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Working Paper Series
From Individual to Aggregate Labor
Supply: A Quantitative Analysis based
on a Heterogenous Agent
Macroeconomy
WP 03-05
Yongsung Chang
Federal Reserve Bank of Richmond
Sun-Bin Kim
Concordia University
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
From Individual to Aggregate Labor Supply:
A Quantitative Analysis based on a Heterogeneous Agent
Macroeconomy *
Federal Reserve Bank of Richmond Working Paper 03-05
July 2003
Yongsung Chang
Research Department, Federal Reserve Bank of Richmond
and
Sun-Bin Kim,
Department of Economics, Concordia University
Abstract:
We investigate the mapping from individual to aggregate labor supply using a general
equilibrium heterogeneous-agent model with an incomplete market. The nature of
heterogeneity among workers is calibrated using wage data from the PSID. The gross
worker flows between employment and nonemployment and the cross-sectional earnings
and wealth distributions in our model are comparable to those in the micro data. We find
that the aggregate labor supply elasticity of such an economy is around 1, bigger than
micro estimates but smaller than those often assumed in aggregate models.
Keywords: Aggregate Labor Supply Elasticity, Heterogeneity, Indivisible Labor
JEL Classification: E24, E32, J21, J22
* We thank Robert Shimer for sharing his monthly worker transition-rate data with us. We thank Steven
Davis, John Kennan, Narayana Kocherlakota, Thomas MaCurdy, Cesaire Meh, Richard Rogerson, JoséVíctor Ríos-Rull, and Randy Wright for their helpful comments. The views expressed herein are those of
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; Kim: sunbink@alcor.concordia.ca.
1
Introduction
Despite enormous heterogeneity in the workforce, economists, for simplicity and tractability, often
postulate and analyze an economy populated by identical agents. A fully specified representativeagent model has become a workhorse in macroeconomics, and it is common practice to rely on micro
evidence to pin down the key parameters of highly aggregated models (Kydland and Prescott, 1982;
King, Plosser, and Rebelo, 1988).
However, this practice often creates a tension between micro and macro observations.1 A
prominent example is the differences in measurement of labor supply elasticity. One of the stylized
facts in aggregate fluctuations is that total hours worked varies greatly over the business cycle
without much variation in wages.2 To be consistent with the observed movement in hours and wages,
a representative-agent model emphasizing the intertemporal substitution of leisure, pioneered by
Lucas and Rapping (1969), requires the use of labor supply elasticity which is beyond admissible
estimates based on micro data (Ghez and Becker, 1975; MaCurdy, 1981; Altonji, 1986; Abowd and
Card, 1989).3
1
Browning, Hansen, and Heckman (1999) raise warning flags about the current use of micro evidence in calibrating
macro models.
2
The importance of labor supply elasticity is not limited to the business cycle analysis; for example, it plays a key
role for the timing and effect of fiscal policy (Aucherbach and Kotlikoff, 1987; Judd, 1987).
3
In his survey paper, Pencavel (1986) reports that most estimates are between 0 and 0.45 for men. In their
parallel survey of research on the labor supply of women, Killingsworth and Heckman (1986) present a wide range
of estimates, from -0.3 to 14; they do not venture a guess as to which is correct but conclude that the elasticity is
probably somewhat higher for women than men. See Blundell and MaCurdy (1999) for a more recent review of the
literature. See also Mulligan (1998) or Rupert, Rogerson, and Wright (2000) on how the current micro estimates may
underestimate workers’ willingness to substitute leisure over time. An alternative equilibrium approach is to introduce
shifts in labor supply through shifts in preference (Bencivenga, 1992), home technology (Benhabib, Rogerson, and
Wright, 1991; Greenwood and Hercowitz, 1991) or government spending (Christiano and Eichenbaum 1992). Even
1
The participation margin, the so-called extensive margin, has been recognized as a potential
resolution. Hours fluctuations are accounted for mainly by movement in and out of employment by
workers (Coleman, 1984; Heckman, 1984) with different reservation wages. Under this environment,
the slope of the aggregate labor supply curve has little to do with intertemporal substitution but
rather with the distribution of reservation wages across workers. The well-known lottery economy
by Rogerson (1988) and Hansen (1985) is a special case where the reservation wage distribution is
degenerate, yielding a very high elasticity, which is in fact infinity.
In this paper, we investigate the mapping from individual to aggregate labor supply using a
general equilibrium model economy in which workers face idiosyncratic productivity shocks and the
capital market is incomplete.4 The heterogeneity of the workforce, more precisely the stochastic
process of idiosyncratic productivity, is calibrated to be consistent with the wage data from the
Panel Study of Income Dynamics (PSID) for 1971-1992. As the reservation wage distribution is
crucial – but cannot be observed in practice – for our analysis, we test the model heterogeneity
indirectly. The gross worker flows in and out of employment in the model are comparable to
those in the Current Population Survey (CPS) for 1967:II-2000:IV. Cross-sectional distributions
of earnings and wealth are comparable to those from the PSID and Survey of Consumer Finance
(SCF) although the wealth distribution in the model is somewhat less skewed than that in the data.
From the reservation wage distribution, we uncover the upper bound of aggregate labor supply
elasticity of our heterogeneous-agent economy. We find that elasticities range from .89 to 1.97
in a disequilibrium approach where the role of labor supply is dismissed in the short run, its slope is still important
for the welfare cost departing from an equilibrium.
4
An economy with indivisibility at the micro level may be approximated by a representative-agent economy with
divisible labor, as the indivisibility is smoothed by an aggregation over heterogeneous agents. While this point is well
illustrated in Mulligan (2001), we have yet to investigate its quantitative implications because the mapping from the
micro to the macro function depends crucially on the heterogeneity of the workforce.
2
depending on the nature of heterogeneity. These values are bigger than typical micro estimates,
but smaller than those often assumed in aggregate models. We also show that our model implies
a small compensated labor-supply elasticity (between .37 and .69) at the individual level and a
moderate elasticity (between .86 and 1.13) at aggregate level. In reference to the real-business-cycle
analysis, our heterogeneous model economy is comparable to the representative-agent economy with
the compensated labor-supply elasticity of 2.
The closest to our work are Kydland (1984), Cho and Rogerson (1988), Cho (1995), and
Gomes, Greenwood and Rebelo (2001). Kydland constructs an economy with two types of workers,
skilled and unskilled, and reproduces some labor-market regularities in relative wages and hours.
However, this approach does not reflect the participation margin, a dominant source of the variation
in total hours. Cho and Rogerson consider an economy which is populated by a continuum of
identical families consisting of two members and show that the aggregate labor supply depends
on the relative productivity among family members. While the female labor supply is indeed an
important source of variation in aggregate hours, our analysis extends to a more general crosssectional heterogeneity. Cho incorporates ex post heterogeneity into a standard real-business cycle
framework. This considerably simplifies the computation as consumption is shared among workers.
