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Measuring discrimination against females: is the "non-discriminatory" wage the male or the female wage?


Typically, when using Oaxaca's decomposition (7) to measure discrimination against females, (or for that matter any group) either the wage regression for males is used to predict female earnings which can be compared to actual earnings or the wage regression for females is used to predict male earnings which can be compared to male earnings. These two methods will produce different estimates of discrimination. (6) The appropriate method for estimating discrimination depends upon whether discrimination consists of a bias against females (in which case the former method is correct), a bias for males (in which case the latter method is correct), or some combination of both (in which case neither method is correct). Neumark (6) attempts to address this problem by estimating the true "no-discriminatory wage coefficients" under the assumption that employers' utility functions are homogeneous of degree zero with respect to male and female labor. He finds that this approach leads to lower estimates of discrimination than the standard techniques.

This paper uses a wage frontier approach to estimate the "true" wage regression for both males and females(1). This approach has a number of advantages. First it allows a determination to be made of to what extent discrimination raises male wages or lowers female wages without making assumptions about the utility functions of employers(2). If the benchmark "non-discriminatory" wage regression is the male wage regression, it is the case that female earnings are lowered by discrimination. On the other hand, if the "non-discriminatory" wage regression is the female wage regression then male earnings are raised by discrimination. Neumark's approach places the non-discriminatory wage line rather arbitrarily in the middle of the male and female wage regression lines. Secondly, the approach provides a comparison with that taken by Neumark and provides a framework for testing his assumptions about the "non-discriminatory" wage regression coefficients. And finally, this approach allows estimation of discrimination when discrimination is applied differentially to members of the group being discriminated against.

This research extends the important work of measuring discrimination that has been conducted by many economists over the past twenty years by using a new approach for measuring discrimination. The frontier approach introduced in this paper represents a more general alternative to Oaxaca's and Neumark's attempt to estimate the "non-discriminatory" wage frontier. It is more general in the sense that it makes no assumptions about the nature of discriminatory preferences. There are two major findings of this study. The first is that sex discrimination seems to raise male earnings by less than 4 percent and seem to lower female earnings by more than 20 percent(3). Clearly, the majority of discriminatory preferences seem to lower female wages rather than to raise male wages. Secondly, using a frontier approach results in estimates of unexplained differentials that are higher than those obtained by Neumark, though smaller than those obtained by the traditional method. This confirms Neumark's conclusion that the traditional approach overestimates the amount of the unexplained differential, but perhaps by less than his research indicates.

Measuring Discrimination

The standard procedure for estimating discrimination against females (or blacks) is to estimate two wage equations: one for the females and a second for the males. The total wage differential can then be decomposed into two portions-an unexplained and an explained portion. One of two assumptions must be used to complete the decomposition. One must assume either that in the absence of discrimination all female workers would be treated like males, or that discrimination consists entirely of nepotism for male employees. The decomposition formulas are as follows:

log (|w.sub.m~) - log (|w.sub.f~) = |B.sub.m~(|X.sub.m~ - |X.sub.f~) + (|B.sub.m~ - |B.sub.f~)|X.sub.f~ (1)


log (|w.sub.m~) - log (|w.sub.f~) = |B.sub.f~(|X.sub.m~ - |X.sub.f~) + (|B.sub.m~ - |B.sub.f~)|X.sub.m~ (2)

Here |B.sub.m~ is the male coefficient and |X.sub.m~ is the mean over the males of human capital characteristic X. The explained differential is to the left of the plus sign and the unexplained to the right. The first formula assumes discrimination is against females. These two approaches quite obviously will produce different estimates of the amount of discrimination and neither shed any light on the nature of discriminatory preferences.

Neumark (6) has attempted to address this issue. He shows that both of the above decompositions are special cases of a more general decomposition:

log (|w.sub.m~) - log (|w.sub.f~) = B(|X.sub.m~ - |X.sub.f~) + (|B.sub.m~ - B)|X.sub.m~ - (|B.sub.f~ - B)|X.sub.f~ (3)

where B is the non-discriminatory coefficient. This decomposition reduces to the previous ones if B = |B.sub.f~ or B = |B.sub.m~. The only remaining problem is to estimate B. Neumark shows that under the assumption that preferences against females (or for males) depend only on the ratio of male to female employees the full-sample OLS results can be used for B. Neumark's approach along with either version of Oaxaca's decomposition is forced to make some assumption about the nature of discriminatory preferences. This precludes the possibility of testing hypotheses about or estimating characteristics of the preferences. In the remaining sections of this paper I propose a frontier approach as an alternative to measuring discrimination. The frontier approach does not require the assumptions about the nature of discriminatory preferences and will allow a direct estimate to be made of the nature of the discriminatory preferences. In addition this approach provides an alternative measure of the amount of the total male-female differential that is due to discrimination.


