria number two. Hence to be strictly accurate credit is a money substitute and not money proper.

This distinction between money and credit causes much confusion in discussions of monetary theory. In lay terms credit and money are frequently used interchangeably. Even in economics, credit is often referred to as money. For example, bank deposits are generally included in summations of the national broad money supply. However any detailed study of monetary theory needs to recognize the proper distinction between money and credit.

Money distribution models

Money exchange model

We consider a closed economy of N agents and total money X.

Each agent j has money x .j

Money exchange model

Money exchange model

Gibb’s distribution

Money exchange model

Saving propensity data

UK data: 1963-2002

Gamma distribution

The equilibrium distribution is

a Gamma distribution

where ‹x› is the average x,

The normalization constant is

Gamma distribution parameter

Gamma distribution Normalization constant

Variation of Mode

Maxwell-Boltzmann distribution

There is an obvious connection with the

And this will be dealt in details by Marco Patriarca on Friday.

Ludwig Boltzmann

Ludwig Boltzmann (February 20, 1844 – September 5, 1906) was an Austrian physicist famous for the invention of statistical mechanics.Boltzmann was born in Vienna, Austria.Boltzmann committed suicide in 1906 by hanging himself while on holiday in Duino near Trieste in Italy. The motivation behind the suicide remains unclear, but it may have been related to his lingering resentment over the scientific establishment's rejection of his theories. Today, his formula for entropy S is famous:

where kB = 1.3806505(24) × 10−23 J K−1 is the Boltzmann constant and P is the number of possible microscopic states which give the same thermodynamical state that a system may be in.

J.C. Maxwell

James Clerk Maxwell (June 13, 1831 - November 5, 1879) was a Scottish physicist, born in Edinburgh. Maxwell developed a set of equations expressing the basic laws of electricity and magnetism as well as the Maxwell distribution in the kinetic theory of gases. Maxwell is generally regarded as the nineteenth century scientist who had the greatest influence on twentieth century physics, making contributions to the fundamental models of nature.

j

Maxwell-Boltzmann distribution

• The probability density function for the speed is thus:

• The following slide has a chart displaying the density functions for a few noble gases speeds at a temperature of 298.15K (25 degree Celsius).

Maxwell-Boltzmann distribution

Income distributions

Income distribution

Poverty, Inequality and the Distribution of Income in the G20Xavier Sala-i-Martin and Sanket Mohapatra, Discussion Paper #:0203-10Department of Economics, Columbia University, New York, NY 10027November 2002

Income distribution

Income distribution

Income distribution

Income distribution

Income distribution

Income distribution

Evolution of Income distribution

Poverty rates

Gini coefficient estimates

Variant money exchange models

Constant savings model

Constant savings model

Minimum money exchange model

This leads to a complete instability, with all the money goingto ONE person and all others left with ZERO money!

Gini coefficient = 1!This is actually a Pareto optimal state (it will be impossible to raise the well-being of anyone except the WINNER, and vice versa ) but the situationis economically undesirable!

Commodity model

Commodity model: unfrustrated

g=q /<q>and g<1

0

Commodity model: frustrated

g=q /<q>and g>1

0

Role of money

Money distribution for agents follow Gibb’s distribution in all cases

Minimum amount ofmoney is required forthe smooth functioningof the Economy!

Future directions: Influence of networks

Erdös-Rényi model for Random networks

Pál ErdösPál Erdös (1913-1996)

Connect with probability p

p=1/6 N=10

k ~ 1.5 Poisson distribution

Random & Gradient networks

Random substrate network Resultant Gradient network

We assign a utility function (between 0 & 1)to each agent and an agent trades with another agent which has maximum utilityand stops trading with all other agents!

Random & Gradient networks

Random substrate network Resultant Gradient network

Gradient network

Bala-Goyal networks: star

Bala-Goyal networks: wheel

Bala-Goyal networks: wheel

And we ask the question, what happens when we exchange money on these networks of agents?

References

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Stauffer & T. Lux, to be published in Springer (2005).25. L. Adamic, http://www.hpl.hp.com/research/idl/papers/ranking/ranking.html26. http://en.wikipedia.org/wiki/Main_Page27. Z. Toroczkai, K.E. Bassler, Nature 428, 716 (2004).28. V. Bala and S. Goyal, A Noncooperative Model of Network Formation, Econometrica 68, 1181-1229 (2000).