Manajemen | Fakultas Ekonomi Universitas Maritim Raja Ali Haji 350.full
Structural Estimation of Family
Labor Supply with Taxes
Estimating a Continuous Hours Model Using a
Direct Utility Specification
Bradley T. Heim
abstract
This paper proposes a new method for estimating family labor supply in the
presence of taxes. This method accounts for continuous hours choices,
measurement error, unobserved heterogeneity in tastes for work, the
nonlinear form of the tax code, and fixed costs of work in one comprehensive
specification. Estimated on data from the 2001 PSID, the resulting elasticities
for married males are consistent with those found elsewhere in the literature
but female wage elasticities are substantially smaller than those found in
most of the literature. Simulations of recent tax acts predict small effects on
the labor supply of married couples.
I. Introduction
Changes in the income tax code, including the recent changes that
were enacted as part of the Economic Growth and Tax Relief Reconciliation Act
(EGTRRA) of 2001 and the Jobs and Growth Tax Relief Reconciliation Act
(JGTRRA) of 2003, generally take a complex form. Predicting the possible effects
of such policies, both actual and prospective, requires estimates that can be used
to simulate the effects of complex changes in the tax code. In addition, the fact that
Bradley Heim is a Financial Economist in the Office of Tax Analysis, U.S. Department of the Treasury.
Substantial work on this paper was completed when the author was an Assistant Professor at Duke
University. He wishes to thank Joe Altonji, Peter Arcidiacono, Eric French, Hilary Hoynes, Marjorie
McElroy, Bruce Meyer, Robert Triest, seminar participants at the University of Virginia, the 2003 North
American Summer Meetings of the Econometric Society, and the 2004 Annual Meetings of the National
Tax Association, and two anonymous referees for helpful advice and comments. He takes responsibility
for all remaining errors. The views expressed are those of the author and do not necessarily reflect those
of the Department of the Treasury. The data used in this article can be obtained beginning October
2009 through September 2012 from Bradley T. Heim, Office of Tax Analysis, U.S. Department of
Treasury, Room 4036B, 1500 Pennsylvania Ave NW, Washington, DC 20220, or
Bradley.Heim@do.treas.gov.
½Submitted March 2007; accepted February 2008
ISSN 022 166X E ISSN 1548 8004 8 2009 by the Board of Regents of the University of Wisconsin System
T H E JO U R NAL O F H U M A N R ES O U R C ES
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Heim
couples are taxed jointly implies that it is important to evaluate both husbands’ and
wives’ labor supply responses to changes in the tax code, particularly if there are
cross-substitution effects. A structural model of joint labor supply is required to fully
capture these responses. This paper proposes a new method to structurally estimate
family labor supply in the presence of taxation.
Starting with Hausman and Ruud (1984), several studies have attempted to estimate structural models of husbands’ and wives’ joint labor supply choices in the
presence of taxes. As noted in Blundell and MaCurdy (1999), these approaches generally fall into two groups.
In the first group, the structural model allows for continuous hours choices of husbands and wives, with a nonnegativity constraint on the wife’s hours choice.1 Examples
of this type of study include Hausman and Ruud (1984), Ransom (1987), Kapteyn
et al. (1990), Kooreman and Kapteyn (1986), and Blundell and Walker (1986). However, estimating continuous family labor supply while allowing for both unobserved
heterogeneity in tastes for work and measurement error in hours requires accounting
for four stochastic elements in the model, and was thought to require integrating
over complex regions, which greatly complicates estimation. As a result, these studies
generally chose a subset of the desirable elements of a family labor supply model:
accounting for unobserved heterogeneity in tastes for work, accounting for measurement error in hours of work, or accounting for the endogeneity of the after tax wage
due to the structure of the tax code. Further, such studies generally required functional
forms for utility for which both the unconditional and conditional labor supply functions exist in closed form. As noted in Kapteyn et al. (1990), such a requirement
greatly limits the number of functional forms that can be used in estimation.
In the second group, the structural model is specified as a choice of the husband
and wife among a discrete set of hours combinations. For example, in van Soest
(1995), husbands and wives are modeled as choosing between either 25 or 36 discrete choices of hours combinations. Such an approach also has been used by Hoynes
(1996), and others. This approach allows for the incorporation of heterogeneity of
preferences, measurement error in wages, and a very detailed treatment of the budget
constraint. However, in order to maintain computational feasibility, errors in evaluating the utility of each discrete choice are often assumed to be distributed i.i.d. extreme value, and this assumption of independent error terms becomes untenable as
the number of hours choices increases. In addition, many tax law changes would
be expected to cause nontrivial changes in hours, but changes that would be too small
to appear in a model in which individuals choose between a small number of hours
choices. Finally, although it is generally assumed that annual hours peak at full time,
part time, and nonparticipation, with little spread around these points, in reality there
is considerable variation in the distribution of reported hours,2 which discrete choice
models are resigned to explain through measurement error.
In this paper, a method is proposed that overcomes the limitations in both of these
approaches, in that it allows for continuous hours choices of the husband and wife,
while still incorporating measurement error in hours and heterogeneity in tastes for
1. The husband is assumed to be working.
2. The variation in hours can be seen in Figure 4 below, which presents data on hours worked among husbands and wives using the 2001 Panel Study of Income Dynamics.
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work in the stochastic specification, accounts for the nonlinear structure of the tax
schedule, and accounts for fixed costs of working, all in one comprehensive model.
The method estimates parameters of a direct utility function using maximum likelihood techniques by using numerical methods to explicitly solve for the utility maximizing hours of husbands and wives, given parameters, and then calculating the
likelihood given the stochastic specification.
This model is estimated using data from the 2001 Panel Study of Income Dynamics. The key empirical findings include elasticities for married males consistent with
those found elsewhere in the literature, but female wage elasticities that are substantially smaller than those found in most of the literature. Simulations of the two most
recent major tax changes show that these smaller elasticities, in turn, imply small
responses of married couples’ labor supply.
The paper proceeds as follows. Section II outlines the assumed model of family
labor supply and provides an overview of the estimation method. Section III
describes in detail the particular features of the estimation method, and demonstrates
how it may be altered to account for heterogeneity in tastes for work, the missing
wages of nonworkers, and fixed costs of work. Section IV describes the data and
functional forms used in the estimation, and the estimates are reported in Section
V. In Section VI, these results are used to predict the labor supply effects of the
2001 and 2003 tax law changes. Section VII concludes.
II. An Overview of the Estimation Method
Couples are modeled in this paper as maximizing a joint utility function, Uðhh ; hw ; CÞ,3 over hours of work for the husband, hh , hours of work for the
wife, hw , and total consumption, C, subject to a joint budget constraint, given by
J
ð1Þ
C # Y + Wh hh + Ww hw 2 + tj ðIj 2 Ij21 Þ1ðY TI . Ij Þ
j¼1
J
2 + tj ðY TI 2 Ij21 Þ1ðIj $ Y TI . Ij21 Þ;
j¼1
where Wh and Ww denote the husband’s and wife’s wages, respectively, Y denotes the
the family’s taxable income (income less
family’s nonlabor income, Y TI denotes
J
deductions and exemptions), Ij j¼1 denotes the income tax bracket endpoints, tj
denotes the tax rate on income between Ij and Ij+1 , and 1ðÞ denotes the indicator
function. In this case, the budget constraint consists of the union of several plane segments, with kink lines at the intersection of these segments. (See Figure 1.)
3. There has been a fair amount of criticism of this unitary model of labor supply. (See McElroy and Horney
1981, Chiappori 1988, Schultz 1990, and Fortin and Lacroix 1997.) Donni (2003) examines the collective
model in the presence of taxes, but does not estimate a model. Blundell, Chiappori, Magnac and Meghir
(2007) and Bloemen (2004) estimate collective models labor supply, but do not account for taxes. This approach could plausibly be adapted to estimate a collective model of labor supply in the presence of taxes, since
implicit in the calculation of the likelihood is the explicit solution of the structural model, which could be done
analogously for other assumed theoretical models. Such extensions are left for future work.
Heim
Figure 1
Married Couple’s Nonlinear Budget Constraint
Let the gradient of a particular budget segment, j, in the direction of the husband’s
hours be wjh , the gradient in the direction of the wife’s hours be wjw , and the point at
which this segment of the budget constraint would intersect the origin if extended
(the ‘‘virtual income’’) be yj .
To understand the estimation approach taken in this paper, start with the estimation
method used in Hausman and Ruud (1984). In that paper, husbands’ and wives’ labor
supply functions given a linear budget constraint with wages wjh and wjw and virtual
income yj are given by
ð2Þ
hjh ¼ hh ðwjh ; wjh ; yj ; uÞ
hjw ¼ hw ðwjh ; wjh ; yj ; uÞ
To solve for the desired hours of the couple on their actual, nonlinear budget constraint, Hausman and Ruud use an algorithm that first calculates the set of hours that
result when the wages and virtual incomes associated with each budget segment are
substituted into these labor supply equations. If a member of this set lies on a plane
of the actual budget constraint, then the desired hours of the couple, given parameters, have been found. If not, the desired hours lie either on a kink of the budget set or
the boundary, and a further algorithm is used to identify which is the case. Once the
desired hours have been solved for, given a stochastic structure that maps desired
hours to observed hours, the likelihood of the sample is calculated, and maximum
likelihood techniques are used to solve for the likelihood maximizing parameters.
This paper follows Hausman and Ruud’s specification directly, with one crucial
difference. Instead of starting with a closed form for the husbands’ and wives’ labor
supply functions and solving for the desired hours using their algorithm, the method
in this paper starts with a functional form for the husband’s and wives’ utility function and solves for the desired hours using numerical methods. In a specification with
no heterogeneity, no fixed costs, and the direct utility function associated with the
labor supply functions used in Hausman and Ruud, the two methods would be
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The Journal of Human Resources
identical, because the two methods would simply be using two alternative ways of
solving for the same set of desired hours. Given the same stochastic specification,
the likelihoods would be the same, resulting in the same likelihood maximizing
parameters.
