Fig. 10 is like Fig. 8, only it deals with income to persons. It shows annual percentage changes in median income to persons from 1948 to 1967.[4] The percentage changes for the 19-year period and three subperiods are shown in Fig. 9(b). The pattern of movement for persons is very much like that for families, except that both the long-term percentage gains and the cyclical fluctuations are even wider for income to nonwhite persons. These points are rather plain in the graphs for the total postwar period and the three subperiod trends.
The greater cyclical instability of nonwhite income can be seen more precisely by removing the trend effect, and comparing the residuals of the year-to-year percentage changes in median income. When the income levels are taken into account, the deviations from the trend (measured by the sum of the squares of the residual over the predicted income) are about 4 times higher for nonwhites in the case of persons, and about 4-1/3 times higher for nonwhites in the case of families. These greater deviations for nonwhites exceed sampling error easily. The great instability of nonwhite family income, and even more, of income to nonwhite persons has, of course, direct implications for hardship -- an aspect of welfare that is not adequately reflected in the usual measurement of the present value of earnings or the smoothed curves of lifetime earnings.
The greater instability of nonwhite income may be related to the fragmentary data often referred to in relative income or in permanent income theories of the consumption function.[5] These data indicate that nonwhites at any given income level save a larger percentage of their income than whites. Joseph Newhouse has suggested to us that the greater instability of nonwhite measured income biases downward estimates of marginal propensity and average propensity to consume more for nonwhites than for whites.[6] We would add that, even where a nonwhite and a white have identical lifetime incomes and therefore the same permanent incomes, the fact that the nonwhite receipts fluctuate much more widely would make it reasonable for the nonwhite as a rule to consume less in order to even out the sequence of feast and famine. This is so given the imperfection of capital markets, particularly for nonwhites, and the uncertainties as to the periods and amplitudes of the fluctuations. Nonwhites who want to avoid the risks of very low consumption would have to maintain reserves that would reduce their lifetime consumption. If the nonwhites do not steadily save more and consume less (and the data, despite their very frequent use in analyses of the consumption function, are piecemeal and ambiguous) they are worse off than whites, not merely because their lifetime income and consumption are generally lower, but because these are also much unsteadier.
The upshot of these considerations is to reinforce the judgment that the greater instability of nonwhite income makes nonwhites worse off compared with whites than the comparisons of their averages and totals accumulated over time would indicate.
For both nonwhites and whites, the growth rates of income to persons are generally lower than those of family income since both white and nonwhite women increased their participation in the labor force and their contribution to family income. White women participate less, but in the postwar period increased their participation more than nonwhites. In March 1969, it was 12 percentage points lower for white women, but was as much as 20 points lower in past decades.[7] As a result, the relative (to white) growth rate of income to nonwhite persons (the aggregate of men and women) is higher than the relative growth of non-white family income. The growth rate of nonwhite income to persons was more than double that of white (3.1 percent per year compared with 1.5). The growth rate of nonwhite family income was about one-third higher than white. (3.8 percent per year compared with 1.8 percent.)
Short-term ups and downs in the labor market have long-term effects on race differences in income. Slack markets reduce incentives for employers to train unskilled labor for more highly skilled, better paying jobs; and reduce the rewards to employees for investing time and money in acquiring skills.[8] Labor markets that are tight enough and last long enough have the reverse effect. Employer prejudice may be overridden by the gap between the demand for and the short supply of the favored kind of labor. And there are net incentives for minorities to migrate to higher income regions, occupations, and industries. Tight labor markets not only alter and increase a minority's productive potential by offering it job training and experience in higher level skills, they also, as John McCall suggests, provide employers with information as to what that potential is and so can modify stereotyped underestimates.[9] The Korean War and again the Vietnam War were the times of greatest improvement since World War II in nonwhite median income. (The more fragmentary evidence on World War II gains for nonwhites shows the same. The changes for white as well as nonwhite women during World War II may be the most familiar example of the immediate short-term and the residual long-term effects of a very tight labor market.) Income to nonwhite persons has not returned to pre-Korean troughs either in absolute terms or relative to white.
