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10.1177/1091142104269657PUBLIC FINANCE REVIEWBelman, Heywood / EARNINGS DISPERSION

PUBLIC-SECTOR WAGE COMPARABILITY:

THE ROLE OF EARNINGS DISPERSION

DALE BELMANMichigan State University

JOHN S. HEYWOODUniversity of Wisconsin–Milwaukee

Economists use average wage differentials to examine whether public- and private-sec-tor workers have comparable earnings. Such average differentials, originally developedfor other purposes, fail to measure the true distance from comparability. In short, if aver-age earnings in the public and private sectors are identical, earnings need not be compa-rable. The authors develop alternative statistical measures of comparability that demon-strate that differences in average earnings contribute only modestly to deviations fromcomparability and that state and local governments in the United States deviate morefrom comparability than does the federal government.

Keywords: comparability; mean squared deviation; earnings differentials

1. INTRODUCTION

Three decades of econometric research have investigated whetherthe earnings of public- and private-sector workers are comparable.This work has used large national samples to compare the earnings offederal, state, and local workers to their private-sector counterparts(for extensive reviews, see Gregory and Borland 1999; Bender 1998;Belman and Heywood 1996). The literature attempts to control forself-selection (Robinson and Tomes 1984; Gyourko and Tracy 1988;

AUTHORS’NOTE: The authors thank Morley Gunderson, Douglas Hyatt, Xiandong Wei, semi-nar participants at Lingnan University, and a reviewer for valuable comments. This article is thelatest is a series of joint works that compare public- and private-sector labor markets. Previouswork examines differences in fringe benefit provision, changes in pension provision over time,and the role of sample selection in estimating earnings differences. Additional joint work has fo-cused on detailed state-by-state public earnings differentials, the U.S. Postal Service, and themethodology for determining comparability.

PUBLIC FINANCE REVIEW, Vol. 32 No. 6, November 2004 567-587DOI: 10.1177/1091142104269657© 2004 Sage Publications

567

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Belman and Heywood 1989), it investigates substantial variations inspecification (Moulton 1990; Linneman and Wachter 1990; Belmanand Heywood 1990), and it uses samples specific to narrow jurisdic-tions (Moore and Newman 1991; Belman and Heywood 1995). De-spite these differences, the literature unanimously uses the averageearnings differential to measure the degree of comparability.

The average earnings differential measures whether characteristicstypical of one sector will be rewarded differently in the other. Thus, ifthe characteristics of a public-sector worker result in an estimated pri-vate-sector wage different from his or her actual earnings, that workeris not being paid comparably. Yet the average of such comparisonsshould not be taken as an estimate of comparability. If half of public-sector workers are “overpaid” by 20 percent and half are “underpaid”by 20 percent, the average differential will be close to zero, suggestingcomparability when, in truth, no workers are being paid comparably.

This is more than just the point that averages may not apply to eachworker. Instead, it is a contention that the dispersion in individualearnings comparability must be as important as (and in many casesmore important than) the average, the statistic estimated and debated.For instance, a circumstance in which every public-sector worker hasa 5 percent positive differential is actually much closer to comparabil-ity than the bifurcated case presented earlier in which all workersdeviate by 20 percent.

This point is crucial for at least three related reasons. First, compa-rability is fundamentally not an average concept. Both the relevant le-gal statutes and economic theory make clear that the crucial underly-ing notion is how many workers are how close to individualcomparability (see Belman and Heywood 1996). Yet this is the veryconcept on which the average differential sheds little light. Second,the potential for serious mismeasurement is systematic. It is wellknown that public-sector earnings show less dispersion than private-sector earnings do. Thus, individual earnings differentials favor thepublic sector at the bottom of the earnings distribution and the privatesector at the top of the distribution. Third, the federal sector differen-tial has shown a sizable advantage in favor of the public sector whilethe state and local differentials have been very small, suggestingrough comparability (Smith 1977; Quinn 1979; Moore and Raisian

568 PUBLIC FINANCE REVIEW



1991). Indeed, this suggestion seems so strong that many (Venti 1987;Moulton 1990; Linneman and Wachter 1990; Belman and Heywood1993; Heywood and Mohanty 1995) focus exclusively on the federalsector. This focus is inappropriate if the small average differential inthe state and local sectors hides greater dispersion in the individualdifferentials.

