Firms and Labor Market Inequality: Evidence and Some Theory

A. Measuring Rents

The empirical rent-sharing literature is motivated by an assumed structural relationship between wages and either profit per worker or a measure of quasi rent per worker. To facilitate discussion, suppose that there is a single type of labor at a firm j and that the wage (w_j) is determined by a structural relationship of the form

We assume that TFP_j is the driving source of variation that researchers are implicitly trying to model in the rent-sharing literature. Under this interpretation, firm-specific TFP shocks lead to changes in quasi rent per worker that cause wages to fall or rise relative to the alternative wage. The elasticity of wages with respect to an exogenous change in quasi rent per worker is

An important confounding factor in the rent-sharing literature is variation in worker quality. Firms that employ more highly skilled workers would be expected to have higher revenue per worker and value added per worker and also pay higher wages, leading to a potential upward bias in the measured rent-sharing elasticity in cross-sectional studies that compare different firms at a point in time. A similar bias can also arise in longitudinal studies that compare changes in firm-specific wages and profitability over time if there are unobserved changes in the skill characteristics of workers. For this reason, when we summarize the studies in the literature, we classify studies based on whether the research design includes controls for unobserved worker skills.

B. A Summary of the Rent-Sharing Literature

Table 1 synthesizes the estimated rent-sharing elasticities from the 22 studies listed in Table A1, extracting one or two preferred specifications from each study and adjusting all elasticities to an approximate value added per worker basis.⁵ We divide the studies into three broad generations on the basis of the level of aggregation in the measures of rents and wages.

The first group of studies, which includes two influential papers from the early 1990s, uses industry-wide measures of productivity and either individual-level or firm-wide average wages. The average rent-sharing elasticity in this group is 0.16. A second generation of studies includes five papers, mostly from the mid-1990s, that use firm- or establishment-specific measures of rents but measure average wages of employees at the workplace level. The average rent-sharing elasticity in this group is 0.15, although there is a relatively wide range of variation across the studies. Given the likely problems caused by variation in worker quality, we suspect that most first-generation and second-generation studies yield upward-biased estimates of the rent-sharing elasticity.

A third generation of studies consists of 18 relatively recent papers that study the link between firm- or establishment-specific measures of rents and individual-specific wages. Many of these studies attempt to control for variation in worker quality, in some cases by studying the effect of changes in measured rents on changes in wages. In this group the mean rent-sharing elasticity is 0.08, although a few studies report rent-sharing elasticities that are 0.05 or smaller. To quantify the potential role of rent sharing in the overall dispersion of wages, note that the standard deviation of average value added per worker across firms in the Portuguese data that we analyze in the next section is about 0.70, with a spread between the 90th and 10th percentiles of around 1.6. An elasticity of 0.08 implies that the variation in productivity across firms creates a Lester range of wage variability (Lester 1952) of about 13 log points between firms at the 90th and 10th percentiles.

Although significant progress has been made in this literature, none of these studies is entirely satisfactory. Very few studies have clear exogenous sources of variation in productivity. Most papers (e.g., Guiso, Pistaferri, and Schivardi 2005; Carlsson, Messina, and Skans 2014; Card, Cardoso, and Kline 2016) rely on timing assumptions about the stochastic process driving productivity to justify using lags as instruments. A notable exception is Van Reenen (1996), who studies the effects of major firm innovations on employee wages. He finds a very large rent-sharing elasticity of 0.29, but this figure may be upward biased by skill upgrading on the part of innovative firms, a concern he could not address with aggregate data. Other studies (e.g., Abowd and Lemieux 1993; Card et al. 2014) use industry-level shocks as instruments for productivity. However, these instruments may violate the exclusion restriction if labor supply to the sector is inelastic, since even fully competitive models predict that industry-level shocks can yield equilibrium wage responses. Moreover, industry-level shocks might yield general equilibrium responses that change worker’s outside options (Beaudry, Green, and Sand 2012). Finally, with the move to matched employer-employee micro data, economists have had to contend with serious measurement error problems that emerge when constructing plant-level productivity measures. It remains to be seen whether instrumenting using lags fully resolves these issues.

