Earnings by Size: A Tale of Two Distributions

The Review of Economic Studies, Ltd.

Earnings by Size: A Tale of Two DistributionsAuthor(s): Alan HarrisonSource: The Review of Economic Studies, Vol. 48, No. 4 (Oct., 1981), pp. 621-631Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2297201 .

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Review of Economic Studies (1981) XLVIII, 621-631 0034-6527/81/00540621$02.00

? 1981 The Society for Economic Analysis Limited

Earnings by Size: A Tale of Two

Distributions ALAN HARRISON McMaster University

Part of the "conventional wisdom" on the subject of the size distribution of earnings is that earnings are approximately lognormal, but with an upper tail which is better described by the Pareto function.1 The primary purpose of this paper is to re-examine these propositions, using British New Earnings Survey data disaggregated by occupational groups.

We have several reasons for feeling that such a re-examination is timely. First, most previous studies have considered the distribution among all employees, while theoretical considerations suggest that disaggregated data might be more appropriate. We elaborate this point in Section 1 of the paper. Second, the method of estimating the parameters of the Pareto and lognormal functions has not always been given the attention it deserves. In Section 2 we address this issue, and also refer briefly to the related one of goodness of fit.

The remainder of the paper presents parameter estimates for the Pareto and lognormal functions, using first aggregate data (Section 3), and then distributions within occupational groups (Section 4). Section 5 compares these two sets of results, and at the same time re-assesses the conventional wisdom of the superiority of the Pareto function in the upper tail. Section 6 summarizes our findings.

1. THEORETICAL CONSIDERATIONS

The history of the Pareto function dates back to the original discovery by Pareto in 1897 that a number of observed earnings distributions, principally from European countries, were well described by:

F(y) = > y-<1 (1)

where a >0, and y = Y/ YL, YL being, in Pareto's original version, some minimum necessary to sustain life. Over time, however, has come the realization that only the upper tail is Pareto in form, and a number of theories have been proposed to explain this phenomenon. Perhaps the best known work in this area is that of Lydall (1968). Proceeding from the observation that the "standard distribution" (the distribution of earnings among all male employees other than farm workers) has a Pareto tail for the top 15-20% of employees, he advances a "model of hierarchical earnings" based on the notion that "large organisations-which dominate the upper tail of the distribution-are organised on a hierarchical principle" (1968, p. 8).

The important point about Lydall's theory, in the context of this paper, is that it applies to an individual enterprise. If the theory is correct therefore, we should expect to find Pareto upper tails within the occupational groups which are comprised predominantly of the employees of these organizations, so that a full examination of the theory requires the use of appropriately disaggregated data, as well as a consideration of the standard distribution.

621



622 REVIEW OF ECONOMIC STUDIES

Equally compelling reasons supporting the use of disaggregated data can be found in the case of the lognormal function. Aitchison and Brown, for example, note that the law of proportionate effect, postulated by models predicting lognormality, is less appropriate when we are considering a heterogeneous group of workers than if the population is divided into sectors, within each of which the postulate applies (1954, p. 95). The distribution of earnings among the employees in each sector will then be lognormal; the proposition that the distribution among all employees will also be lognormal relies on further assumptions however. Two which together yield this result, and for which Aitchison and Brown cite empirical support, are that:

(i) the variance of the logarithm of earnings is constant for all sectors; (ii) the number of sectors is sufficiently large for the distribution of the arithmetic

mean earnings levels of the sectors to be approximately continuous, and this distribution is lognormal.2

Rejection of lognormality at the aggregate level may therefore imply a denial of one of these assumptions rather than a denial of the law of proportionate effect, and the use of disaggregated data allows us to avoid this potential confusion.3

2. ESTIMATING PARAMETERS OF FREQUENCY FUNCTIONS

A common technique for estimating the Pareto constant, a, in equation (1) is to linearize the cumulative function by taking logarithms, and apply ordinary least squares. Aigner and Goldberger (1970) demonstrate, however, that use of the cumulative function violates the requirement that the residuals be identically and independently distributed. They therefore propose a number of alternative estimation methods, one of which is a generalized least squares regression of interval frequencies, that is, frequencies in different earnings ranges. This approach can, furthermore, be applied to a wide class of frequency functions.