The next section argues in more detail that comparability requiresmeasures beyond the traditional average differential. This is illus-trated by estimating the average differential for each level of govern-ment and demonstrating the dispersion behind those differentials. Thethird section presents alternative measures of comparability that ac-count for this dispersion, ultimately focusing on a mean squared devi-ation (MSD) criterion. These alternative measures demonstrate thatthe state and local sectors are further from comparability than the av-erage differential implies and, contrary to earlier work, are actuallyless comparable than the federal sector. The third section also con-trasts the importance of accounting for dispersion when estimatinggovernment earnings comparability with its near irrelevance when es-timating differentials for race, gender, and union status. A finalsection presents a discussion and conclusions.

2. COMPARABILITY AND AVERAGE DIFFERENTIALS

THE STANDARD OF COMPARABILITY

The level of public-sector compensation plays a role in determin-ing the effectiveness and efficiency of government services. Too higha level wastes resources better spent on other objectives or on reducingtaxes. Too low a level precludes attracting the workers needed to pro-vide the quality of services the public demands. Thus, the standard ingovernmental wage setting has been one of comparability with the pri-vate sector. This standard has a 130-year history in the United Statesand is detailed in a wide variety of federal and state legislation and innumerous court rulings (see Belman and Heywood 1996). While notuniversal, it is far and away the dominant criterion in the determina-tion and evaluation of U.S. public-sector earnings.

Belman, Heywood / EARNINGS DISPERSION 569



The objectives the federal government hopes to achieve from com-parability were outlined in the Reform Act of 1962, which provided amechanism for implementing comparability. These objectives in-cluded using a “logical and factual” basis for the setting of federal pay,ensuring equality between otherwise similar workers in the privateand federal sectors, and ensuring fairness to private-sector employersby neither underpaying nor overpaying federal workers. Economistshave interpreted this and other federal government language as a gov-ernmental objective of ensuring efficiency in the employment ofworkers.1 Thus, while fairness is clearly an objective, there is at leastsome appeal to the private-sector wage being an appropriate or at leasta market wage.

Even this cursory review of the objectives of comparability makesclear that the objectives apply position by position and worker byworker. Taking the example of half the public sector being overpaid by20 percent and half being underpaid by 20 percent helps make thispoint. If, as a private-sector employer, you find yourself competingwith a public sector that hires workers in your field at a 20 percent pre-mium, you surely do not care that somewhere else other private em-ployers easily pay more than the public sector. Equally compelling,public workers who enjoy a 20 percent premium do not compensatethose who are at a 20 percent disadvantage. Moreover, when com-pared to the private sector, no public-sector workers meet the objec-tive of equality of pay with otherwise equal private workers. Further-more, any claim to efficiency or “the market” is surely irrelevant whenyou can find no workers in the public sector actually earning the pri-vate-sector wage. Even the hope of a logical and factual basis for paysetting seems not to be achieved as it seems arbitrary which workersshould enjoy advantages over the private sector and which shouldsuffer disadvantages or how large those differences should be.

In short, the concept of comparability is not an average concept.The objectives of comparability are more nearly met with a greaternumber of workers who are closer to receiving the same earnings in ei-ther sector. While achieving an average differential of zero indicatesthat the sum of log dollars spent on public compensation is identical tothat which would be spent achieving comparability, it does not indi-cate comparability. It remains possible that some, most, or even all




public employees do not receive the earnings of their private-sectorcounterparts. Thus, it becomes critical to evaluate the extent to whichaverage differentials are misleading and to provide a more revealingand useful alternative measure.

These points are increasingly recognized in the literature as re-searchers move away from simple average wages to consider other as-pects of the wage distribution. Thus, quantile regressions have beenused to examine the public-private earnings differential in differentportions of the wage distribution (Poterba and Rueben 1994; Mueller1998; Blackaby, Murphy, and O’Leary 1999). Alternatively, differen-tials have been estimated within different skill levels (Elliot andDuffus 1996) or for different occupational groups (Belman and Hey-wood 2004). Yet none of these studies have developed a single de-scriptive measure that recognizes the individual nature of the compa-rability standard.2

ESTIMATING AVERAGE DIFFERENTIALS

In what follows, we demonstrate that average differentials conceala degree of dispersion that renders the estimates misleading. This setsthe stage for formulating alternative measures of comparability. Fol-lowing the public-sector earnings literature, this research uses the de-composition approach (Oaxaca 1973) in which natural log earningsequations are estimated separately for each sector: private, federal,state, and local. The Oaxaca decomposition is based on equationsestimated as

ln �W X eig

ig g

ig= +′β

ln �W X eip

ip p

ip= +′β

where

g indexes government employeesp indexes private-sector employeesi indexes individualslnWi

s is the log of the wage for individual i in sector s (either g or p)