C. Specification Issues: A Replication in Portuguese Data

To supplement the estimates in the literature and probe the impact of different design choices on the magnitude of the resulting elasticities, we conducted our own analysis of rent-sharing effects using matched employer-employee data from Portugal. The wage data for this exercise come from Quadros de Pessoal (QP), a census of private sector employees conducted each October by the Portuguese Ministry of Employment. We merge these data to firm-specific financial information from the Sistema de Analisis de Balances Ibericos (SABI) database, distributed by Bureau van Dijk.⁶ We select all male employees observed between 2005 and 2009 who work in a given year at a firm in the SABI database with valid information on sales per worker for each year from 2004 to 2010 and on value added per worker for each year from 2005 to 2009.

Panel A of Table 2 presents a series of specifications in which we relate the log hourly wage observed for a worker in a given year (between 2005 and 2009) to mean log value added per worker or mean log sales per worker at his employer, averaged over the sample period. These are simple cross-sectional rent-sharing models in which we use an averaged measure of rents at the employer to smooth out the transitory fluctuations and measurement errors in the financial data. In row 1 we present models using mean log value added per worker as the measure of rents; in row 2 we use mean log sales per worker; and in row 3 we use mean log value added per worker over the 2005–2009 period but instrument this with mean log sales per worker over a slightly wider window (2004–2010). For each choice we show a basic specification (with only basic human capital controls) in column 1, a richer specification with controls for major industry and city in column 2, and a full specification with dummies for 202 detailed industries and 29 regions in column 3.

Two main conclusions emerge from these simple models. First, the rent-sharing elasticity is systematically larger when rents are measured by value added per worker than by sales per worker.⁷ Second, the rent-sharing elasticities from this approach are relatively large. Interestingly, the 0.20–0.30 range of estimates is comparable to the range of the studies in groups 1 and 2 of Table 1.

An obvious concern with the specifications used in panel A is that they fail to fully control for variation in worker quality. As discussed above, this is likely to lead to an upward bias in the relationship between wages and value added per worker. The specifications in panel B of Table 2 partially address this by examining the effect of changes in firm-specific rents on changes in wages for workers who remain at the firm over the period from 2005 to 2009, a within-job or stayers design. We present three sets of specifications of this design. The models in row 4 measure the change in rents by the change in log value added per worker. The models in row 5 use the change in log sales per worker. The models in row 6 use the change in value added per worker as the measure of rents but instrument the change using the change in sales per worker over a slightly wider interval to reduce the impact of measurement errors in value added.⁸

Relative to the cross-sectional models, the within-job models yield substantially smaller rent-sharing elasticities. This difference is likely due to some combination of unobserved worker quality in the cross-sectional designs (which leads to an upward bias in these specifications), measurement error (which causes a larger downward bias in the stayer designs), and the fact that value-added fluctuations may include a transitory component that firms insure workers against (Guiso et al. 2005).⁹ The discrepancy is particularly large for ordinary least squares (OLS) models using sales per worker (compare rows 2 and 5 of Table 2): the elasticity for stayers is only about one-tenth as large as the cross-sectional elasticity. We suspect that measurement errors and transitory fluctuations in annual sales are relatively large, and the impact of these factors is substantially magnified in the within-job specifications estimated by OLS. Given the presence of errors and idiosyncratic fluctuations, we prefer the instrumental variables (IV) estimates in row 6, which point toward a rent-sharing elasticity of approximately 0.06.

An interesting feature of both the OLS and the IV within-job estimates is that the addition of detailed industry controls reduces the rent-sharing elasticity by 10%–20%. Since these industry dummies absorb industry-wide productivity shocks that are shared by the firms in the same sector, we conclude that the rent-sharing elasticity with respect to firm-specific productivity shocks (which is estimated by the models in col. 3) is somewhat smaller than the elasticity with respect to sector-wide shocks (which are incorporated in the elasticities in the models in col. 1). If true more generally, this suggests that the use of industry-wide rent measures will lead to somewhat larger rent-sharing elasticities than would be obtained using firm-specific productivity measures and controlling for industry-wide trends. A similar conclusion is reported by Carlsson et al. (2014).

Overall, we conclude that the estimated elasticities of wages with respect to value added per worker in Portugal are comparable in magnitude to the estimates in the existing rent-sharing literature summarized in Table 1. A typical estimate in our data and the literature from specifications that control for worker quality is between 0.05 and 0.10, although there are a few estimates on each side of this range.