However, there are at least four advantages to specifying the estimation method
in terms of a utility function and using numerical methods to solve for desired
hours. First, if one is working with a direct utility function specification of the
problem, then closed form solutions for the conditional and unconditional labor
supply functions are not required, which expands the set of functional form
choices. Second, heterogeneity in tastes for work can be incorporated by assuming that couples are drawn from a finite set of types. Third, one can account for
missing wages of nonworkers by assuming that unobserved wages are drawn
from a distribution of wages, instead of assuming that missing wages are known
with certainty.4 Finally, fixed costs can be incorporated into the specification of
the budget constraint.5 On the other hand, a computationally intensive numerical
algorithm must still be used in this approach, since one must simply solve explicitly for the utility maximizing hours for the husband and wife by maximizing the
utility function subject to the actual budget constraint using a numerical algorithm.6
The overall estimation method works as follows. Given the hours-finding algorithm, the couple’s desired hours conditional on parameters of the utility function are solved for numerically. Then, given the stochastic specification for
measurement error in hours, the likelihood is calculated. Finally, standard maximum likelihood techniques are used to solve for the likelihood maximizing
parameters.7
The construction of the hours-finding algorithm, and the derivation of the likelihood, are described in the next section.
4. Since both of these extensions involve weighted sums of likelihoods that are conditional on particular
heterogeneity and wage draws, the resulting overall likelihood is as well behaved as the underlying conditional likelihoods.
5. So long as there is still a one-to-one correspondence between parameters and desired hours, the model is
identified. A tiebreaking rule ensures that this is so.
6. One might be concerned that this method suffers from the critique in MaCurdy et al. (1990) of continuous hours structural estimation methods, that estimates in Hausman-type labor supply estimation methods
are constrained to globally satisfy Slutsky positivity, which in turn biased estimated parameters. However,
as noted in Heim and Meyer (2004), it is the assumption of a continuous distribution of heterogeneity with
infinite support that forces parameters to satisfy Slutsky positivity globally. In this paper, the specification
of heterogeneity is discrete, and so this critique does not apply. As evidence for this claim, the estimated
parameters in this paper do not satisfy Slutsky positivity at high levels of hours. However, the levels at
which Slutsky positivity is not satisfied are above the levels of hours at which couples are observed, and
so deadweight loss calculations are not affected.
Further, Heim and Meyer note that the constraint in the Hausman method amounts to constraining preferences to be globally convex, and that a way to allow for the estimation of nonconvex preferences is to
estimate parameters of a direct utility function. With a sufficiently robust search algorithm, one could allow
estimated preferences to be nonconvex in this model; however, it would be extremely computationally intensive, and so is not attempted here.
7. Like other papers in this literature, since parameters are estimated using cross-sectional data, identification comes from variation in wages and incomes across individuals. In addition, since state tax schedules
are incorporated, additional identifying variation comes from differences across states in tax schedules.
Heim
III. Details of the Estimation Method
A. Hours-finding Algorithm
For simplicity, denote the budget constraint as
ð3Þ C # Bðhh ; hw ; Þ;
where hh denotes the husband’s hours of work, hw denotes the wife’s hours of work,
and C denotes consumption. Since, in this static model, there is no saving or dissaving, the individual will consume all of their income, and so
ð4Þ C ¼ Bðhh ; hw ; Þ:
For now, suppose that the budget constraint is a convex budget constraint. It is clear
that the unique utility maximizing hours or work, hh and hw , satisfy
ð5Þ hh ¼ arg max fUðhh ; hw ; Bðhh ; hw ; ÞÞ: H $ hh $ 0g
hh
and
ð6Þ hw ¼ arg max fUðhh ; hw ; Bðhh ; hw ; ÞÞ: H $ hw $ 0g
hw
where H denotes the endowment of hours each period. Hence, a straightforward way
to find the utility maximizing combination of hours is to maximize utility over the
husband’s hours conditional on the wives hours using a line search technique,8 then
maximize utility over the wife’s hours conditional on the husband’s hours, and iterate
in this way until the hours coordinates are changing less than a specified tolerance.
When this happens, a fixed point of the system of equations
ð7Þ hh ¼ arg max fUðhh ; hw ; Bðhh ; hw ; ÞÞg
hh
and
ð8Þ hw ¼ arg max fUðhh ; hw ; Bðhh ; hw ; ÞÞg
hw
has been found. This algorithm is known as a coordinate direction method, and is
guaranteed to find a fixed point to this system.9
If the budget constraint is linear, and the utility function is quasiconcave and twice
differentiable, then the fixed point of this system must be the unique utility maximizing hours on the budget constraint. In addition, if the fixed point of this system lies on
a plane of a kinked, but convex, budget constraint, the fixed point of this system still
8. For this paper, a bracketing method is used to search along a line. For more information on bracketing
methods, see Judd (1998). Though slow compared to other methods, this type of search method does not
require the calculation of derivatives and guarantees that a maximum will be found.
9. See Judd (1998).
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The Journal of Human Resources
must be the unique utility maximum on the budget constraint. Panel (a) of Figure 2
illustrates this case. Suppose that the search algorithm starts at (hh’, 0). Maximizing
utility keeping hh fixed at hh’, yields hw’ as the solution to Equation 8. The rounded
shape in the figure consists of the isoutility curve on the boundary of the budget constraint that goes through this point, where points inside this shape yield higher utility.
Maximizing utility keeping hw fixed at hw’ yields a value for hh inside of the rounded
shape, and the utility at this point will be higher than at the previous point. This process will continue to iterate toward a fixed point that lies on the middle plane of this
budget constraint, which is the utility maximizing point.
However, if the fixed point that this algorithm finds lies along a kink, it is possible
that the fixed point is not the utility maximum. This is because it is possible that, due
to the kinks in the nonlinear budget constraint, the intersection of the indifference
surface with the budget constraint may itself have a ‘‘kink’’ in it, and so searching
along the coordinates may miss points of higher utility that lie in the interior of this
intersection. This second case is shown in Panel (b). In this case, maximizing utility
keeping hh fixed at hh’ yields hw’, and maximizing utility keeping hw fixed at hw’
yields hh’, where (hh’, hw’) is a fixed point of the system that lies on the kink of
the budget constraint. However, there are points inside the isoutility curve in the figure that yield higher levels of utility, and hence the fixed point is not the global utility
maximum.
If the algorithm finds a fixed point at a kink, in order to ensure that the fixed point
is indeed the optimum, one must search along the kink for a higher point. If such a
point is found, the method returns to searching along the coordinate directions until
another fixed point is found. If not, then the algorithm has found the utility maximizing hours.
The foregoing discussion assumed that the budget constraint was convex. If the
budget constraint is nonconvex, however, there is the possibility of multiple local optima, and hence multiple fixed points of the system in Equations 7 and 8. Such a nonconvexity might arise, for example, due to the presence of fixed costs of work, or due
Figure 2
Search Algorithm—Two Cases
Heim
to the Earned Income Tax Credit. The alterations to the algorithm necessary to deal
with the fixed costs of work are discussed in Section III.E. To deal with nonconvexities due to the EITC, one can use an insight in Hausman (1985) and treat the nonconvex budget constraint as the union of convex budget constraints. One searches for
the utility maximizing hours on each constituent budget constraint, and compares the
utility levels at each optimum to find the global optimum. However, this procedure
increases the execution time of the search algorithm in proportion to the number of
convex budget constraints that make up the nonconvex budget constraints. Since only
a small proportion, less than 6 percent, of the respondents in the sample used here
earn an amount that would make them eligible to receive the EITC, the EITC is omitted from the budget constraints used in the estimation below.10
B. Derivation of Likelihood Function
For the stochastic specification, the specification in Hausman (1981) is generalized,
in assuming that when the utility maximizing hours of a member of a couple are positive, then the observed hours for this member are these utility maximizing hours plus
additive measurement errors, subject to a zero hours lower bound. On the other hand,
if the utility maximizing hours of a member of a couple are zero, then that member’s
observed hours are also zero.11 Thus, in effect, this specification assumes that a survey respondent may have forgotten about working positive hours, and misreport that
they worked zero hours, but that if the respondent worked 0 hours, they will accurately report this.12 Formally, the assumed data generating process is given by
maxfhhi ðu0 Þ + ehi ; 0g if hhi ðu0 Þ . 0
hhi ¼
0
if hhi ðu0 Þ ¼ 0
ð9Þ
maxfhwi ðu0 Þ + ewi ; 0g if hwi ðu0 Þ . 0
hwi ¼
0
if hwi ðu0 Þ ¼ 0
where hhi and hwi denote observed hours of the husband and wife in couple i, and
hhi ðu0 Þ and hwi ðu0 Þ denote the husband and wife’s utility maximizing hours of work
given the true parameters u0 .13
10. The Earned Income Tax Credit is, however, incorporated in this same type of model in Heim (2008),
where the sample consists of married couples in which the wife has less than a high school degree.
11. A possibly more desirable stochastic specification would be that used in Hoynes (1996), which assumes
that respondents correctly report their participation status, but may report positive hours with error. However, because the distribution of heterogeneity is discrete in the current specification, for the log likelihood
to not be negative infinity, participation status would need to be correctly predicted for at least one heterogeneity node for all observations, which would introduce numerous discontinuities into the likelihood and
may constrain parameters.
12. Note that this stochastic specification differs from other papers in this literature. For example, Hoynes
(1996), assumes that labor force participation and nonparticipation are accurately observed, but that positive hours amounts may be observed with error. Van Soest (1995), on the other hand, does not account for
measurement error in hours of work.
13. A specification was also estimated in which respondents can misreport participation in both directions.
Results from this specification are presented in an appendix available from the author, and are quantitatively
and qualitatively similar to the base specification.