It is, however, clear in the data that persons with greater hours or greater earnings per hour
consume more.5 Gomes et al. also analyze the non-convexity of labor supply in an incomplete
market with aggregate fluctuations. They focus on the cyclical behavior of unemployment rates,
whereas we look into the mapping from individual to aggregate labor supply functions.
Other important works on the labor-market heterogeneity in the context of stochastic general
5
For example, according to the Consumer Expenditure Survey data for 1990-1994, for single households, a one
percent increase in hourly wage is associated with .6 percent increase in total consumption cross-sectionally; for
married households, a one percent increase in household wage (the average wages of husband and wife if both are
working) is associated with .29 percent increase in total consumption.
3
equilibrium include those of Andolfatto and Gomme (1996), Castañeda, Dı́az-Giménez, and Rı́osRull (1998), Merz (1999), and den Haan, Ramey, and Watson (2000). Andolfatto and Gomme study
the unemployment insurance policy; Castañeda et al., the income distribution and unemployment
spells; Merz, the cyclical behavior of labor turnover; den Haan et al., the propagation mechanism
under labor-market matching and job destruction.
The paper is organized as follows. Section 2 lays out the model economy. In Section 3 we
calibrate the model parameters consistent with various micro data. In Section 4, we investigate the
aggregate labor supply of the model in both steady state and fluctuations. We also provide com
parison with the representative-agent model. The conclusion, Section 5, summarizes our findings.
Appendix collects the computational details and data sources.
2
The Model
2.1
Environment
The model economy is a version of the stochastic-growth model with a large (measure one) popu
lation of infinitely lived workers. Individual workers differ from each other in productivity.
6
Each worker maximizes the expected discounted lifetime utility:
U=
max
{ct ,ht }∞
t=0
E0
��
∞
�
β u(ct , ht ) ,
t
t=0
with
u(ct , ht ) = ln ct + B
6
(1 − ht )1−1/γ
,
1 − 1/γ
Ideally, one would allow for heterogeneity both in the market and preference (or non-market productivity).
However, it would be necessarily controversial to make an assumption about preference heterogeneity. We focus on
the heterogeneity in the market productivity only whose process is inferred from the individual earnings data.
4
where E0 [·] denotes the expectation operator conditional on information available at time 0, β is
the discount factor, ct consumption, and ht hours worked at time t. The utility is separable across
times and between consumption and leisure. This assumption about the form of utility is popular
in both business cycle analysis and empirical labor supply literature. The parameter γ denotes
the intertemporal substitution elasticity of leisure. Log utility in consumption supports a balanced
growth path.
According to our production technology, which will be specified below, labor input enters
simply as efficiency units. Thus, a worker who supplies ht units of time earns wt xt ht , where wt is
the market wage rate for an efficiency unit of labor, and xt represents the worker’s productivity.
We assume that individual productivity xt exogenously varies over time according to a stochastic
process with a transition probability distribution function πx (x� |x) = Pr(xt+1 ≤ x� |xt = x). Since
participation is the dominant source of variation in total hours worked in the data, we abstract
from an intensive margin and assume that labor supply is indivisible; i.e., ht takes either zero or
¯
h(<
1). A worker can save and borrow by trading a claim for physical capital, which yields the
rate of return rt and depreciates at rate δ. The capital market is incomplete; the physical capital
is the only asset available to insure against idiosyncratic risks in x, and workers face a borrowing
constraint; the asset at can take a negative value but cannot go below ā at any time. A worker’s
budget constraint is:
ct = wt xt ht + (1 + rt )at − at+1 ,
and
at+1 ≥ a.
¯
Firms produce output according to a constant-returns Cobb-Douglas technology in capital, Kt ,
and efficiency units of labor, Lt :
Yt = F (Lt , Kt , λt ) = λt Lαt Kt1−α ,
5
where λt is aggregate productivity, following a stochastic process with a transition probability
distribution function, πλ (λ� |λ) = Pr(λt+1 ≤ λ� |λt = λ).
It is useful to consider a recursive equilibrium. Suppose µ(a, x) denotes the distribution (mea
sure) of workers.7 Let V E and V N denote the values of being employed and nonemployed, respec
tively. If a worker decides to work, she solves the following Bellman equation by choosing the next
period asset holding a� :
�
¯
V (a, x; λ, µ) = max
u(c, 1 − h)
�
E
a ∈A
��
� E � � � �
��
N � � � � �
+ βE max V (a , x ; λ , µ ), V (a , x ; λ , µ ) x, λ
�
(1)
subject to
¯ + (1 + r)a − a� ,
c = wxh
a� ≥ ā,
and
µ� = T(λ, µ).
where T denotes a transition operator for µ.
If the worker decides not to work, her Bellman equation is:
�
N
V (a, x; λ, µ) = max
u(c, 1)
�
a ∈A
��
� E � � � �
��
N � � � � �
+ βE max V (a , x ; λ , µ ), V (a , x ; λ , µ ) x, λ
�
(2)
subject to
c = (1 + r)a − a� ,
a� ≥ ā,
7
Let A and X denote sets of all possible realizations of a and x, respectively. The measure µ(a, x) is defined over
a σ-algebra of A × X .
6
and
µ� = T(λ, µ).
Having solved (1) and (2), it is straightforward to deal with worker’s labor supply decision:
V (a, x; λ, µ) = max
¯
h∈{0,h}
2.2
�
�
V E (a, x; λ, µ), V N (a, x; λ, µ) .
(3)
Equilibrium
Equilibrium consists of a set of value functions, {V E (a, x; λ, µ), V N (a, x; λ, µ), V (a, x; λ, µ)}, a set of
decision rules for consumption, asset holdings, and labor supply, {c(a, x; λ, µ), a� (a, x; λ, µ), h(a, x; λ, µ)},
aggregate inputs, {K(λ, µ), L(λ, µ)}, factor prices, {w(λ, µ), r(λ, µ)}, and a law of motion for the
distribution µ� = T(λ, µ) such that:
1. Individual optimization:
Given w(λ, µ) and r(λ, µ), the individual decision rules c(a, x; λ, µ), a� (a, x; λ, µ), and h(a, x; λ, µ)
solve (1), (2), and (3).
2. The firm’s profit maximization:
�
�
w(λ, µ) = F1 L(λ, µ), K(λ, µ), λ
(4)
�
�
r(λ, µ) = F2 L(λ, µ), K(λ, µ), λ − δ
(5)
for all (λ, µ).
3. The goods market clears:
�
� �
�
�
�
a (a, x; λ, µ) + c(a, x; λ, µ) dµ = F L(λ, µ), K(λ, µ), λ + (1 − δ)K
for all (λ, µ).