A frequency function to be estimated predicts a relative frequency p(x; 0) where 0 is a vector of parameters. Since the data on size distributions are typically grouped, we use the predicted relative internal frequency in the i-th range (bounded by xi and xi+,):

xi_+_

p(x; 0) dx =Xi(0). (2) Xi

Defining fi = ni/n, where n is the sample size and ni is the number of observations in the i-th range, the regression model of Aigner and Goldberger is given by:

fi = Xi(o) + si. (3)

Now our observations, ni, are assumed to be based on a random sample from a multinomial distribution so that the dispersion matrix of the observed numbers is known (Kendall and Stuart, 1969, Vol. 1, p. 355). This in turn means that we know the dispersion matrix of the fi's, and hence of the Ei's. The natural extension is therefore to make use of this information in a generalized least squares framework, although the dispersion matrix is a function of the /i's, the predicted frequencies, so that the efficient estimation of 0 appears to require an iterative technique. As Aigner and Goldberger note, however, the

2 generalized least squares procedure is equivalent to what is elsewhere called minimum x estimation, and direct x2 minimization is computationally much simpler than the iterative routine outlined above; hence, this is the method we adopt here. The work of McDonald and Ransom (1979), who evaluate alternative estimation methods, lends support to this choice.

Once 0 is estimated, the question of goodness of fit of the frequency function must 2 be faced. The use of minimum x2 estimation allows a direct test of the null hypothesis,

but the x2 values we obtain are, for the most part, extremely high. Chesher (1979, p. 8),



HARRISON EARNINGS BY SIZE 623

Kloek and van Dijk (1978, p. 72), and McDonald and Ransom (1979, p. 1521) report similar problems, and all blame large sample sizes. Specifically, when the sample is large, even minor departures from the null hypothesis are almost always detected (see Cochran, 1952, p. 335). Kendall and Stuart (1969, Vol. 2, p. 190) suggest that this can be overcome by an appropriate reduction in the power of the test, to correct an imbalance between the possibilities of Type I and Type II error. This can be done by reducing the significance level as the sample size increases. The choice of the significance level such that the two types of error are balanced is not however an easy matter. As an alternative, one can compute the critical level of the x2 statistic;4 if a number of functional descriptions are being considered, one can then compute critical levels (the highest indicating the "best" fit), a procedure adopted by Kloek and van Dijk (1978). By so doing, however, we may merely be choosing the best from a number of poor functional descriptions, since we have deliberately avoided the selection of that critical level which represents a reasonable significance level for balancing the two types of error.

Another, and entirely new, test of goodness of fit has been proposed by Gastwirth and Smith (1972). This is based on inequality measures, and compares the theoretical measure (under the null hypothesis) with calculated upper and lower bounds. This measure is also not without problems. While it may be reasonable to assert that "any fixed distribution whose theoretical ... [measure] falls outside the bounds by a significant amount should be declared to fit the data inadequately" (Gastwirth and Smith, 1972, p. 320), it is less than clear that the reverse is true; in other words, the test may only detect some instances of badness of fit. Notwithstanding this, the test does seem to represent an improvement over the simple x2 test, and we therefore adopt it, together with the calculation of the critical levels of the x statistics, in what follows.

The inequality measure adopted for the Gastwirth-Smith test was the generalized entropy measure proposed by Cowell and Kuga (1981):

I ,(, +6 l) [= [Yi/ Y)+ -1 (4)

where Y is mean earnings, and /3 plays a role similar to that of a parameter of inequality aversion. Seven values of 13 betweem -0.5 and -6 5 were used, in the hope that this would minimize the problem of inappropriately accepting a frequency function; whenever we refer below to meeting the Gastwirth-Smith criterion, this should be taken to mean that goodness of fit is established, according to the criterion, for all tested values of ,B.

3. RESULTS FOR THE DISTRIBUTION AMONG ALL EMPLOYEES

In this section we report results for alternative functional descriptions of what is, except for our inclusion of farm workers, Lydall's standard distribution. Our data were taken from the 1972 New Earnings Survey: a distribution of gross weekly earnings of 91,968 full-time male workers aged 21 and over, whose pay was not affected by absence, in Great Britain in April 1972. The data used here are a slightly more detailed version of the figures published in Department of Employment (1972, Table 8), dividing the sample into 34 rather than 32 earnings ranges.

We first fitted the data to the lognormal distribution, finding the minimum x2 of 5170X7 at a mean of logarithms (,) of 3 52 (1X69x 10-6), and a standard deviation of logarithms (o-) of 0 39 (8 57 x 10-7), where the figures in brackets are asymptotic standard errors.5 Although the extremely high x2 can be attributed to the sample size, the correspondence between observed and predicted frequencies is generally low. There is however a distinct contrast between the main body of the distribution, from ?19 to ?50, comprising 85% of the total number of employees, and the upper and lower tails. The former is tolerably well described, while in the latter there is a substantial tendency for the predicted frequencies to exceed the observed frequencies, except at the extremes




(Y -'?8 and Y_? ?80) where the reverse is true. In spite of this, the Gastwirth-Smith criterion, using the generalized entropy measure, was satisfied.6

Turning to the description of the upper tail alone, and considering instead the Pareto function, improves matters quite considerably. The Pareto upper tail was estimated for all possible value of YL (see equation (1)) between ?35 and ?80, and the results are presented in Table I. Acceptance or rejection of the null hypothesis is shown, in the column headed GS, by a tick or a cross.

