-
Reservation wages and working hours for
recently unemployed US women
Tue Grgens *
Economics RSSS, Australian National University, Canberra ACT
0200, Australia
Received 1 September 2000; received in revised form 29 January
2002; accepted 29 January 2002
Abstract
This paper estimates a structural model of job search behavior
where jobs are characterized by
both wages and working hours. Worker heterogeneity is taken into
account by including covariates in
the job offer distribution and the utility function. The
estimates are bias-corrected using a statistical
model of measurement error designed to accommodate recent
evidence on the empirical distribution
of measurement errors. The results suggest that the difference
between full time and part time
reservation wages vary from 16% to 31%, depending on the
characteristics of the individual. Thehypothesis that full and part
time reservation wages are identical is rejected. D 2002 Elsevier
Science
B.V. All rights reserved.
JEL classification: C24; J22; J64
Keywords: Job search; Reservation wage; Female labor supply;
Measurement error; Structural modeling
1. Introduction
This paper estimates a structural model of job search behavior
where jobs are
characterized by wages and working hours. Firms are assumed to
offer take-it-or-leave-
it contracts stipulating both the wage and working hours. The
model implies that workers
have conditional reservation wages which depend on the hours
offered, and the main
objective of the paper is to investigate empirically how much
conditional reservation
wages depend on hours. The data, taken from the Current
Population Survey, consist of
accepted wagehours pairs for a representative sample of US women
who were working
in March 1990 and who experienced a spell of unemployment in
1989.
0927-5371/02/$ - see front matter D 2002 Elsevier Science B.V.
All rights reserved.
PII: S0927 -5371 (02 )00007 -6
* Tel.: +61-2-6125-3369; fax: +61-2-6125-0182.
E-mail address: [email protected] (T. Grgens).
www.elsevier.com/locate/econbase
Labour Economics 9 (2002) 93123
-
Most models of workers job search focus exclusively on the wage
rate and ignore
working hours and other nonwage job characteristics.1 They are
wage search models in
the sense that the workers optimal search behavior is
characterized by a reservation wage
strategy: a job offer is accepted if and only if the offered
wage exceeds the reservation
wage. For working hours, the underlying assumption is typically
that workers may freely
choose their working hours given the wage. This paper assumes
that firms offer contracts
which fully specify the job. Workers have preferences defined
over jobs, and it is shown
that their optimal search behavior is a reservation utility
strategy. Corresponding to the
reservation utility, there is a reservation wage function which
yields a conditional
reservation wage given the nonwage characteristics of a job. If
the reservation wage
function is constant, then the conditional reservation wages are
identical, and thus the
model reduces to a wage search model with a single reservation
wage.
Ignoring the relationship between conditional reservation wages
and working hours can
lead to incorrect inference and misleading policy advice. For
example, in wage search
models, the reservation wage is often estimated by the minimum
observed wage in the
sample. If the conditional reservation wages vary with nonwage
job characteristics, then
the minimum observed wage will estimate the lowest of the
conditional reservation wages.
Since some conditional reservation wages are higher, the model
will underestimate the
probability of an offer being rejected. This could lead to
incorrect conclusions about such
questions as whether unemployment is caused mainly by infrequent
job offers or by high
offer rejection probabilities.
It is therefore necessary to investigate the effect on
reservation wages of all job
characteristics such as working hours, health insurance, paid
vacation and sick days,
pensions and so forth. Here the focus is on working hours,
because data on working hours
are readily available and the importance of hours in other
contexts is already well
documented.2 The empirical model distinguishes only between part
time and full time
jobs. Since many of the nonwage job characteristics mentioned
are only available to full
time workers, one of the benefits of this simplification is that
the model may implicitly
capture some of the effects of job characteristics excluded from
the analysis.
The underlying theoretical model is based on wage search models
with infinite search
horizon, such as those by Mortensen (1986) and Mortensen and
Neumann (1988),
extended to incorporate nonwage job characteristics as in Kiefer
(1987). The structural
parameters of the model are estimated by maximum likelihood
allowing for worker
heterogeneity and measurement error. Since it is computationally
easier to estimate a
search model on a sample of homogeneous workers, some authors
choose to split their
sample into approximately homogeneous groups and estimate the
model separately on
each subsample. For efficiency reasons, worker heterogeneity is
here accounted for by
including covariates in the model, as done for instance by
Wolpin (1987) and Blau
(1991).
1 See the survey by Devine and Kiefer (1991).2 See the surveys
by Killingsworth and Heckman (1986), Nickell (1986) and Pencavel
(1986). The
importance of other nonwage benefits has been studied, for
example, by Woodbury (1983).
T. Grgens / Labour Economics 9 (2002) 9312394
-
It is well known that most labor market data contain large
measurement errors. This is a
problem in any empirical work, but search models are
particularly sensitive to measure-
ment error. In wage search models this is often dealt with in an
ad hoc manner, for example
by removing observations of low wages from the sample. This
paper follows the example
of Wolpin (1987), Blau (1991) and Christensen and Kiefer (1994)
among others, and
incorporates a statistical model of measurement error. However,
whereas previous studies
have based their measurement error model on computational
practicability, the model
proposed here is designed to fit recent findings on the
empirical distribution of measure-
ment errors by Rodgers et al. (1993) and Bound et al. (1994).
This paper also investigates
how robust the empirical results regarding working hours and
conditional reservation
wages are to changes in the assumptions about the measurement
errors.
This paper appears to be the first to examine the relationship
between conditional
reservation wages and working hours for women. The relationship
for recently unem-
ployed US men was studied by Blau (1991). Although Blau found
that conditional
reservation earnings clearly depend on hours, he did not reject
the hypothesis of constant
conditional reservation wages for recently unemployed US men.
The finding of this paper
is that the reservation wage function is not constant for
women.
The main conclusion of the paper is that the conditional
reservation wages of the
women in the sample are from 16% to 31% higher for full time
than for part time jobs,depending on the characteristics of the
worker. Women whose expected wage offers are
relatively high, e.g. older, well-educated women living in a
central city in the Northeast,
have higher part time than full time reservation wages. Women
whose expected wage
offers are relatively low, e.g. younger women with less
schooling living in the rural
Midwest, have higher full time than part time reservation wages.
Receipt of benefits while
unemployed increases part time reservation wages substantially,
but has negligible effects
on full time reservation wages. The presence of young children
and husbands in the
household has a positive, but very small effect on the womens
reservation wages. Only if
the husband earns a high hourly wage (US$15 or more) do the
reservation wages increase
significantly.
Observed wages and hours reflect both opportunities (the job
offer distribution) and
individual choices (preferences). The structural framework
allows one to disentangle such
demand and supply factors. For instance, it is found that the
mean wage offers
increase monotonically with the level of education. The implied
returns to schooling
suggest that there are substantial premiums to graduating from
high school and college,
whereas the premium for attending college but leaving before
graduation is smaller. It is
also found that differences in observed wages misrepresent
returns to schooling. Women
with less education tend to receive lower wage offers, but they
also reject a larger
proportion of them. The difference in wages across education
groups is therefore smaller
when measured on observed wages, rather than offered wages. This
result is similar to the
findings by Eckstein and Wolpin (1995) for young men.