7
(6)
4. Factor markets clear:
�
L(λ, µ) =
xh(a, x; λ, µ)dµ
(7)
�
K(λ, µ) =
adµ
(8)
for all (λ, µ).
5. Individual and aggregate behaviors are consistent:
�
�
0
��
0
µ (A , X ) =
A0 ,X 0
A,X
1a� =a� (a,x;λ,µ)
�
dπx (x |x)dµ da� dx�
�
(9)
for all A0 ⊂ A and X 0 ⊂ X .
3
Calibration
Individual productivity x is assumed to follow an AR(1) process in logs:
ln x� = ρx ln x + εx ,
εx ∼ N (0, σx2 ).
(10)
As we view x to reflect a broad measure of earnings ability in the market, the stochastic process
of x is estimated by the individual wages from the PSID for 1971-1992. Appendix A.1 describes in
detail the data we use. According to the model, the log wage for individual i at time t, denoted by
ln wti , can be written as ln wti = ln wt + ln xit . When quasi-differenced, individual wage evolves as:
ln wti = ρx ln wti−1 + (ln wt − ρx ln wt−1 ) + εix,t .
(11)
Equation (11) is estimated by the OLS with year dummies in the regression, capturing aggregate
effects including ln wt − ρx ln wt−1 . The annual estimates are ρ�x = .818 (with a standard error of
�x = .291.8 According to the frequency conversion procedure described in Appendix
.0025) and σ
8
Our estimate is slightly lower than, but comparable to, the persistence of idiosyncratic earnings risks in Storeslet
ten, Telmer, and Yaron (1999). The difference is due to their decomposition of idiosyncratic shocks into a persistent
AR(1) and purely temporary i.i.d. components, whereas we assume a single AR(1) process.
8
A.2, the corresponding quarterly values are ρx = .95 and σx = .225, which we refer to as the
benchmark economy.
We consider two possible deviations from these values. The dispersion of productivity distri
bution may be larger than that of σx obtained from the wage distribution because the workers at
the very low end of the productivity distribution are less likely to participate. The second model
we consider has a larger dispersion in productivity: σx is magnified by 25%, yielding σx = .28125.9
In our model, x reflects the heterogeneity in earnings – both permanent and temporary – in
the population. Thus, we have not controlled for individual characteristics in the regression. The
persistence of wage may differ across population, especially for women who play an important role
in the variation of total hours. Table 1 presents the estimate of ρx for various groups. Wages exhibit
a smaller persistence for women. For example, the persistence is .722 for married women whereas it
is .809 for married men. When we control for individual characteristics – gender, age, age squared,
and years of schooling – in the regression, the persistence also decreases. For example, the annual
estimate decreases from .818 to .743 for all workers. According to Pesaran and Smith (1995), when
coefficients differ across groups, pooling and aggregation tend to lead to a higher estimates. Also, as
wages tend to reflect good realizations of productivity, they appear more persistent than underlying
productivity. Thus, we consider a less persistent productivity, ρx = .92, for our third model. In
fact, with ρx = .92 (and σx = .225), the model shows the persistence of .816 and the standard
deviation of innovation of .291 for individual wage regression in (11), which are almost identical to
9
To understand the magnitude of selection bias, consider an extreme case where employment is completely ordered
by the current productivity, that is, a worker with the highest productivity is hired first and so forth. In this case,
the (observed) wage distribution is a truncated distribution of x. Under log-normality, when the bottom 40 percent
(the average nonemployment rate in the CPS for 1967-2000) is truncated, the standard deviation of the underlying
distribution is 1.5 times larger than that of the truncated distribution (See Maddala 1983). Given that labor supply
depends on preference and wealth as well as wages, we magnify the estimate by 25%.
9
those in the PSID.
Other parameters of the model economies are in accord with business cycle analysis and em
pirical labor supply literature. According to the Michigan Time-Use Survey, a typical household
allocates about 33 percent of its discretionary time for paid compensation (Hill, 1984; Juster and
¯ = 1/3 . Most micro estimates of intertemporal substitution elasticity of leisure
Stafford, 1991): h
fall between 0 and .5: we use γ = .2. With a discrete choice of hours of work (i.e., labor is indi
visible), the value of γ is not so important for the aggregate labor supply elasticity since it mostly
depends on the shape of the reservation wage distribution. The labor share, α, is .64, and the quar
terly depreciation rate, δ, is 2.5 percent. We search for the weight parameter on leisure, B, such
that the steady state employment rate is 60 percent, the average from the CPS for 1967:II-2000:IV.
The discount factor β is chosen so that the quarterly rate of return to capital is 1 percent.10 The
borrowing constraint ā is set to –2 which is approximately two quarters’ earnings for a worker with
the average productivity in our model economy.11 Table 2 summarizes the parameter values. Fi
nally, when we investigate the model economy with aggregate fluctuations, we introduce exogenous
shifts in labor demand through aggregate technology shocks λt . We assume that ln λt follows an
AR(1) process of which persistence is .95 and the standard deviation of innovation is .7 percent,
which is consistent with the linearly de-trended post-war total factor productivity. We solve the
equilibrium of the model economy in a discrete state space. Appendix A.4 provides a detailed
description of the computational procedure.
10
The discount factor is lower than that in the representative-agent model, because market incompleteness increases
savings as noted in Aiyagari (1994).
11
Given the persistent idiosyncratic earnings process, the size of the borrowing constraint itself does not affect the
main result of the paper; for example, we obtain a similar aggregate labor supply elasticity with ā = 0.
10
4
Results
4.1
Steady State
We first characterize the steady state of the model economy where µ(x, a) is invariant. As an indirect
diagnostic test, we ask whether the model generates a reasonable labor market mobility and crosssectional distributions in wealth and earnings. Even in the absence of aggregate fluctuations there
are constant flows of workers in and out of employment due to individual productivity shocks.
Table 3 presents employment rate, gross-worker flows, and hazard rates from the model and the
CPS. The statistics for the CPS are quarterly averages for 1967:II-2000IV.
As described in the previous section, the utility parameter B is calibrated to match the average
employment rate of 60 percent. The quarterly gross-worker flows in the CPS are computed using
Abraham and Shimer’s (2001) monthly hazard rates as described in Appendix A.3. On average,
7.07 percent of the population moved from employment to nonemployment each quarter; 6.88
percent of the population moved in the opposite direction, from nonemployment (unemployment
plus non-labor force) to employment.12 In our first model, these flows are 5.92 percent, somewhat
lower than those in the CPS data. With different degrees of idiosyncratic shocks, they are 5.64
(σx =.28125) and 6.85 (ρx =.92).13 Overall, the worker flows and hazard rates are somewhat lower
than, but comparable to, those in the CPS.