The figures in Table I can be interpreted in two ways. If the extremely low critical levels are ignored, there does appear to be some evidence of a fairly stable Pareto upper tail above ?45 with a coefficient of 3 85, a conclusion reinforced by the fact that all of the fitted Pareto tails meet the Gastwirth-Smith criterion. Notwithstanding this, the values of a for YL= ?70 and YL= ?80 cast some doubt on this interpretation, and if the highest critical level is used as the criterion for selection of the starting point of the Pareto upper tail, the fit for YL = ?70, with a = 3 66, is superior to all others.

TABLE I

Estimates of Pareto's a for the upper tail of the distribution of earnings-Great Britain 1972

2 Sample

YL a SE X CL GS size

35 3 406 2 90 x 10-4 657 77 0 1 40,767 37.5 3 592 3 91 X 10-4 238 77 0 1 33,703 40 3 717 5 15 x 10-4 93*34 0 1 27,519 42 5 3.787 6 62 x 10-4 55 64 3 31 x 10-9 1 22,327 45 3 854 8 44 x 10-4 26 87 3 52 x 10-4 1 18,264 47.5 3.875 1.05 X 10-3 24 59 4 06 x 10-4 1 14,898 50 3 851 1*29X10-3 22 19 4*81X10-4 1 12,145 55 3 893 1 92 x 10-3 18 96 801 X 10-4 8487 60 3 848 2 73 x 10-3 16 22 102 x 10-3 5988 70 3 659 4 91 X 10-3 1.23 0*54 1 3200 80 3 695 8 96 x 10-3 0 89 0 35 1975

Notes: 1. CL is the critical level of the X2 statistic. See text for details. 2. In the column headed GS, a tick or cross indicates acceptance or rejection of the null hypothesis according to the Gastwirth-Smith criterion. See text for details.

Adopting the former interpretation for the moment prompts three observations. First, the number of employees with earnings in excess of ?45 is 19 9% of the total sample, within the 15-20% range Lydall mentions (see Section 1). Second, in spite of the apparently poor showing of the null hypothesis, when judged by the very low critical level of the x2 statistic, the percentage deviations between predicted and observed frequencies, shown in Table II, are generally small. Finally, and in support of the conventional wisdom, the Pareto function's description of the distribution above ?45 is distinctly superior to that of the lognormal, as can be seen from Table II which shows the percentage deviations for both functional descriptions.

When the distribution among all employees is considered therefore, we have two qualifications to the conventional wisdom. The lognormal performs less well, even in the main body of the distribution, than is usually believed, and the function is rejected by a X2 test without question; and a strict interpretation of the x 2 statistics' critical levels for the Pareto upper tail suggests that it applies to only a small part of the distribution rather than to the top 20% of all employees.

The inability of the lognormal to describe the body of the distribution is to some extent a consequence of the attempts of the estimated function to describe, albeit poorly, the tails of the distribution. This became apparent from two experiments we conducted. First we computed the increase in the x2 statistic one earnings range at a time, which demonstrated that 43% of the final figure of 5171 occurs because of deviations between




TABLE II

Percentage Deviations of Predicted from Observed Frequen - cies in the Upper Tail for the Pareto (YL = ?45) and Lognor-

mal Functions

Lower Limit on Percentage Deviation Earnings Range

(?) Pareto Lognormal

45 2*07 9*18 47.5 -3.28 12-86 50 2-27 30*55 55 -3-91 31-29 60 -3-17 32-86 70 9-26 34.33 80 2-33 -6*67

100 0-16 -65*65 200 -19-19 -73 96

observed and predicted frequencies below ?19, and a further 37% because of deviations above ?50. Between these earnings level the "partial" x2 statistic is 1056. The second experiment was therefore to aggregate the observed frequencies in each tail into a single frequency and re-estimate the lognormal function. The mean is unchanged, at 3 52, but the standard deviation, relieved of the need to cope with elongated tails, falls from 0 39 to 0 34. Furthermore, the x2 statistic is reduced to 461, of which 29% occurs in the lower tail and only 15% in the upper tail; and there is a significant improvement in the correspondence between predicted and observed frequencies in the body of the distribution, where the "partial" x2 statistic falls to 259.