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The Journal of Human Resources
Assuming that the errors are distributed bivariate normal with standard deviations
sh and sw , and correlation r, the overall likelihood for couple i is
"
1ðhhi ðuÞ.0;hwi ðuÞ.0Þ #1ðhhi .0;hwi .0Þ
1
hhi 2hhi ðuÞ hwi 2hwi ðuÞ
b
;
;r
ð10Þ li ðuÞ ¼
sh sw
sh
sw
2
2
02h ðuÞ h 2h ðuÞ
131ðhhi ðuÞ.0;hwi ðuÞ.0Þ 31ðhhi .0;hwi ¼0Þ
hi hi
wi
sh
sw 2r
hhi 2hhi ðuÞ
6
7
1
4
@
A5
ffiffiffiffiffiffiffiffi
p
F
7
6 sh f
sh
12r2
7
6
36
7
6
7
4
5
h
i1ðhhi ðuÞ.0;hwi ðuÞ¼0Þ
hhi 2hhi ðuÞ
1
+ sh f
sh
22
02h ðuÞ h 2h ðuÞ
131ðhhi ðuÞ.0;hwi ðuÞ.0Þ 31ðhhi ¼0;hwi .0Þ
wi wi
hi
sw
64 1 f hwi 2hwi ðuÞ F@ sh 2r
7
A5
ffiffiffiffiffiffiffiffi
p
6 sw
7
sw
2
12r
6
7
36
7
6
7
4
5
h
i
1ðhhi ðuÞ¼0;hwi ðuÞ.0Þ
h
2h
ðuÞ
+ s1w f wi swwi
2h
i1ðhhi ðuÞ.0;hwi ðuÞ.0Þ 31ðhhi ¼0;hwi ¼0Þ
2h ðuÞ 2h ðuÞ
B shih ; swiw ; r
7
6
7
6
h
i1ðhhi ðuÞ¼0;hwi ðuÞ.0Þ
7
6
7
6 + F 2hswi ðuÞ
w
7
6
36
7
7
6
h
i1ðhhi ðuÞ.0;hwi ðuÞ¼0Þ
7
6
7
6 + F 2hhi ðuÞ
5
4
sh
+1ðhhi ðuÞ ¼ 0; hwi ðuÞ ¼ 0Þ
where bðk; l; rÞ denotes the p.d.f. of the standard bivariate normal distribution with
correlation r, and Bðk; l; rÞ denotes the c.d.f. of the standard bivariate normal distribution, cumulative over ð2N; k3ð2N; l. The sample likelihood is
Y
ð11Þ L ¼
li ðuÞ:
i
This likelihood consists of four terms. The first term, for a couple in which both
members are observed working positive hours (hhi . 0; hwi . 0), denotes the probability
that couple i is observed working positive hours in the amount of hhi and hwi , given that
the utility maximizing hours of work of both members of the couple, given parameters,
u, are hhi and hwi , respectively.14 The second term, for a couple in which the husband
works positive hours and the wife does not work (hhi . 0; hwi ¼ 0), consists of two
parts. The first is the probability that couple i is observed working hhi and 0 hours, respectively, given that the utility maximizing hours of the couple are both positive. The
second is the probability that couple i is observed working hhi and 0 hours, respectively,
given that the husband’s utility maximizing hours are positive, and the wife’s utility
14. Note that if either or both of these optimal hours are zero, given the stochastic specification, the probability that both members of the couple would be working at positive hours is zero.
Heim
maximizing hours are 0. The third term, for a couple in which the husband does not
work and the wife works positive hours, is analogous to the second. The fourth term,
for a couple in which both members do not work (hhi ¼ 0; hwi ¼ 0) consists of four
parts. The first part is the probability that the husband and wife are both observed working 0 hours, given that the utility maximizing hours of work for both of them are positive. The second part is the probability they are observed working 0 hours, given that
the husband’s utility maximizing hours are positive, and the wife’s are 0. The third part
is the probability that they are observed working 0 hours, given that the husband’s utility
maximizing hours are 0, and the wife’s are positive. The fourth part is the probability
that they are observed working 0 hours, given that the utility maximizing hours of the
couple are 0. This probability, given the stochastic specification, is 1.
Of course, the difficulty of implementing the likelihood function in this particular
form is that, if the utility maximizing hours of work is zero for any member of any
couple observed working positive hours, the likelihood is zero, and hence the log
likelihood is negative infinity. This, however, is ameliorated somewhat if unobserved
heterogeneity in tastes for work is incorporated in the stochastic specification, which
is described next.
C. Incorporating Unobserved Heterogeneity
Labor supply studies typically incorporate unobserved heterogeneity in tastes for
work in part to explain why observably identical individuals are observed working
different hours.15 Further, such studies invariably find that unobserved heterogeneity
enters significantly in the stochastic specification.16
Customarily, when continuous hours choices are allowed for in a structural labor
supply model, a continuous distribution of heterogeneity in tastes for work also is assumed. Unfortunately, that approach runs into a number of problems in this setting,17
but unobserved heterogeneity
J can be incorporated by assuming that it has a discrete
distribution.18 Let vhj ; vwj j¼1 denote the nodes of a discrete distribution of heterogeJ
neity in taste for work, and fpj gj¼1 denote their respective probabilities. The likelihood function may then be adapted in two steps. First, replace hhi ðuÞ and hwi ðuÞ
with hhi ðu; vhj ; vwj Þ and hwi ðu; vhj ; vwj Þ, respectively, in the above likelihood function.
This is the likelihood given that the husband and wife’s values from the heterogeneity
distribution are vhj and vwj , respectively. Denote this likelihood li ðu; vhj ; vwj Þ. The overall
15. Incorporation of observable heterogeneity is discussed in Section IV.B.
16. See, for example, Hausman (1981), MaCurdy et al. (1990), Triest (1990), and many others in the case
of individual labor supply, and van Soest (1995) and Hoynes (1996) in the case of family labor supply.
17. The difficulty arises from the need to integrate the above likelihood over this distribution, the solution
of which would not have a closed form. This is not necessarily insurmountable, however, since numerical
integration could still be done in one of two ways. First, one could take many draws from the heterogeneity
distribution, and calculate a weighted sum of the likelihood evaluated at each of these draws. However, this
approach is computationally infeasible in this setting, due to the large number of draws that are necessary to
acquire a close approximation to the likelihood. The second approach would be to use Gaussian quadrature
to integrate over the heterogeneity distribution, by again taking a weighted sum, but at a smaller number of
carefully chosen nodes. However, this approach is typically taken when the heterogeneity is unidimensional, or when the heterogeneity terms are independently distributed. Since neither of these assumptions
is tenable, and no way was found of using Gaussian quadrature to evaluate integrals over a standard bivariate normal distribution with nonzero correlation, this approach also is not feasible.
18. Hoynes (1996) also uses this approach.
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The Journal of Human Resources
likelihood, then, is calculated by weighting each of these j conditional likelihoods by
the probability that the heterogeneity nodes take this value, and summing over all
nodes of the heterogeneity distribution.19 Formally, the likelihood for couple i is
J
ð12Þ li ðuÞ ¼ + pj li ðu; vjh ; vjw Þ
j¼1
These nodes and weights are estimated along with the rest of the parameters, subject
to some straightforward normalizations.20
D. Accounting for Missing Wages
Obviously, if either or both individuals do not work, a wage is not observed for the
nonworking spouse or spouses, and one must account somehow these missing wages.
The approach taken in this paper follows a specification in van Soest (1995). Namely,
a selection corrected auxiliary wage equation is first estimated. Then, the likelihood
is evaluated at each wage in the predicted distribution of wages for each labor force
nonparticipant, and these likelihoods are weighted by their respective probabilities.21
Formally, the likelihood function is thus augmented as follows. Let
li ðu; vjh ; vjw ; wph ; wpw Þ be the likelihood for a couple who both do not work, given predicted wages wph and wpw , and heterogeneity draws vjh and vjh . The overall likelihood
for this couple, given the heterogeneity draws, is then
Z NZ N
ð13Þ li ðu; vjh ; vjw Þ ¼
li ðu; vjh ; vjw ; wph ; wpw ÞdF h ðwphi ÞdF w ðwpwi Þ
0
0
where F h ðwphi Þ denotes the estimated wage offer distribution for the husband of couple i,
and F w ðwpwi Þ denotes the estimated wage offer distribution for the wife of couple i. To
19. One may note that incorporating unobserved heterogeneity helps to alleviate the problem of zero probabilities noted above. With unobserved heterogeneity incorporated, for an observation to have zero likelihood, it must be that either or both utility maximizing hours are zero at all heterogeneity nodes. In practice,
this has not posed a problem.
20. These normalizations assure that all weights are between 0 and 1 and sum to 1, and that all the nodes
are identified by constraining the husbands’ nodes to be increasing in j.
21. Two other approaches have been used, but both have their shortcomings in this setting.
The first is to simply impute a predicted wage from an auxiliary wage regression for either all nonworkers, or all individuals, both working and nonworking. (See, for example, Hausman 1981, the initial
specification in van Soest 1995, and Hoynes 1996.) This option has the undesirable property that wages
are estimated outside of the structural model. In addition, as noted in MaCurdy et al. (1990) this approach
misspecifies the sample likelihood to some extent.
The second, and likely best approach, is to follow a suggestion in MaCurdy et al. (1990). In this approach, one assumes a joint distribution for tastes for work and wages. The likelihood of working 0 hours,
then, is the joint probability that the wage is low enough and the heterogeneity term is such that the individual desires to work zero hours in the structural model, given parameters. However, this approach runs
into some practical difficulties in the current context. Since this specification does not have a continuous
distribution of heterogeneity, one cannot integrate over such a joint distribution, but would instead have
to specify a joint discrete distribution. As such, the gradient for these individuals would likely be zero
in most regions of the parameter space, and the likelihood would have large discontinuous in others. Further, even if there was a continuous distribution, one would need to solve for the possibly two dimensional
region in ðvh ; vw Þ space over which the desired hours are zero.
Heim
evaluate this double integral, Gaussian quadrature techniques are used, which approximates the double integrals by replacing them with a weighted double sum of the form
K
L
p
p
ð14Þ li ðu; vhj ; vwj Þ ¼ + + pk pl li ðu; vhj ; vwj ; whk
; wwl
Þ
k¼1 l¼1
p L
p K
are the nodes at which the likelihood is evaluated, and
and wwl
where whk
l¼1
k¼1
fpk gKk¼1 and fpl gLl¼1 are the weights.22 In practice, ten nodes are used.
For couples with only one nonworking member, the likelihood is defined and calculated analogously to the method described above.
This approach has the benefit that one does not need to assume that wages are predicted exactly in order for the likelihood to be specified correctly, but has the undesirable property that the wage distribution is predicted outside of the structural
model.
