Are changes in the dispersion of hours worked a cause of increased earnings inequality?

Earnings are the product of wages and hours of work, hence the dispersion of hours can magnify or dampen a given distribution of wages. This paper examines how earnings inequality is aﬀected by the dispersion of working hours using data for the US, the UK, Germany and France over the period 1989-2012. We ﬁnd that hours dispersion can account for over a third of earnings inequality in some countries and that its contribution has been growing over time. We interpret the expansion in hours inequality in European countries as being the result of weaker union power that led to less successful bargaining concerning working hours.


Introduction
A vast literature has examined the evolution of wage and earnings inequality over the past three decades and, despite substantial heterogeneity across countries, has identified a major increase in the dispersion of both in many industrial economies. 1 An orthogonal research agenda has focused on the evolution of working hours, and in particular on the divergence in working patterns between the US and Europe since the 1970s. 2 Given such differences across countries, it is conceivable that hours of work also differ in terms of their dispersion, raising the question of whether hours inequality has contributed to the increase in earnings inequality. Understanding the role and source of hours dispersion is crucial for the design of policies aimed at reducing inequality. On the one hand, policies that try to increase hours for those with the lowest skills and wages and reduce them for those at the top of the distribution can be alternatives to ex post redistribution. On the other, the source of the dispersion is important for policy as it could be due to imposed constraints or the result of certain groups, such as women with young children, choosing to spend less time at work. This paper represents a first step in trying to understand those questions.
We use data for the US, the UK, Germany and France to decompose earnings inequality and asses the roles played by the dispersion of wage and by inequality in hours in explaining cross-country differences and changes over time. By definition, an individual's earnings are the product of her hourly wage rate and her hours of work. Using as our inequality index the mean log deviation (MLD), an inequality index belonging to the general entropy family, we decompose earnings inequality into the dispersion of hourly wages and a component capturing the contribution of hours. This term has itself two elements, a measure of inequality of hours of work and a term capturing the correlation between hours and hourly wages. Hours of work can as a result magnify or dampen wage inequality depending on how dispersed hours are and of whether they are positively or negatively correlated with wages.
Our sample covers the period 1989 to 2012 and considers both aggregate behavior as well as that of subgroups defined by gender and skill category. When we look at the distribution of hours we find a surprising pattern. Figure 1 plots the distribution of hours worked by employed individuals in our four sample countries. The left-hand panels depict the US and France, while the right-hand ones present the distributions in the UK and Germany. In order not to focus on a single, potentially un-representative, year, the data cover two 5-year periods, 1995-2000 and 2007-2012. The US and France exhibit fairly concentrated distributions, with about 30% of individuals declaring to work around 40 hours a week in [1989][1990][1991][1992][1993][1994]. For the latter period, this fraction falls slightly in the US, while France exhibits twin-peaks due to the introduction of the 35-hour week. In sharp contrast, Germany and the UK present fairly dispersed distributions, with much thicker tails at the bottom and, especially, at the top.
These differences in dispersion are reflected in the way in which wages and hours affect the distribution of earnings, as captured by our decomposition of the mean log deviation. Although our results indicate that in all countries the dispersion of working time is an unequalising factor that increases earnings inequality over and above that implied by wages, differences across countries are substantial. In the US and France the overall contribution of hours to earnings inequality is moderate, with wages accounting for at least two thirds of inequality in earnings.
In contrast, hours play a crucial role in the UK and Germany, being responsible for between 28 and 40 percent of the dispersion in earnings.
We pay particular attention to the covariance between wages and hours, which exhibits very different patterns across countries and over time. Wages and hours move together in the Anglo-Saxon economies, while they are negatively correlated in France and Germany at the start of the sample period, implying that part of the dispersion in wages was offset by the fact that those with the lowest earning potential spend more hours at work, the effect being particularly strong in France. These countries exhibit, however, an increase in the covariance over time and by the end of the period, those with higher wages also work longer hours. In the case Germany, this change has accounted for half of the increase earnings inequality, in France it is the major culprit.
