ISSN (Print) 0473-453X

Discussion Paper No. 1063

ISSN (Online) 2435-0982

CROSS-SECTIONALAND
AGGREGATE LABOR SUPPLY

Yongsung Chang
Sun-Bin Kim
Kyooho Kwon
Richard Rogerson

October 2019

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

Cross-Sectional and Aggregate Labor Supply∗
Yongsung Chang
Seoul National University

Sun-Bin Kim
Yonsei University

Kyooho Kwon
Korea Development Institute

Richard Rogerson
Princeton University
NBER

October 9, 2019

Abstract
Standard heterogeneous agent macro models that highlight idiosyncratic productivity
shocks do not generate the near zero cross-sectional correlation between hours andwages found in the data. We ask whether matching this moment matters for businesscycle properties of these models. To do this we explore two extensions of the modelin Chang et al. (2019) that can match this empirical cross-section correlation. Oneof these departs from the assumption of balanced growth preferences. The otherintroduces an idiosyncratic shock to the opportunity cost of market work that is highly
correlated with the shock to market productivity. While both extensions can matchthe empirical correlation, they have large and opposing effects on the cyclical volatilityof the labor market. We conclude that the cross-sectional moment is important forbusiness cycle analysis and that more work is needed to distinguish the potential
mechanisms that can generate it.
Keywords: Hours, Employment, Cross-section, Business Cycles, Comparative Advantage∗
We thank Hyun-Tae Kim for excellent research assistance.
This work is supported by
grants from the National Research Foundation of Korea funded by the Korean Government (NRF2016S1A5A2A03926178). Part of this work has been accomplished while Sun-Bin Kim was visiting theInstitute of Social and Economic Research at Osaka University, who is grateful for their hospitality.
Correspondence: Chang: yohg@snu.ac.kr; Kim: sunbin.kim@yonsei.ac.kr; Kwon: kwonkh98@gmail.com;
Rogerson: rdr@princeton.edu.

11. Introduction
The modern business cycle literature emphasizes the need to construct models that containan explicit description of agents’ objective functions and the constraints that they face,
as well as a notion of equilibrium. One of the benefits of following this approach was thepromise that such models could be connected to micro data to provide discipline regarding
functional forms and parameter values. The early models in this literature, such as theseminal work of Kydland and Prescott (1982), assumed a representative household, andthus precluded any possibility to connect the model with cross-sectional micro data. But
in the last two decades, advances in computational methods have allowed researchers
to develop aggregate business cycle models that feature heterogeneous households, thus
allowing researchers to study aggregate phenomena while simultaneously connecting withrich micro data sets. In this paper we take advantage of these advances to address asomewhat long standing tension between “ micro” and “ macro” models of labor supply
related to the relative magnitude of income and substitution effects.
Dating back at least to the work of Lucas and Rapping (1969), macroeconomists haveargued that aggregate time series data on hours of work and real wages suggest offsetting
income and substitution effects. In particular, it is argued that because average hours per
individual have displayed little or no trend at the same time that average real wages haveincreased by several multiples, that the real wage must have no effect on desired labor
supply, i.e., that income and substitution effects must offset each other.
Because this condition–offsetting income and substitution effects–also allows one tofind a balanced growth path equilibrium in the one sector growth model, the macro literature typically refers to preferences with this property as “ balanced growth preferences”.1
1

A recent paper by Boppart and Krusell (2016) argues that hours have been trending down and that
one can have a balanced growth path with this property if income effects dominate. Ohanian et al. (2008)
argue that most of the decline in hours can be attributed to changes in tax and transfer policies without
departing from the assumption of offsetting income and substitution effects.

2Preferences with this property are a standard feature of modern business cycle analyses. At the same time, labor supply analyses based on micro cross-sectional data often
conclude that income and substitution effects are not perfectly offsetting, instead finding
evidence for specifications in which income effects dominate substitution effects. See forexample, the analyses in Pijoan-Mas (2006) and Heathcote et al. (2014).
In this paper we revisit this tension in a state of the art heterogeneous agent businesscycle model. The starting point for our analysis is the recent model of Chang et al.
(2019). This paper extends the heterogeneous agent-indivisible labor models of Chang andKim (2006, 2007) to allow for active adjustment along both the intensive and extensive
margins, implying that in equilibrium there are non-degenerate distributions for both
hours worked and wages for workers in the cross-section. As is common in much of theheterogeneous agent macro literature, Chang et al. (2019) assumes a single dimension ofexogenous heterogeneity, namely idiosyncratic productivity shocks. However, assumingbalanced growth preferences in this environment leads to the prediction of a large positive
correlation between hours and wages in the cross-section. This correlation is at odds withthe data–which reveals a correlation near zero–and is intimately related to the tension
between “ micro” and “ macro” studies of labor supply noted above. As we discussin detail below, one common practice to achieve a near zero correlation between wages
and hours in this setting is to adopt preferences under which income effects dominates
substitution effects. This is the route taken by both Pijoan-Mas (2006) and Heathcote etal. (2014).
It is perhaps not surprising that a model in which idiosyncratic productivity shocks
are the sole source of randomness generates a cross-sectional correlation between wages
and hours that is at odds with the data. For this reason we extend the model of Chang etal. to include an additional source of randomness–idiosyncratic shocks to the opportunity
cost to market work, which we model as a shock to preferences. This shock could proxy3

