# The Ethical Rate of Unemployment: A Technical Note

Galbraith, James K., Journal of Economic Issues

Assuming that there exists a stable long-term, positive relationship between the rate of unemployment and the rate of change of inequality in the wage structure, the ethical rate of unemployment is defined as that rate of unemployment above which inequality tends to rise and below which inequality tends to decline. The questions addressed in this note are, first, does such a rate exist, and second, if the historical evidence suggests that it does, where is it located?

Measurement of the Evolution of Inequality from Grouped Data(1)

Originally drawn from information theory, Theil's T has the following formula:

T = (1/n)[Sigma]([Y.sub.i]/[Mu])log([Y.sub.i]/[Mu]) (1)

Here, n is the number of individuals, [Y.sub.i] is each person's income, and [Mu] is average income for the whole population. Notice that, when a group population consists of equal individuals, the final terms in T all reduce to log ([y.sub.i]/[Mu]) = log(l). Thus,T overall is zero for the case of perfect equality, and, since deviations of ([y.sub.i[/[Mu]) below the mean have values between zero and one, whereas deviations above the mean are unbounded, T increases as deviations away from the average value increase.

The formula for computing T from grouped data is this:

T = [Sigma]([p.sub.i][[Mu].sub.i]/[Mu])log([[Mu].sub.i]/[Mu]) + [Sigma]([p.sub.i][[Mu].sub.i]/[Mu])[T.sub.i] (2)

where now [p.sub.i] is the proportion of workers employed in the i-th group, [[Mu].sub.i] represents the average income for the i-th group, g represents average overall income, and [T.sub.i] is the Theft T as measured strictly within the i-th group. Thus, the grouped Theil statistic is the sum of that part of inequality that occurs between groups (on the left of the above expression) and a part that occurs within groups (on the right).

The formula for T[prime], the between-group-Theil statistic, is just the first element in the formula for computing the Theil T from grouped data:

T[prime] = [Sigma]([p.sub.i][[Mu].sub.i]/[Mu])log([[Mu].sub.i]/[Mu]) (3)

Since the within-group element in variation is omitted, this is obviously a lower-bound estimate of dispersion. However, T[prime] must converge to T as the group structure becomes more finely disaggregated. It follows that for a consistently observed and reasonably fine structure of groups, the movement of T[prime] through time must bear a close relationship to the movement of T, and the movement of T[prime] can serve as a proxy measure for the movement of T.

Measuring Inequality in U.S. Wages, 1920-1992

Measures of average wages by industry and occupation meet the criteria for disaggregation of the Theil statistic into between-group and within-group components, insofar as they form a mutually exclusive, exhaustive categorization. Thus, a consistently observed industrial structure should yield estimates of T[prime] through time whose movements resemble that of the unobservable and unrecoverable T for the distribution of wages. This permits estimation of the evolution of inequality in the industrial wage structure for time periods (and countries) where direct measurement of the evolution of inequality did not occur. See the references for examples.

For the period 1958-1992, I calculate T[prime] for the structure of hourly wages in manufacturing from a three-digit standard industrial classification (SIC) for 139 industries as reported by the Economic Census Reports.(2) Approximately 70 percent of manufacturing employment is covered in this measurement. As I have reported [Galbraith 1998], the resulting series is closely related to the Gini series for total household income inequality as computed from the Current Population Surveys (r = .86), although the two series diverge after the mid-1980s, possibly because income inequality continued to increase, while inequality in hourly wage structures did not. Figure 1 presents T[prime] for this time period. …

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