Employers are assumed to have a "distaste" for employing female workers and/or a preference for employing male employees. Employers maximize utility which is a function of profits and number of male and female employees by determining the appropriate number of male and females to employ. Thus, employer's utility, under the assumptions that distaste is measurable in dollars and utility is linear in dollars, is given by:

U = h(|Pi~,|L.sub.f~,|L.sub.m~) (4)


U = f(|L.sub.m~,|L.sub.f~) - |w.sub.m~|L.sub.m~ - |w.sub.f~|L.sub.f~ - ||Gamma~.sub.f~|L.sub.f~ + ||Gamma~.sub.m~|L.sub.m~. (5)

Here h, is the employers utility function, |Pi~, profits f, the production function, |Gamma~, the employers marginal distaste for females (or preference for males) in dollars, |w.sub.i~, the wages of group i, and |L.sub.i~, employment of group i. The first order conditions for profit maximization are:

dU/d|L.sub.m~ = |f.sub.m~ - |w.sub.m~ + ||Gamma~.sub.m~ = 0, and (6)

dU/d|L.sub.f~ = |f.sub.f~ - |w.sub.f~ - ||Gamma~.sub.f~ = 0. (7)

Here |f.sub.i~ is the marginal productivity of factor i. These can be solved for male and female wages:

|w.sub.m~ = |f.sub.m~ + ||Gamma~.sub.m~, and (8)

|W.sub.f~ = |f.sub.f~ - ||Gamma~.sub.f~. (9)

This represents the standard distaste for females model.

Under the assumption that male and female productivity are determined by human capital and other worker characteristics and that males and females with identical human capital are equally productive (8) and (9) can be written as:

|w.sub.i~ = B|prime~H|C.sub.i~ - || (1 - |Male.sub.i~) + ||Gamma~.sub.mi~ (|Male.sub.i~) (10)

where Male is a dummy variable taking the value 1 for male workers and 0 otherwise, HC is a vector of human capital characteristics, and B is a parameter vector. The tastes for males and the distaste for females can be modeled in several ways: they could be intercept shift terms (in which case OLS estimation of the wage equation with a male dummy is appropriate) or they could be interactively related to the parameter vector B (in which case separate male and female equation should be estimated). Neither of these approaches allow for a determination of wage determination in the absence of discrimination, nor provide estimates of whether discrimination is a bias for males or against females.

A third possibility introduced in this paper is to treat the employer taste terms as being employer specific. Each male and female accepts a job offer with an individual employer's taste built into the wage. If we assume that no employers are biased against males and none biased for females, the employers tastes can be thought of as truncated random variables.(4) If we make the further assumption that these distributions are truncated normal and add a normally distributed error term the likelihood function for (10) can be specified and maximum likelihood estimation employed. Under these assumptions the estimation procedure will give estimates of the "true" wage regression line, as well as, estimates of the average bias for males and against females. While this approach does not escape all the criticisms that have been leveled against Oaxaca's decomposition, it does aid in the problem of determining the nature of discrimination.(5) In addition it relaxes the assumption made in the traditional methods of measuring discrimination that an identical amount of discrimination is applied to each individual. This assumption is replaced by the assumption that the amount of discrimination faced by each individual comes from the same distribution of discriminatory tastes. This means however that the actual amount of discrimination will vary across individuals. This is consistent with the different perceptions females may have about the amount of labor market discrimination that exists.

After adding the normal error term to (10) we have:

|w.sub.i~ = B|prime~H|C.sub.i~ - | (1 - |Male.sub.i~) + |u.sub.mi~ (|Male.sub.i~) + ||Epsilon~.sub.i~ (11)

The likelihood function for (11) under the assumption of truncated half-normal distributions for the employers' tastes for males and females can be shown to be:

|Mathematical Expression Omitted~

where F is the normal distribution function, F* is the cumulative normal distribution function, |Sigma~ is equal to the sum of the normal and truncated variances and, ||Lambda~.sub.f~ = ||Sigma~.sub.uf~/|Sigma~ and ||Lambda~.sub.m~ = |||Sigma~, are the ratios of the standard deviations of male (female) preferences of employers and the standard deviation of the normal wage error.(6)

Data and Results

Equation (12) was estimated using 1983 CPS data for white full-time, non-union, non-agricultural, private sector workers. Three versions were estimated.(7) Each model contained the human capital variables of education, estimated experience (age-tenure-6), and tenure. Model 3 contains only these variables, while Model 2 contains industrial dummy variables and Model 1 controls for both occupation and industry. The estimation results are presented in Table 1. The coefficients on the human capital variables are all significant and of the correct sign. Since LAMBAF and LAMBAM are the ratios of the truncated discrimination distributions to the normal distributions we can conclude from the fact that both are significantly different from zero that the preferences both against females and for males are significant.(8) Since we know that the expected value of the half-normal distribution is equal to -|(2/|Pi~).sub.1/2~||Sigma~.sub.f~ or -|(2/|Pi~).sub.1/2~ ||Sigma~.sub.m~ the expected levels of discrimination can be easily computed.