A. AKM Models

In their seminal study of the French labor market, Abowd et al. (1999) specified a model for log wages that includes additive effects for workers and firms. Specifically, their model for the log wage of person i in year t takes the form

Estimates of AKM-style models on population-level administrative data sets from a variety of different countries have found that the firm effects in these models typically explain 15%–25% of the variance of wages, less than the person effects but enough to indicate that firm-specific wage setting is important for wage inequality.¹⁰ One problem with this assessment is that the person and firm effects are estimated with considerable imprecision, which means that the explanatory power of firms will typically be somewhat overstated, a problem that was also recognized in the earlier literature on industry wage differentials (Krueger and Summers 1988). Andrews et al. (2008) provide an approach to dealing with this problem that we discuss in more detail below.

If different firms pay different wage premiums, the pattern of sorting of workers to firms will also matter for overall wage inequality. In particular, the variance of log wages is

An alternative decomposition uses the fact that

B. Identifying Age and Time Effects

A technical issue that arises with the AKM model is appropriate specification of the effects of age. Following Mincer (1974), it is conventional to include a polynomial in age or potential experience (age minus education minus 6) in X_it. However, it is also standard to include a set of year indicators in X_it to adjust for changing macroeconomic conditions. This raises an identification problem because age (a_it) can be computed as calendar year (t) minus birth year (b_i). Hence, we face the classic problem of distinguishing additive age, year, and cohort effects, where cohort effects are understood to load into the person effects.

In their original paper, Abowd et al. (1999) solved this problem by using actual labor market experience (i.e., the number of years the worker had positive earnings since entering the labor market), which, if some employment histories have gaps, will not be perfectly collinear with year and person dummies. While in some respects this provides a simple fix to the problem, there are two important drawbacks. First, it is not always possible to reconstruct a worker’s employment history both because some data sets do not always go far enough back to cover the cohorts of interest and because some data sets report only point-in-time measures of employment (e.g., who was on the payroll in October) rather than a complete history of all employment spells in all years. Second, it is not clear that employment gaps are exogenous, even conditional on a person effect. For example, leaving employment for an entire year could reflect severe health shocks that directly influence earnings ability and confound estimation of relative firm pay.

An alternative approach to dealing with this problem is to impose a linear restriction on the effects of age or time. While the firm effects are invariant to how age and time effects are normalized, different normalizations will yield different values of the person effects and the covariate index ${X_{it}}^{\prime }\beta$. Card et al. (2013) allow for separate third-order polynomials in age by education group along with unrestricted year effects. To obtain identification, they restrict the age profile to be flat at age 40. This is accomplished by omitting the linear age term for each education group and using a cubic polynomial in $a_{\mathit{it}}-40$. The same restriction is used by Card et al. (2016). While this restriction is unlikely to hold exactly, there is reason to believe it provides a good approximation to the shape of the age-earnings profile.¹¹

Table 3 examines the sensitivity of the results of Card et al. (2016) to four alternate normalizations of the age effects. The first column shows the baseline normalization, which attributes a relatively small fraction of the overall variance of wages to the time-varying individual component of wages. Renormalizing the age profile to be flat at age 50 (col. 2) has little effect on this conclusion, whereas renormalizing the profile to be flat at age 30 leads to a slightly larger variance share for the time-varying component and also implies a relatively strong negative correlation between the person effects and the index ${X_{it}}^{\prime }\beta$. Normalizing the age profile to be flat at age 0—which is what is being done by simply omitting the linear term from an uncentered age polynomial—exacerbates this pattern and leads to a decomposition that suggests that the variances of α_i and ${X_{it}}^{\prime }\beta$ are both very large and that the two components are strongly negatively correlated.¹²Figure 2 contrasts the implied age profiles for four single year-of-birth cohorts of low-education men from this naive specification, with the implied profiles for the same groups under the baseline normalization. Evidently, the strong negative correlation between the person effects and the covariate index reported in column 4 of Table 3 is driven by implausibly large cohort effects, which trend in a way to offset the imposed assumption that the cubic age profile is flat at age 0.