Finally, the estimates suggest that measurement errors in
observed wages are large, to
the extent that there is as much noise as information in the
data, and that the distribution of
measurement errors is strongly leptokurtic (fat-tailed). These
results agree with empirical
findings from data validation studies and imply that the common
assumption of
lognormally distributed measurement errors in wages is
inadequate. The estimated
T. Grgens / Labour Economics 9 (2002) 93123 95
-
reservation wages are somewhat sensitive to the specification of
the distribution of
measurement errors, but the qualitative conclusion that hours
matter is robust.
2. A job search model
This section presents a general model of workers job search
where workers have
preferences defined over wage and nonwage characteristics. The
focus is on the optimal
search behavior of workers. The behavior of firms is taken as
given and equilibrium issues
are not discussed. As mentioned, the basic structure is taken
from wage search models by
for example Mortensen (1986) and Mortensen and Neumann (1988),
extended to
incorporate nonwage job characteristics as in Kiefer (1987).
2.1. Workers reservation utility
Consider an environment in which workers are either employed or
unemployed. A job
is represented by a pair (w,h), where w is the wage rate and h
is a vector of nonwage job
characteristics such as working hours, working conditions,
fringe benefits, pensions, and
so on. Firms design take-it-or-leave-it job offers (wo,ho);
there is no negotiation between
employers and workers about job contracts. Workers receive job
offers according to a
Poisson process with a state-dependent parameter: unemployed
workers receive offers at
constant rate k0, while employed workers receive offers at
constant rate k1. There is norecall; that is, once refused, a job
offer is no longer available. Accepted jobs terminate
randomly according to a Poisson process with parameter d. The
distribution of job offers,offer arrival times and job termination
times is constant over time and known to
everybody.
Workers live forever and maximize expected discounted lifetime
utility. The rate of
time preference, q, is constant over time. The instantaneous
utility when employed at a job(w,h) is U(w,h), where U is strictly
increasing and differentiable with respect to w.3 The
instantaneous utility when unemployed, denoted by U0, is the net
of search costs and does
not vary over the duration of the unemployment spell. As utility
functions are determined
only up to an affine transformation, a normalization is needed.
A convenient choice is
U(0,h1) = 0 and U(1,h1) = 1 for some h1.
The defining characteristic of a wage search model is that the
optimal search behavior
is a reservation wage strategy: a worker should accept a job
offer if and only if the offered
wage is larger than a threshold, termed the reservation wage. In
the wage search literature,
it is customary to assume that workers maximize the expected
present value of the wage
stream. In other words, the instantaneous utility of work is
proportional to the wage. This
special case will be referred to as the Constant Marginal
Utility with respect to the Wage
(CMUW) wage search model.
3 A similar specification was used by Kiefer (1987). The
function U can be formed by assuming that income
is fully consumed in each period and substituting the budget
constraint into a direct instantaneous utility function
defined over consumption, leisure and other nonwage job
characteristics.
T. Grgens / Labour Economics 9 (2002) 9312396
-
Results derived for the CMUW wage search model can be used to
characterize the
optimal search behavior of workers in the present, more general
model. From a workers
perspective, all relevant information about a job is summarized
in her (instantaneous)
utility function. Therefore, job search can be viewed as search
for utility. In the CMUW
wage search model, workers receive wage offers and maximize the
expected present value
of their wage stream. Similarly, in the job search model,
workers receive utility offers and
maximize the expected present value of the utility stream.
Mathematically, the two
models, and hence their solutions, are identical. The optimal
search behavior in the
CMUW wage search model is a reservation wage strategy.
Equivalently, the optimal
search behavior in the utility search model is a reservation
utility strategy, where a job
offer should be accepted if and only if its associated
instantaneous utility exceeds the
reservation utility.
To be specific, associate with each job offer (wo,ho) a utility
offer uo =U(wo,ho), and let
Q be the distribution function of the utility offers; that is,
define Q by
Qu PruoVu PrUwo,hoVu: 1
Results derived for the CMUW wage search model show that for
employed workers the
reservation utility trivially equals the instantaneous utility
of the current job, whereas for
unemployed workers the reservation utility r is the unique
solution to the equation
r U0 kZ lr
1 Qu1 j1 Qu du, 2
where k=(k0 k1)/(q + d) and j = k1/(q + d). Notice that k and j
are unitless parameters. InEq. (2), rU0 represents the marginal
cost of rejecting an offer of utility r, while thesecond term on
the right-hand side is the expected marginal benefit of continued
search,
given that only offers of utility r or more are accepted.
2.2. Distribution of accepted jobs
The implied conditional distribution of accepted jobs is simply
the job offer distribution
truncated at the indifference curve corresponding to the
reservation utility level. Specif-
ically, for a job offer (wo,ho) to be acceptable to an
unemployed worker, it must satisfy
U(wo,ho)z r. The probability of this event is 1Q(r). In the case
where the nonwagecharacteristics are all discrete, let w(wjh)
denote the conditional density of wo at w givenho = h and let p(h)
be the marginal probability of ho = h. The (mixed) density w* of
the firstjob accepted by an unemployed worker is then
w*w,h ww j hph1 Qr 1Uw,hzr, 3
where 1() denotes the indicator function (that is, 1(A) = 1 if
the event A is true and 1(A) = 0otherwise). The maximum likelihood
estimates presented in Section 4 are based on a
version of this density adjusted for measurement errors.
T. Grgens / Labour Economics 9 (2002) 93123 97
-
2.3. Wage search
There are several ways job search can reduce to wage search. A
possibility, which is
nested within the present model, is that workers do not care
about job characteristics other
than the wage. This case is easily tested empirically. Of
course, since some wage search
models are not included under the null, such a test is not a
definitive test of the wage
search model.4
Define the conditional reservation wage function W by
W h U1r,h, 4
where the function U 1, defined by U(U 1(u,h),h) = u, represents
an indifference curve;that is, U 1(u,h) is the wage that yields
utility u when combined with nonwagecharacteristics h. The function
W is the indifference curve corresponding to the reservation
utility level r, and a job offer (wo,ho) is acceptable if and
only if wozW(ho). By definition,job search reduces to wage search
when W is constant for all offered h. This hypothesis is
tested in Section 4.
3. Parameterization and measurement error
The parameterization is guided by several considerations
including the availability and
limitations of data, heterogeneity among workers, and
measurement errors. The model is
estimated by the maximum likelihood method. The likelihood
function is based on the
density given in Eq. (3) of the first job accepted by an
unemployed worker.
3.1. Data and dependent variables
The data set is a subsample of the March 1990 Current Population
Survey (CPS)
consisting of 730 females who were working at the time of the
interview and who
experienced a spell of unemployment in 1989. Details are
provided in Appendix A. The
CPS was chosen because it is representative of the entire US
population and because it is
large. A minor drawback of the CPS is that while it has
information on whether a worker
had a spell of unemployment during the previous year and on the
current job, it is not
known whether the current job is the first after the
unemployment spell, as assumed in
deriving the density in Eq. (3). However, given that the maximum
period between the end
of the unemployment spell and the beginning of the current job
is only about 15 months,
assuming the current job is similar to the first seems rather
harmless.