X is is the observed characteristics of individual i in sector s

βs is the vector of coefficients for the s sectorei

s is the residual for individual i in sector s

Two sets of differentials are then computed by taking a base group,the observations from either the public or government samples, andcomparing the estimated earnings of that group using the other sec-tor’s equation to their estimated earnings using their own sector equa-tion. The differential for a private employee is computed as

θ β βiP

ip g pX= −

′ � � , (1)

while a government employee’s differential is computed as

θ β βig

ig g pX= −

′ � � . (2)

Both of these estimates use only the deterministic parts of the equa-tions and do not incorporate the error terms. These individual differ-entials are then averaged within the sector to provide the typical pri-vate and public base Oaxaca differential.

θ β β β βp

pip

i

ng p p g p

nX X

p

= − = −

′

=

′∑11

� � � �

θ β β β βg

gig

i

ng p g g p

nX X

g

= − = −

′

=

′∑11

� � � � .(3)

Our analysis uses the May 1993 Current Population Survey. Welimit our sample to employed nonfarm males. This results in a finalsample of 7,897 private-sector workers, 409 federal workers, 458 stateworkers, and 779 local workers. The specification adopted for this ar-ticle is as standard as we could make it including as regressors com-pleted education, age and age squared, region of the country, maritalstatus, union status, race, urban residency, broad occupation, job ten-ure, part-time status, and establishment and firm size.3 While thesevariables match those used elsewhere, the tenor of our results does not




depend greatly on the inclusion or exclusion of any particularregressor.

The estimated coefficients for educational achievement, age, andother key demographic variables are consistent with past research onpubic and private wage formation (see Table A2 in the appendix). Theage/earnings profile is similar across sectors as are the roles of race,job tenure, residence in a metropolitan area (except for state employ-ees), marital status (again, the return for state employees is lower thanfor the other sectors), and working part-time. Yet the pattern of returnsto education varies considerably by sector. Using employees who didnot complete eighth grade as the base, returns to completing a highschool education are higher in federal and state government than inthe private sector or local government. Returns to completing a bache-lor’s degree are similar across government employment, ranging from.147 to .176 log points, but the private sector return for a university de-gree is a much larger at .246 log points. Rewards to master’s degreesare similar across the four sectors while professional degrees are mosthighly rewarded in federal and local government. Returns to manage-rial and professional occupations are highest in federal employment,similar in private and state employment, and lowest in local govern-ment. While there are substantial returns to working in larger estab-lishments in the private sector, the coefficients on establishment sizeare not significant in the federal equation and, except for the large neg-ative effects of working in very small establishments, not significantin the state and local government equations. Firm size has large posi-tive effects on wages in the private sector but does not have a statisti-cally meaningful effect on wages in local government. Firm size ef-fects cannot be estimated for federal and state governments as thefederal government and all state employers are in the largest size cate-gory. The fit of the equations is quite good for ordinary least squaresmicro-data equations: the r 2 ranges from .49 to .51.

Table 1 shows private and public base differentials presenting thefamiliar pattern that the federal sector differences are positive (a pub-lic-sector advantage) and of reasonably large size while the state andlocal sector differences are smaller in magnitude and negative.

We then divide our sample into three separate occupationalgroupings—blue-collar, white-collar, and service workers—and re-




peat the estimation within each grouping. In Table 1, we present theprivate-sector base estimates. They indicate a large public-sector ad-vantage for service workers at the local and federal level (especiallylarge at the federal level) and a small disadvantage for service workersat the state level. At the same time, white-collar workers appear sub-stantially underpaid in the local sector and blue-collar workers appearsubstantially underpaid in the state sector. Thus, we easily identifygroups of public-sector workers who appear substantially overpaidand groups who appear substantially underpaid. This happens evenwithin the local sector where the overall differentials indicate nearcomparability. This is exactly the scenario we described in the intro-duction and makes compelling the need to develop a measure of com-parability that does not allow the averaging of over- and underpay-ment.

