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APPENDIX B Income Differences or Income Ratios? SCHOLARS HAVE developed many statistical measures of inequality.1 The standard procedure is to use some version of a ratio measure that essentially treats the comparison proportionally—the difference in logged dollars , for example, or the proportion of total dollars garnered by each quintile , or some yet more complex measure, such as Gini orTheil coefficients. All of these show a trend toward a more equal distribution of income before 1970 and a steady trend toward a less equal one between 1970 and at least the middle 1990s.2 Why ratio measurement is the standard is not obvious .There is some intuitive appeal to using an arithmetic difference instead . Take, for example, the numbers in figure 5.9: in 1999 full-time workers with a BA degree made $49,250, while high school graduates made $26,100, for a difference of $23,150.At the typical hourly wage of a high school graduate in 2000 ($12.30), he or she would have to find an additional thirty-six hours of work a week to close the gap.3 The result is intuitive and is sometimes different from that generated by ratios. Similar calculations based on arithmetic differences suggest that income inequality started growing in the United States starting in 1960 rather than 1970, as suggested by ratio measures.4 Our efforts to pin down the logic behind the consensus on using ratio measures, as we have done in this book, were not fully successful. Different sources and different authorities provide different rationales. Some authorities prefer the ratio measure because of the empirical observation that income and wealth data are right-skewed, or because research suggests that there are declining marginal returns to additional dollars, or because statistical models assuming proportionality better fit the data. Others prefer it because of the formal requirement that the inequality measures be “scale- invariant” (not changing if all the measures are multiplied by a constant) or because of some other technical consideration.5 YoramAmiel and Frank Cowell introduce popular opinion to the issue.In questionnaires administered to four thousand college students from several countries, they found that only about half clearly endorsed the notion of “scale independence,” that is, proportionality, a key principle behind using ratios.6 Fewer still endorsed “translation independence,” that is, arithmetic differences.But many believed that evaluations of differences rest on the initial level of affluence.When incomes are low, absolute increases—everyone gets a $100 raise—promote equality; when incomes are high, proportional increases—everyone gets a 3 percent raise—do. (The students’ judgments also failed to conform to other basic principles in economic models of inequality , a noteworthy problem considering that most of them were economics students.) In the end, the essential rationale for assessing income differences in ratio terms is psychological:the more dollars someone has,the less each new dollar “means.”A $1,000 raise to someone earning $100,000 means less—psychologically and practically—than it does to someone with a $10,000 income ; it may yield perhaps only one-tenth as much utility to the wealthier as to the poorer person.The rationale also draws on the consensus that humans judge fairness as a ratio; we see people’s deserts as “just deserts” when they are rewarded proportionally to their efforts.7 In the end, we adopted the conventional approach, measuring differences in dollar earnings, income , and wealth, all as proportions.8 One modest test of these two standards of comparison—arithmetic difference versus ratio—can be constructed by looking at trends in black and white family incomes.A difference measure—white family-of-four median income minus black family-of-four median income (figure 6.3)—shows that the income gap between blacks and whites widened between 1970 and 2000 (by about 8 percent), while a ratio measure—white income divided by black income—shows that the gap narrowed (by about 14 percent). How did African Americans’ subjective evaluations of their financial position change in that same period? From 1972 through 2000, the General Social Survey asked respondents to rate their family’s financial position compared to the “average.” Over those years, African-American respondents became somewhat more positive about their financial situations.(The proportion saying that their family’s situations was below average dropped from 50 percent in the mid-1970s to 40 percent at the end of the century,and the proportion saying above average grew from 5 percent to over 10 percent.)The...

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