A more important drawback of the CPS is that the information
about the duration of
previous unemployment spells is very limited. It has information
about the total number of
4 Burdett and Mortensen (1978) among others considered models
where firms do not care about working
hours and workers choose their hours freely given the offered
wage rate. Bloemen (1997) estimated a wage search
model with unrestricted hours choice for a sample of recently
unemployed Dutch men. These are examples of
wage search models which are not nested within the present
model.
T. Grgens / Labour Economics 9 (2002) 9312398
-
weeks of unemployment in the previous calendar year, but it is
not known whether this
constitutes one or several spells, and it is not known if spells
were ongoing at the
beginning or the end of the calendar year. In search models
estimated on homogeneous
samples, identification of offer arrival rates often depends
crucially on the availability of
duration data (see for example Flinn and Heckman, 1982).
However, it can be shown that
data on accepted jobs are sufficient for identification in
heterogeneous samples, where the
offer arrival rates are identified from the functional form of
the relationship between the
reservation utility and worker characteristics. Since the
duration data available are
extremely imperfect and since duration data are not needed for
identification, duration
data are not used in this paper. Some consequences of this are
discussed later.
The only nonwage job characteristic considered is working hours.
Hours are usual
weekly working hours, as defined in the CPS. As can be seen in
Fig. 1, the marginal
distribution of hours in the sample is very irregular. It is
difficult to find a satisfactory
parametric distribution capable of generating this structure
without introducing a param-
eter vector of very high dimension. Consequently, in order to
economize on the number of
parameters in the model, jobs are assumed to be either part
time, h = h1, or full time, h = h2.
As a positive side effect, this simplification may capture the
difference in other nonwage
job characteristics which exists between part and full time
jobs. For example, part time
jobs typically do not qualify for health and retirement
benefits. Although these nonwage
characteristics are unavailable in the CPS data, their effect
may show up in the empirical
results, for example as workers having strong preferences for
full time jobs. Following the
CPS convention, part time is defined as 134 hours of work per
week, and full time is 35
or more hours per week. Under this definition, there are 269
part time jobs (37%) and 461
full time jobs (63%) in the sample.
The other dependent variable, wages, is defined as usual weekly
earnings divided by
usual hours. Earnings include overtime pay, commissions, tips,
and so forth.
Fig. 1. Marginal distribution of hours.
T. Grgens / Labour Economics 9 (2002) 93123 99
-
3.2. Worker heterogeneity
In the literature, the usual method of accounting for
heterogeneity is to group workers
according to their characteristics and estimate the model
separately for each group. In
wage search models, this approach has the advantage that workers
within a group have the
same reservation wage, which can be estimated consistently by
the smallest observed wage
in the group. There is then no need for solving the reservation
wage equation, and this
eases the computational burden considerably. The disadvantages
are, of course, that
grouping becomes impractical for even a modest number of
discrete covariates and
impossible if a covariate is continuous. Furthermore, grouping
is inefficient if parameter
restrictions across groups are ignored. In this paper, worker
heterogeneity is incorporated
in the traditional parametric way for efficiency reasons. The
resulting problem of solving
the reservation utility equation for each observation in each
evaluation of the likelihood
function is not impractical, given todays computing power.
The parameterization outlined below is a compromise between the
desire to include
many individual effects and the need to limit the number of
parameters. First, assume that
everyone faces the same marginal distribution of hours offers,
so
Prho hj p1 if j 1
p2 if j 2,
8 0 0.53 0.89 0.35 1.77g23, Husbands wage > 15 7.38 7.20
16.29 27.34c2 4.09 3.24 8.61 13.07
Unitless parameters
k 0.04 0.07 0.03 0.05j 0.00 0.00
Measurement errors
s 0.32 0.02 0.29 0.02g 0.38 0.18 0.00d 1.52 0.22 lcw 0.00
0.00
cc 0.00 0.00
log L 2030.27 2046.77est: estimate; se: standard error. If no
standard error is given, the parameter is fixed.
T. Grgens / Labour Economics 9 (2002) 93123106
-
living in a central city on observed wages is only 13%, compared
to the 30% effect on
offered wages mentioned earlier, and the effect of union
contracts is only 20%, whereas
the effect on offered wages is 34%.
Eckstein and Wolpin (1995) argued that in order to measure the
return to schooling it is
appropriate to compare wage offer distributions rather than
distributions of accepted or
observed wages. The wage offer distribution reflects the
opportunities available to
workers. Since not all offers are accepted, the distribution of
accepted wages reflects
available opportunities as well as individual choices. To the
extent that reservation wages
differ across education groups, differences in accepted wages
may therefore provide a
distorted picture of the return to schooling. The estimates
presented in columns I and
OLS in Table 1 confirm that the differences are substantial. The
offered wages are 34%
higher for high school graduates than for high school drop-outs.
Women, who have
attended, but not completed college, are offered 16% higher
wages than high school
graduates, and the wage offers for women, who graduated from
college, are additionally
54% higher.13 The percentages based on observed wages are much
lower, 12%, 7% and
36%, respectively, because less educated workers tend to receive
lower wage offers and
reject a larger proportion of their offers.
Estimates of annual rates of return are presented in Table 2,
together with Eckstein and
Wolpins (1995) estimates for young men.14 The estimated returns
to high school
graduation for women are remarkably similar to the return for
white young men and
much larger than the return for black young men. The return to
college education without
graduation is much lower for recently unemployed women than for
young men, while the
return to graduating from college is much higher. Part of these
differences are no doubt due
to factors such as occupational choice. Since the wages observed
by Eckstein and Wolpin
relate to the first accepted full time job after leaving school
and since wages tend to
increase over the life cycle, it is also possible that the
difference in wage measures account
for some of the difference in the estimated return to graduating
from college.
Table 2
Annual rates of return to schooling
Offered wages Observed wages
W BYM WYM W BYM WYM
Some high school to high school graduate 0.25 0.03 0.27 0.08
0.06 0.05
High school graduate to some college 0.08 0.32 0.17 0.04 0.09
0.12
Some college to college graduate 0.31 0.06 0.20 0.05
Estimates for women (W) based on model I in Table 1, estimates
for black young men (BYM) and white young
men (WYM) from Eckstein and Wolpin (1995).
13 All of these estimates are biased upward to the extent that
education is correlated with unmeasured
characteristics such as ability.14 Following Eckstein and Wolpin
(1995), the estimates are calculated assuming that jobs last
forever (i.e.
wage and hours remain constant forever), and assuming that it
takes six quarters to complete high school, eight
quarters to complete some college and another eight quarters to
graduate from college. Direct costs of schooling
are ignored. Under these assumptions, the annual return to the
next level of education are exp(4b3/6) 1,exp(4(b4 b3)/8) 1 and
exp(4(b5 b4)/8) 1, respectively.