These are shown in Table 1 as well. We find that male earnings will be above the non-discriminatory wage line by less than 4 percent on average and female earnings below the non-discriminatory wage line by more than 20 percent. This would appear to indicate that sex discrimination predominantly lowers female wages and only marginally increases male wages.(9)
Estimation of Equation 8
Dual Frontier Estimation
PARAMETER Model 1 Model 2 Model 3
Constant 0.702 0.566 0.712
 (19.13) (20.80) (26.19)
Education 0.051 0.077 0.077
 (27.36) (48.87) (49.37)
Age 0.014 0.016 0.017
 (15.83) (17.26) (18.14)
Age**2/100 -0.029 -0.032 -0.036
 (-13.56) (-14.36) (-15.50)
Tenure 0.030 0.035 0.037
 (24.44) (27.54) (28.24)
Tenure**2/100 -0.059 -0.066 -0.069
 (-15.27) (-16.74) (16.67)
Large Plant(1) 0.150 0.158 0.187
 (16.57) (17.78) (20.43)
Construction 0.276 0.289
 (12.74) (13.78)
Manufacturing 0.209 0.169
(Non-Durables) (12.33) (10.50)
Manufacturing 0.276 0.251
(Durables) (16.66) (15.91)
Transportation 0.363 0.364
 (18.34) (17.98)
Wholesale Trade 0.228 0.225
 (13.28) (12.85)
FIRE 0.264 0.277
 (15.67) (16.23)
Services 0.138 0.123
 (10.82) (10.20)
Mining 0.484 0.476
 (13.76) (13.03)
Managerial 0.352
Professional 0.323
Technical 0.272
Sales 0.169
Clerical -0.472
Household -0.027
Protective -0.000
Service (-0.00)
Other Service 0.031
Farming 0.243
Craft 0.023
Operators 0.085
Transportation 0.398
Construction 0.276
SIGIM 0.401 0.417 0.432
 (120.03) (133.24) (124.69)
LAMBDAM 0.109 0.048 0.093
 (1.96) (0.99) (1.91)
SIFIG 0.398 0.444 0.465
 (55.43) (60.48) (58.89)
LAMBDAF 0.891 1.072 1.106
 (11.72) (14.48) (14.51)
Expected Value
of ||Tau~.sub.m~ 0.034 0.016 0.032
of ||Tau~.sub.f~ -0.211 -0.259 -0.272
N = 9775 9775 9775
Log-Likelihood = -4209.68 -4704.01 -5744.78
1. Large Firms have more than 100 employees
2. Omitted Industry is Retail Trade
3. Omitted occupation is Laborers

For purposes of comparison, both the traditional decomposition approach, using both the male and female coefficients as the non-discrimination coefficients (equations (1) and (2)) and Neumark's approach (equation (3)), using the full sample OLS results as the non-discriminatory wage coefficients and using the frontier coefficients were used to generate unexplained differentials. These results appear in Table 2. The traditional approach finds the highest levels of unexplained differentials from 62.7 to 85.3 percent. On the other hand Neumark's technique estimates the lowest unexplained differentials from 46.6 to 83.4 percent. While the frontier method falls in the middle, somewhat closer to the traditional approach, finding unexplained differentials of between 68.2 and 82.4 percent.


Table 2 also reports the relationship of male and female actual wages to their estimated non-discriminatory level from the different models.(10) The traditional approach using the female coefficients diverges greatly from the other approaches. Both the frontier approach and Neumark's approach place the non-discriminatory wages somewhere between what is actually observed for males and females, with the frontier approach coming closer to the traditional approach using male coefficients. The estimates from both non-traditional approaches seem to indicate that using the female wage regression coefficients to estimate discrimination is inappropriate and will result in estimates of non-discriminatory wages that are too low.