3. ALTERNATIVE MEASURESOF THE COMPARABILITY

MODIFYING THE PUBLIC-SECTOR MEASURE

Perfect earnings comparability should be identified as the circum-stance in which every individual differential (those traditionally aver-aged) is zero. Thus, a first approach might measure the differencefrom comparability by averaging the absolute value of the differencein the individual predictions. This measure indicates how far away onaverage worker wages are from comparability, the positives and nega-tives not canceling out. This is easily constructed from the estimates


TABLE 1: Public-Sector Earnings Differentials

Private Public Blue- White-Sector Base Base Collar Collar Service

Federal .096 .048 .142 .086 .315State –.047 –.053 –.133 .012 –.042Local –.023 –.002 .041 –.104 .143

NOTE:The wage difference between the public and private sectors is presented in eachentry.



of each worker’s wage in their current position and what they wouldearn in the alternative sector.4

θ β βp

pip g p

i

n

nX

p

= −

′

=∑1

1

� �

θ β βg

gig g p

i

n

nX

g

= −

′

=∑1

1

� � .(4)

This new measure, the absolute differential, is presented in the firsttwo columns of Table 2. Averaging across the two bases, the federaldifferential is 13.2 while the state and local differentials are actuallylarger, each at 15.5. This pattern contrasts with the consistently largerfederal average differential in the literature and indicates that the as-sumption that attention should be paid primarily to the federal differ-ential is misplaced. While all sectors evidence canceling out in the av-erage differential, the federal differential is subject to less than theothers.

A related alternative measure establishes an a priori band that ap-proximates comparability and determines what share of workers fallwithin that band. As an illustration, imagine a band of 5 percent eitherside of exact comparability. Any individual worker with an earningsadvantage or disadvantage of smaller than 5 percent would be in-cluded among those paid in accord with comparability. Using the indi-vidual differentials, we compute just this measure for all three publicsectors. As the third and fourth columns of Table 2 show, only a smallminority of public employees are paid wages approximately compara-ble. Again, the impression that the federal sector is the least compara-ble receives no support. Averaging across the bases, the shares identi-fied as close to comparable are .192 in the state sector, .223 in the localsector, and .254 in the federal sector. Obviously, there is nothingunique about a band of 5 percent. The point is that creating an a prioriband independent of the distribution of the individual differential al-lows for a measure that identifies the share of the workforce that ispaid a roughly comparable wage.5

We suggest as a final alternative a MSD criterion that isolates boththe information contained in the average differential and information




that concerns us, the dispersion in the individual differential. Puttingaside the distinction between public and private base differentials forthe moment, the differential for any given worker can be identified as

( )θ β βi ig i p is g pW W X= − = −′ln � ln � � � , (5)

where the first part of the center term is the estimated earnings of thepublic-sector worker and the second part of the center term is the esti-mated private-sector earnings of that same worker.6 The typicalOaxaca estimate can use either an actual or predicted value for an indi-vidual’s “own” sector wage. We use estimates of the own and othersector log wage. Thus, by design, θi is purely a prediction and does notinclude an error term.7

At issue is how far that value is from comparability. Identify θc asthe ideal value of equation (1), which reflects full comparability and isthus zero. Then the MSD criterion can be expressed as8

MSD(θ) = 1/nΣi(θi – θc)2 (6)

and, as with the typical Oaxaca, can be estimated separately with pri-vate and government bases.9 As θc is zero, this amounts to simply theaverage squared value of the individual public-sector worker differen-tials and will be strictly positive. The advantage of this measure com-pared to the absolute differential is in interpretation. By adding andsubtracting the mean value of θi and by expanding the square identi-fied in equation (6), the criterion can be rewritten as

MSD(θ) = 1/nΣi(θi – θ)2 + θ2, (7)


TABLE 2: Alternative Measures of Comparability

Absolute Absolute Observed ObservedDifferential Differential in 5 Percent in 5 Percent

Sector Private Base Public Base Band Private Band Public

Federal .161 .104 19.5% 31.3%State .155 .154 17.8 19.2Local .134 .175 26.9 17.6

NOTE: Note that the absolute differential is measured in ln wages and that the bandingmeasure is the percentage of the workforce that falls within the 5 percent band.



where θ is the mean of the θis, the conventional public sector differen-tial, in the sample of size n. This allows the criterion to be thought of asconsisting of two components. The second component, the square ofthe average differential, is analogous to the bias in an estimator. Thisvalue will be zero whenever the log dollar volume of earnings in thepublic sector is identical to that which would be paid in the private sec-tor for the same workers. The first component is the variance in indi-vidual differentials from the sample mean. This component reflectsthe dispersion in the comparability treatment of public-sector work-ers. If workers all have the same value of θ, whether large or small, thisfirst component will be zero. Thus, another way of phrasing the objec-tion to the average differential as a measure of comparability is that itexcludes the potentially crucial role played by the variance in individ-ual comparability.10