Wealth and earnings, excluding preference and non-market opportunity which are hard to
12
These numbers are slightly higher than those in Blanchard and Diamond (1990) due to a different sample period
and adjustment method. Also, we do not make a distinction between nonemployment and unemployment. According
to Shimer, as well as Blanchard and Diamond, the flows between employment and non-labor force are as big as those
between employment and unemployment.
13
For each model, we adjust the utility parameter B and the discount factor β so that the employment rate is 60
percent and the quarterly rate of return to capital is 1 percent in steady state.
11
measure, are probably the most important factors for labor-market participation decision. Figure
1 exhibits the Lorenz curves of the wealth distributions from the 1984 PSID and three model
economies.14 Family wealth in the PSID reflects the net worth of house, other real estate, vehicles,
farms and businesses owned, stocks, bonds, cash accounts, and other assets. According to Table
4, the Gini coefficient of wealth is .76 in the PSID, whereas those from the models are .64 for the
benchmark, .65 (σx =.28125), and .58 (ρx =.92).
Figure 2 shows the Lorenz curves of earnings. The data is based on family earnings (earnings of
head of household and spouse) also from the 1984 PSID. The model and the PSID exhibit similar
inequality, but there are more zero earners at the bottom of the distribution in the model; 40
percent of population in the model and 20 percent in the PSID. In the PSID, a family with at
least one family member working at some point during the survey year recorded positive earnings,
whereas the model is calibrated to match the average employment rate of 60 percent. This makes
the Gini index of the model, between .57 and .68, somewhat higher than .53, the Gini index in
the PSID (See Table 4). However, when we use positive earnings only, the Gini indices from the
models are .39 (benchmark), .47 (σx =.28125), and .29 (ρx =.92), comparable to .42 in the PSID.
Table 5 summarizes the detailed information on wealth and earnings from the SCF, PSID,
and benchmark model.15 Since the wealth-earnings distributions between the PSID and SCF are
similar, we discuss the comparison between the model and PSID only. For each quintile group
of wealth distribution, we calculate the wealth share, the ratio of group average to economy-wide
14
In the PSID, information on family wealth is available for 1984, 1989, and 1996 survey years. We use the 1984
survey because the date falls in the mid point of our sample period. The degree of inequality does not vary significantly
across the three surveys.
15
In terms of Gini indices, the wealth and earnings distributions from the PSID are slightly less concentrated than
those in the SCF. According to Dı́az-Giménez, Quadrini, and Rı́os-Rull, Gini indices are .78 and .63 for wealth and
earnings, respectively, in the 1992 SCF.
12
average, and the earnings share. Both in the data and model, the poorest 20 percent of the wealth
distribution owns almost nothing. In fact, households in the first quintile of the wealth distribution
are in debt both in the model and data. On the contrary, households in the 4th and 5th quintile
of the PSID own 18.74 and 76.22 percent of total wealth, respectively. According to the model,
they own 24.48 and 66.31 percent, respectively. The average wealth of the 4th and 5th quintile
are, respectively, .93 and 3.81 times larger than that of a typical household, while these ratios are
1.22 and 3.33 according to our model. The 4th and 5th quintile group of the wealth distribution
earn, respectively, 24.21 and 38.23 percent of total earnings in the PSID. The corresponding groups
earn 23.44 and 30.39 percent, respectively, in the model. Overall, the wealth distribution is more
skewed in the data. In particular, the model fails to match the highly concentrated wealth in the
right tail of the distribution. About half of total wealth—43 and 53 percent in the PSID and SCF,
respectively—is held by the top 5 percent of the population (not shown in the Table). In our model,
only 20 percent of total wealth is held by them. However, our primary objective is not to explain the
behavior of the top 1 or 5 percent of the population.16 We argue that the model economy possesses
a reasonable heterogeneity—especially for the 2nd and 3rd quintile of the distribution, probably
the most relevant group for the business cycle fluctuations—to study the average response of hours,
as the stochastic process of productivity is estimated from the panel data, and the cross-sectional
earnings distribution are, by and large, consistent with the data counterparts.
The shape of reservation wage distribution is crucial for the mapping from individual to aggre
gate labor supply. In Figure 3, we plot the reservation wage schedule of the benchmark model for
all asset levels (Panel A) and for assets less than $200,000 (Panel B). At a given asset level, workers
16
As is well known, accounting simultaneously for the earnings and wealth in the U.S. economy is no easy task
given the extreme wealth concentration observed in the data. For studies on the wealth distribution in a dynamic
general equilibrium environment, see among others Huggett (1996), Krusell and Smith (1998), Quadrini (2000), or
neda, Ana, Javier Dı́az-Gim´enez, and Jos´e-Vı́ctor Rı́os-Rull (2000).
Casta˜
13
with the wage (productivity) above the line choose to work. The reservation wage increases as
the asset level increases. To illustrate, we adjust the units such that the mean asset of the model
matches the average asset in the 1984 PSID survey, $60,524.17 Thus, the values are in 1983 dollars.
Consider a worker whose assets are $29,234, the median of the wealth distribution from the model.
According to the model, he is indifferent between working and not working at quarterly earnings of
$3,814. A worker whose assets are equivalent to the average asset holding of the economy, $60,524
(which belongs to the 66th percentile of the wealth distribution in our model and to the 72th
percentile in the PSID) is indifferent in working at $4,871 per quarter.
Based on the reservation wage schedule and invariant distribution µ(x, a), we infer the elasticity
of aggregate labor supply. In Figure 4 we plot the inverse cumulative distribution of reservation
wages for three model economies. In practice, the reservation wage distribution is neither observed
nor constant over time. In Table 6 we compute the elasticities of employment with respect to
the reservation wage around the steady state employment rate of 60 percent. These values may
be viewed as upper bounds for aggregate employment response as they assume that the entire
wealth distribution is held constant. For the benchmark case, the elasticities are 1.19, 1.09, and
1.0, respectively, at the employment rates of 58, 60, and 62 percent. The elasticities are somewhat
smaller with a bigger heterogeneity (σx = .28125) as the reservation wage distribution is more
dispersed. With a lower persistence (ρx = .92) – which generates a wage process similar to those
in the PSID – the elasticities are 1.88, 1.78, and 1.61, respectively, at the employment rate of 58,
60 and 62 percent. Although these values are bigger than typical micro estimates, they remain at
moderate range. In particular, a very high elasticity – in fact, infinity – generated by a lottery
economy with a homogeneous workforce examined by Hansen and Rogerson, does not survive a
serious heterogeneity.
17
The mean asset in our model is 12.63 units. The reservation wages in the vertical axis reflect quarterly earnings
¯
(the reservation wage rate multiplied by h).
