Home > Wealth, inequality and employment > A tentative model of revenues distributions

A tentative model of revenues distributions


Monday 19 December 2016, by Bernard Zimmern

Though Paul Krugman, a Nobel price, could write in 1995: « We have to say that the rank-size rule is a major embarrassment for economic theory: one of the strongest statistical relationships we know, lacking any clear basis in theory », scientists have tried to explain Pareto distributions by building models.
Benoît Mandelbrot has given an understanding of the Zipf law, a Pareto with discrete distribution on frequency of words in the language, and shown why this frequency is the consequence of the theory of information from Shannon.

Others have shown that the size of cities follows a Pareto distribution by using the law of mass attraction as the factor explaining their growth speed. [1]
A model is not the reality but a way of offering factors that intermingle to mimick Pareto reality.
That is what we are attempting to do now for incomes of the whole population and of entrepreneurs.

Executive summary

Income distributions are typically governed by Pareto distributions. They mimick atmospheric pressure distribution, income replacing the height. Basic thermodynamic suggest that air pressure, built from the choc of molecules, has its equivalent in economics in competition And the role of entrepreneurs can be paralleled with the role of pocket of air, heated by the sun, that lift the ceiling of the atmopsheric boundary, in the same way that entrepreneurs lift the whole population income.

More competition is increasing the total economy.

More heat leads to a lower α that is the basic index of the Pareto law but a lower α is also associated with more inequality in revenue.

Transposing thermodynamic in economics as most economist have done with other concepts of mechanics such as forces, equilibrium, leads to the basic idea that competition lifts the ceiling of the economy but also leads to more inequality.

This model suggests that for centuries, the whole economy was made of a superposition of layers, that exchanges between layers were laminar flows for most of human history with very low increase in productivity, until the industrial revolution where exchanges between layers became turbulent.

Turbulence is what one finds behind Schumpeter creative destruction. It is most probably a result of increased human density with villages becoming towns and then megalopoles.
Though the height of the atmosphere can vary widely around the Earth, this model does not allow for an increase in the height of the ceiling of the atmosphere that would mimick the expansion of economy through income and wealth.

Income (or wealth) distribution and atmospheric pressure

There is a great similarity between an income Pareto distribution and atmospheric pressure.

To demonstrate this, just take a representation of the Pareto used to calculate the Gini index, the Lorenz distribution, of which the mathematical formulation is L (F) = 1- (1-F)1-1/α where F indicates position in the income scale (or wealth if the law covers wealth), measured from 0 to 1.

Exemple : for F = 0.99, the Lorenz law gives the share of the total revenue of the first poorest 99% of the population ; and the difference with 1, the share of the richest centile. For F = 0.999, it gives the share of the poorest 99,9%, etc .

1-L(F)= (1-F) 1-1/α gives the share of the total income (or wealth) of the percentiles above F.

This formulation is the same in form as that of the air pressure p as a function of the altitude z:

p (z) = 1013.25 (1- 0,0065.z / 288.15)5,255

It suffices to replace L (F) by p (z) / 1013.25, thus replacing the accumulated Lorenz by the pressure measured in atmospheres [p (z) / 1013.25], the fortunes F by altitude in meters divided by 44,330 (the altitude where, from the formula, the air layer ends), and the exponent 5,255 by (1-1/α).

The latter becomes the pressure reached at altitude 0.99 of the total height of the layer of air or more precisely the gas layer whose exponent is (1-1/α), a gas lighter than air.

This exponent 5,255 comes from Mg / Rβ wherein M is the molecular weight of air (28.9 grams / molecular volume), g is the acceleration of gravity, R is the universal gas constant and β is the temperature drop gradient with height (related to the pressure drop) in degrees ° K per meter.

To go from the formula of the atmosphere to the distribution of Lorenz derived from a Pareto, simply replace p and z as we have seen but also reduce the exponent, (conceptually) by reducing Mg/Rβ.

For an α of 1.5, which is a current value of income distributions, (1-1 /α) = 0.33 and you have to imagine a planet where g, the attraction of gravity, is reduced by 3 x 5,255 or about 15 time (the other parameters are physical constants well defined for gases and β being also defined within a short range that varies with the content of humidity).

A remark is that in Pareto distributions, α cannot be below 1 and that, therefore, (1-1/α) is a quantity below 1 with the property that the smaller it is, the larger is the value 1-L(F) :

Exemple : square root of 0,16 is 0,4 ; but power ¼, i.e.of the square root of the square root is 0,63.

Thermodynamics and competition

It is possible to imagine what are the forces that define α, the proof being impossible, the only usefulness being to give a meaning to the results.

When doing so, we’ll use the thermodynamic theory that allows to calculate atmospheric pressure. We are following a common pattern as most economic models have been derived from common physics, starting with forces, equilibrium, speeds, momentums.

We know that the pressure at a given air point is given by the shock of air molecules, in Brownian motion, with speeds related to temperature (thermodynamic and kinetic theory of gases: see wikipedia), the limit to the infinite expansion of the air layer being due to gravity. However, a finite limit exists only because there is a drop in temperature from the surface defined by a zero altitude, which transforms an exponential pressure according to altitude, in Pareto (by reducing its exponent by one).