Although some of the differences are related to the skill and gender composition of employment, between group inequality in hours is only a small part of the story. Observed changes over time are largely due to the behavior of unskilled men and skilled women. Hours inequality has increased for the two groups, and both exhibit a marked increase in the covariance which has gone from being negative at the start of the period to null or positive. As a result, the equalizing force due to those with lower wages working longer hours seems to have been eroded over time.
Our paper contributes to two strands of literature. As mentioned before, there is a substantial literature on cross-country differences in working hours, and we add to this a new dimension by focusing on hours inequality. We also contribute to our understanding of what drives earnings dispersion by focusing on the neglected role of hours. Our paper is closely related to the analysis by Gottschalk and Danzinger (2005) of the relationship between individual wage rate inequality and household income inequality in the US. They examine the various elements that determine household income inequality and emphasize, amongst other things, the importance of considering the distribution of hours. As in our results, they find an important role for changes in the hours worked by women. Our analysis has a very different focus as we provide an international comparison rather than the more detailed analysis of a single country that they consider.
Our analysis is also related to Bell and Freeman (2001) and Bowles and Park (2005) who argue that greater wage inequality is associated with higher average hours of work, implying that the increase in wage inequality that occurred over the last decades is likely to have spurred an increase in hours worked. We argue that the impact of this mechanism on overall inequality depends on two channels, how unequal the hours response is and on the correlation of hours worked and hourly wages.
The paper is organized as follows. Section 2 describes our empirical approach and presents the decomposition that we use, and is followed by a section describing the data. Section 4 presents our main findings, while we concludes in section 5.

Decomposing Earnings Inequality
A vast literature has examined the decomposition of inequality indices by factor components. 3 As is well known, the various inequality indices have different merits and drawbacks, and the choice of index is consequently not trivial for the results. Two common measures are the half squared coefficient of variation (CV), which is particularly tractable, and the Gini coefficient, with the latter providing a less tractable decomposition but being less sensitive to extreme observations than the former. 4 Moreover, recent work, such as Jenkins and van Kerm (2005), proposes density function decompositions which have the advantage of being independent of the choice of inequality index.
These approaches have focused on decompositions over additive terms and are hence easily applicable to income factors. In the case of earnings, we are interested in decomposing total earnings inequality into a term due to wage rate inequality and one capturing hours inequality.
4 See, for instance, Jenkins (1995) and García-Peñalosa and Orgiazzi (2013) for decompositions of the CV and Lerman and Yitzhaki (1985), Garner (1993) and Podder (1993) on Gini decompositions. wage, w i , and the number of hours worked, h i . That is, (1) Our two terms of interest appear multiplicatively and as a result there are few inequality indices that can be satisfactorily decomposed. We have chosen to employ the mean log deviation (MLD), an index belonging to the general entropy (GE) family. The MLD, also called Theil's L index, is the general entropy index for α =0, and shares a number of desirable properties of this class of indices. 5 The parameter α in the GE class of indices captures the weight given to income differences at various parts of the income distribution. For lower values of α, such as α =0, GE is particularly sensitive to changes in the lower tail of the distribution.
The MLD is defined as the standard deviation of the logarithm of the variable of interest and has been shown by Esteban and Duro (1998) to be decomposable. Overall inequality in earnings can hence be expressed as the sum of three components: inequality in hourly wages, inequality in hours worked, and a component capturing the correlation between hours worked and hourly wages. Denote by I y the MLD of earnings, which is defined as where N is the number of observations andȳ is average earnings. We can also define the index for hourly wages and hours worked, namely, wherew andh denote the average levels of the two variables.
Using the fact that the covariance between hourly wages and hours worked, cov, can be shown to be given by cov = y − wh, equation (2) can be expressed as the sum of (3) and (4) plus a third term capturing the correlation between hours worked and hourly wages. That is, These three terms represent the absolute contributions to inequality of the various elements. The first two terms are simply inequality in hourly wages and hours worked, and both are positive.
The third term, denoted ρ, captures the correlation between hours and wages. If the covariance is negative, this term will be negative too, reducing earnings dispersion. The total contribution of hours to overall inequality hence depends on the value of the sum of the dispersion in hours and the correlation term. Whenever this sum is equal to zero, then earnings inequality equals the dispersion in the hourly wage rate and hours play no role. If, instead, the sum is positive, then hours magnify the impact of wage inequality on earnings dispersion. When the correlation term is sufficiently negative, I h + ρ may be negative, implying that hours reduce overall inequality, and the more dispersed hours are, the more they will reduce inequality.
A convenient way of expressing equation (5) is to consider the relative contributions of the three terms, obtained when dividing equation (5) by I y , that is The terms RC w , RC h , and RC ρ are the relative contribution of inequality in hourly wages, of the dispersion of hours, and of the correlation term to inequality in earnings, respectively. In other words, they measure the share of earnings dispersion due to each of the three components.
There are two key questions that we want to address that can be framed in terms of these contributions. The first one is how close RC w is to 1. If the relative contribution of wages is close to 1, it would indicate that most of earnings inequality is due to differences in the hourly wage rates received by individuals and that hours play a small role. In contrast, a small value of RC w would imply that differences in hours worked magnify the dispersion of wages. Second, note that a high RC w does not imply that there is little inequality in hours. In fact, it may be due to hours exhibiting little dispersion or to hours being unequally distributed but negatively correlated with hourly wages. In the second case, individuals will be partly offsetting the impact of wage inequality by working more the less well-paid they are. It is in fact possible that, if the correlation is sufficiently negative, the term RC ρ totally offsets RC h , implying that hours inequality makes the distribution of earnings less unequal than that of wages.
The MLD index allows us to further decompose equation (5) into a term capturing withingroup (W ) and one measuring between-group (B) inequality. If the total population is divided into J exhaustive groups, with group j ∈ {1, ..., J}, then the inequality index for earnings takes the form where p j = N j N denotes the proportion of individuals belonging to group j,ȳ j is the mean income of group j and I yj refers to the inequality index computed over the members of group j.
This decomposition can be performed over earnings, wages and hours, therefore the correlation term ρ can also be written as a sum of within-group and between-group inequality.
Equation (5) implies that both the within-group and the between-group terms of inequality in wages, of hours dispersion and of the correlation term, have to sum up to within and between inequality of earnings, implying that the within and between components of ρ can be calculated respectively. Combining equations (5) and (7) gives a nested decomposition of overall inequality, which takes the form Using a nested approach allows us to differentiate the contribution of inequality in wages, hours dispersion and the correlation term to inequality within and between each group.
3 The Data

Databases
The harmonized dataset constructed for this paper is based on different national surveys collected from national statistical institutes. We use household or labor surveys for the US, the UK, Germany and France, covering two decades starting around 1990. In particular, we use the

Variable definitions
Our two key variables of interest are earnings and hours worked, from which we then compute the hourly wage. We focus on prime-age workers, i.e. those who are at least 25 years-old and at most 54, who are (dependent) employees in either the private or the public sector. As is well established, employment patterns for young and for mature workers differ substantially across countries, much more than for prime-age workers. Focusing on this age group allows us to abstract for differences in the education system and in retirement possibilities. Details on sample sizes by country and year are provided in the appendix.
Both variables are measured at a particular point in time, that is, we use questions referring to the current job of the individual. This contrasts with papers that use annual hours and earnings and compute wages from those. There are good reasons for not pursuing this path, since both unemployment rates and vacation patterns vary substantially across countries and would have a major impact on measured hours. Focusing on a snapshot of weekly hours/earnings implies greater comparability of the data.

Earnings
The measure of earnings that we employ is the usual gross income from labor that the individual receives over a week from the main current job. For employees this means contractual wages plus overtime pay. This variable is present in all the datasets, yet some important differences need to be highlighted. Our main concern is that income from self-employment is difficult to measure in household surveys, mostly because the self-employed tend to have high non-response and under-reporting rates; in addition, income from self-employment varies considerably over time.
For these reasons, the self-employed are not asked about current usual earnings in the CPS, and in the BHPS over one fifth of self-employed respondents either refuse to give information or do not know how much they earn. 7 We therefore decided to remove the self-employed from our sample. 8 A second concern is that three countries report gross earnings, while France only provides earnings net of social security contributions (but not of income taxes). However, since such contributions are roughly proportional to gross earnings, this difference should have little effect on measured inequality. 9 Survey frequency and the period of time covered by the questions also varies across datasets.
The US and France, for instance, collect data monthly and quarterly respectively and the questions concern current employment. Instead, the UK and Germany survey once a year, asking questions about current earnings and also about the employment situation during the previous year. Note that although there are differences in survey frequency (monthly, quarterly, and annual), we always use questions concerning the same reference period -current job -and not questions concerning earnings last year, available in some of the surveys. Therefore the periodicity over which the survey is conducted does not create comparability problems in terms of the variable we are using.
Our selection rule is to select the month of March or the first quarter of the year, and we do so for the US and for France. However, in the case of Germany such a choice implies a small number of observations. Since keeping the largest possible sample is crucial given our intention of decomposing the population by gender and educational groups, we use data for the entire year. For the UK, BHPS data was collected in October/November and thus we are forced to use this period. In all cases we checked that at the aggregate level (i.e. before dividing into 7 See Bardasi et al. (1999) for further discussion. 8 The results including the self-employed can be obtained from the authors. 9 The current social contributions stand at an average of 23% of the gross salary, and vary over a range of 20 to 26%, the main difference steming from complementary pensions. Inequality in earnings net of these contributions will hence be slightly lower than in gross earnings. population subsamples) annual data and first-quarter/March data gave results that were not significantly different.
Finally, a more technical concern is the different policy of top-coding high incomes across countries. The US, for instance, top-codes systematically, with a top-coding value of 2,885$ per week for the most recent years. We decided to follow the recommendation of the LIS project and we top-code earnings at 10 times the weighted median of earnings. For those observations for which earnings were top-coded, the hourly wage was calculated after the top-coding was performed. Since we are interested in hourly wages, we also consider extreme values for this variable. Whenever hourly wages were above 10 times the weighted median of wages, we removed those observations. 10

Hours worked
Hours worked can be measured in different ways, capturing contract hours, actual hours or usual hours. For most of the databases we use the question concerning "usual hours worked in the main current job". Some databases also ask about the number of hours actually worked during the previous week. Although this variable may have less measurement problems, we were concerned with seasonality and we hence decided against its use. 11 The harmonization of this variable was not straightforward due to coding problems. First, we had to make sure that it included both contractual hours and overtime. Second, it is a variable that is often truncated. In particular, Germany truncates at 80 hours per week and the US at 99 hours. Given the issue we are interested in, this may be a concern as truncation affects the upper tail of the distribution of hours worked. Inspection of the data indicates that this is not the case since we did not find a concentration of observations at the truncation points.
Nevertheless, we decided to drop extreme observations and consider only workers that spend between 2 and 90 hours a week working on their main job.

The United States: data sources and definitions
Before proceeding to examine the data for the four countries, we consider in detail data sources for the US. Figure 1 above presents the distribution of hours of work for the United States, with hours being highly concentrated around 40 and both the upper and lower tails being rather thin.
This pattern did not match our expectations, our prior being that the US would exhibit a fat upper tail capturing the workaholic culture that we often find discussed in the popular press; see Schor (2008). Our results raise the question of whether the data we are using is the most suitable one, and if other variable definitions or data sources would yield a different picture.
In order to address this concern, we consider a number of additional measures. First, we consider the CPS data and our core variable of weekly hours but do not restrict our sample by age, and compute the distribution of hours for this group. Second, we consider PSID as an alternative dataset. In order to asses the accuracy of our chosen data, we compute measures of hours inequality for both CPS and PSID for the years 2003, 2005, and 2007. These are the only three years for which we have data for the question 'how many hours do you work per week in your current or most recent job' in PSID. 12 We use two measures of hours. The first are hours worked the previous week, i.e. the most comparable measure to the one in CPS, denoted J1 in the survey. We also use the variable Hann which are total annual hours worked during the previous year. We divide the data by 50 weeks (the modal working weeks in a year in the US) in order to get a measure comparable with the others. consider the entire CPS sample we find the same median but the mean is almost two hours lower, 12 The question is also asked in 2009 and 2011 but the response rate is extremely low.
probably reflecting the fact that very young individuals work less. We find a substantially higher degree of inequality, as captured by the two measures of dispersion. In particular I h goes from 0.04 to 0.07, probably due to the low working hours of young individuals (notably, students).  figure 2 depicts the distribution of this variable, and we can see that is much more dispersed and much smoother than any of our other series. Mean working hours are somewhat lower than for weekly data (one and a half hours less) and the median is slightly higher, while dispersion is substantially increased. The standard deviation for 2007 is 13.54, compared to 11.20 and 9.33 for the weekly data from PSID and CPS, respectively. The MLD is almost twice as large as for weekly data.
These differences raise a fundamental question about which is the most suitable data to use.
The US labor market implies much more frequent flows into and out of employment as well as more job-to-job transitions than European ones; as a result, comparing annual working hours across countries implies allowing for the fact that some of the observed differences are due to distinct patterns of employment. This problem is particularly acute in our context since we are interested in inequality in hours worked, and the much larger dispersion of annual hours implies that if we were to use this measure we would be comparing differences in earnings inequality caused by unemployment spells rather than those due to different choices of regular working hours. We have hence chosen to employ the measure of hours based on reported weekly hours.
This measure ignores a difference between the US and Europe, namely the fact that Europeans tend to take more weeks of paid holiday. Note, however, that weeks of holiday are paid at the same rate as weeks of work; consequently, if all workers are taking their mandatory holiday, this should affect the difference in mean hours worked across the continents but not their dispersion or that of earnings. Table 2 presents the values of several inequality indices for earnings, hourly wages and hours worked. We report the index that we employ in our analysis, the mean log deviation, and some common inequality measures often used in the literature. The dispersion in earnings is often measured by the Gini coefficient, while for wage inequality we compute the standard deviation of the log of wages. 13 For hours, we compute for each year average hours and the MLD and report the highest and the lowest value of each for each country.

Key magnitudes
As is well-established, earnings inequality measured by the Gini index is highest in the US and the UK, followed by Germany and France. Interestingly, the range of the Gini coefficient for the period 1990-2012 is largest for the US and for Germany, indicating that both countries have experienced substantial changes during our period of study. The MLD gives a slightly different picture, with the UK exhibiting the greatest earnings inequality, and the US being somewhere in between the UK and Germany, while France lags well behind. Behind this pattern lies the fact that the MLD is particularly sensitive to inequality at the bottom of the distribution and less so to that at the top, and that a substantial fraction of inequality in the US has been driven by the behavior of top incomes. 14 In terms of hourly wages, the US and the UK are the most unequal countries, with the MLD ranging between 0.11 and 0.19, while France exhibits the lowest degree of wage dispersion, with a minimum of 0.08 and a maximum of 0.11.
We report average hours worked by country, which have been widely discussed in the litera-13 A common measure of wage inequality is the standard deviation of log wages; see for example Juhn et al. (1993) and Lemieux (2008).
14 See Piketty and Saez (2003)   week. The dispersion of hours is lower than that of wages, as expected, with the MLD ranging between 0.03 and 0.10, roughly half of the dispersion we observe for wages. In France and the US, hours dispersion is low and relatively stable over the period, fluctuating between 0.03 and 0.04 in both countries. In contrast, hours inequality in Germany and the UK is substantially higher and has changed markedly over time, with the MLD of hours peaking at 0.082 and 0.94, respectively, figures that are comparable to the dispersion of hourly wages.
The evolution over the past decades of earnings and wage inequality are by now well-known. hours. These changes are apparent in figure 3 where both average hours worked and the MLD of hours exhibit a jump. Unfortunately, there is no way to deal with this change in sample composition. In what follows we will report figures for the entire period for which we have data for the US, 1989 to 2012, but the reader should bear in mind that comparisons with the pre-1993 data are to be interpreted with great care.

General trends
To understand the broad patterns described above we consider in detail the evolution of the various magnitudes, as well as differences across groups defined by gender and skill. We start with the general time trends of hours worked and hours inequality in our sample period.  Note: Inequality in earnings, wages and hours is measured by the MLD and denoted by Iy, Iw, and Ih respectively. ρ denotes the correlation term, while RC i is the relative distribution of wages, hours and the correlation term. explain a large fraction of inequality in earnings, reaching 75%, while in the UK it only represents half of it. Germany exhibits an upward trend in wage inequality, although the increase is smaller than that observed for earnings inequality. France is the least unequal country, its dispersion in hourly wages falls and reaches values below 0.10, with the corresponding contribution to earnings inequality falling from 75% to 60%.
The middle graphs of figure 4 depict the absolute contribution of the dispersion of hours worked and of the correlation between hours and hourly wages, respectively. Hours inequality increase markedly in both France and Germany. Concerning the contribution to earnings inequality of the correlation between hours and wages, we identify two different patterns: the Anglo-Saxon countries exhibit a mildly positive correlation, constant over time, explaining 10% of overall inequality in those countries; Germany and France both exhibit negative/nil values in the 90's and positive values after the year 2000. The bottom panels, included as a robustness check, report hours inequality as measured by the Gini coefficient and the covariance between hours and wages. We can see that the Gini delivers the same evolution of hours that we obtained with the MLD, and that the coefficient of correlation for France and Germany exhibits sharp changes, which in turn are the main effect behind changes in ρ.
The overall evolution of these four countries is summarized in figure 5, where we report inequality in wages and inequality in hours for the years reported in table 3. It can be clearly seen that each country follows a specific pattern: the UK with the highest inequality in hours vis a vis the US with the highest inequality in wages, France with the lowest inequality along both dimensions, and Germany moving from the French "model" of labor market to the British one over the two decades.

Differences in hours worked across skill and gender groups
The different dynamics that we observe in the four countries may reflect compositional effects.
For example, if part-time employment is a major source of hours dispersion and if this type of employment concerns mainly women, then observed cross-country differences could be the result of differences in the proportions of working women. Similarly, long-working weeks may concern only high-skilled males (the workaholic trader we find in the popular press) and consequently the share of skilled employment may be an important determinant of hours dispersion. To address these issues we decompose earnings inequality for four population subgroups, dividing the sample by gender and educational levels, low-skilled and high-skilled, the threshold being having at least some university education. In all countries but the US the share of low-skilled men declines and the share of high-skilled women is on the rise during our sample period. For France and Germany the former group remains the largest (reaching a slightly less than 40% at the end of the sample period), while neither of the two groups of high-skilled groups do pass the threshold of 20% each (see figure A.1 in the Appendix). repeat the exercise for wages, hours and the wage-hour covariance (the corresponding figures for selected years are reported in tables A.3 to A.9 in the appendix). It is interesting to observe that US inequality is pulled by the male component, while in the UK is women that exhibit the highest dispersion. Note also that the rise of inequality in Germany is mainly attributable to low-skilled workers, with inequality among high-skilled men remaining constant and that among high-skilled women exhibiting an inverse U-shaped pattern. Lastly, for all countries and for both skill levels the female component is characterized by higher inequality when compared to its male counterpart. Our main interest lies in figures 8 and 9, depicting the evolution of inequality in hours and of the covariance term. Figure 8 highlights gender differences in working hours: while male groups experience constant patterns of hours, likely centered on full time employment, female working hours are much more dispersed, especially in the UK and gradually also in Germany.
The picture is completed by figure 9 reporting the covariance contribution to earnings inequality. Various comments are in order. There are striking differences between the high-and the low-skilled, with the latter exhibiting a smaller covariance term. In some groups (low-skilled men in the US and France) hours are likely to be fixed and therefore independent from wages.
When deviating from zero, the covariance between hours and wages of the low-skilled tends to be negative for males and positive for females, although for low-skilled males, both the UK and Germany exhibit a substantial negative correlation at the start of our sample period that disappears over time. The changes for the high-skilled are striking: in both gender groups we find a move from highly negative covariance terms to nil or positive ones, with the exception of the US where the term is positive throughout the period. Concerning high-skilled males in the UK and France, a highly negative term reaches the same (positive) level as in the US by the end of the period. This implies that the equalising effect stemming from the fact that those with lower wages worked more hours has been eroded over the past two decades.
The case of Germany is particularly interesting, with the transition from low to high earnings inequality being in part driven by low-skilled workers: the equalising negative correlation that used to be apparent for men disappears while women become more and more responsive to labor market, moving from a zero covariance to a positive one, i.e. working more hours for higher wages. Skilled individuals experienced also a substantial change in the covariance term that was particularly marked for women. Note: The within-between index decomposition for each component is reported as a percentage of inequality in earnings.
These observations are confirmed by table 4 where we propose a standard between-within decomposition of each variable under analysis (earnings, wages, hours and covariance among the last twos). Equation (8) is divided by the inequality index, which yields the relative contributions of the within-group and between-group components that are reported in the table. The table hence has a double reading. The between-group and within-group components of earnings inequality add up to one, while the between-group (within-group) components of wages, hours and the covariance reported add up to the between-group (within-group) components of earn-ings. Not surprisingly, the largest share of earnings inequality is attributable to within-group differences, with the between-group component ranging between 15 and 27%. Note, also, that as far as hours are concerned, between-group inequality is particularly small, accounting for only around 10% of overall inequality in hours. This indicates that attributing the dispersion of hours to, say, female part-time employment ignores most of the sources of variation. When we consider the covariance term, its between-group component exhibits different patterns across countries, increasing substantially in France, falling in the UK, and fluctuating slightly in the US and Germany, while the within-group component is substantially larger at the end than at the start of the period for all four countries.

Discussion
Our results indicate that the overall contribution of hours worked to earnings inequality can be substantial, accounting for over a third of overall dispersion in some instances. Moreover, if we consider together the dispersion of hours worked and the covariance between wages and hours, they are responsible, in some countries, for half of the overall earnings dispersion. Inequality in hours of work seems to be largely driven by the female component of employment, possibly by part-time working regimes. In the two countries with the highest inequality, the UK and Germany, women account for at least 40% of employment and both countries are characterized by substantial part-time employment. Nevertheless, our within-between group decomposition indicates that this is only part of the story, with those two countries also exhibiting very substantial inequality in hours within groups.
