11_ab_gorgens_reservation wages and working hours for

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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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.


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


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


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).


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.


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


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)].


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.

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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

11_ab_gorgens_reservation wages and working hours for

Documents

reservation wage function

wage search model

single reservation wage

time reservation wages

offered wage

job offer

reservation wagestrategy

reservation wagefunction