This again helps to clarify and distinguish possible objectives. We observe that nonwhite median income has improved at a greater rate than white median income for both persons and families and that in relative terms the improvement for nonwhites has been greatest at the low percentiles for persons. Thus, efforts to achieve the distinct though related and useful goals of reducing poverty and hard-core unemployment would be directed at those parts of the income distribution where the greatest nonwhite relative improvement has already occurred, leaving the higher percentiles relatively unaffected. Notice that in Fig. 13 it is only in the upper 10 percent of the income distribution that we are not at least 95 percent confident that nonwhite income has improved relative to white income. And in the upper 10 percent, we are talking not just about millionaires: income to nonwhite persons at the 90th percentile in 1967 was only about $6,796. Thus, a substantial effort aimed at the higher and middle percentiles as well as at the lower ones will be required in order to equalize white and nonwhite income distributions.
The pattern of relative nonwhite improvement in income position for families is more dependent on the choice of years than in the case of persons. The trend of greater improvement for lower percentiles is clear for persons when any earlier year for which CPS data are available is compared with either 1966, 1967, or 1968. However, in the case of families, we find this pattern occurring in a comparison of either 1945 or 1949 with 1967, but not for either 1947 or 1948 with 1967. Fig. 14 illustrates the 1947 to 1967 comparison for families. We find a horn-like opening at the lowest percentiles, but there is roughly equal improvement from about the 50th to the 80th percentiles, so that overall the improvement is slightly greater for the upper half of the distribution than for the lower half for families since 1947.[10]
The greater relative improvement in total money income at the lower percentiles displays the effect of the increase in transfer payments since World War II (an increase that has, of course, been neither steady over time nor uniform in its effect on all categories of low income groups). In fact, as Fig. 15 shows, the personal income ratio-at-quantiles curves have tended to rotate clockwise since the late 1940s, with the upper percentiles as a fixed pivot.
The short run behavior at the low percentiles might be expected to exhibit the combined effect of this nonuniform trend of increase in transfer payments and a cyclical instability even greater than that at the median. And, in fact, in periods of cyclical expansion such as the long period of growth from 1958 to 1967, the nonwhite to white income ratios grew more rapidly at the lower percentiles than at the median but declined less, or even grew, in comparison with the median at the lower quantiles during contractions such as that between 1948 and 1949.
Using unpublished census data, we have been able to make a comparison between white and nonwhite wage and salary distributions in 1939 and 1967. This 28-year span seems the longest permitting any confident comparison. Fig. 16 shows the wage and salary ratios for the two years, both for full-time and part-time workers.
It is quite clear from this graph that for about 80 to 90 percent of the wage and salary distribution, the nonwhite to white ratios have roughly doubled in the 28-year period. The trend of greater improvement in lower percentiles noted earlier in the case of total money income to persons is not present in the case of wage and salary income to persons -- a confirmation that improved welfare arrangements benefiting nonwhites in the lowest percentiles rather than higher earnings account for the greater closing of the gap at that end of the distribution. (Detailed data recently made available by color on types of income other than earnings for 1968 strengthen this conclusion.)
For one thing, the changes over time observed in such statements depend on the time covered, and many of the comparisons measure the difference between two rather accidentally chosen years. Given the relatively extreme fluctuations of nonwhite income, this has a very large effect. The most pessimistic conclusions stem from comparing the Korean highs with pre-Vietnam low points. But even the two most recent decennial censuses can mislead. They deal with income in 1949 and 1959, two recession years, and this has strongly influenced the results of many investigations. The censuses are a unique, invaluable, and much used source of detailed income data. But are not unaffected by their arbitrary position in the business cycle. This has biased not only longitudinal studies but cross-sectional ones as well.
To illustrate, consider the three ratio-at-quantiles curves for persons in 1949, 1959, and 1966 shown in Fig. 15. Here we can see that the improvement for nonwhites relative to whites between 1959 and 1966 greatly exceeds that between 1949 and 1959. The use of 1959 data without corrections based on other years may bias conclusions from cross-sectional studies as well. To take an example, prospective lifetime earnings and relative marginal returns to schooling of whites and nonwhites are often estimated by using only the 1959 income of various age cohorts in that year. This neglects trends as well as cyclical effects -- and, as we have seen, the long-term percentage rate of growth in median income for nonwhites substantially exceeds that for whites; and nonwhite income relative to white is particularly low in recession years.
2. Focus on Income to Men
A second thing that has frequently misled inference in the field is the focus on income to men. A good many studies deal with male income but draw conclusions about the total nonwhite population. Fig. 17 suggests the combined biases that can result from the arbitrary choice of years and an exclusive focus on men. It shows the ratio-at-quantiles curves for men in 1949, 1959, and 1966. The improvement to 1966 for males is clearly less dramatic than for aggregate persons, and from 1949 to 1959 the income ratios decline for many percentiles. This indicates that the improvement for females would be even greater than that for aggregate persons. Unfortunately, the median income for nonwhite females in 1949 fell within the lowest income interval published so that no income ratios could be computed for the lower half of the distribution.