4. RESULTS FOR THE DISTRIBUTIONS WITHIN OCCUPATIONAL GROUPS

The next step in our re-examination of the conventional wisdom was to consider the distributions of earnings within occupational groups. Again the data are more detailed than those published (Department of Employment, 1973, Table 80), dividing the sample into 34 rather than 11 earnings ranges, and providing actual numbers of employees in each range (like the data on the distribution among all employees) rather than rounded cumulative percentages. The disaggregation of the total sample yields sixteen broad groups with sample sizes ranging from 1021 (Medical, Dental, Nursing, and Welfare) to 22,170 (Building, Engineering, etc.), and there is also a simple division into manual and non-manual occupations. The full list of groups is given in Table III which presents the results of estimating the parameters of the lognormal for each of the sample distributions.

Since the x2 statistic is so sensitive to sample size, it is difficult to assess the correspondence between this test and the Gastwirth-Smith criterion7 from Table III. We therefore divided each x2 statistic by the appropriate sample size, and compared instead values of X2/n and the Gastwirth-Smith criterion,8 in the hope that this would better allow us to evaluate the null hypothesis of lognormality for the various occupational groups. Inevitably, some contradictions are evident. The fit for the occupational group "managers" does not meet the Gastwirth-Smith criterion, even though its value of X2/n is less than that for each of six occupational groups in which the criterion is satisfied. A similar instance is found in the case of "all manual workers", although this time the value of X2/n is exceeded by values for only two of the groups where the Gastwirth-Smith criterion is met. On the other hand the distribution for "medical, dental, nursing, and welfare" workers is out of line in the other direction, its value of X2/n being the second largest of any group despite the fit of the lognormal satisfying the Gastwirth-Smith criterion.




TABLE III

Lognormal Parameter Estimates for the Distributions of Earnings Within Occupational Groups-Great Britain, 1972

Occupational a 2 Sample

Group (SE) (SE) x CL GS Size

Managers 3 895 0 504 280 97 0 x 6697 (4.06 x 10-5) (2.60 x 10-5)

Supervisors and Foremen 3-641 0-293 298 29 0 1 6174 (1.40 x 10-5) (7.11 x 10-6)

Engineers, Scientists, etc. 3-786 0-476 946 01 0 x 3228 (7-19 x 10-5) (4*06 x 10-5)

Technicians 3 589 0 297 117 50 5-37x1012 1 3190 (2 77 x 10-5) (1-40 x 10-5)

Academic and Teaching 3 787 0 367 150 24 0 1 2692 (5-05 x 10-5) (2.62 x 10-5)

Medical, Dental, etc. 3-585 0 533 244 08 0 1 1021 (2.84 x 10-4) (1.56 x 10-4)

Other Professional 3 765 0-480 204-36 0 1 3324 and Technical (7.09 x 10-5) (3.98 x 10-5)

Office and 3-371 0-366 870-70 0 x 8263 Communications (1.63 x 10-5) (8.20 x 10-6)

Sales 3 435 0 393 205-41 0 1 4043 (3-84 x 10-5) (1.94 x 10-5)

Security 3 504 0-438 310-16 0 x 2485 (7-74 x 10-5) (3.95 x 10-5)

Catering, Domestic, etc. 3-126 0-389 109-44 1-11 x 10 -l 1634 (9-30 x 10-5) (4-70 x 10-5)

Farming, Forestry, etc. 3-084 0-346 414-22 0 x 1647 (7.28 x 10-5) (3-68 x 10-5)

Transport 3-464 0 309 218-92 0 J 6783 (1.41 x 10-5) (7-13 x 10-6)

Building, Engineering, 3-487 0-290 319-80 0 1 22,170 etc. (3.80x10-6) (1.92x10-6)

Textiles, Clothing, 3-358 0 285 25 56 0-742 1 1568 etc. (5-21 x 10-5) (2.62 x 10-5)

Other Occupations 3 407 0-325 650-16 0 1 17,049 (6-24 x 10-6) (3.14 x 10-6)

All Manual 3-423 0-356 3784-88 0 x 58,170 (2.19 x 10-6) (1. 1 x 10-6)

All Non-manual 3-666 0-436 1424-31 0 1 33,798 (5.65 x 10-6) (2-94 x 10-6)

All Occupations 3 521 0 393 5170-70 0 v 91,968 (1.69 x 10-6) (8-57 x 10-7)

Note. See Notes to Table I.