T. Grgens / Labour Economics 9 (2002) 93123 107
-
Almost half the women in the sample have no young children, no
wage-earning
husband in the household, and did not receive benefits in the
previous year. This is a very
diverse group, slightly younger and slightly more likely to have
dropped out of high
school or college, but equally likely to have completed college,
than the population as a
whole. For these women, the estimated intercept of the
instantaneous utility function is
smaller for full time (g20 = 16.14) than for part time jobs (g1
= 0), so at very low wagesshe prefers part time over full time
jobs. However, since the marginal instantaneous utility
of the wage is higher for full time (c2 = 4.09) than for part
time jobs (c1 = 1) the ranking isreversed for all sufficiently
large wage rates. This is consistent with the common finding
that female labor supply is relatively elastic. Of course, the
results are not directly
comparable since the labor supply literature assumes that
workers can choose their
working hours freely at a given wage rate.
Fig. 2 shows the estimated utility function for these women,
with Uj(w) =U(w,hj). The
solid horizontal line represents the utility when unemployed for
a woman who did not
receive benefits. Her utility when unemployed (net of search
costs) is 3.76, so for wages
lower than US$3.76 she prefers not to work. For wages between
US$3.76 and US$5.22
she prefers a part time job over a full time job. At any wage
higher than US$5.22, a full
time job is preferred. Receipt of unemployment benefits
increases the utility when
unemployed to 4.90, represented by the dashed line in Fig. 2.
This almost eliminates
the range of wages where part time employment is preferred.
In general, an increase in working hours has two effects on
utility: a negative direct
effect, since the marginal utility of work is negative, and a
positive indirect effect, since
more working hours implies higher earnings, and higher earnings
implies higher
consumption, and the marginal utility of consumption is
positive. In the present context,
this decomposition is less useful, because it is difficult to
measure the marginal utility of
work when hours are discrete. However, the vertical distance
between the curves U2 and
Fig. 2. No husband, no children.
T. Grgens / Labour Economics 9 (2002) 93123108
-
U1 in Fig. 2 represents the net effect of increasing working
hours from part time to full
time. The estimates imply that the negative direct effect
dominates the positive indirect for
wages smaller than US$5.22, while the positive dominates the
negative for wages larger
than US$5.22.
The presence of young children or a husband earning a moderate
hourly wage (less than
US$15) increase the utility when unemployed by 0.15 and 0.12 and
the utility of a full
time job by 0.14 and 0.53, respectively. The implications are
small reductions in the range
of wages where part time employment is preferred. For example,
young children reduce
the range from [3.76, 5.22] to [3.91, 5.18]. However, these
coefficients (a1, a2, g21 and g22)are relatively small and not
statistically significant according to a joint likelihood ratio
test
( p-value 0.613).
The presence of a husband earning a high hourly wage (more than
US$15 per hour) has
substantial effect, however. Fig. 3 shows the utility function
for a woman with a husband
earning more than US$15 per hour, but no children. Her utility
when unemployed goes up
by 1.63 and her utility of full time employment falls by 7.38.
As a consequence, she
prefers unemployment over employment for all wages less than
5.39, part time employ-
ment for wages between 5.39 and 7.61, and full time employment
for all wages larger than
7.61. The effect of receiving unemployment benefits (dashed
horizontal line) is to narrow
the range where part time employment is preferred, from [5.39,
7.61] to [6.53, 7.61].
The estimated values of k and j are 0.04 and 0, respectively,
both of which seem low(the restriction j = 0 is imposed in Table 1,
but the likelihood function is in fact maximizedat this value).
Recall that j = k1/(q + d) and k=(k0 k1)/(q + d). Unless the time
preferenceand the job termination rate are very large, the
estimates imply that employed workers
have zero probability of receiving a job offer and that there is
little difference between the
job offer arrival rates for unemployed and employed workers.
However, as mentioned
earlier, while k and j are theoretically identified by
nonlinearities in the reservation utility
Fig. 3. Husband earning US$15 or more.
T. Grgens / Labour Economics 9 (2002) 93123 109
-
function, practical estimation can be difficult, and a strong
positive correlation between the
estimates is to be expected. Reestimation of the model with j
fixed at other valuesconfirmed this. It also showed that
correlation between k and j on the one hand, and theremaining
parameters in the model on the other, is extremely low and that the
effect on the
conditional reservation wages is negligible. The conclusion is
therefore that the estimates kand j are unreliable, but that this
has little effect on the remaining parameters. Notsurprisingly
perhaps, it appears that duration data (in the absence of data on
job offers) are
needed to get reliable estimates of k and j.Figs. 4 and 5 show
the sample15 together with the estimated mean log wage offer
functions lj and reservation wage functionsWj, as a function of
the index xoVb. SinceWj is
a function of xuVa and xeVg2 as well as of xoVb it is not
possible to plot the entire Wj-surface.Therefore, reservation wages
functions are shown in the figures for the same four cases
represented in Figs. 2 and 3. The lowest lines represent a woman
with no husband and no
young children who did not receive benefits. The three lines
above represent women who
either received unemployment benefits or have a husband earning
more than US$15 per
hour or both. The effect of benefit receipt on part time
reservation wages is quite substantial,
while the effect on full time reservation wages is negligible.
This is because the marginal
utility of wages, cj, is much higher for full time than for part
time jobs. The effect of ahusband earning more than US$15 per hour
is large for both part time and full time jobs.
The lowest estimated conditional reservation wage for the women
in the sample is
US$3.76 for part time and US$4.76 for full time jobs. By way of
comparison, the legal
minimum wage in March 1990 was US$3.35, but a new minimum of
US$3.80 took effect
on April 1, 1990. The largest estimated conditional reservation
wages are US$7.46 for part
Fig. 4. Part time jobs.
15 One observation with a wage of about US$55 is omitted in Fig.
5. The observation is included in the
estimation.
T. Grgens / Labour Economics 9 (2002) 93123110
-
time and US$7.57 for full time jobs. The part time reservation
wage is lower than the full
time for women whose index xoVb is small, while the reverse is
the case for women whoseindex is large. The difference between the
full time and part time reservation wages for the
women in the sample ranges from US$ 1.03 to US$1.67, with an
average of US$0.77.Relative to the part time reservation wage, the
difference ranges from 16% to 31%, withan average of 19%.
These results suggest that women are not indifferent to working
part time or full time.
Depending on their characteristics and on the offered wages,
some women prefer part time
over full time while others prefer full time over part time. As
mentioned earlier, if workers
do not care about hours (g20 = 0, g2 = 0 and c2 = 1), then job
search reduces to wage search.The likelihood ratio test statistic
for this hypothesis is 78.44, which is statistically
significant at almost any level (there are five degrees of
freedom and the p-value is
0.000). A comparison of the estimated reservation wages in model
I and in the
corresponding wage search model shows that the reservation wages
are generally lower
in the wage search model. As a percentage of the reservation
wages in the wage search
model, the part time reservation wages in model I are 1.5%
higher, ranging from 9% to79% among the women in the sample, and
the full time reservation wages are 20% higher,
ranging from 3% to 58%. This confirms the claim made in the
introduction that reservation
wages tend to be underestimated when nonwage job characteristics
are ignored.
The estimated parameters relating to the measurement error model
are as expected. The
standard deviation of the measurement errors, s, is of similar
magnitude to the standarddeviation of the wage offers, rj; this
suggests that there is as much noise as information indata, in
accordance with validation studies. The parameter g is near zero
and d is small,
implying near-symmetric and leptokurtic measurement errors in
wages, also in accordance
with validation studies.