for either a shock to the value of leisure or to home productivity. Although we cannot
measure this shock directly, we infer the size of this shock by calibrating it to match thecross-sectional dispersion in hours worked. Adding a source of idiosyncratic randomness
that is uncorrelated with productivity does serve to reduce the cross-sectional correlation
between wages and hours worked, but it remains large and positive relative to the data.
Starting from this benchmark with two uncorrelated sources of idiosyncratic shocks wedevelop two extensions that can generate a cross-sectional correlation between hours andwages that is near zero. The first extension maintains a zero correlation between the two
idiosyncratic shocks but departs from balanced growth preferences. The second extension
maintains balanced growth preferences but allows the two idiosyncratic shocks to becorrelated. We show that if the two shocks are sufficiently highly positively correlated
one can generate a cross-sectional correlation near zero. Intuitively, these results suggestthat there is a continuum of specifications that can generate a near zero cross-sectionalcorrelation between wages and hours, with there being a trade-off between a more positive
correlation between the two shocks and the extent to which the income effect dominates
the substitution effect.
Having isolated specifications that are consistent with the cross-sectional relationshipbetween wages and hours, we next ask whether matching this moment is important forbusiness cycle properties. To answer this question we expose the benchmark model as well
as both extensions just described to the same stochastic process for aggregate productivity shocks and compare the implied business cycle properties across the three models.
Both extensions imply first order effects for the magnitude of labor market fluctuations,
implying that matching the cross-sectional correlation between wages and hours is indeed
important. But most importantly, the two extensions imply affect the magnitude of labormarket fluctuations in opposite directions: whereas maintaining balanced growth preferences but imposing a high positive correlation between the two shocks implies significantly4larger fluctuations, maintaining a zero correlation between the shocks and departing frombalanced growth preferences results in significantly smaller fluctuations. Comparing thetwo extensions, we find that labor market fluctuations are more than twice as large in thecase with balanced growth preferences and correlated shocks.
The correlation of the two shocks offers a simple economic interpretation regarding theelasticity of aggregate labor supply. A high correlation between the two shocks (productivities in the market and non-market activities) leads to a weak cross-sectional comparativeadvantage in market work relative to non-market work, whereas a negative correlation
between the two shocks represents an economy with a strong comparative advantage.
The economy with strong comparative advantage exhibits much less movement of labor
between the market and home.
As noted above, our paper is closely related to several earlier papers, including Chang
and Kim (2006, 2007), Pijoan-Mas (2006), Heathcote et al. (2014), Erosa et al. (2016)
and Chang et al. (2019). Here we note the key differences not already noted. Both
Pijoan-Mas (2006) and Heathcote et al. (2014) assume that all hours adjustment occurs
along the intensive margin, and neither studies business cycle fluctuations. Pijoan-Mas(2006) features a single idiosyncratic shock. While Heathcote et al. (2014) allow for
two idiosyncratic shocks, they only study the case in which they are assumed to be
uncorrelated. Like us, Erosa et al. (2016) study a model that features labor adjustment
along the intensive and extensive margin as well as with multiple sources of idiosyncratic
shocks. Their model is richer than ours along several dimensions but they do not focus on
the cross-sectional correlation between wages and hours and its significance for aggregate
fluctuations.
An outline of the paper follows. In Section 2, we develop our benchmark model.
We start by introducing the model from Chang et al (2019) where the cross-sectional
heterogeneity solely arises from the realization of idiosyncractic productivity shocks. We
5refer to this as a one-shock model. A calibrated version of this model implies a large
positive cross-sectional correlation between wages and hours. We then extend the Changet al. (2019) model to include a shock that is orthogonal to productivity and that affects
hours. We refer to this as a two-shock model and identify it as our benchmark model.
In Section 3, we extend the benchmark model in two different directions to match thecross-sectional correlation between hours and wage in the data. Section 4 compares thebusiness-cycle statistics generated from the benchmark model as well as both extensions
from Section 3. Section 5 concludes.

2. Aggregate Labor Supply in a Heterogeneous Agent Model
We develop a benchmark model for simultaneously studying labor supply at both theaggregate level and in the cross-section. We proceed in two steps. We begin by describing
the one-shock model of Chang et al. (2019), which is a generalization of the standard
heterogeneous agent-incomplete markets model that has become one of the benchmark
models in macroeconomics. Notably, this model has a single source of exogenous randomness at the individual level, namely productivity shocks. Having introduced this model wenote its inability to account for the cross-sectional correlation between wages and hours
of work found in the data. This failure motivates us to extend the model to allow for asecond idiosyncratic shock that influences hours of work and is orthogonal to productivity(two-shock model in Section 3). While this extension generates a cross-sectional correlation between wages and hours that is closer to that found in the data, the gap between
model and data remains very substantial.

