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Working Paper Series
Labor Supply Shifts and Economic
Fluctuations
WP 03-07
Yongsung Chang
Federal Reserve Bank of Richmond
Frank Schorfheide
University of Pennsylvania
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Labor Supply Shifts and Economic Fluctuations *
Federal Reserve Bank of Richmond Working Paper 03-07
July 2003
Yongsung Chang
Research Department, Federal Reserve Bank of Richmond
and
Frank Schorfheide
Department of Economics, University of Pennsylvania
Abstract:
We propose a new VAR identification scheme that distinguishes shifts of and movements
along the labor demand schedule to identify labor-supply shocks. According to our VAR
analysis of post-war U.S. data, labor-supply shifts account for about 30 percent of the
variation in hours and about 15 percent of the output fluctuations at business cycle
frequencies. To assess the role of labor-supply shifts in a more structural framework,
estimates from a dynamic general equilibrium model with stochastic variation in home
production technology are compared to those from the VAR.
JEL Classification: E32, C52, J22
Key Words: Labor-Supply Shifts, VAR, Home Production, Bayesian Econometrics
* Marco Airaudo provided excellent research assistance. We wish to thank Larry Christiano, Frank
Diebold, Martin Eichenbaum, John Geweke, Michael Kiley, Richard Rogerson, and Chris Sims for helpful
comments and suggestions. Thanks also to seminar participants at the NBER Summer Institute, University
of Pennsylvania, Princeton, Rochester, ISBA Regional Meeting, USC, Econometric Society Meetings, the
Federal Reserve Bank of Cleveland, and the Board of Governors. The second author gratefully
acknowledges financial support from the University of Pennsylvania Research Foundation. The GAUSS
programs to implement the empirical analysis are available at http://www.econ.upenn.edu\~schorf. The
views expressed herein are those of the authors and do not necessarily reflect those of the Federal Reserve
Bank of Richmond or the Federal Reserve System. Chang: Yongsung.Chang@rich.frb.org; Schorfheide:
schorf@econ.upenn.edu.
2
1
Introduction
A leading question in macroeconomics is the identification of forces that cause the
cyclical allocation of time. Modern dynamic general equilibrium analysis empha
sizes shifts in labor demand due to technological change. Empirical studies on the
decomposition of sources of business cycles by Shapiro and Watson (1988) and Hall
(1997) have called for an attention to labor-supply shifts. This paper examines the
importance of labor-supply shifts as a source of economic fluctuations.
First, we develop and apply a new identification procedure for vector autore
gressions (VAR). It decomposes the fluctuations of aggregate hours and output into
movements along the labor demand schedule and shifts of the schedule itself. The
former is interpreted broadly as response to a labor-supply shock. Our VAR iden
tification is based on the notion that an increase in hours due to a labor-supply
shock leads to a fall in labor productivity, as the production capacity is fixed in the
short-run and the economy operates along the decreasing marginal-product-of-labor
schedule. We place a prior distribution on the slope of the short-run labor-demand
curve and on the reduced-form VAR parameters and conduct Bayesian inference.
Second, we impose additional restrictions by estimating a fully-specified dynamic
stochastic general equilibrium model (DSGE). The DSGE model potentially yields a
more precise estimate of the relative importance of labor supply shifts. We consider
an aggregate home production model (Benhabib, Rogerson, and Wright, 1991, and
Greenwood and Hercowitz, 1991)) in which labor-supply shifts are caused by the
stochastic variation in home production technology.
3
The main empirical findings can be summarized as follows. Based on the VAR
variance decomposition, temporary shifts in labor supply are an important source
of hours fluctuations. They account for about 30 percent of the cyclical variation of
hours worked. The DSGE model attributes more than 50 percent of the variation of
hours to temporary labor-supply shifts. This larger estimate, however, may partly
be due to misspecified over-identifying restrictions as the time series fit of the DSGE
model is significantly worse than the VAR fit. According to both VAR and DSGE
models, labor-supply shocks are less important for aggregate output as they explain
only about 15 percent of its variation at business cycle frequencies.
Our estimates of the contribution of labor-supply shifts to economic fluctuaΩ
tions are somewhat smaller than those reported by Shapiro and Watson (1988) and
Hall (1997). Shapiro and Watson (1988) identify labor-supply shocks through the
stochastic trend in hours worked. While the empirical evidence on the stationarity
of hours worked is not conclusive, we assume hours are stationary, which is conΩ
sistent with a large class of dynamic equilibrium models. In Hall (1997), Parkin
(1988), and Baxter and King (1991), labor-supply shocks (or preference shocks) are
identified as deviations from the optimality condition associated with the labor supply of competitive households. However, these residuals also reflect the extent to
which a representative-agent model is inconsistent with aggregate hours and wages,
potentially leading to a bigger estimate of labor-supply shifts. While we exploit the
labor-market equilibrium as Hall, our identification scheme does not rely on a speΩ
cific form of households’ preferences. Instead, it is based on the firms’ production
technology through the slope of the marginal-product-of-labor schedule.
4
The paper is organized as follows. Section 2 develops the VAR identification
scheme. The home production model is presented in Section 3. Section 4 discusses
the econometric estimation and inference. The empirical findings are summarized
in Section 5 and Section 6 provides a conclusion.
