Wage gaps large and small.

1. Introduction

In the Wealth of Nations, Adam Smith posited theory and institutional explanations for why wages differ. Modern labor economics has retained a focus on wage differentials. Although much of the literature is unabashedly empirical, it is informed by theory. Neoclassical economics, including human capital theory, remains the principal approach, although labor economists recognize the role played by human nature, workplace incentives, institutions, and public policy in wage determination.

In this address, I highlight several topics I have studied, all involving wage gaps. (1) These include Census imputation methods, union premiums, product market regulation, wages in male and female jobs, the wage effects of military service, and interarea wages and cost of living. The purpose is not to trumpet my work, although it may appear as such, but to draw broader conclusions about labor economists' understanding of wage determination.

2. Equalizing Differentials and the Law of One Wage

The theory of equalizing differentials states that in competitive labor markets, workers with similar skills working in similarly attractive jobs and locations should receive similar compensation. Long-run wage differentials are explained by differences in skill and job disamenities, with a single "price" (wage) conditional on worker and job attributes. (2) The law of one wage--equal compensation for equivalent workers and jobs--follows naturally from competitive theory.

Empirical studies attempt to test the law of one price in product markets. Industrial organization economists focus on a narrow set of homogeneous goods in markets with low-cost information and similar transportation costs, i.e., purchases of electronics and books at Internet sites. International economists test for purchasing power parity for homogeneous goods, i.e., Big Mac prices across countries (for a recent paper, see Parsley and Wei 2007). Even in these markets, there is considerable deviation from the law of one price.

One does not see an equivalent literature in labor economics. Yes, there is a vast literature on wage gaps, but authors rarely characterize such work as a test of a "law of one wage" because there is little expectation that there should be wage equality, even conditional on controls. Why not? First, as emphasized in personnel economics, pay schemes that maximize profits often involve wages deviating from spot marginal products, thus creating competitive wage differences across similar workers in similar jobs (Lazear and Oyer 2007). Second, unions or other institutions can affect wages. Third, there are rigidities in labor markets owing to imperfect mobility, say from information and search costs, firm-specific training, personal job attachment, or tied household decisions regarding jobs, location, and housing (Manning 2003; Mortensen 2003). Fourth are problems of measurement, leading to apparent variation in wages even when there is none. And fifth, even if data are error free, we cannot hope to measure all the multitude of worker and job attributes that influence wages.

How bad is it? It depends on whether you see the glass as half empty or half full. The Mincerian human capital earnings equation (Chiswick 1974; Mincer 1974) serves as the workhorse of wage gap studies. (3) Wages are modeled as a multiplicative function of time investments in human capital, in its most basic form with the natural logarithm of earnings a linear function of schooling and a quadratic of potential work experience. With the help of a few heroic assumptions, the schooling coefficient is interpreted as a rate of return to schooling investments ([R.sub.s]), and coefficients on the experience profile reflect a combination of postschool investment intensity ([K.sub.0]), investment length ([T.sup.X]), and the returns to postschool training ([R.sub.p]).

For example, estimating this canonical wage equation using the 2004-2006 Current Population Survey (CPS) earnings files and a combined sample of men and women, I obtain

In(Wage) = 0.887 + 0.107 School + 0.039 Exp--0.067 [Exp.sup.2]/100 [R.sup.2] = 0.339 n = 354,132

[R.sub.s] = 10.7% rate of return to schooling investment

[t.sup.*] = 29.2 years experience at peak of earnings-experience profile

[T.sup.*] = 19.2 years of positive postschool investment (assumed to equal [t.sup.*]--10)

[K.sub.0 ]= 0.257 initial postschool training investment ratio

[R.sub.p ]= 8.7% rate of return to postschool investment in human capital.

The coefficient on School, 0.107, is interpreted as an average rate of return (ignored are important issues such as ability bias and selection); more accurately, it simply says that on average an additional year of schooling is associated with approximately 10.7% higher hourly earnings. The coefficients on potential experience, Exp, and its square, [Exp.sup.2], imply a peak of earnings at 29.2 years of experience (age 49), an initial investment ratio of 0.257 that declines linearly over an investment span of 19.2 years, and a rate of return on postschool training investments of 8.7% (for calculation details, see Mincer 1974; Freeman and Hirsch 2001).

Although this is a highly simplistic model, I find it remarkable that a specification using information on only two worker attributes, schooling and age, accounts for a third of the total variation in individual worker earnings and allows us to infer very roughly key human capital investment parameters. (4)

I next estimate a dense specification of the Mincerian wage equation of the sort seen widely in the literature. This includes 77 rather than three explanatory variables, many of these being dummy variables for such things as schooling degree, broad occupation and industry, region, city size, and a host of demographic variables. There is a labor economics literature surrounding most of these variables. Introduction of 74 additional covariates raises the [R.sup.2] to only 0.528, or from about a third to a half. Indeed, it is rare that one accounts for over half of the individual variation in earnings in a wage equation.

Is one half high or low? Some argue this is low and suggest that much about earnings determination is inexplicable, i.e., the result of luck or randomness in the labor market. I take the half-full rather than half-empty view, for those same reasons stated earlier as to why we do not test a law of one wage. For example, while we account for schooling, potential experience, and other personal and location attributes, these are imperfect measures of human capital, failing to measure the quality of training, worker ability, and personal motivation, all of which affect productivity and earnings. Mismeasurement of earnings and other attributes is likely to increase the residual variation. For example, I excluded from the estimation samples the roughly 30% of workers who do not report their earnings and instead have them imputed (i.e., assigned) by the Census, a topic I return to shortly. Had I included imputed earners, as is standard in the literature, this would lower the [R.sup.2] by 6 points, from 0.528 to 0.466.

No apology is needed for accounting for only half of measured earnings. That said, measurement and specification issues make it difficult to interpret the residual variance and thus say just how large is the deviation from the law of one wage. (5) Where the Mincer equation has proved itself invaluable is in the study of key wage determinants. Much of my work, at least what I focus on in this lecture, examines labor market wage gaps that shed light on specific topics, for example, union wage premiums or interarea wage differences. I will argue that such studies also tell us something about the competitiveness of U.S. labor markets and whether the law of one wage provides a reasonable first approximation of how wages are determined.

3. Match Bias from Imputation: Wage Gaps Larger Than We Think?

A theme running through empirical labor studies is that better control for and measurement of worker and job attributes will lessen the magnitude of what appear to be noncompetitive wage gaps. Estimated wage gaps due to, say, unions, employment in large firms, industry, marriage, etc., would be smaller if only we had better measures of relevant worker skills and job attributes correlated with these regressors. Such reasoning is logical and often correct. For example, Hirsch (2005) shows how part-time/full-time wage gaps for women and men decline as one controls for detailed worker, location, and job characteristics. (6)

In this section, I discuss my research on Census earnings imputation and what I have dubbed "match bias" (Hirsch and Schumacher 2004; Bollinger and Hirsch 2006). Routine inclusion of imputed earners in wage regressions not only increases residual wage dispersion, as one would expect, but severely attenuates (biases toward zero) wage gap estimates for attributes that are not imputation match criteria. This bias affects no small portion of the large literature using the CPS to estimate wage gaps. Many labor market wage differentials are not smaller but, rather, larger than we think. Let me explain.