14
4.2
Fluctuations
While the labor supply elasticity plays an important role in many issues such as timing of taxes and
government spending (Auerbach and Kotlikoff, 1987; Judd, 1987), it is one of the most extensively
debated parameter in business cycle literature. In this section, we examine the fluctuations of the
model economy in the presence of exogenous shifts in aggregate productivity. We do not take a
stand on the sources of the business cycle here. However, we intentionally exclude other types of
aggregate disturbances, especially those that shift the labor supply curve, as we are interested in
the response of labor supply. Aggregate productivity shocks serve as an instrument, exogenously
shifting the labor-demand curve, to identify the response of labor supply.
Computing the equilibrium fluctuations of an economy of this sort requires a considerable
degree of approximation. We use the so-called “bounded rationality method” developed by Krusell
and Smith, in which agents are assumed to make use of a finite set of moments of the distribution
µ. The justification of this method is that by using partial information about µ, households do
almost as well as by using all the information in µ when predicting future prices. In fact, Krusell
and Smith show that use of the first moment provides a good approximation in a stochastic-growth
model. Gomes et al. and Castañeda et al. have shown that this method can be applied to various
economic environments. The procedure used in this section closely follows those suggested in these
works. The details of computation are provided in Appendix A.4.
We provide a comparison with the representative-agent (with divisible labor) model in terms
of second moments of key aggregate variables. A common way to characterize the behavior of labor
supply is using a Frisch function (Frisch, 1959).18 For a representative agent-economy with the
same type of utility function as ours, the Frisch labor supply function linearized around the steady
18
See Heckman (1974), MaCurdy (1982) or McLaughlin (1995) for relationships among various measures of labor
supply elasticities.
15
state is
�
�t − �
ht = ψ(w
ct ),
ψ=
¯
1−h
γ,
¯
h
(12)
where the circumflex denotes the variable’s percentage deviation from its steady state value. The
compensated labor supply elasticity (Frisch elasticity) ψ represents the elasticity of hours with
respect to wages, holding consumption (or wealth) constant. We consider four representative
economies with the compensated labor supply elasticity, ψ, equaling to .4, 1, 2, and 4. With
¯ = 1/3, these values correspond to γ of .2, .5, 1, and 2, respectively. In reference to the realh
business cycle analysis, Prescott (1986) corresponds to ψ = 2, King, Plosser and Rebelo (1988) to
ψ = 4, and Hansen (1985) to ψ = ∞. The representative-agent economies have the same parameter
values as our benchmark model summarized in Table 2 except for the intertemporal substitution
elasticity γ and B.19
Table 7 displays the statistics of five model economies (our benchmark economy and four
representative-agent economies) and the U.S economy. The upshot is that the response is similar to
those of the representative-agent economies with ψ = 2. The volatility of output of our benchmark
economy is 1.50, slightly smaller than that of the economy with ψ = 2 (1.53). The volatility of
consumption is bigger in our model, but the investment volatility is close to that with ψ = 2.
Hours are fairly volatile: the standard deviation of hours both in its absolute term and relative to
output and labor productivity are close to those of the representative agent model with ψ of 4. This
is partially due to a compositional bias. Typically new workers are less productive than existing
workers, making employment more volatile than total hours in efficiency units. In our model,
the volatility of hours in efficiency unit is .78, 30% smaller than that of employment, comparable
to previous models with a complete market (Cho and Rogerson, 1988; Chang, 2000). Likewise,
the standard deviation of marginal product of labor is .88, 15% bigger than the average labor
19
For each representative-agent economy, the parameter B is adjusted to yield h̄ = 1/3 in the steady state.
16
productivity (.77) comparable to Bils (1985).
The response of aggregate hours to shifts in demand is moderate as the reservation wage
distribution is scattered. For example, the dispersion of individual productivity, measured by the
cross-sectional standard deviation of log wages in the PSID (.549), is larger than that of aggregate
productivity, measured by the time-series standard deviation of aggregate TFP (.0224) by a factor
of nearly 23.20
Due to incomplete financial market and borrowing constraint, the aggregation theorem cannot
be applied in our model. However, it still might be of interest to estimate the equation (12) as if the
aggregate time series were generated by a representative-agent model. When we use 3,000 periods
of aggregate time series from our benchmark economy, the estimate for ψ is 1.03 by the OLS.21
The estimates are .86 and 1.13, respectively, for the models with σx = .28125 and ρx = .92.22
These numbers are not far from those we obtained based on the reservation wage distribution. For
comparison to micro labor supply elasticities, we also construct annual panel data from our model
and estimate the ψ for individual workers. Since we have an extensive margin only, our estimates are
approximations relying on time aggregation: the variation of annual hours stems from the changes
in quarterly labor-market participation. The OLS estimate based on the panel data consisting of
10,000 workers for 30 years from the benchmark model is .37. Table 8 summarizes the estimate of
ψ for individual and aggregate labor supply for each model.23 The aggregate elasticities tend to be
20
We abstract from the variation of hours per worker to isolate the effect of participation margin only. Allowing
for an intensive margin may generate a bigger response of labor supply. However, under the small intertemporal
substitution elasticity of leisure, the effect on aggregate labor supply would be small.
21
In general, estimation of labor supply using aggregate time series data suffers an identification problem due to
difficulty in finding a good instrument. Hall (1980) estimates the aggregate labor supply elasticity with instruments
such as military spending, political party of the president, and oil prices.
22
R2 ’s of the regressions are between 0.33 and 0.40.
23
With a discrete choice of labor supply, R2 ’s of panel data regressions are very low. They are between 0.03 and
17
bigger than individual elasticities, which is in line with typical micro estimates in the literature.
5
Conclusion
Labor supply elasticity is at the heart of macroeconomic research. It is a cornerstone of the equilib
rium approach that relies on intertemporal substitution of leisure. In a disequilibrium approach, in
which the role of labor supply is dismissed in the short run, its slope is still crucial for the welfare
loss of the economy departing from the equilibrium.
Aggregate models based on the intertemporal substitution of leisure often assume a high ag
gregate labor supply elasticity, despite the low estimates from empirical studies based on individual
data. The fact that fluctuations of hours are mainly accounted for by participation suggests that
the aggregate labor supply has little to with the intertemporal substitution, but rather with the
distribution of reservation wages among heterogeneous workers.
We investigate the mapping from individual to aggregate labor supply using a general equilib
rium where heterogeneous agents decide on labor-market participation and the capital market is
incomplete. The nature of heterogeneity among workers is calibrated using panel data on individual
wages. Worker flows between employment and nonemployment, and cross-sectional distributions
of earnings and wealth in our model are comparable to those in the U.S. data. While the model
economy is parsimonious, we find that the aggregate labor supply elasticity of such an economy is
around 1, bigger than micro estimates but smaller than those often assumed in aggregate models.