3. The Use of Averages Rather than Distributions
A third difference between the results we have presented and many more familiar statements is that the latter tend to use only the median or mean or just a few points on the distribution. The possible misleading effects of such a limitation are illustrated neatly by Fig. 18, the ratio-at-quantiles comparison for families in 1945 and 1952. The two curves cross just about at the median, so that a comparison of ratios of median incomes only for the two years would lead one to conclude that there had been no change in the income position of nonwhites relative to whites. However, as can be seen from this graph, there were relative changes at all parts of the distribution except the median. The ratios improved in the lower half of the distribution, and declined in the upper half. This is, of course, an extreme case, but it illustrates nicely the need to consider the complete distribution of income rather than only the median.
The 1945 to 1952 comparison serves as a reminder also that the starting points for the continuous series on income after the war -- in 1947 in the case of families and in 1948 in the case of income to persons -- are accidental. This puts at hazard some of the familiar generalizations about subtrends within the postwar period. The Current Population Survey of income got underway in 1944 and information by color has been put out regularly only since 1947. However, distributions can be obtained for total income to families in 1945, and we have used them in Fig. 18. The CPS sample at that time was much smaller than the current one (8,000 households compared with about 50,000 households in 1968); and the sampling error is larger. More serious (since we can take account of the sampling errors), there may be some undisclosed systematic bias limiting the use of the 1946 survey of 1945 income. For example, we do not know what the effects of the inclusion and exclusion of members of the armed forces were, and this could be important for the year 1945. Nonetheless, this and other evidence suggest that most of the improvement from 1939 to 1947 took place between 1941 and 1945.[12] Statements that divide the postwar period into two subperiods -- a Golden Age followed by a Fall[13] -- are dubious for their start as well as their finish.
4. Problems in Exclusive Use of Family Income
Fourth, most of the common statements refer to nonwhite relative family income, and as we have seen, family income is a more rubbery unit subject to alteration with changes in family structure. Nonwhite relative family income grew more slowly than income to persons.
5. Income Differences Versus Income Ratios
Fifth, several of the familiar statements refer to increasing dollar differences between whites and nonwhites.[14] We have taken ratios or percentage differences or logarithms or the like as more appropriate measures from a welfare standpoint.[15] If one regards the absolute dollar difference as defining the problem, then it would appear that Negro male wage and salary earners were much better off in 1939 compared with whites than they are today. The median of their wages and salaries was $460, 41 percent of whites, and $652 less. In 1967 Negro median wages and salaries were about 64 percent of whites, and $2,464 less ($4,369 compared with $6,833). If we adjust the 1939 wage and salary income to 1967 dollars,[16] we get median nonwhite and white incomes of approximately $1,095 and $2,650, for an income difference of about $1,555. If the nonwhite median had been about 77 percent of the white median in 1967, the dollar difference would have been about the same as in 1939. Eliminating a $1,550 difference would then mean a 29 percent increase in income for nonwhites in 1967, whereas in 1939 it would have meant more than doubling nonwhite income. It seems implausible to suppose the marginal utility of the dollar difference is the same in both cases. In fact, while appropriate in some circumstances, the use of absolute or dollar differences here (as in the case of discussions of the economic development of The Third World, and the gap between the rich and poor countries), tends to serve a hortatory rather than an analytic function. (The Japanese per capita GNP tripled in the decade beginning 1953 while that of the United States increased by less than 50 percent. The Japanese lost ground in dollars but most of us, including the Japanese, think they are catching up.) Welfare considerations aside, even for purposes of prediction, percentage rates of change have an obvious relevance to economic growth. One would expect that a $1,550 increase would take a good deal longer starting from a $1,095 base than starting from a $4,400 base. It is because the rate of change or amount of growth at a given date is related to size at that date that exponential functions are generally useful in forming our expectations about growth.