Virtually all structural studies of job (or wage) search assume
that measurement errors
in wages are lognormally distributed, and it is therefore of
considerable interest to compare
Fig. 5. Full time jobs.
T. Grgens / Labour Economics 9 (2002) 93123 111
-
the flexible leptokurtic model and the standard lognormal model.
Results for the lognormal
model ( g = 0 and d =l) are reported in column II in Table 1.
The first thing to notice isthat the lognormal model is strongly
rejected against the more flexible leptokurtic model.
The likelihood ratio statistic is 32.99 with a p-value of 0.000
(2 df ). On the other hand,
while there are some differences, most of the parameter
estimates are fairly similar. The
large differences in g2j and c2 do not imply large difference in
the conditional reservationwages, because the ratios g2j/c2 are
similar. Measurement errors and robustness are furtherdiscussed in
the next subsection.
Finally, to get an idea of how well the benchmark model fits the
data, the estimated
model was simulated to generate a wage distribution which can be
compared to the
empirical wage distribution. In these simulations, the
characteristics of the 730 females
were taken as given. A sample of observed wages was constructed
by drawing an
acceptable hourswage pair and a random wage measurement error
from the appropriate
distributions for each person in the sample. Five thousand such
samples were drawn, and
the resulting density is shown in Fig. 6, along with a histogram
of the empirical wage
distribution.16 The value of the corresponding v2 statistic is
185.92, which according to av2 distribution with 34 df (64
categories and 29 estimated parameters) has a p-value of0.000. The
model is thus formally rejected. However, as can be seen from Fig.
6, the main
reason for the rejection is that the model does not pick up the
spikes in the empirical wage
distribution at whole dollar amounts (especially at US$4, US$5,
US$9 and US$10 per
hour). Otherwise, the model seems to fit the empirical
distribution very well. The
simulated mean and standard deviation (with observations over
US$25 censored), 6.84
and 3.67, are virtually identical to their empirical
counterparts, 6.85 and 3.66 (the
simulated frequencies of observed part and full time jobs also
match the empirical
frequencies, 36.8% and 63.2%). Since modeling spikes is not a
priority in this paper,
the formal rejection of the model is not a cause for much
concern.
4.2. Measurement errors and robustness
The main conclusion so far is that the difference between full
time and part time
conditional reservation wages is significant. For the women in
the sample, the difference
ranges from 16% to 31%, depending on her characteristics, with
an average of 19%. Asargued earlier, estimates of reservation wages
are sensitive to measurement error. The
purpose of this section is to investigate how robust these
findings are to changes in the
measurement error assumptions. The strategy is to compare a
variety of models for which
the implied moments and correlations of the measurement errors
are similar to those found
by Rodgers et al. (1993) for the PSID validation study.
16 The histogram is constructed using 61 intervals of US$0.25
from US$0.875 to US$16.125, one lower
interval [0, 0.875] and two upper intervals [16.125, 20] and
[20, 25]. Three observations larger than 25 were
censored. Note that with a sample of 730, the expected number of
observations is less than 5 in cells from 0 to
2.875 and from 10.875 to 16.275. If cell boundaries are chosen
such that the expected number of observations in
each cell is 10, the v2 statistic is 698.40 with 44 df and a
p-value of 0.000. Again, the main reason for rejection isspikes at
isolated values, most notably in the cells containing US$4.00 and
US$5.00 per hour.
T. Grgens / Labour Economics 9 (2002) 93123112
-
Rodgers et al. consider several definitions of the hourly wage:
an annual average, the
average wage in previous pay period, and the usual hourly wage.
Their findings can be
summarized as follows. The measurement errors in log wages have
approximately zero
mean; depending on the wage concept the skewness and kurtosis
coefficients range from
1.59 to 0.64 and 5.66 to 9.05, respectively. In addition, the
correlation between themeasurement errors and the true levels
ranges from 0.19 to 0.07 and the correlationbetween the observed
and the true wages ranges from 0.24 to 0.61. The wages explained
in
this paper are so-called usual hourly wages, and the skewness
and kurtosis coefficients and
the correlations for usual hourly wages are 0.65, 5.66, 0.19 and
0.24, respectively.The correlation between the true and the
observed usual wages is very low. However, the
usual wage rate is not a well defined concept, and thus the true
usual wage in the
validation study is unlikely to be measured without error.
These numbers are merely indicative of the possible range of the
moments. The PSID
validation sample is not representative of the population of
working women in the US. In
fact, almost all of the employees of the company in the
validation study were male and
results by Bound and Krueger (1991) using CPS data suggest that
reporting errors are
smaller for women than for men. Furthermore, the workers
included in the validation study
were all paid by the hour and the wage rate varied with the
particular task being
performed. If hourly wages and working hours fluctuate less for
the women in the CPS
sample, they may be able to provide more precise information. If
so, measurement errors
will also be smaller in the CPS than in the validation sample.
On the other hand, the fact
that the validation data contain few or no coding errors pulls
in the opposite direction.
Recall that the parameters of the measurement error model are s,
g, d, cw and cc. Byfixing s, g, d or cw and reestimating the model,
it is possible to generate distributions ofmeasurement errors in
wages which cover the range found by Rodgers et al. (1993). In
addition, the effect of classification errors in job type can be
investigated by varying cc.
Fig. 6. Goodness-of-fit.
T. Grgens / Labour Economics 9 (2002) 93123 113
-
Selected estimation results are presented in Table 3, where each
column represents a
different model (that is, a different set of assumptions).
Models I and II are the benchmark
model and the lognormal model from Table 1. Models IIIVII show
the effect of
increasing the variance, skewness and kurtosis parameters in the
wage measurement error
distribution. Models VIIIXI investigate the effect of
correlation between the wage
measurement error and accepted wages. Model XII shows the effect
of increasing the
probability of misclassifying a job. The first five rows in
Table 3 show the values of the
measurement error parameters for that particular model (some
fixed, some estimated), the
next four, the implied moments and correlations, and the
following nine rows show
summary statistics for the sample distribution of part time and
full time reservation
wages.17 Finally, the last row shows the value of the
log-likelihood function.
Consider first the benchmark model I, where cw = 0 and cc= 0.
The implied correlation
between true and observed wages in the benchmark model I is
0.73, which is high relative
to the finding by Rodgers et al. (1993). The skewness
coefficient is also slightly higher
than expected, but the kurtosis coefficient and the correlation
between accepted wages and
measurement error are well within Rodgers et al.s ranges. As
mentioned earlier, the
average part time reservation wages is US$4.33 and the average
difference between full
time and part time reservation wages is 77 US cents, or 19%.
Turning now to the other models presented in Table 3, notice
that the moments and
correlations of the measurement errors cover virtually the
entire ranges given by Rodgers
et al. (1993). The only exception is the persistent high
correlation between the true and the
observed log wages. A possible explanation is that the women in
the CPS sample are better
at reporting their wages than the men in the validation study.