Rather than restricting the age profile to be flat at a point, we can also achieve identification by assuming that the true profile is everywhere nonlinear. Column 5 shows the results of using a linear combination of normal density functions in age (with 5-year bandwidths) to approximate the age profile.¹³ Because each Gaussian component is nonlinear, we do not need restrictions on the parameters to avoid collinearity with cohort and time effects. Nevertheless, using Gaussian basis functions will solve the identification problem only if the true age profile has no linear segments. As shown in column 5, the Gaussian approximation yields results somewhere between our baseline normalization and the specification in column 3: although the estimated variability of the worker, firm, and time-varying components is very close to baseline, the correlation of the person effects and ${X_{it}}^{\prime }\beta$ becomes slightly negative. Fortunately, the covariance of the person and firm effects is essentially the same under our baseline normalization and the Gaussian specification, leading us to conclude that most of the statistics of interest in this literature found under an age 40 normalization are robust to alternate identifying assumptions.

To summarize: in comparing results from different applications of the AKM framework researchers should pay close attention to the choice of normalization. The values of the person effects (i.e., the α_i’s) and the time-varying controls (i.e., ${X_{it}}^{\prime }\beta$) are not separately identified when X_it includes both year effects and a linear age term. The choice of normalization has no effect on estimates of $\mathrm{var}\left({\psi }_{J\left(i,t\right)}\right)$, $\mathrm{var}\left({\alpha }_i+X_{\mathit{it}}\beta \right)$, or the covariance term $\mathrm{cov}\left({\psi }_{J\left(i,t\right)},{\alpha }_i+X_{\mathit{it}}\beta \right)$, but as shown in Table 3, it will affect the estimated covariance of the person and firm effects and the relative magnitudes of $\mathrm{var}\left({\alpha }_i\right)$ and $\mathrm{var}\left({\psi }_{J\left(i,t\right)}\right)$.

C. Worker-Firm Sorting and Limited Mobility Bias

In their original study, Abowd et al. (1999) reported a negative correlation between the estimated worker and firm effects, suggesting that sorting of workers to different firms tended to reduce rather than increase overall wage inequality. Subsequent research, however, has typically found positive correlations. For example, Abowd et al. (2003) report a correlation of 0.08 for US workers, while Card et al. (2013) report a correlation of 0.23 for male German workers in the 2000s. As discussed by Abowd et al. (2004) and Andrews et al. (2008), these correlations are biased down in finite samples with the size of the bias depending inversely on the degree of worker mobility among firms. Maré and Hyslop (2006) and Andrews et al. (2012) show convincingly that this limited-mobility bias can be substantial. In sampling experiments they find that the correlation of the estimated effects becomes more negative when the AKM model is estimated on smaller subsets of the available data. While Andrews et al. (2008) and Gaure (2014) provide approaches to correcting for this downward bias in the correlation (and the upward biases in the estimated variances of person and firm effects), their procedures require a complete specification of the covariance structure of the time-varying errors, which makes such corrections highly model dependent.¹⁴ The development of corrections that are robust to unmodeled dependence and heteroscedasticity is an important priority for future research.

D. Exogenous Mobility

Abowd et al.’s (1999) additive worker and firm effect specification is simple and tractable. Nevertheless, it has been widely criticized because OLS estimates of worker and firm effects will be biased unless worker mobility is uncorrelated with the time-varying residual components of wages. In an attempt to provide some transparent evidence on this issue, Card et al. (2013) develop a simple event study analysis of the wage changes experienced by workers moving between different groups of firms. Rather than rely on a model-based grouping, Card et al. (2013) define firm groups on the basis of the average pay of coworkers. If the AKM model is correct and firms offer proportional wage premiums for all their employees, then workers who move to firms with more highly paid coworkers will on average experience pay raises, while those who move in the opposite direction will experience pay cuts. Moreover, the gains and losses for movers in opposite directions between any two groups of firms will be symmetric. In contrast, models of mobility linked to the worker- and firm-specific match component of wages (e.g., Eeckhout and Kircher 2011) imply that movers will tend to experience positive wage gains regardless of the direction of their move, violating the symmetry prediction.

Figures 3 and 4 present the results of this analysis using data for male and female workers in Portugal, taken from Card et al. (2016). The samples are restricted to workers who switch establishments and have at least 2 years of tenure at both the origin and destination firm. Firms are grouped into coworker pay quartiles (using data on male and female coworkers). For clarity, only the wage profiles of workers who move from jobs in quartile 1 and quartile 4 are shown in the figures. The wage profiles exhibit clear steplike patterns: when workers move to higher-paying establishments, their wages rise; when they move to lower-paying establishments, their wages fall. For example, males who start at a firm in the lowest quartile group and move to a firm in the top quartile have average wage gains of 39 log points, while those who move in the opposite direction have average wage losses of 43 log points. The gains and losses for other matched pairs of moves are also roughly symmetric, while the wage changes for people who stay in the same coworker pay group are close to 0.