6. Comparison of Population Proportions at Fixed Income Levels
Sixth, the ratio of proportions[17] at a given income is frequently used in comparing white and nonwhite proportions above or below a poverty line or some given income levels. However, aside from the troubles that can stem from focusing on one or two arbitrary dividing points in the distribution, such ratios of cumulative percentages do not directly measure relative income and, as we suggested earlier, can be quite misleading. Consider the case of the poverty line for families: 56.0 percent of nonwhite families and 16.5 percent of white families were below the poverty line in 1959, giving a ratio of .295 for the white-to-nonwhite incidence of poverty. In 1968, 32.4 percent of nonwhite families and 8.4 percent of white families were below the poverty line, giving a white to nonwhite ratio of .259. The white to nonwhite ratio then, decreased, suggesting that in relative terms things improved more for whites than for nonwhites. However, if we compare proportions above the poverty line, we find that the nonwhite to white ratio increased from .527 in 1959 (44.0 percent for nonwhites and 83.5 percent for whites) to .738 in 1968 (67.6 percent and 91/6 percent for nonwhites and whites respectively). The nonwhite to white ratio of proportions of families above the poverty line, then, increased, suggesting that in relative terms things improved more for nonwhites than for whites.[18]
The results are similar when we compare proportions above and below $8,000. The nonwhite to white ratio of proportions of families above $8,000 (price adjusted, in 1967 dollars) was .278 in 1947 (5 percent of nonwhite families and 18 percent of white families) and .509 in 1967 (27 percent of nonwhite families and 53 percent of white families). This suggests that in relative terms the situation improved more for nonwhites. The white to nonwhite ratio of proportions below $8,000 decreased from .863 in 1947 to .644 in 1967, indicating greater relative improvement for whites.[19]
The reason for this apparent contradiction is indicated by the curves shown in Fig. 19. If we have two curves of identical shape but differing only by a translation along the horizontal (income) scale, we would find that income differences would be the same for all quantiles (Fig. 19a). Or, if the horizontal scale were log income, we would find that income ratios would be the same for all quantiles. But with either income or log income, the use of the ratio of proportions measure would indicate increasing nonwhite to white ratios if we compare proportions (or quantiles) below incomes, and decreasing nonwhite to white ratios for proportions above incomes. Extending this to changes over time, even if the translations over time for the income distributions were the same for whites and nonwhites (Fig. 19b) on the income scale (or on the log income scale), we would still find that white to nonwhite ratios of proportions below any income figure (low income or high) would decrease, and nonwhite to white ratios of proportions above any income figure would increase. Of course, the nonwhite and white curves do not differ only by a translation on either the income scale or the log income scale, nor have changes over time preserved the shapes of either income curve. But the point here is that ratios of proportions above or below given income levels may reveal little or nothing about relative improvement over time of white and nonwhite incomes.
7. Migration and "The Reverse Harvard-Yale Paradox"
A seventh point that has sometimes misled inference has to do with a phenomenon we have thought of as "The Reverse Harvard-Yale Paradox," a title suggested by the familiar story of a student who transferred from Yale to Harvard (or vice versa, depending on whether a Yale or Harvard man tells the story) and raised the average in both places. The reverse statistical phenomenon may work to reduce income ratios when we infer them from subgroup changes over time. And, in fact, it may have something essential to do with the dynamics of relative growth, with shifts of minorities from low income regions, occupations, and industries to higher income regions, occupations, and industries. For one might expect to find: (1) that the ones who make the shift are in general better educated, more enterprising, and have higher incomes than the average in the places that they leave; (2) that, as a result of the shift, on the average the move will raise their own incomes; but (3) not immediately to the average level at their destination. From a national standpoint, that is from a standpoint including all those involved both at the starting point and at the destination, there is then a net relative improvement. But looking at the starting point and at the end point separately, relative income at each place may decline. More generally, the relative improvement in the national aggregate could be expected to be larger than those displayed in the subtotals.
This has happened in the case of geographical migration. There has been a very large migration of nonwhites from the South to the North and West. There is a substantial evidence that the migrants were younger and better educated than the average.[20] Between 1955 and 1960 the South lost a fifth of all the nonwhite men 25 to 29 years old who had some college training, but only 6 percent with elementary schooling. And, in fact, the regional ratios between the two census years show declines in the income of Negro men while the overall ratio was static,[21] and in the surge since 1959 the improvement has been larger for the total than would be inferred from the subgroups. The smaller the geographical subunit, the more misleading the subunit income ratios may be; so the trouble is likely to be even more acute for cities or neighborhoods than for regions.
