Exploring A New Class of Inequality Measures and Associated Value Judgements: Gini and Fibonacci-Type Sequences

Abstract

This paper explores a single-parameter generalization of the Gini inequality measure. Taking the starting point to be the Borda-type social welfare function, which is known to generate the standard Gini measure, in which incomes (in ascending order) are weighted by their inverse rank, the generalisation uses a class of non-linear functions. These are based on the so-called ‘metallic sequences’ of number theory, of which the Fibonacci sequence is the best-known. The value judgements implicit in the measures are explored in detail. Comparisons with other well-known Gini measures, along with the Atkinson measure, are made. These are examined within the context of the famous ‘leaky bucket’ thought experiment, which concerns the maximum leak that a judge is prepared to tolerate, when making an income transfer from a richer to a poorer person. Inequality aversion is thus viewed in terms of being an increasing function of the leakage that is regarded as acceptable.

Appendix: Derivation of the Pell Index of Inequality

This appendix provides a short derivation of the metallic ratio expression for the ‘Pell Index’ of inequality, denoted G₂ above and shown in equation (3.10). The derivation makes use of the following two standard results relating to the Pell number sequence: ¹⁵

(a):

The n th term of the Pell sequence, $P\left (n\right )$, can be approximated by $\delta ^{n}/\left (2\sqrt {2}\right )$, where δ is the ‘silver ratio’, $\delta =1+\sqrt {2}$.

(b):

The sum of the first n Pell numbers, ${\sum }_{i=1}^{n}P\left (i\right )$, is $[ 3P\left (n\right ) +P\left (n-1\right ) -1] /2$. In terms of the approximation in (a) and after some manipulation, this can be written as:

$$ {\sum}_{i=1}^{n}P\left( i\right) =\frac{\delta^{n-1}\left( 3\delta +1\right) -2\sqrt{2}}{4\sqrt{2}} $$

(A.1)

Using (16), the Pell inequality index is derived from the function, W₂. Write W_P = W₂, and using the notation introduced in Section 3, for any ordered n-vector of incomes $\mathbf {x=}\left (x_{1},...,x_{n}\right )$:

$$ W_{P}\left( \mathbf{x}\right) =\sum\limits_{i=1}^{n}S_{2}\left( i\right) x_{i}={\sum}_{i=1}^{n}P\left( n+1-i\right) x_{i} $$

(A.2)

Making use of result (b) stated above, this becomes:

$$ W_{P}\left( \mathbf{x}\right) =\frac{\left( 3\delta +1\right) {\sum}_{i=1}^{n}\delta^{n-i}x_{i}-\left( 2\sqrt{2}\right) n\bar{x}}{4\sqrt{2}} $$

(A.3)

The equally distributed equivalent, x_E,P, using (17), is given by:

$$ x_{E,P}=\frac{W_{P}\left( \mathbf{x}\right) }{{\sum}_{i=1}^{n}P\left( n+1-i\right) } $$

(A.4)

Using, again, result (b) stated above:

$$ \sum\limits_{i=1}^{n}P\left( n+1-i\right) =\frac{1}{4\sqrt{2}}\left[ \left( 3\delta +1\right) \sum\limits_{i=1}^{n}\delta^{n-i}-2n\sqrt{2}\right] $$

(A.5)

But ${\sum }_{i=1}^{n}\delta ^{n-i}=\left (\delta ^{n}-1\right ) /\sqrt {2}$, and making the appropriate substitution yields:

$$ \sum\limits_{i=1}^{n}P\left( n+1-i\right) =\left[ \left( 3\delta +1\right) \left( \delta^{n}-1\right) -4n\right] /8 $$

(A.6)

Using (A.3), (A.4) and (A.6), the equally distributed equivalent is:

$$ x_{E,P}=\left( \frac{8}{4\sqrt{2}}\right) \frac{\left( 3\delta +1\right) {\sum}_{i=1}^{n}\delta^{n-i}x_{i}-2n\bar{x}\sqrt{2}}{\left( 3\delta +1\right) \left( \delta^{n}-1\right) -4n} $$

(A.7)

This simplifies to:

$$ x_{E,P}=\frac{\left( 3\delta +1\right) \sqrt{2}{\sum}_{i=1}^{n}\delta^{n-i}x_{i}-4n\bar{x}}{\left( 3\delta +1\right) \left( \delta^{n}-1\right) -4n} $$

(A.8)

The Pell index is the proportional deviation of the equally distributed equivalent income from the arithmetic mean, $G_{P}=1-x_{E,P}/\bar {x}$. Using (A.8), this gives the result in equation (3.10).

As a check, the exact value of the Fibonacci and Pell inequality indices, as given in equation (3.16), can be compared with the ‘metallic ratio’ approximations as given by equations (3.9) and (3.10) respectively, for a hypothetical income distribution. Consider the ten-person ordered distribution $\left (20,30,50,55,60,75,90,120,140\right )$ used in Section 3.3 above. It turns out that, for this distribution, the exact and approximate values, as given by (3.6), for k = 1, and (3.9) respectively for the Fibonacci index are 0.5325 and 0.5255. The corresponding exact and approximate values for the Pell index, given by (3.6), for k = 2, and (3.10) respectively, are 0.6404 and 0.6441.