Aside from these peculiarities, the fit to the lognormal of some of'the individual groups' distributions improve quite noticeably over the aggregate lognormal fit when judged in terms of relative sizes of X2/n. This is particularly true of the two groups "building, engineering, etc.", and "textiles, clothing and footwear". For the latter group, even the x2 is very low, with a critical level of 0742, indicating a very strong showing by the null hypothesis of lognormality, although it is the former group which records the smallest value of X2/n. In spite of this improvement, problems still occur in the tails of some distributions, but in a number of cases the difficulties are more persistent in the lower than in the upper tail. For instance, it was often possible to reduce the x2 statistic considerably by aggregating only lower tail frequencies,9 whereas for the distribution among all workers, a similar experiment proved far less successful.




In Table IV we present the results of fitting the upper tails of the distributions within occupational groups to a Pareto function. A version of the table equivalent to that given in Table I would stretch to some eight pages, and we therefore include only some of the estimates, although we have attempted to make a choice which is representative of the full table.10 Rather than comment in detail on Table IV now, it seems sensible to postpone the discussion until Section 5, in which the conventional wisdom is re-examined in the light of the evidence of Tables III and IV. It is to this section that we now proceed.

TABLE IV

Pareto Upper Tail Estimates for the Distributions of Earnings Within Occupational Groups

Occupational Sample Group YL a SE X CL GS Size

Managers 50 2 496 2 12 x 10-3 81 80 0 / 3169 55 2 685 2 97 x 10-3 27 05 1 95 x 10-5 x 2646 60 2*815 4 06 x 10-3 8 59 3 53 x 10-2 x 2173 70 2 922 6 95x10-3 4 52 0105 v 1442 80 3 060 118x 10-2 3 24 x 10-4 0 985 1006

Supervisors and 45 5 867 2 22x10-2 12 25 9 28x10-2 1595 Foremen 50 6 180 4 53x10 2 7.55 0o183 1 889

55 6 554 9 01 x 10-2 3 43 0 488 1 515 70 5 589 0 371 0 34 0 845 1 100

Engineers, 50 4 055 1.32x10-2 88 63 0 1 1568 Scientists, and 60 5 367 4 46x10-2 11 12 1.11x 10-2 717 Technologists 70 5 317 0 108 11.15 3 79x10-3 1 309

Technicians 45 5 709 5 07 x 10-2 12 53 844 x 10-2 664 55 6 783 0221 5 04 0 283 1 225 60 6 727 0415 5.05 0*168 1 123

Academicand 50 4 080 1 83x10-2 13-42 1P97x10-2 960 Teaching 55 4424 3*04x10 2 159 0*811 1 691

60 4*551 4 77 x 10-2 0 60 0896 1 480 Medical, Dental, 35 2 076 9 88 x 10-3 49 83 6 72 x 10-7 456

Nursing, 45 1 764 1*46x10 2 37 32 4 08x10-6 x 236 and Welfare 50 1 724 1*77 x 10-2 35 90 9 93 x 10-7 x 191

Other 50 2 984 7*87x10-3 43 06 3*59x10-8 1 1204 Professional 60 3 486 1 76 x 10-2 4 35 0 226 1 764 and Technical 70 3 682 3 45 x 10 2 2 04 0 360 x 461

Office and 40 4 989 1 81 x 10-2 31 22 2*72 x 10-4 1406 Communications 50 5 765 7 28x10-2 9 36 9 55x10 2 481

60 5 527 0 218 6 28 9 86 x 10-2 156 70 4 832 0484 2 85 0 240 x 57

Sales 40 4 077 1 87x10-2 7 86 0548 1 909 45 4 251 3 23 x 10-2 5 12 0 646 1 580 50 4 201 5 08 x 10-2 4 96 0 421 1 366 60 3 849 0 102 2 98 0 394 1 161

Security 40 4 010 2 00x 10-2 33 36 1 16x10-4 825 45 4 570 3*84x10-2 11 46 0120 1 562 55 5 016 0*117 8 75 6 75 x 10-2 230 70 6 609 0 696 2*47 0 292 1 75

Catering, Domestic 30 4 449 5 58x 10-2 21*30 6 73x 10-2 x 359 and Other Service 35 5 528 0 152 4 73 0 944 1 204

40 5 653 0333 464 0 865 1 98

Farming, Forestry 25 4 934 4 20 x 10-2 19 93 0 223 x 586 and Horticultural 30 5 830 0 137 11 30 0 586 x 253

37*5 7 570 0 803 5 64 0*845 x 73




TABLE IV-cont.