E. Fixed Costs
It is typically thought that workers, and especially married women with children,
face substantial fixed costs of labor market entry due to the need to pay for transportation, child care, work-related clothing, and a variety of work-related costs. In addition, such fixed costs may help to explain the lack of individuals working small
number of hours. Further, fixed costs have often been found to be significant in
the estimation of structural labor supply models.23 Thus, the estimation method is
augmented to allow for the estimation of fixed costs of working for both the husband
and wife.
In the specification used in this paper, fixed costs are modeled as monetary costs of
labor force participation. The difficulty of implementing this, however, is that when
fixed costs are incorporated into the budget constraint, there may be multiple local
maxima. Note that the hours-finding algorithm described above is only guaranteed
to find a local optimum, and the local optimum is only guaranteed to be the global
optimum, when the budget constraint is convex. Hence, to find the global optimum
on the nonconvex budget constraint when fixed costs are incorporated, the hoursfinding algorithm must be augmented as follows.
First, the hours-finding algorithm described above locates the utility maximizing
hours over the entire hours space, assuming that the fixed costs of both members
of the couple working are paid over this entire region. Second, a one dimensional
search algorithm locates the utility maximizing hours of the husband, given that
the wife does not work, and that only the fixed costs of the husband working are paid.
Third, a one dimensional search algorithm locates the utility maximizing hours of the
wife, given that the husband does not work, and only the fixed costs of the wife working are paid. Fourth, the utility of both members of the couple working zero hours is
22. As noted in Butler and Moffitt (1982), such an approximation is exact if the integrand is a function of
the form e2Z gðZÞ, where gðZÞ is a polynomial of degree less than 2K21. Since the normal probability density function is of this form, this approach can calculate the likelihood with a higher degree of accuracy,
given a number of evaluation nodes, than would a straightforward random draws approach.
23. See, for example, Heckman (1974), Hausman (1980), Heckman (1980), Cogan (1981), Blau and
Robins (1988), Bourguignon and Magnac (1990), Ribar (1992), and Hoynes (1996).
361
362
The Journal of Human Resources
evaluated, when neither pays their fixed cost. Finally, the utility values at each of
these local maxima are compared, and the utility maximizing hours on the nonconvex budget constraint is the argmax of this set of local maxima.24
Fixed costs are specified as being possibly different for husbands and wives, and
the fixed costs do not depend on the demographic characteristics of the couple. The
fixed costs when both work are assumed to be the sum of the husband’s and wife’s
fixed costs. Though this is likely a gross simplification, as there are probably economies of scale in work costs, and those work costs likely depend on the characteristics of the couple, the estimated fixed costs turn out to be quite small. Hence, it
seems unlikely that the choice of the specification of fixed costs has much effect
on the estimates of labor supply parameters in the current study.
IV. Data and Functional Form
A. Data
The data for this study come from the 2001 wave of the Panel Study of Income Dynamics. These data contain information on hours of work during the year 2000, and
so contain information on labor supply during the year immediately before the tax
changes that were implemented in 2001.25
In this wave of the PSID, individuals were asked questions about their hours of
work in the past year, their hourly wage, and several components of their nonlabor
income. In addition, data was collected on the respondents’ education, their family
size, and whether or not they had children.
Since this paper concerns the estimation of a joint model of labor supply, observations in which the respondent is not married are excluded. In addition, the sample
is restricted to couples in which the husband is between 25 and 55 years of age, to
focus on the labor force behavior of couples in their prime working years.26 The estimation sample excludes those who are nonworking due to being retired, a student,
or permanently disabled, those who report working more than 4000 hours in a year,
and couples for whom any data is missing.
For a measure of individuals’ hourly wages, individuals’ direct reports of their hourly
wage are utilized. For those that are salaried, the reported salary is divided by a standard
number of hours per period.27 This has the benefit of not being contaminated by measurement error in hours, as would a wage measure created by labor income divided
24. Note, however, that the utility of two or more of these options could be equal at some parameter values.
Hence, there needs to be some tie-breaking rule in order for the likelihood function to be well defined.
When the local utility maxima are equal, it is assumed that couples will choose to both work over either
or both not working, and will choose the husband alone working over the wife alone working. Finally, they
will choose any of these over both husband and wife not working. Such ties were not a problem in practice,
however.
25. The PSID was chosen because of its wide use in structural labor supply studies, and particularly because of the availability of direct wage and hours reports. Since the model estimated in this paper is static,
however, the panel nature of the PSID was not used.
26. The sample is not cut to only include women 25 to 55, however. This is because, since husbands tend to
marry younger wives, such a cut would tend to exclude younger men and older women.
27. If they receive a weekly salary, their reported salary is divided by 40. If they receive a monthly salary, it
is divided by 160. If they receive a yearly salary, it is divided by 2000.
Heim
by observed hours, which may lead to division bias in the estimated wage elasticities.28 It
may, however, introduce some measurement error for salaried workers.29
For the nonlabor income variable, the couple’s reported total income (before
deductions and exemptions) is used, less the reported labor incomes of the husband
and wife.30 This contrasts with the nonlabor income variables used in many of the
classic labor supply studies, which used previous waves of the PSID in which data
on the taxable income of the couple was not collected.31
Finally, to control for heterogeneity in tastes for leisure due to observable factors,
data on the age, race, and education of the husband and wife, the number of children
in the family, and the age of the youngest child (to identify the presence of a child
under the age of 6) are utilized.
Summary statistics for these variables are presented in Table 1. In this sample, almost all men, and 81 percent of women, report working some hours. The mean annual hours of work are 2244 for men and 1436 for women.
Two auxiliary Heckman selection corrected wage regressions with bivariate normal disturbances are estimated to estimate a distribution of missing wages for nonworkers, one for husbands and one for wives. A full maximum likelihood estimator is
used for these. The results are presented in Table 2. Coefficients from these regressions are generally of the expected sign and of plausible magnitude.
In the estimation, fairly detailed versions of federal and state tax systems are incorporated. The standard federal income tax schedule is utilized, and it is assumed that those
who report not itemizing deductions take the standard deduction. For those who report
having itemized deductions, an approximation to their deductions is imputed by adding
their reported deductible medical expenses, charitable contributions, and interest paid on
a home mortgage, subject to this sum being at least as large as the standard deduction.
Everyone is assumed to takes exemptions of $2800 per family member. Finally, state income tax schedules are incorporated, as is the employee’s share of the payroll tax.32
B. Functional Forms
For the specification of the couple’s preferences, a quadratic utility specification is
used,33 in which
28. See Borjas (1980) and Eklof and Sacklen (2000).
29. Wage observations and predications are also truncated at the federal minimum wage of $5.15 per hour.
30. This variable still isnÔt ideal, however, since people may earn money from side jobs and investments
that either arenÕt taxable or aren’t reported. The amount of error this induces, however, should be small,
and likely is much smaller than the error in other available measures.
31. See, for example, Hausman (1981), who had to define nonlabor income as an 8 percent return on home
values.
32. The major parts of the tax system that are not incorporated are the Earned Income Tax Credit and the
Alternative Minimum Tax. The AMT is omitted because whether a couple was required to pay the AMT is
not observed. The EITC is omitted for two reasons. First, only 5.7 percent of couples in the sample have
earnings that make them eligible for the EITC, so the bias that results from omitting it is likely negligible.
Second, incorporating the EITC requires altering the hours-finding algorithm to deal with the nonconvex
budget constraints that result. These issues are addressed in Heim (2008), which estimates the model used
here on a sample of married couples with wives that have less than a high school education, and so are more
likely to qualify for the EITC.
33. For good descriptions of the properties of the quadratic utility function, see Stern (1986) and
Goldberger (1987).
363
364
The Journal of Human Resources
Table 1
Sample Statistics
Mean
Standard
Deviation Minimum Maximum
Annual hours of work of husband
2,244.18
556.02
0
Annual hours of work of wife
1,436.69
872.38
0
Husband annual hours positive?
0.99
0.11
0
Wife annual hours positive?
0.81
0.39
0
Wage of husband
25.25
24.47
5.15
Wage of wife
16.02
13.22
5.15
Taxable income of husband and wife 84,745.38 87,965.10
0
Deductions
9,788.62 6,318.26 7,350
Labor income of husband
56,183.89 65,783.17
0
Labor income of wife
23,856.02 22,403.30
0
Nonlabor income of couple
4,705.48 33,636.09
0
Age of husband
41.45
7.96
25
Age of wife
39.74
8.12
20
Number of children
1.34
1.16
0
Presence of children younger than 6
0.28
0.45
0
Years of education of husband
13.52
2.41
3
Years of education of wife
13.57
2.30
2
Husband nonwhite
0.29
0.45
0
Wife nonwhite
0.28
0.45
0
ð15Þ
3,920
3,920
1
1
375
250
2,112,300
134,250
1,156,700
250,000
890,600
55
62
7
1
17
17
1
1
Uðhh ; hw ; CÞ ¼ bhh h2h + bww h2w + bCC C2
+ bhw hh hw + bhC hh C + bwC hw C
+ bh hh + bw hw + bC C
and where bC is normalized to 1. The quadratic utility function can be thought of as a
second order Taylor approximation to an arbitrary utility function. It has the desirable
features that labor supplies can be backward bending, and hours endowments do not
have to be specified or separately estimated. However, nothing in this utility function
constrains it from displaying a positive marginal utility of working at low hours levels.
This functional form also easily allows for the incorporation of heterogeneity due to both
observable and unobservable factors by specifying the coefficients on the linear terms as
ð16Þ
bh ¼ X#h ah + vh
bw ¼ X#w aw + vw
where the vectors Xh and Xw contain observable preference shifters,34 and vh and vw
denote unobservable heterogeneity in taste for work.
34. This method of incorporating observable preference shifters is analogous to the demographic translating in Pollak and Wales (1992).
