giniindex {ClaimsProblems}

Gini index

Description

This function returns the Gini index of any rule for a claims problem.

Usage

giniindex(E, d, Rule)

Arguments

E

The endowment.

d

The vector of claims.

Rule

A rule: AA, APRO, CE, CEA, CEL, DT, MO, PIN, PRO, RA, Talmud.

Details

Let E> 0 be the endowment to be divided and d\in \mathcal{R}^n the vector of claims with d\ge 0 and such that D=\sum_{i=1}^{n} d_i\ge E, the sum of claims D exceeds the endowment.

Rearrange the claims from small to large, 0 \le d_1 \le...\le d_n. The Gini index is a number aimed at measuring the degree of inequality in a distribution. The Gini index of the rule R for the problem (E,d), denoted by G(R,E,d), is the ratio of the area that lies between the identity line and the Lorenz curve of the rule over the total area under the identity line.

Let R_0(E,d)=0. For each k=0,\dots,n define X_k=\frac{k}{n} and Y_k=\frac{1}{E} \sum_{j=0}^{k} R_j(E,d). Then

G(R,E,d)=1-\sum_{k=1}^{n}(X_{k}-X_{k-1})(Y_{k}+Y_{k-1}).

In general 0\le G(R,E,d) \le 1.

Value

The Gini index of a rule for a claims problem and the Gini index of the vector of claims.

References

Ceriani, L. and Verme, P. (2012). The origins of the Gini index: extracts from Variabilitá e Mutabilitá (1912) by Corrado Gini. The Journal of Economic Inequality, 10(3), 421-443.

Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2022). Deviation from proportionality and Lorenz-domination for claims problems. Rev Econ Design. doi: 10.1007/s10058-022-00300-y

See Also

lorenzcurve, cumawardscurve, deviationindex, indexgpath, lorenzdominance.

Examples

E=10
d=c(2,4,7,8)
Rule=AA
giniindex(E,d,Rule)
# The Gini index of the proportional awards coincides with the Gini index of the vector of claims
giniindex(E,d,PRO)