Lc {ineq}

Lorenz Curve

Description

Computes the (empirical) ordinary and generalized Lorenz curve of a vector x

Usage

Lc(x, n = rep(1,length(x)), plot = FALSE)

Arguments

x

a vector containing non-negative elements.

n

a vector of frequencies, must be same length as x.

plot

logical. If TRUE the empirical Lorenz curve will be plotted.

Details

Lc(x) computes the empirical ordinary Lorenz curve of x as well as the generalized Lorenz curve (= ordinary Lorenz curve * mean(x)). The result can be interpreted like this: p*100 percent have L(p)*100 percent of x.

If n is changed to anything but the default x is interpreted as a vector of class means and n as a vector of class frequencies: in this case Lc will compute the minimal Lorenz curve (= no inequality within each group). A maximal curve can be computed with Lc.mehran.

Value

A list of class "Lc" with the following components:

p

vector of percentages

L

vector with values of the ordinary Lorenz curve

L.general

vector with values of the generalized Lorenz curve

References

B C Arnold: Majorization and the Lorenz Order: A Brief Introduction, 1987, Springer,

F A Cowell: Measurement of Inequality, 2000, in A B Atkinson / F Bourguignon (Eds): Handbook of Income Distribution, Amsterdam,

F A Cowell: Measuring Inequality, 1995 Prentice Hall/Harvester Wheatshef.

See Also

plot.Lc, Lc.mehran, plot.theorLc

Examples

## Load and attach income (and metadata) set from Ilocos, Philippines
data(Ilocos)
attach(Ilocos)
## extract and rescale income for the provinces "Pangasinan" und "La Union"
income.p <- income[province=="Pangasinan"]/10000
income.u <- income[province=="La Union"]/10000

## compute the Lorenz curves
Lc.p <- Lc(income.p)
Lc.u <- Lc(income.u)
## it can be seen the the inequality in La Union is higher than in
## Pangasinan because the respective Lorenz curve takes smaller values.
plot(Lc.p)
lines(Lc.u, col=2)
## the picture becomes even clearer with generalized Lorenz curves
plot(Lc.p, general=TRUE)
lines(Lc.u, general=TRUE, col=2)
## inequality measures emphasize these results, e.g. Atkinson's measure
ineq(income.p, type="Atkinson")
ineq(income.u, type="Atkinson")
## or Theil's entropy measure
ineq(income.p, type="Theil", parameter=0)
ineq(income.u, type="Theil", parameter=0)



# income distribution of the USA in 1968 (in 10 classes)
# x vector of class means, n vector of class frequencies
x <- c(541, 1463, 2445, 3438, 4437, 5401, 6392, 8304, 11904, 22261)
n <- c(482, 825, 722, 690, 661, 760, 745, 2140, 1911, 1024)

# compute minimal Lorenz curve (= no inequality in each group)
Lc.min <- Lc(x, n=n)
# compute maximal Lorenz curve (limits of Mehran)
Lc.max <- Lc.mehran(x,n)
# plot both Lorenz curves in one plot
plot(Lc.min)
lines(Lc.max, col=4)

# add the theoretic Lorenz curve of a Lognormal-distribution with variance 0.78
lines(Lc.lognorm, parameter=0.78)
# add the theoretic Lorenz curve of a Dagum-distribution
lines(Lc.dagum, parameter=c(3.4,2.6))