Heim
Table 2
Auxiliary Log Wage Regressions
Women
Log Wage
Equation
Age
0.0204
(0.0156)
20.0001
(0.0002)
0.1250
(0.0605)
0.3041
(0.0617)
0.7086
(0.0621)
20.0309
(0.0371)
20.0370
(0.0587)
Age squared
High school graduate
Some college
College graduate plus
Black
Other
Presence of children
younger than 6
Number of children
Constant
r
s
l
Log likelihood
N
1.7089
(0.3028)
20.3059
(0.1362)
0.5046
(0.0128)
20.1544
(0.0712)
21,530.32
1,501
Men
Selection
Equation
20.0109
(0.0452)
20.0013
(0.0006)
0.3893
(0.1335)
0.6351
(0.1369)
0.6862
(0.1399)
0.1185
(0.1075)
20.1957
(0.1391)
20.6095
(0.1051)
20.1370
(0.0389)
1.5015
(0.8748)
Log Wage
Equation
Selection
Equation
0.0901
(0.0196)
20.0010
(0.0002)
0.2459
(0.0533)
0.4325
(0.0550)
0.8434
(0.0536)
20.2573
(0.0393)
20.1984
(0.0576)
0.0321
(0.1391)
20.0006
(0.0017)
0.1312
(0.9123)
0.1940
(0.3046)
0.8108
(0.4345)
20.6978
(0.2494)
20.5477
(0.3434)
20.1194
(0.2611)
0.6249
(0.3895)
20.5511
(0.1455)
0.5844
(0.0111)
20.3220
(0.0869)
21,372.85
1,501
2.0911
(2.8728)
Note: Asymptotic standard errors in parentheses.
V. Results
Table 3 presents the results from three specifications of the model.
In Column 1, heterogeneity in tastes for work is accounted for, but these tastes do
not depend on the demographic characteristics of the couple. The first panel of the
table contains the estimated coefficients on the squared terms in the utility function.
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The Journal of Human Resources
Table 3
Estimation Results: 2001 PSID Sample
(1)
Squared and interacted terms
210.994
bhhx102
(0.775)
bwwx102
20.926
(0.052)
0.032
bCCx105
(0.015)
20.854
bhwx104
(0.148)
bhCx104
20.259
(0.133)
20.828
bwCx104
(0.059)
Heterogeneity nodes
Weights
0.033
p1
(0.005)
0.701
p2
(0.015)
0.161
p3
(0.011)
0.029
p4
(0.007)
p5
0.076
Husband nodes
84.271
bh1
(11.828)
470.410
bh2
(32.062)
472.470
bh3
(33.370)
bh4
679.764
(49.689)
694.488
bh5
(47.036)
Wife nodes
17.704
bw1
(2.034)
29.984
bw2
(2.118)
(2)
(3)
25.5441
(0.486)
20.780
(0.050)
20.046
(0.014)
20.801
(0.183)
20.019
(0.142)
20.463
(0.059)
25.613
(0.518)
20.799
(0.057)
20.048
(0.014)
20.796
(0.194)
20.048
(0.152)
20.497
(0.062)
0.031
(0.005)
0.711
(0.015)
0.166
(0.011)
0.022
(0.005)
0.070
0.031
(0.005)
0.712
(0.015)
0.163
(0.011)
0.023
(0.005)
0.072
32.1089
(5.899)
230.5252
(20.978)
230.6359
(21.531)
343.6533
(31.882)
344.5299
(30.801)
35.096
(9.436)
232.781
(23.987)
233.052
(24.518)
345.730
(35.574)
347.737
(34.550)
17.8059
(1.880)
24.9652
(1.910)
19.077
(3.695)
25.977
(3.490)
(continued )
Heim
Table 3 (continued)
bw3
bw4
bw5
Stochastic elements
sh
sw
r
Demographic characteristics
ah
Husband’s education
Wife’s education
Husband’s age
Wife’s age
Husband’s age squared/1,000
Wife’s age squared/1,000
Number of children
Presence of children younger than 6
Nonwhite
aw
Husband’s education
Wife’s education
Husband’s age
(1)
(2)
(3)
212.267
(5.122)
2119.885
(8.737)
31.292
(2.470)
28.377
(3.277)
2121.284
(148.822)
29.125
(2.293)
211.441
(6.048)
225.362
(216.701)
29.772
(3.846)
351.96
(7.99)
571.53
(10.57)
0.134
(0.039)
338.45
(7.19)
557.69
(10.35)
0.111
(0.036)
338.82
(7.29)
557.26
(10.18)
0.109
(0.037)
0.577
(0.193)
20.397
(0.197)
20.268
(0.143)
20.137
(0.145)
0.275
(0.185)
20.214
(0.180)
0.044
(0.205)
0.218
(0.311)
23.504
(0.979)
0.736
(0.196)
20.0
Labor Supply with Taxes
Estimating a Continuous Hours Model Using a
Direct Utility Specification
Bradley T. Heim
abstract
This paper proposes a new method for estimating family labor supply in the
presence of taxes. This method accounts for continuous hours choices,
measurement error, unobserved heterogeneity in tastes for work, the
nonlinear form of the tax code, and fixed costs of work in one comprehensive
specification. Estimated on data from the 2001 PSID, the resulting elasticities
for married males are consistent with those found elsewhere in the literature
but female wage elasticities are substantially smaller than those found in
most of the literature. Simulations of recent tax acts predict small effects on
the labor supply of married couples.
I. Introduction
Changes in the income tax code, including the recent changes that
were enacted as part of the Economic Growth and Tax Relief Reconciliation Act
(EGTRRA) of 2001 and the Jobs and Growth Tax Relief Reconciliation Act
(JGTRRA) of 2003, generally take a complex form. Predicting the possible effects
of such policies, both actual and prospective, requires estimates that can be used
to simulate the effects of complex changes in the tax code. In addition, the fact that
Bradley Heim is a Financial Economist in the Office of Tax Analysis, U.S. Department of the Treasury.
Substantial work on this paper was completed when the author was an Assistant Professor at Duke
University. He wishes to thank Joe Altonji, Peter Arcidiacono, Eric French, Hilary Hoynes, Marjorie
McElroy, Bruce Meyer, Robert Triest, seminar participants at the University of Virginia, the 2003 North
American Summer Meetings of the Econometric Society, and the 2004 Annual Meetings of the National
Tax Association, and two anonymous referees for helpful advice and comments. He takes responsibility
for all remaining errors. The views expressed are those of the author and do not necessarily reflect those
of the Department of the Treasury. The data used in this article can be obtained beginning October
2009 through September 2012 from Bradley T. Heim, Office of Tax Analysis, U.S. Department of
Treasury, Room 4036B, 1500 Pennsylvania Ave NW, Washington, DC 20220, or
Bradley.Heim@do.treas.gov.
½Submitted March 2007; accepted February 2008
ISSN 022 166X E ISSN 1548 8004 8 2009 by the Board of Regents of the University of Wisconsin System
T H E JO U R NAL O F H U M A N R ES O U R C ES
d
44
d
2
Heim
couples are taxed jointly implies that it is important to evaluate both husbands’ and
wives’ labor supply responses to changes in the tax code, particularly if there are
cross-substitution effects. A structural model of joint labor supply is required to fully
capture these responses. This paper proposes a new method to structurally estimate
family labor supply in the presence of taxation.
Starting with Hausman and Ruud (1984), several studies have attempted to estimate structural models of husbands’ and wives’ joint labor supply choices in the
presence of taxes. As noted in Blundell and MaCurdy (1999), these approaches generally fall into two groups.
In the first group, the structural model allows for continuous hours choices of husbands and wives, with a nonnegativity constraint on the wife’s hours choice.1 Examples
of this type of study include Hausman and Ruud (1984), Ransom (1987), Kapteyn
et al. (1990), Kooreman and Kapteyn (1986), and Blundell and Walker (1986). However, estimating continuous family labor supply while allowing for both unobserved
heterogeneity in tastes for work and measurement error in hours requires accounting
for four stochastic elements in the model, and was thought to require integrating
over complex regions, which greatly complicates estimation. As a result, these studies
generally chose a subset of the desirable elements of a family labor supply model:
accounting for unobserved heterogeneity in tastes for work, accounting for measurement error in hours of work, or accounting for the endogeneity of the after tax wage
due to the structure of the tax code. Further, such studies generally required functional
forms for utility for which both the unconditional and conditional labor supply functions exist in closed form. As noted in Kapteyn et al. (1990), such a requirement
greatly limits the number of functional forms that can be used in estimation.
In the second group, the structural model is specified as a choice of the husband
and wife among a discrete set of hours combinations. For example, in van Soest
(1995), husbands and wives are modeled as choosing between either 25 or 36 discrete choices of hours combinations. Such an approach also has been used by Hoynes
(1996), and others. This approach allows for the incorporation of heterogeneity of
preferences, measurement error in wages, and a very detailed treatment of the budget
constraint. However, in order to maintain computational feasibility, errors in evaluating the utility of each discrete choice are often assumed to be distributed i.i.d. extreme value, and this assumption of independent error terms becomes untenable as
the number of hours choices increases. In addition, many tax law changes would
be expected to cause nontrivial changes in hours, but changes that would be too small
to appear in a model in which individuals choose between a small number of hours
choices. Finally, although it is generally assumed that annual hours peak at full time,
part time, and nonparticipation, with little spread around these points, in reality there
is considerable variation in the distribution of reported hours,2 which discrete choice
models are resigned to explain through measurement error.
In this paper, a method is proposed that overcomes the limitations in both of these
approaches, in that it allows for continuous hours choices of the husband and wife,
while still incorporating measurement error in hours and heterogeneity in tastes for
1. The husband is assumed to be working.
2. The variation in hours can be seen in Figure 4 below, which presents data on hours worked among husbands and wives using the 2001 Panel Study of Income Dynamics.
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The Journal of Human Resources
work in the stochastic specification, accounts for the nonlinear structure of the tax
schedule, and accounts for fixed costs of working, all in one comprehensive model.
The method estimates parameters of a direct utility function using maximum likelihood techniques by using numerical methods to explicitly solve for the utility maximizing hours of husbands and wives, given parameters, and then calculating the
likelihood given the stochastic specification.
This model is estimated using data from the 2001 Panel Study of Income Dynamics. The key empirical findings include elasticities for married males consistent with
those found elsewhere in the literature, but female wage elasticities that are substantially smaller than those found in most of the literature. Simulations of the two most
recent major tax changes show that these smaller elasticities, in turn, imply small
responses of married couples’ labor supply.