62.1. One-Shock Model: Chang et al. (2019)
There is a unit measure of ex-ante identical infinitely lived individuals. Each individual
has preferences over streams of consumption (cit ) and hours of work (hit ) given by:
∞
X

1+1/γ

β t [log(cit ) − B

t=0

hit
]
1 + 1/γ

where 0 < β < 1, B > 0 and γ > 0. Each individual is endowed with a unit of time ineach period.
Individuals are subject to idiosyncratic productivity shocks, denoted by zit . The
stochastic evolution of zit is described by the same transition probability distributionfunction for all individuals, but realizations are iid across individuals. In particular, we
will assume that zit follows an AR(1) process in logs:
ln zit+1 = ρz ln zit + εzit , εzit ∼ N (0, σz2 ).
More generally, one might want to consider a specification which features both permanent and multiple random components with varying persistence, as well as a deterministic
time varying components. Our more parsimonious specification is attractive in part for
its tractability, but it is also able to capture elements of such a more general specification. If ρz is large, implying a large amount of persistence, then in a short panel someof the cross-sectional variation would be identified as permanent fixed effects. The meanreversion process would imply a certain amount of deterministic dynamics. While we
could feasibly consider more complicated processes, we follow much of the literature inchoosing this simple specification as we feel it facilitates transparency.
In order to generate adjustment along both intensive and extensive margins we assume
a nonconvexity in the mapping from time devoted to work to the resulting efficiency unitsof labor. We view this non-convexity as capturing factors such as set-up costs, supervisory
time and/or the need to coordinate with other workers. If an individual with idiosyncratic
7

productivity zit devotes hit units of time to market work, this will generate zit g(hit ) unitsof labor services. Following Prescott et al. (2009) and Rogerson and Wallenius (2009),
we assume that g(·) takes the following simple form:2nog(ht ) = max 0, ht − ĥ , ht ∈ [0, 1].
where 0 < ĥ < 1.
There is an aggregate Cobb-Douglas production function that produces output using
inputs of labor services (Lt ) and capital services (Kt ):Yt = Lαt Kt1−α .
Output can be used for either consumption or investment, and capital depreciates at rateδ each period.
We follow Bewley (1986), Huggett (1993), and Aiyagari (1994) in assuming that markets are incomplete. Specifically, we assume that there are no markets for insurance andthe only asset is physical capital. Individuals trade claims to physical capital, denoted bya, but trade in these claims is subject to an exogenous borrowing constraint that limits
the amount of debt that an individual can acquire:at ≥ aIn each period t there is a market for units of labor services, with price wt , and a rental
market for capital services, with price rt + δ, so that rt is the rate of return to capital.
When a worker of productivity zit devotes hit units of time to market work, the resulting
labor earnings are wt zit g(hit ). The period budget equation for an individual is:

cit + ait+1 = wt zit g(hit ) + (1 + rt )ait
2
French (2005) considers an alternative and smoother specification in which the wage per unit of timeis smoothly increasing in the number of hours worked.

8

In this section we focus on a steady state equilibrium, in which the two prices wt andrt are constant, as is the distribution of individual state variables (ait , zit ).

2.2. Calibration
In this subsection we describe a simple calibration procedure that can be used to assign
all of the model’s parameters. As is standard in the business cycle literature, we set atime period equal to one quarter. Our calibration procedure will not pin down a value
for the preference parameter γ; instead this parameter will be set up front and then thecalibration procedure will identify all of the other parameters given this choice.
A sizable literature has estimated processes for idiosyncratic wage shocks using annual
data, typically for males, and we use estimates from this literature to parameterize ouridiosyncratic productivity shock process. Papers from this literature include Card (1994),
Floden and Linde (2001), French (2005), Chang and Kim (2006), and Heathcote et al.
(2008). While there is some variation across studies, the consensus is that these shocks
are large and persistent. Guided by these empirical studies, we set ρz = 0.975 andσz = 0.165.3 The Cobb-Douglas parameter α and the depreciation rate are set to standard
values in the literature: α is set to 0.64 and δ is set to .025, effectively targeting labor’s
share of income and the investment to output ratio. Note that none of these values are
influenced by the choice of γ.
There are four additional parameters to calibrate: β, B, a, and ĥ. We set a = 0
independently of γ. Given a value for γ and our previous choices, we choose the values of
the remaining three parameters to match three moments in the steady state equilibrium:a (quarterly) rate of return to capital of 1%, an employment rate of 70%, and average
3

The specific values that we adopt correspond to the estimates in Floden and Linde (2001). Because
these values are based on on annual data, we need to convert them to quarterly values for our calibration.
We choose values for the quarterly process such that when we estimate an annual process it delivers theappropriate values for persistence and the standard deviation of the innovations.

9hours (conditional on working) equal to 1/3.4
The steady state properties of the above model were analyzed extensively in Changet al. (2019), so we refer the reader there for a more detailed analysis. For each of
several values of γ, they examined the implications for the steady state distributions ofhours worked, earnings and wealth, and the transition of individuals both between nonemployment and employment and within the annual hours worked distribution. The main
message was that for a wide range of values of γ, going from 0.25 to 1.5, the calibrated
model did a reasonable job of accounting for the patterns in the data.
For concreteness and future reference, we reproduce the results from Chang et al.
(2019) for the case in which γ = 1. Table 1 shows the full set of calibrated parameters
and Table 2 shows the ability of the model to account for the cross-sectional dispersion
in hours, earnings and wealth.5
Table 1: Calibration for One-Shock Model
α
0.36

β
0.97656

γ
1.00

δ
0.25

B
18.9

ρz
0.975

σz
0.165

ĥ
0.151

Table 2: Cross-Sectional Dispersion Measures
Model
0.32
0.59
0.71

S.D. of Annual Hours
Earnings Gini
Wealth Gini

Data
0.45 (CPS)
0.63 (SCF)
0.78 (SCF)

Note that this one-shock model in which productivity shocks are calibrated to thedata does not generate sufficient cross-sectional dispersion in hours worked relative to thedata. We will return to this issue in the next section when we introduce an additional
4With a quarterly employment rate of 70%, the average annual employment rate in our model (i.e.,fraction of individuals who work at least one quarter during a year) is 76.7%. This corresponds to theaverage annual employment rate in the PSID over the period 1968-1998 for household heads and spouses
with ages between 18 and 65. We use a cutoff of 240 annual hours as the threshold for employment, i.e.,
we treat individuals with less than 240 annual hours worked as not employed.
5
When we compute the standard deviation of hours in the model we normalize the level of hours tomatch the mean level of annual hours in the data.