Another important feature of the wage profiles in figures 3 and 4 is that wages of the various groups are all relatively stable in the years before and after a job move. Workers who are about to experience a major wage loss by moving to a firm in a lower coworker pay group show no obvious trend in wages beforehand. Similarly, workers who are about to experience a major wage gain by moving to a firm in a higher pay group show no evidence of a pretrend. By contrast, if worker mobility were driven by gradual employer learning, we would expect wage changes to precede moves between firm quality groups over the time horizons examined (Lange 2007).

Card et al. (2016) also present simple tests of the symmetry restrictions imposed by the AKM specification, using regression-adjusted wage changes of males and females moving between firms in the four coworker pay groups. Comparisons of upward and downward movers are displayed visually in figure 5 and show that the matched pairs of adjusted wage changes are roughly scattered along a line with slope of −1, consistent with the symmetry restriction.

Similar figures can be constructed using firm groupings based on the estimated pay effects obtained from an AKM model. As shown in Card et al. (2013, their fig. VII), applying this approach to data for German males yields the same conclusions as an analysis based on coworker pay groups. Macis and Schivardi (2016) report such diagnostics using social security earnings data for Italian workers and confirm that wage profiles of movers exhibit the same steplike patterns found in Germany and Portugal.

E. Additive Separability

Another concern with the AKM model is that it presumes common proportional firm wage effects for all workers. One way to evaluate the empirical plausibility of the additive AKM specification is to examine the pattern of mean residuals for different groups of workers and firms. Figures 6 and 7, taken from Card et al. (2016), show the mean residuals for 100 cells on the basis of deciles of the estimated worker effects and deciles of the estimated firm effects. If the additive model is correct, the residuals should have mean 0 for matches composed of any grouping of worker and firm effects, while if the firm effects vary systematically with worker skill, we expect departures from 0. Reassuringly, the mean residuals are all relatively close to 0. In particular, there is no evidence that the most able workers (in the 10th decile of the distribution of estimated person effects) earn higher premiums at the highest-paying firms (in the 10th decile of the distribution of estimated firm effects). The largest mean residuals are for the lowest-ability workers in the lowest paying firms, an effect that may reflect the impact of the minimum wage in Portugal. Residual plots for workers and firms in Germany (reported by Card et al. [2013]) and in Italy (reported by Macis and Schivardi [2016]) also show no evidence of systematic departures from the predictions of a simple AKM-style model.

A different approach to assessing the additive separability assumption comes from Bonhomme, Lamadon, and Manresa (2015), who estimate a worker-firm model with discrete heterogeneity where each pairing of worker and firm type is allowed a different wage effect. Their results indicate that an additive model provides a very good approximation to Swedish employer-employee data; allowing interactions between worker and firm type yields a trivial (0.8%) increase in explained wage variance. Lochner and Schulz (2016) reach a similar conclusion using information on relative wage rankings inside the firm and firm value added to infer worker and firm types. They find, using German data, that additive separability provides a good approximation to the wage structure, except for the lowest-skilled workers.

Although these results suggest that firm effects are, on average, similar for different types of workers, there is of course scope for differences to emerge in selected subpopulations. For example, Goldschmidt and Schmieder (2015) find in large German firms that food, cleaning, security, and logistics (FCSL) workers exhibit different wage fixed effects than other occupations. Specifically, the firm wage effects of FCSL workers are attenuated relative to non-FCSL workers. Likewise, Card et al. (2016) find that Portuguese women exhibit slightly attenuated firm effects relative to men, which they argue reflects gender differences in bargaining behavior.

A. Differential Rent Sharing

We can use the AKM framework to examine another interesting question: to what extent do different groups of workers receive larger or smaller shares of the rents at different firms? To do this, we fit separate AKM models for less educated men (with less than a high school education) and more educated men (with a high school education or more) to our Portuguese wage sample. We then reestimated the same rent-sharing specifications reported in panel A of Table 4 separately for the two groups. The results are reported in panels B and C of Table 4.