The transposition between the level of molecular collisions and the economy is the level of competition. What prevents the economic world to explode and plays the role of g, the acceleration of gravity, is likely all the rules and regulations, including myths, that humanity invented to organize itself.

What differentiates the periods of history are the changes in heat transfers, the β of the formula.

In the earlier period of humanity the β was high, with little exchanges between the different layers of the human society. Initially, the heat transfer coefficients between the different layers of the human society were low, due in great part to the low density of inhabitants. Economists have noticed that productivity has not much changed until the Middle Age and starts to explode with industrial age.
What prevented productivity to increase is probably the corpus of laws, myths, traditions, regulations, clan rules. The take-off was possible only when a certain individual freedom emerged authorizing competition between producers, allowing the emergence of technological innovations and such individual freedom appeared when human concentration in villages becoming town became high enough to allow competition between producers to accelerate innovations; this changed the transmission of income through the layers of humanity, possibly from laminar flows with low Reynolds number [2] to higher numbers leading to turbulence.

David Birch, the founder of firm demography, was the first to describe the turbulent Birch cloud [3] between producers, allowing the emergence of technological innovations that match the idea of Schumpeter creative destruction.

The parallell to the molecule thermodynamic properties is to replace the brownian motion by competition : this is what makes people hit each other to improve ones lot and climb in the social ladder.

The more competition, the hotter the gas and the lower is the α

Hot spots

One well known rule by meteorologist is that when air is heated by the earth itself receiving sun heat radiations, this creates a bubble of hot air, of which density is lower than surrounding cold air and that bubble starts climbing because of the Archimede principle. If there were no friction with sourrounding masses, il would go up and establish a column of hotter and lighter air that would culminate higher than the rest if there was no friction with the surrounding atmosphere…

Such column would have an a, the drop of temperature in °K per 1000 meter, lower than usual, such as 3,5 °K/100 meters instead of the usual 5 to 6. And, hence, Mg/aR would increase and (1- uz) 1-1/α has to increase (where z is the altitude and u a coefficient) which means since 1-uz is smaller than 1 that the exponent has to decrease, i.e. α has to decrease.

This means that an increase in temperature, an increase in brownian air molecule motion, i.e. increase in competition in the economic field, translates into distributions that have a lower α, i.e. have more inequality.

It can indeed be easily seen that α = ∞ means income equal for all and that dimishing α are linked to higher inequality.

Not surprisingly, a bubble with lower α means increasing competition, increased inequality but lifting ceiling of the whole column of air, i.e. of the whole economy.

In summary, if an inside portion of the air has a thermal gradient β that is higher than the thermal gradient of the adiabatic outside air, this portion will raise and create an unstable atmosphere and turbulences.

Turbulences in economy are raising the economy ceiling.

This is exactly what happens in the economic world where the success of entrepreneurs in the 1% is shaking the established order, leading to the Schumpeter creative destruction.

The α is lower when looking to entrepreneurs households that succeeded in getting to the 1%, the long term businesses creations, than the service business that make most of the 99%.

A consequence is that a lower α is a sign the economy is more unequal but also more turbulent and leads to higher growth.

Real alpha curves in accordance with theory

We are reproducing below the Log-Log curves derived from the SCF showing the whole population of households distribution and the ones of entrepreneurs extracted from the SCF but also from the HFCS.
In practice, a real Lorenz curve is the curve achieved with varying α along the income.

The Log-Log curves derived from the SCF shown below are fairly consistant from one to another and the change of slope indicates a lowering of the α, that is also consistant with the increase in inequality that happened.

But the curves below presenting the Log-Log curves of the whole population and those of entrepreneurs with lower α explains why αhas dzecreased with time as the number of entrepreneurs and the number of people employed has increased.

As hot parts of air, the entrepreneurs have lifted the whole economic atmosphere and raised inequality but created employment.

Each of these curves is obtained by ordering all people covered by the survey according to their total yearly income (in fact the year preceding the survey i.e. 2012 for survey 2013) and then cumulating their weights starting with the highest income ; the logarithm of this cumulation is shown in ordinate as a function of the logarithm of the income in abcissae.
We can see that the curves below an abcissae of 6 (one million dollars income) for year 1989- light green- up to below 7 (ten million dollars) for year 2013 – light blue- are straight lines. The change from 6 to 7 comes from the fact we use SCF data in current $.
The slopes of these straight parts drop from around α # 1,9 to α # 1,5.
It indicates a raise in inequality but also raise of economic activity and, as we’ll see in part 6, an increase in employment.

One sees that France is the most egalitarian contry with the green line descending the fastest, followed by the blue line that shows German population.
The French entrepreneurs (in violet)are more equal (but not as competitive) than the German entrepreneurs (red) that are less competitive than US population (lighter blue line) that is below the US entrepreneurs curve (in orange).

[1Eventhough there has been many studies to prove that in reality, the words frequency departed from a pure Zifp law, all the same for the size of the cities.

[2A Reynolds number is a number associated with a fluid, its mass and viscosity that tells when the fluid flow is changing from laminar to turbulent.

[3A Birch cloud describes a business economy by splitting the businesses sizes by layers of sizes (number of employees or turnover) and measuring the changes of height from one year to the next. From appart, the economy looks as a cloud, very steady , increasing just of a few % every year ; but, inside the cloud, there are poweful ascending and descending currents, with businesses doubling or tripling their sizes and others dropping by half or more.