10

shock.6 The earnings Gini in the model is modestly smaller than in the data; additional
dispersion in hours would potentially address this problem as well. While this modelcannot generate the extreme concentration of wealth among the top 1% that we see inthe data, it nonetheless is able to capture a large amount of the wealth dispersion as
measured by the Gini coefficient. As noted earlier, Chang et al. (2019) reports on theability of the model to account for the data along several other dimensions. For example,
it does a reasonable job of capturing the flow of workers between employment and nonemployment, as well as within the hours worked distribution.

2.3. Hours and Wages in the Cross-Section
Although the calibrated model does a reasonable job of capturing many salient featuresof the cross-sectional data, there is one dimension along which it performs poorly: thecross-sectional correlation between hours worked and wages. In the calibrated economy
with γ = 1 this value is 0.78, but in the data this value is around zero. Controlling forage and education, we estimate that this correlation is equal to −0.04 using data fromthe PSID and equal to 0 using data from the SIPP.7
In considering this correlation in the data it is important to be aware of the potential
effects of measurement error, since measurement error will tend to bias the correlation
to zero. Heathcote et al. (2014) estimate the amount of measurement error in earnings
and hours as part of their exercise and find that it does not create a sufficiently large
negative bias to reconcile the model with the data. Given the raw correlations reported
above, the presence of measurement error suggests that the “ true” correlation is modestly
positive. Given the model implied value of 0.78 for this correlation, it seems of second
6

Chang et al. (2019) shows that one can match the standard deviation of hours worked in the datawith a one-shock model by some combination of increasing the standard deviation of the shock innovations
and increasing the magnitude of γ.
7

Pijoan-Mas (2006) and Heathcote et al. (2014) also report values for this correlation that are close to
zero.

11

order importance to consider the issue of whether the true correlation is 0, 0.1 or 0.2. For
concreteness, in what follows we will take 0 as the target value for this correlation, since
this represents the most challenging case.
Although the value for the cross-sectional correlation between wages and hours justreported was for the specific case of γ = 1, we note that the model implied correlationreflects a very robust mechanism. With so-called balanced growth preferences there isno intratemporal effect of wages on hours of work controlling for wealth. But, transitory
fluctuations in individual productivity induce intertemporal substitution effects that generate a positive correlation between wages and hours controlling for wealth. So in fact,
the strong positive relation between wages and hours in the cross-section is not specific toany particular value of γ. For example, if we instead chose γ = 0.5 or γ = 1.5 the impliedvalues for this correlation would be 0.82 and 0.76 respectively.

2.4. A Two-Shock Benchmark Model
The model of Chang et al. (2019) assumed that market productivity was the only sourceof exogenous heterogeneity across individuals. It is natural to think that allowing for an
additional source of randomness that influences hours and is orthogonal to productivity
could lead to substantial improvement in terms of the model’s ability to account forthe cross-section correlation between hours and wages. In this subsection we extend theprevious model to include a second dimension of exogenous cross-sectional heterogeneity,
namely heterogeneity in the opportunity cost of market work.8
The only change relative to the previous model is that we will now write preferencesas:8
Heathcote et al. (2014) also study a two shock model. While their model is richer than ours alongsome dimensions (e.g., they model life cycle choices), all of the labor supply adjustment in their model
occurs along the intensive margin. Erosa et al. (2016) consider a life cycle model with multiple shocks
and adjustment along the intensive and extensive model.

12

∞
X

1+1/γ

β t [log(cit ) − Bxit

t=0

hit
]
1 + 1/γ

where all parameters are as before, and xit is an idiosyncratic preference shock affecting
the disutility of working. We assume that this shock follows the same AR(1) process inlogs for all individuals, with realizations iid across individuals:

log xit+1 = ρx log xit + εxit+1 , εxit ∼ N (0, σx2 ).
This shock will capture the role of non-wage factors in generating cross-sectional dispersion
in hours of work. These non-wage factors could reflect heterogeneous preferences forleisure, or more generally heterogeneous value for time spent in home production. At thispoint we will impose that the two shocks are uncorrelated, and will refer to this economy
with two uncorrelated idiosyncratic shocks as our benchmark model.