The estimates reveal several interesting patterns. Most importantly, although the correlation between wages and value added per worker is a little higher for the more educated men, virtually all of this gap is due to a stronger correlation between the worker quality component of wages and value added. The correlations with the firm-specific pay premiums are very similar for the two education groups. Thus, we see no evidence of differential rent sharing.

This finding is illustrated in figure 8, which shows a binned scatterplot of mean log value added per worker at different firms (on the horizontal axis) versus the relative wage premium for high-educated versus low-educated men at these firms. We also superimpose a bin-scatter of the relative share of higher-education workers at different firms (including both men and women in the employment counts for the two education groups). The relative wage premium is virtually flat, consistent with the regression coefficients in rows 7 and 11 of Table 4, which show nearly the same effect of value added per worker on the wage premiums for the two education groups. In contrast, the relative share of highly educated workers is increasing with value added per worker, a pattern we interpret as largely driven by the labor quality component in value added per worker.¹⁷

A. Market Structure

There are J firms and two types of workers: lower skilled (L) and higher skilled (H). Each firm $j\in \left\{1,\dots ,J\right\}$ posts a pair $(w_{Lj},w_{Hj})$ of skill-specific wages that all workers costlessly observe. Hence, in contrast to search models, workers are fully informed about job opportunities. As in many search models, however, we assume that firms will hire any worker (of appropriate quality) who is willing to accept a job at the posted wage.

Firms exhibit differentiated work environments over which workers have heterogeneous preferences. For worker i in skill group $S\in \left\{L,H\right\}$, the indirect utility of working at firm j is

B. Firm Optimization

Firms have production functions of the form

The firm’s problem is to post a pair of skill-specific wages that minimize the cost of labor services given knowledge of the supply functions (5) and (6). Firms cannot observe workers’ preference shocks {ϵ_iSj}, which prevents them from perfectly price discriminating against workers according to their idiosyncratic reservation values. The firm’s optimal wage choices solve the cost minimization problem

Note that firms post wages with knowledge of the shape of the skill-specific supply schedules but not the identities of the workers who comprise them. The last worker hired is indifferent about taking the job, but the other employees strictly prefer their job to outside alternatives. These inframarginal workers capture rents by means of an information asymmetry: they hide from their employer the fact that they would be willing to work for a lower wage.

We now derive results under various specifications of the production function and the firm’s marginal revenue function. On the technology side, we start with a simple baseline case where $f(\dots )$ is linear in L_j and H_j. This corresponds to a standard efficiency units model of the labor market in which lower- and higher-skilled workers are perfect substitutes. We then consider the more general case where $f(\dots )$ is a constant elasticity of substitution (CES) production function. On the revenue side, we initially assume that the firm faces a fixed output price. We then consider the more general case where the firm faces a constant elasticity product demand function.

C. Baseline Case: Linear Production Function and Fixed Output Price

To develop intuition, we begin with the simplest possible example, where the firm faces a fixed output price $P_j^0$ and has a linear production function

To understand the implications of this model for the relative wage structure, suppose that the reference wages of the two skill groups are proportional to their relative productivities, so that

Note that under the linear technology assumption, value added per standardized unit of labor is $v_j\equiv P_j^0{Y}_j/N_j=P_j^0{T}_j$, so $R_j=v_j/b$ is the ratio of value added per standardized unit of labor to the outside wage for a worker with 1 efficiency unit of labor. Equations (12) and (13) therefore imply that the elasticity of wages of skill group S with respect to value added per worker is

The estimated rent-sharing elasticities in Table 1 (and in our reanalysis of Portuguese data) indicate that a typical value of this elasticity is around 0.10, which suggests that the average value of β_SR_j is also around 0.10.²³ Hence, an average worker earns about 10% higher wages than he or she would earn at the lowest-wage firms in the economy that have $v_j=b$. This estimate of average rents earned per worker is remarkably consistent with the estimates of Card et al. (2016, their Table III) obtained by benchmarking wage premiums at firms in the Portuguese economy relative to the premiums paid by the least profitable firms in the country. By contrast, Hornstein et al. (2011) show that a wide class of search models have difficulty generating mean wages more than 5% above the lowest offered wage.

