Anirban Chakraborti

1

A Physicists attempt to model wealth
distributions in Economic systems

• Anirban Chakraborti
• Brookhaven National Laboratory,
• Department of Physics,
• Upton, New York 11973, USA.

SINP, Kolkata, 2005.
2
Acknowledgements

• Collaborators
• Rui Carvalho, UCL, London, UK
• Bikas K. Chakrabarti, SINP, Kolkata, India
• Giulia Iori, CU, London, UK
• Kimmo Kaski, LCE, HUT, Finland
• Marco 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.

3
Econophysics of Wealth distributions
Conference title
4
What is Econophysics?
5
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.

6
What is Economics?
Economics deals with the real life around us
market place, environment, family life!
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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.

8
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 properties of a
macroscopic system from the microscopic dynamics
of the system
9
What is Econophysics?
EconomicsPhysicsEconophysics

• It is the interdisciplinary field consisting of
  various
• conceptual approaches (originating from the
  physical
• sciences) 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.

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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.
11
What is Wealth?
12
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.

13
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.

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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.

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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.

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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.

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Anthropological view of Wealth

• Wealth as control of arable land
• Protection 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.

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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.

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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.

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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).

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Distribution of Wealth

• The distribution of wealth
• From 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.

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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).

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Gini Coefficient

• 2004 Gini coefficients in selected countries
  (UNHDR 2004)
• Denmark 0.247
• Japan 0.249
• Sweden 0.250
• Germany 0.283
• India 0.325
• France 0.327
• Australia 0.354
• UK 0.360
• USA 0.408
• China 0.447
• Russia 0.456
• Mexico 0.546
• Chile 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.
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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.

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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.

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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."

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Paretos law
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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 individuals choices.

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Vilfredo Pareto

• He introduced the concept of Pareto efficiency
  and helped develop the field of microeconomics
  with ideas such as indifference curves. Paretos
  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.

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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.

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Pareto distribution Zipfs 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.

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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 (11/b). This also implies that any
  process generating an exact Zipf rank
  distribution must have a strictly power-law
  probability density function.

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What is Money?
34
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.

35
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.

36
Characteristics of Money

• 3. Store of value
• When 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.

37
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.

38
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.

39
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).

40
Money

• Representative money
• An 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.

41
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.

42
Money

• Fiat money
• An 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.

43
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.

44
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.

45
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.

46
Money distribution models
47
Money exchange model

• We consider a closed economy of N agents and
  total money X.
• Each agent j has money x .

j
48
Money exchange model
49
Money exchange model
50
Gibbs distribution
51
Money exchange model
52
Saving propensity data
UK data 1963-2002
53
Gamma distribution
The equilibrium distribution is a Gamma
distribution
where x is the average x,
The normalization constant is
54
Gamma distribution parameter
55
Gamma distribution Normalization constant
56
Variation of Mode
57
Maxwell-Boltzmann distribution
There is an obvious connection with the
And this will be dealt in details by Marco
Patriarca on Friday.
58
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.

59
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
60
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).

61
Maxwell-Boltzmann distribution
62
Income distributions
63
Income distribution
Poverty, Inequality and the Distribution of
Income in the G20 Xavier Sala-i-Martin and Sanket
Mohapatra, Discussion Paper 0203-10 Department
of Economics, Columbia University, New York, NY
10027 November 2002
64
Income distribution
65
Income distribution
66
Income distribution
67
Income distribution
68
Income distribution
69
Income distribution
70
Evolution of Income distribution
71
Poverty rates
72
Gini coefficient estimates
73
Variant money exchange models
74
Constant savings model
75
Constant savings model
76
Minimum money exchange model
This leads to a complete instability, with all
the money going to 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
situation is economically undesirable!
77
Commodity model
78
Commodity model unfrustrated
gq /ltqgt and glt1
0
79
Commodity model frustrated
gq /ltqgt and ggt1
0
80
Role of money
Minimum amount of money is required for the
smooth functioning of the Economy!
Money distribution for agents follow Gibbs
distribution in all cases
81
Future directions Influence of networks
82
Erdös-Rényi model for Random networks
Pál Erdös (1913-1996)
83
Random Gradient networks
We assign a utility function (between 0 1) to
each agent and an agent trades with another
agent which has maximum utility and stops trading
with all other agents!
Resultant Gradient network
Random substrate network
84
Random Gradient networks
Resultant Gradient network
Random substrate network
85
Gradient network
86
Bala-Goyal networks star
87
Bala-Goyal networks wheel
88
Bala-Goyal networks wheel

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

89
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