As the model abstracts from other important factors affecting labor supply decisions, it would be
interesting to incorporate preference heterogeneity (e.g., life-cycle effect or home production), par
tial insurance (joint labor supply of the married), or returns to working other than current wages
(learning by doing) into the model.
0.08 whereas t-statistics are well over 200.
18
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24
A
Appendix
A.1
The PSID Data
We use the non-poverty sample of heads of households and spouses for 1971-1992. The wage data
for spouses are available only since 1979. Wages are annual hourly earnings (annual labor incomes
divided by annual hours). Nominal wages are deflated by the Consumer Price Index. The base year
is 1983. Workers who worked less than 100 hours per year or whose hourly wage rate was below $1
(in 1983 dollars) are viewed as nonemployed even though their employment status is reported as
employed in the survey. We use workers who were employed in non-agricultural sectors and were
not self-employed. We also restrict the sample to hourly earnings less than or equal to $500. In
the PSID, the wealth data are available for 1984, 1989 and 1996 only. We use the 1984 data as
the date falls around the mid point of our sample. The distributions are similar across the three
surveys. The wealth is defined as the sum of net worth of all family members resulting from the
aggregation of the following components: house (main home), other real estate, vehicles, farms and
businesses, stocks, bonds, cash accounts, and other assets. Family earnings is the sum of earnings
of head and spouses. The descriptive statistics for our PSID data are summarized in Table A.1.
Table A.1: Summary Statistics for the PSID Data
Variable
Real Wage Rate (in 1983 $)
Log Real Wage
Annual Hours of work
Age
Years of schooling
Gender (male =1)
Family Wealth
Family Earnings
Mean
S.D.
Min
Max
Obs.
10.96
2.24
2062.1
38.2
13.03
.615
60,524
17,485
6.81
.549
551.7
11.17
2.51
.48
231,103
19,181
1.00
.00
100
18
0
0
-134,556
0
175.19
5.16
5,000
65
17
1
9,560,000
530,000
44,717
44,717
44,717
44,717
44,717
44,717
5,515
5,515
Note: Statistics are based on 1971-1992 surveys. Family wealth and earnings are based on those
reported in the 1984 survey.
25
A.2
Conversion between Annual and Quarterly Variances
After controlling for aggregate effect, the individual wage evolves according to x. Since the wages
in the PSID are annual averages:24
�τ =
ln x
�
1�
ln x(τ,1) + ln x(τ,2) + ln x(τ,3) + ln x(τ,4) ,
4
�τ is annual average and x(τ,q) denotes the wage of the qth quarter in year τ . With AR(1)
where x
process for quarterly x, the stochastic process for the annual average is:
�
�
4−j
4
�
1�
4
k
�τ = ρ ln x
�τ −1 + ε�x,τ , ε�x,τ =
ln x
εx,(τ,j)
ρ .
4
j=1
k=0
The quarterly values of ρx and σx are computed from the annual estimates using ρ�x = ρ4x and
�
�4 ��4−j k �2
1
�x = σx 4
σ
.
j=1
k=0 ρ
A.3
The Worker-Flow Data
We compute the quarterly worker flows from the seasonally adjusted monthly hazard rates in
the CPS for 1967:II-2000:IV, obtained from Robert Shimer, as follows. There are three possible
labor-market states: employment, unemployment, and non-labor-force, denoted by e, u, and n,
respectively. The flow of workers from labor-market status i to j during the quarter fij is computed
as:
fij = ī ×
�
h1ik h2kl h3lj ,
k,l∈{e,u,n}
i, j ∈ {e, u, n},
where ī denotes the number of workers in status i in the beginning of the quarter, and hm
kl is the
monthly hazard rate from status k to l in the m-th month of the quarter. This takes into account all
possible paths, direct and indirect, from i to j during a quarter. This also avoids a potential double
counting in a simple sum of monthly flows. Because of survey redesigns and privacy restrictions,
hazard rates are not available in January 1976, January 1978, July 1985, October 1985, January
�4
Note that 41 q=1
ln x(τ,q) can be interpreted as a log-linear approximation of the arithmetic average ln x
�τ =
�
ln[ 41 4q=1 x(τ,q) ].
24
26
1994, and June to October 1995. For these months we interpolate the values from the nearby
periods.
A.4
Computational Procedures
A.4.1
Steady State Equilibrium
The distribution of workers µ(x, a) as well as factor prices are invariant in the steady state. In
finding the invariant µ, we use the algorithm suggested by Rı́os-Rull (1999). We search for the
discount factor β that clears the capital market given the quarterly rate of return of 1 percent.
Computing the steady state equilibrium amounts to finding the value functions, the associated
decision rules, and time-invariant measure of workers. Details are as follows:
1. First, we choose the grid points for x and a. The number of grids are denoted by Nx and
Na . For the benchmark model, Nx = 17 and Na = 1, 163. The asset holding ai is in the
range of [0, 250], where the average asset holding is 12.63. The grid points of assets are not
equally spaced. We assign more points on the lower asset range to better approximate savings
decisions of workers with lower assets. For example, at the asset range close to the borrowing
constraint, the grid points are as fine as .02, which is approximately 2.5% of the average
quarterly labor income (these individuals have negligible interest income); at a high end of
the asset range, the grid increases by .4, which corresponds to 10 – 20 percent of the average
quarterly total income (these individuals have larger interest income than labor income). For
productivity, xj , we construct grid vectors of Nx equally spaced points in which ln xj ’s lie on
�
the range of ±3σx / 1 − ρ2x .
2. Given β, we solve the individual optimization problem in (1), (2), and (3) at each grid point of
the individual states. In this step, we also obtain the optimal decision rules for asset holding
a� (ai , xj ) and labor supply h(ai , xj ). This step involves the following procedure:
27
(a) Initialize value functions V0E (ai , xj ), V0N (ai , xj ), and V0 (ai , xj ).
(b) Update value functions by evaluating the discretized versions of (1), (2), and (3):
V1E (ai , xj )
�
�
�
¯ j + (1 + r)ai − a� , 1 − h
¯
= max u whx
+β
Nx
�
(A.4.1)
�
�
�
E �
N �
�
�
�
max V0 (a , xj ), V0 (a , xj ) πx (xj |xj ) ,
j � =1
V1N (ai , xj )
�
�
�
= max u (1 + r)ai − a� , 1
+β
Nx
�
�
max
(A.4.2)
�
V0E (a� , xj � ), V0N (a� , xj � )
�
πx (xj � |xj ) ,
j � =1
and
�
�
V1 (ai , xj ) = max V1E (ai , xj ), V1N (ai , xj ) ,
(A.4.3)
where πx (xj � |xj ) is the transition probabilities of x, which is approximated using Tauchen’s
(1986) algorithm.
