Lc                   package:ineq                   R Documentation

_L_o_r_e_n_z _C_u_r_v_e

_D_e_s_c_r_i_p_t_i_o_n:

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

_U_s_a_g_e:

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

_A_r_g_u_m_e_n_t_s:

       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.

_D_e_t_a_i_l_s:

     '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'.

_V_a_l_u_e:

     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

_A_u_t_h_o_r(_s):

     Achim Zeileis zeileis@ci.tuwien.ac.at

_R_e_f_e_r_e_n_c_e_s:

     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.

_S_e_e _A_l_s_o:

     'plot.Lc', 'Lc.mehran', 'plot.theorLc'

_E_x_a_m_p_l_e_s:

     ## 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))