Occupational Sample Group YL a SE X2 CL GS Size

Transport 42 5 6 147 3 64x10-2 7 32 0 502 1 1065 47 5 6 602 8 05 x 10-2 1 46 0*962 1 565 55 6 544 0 217 1*43 0 839 1 213 70 5 738 0977 0 13 0957 1 40

Building, 50 6 919 3 26x10-2 7 11 0213 1 1556 Engineering, etc. 55 7 393 7.09x10-2 0 76 0 944 x 839

60 7 504 0 144 0 589 0 899 1 445 70 7 301 0 464 0 475 0 789 1 138

Textiles, Clothing, 35 6 551 0 112 7 80 0 731 1 390 and Footwear 42 5 8 085 0 581 2 09 0 978 1 116

45 8272 0961 200 0956 1 74 Other 42 5 6 107 1 74x10-2 13 18 0 106 1 2198

Occupations 55 6 595 0 103 2 34 0 673 1 456 70 7 451 0 696 0 89 0 641 1 96

AllManual 45 6 365 6 34x 10-3 29 46 1 19x 10-4 6604 Occupations 50 6 714 1 37x10-2 11 28 4 61x10-2 1 3476

60 7 192 5 69x10-2 0 23 0972 1 1033 70 7 158 0 181 0 22 0 895 1 340

All Non-Manual 50 3-277 1-31 x 10-3 59-54 1-51 x 10-4 1 8669 Occupations 60 3 500 2 73x 10-3 5 54 0 136 1 4955

80 3 571 8 92 x 10-3 0 14 0710 1 1845

All Occupations 50 3-851 1-29x 10-3 22-19 4-81 x 10-4 1 12,145 60 3 848 2*73 x 10-3 16 22 102 x 10-3 5988 70 3 659 4 91 x 10-3 1 23 0 54 1 3200 80 3 695 8 96 x 10-3 0 89 0 35 1 1975

Note. See Notes to Table I.

5. THE CONVENTIONAL WISDOM RE-ASSESSED

We have earlier briefly discussed certain reservations about the conventional wisdom, based on the estimates reported in Section 3. Now we wish to consider more fully the results from Section 4, derived from less aggregated data, and to compare the findings with our earlier observations. Additionally, the use of these data, prompted by the theoretical considerations of Section 1, allows us to assess empirically certain aspects of the theories discussed there.

We have noted already that, if standard levels of significance are used in x2 tests of goodness of fit, the conventional wisdom is not supported by the observed distribution of earnings among all workers. The distribution is not lognormal, nor is there evidence of a Pareto upper tail among the top 15-20% of the workers. If the results of applying the Gastwirth-Smith criterion are considered instead, however, both hypotheses find some support, in addition to which considerable reductions in the x2 statistics for the lognormal can be achieved by aggregating observed frequencies in the tails of the distribution. At the very least then, it seems reasonable to suggest that the main body of the distribution is approximately lognormal, and that there is support for a fairly stable Pareto upper tail, covering nearly 20% of the workers.

Similar remarks can be made of the results from distributions within most occupational groups. The lognormal descriptions are still only approximate when judged by x values, except in the case of 1568 "textiles, clothing, and footwear" workers, where the relatively low sample size favours the null hypothesis. Nevertheless, many of these descriptions are deemed acceptable by the Gastwirth-Smith criterion, as are many of the Pareto upper tails. There are, however, at least two differences between the results from




Sections 3 and 4, the second of which leads us to question the validity of the conventional wisdom more closely than we did when faced with the results of Section 3 alone.

First, on the positive side, the upper tails of the distributions within occupational groups seem to present less of a problem to the lognormal than does the upper tail of the aggregate distribution. Notwithstanding this, we still find support for the Pareto function in the upper tail. This is even true of the group which, on the basis of the 2 test, provides the strongest evidence of lognormality: the "textiles, clothing and footwear" workers mentioned above. For this group, the percentage deviations of expected from observed frequencies in the upper tail are typically smaller for the Pareto (YL = ?42X5) than for the lognormal, although the number of workers concerned is only 116, a little over 7% of the total sample.

Second, and less favourably for the conventional wisdom, the stability of the Pareto upper tail, so apparent at the aggregate level, often disappears when the data are disaggregated and x2 tests suggest that we cannot reject the hypothesis of a different Pareto tail for different lower limits within the same group. Partly, of course, the non-significant x2 statistics reflect the much smaller samples; also the changes in a, as YL changes, are not always statistically significant, since a further effect of the smaller samples is to increase the standard errors of the estimates. Nevertheless, there is no doubt that the results of Table I afford the Pareto upper tail more support than do those of Table IV. Furthermore, this weakening of support is particularly noticeable in the case of, among others, "managers", and "other professional and technical" workers, groups which, on the basis of Lydall's theory, might be expected to contribute significantly to the aggregate Pareto upper tail, and therefore to reflect its stability.

