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New evidence for the power-law distribution of wealth
PHYSiCA ELSEVIER
Physica A 242 (1997) 90-94
New evidence for the power-law distribution of wealth Moshe Levy*, Sorin Solomon Department of Theoretical Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Received I I March 1997
Abstract
We present a non-conventional approach for studying the distribution of wealth in society. We analyze data from the 1996 Forbes 400 list of the richest people in the US. Our results confirm that wealth is distributed according to a power law. The measured exponent of the power-law is 1.36. As theoretically predicted, this value is in close agreement with the exponent of the Lrvy distribution of stock market fluctuations.
A century ago the economist Vilfredo Pareto has discovered the first power-law. Pareto has discovered that the distribution of incomes obeys a power-law [1]. Namely, the probability density function describing the distribution of incomes is of the form: P ( I ) = C I -(l+'~) ,
where I is the income, the coefficient a is known as the Pareto exponent and C is a normalization constant. It was later found that wealth is also distributed according to a power-law (e.g., see [2]). The power-law distribution of wealth has important implications as to the degree of inequality in the society [3], and as to the distribution of stock market fluctuations. Mantegna and Stanley [4] have shown that the fluctuations in the price of the S&P index are distributed according to a (truncated) L r v y distribution. In [5] it is argued that the exponent o f the Lrvy distribution o f fluctuations should be identical to the exponent ~ of the power-law wealth distribution. The two main methods for measuring the distribution o f wealth are: (I) direct measurement o f the probability density function and (If) measurement of the cumulative
* Corresponding author. 0378-4371/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved. PII S0378-437 1(97)002 17-3
91
M. Levy, S. Solomon/Physica A 242 (1997) 90-94
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Fig. 1. Direct measurement of the probability density function. This measurement was done in UK. The solid line represents the empirical data. A power-law distribution should appear linear on a log-log scale, with slope -(1 + cQ. The dashed line is a power-law fit to the data. One can see that the empirical distribution is fitted rather well by the power-law distribution, however, since the number of wealth ranges is small this measurement is not very definitive (Source: National Income and Expenditure, 1970).
distribution. Method (I) is performed by dividing the wealth scale into ranges, and counting for each range the number of persons whose wealth is within that range (see Fig. 1). The problem with this method is that for practical reasons the number of ranges is small and thus one does not get a clear picture of the distribution. In method (II) one selects a few wealth levels, and for each wealth level one counts the number o f people with wealth exceeding that level (see Fig. 2). This method is more accurate and definitive than (I). A drawback of this method is that it is indirect in the sense that one does not measure the probability distribution but rather the integral o f this function, and thus there is to some extent an averaging effect. Both methods (I) and (I|) have the disadvantage o f giving a relatively poor description of the distribution at extremely high wealth levels since the number of persons at these levels is very small, and therefore the statistics are not good. The point is that the exact distribution at the extremely high wealth levels is very important, since a large percentage o f the total wealth is concentrated at these levels. For example, in the US the top 1% o f the population holds over 40% of the wealth [6]. We have employed a different method for describing the wealth distribution at very high wealth levels. We rank the 400 richest people in the US according to their wealth,
92
M. Levy, S. Solomon/Physica A 242 (1997) 90-94
100 CO t-"-
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10 e,-
tO
"5 Q. 0
1 O0
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w (thousands of Crowns)
Fig. 2. Measurement of the percentage of the population with wealth exceeding different wealth levels. This measurement was done in Sweden. A power-law distribution of wealth yields a straight line on a log-log scale, with slope -~. The empirical data is represented by dots, the solid line is the power-law fit. The empirical distribution is in excellent agreement with the power-law fit (Source: Steindl, 1965).