2
A VAR Model of Labor Market Fluctuations
In this section we describe our identification scheme for the VAR using a labor
demand and supply framework. The labor-market fluctuations are viewed as a se
ries of equilibria generated by competitive households and firms whose tastes and
technologies are perturbed by stochastic disturbances. To identify the sources of
fluctuations we will fit a VAR and a DSGE model (specifically, an aggregate home
production model) to three macroeconomic time series: hours worked, labor pro
ductivity, and expenditure on consumer durable goods. Expenditure on consumer
durables serves as a proxy for the household’s permanent income. In the context of
the home production model, it represents the investment in home capital.
2.1
Identifying Assumptions
During the past four decades, labor productivity, spending on consumer durables,
and aggregate output exhibited a pronounced trend, whereas aggregate hours and
the consumption share did not show an apparent trend. Based on this observation,
many dynamic macroeconomic models have been designed to evolve along a balanced
growth path. A common stochastic trend in output, consumption, investment, cap-
5
ital, and labor productivity is induced by a labor augmenting technology and hours
worked are stationary around this path. The VAR innovations are decomposed into
three orthogonal shocks, denoted by �a,t , �b,t , and �z,t .
Assumption 1 The shock �z,t has a permanent effect on labor productivity and
consumer durables whereas it has no effect on hours in the long run. The shocks �a,t
and �b,t have transitory effects on hours, labor productivity, and consumer durables.�
The shock �z,t induces a common stochastic trend, and it is subsequently interpreted
as permanent technology shock in a DSGE model.
At time t, the competitive firms’ inverse labor demand can be written as:
Wt = M P Lt = ϕD (Lm,t |Km,t , St ),
(1)
where Wt represents the real wage rate, M P Lt the marginal product of labor, Lm,t
hours employed, Km,t capital stock, and St state of technology at time t.1 Similarly,
the inverse labor supply of a competitive household is:
Wt = ϕS (Lm,t |Ω(St , Tt )),
(2)
where Ωt represents endogenous variables that influence the labor supply (e.g., real
interest rate, consumption, and wealth) and Tt the exogenous stochastic component
of tastes.
1
Subscripts m and h, respectively, denote the market and home sector, consistent with the DSGE
model introduced in Section 3.
6
Assumption 2 The shock �b,t has a contemporaneous impact on Tt , but not on St
and Km,t . Thus, upon impact this shock shifts the labor-supply curve, but not the
labor-demand curve (marginal-product-of-labor schedule). �
We will interpret the shock �b,t broadly as a labor-supply shock, such as an
unanticipated shift of the preference for leisure or the productivity of non-market
activities. The capital stock Km,t is inherited from the previous period and therefore not affected by the current period’s labor-supply shock. Given that produc
tion capacity is fixed in the short-run, firms operate along the downward sloping
marginal-product-of-labor schedule.
The responses of the marginal product of labor and hours worked (both in logs)
to a labor-supply shock �b,t have to satisfy the following relationship
∂ ln M P Lt
=
∂�b,t
�
1
∂ϕD
·
ϕD ∂ ln Lm,t
�
·
∂ ln Lm,t
.
∂�b,t
(3)
The factor in parentheses is the slope of the inverse labor demand function. For
example, under a Cobb-Douglas production technology with labor share α one ob
tains:
∂ ln Lm,t
∂ ln Pt
= (α − 1)
,
∂�b,t
∂�b,t
(4)
where Pt = M P Lt is the labor productivity. Roughly speaking, conditional on
the slope of the labor demand function, it is possible to identify �b,t through its
joint effect on hours and productivity. The slope of the labor demand schedule
itself, however, is imposed and not identifiable. Finally, �a,t , namely a temporary
labor demand shock, is identified by assuming that the three structural shocks are
7
orthogonal.2 .
2.2
VAR Specification
Define the vector of stationary variables Δyt = [Δ ln Pt , Δ ln Ih,t , ln Lm,t ]� . Moreover,
let �t = [�z,t , �a,t , �b,t ]� . The VAR can be expressed in vector error correction form as
Δyt = Φ0 + Φvec yt−1 +
p−1
�
Φi Δyt−i + ut ,
ut ∼ iid N (0, Σu ).
(5)
i=1
The reduced form disturbances ut are related to the structural disturbances �t by
ut = Φ∗ �˜t , where �˜t is a standardized version of �t with unit variance.
According to Assumption 1, the shock �z,t generates a stochastic trend in pro
ductivity and expenditures on consumer durables. The two series are cointegrated
with cointegration vector λ = [1, −λ21 , 0]� . Instead of restricting λ21 to one we esti
mate the parameter in the VAR analysis to allow for a possibly steeper Engle curve
for expenditures on durable goods. We do not impose a cointegration relationship
�
between the cumulative hours of work ( tτ =0 Lm,t ) and labor productivity or con
sumer durables. Hence, the rank of Φvec = µλ� is one, which is in fact confirmed by
a formal selection based on Bayesian posterior odds. The stochastic trend of yt has
2
Our analysis does not consider other disturbances such as monetary and fiscal policy shocks.
For post-war U.S. data, government policy shocks are often considered to be of secondary impor
tance in business-cycle analysis. For example, according to King, Plosser, Stock and Watson (1991),
permanent nominal shocks account for only a small variation of real variables. The cyclical compo
nents of government spending is not highly correlated with output measures – it is less than 0.2 for
Hodrick-Prescott filtered data. Also, expanding the list of shocks often invites arbitrary identifying
restrictions in the VAR analysis.