The elasticity of labor supply for skill group S when wages are determined by the first-order conditions (12) and (13) is

A key implication of equations (12) and (13) is that when ${\beta }_L={\beta }_H$, the relative wages of the two skill groups are independent of firm-specific productivity. To simplify the discussion, assume that β_LR_j and β_HR_j are both relatively small (i.e., on the order of 0.10). In such a case, the Taylor approximation

When ${\beta }_L={\beta }_H=\beta$, wages can be written in the form

When one group has a higher value of the supply parameter β, the log wage gap between workers in different skill groups will be higher at more profitable firms. In this case, the data will be described by an AKM-style model with skill group–specific firm effects. The wage premium for skill group S at firm j will be

1. Between-Firm Sorting

Even when ${\beta }_L={\beta }_H$ and the wage gap between workers in the two skill groups is constant at any given firm, the market-wide average wage for each skill group will depend on their relative distribution across firms. In particular, equation (15) implies that the expected log wage for workers in skill group S is

$$E[\ln w_{\mathit{Si}}]={\alpha }_S+\sum _j{\psi }_j{\pi }_{\mathit{Sj}},$$

where π_Sj is the share of workers in skill group S employed at firm j. Thus, the market-wide wage differential between high- and low-skilled workers depends on their relative productivity, their relative supply elasticities, and the relative shares of the two groups employed at firms with higher or lower wage premiums:

$$\begin{array}{}\text{(16)}& E[\ln w_{\mathit{Hi}}]-E[\ln w_{\mathit{Li}}]={\alpha }_H-{\alpha }_L+\sum _j{\psi }_j({\pi }_{\mathit{Hj}}-{\pi }_{\mathit{Lj}}).\end{array}$$

The third term in this expression represents a between-firm sorting component of the average wage gap. Card et al. (2016) show that 15%–20% of the wage differential between men and women in Portugal is explained by the fact that males are more likely to work at firms that pay higher wage premiums to both gender groups. Similarly, Card et al. (2012) show that an important share of the rising return to education in Germany is explained by the increasing likelihood that higher-educated workers are sorted to establishments with higher pay premiums.

Some simple evidence on the importance of the sorting component for the structure of wages for Portuguese male workers is presented in figure 10. Here, we plot the mean firm effects by age for Portuguese men in five different education groups. We normalize the estimated firm effects using the procedure described by Card et al. (2016), which sets the average firm effect to 0 for firms in (roughly) the bottom 15% of the distribution of log value added per worker. The figure shows two important features. First, within each education group, the mean firm effect associated with the jobs held by workers at different ages is increasing until about age 50 and then slightly decreasing.²⁵ Thus, the life-cycle pattern of between-firm sorting contributes to the well-known shape of the life-cycle wage profile. Second, at all ages more highly educated workers are more likely to work at firms that pay higher wage premiums to all their workers. A significant share of the wage gap between men with different education levels is therefore attributable to differential sorting.

When the supply parameter β varies across groups, the wage decomposition will contain an additional term, reflecting a weighted average across firms of the rent-sharing components of the two skill groups:

$$\begin{array}{rl}E[\ln w_{\mathit{Hi}}]-E[\ln w_{\mathit{Li}}]& ={\alpha }_H-{\alpha }_L+\sum _j{\psi }_j^L({\pi }_{\mathit{Hj}}-{\pi }_{\mathit{Lj}})+\sum _j({\psi }_j^H-{\psi }_j^L){\pi }_{\mathit{Hj}}\\ & ={\alpha }_H-{\alpha }_L+\sum _j{\psi }_j^H({\pi }_{\mathit{Hj}}-{\pi }_{\mathit{Lj}})+\sum _j({\psi }_j^H-{\psi }_j^L){\pi }_{\mathit{Lj}}\mathrm{.}\end{array}$$

As in a traditional Oaxaca (1973)-style decomposition, these expressions give alternative ways to evaluate the contributions of differences in the distributions of the two skill groups across firms and differences in the return to working at a given firm between the two groups.