The paper proceeds as follows. Section II outlines the assumed model of family
labor supply and provides an overview of the estimation method. Section III
describes in detail the particular features of the estimation method, and demonstrates
how it may be altered to account for heterogeneity in tastes for work, the missing
wages of nonworkers, and fixed costs of work. Section IV describes the data and
functional forms used in the estimation, and the estimates are reported in Section
V. In Section VI, these results are used to predict the labor supply effects of the
2001 and 2003 tax law changes. Section VII concludes.
II. An Overview of the Estimation Method
Couples are modeled in this paper as maximizing a joint utility function, Uðhh ; hw ; CÞ,3 over hours of work for the husband, hh , hours of work for the
wife, hw , and total consumption, C, subject to a joint budget constraint, given by
J
ð1Þ
C # Y + Wh hh + Ww hw 2 + tj ðIj 2 Ij21 Þ1ðY TI . Ij Þ
j¼1
J
2 + tj ðY TI 2 Ij21 Þ1ðIj $ Y TI . Ij21 Þ;
j¼1
where Wh and Ww denote the husband’s and wife’s wages, respectively, Y denotes the
the family’s taxable income (income less
family’s nonlabor income, Y TI denotes
J
deductions and exemptions), Ij j¼1 denotes the income tax bracket endpoints, tj
denotes the tax rate on income between Ij and Ij+1 , and 1ðÞ denotes the indicator
function. In this case, the budget constraint consists of the union of several plane segments, with kink lines at the intersection of these segments. (See Figure 1.)
3. There has been a fair amount of criticism of this unitary model of labor supply. (See McElroy and Horney
1981, Chiappori 1988, Schultz 1990, and Fortin and Lacroix 1997.) Donni (2003) examines the collective
model in the presence of taxes, but does not estimate a model. Blundell, Chiappori, Magnac and Meghir
(2007) and Bloemen (2004) estimate collective models labor supply, but do not account for taxes. This approach could plausibly be adapted to estimate a collective model of labor supply in the presence of taxes, since
implicit in the calculation of the likelihood is the explicit solution of the structural model, which could be done
analogously for other assumed theoretical models. Such extensions are left for future work.
Heim
Figure 1
Married Couple’s Nonlinear Budget Constraint
Let the gradient of a particular budget segment, j, in the direction of the husband’s
hours be wjh , the gradient in the direction of the wife’s hours be wjw , and the point at
which this segment of the budget constraint would intersect the origin if extended
(the ‘‘virtual income’’) be yj .
To understand the estimation approach taken in this paper, start with the estimation
method used in Hausman and Ruud (1984). In that paper, husbands’ and wives’ labor
supply functions given a linear budget constraint with wages wjh and wjw and virtual
income yj are given by
ð2Þ
hjh ¼ hh ðwjh ; wjh ; yj ; uÞ
hjw ¼ hw ðwjh ; wjh ; yj ; uÞ
To solve for the desired hours of the couple on their actual, nonlinear budget constraint, Hausman and Ruud use an algorithm that first calculates the set of hours that
result when the wages and virtual incomes associated with each budget segment are
substituted into these labor supply equations. If a member of this set lies on a plane
of the actual budget constraint, then the desired hours of the couple, given parameters, have been found. If not, the desired hours lie either on a kink of the budget set or
the boundary, and a further algorithm is used to identify which is the case. Once the
desired hours have been solved for, given a stochastic structure that maps desired
hours to observed hours, the likelihood of the sample is calculated, and maximum
likelihood techniques are used to solve for the likelihood maximizing parameters.
This paper follows Hausman and Ruud’s specification directly, with one crucial
difference. Instead of starting with a closed form for the husbands’ and wives’ labor
supply functions and solving for the desired hours using their algorithm, the method
in this paper starts with a functional form for the husband’s and wives’ utility function and solves for the desired hours using numerical methods. In a specification with
no heterogeneity, no fixed costs, and the direct utility function associated with the
labor supply functions used in Hausman and Ruud, the two methods would be
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The Journal of Human Resources
identical, because the two methods would simply be using two alternative ways of
solving for the same set of desired hours. Given the same stochastic specification,
the likelihoods would be the same, resulting in the same likelihood maximizing
parameters.
However, there are at least four advantages to specifying the estimation method
in terms of a utility function and using numerical methods to solve for desired
hours. First, if one is working with a direct utility function specification of the
problem, then closed form solutions for the conditional and unconditional labor
supply functions are not required, which expands the set of functional form
choices. Second, heterogeneity in tastes for work can be incorporated by assuming that couples are drawn from a finite set of types. Third, one can account for
missing wages of nonworkers by assuming that unobserved wages are drawn
from a distribution of wages, instead of assuming that missing wages are known
with certainty.4 Finally, fixed costs can be incorporated into the specification of
the budget constraint.5 On the other hand, a computationally intensive numerical
algorithm must still be used in this approach, since one must simply solve explicitly for the utility maximizing hours for the husband and wife by maximizing the
utility function subject to the actual budget constraint using a numerical algorithm.6
The overall estimation method works as follows. Given the hours-finding algorithm, the couple’s desired hours conditional on parameters of the utility function are solved for numerically. Then, given the stochastic specification for
measurement error in hours, the likelihood is calculated. Finally, standard maximum likelihood techniques are used to solve for the likelihood maximizing
parameters.7
The construction of the hours-finding algorithm, and the derivation of the likelihood, are described in the next section.
4. Since both of these extensions involve weighted sums of likelihoods that are conditional on particular
heterogeneity and wage draws, the resulting overall likelihood is as well behaved as the underlying conditional likelihoods.
5. So long as there is still a one-to-one correspondence between parameters and desired hours, the model is
identified. A tiebreaking rule ensures that this is so.
6. One might be concerned that this method suffers from the critique in MaCurdy et al. (1990) of continuous hours structural estimation methods, that estimates in Hausman-type labor supply estimation methods
are constrained to globally satisfy Slutsky positivity, which in turn biased estimated parameters. However,
as noted in Heim and Meyer (2004), it is the assumption of a continuous distribution of heterogeneity with
infinite support that forces parameters to satisfy Slutsky positivity globally. In this paper, the specification
of heterogeneity is discrete, and so this critique does not apply. As evidence for this claim, the estimated
parameters in this paper do not satisfy Slutsky positivity at high levels of hours. However, the levels at
which Slutsky positivity is not satisfied are above the levels of hours at which couples are observed, and
so deadweight loss calculations are not affected.
Further, Heim and Meyer note that the constraint in the Hausman method amounts to constraining preferences to be globally convex, and that a way to allow for the estimation of nonconvex preferences is to
estimate parameters of a direct utility function. With a sufficiently robust search algorithm, one could allow
estimated preferences to be nonconvex in this model; however, it would be extremely computationally intensive, and so is not attempted here.
7. Like other papers in this literature, since parameters are estimated using cross-sectional data, identification comes from variation in wages and incomes across individuals. In addition, since state tax schedules
are incorporated, additional identifying variation comes from differences across states in tax schedules.
Heim
III. Details of the Estimation Method
A. Hours-finding Algorithm
For simplicity, denote the budget constraint as
ð3Þ C # Bðhh ; hw ; Þ;
where hh denotes the husband’s hours of work, hw denotes the wife’s hours of work,
and C denotes consumption. Since, in this static model, there is no saving or dissaving, the individual will consume all of their income, and so
ð4Þ C ¼ Bðhh ; hw ; Þ:
For now, suppose that the budget constraint is a convex budget constraint. It is clear
that the unique utility maximizing hours or work, hh and hw , satisfy
ð5Þ hh ¼ arg max fUðhh ; hw ; Bðhh ; hw ; ÞÞ: H $ hh $ 0g
hh
and
ð6Þ hw ¼ arg max fUðhh ; hw ; Bðhh ; hw ; ÞÞ: H $ hw $ 0g
hw
where H denotes the endowment of hours each period. Hence, a straightforward way
to find the utility maximizing combination of hours is to maximize utility over the
husband’s hours conditional on the wives hours using a line search technique,8 then
maximize utility over the wife’s hours conditional on the husband’s hours, and iterate
in this way until the hours coordinates are changing less than a specified tolerance.
When this happens, a fixed point of the system of equations
ð7Þ hh ¼ arg max fUðhh ; hw ; Bðhh ; hw ; ÞÞg
hh
and
ð8Þ hw ¼ arg max fUðhh ; hw ; Bðhh ; hw ; ÞÞg
hw
has been found. This algorithm is known as a coordinate direction method, and is
guaranteed to find a fixed point to this system.9
If the budget constraint is linear, and the utility function is quasiconcave and twice
differentiable, then the fixed point of this system must be the unique utility maximizing hours on the budget constraint. In addition, if the fixed point of this system lies on
a plane of a kinked, but convex, budget constraint, the fixed point of this system still
8. For this paper, a bracketing method is used to search along a line. For more information on bracketing
methods, see Judd (1998). Though slow compared to other methods, this type of search method does not
require the calculation of derivatives and guarantees that a maximum will be found.
9. See Judd (1998).
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The Journal of Human Resources
must be the unique utility maximum on the budget constraint. Panel (a) of Figure 2
illustrates this case. Suppose that the search algorithm starts at (hh’, 0). Maximizing
utility keeping hh fixed at hh’, yields hw’ as the solution to Equation 8. The rounded
shape in the figure consists of the isoutility curve on the boundary of the budget constraint that goes through this point, where points inside this shape yield higher utility.
Maximizing utility keeping hw fixed at hw’ yields a value for hh inside of the rounded
shape, and the utility at this point will be higher than at the previous point. This process will continue to iterate toward a fixed point that lies on the middle plane of this
budget constraint, which is the utility maximizing point.
However, if the fixed point that this algorithm finds lies along a kink, it is possible
that the fixed point is not the utility maximum. This is because it is possible that, due
to the kinks in the nonlinear budget constraint, the intersection of the indifference
surface with the budget constraint may itself have a ‘‘kink’’ in it, and so searching
along the coordinates may miss points of higher utility that lie in the interior of this
intersection. This second case is shown in Panel (b). In this case, maximizing utility
keeping hh fixed at hh’ yields hw’, and maximizing utility keeping hw fixed at hw’
yields hh’, where (hh’, hw’) is a fixed point of the system that lies on the kink of
the budget constraint. However, there are points inside the isoutility curve in the figure that yield higher levels of utility, and hence the fixed point is not the global utility
maximum.