8
the form CLR Φ∗
�t
˜t .
τ =0 �
Since productivity and consumption expenditures have a
common trend, the first two rows of the 3 × 3 long-run multiplier matrix CLR are
proportional.
The structural shocks �˜t are identifiable if the elements of the 3 × 3 matrix Φ∗
can be uniquely determined based on Φ0 , . . . , Φp , Φvec , and Σu . Let Ψ∗ denote the
unique lower triangular Cholesky factor of Σu . Any matrix Φ∗ such that Φ∗ Φ�∗ = Σu
is an orthonormal transformation of Ψ∗ , that is, Φ∗ = Ψ∗ B for some orthonormal
matrix B. Let [A]ij denote the i’th row and j’th column of a matrix A. According to
Assumption 1, the shocks �a,t and �b,t only have transitory effects on productivity and
consumer expenditures. Thus, the elements [(CLR Ψ∗ )B]12 and [(CLR Ψ∗ )B]13 have
to be zero. The contemporary effects of the labor-supply shock �b,t on productivity
and hours worked are given by ∂Pt /∂�b,t = [Φ∗ ]13 and ∂Lm,t /∂�b,t = [Φ∗ ]33 . Define
C∗ = [1, 0, −(α − 1)]. According to Assumption 2 and Equation (4) the value of
[(C∗ Ψ∗ )B]13 has to be zero. These three orthogonality conditions uniquely determine
the orthonormal transformation B.
3
A Fully Specified Model Economy
The DSGE model provides a more specific interpretation of the three structural
shocks and their propagation. It also assists the understanding of the economic
intuition behind our identification scheme. The model economy consists of identical,
infinitely lived households who maximize the expected discounted lifetime utility
9
defined over consumption Ct and pure leisure
IE t
�∞
�
�
β s−t (log Cs + κ log(1 − Lm,s − Lh,s )) ,
s=t
where Lm,t is the fraction of time supplied to the labor market and Lh,t is hours
spent on home production. IE t is the time t conditional expectation and β is the
discount factor. Consumption is an aggregate of market consumption Cm,t and the
consumption of home produced goods Ch,t :
υ−1
υ−1
υ
υ
C(Cm,t , Ch,t ) = [χCm,t
+ (1 − χ)Ch,tυ ] υ−1 ,
(6)
The substitution elasticity υ captures the households’ willingness to substitute mar
ket and home-produced goods. The home production technology exhibits constantreturns-to-scale in home capital Kh,t and labor Lh,t :
Ch,t = [ψ(Xh,t Lh,t )
τ −1
τ
τ −1
τ
+ (1 − ψ)Kh,tτ ] τ −1 ,
(7)
where Xh,t is a labor-augmenting home productivity process. We do not restrict
the home technology to a Cobb-Douglas function (unlike the market technology
below) because some consumer durable goods (e.g., dishwashers and microwaves)
are substitutes for time, whereas others (e.g., DVD players) are complements. Our
specification reduces to a separable-in-logs utility if ν = τ = 1. The households own
Kh,t and the market capital stock Km,t ; and their budget constraint is:
Cm,t + Im,t + Ih,t = Wt Lm,t + Rt Km,t ,
(8)
where Rt is the rental rate of market capital and Im,t and Ih,t are capital invest
ments. To avoid unreasonably volatile investments in a multi-sector model, capital
10
accumulation is subject to a convex adjustment cost as in Baxter (1996):
Kj,t+1 = φ(Ij,t /Kj,t )Kj,t + (1 − δ)Kj,t ,
j = h, m
(9)
where δ is the depreciation rate of capital and φ� > 0, φ�� ≤ 0.
The representative firm produces the market output Yt according to a CobbDouglas technology in market capital and labor and maximizes the profit each pe
riod:
max
Lm,t ,Km,t
1−α
Km,t
(Xm,t Lm,t )α − Wt Lm,t − Rt Km,t ,
(10)
where Xm,t represents a labor-augmenting market productivity process. The goods
market equilibrium condition is:
Yt = Cm,t + Im,t + Ih,t .
(11)
Market and home productivity are, respectively, Xm,t = exp[zt + at ] and Xh,t =
exp[zt + bt ], where zt represents a common trend that follows a random walk with
drift:
zt = γ + zt−1 + �z,t .
(12)
The temporary components, at and bt , follow stationary first-order autoregressions:
at = ρa at−1 + �a,t ,
bt = ρb bt−1 + �b,t .
(13)
We assume that �t = [�z,t , �a,t , �b,t ]� is serially uncorrelated with diagonal covariance
matrix Σ� . The diagonal elements are denoted by σz2 , σa2 , and σb2 , respectively.
As the VAR in Section 2, the log-linearized DSGE model provides a probabilistic
representation for Δyt = [Δ ln Pt , Δ ln Ih,t , Lm,t ]� . The model economy also satisfies
11
all the identifying assumption of the VAR described above. Moreover, the DSGE
model imposes additional restrictions which potentially yield more precise estimates
of variance decompositions and impulse responses.
