# Gini says: measuring income inequality

### an LBO report, October 18, 1993

There are several ways to express the degree of income inequality in a society. The simplest way is to arrange whatever units you choose persons, families, or households in rank order, from poorest to richest; divide the hierarchy into fifths (quintiles) or tenths (deciles); and compute either the average income by decile or quintile or the share that each grouping has of the society's total income. Then, the shares or averages of rich and poor can be compared.

Here are examples for the U.S. for two important years 1968, when the U.S. income distribution was the most equal it has been in modern times, and 1992, when it was the most unequal (so far!). The five columns on the left show the share of pretax income earned by each quintile of households, from the poorest to the richest; columns six through nine show the ratios of those shares for the richest to poorest, the middle to the poorest, and the richest to the middle. Note that from 1968 to 1992, the increase in inequality was almost entirely the result of the rich getting richer at the expense of the lower-middle and middle ranks.

This technique is simple and revealing, but not without awkwardness: which comparison to choose? Economists have devised several ways of making such comparisons with a single index number. The most popular of these is the Gini index (or coefficient or ratio or number). While it simplifies com parisons, however, the Gini is not easy to explain. Here's an attempt to do so.

### Plotting distribution: Lorenz curves

We can start with the nearby graph, derived from the numbers in the table. The horizontal axis shows the quintiles; the vertical, the cumulative share of income earned by the plotted quintile and the ones below it. For example, for 1968, the cumulative value for quintile 1 is the share earned by quintile 1, or 4.2%; that for quintile 2, 11.1%+4.2%, or 15.3%; and so on, up to quintile 5, when the sum is 100%. The graph plots these values for 1968 and 1992. These lines are called Lorenz curves.

In a society with perfectly equal income distribution, the cumulative share of income would be equal to the cumulative population share: e.g., the value for quintile 2 would be 40% (20%+20%), instead of 1968's actual 15.3%. This mythical egalitarian society is represented by the heavy 45° line.

The gap between the actual lines and the mythical line is a function of the degree of inequality; the 1992 line is further away from the heavy line than 1968's is. The Gini index measures the gap between the actual line and the 45° line. In the egalitarian society, the Gini would be 0.000, since the Lorenz curve would match the 45° line perfectly; the higher the Gini, then, the greater the distance, and the more unequal the distribution of income. In a perfectly unequal society, in which one person (or household or family) had all the income, the Lorenz curve would look like a backwards "L," and its value would be 1.000. In practice, the Gini usually falls between 0.200 and 0.450.

This simple Lorenz chart is made only using quintile figures, so a Gini computed from it would be inexact. For the U.S., real Ginis are computed using a sample of some 60,000 households polled by the Census Bureau every March as part of its Current Population Survey. The Gini formula is pro vided for connoisseurs at the end of this screed.

### American Ginis

Shown nearby are the Gini indexes for families and households, as reported by the U.S. Bureau of the Census, for cash incomes before taxes. (Cash income means noncash benefits like food stamps and the value of employer-paid health insurance are excluded.) For the first two decades after World War II, inequality trended raggedly downwards. Since the late 1960s, however, inequality has been increasing with much less raggedness and more thrust than the earlier decline. Note that this polarization long predates the Reagan years, despite a lot of partisan rhetoric to the contrary. In fact, the Gini has risen at a fairly constant rate over the last 25 years no matter which party was in power.

Though the increase in the Gini between 1991 and 1992 has brought the index to its highest level on record, the change, the Census Bureau tell us, is not statistically significant, meaning that it is within the sample's margin of error.

Two Ginis are shown, one for families and the other for households. Household is a broad concept that includes single people living alone and unrelated individuals living together. (Unrelated, that is, by legal definitions; millions of cohabitants consider themselves families, even if the law doesn't agree.) The average income of the nation's 68.1 million families is higher, and their income distribu tion more equal, than the 28.2 million nonfamily households, since nonfamily households include a lot of poorer people who live alone. Unfortunately, the Census Bureau only began reporting income data for households in 1967, so longer-term comparisons must use the family figures.

## Sources

#### U.S. Ginis

U.S. Bureau of the Census. Current Population Reports, Series P60-184, Money Income of Households, Families, and Persons in the United States, 1992 (September 1993).

#### Lorenz curves

Joseph E. Stiglitz. Economics (New York: W.W. Norton & Co., 1993).

#### Gini formula

Hawyard Alker Jr. Mathematics and Politics (New York: Macmillan Company, 1965), as cited in a 1989 computer program by Tom Finholt, Department of Social and Decision Sciences, Carnegie Mellon University.

#### sensitivity of various inequality indexes

Markus Jantti. "Changing Inequality in Five Countries," Luxembourg Income Study Working Paper #91 (February 1993).