ECONOMIC

the recipients. The data suggest that social security and public assistancedramatically decreased the measured relative inequality of family income.The combined effect of the other transfers is a small decrease in incomeinequality.The Tax Transfer SystemThe success of the Government's programs for the redistribution of incomecannot be judged from any one program or from an examination of taxesor transfers separately. Primarily because of public assistance, social security,medicaid, and food stamps, the transfer system is highly progressive in redistributingincome to low-income families. It was shown above (Table 35)that the net effect of personal income and payroll taxes on cash and imputedincome appears to be progressive. Several studies have examined the effectof the tax system on the distribution of income when accrued capital gainsand losses are included in the income concept. These studies suggest thatthe tax system is roughly proportional over the income intervals in whichmost families belong, regressive for those with very low incomes, and progressiveat the upper end. However, the combined direct effects of the taxand transfer systems clearly appear to be progressive.SUPPLEMENTThe Variance of the Natural Logarithm of IncomeThe "variance of the natural logarithm of incom'e" is the measure of overall incomeinequality used in the analysis of the distribution of income in this chapter. It can bewritten as:N£(lnY,-lnY)«i=l (1)S»(lnY) = JJ-~where Yj is the income of the i th observation (individual or family), In designatesnatural logarithm, ancj there are N observations in the data. Larger values of S 2mean greater inequality of income, and S 2 equals zero if there is no inequality. Whilea reduction of the measure from 0.7 to 0.6 conveys an acceptable suggestion about adecline in inequality, and a decline from 0.7 to 0.5 an acceptable suggestion about agreater decline, the statement that the second of these two declines is twice the firstwould not be meaningful.The variance of the natural logarithm of income is a commonly used simple measureof relative inequality. A measure of relative inequality does not change in valueif all of the observations have the same percentage change in income. IfY*,=Y,(l+k), (2)where k is the percentage change in income, the natural logarithm of both sides ofequation (2) islnY* i = lnY i + ln(l + k). (3)Computing the mean of both sides of equation (3),E I n Y + ln(l-f k). (4)179