Table 3 also shows that the
estimated reservation wages do vary with the measurement error
assumptions. The average
part time conditional reservation wage ranges from US$3.74 to
US$4.57 and the standard
deviation from 0.61 to 0.79. The benchmark model is in the
middle, with values US$4.33
and 0.72, respectively. The average differences between full
time and part time reservation
wages ranges from 72 to 86 US cents and the standard deviation
from 0.40 to 0.61. Again,
the benchmark model is in the middle, with values 77 US cents
and 0.46. In percentage
terms, the average difference between full time and part time
reservation wages ranges
from 18% to 25% and the standard deviation from 0.09 to 0.15.
The values for the
benchmark model are 19% and 0.11. The overall lowest estimate of
the individual
difference between the conditional reservation wages for full
and part time jobs is
US$ 2.40 or 33% and the largest is US$1.86 or 36%.The conclusion
that women care about hours therefore seems quite robust. The size
of
the differences in part and full time reservation wages vary
somewhat with the
assumptions about the measurement errors, but the general
pattern is the same in all the
estimated models: some womens part time reservation wages are
significantly higher than
17 The moments reported by Rodgers et al. (1993) are aggregate
moments for the PSID validation sample.
Equivalent aggregate moments for the present sample, as implied
by the estimated models, are computed the
following way: first the conditional moments given x and h are
computed for each observation in the sample, then
the aggregate moments are computed by summing over all
observations. It is implicitly assumed that the sample
distribution of x and h equals the population distribution.
Formulae for the conditional moments given x and h as
well as for the unconditional moments are given in Appendix
C.
T. Grgens / Labour Economics 9 (2002) 93123114
-
Table 3
Estimated reservation wages
I II III IV V VI VII VIII IX X XI XII
s 0.32 0.29 0.40 0.25 0.32 0.23 0.21 0.32 0.33 0.31 0.25 0.32g
0.38 0.00 0.39 1.00 0.30 1.00 2.00 0.38 0.39 0.37 2.00 0.39d 1.52 l
1.18 2.73 1.30 2.00 2.00 1.55 1.58 1.49 3.96 1.53cw 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.10 0.20 0.10 0.20 0.00cc 0.00 0.00 0.00 0.00
0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.05
Skewness (log ew) 0.75 0.00 1.70 0.44 0.96 0.87 1.39 0.72 0.68
0.77 0.36 0.75Kurtosis (log ew) 7.56 3.00 19.40 3.93 11.54 5.59
7.25 7.20 6.79 7.85 3.42 7.47
Cor(log wa, log ew) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.03 0.07
0.03 0.19 0.00Cor(log wa, log w) 0.73 0.78 0.62 0.84 0.73 0.87 0.89
0.72 0.70 0.73 0.83 0.72
E(W1) 4.33 4.25 4.57 3.99 4.25 3.86 3.74 4.25 4.17 4.41 3.67
4.34
SD(W1) 0.72 0.79 0.76 0.75 0.70 0.70 0.66 0.68 0.66 0.76 0.61
0.73
MIN(W1) 3.76 3.60 3.97 3.38 3.71 3.30 3.23 3.72 3.69 3.79 3.22
3.77
MAX(W1) 7.46 8.57 8.30 8.10 7.19 7.50 6.90 7.09 6.76 8.12 6.12
7.38
E(W2W1) 0.77 0.72 0.83 0.74 0.77 0.74 0.75 0.81 0.86 0.73 0.84
0.85SD(W2W1) 0.46 0.61 0.40 0.60 0.46 0.56 0.52 0.45 0.45 0.47 0.51
0.52MIN(W2W1) 1.03 2.40 0.83 2.36 0.91 1.89 1.39 0.75 0.43 1.27
0.84 1.06MAX(W2W1) 1.67 1.76 1.71 1.65 1.65 1.54 1.41 1.69 1.71
1.66 1.56 1.86E((W2W1) /W1) 0.19 0.19 0.19 0.21 0.19 0.21 0.22 0.20
0.22 0.18 0.25 0.21SD((W2W1) /W1) 0.11 0.15 0.09 0.15 0.11 0.15
0.15 0.11 0.11 0.11 0.15 0.13MIN((W2W1) /W1) 0.16 0.31 0.12 0.33
0.15 0.29 0.24 0.13 0.08 0.19 0.16 0.17MAX((W2W1) /W1) 0.31 0.35
0.30 0.35 0.31 0.36 0.36 0.32 0.33 0.30 0.36 0.35LOG L 2030 2047
2036 2043 2031 2044 2046 2031 2032 2030 2047 2031Values for s, g,
d, cw and cc in roman font indicate fixed parameters, while values
in italics indicate maximum likelihood estimates.
T.Grgens/LabourEconomics
9(2002)93123
115
-
their full time, while for others, the relationship is reverse.
On average, the full time
reservation wages are 19% higher than the part time. The
benchmark model is in the
middle of the spectrum and fits the data reasonably well. Until
more precise evidence on
the distribution of measurement errors become available, it may
therefore well be the
preferred model.
5. Concluding remarks
This paper has estimated a model of job search for a sample of
women who were
unemployed in 1989 and who were working as of March 1990. It
appears that this is the
first paper to study the reservation wages for women using a
structural search framework.
It also appears to be the first paper to incorporate an
empirically appropriate, rather than a
convenient model of measurement errors.
The main finding of this paper is that the difference between
full time and part time
conditional reservation wages is significant. The hypothesis of
constant conditional
reservation wages is strongly rejected by a statistical test.
Moreover, the conclusions are
reasonably robust to changes in the assumptions about the
distribution of the measurement
errors.
Acknowledgements
I thank Martin Appel, Denise Doiron, Catherine de Fontenay, Joel
Horowitz, Forrest
Nelson, George Neumann, Gene Savin, Allan Wurtz and referees for
helpful comments.
Appendix A. The data
The data is a subset of the Current Population Survey (CPS):
Annual Demographic
File, 1990 (i.e. the March 1990 CPS). Only rotation groups 4 and
8 were asked about their
current wage and working hours. Out of the 16225 women in these
rotation groups, 731
experienced a spell of unemployment in 1989 and were currently
working as of March
1990. Of the 731 observations, the one with the lowest hourly
wage (US$1.30) was
dropped.
Table 4
Means and standard deviations
Variable Part time Full time Total
Mean SD Mean SD Mean SD
Wage 5.5 2.6 7.7 4.5 6.9 4.1
Log wage 1.6 0.4 1.9 0.5 1.8 0.5
Hours 20.9 7.2 40.4 4.2 33.2 10.9
Age 30.8 13.4 34.4 11.5 33.1 12.3
Age2/1000 1.1 1.0 1.3 0.9 1.2 1.0
T. Grgens / Labour Economics 9 (2002) 93123116
-
In terminology of the CPS March 1990 Technical Documentation
Data Dictionary, the
variables used in the analysis are constructed as follows. Weeks
unemployed is LWEEKS,
and a person experienced a spell of unemployment in 1989 if
LWEEKS > 0. Hours is usual
weekly hours worked, A-USLHRS, and a person was working as of
March 1990 if A-
USLHRS > 0. The hours categories are 134 hours and 35 or more
hours per week. The
wage rate is usual weekly earnings before deductions, A-GRSWK,
divided by usual hours
worked, A-USLHRS. A-GRSWK includes overtime pay, commissions,
tips etc. One
observation of A-GRSWK is topcoded at US$1923; the second
largest value is US$1200.