A migration that means a net improvement from the standpoint of the aggregate of those involved at the starting and end points of the migration is all to the good if one is interested in people. If one is interested in places rather than people, on the other hand, the story may be very different. And, as Harry Johnson points out, politicians tend to come attached to places.
As our formulation above suggests, this problem in comparing sub-groups affects not only geographical migration, but also shifts to higher income job categories or higher income industries. To take the case of occupations, the lower relative income of nonwhites is a product of two components: an under-representation of nonwhites among job categories that pay high average incomes, and a lower than average income for nonwhites within the job categories.[22] A shift in the proportion of nonwhites toward the higher paying job categories then can be accompanied by a transient lowering of nonwhite relative pay within the job category. To overcome the large remaining gap between nonwhite and white income, however, it is plain that ultimately there will have to be large improvements in both components of the product, that is, in the relative distributions of nonwhites in jobs and their relative rate of pay within the job categories.
8. Occupational Shifts Understate Income Changes
An eighth point about estimates of changes over time in nonwhite relative income specifically affects those estimates of very long term trends that have been made using the occupational distributions in the decennial censuses with fixed wage and salary or other income weights. Such estimates do not reflect the relative improvements in nonwhite real incomes within job categories, but only the improvements in the distribution of nonwhites among jobs. They estimate only one of the two components of the product that measures changes in relative nonwhite income. Moreover, some of the best known of these use extremely broad categories.[23] However, the effects of past and current discrimination in the job market are displayed not only in the distribution among jobs, but also in differences in the rate of pay within jobs. The analogy sometimes suggested with indexes of output where one wants to eliminate the effects of price change does not apply. In estimating the changes in relative nonwhite income, one needs to take into account changes in real income relative to white within occupations. Gary Becker notes,
A decrease in discrimination could increase merely the relative income of Negroes within an occupational category and not change their relative occupational distribution. However, since discrimination against Negroes has been greater in the more skilled occupations, a large decrease in discrimination would probably also increase their opportunities in these occupations.[24]Supposing this is true, the magnitude of a total reduction in discrimination has nonetheless to be measured as a product of the two components.
Second, if we project the rate of convergence of nonwhite and white median personal incomes since 1948 into the future, it will take until the end of the century before nonwhites overtake whites. And longer than that for median family income. Extrapolations at the lower end of the distributions are somewhat more promising. As for the top end of the distribution, simple extrapolations would show no future convergence at all.
Given the present large disparities, convergence will require a drastic improvement in the relative distribution of nonwhites among occupations and an improvement of their relative income within occupations. This in turn will require changes in education. But not only that. In the next section we consider the effects of job distribution and years of schooling, as well as the effects of the age structures of the two populations.
[1]The business cycle peaks and troughs as set by the National Bureau of Economic Research for the period are: peak November 1948, through October 1949, peak July 1953, through August 1954, peak July 1957, through April 1958, peak May 1960, through February 1961. See Moore and Shiskin, 1967, p. 25. The income figures are, of course, annual totals and therefore tend to hide or distort short-term fluctuations that peak at various other times of the year.
[2]Creamer, 1956.
[3]See Gilman, Chapter 3, 1963. Barbara Bergmann and David Kaun, on the other hand, did find support for the "last hired" hypothesis in their study of Negro unemployment. See Bergmann and Kaun, 1966, pp. 90-99.
[4]Here and in the analyses of the changes of income distribution over time, we have used unpublished census data to extend the CPS income to persons series back from 1953 to 1948.
[5]See Brady and Friedman, 1947; Duesenberry, 1948; Tobin, 1951; Duesenberry, 1962; and Friedman, 1957.
[6]The marginal propensity to consume, k, can be estimated from income, y, and consumption, c, with the equation c = k y + e. If we take account of cyclical variation in income, the equation becomes
c = k' (yp + yt) + e', where yp represents permanent income and yt represents transient income. yt has zero mean and some positive variance and is independent of yp. The regression coefficient for the second equation is estimated as
.
is therefore an unbiased estimate of k only when Var (yt) = 0, but otherwise it will underestimate marginal propensity to consume.[7]Monthly Labor Review, June 1970, p. 12.
[8]See Johnson, 1966.
[9]McCall, 1970. McCall's work, done independently and in parallel with our own, used different methods (a Markov mover-stayer model) and different data (Social Security Administration work history information). It reaches a similar conclusion about the greater instability of nonwhite incomes.
[10]From 1945 to 1967 the improvement in family income in the lower half is triple that in the top half. For possible limitations in the 1945 data, see page 48.