D. Downward-Sloping Firm-Specific Product Demand

So far we have assumed that the firm is a price taker in its output market. Suppose now that the firm faces an inverse demand function $P_j=P_j^0{Y}_j^{-1/\epsilon }$, with $\epsilon >1$ giving the elasticity of product demand. This yields the marginal revenue function

The implied elasticity of wages of skill group S with respect to value added per standardized unit of labor is

When the firm faces a downward-sloping product demand, value added per efficiency unit of labor (v_j) depends on the endogenous choice of output. In the appendix, we show that the elasticities of v_j with respect to an exogenous shift in output demand (indexed by $P_j^0$) or an exogenous increase in productivity (indexed by T_j) are

E. Imperfect Substitution between Skill Groups

A limitation of our baseline model is that it assumes perfect substitutability between the two skill groups. We now extend the model by assuming that the firm’s output is a CES aggregate of high- and low-skilled labor:

Note that even when ${\beta }_L={\beta }_H$, log wages in this model are nonseparable in worker and firm heterogeneity. This is because firm heterogeneity in factor proportions leads to firm-specific wage-skill gaps of the sort usually analyzed at the market level (e.g., Katz and Murphy 1992). If skill types were observable, it would be natural to estimate such a model via nonlinear least squares using data on firm value added. With unobserved skill types, an interactive fixed effects specification would be required that allows the firm effects to depend on the unobserved skill ratio at the firm.

To derive the rent-sharing elasticities in this model, we define

The general effect of allowing imperfect substitution between L and H labor is to generate value added elasticities that are smaller than the rent shares τ_Lj and τ_Hj but of similar magnitudes. For example, consider a calibration with $\theta =0.6$, $\sigma =1.4$, and an average workforce comprised of 60% L workers and 40% H workers. In this setup, a value of $R_j=1.45$ combined with ${\beta }_L=0.09$ and ${\beta }_H=0.15$ yields ${\tau }_L=0.10$ and ${\tau }_H=0.20$, implying that lower-skilled workers earn a 10% wage premium relative to their outside wage while higher-skilled workers earn a 20% premium. The implied firm-specific labor supply elasticities of the two groups are 5 and 2, respectively, while the implied elasticities of their wages with respect to value added per worker are 0.05 and 0.11, respectively.

To summarize, a CES technology will give rise to generalized AKM models in which the wage differentials between different skill groups vary across firms, and the elasticity of wages with respect to value added per worker will also vary across groups and across firms, depending on the firm-specific skill employment shares at the firm. Nevertheless, the basic intuition of our benchmark model remains. In particular, a simple model of monopsonistic wage setting can explain the existence and quantitative magnitude of systematic firm wage premiums that are highly correlated across skill groups and highly correlated with measures of firm-specific productivity.

F. Relationship to Other Models and Open Questions

Although we have worked with a static model of employer differentiation, there are obvious benefits to considering more realistic dynamic models, not least of which is that they provide a rationale for the worker mobility typically used to estimate firm wage effects. Section A2 considers a simple dynamic extension of our framework that yields random mobility between firms and has essentially identical steady-state implications for wages and employment. However, it would be interesting to consider richer models where workers systematically climb a productivity job ladder and can spend some time unemployed. Another interesting extension would be to allow incumbent workers to face switching costs that lead firms to price discriminate against them. This could lead to offer-matching behavior (as in Postel-Vinay and Robin 2002) and to new predictions about recruitment and retention policies.

By assuming that the number of employers is very large, we have adopted a partial equilibrium framework with no strategic interactions between employers. With a finite number of firms, a shock to one firm’s productivity will affect the equilibrium employment and wages of competitor firms. Staiger, Spetz, and Phibbs (2010) provide compelling evidence of such responses in the market for nurses. As in the oligopoly literature, analysis of a finite employer model with strong strategic dependence may be complicated by the presence of multiple equilibria, which requires different methods for estimation (e.g., Ciliberto and Tamer 2009) but may also yield interesting policy implications.

Finally, it is worth noting some links between our modeling of workplace differentiation with the literature on compensating differentials for nonwage amenities (Rosen 1986; Hwang et al. 1998). In our model, nonwage amenities that are valued equally by all workers simply shift the intercept of the labor supply curve to the firm. But a monopsonist firm sets wages on the basis of the elasticity of labor supply to the firm, which is governed entirely by the distribution of taste heterogeneity. For this reason, our model exhibits no compensating differentials of the standard sort. Amenities affect firm effects only through their influence on TFP: a firm with attractive nonwage amenities will grow large, which should depress its revenue productivity and therefore lower its firm wage effect. Empirically distinguishing this effect, which is mediated through product prices, from the standard compensation mechanism is policy relevant, since the monopsony model will tend to imply a different incidence of, for example, employer-provided health benefits on workers than a compensating differentials model.