Age is A-AGE. The education categories are based on highest
grade attended, A-HGA,
and whether or not the grade was completed, A-HGC. The highest
grade completed is
defined as A-HGA if the grade was completed and A-HGA minus 1 if
not. People whose
highest grade completed is between 1 and 11 years are
categorized as having less thanhigh school education, 12 is high
school education, 1315 is more than high school but
not a college degree and 16 years or more is a college degree.
The central city dummy
variable is 1 if the person lives in a central city statistical
area, HCCC-R = 1, and 0 else.
The region dummies are 1 for the Northeast, 11VHG-ST60V 23, 2
for the Midwest,31VHG-ST60V 47, 3 for the South, 41VHG-ST60V 74,
and 4 for the West, 81VHG-ST60V 95. The union dummy is 1 if the
person is a union member, A-UNMEM=1, or ifshe is covered by a union
contract, A-UNCOV= 1. The children dummy variable is 1 if
there are own children under 6 in the family, FOWNU6 >1, and
0 else. The husbands wage
rate, if any, is defined similarly to the womans, and the
husbands wage group is 0 if the
woman is not married or if no wage is reported for the husband,
1 if the husbands wage is
between US$0 and US$15 and 2 if it is above US$15 a week. The
dummy for receipt of
unemployment benefits is 1 if UC-YN= 1 and 0 otherwise. Summary
statistics of the data
are given in (Tables 4 and 5).
Table 5
Number of observations
Variable Part time 134 hour Full time 3599 hour Total
High school drop-out 65 84 149
High school 123 205 328
Some college 56 93 149
College 25 79 104
Not central city 225 347 572
Central city 44 114 158
Northeast 63 102 165
Midwest 81 87 168
South 64 168 232
West 61 104 165
Not union covered 246 409 655
Union covered 23 52 75
No children under age 6 229 379 608
Children under age 6 40 82 122
No wage-earning husband 196 299 495
Husbands wage > 0 52 128 180
Husbands wage > 15 21 34 55
Number of observations 269 461 730
T. Grgens / Labour Economics 9 (2002) 93123 117
-
Appendix B. Density of observed jobs
Let / denote the standard normal density function. The
assumptions of Section 3 implythat the density of mj is (for v >
0)
fmjv /log v
rj
1
vrj, 16
and the density of is (for e > 0)
fe
/ g dlog loge kl
log k
l
21
s24
35
0@
1A d
l
e
loge k
l
21
s if d
>>>>>>>>>>>>>>>>:
17
where l and k are functions of the unknown parameters d, g, and
s given after Eq. (11).Recall that (wo,ho), (wa,ha) and (w,h)
denote offered, accepted, and observed jobs and
ew the measurement error in observed wages. Since mj is
independent of ha, the conditional
density of the true accepted wage wa given x = xi and ha = hj is
the density of offered wages
truncated at the conditional reservation wage Wij,
fwaw j x xi,ha hj 1wzWij
fmjw
lij
!
1 FmjWij
lij
! if Wij > 0
1wz 0fmjw
lij
!if WijV0:
8>>>>>>>>>>>>>>>>>>>:
18
Eqs. (16) and (18) imply that18
Elog wa j x xi,ha hj log lij
rj/logWij log lij
rj
1 U logWij log lijrj
if Wij > 0log lij if Wij V 0,
8>>>>>>>>>:
19
18 See Johnson et al. (1994).
T. Grgens / Labour Economics 9 (2002) 93123118
-
where U is the standard normal distribution function. This gives
a closed form ex-pression for sij = exp(E[log w
ajx = xi, ha = hj]), which appears in the likelihood
function(15).
An observation of (w,h) presupposes a job offer (wo,ho), which
is acceptable, that is,
an offer for which wozW(ho). Let Ai denote the event wozW(ho)
for a worker with
characteristics x = xi. The conditional density of (w,h) at
(wi,hi) given x = xi is the
derivative with respect to wi of Pr(wV wi,h = hijx= xi, Ai). For
simplicity, the con-ditioning on x = xi is suppressed in the
notation in the remainder of this section. Since
ho is discrete and since w is conditionally independent of h
given x and ho, it follows
that
PrwVwi,h hi j Ai
X2j1
Prh hi,ho hj j AiPrwVwi j h h,ho hj,Ai
X2j1
Prh hi,ho hj j AiPrwVwi j , ho hj,Ai
X2j1
Prh hi j ho hj,AiPrho hjPrwVwi,Ai j ho hjPrAi : 20
The last line follows from properties of conditional
probability.
Since observing a job presupposes an acceptable job offer, the
assumption that the
probability of a classification error in hours is a constant
parameter, cc, implies
Prh hi j ho hj,Ai Prh hi j ha hj c1Iijc 1 ccIij , 21
where Iij = 1 if hi = hj and Iij = 0 otherwise.
Since mj and are independent of x and ho and of each other,
PrwVwi,Ai j ho hj Prl1cwij m
1cwj
scwijVwi,lijmjzWij
!
Z lmax 0,
Wij
lij
Pr V wiscwijl1cwij v1cw
!fmjvdv
Z lmax0,Wij
Pr Vwis
cwij
v1cw
1
lijfmj
v
lij
!dv, 22
where the last line follows after a change of variables.
T. Grgens / Labour Economics 9 (2002) 93123 119
-
Substituting Eqs. (21) and (22) into Eq. (20) and
differentiating with respect to wi gives
the density of (w,h) at (wi,hi),
f wi,hi X2j1
pjc1Iijc 1 ccIij1 Qiri
Z lmax0,Wij
fwis
cwij
v1cw
scwij
v1cwlijfmj
v
lij
!dv, 23
where pj= Pr(ho = hj) and 1Qi(ri) = Pr(Ai) are defined in Eqs.
(12) and (13).
Appendix C. Measurement error moments and correlations
Define log D = log waE(log wajx,ha), where E(log wajx,ha) is
given in Eq. (9). FromEq. (10) and the assumptions that log has
mean zero and is conditionally independent oflog wa given (x,ha),
the first four moments of the measurement errors in log wages
are
Elog ew 0, 24
V log ew V log c2wV log D, 25
Elog ew3 Elog 3 c3wElog D3, 26
Elog ew4 Elog 4 c4wElog D4 6c2wV log V log D: 27
The correlations are
Corlog wa,log ew cwV log DV log wap V log ewp , 28
Corlog wa,log w V log wa cwV log D
V log wap V log wp : 29The remainder of this section provides
expressions for the terms on the right-hand sides.
The moments of the distribution of (log k)/l as defined in Eq.
