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....