Although it is possible to calculate a combined Oaxaca decomposi-tion for public and private employees, it has been more typical to cal-culate a private base differential using the characteristics of privateemployees and a public base differential using the characteristics ofpublic employees. This approach recognizes that differences betweenthe characteristics of typical public and private employees results indifferent measures of comparability and that the increase in the num-ber of measures is balanced by the greater insight into nature of com-parability. We follow this tradition in Table 3, which presents the val-ues for the MSD criterion for both the private and public sector base.11

In all cases, the log differences have been multiplied by one hundredso the numbers presented are not extremely small. Each value of thecriterion is broken into its components, the variance and the aggregatedifferential squared. Thus, starting with the federal private base esti-mate, the variance in the individual differential is 319 and the averagedifferential squared is 91 (9.55, the value from Table 1, squared) for atotal MSD criterion of 410. This procedure is repeated for the othersectors and for the public-sector bases.

Table 3 makes clear two fundamental points. First, the bias compo-nent represents only a small portion of the deviation from comparabil-ity. Even in the federal sector, the bias component accounts for lessthan a quarter of the MSD. Thus, if policy makers adjusted all federalwages by the average differential, they would not come close to




achieving comparability. Second, the MSD continues to suggest thatthe federal sector is, if anything, closer to comparability. If the twobases are averaged, the MSD is 296 in the federal sector, 338 in thestate sector, and 365 in the local sector. At minimum, there is no evi-dence that federal government pay-setting policies deserve more scru-tiny than those of state and local governments.12

As a further illustration of the MSD criterion, we return to the occu-pational breakdown that we explored in section 2. Recall that white-collar workers had very disparate measures of the average differential,with the federal sector appearing to overpay between 8 and 9 percent,the state sector appearing comparable, and the local sector underpay-ing about 10 percent. Again, using the private base estimates, theMSD mimics the full sample. The average differential contributesonly a small share of the deviation from comparability. The state sec-tor, which had an average white-collar differential of essentially zero,has such a large variance that its MSD exceeds that in the federal sec-tor. The pattern again shows the federal sector closest to comparabil-ity. The policy implication is that governments, state and local in par-ticular, should emphasize the pattern of earnings rather than theaverage level. To achieve comparability, the large set of workers paidmore than a comparable wage and the large set of workers paid less


TABLE 3: Mean Squared Deviation Criterion

Private Base Public Base

MSD Variance Bias2 MSD Variance Bias2

All workersFederal 410 319 91 182 161 22State 345 322 22 331 297 28Local 302 296 5 428 428 0

White-collar workersFederal 328 253 74State 359 358 1Local 418 309 109

Blue-collar workersFederal 879 676 202State 570 392 178Local 336 319 16

NOTE: Mean squared deviation (MSD) = variance + bias2.



than a comparable wage should receive separate attention. The use ofpredictions in the measures described above implies that the variancehas both a population and sample component. As the appendix andTable A2 show, adjusting for the effect of sampling error does notchange the basic results or this policy implication.

COMPARISONS WITH MEASURESFOR RACE, GENDER, AND UNIONS

This subsection provides a short illustration of why earnings dis-persion is particularly relevant in estimating government earningscomparability. In short, many of the other differentials estimated withthe average differential are more appropriately done so, and moreover,accounting for the individual variance in the differential does notgreatly alter the pattern presented by the average differential. To showthis point, we compare our previous estimates for public-sector earn-ings comparability with analogous estimates for earnings compara-bility by race, gender, and union status.

The Oaxaca decomposition was originally used to estimate earn-ings differentials by race and gender. At issue was whether a protectedgroup earned less on average after holding human capital constant. In-deed, many labor economics texts virtually define earnings discrimi-nation in this fashion.13 Yet while an average race or gender differen-tial of zero indicates the absence of group discrimination, an averagepublic-sector differential of zero does not indicate comparability. Dis-crimination is a group concept, but comparability is an individual con-cept that, as an objective of public policy, can be achieved only at a lessaggregate level.

Table 4 presents gender, race, and union differentials estimated forour sample.14 As is apparent, in each case, the average differential iscloser to those that account for dispersion than is the case with the av-erage government differential. The absolute differential that accountsfor canceling out is very similar to the average differential. Unlike thepublic-sector mean squared error measures, those in Table 4 arelargely driven by the bias showing that dispersion is a less importantcomponent. Thus, the very case of public-sector comparability inwhich dispersion is theoretically pertinent is the case in which theindividual dispersion in the differential plays the greatest role.