Other features of Table IV prompt further comments. First, we find once again a number of contradictions between the x2 test and the Gastwirth-Smith criterion, and very occasionally a steadfast refusal of the Gastwirth-Smith criterion even to admit that a Pareto upper tail exists at all. Second, as we have mentioned, a good showing by the null hypothesis of lognormality does not preclude acceptance of a Pareto upper tail for the same distribution. This is only true for "textiles, clothing, and footwear" workers if the X2 test is considered, but applies also to other distributions if the Gastwirth-Smith criterion is considered instead. Third, the lognormal description of the distribution of earnings of workers in "all manual occupations" is rejected by the Gastwirth-Smith criterion, while the same distribution has a well-defined Pareto upper tail, precisely the reverse of what is usually believed; acceptance of lognormality in the case of workers in "all non-manual occupations" only serves to confound the issue still more.

If, then, it is accepted that the conventional wisdom should be examined using data disaggregated by occupation, the results lead us to question its validity. Also, of course, they question some of the theoretical underpinnings of the conventional wisdom, consideration of which initially suggested the use of disaggregated data. First, and probably most significantly, the variability of the Pareto coefficients within occupational groups is undoubtedly at odds with Lydall's model of hierarchical earnings. The model applies to individual enterprises, and explains the overall Pareto tail as the outcome of aggregation across enterprises. This explanation is, however, inconsistent with the instability of Pareto's a which we find at a lesser level of aggregation, so that while the model may be appropriate for certain enterprises, the extension of it to a theory of the aggregate Pareto tail appears less appropriate.

Turning next to the theories which predict lognormality, we can see immediately from Table III that the estimates of o- for the different groups vary quite significantly. This is, of course, a violation of the condition noted by Aitchison and Brown for a mixture of lognormal distributions to be itself lognormal. The variation in 0C should not be treated with too much respect, however, since we have not established that all of the distributions within occupational groups are lognormal. Nevertheless, improvement in the descriptive ability of the lognormal for some of the distributions within occupational groups surely




suggests, at the very least, that further consideration of these disaggregated data is appropriate.

6. SUMMARY

This paper has pursued two related themes. Our primary concern has been to re-examine the conventional wisdom of the literature on the distribution of earnings, using disaggregated data, but the considerations of economic theory which prompted us to use these data have also meant that certain aspects of the theories can be assessed. Along the way we have additionally raised, without fully resolving, the question of goodness of fit.

We have demonstrated that certain aspects of the conventional wisdom begin to break down when examined instead with disaggregated data. In particular the Pareto upper tail, stable in the aggregate, becomes unstable for the distributions within a number of occupational groups. The lognormal, on the other hand, registers only a small improvement over the fit of the aggregate data, and then only in some cases, although for one group at least strong evidence of lognormality is found.

These findings refer us back to the theoretical considerations, and suggest that Lydall's model of hierarchical earnings cannot easily be advanced as an explanation of the stable Pareto upper tail in the overall distribution. Less convincingly perhaps, the variation in the estimates of o-, the standard deviation of logarithms, across the occupational groups, may serve as a pointer to the poor performance of the lognormal function as a description of the aggregate distribution.

First version received August 1979; final version accepted January 1981 (Eds.).

I have benefited considerably from the help, advice, and suggestions of Tony Atkinson, Andrew Chesher, Frank Cowell, Frank Denton, Alan Gelb, John Kennan, Les Robb and Tony Shorrocks. I have also received useful comments on an earlier draft from an anonymous referee, and from the participants in seminars at the Universities of British Columbia, Essex, and Western Ontario. Finally, I should like to thank the Director of Statistics at the Department of Employment, G. Penrice, who very kindly made available unpublished data from the New Earnings Survey. The department is not responsible in any way for the use to which the data have been put, neither are any of the above implicated in errors which might remain in the paper.

NOTES 1. See, for example, Lydall (1968, p. 7). 2. Where the number of sectors is small this condition becomes the stronger one that the means be constant

across sectors. 3. Similar remarks apply to Roy's (1950) theory predicting lognormality, since it deals explicitly with

distributions within occupations. 4. The critical level is the significance level at which the null hypothesis is just rejected; see Lehmann

(1959, p. 62). 5. The x was minimized using an algorithm of O'Neill (1971), based on a simplex method for function

minimization due to Nelder and Mead (1965). The relative interval frequencies (equation (2)) were numerically integrated by DCADRE, an algorithm for numerical quadrature from the IMSL library. For the minimization, the terminating limit for the standard deviation of vertex function values was set at 1.0 x 10-6; the number of function evaluations performed before the minimum was reached was usually less than 200 and always less than 300, and each minimization took well under 10 seconds central processor time on the McMaster University CDC CYBER 170/730. Standard errors for the parameter estimates were obtained by computing and inverting the information matrix; see Theil (1971, pp. 384-391). This method seemed appropriate, given the asymptotic