2.5. The Role of Comparative Advantage
It is useful to develop some intuition to better understand the results that will be presented
later in the paper. In particular, consider an infinitely lived individual with period utility
function:log(ct ) −

1
1+1/γ
xt ht
.
1 + 1/γ

Suppose that the discount factor is β and period labor income is equal to wht zt . Assume
that w is constant over time, that the individual can borrow and lend at an interest rateequal to (1/β)−1, and that the values of x and z vary over time in a deterministic fashion.
Suppose also that there is a utility cost associated with participation in each period andthat this cost is represented as ūxt , so that the cost is scaled up by the preference shock
xt .
13

This utility maximization problem is not identical to our model. But it has some of
the key features from our model–a nonconvexity that generates adjustment along both theintensive and extensive margins, and two sources of idiosyncratic variation–and permits
an analytic characterization of the optimal decision rules for labor supply. For this reason
it is useful in providing some valuable intuition about how the two sources of idiosyncratic
variation affect labor supply decisions.
Deriving first order conditions for the individual’s optimal choices of consumption,
hours and participation over time, straightforward manipulation yields that consumption
will be constant over time, that the optimal choice for ht conditional on working satisfies:ztw
ht = ( )γ ( )γ
c
xt

(1)

and the decision about whether to work is determined by a reservation value for the ratio
zt /xt .
A key property is that the ratio zt /xt serves as a sufficient statistic to characterize labor
supply decisions along both the intensive and extensive margin. The ratio zt /xt can beinterpreted as a measure of comparative advantage in market work. In contrast, absoluteadvantage in market work is captured by zt , given that the wage per efficiency unit w isassumed to be constant. Importantly, this model says that labor supply decisions along
both margins (intensive and extensive) are driven by comparative advantage in market
work rather than absolute advantage in market work. In the special case in which zt isalways just a positive multiple of xt , the value of zt /xt is constant and hours of work will
be constant despite changes in the observed wage.
A key insight from the indivisible labor literature (see, for example, Chang and Kim(2006)) is that the responsiveness of the aggregate employment rate depends upon thedensity of individuals in the vicinity of the reservation wage. Loosely speaking, the moredispersion there is in the relevant individual state variables, the less responsive is the14extensive margin. The fact that zt /xt is a sufficient statistic for the extensive margin
implies that the extent of dispersion in zt /xt is key to determining responsiveness along
the extensive margin. Importantly, if zt and xt move in the same direction then thedispersion in zt /xt will tend to be less than the dispersion in zt alone. Conversely, if xt
tends to move in the opposite direction of zt then the dispersion in zt /xt will tend to belarger than the dispersion in zt alone.

2.6. Calibration of the Two-Shock Model
Much of the calibration proceeds as before, but there are now two new parameters tobe determined: ρx and σx . In what follows we will set the persistence parameter for theidiosyncratic preference shock equal to the same value that we used for the idiosyncratic
productivity shock: ρx = 0.975.9 This leaves us with one additional parameter that needs
to be calibrated, namely σx . To accomplish this we add one additional moment to ourlist of moments to be matched, as we now require that the model generates the samedispersion in annual hours worked as found in the data. Importantly, although it is notpossible to directly measure the size of the second shock, we can discipline the size of thisshock by requiring that the model generates the appropriate dispersion in hours of work.
Table 3 shows the calibrated parameter values for this model. In what follows we willrefer to this as our benchmark model.
Table 3: Calibration for Two-Shock Model (Benchmark)
α
0.36

β
0.97678

γ
1.00

δ
0.25

B
20.6

ρz
0.975

9

σz
0.165

ρx
0.975

σx
0.103

ĥ
0.133

There is nothing special about the case in which ρz = ρx . We choose a highly persistent process basedon the fact that empirical studies of micro data tend to find evidence for fixed effects, and in a short panel
our idiosyncratic shock would at least partially be captured by fixed effects.

15

2.7. Steady State Properties of the Benchmark Model
The model with two dimensions of exogenous heterogeneity looks very similar to thebenchmark model along the dimensions previously reported in Table 2, though by construction it now achieves the same dispersion of hours worked as found in the data. The
two-shock model also performs quite similarly to the one shock model along the otherdimensions that Chang et al. (2019) documented. In particular, the transition of individuals between employment and non-employment, as well as within the hours worked
distribution capture some key patterns from the data, as Table 4 below shows.
Table 4: Annual Hours Transition
PSID

t

Not Work
1st
2nd
3rd
4th
5th

Not Work
83.57
21.08
4.77
2.81
2.26
1.81

1st
12.25
49.45
15.40
6.75
5.14
3.41

t+1
2nd
3rd
1.69
0.91
14.91
6.15
45.77 18.27
19.77 46.24
10.63 19.91
4.69
6.80

4th
0.99
5.29
11.15
17.88
42.42
19.77

5th
0.60
3.12
4.63
6.54
19.64
63.52

4th
0.81
5.61
7.88
24.68
39.95
20.49

5th
0.14
1.72
2.39
6.09
24.48
66.25

Model

t

Not Work
1st
2nd
3rd
4th
5th

Not Work
72.49
21.12
8.46
3.24
1.69
0.49

1st
17.64
34.43
20.54
11.03
6.99
3.57

t+1
2nd
3rd
6.53
2.39
24.53 12.60
37.54 23.19
17.67 37.28
7.57 19.31
3.83
5.37

Notes: Transition matrix of annual hours worked by quintile groups and non-employment(‘Not Work’).