4
Econometric Approach
The goal of the econometric analysis is to assess the relative importance of laborsupply shocks for the cyclical variation of output and hours.3 The VAR is denoted
by M0 and the over-identified log-linearized DSGE model by M1 . To be consistent
with the Cobb-Douglas production technology used in the DSGE model, we will
assume that under the VAR specification the slope of the inverse labor demand
function is also α − 1. Hence, the parameter α appears in both M0 and M1 . The
parameters of model Mi , except for α, are stacked in the vector θ(i) , i = 0, 1. θ(0)
contains the cointegration parameter λ12 and the non-redundant elements of the
reduced-form matrices Φ0 , . . . , Φp , Σu in Equation (5).
Variance decompositions and truncated impulse response functions, denoted by
the m × 1 vector ϕ, are transformations of the parameters θ(i) and α, that is,
ϕ = ϕ˜i (θ(i) , α). Under both M0 and M1 the vector process Δyt has a movingaverage (MA) representation in terms of the standardized structural shocks �˜t :
Δyt = µ(θ(i) , α) +
∞
�
Cj (θ(i) , α)�˜t−j .
(14)
j=0
3
A Technical Appendix that summarizes the computational details is available from authors
upon request.
12
The population mean µ and the moving average coefficients Cj are model-specific
functions of θ(i) and α. Define the vectors Mz = [1, 0, 0]� , Ma = [0, 1, 0]� , and
Mb = [0, 0, 1]� . The impulse responses to the shock �˜s,t are given by Cj Ms .
The h-th order autocovariance matrix of Δyt can be decomposed into the con
tributions of the three structural shocks:
�
ΓΔy (h) =
s∈{z,a,b}
∞
�
�
(s)
ΓΔy (h) =
Cj Mk Mk� Cj� +h .
(15)
s∈{z,a,b} j=max{0,−h}
The relative contribution of shock s to the unconditional variance of the j’th element
(s)
of Δyt is given by the ratio [ΓΔy (0)]jj /[ΓΔy (0)]jj . The spectrum of the stationary
process Δyt is
SΔy (ω) =
�
(s)
SΔy (ω)
=
�
∞
�
(s)
ΓΔy (h)e−ihω
(16)
s∈{z,a,b} h=−∞
s∈{z,a,b}
and represents the contribution of frequency ω to the variance of Δyt . To assess
the relative importance of the three shocks at business cycle frequencies we consider
the decomposition of
�ω
ω
SΔy (ω)dω, where ω and ω correspond to cycles of 32 and
6 quarters, respectively.4
The likelihood functions for the two models are denoted by p(YT |θ(i) , α, Mi ).
4
According to M0 and M1 the level of output is integrated of order one and its autocovariances
(s)
do not exist. Let SΔy (ω) denote the three components of the spectrum of output growth. We define
the spectrum of the level of output at frequencies ω > 0 as
(s)
Sy(s) (ω) = lim
φ→1
SΔy (ω)
.
1+
− 2φcos(ω)
φ2
(17)
The term 1/[1 + φ2 − 2φcos(ω)] is the power transfer function of the AR(1) filter [1 − φL]−1 , where
L denotes the temporal lag operator.
13
We adopt a Bayesian approach and place a prior distribution with density
p(θ(i) , α|Mi ) = p(θ(i) |Mi )p(α),
i = 0, 1
(18)
on the parameters. Equation (18) reflects the assumption that α is a priori inde
pendent of θ(0) and θ(1) . Moreover, the prior distribution of α is the same for both
models. According to Bayes Theorem the posterior density of the parameters is
proportional (∝) to
p(θ(i) , α|YT , Mi ) ∝ p(YT |θ(i) , α, Mi )p(θ(i) |Mi )p(α).
(19)
The likelihood function of the VAR is uninformative about the slope of the
inverse demand schedule α − 1 and depends only on the reduced form parameters
θ(0) :
p(YT |θ(0) , α, M0 ) = p̃(YT |θ(0) , M0 ).
(20)
Straightforward manipulations using Bayes Theorem can be used to verify that the
VAR posterior is the product of the posterior density of the identifiable reduced-form
parameters obtained from p̃(YT |θ(0) , M0 ) and the prior density of α:5
p(θ(0) , α|YT , M0 ) = p̃(θ(0) |YT , M0 )p(α).
(21)
Since ϕ = ϕ˜i (θ(i) , α), Equations (18) and (19) implicitly determine the prior and
posterior of variance decompositions and impulse response functions. Rather than
attempting to specify a prior on ϕ directly, as in Gordon and Boccanfuso (2001), we
5
Our VAR based inference is a specific example of Bayesian analysis of a nonidentified econo
metric model. Poirier (1998) provides a comprehensive survey and many additional examples.
14
use economic intuition derived from assumptions on aggregate preferences, produc
tion technologies, and equilibrium relationships to specify the prior for ϕ indirectly.
Since the distribution of reduced-form parameters θ(0) is updated based on the sam
ple information YT , the implied distribution of ϕ is updated with every observation
and we learn about the relative importance of structural shocks and the response of
the economy.
Our method is explicit about the direction of the parameter space in which
learning does not occur. If the dimension of the nonidentifiable component of the
parameter vector is low, as in our application, we can assess the robustness of our
conclusion by tracing out, for instance, the relative importance of the labor-supply
shock as a function of α. A similar approach was used by King and Watson (1992)
who plotted their statistics of interest against a one-dimensional variable indexing
VAR identification schemes.