and we analyze wealth as a function o f rank. Our data source is the 1996 Forbes 400 list. For a power-law distribution o f wealth with exponent ~ the expected relation between rank and wealth is: W = A n -I''~ ,
where W is the wealth, A is a constant n is the rank (i.e. for the person ranked 200 in the wealthiest people list n = 200). (For derivation o f this relation see, e.g. [7]). Fig. 3 shows our results. The dots represent the empirical data. The solid line is a power-law fit with slope - 0 . 7 3 5 , which corresponds to a value o f c~ = 1.36. The agreement o f the empirical data with a power-law is very good. Notice that most o f the deviations from the theoretical fit occur at round values o f wealth, especially at $2 Billion and $1 Billion. This is probably due to the rounding-off o f estimated wealth values. The measured value o f c~ is in very good agreement with the exponent o f the Ldvy stock market fluctuation distribution measured by Mantegna and Stanley, which is 1.40. This supports the conjecture brought forth in [5] regarding the connection between the distribution o f wealth and the distribution o f stock market fluctuations.
93
M. Levy, S. Solomon/Physica A 242 (1997) 90-94 I
-6 "a "6 0 . B
i
1000
I O0 rank
Fig. 3. A power-law distribution of wealth implies a relation between the rank of a person in the wealth hierarchy and his wealth. For a power-law distributionwith exponent ct the expected relation is W ~ An -1/~, where W is the wealth, A is a constant n is the rank (i.e. for the person ranked 200 in the wealthiest people list n = 200). The figure also shows the wealth of the richest people in the US as a function of their rank. The solid line is a power-law fit with slope -0.735, which corresponds to ~ = 1.36 (Source: Forbes 400 list, April 1996).
Acknowledgement The authors would like to thank Dietrich Stauffer for his ideas and his help.
Note added in Proof In [5] we proposed an explanation to the fact that the stock price fluctuations do not follow a guassian (normal) distribution but rather a wider Levy distribution Lu of index/~ = 1.4. We claimed that this is related to the Pareto law in the investors wealth. This was based on the assertion (supported by data) that each investor induces random price variations proportional to his/her current wealth. Mathematically, this is a particular case of the well known result by P. Levy: a Levy fluctuations distribution Lu of index p arises from a random walk with random step sizes l distributed according the Pareto power law P u ( I ) ~ 1 - 1 - ~ . The phenomenological data confirmed our claim: the Pareto exponent /~ = 1.4 of the individual wealth distribution Pu equals the index of the Levy distribution L~ characterising of the stock price fluctuations.
94
M. Levy, S. Solomon/Physica A 242 (1997) 90-94
It is surprising and amusing that such a relation providing such highly nontrivial information and obeyed to such a high precision by the data was not noticed until [5]. In a recent paper ("Price variations in a stock market with many agents", to appear in Physica A"), P. Bak, M. Paczuski and M. Shubik (BPS) propose a different mechanism to explain the widening of the distribution of stock market price fluctuations: they attribute the departure from the gaussian (normal) distribution to correlations between the investments of large sets of individual investors (of equal wealth). According to the analysis above, the mechanism proposed by BPS, implies a relation between the fluctuations distribution L~ and the distribution of the sizes of the sets of correlated investors. No such relation has been actually observed. We conclude that the data support the mechanism proposed in [5] relating stock fluctuations to individual investors wealth distribution.
References [1] V. Pareto, Cours d'Economique Politique, vol. 2, Macmillan, London, 1897. [2] A.B. Atkinson, A.J. Harrison, Distribution of Total Wealth in Britain, Cambridge University Press, Cambridge, 1978. [3] D.J. Slottje, The Structure of Earnings and the Measurement of Income Inequality in the U.S., Elsevier, New York, 1989. [4] R.N. Mantegna, H.E. Stanley, The scaling behavior of an economic index, Nature 376 (1995) 46. [5] M. Levy, S. Solomon, Power laws are logarithmic Boltzmann laws, Int. J. Modern Phys. C 7 (1996) 595. [6] E.N. Wolff, The American Prospect 22 (1995) 58. [7] H. Takayasu, Fractals in the Physical Sciences, Wiley, New York, 1990.