If the algorithm finds a fixed point at a kink, in order to ensure that the fixed point
is indeed the optimum, one must search along the kink for a higher point. If such a
point is found, the method returns to searching along the coordinate directions until
another fixed point is found. If not, then the algorithm has found the utility maximizing hours.
The foregoing discussion assumed that the budget constraint was convex. If the
budget constraint is nonconvex, however, there is the possibility of multiple local optima, and hence multiple fixed points of the system in Equations 7 and 8. Such a nonconvexity might arise, for example, due to the presence of fixed costs of work, or due
Figure 2
Search Algorithm—Two Cases
Heim
to the Earned Income Tax Credit. The alterations to the algorithm necessary to deal
with the fixed costs of work are discussed in Section III.E. To deal with nonconvexities due to the EITC, one can use an insight in Hausman (1985) and treat the nonconvex budget constraint as the union of convex budget constraints. One searches for
the utility maximizing hours on each constituent budget constraint, and compares the
utility levels at each optimum to find the global optimum. However, this procedure
increases the execution time of the search algorithm in proportion to the number of
convex budget constraints that make up the nonconvex budget constraints. Since only
a small proportion, less than 6 percent, of the respondents in the sample used here
earn an amount that would make them eligible to receive the EITC, the EITC is omitted from the budget constraints used in the estimation below.10
B. Derivation of Likelihood Function
For the stochastic specification, the specification in Hausman (1981) is generalized,
in assuming that when the utility maximizing hours of a member of a couple are positive, then the observed hours for this member are these utility maximizing hours plus
additive measurement errors, subject to a zero hours lower bound. On the other hand,
if the utility maximizing hours of a member of a couple are zero, then that member’s
observed hours are also zero.11 Thus, in effect, this specification assumes that a survey respondent may have forgotten about working positive hours, and misreport that
they worked zero hours, but that if the respondent worked 0 hours, they will accurately report this.12 Formally, the assumed data generating process is given by
maxfhhi ðu0 Þ + ehi ; 0g if hhi ðu0 Þ . 0
hhi ¼
0
if hhi ðu0 Þ ¼ 0
ð9Þ
maxfhwi ðu0 Þ + ewi ; 0g if hwi ðu0 Þ . 0
hwi ¼
0
if hwi ðu0 Þ ¼ 0
where hhi and hwi denote observed hours of the husband and wife in couple i, and
hhi ðu0 Þ and hwi ðu0 Þ denote the husband and wife’s utility maximizing hours of work
given the true parameters u0 .13
10. The Earned Income Tax Credit is, however, incorporated in this same type of model in Heim (2008),
where the sample consists of married couples in which the wife has less than a high school degree.
11. A possibly more desirable stochastic specification would be that used in Hoynes (1996), which assumes
that respondents correctly report their participation status, but may report positive hours with error. However, because the distribution of heterogeneity is discrete in the current specification, for the log likelihood
to not be negative infinity, participation status would need to be correctly predicted for at least one heterogeneity node for all observations, which would introduce numerous discontinuities into the likelihood and
may constrain parameters.
12. Note that this stochastic specification differs from other papers in this literature. For example, Hoynes
(1996), assumes that labor force participation and nonparticipation are accurately observed, but that positive hours amounts may be observed with error. Van Soest (1995), on the other hand, does not account for
measurement error in hours of work.
13. A specification was also estimated in which respondents can misreport participation in both directions.
Results from this specification are presented in an appendix available from the author, and are quantitatively
and qualitatively similar to the base specification.
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The Journal of Human Resources
Assuming that the errors are distributed bivariate normal with standard deviations
sh and sw , and correlation r, the overall likelihood for couple i is
"
1ðhhi ðuÞ.0;hwi ðuÞ.0Þ #1ðhhi .0;hwi .0Þ
1
hhi 2hhi ðuÞ hwi 2hwi ðuÞ
b
;
;r
ð10Þ li ðuÞ ¼
sh sw
sh
sw
2
2
02h ðuÞ h 2h ðuÞ
131ðhhi ðuÞ.0;hwi ðuÞ.0Þ 31ðhhi .0;hwi ¼0Þ
hi hi
wi
sh
sw 2r
hhi 2hhi ðuÞ
6
7
1
4
@
A5
ffiffiffiffiffiffiffiffi
p
F
7
6 sh f
sh
12r2
7
6
36
7
6
7
4
5
h
i1ðhhi ðuÞ.0;hwi ðuÞ¼0Þ
hhi 2hhi ðuÞ
1
+ sh f
sh
22
02h ðuÞ h 2h ðuÞ
131ðhhi ðuÞ.0;hwi ðuÞ.0Þ 31ðhhi ¼0;hwi .0Þ
wi wi
hi
sw
64 1 f hwi 2hwi ðuÞ F@ sh 2r
7
A5
ffiffiffiffiffiffiffiffi
p
6 sw
7
sw
2
12r
6
7
36
7
6
7
4
5
h
i
1ðhhi ðuÞ¼0;hwi ðuÞ.0Þ
h
2h
ðuÞ
+ s1w f wi swwi
2h
i1ðhhi ðuÞ.0;hwi ðuÞ.0Þ 31ðhhi ¼0;hwi ¼0Þ
2h ðuÞ 2h ðuÞ
B shih ; swiw ; r
7
6
7
6
h
i1ðhhi ðuÞ¼0;hwi ðuÞ.0Þ
7
6
7
6 + F 2hswi ðuÞ
w
7
6
36
7
7
6
h
i1ðhhi ðuÞ.0;hwi ðuÞ¼0Þ
7
6
7
6 + F 2hhi ðuÞ
5
4
sh
+1ðhhi ðuÞ ¼ 0; hwi ðuÞ ¼ 0Þ
where bðk; l; rÞ denotes the p.d.f. of the standard bivariate normal distribution with
correlation r, and Bðk; l; rÞ denotes the c.d.f. of the standard bivariate normal distribution, cumulative over ð2N; k3ð2N; l. The sample likelihood is
Y
ð11Þ L ¼
li ðuÞ:
i
This likelihood consists of four terms. The first term, for a couple in which both
members are observed working positive hours (hhi . 0; hwi . 0), denotes the probability
that couple i is observed working positive hours in the amount of hhi and hwi , given that
the utility maximizing hours of work of both members of the couple, given parameters,
u, are hhi and hwi , respectively.14 The second term, for a couple in which the husband
works positive hours and the wife does not work (hhi . 0; hwi ¼ 0), consists of two
parts. The first is the probability that couple i is observed working hhi and 0 hours, respectively, given that the utility maximizing hours of the couple are both positive. The
second is the probability that couple i is observed working hhi and 0 hours, respectively,
given that the husband’s utility maximizing hours are positive, and the wife’s utility
14. Note that if either or both of these optimal hours are zero, given the stochastic specification, the probability that both members of the couple would be working at positive hours is zero.
Heim
maximizing hours are 0. The third term, for a couple in which the husband does not
work and the wife works positive hours, is analogous to the second. The fourth term,
for a couple in which both members do not work (hhi ¼ 0; hwi ¼ 0) consists of four
parts. The first part is the probability that the husband and wife are both observed working 0 hours, given that the utility maximizing hours of work for both of them are positive. The second part is the probability they are observed working 0 hours, given that
the husband’s utility maximizing hours are positive, and the wife’s are 0. The third part
is the probability that they are observed working 0 hours, given that the husband’s utility
maximizing hours are 0, and the wife’s are positive. The fourth part is the probability
that they are observed working 0 hours, given that the utility maximizing hours of the
couple are 0. This probability, given the stochastic specification, is 1.
Of course, the difficulty of implementing the likelihood function in this particular
form is that, if the utility maximizing hours of work is zero for any member of any
couple observed working positive hours, the likelihood is zero, and hence the log
likelihood is negative infinity. This, however, is ameliorated somewhat if unobserved
heterogeneity in tastes for work is incorporated in the stochastic specification, which
is described next.
C. Incorporating Unobserved Heterogeneity
Labor supply studies typically incorporate unobserved heterogeneity in tastes for
work in part to explain why observably identical individuals are observed working
different hours.15 Further, such studies invariably find that unobserved heterogeneity
enters significantly in the stochastic specification.16
Customarily, when continuous hours choices are allowed for in a structural labor
supply model, a continuous distribution of heterogeneity in tastes for work also is assumed. Unfortunately, that approach runs into a number of problems in this setting,17
but unobserved heterogeneity
J can be incorporated by assuming that it has a discrete
distribution.18 Let vhj ; vwj j¼1 denote the nodes of a discrete distribution of heterogeJ
neity in taste for work, and fpj gj¼1 denote their respective probabilities. The likelihood function may then be adapted in two steps. First, replace hhi ðuÞ and hwi ðuÞ
with hhi ðu; vhj ; vwj Þ and hwi ðu; vhj ; vwj Þ, respectively, in the above likelihood function.
This is the likelihood given that the husband and wife’s values from the heterogeneity
distribution are vhj and vwj , respectively. Denote this likelihood li ðu; vhj ; vwj Þ. The overall
15. Incorporation of observable heterogeneity is discussed in Section IV.B.
16. See, for example, Hausman (1981), MaCurdy et al. (1990), Triest (1990), and many others in the case
of individual labor supply, and van Soest (1995) and Hoynes (1996) in the case of family labor supply.
17. The difficulty arises from the need to integrate the above likelihood over this distribution, the solution
of which would not have a closed form. This is not necessarily insurmountable, however, since numerical
integration could still be done in one of two ways. First, one could take many draws from the heterogeneity
distribution, and calculate a weighted sum of the likelihood evaluated at each of these draws. However, this
approach is computationally infeasible in this setting, due to the large number of draws that are necessary to
acquire a close approximation to the likelihood. The second approach would be to use Gaussian quadrature
to integrate over the heterogeneity distribution, by again taking a weighted sum, but at a smaller number of
carefully chosen nodes. However, this approach is typically taken when the heterogeneity is unidimensional, or when the heterogeneity terms are independently distributed. Since neither of these assumptions
is tenable, and no way was found of using Gaussian quadrature to evaluate integrals over a standard bivariate normal distribution with nonzero correlation, this approach also is not feasible.