(11) are providedby Johnson et al. (1994, p. 35). Define x = exp(d
2) and X = g/d. If d is finite, thenE(log ) = 0 and
V log s2, 30
T. Grgens / Labour Economics 9 (2002) 93123120
-
Elog 3 l3 1
4x1=2x 12xx 2sinh3X 3sinhX, 31
Elog 4 l4 1
8x 12x2x4 2w3 3x2 3cosh4X
4x2x 2cosh2X 32x 1: 32If d =l, then log is normally distributed,
so E(log ) = 0 and
V log s2, 33
Elog 3 0, 34
Elog 4 3s4: 35
If WijV 0, then there is no truncation and the conditional
moments of log D given x = xiand ha = hj equal the conditional
moments of log w
o, that is,
Elog D2 j x xi,ha hj r2j , 36
Elog D3 j x xi,ha hj 0, 37
Elog D4 j x xi,ha hj 3r4j : 38
If Wij > 0, then the conditional distribution of log D given
x = xi and ha = hj is a truncated
normal. The higher order moments of that can be found in Sugiura
and Gomi (1985), and
they are
Elog D2 j x xi,ha hj r2j kij zijzij 1, 39
Elog D3 j x xi,ha hj r3j 2z3ij 3kijz2ij k2ij 1zij, 40
Elog D4 j x xi,ha hj r4j 3z4ij 6kijz3ij 22k2ij 1z2ij
k3ij 3kijzij 3, 41
where kij=[log Wij log lij]/rj and zij =/(kij)/[1/(kij)].
T. Grgens / Labour Economics 9 (2002) 93123 121
-
To obtain the unconditional moments of log D, notice that
Elog Dk j x xi,h hiXJj1
Elog Dk j x xi,ha hjPrha hj j x xi,h hi, 42
where
Prha hj j x xi,h hi
~Pr
ha hj,h hi j x xi
Pr
h hi j x xi,ha hj
Prha hj j x xi
Pr
h hi j x xi,ha hj
Prho hj,Ai j x xi
Pr
h hi j x xi,ha hj
PrAi j x xi,ho hj
Prho hj j x xi
c1Iijc 1 ccIij
1 Qiri
pj: 43
Now estimate E[(log D)k] by the sample average n1Xni1
ElogDk j x xi,h hi:Finally, the variance of log w is
V log w V log wa V log ew 2Covlog wa,log ew, 44and the variance
of log wa is
V log wa EElog D2 j x,ha VElog wa j x,ha, 45
where the quantities on the right-hand side in Eq. (45) are
estimated by the appropriate
sample averages.
References
Blau, D.M., 1991. Search for nonwage job characteristics: a test
of the reservation wage hypothesis. Journal of
Labor Economics 9 (2), 186205.
Bloemen, H.G., 1997. Job search theory, labour supply and
unemployment duration. Journal of Econometrics 79,
305325.
Bound, J., Brown, C., Duncan, G.J., Rodgers, W.L., 1990.
Measurement error in cross-sectional and longitudinal
labor market surveys: validation study evidence. In: Hartog, J.,
Ridder, G., Theeuwes, J. (Eds.), Panel Data
and Labor Market Studies. North-Holland, Amsterdam, pp. 119.
Bound, J., Brown, C., Duncan, G.J., Rodgers, W.L., 1994.
Evidence on the validity of cross-sectional and
longitudinal labor market data. Journal of Labor Economics 12
(3), 345368.
Bound, J., Krueger, A.B., 1991. The extent of measurement error
in longitudinal earnings data: do two wrongs
make a right? Journal of Labor Economics 9 (1), 124.
Burdett, K., 1979. Unemployment insurance payments as a search
subsidy: a theoretical analysis. Economic
Inquiry 17, 333343.
Burdett, K., Mortensen, D.T., 1978. Labor supply under
uncertainty. Research in Labor Economics 2, 109157.
Christensen, B.J., Kiefer, N.M., 1994. Measurement error in the
prototypical job-search model. Journal of Labor
Economics 12 (4), 618638.
T. Grgens / Labour Economics 9 (2002) 93123122
-
Devine, T.J., Kiefer, N.M., 1991. Empirical Labor Economics. The
Search Approach. Oxford Univ. Press, New
York.
Duncan, G.J., Hill, D.H., 1985. An investigation of the extent
and consequences of measurement error in labor-
economic survey data. Journal of Labor Economics 3 (4),
508532.
Eckstein, Z., Wolpin, K.I., 1995. Duration to first job and the
return to schooling: estimates from a search-
matching model. Review of Economic Studies 62, 263286.
Flinn, C., Heckman, J., 1982. New methods for analyzing
structural models of labor force dynamics. Journal of
Econometrics 18, 115168.
Johnson, N.L., Kotz, S., Balakrishnan, N., 1994. Continuous
Univariate Distibutions. second ed. Wiley, New
York.
Kiefer, N.M., 1987. A proposition and an example in the theory
of job search with hours constraints. Journal of
Labor Economics 5 (2), 211220.
Killingsworth, M.R., Heckman, J.J., 1986. Female labor supply: a
survey. In: Ashenfelter, O., Layard, R. (Eds.),
Handbook of Labor Economics, vol. 1. North-Holland, Amsterdam,
pp. 103204, chapter 2.
Mortensen, D.T., 1977. Unemployment insurance and job search
decisions. Industrial and Labor Relations Re-
view 30 (4), 505517.
Mortensen, D.T., 1986. Job search and labor market analysis. In:
Aschenfelter, O., Layard, R. (Eds.), Handbook
of Labor Economics, vol. 11. North-Holland, Amsterdam, pp.
849919, chapter 15.
Mortensen, D.T., Neumann, G.R., 1988. Estimating structural
models of unemployment and job duration in
dynamic econometric modeling. In: Barnett, W.A., Berndt, E.R.,
White, H. (Eds.), Dynamic Econometric
Modelling. Proceedings of the Third International Symposium in
Economic Theory and Econometrics. Cam-
bridge Univ. Press, Cambridge, pp. 335355, chapter 15.
Nickell, S.J., 1986. Dynamic models of labour demand. In:
Aschenfelter, O., Layard, R. (Eds.), Handbook of
Labor Economics, vol. 1. North-Holland, Amsterdam, pp. 473522,
chapter 9.
Pencavel, J., 1986. Labor supply of men: a survey. In:
Aschenfelter, O., Layard, R. (Eds.), Handbook of Labor
Economics, vol. 1. North-Holland, Amsterdam, pp. 3102, chapter
1.
Rodgers, W.L., Brown, C., Duncan, G.J., 1993. Errors in survey
reports of earnings, hours worked, and hourly
wages. Journal of the American Statistical Association 88 (424),
12081218.
Sugiura, N., Gomi, A., 1985. Pearson diagrams for truncated
normal and truncated Weibull distributions. Bio-
metrika 72 (1), 219222.
Wolpin, K.I., 1987. Estimating a structural search model: the
transition from school to work. Econometrica 55,
801817.
Woodbury, S.A., 1983. Substitution between wage and nonwage
benefits. American Economic Review 73 (1),
166182.
T. Grgens / Labour Economics 9 (2002) 93123 123
Reservation wages and working hours for recently unemployed US
womenIntroductionA job search modelWorkers' reservation
utilityDistribution of accepted jobsWage search
Parameterization and measurement errorData and dependent
variablesWorker heterogeneityMeasurement errorLikelihood function
and identification
Empirical resultsBenchmark modelMeasurement errors and
robustness
Concluding remarksAcknowledgementsThe dataDensity of observed
jobsMeasurement error moments and correlationsReferences