5. CONCLUSIONS

Comparability is a disaggregate concept that should be evaluatedby indicating how many workers are how close to individual differen-tials of zero. The average differential used in the literature does not in-dicate this information.

We presented three initial alternative measures of comparability:the absolute differential, the percentage of the workforce that fallswithin a comparability band, and the MSD criterion. Ultimately, allthree alternative measures reveal a pattern not evident in the tradi-tional measure. The state and local sectors that appear closer to com-parability with the average differential are not in reality any closer.

The emphasis in past literature on the federal sector’s deviating themost from comparability appears wrong. While the federal sector maypay positive premiums to a larger share of its workforce, those premi-ums are small. More important, the state and local sector pay bothlarger positive and larger negative premiums as revealed in the largervariance in their individual comparability measures. As a conse-quence, it may well be the state and local sectors that deviate mostfrom comparability. While our estimates are dependent on a singledata source and need to be replicated before they should be taken forgranted, the point is not the specific estimates but the need to moveaway from average differentials.


TABLE 4: Alternative Measures Applied to Other Differentials

Gender Race Union

Average differential –.204 –.136 .134Absolute differential .205 .154 .167Mean squared deviation 522 365 463Bias2 436 187 286Variance 86 178 177



APPENDIXRemoving the Effects of Sampling Error

from the Mean Squared Deviation Criterion

As with all predictions, the variance of the predicted earnings gap in the meansquared deviation (MSD) is composed of a population and a sample component. Thelatter results because estimated rather than actual coefficients are used to form theprediction. Ideally, the MSD measure would use only the population component ofvariance.

Consider the variance component of the MSD:

( ) ( ) ( )( )Var E X E Xs n s s s si

θ θ θ= −=∑1

1

2( ,

where the subscript s refers to the sector for which the MSD is constructed, p or g.Substituting the regression equations for zero,

( )Var X X X Xs n si g si pi

s g s pθ β β β β= ′ − ′

−

′ − ′=∑1

1

� � � �

2

,(1)

which is rearranged to

( )Var X X X Xs n si si

g si s pθ β β= ′ − ′

−

′ − ′

=∑1

1

� �

2 (2)

using ′ ′X ds to represent X Xsi s′ −

′.

( )Var X d X d X d X d X ds n s g g s s p p s s g pθ β β β β β β=′ ′ + ′ ′ + ′1 2( � � � � � � ′

=∑ X dsi 1

.(3)

Consider the first term:

X d X d X d X X X usg g s

sg g g g i g

′ ′ = ′ + ′

′

+

−� � �β β β β

1

X X X u X d

X d X d X d V X d

g g g i s

sg g s

s g

′

′

′

= ′ ′ + ′

−1

β β s

,

(4)

where V is the variance matrix of the government regression. The first term is the pop-ulation component, and the second is the sample component. The second term ofequation (3) can analogously be expressed as




X d X d X d X d X d V X dsp p s

sp p s

s p s′ ′ = ′ ′ + ′� �β β β β . (5)

Although the third term of equation (3) can be similarly decomposed, the sample termcannot be formed as term uiguip is viewed as either undefined, as individuals can haveonly errors in their actual sector, or as inherently unobservable. The population com-ponent of the variance term of the MSD can then be calculated using observed data as

Var(θs) = X dsVg X ds + X dsVpX ds. (6)

This forms the basis of the estimates in Table A1.

TABLE A1: Mean Squared Deviations Using the Correction Implied by Equation (6)

Private Base Public Base

Workers MSD Variance Bias2 MSD Variance Bias2

Federal 218 126 91 101 79 23State 29 7 22 220 201 19Local 94 89 5 278 278 0

NOTE: Mean squared deviation (MSD) = variance + bias2.




583

TAB

LE

A2:

Reg

ress

ion

Est

imat

es o

f L

og

Wag

e E

qu

atio

ns

for

Pri

vate

,Fed

eral

,Sta

te,a

nd

Lo

cal E

mp

loye

es

Priv

ate

Fede

ral

Sta

teLo

cal

Age

0.04

44(1

7.31

)0.

0433

(3.6

8)0.

0440

(3.3

5)0.

0427

(5.0

0)A

ge2

–0.0

005

(–15

.66)

–0.0

004

(–3.

04)

–0.0

004

(–2.