The VAR identification proposed in this paper is based on the notion that pro
ductivity and hours worked move in opposite directions in response to a labor-supply
shock. Equation (4) can be qualitatively interpreted as an inequality restriction on
the impulse responses:
∂ ln Pt
> 0 and
∂�b,t
∂ ln Lm,t
<0
∂�b,t
(22)
Canova and DeNicolo (2002), Faust (1998), and Uhlig (2003) develop identification
and inference procedures based on such inequality constraints. Our approach places
a prior distribution on identification schemes that are consistent with (22) and av
erages the posterior distribution of population characteristics ϕ over a priori likely
15
values of the unidentifiable parameter α that indexes the identification schemes.
5
Empirical Analysis
Both VAR and DSGE models are fitted to post-war quarterly U.S. data on labor
productivity, expenditure on consumer durables, and hours worked.6 The sample
period ranges from 1964:I to 1997:IV. The overall sample size is T = 136 and the first
T∗ = 20 observations are used as training sample to initialize lags and parameterize
the prior distributions.
5.1
Priors
The prior distribution used in the estimation of the DSGE model is summarized
in columns 3 to 5 of Table 1. The shapes of the densities are chosen to match the
domain of the structural parameters. The prior means for labor share, discount
rate, productivity growth, capital depreciation, and the steady state ratio of home
to market investment are respectively α
¯ = 0.666, β¯ = 0.993, γ¯ = 0.004, δ¯ = 0.025,
and I¯h /I¯m = 0.7. These values can be justified based on the training sample and
6
Real gross domestic product (GDPQ), consumption of consumer durables (GCDQ), employed
civilian labor force (LHEM), civilian noninstitutional population 20 years and older (PM20 and
PF20) are extracted from the DRI·WEFA database. Define P OP Q = 1E6 ∗ (P F 20 + P M 20),
Yt = GDP Q/P OP Q and Ih,t = GCDQ/P OP Q. Average weekly hours, private non-agricultural
establishments (EEU00500005) are obtained from the Bureau of Labor Statistics. Annual hours
worked at monthly frequency are Lm,t = 52 ∗ EEU 00500005 ∗ LHEM / P OP Q and converted to
quarterly frequency by simple averaging. Labor productivity is Pt = Yt /Lm,t .
16
are commonly used in the literature. Hence, we use fairly small standard deviations
for the distributions of these parameters. Prior means for the steady state hours,
¯ m = 0.33 and L
¯ h = 0.25, are obtained from the Michigan Time Use Survey. A
L
larger standard deviation is allowed for Lh , as the hours spent on home work may
be measured with a greater uncertainty.
We allow for large standard deviations in the prior distributions of home tech
nology parameters because they are not easy to infer. The prior means for the
substitution elasticities, ν̄ = 1 and τ̄ = 1, correspond to a one-sector model with
separable-in-logs utility. The prior mean of the labor share ψ in the home produc
tion function is also set to 0.666. The weight on leisure χ in the utility function is
implicitly determined by the other parameters. The steady state adjustment costs
are assumed to be zero and the elasticity of the investment/capital ratio with re
spect to Tobin’s q, η = (|(I ∗ /K ∗ )φ�� /φ� |−1 ) is estimated. The prior mean for η is
100, implying a small adjustment cost, with a large standard deviation of 100. We
use diffuse priors for the exogenous technology processes at , bt , and zt . Finally, we
introduce two additional parameters ξ1 and ξ2 to adjust the normalization of total
hours to one and to capture the average growth rate differential between labor pro
ductivity and home investment in the data. The structural parameters are assumed
to be a priori independent of each other.
The training-sample is used to construct a conjugate prior for the VAR parame
ters conditional on the cointegration parameter λ21 . While the DSGE model implies
that λ21 = 1, we relax that restriction and choose the prior λ21 ∼ N (1, 0.0252 ). The
prior for α is the same as in the DSGE model analysis (Table 1). Posterior odds
17
were used to select the lag-length p = 2.
5.2
Parameter Estimation and Time Series Fit
The posterior means and standard errors of the parameters of the DSGE model
are reported in columns 6 and 7 of Table 1.7 The estimated substitution elastic
ity between market goods and home goods, υ, is 2.302, slightly higher than those
of Rupert, Rogerson, and Wright (1995) and McGrattan, Rogerson, and Wright
(1997). The substitution elasticity between capital and labor in home production,
τ , is 2.446 suggesting that goods and time are substitutes in home production activ
ity. The estimated labor share ψ in the home technology is 0.753 and the fraction
of hours spent on home production activity Lh is 0.170. The temporary home pro
duction shock is somewhat more persistent than the market shock: ρ̂a = 0.745 and
ρ̂b = 0.865. The 90 percent posterior confidence interval for the correlation (con
ditional on time t − 1 information) between market productivity ln Xm,t and home
productivity ln Xh,t ranges from 0.18 to 0.37, somewhat lower than the values used
in the literature (e.g., 0.67 in Benhabib, Rogerson and Wright, 1991 and 1 in Greenwood and Hercowitz, 1991). Finally, the adjustment cost parameter estimate η̂ is
30.70, implying a small adjustment cost in capital accumulation.
