Physica A 387 (2008) 3847–3851www.elsevier.com/locate/physa

Double power laws in income and wealth distributions

Ricardo Coelho∗, Peter Richmond, Joseph Barry, Stefan Hutzler

School of Physics, Trinity College Dublin, Dublin 2, Ireland

Available online 15 January 2008

Abstract

Close examination of wealth distributions reveal the existence of two distinct power law regimes. The Pareto exponents of thesuper-rich, identified, for example in rich lists such as provided by Forbes, are smaller than the Pareto exponents obtained for topearners in income data sets. Our extension of the Slanina model of wealth is able to reproduce these double power law features.c© 2008 Elsevier B.V. All rights reserved.

PACS: 89.65.Gh

Keywords: Econophysics; Wealth distribution; Power laws

1. Introduction

The first person to study the topic of wealth distributions in a quantitative manner, Pareto, was trained as anengineer [1]. In recent years, it is the physics community that has made significant contributions to the topic, againby focusing not only on theoretical methodologies [2–6] but also making comparisons of their results with empiricaldata [7–11]. For a recent detailed review of the subject see Ref. [12].

What does seem clear from the mounting evidence is that income and wealth distributions across societieseverywhere follow a robust pattern and that the application of ideas from statistical physics can provide understandingthat complements the observed data. The distribution rises from very low values for low income earners to a peakbefore falling away again at high incomes. For very high incomes it takes the form of a power law as first noted byPareto. The distribution is certainly not uniform. Many people are poor and few are rich.

The cumulative probability, P(>m) corresponds to the probability of finding earners that have an income biggeror equal to a certain amount of income, m. For values of m less than the average income it decreases slowly from itsmaximum value 1. For values roughly higher than the average it follows a power law

P(>m) = m−α (1)

where α is the Pareto exponent.Looking closely at results for income and wealth distributions around the world (Table 5.2 in Ref. [12]) we see

that the values for the exponents for wealth/income data sets and data that concern only the top wealthiest people in

∗ Corresponding author.E-mail address: rjcoelho@tcd.ie (R. Coelho).

0378-4371/$ - see front matter c© 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physa.2008.01.047

3848 R. Coelho et al. / Physica A 387 (2008) 3847–3851

Fig. 1. Distribution of the Pareto exponents found by different authors in the last decade. The black curve is from datasets taken from tax/incomedatabases. The grey curve is from super-rich lists, such as Forbes. The Pareto exponent for the top richest is around one, while for the “normal”rich people it is around two (data taken from Table 5.2 in Ref. [12]).

Fig. 2. Distribution of the cumulative weekly income in UK for 1995. Data for income <104.5 pounds represents income for 1995 from UKRevenue Commissioners and UK New Income Survey and a similar Pareto exponent is achieved for the high end of these curves, ∼3.2–3.3. Thedata for income >105.5 pounds represents an estimation of the income, in 1995, for the top richest in UK. In this case the Pareto exponent is lowerand around 1.3.

society differ. Fig. 1 shows the distribution of the Pareto exponents when we take these different origins of the datainto account. The average Pareto exponent is approximately 2.0 for the top earners in tax/inheritance statistics, and just1.1 for the super-rich. Using test statistics to compare the differences between both samples we reject the hypothesisof there being no difference between the mean of both populations.

We believe that the studies of wealth that are based on tax/income generally do not include the wealth of veryrich people. A further indication of two power law regimes is the study of Souma [7]. In Figure 1 of his paper [7],Souma found a Pareto exponent of 2.06 in the high end. However, we see an indication of a second power law forthe top richest (higher than 3000 million yen) which we estimate as an exponent below 1.0 based on his figure. Yet afurther indication of two power laws comes from our analysis of UK data. In Fig. 2 we show data for the cumulativedistribution of incomes in the UK for the year 1995. The upper curve is calculated from survey data and tends toa power law which was confirmed by data obtained by Cranshaw [13] from the UK Revenue Commissioners. Thelower curve is calculated using the UK New Income Survey data, which takes a 1% sample of all employees in GreatBritain. The slight shift in the two curves is due to uncertainty in a normalization factor but the power law is clearlyseen and extends from weekly incomes of just under £1000 per week up to around £30 000 per week. Over this regionthe Pareto exponent is ∼3.3. This might be assumed to be the end of the story with the power law being associatedwith Pareto’s law. However, from data published by Sunday Times [14] for the wealth of billionaires in UK for 2006,we can make an estimate of the income in 1995 generated by the wealth. In order to move from 2006 back to 1995,we made some creative estimations. First, we said that probably the wealth of the top richest group had increased in

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proportion to the stock market over the period 1995–2006. This index has roughly doubled in that time, so in 1995,the wealth is roughly 50% of the 2006 value. Then we assumed this wealth generated an income from being investedand the interest rate was around 4% per annum. This yields a second power law with Pareto exponent ∼1.26. Thissuggests what many people believe to be true, namely that the super-wealthy pay less tax as a proportion of theirincome than the majority of earners in society!

2. Wealth models

A number of models have been proposed to account for the distributions of wealth in society. One class that mightbe considered to constitute a mesoscopic approach is based on a generalized Lotka Volterra models [2,15,16]. Othermicroscopic models invoke agents that exchange money via pairwise transactions according to a specific exchangerule. The results from these latter models depend critically on the nature of the exchange process. Two quite differentexchange processes have been postulated. The first by Chakraborti and colleagues [17,18] conserves money duringthe exchange and allows savings that can be a fixed and equal percentage of the initial money held by each agent. Thisyields the Boltzmann distribution. Allowing the saving percentage to take on a random character then introduces apower law character to the distribution for high incomes. The value of the power law exponent, however, can be shownto be exactly one [4]. Only a slight variation of the exponent is achieved by attributing memory to the agents [11].