97)

–0.0

005

(–5.

06)

Mar

ried

0.08

79(8

.19)

0.08

15(2

.23)

0.04

00(0

.88)

0.08

64(2

.66)

Bla

ck–0

.125

2(–

6.36

)–0

.123

5(–

2.16

)–0

.126

8(–

1.83

)–0

.158

7(–

3.48

)U

nion

0.14

08(1

0.10

)0.

0044

(0.1

1)0.

0791

(1.7

2)0.

0550

(1.7

1)Ju

nior

hig

h sc

hool

0.15

53(5

.46)

0.56

34(1

.69)

0.09

60(0

.43)

0.23

79(2

.27)

Hig

h sc

hool

0.08

03(4

.93)

0.16

20(1

.53)

0.23

26(2

.00)

0.06

87(1

.19)

Ass

ocia

te’s

deg

ree

0.07

17(3

.99)

0.03

38(0

.65)

0.01

47(0

.18)

0.15

27(3

.45)

Bac

helo

r’s d

egre

e0.

2462

(17.

56)

0.17

59(3

.88)

0.14

71(2

.51)

0.15

27(3

.45)

Mas

ter’s

deg

ree

0.05

12(2

.23)

0.05

34(0

.93)

0.04

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NOTES

1. For more on the objectives of comparability and their interpretation by economists, seeEhrenberg and Schwartz (1986), Venti (1987), and Belman and Heywood (1996).

2. Thus, while quantile regression reveals that the differential is largest in the lower portionsof the distribution, it does not incorporate the individual variation around this new largerdifferential.

3. Including the firm size variable in the private-sector regressions has been shown to play asubstantial explanatory role and to reduce the size of the federal aggregate differential (Belmanand Heywood 1990). We stress that while the exact differentials vary modestly, the general re-sults presented in this article do not hinge on its inclusion.

4. Note that altering the average differential to estimate the wage based on average charac-teristics (a variant on the Oaxaca decomposition) does not solve the fundamental shortcoming.Such a procedure still allows positive and negative individual differentials to cancel out, but theywould do so in the averaging of the characteristics rather than the averaging of the actualdifferentials.

5. Our experiments with alternative bands serve merely to reinforce the point illustratedwith the 5 percent band.

6. The typical Oaxaca decomposition has two components: one similar relating to the effectof differences in the public and private equations, holding characteristics constant, and the sec-ond related to the effect of differences between the mean characteristics of the public and privatesector, holding the estimated coefficients constant. This latter term does not appear when the dif-ferentials of individuals are considered as, unlike the averages across the public and privatesectors, individuals’ characteristics do not change.

7. It is common practice to calculate the Oaxaca differential using the actual wage for an in-dividual’s base group rather than the prediction. For example, using the public base, this ap-

proach would calculate each individual’s differential as ln Wig– ln �W i

p= Xi

s ′(β β� �g − p) + ei.

The inclusion of the error term ei does not affect the average differential as

1 11 1

/ ln ln � / ( ( � � ) )n W W n X eig

i

n

ip

is

i

n

i=

′

=∑ ∑− = − +β βg p but ei

i

n

=∑ =

1

0. The Oaxaca using the ac-

tual own wage is then equivalent to using predictions for the own and other group wage. Thisproperty does not hold for the absolute value or the mean squared deviation (MSD) measure wepropose. If actual rather than predicted wages were used in the absolute value measure, it would

include the term 1/n eii

n

=∑

1

, which is certain not to be equal to zero. Similarly, use of an actual

value in the MSD measure would add the own sector variance of wages, σ g2 for the public em-

ployee base estimate, to the measure. Use of predicted values for both the own as well as the otherwage is essential to obtaining a useful measure.

8. Although the MSD would likely be computed separately for public and private base sam-ples, we suppress sector notation indicating base to promote clarity.

9. An alternative to creating separate public- and private-sector bases would be to use eachpaired set of employees (private/federal, private/state, private/local) in these calculations. Al-though combined bases have occasionally been used for Oaxaca calculations, we follow the moreconventional practice of presenting distinct private- and public-sector calculations.

10. Another way of illustrating this is to rewrite a general version of equation (2) as MSD(θ) =α[1/nΣ(θi – θ)] + βθ

2. The aggregate differential sets aα to zero, putting all the weight on the




“bias” component while the version in equation (2) gives the two components equal weight,α = β = 1.

11. Clearly, equations (1) through (3) could have been expressed with θ defined as the differ-ence between an estimated public wage and the actual private wage. This generates the alterna-tive private-sector base estimate of the MSD criterion.