Although the model captures the key patterns found in the empirical transition matrix,
we note that overall, it does not exhibit as much persistence as in the data–relative to16the data the model has too many transitions from non-employment to employment, andtoo many transitions out of each hours quintile.10
But, most importantly for our purposes is the implication of the extended model forthe cross-sectional correlation between wages and hours of work. This value is now equal
to 0.53. Although the effect of adding an idiosyncratic shock orthogonal to individual
productivity has the intuitive effect of decreasing this correlation relative to the modelwith only idiosyncratic productivity shocks, the quantitative magnitude of this effect is
not sufficiently large to allow the model to match the data. We note again that this would
remain true if we adopted a modestly positive correlation as the true target. In the next
section we explore two extensions to this benchmark model that can help us to match
this empirical correlation.

3. Two Extensions
In this section we generalize the benchmark model along two dimensions. First, we
generalize the utility function to now read:

∞
X
t=0

1+1/γ

βt[

h
c1−σ
it
− Bxit it].
1−σ
1 + 1/γ

Our benchmark model corresponds to the limiting case in which σ tends to one. Given
this class of preferences, this limiting case is the only specification that is consistent withbalanced growth, i.e., that delivers perfectly offsetting income and substitution effects.
If σ is greater than one, then the income effect dominates the substitution effect, but if
σ is less than one, then the substitution effect dominates the income effect.
The second generalization is to allow the two idiosyncratic shocks, zit and xit , to becorrelated. In particular, we assume that the two innovations εzt and εxt are correlated
10
Allowing for greater persistence of the two shocks would presumably improve the fit of the model
along this dimension.

17

and denote the value of this correlation as ρzx . In this section we show that each ofthese departures individually can reconcile the model-implied cross-sectional correlation
between wages and hours with the value from the data.

3.1. Extension I: Departure from Balanced Growth Preferences
In this subsection we ask whether maintaining ρzx = 0 as in the benchmark model, we
can find a value of σ such that the model will match all of the previous targets as wellas generate a cross-sectional correlation between wages and hours of approximately zero.
Pijoan-Mas (2006) and Heathcote (2014) pursue this same strategy for targeting the crosssectional correlation, though Pijoan-Mas (2006) had only productivity shocks and both
papers assumed that all hours adjustment occurred along the intensive margin. Assuming
γ = 1 as previously, it turns out that setting σ = 2 will achieve this in our model economy.
This is slightly larger than the 1.45 value adopted by Pijoan-Mas (2006) or the 1.78 value
adopted by Heathcote et al. (2014) to achieve the same goal–a near zero correlation.11
Table 5 shows the full set of calibrated parameters.
Table 5: Calibration of Extension I Model
α
0.36

β
0.96946

γ
1.00

δ
0.25

B
77.0

ρz
0.975

σz
0.165

ρx
0.975

σx
0.144

ĥ
0.144

σ
2

This result that σ needs to be greater than one is qualitatively intuitive. We observed
previously that when ρzx = 0, our two-shock model still generates a significantly positive
correlation between hours and wages in the cross-section. When we deviate from balanced
growth preferences, the effect on the cross-sectional correlation between wages and hours
depends on the value of σ. If σ is less than one then substitution effects dominate income11

As noted earlier, there are various other differences between our paper and these two which makecould potentially impact the value of σ. For example, Heathcote et al (2014) study a life cycle modelwith insurable and uninsurable shocks. Pijoan-Mas (2006) has a different borrowing constraint but only
productivity shocks. And both of these papers assume that all labor supply adjustment occurs along theintensive margin.

18

effects, thereby generating a positive correlation. But, if σ is greater than one, income
effects dominate substitution effects and the effect on the correlation is negative.
In addition to hitting all of the targeted moments, this specification performs about
the same as the benchmark model in terms of the other moments we have considered.
(See Appendix A2).

3.2. Extension II: Departure from Uncorrelated Shocks
In this subsection we ask whether maintaining balanced growth preferences as in thebenchmark model, we can find a value of ρzx such that the model will match all of theprevious targets and also generate a cross-sectional correlation between wages and hoursof roughly zero. Assuming γ = 1 as previously, it turns out that setting ρzx = 0.9 willachieve this. Table 6 shows the full set of calibrated parameters.
Table 6: Calibration of Extension II Model
α
0.36

β
0.9814

γ
1.00

δ
0.25

B
26.2

ρz
0.975

σz
0.165

ρx
0.975

σx
0.147

ĥ
0.133

ρzx
0.9

This result is also qualitatively intuitive. To see why recall equation (1). This equation
illustrates that a positive correlation between z and x will serve to dampen the positiveeffect of wages on hours of work. Put somewhat differently, in our simple example it iscomparative advantage that matters for market hours, and a high positive correlation
between the two shocks tends to decrease the correlation between market wages andcomparative advantage. Conversely, if the two shocks are negatively correlated then thecorrelation between market wages and comparative advantage in market work tends tobe very high.