To assess the relative time series fit of the VAR and the DSGE model we compute
7
While McGrattan, Rogerson, and Wright (1997) also estimate home production models based
on aggregate time series, our analysis distinguishes itself from theirs in several dimensions. First,
we focus on the comparison to a structural VAR, particularly, the variance decomposition. Second,
microeconomic evidence is incorporated through prior distributions in our Bayesian estimation.
Third, we are able to uncover the comovement of innovations to market and home productivity.
18
marginal data densities
�
p(YT |Mi ) =
p(YT |θ(i) , α, Mi )p(θ(i) , α|Mi )d(θ(i) , α)
(23)
conditional on the training sample 1964:I to 1968:IV. The log-marginal data density
can be interpreted as a measure of one-step-ahead predictive performance ln p(YT |Mi ) =
�T
t=T∗
p(yt |Yt , Mi ). The values are ln p(YT |M0 ) = 1087.2 for the VAR and ln p(YT |M1 ) =
999.6 for the DSGE model, implying that for a wide range of prior probabilities the
posterior probability of the DSGE model is essentially zero.8 To shed more light
on the poor fit of the DSGE model, we computed in-sample, root-mean-squarederrors (RMSE) at the posterior mode estimates. The RMSE’s for the growth rates
of output and consumer durable expenditures are very similar for the two models,
whereas the RMSE of hours is substantially higher for the DSGE model: 0.0080
versus 0.0059 for the VAR.
5.3
Variance Decompositions and Impulse Responses
Our main interest is to unveil the sources of cyclical variation in hours and output.
Table 2 presents the variance decomposition (posterior means and 90 percent confi
dence intervals) for output and hours at business cycle frequencies, namely cycles of
6 to 32 quarters. Labor-supply shifts play an important role for the fluctuations of
hours. The shock �b accounts for almost 30 percent (posterior mean) of the fluctu
ations in hours according to the VAR, and more than half according to the DSGE
model. The relative contribution of the labor-supply shocks to output fluctuations
8
The posterior probability of model Mi is πi,T =
probability.
�
πi,0 p(YT |Mi )
,
πj,0 p(YT |Mj )
j=0,1
where πi,0 is its prior
19
is somewhat small albeit non-negligible; they account for about 15 percent of the
output variation.
The posterior confidence intervals indicate that the VAR model based variance
decompositions are associated with more posterior uncertainty than the DSGE decompositions. For instance, the 90 percent confidence intervals for the contribution
of �b,t to hours fluctuations are [0, 0.277] (VAR) and [0.083, 0.186] (DSGE), respec
tively. As pointed out by Faust and Leeper (1997), VAR variance decompositions
based on long-run restrictions are associated with a high degree of uncertainty. Con
ditional on the over-identifying restrictions embodied in the DSGE model, however,
one obtains fairly sharp estimates.
While VAR and DSGE model analysis broadly agree upon the decomposition of
output fluctuations, there are discrepancies in terms of the variance decomposition
of hours. According to the marginal data densities and the RMSEs, the time series
fit of the VAR is much better than the fit of the DSGE model, in particular in terms
of hours worked. Hence, the DSGE-based decomposition probably overestimates the
contribution of labor-supply shocks to the fluctuations of hours. The finding that
the DSGE model analysis attributes only about 7 percent of the hours fluctuation
to the permanent technology shock �z,t is partly due to the balanced-growth-path
property of this class of models; common technology shocks tend to shift both labor
demand and supply in a similar magnitude, leaving hours almost unaffected.
The VAR-based point estimates and confidence intervals reported in rows 1-3
and 10-12 of Table 2 were computed based on the reduced form parameters θ(0) and
the slope of the labor demand schedule, α − 1. Conditional on the VAR, because
20
the data provide no information about α, the inference is potentially sensitive to
the choice of the prior p(α). Moreover, our identifying assumption for labor-supply
shocks exploits the notion that the production capacity is fixed in the short run
and that there exists a stable relationship between labor productivity and hours
employed. This premise may be violated if firms heavily rely on the factor utilization.
For example, if there is a significant variation in the level workers’ effort (e.g., labor
hoarding in Burnside, Eichenbaum, and Rebelo, 1993) in the face of exogenous
shifts in labor supply, our identifying restriction is no longer valid. However, if the
capital utilization is time-varying and the cost of intense utilization results in a faster
depreciation of capital, our identifying restriction is still appropriate. In this case,
the restriction has to be modified to accommodate the systematic capital utilization;
the slope of the labor demand schedule is smaller in absolute value than α − 1 (see
Appendix). To assess the robustness of our VAR-based analysis we also report the
contribution of the labor-supply shock to output and hours variation conditional on
α = 0.3, 0.6, and 0.9 in Table 2. The share of �b lies between 23 and 38 percent
for hours and 12 to 20 percent for output. As we move to a higher value of α,
i.e., utilization of capital is less costly, the importance of the labor-supply shock is
reinforced.
Overall, both the VAR and DSGE model analysis suggests that the labor-supply
shocks play an important role for economic fluctuations, especially for hours worked.
