A CLASS OF GENERALIZED MEASURES OF MOBILITY WITH APPLICATIONS Esfandiar MAASOUMI and Sourushe ZANDVAKILI

A class of anonymous measures of mobility is defined that considers the relationship between the long-run and the short-run distributions of income or any other attribute. Several measures of long-run income are explored, and distributions are compared on the basis of several inequality measures. The data from the Michigan Panel Study of Income Dynamics is used to demonstrate. The basic concept of mobility adopted here was introduced by Shorrocks (1978a). A comparison of inequality in the distribution of long-run incomes, measured over T periods, t = 1, 2,. . _, T, with some representative value of inequality in the short-run (single period) distributions reveals the degree of mobility, stability, or equalization over time. Measurement of inequality in long-run income is of interest in itself since such measurements can be free of short-run transitory effects. For instance, the long-run measures are free of life-cycle income effects, particularly when long periods of accounting are used. Defining y,, as the income of individual i; i = 1,. . . , N, in period t; t = 1,. . . , T, the simplest measure of long-run income is y, = CT=iu,,. To evaluate the inequality in the distribution y = ( y,, . . . , yN) is equivalent to assessing equality preferring Social Welfare Functions (SWF’s) which (i) are individualistic, and (ii) represent each individual by his (her) total income y, [see, e.g., Atkinson (1970)]. A more general approach replaces yi of (ii) with individual utility functions. Of course y, = X,u,, can also be interpreted as a particularly simple, additive utility function over the T periods. In this vein, we propose more general utility (or aggregator) functions, S, = S,(y,,, y,*, . . . , v,~), which include the simple additive form as a special case. We do this in order to study the effects of different levels of income substitution over time. Of course, S, can also be viewed as a measure of aggregate income over the accounting period. However S, is interpreted and measured, we define I($, . . . , S,) = I(S), say, as a measure of long-run inequality. Similarly, 1( yit, yzr,. . . , yN,) = I,, say, is the same measure of inequality for period t. Let I = XTwZ, be a weighted average of the short-run inequalities. We then define the following measure of stability: