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Hours worked and the U.S.\ distribution of real annual earnings 1976--2019
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This paper examines the role of annual hours of work in the changing distribution of annual labor earnings in the U.S.\ from 1976 to 2019. This period experienced substantial fluctuations in the level of annual hours of work due to an upward trend in female labor force attachment and three substantial recessions. Identifying the impact of annual hours worked on annual labor earnings requires accounting for the simultaneity of hours and wages and the selection bias contaminating observed wages. We do so by extending the Fern\'{a} ndez-Val et al. (2021a), hereafter FVV, estimator for nonseparable models with censored selection rules. Whereas FVV employ the estimated hours equation to account for the selection bias in wages we also use it to predict counterfactual labor earnings.
Empirical investigations of inequality typically focus on wages. Notable examples include Katz and Murphy (1992), Murphy and Welch (1992), Juhn, et al. (1993), Welch (2000), Autor, et al. (2008), Acemoglu and Autor (2011), Autor, et al. (2016), and Murphy and Topel (2016). The distribution of annual labor earnings provides a complementary description of inequality as it incorporates the number of annual hours worked by those at different points of the wage distribution. Changes in earnings incorporate variation on earnings from movements in both labor supply and demand not captured in wage inequality. For example, did the drastic changes in female labor supply exacerbate or mitigate the increases in wage inequality? Have the increased education levels accompanying the rising skill premia exacerbated earnings inequality by increasing the level of annual hours of the more educated? Education effects may also reflect changes in the demand for highly skilled labor. Another demand consideration is the impact of recessions. These episodes witness decreases in labor demand. Understanding their impact on earnings at different points of its distribution assists policy makers in forming policies to, for example, offset earnings losses or to encourage additional educational investment. These issues motivate our focus on understanding the impact of hours on the earnings distribution. We do not provide a detailed examination of the factors generating the changes in the hours distribution.
To evaluate the sources of changes in earnings inequality we construct counterfactual earnings distributions which exploit that earnings are the product of annual hours and hourly wages. We decompose differences across counterfactual distributions into four components. The first two are the structural and composition effects identified in wage decomposition exercises, although their interpretation here is more complicated. The remaining two are hours effects. The extensive effect captures the impact of movement from nonemployment to positive annual hours worked. The intensive effect reflects the impact of changes in the annual hours of those working. The combined effect of hours can be estimated without imposing a \textquotedblleft rank invariance" assumption, but we employ it to separate the hours effects.
This paper makes the following contributions. First, while most previous studies treat annual hours worked as exogenous when evaluating their impact on annual earnings we follow Altonji et al. (2013) and model hours as endogenous. Second, we extend the existing decomposition methodologies by quantifying the hours effects on the earnings distribution and inequality. Finally, we apply our methodology to data from the Annual Social and Economic Supplement of the Current Population Survey, or March CPS, for the survey years 1976--2019. We rely on existing evidence on wages, estimated for the same sample and reported in Fern\'{a}ndez-Val et al. (2020), hereafter FPVV, when interpreting our results.
Section (ref) summarizes the relevant literature. Section (ref) discusses the data and Section (ref) presents the modelling framework. The counterfactual distributions and decomposition methodology are discussed in Section (ref). Preliminaries with respect to the empirical investigation are provided in Section (ref) and Section (ref) reports and discusses the empirical results. Section (ref) concludes.
Gottschalk and Danziger (2005) examine the ratio of the upper and lower deciles of the earnings distributions of working individuals using the CPS for 1975 to 2002 and conclude it increased for males but decreased for females. They attribute the decrease for females to increased working hours of those at the bottom of the female wage distribution. Blau and Kahn (2009) decompose changes in annual earnings into hours and wages effects and model the variance of log annual earnings as the sum of the variances of the log hourly wage and working hours plus twice their covariance. Examining data for the U.S. and seven other countries for the mid to late 1990s they find that the variance in working hours plays an important role in explaining the differences in the variance of annual earnings. However, it is less important for the U.S.. Checchi et al. (2016) use a similar approach in examining the mean log deviation of earnings. Biewen and Pl\"{o}tze (2019) investigate the role of working hours in the increased earnings inequality in Germany. They follow a similar methodology to Blau and Kahn (2009) and conclude, via the decomposition method of Dinardo et al. (1996), that changes in working hours increased both male and female earnings inequality.
While the majority of papers treat working hours as exogenous to earnings Altonji et al. (2013) estimate a semi-structural model for the labor market behavior of male household heads with endogenous labor market transitions, wages and working hours. Their main objective is to evaluate the impact of shocks in, for example, unemployment on the variance in the working hours, wages and earnings. This approach could also assess the impact of working hours on the earnings distribution. Our non-structural approach is complementary to theirs as we are able to directly change the hours distribution without changing shocks in the underlying economic model. Moreover, our nonparametric and nonseparable model provides a more flexible approach to producing counterfactual earnings distributions. Card and Hyslop (2021) also consider a model with an endogenous work decision. Examining three different PSID cohorts of females they decompose changes in the variance into extensive and intensive margins. Their dynamic model allows a consideration of both within and between cohort variances in earnings. They conclude that the substantial reduction in inequality for more recent cohorts is the result of increased employment rates.
We employ March CPS data from 1976 to 2019 for annual earnings, weeks worked, and usual hours of work per week in the previous calendar year. We restrict our analysis to those aged 24--65 years in the survey year. This produces a sample of 1,933,659 males and 2,098,894 females. Annual sample sizes range from a minimum of 30,767 males and 33,924 females in 1976 to a maximum of 55,039 males and 59,622 females in 2001. Annual earnings are reported as wage and salary incomes. Hourly wages are defined as annual earnings divided by annual hours worked. Annual hours worked last year (hereafter annual hours) are the product of weeks worked and usual hours of work per week. The same level of annual hours can reflect different combinations of hours and weeks and this has implications for evaluating changes along the extensive and intensive margins. We discuss this in Section (ref).
Individuals reporting zero hours worked last year generally respond as not being in the labor force in the week of the March survey. We refer to these individuals as nonemployed and note that the movement between years, especially for males, could reflect unemployment. As hourly wages are defined as the ratio of reported annual earnings and annual hours they are only available for those in the labor force. This has implications for those in the Armed Forces, the self-employed, and unpaid family workers as their annual earnings and annual hours tend to be poorly measured. Accordingly, we confine attention to civilian dependent employees with hourly wages and those out of the labor force last year. This sample contains 1,676,014 males and 1,975,013 females (respectively 86.7% and 94.1% of the original sample aged 24--65). The subsample of civilian dependent employees with positive hourly wages contains 1,450,941 males and 1,377,718 females.
There are two additional data issues. The first is the top coding of earnings. This affects mean earnings but not quantiles below the top coded values. The second is the difference between actual and usual working hours. Although the CPS reports actual hours worked in the survey week we use usual hours as they are closely related to annual earnings. The two measures have similar means and their correlation is high at 0.7 in 1976 and 0.65 in 2019.
\pgfplotstableread{results_earnings_quantiles_raw_0.10_males.txt}{\resultsa} \pgfplotstableread{results_earnings_quantiles_raw_0.25_males.txt}{\resultsb} \pgfplotstableread{results_earnings_quantiles_raw_0.50_males.txt}{\resultsc} \pgfplotstableread{results_earnings_quantiles_raw_0.75_males.txt}{\resultsd} \pgfplotstableread{results_earnings_quantiles_raw_0.90_males.txt}{\resultse} \pgfplotstableread{results_earnings_quantiles_raw_0.10_females.txt}{ \resultsf} \pgfplotstableread{results_earnings_quantiles_raw_0.25_females.txt}{ \resultsg} \pgfplotstableread{results_earnings_quantiles_raw_0.50_females.txt}{ \resultsh} \pgfplotstableread{results_earnings_quantiles_raw_0.75_females.txt}{ \resultsi} \pgfplotstableread{results_earnings_quantiles_raw_0.90_females.txt}{ \resultsj} \pgfkeys{/pgf/number format/.cd,1000 sep=}
Figure (ref) presents the time series of various quantiles of the annual earnings distribution for the full sample. We use the CPI to transform these quantiles to 2019 dollars. We present the lower quartile (Q1), the median (Q2), the upper quartile (Q3), and the last decile (D9). Shaded vertical bars denote recessions as identified by the NBER.
Mean female earnings increase by over 100% and reflect the drastic changes in both employment rates and annual hours. Female employment rates are under 75% for almost all years of our analysis and the lowest quartile of female earnings is nearly always zero. The quantiles we examine are generally monotonically increasing throughout the period although there is evidence of cyclicality. The extraordinary increase at the median of 512% reflects the drastic changes in the female employment behavior. There are also substantial increases of 77% and 70% at Q3 and D9.
Mean male earnings decrease by 7%. The changes are cyclical and variable. The mean decreases substantially until around 2010 but this decrease is subsequently partially offset. A sharp decline at the beginning of the sample period is followed by a modest increase and a comparable decrease. A large decrease occurs during the Great Recession. The lower quartile paints a remarkably bleak portrait of the labor market experience at this quantile. There are three instances of large decreases. The first is the beginning of the 1980s, the second is the early 1990s, and the final, and most dramatic, corresponds to the Great Recession. This produces a reduction of 53% from 1976 to 2012. Despite the sharp rebound, earnings are 30% lower in 2019 than in 1976. Median earnings follow a similar pattern to mean earnings. They are variable and cyclical and decrease by 17% over the sample period. Earnings at Q3 and D9 show similar cyclical patterns although the reduction for the period at Q3 is only 0.1% while there is an increase of 11% at D9. The remarkably contrasting experiences of females and males is reflected in earnings dispersion. Using the ratio (D9/Q2) as a measure of inequality, male earnings dispersion has increased from 1.84 to 2.48, while that of females decreased from 10.3 to 3.05. To highlight the impact of non participation consider the corresponding quantiles of earners. Females have gains of 90% at Q1, 52% at Q2, and 50% at Q3. Males annual earnings decrease by 15% at Q1, 9% at Q2, and increase by 8% at Q3.
Figure (ref) of Fern\'{a}ndez-Val et al. (2021b) shows that the male employment rate for our data decreased from about 89.9% in 1976 to 83.3% in 2019. The decline is relatively steady although the larger falls occur during cyclical downturns and the small increases follow these recessionary periods. The female employment rate increases from 56.0% in 1976 to 74.9% by 2001. It falls over the remainder of our sample period with the biggest declines in recessionary years. Beaudry et al. (2016) conjecture that the recent decrease of employment of both males and females reflects the diminished demand for cognitive skills which reduced the number of high-skilled workers in well-paying jobs (see also Dillon and Veramendi, 2018). This resulted in these workers taking jobs not requiring high levels of cognitive skills and the crowding out of lower educated workers that had held them.
\pgfplotstableread{results_hours_quantiles_raw_0.10_males.txt}{\resultsa} \pgfplotstableread{results_hours_quantiles_raw_0.25_males.txt}{\resultsb} \pgfplotstableread{results_hours_quantiles_raw_0.50_males.txt}{\resultsc} \pgfplotstableread{results_hours_quantiles_raw_0.75_males.txt}{\resultsd} \pgfplotstableread{results_hours_quantiles_raw_0.90_males.txt}{\resultse} \pgfplotstableread{results_hours_quantiles_raw_0.10_females.txt}{\resultsf} \pgfplotstableread{results_hours_quantiles_raw_0.25_females.txt}{\resultsg} \pgfplotstableread{results_hours_quantiles_raw_0.50_females.txt}{\resultsh} \pgfplotstableread{results_hours_quantiles_raw_0.75_females.txt}{\resultsi} \pgfplotstableread{results_hours_quantiles_raw_0.90_females.txt}{\resultsj} \pgfkeys{/pgf/number format/.cd,1000 sep=}
Figure (ref) presents the distribution of annual hours for our sample period. Male annual hours display cyclical behavior which is particularly important at lower quantiles. There has been a large increase in median female annual hours while higher quantiles are relatively constant. Cyclical effects for females are weaker and are manifested as occasional dips in otherwise upward trends. The median for males and Q3 for females show little or no variation due to bunching in hours. Approximately 40% of males and 30% of females report 2080 hours in 1976. This corresponds to 52 weeks of a 40-hour work week. Almost 50% of males and 40% of females report 2080 annual working hours by 2019. While this increased bunching of annual hours has received some attention (see Bick, Blandin and Rogerson 2021) we do not investigate it here. As the impact of movements on the extensive margin depends on where entering workers locate in the hours distribution, Figure (ref) of Fern\'andez-Val et al. (2021b) reports that almost all working males are working at least 500 annual hours while a substantial fraction of employed females are working less than 500 hours.
\pgfplotstableread{results_hours1_quantiles_raw_0.25_males.txt}{\resultsb} \pgfplotstableread{results_hours1_quantiles_raw_0.50_males.txt}{\resultsc} \pgfplotstableread{results_hours1_quantiles_raw_0.75_males.txt}{\resultsd} \pgfplotstableread{results_hours1_quantiles_raw_0.25_females.txt}{\resultsg} \pgfplotstableread{results_hours1_quantiles_raw_0.50_females.txt}{\resultsh} \pgfplotstableread{results_hours1_quantiles_raw_0.75_females.txt}{\resultsi} \pgfkeys{/pgf/number format/.cd,1000 sep=}
Figure (ref) presents mean hours by different quartiles of the earnings distribution. Q1 denotes mean hours for workers with labor income less than the lower quartile of earnings. Q2, Q3 and Q4 are defined similarly. Male hours have not substantially increased at any point in the earnings distribution and their cyclicality is more evident at lower earnings. There are small increases at higher female earnings and much larger increases at the median and below. The number of female annual hours worked has doubled over the sample period at the lowest quartile (see also Gottschalk and Danziger, 2005).
We also consider if movements in annual hours and employment rates reflect notable changes in their likely determinants. Education levels, particularly of females, increased dramatically. The large increases in those obtaining college degrees are sometimes attributed to the decreasing selectivity of colleges although the effects appear too small to fully explain these observed increases (Hoxby 2009). Figures (ref) and (ref) of Fern\'{a} ndez-Val et al. (2021b) show that college graduates have the highest employment rates and average working hours over the whole sample period for both genders while the difference between education levels is cyclical.
We also explore the relationship between annual hours and some demographic characteristics which influence labor supply. Figure (ref) of Fern\'{a}ndez-Val et al. (2021b) illustrates that males work more when they are married or have children while the opposite is observed for females. While the magnitude of these effects for females has decreased over time, their relative ordering has remained stable. We examined NLSY data to investigate the relationship between working hours and cognitive skills at different periods and find a consistently positive relationship between working hours and the AFQT test score for both males and females.
This section highlights that the implied drastic increases in males' earnings inequality are not accompanied by corresponding changes in annual hours worked although the annual hours of those at lower annual earnings are more cyclical. The drastic changes in annual earnings for females are accompanied by substantial changes in their annual hours. Females' hours are increasing throughout their earnings distribution with the more drastic increases at lower earnings. The data also suggest that the nature of relationships between an individual's observed characteristics and their employment decision and annual hours has not changed. This supports our modeling approach which assumes the types of individuals, as captured by both their observed and unobserved characteristics, likely to work many hours in each cross section do not change over the sample period.
Consider the following model describing earnings, wages and hours:
where $Y_{t}$, $W_{t}$, and $H_{t}$ denote annual earnings, hourly wages, and annual hours of work for an individual in year $t$, respectively. Equation ((ref)) specifies that $Y_{t}$ is the product of $W_{t}$ and $ H_{t}$. Equation ((ref)) states that $W_{t}$ is a function of observed individual characteristics, $X_{t}$, and unobserved characteristics, $E_{t}$ . The function $g_{t}$ is assumed to be unknown and nonseparable with respect to its arguments. $W_{t}$ is only observed for individuals reporting positive annual hours. The process generating observed annual hours is defined in (ref). $H_{t}$ is a function of $X_{t}$ and additional observed characteristics, $Z_{t}$, that influence the individuals' preference for work but not their wage. The random variable $V_{t}$ represents unobserved characteristics that affect $H_{t}$. The relationship between these determinants and hours is represented by the unknown and nonseparable function $k_{t}$. $H_{t}$ is censored from below at zero.
The support of random variables and vectors is denoted by calligraphic letters; e.g., $\mathcal{XZ}_{t}$ is the joint support of $(X_{t},Z_{t})$. Since this can depend on whether the individual belongs to the \textquotedblleft selected population\textquotedblright\ with $H_{t}>0,$ an asterisk indicates support in the selected population. For example, $ \mathcal{X}_{t}^{\ast }$ is the support of $X_{t}\mid H_{t}>0$. Small letters in parentheses indicate the support is conditional on other variable(s) taking a particular value; e.g. $\mathcal{Z}_{t}(x)$ is the support of $Z_{t}\mid X_{t}=x$.
Equations (ref)-(ref) comprise the model of FVV. We augment it with ((ref)). Two assumptions are required to continue. First, the vector $(X_{t},Z_{t})$ is independent of the vector $(E_{t},V_{t})$ in the entire population. This excludes additional issues related to endogeneity. This assumption may be strong for human capital variables but we do not address this here. Second, $V_{t}$ is continuously distributed, and the distribution of $V_{t}$ and $v\mapsto k_{t}(\cdot ,\cdot ,v)$ are strictly increasing on the support of $V_{t}$. Given these conditions, we can normalize $V_{t}\sim U(0,1)$.
We allow for dependence between $E_{t}$ and $V_{t}$, implying that unobserved characteristics affecting the hourly wage are correlated with those contributing to annual hours. This arises in the labor supply literature (Killingsworth, 1983) where working hours are chosen according to the maximization of a utility function, denoted by $U(c, l)$, which depends on consumption $c$ and leisure $l$. Then $H =\arg \max_{h}U(g_{t}(X,E)h+S,\bar H-h)$ where $S$ and $\bar H$ denote non-labor income and total time available. The restrictions for participation are $H>0$ and $U(g_{t}(X,E)H+S,\bar H-H)>U(S,\bar H)$ implying that any (un)observed heterogeneity in $g_{t}$ should also be in $k_{t}$.
The correlation between $E_{t}$ and $V_{t}$ produces a selection bias in estimating $g_{t}$ over the sample for $H_{t}>0$ even when $E_{t}$ and $ X_{t} $ are independent over the entire population. We address this via the FVV control function approach based on the Imbens and Newey (2009) estimator but allowing for censored selection. The appropriate variable is $V_{t}$, which captures the \textquotedblleft rank\textquotedblright\ of the individual's annual hours relative to observationally equivalent individuals. $V_{t}$ is a control function since conditioning on $X_{t}$, $Z_{t}$ and $V_{t}$ makes selection, i.e. $ k_{t}(X_{t},Z_{t},V_{t})>0$, deterministic. Thus, the distribution of $E_{t}$ conditional on $X_{t}$, $Z_{t}$ and $V_{t}$ does not stochastically depend on selection. Moreover, conditional on $V_{t}$, $E_{t}$ does not depend on $ X_{t}$ and $Z_{t}$ in the selected population.
As $V_{t}$ is constructed using the movement in $H_{t}$ for observations $ H_{t}>0$, it has variation conditional on $(X_{t},Z_{t}).$ This contrasts with the propensity score, often used as the basis of control functions in dichotomous selection models, which is constant conditional on $(X_{t},Z_{t})$. Moreover, as monotonicity implies $V_{t}$ equals the distribution of $H_{t}\mid X_{t},Z_{t}$ it is identified provided $H_{t}>0$.
To distinguish between extensive and intensive margins we consider a generalization of the model (ref)-(ref) where:
where $k_{t}^{1}$ and $k_{t}^{2}$ satisfy the same conditions as equation (ref), and $k_{t}^{1}(X_{t},Z_{t},V_{t}^{1})\geq k_{t}^{2}(X_{t},Z_{t},V_{t}^{2})$. Here, $k_{t}^{1}$ represents the hours worked equation and $k_{t}^{2}$ captures the participation or employment equation. Model (ref)- (ref) is a special case with $k_{t}^{1}=k_{t}^{2}$ and $ V_{t}^{1}=V_{t}^{2}$. The model (ref)-(ref) is useful for counterfactual scenarios in which, for example, we examine how the population from a year $t$ with a high participation rate would participate in a year $s$ with a lower participation rate, i.e. $ k_{t}(X_{t},Z_{t},V_{t})\geq k_{s}(X_{t},Z_{t},V_{t})$, while keeping the hours worked equation as in year $t$. This scenario can be modeled by setting $k_{t}^{1}=k_{t}$, $k_{t}^{2}=k_{s}$ and $V_{t}^{1}=V_{t}^{2}=V_{t}$ . We return to this in Section (ref). The condition $ V_{t}^{1}=V_{t}^{2}$ implies that for every $(X_{t},Z_{t})$ combination, $ k_{t}^{1}(X_{t},Z_{t},V_{t})=k_{t}^{2}(X_{t},Z_{t},V_{t})+\Psi _{t}(X_{t},Z_{t}),$ for some positive function $\Psi _{t}$. This enables an investigation of when workers decide not to work if their \textquotedblleft potential\textquotedblright\ working hours are lower than a certain cutoff value $\Psi (X_{t},Z_{t})\geq 0$. It also imposes rank invariance in the hours worked and participation equations conditional on $(X_{t},Z_{t})$ (e.g., Chernozhukov and Hansen, 2005). In the previous example, it implies the number of hours worked does not change for those who remain working after the counterfactual change in the participation equation.
Our main objects of interest are the observed and counterfactual earnings distributions. The counterfactual distributions are constructed from (ref)-(ref) and (ref)-(ref) by combining the functions $g_{t}$ and $k_{t}$ and the distribution of $(X_{t},Z_{t},V_{t})$ from different years. A critical component of these distributions is the relationship between earnings, wages and hours. For any $y>0:$
where the second term reflects that the probability that $Y_{t}\leq y$ equals unity when $H_{t}=0$, and $\mathrm{P} (Y_{t}\leq 0)=\mathrm{P} (H_{t}=0).$ The same relationship holds when we condition on $(X_{t},Z_{t},V_{t})$.
We write the observed distribution of earnings in terms of the functions of the model. By the law of iterated probability:
Using the model (ref)-(ref), for any $y>0$:
with $G_{t}(w,x,v):=\mathrm{P} (g_{t}(x,E_{t})\leq w\mid V_{t}=v)$. We drop the conditioning of $X_{t}$ and $Z_{t}$ in the probability of the first integral since $E_{t}$ is independent of these variables conditional on $V_{t}$. FVV referred to $G_{t}(w,x,v)$ as the local distribution structural function (LDSF). It represents the distribution of potential hourly wages $ g_{t}(x,E_{t})$ corresponding to $X_{t}=x$ conditional on $V_{t}=v$. FVV show that identification of the LDSF requires that the specific combination of $x$ and $v$ being considered belongs to the support of $ (X_{t},V_{t})$ in the selected population, i.e. $(x,v)\in \mathcal{XV} _{t}^{\ast }$. This requires a realization of $(X_{t},Z_{t},V_{t})$ equal to $(x,z,v)$ such that $ k_{t}(x,z,v)>0$. The exclusion restriction $Z_{t}$ plays two roles. First, it is needed to guarantee that $(X_{t},Z_{t})$ is independent of $(E_{t},V_{t})$. \ That is, $Z_{t}$ must be correctly specified to ensure the control function is correctly estimated. Second, it expands the set $\mathcal{XV}_{t}^{\ast }$ if there is even only one value $z\in Z_{t}$ for which $k_{t}(x,z,v)$ is positive. This expansion is useful if, for example, we are interested in identifying the marginal distribution of $g_{t}(x,E_{t})$ in the selected or entire populations.
Equation ((ref)) is the basis of the counterfactual distributions. We define a counterfactual earnings distribution as:
which corresponds to the resulting earnings distribution if the hours worked equation were as in year $r$, the distribution of explanatory variables and control function as in year $p$, and the wage structure (LDSF) as in year $s$ . For example, the observed distribution of earnings in year $t$\ corresponds to $G_{\langle t,t,t\rangle }=F_{Y_{t}}$.
Similarly, we can define counterfactual distributions of earnings from model (ref)-(ref). In this case:
which corresponds to the resulting earnings distribution if the participation equation were as in year $q$, hours worked equation as in year $r$, distribution of explanatory variables and control function as in year $ p $, and wage structure as in year $s$. This expression yields the counterfactual distribution (ref) if $k_{q}^{2}=k_{r}^{1}$ and $ V_{r}^{1}=V_{q}^{2}$.
A first step in the empirical analysis using counterfactual distributions is to establish their nonparametric identification. The distributions $G_{\langle r,s,t\rangle }$ and $G_{\langle q,r,s,t\rangle }$ are identified if all the integrands are well-defined over supports that contain the regions of integration. The following proposition gathers support conditions that guarantee identification and is proven in Appendix (ref) of Fern\'{a}ndez-Val et al. (2021b).
The conditions of Proposition (ref) simplify when we only consider two years for $q$, $r$, $s$ and $p$, such as $0$ and $t$, which is relevant for the earnings decompositions in the next section. For example, we only need $\mathcal{XV}_{t}\cap \mathcal{XV}_{0}^{\ast }\subseteq \mathcal{XV} _{t}^{\ast }$ and $\mathcal{XZ}_{t}\subseteq \mathcal{XZ}_{0}$ to identify $ G_{\langle 0,t,t,t\rangle }$ and $G_{\langle 0,0,t,t\rangle }$, and $ \mathcal{XZ}_{t}\subseteq \mathcal{XZ}_{0}$ to identify $G_{\langle 0,0,0,t\rangle }$. A sufficient condition for $\mathcal{XV}_{t}\cap \mathcal{ XV}_{0}^{\ast }\subseteq \mathcal{XV}_{t}^{\ast }$ is that the employment rates in year $0$, conditional on $X$, be lower than those in year $t$. For females, this is satisfied if the year $t$ is after year $0$. A sufficient condition for $\mathcal{XZ}_{t}\subseteq \mathcal{XZ}_{0}$ is that any $ (x,z) $ observed in year $t$ be also observed in year $0$.
The impact of $Z$ on the plausibility of the condition $\mathcal{XV}_{p}\cap \mathcal{XV}_{q}^{\ast }\subseteq \mathcal{XV}_{s}^{\ast }$ is ambiguous. Without further assumptions, $Z$ might expand each of the sets $\mathcal{XV} _{p}$, $\mathcal{XV}_{q}^{\ast }$ and $\mathcal{XV}_{s}^{\ast }$ differently. However, if $Z$ has sufficient enough variation to ensure $ \mathcal{XV}=\mathcal{X}\times \lbrack 0,1]$ in each period the condition simplifies to $\mathcal{X}_{p}\cap \mathcal{X}_{q}^{\ast }\subseteq \mathcal{ X}_{s}^{\ast }$.
Let $Q_{\langle q,r,s,p\rangle }(\tau )$\ denote the counterfactual $\tau $ -quantile of earnings corresponding to the left-inverse of the distribution $ y\mapsto G_{\langle q,r,s,p\rangle }(y)$.\footnote{ \ That is, $Q_{\langle q,r,s,p\rangle }(\tau )=\inf \{y\in \mathbb{R}\colon G_{\langle q,r,s,p\rangle }(y)\geq \tau \}$, for $\tau \in (0,1)$.} The definitions above allow the following decomposition of the observed change in the $\tau $-quantile of earnings between the baseline year (labeled as year 0) and any other year $t$:
provided that $Q_{\langle 0,0,0,0\rangle }(\tau )>0$.
Term [1] is the \textquotedblleft extensive hours effect\textquotedblright\ capturing how individuals moving from working to non working reduce their earnings to zero. Term [2] is the \textquotedblleft intensive hours effect\textquotedblright\ and measures earnings movements resulting from shifts in the distribution of annual hours for those who work conditional on their characteristics (i.e. $X_{t}$ and $V_{t}$). The sum of [1] and [2] is the \textquotedblleft total hours effect\textquotedblright , $[1]+[2]=\frac{ Q_{\langle t,t,t,t\rangle }(\tau )-Q_{\langle 0,0,t,t\rangle }(\tau )}{ Q_{\langle 0,0,0,0\rangle }(\tau )}.$ Term [3] is the \textquotedblleft structural wage effect\textquotedblright\ reflecting earnings changes from movements in the hourly wage distribution for workers conditional on their characteristics and captures the changing market evaluation of these characteristics. Term [4] is the \textquotedblleft composition effect\textquotedblright\ and represents changes in the earnings distribution due to changes in the distribution of worker characteristics.
The extensive hours effect captures the difference between the quantiles of observed earnings in year $t$\ and those that would prevail if the participation equation was as in year $0$. As this reflects the mapping of the $X$ and $Z$ into participation via the hours equation for year $0$, $k^{2}$, it corresponds to a structural effect. Due to the identification assumptions for $G_{\langle 0,t,t,t\rangle }$, it is always non-negative. That is, an individual with a positive probability of working in base year $0$ should have a positive probability in year $t$, i.e. $ k_{0}(X_{t},Z_{t},V_{t})\leq k_{t}(X_{t},Z_{t},V_{t})$. By the monotonicity and rank invariance assumptions, those who stop working under the participation equation $k_{0}$ will have low levels of $V_{t}$ and small numbers of annual hours before the change.
The intensive hours effect is the difference between the quantiles of earnings from changing the hours equation from year $t$\ to year $0$, while keeping the functions $G$\ and $F_{X,Z,V}$ as in year $t$\ and the participation equation as in year $0$. As we go from the base year to year $ t $, this will capture differences in the hours distribution of those working. This is also a structural effect on hours worked as it results from changing the hours equation $k^{1}$ which maps the $X$ and $Z$ into annual hours from $k_{t}$ to $k_{0}$. This effect is also non-negative by construction as individuals work less under the hours worked equation of year $0$ and we are keeping hourly wages fixed. We highlight again that these differences may reflect either changes in weekly hours, weeks worked or unemployment spells.
The composition effect contains the impact on earnings of changes via movements in the composition of both the observable and unobservable characteristics that affect wages and/or working hours. For example, increases in female educational attainment could shift the distribution of earnings to the right due to increases both in the hourly wages and annual hours. Due to the nonseparable and nonparametric nature of the model, it does not seem possible to disentangle these two effects. The higher educational levels which increased hours, and thus higher annual earnings (on both the extensive and intensive margin), are interpreted as composition and not hours effects.
The structural wage effect captures how the individual's characteristics map into the distribution of hourly wages and annual earnings. The structural effects operating on earnings through hours are assigned either the interpretation of extensive or intensive hours effects. Studies that estimate parametric selection models generally attribute changes in the values of the control function and its coefficient as selection effects. However, FPVV show that in a nonseparable model this attribution is not possible. We interpret the impact of changes in the value of the control function as a composition effect and a change in the impact of the control function on earnings, operating through hourly wages, as a structural effect.
As with the Oaxaca-Blinder decomposition, the baseline year $0$ and the order of the decomposition have implications for the interpretation and magnitude of the effects. They also matter here for the plausibility of the support conditions needed for nonparametric identification of the counterfactual distributions involved in the decomposition. This makes it convenient to choose the year with the lower participation as the baseline year.
The decomposition in (ref) requires estimation of the counterfactual distributions in equation (ref) comprising components from different years. We start by rewriting this equation as: \footnote{ To rewrite the bound $k_{r}(x,z,v)>0$, we use that $F_{H_{r}\mid X_{r},Z_{r}}(k_{r}(x,z,v)\mid x,z)>F_{H_{r}\mid X_{r},Z_{r}}(0\mid x,z)$ and $\mathrm{P} (k_{r}(x,z,V)\leq k_{r}(x,z,v)\mid x,z)=v$ by the assumptions on the hours worked equation.}
where $v\mapsto Q_{H_{r}\mid X_{r},Z_{r}}(v\mid x,z)$\ is the quantile (left-inverse) function of $h\mapsto F_{H_{r}\mid X_{r},Z_{r}}(h\mid x,z)$. Equation ((ref)) describes the mechanics for the construction of the counterfactual distributions. Based on the numerator in the first argument of the LDSF, it computes the hours of the individuals in the population with composition as in year $p$ with the hours equation of year $ r $ by keeping the individuals' ranks identical conditional on their observed characteristics $X_{p}$ and $Z_{p}$. For example, an individual working median annual hours in the subpopulation of individuals with the same $X_{p} $ and $Z_{p}$ is assigned the median working hours in the same subpopulation in year $r$. We do this for every individual and at each quantile. An individual working in year $p$ \ may not work in year $r$ if the corresponding quantile equals zero in year $r$. An interesting aspect of ((ref)) is the calculation of the area of integration. The condition that defines this area compares the realization of the control function in year $p$ with the fraction of individuals that do not work in year $q,$ conditional on their values of $X_{p}$ and $Z_{p}$. If the value of the control function is higher, then the individual would also work in year $q$. Otherwise, that individual would not work.
We assume a random sample of size $n_{t}$, $\{(Y_{i,t},H_{i,t}\times W_{i,t},H_{i,t},X_{i,t},Z_{i,t})\}_{i=1}^{n_{t}}$, from $(Y_{t},H_{t}\times W_{t},H_{t},X_{t},Z_{t})$ for every year $t$, where $H_{t}\times W_{t}$ indicates that $W_{t}$ is only observed when $H_{t}>0$. We estimate the control function via logistic distribution regression. For every observation in the selected sample with $H_{i,t}>0$, we set:
where $\Lambda $ is the standard logistic distribution function, $ P_{i,t}:=p(X_{i,t},Z_{i,t})$\ is a $d_{p}$-dimensional vector comprising of transformations of $X_{i,t}$\ and $Z_{i,t}$, and $\widehat{\pi }_{t}(h)$\ is the coefficient estimate in the logistic distribution regression of $\mathbf{ 1}(H_{i,t}\leq h)$ on $P_{i,t}$. With $\widehat{V}_{i,t}$ we estimate the LDSF of hourly wages as $\widehat{G}_{t}(w,x,v)=\Lambda (m(x,v)^{\top } \widehat{\theta }_{t}(w)), $ where $m(x,v)$\ is a $d_{m}$-dimensional vector comprising of transformations of $(x,v)$\ and the $\widehat{\theta }_{t}(w)$ \ is the coefficient estimate in the logistic distribution regression of $ \mathbf{1}(W_{i,t}\leq w)$ on $m(X_{i,t},\widehat{V}_{i,t})$.
With these different components we estimate $G_{\langle q,r,s,p\rangle }(y)$ \ by the empirical counterpart of (ref), namely:
where $\widehat{Q}_{H_{r}\mid X_{r},Z_{r}}(v\mid X_{i,s},Z_{i,s})=\int_{0}^{\infty }1\{\Lambda (P_{i,s}^{\top }\widehat{\pi } _{r}(h))\leq v\}dh$\ is the generalized inverse of $h\mapsto \Lambda (P_{i,s}^{\top }\widehat{\pi }_{r}(h))$.
The standard errors and confidence intervals of the components of the decomposition can be calculated using weighted bootstrap. We report standard errors for the decompositions in Appendix (ref) of Fern\'{a} ndez-Val et al. (2021b). The estimates are very precise due to the large sample sizes.
Following Mulligan and Rubinstein (2008), we use 6 indicators to capture the highest level of education: (i) 0-8 years of completed schooling, (ii) high school dropouts, (iii) high school graduates or 12 years of schooling (including GEDs), (iv) some college, (v) college, and (vi) advanced degree. Following Autor et al. (2016), we employ a quadratic polynomial in potential experience fully interacted with the education indicators. We include 4 indicators of marital status, and indicators for African-American and Hispanic origin. We capture regional variation with indicators for the 4 different regions defined in the CPS (North East, North, West and South). The variables in $Z_{t}$ are the four indicators capturing marital status interacted with an indicator for the presence of children under the age of five. These household characteristics are frequently used as exclusion restrictions (see, for example, Mulligan and Rubinstein, 2008 for the CPS and Card and Hyslop, 2021 for the PSID) but not without controversy. For example, Blundell et al. (2007) argued that highly talented women delay childbirth implying that conditional on age the family characteristics may be correlated with unobserved ability. We explore the sensitivity to the exclusion restrictions by including these variables in $X_{t}$ rather than $ Z_{t}$ and find no notable change in the results. We accordingly employed the above division of the variables into $X_{t}$ and $Z_{t}$.
Figure (ref) revealed substantial bunching of reported working hours. This bunching, particularly at 2080 working hours per year, probably reflects the convenience to respond 40 hours per week for 52 weeks per year. However, it creates a difficulty in predicting an individual's working hours. Accordingly, we modify our method as follows.
Suppose $H_{t}^{\ast }$\ is the actual number of the individuals' annual hours and $H_{t}$\ is their observed annual hours. Hence, $H_{t}^{\ast }$\ is the left-hand side variable in ((ref)). Furthermore, assume $H_{t}=j \text{ if }h_{j-1}<H_{t}^{\ast }\leq h_{j}$. This implies that workers with hours between cutoff values $h_{j-1}$ and $h_{j}$ report hours to be exactly equal to $j$. The random variable $H_{t}$ will have a discrete distribution although $H_{t}^{\ast }$ is continuous. From monotonicity $ h_{j-1}<H_{t}^{\ast }\leq h_{j}$\ implies $V_{t}$\ is between the values $ F_{H\mid X,Z}(h_{j-1}\mid x,z)$\ and $F_{H\mid X,Z}(h_{j}\mid x,z)$. Also, $ V_{t}$ is uniformly distributed between these values although we are unable to estimate the control function for the individual observations. A solution is to estimate the values $F_{H\mid X,Z}(h_{j}\mid x,z)$ for every $j$ based on the observed levels of working hours, and to randomly sample values between the estimated $F_{H\mid X,Z}(h_{j-1}\mid x,z)$\ and $F_{H\mid X,Z}(h_{j}\mid x,z)$ to obtain draws of $V_{t}$. We use these sampled values rather than the estimated values based on $H_{t}$ to calculate the counterfactual earnings distributions.
The chosen base years are 1976 for females and 2010 for males. The base year should have the lowest employment rates as this increases the likelihood that the identification conditions are satisfied. The female employment rate is at its lowest in 1976 and we assume that females with a certain combination of $(x,z,v)$ working in 1976 will have a positive probability of working in subsequent years. We choose 2010, the bottom of the financial crisis, for males. As we employ different base years we can compare trends but not hours and earnings across gender. Irrespective of the base year, the results in Section (ref) are relative to 1976. This has implications, discussed below, for the decomposition figures.
We estimate the hours equation (ref) separately by gender and survey year and construct the control function $\widehat{V}_{t}$. We do not report these results but highlight their implications for the earnings decompositions. We estimate the hourly wage equation (ref) for the selected sample using $X_{t},$ $\widehat{V}_{t}$ and $\widehat{V}_{t}^{2},$ in addition to the product of $\widehat{V}_{t}$ with $X_{t}$, as conditioning variables.
Recall what is required for identification and the correct interpretation of our results. First are the support restrictions in Proposition (ref). The most important is $\mathcal{XV}_{p}\cap \mathcal{XV}_{q}^{\ast }\subseteq \mathcal{XV}_{s}^{\ast }$ which is satisfied whenever $\mathcal{XV}_{q}^{\ast }\subseteq \mathcal{XV}_{s}^{\ast }$. This implies that the support of $V_{t}$ for any $X_{t}$ in the year $t$ should be at least as large as the base year support. To investigate this note that for any $x\in \mathcal{X}_{t}^{\ast }$ the lower bound of $ \mathcal{V}_{t}^{\ast }(x)$ equals $\min_{z\in \mathcal{Z}_{t}^{\ast }(x)}F_{H_{t}|X_{t},Z_{t}}(0|x,z). $ As monotonicity imposes that the upper bound always equals 1, Figure (ref) of Fern\'andez-Val et al. (2021b) reports the percentage of observations in year $t$ that have lower bounds higher than the expected lower bounds in the base year. These are zero by construction in 1976 for females and 2010 for males. For both males and females the number of violations is very small.
Rank invariance is required to distinguish between extensive and intensive effects. A necessary condition is that the conditional distributions of the control function do not differ between the years. We investigate this through the stability of the estimated parameters for the hours equation across years. We find the coefficients for two important determinants of hours, education and age, to be stable over time. This is supported by Figure (ref) of Fern\'{a}ndez-Val et al. (2021b).
We discuss the results for males and females separately but present the corresponding graphs together. Our primary focus is on the hours effects but we also discuss the structural and composition effects. As we employ different base years it is important to note the following. For females the base year is 1976 so the interpretation is straightfoward. For males the base year is 2010 and this imposes that the total effect, and each individual component, is zero in that year. For presentational purposes we shift figures illustrating the various effects to impose that they are all set to zero in 1976.\footnote{ For example, for the extensive effect, we use the following formula: $\left( [Q_{\langle t,t,t,t\rangle }(\tau )-Q_{\langle 1976,1976,1976,1976\rangle }(\tau )-Q_{\langle 2010,t,t,t\rangle }(\tau )+Q_{\langle 1976,t,t,t\rangle }(\tau )]\right) /Q_{\langle 1976,1976,1976,1976\rangle }(\tau )$} This facilitates an easier comparison across gender and a simpler representation of changes over the sample period. One feature of the results which we do not pursue but which merits noting is the impact of wages on hours. While we do not impose sufficient structure to estimate labor supply elasticities we do investigate the relationship between mean wages and the control function. The estimates are consistent with a positive relationship between wages and annual hours and a positive wage elasticity for both males and females.
\paragraph{Structural Wage Effects}
Figure (ref) shows changes in mean earnings are generally driven by the large and negative structural wage effects. This partially reflects the lower wage profiles of younger generations (see, for example, Kambourov and Minovskii, 2009). The structural wage effect increases from the mid 1990s but remains negative until the end of the sample period. Figure (ref) examines the lower quartile of annual earnings noting this corresponds to the first decile of earners. The structural wage effect is an important component of the decreases and exacerbates the earnings reductions from nonemployment during cyclical downturns discussed below. There are negative structural wage effects throughout the entire sample period at this quantile. Figure (ref) reveals that negative structural wage effects are also the driving influence in the evolution in median earnings. This is consistent with existing work on the evaluation of workers' characteristics in the middle of the male wage distribution. The large negative structural effects at Q3 and the small effects at D9, shown in Figures (ref) and (ref) , are also consistent with existing empirical work.
\pgfplotstableread{results_mean_male_1_2020_2.txt}{\resultsa} \pgfplotstableread{results_mean_male_2_2020_2.txt}{\resultsaa} \pgfplotstableread{results_mean_male_3_2020_2.txt}{\resultsab} \pgfplotstableread{results_mean_male_4_2020_2.txt}{\resultsac} \pgfplotstableread{results_mean_male_5_2020_2.txt}{\resultsad}
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\paragraph{Composition Effects}
As there is no evidence of composition effects in the male hours distribution the reported effects are operating through their impact on hourly wages. At the mean and the considered quantiles the composition effect is positive and frequently large. This reflects the increased educational attainment of males. This is consistent with earlier studies examining the role of composition effects on the wage distribution.
\paragraph{Hours Effects}
The hours effects at mean earnings are negative and economically important. \footnote{ The hours effects are constructed to be positive relative to the base year. For males we choose 2010 as the base year but normalize the total effect to be zero in 1976 for easy comparisons across gender. This allows the observed hours effects to be negative.} The extensive margin is generally the more important although the intensive effect is frequently of a similar magnitude. The intensive effect appears to be particularly important in cyclical downturns. Figure (ref) indicates that the substantial reductions in earnings at Q1 in recessionary periods reflect movements at both the extensive and intensive margins. The biggest effects are at the intensive margin during the recessionary periods although movements at the extensive margin also substantially decrease earnings. The very large intensive margin effect during the Great Recession probably includes spells of unemployment in reduced weeks worked. The total hours effect is large and economically important and represents the largest component of the change in earnings. The behavior of male earnings at this quantile is one of this paper's more important findings and supports the following conclusions. First, while previous studies have documented the cyclical sensitivity of wages at this quantile (see, for example Elsby et al, 2016), we find that these wage decreases are accompanied by non employment and large reductions in hours worked. Hoynes et al. (2012) report that the unemployment and employment levels of lowly educated males of Black and Hispanic ethnicity display large cyclical patterns.\footnote{ We conducted our empirical analysis separately by racial subgroups and found that the earnings behavior of Hispanics and Blacks displayed greater cyclicality than Whites. However, this appears to reflect their location in the earnings distribution rather than heterogeneous impacts by race.} Our findings support this result given these are frequently the lowest wage workers. Similar to Hoynes et al. (2012), we find evidence of this relationship in every cyclical downturn with a substantially greater impact in the Great Recession. Second, we find that the intensive hours effect recovers quickly after each recession. This is consistent with Altonji et al. (2013) who find that unemployment spells are associated with large reductions in working hours in the first year following unemployment although wage decreases do not persist. Third, although the intensive hours effects are highly cyclical, the structural wage effect, especially during the Great Recession, is less so. That is, the large earnings reduction experienced at this quantile during this recession are due to decreases in annual hours. This supports the findings of Altonji et al. (2013), Beaudry et al. (2016) and Valetta (2017) who find that highly educated workers may have substituted workers at the bottom of the distribution. This reduced the employment levels of these displaced workers but did not greatly affect the wages of those employed. It is also consistent with Elsby et al. (2016) who find that wage declines during the recessions in the early 1980s were larger than those of the Great Recession. Finally, this reduction in hours exacerbates the inequality generated by wage differences.
Figure (ref) presents the decomposition for male median annual earnings and reveals that the intensive hours effects are relatively small with the exception of notable contributions during recessionary periods. There is no evidence of extensive hours effects recalling part of the intensive hours effect reflects an increase in the number of weeks of positive hours. However, a comparison with Figure (ref) reveals the large hours effects reinforcing the negative structural wage effects at Q1 are smaller at Q2. This suggests that the reductions in hours are disproportionately borne by those at the lower part of the wage and earnings distributions. The decompositions for Q3 and D9, shown in Figures (ref) and (ref) respectively, support this view. At neither quantile is there evidence of substantial hours effects. Q3 of male annual earnings is reasonably flat and the substantial increase at D9 reflects the growth in hourly wage rates at this point of the wage distribution.
\paragraph{Structural Wage Effects}
Figure (ref) reveals that\ while mean female earnings increased by over 100% a relatively small fraction is due to the structural wage effect. Previous work has established that the structural wage effect increases female hourly wages at the upper end of the female wage distribution but reduces wages at lower quantiles. The positive effects here appear to dominate towards the end of the time period. Figure (ref) also reveals that virtually none of the dramatic increase in median earnings over the sample period can be attributed to the structural wage effect. In fact, for a substantial part of the sample period the structural wage effect is negative. A similar result is found for Q3. At D9 the structural wage effect is initially negative but grows through the sample period and is large and positive by 2019.
\paragraph{Composition Effects}
The composition effect, operating both through hours and wages, is the driving influence behind the growth in mean earnings and accounts for over half of the increase by 2019. It reflects the large increase in female educational attainment which increased both annual hours and wage rates. Mean annual hours increase by around 50% over the sample period and nearly half of this is due to the composition effect.\footnote{ Female mean annual hours worked increase from 1515 to 1867 over the sample period.} The composition effect at Q2 is large and operates through both wages and hours.\footnote{ Those with median earnings are not necessarily working median hours. However, it is worth noting that median hours increase by 150 percent and 50 percent of this change is due to composition effects.} The composition effect is known to increase wages at all points of the female wage distribution (see, for example, FPVV). However it also captures the impact of higher education levels on annual hours. It contributes more than half of the observed growth at Q2. A similar result is observed at Q3 although the growth in earnings is less dramatic. At D9 the contribution falls to slightly less than 50% of the total growth in earnings.\footnote{ The decompositions of hours distributions revealed no composition effects above the median. While we continue to acknowledge that observations at the upper quantiles of earnings are not necessarily the same as those in upper quantiles of hours distribution it is very likely that there is substantial overlap.}
\paragraph{Hours Effects}
For mean earnings the intensive hours effect is substantial and reflects the drastic shift in the females hours distribution. This has increased females earnings by around 25%. The extensive hours effect is slightly smaller and is similar in magnitude to the structural wage effect by 2019. This differs somewhat from the conclusion of Card and Hyslop (2021) who find large extensive effects on inequality. We pursue the causes of this relatively modest increase below.
Movement on the intensive margin at the median is producing an increase of around 125% and this represents around a quarter of the total increase. This reflects the large increase in annual hours for those at the lower part of the hours distribution. The contribution of movements at the extensive margin is surprisingly small and probably captures our assumption that those who entered the labor market are those with the lowest number of annual hours.
The estimated hours effects are interesting from a number of perspectives. The intensive margin is generally increasing from 1976 to 2000 before leveling off and decreasing for the remainder of the sample period. The large cyclical effects operating through the intensive margin for males are much smaller for females. While some of the recessionary periods witness declines, the impact on median earnings is relatively small. The exception is the Great Recession which has a remarkable decline via the intensive hours effect. This supports Hoynes et al. (2012) who conclude that female minority groups suffered substantially during this period. An even more striking result is the smaller extensive margin effect although by\ 2019 it is similar in magnitude to the intensive margin effect and the total hours effect is around 200%. This represents approximately 40% of the total change in median earnings. Recall that, due to the rank invariance assumption, the decomposition exercise compares different years to the base year and this deletes the individuals with very few annual hours, and very low annual earnings, in calculating the extensive hours effects. Although there are points in the distribution where this will make a difference, generally only the bottom part of the hours distribution is relevant. The evidence here suggests that these deletions are of secondary importance in comparison to the changing hours distribution of those working. However, this requires the rank invariance assumption and we further investigate this below.
While the hours effects decrease as we examine the higher levels of earnings, they remain important at both Q3 and D9. At each of these quantiles the increase in earnings is substantial and a relatively large contribution comes from movements on the intensive margin. This illustrates how drastic the change in the female hours distributions has been. At both Q3 and D9 the movements at the intensive margin explain around one quarter of the total increases in earnings. The movements in the extensive margin are not substantial at either quantile.
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The small contribution of the extensive effects seems puzzling. One reason is that the component of the extensive margin capturing the impact of increased education on employment rates is attributed here to the composition effect. A second cause is the monotonicity assumption. Recall that the extensive effect is calculated by setting the working hours of the marginal workers equal to zero and evaluating the impact of such an exercise on the earnings distribution. Monotonicity imposes that the marginal workers are those with the lowest working hours. As these workers have generally low earnings, the impact of their exclusion will be small. This is especially true if the distribution of annual hours features a large fraction of workers with low annual hours. Another possibility is our use of a year as the unit of measurement. This implies that short spells of unemployment are captured in the intensive rather than the extensive hours effect. It is also responsible for the small extensive hours effects for males during the Great Recession. Other empirical studies have experienced related problems associated with short spells of unemployment (Altonji, et al., 2013). The use of cross sectional data requires a definition of movement on the extensive margin and this has implications for the intensive margin. This is also noted by Blundell et al. (2013).
To explore the empirical impact of these lower hour individuals we redefined individuals with under 500 annual hours as non participants. We re-estimate our model and reproduce the decompositions for the mean earnings. Because lower-hours individuals are not classified as workers, the entering marginal workers enter at a higher level of hours (500 instead of 1). Figure (ref) shows that the results for males are almost identical to those for the zero hour cutoff.\footnote{ This is explained by Figure (ref) of Fern\'{a}ndez.Val et al. (2021b) which indicates that the majority of males are working a substantial numbers of annual hours.} Thus, the choice of cut-off hours is less influential in reassigning the impact across the two hours effects as the \textquotedblleft entrants" are likely to be assigned a relatively high number of annual hours and, more importantly, a number similar to those already in employment. The corresponding figure for females reveals a different story. For the zero cut-off approximately a quarter of the 80% increase in earnings over the sample period was due to the hours effects and by 2019 approximately three quarters of this was due to movements on the intensive margin and one quarter to the extensive margin. As the minimum of an entrant is 500 hours the extensive margin effect increases and is generally equal to that of the intensive margin.\footnote{ This is explained by Figure (ref) which shows that a large number of females work less than 500 hours.} In fact, by the end of the sample period it is the most important of the two. Clearly, an exercise with a higher cut-off point would attribute an even higher share to the extensive margin. The sensitivity to the \textquotedblleft cut-off" point highlights the importance of the definition of the extensive margin and intensive margin effects. We prefer to conduct our analysis over the entire working sample as the lower hours workers need to be included to appropriately describe earnings inequality. They also provide a clearer understanding of the role of hours on earnings inequality.
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We employ two additional approaches to further explore this issue. First we re-estimated the model and conducted the decompositions using the CPS Merged Outgoing Rotation Group (MORG) data. The dependent variable is now weekly earnings and the censoring variable is hours worked in the survey week. The use of this measure of hours avoids the issue raised in our discussion of the ASEC data which may inappropriately treat individuals working the same annual hours, but not the same weekly hours, as identical. These data assign a higher weight to the extensive margin as an individual not working in the survey week will be recorded as non employed. For the ASEC data an individual is recorded as employed even if they work only one week of the year. The decomposition for male mean earnings is shown in Figure (ref) of Fern\'{a}ndez-Val et al. (2021b). Over the entire period there is a reduction in weekly earnings of approximately 8% which is similar to the reduction in annual earnings. As expected the role of the extensive margin is more important. For example, the biggest reduction in weekly earnings occurs over the Great Recession. During this period there is a large increase in contribution of the extensive margin. However the impact of the two hours effects are approximately equal. Also the magnitude is larger and for the weekly data there appears to be more drastic movement in the extensive effect whereas for the annual data the response is flatter. However, more interesting is that the dramatic effect in the intensive margin effect exists for both data sets. The hours effects for female weekly earnings shown in Figure (ref) of Fern\'andez-Val et al. (2021b) are very similar to those for annual earnings. The trends and the magnitudes are similar for both data sets and the primary difference is that the annual data attributes more of the total hours effects to the intensive margin.
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Another approach to evaluate the impact of non employment during the year on the attribution to the two hours effects is to treat an observation working part of the year as two different observations with weights reflecting the fraction of the year worked.\footnote{ We are grateful to a referee for this suggestion.} This also assigns a greater weight to the extensive margin as any individual not employed 52 weeks per year is contributing to the extensive margin. This definition is similar to that employed in Blundell et al. (2013). The associated decomposition is shown in Figure (ref) and while the differences with the earlier figures can be anticipated the relative magnitude of the hours effects are interesting. For males the intensive margin is essentially zero for the whole sample except for a notable negative impact during the Great Recession. In contrast the extensive margin is larger and more closely tracks the overall decrease in annual earnings. The impact at the extensive margin is much larger for the Great Recession than the earlier figures although there remains a substantial intensive margin effect for this period. For females the extensive margin effect is now larger than the intensive margin effect for the entire sample period although by 2019 the effects are close in magnitude. This figure supports our earlier conjectures that while some of the extensive margin is being allocated to the intensive margin there appears a contribution from each.
These three alternative approaches highlight the nature of the sensitivity of the relative magnitude of the extensive and intensive hours effects to the modelling approach, and particularly the monotonicity assumption, and the time unit associated with the measurement of employment. We acknowledge that our approach is likely to underestimate the extensive margin effects but substantial evidence of intensive margin effects remain even if we reassign some part of them to the extensive margin.
\pgfplotstableread{results_quantile_male_1_mean_1.txt}{\resultsa} \pgfplotstableread{results_quantile_male_2_mean_1.txt}{\resultsaa} \pgfplotstableread{results_quantile_male_3_mean_1.txt}{\resultsab} \pgfplotstableread{results_quantile_male_4_mean_1.txt}{\resultsac} \pgfplotstableread{results_quantile_male_5_mean_1.txt}{\resultsad} \pgfplotstableread{results_quantile_male_6_mean_1.txt}{\resultsae}
\pgfplotstableread{results_mean_male_1_2020_alternative1.txt}{\resultsa} \pgfplotstableread{results_mean_male_2_2020_alternative1.txt}{\resultsaa} \pgfplotstableread{results_mean_male_3_2020_alternative1.txt}{\resultsab} \pgfplotstableread{results_mean_male_4_2020_alternative1.txt}{\resultsac} \pgfplotstableread{results_mean_male_5_2020_alternative1.txt}{\resultsad}
\pgfplotstableread{results_mean_female_1_2020_alternative1.txt}{\resultsb} \pgfplotstableread{results_mean_female_2_2020_alternative1.txt}{\resultsba} \pgfplotstableread{results_mean_female_3_2020_alternative1.txt}{\resultsbb} \pgfplotstableread{results_mean_female_4_2020_alternative1.txt}{\resultsbc} \pgfplotstableread{results_mean_female_5_2020_alternative1.txt}{\resultsbd}
Studies of inequality typically examine the ratio of quantiles above and below the median. Given the inclusion of those working zero hours in our sample, the female participation rate of 56.5% in 1976 does not allow comparisons for females involving earnings below the fourth decile. Accordingly, we begin in Figure (ref) by focusing on relative changes in the D9/D5 ratio.
\pgfplotstableread{results_quantile_male_1_d9d5.txt}{\resultsaa} \pgfplotstableread{results_quantile_male_2_d9d5.txt}{\resultsab} \pgfplotstableread{results_quantile_male_3_d9d5.txt}{\resultsac} \pgfplotstableread{results_quantile_male_4_d9d5.txt}{\resultsad} \pgfplotstableread{results_quantile_male_5_d9d5.txt}{\resultsae}
\pgfplotstableread{results_quantile_female_1_d9d5.txt}{\resultsa} \pgfplotstableread{results_quantile_female_2_d9d5.txt}{\resultsb} \pgfplotstableread{results_quantile_female_3_d9d5.txt}{\resultsc} \pgfplotstableread{results_quantile_female_4_d9d5.txt}{\resultsd} \pgfplotstableread{results_quantile_female_5_d9d5.txt}{\resultse}
The 28% increase for males is substantial and economically significant. There are no hours effects and the increase largely reflects the structural wage component. Existing empirical work has found substantial and positive real wage growth at the higher quantiles while real wages below Q3 have decreased. This reflects the increase in the returns to education at the higher quantiles of the wage distribution. There is evidence of small composition effects as the effects operating at the two quantiles offset each other. The hours effects are small and consistent with earlier research which finds working hours have a relatively modest impact in explaining the changes in males earnings inequality in the United States (see, for example, Blau and Kahn, 2009 and Checchi et al, 2016). Movements on the extensive margin at both the median and the ninth decile are negligible and have no effect on this measure of inequality.
The corresponding figure for females differs in several ways. First, there is a drastic decrease of over 60% in this measure of inequality. The fall occurs primarily from 1976 to 1990 and is maintained for the remainder of the sample period. This large effect is initially due to the composition and intensive hours effects. For the first part of the sample period both effects are contributing to the large decrease in inequality. However, by the mid 1980s the intensive margin effect begins to diminish and the large decrease in inequality observed at the end of the sample period is almost entirely due to the composition effect. The intensive margin appears to capture the increased hours of those working relatively few hours early in the sample period who subsequently increase their hours in latter years. The presence of an hours effect differs from the conclusions of Blau and Kahn (2011) and Checchi (2016) who report that the variance in working hours is relatively small in the U.S. and the change in the correlation between working hours and wages is small. This reflects the benefit of our approach which isolates movements in hours at specific points of the earnings distribution rather than employing the variance based on the whole distribution. Also note that these studies focus only on those working positive hours. Second, the structural wage effect is relatively unimportant for this measure of female earnings inequality. The evidence in FPVV suggests the structural wage effects should be increasing inequality as those at the upper quantiles of the wage distribution have experienced much larger wage increases from this source than those lower in the wage distribution. There is some evidence in support of this phenomena. The extensive hours effect appears to be non existent.
We conclude our empirical work by examining the interquartile range for males shown in Figure (ref). While the magnitude of its increase is not greatly different to the earlier measure its evolution over the time period is drastically different. While the D9/D5 ratio shows a steady increase, the interquartile ratio increases in large jumps during the recessionary period. The Great Recession generates an increase in the Q3/Q1 from 20% to 60%. This has been followed by an equally dramatic decrease in inequality for the remainder of our sample period although it remains well above that before the Great Recession. The trend in the inequality can be almost fully described by the structural effect although it is partially explained by the composition effect since the 2000s. This is consistent with the earlier figures which illustrate the negative impact of the composition effect on earnings at Q1 for the same period. The cyclicality in the Q3/Q1 ratio is also partially explained by the two hours effects. This is most apparent in the Great Recession which sees a large increase in the intensive hours effect.
\pgfplotstableread{results_quantile_male_1_q3q1.txt}{\resultsa} \pgfplotstableread{results_quantile_male_2_q3q1.txt}{\resultsaa} \pgfplotstableread{results_quantile_male_3_q3q1.txt}{\resultsab} \pgfplotstableread{results_quantile_male_4_q3q1.txt}{\resultsac} \pgfplotstableread{results_quantile_male_5_q3q1.txt}{\resultsad} \pgfplotstableread{results_quantile_male_6_q3q1.txt}{\resultsae}
The inequality decompositions illustrate the benefit of examining earnings inequality and specifically the impact operating through hours. In contrast to wage inequality, the results differ greatly by gender. The most striking finding for males is the remarkable sensitivity of inequality to recessionary factors. While there is evidence of this also in wages it is far more dramatic in earnings. The total hours effects in the inequality decomposition are greater than the structural effects responsible for wage inequality. The increases in earnings inequality are primarily generated by reduction in hours worked by the lower wage males during recessions. While it is beyond the scope of this paper to provide policy prescriptions to address earnings inequality the implications of our results are clear. Lower wage males suffer greatly in recessionary periods through both lower wages and lower demand for their labor. The lower level of demand is reflected in spells of unemployment and reduction in working hours for those employed. This suggests that the appropriate policies for combatting earnings inequality should focus on supplementing the lower wage males in these periods. This could take the form of targeted unemployment programs or through incentives for employers to maintain the hours of these workers through recessions. The results for females are remarkably different. Although female wage inequality has increased in a manner similarly to males, female earnings inequality has decreased dramatically. This has occurred despite changes in hours affecting both the earnings of low and high earning females. However, the larger decreases at the bottom have reduced earnings inequality. This also has very clear policy implications for reducing inequality. Policies should focus on enabling females to increase their level of labor supply.
While our reduced form approach highlights the relationship between hours and earnings, we acknowledge that effective public policy is guided by a deeper understanding of individuals' economic motivations. While we establish that movements on both the extensive and intensive margins are important, understanding why they occur requires an approach imposing greater economic structure. For example, Keane and Wasi (2016) estimate supply elasticities which incorporate both margins and investigate how they vary by individual characteristics such as age and education. Policies based on our findings in conjunction with those from studies such as Keane and Wasi (2016) are more likely to achieve goals such as reducing earnings inequality.
This paper examines the relationship between annual hours of work and annual earnings by decomposing movements in the annual earnings distribution into composition, structural and hours effects. This requires the estimation of a nonseparable model of hours and wages in the presence of a censored selection rule. Our empirical work has a number of interesting results. Changes in the hours of annual work are important for the lower quartile of male annual earnings but their importance quickly diminishes as we examine higher quantiles. Using the interquartile ratio as an alternative measure of inequality reveals that hours effects are very important and particularly notable in recessionary periods. Changes in the female distribution of annual hours have increased earnings at all quantiles of the female annual earnings. These effects are generally operating through the intensive margin although there is also an important role for the extensive margin. The female upper decile to median ratio decreases dramatically indicating a drastic reduction in earnings inequality. This reduction is due to large effects at the intensive margin of annual hours of work but also composition effects, such as increased levels of education, which increase wages at all quantiles and increase the level of annual hours of work.