19

3.3. Discussion
The two previous subsections demonstrate two different extensions of the benchmark
model that are able to generate a near zero correlation between wages and hours in the
cross-section while still accounting for the cross-sectional dispersion in hours and earnings.
Here we draw two generalizations that follow.
First, we introduced two new parameters relative to the benchmark model and showedthat each of them individually could be adjusted to as to hit the desired targets. It
necessarily follows that one could find a continuum of parameterizations in which the two
new parameters are varied together such that all of the desired targets are hit. In thenext section we will continue to focus on the two specifications in which only one of theparameters is varied in order to highlight the effects of the two distinct channels.
Second, although we have chosen to target a cross-sectional correlation of zero between
wages and hours, it should be clear that targeting a modest positive value for this correlation would simply require a modestly smaller adjustment for the two parameters being
varied. For example, we note that for Extension II, reducing the targeted correlation
to 0.34 would reduce the required value of ρzx from 0.9 to 0.5. Table 7 provides information on the relationship between the correlation of the two shocks and the resulting
cross-sectional correlation between wages and hours. 12
Table 7: ρzx and corr(w, h)ρzxcorr(w, h)

−0.90
0.78

−0.50
0.66

12

0
0.53

0.50
0.34

0.90
−0.004See Appendix Tables A1 and A2 for the calibrated parameters and other cross-sectional momentsfrom the models with various ρzx .

20

4. Implications for Business Cycles
In this section we examine the extent to which the cross-sectional correlation between
wages and hours is an important moment from the perspective of business cycle fluctuations, and if so, if it matters how one chooses to match this moment. That is, we examine
the extent to which our three models (Benchmark, Extension I, and Extension II) differ
in their implications for business cycles.
We pursue this by considering how all three models behave when confronted withthe same process for aggregate productivity shocks. That is, we now assume that theaggregate production function is:

Yt = Zt Ktα Lt1−αwhere Zt is the aggregate productivity shock and is assumed to follow an AR(1) process
in logs:log Zt+1 = ρZ log Zt + εZt+1
and εZt is assumed to be log normally distributed with mean zero and standard deviation
σZ . We adopt commonly used parameter values for the technology shock process, ρZ =0.95 and σZ = 0.007.
A recursive representation of equilibrium implies that the aggregate state variable
will include the measure of individuals over idiosyncratic shocks and asset holdings. To
solve for the equilibrium with such a high dimensional aggregate state variable we employ Krusell and Smith’s (1998) “ bounded rationality” method, which approximates the
distribution of workers over individual states by a limited number of its moments. In
particular, we assume that agents make use of the average asset holdings of the economy

21

as well as the aggregate technology shock Z.13 We generate 3,000 quarterly periods fora model economy. After dropping the first 1,000 observations, we take logs and apply a
Hodrick-Prescott filter (with smoothing parameter 1, 600 to be comparable to those fromthe data) to produce the business cycle statistics.
Table 8 displays the results for several business cycle statistics of interest. Apart fromaggregate output we focus entirely on labor market statistics for the simple reason thatthe models all imply similar properties for the behavior of consumption and investment
relative to output. We also include results from the Chang et al. (2019) specification that
we described earlier, in which the idiosyncratic productivity shock is the sole source ofidiosyncratic randomness.
Table 8: Business Cycle Statistics
σY

σH

σH
σY

σE

σh

σL

σw

Data (BLS)

2.01

1.80

0.89

1.51

0.48

−

0.98

Chang et al. (2019)
Benchmark (σ = 1, ρzx = 0)
Extension I (σ = 2, ρzx = 0)
Extension II (σ = 1, ρzx = 0.9)

1.74
1.65
1.46
1.92

0.79
0.72
0.39
0.93

0.45
0.44
0.28
0.48

0.69
0.58
0.44
0.95

0.12
0.17
0.09
0.11

1.09
0.93
0.72
1.47

0.83
0.85
0.66
0.84

Notation in this table is as follows: Y denotes aggregate output, H denotes aggregate
hours, E denotes aggregate employment, h denotes hours per worker, L denotes aggregate
efficiency units of labor, and w denotes average wage per hour worked. There are several
messages to point out. First, we note that that introducing a second shock into theframework of Chang et al. (2019) has a modest but not insignificant effect on the extent oflabor market fluctuations. Fluctuations in both aggregate hours and aggregate efficiency
units are reduced by around ten percent relative to the one shock model of Chang et al.
(2019). This is consistent with the intuition that we described previously. Adding an
additional shock that is uncorrelated with the individual productivity shock effectively
13
Note that because we have a continuum of agents, the joint distribution of idiosyncratic shocks (z, x)
is necessarily stationary, which is why this distribution does not enter the aggregate state vector.

22

increases the amount of heterogeneity in the cross-section and as a result leads to fewer
marginal individuals. This results in smaller fluctuations along the extensive margin, andsmaller fluctuations in the aggregate, though the effect of smaller fluctuations along theextensive margin is at least partially offset by greater fluctuations along the intensive
margin.
The second message from Table 8 is that modifying the benchmark model so as toaccount for the cross-sectional correlation between hours and wages has first order effects
on the resulting business cycle statistics. When we move from the benchmark model toExtension I there is a substantial decrease in the size of labor market fluctuations. In
this case the decrease occurs along both the intensive and the extensive margin. The
intuition for this effect is straightforward: with σ = 2 the income effect dominates thesubstitution effect in terms of intratemporal effects of wages on labor supply. As a result,the intertemporal substitution effects are now dampened, and there is less response oflabor supply to changes in wages. This effect operates on both the intensive and extensive
margins.
Next consider moving from the benchmark model to Extension II. In this case weobserve a first order increase in the magnitude of fluctuations in labor market variables.
For example, the standard deviation of fluctuations in total hours increases from 0.72 to
0.93 and the standard deviation of efficiency units of labor increases from 0.93 to 1.47.
These increases are accounted for by greater fluctuations along the extensive margin, as
fluctuations along the intensive margin actually decrease somewhat.
The intuition for why we observe increased fluctuations in this case is straightforwardand draws on our earlier heuristic analysis of the two shock model that suggested it is
the ratio of the two shocks that matters. Fluctuations along the extensive margin aredetermined by the mass of individuals who are close to indifference between working
and not working. Our earlier analysis argued that when the two shocks are negatively
23

correlated, the ratio of the two shocks has much larger variance than the z shock alone.
And if the two shocks are positively correlated the effective shock has smaller dispersion
than the z shock alone.
To better illustrate this intuition, Table 9 below repeats the business cycle analysis
for several alternative values of ρzx .
Table 9: Effect of ρzx on Business Cycle Statistics