Physica A 242 (1997) 90-94
New evidence for the power-law distribution of wealth Moshe Levy*, Sorin Solomon Department of Theoretical Physics, The Hebrew University of Jerusalem, Jerusalem 91904, Israel
Received I I March 1997
Abstract
We present a non-conventional approach for studying the distribution of wealth in society. We analyze data from the 1996 Forbes 400 list of the richest people in the US. Our results confirm that wealth is distributed according to a power law. The measured exponent of the power-law is 1.36. As theoretically predicted, this value is in close agreement with the exponent of the Lrvy distribution of stock market fluctuations.
A century ago the economist Vilfredo Pareto has discovered the first power-law. Pareto has discovered that the distribution of incomes obeys a power-law [1]. Namely, the probability density function describing the distribution of incomes is of the form: P ( I ) = C I -(l+'~) ,
where I is the income, the coefficient a is known as the Pareto exponent and C is a normalization constant. It was later found that wealth is also distributed according to a power-law (e.g., see [2]). The power-law distribution of wealth has important implications as to the degree of inequality in the society [3], and as to the distribution of stock market fluctuations. Mantegna and Stanley [4] have shown that the fluctuations in the price of the S&P index are distributed according to a (truncated) L r v y distribution. In [5] it is argued that the exponent o f the Lrvy distribution o f fluctuations should be identical to the exponent ~ of the power-law wealth distribution. The two main methods for measuring the distribution o f wealth are: (I) direct measurement o f the probability density function and (If) measurement of the cumulative
* Corresponding author. 0378-4371/97/$17.00 Copyright @ 1997 Elsevier Science B.V. All rights reserved. PII S0378-437 1(97)002 17-3
91
M. Levy, S. Solomon/Physica A 242 (1997) 90-94
10000.0
1000.0
tO
.m
100.0
0..
"5 10.0
\
E t-
1.0
-( 0.1
I
,
1000
,
,
\
I
10000 wealth (pounds)
100000
Fig. 1. Direct measurement of the probability density function. This measurement was done in UK. The solid line represents the empirical data. A power-law distribution should appear linear on a log-log scale, with slope -(1 + cQ. The dashed line is a power-law fit to the data. One can see that the empirical distribution is fitted rather well by the power-law distribution, however, since the number of wealth ranges is small this measurement is not very definitive (Source: National Income and Expenditure, 1970).
distribution. Method (I) is performed by dividing the wealth scale into ranges, and counting for each range the number of persons whose wealth is within that range (see Fig. 1). The problem with this method is that for practical reasons the number of ranges is small and thus one does not get a clear picture of the distribution. In method (II) one selects a few wealth levels, and for each wealth level one counts the number o f people with wealth exceeding that level (see Fig. 2). This method is more accurate and definitive than (I). A drawback of this method is that it is indirect in the sense that one does not measure the probability distribution but rather the integral o f this function, and thus there is to some extent an averaging effect. Both methods (I) and (I|) have the disadvantage o f giving a relatively poor description of the distribution at extremely high wealth levels since the number of persons at these levels is very small, and therefore the statistics are not good. The point is that the exact distribution at the extremely high wealth levels is very important, since a large percentage o f the total wealth is concentrated at these levels. For example, in the US the top 1% o f the population holds over 40% of the wealth [6]. We have employed a different method for describing the wealth distribution at very high wealth levels. We rank the 400 richest people in the US according to their wealth,
92
M. Levy, S. Solomon/Physica A 242 (1997) 90-94
100 CO t-"-
:5 0 X 0,)
10 e,-
tO
"5 Q. 0
1 O0
1000
w (thousands of Crowns)
Fig. 2. Measurement of the percentage of the population with wealth exceeding different wealth levels. This measurement was done in Sweden. A power-law distribution of wealth yields a straight line on a log-log scale, with slope -~. The empirical data is represented by dots, the solid line is the power-law fit. The empirical distribution is in excellent agreement with the power-law fit (Source: Steindl, 1965).