(c) If V1 and V0 are close enough for all grid points, then we found the value functions.
Otherwise, set V0E = V1E and V0N = V1N , and go back to step 2-(b).
3. Using a� (ai , xj ), πx (xj � |xj ) obtained from step 2, we obtain time-invariant measures µ∗ (ai , xj )
as follows:
(a) Initialize the measure µ0 (ai , xj ).
(b) Update the measure by evaluating the discretized version of (9):
µ1 (ai� , xj � ) =
Na �
Nx
�
1ai� =a� (ai ,xj ) µ0 (ai , xj )πx (xj � |xj )
(A.4.4)
i=1 j=1
(c) If µ1 and µ0 are close enough for all grid points, then we found the time-invariant
measure. Otherwise, replace µ0 with µ1 , and go back to step 3(b).
28
�
�1−α
4. We calculate the real interest rate as a function of β, i.e., r(β) = α K(β)/L(β)
− δ,
where K(β) =
�Na �Nx
i=1
∗
j=1 ai µ (ai , xj )
and L(β) =
�Na �Nx
i=1
j=1 xj h(ai , xj )µ
∗ (a , x ).
i j
Other
aggregate variables of interest are calculated using µ∗ and decision rules. If r(β) is close
enough to the assumed value of the real interest rate, we find the steady state. Otherwise,
we choose a new β and go back to step 2.
A.4.2
Equilibrium with Aggregate Fluctuations
Approximating the equilibrium in the presence of aggregate fluctuations requires us (i) to include
the measure of workers and the aggregate productivity shock in the list of state variables, and (ii)
to keep track of the evolution of the measure µ over time. Since µ is an infinite dimensional object,
it is almost impossible to implement these tasks as they are. We follow the procedure suggested by
Krusell and Smith (1998); agents are assumed to make use of its first moment only in predicting the
law of motion for µ. Therefore, computing the equilibrium with aggregate fluctuations amounts to
finding the value functions, decision rules, and law of motion for the aggregate capital within the
class of log-linear functions in K and λ. The same method is used in Gomes et al. in their analysis
on equilibrium unemployment rates. Details are as follows:
1. In addition to the grids for individual state variables specified above, we choose 11 grid points
for the aggregate capital K in the range of [.9K ∗ , 1.1K ∗ ], where K ∗ denotes the steady state
aggregate capital. In our numerous simulations, the capital stock has never reached the upper
or lower bound. The aggregate productivity λ has 9 grid points and its transition probability
πλ (λ� |λ) is calculated using Tauchen (1986)’s algorithm.
2. Let the parametric law of motion for the aggregate capital take a log linear in K and λ:
ln Kt+1 = κ00 + κ01 ln Kt + κ02 ln λt .
(A.4.5)
In order for individuals to make their decisions on savings and labor supply they have to
29
know (or predict) the interest rate and wage rate for an effective unit of labor. While the
factor prices depend on aggregate capital and labor, aggregate labor input is not known to
individuals at the moment when they make decisions. Thus, individuals need to predict
the factor prices. These predictions on factor prices, in turn, must be consistent with the
outcome of individual actions, the factor market clearing in (7) and (8). We also assume that
individuals predict the market wage and the interest rate using a log-linear function of K and
λ:
ln wt = b00 + b01 ln Kt + b02 ln λt .
(A.4.6)
ln (rt + δ) = d00 + d01 ln Kt + d02 ln λt
(A.4.7)
and
3. We choose the initial values for the coefficients κ0 ’s, b0 ’s and d0 ’s. Good initial values may
come from a representative-agent model.
4. Given the law of motion for the aggregate capital and the prediction functions for factor prices,
we solve the individual optimization problem in (1), (2), and (3). This step is analogous to
step 2 in the steady state computation:
(a) We have to solve for the value functions and the decision rules over a bigger state space.
Now the state variables are (a, x, K, λ).
(b) Computation of the conditional expectation involves the evaluation of the value functions
not on the grid points along the K dimension since K � predicted by (A.4.5) need not be
a grid point. We polynomial-interpolate the value functions along K dimension when
necessary.
5. Using a� (ai , xj , Kl , λm ), h(ai , xj , Kl , λm ), πx (xj � |xj ), πλ (λm� |λm ), and the assumed law of
motion for the aggregate capital, we generate a set of artificial time series data {Kt , wt , rt }
30
of the length of 3,000 periods. Each period, {Kt , wt , rt } is calculated by aggregating labor
supply and assets of 50,000 individuals.
6. We obtain new values for coefficients κ1 ’s, b1 ’s and d1 ’s by the OLS from the simulated data.
If κ1 ’s, b1 ’s and d1 ’s are close enough to κ0 ’s, b0 ’s, and d0 ’s, respectively, we find the law of
motion. Otherwise, we update coefficients by setting κ0 = κ1 , b0 = b1 ’s and d0 = d1 ’s, and
go back to step 4.
The estimated law of motion for capital and prediction functions and their accuracy, measured
by R2 for the prediction equations are as follows.
• the law of motion for aggregate capital in equation (A.4.5):
ln Kt+1 = .1247 + .9508 ln Kt + .0997 ln λt ,
R2 = .9999
• the market wage rate in equation (A.4.6):
ln wt = −.2621 + .4442 ln Kt + .8068 ln λt ,
R2 = .9940
• the interest rate in equation (A.4.7):
ln (rt + δ) = −1.3491 − .7897 ln Kt + 1.3434 ln λt ,
R2 = .9691
The law of motion for aggregate capital provides the highest accuracy. The wage function is
more accurate than the interest rate function. Overall, predictions functions are fairly precise
as R2 ’s are close to 1. Finally, as the agents make decisions based on the predicted prices,
the actual employment may not be necessarily consistent with the predicted prices. We also
used the method suggested in Rı́os-Rull in which labor market clearing is imposed as an extra
step. (See Rı́os-Rull (1999) for details.) The result with a two-step process was very similar
to the one reported here as the predicted prices approximate the actual prices very closely.
31
Table 1: Estimates of Stochastic Process for Idiosyncratic Shocks
Annual Estimates
ρ�x
σ
�x
All
All - with gender dummy
Male
Married
Single
Female
Married
Single
With individual characteristics
All
Male
Married
Single
Female
Married
Single
Quarterly Values
ρx
σx
obs
.818 (.002)
.792 (.002)
.291
.288
.951
.944
.225
.225
47,114
47,114
.809 (.003)
.760 (.010)
.265
.328
.948
.934
.206
.259
24,294
3,265
.722 (.006)
.773 (.008)
.311
.298
.922
.938
.249
.234
11,684
5,525
.743 (.003)
.284
.928
.225
47,114
.761 (.004)
.760 (.011)
.261
.321
.934
.915
.206
.259
24,294
3,265
.717 (.006)
.708 (.009)
.311
.291
.920
.917
.249
.234
11,684
5,525
Note: The annual estimates are based on OLS of equation (11) using the wage data from
the PSID 1971-1992. Numbers in parenthesis are standard errors. The corresponding quar
terly values are calculated as described in the Appendix. For the second set of estimates,
individual characteristics are controlled by including gender, age, age square, and years of
schooling in the regression.