Our estimates are, however, smaller than the previous estimates. Shaprio and Wat
son (1988) find that 60 percent of the cyclical variation in hours is due to permanent
shifts in labor supply. In Shapiro and Watson, labor-supply shocks are identified by
21
the stochastic trends in hours worked. While the empirical evidence on the station
arity of hours is not conclusive, we assume hours to be stationary. This is consistent
with a large class of DSGE models. Hall (1997) attributes almost the entire variation
of hours to preference shocks, which he identifies as deviations from the optimal
ity condition associated with the labor supply of a competitive household. Hall’s
finding is broadly in line with our DSGE model analysis as both approaches impose
more restrictions on the short-run labor market equilibrium and, as a result, require
bigger shifts in labor supply. However, as mentioned above, the weak time series
fit of the DSGE model indicates that some of these over-identifying restrictions are
potentially misspecified and that the corresponding findings have to be interpreted
with caution. Our VAR identification scheme does not rely on a specific form of
household’s preference. Instead, it is based on the firm’s production technology
through the slope of marginal-product-of-labor schedule.
To assess whether the structural shocks identified from the VAR conform with
our economic intuition, Figure 1 depicts one-standard-deviation impulse responses
of labor productivity, expenditure on consumer durable goods (investment in home
capital), and market hours to the three structural shocks.9 The graphs show the re
sponses from the DSGE model (solid lines) and those from the VAR along with the
90 percent confidence interval (dotted lines). Looking at the first row, in response
to a permanent shock, labor productivity both in the DSGE model and VAR ap
proach the new steady state at a similar pace. Spending on consumer durables also
9
The signs of the responses are normalized as follows: the initial responses of productivity to
�z,t , productivity to �a,t , and hours to �b,t are positive, positive, and negative, respectively.
22
increases permanently. Market hours rises instantaneously and slowly returns to its
steady state in the DSGE model, whereas its response is hump-shaped in the VAR.
The VAR responses to a temporary market productivity shock (second row of the
Figure) closely traces those from the DSGE model except for the delayed response
of hours in the VAR. Finally, in response to a temporary home productivity shock,
labor productivity rises in both the VAR and DSGE model. Home investment rises
immediately in the DSGE model whereas it exhibits a slow hump-shape response.
In sum, the VAR responses, by and large, conform with the economic intuition pro
vided by the DSGE model. However, hours responses are delayed in the VAR for
about 2 quarters, and the short-run dynamics of expenditure on consumer durables
are somewhat different from the DSGE model prediction.
Based on the competitive labor market equilibrium, we identify exogenous shifts
in labor supply. Yet the proposed identification scheme allows a more general – also
an alternative – interpretation than labor-supply shocks. As an illustrative example,
consider a model economy with sticky prices where firms have to produce goods
to meet their demand. In this economy, the labor demand is no longer a simple
reflection of the marginal product of labor. It is instead jointly determined by the
demand for goods and the output-labor elasticity from the production technology.
Suppose now there is an increase in the demand for goods that is not caused by a
productivity shift. This will lead to an increase in the demand for labor at a given
level of production capacity. The joint behavior of labor productivity and hours is
still dictated by the marginal-product-of-labor schedule.10
10
In this event, the real wage will increase given the upward sloping labor supply curve. However,
23
5.4
Evolution of Latent Technology Processes
According to the home production model, recessions may occur because agents find
it optimal to allocate more time in non-market activities. In our DSGE model the
attractiveness of non-market activity is measured by the latent home technology
process.11 We plot three technology indices in Figure 2 together with the NBER
business cycle peaks and troughs. All five recessions during the sample period
are associated with low levels of market productivity. Two business cycle troughs,
March 1975 and November 1982, coincide with unusually high productivity of nonmarket activities. The strong interpretation of this finding is that unusually high
non-market productivity or preference shift has contributed to a low employment
and output. A weaker interpretation is that the economic downturns in March 1975
and November 1982 cannot be solely explained by an adverse technology shock in
the market.
6
Conclusion
We investigate the sources of economic fluctuations in the context of a dynamic
general equilibrium. A new VAR identification scheme which distinguishes between
shifts of and movements along the marginal product of labor is proposed to identify
labor productivity falls as employed hours increases, justifying our use of labor productivity instead
of a wage series under this more general interpretation.
11
Ingram, Kocherlakota and Savin (1997) recover non-market variables (non-market consumption,
non-market hours, and leisure) based on the households’ optimality conditions and times series of
market variables, whereas we obtain the time series of the latent home technology.
24
three structural disturbances: temporary labor-supply shifts, temporary labor de
mand shifts, and permanent productivity shocks that eventually move both demand
and supply. According to the variance decomposition from the VAR, the laborsupply shifts are an important driving force of the cyclical fluctuation of hours,
as they account for about 30 percent of the variation. For output fluctuations at
business cycle frequencies, the role of labor-supply shifts is modest as their relative
contribution is about 15 percent. To assess the importance of labor-supply shifts
in the context of an equilibrium model, a home-production model with stochastic
variation in non-market technology is estimated. While the DSGE model based
decomposition of output resembles the VAR results, the structural model attributes
a higher fraction of hours fluctuations to the labor-supply shifts than the VAR.
This result, however, is partly due to misspecified over-identifying restrictions of
the DSGE model.