On the other hand, the model of Slanina [19] assumes a different exchange rule. It also allows creation of moneyduring each exchange process and the solution is not stationary. We must normalize the amount of money held by anagent with the mean value of money within the system at any time. In this way a stationary solution for the distributionof the normalized money can be obtained. Such a procedure must also be applied to obtain a solution from the LotkaVolterra approach and it is interesting to see that the final results for both methods yield distribution functions of thesame form. Detailed numerical comparisons with the data suggest that this form gives a good fit to the data below thesuper-rich region [12].

3. Expansion of Slanina’s model

Slanina’s model [19] involves the pairwise interaction of agents, which at every exchange process also receivesome money from outside. The time evolution of trades is represented as(

vi (t + 1)

v j (t + 1)

)=

(1 + ε − β β

β 1 + ε − β

) (vi (t)v j (t)

)(2)

where vi (t) is the wealth of agent i at time t (i = 1, . . . , N , where N is the total number of agents), β quantifiesthe percentage of wealth exchanged between agents and ε measures the quantity of wealth injected into the system atevery exchange. In the simplest case, the values of β and ε are kept constant for all trades. This results in a power lawfor the rich end at the normalized distribution of wealth, i.e. the distribution of xi (t) = vi (t)/v(t) where v(t) is themean wealth at time t (v(t) =

∑Ni=1 vi (t)/N ). An approximation of the Pareto exponent is given by Slanina [19] as:

α ∼2β

ε2 + 1 (3)

apart from some correction in the ε term. To check the accuracy of this approximation, we performed simulations for104 agents trading 103

× N times and averaged over 103 realizations. The percentage of wealth exchanged (β) wasset to 0.005 and the percentage of wealth injected into the system (ε) to 0.1. Fitting a power law to the high end of ourdistribution in Fig. 3, we find an exponent of 2.0 in excellent agreement with the value of 2.0 of Eq. (3)

Our expansion of Slanina’s model is given by making β a function of v, β(v). The main conclusion that we cantake from this wealth dynamic is that a double power law arrives from the difference between the percentage of moneythat agents put into society for trade. This difference can be related with different levels of fear to risk or from someeconomic issues related with taxation. This results in the following update rule:(

vi (t + 1)

v j (t + 1)

)=

(1 + ε − β(vi ) β(v j )

β(vi ) 1 + ε − β(v j )

) (vi (t)v j (t)

). (4)

3850 R. Coelho et al. / Physica A 387 (2008) 3847–3851

Fig. 3. Cumulative distribution of wealth in a simple Slanina model, for 104 agents trading 103× N times and averaged over 103 realizations. The

percentage of wealth exchanged (β) is set to 0.005 and the percentage of wealth injected into the system (ε) is 0.1. The Pareto exponent for thehigher end is ∼2.0.

Fig. 4. Cumulative distribution of wealth in expanded Slanina model. The values for number of agents, time steps, realizations and percentage ofwealth injected into the system (ε) are the same as used in Fig. 3. The percentage of wealth exchanged (β) if the agent has wealth smaller than10v(t) is 0.01 and if the agent has wealth higher or equal to 10v(t) is 0.00125. Two different Pareto exponents appear in different parts of thedistribution. One for what we call rich people is around ∼2.5 and a second one for the top richest is around ∼1.3. The vertical dashed line showsthe threshold that we choose for different values of β.

Here we consider the simplest case, i.e.

β(v) =

{β1, v < nv(t)β2, v ≥ nv(t)

, β1 > β2. (5)

If an agent has wealth higher than a threshold (n times the average v(t)), the second parameter (β2) will be used. Thethreshold adopted in these simulations is 10v(t), so if agent i or j have a mean wealth higher than this threshold theywill trade a different percentage than if they had a smaller amount.

To simulate a society like the UK, where two Pareto exponents exist, one for the top earners around 3.0 andanother one for the super-rich around 1.5, we have chosen the parameters β and α according to Eq. (3), i.e. β1 = 0.01,β2 = 0.00125 and ε = 0.1. Fig. 4 shows the result of our simulations. Two distinct power laws are visible, one inthe regime between v(t) and 10v(t) and another one for wealth larger than 10v(t). The Pareto exponents are 2.51and 1.29, respectively, and thus differ from the prediction of Eq. (3). However in our case, this prediction shouldonly be taken as a first order approximation, since we are essentially dealing with two societies (each specified byits respective β values) which are interacting. Agents switch between their interaction parameters according to theirrelative wealth.

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The main success of the modified Slanina model is thus the reproduction of two power laws regimes. We havesought to modify the Lotka Volterra approach in an attempt to model this double power law; however, thus far ourefforts have been unsuccessful.

An improved accuracy of the theoretical results should be achieved in future work, where we intend to find ananalytical solution for the case of two Pareto exponents in the same wealth distribution.

4. Conclusion

As was discussed in Ref. [12], progress in understanding the details of wealth distribution is invariably linkedto obtaining datasets that encompass the entire population of a country. It appears that at present, this informationis only available for a few countries, for example Japan (Souma [7]). Generally, the super-rich are not included inincome data. Published wealth lists are estimates, but for the moment might well remain the only public source forthe information on these top earners. We hope that analyses of the kind we have made in this paper encourage therelease of more detailed income data over the entire income range. Only with more complete datasets will we be ableto properly understand these complex economic systems.

Acknowledgements

R. Coelho acknowledges the support of the FCT/Portugal through the grant SFRH/BD/27801/2006. J. Barry isfunded by IRCSET-Embark (Irish Research Council for Science, Engineering and Technology). The authors alsoacknowledge the help of COST (European Cooperation in the Field of Scientific and Technical research) Action P10.

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