Table 2: Parameters of the Benchmark Economy
Parameter
α = .64
β = .979852
γ = .2
B = 1.025
h = 1/3
ρx = .95
σx = .225
ā = −2.0
Description
Labor share in production function
Discount factor
Intertemporal substitution elasticity of leisure
Utility parameter
Steady state hour
Persistence of idiosyncratic productivity shock
Standard deviation of innovation to idiosyncratic productivity
Borrowing constraint
32
Table 3: Labor-Market Steady States
Variable
CPS
Employment rate
Flow out of employment
Flow into employment
Hazard rate out of nonemployment
Hazard rate out of employment
σx = .225
ρx = .95
Model
σx = .28125
ρx = .95
σx = .225
ρx = .92
60.21
5.92
5.92
14.89
9.84
60.20
5.64
5.64
14.16
9.36
60.21
6.85
6.85
17.22
11.38
60.15
7.07
6.88
17.75
11.80
Note: All variables are in percentage. The CPS statistics are quarterly averages for 1967:II2000:IV as described in Appendix A.3. σx and ρx denote, respectively, the standard devia
tion of innovations and persistence of idiosyncratic productivity shocks.
Table 4: Gini Indices for Wealth and Earnings
Variable
Wealth
Earnings
Earnings (non-zeros)
PSID
σx = .225
ρx = 95
Models
σx = .28125
ρx = .95
σx = .225
ρx = .92
.64
.63
.39
.65
.68
.47
58
.57
.29
.76
.53
.42
Note: The PSID statistics reflect the family wealth and earnings in the 1984 survey.
33
Table 5: Characteristics of Wealth Distribution
Quintile
3rd
4th
1st
2nd
5th
Total
SCF
Share of wealth
Group average/population average
Share of earnings
-.39
-.02
7.05
1.74
.09
14.50
5.72
.29
16.48
13.43
.67
20.76
79.49
3.97
41.21
100
1
100
PSID
Share of wealth
Group average/population average
Share of earnings
-.52
-.02
7.51
.50
.03
11.31
5.06
.25
18.72
18.74
.93
24.21
76.22
3.81
38.23
100
1
100
Benchmark Model
Share of wealth
Group average/population average
Share of earnings
-2.68
-.13
10.79
1.78
.09
15.94
10.11
.51
19.43
24.48
1.22
23.44
66.31
3.33
30.39
100
1
100
Note: The SCF statistics are from Dı́az-Giménez, Quadrini, and Rı́os-Rull (1997). The
PSID statistics reflect the family wealth and earnings from the 1984 survey.
Table 6: Implied Elasticity from the Reservation Wage Distribution
Model
σx = .225, ρx = .95
σx = .28125, ρx = .95
σx = .225, ρx = .92
Employment Rate
E = 58% E = 60% E = 62%
1.19
.96
1.88
1.09
.89
1.78
1.00
.82
1.61
Note: The numbers reflect the elasticity of labor-market participation rate with respect to
reservation wage (evaluated at employment rates of 58, 60, and 62 percent) based on the
reservation wage distribution in the steady state.
34
Table 7: Comparison with Representative-Agent Economies
Benchmark
σ(Y )
σ(C)
σ(I)
σ(N )
σ(N )/σ(Y )
σ(N )/σ(Y /N )
1.50
.57
4.61
1.00
.67
1.31
Divisible Labor
ψ = .4 ψ = 1 ψ = 2 ψ = 4
1.22
.41
3.72
.25
.20
.23
1.38
.45
4.26
.50
.36
.55
1.54
.49
4.81
.75
.49
.91
1.71
.52
5.39
1.01
.59
1.37
U.S. Data
1948:I-2000:IV
2.22
.96
4.67
1.78
.80
1.61
Note: All variables are de-trended by the H-P filter. “Divisible Labor” denotes the
representative-agent model with different Frisch labor supply elasticities (ψ). The statis
tics for data are based on per capita values (divided by civilian noninstitutional population
over 16) from the Citibase: Y = nonfarm business GDP (GPBUQF); C = consumption of
non durables and services (GCNQ+GCSQ); I = non-residential fixed private investment
(GIFQ); N = total employed hours in private non-agricultural sector based on the establish
ment survey (LPMHU).
Table 8: Compensated Labor Supply Elasticities From the Model-Generated Data
Model
σx = .225, ρx = .95
σx = .28125, ρx = .95
σx = .225, ρx = .92
Individual
Aggregate
.37
.44
.69
1.03
.86
1.13
Note: All estimates are based on the OLS of equation (12) using model-generated data. The
individual labor supply elasticities are based on the annual panel data of 10,000 workers for
30 years. The aggregate estimates are based on the quarterly time series of 3,000 periods.
35
100
o
90
45
σx=.225 ρ=.95
σ =.28125 ρ=.95
x
σ =.225 ρ=.92
80
U.S.
x
70
60
50
40
30
20
10
0
0
10
20
30
40
50
60
70
80
90
100
90
100
Figure 2: Lorenz Curves for Earnings
100
90
45o
σx=.225 ρ=.95
σx=.28125 ρ=.95
σx=.225 ρ=.92
80
U.S.
70
60
50
40
30
20
10
0
0
10
20
30
40
50
36
60
70
80
Figure 3: Reservation Wage Schedule
A: All Asset Levels
4
3.5
x 10
Reservation Wages
3
2.5
2
1.5
1
0.5
0
−2
0
2
4
6
8
10
12
5
x 10
B: Assets less than $200,000
Reservation Wages
10000
8000
6000
4000
2000
0
−5
0
5
10
Assets in 1983 dollars
15
20
4
x 10
Note: The graph denotes the reservation wage schedule of the benchmark model. Wages
(quarterly earnings) and assets are in 1983 dollars.
37
Figure 4: Reservation Wages and Participation Rates
18000
16000
Reservation Wages
14000
12000
10000
8000
6000
4000
σx = .225 ρ = .95
σ = .28125 ρ = .95
x
σ = .225 ρ = .92
2000
x
0
0
10
20
30
40
50
60
70
80
90
100
Participation Rates (%)
Note: The graph denotes the inverse cumulative distribution functions of reservation wages.
Wages are quarterly earnings in 1983 dollars.
38
@FedFRASER