and we analyze wealth as a function o f rank. Our data source is the 1996 Forbes 400 list. For a power-law distribution o f wealth with exponent ~ the expected relation between rank and wealth is: W = A n -I''~ ,
where W is the wealth, A is a constant n is the rank (i.e. for the person ranked 200 in the wealthiest people list n = 200). (For derivation o f this relation see, e.g. [7]). Fig. 3 shows our results. The dots represent the empirical data. The solid line is a power-law fit with slope - 0 . 7 3 5 , which corresponds to a value o f c~ = 1.36. The agreement o f the empirical data with a power-law is very good. Notice that most o f the deviations from the theoretical fit occur at round values o f wealth, especially at $2 Billion and $1 Billion. This is probably due to the rounding-off o f estimated wealth values. The measured value o f c~ is in very good agreement with the exponent o f the Ldvy stock market fluctuation distribution measured by Mantegna and Stanley, which is 1.40. This supports the conjecture brought forth in [5] regarding the connection between the distribution o f wealth and the distribution o f stock market fluctuations.
93
M. Levy, S. Solomon/Physica A 242 (1997) 90-94 I
-6 "a "6 0 . B
i
1000
I O0 rank
Fig. 3. A power-law distribution of wealth implies a relation between the rank of a person in the wealth hierarchy and his wealth. For a power-law distributionwith exponent ct the expected relation is W ~ An -1/~, where W is the wealth, A is a constant n is the rank (i.e. for the person ranked 200 in the wealthiest people list n = 200). The figure also shows the wealth of the richest people in the US as a function of their rank. The solid line is a power-law fit with slope -0.735, which corresponds to ~ = 1.36 (Source: Forbes 400 list, April 1996).
Acknowledgement The authors would like to thank Dietrich Stauffer for his ideas and his help.
Note added in Proof In [5] we proposed an explanation to the fact that the stock price fluctuations do not follow a guassian (normal) distribution but rather a wider Levy distribution Lu of index/~ = 1.4. We claimed that this is related to the Pareto law in the investors wealth. This was based on the assertion (supported by data) that each investor induces random price variations proportional to his/her current wealth. Mathematically, this is a particular case of the well known result by P. Levy: a Levy fluctuations distribution Lu of index p arises from a random walk with random step sizes l distributed according the Pareto power law P u ( I ) ~ 1 - 1 - ~ . The phenomenological data confirmed our claim: the Pareto exponent /~ = 1.4 of the individual wealth distribution Pu equals the index of the Levy distribution L~ characterising of the stock price fluctuations.
94
M. Levy, S. Solomon/Physica A 242 (1997) 90-94
It is surprising and amusing that such a relation providing such highly nontrivial information and obeyed to such a high precision by the data was not noticed until [5]. In a recent paper ("Price variations in a stock market with many agents", to appear in Physica A"), P. Bak, M. Paczuski and M. Shubik (BPS) propose a different mechanism to explain the widening of the distribution of stock market price fluctuations: they attribute the departure from the gaussian (normal) distribution to correlations between the investments of large sets of individual investors (of equal wealth). According to the analysis above, the mechanism proposed by BPS, implies a relation between the fluctuations distribution L~ and the distribution of the sizes of the sets of correlated investors. No such relation has been actually observed. We conclude that the data support the mechanism proposed in [5] relating stock fluctuations to individual investors wealth distribution.
References [1] V. Pareto, Cours d'Economique Politique, vol. 2, Macmillan, London, 1897. [2] A.B. Atkinson, A.J. Harrison, Distribution of Total Wealth in Britain, Cambridge University Press, Cambridge, 1978. [3] D.J. Slottje, The Structure of Earnings and the Measurement of Income Inequality in the U.S., Elsevier, New York, 1989. [4] R.N. Mantegna, H.E. Stanley, The scaling behavior of an economic index, Nature 376 (1995) 46. [5] M. Levy, S. Solomon, Power laws are logarithmic Boltzmann laws, Int. J. Modern Phys. C 7 (1996) 595. [6] E.N. Wolff, The American Prospect 22 (1995) 58. [7] H. Takayasu, Fractals in the Physical Sciences, Wiley, New York, 1990.