2 equivalence of minimum x and maximum likelihood estimation. 6. The standard procedure for computing bounds for the Gastwirth-Smith criterion requires knowledge

of the interval means (Gastwirth, 1972). Our data on the aggregate distribution lacks this information, however, and only the overall mean is available. Cowell (1980) has derived alternative bounds for this case (as well as for the case where not even the population mean is known), and these were the bounds used to apply the criterion. It should be borne in mind, though, that these bounds are wider so that basing a goodness-of-fit test on them is inevitably less precise than if the interval means were known.

7. Information on the distributions within occupational groups is still more restricted than is that on the aggregate distribution, since the overall mean is not even available. Cowell (1980) provides details of the bounds




in this case, and these are still more coarse than before, so that a tick in Table III need not be equivalent to acceptance of lognormality in the aggregate case. The chance of such a reversal are small however; computing the bounds for the aggregate distribution as if the population mean is not known, for example, widens the bounds only very slightly.

8. We are not intending to claim that such a comparison in any sense constitutes a rigorous test. It does however allow us to isolate the effect of sample size in the x2 statistic, since we are comparing _ (fi -_<hi)2/i with the Gastwirth-Smith criterion, which itself pays no attention to sample size.

9. In an earlier draft of this paper, some results from which are contained in Harrison (1979), we computed parameter estimates for the lognormal from the published distributions within occupational groups. These data report only eleven earnings range, and aggregate observed frequencies below ?18 and above ?80. The outcome was, in all cases a considerable reduction in the x2 statistic which for some groups could also be identified with an increase in the critical level of the x2 value. This illustrates the need to take account of the number of earnings ranges, as well as sample size, when comparing results from studies using different data. Some of Kloek and van Dijk's results, for example, are for observed distributions divided into only 6 earnings ranges (1978, p. 73).

10. A copy of the complete table is available from the author on request.

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Journal of the American Statistical Association, 65, 712-723. AITCHISON, J. and BROWN J. A. C. (1954), "On Criteria for Description of Income Distribution",

Metroeconomica, 6, 88-107. CHESHER, A. (1979), "An Analysis of the Distribution of Wealth in Ireland", Economic and Social Review,

11, 1-17. COCHRAN, W. G. (1952), "The x Test of Goodness of Fit", Annals of Mathematical Statistics, 23, 315-345. COWELL, F. A. (1980), "Grouping Bounds for Inequality Measures Under Alternative Informational

Assumptions" (unpublished paper). COWELL, F. A. and KUGA, K. (1981), "Inequality Measurement: An Axiomatic Approach", European

Economic Review (forthcoming). DEPARTMENT OF EMPLOYMENT (1972), "New Earnings Survey, 1972", Department of Employment

Gazette, 80, 978-1021. DEPARTMENT OF EMPLOYMENT (1973), "New Earnings Survey, 1972: Some Further Results", Depart-

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and Statistics, 54, 306-316. GASTWIRTH, J. L. and SMITH, J. T. (1972), "A New Goodness-of-Fit Test", Proceedings of the American

Statistical Association, Business and Economic Statistics Section, 320-322. HARRISON, A. J. (1979), "The Upper Tail of the Earnings Distribution: Pareto or Lognormal?", Economics

Letters, 2, 191-195. KENDALL, M. G. and STUART A. (1969) The Advanced Theory of Statistics, 3rd edition (London: Griffin). KLOEK, T. and VAN DIJK H. T. (1978), "Efficient Estimation of Income Distribution Parameters", Journal

of Econometrics, 8, 61-74. LEHMANN, E. L. (1959) Testing Statistical Hypotheses (New York: John Wiley and Sons). LYDALL, H. F. (1968) The Structure of Earnings (London: Oxford University Press). MCDONALD, J. B. and RANSOM, M. R. (1979), "Functional Forms, Estimation Techniques and the

Distribution of Income", Econometrica, 47, 1513-1525. NELDER, J. A. and MEAD, R. (1965), "A Simplex Method for Function Minimization", Computer Journal,

7, 308-313. O'NEILL, R. (1971), "Function Minimization Using a Simplex Procedure", Applied Statistics, 20, 338-345. ROY, A. D. (1950), "The Distribution of Earnings and of Individual Output", Economic Journal, 60, 489-505. THEIL, H. (1971) Principles of Econometrics (New York: John Wiley and Sons).



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Earnings by Size: A Tale of Two Distributions

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