In order to test whether the estimates from the different models were significantly different, equation (12) for Model 1 was reestimated under three sets of restrictions about the "non-discriminatory" wage differentials: 1) that Neumark is correct; 2) that the OLS estimates using the males is correct; and 3) that the OLS estimates using the females is correct. The log-likelihood estimates from these estimations are reported in Table 3. These results indicate the unrestricted frontier model is significantly different from the other models, while Neumark's model has the second highest log-likelihood and the model using the female OLS coefficients is worst.
Comparison of Models
Log-Likelihood Ratio Test
Model 1 Estimates
Restrictions Log-Likelihood F-Statistic
None -4209.70 -
Neumark's -4338.95 127.2
OLS from Males -4345.10 270.8
OLS from Females -4355.65 291.9
Note: 27 restrictions, N = 9775
The F-Statistic comparing Neumark's restriction with the male
OLS estimates is 12.3.


This paper presents a technique for estimating discrimination through unexplained differentials that has two attractive qualities: it does not depend on assumptions about the nature of discrimination, while it identifies the "non-discriminatory" wage regression. The results shown here indicate that when using the frontier approach with 1983 CPS data 68 to 84 percent of the male-female earnings differential is unexplained. The majority of this is due to the reduction in female earnings. So sex discrimination appears to primarily lower female wages. Estimates of the unexplained differential computed using the frontier coefficients lead me to agree with Neumark's conclusion that using Oaxaca's decomposition with either the male or female regression coefficients as non-discriminatory results in higher estimates of the unexplained differential. However, the frontier approach provides estimates of this differential that are closer to the traditional method than Neumark's approach.


1. Boehm and Hofler (2) explore this approach in the housing market. Hofler and Polachek (4), Polachek and Yoon (9), and Robinson (10) explore related labor market information issues.

2. In the case of pure wage discrimination, it would be possible to attribute the increase in male wages to nepotism and the decrease in female wages to discrimination, though this would not be the case with other types of discriminatory behavior.

3. Male hourly wages.

4. This is a much weaker assumption than that employed by Neumark. He assumes employers all have identical utility functions, these utility functions are homogeneous of degree one in male and female labor, and no employers prefer females or discriminate against males.

5. This approach along with that of Neumark and Oxaca must assume that the characteristics controlled for in the estimation are measured identically across male and females. In other words, we must assume that a year of male education must be the same as a year of female education.

6. See Aigner, Lovell and Schmidt (1) on stochastic frontier estimation.

7. Since occupational and industrial segregation of females may be part of the discriminatory labor market models with and without controls for occupation and industry were estimated for comparative purposes.

8. With the exception of LAMBDAM in Model 2.

9. Of course in the pure wage discrimination case, we could interpret these results to indicate there was little nepotism and more discrimination.

10. Since the frontier estimates from Table 1 are the expected values of the discrimination error term, while the estimates from Table 2 are actual differences computed using the frontier coefficients, the estimated differentials are slightly different.


Aigner, D., Lovell, C., and P. Schmidt, "Formulation and Estimation of Stochastic Frontier Production Models," Journal of Econometrics, 6, 1977, 21-37.

Boehm, T. and R. Hofler, "A Frontier Approach to Measuring the Effect of Market Discrimination: A Housing Illustration," Southern Economic Journal, October 1987, 301-315.

Butler, R. J., "Estimating Wage Discrimination in the Labor Market," Journal of Human Resources, Fall 1982, 606-21.

Hofler, R. and S. Polachek, "Ignorance in the Market: A New Approach to Measuring Information Content," Proceedings of the 1982 Business and Economic Statistics Section, Washington D.C.: American Statistical Association, 1983, 442-425.

Low, S. and D. Villegas, "An Alternative Approach to the Analysis of Wage Differentials," Southern Economic Journal, Summer 1987, 449-462.

Neumark, David, "Employer's Discriminatory Behavior and the Estimation of Wage Discrimination," Journal of Human Resources, Vol. XXIII, 1988, 279-295.

Oaxaca, R., "Male-Female Wage Differentials in Urban Labor Markets," International Economic Review, October 1973, 693-709.

Polachek, S., "Potential Biases in Measuring Male-Female Discrimination," Journal of Human Resources, Vol. 10, Spring 1975, 205-29.

Polachek, S. and B. Yoon, "A Two-Tiered Earnings Frontier Estimation of Employer and Employee Information in the Labor Market," Review of Economics and Statistics, Vol. 69, 2, May, 1987, 296-302.

Robinson, M. "A Regional Analysis of Male-Female Earnings Differentials," Dissertation, University of Texas at Austin, 1987.

Michael D. Robinson Assistant Professor of Economics, Mount Holyoke College.
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Author:Robinson, Michael D.
Publication:American Economist
Date:Mar 22, 1993
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