Inequality possibility frontier

From: Equitable payment and social inequalities – Paper presented to the ICEC World Congress in Rio de Janeiro, October 2016 –  Ing. Gianluca di Castri, DIF, EIE/ICEC.A

We must understand why a high differential in equitable payment is normally accepted in highly developed societies, while it is badly considered in low income or poor countries. This bring the need to understand something more about the inequality in income distribution and to the general problem of the so called “social justice”.

We shall use as measure of the wealth the GNP per capita adapted to purchasing power (PPP) and the Gini coefficient as measure of the inequality: although we are aware that the Gini coefficient has some limits, we believe that it is still a method of measurement easy to manage and to understand.

In recent time, important studies on that matter have been made by Angus Maddison first (please refer to Maddison Project) and then by Branko Milanovic.

The purpose of Milanovic’s research was to measure “how close is measured inequality to the maximum inequality that can exist in a given society”, that he calls maximum feasible inequality.

The first situation to be considered is the case of a pre-industrial, subsistence economy where all people but an extremely small minority live at subsistence level. Taking into consideration.

  • Y= total income,
  • n= number of people,
  • s=  level of subsistence (at purchase power parity)
  • e= minority

The surplus over the level of subsistence shall be

S = Y – n (1 – e) s

Then the income of the minority

M = S / e

The level of subsistence is to be calculated in physiological terms, without any social consideration. In theory, it should not have any variability.  However, we must accept that some variability can be due to different climates or other environmental conditions.

If we calculate the mean income

m = Y / n

And then its ratio to the income corresponding to the level of subsistence

a = m / s

the coefficient of Gini, that actually corresponds to the inequality frontier,  shall be

G0 = (a – 1) / a

We can define an inequality possibility frontier that is the locus of maximus possible Gini (G0) coefficients and then the inequality extraction ratio R = G/G0 that is the ratio of the real Gini coefficient to the maximum possible Gini. R gives an estimate of how close a society is to its inequality frontier.

A further consideration is given by the fact that, when the general level of the country increases, people will not accept anymore to live at subsistence level, then the physiological minimum becomes a social minimum S0 (relative poverty line), that tend to increase with the mean income.

A more affluent society requires a higher social minimum.

We must consider that the definition of poverty is not limited to the inability to satisfy basic needs, but should take into consideration the capability to operate without shame in a society: this definition can be found in the works of several scholars throughout the centuries, the most recent one are Amartya Sen and Ravallion.

The social minimum S0 can be defined as

S0 = s a^x

Where x is the elasticity



The elasticity[i] of the social minimum compared to the mean income[ii] is a number between 0 and 1, that is to say that its increase is less than proportional to the increase of the mean income.

According to Chen and Ravaillon the elasticity of the official poverty line with respect to the mean income is 0.33.

However, taking into consideration that the socially accepted minimum (subjective poverty) is well above the poverty line, a more reasonable value of the elasticity, seems to in the range from 0,40 to 0,70 (Flik, van der Praag)

Milanovic has calculated a general expression that links the maximum feasible Gini, the average income and the elasticity of the minimum with respect to the average income.

G = 1 – (1/a) a^x

  • If the elasticity b=1, it means that the social minimum increases as the average income, namely that to all members of the community has to be guaranteed the mean income. Gini is equal to zero.
  • If the elasticity b=0, the social minimum is corresponding to the subsistence level, Gini can be close to 1
  • Id b=0,5, a reasonable value, the social minimum has an increase of 50% of the increase of the mean income. This is one among the definitions used for relative poverty level[iii]

Useless to say, the real problem when trying calculations on real data is the correct identification of the purchase power parity coefficient.

As far as the relationship between equitable payment and social inequality is concerned, it has been understood that a high “general income” (that is can be identified with the arithmetic mean or with the median income) allows for higher “differential incomes”, that, as far as work income is concerned, can be identified with the wage differential ratio.

According to Jaques, the fundamentals of that are to be found in the relationship between fair pay and fair expenditure capacity, for a person whose potential capability of work be congruent with the work he’s actually doing.

A similar relationship between the PPP pro capite and the maximum Gini has been found by Paolo Malanima in his historical research on the development of Italy.


[i] Elasticity can be defined correctly in term of differential equations. However a simplified definition  is “ the ratio of the percentage change in one variable to the percentage change in another variable, when the latter variable has a causal influence on the former”. The correct definition of elasticity of the function y=f(x) in point x is

e=(dy/dx)/(y/x)=y’ x/y

[ii] Median income is the amount that divides the income distribution into two equal groups, half having income above that amount, and half having income below that amount. Mean income (average) is the amount obtained by dividing the total aggregate income of a group by the number of units in that group.

[iii] The relative poverty level is calculated as 50% or 33% of the mean income. According to other scholars, it should be 50% of the median income.

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