25
Appendix: Labor Demand with Variable Capital Utiliza
tion
Consider a Cobb-Douglas production function with inputs in capital services and
hours:
Yt = (ut Km,t )1−α (Xm,t Lm,t )α ,
where ut represents the utilization of the capital stock. Suppose the intensive use of
capital results in a fast depreciation. At the cost of a more complicated notation, we
could work with an alternative decentralization scheme in which firms make decisions
on accumulation. However, since both decentralizations are essentially identical, as
in the main text, suppose the firm rents the capital from the households. Yet the
firm has to compensate households for faster depreciation when the capital is utilized
more intensively:
max
Lm,t ,Km,t ,ut
(ut Km,t )1−α (Xm,t Lm,t )α − Wt Lm,t − (Rt + δ(ut ))Km,t .
For illustrative purposes, assume that the elasticity of depreciation is constant:
δ(ut ) = δ0
uλ+1
t
λ+1 ,
where λ > 0. As λ → ∞, the utilization is held constant and
the depreciation rate is fixed. The first order conditions of the profit maximization
problem with respect to Lm,t and ut imply that the inverse labor demand schedule
still depends on the predetermined capital stock and the market productivity shocks
only. However, its slope changes:
∂ ln Lm,t
∂ ln Wt
= µ(α − 1)
,
∂�b,t
∂�b,t
µ=
λ
≤ 1.
λ+α
26
Therefore, the proposed identification scheme is still valid but the slope of the labor
demand schedule is smaller than in the constant utilization case, reflecting an extra
margin for the firm to exploit.
27
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30
Table 1: Prior and Posterior Distribution for DSGE Model Parameters
Parameters
Name
Range
α
Prior
Posterior
Density
Mean
S.D.
Mean
S.D.
[0,1]
Beta
0.666
0.020
0.741
0.018
β
[0,1]
Beta
0.993
0.002
0.979
0.003
γ
IR
Normal
0.004
0.0005
0.003
0.0004
δ
[0,1]
Beta
0.025
0.002
0.017
0.002
Lm
[0,1]
Beta
0.330
0.020
0.335
0.021
ρa
[0,1]
Beta
0.800
0.100
0.745
0.034
ρb
[0,1]
Beta
0.800
0.100
0.865
0.034
Lh
[0,1]
Beta
0.250
0.050
0.183
0.045
η
IR+
Gamma
100.0
100.0
30.70
5.546
ψ
[0,1]
Beta
0.666
0.100
0.753
0.080
ν
IR+
Gamma
1.000
2.000
2.302
0.388
Ih /Im
IR+
Gamma
0.700
0.020
0.686
0.020
τ
IR+
Gamma
1.000
2.000
2.446
0.536
ξ1
IR
Normal
2.960
1.000
3.145
0.006
ξ2
IR
Normal
0.000
0.020
0.005
0.0003
σz
IR+
InvGamma
0.01∗
2.000∗
0.007
0.001
σa
IR+
InvGamma
0.01∗
2.000∗
0.009
0.001
σb
IR+
InvGamma
0.015∗
2.000∗
0.021
0.007
Notes: For the Inverse Gamma (u, s) priors we report the parameters u and s.
For u = 2 the standard deviation (S.D.) is infinite. The posterior moments are
calculated from the output of the Metropolis algorithm.
31
Table 2: Variance Decomposition at Business Cycle Frequencies
Variable
Model
Shock
Mean
ln Lm,t
VAR
�z
0.205
[ 0.000
0.529 ]
VAR
�a
0.510
[ 0.069
0.905 ]
VAR
�b
0.285
[ 0.005
0.588 ]
VAR (α = 0.3)
�b
0.230
[ 0.006
0.454 ]
VAR (α = 0.6)
�b
0.268
[ 0.007
0.551 ]
VAR (α = 0.9)
�b
0.376
[ 0.021
0.738 ]
DSGE
�z
0.066
[ 0.006
0.124 ]
DSGE
�a
0.268
[ 0.132
0.403 ]
DSGE
�b
0.666
[ 0.526
0.805 ]
VAR
�z
0.491
[ 0.089
0.911 ]
VAR
�a
0.367
[ 0.000
0.703 ]
VAR
�b
0.142
[ 0.000
0.277 ]
VAR (α = 0.3)
�b
0.118
[ 0.000
0.250 ]
VAR (α = 0.6)
�b
0.130
[ 0.000
0.256 ]
VAR (α = 0.9)
�b
0.204
[ 0.000
0.388 ]
DSGE
�z
0.448
[ 0.339
0.560 ]
DSGE
�a
0.417
[ 0.311
0.519 ]
DSGE
�b
0.135
[ 0.083
0.186 ]
ln Lm,t
ln Ym,t
ln Ym,t
Conf. Interval
Notes: Decomposition of aggregate output ln Yt and market hours ln Lm,t at business
cycle frequencies (6 to 32 quarters per cycle). The table reports posterior means
and 90 percent confidence intervals. VAR (α = x) signifies that α was fixed at the
value x rather than integrated out with respect to its prior distribution.
32
Figure 1: Impulse-response Functions
Notes: Figure depicts posterior mean responses for VAR (dashed) and DSGE model
(solid). The dotted lines represent pointwise 90 percent Bayesian confidence intervals
based on the VAR posterior.
33
Figure 2: Filtered technology processes at , bt , and xt
Notes: The posterior mean estimates of the latent technology processes are based
on the DSGE model. Solid vertical lines correspond to business cycle peaks, dashed
lines denote business cycle troughs (NBER Business Cycle Dating).
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