This function returns the Lorenz curve of any rule for a claims problem.
lorenzcurve(E, d, Rules, col = NULL, legend = TRUE)The graphical representation of the Lorenz curve of a rule (or several rules) for a claims problem.
The endowment.
The vector of claims.
The rules: AA, APRO, CE, CEA, AV, DT, MO, PIN, PRO, RA, Talmud, RTalmud.
The colours. If col=NULL then the sequence of default colors is:
c("red", "blue", "green", "yellow", "pink", "orange", "coral4", "darkgray", "burlywood3", "black", "darkorange", "darkviolet").
A logical value. The colour legend is shown if legend=TRUE.
Let \(N=\{1,\ldots,n\}\) be the set of claimants, \(E\ge 0\) the endowment to be divided and \(d\in \mathbb{R}_+^N\) the vector of claims such that \(\sum_{i \in N} d_i\ge E\).
Rearrange the claims from small to large, \(0 \le d_1 \le...\le d_n\). The Lorenz curve represents the proportion of the awards given to each subset of claimants by a specific rule \(\mathcal{R}\) as a function of the cumulative distribution of population.
The Lorenz curve of a rule \(\mathcal{R}\) for the claims problem \((E,d)\) is the polygonal path connecting the \(n+1\) points, $$(0,0), \Bigl(\frac{1}{n},\frac{\mathcal{R}_1(E,d)}{E}\Bigr),\dots,\Bigl(\frac{n-1}{n},\frac{\sum_{i=1}^{n-1}\mathcal{R}_i(E,d)}{E}\Bigl),(1,1).$$ Basically, it represents the cumulative percentage of the endowment assigned by the rule to each cumulative percentage of claimants.
Lorenz, M. O. (1905). Methods of measuring the concentration of wealth. Publications of the American statistical association 9(70), 209-219.
Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez Rodríguez, E. (2023a). Deviation from proportionality and Lorenz-domination for claims problems. Review of Economic Design 27, 439-467.
Mirás Calvo, M.Á., Núñez Lugilde, I., Quinteiro Sandomingo, C., and Sánchez-Rodríguez, E. (2023b). Refining the Lorenz‐ranking of rules for claims problems on restricted domains. International Journal of Economic Theory 19(3), 526-558.
cumawardscurve, deviationindex, giniindex, indexgpath, lorenzdominance.
E=10
d=c(2,4,7,8)
Rules=c(AA,RA,Talmud,CEA,CEL)
col=c("red","blue","green","yellow","pink")
lorenzcurve(E,d,Rules,col)
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