[11]U.S. National Resources Committee, 1938.
[12]In fact by 1944: 1945 was a year of recession from the wartime peak.
[13]See, for example, the statements of Tobin and of Ross quoted in the introduction.
[14]Fein and Michelson hold the problem is that "the absolute spread between white and nonwhite median family income is rising." Washington Post, January 14, 1968. Vivian Henderson writes, "The real predicament" of Negroes is that they are "losing rather than gaining ground in reaching dollar parity," and "People do not spend or save percentages. They spend and save dollars." Henderson, 1967, pp. 76-104.
[15]The use of differences in this context seems to rest on the assumption that utility is proportional to income, or at least that the slope of utility as a function of income is constant and that prudent persons maximize expected income. However, at least since Daniel Bernoulli and Gabriel Cramer in the 18th century, it has been clear that such a rule of behavior is inconsistent with observed practices both of insurance and fair gambling. Cramer and Bernoulli indicated that not expected income but the expected utility of income is maximized. And the utility of an increment in income depends on the amount of income to which the increment is added. While plausible utility functions may differ from each other for extreme values of income, the logarithm of income is the most widely accepted approximation for the moderate ranges of income that include all the income distributions considered in this study. For an excellent historical and critical comment on these matters, see Savage, 1954, pp. 91ff.
[16]The adjustment is actually based on the change in the price index between 1940 and 1967, since the value of the index for 1939 could not be found.
[17]See footnote 1, p. 19.
[18]U.S. Bureau of the Census, 1969, Table D.
[19]U.S. Bureau of the Census, 1968, p. 8.
[20]See Newman, 1966.
[21]See Batchelder, 1964.
[22]The nonwhite to white income ratio R = 
. 
, where Gn and Gw are nonwhite and white incomes respectively, and 
represents the income of a population that has the white job distribution, but the nonwhite rates of pay within each job category. Thus, the factor 
is an index of relative job distributions with fixed nonwhite income weights, and the factor 
is an index of relative income within job categories, using fixed white job distribution as weights.
[23]See, for example, Becker, 1957, Chapter 9. In this fruitful and pioneering theoretical work, Becker presented an index constructed by applying 1940 census estimates of 1939 wage and salary weights to three broad categories for whites and nonwhites: skilled, semiskilled, and unskilled.
Elton Rayack has criticized Becker's index for the use of fixed income weights and constructed an index of his own designed to reflect the fact that there is a narrowing of differentials between the average earnings of skilled workers of both races and the average of both races in unskilled occupations. (Rayack, 1961, p. 109. Becker, 1962, p. 214.) Rayack's point, which Becker answers adequately, is quite different from our own. Becker's index measures improvements in job distribution among nonwhites, but it says nothing about changes in nonwhite relative income within job categories. neither, however, does Rayack's index.
Some misunderstanding may be created by references to the supposed stability from 1910 to 1950 of such an index of the relative changes in the distribution of nonwhites among very broad job categories as implying that there was no change in market discrimination in those 40 years. "The average occupational position of Negroes has risen quite strikingly in both the North and South, but their position relative to whites has been remarkably stable; in the North this was only slightly higher in 1950 than in 1910, and in the South it was slightly lower in 1950 than in 1910. While many important and relevant changes may have taken place in both regions, a very tentative conclusion from this stability would be that neither striking increases nor striking decreases in discrimination against Negroes have occurred during the last four decades." See Becker, 1957, p. 125. Cf. also Batchelder, 1964, p. 527.
However, for the reasons given, as well as several others connected with problems of separating the effects of past discrimination in schooling, etc., from current market discrimination, that index does not measure Becker's "Market discrimination coefficient," which Becker defines as the proportional difference between the equilibrium wage rates for whites and for nonwhites with and without discrimination -- assuming white and nonwhite labor are perfect substitutes. A good many such inferences have been much less tentative than Professor Becker's.
[24]Ibid., p. 114.
[25]As an indication of the possible effect, consider the relationship between the rate of change in GNP with the rate of change of white and nonwhite median incomes. Although the relationship is not very strong using only one independent variable, the regression results are nevertheless interesting. Letting x = the rate of change in GNP, and Y1 = rate of change in median income to nonwhite persons, Y2 = rate of change in median income to white persons, Y3 = rate of change in median income to nonwhite families, and Y4 = rate of change in median income to white families, we find the following equations:
