Journal of Monetary Economics 72 (2015) 114–130

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Journal of Monetary Economics journal homepage: www.elsevier.com/locate/jme

Revisiting wage, earnings, and hours profiles Peter Rupert a, Giulio Zanella b,n a b

University of California, Santa Barbara, United States Department of Economics, University of Bologna, Piazza Scaravilli 2, 40126 Bologna (BO), Italy

a r t i c l e in f o

abstract

Article history: Received 28 November 2012 Received in revised form 3 February 2015 Accepted 3 February 2015 Available online 12 February 2015

For the youngest cohorts whose entire working life can be observed, hours start falling much earlier than wages. Wages do not fall (if they fall at all) until one's late 60s. The data suggest that many workers start a smooth transition into retirement by working progressively fewer hours while still facing an upward-sloping wage profile. This pattern is not an artifact of staggered abrupt retirement or selection. This evidence imposes restrictions on dynamic models of the aggregate economy, and provide updated numerical profiles that can be readily used in quantitative macroeconomic analysis to incorporate this new pattern into aggregate models. & 2015 Elsevier B.V. All rights reserved.

Keywords: Life cycle Wage profile Labor supply Human capital Pre-retirement

1. Introduction An empirical investigation into life cycle profiles of wage rates and hours of market work is undertaken, from the vantage point of consistent, four-decades-long data, to examine their behavior over the life-cycle. Theoretical and empirical investigation of such profiles has a long tradition in labor economics and macroeconomics, because of their importance in understanding a variety of phenomena such as labor supply, retirement, incentive contracts, human capital, and inequality.1 The March Supplement of the Current Population Survey (March CPS) and the Panel Study of Income Dynamics (PSID) now cover in a consistent way more than 40 years, and so allow one to observe the entire working life or substantial portions of it for several cohorts. Older cohorts (workers born before WW2 and who entered the labor market before the 1960s) are described by the traditional tracking between wages and hours that motivates the use of parallel, hump-shaped wage and hours profiles. However, for younger cohorts (individuals born during or after WW2 and who entered the labor market during the 1960s) such tracking disappears: hours per worker fall substantially (and so do earnings) beginning shortly after age 50, but wages do not fall—if they fall at all—until these workers are in their late 60s. Moreover, this pattern is not an artifact of selection out of employment, and we argue that this substantial drop in hours while wages are still growing is generated by a process that resembles a smooth transition into retirement, in the form of less overtime work and passage from full- to part-time. The pattern observed imposes restrictions on a benchmark life cycle model. Alternative departures from the benchmark are explored that may reconcile theory and data. A departure that seems able to reproduce the fall in hours despite non-declining

n

Corresponding author. E-mail addresses: [email protected] (P. Rupert), [email protected] (G. Zanella). 1 The human capital model initiated by Ben-Porath (1967) and further developed by Ghez and Becker (1975), Blinder and Weiss (June 1976), Ryder et al. (1976), Heckman (1976) and Rosen (1976) occupies a prominent position among the theories developed to characterize life cycle profiles. Weiss et al. (1986) is still an excellent review. http://dx.doi.org/10.1016/j.jmoneco.2015.02.001 0304-3932/& 2015 Elsevier B.V. All rights reserved.

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wages for low educated workers is an explicit role for human capital, such as endogenous wages from on-the-job training. For highly educated workers, increasing disutility of work shortly after age 55 (not because of health reasons) seems an important part of the story. The life cycle profiles estimated are straightforward to interpret, and can be readily used in quantitative macroeconomic analysis. In particular, our wage profiles are an up-to-date, more direct description of the life cycle than the corresponding profile based on efficiency units weights provided by Hansen (1993). The latter are hump-shaped, with the decline beginning around age 55—i.e., 10–15 years earlier than they actually start declining for the younger cohorts in our samples.2 There may be a host of reasons why wages fall much later than hours for these cohorts. We do not take a stand on these reasons, which go beyond the scope of this paper.3 The focus in this paper is on the intensive margin of labor supply—hours per worker. The reason the extensive margin is not dealt with is that retirement choices are already well understood within the life cycle model. There is related research that show technological features such as nonconvexities in the mapping between hours and labor services (Rogerson and Wallenius, 2009; Erosa et al., 2014) and institutional features such as social security rules (French, 2005; Erosa et al., 2012) allow one to match extensive margin behavior late in the life cycle. It is the intensive margin pattern documented here that is less explored. Revisiting wage and hours profiles is important because data long enough to allow one to observe the entire life cycle in a consistent way were unavailable until recently. An early stratagem to overcome this limitation was synthetic cohorts from single cross-sections—i.e., using the a þj year-olds at time t as a counterfactual for the a year-olds at time t þ j. However, such synthetic cohorts may produce a biased picture of the life cycle when productivity changes over time. Thornton et al. (1997) and Rubinstein et al. (2006) offer an illustration for annual and weekly earnings, respectively. An alternative method was the construction of pseudo-panels from repeated cross-sections. For instance, Mincer (1974) analyzed census data and observed that the weekly wage rate was not declining at the end of working life, while annual earnings were. This method is also the starting point of our analysis, which confirms Mincer's observation. The increasing availability of longitudinal data has allowed a more direct look at actual portions of workers' careers by following them over time. Johnson and Neumark (1996) used panel data from the National Longitudinal Survey of Older Men to infer the dynamics of male wages during the late portion of the working life, and did not find clear evidence of negative wage growth until workers in their sample were in their 60s. They did not consider the associated labor supply behavior, however. Recent research in macroeconomics, based on micro panel data, has focused on earnings profiles, finding that they are hump-shaped. Examples include Huggett et al. (2011) and Heathcote et al. (2010b). The dynamics of earnings, however, results from both wages and hours, and it is of interest to look separately at these two processes. A number of additional recent papers in macroeconomics are directly or incidentally concerned with life cycle profiles. Among them, French (2005) studies the effect of health and social security rules on hours and retirement behavior of American male household heads. As a preliminary step in his structural investigation, French uses the annual portion of the PSID (1968–1997) to estimate wage and hours profiles. He finds that the wage profile is hump-shaped, with the turning point between the ages of 55 and 60. The hours profile he estimates is also declining, with the trend changing at about these same ages. French notes that “Most of the variation in the wage and labor supply profiles is from individuals aged 55–65 […]. [The] decline in hours coincides closely with the decline in wages” (p. 412). The apparent contrast with our results has a simple explanation: the different pattern that we document here emerges with cohorts that are younger than those considered by French. Imai and Keane (2004) study a human capital model to reconcile small and large intertemporal elasticities of labor supply at the individual and aggregate level. The model is estimated using data from the NLSY, and produces out-of-sample (i.e., after age 40) predictions that result in markedly humpshaped profiles. Nonetheless, as illustrated below, Imai and Keane (2004) offer a key insight: investment in human capital increases substantially the opportunity cost of time early in the life cycle, which offers an avenue for reconciling a non-declining wage profile and a declining hours profile. Rogerson and Wallenius (2009), like Imai and Keane (2004), are interested in understanding the discrepancy between micro and macro elasticities of labor supply. The main feature of their model is a nonconvexity (a flat initial portion in the mapping between hours of work and labor services) that generates motives for entering and leaving the labor force at specific points in the life cycle. Their model takes the wage profile as exogenous and assumes it is hump-shaped. This implies that hours decline before retirement. In this model, the only way one can generate the decline in hours (and the other results) when wages do not decline is by assuming that the disutility of working increases at later stages in the life cycle. This is one of the variants of the benchmark model explored below. The remainder of the paper is organized as follows. Section 2 describes the data. Section 3 illustrates how these data restrict the theory. Section 4 concludes. In a Supplemental Web Appendix additional data is provided, as well as tables containing the numerical wage and hours profiles that can be readily used in quantitative analysis.

2

Abbasoglu (2012) compares the profiles in Hansen (1993) to ones generated from the PSID, similar to ours. One possibility is that there are pure time and cohort effects at work. Another is that these effects reflect higher education and higher participation rates at older ages for these cohorts relative to older ones. It is also possible that human capital depreciates only slowly (or not at all) for these younger cohorts, or that the training content of an hour of work now decreases fast enough to compensate a declining productivity profile before retirement. Yet another explanation is that incentive contracts are in place. For instance, the work, among others, of Lazear (1979, 1981), Freeman (1977), Medoff and Abraham (1980), and Harris and Holstrom (1982) show that individual wage growth is possible even in the absence of individual productivity growth if dynamic incentives matter. Gibbons et al. (1999) provide a thorough review of this literature. A related question not addressed is what is the underlying productivity profile; as the focus is on observed wages rather than unobserved productivity because hours respond to the former, but of course these two are not necessarily the same. 3

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2. Data Two different data sources are used, the March CPS and the PSID. These two data sets complement each other with respect to the analysis and so there are advantages from using them jointly in this paper. The CPS offers the advantage of a large sample size. Its main limitation is that it is essentially a repeated cross-section, not a proper panel. One can study the life cycle in the CPS by following a cohort over time and relying on the fact that in a sufficiently large sample, cohort sample means are consistent estimates of the underlying population means in any given cross-section. Just to mention one difficulty with this procedure, because of immigration, later CPS cross-sections for a given cohort are non fully comparable with earlier ones for that same cohort. An obvious solution is to exclude observations that migrated to the US after the initial cross-section of interest, but the March CPS records year of immigration only since 1995. Furthermore, no analysis requiring the observation of the same individuals over time is possible with this pseudo-panel approach. The PSID has a relatively small sample size, but it offers all of the advantages of a rich and long panel. In particular, after the full release of the 2011 wave at the end of 2013, this data set offers the unique opportunity to observe individual life cycle profiles spanning 43 years—the first wave dates back to 1968. This amounts to the entire working life of the cohorts who entered the labor market at the end of the 1960s. Therefore, the CPS is used as the base data and the PSID when needed to exploit the panel dimension. This strategy is justified by the fact that—as documented below—the wage and hours profiles follow a virtually identical pattern in the two data sets. For the CPS, data are pulled from the IPUMS-CPS data base (King et al., 2010), which provides standardized CPS data from 1963 to 2013. However, only the 1976–2013 portion of these data is used because prior to 1976 the CPS did not ask to report usual weekly hours and weeks worked the previous year, which one needs to compute annual hours (and so wage rates) in a consistent way over the life cycle. For the PSID, data are from the PSID Data Center from 1968 to 2011—readers should keep in mind that the PSID was at annual frequency until 1997 and at a biennial frequency thereafter. For both the CPS and the PSID individual-level data are used that contain demographic, socioeconomic and labor market information from income year 1967 to income year 2012 and for different cohorts.4 Averages of hourly wages and annual hours are computed for workers of a given cohort and all available years to trace the life cycle profiles of individuals in that cohort. This procedure raises a fundamental selection problem: only wages for workers are observed, an issue addressed in Section 2.2. Cohorts are defined over 5-year bins and the central value of the bin is used to keep track of the age of the cohort. For instance, workers who where between 21 and 25 years old in 1967 are defined as the “23 years old” cohort, those who where between 26 and 30 years old in 1967 are the “28 years old” cohort, etc. In order to help the reader, these cohorts are referred to by their birth years. For instance, individuals in cohort 23 are born between 1942 and 1946, individuals in cohort 28 are born between 1937 and 1941, etc. The youngest cohort here is the 23 years old cohort, and the oldest is the 48 years old cohort (i.e., those born between 1917 and 1921). Wage rates are constructed as the ratio of annual hours to annual earnings. It is well known that this procedure, albeit standard, leads to measurement error in hourly wages, but in our study this is vastly mitigated by averaging across individuals in a given cohort. Wages and earnings are before-tax and are expressed in constant 2010 dollars using the CPI-U as deflator. In constructing our final sample the following selection criteria are applied. First, self-employed individuals are excluded from the analysis because their earnings are a mix of labor and capital income. Second, following Heathcote et al. (2010a) we exclude individuals whose nominal wage rate in a given year is less that 50% of the federal minimum wage in force that year. Third, again following Heathcote et al. (2010a) we exclude in any given year individuals who report working less than 260 hours during that year—this helps getting rid of implausibly large wage rates, which may result when too few hours are reported. Fourth, top-coded wages are removed for those individuals at or above the top 1% every year. This eliminates all of the top-coded observations in the CPS and the PSID, and a few more. We prefer this “trimming” procedure to the alternative popular procedure of fitting a Pareto distribution to infer top-coded values because the interest is in rendering the life cycle of a representative worker. It makes sense in this perspective to concentrate on the central part of the wage distribution. In addition, when using the PSID, the analysis is restricted to households belonging to the SRC sample.5 When using the CPS sampling weights are applied to produce representative statistics. The analysis is restricted to men because the labor market history of American women underwent major changes during the period under consideration. However, the analysis is replicated for women in the Supplemental Web Appendix. Because of such changes, the profiles of women should be interpreted with caution. Also, when comparing the CPS and the PSID (but not when analyzing CPS data alone) we will restrict to household heads and their spouses—these two categories comprise almost 90% of the population 23 years of age and older. Table 1 reports summary statistics of core demographic and socio-economic characteristics for 6 different cohorts of males in the CPS and the PSID, pooled over the 1975–2010 period (i.e., the period of overlap between the two

4 A year in our data set refers to the “income year”, not the period in which the data were actually collected. For instance, the 1980 CPS and the 1980 wave of the PSID collected wage and hours information referring to income year 1979. 5 The core sample of the PSID originates from two distinct samples: a cross-sectional national representative sample of about 3000 households (the 1968 SRC sample) and a non-representative sample of about 2000 low-income families located in metropolitan areas (the SEO sample). The SRC sample was representative of the U.S. population back in 1967, so there is no guarantee that it is representative during the following 41 years. However, this is the group that maximizes the representativeness of our sample across the four decades. Another way to make the PSID representative of the U.S. population is to apply individual sampling weights to the full core sample. Unfortunately, this cannot be done in a consistent way across all of the 43 waves. Fiorito and Zanella (2012) illustrate how representative the components of the PSID core sample are.

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Table 1 Cohort characteristics in the CPS and in the PSID, men. Cohort Year born

23 1942–1946

28 1937–1941

33 1932–1936

38 1927–1931

43 1922–1926

48 1917–1921

CPS

PSID

CPS

PSID

CPS

PSID

CPS

PSID

CPS

PSID

CPS

PSID

N. in 1975

4121

215

2954

177

2702

167

2791

223

2915

180

2474

156

White

0.87 (0.33)

0.95 (0.22)

0.87 (0.34)

0.88 (0.32)

0.88 (0.33)

0.87 (0.34)

0.90 (0.31)

0.92 (0.27)

0.90 (0.30)

0.92 (0.27)

0.90 (0.30)

0.92 (0.27)

Black

0.09 (0.29)

0.02 (0.15)

0.10 (0.29)

0.07 (0.25)

0.09 (0.29)

0.07 (0.26)

0.08 (0.27)

0.06 (0.23)

0.08 (0.28)

0.05 (0.21)

0.08 (0.28)

0.04 (0.19)

Married

0.83 (0.38)

0.82 (0.38)

0.84 (0.37)

0.84 (0.36)

0.84 (0.37)

0.86 (0.35)

0.84 (0.37)

0.89 (0.32)

0.84 (0.37)

0.85 (0.36)

0.84 (0.37)

0.85 (0.36)

Widowed

0.01 (0.11)

0.01 (0.10)

0.02 (0.15)

0.03 (0.16)

0.04 (0.19)

0.04 (0.19)

0.05 (0.22)

0.05 (0.22)

0.06 (0.24)

0.08 (0.26)

0.07 (0.26)

0.12 (0.33)

Divorced

0.09 (0.29)

0.11 (0.32)

0.08 (0.27)

0.11 (0.31)

0.07 (0.25)

0.07 (0.26)

0.06 (0.24)

0.04 (0.20)

0.05 (0.22)

0.04 (0.21)

0.04 (0.20)

0.01 (0.10)

Separated

0.02 (0.14)

0.02 (0.13)

0.02 (0.14)

0.01 (0.09)

0.02 (0.13)

0.01 (0.09)

0.01 (0.12)

0.01 (0.09)

0.01 (0.12)

0.00 (0.06)

0.01 (0.11)

0.00 (0.07)

College

0.29 (0.46)

0.35 (0.48)

0.25 (0.43)

0.28 (0.45)

0.23 (0.42)

0.28 (0.45)

0.21 (0.40)

0.25 (0.44)

0.18 (0.38)

0.21 (0.41)

0.13 (0.34)

0.18 (0.38)

High school

0.34 (0.47)

0.35 (0.48)

0.36 (0.48)

0.32 (0.47)

0.35 (0.48)

0.34 (0.47)

0.33 (0.47)

0.34 (0.47)

0.33 (0.47)

0.35 (0.48)

0.33 (0.47)

0.35 (0.48)

Less than HS

0.16 (0.36)

0.11 (0.31)

0.21 (0.41)

0.16 (0.37)

0.26 (0.44)

0.26 (0.44)

0.32 (0.47)

0.27 (0.44)

0.36 (0.48)

0.34 (0.48)

0.41 (0.49)

0.34 (0.47)

Notes: The table reports cohort size in 1975 and mean demographic and socioeconomic characteristics of men (household heads and their spouses) in the CPS and the PSID. Standard deviations in parentheses. Each cohort is pooled over the 1975–2010 period, i.e., the time interval where the CPS and the PSID overlap in our data set. CPS means are weighted by sample weights. The PSID sample includes individuals belonging to the SRC subsample only. A cohort is labelled by age in 1967, and results from pooling into a five-year bin individuals belonging to five adjacent one-year bins. For example, cohort 23 is composed of individuals who where between 21 and 25 years old in 1967, so individuals in this cohort were born between 1942 and 1946. “White” includes whites of any ethnicity.

data sources in our final data set). Beyond the fact that the CPS cohorts analyzed are much larger that the corresponding PSID ones, this table shows that the two data sources agree, by and large, on the characteristics of the cohorts under investigation. 2.1. Empirical profiles Fig. 1 reports the wage and hours profiles of all men belonging to these cohorts, in the CPS pseudo-panel. This picture shows both the raw data and interpolations based on the best second-order and fourth-order fractional polynomials for wages and hours, respectively.6 NBER recessions spanned by the data are also indicated in the figure—a specific recession can be identified by combining age and birth interval. For the four cohorts born before 1937, there is a close tracking between hours and wages, as implied by the benchmark life cycle model. For workers in these cohorts both hours and wages start declining between the ages of 55 and 60. However, no such tracking is visible for the two youngest cohorts. In particular, at the last data point, workers born between 1937 and 1941 are between 68 and 72 years old; and those born between 1942 and 1946 are between 63 and 67 years old. There is no visible decline in wages in the late stages of the working life for individuals in these cohorts. The shape of the hours profile is invariant, though: these are almost perfectly overlapped across cohorts, and they all start falling around age 55.7 For all of the six cohorts, of course, the implied earnings profile (not shown) is hump-shaped.8 The profiles are broken down by education, considering two groups: workers with at least a four-year college degree “college”, and workers with a lower educational attainment “no college”. The decomposition is reported in Fig. 2. This shows a pattern similar to the one in Fig. 1: for the older cohorts, both the wage and hours profiles are hump-shaped, with wages generally declining at a similar rate across the two groups; however, workers in the two young cohorts are still reducing hours beginning shortly after age 50 despite their wage profiles fall much later. There are important differences across the two groups, too: for the younger cohorts, the wage profile of the less educated is virtually flat, while that of college 6 A fourth order polynomial is more appropriate for hours, given the sharp turning point. Since no such turning point is obvious for wages, a second order polynomial is preferable. 7 In Fig. 1 there is some increase in hours for the youngest cohorts relative to the older ones. This is consistent with the evidence reported in papers that analyze time use trends across different cohorts. See, for instance, figures 21 and 22 in Attanasio et al. (2014). 8 Notice that the wage profiles tend to shift upward from older to younger cohorts. This, of course, reflects productivity growth.

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Annual hours per worker 1600 1300 2200 1900 70

Cohort: 1917-1921

Annual hours per worker 2200 1900 1600 1300

Annual hours per worker

70

60

16

60

50 Age

25 22 19 Hourly Wage

50 Age

40

28

40

16

Hourly wage

30

25 22 19 Hourly Wage

70

Cohort: 1922-1926

30

Annual hours per worker 2200 1300 1900 1600

60

70

16

16 50 Age

50 60 Age

25 22 19 Hourly Wage

25 22 19 Hourly Wage 40

40

28

28

Annual hours per worker 2200 1900 1600 1300

Cohort: 1927-1931

30

30

70

Cohort: 1932-1936 28

60

Annual hours per worker 2200 1900 1600 1300

50 Age

16

16 40

25 22 19 Hourly Wage

25 22 19 Hourly Wage 30

Cohort: 1937-1941 28

28

Annual hours per worker 2200 1900 1600 1300

Cohort: 1942-1946

30

40

50 Age

60

70

Recession

Fig. 1. Wage and hours profiles in the CPS, men. Notes: The figure reports the profiles of hourly wages and annual hours per worker of men in a pseudopanel created from CPS data. The interpolating line is the best second-order fractional polynomial for wages, and the best fourth-order fractional polynomial for hours. Shaded areas are NBER recessions. The Supplemental Web Appendix contains the corresponding picture for women.

graduates keeps growing at a sustained rate until at least age 60. The hours profile of college graduates is above the corresponding profile of non-graduates (before age 55 the difference is about 200 hours), with some convergence towards the end of the working life, at least for the three younger cohorts. How do workers reduce hours along the life cycle? The PSID allows us to observe the same individuals at the life cycle peak of hours as well as at later stages, when their hours profile declines. Before switching to this alternative data source, notice that the pattern documented in the CPS is also found in the PSID. Fig. 3 reports the comparison for the two youngest cohorts. This picture shows that the shape of the wage and hours profiles is the same in the two data sets. There are differences in levels, though, especially for the wage profile: the average gap is about $4–$5 per hour after age 40. To be clear, this does not mean that the CPS and the PSID do not agree on the level of the male hourly wage in the US. Average wages in each full cross-section (i.e., pooling all cohorts instead of following a single cohort over time), shows the levels are essentially the same. The gap observed in Fig. 3 is instead a manifestation of the problems briefly discussed above, notably the fact that the PSID profiles depict the life cycle of a constant group of individuals in a given cohort (up to attrition, which is not an issue, as shown in Rupert and Zanella (2012)) who were representative of this cohort in 1967, while the CPS profiles depict the life cycle of a varying group of individuals in a given cohort who are representative of this cohort every year. Fig. 4 answers the question of how workers reduce hours by showing, from two different angles, the evolution of hours during the working life of employed individuals in these two PSID cohorts, for all men and by education. First, kernel density estimates of the hours distribution at three different points of the working life: when individuals are between 48 and 52 year old (that is, at the peak of hours), ten years later, when they are between 58 and 62 years old, and when they are between 64 and 68 years old. Second, scatter plots of hours at the last of these three points in time (age 64–68) against hours at the first (age 48–52). The density estimates show that the distribution at the early stage is concentrated around the full-time level of hours, and has a fat right tail indicating that many workers—especially among college graduates—engage in overtime work. As these workers age, the highly educated reduce drastically these overtime hours and tend to part-time levels. That is, for these workers the hours distribution at the later stages has a much thinner right tail and looks bimodal with a second peak around part-time hours. For the

P. Rupert, G. Zanella / Journal of Monetary Economics 72 (2015) 114–130

60

70

60

70

Cohort: 1922-1926

70

30

40

50 Age

60

30

70

40

50 Age

60

70

Cohort: 1917-1921

Annual hours per worker 2400 2000 1600 1200

50 Age

60

40 35 30 25 20 15 Hourly Wage

40

50 Age

Annual hours per worker 2400 2000 1600 1200

30

40

40 35 30 25 20 15 Hourly Wage

40 35 30 25 20 15 Hourly Wage

Annual hours per worker 2400 2000 1600 1200

Cohort: 1927-1931

30

Annual hours per worker 2400 2000 1600 1200

50 Age

Cohort: 1932-1936 40 35 30 25 20 15 Hourly Wage

40

Annual hours per worker 2400 2000 1600 1200

30

Cohort: 1937-1941 40 35 30 25 20 15 Hourly Wage

40 35 30 25 20 15 Hourly Wage

Annual hours per worker 2400 2000 1600 1200

Cohort: 1942-1946

119

30

40

Hourly wage, college

Annual hours per worker, college

Hourly wage, no college

Annual hours per worker, no college

50 Age

60

70

Fig. 2. Wage and hours profiles in the CPS, by education, men. Notes: The figure reports the profiles of hourly wages and annual hours per worker of men in a pseudo-panel created from CPS data, by education. The “college” group are workers who have a four-year college degree or a higher level of education; the “no college” group are all of the remaining workers. The interpolating line is the best second-order fractional polynomial for wages, and the best fourth-order fractional polynomial for hours. Shaded areas are NBER recessions. The Supplemental Web Appendix contains the corresponding picture for women.

low educated, there is a large decrease in the density at full-time and a corresponding increase at part-time levels, as well as a thinner right tail. The scatter plots confirm this pattern by representing the same individuals at two different points in time. Therefore, ignoring for a moment selection into employment (an issue tackled below), this picture suggests that many senior workers switch to part-time mode and reduce overtime work, a pattern confirmed when decomposed by education. Such a “preretirement” process conforms to the more general notion of retirement advocated by Heckman (1976): “Retirement can more generally be defined as a period with few hours of work supplied to the market.” (p. S15).9 It is of interest to look at the characteristics of workers who are on this pre-retirement path. Again, the PSID allows us to observe the same workers at different stages of the working life. Table 2 summarizes the key economic and demographic characteristics: health, net wealth, non-labor income, marital status, and children below 18 living in the household. This table reveals a clear pattern: workers who remain in employment but reduce hours after the peak at age 50 are in worse health, have higher net wealth, and live in families with higher total income relative to workers who do not reduce hours. However, these differences are not as large as those between workers who reduce hours and those who leave the labor market altogether at these later stages of the working life. In other words, health and income effects are potentially important forces driving senior workers into a pre-retirement path, although not as important as in driving them out of the labor market. To sum up the finding documented in this section, the empirical regularity that emerges from our analysis shows that, for the younger cohorts, the wage profile increases over the life cycle with no clear tendency to decline before one's late 60s. At the same time, there is a sizable reduction in hours of market work along the intensive margin. Such a reduction constitutes a smooth transition into retirement via reduction of overtime work and switch to a part-time work mode, a process we refer to as “pre-retirement”. A concern is that this pattern may be an artifact of staggered retirement, and selection out of employment.

9 Given the possibly relevant role of moves from full- to part-time during the pre-retirement years, the existence of a negative part-time premium on hourly wages documented by Aaronson and French (2004) suggests that pre-retirement adjustments may lead to underestimating productivity growth late in the working life.

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Cohort: 1942-1946

Cohort: 1937-1941 2400

35

2400

35

1600 20

2000

25

1600

Annual hours per worker

25

30

Hourly Wage

Hourly Wage

2000

Annual hours per worker

30

20

1200

15 20

30

40

50

60

70

1200

15 20

30

Age

Wage, CPS

40

50

60

70

Age

Wage, PSID

Hours, CPS

Hours, PSID

Fig. 3. Wage and hours profiles in the CPS and the PSID, men. Notes: The figure reports the profiles of hourly wages and annual hours per worker of men in a pseudo-panel created from CPS data and the actual PSID panel (Core sample, SRC subsample) in two different cohorts. The interpolating line is the best second-order fractional polynomial for wages, and the best fourth-order fractional polynomial for hours. Shaded areas are NBER recessions. The Supplemental Web Appendix contains the corresponding picture for women.

2.2. Is this pattern an artifact? There are two important processes that may give the false appearance of a smoothly declining hours profile and a nondeclining wage profile, and they are both manifestations of the same selection problem: the fact that wages are observed only for individuals who work in a given year and, for the sub-sample of these that leave employment during that year, hours are measured only up to the day they stop working. First, the pattern may result from staggered, abrupt retirement at points different from the end of the year. The following example illustrates. Suppose that all individuals in a cohort either work full-time at the same, given hourly wage or do not work at all, and that individuals can retire only on July 1st of any year—i.e., at mid-year. In this case the true individual hours profile is flat at 2000 hours until the year before retirement, has an intermediate point at 1000 hours in the retirement year, and is flat at zero hours afterwards. The wage profile, instead, is flat. Now suppose that an increasing fraction of workers in the cohort retire every year, starting at age 58. This would induce a smoothly declining aggregate hours profile (in correspondence to non-declining wages) while the true underlying individual profiles are discontinuous. Second, our sample changes over time because of selection into and out of employment. The selection problem in this context is easy to see: at the late stages of the working life workers begin self-selecting into retirement. This process is very likely non-random with respect to wages and hours. Suppose that low-wage workers retire first, or that it is precisely a negative wage shock that induces workers to retire. Then our sample would change in a way that leads to overestimating wage growth late in the working life: this renders a non-declining selected wage profile while the true underlying profile is actually declining.10 Is the empirical regularity isolated above an artifact of these two processes? The answer is, as shown below, is no. 2.2.1. Pre-retirement vs. abrupt retirement during the year Both Rogerson and Wallenius (2013) and Fan (2013) claim on the basis of PSID data that “abrupt retirement” (i.e., moving from, say, full-time hours to zero hours with an intermediate level of hours in the year a worker retires) is by far the most important retirement mode in the U.S.—more than 70% of workers. This, apparently, contrasts with the pre-retirement pattern documented above. To understand the difference between these two retirement modes, consider Fig. 5. This picture reproduces the individual profiles of two workers in the 23 years old cohort (individuals born between 1942 and 1946). Panel A refers to a man who is retiring along a pre-retirement pattern. He works at a full-time level (i.e., about 2000 hours per year) every year until age 55, then switches to a part-time level (i.e., about 1000 hours) and further reduces hours until he is at least 64, before retiring at 65 or 66. Panel B, instead, refers to a man who moves abruptly from full-time work to zero 10 A third, potential problem when employing the PSID is that the sample changes over time because of attrition. However, given the agreement of the PSID pattern with the CPS one (i.e., a data set where attrition is not an issue) this is not a concern. The working paper version, Rupert and Zanella (2012) contains informal and formal tests that confirm this conjecture.

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College

.001

.001

.0015

All men

0

No college

0

0

Hours, age 48-52

4000 3000 2000 1000

Hours, age 64-68

4000 3000 2000 1000

0

Hours, age 48-52

1000 2000 3000 4000

College Hours, age 64-68

4000 3000 2000 1000

Hours, age 64-68

1000 2000 3000 4000

.0005

0

0

1000 2000 3000 4000

All men

No college

.001

Density

.0005

Density

.0005

Density

1000 2000 3000 4000

121

1000 2000 3000 4000

Hours, age 48-52

Hours, age 58-62

1000 2000 3000 4000

Hours, age 48-52

Hours, age 64-68

Fig. 4. Hours at different points of the life cycle: PSID cohorts 1937–1946, men. Notes: Top row: kernel density estimate (Epanechnikov kernel) of hours worked by PSID men born between 1937 and 1946, by education, at three different points in time: when they are 48–52 years old, when they are 58–62 years old, and when they are 64–68 years old; zeros are excluded. Bottom row: hours worked by an individual when he is between 64 and 68 years old as a function of hours worked by that same individual when he was between 48 and 52 years old; zeros are included in the second period. The Supplemental Web Appendix contains the corresponding picture for women.

hours at age 58, through an intermediate step of 330 hours—these are presumably worked during the first two months of the retirement year, which is year 2000. There are 87 individuals in cohort 23 for whom we can inspect individual hours like the two as just illustrated. These are 87 individuals in the cohort who work every year between 40 and 50 years of age, and who remain in the survey and may move to zero hours afterwards. Of these 87 profiles, 34 look like case A in Fig. 5, 30 look like case B, 12 are non-declining as of cohort age 66 (i.e., when individuals in the cohort are between 64 and 68 years old), and 11 are hard to interpret. This indicates that pre-retirement and abrupt retirement are equally important in the data. The contrast with Rogerson and Wallenius (2013) and Fan (2013) is in fact only apparent. The former restrict the sample to individuals who worked at least 1750 hours after age 56, which may remove workers who are already on a pre-retirement path like the worker in panel A of Fig. 5. The latter looks at labor supply two years before “retirement”, defined as a non-absorbing state, which may misclassify workers who go through non-employment spells as part of their pre-retirement behavior. At any rate, because abrupt retirement accounts for a non-negligible share of cases, one may still worry that the pattern of a smoothly declining aggregate hours profile and non-declining wage profile may be an artifact of extensive margin movements, as the example given at the beginning of this section illustrates. There is a simple way to demonstrate that this is not the case: shut off the extensive margin entirely. This can be done by considering the group of individuals in the cohort who are employed every year since they entered the labor market until the late stages of the working life. These are referred to as the “continuously employed”: for them, there are no extensive margin movements at the annual frequency in the period of observation. Four groups are examined, nested in the 23 years old cohort: those who work every year from 1967 to 2004, 1967 to 2006, 1967 to 2008, and 1967 to 2010. Now numbers are of course tiny in a single cohort: 48, 42, 32, and 23 workers in the four groups, respectively. Despite such tiny sample size, if the shape of the wage and hours profiles of these groups do not diverge substantially from those of the full sample, one has evidence suggesting that the pattern isolated is not an artifact of self-selection out of employment towards the end of the life cycle. Fig. 6 illustrates that this is indeed the case. In particular, workers with an inactive extensive margin are clearly moving along a pre-retirement path beginning shortly after age 50 despite no clear downward trend in wages until after age 60, similar to the pattern in the full sample.

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Table 2 Characteristics by hours reduction at different stages in the PSID, men. Age 58–62 relative to 48–52, characteristics at 58–62

Age 64–68 relative to 48-52, characteristics at 64–68

Variable

Work more

Work less

Don't work

Work more

Work less

Don't work

Health condition

0.095 (0.257)

0.155 (0.334)

0.458 (0.458)

0.067 (0.176)

0.197 (0.355)

0.309 (0.413)

Net financial wealth

409.1 (1092.7)

464.7 (1271.4)

762.3 (1945.5)

267.9 (284.8)

336.5 (408.7)

717.0 (2000.8)

Net total wealth

586.3 (1219.4)

650.0 (1407.1)

922.0 (2000.8)

507.2 (446.7)

537.6 (525.0)

937.0 (2158.8)

Non-labor income

28.0 (37.2)

44.6 (55.4)

87.8 (152.8)

39.5 (37.1)

49.7 (34.5)

88.9 (105.4)

Married

0.826 (0.364)

0.853 (0.334)

0.773 (0.407)

0.878 (0.285)

0.801 (0.389)

0.808 (0.393)

Widowed

0.040 (0.159)

0.018 (0.091)

0.064 (0.219)

0.022 (0.086)

0.088 (0.267)

0.053 (0.214)

Divorced

0.104 (0.308)

0.092 (0.278)

0.164 (0.373)

0.033 (0.129)

0.097 (0.298)

0.140 (0.345)

Separated

0.000 (0.000)

0.007 (0.066)

0.000 (0.000)

0.000 (0.000)

0.000 (0.000)

0.000 (0.000)

Children in household

0.124 (0.431)

0.045 (0.222)

0.076 (0.294)

0.044 (0.172)

0.053 (0.302)

0.047 (0.242)

Individuals

67

101

55

15

72

111

Notes: The table reports (at two different points of the life cycle: age 58–62 and age 64–68) the average characteristics of men born between 1937 and 1946 who were employed at the baseline (age 48–52) in three different groups: those who work more hours relative to the baseline (column “work more”), those who still work but work fewer hours relative to the baseline (column “work less”) and those who report zero hours (column “don't work”). Standard deviations in parentheses. “Health condition” takes value 1 if a person responded “yes” to the question “Do you have any physical or nervous condition that limits the type of work or the amount of work that you can do?”, and 0 otherwise; “Net financial wealth” is total family wealth net of equity (which is included in “Net total wealth”) and debts; “Non-labor income” is total family income from all sources, net of one's own labor income.

2.2.2. Self-selection To check the possible bias arising from selection into employment we estimated selection-corrected wage profiles following the methodology of Wooldridge (1995), an extension of standard two-stage methods valid for cross sectional data such as “Tobit” and “Heckit”. Specifically, predict wages of individuals who are not employed in a given year—but for whom covariates are observed in that year as well as in other periods—using the coefficients of a wage regression consistently estimated. Consistency is achieved as follows. The model consists of a pair of equations, one for the wage rate (a Mincer-type wage equation) and one for selection into employment; the latter takes the Tobit form because selection is determined by a corner solution in the labor supply problem: wit ¼ xit β þ θi þ εit ;

ð1Þ

hit ¼ maxð0; Zi γ þ υit Þ;

ð2Þ

where wit is the wage rate of individual i in year t time-varying covariates (and a constant), θi an individual fixed effect, hit hours, and Zi contains vectors xit at all leads and lags, as well as additional variables excluded from xit , at all leads and lags; in practice, feasibility requires us to replace Zi with the corresponding longitudinal averages. Assume that, conditional on Zi , υit is normally distributed. It is understood that wit is observed if and only if hit 40. Under the assumption that conditional on the unobservables determining selection and the individual fixed effect the error term in the wage equation is mean independent of xit , i.e., E½εit jυit ; θi  does not vary with xit , β can be consistently estimated via a fixed effect regression after including in Eq. (1) the Tobit residuals from Eq. (2) estimated for each period t as additional regressors. We included in xit the following variables: second-order polynomials in age, total family financial wealth, total family equity (i.e., the difference between total wealth and financial wealth), total non-labor income (i.e., total family income net of one's own labor income), reported health problems limiting work, whether married, and number of children below 18 living in the household. As is common when dealing with selection into employment in the US, the excluded variables (i.e., variables in the selection equation that are excluded from the wage equation) are two dummies that capture incentives to reduce labor supply at the social security early and full retirement ages. These are dummies taking value 1 if an individual is over 62 and 65 years of age, respectively, and zero otherwise. The rationale for this restriction is that while these incentives affect labor supply, they are unlikely to have a direct effect on offered wages. This procedure produces a counterfactual wage profile—i.e., the profile observed under the maintained identifying assumptions had those workers who have missing wages

3000

3000

2000

2000

Annual hours

Annual hours

P. Rupert, G. Zanella / Journal of Monetary Economics 72 (2015) 114–130

1000

123

1000

0

0 20

30

40

50

60

70

20

30

40

50

60

70

Fig. 5. Retirement modes: abrupt retirement vs. preretirement path. Notes: The figure shows two individual profiles of male workers who where between 21 and 25 years old in 1967. Worker in panel A moves into retirement through a pre-retirement stage around part-time hours. Worker in panel B moves into retirement by switching abruptly to zero hours. The interpolating line is the best fourth-order fractional polynomial.

accepted their best wage offer.11 The observed and counterfactual wage profiles of men in cohort 23 are reported in Fig. 7. Notice that the correction is possible only for those years when information on total family wealth is available in the PSID (1984, 1989, 1994, and from 1998 onward). This figure suggests that the non-declining wage profile is not an artifact of selection. During the late years of the working life, most notably for low-educated men, the counterfactual wage profile tends to be above the observed one, suggesting “positive selection” into retirement: workers with higher wages leave employment first. This is consistent with the previous comparison between all working men and continuously employed men, Fig. 6. That comparison showed that towards the end of the working life (but not earlier), the wage profile of continuously employed workers is below the wage profile of all workers. This suggests that those who remain continuously attached to the labor market until the late stages of the life cycle have lower wage rates. In turn, this is consistent with the potentially important role of wealth effects in retirement suggested by Table 2. The evidence in Fig. 7 is also consistent with the findings of Casanova (2013), who studies the wage process of older workers using panel data from the Health and Retirement Study. She, too, finds evidence of positive selection into retirement. Our findings in this respect, however, are at odds with French (2005), who finds that selection-corrected wages are 7% and 11% lower at age 62 and 65, respectively, relative to the uncorrected ones. In addition to the key difference already pointed out (we are considering a younger cohort than those considered by French), French uses a very different correction procedure, based on assuming that selection bias in the PSID is the same as in his model, and comparing the wages of workers and non-workers in the model. But we do not claim that the selection-correction procedure employed here is the most appropriate one or provides the right counterfactual wage profile. The limitations of such a procedure are well known. The point, is that even with negative wage growth on the order of 10% at the end of the working life, the observed fall in hours can hardly be replicated in a benchmark life cycle model unless the intensive margin elasticity of labor supply is calibrated to an implausibly large value, as shown next. 3. A benchmark life cycle model This section undertakes a very simple quantitative exercise aimed at understanding how accurately a basic life cycle model explains the empirical pattern. The model is intentionally simple. Not surprisingly, in its basic version it is unable to replicate the pattern. Variants of the benchmark model, among those more frequently considered in the macroeconomic literature, are able to reconcile facts and theory. The representative worker enters the labor market at time 0 and dies at time T. For the moment, assume there is no uncertainty. The problem is to choose sequences of consumption, fct g, labor supply, fht g, and assets, fat þ 1 g, given an exogenous wage profile, fwt g, a sequence of real interest rates, fr t g, and initial and terminal assets, to maximize utility over the life cycle: " # 1 þ 1=ϵ T X ht t max β vðct Þ  γ ð3Þ fct ;ht ;at þ 1 g 1 þ 1ϵ t¼0 s:t:

ct þ at þ 1 rat ð1 þ r t Þ þwt ht

ht rh

ð4Þ ð5Þ

11 Correcting the hours profile is harder with this procedure: one cannot maintain credibly that the unobservable determinants of hours are uncorrelated with the unobservable determinants of the decision to supply a positive amount of hours.

1200 1600 2000 2400

15 20 25 30 35

Hourly wage

Continuously employed 1967-2008

20

30

40 50 Age

60

70

Hourly wage

15 20 25 30 35 20

30

40 50 Age

60

Annual hours per worker

70

Annual hours per worker

60

1200 1600 2000 2400

40 50 Age

1200 1600 2000 2400

30

70

Continuously employed 1967-2010 Hourly wage

20

Continuously employed 1967-2006

15 20 25 30 35

1200 1600 2000 2400

15 20 25 30 35

Hourly wage

Continuously employed 1967-2004

Annual hours per worker

P. Rupert, G. Zanella / Journal of Monetary Economics 72 (2015) 114–130

Annual hours per worker

124

20

30

40 50 Age

60

Wage, continuously employed

Hours, continuously employed

Wage, full cohort 23

Hours, full cohort 23

70

Fig. 6. Wage and hours profiles: cohort 23, men, full sample vs. continuously employed. Notes: The figure reports wage and hours profiles of men in the PSID who where between 21 and 25 years old in 1967 (cohort 23, born between 1942 and 1946), for the full sample and for the sample of those who are employed every year from 1967 to 2004, 1967 to 2006, 1967 to 2008, and 1967 to 2010 (i.e., the “continuously employed”). The interpolating line is the best second-order fractional polynomial for wages and the best fourth-order fractional polynomial for hours.

a0 ¼ a

ð6Þ

aT þ 1 ¼ 0

ð7Þ

where β is the discount factor, vð:Þ a quasi-concave increasing function, ϵ the intertemporal elasticity of substitution of labor (Frisch), and h the time endowment. From the first-order conditions, the dynamics of hours in this model, for t o T, is given by   ht þ 1 1 wt þ 1 ϵ ¼ : ð8Þ ht βð1 þr t þ 1 Þ wt The term 1=βð1 þr t þ 1 Þ is the ratio between the multipliers on the budget constraint in two adjacent periods and captures the income effect: the worker is moving along a known wage profile and is accumulating assets optimally. The term wt þ 1 =wt , on the other hand, captures the substitution effect. If the income effect is not too strong (i.e., βð1 þ r t þ 1 Þ r 1) then hours rise if wages do, or fall if wages fall—i.e., a tracking between hours and wages. In order to look at the life cycle until at least age 70, the focus is on men in the CPS cohort of those born between 1937 and 1941. Eq. (8) is calibrated as follows: β ¼ 0:98; ϵ ¼ 0:314, a value representing the pure intensive margin responses of men in the PSID, and computed using the data and method in Fiorito and Zanella (2012).12 rt is estimated as the difference between the annualized 6-Month Treasury Bill interest rate and the CPI-U inflation rate from 1967 to 2010; finally, h0 is set to the actual data point and hours are simulated forward based on the actual CPS wage profile. As shown in Fig. 8, under this parameterization the model generates a non-declining hours profile that reproduces the data fairly well until the cohort is in its early 50s, and very badly thereafter. In particular, given a non-declining wage profile, the theoretical tracking between hours and wages prevents the model from generating a fall in hours late in the working life.13 A measure of the mismatch between the model and the data is provided by the answer to the following question: by how much should the wage rate fall after age 50 for Eq. (8) to reproduce the observed hours profile? The answer is that relative to age 50 wages should fall, cumulatively, by about 21% by age 60, 56% by age 65, and 71% by age 70, for men with a college degree. For men without a college degree these figures are about the same: 19% by age 60, 55% by age 65, and 66% by age 70. As Fig. 2 shows, for men in the cohort born between 1937 and 1941 with a college degree the cumulative rate of change in 12 13

For ease of exposition, this value is applied to both college graduates and non-graduates. Gomme et al. (2004) also find that a standard life cycle model does not perform well when trying to explain the hours of older workers.

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All men

College 40 Hourly wage

35 Hourly wage

125

30 25 20

35 30 25 20

15 20

30

40

50

60

70

20

30

40

Age

50

60

70

Age

No college

Hourly wage

30 25 20 15 20

30

40

50

60

70

Age

Observed wage

Counterfactual wage

Recession

Fig. 7. Selection-corrected wage profile, men. Notes: The figure reports the observed average wage profile of men in the PSID who where between 21 and 25 years old in 1967 (cohort 23, born between 1942 and 1946), and the corresponding profile after prediction of missing wages through a selectioncorrection procedure. The interpolating line is the best second-order fractional polynomial. The Supplemental Web Appendix contains the corresponding picture for women.

hourly wage between 50 and 70 is essentially zero. For men without a college degree it is negative, about 17%, but the magnitude is far from what the model requires. Clearly, some departure from this benchmark model is needed in order to match the data. Four possible departures are considered below: (1) uncertainty about future wages, (2) borrowing constraints, (3) human capital investment via on-thejob training and (4) time-varying disutility of labor. In all these cases hours can potentially fall even if wages do not.14 3.1. Uncertainty As illustrated by Low (2005), if one introduces wage uncertainty in the benchmark model, then it is possible to have declining hours in spite of an increasing wage profile. That is, it is possible to observe high labor supply early in life when wages are low, and low labor supply later in life when wages are high. Uncertainty breaks the close tracking between hours and wages. The intuition is straightforward: when the wage rate is uncertain, workers are subject to a precautionary effect and delay leisure until the late stages of the life cycle. The point can be illustrated analytically as follows. Without uncertainty, the growth rate of hours, from Eq. (8), is approximately equal to

Δln ht þ 1 ¼  ϵ ln βð1 þ rt þ 1 Þ þ ϵΔln wt þ 1 ;

ð9Þ

where Δln ht þ 1 ¼ ln ht þ 1  ln ht , and similarly for Δln wt þ 1 . If the wage rate is uncertain, then the first-order conditions boil down to " 1=ε # 1=ε h ht ¼ βð1 þr t ÞEt t þ 1 : ð10Þ wt wt þ 1 1=ε

Taking a second-order expansion of ht þ 1 =wt þ 1 around ðht ; wt Þ and rearranging, gives the analog of Eq. (9) under uncertainty: i     1ϵ h   2 Et Δln ht þ 1 ¼  ϵ ln βð1 þ r t þ 1 Þ þ ϵEt Δln wt þ 1  Et Δln ht þ 1  ϵEt Δln w2t þ 1 : ð11Þ 2ϵ The variance of the wage rate and the consequent variance of hours (i.e., the last two terms) counterbalance the effect of positive wage growth on the hours profile. If the variance increases late in the life cycle, then hours may fall even if the wage rate does not. However, it is obvious that such variance should increase substantially. To illustrate, solve Eq. (11) for 14 Incentives embedded in the Social Security system are not considered here. While it is well known that these affect labor supply, they primarily affect the extensive margin. And when they do affect the intensive margin—e.g., by giving incentives to work less hours—they do not do so until age 62.

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Et ½Δln ht þ 1  after replacing the left-hand side with Δln ht þ 1 , and recover the implied series of the variance of the growth rate of hours under the same calibrated parameters.15 Given the variance of wages, such variance in the series for hours growth allows the model to perfectly replicate the CPS hours profile. Quantitatively, this variant of the benchmark model needs the variance of hours growth to increase substantially (by an order of magnitude) shortly after age 50 in order to induce the observed fall in hours. For men with a college degree, the increase is from 0.008 at age 53 to 0.016 at age 60, to 0.040 at age 65, to 0.062 at age 69; for men without a college degree, the corresponding increase is from 0.003 at age 55 to 0.024 at age 60, to 0.035 at age 65, to 0.068 at age 69. The benchmark model with uncertainty, therefore, raises the question of why the variance of hours (or the variance of wages, or both) increase sharply after this age. The simulation implies an implausible degree of non-stationarity late in the life cycle.16 2

3.2. Borrowing constraints Erosa et al. (2014) are able to replicate the fall in hours worked in the PSID by calibrating a model with both uncertainty and a borrowing constraint (a zero lower bound on assets). Workers engage in precautionary savings and work more when young (more than in Low (2005), where there is no borrowing constraint) even if their future wage rate will be higher. Later in the life cycle, the stock of assets is sufficiently large that even small negative wage shocks induce workers not to work in a certain period, although they may re-enter the labor market later. Since the time period in their model is a third of a year, such spells out of employment generate a decline in hours at the annual frequency. This generates a declining wage profile if more and more individuals in a cohort work only a fraction of a year in response to such shocks, as would happen for instance if heterogeneous individuals accumulated assets at different rates. This mechanism, therefore, is consistent with the pre-retirement pattern described above. However, as discussed in Section 2.1, if one looks at individual hours profiles in the PSID, one sees that pre-retirement transitions to fewer hours are actually driven by a substantial portion of individuals working progressively fewer hours towards the end of the working life rather than an increasing number of individuals working at a given, below-full-time level of hours. It is nonetheless of interest to understand whether a borrowing constraint improves the empirical performance of the benchmark model along the intensive margin of labor supply. To isolate the role of such a constraint, we assume that there is no uncertainty. Adding inequality at þ 1 Z 0 to the constraint set and denoting by μt the associated multiplier, the Euler equation becomes

λt ¼ βð1 þ rt þ 1 Þλt þ 1 þ μt ;

ð12Þ

meaning that when the borrowing constraint is binding (μt 40) a worker cannot smooth consumption, so λt 4 βð1 þ rt þ 1 Þλt þ 1 and consumption at time t is lower than it would have been otherwise. The dynamics of hours is now given by " #ϵ 1  μλtt ht þ 1 wt þ 1 ¼ : ð13Þ ht βð1 þ rt þ 1 Þ wt That is, hours grow at a slower (possibly negative) rate when the borrowing constraint is binding, and at the same rate as in the basic model when it is not.17 As in the previous exercise, it is possible to back out from Eq. (13) the series of μt =λt that allows the model to reproduce the data. Not surprisingly, the same difficulty as in the previous case arises with uncertainty and no borrowing constraints. Specifically, the μt =λt ratio is required to increase sharply after age 50: for men with a college degree, from 0.017 at age 53 to 0.082 at age 60, to 0.154 at age 65, to 0.237 at age 69; for men without a college degree, from 0.004 at age 55 to 0.086 at age 60, to 0.117 at age 66, to 0.139 at age 69. Given λt , this increasing value of the μt =λt ratio means that the borrowing constraint becomes more and more binding late in the working life. Given the empirical fact that after older workers dissave, this is an implausible explanation for the observation that hours fall much earlier than wages along the intensive margin. Erosa et al. (2014), in fact, show that their model reproduces the fall in hours per worker worse than the fall in hours per person. 3.3. Human capital A falling hours profile can be generated out of a non-declining wage profile in the benchmark model if there is accumulation of human capital via learning-by-doing or on-the-job training. In this case it is by working that an individual obtains wage growth over the life cycle, and current wages depend on past labor supply. This breaks the tracking between hours and wages. However, the extent of the fall in hours late in the working life depends on how fast the effect of current hours of work on future earnings declines as the worker approaches the end of working life. To illustrate, assume that the Via the formula Vt ½Δln ht þ 1  ¼ Et ½Δln ht þ 1   Et ½Δln ht þ 1 2 . Notice that this conclusion does not depend on the assumption of intratemporal separability between consumption and leisure. Such an assumption, as Low (2005) makes clear, has consequences for the shape of the consumption profile, which is not the focus of this paper. 17 This may sound counterintuitive: an individual for whom the borrowing constraint is active—i.e., for whom the marginal utility of consumption is higher than in the unconstrained case and so an extra hour of work is more valuable—may be expected to work more. However, this is a statement about growth rates, not levels. A borrowing-constrained worker does, in fact, work more hours in the model when the constraint is binding compared to when it is not. As a consequence, the hours growth rate is defined relative to a higher level and so is lower. 15 16

2

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College Annual hours per worker

Annual hours per worker

All men 2200 2000 1800 1600 30

40

50

127

60

2400

2000

1600

70

30

40

50

Age

60

70

Age

Annual hours per worker

No college 2200 2000 1800 1600 30

40

50

60

70

Age

CPS, cohort 1937-1941

Model

Fig. 8. Actual and simulated hours profiles in the CPS, by education, men. Notes: The figure reports the hours profiles of men in a pseudo-panel created from CPS data, by education, and the corresponding profiles generated by Eq. (8). See text for calibration details. The “college” group are workers who have a four-year college degree or a higher level of education; the “no college” group are all of the remaining workers. Shaded areas are NBER recessions. The Supplemental Web Appendix contains the corresponding picture for women.

current wage rate depends on past hours:  t  1 t 1 t1 wt ¼ wt ðh Þ; where h ¼ hj j ¼ 0 : Such a wage process implies that the opportunity cost of time is higher than in the benchmark model, by an amount equal to the present value of the cumulative effect of current labor supply on all future wage rates—see, for instance, Keane 1=ϵ (2011). To see this, notice that now the first-order condition for hours is no longer γ ht ¼ v0 ðct Þwt , but ϵ γ h1= ¼ v0 ðct Þwt þ t

T X

 ∂wj ; ∂ht

βj  t v0 cj hj

j ¼ t þ1

ð14Þ

where the second term on the right-hand side is the present value (in utility terms) of the lifetime increase in earnings brought about by an extra hour of work at time t. That is, the “shadow” component of the wage rate the worker is actually responding to. Then Eq. (8) becomes   ~ t þ1 ϵ ht þ 1 1 w ¼ ; ð15Þ ht βð1 þ rt þ 1 Þ w~ t where w~ t ¼ wt þ

T X j ¼ tþ1

1 j

∏ ð1 þ r k Þ

hj

∂wj ∂ht

ð16Þ

k ¼ tþ1

is the full opportunity cost of time, the sum of the observed current wage rate and the unobserved, shadow component (in consumption terms). Because hours respond to the unobserved profile of the full opportunity cost of time (not the observed ~ t  wt declines fast enough late in the working life, even if the wage profile), hours decline if the shadow component w current wage rate wt does not. One problem with this explanation is that the benchmark model fits the data quite well until age 55–60, as Fig. 8 shows. ~ t , one needs the second term on Therefore, in order to explain the dynamics of the empirical hours profile via responses to w the right-hand side of Eq. (16) to grow at the same rate as the first term (i.e., wt) until age 55–60 (so to fit the data like in Fig. 8 until this age), and to decline sharply thereafter. The first of these two requirements is at odds with a non-declining wage profile. The reason is that in the Ben-Porath (1967) model the net return from on-the-job investment in human capital declines monotonically with age, because that additional human capital becomes less and less valuable as the worker approaches the end of the working life. Therefore, the shadow component of the full opportunity cost of time typically grows at a negative rate, which would worsen the fit before age 55. The second requirement, instead, is at odds with

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evidence on the technology of on-the-job-training. Kuruscu (2006) provides a simple, non-parametric way of estimating the shadow component out of the observed wage profile in the Ben-Porath (1967) model. We replicated Kuruscu's computations in our CPS data set for the cohort of those born between 1937 and 1941 (the youngest cohort possible to follow until age 70), and found, as he did, that the shadow component is convex in age; i.e., it decreases fast early in the life cycle and slowly thereafter. This is the opposite of what the benchmark model needs in order to replicate the observed fall in hours after age 55. In other words, the hours profile in the model with human capital would start falling much earlier than in the data. In order to make some progress, set the problem of fitting the early part of the working life aside for a moment and ask: based on estimates of the returns to on-the-job training that are consistent with the empirical wage profile—i.e., estimates obtained employing the method of Kuruscu (2006)—how well does the human capital variant of the benchmark model replicate the fall in hours after age 50? This question is answered by computing the growth rate of the estimated full ~ t , after age 50, and by comparing this rate with the one that makes the benchmark model opportunity cost of time, w consistent with the observed hours profile. An important caveat in this exercise is that the value of ϵ used so far (0.314) is not the correct elasticity of intertemporal substitution of labor in the human capital variant of the model. This leads to a higher calibrated value of ϵ because, as first illustrated by Imai and Keane (2004), the full opportunity cost of time elasticity is larger than the wage elasticity. Set ϵ ¼ 1:25, consistent with the findings in Imai and Keane (2004) Table 6, that the corrected elasticity is about four times the underlying elasticity with respect to wt. This value is also in the range estimated by Wallenius (2011), 1.1–1.4, as a plausible estimate of ϵ in a model with human capital accumulation.18 Under this new parameterization, the benchmark model fits the data if, relative to age 50 wages fall, cumulatively, by about 5% by age 60, 25% by age 65, and 37% by age 70, for men with a college degree. For men without a college degree these figures are: 4% by age 60, 25% by age 65, and 35% by age 70. The fall in the full opportunity cost of time generated by the model when employing the Kuruscu (2006) method are about 4% by age 60, 5% by age 65, and 22% by age 70, for men with a college degree; and about 18% by age 60, 29% by age 65, and 35% by age 70, for men without a college degree. Therefore, the model with human capital does a remarkably good job at replicating the rate of fall in hours during the very late stages of the working life (65–70 years old) for low educated men. For their highly educated counterparts, the gap between model and data is still sizable, although much smaller than in the other models considered so far. However, a discrepancy between data and model remains with respect to the hours level over the life cycle.19

3.4. Increasing disutility of work/utility of leisure One way to fill the residual gap is to allow the disutility of labor to increase towards the end of the working life. If γ in (3) is not constant, then Eq. (8) becomes   ht þ 1 1 wt þ 1 γ t ϵ ¼ : ht βð1 þr t þ 1 Þ wt γ t þ 1

ð17Þ

Clearly, if γt increases sufficiently fast after age 55 then hours fall even if wages are growing. For example, when analyzing wage and hours profiles in the March CPS between 1977 and 1989 and finding profiles in line with those reported in Fig. 1, Card (1994) comments: “The life-cycle profiles of wages and hours are far from parallel. Of course this does not refute the life-cycle model, because tastes may vary systematically with age” (p. 56). How should the taste parameter, γt, vary with age in this model for Eq. (17) to reproduce the observed hours profile? The answer is that this γt should be roughly constant until about age 55 for men with a college degree and until about age 60 for men without a college degree, as is clear from Fig. 8. Afterward, γt should increase by the following factors, approximately: 1.3 by age 60, 2.3 by age 65, and 3.5 by age 70, for men with a college degree; 1.2 by age 62, 1.8 by age 65, and 2.4 by age 70, for men without a college degree. Such an increase may reflect factors such as health deterioration, stress, or a desire to work less and devote time to other activities. Health, however, explains little of the discrepancy between the actual and simulated profiles, despite the pattern reported in Table 2. A fixed-effects regression (using the PSID sample, that is) of annual hours on a quartic polynomial in age and the “Health condition” dummy (taking value 1 if a person responded “yes” to the question “Do you have any physical or nervous condition that limits the type of work or the amount of work that you can do?”, and 0 otherwise), indicates that deteriorating health (as captured by this dummy) has no effect on men without a college degree, and induces a decline of about 270 annual hours for men with a college degree. Given that, in this latter group, the share reporting a “health condition” increases by 17 percentage points between 55 and 70, this channel accounts for a decline of 270  :17  50 hours in the average profile. This negligible effect is consistent with French (2005), who concludes that “health status alone must have a small causal role in the decline in the number of hours worked by workers near retirement.” (p. 410).

18 We do not employ the much higher value estimated by Imai and Keane (2004), ϵ ¼ 3:82, because they derive such a value under the assumption of a markedly hump-shaped wage profile with peak at age 45, contrary to what is documented in this paper. 19 The tension in matching the early and the late portions of the life cycle in a model in which future wages depend on current labor supply is well illustrated by Wallenius (2011).

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4. Conclusions For individuals in the youngest cohorts that, to date, can be followed for the entire working life (i.e., those born between 1937 and 1946), life-cycle wage profiles show no tendency to decline—if they decline at all—until very late into one's 60s, while hours begin to fall shortly after age 50. This result challenges a common assumption in macroeconomic models; namely, that the wage profile tracks the hours profile closely and so is hump-shaped, with the decline late in the working life that induces a reduction of labor supply along the intensive margin. A benchmark life cycle model is unable to replicate the pattern for younger cohorts. Models and data can be reconciled by departing from such benchmark. The mere presence of uncertainty and borrowing constraints seem unable, by themselves, to do so. However, the life cycle model is able to replicate at least some features of the empirical pattern under two circumstances, in particular. First, the benefits of accumulating human capital via on-the-job training are an important component of the opportunity cost of time and decline relatively fast after age 50—in line with previous empirical research. This mechanism accounts for the negative growth rate of hours of men without a college degree during the very late stages of the working life particularly well. Second, the value of alternatives to work for pay increase fast after that same age. Research about this channel is more scant. While deteriorating health plays a role, it seems unlikely that it plays a decisive role. The possible desire by part of older workers to allocate more time to activities different from work for pay seems a promising direction for future research.

Acknowledgments We are grateful to Eric French, Gueorgui Kambourov, Peter Kuhn, and Richard Rogerson for useful suggestions at the very early stages of this project, to Yongsung Chang and an anonymous referee for detailed comments on the first draft, as well as to seminar participants at the universities of Aarhus, Alicante, Bologna, Chicago Booth, Humboldt-Berlin, IMT Lucca, Leicester, Rome-La Sapienza, Siena, the Stockholm School of Economics, and to participants to the EALE/SOLE 2010 meeting in London, the SED 2010 meeting in Montreal, and the EEA 2010 congress in Glasgow. Bonnie Queen provided outstanding research assistance. All errors are ours. Appendix A. Supplementary data Supplementary data associated with this article can be found in the online version at http://dx.doi.org/10.1016/j.jmoneco. 2015.02.001.

References Aaronson, D., French, E., 2004. The effect of part-time work on wages: evidence from the social security rules. J. Labor Econ. 22 (April (2)), 329–352. Abbasoglu, O., 2012. Life-Cycle Analysis with an Increasing Profile, Working Paper, University of Southern California. Attanasio, O., Hurst, E., Pistaferri, L., March 2014. The evolution of income, consumption, and leisure inequality in the US, 1980–2010. In: Improving the Measurement of Consumer Expenditures. NBER Chapters. National Bureau of Economic Research, Inc. Ben-Porath, Y., 1967. The production of human capital and the life cycle of earnings. J. Polit. Econ. 75, 352–365. Blinder, A.S., Weiss, Y., 1976. Human capital and labor supply: a synthesis. J. Polit. Econ. 84 (June (3)), 449–472. Card, D., 1994. Intertemporal labor supply: an assessment. In: Sims, C. (Ed.), Advances in Econometrics, Sixth World Congress, Cambridge University Press, pp. 49–78. Casanova, M., 2013. Revisiting the Hump-Shaped Wage Profile, mimeo, University of California, Los Angeles. Erosa, A., Fuster, L., Kambourov, G., 2014. Towards a Micro-Founded Theory of Aggregate Labor Supply, Working Paper, IMDEA Social Sciences Institute and University of Toronto. Erosa, A., Fuster, L., Kambourov, G., 2012. Labor supply and government programs: a cross-country analysis. J. Monet. Econ. 59 (1), 84–107. Fan, X., 2013. Retiring Cold Turkey, Working Paper, ARC Centre of Excellence in Population Ageing Research (CEPAR), Australian School of Business, University of New South Wales. Fiorito, R., Zanella, G., 2012. The anatomy of the aggregate labor supply elasticity. Rev. Econ. Dyn. 15, 171–187. Freeman, S., 1977. Wage trends as performance displays productive potential: a model and application to academic early retirement. Bell J. Econ. 8, 419–443. French, E., 2005. The effects of health, wealth, and wages on labour supply and retirement behaviour. Rev. Econ. Stud. 72 (2), 395–427. Ghez, G., Becker, G., 1975. The Allocation of Time and Goods over the Life Cycle. NBER, Cambridge. Gibbons, R., Waldman, M., April 1999. Careers in organizations: theory and evidence. In: Ashenfelter, O., Card, D. (Eds.), Handbook of Labor Economics, vol. 3. Elsevier, pp. 2373–2437. (Ch. 36). Gomme, P., Rogerson, R., Rupert, P., Wright, R., 2004. Home production in a life-cycle model. In: NBER Macroeconomics Annual. MIT Press, Cambridge, MA, and London, pp. 415–592. Hansen, G.D., 1993. The cyclical and secular behaviour of the labour input: comparing efficiency units and hours worked. J. Appl. Econom. 8 (1), 71–80. Harris, M., Holstrom, B., 1982. A theory of wage dynamics. Rev. Econ. Stud. 49, 315–333. Heathcote, J., Perri, F., Violante, G.L., 2010a. Unequal we stand: an empirical analysis of economic inequality in the united states: 1967–2006. Rev. Econ. Dyn. 13 (1), 15–51. Heathcote, J., Storesletten, K., Violante, G.L., 2010b. The macroeconomic implications of rising wage inequality in the united states. J. Polit. Econ. 118 (4), 681–722. Heckman, J.J., 1976. A life-cycle model of earnings, learning, and consumption. J. Polit. Econ. 84 (Part 2) (August (4)), S11–S44. Huggett, M., Ventura, G., Yaron, A., 2011. Sources of lifetime inequality. Am. Econ. Rev. 101 (7), 2923–2954. Imai, S., Keane, M.P., May, 2004. Intertemporal labor supply and human capital accumulation. Int. Econ. Rev. 45 (2), 601–641. Johnson, R.W., Neumark, D., 1996. Wage declines among older men. Rev. Econ. Stat. 78 (4), 740–748.

130

P. Rupert, G. Zanella / Journal of Monetary Economics 72 (2015) 114–130

Keane, M., 2011. Income Taxation in a Life Cycle Model with Human Capital, Working Paper, ARC Centre of Excellence in Population Ageing Research (CEPAR), Australian School of Business, University of New South Wales. King, M., Ruggles, S., Alexander, T., Flood, S., Genadek, K., Schroeder, M., Trampe, B., Vick, R., 2010. Integrated Public Use Microdata Series, Current Population Survey: Version 3.0 [Machine-Readable Database]. University of Minnesota, Minneapolis. Kuruscu, B., 2006. Training and lifetime income. Am. Econ. Rev. 96 (3), 832–846. Lazear, E.P., 1979. Why is there mandatory retirement? J. Polit. Econ. 87 (December (6)), 1261–1284. Lazear, E.P., 1981. Agency, earnings profiles, productivity, and hours restrictions. Am. Econ. Rev. 71 (September (4)), 606–620. Low, H., 2005. Self-insurance in a life-cycle model of labor supply and savings. Rev. Econ. Dyn. 8 (4), 945–975. Medoff, J., Abraham, K., 1980. Experience, performance, and earnings. Q. J. Econ. 95, 703–736. Mincer, J., 1974. Schooling, Experience and Earnings. Columbia University Press. Rogerson, R., Wallenius, J., 2009. Micro and macro elasticities in a life cycle model with taxes. J. Econ. Theory 144 (November (6)), 2277–2292. Rogerson, R., Wallenius, J., 2013. Nonconvexities, retirement, and the elasticity of labor supply. Am. Econ. Rev. 103 (4), 1445–1462. Rosen, S., 1976. A theory of life earnings. J. Polit. Econ. 84 (August (4)), S45–S67. Rubinstein, Y., Weiss, Y., April, 2006. Post schooling wage growth: investment, search and learning. In: Hanushek, E., Welch, F. (Eds.), Handbook of the Economics of Education, vol. 1. Elsevier, pp. 1–67. (Ch. 1). Rupert, P., Zanella, G., 2012. Revisiting Wage, Earnings, and Hours Profiles, Working Paper DSE 936, University of Bologna. Ryder, H.E., Stafford, F.P., Stephan, P.E., 1976. Labor, leisure and training over the life cycle. Int. Econ. Rev. 17 (October (3)), 651–674. Thornton, R., Rodgers, J., Brookshire, M., 1997. On the interpretation of age-earnings profiles. J. Labor Res. 18 (2), 351–365. Wallenius, J., 2011. Human capital accumulation and the intertemporal elasticity of substitution of labor: how large is the bias? Rev. Econ. Dyn. 14 (4), 577–591. Weiss, Y., 1986. The determination of life cycle earnings: a survey. In: Ashenfelter, O., Layard, R. (Eds.), Handbook of Labor Economics, North Holland, Amsterdam. Wooldridge, J.M., 1995. Selection corrections for panel data models under conditional mean independence assumptions. J. Econom. 68 (July (1)), 115–132.