12. Bender (2003) adopted our MSD measure to test for comparability of central governmentearnings in the United Kingdom.

13. Wachtel (1992, 226) argued that if low earnings were randomly distributed by race andgender, there would be no possibility of discrimination.

14. Obviously, we added women to complete the gender estimate.

REFERENCES

Belman, Dale, and John S. Heywood. 1989. Government wage differentials: A sample selectionapproach. Applied Economics 21:427-38.

. 1990. The effect of establishment and firm size on public wage differentials. Public Fi-nance Quarterly 18:221-35.

. 1993. Job attributes and federal wage differentials. Industrial Relations 32:148-57.. 1995. State and local government wage differentials: An intrastate analysis. Journal of

Labor Research 16:186-201.. 1996. The structure of compensation in the public sector. In Public sector employment

relations in an age of transformation, ed. Morely Gunderson, Douglass Hyatt, and DaleBelman. Madison, WI: Industrial Relations Research Association.

. 2004. Public wage differentials and the treatment of occupational differences. Journalof Policy Analysis and Management 23:135-52.

Bender, Keith. 1998. The central government: Private sector wage differential. Journal of Eco-nomic Surveys 12:177-200.

. 2003. Testing equality between the public and private sector earnings distributions. Eco-nomic Inquiry 41:62-79.

Blackaby, D., P. Murphy, and N. O’Leary. 1999. The payment of public sector workers in the UK:Reconciliation with North American findings. Economic Letters 65:239-43.

Ehrenberg, R., and J. Schwartz. 1986. Public sector labor markets. In The handbook of labor eco-nomics, Vol. 2, ed. O. Ashenfelter and R. Layard. Amsterdam: North-Holland.

Elliot, Robert, and Keith Duffus. 1996. What has been happening to pay in the public service sec-tor of the British economy? Developments over the period 1970-1992. British Journal of In-dustrial Relations 34:541-85.

Gregory, J., and J. Borland. 1999. Recent developments in public sector labor markets. In Hand-book of labor economics, Vol. 3, ed. Orley Ashenfelter and David Card. Amsterdam:Elsevier.

Gyourko, J., and J. Tracy. 1988. An analysis of public and private sector wages allowing for en-dogenous choice of both government and union status. Journal of Labor Economics 6:229-53.

Heywood, John S., and Madhu Mohanty. 1995. Estimation of the US federal job queue in thepresence of an endogenous union queue. Economica 62:479-93.




Linneman, Peter, and Michael Wachter. 1990. The economics of federal compensation. Indus-trial Relations 29:58-76.

Moore, William, and Robert Newman. 1991. Government wage differentials in a municipal labormarket: The case of Houston metropolitan transit workers. Industrial and Labor RelationsReview 45:145-53.

Moore, William, and John Raisian. 1991. Government wage differentials revisited. Journal ofLabor Research 12:13-34.

Moulton, Brent. 1990. A reexamination of the federal private wage differential in the UnitedStates. Journal of Labor Economics 8:270-93.

Mueller, Richard. 1988. Public-private sector wage differentials in Canada: Evidence fromquantile regressions. Economic Letters 60:229-36.

Oaxaca, Ronald. 1973. Male-female wage differentials in urban labor markets. InternationalEconomic Review 14:693-708.

Poterba, James, and Kim Rueben. 1994. The distribution of public sector wage premia: New evi-dence using quantile regressions methods. NBER Working Paper 4734.

Quinn, Joseph. 1979. Wage differentials among older workers in the public and private sectors.Journal of Human Resources 17:41-62.

Robinson, C., and N. Tomes. 1984. Union wage differentials in the public and private sectors: Asimultaneous equations specification. Journal of Labor Economics 2:106-27.

Smith, Sharon. 1976. Government wage differentials by sex. Journal of Human Resources11:185-99.

. 1977. Government wage differentials. Journal of Urban Economics 4:179-97.Venti, Stephen. Wages in the federal and private sectors. In Public sector payrolls, ed. David

Wise, 147-82. Chicago: University of Chicago Press.Wachtel, Howard. 1992. Labor and the economy. New York: Harcourt Brace Jovanovich.

Dale Belman is an economist and associate professor in the School of Labor and Indus-trial Relations at Michigan State University.

John S. Heywood is professor of economics and director of the graduate program in hu-man resources and labor relations at the University of Wisconsin–Milwaukee.




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