ρzx = −0.9
ρzx = −0.5
ρzx = 0 (Benchmark)ρzx = 0.5
ρzx = 0.9

σY

σH

σH
σY

σE

σh

σL

σw

1.59
1.62
1.65
1.72
1.92

0.66
0.68
0.72
0.77
0.93

0.41
0.42
0.44
0.45
0.48

0.47
0.51
0.58
0.68
0.95

0.22
0.19
0.17
0.14
0.11

0.83
0.87
0.93
1.04
1.47

0.87
0.86
0.85
0.83
0.84

Consistent with the intuition offered, we see that overall fluctuations are monotonicallyincreasing as we move from ρzx = −0.9 to ρzx = 0.9. Note however, that fluctuations along
the intensive margin are actually decreasing at the same time that overall fluctuations are
increasing. The reason for this is that the intensive and extensive margins are substitutes
from the perspective of adjusting overall labor input. When there are lots of marginal
workers, i.e., a large mass of workers who are close to indifferent being working and
not working, then it is relatively easy to adjust total labor input by adjusting along theextensive margin. This reduces the need to make adjustment along the intensive margin.
Returing to Table 8, the third, and we think most dramatic message is that these twodifferent ways of reconciling the benchmark model with the cross-sectional evidence lead todramatically different implications for business cycle fluctuations. If we set the correlationof the two idiosyncratic shocks to zero and depart from balanced growth preferences, theresulting fluctuations in labor market variables are less than half as large as they are in
the case where we maintain balanced growth preferences but allow the two idiosyncratic
shocks to be highly positively correlated.

24

5. Conclusion
Recent advances in modeling aggregate labor supply allow us to now study business cycles
in models that also have a rich set of implications for cross-sectional relationships. In thispaper we leverage these advances in order to address a tension between the implicationsof standard macro models that impose balanced growth preferences and the near zerocross-sectional correlation between wages and hours found in the data. Commonly usedheterogeneous agent macro models imply a large positive value for this correlation.
We ask whether the failure to match this correlation is substantively important for
inferring implied business cycle properties of these models. Our answer to this question
is affirmative. But more importantly, we show that what really matters is the mechanism
through which one achieves a realistic cross-sectional correlation between hours and wages.
We consider two different mechanisms, and find that they have large and opposite effects
on the magnitude of labor market fluctuations that result from a given aggregate shock.
A key parameter that we highlight and that has received little attention in the existing
literature is the correlation between idiosyncratic variation in market productivity and theopportunity cost of time. We think it is important for future work to pay more attention tothis correlation–which reflects the cross-sectional comparative advantage between market
and non-market activities–given its apparent importance for the properties of businesscycles. We note one piece of recent work that speaks to this correlation. Boerma andKarabarbounis (2018) argue that market and home productivity are highly positively
correlated.

25

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28

Appendix: Additional Statistics
Table A1: Calibration of Extension II Model
ρzx

β

B

ĥ

σx

-0.9
-0.5
0.5
0.9

0.975370.976
0.97818
0.9814

19.0
19.2
22.6
26.2

0.128
0.130
0.135
0.139

0.0825
0.09
0.124
0.147

Table A2: Wealth and Earnings
Gini Coefficient: Wealth
PSID=0.76, SCF=0.78
σ = 2, ρzx = 0
ρzx =

0.63

-0.9
0.63

-0.5
0.64

0.0
0.65

0.5
0.67

0.9
0.68

0.5
0.57

0.9
0.53

Gini Coefficient: Earnings
PSID=0.53, SCF=0.63
σ = 2, ρzx = 0
ρzx =

0.53

-0.9
0.60

-0.5
0.59

0.0
0.59

Wealth Share by Quintile
I

II

III

IV

V

PSID
SCF

-0.52
-0.39

0.50
1.74

5.06
5.72

18.74
13.43

76.22
79.49

σ = 2, ρzx = 0

0.07

2.19

9.80

24.77

63.17

ρzx = −0.9
ρzx = 0.0
ρzx = 0.9

0.16
0.08
0.04

2.66
2.10
1.16

9.51
8.75
7.54

23.80
23.20
22.51

63.88
65.87
68.76

Earnings Share by Wealth Quintile
I

II

III

IV

V

PSID
SCF

7.51
7.05

11.31
14.50

18.72
16.48

24.21
20.76

38.23
41.21

σ = 2, ρzx = 0

10.05

14.97

18.79

23.38

32.81

ρzx = −0.9
ρzx = 0.0
ρzx = 0.9

5.56
6.26
10.27

11.14
11.95
13.09

16.91
17.06
18.65

24.87
24.43
23.21

41.52
40.31
34.78

29