Lorenz Curve

Max Lorenz developed the Lorenz curve in 1905 as a graphical representation of income inequality and wealth inequality. This graph shows population percentiles according to income or wealth on a horizontal axis. It plots cumulative income or wealth on the vertical axis, so that an x-value of 45 and a y-value of 14.2 indicates that the bottom 45% of the population controls 14.2% of the total income or wealth. A Lorenz curve is usually determined from incomplete income or wealth observations. It is generally used to show the extent of concentration of income and wealth.

The Lorenz curve represents the distribution of wealth or income within a population. A Lorenz curve represents percentiles of the population versus their cumulative incomes or wealth. It is widely used to measure inequality among a population using Lorenz curves along with their derivative statistics. A Lorenz curve is a mathematical estimate that fits a continuous curve to incomplete and discontinuous data, so it may be an imperfect measure of underlying inequality.

Major Points about Lorenz Curve

• Lorenz curves are often accompanied by a straight diagonal line with a slope of 1, indicating perfect equality in income or wealth distribution; the Lorenz curve lies beneath it, indicating the observed or estimated distribution. As a scalar measure of inequality, the Gini Coefficient involves the difference between the straight line and the curve.
• The Lorenz curve is most commonly used to illustrate economic inequality, but it can also be used to illustrate unequal distribution in any system. By extending the straight diagonal line farther from the baseline, the level of inequality will increase. According to the Lorenz curve, income or wealth distribution is unequal when one has a high income and a low net worth or a high income with a low net worth.
• Lorenz curves are usually derived from an empirical measurement of the distribution of wealth or income across a population, for example, using tax returns with income information for a large proportion of the population. Either the graph of the observed data may be used directly as a Lorenz curve, or statisticians and economists may fit a curve that fills in the gaps in the observed data.
• Compared to summary statistics such as the Gini coefficient or Lorenz asymmetry coefficient, a Lorenz curve provides more detailed information about the distribution of wealth or income across a population. Due to its visual representation of the distribution across each percentile (or other unit), a Lorenz curve can reveal precisely where and by how much the observed distribution varies from equality.

Lorenz Dominance

It is said that the Lorenz curve of a distribution A dominates the distribution of B when the curve A overtakes the curve B at all points along the distribution. In this case, one can say that A is more equal than B. If both distributions have the same mean, A is preferable to B. There can only be an assumption of stretches of a distribution if two Lorenz curves intersect. Whenever such a distribution arises, there are welfare functions that rank it differently. Compared to distributions of B and C, distribution of A dominates distribution of B, but distributions of B do not have dominance.

Generalized Lorenz Curve

It seems Lorenz curves only imply welfare dominance when the same mean is compared to distributions, the most restrictive interpretation of the theory. The criterion developed by Shorerocks and Kakwani (1984) to compare distributions which differ in their means is a useful one. An alternative to the Lorenz Curve is the Generalized Lorenz Curve, which multiplies by the average income of the distribution the accumulated fraction of incomes at each fractional level in the population.

The generalized curve, as a result of this multiplication, provides information about the form and level of the distribution, or the joint first two moments of the distribution, such as the income distribution curve and its congeners of basic statistics. The Lorenz Generalized Curve is represented by the function L(μ,P) = μ L(P). In general, any symmetric welfare function (satisfies anonymity property) and quasi-concave distribution (satisfies Pigu-Dalton property) will lead to higher welfare for A than for B at all points of the graph.