The changing position of Germany in cross-country comparisons of earnings inequality points to the importance of changes in the covariance between hours and wages. From a labor supply point of view, it can be read as an increasing elasticity of hours to wages (which would be consistent with a higher share of women in employment), from a labor demand point of view, it may represent a prevailing intensive margin over the extensive margin. The overall result was that some countries went from a situation in which the least-paid workers had the longest working hours, to one where the best-paid also work hardest. 20 The immediate question that arises from our analysis is to what extent the observed time patterns are correlated with institutional changes within each country. Consider union density as a global measure of the institutionalization of the labour market. 21 Figure 10 plots union density against our measure of the covariance term, and indicates the well-established gradual weakening of labour standards over the past few decades which has been largely driven by the increased labour market participation of women and the up-skilling of the labour force.
The figure indicates that this weakening has affected differently continental Europe (France and Germany) and the Anglo-Saxon countries (UK and US). When unions were strong, the dominant membership of unions consisted of full-time low-skilled male workers, and this was associated with limited variation in hours and reduced inequality in wages. In the UK and US, weaker unions led to an increase in wage inequality, while in the other two European countries they seem to have resulted in a change in the role of hours inequality, captured by the dynamics of the correlation between hours and wages. As we can see in figure 10, this term is negatively correlated with union density for France and Germany but does not display a clear association in the case of UK and US.
One way to rationalise this evidence is the distinction between intensive and extensive margins. When unions are strong, they typically oppose the use (and abuse) of intensive margins by employers, on the expectation of expanding employment opportunities (extensive margins) and increasing their bargaining power. This compresses the distribution of hours around the contractual/legal duration and leaves wages to do the adjustment to excess demand/supply. As unions strength declines, employers become free to choose which margin they prefer to adjust, a decision that will depend on the relative adjustment costs per hour and per head, as well as on their expectations concerning demand. As a consequence, hours become more dispersed, the labour supply elasticity becomes positive and the residual correlation moves from nil to positive. Although more rigorous statistical tests would be required to prove our interpretation, our hypothesis implies that earnings inequality could be attenuated by union presence which would tend to reduce both wage inequality and the positive correlation between hours and wages.

Conclusions
Our paper contributes to the literature on earnings inequality by considering the role of hours worked and not only that of hourly wages, in order to address the question of whether for a given distribution of wages, the dispersion of working time tends to dampen or magnify initial inequalities. To do so, we decompose our inequality index to quantify the contribution of inequality in hourly wages and the dispersion in hours worked to overall inequality in earnings.
Hours dispersion affects overall inequality through two mechanisms, inequality in hours and the correlation between hours worked and hourly wages. As a result, if the latter term is negative (i.e. if poorly-paid workers are those that work most), hours inequality may have a substantial equalizing impact.
Our results uncover a number of surprising patterns. First, we find that hours inequality is moderate in the US and France and much larger in the UK and Germany, with the distributions presenting fat tails both at the bottom (probably associated with female part-time and 'mini-jobs') and at the top. Second, the contribution of hours to inequality is always positive but varies substantially across countries and over time. In 2012, it accounted for only 15% of earnings inequality in the US, 29% in the UK, 30% in France, and 34% in Germany.
In the three European countries we find that there is a substantial change in the covariance between wages and hours and in some case, notably Germany, this has been a major force behind the change in the overall contribution of hours. For several groups, mainly low-skilled males and high-skilled females, the covariance was negative at the start of the period and becomes zero or positive over time. In the case of the UK, a similar pattern is observed for high-skilled males.
As a result, an important equalizing force, the longer hours worked by those with the lowest pay, seems to have disappeared, with important implications for earnings inequality. The US presents a rather stable contribution of wages, hours and the covariance, while Germany has witnessed major changes. Between 1991 and 2012, earnings inequality moved from being par to that of France to being close to the one observed in the US, and this was driven by a change in the contribution of hours and, especially, of the covariance which increased by 13 percentage points.
Our paper indicates that the so-far neglected question of hours inequality can help us understand the evolution of earnings dispersion in certain countries. At the same time, it raises as many questions as it answers. First, we need to understand what drives the broad cross-country patterns that we have uncovered. The fact that the US shares a distribution similar to that of