18. Hoynes (1996) also uses this approach.
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The Journal of Human Resources
likelihood, then, is calculated by weighting each of these j conditional likelihoods by
the probability that the heterogeneity nodes take this value, and summing over all
nodes of the heterogeneity distribution.19 Formally, the likelihood for couple i is
J
ð12Þ li ðuÞ ¼ + pj li ðu; vjh ; vjw Þ
j¼1
These nodes and weights are estimated along with the rest of the parameters, subject
to some straightforward normalizations.20
D. Accounting for Missing Wages
Obviously, if either or both individuals do not work, a wage is not observed for the
nonworking spouse or spouses, and one must account somehow these missing wages.
The approach taken in this paper follows a specification in van Soest (1995). Namely,
a selection corrected auxiliary wage equation is first estimated. Then, the likelihood
is evaluated at each wage in the predicted distribution of wages for each labor force
nonparticipant, and these likelihoods are weighted by their respective probabilities.21
Formally, the likelihood function is thus augmented as follows. Let
li ðu; vjh ; vjw ; wph ; wpw Þ be the likelihood for a couple who both do not work, given predicted wages wph and wpw , and heterogeneity draws vjh and vjh . The overall likelihood
for this couple, given the heterogeneity draws, is then
Z NZ N
ð13Þ li ðu; vjh ; vjw Þ ¼
li ðu; vjh ; vjw ; wph ; wpw ÞdF h ðwphi ÞdF w ðwpwi Þ
0
0
where F h ðwphi Þ denotes the estimated wage offer distribution for the husband of couple i,
and F w ðwpwi Þ denotes the estimated wage offer distribution for the wife of couple i. To
19. One may note that incorporating unobserved heterogeneity helps to alleviate the problem of zero probabilities noted above. With unobserved heterogeneity incorporated, for an observation to have zero likelihood, it must be that either or both utility maximizing hours are zero at all heterogeneity nodes. In practice,
this has not posed a problem.
20. These normalizations assure that all weights are between 0 and 1 and sum to 1, and that all the nodes
are identified by constraining the husbands’ nodes to be increasing in j.
21. Two other approaches have been used, but both have their shortcomings in this setting.
The first is to simply impute a predicted wage from an auxiliary wage regression for either all nonworkers, or all individuals, both working and nonworking. (See, for example, Hausman 1981, the initial
specification in van Soest 1995, and Hoynes 1996.) This option has the undesirable property that wages
are estimated outside of the structural model. In addition, as noted in MaCurdy et al. (1990) this approach
misspecifies the sample likelihood to some extent.
The second, and likely best approach, is to follow a suggestion in MaCurdy et al. (1990). In this approach, one assumes a joint distribution for tastes for work and wages. The likelihood of working 0 hours,
then, is the joint probability that the wage is low enough and the heterogeneity term is such that the individual desires to work zero hours in the structural model, given parameters. However, this approach runs
into some practical difficulties in the current context. Since this specification does not have a continuous
distribution of heterogeneity, one cannot integrate over such a joint distribution, but would instead have
to specify a joint discrete distribution. As such, the gradient for these individuals would likely be zero
in most regions of the parameter space, and the likelihood would have large discontinuous in others. Further, even if there was a continuous distribution, one would need to solve for the possibly two dimensional
region in ðvh ; vw Þ space over which the desired hours are zero.
Heim
evaluate this double integral, Gaussian quadrature techniques are used, which approximates the double integrals by replacing them with a weighted double sum of the form
K
L
p
p
ð14Þ li ðu; vhj ; vwj Þ ¼ + + pk pl li ðu; vhj ; vwj ; whk
; wwl
Þ
k¼1 l¼1
p L
p K
are the nodes at which the likelihood is evaluated, and
and wwl
where whk
l¼1
k¼1
fpk gKk¼1 and fpl gLl¼1 are the weights.22 In practice, ten nodes are used.
For couples with only one nonworking member, the likelihood is defined and calculated analogously to the method described above.
This approach has the benefit that one does not need to assume that wages are predicted exactly in order for the likelihood to be specified correctly, but has the undesirable property that the wage distribution is predicted outside of the structural
model.
E. Fixed Costs
It is typically thought that workers, and especially married women with children,
face substantial fixed costs of labor market entry due to the need to pay for transportation, child care, work-related clothing, and a variety of work-related costs. In addition, such fixed costs may help to explain the lack of individuals working small
number of hours. Further, fixed costs have often been found to be significant in
the estimation of structural labor supply models.23 Thus, the estimation method is
augmented to allow for the estimation of fixed costs of working for both the husband
and wife.
In the specification used in this paper, fixed costs are modeled as monetary costs of
labor force participation. The difficulty of implementing this, however, is that when
fixed costs are incorporated into the budget constraint, there may be multiple local
maxima. Note that the hours-finding algorithm described above is only guaranteed
to find a local optimum, and the local optimum is only guaranteed to be the global
optimum, when the budget constraint is convex. Hence, to find the global optimum
on the nonconvex budget constraint when fixed costs are incorporated, the hoursfinding algorithm must be augmented as follows.
First, the hours-finding algorithm described above locates the utility maximizing
hours over the entire hours space, assuming that the fixed costs of both members
of the couple working are paid over this entire region. Second, a one dimensional
search algorithm locates the utility maximizing hours of the husband, given that
the wife does not work, and that only the fixed costs of the husband working are paid.
Third, a one dimensional search algorithm locates the utility maximizing hours of the
wife, given that the husband does not work, and only the fixed costs of the wife working are paid. Fourth, the utility of both members of the couple working zero hours is
22. As noted in Butler and Moffitt (1982), such an approximation is exact if the integrand is a function of
the form e2Z gðZÞ, where gðZÞ is a polynomial of degree less than 2K21. Since the normal probability density function is of this form, this approach can calculate the likelihood with a higher degree of accuracy,
given a number of evaluation nodes, than would a straightforward random draws approach.
23. See, for example, Heckman (1974), Hausman (1980), Heckman (1980), Cogan (1981), Blau and
Robins (1988), Bourguignon and Magnac (1990), Ribar (1992), and Hoynes (1996).
361
362
The Journal of Human Resources
evaluated, when neither pays their fixed cost. Finally, the utility values at each of
these local maxima are compared, and the utility maximizing hours on the nonconvex budget constraint is the argmax of this set of local maxima.24
Fixed costs are specified as being possibly different for husbands and wives, and
the fixed costs do not depend on the demographic characteristics of the couple. The
fixed costs when both work are assumed to be the sum of the husband’s and wife’s
fixed costs. Though this is likely a gross simplification, as there are probably economies of scale in work costs, and those work costs likely depend on the characteristics of the couple, the estimated fixed costs turn out to be quite small. Hence, it
seems unlikely that the choice of the specification of fixed costs has much effect
on the estimates of labor supply parameters in the current study.
IV. Data and Functional Form
A. Data
The data for this study come from the 2001 wave of the Panel Study of Income Dynamics. These data contain information on hours of work during the year 2000, and
so contain information on labor supply during the year immediately before the tax
changes that were implemented in 2001.25
In this wave of the PSID, individuals were asked questions about their hours of
work in the past year, their hourly wage, and several components of their nonlabor
income. In addition, data was collected on the respondents’ education, their family
size, and whether or not they had children.
Since this paper concerns the estimation of a joint model of labor supply, observations in which the respondent is not married are excluded. In addition, the sample
is restricted to couples in which the husband is between 25 and 55 years of age, to
focus on the labor force behavior of couples in their prime working years.26 The estimation sample excludes those who are nonworking due to being retired, a student,
or permanently disabled, those who report working more than 4000 hours in a year,
and couples for whom any data is missing.
For a measure of individuals’ hourly wages, individuals’ direct reports of their hourly
wage are utilized. For those that are salaried, the reported salary is divided by a standard
number of hours per period.27 This has the benefit of not being contaminated by measurement error in hours, as would a wage measure created by labor income divided
24. Note, however, that the utility of two or more of these options could be equal at some parameter values.
Hence, there needs to be some tie-breaking rule in order for the likelihood function to be well defined.
When the local utility maxima are equal, it is assumed that couples will choose to both work over either
or both not working, and will choose the husband alone working over the wife alone working. Finally, they
will choose any of these over both husband and wife not working. Such ties were not a problem in practice,
however.
25. The PSID was chosen because of its wide use in structural labor supply studies, and particularly because of the availability of direct wage and hours reports. Since the model estimated in this paper is static,
however, the panel nature of the PSID was not used.
26. The sample is not cut to only include women 25 to 55, however. This is because, since husbands tend to
marry younger wives, such a cut would tend to exclude younger men and older women.
27. If they receive a weekly salary, their reported salary is divided by 40. If they receive a monthly salary, it
is divided by 160. If they receive a yearly salary, it is divided by 2000.
Heim
by observed hours, which may lead to division bias in the estimated wage elasticities.28 It
may, however, introduce some measurement error for salaried workers.29
For the nonlabor income variable, the couple’s reported total income (before
deductions and exemptions) is used, less the reported labor incomes of the husband
and wife.30 This contrasts with the nonlabor income variables used in many of the
classic labor supply studies, which used previous waves of the PSID in which data
on the taxable income of the couple was not collected.31
Finally, to control for heterogeneity in tastes for leisure due to observable factors,
data on the age, race, and education of the husband and wife, the number of children
in the family, and the age of the youngest child (to identify the presence of a child
under the age of 6) are utilized.
Summary statistics for these variables are presented in Table 1. In this sample, almost all men, and 81 percent of women, report working some hours. The mean annual hours of work are 2244 for men and 1436 for women.
Two auxiliary Heckman selection corrected wage regressions with bivariate normal disturbances are estimated to estimate a distribution of missing wages for nonworkers, one for husbands and one for wives. A full maximum likelihood estimator is
used for these. The results are presented in Table 2. Coefficients from these regressions are generally of the expected sign and of plausible magnitude.
In the estimation, fairly detailed versions of federal and state tax systems are incorporated. The standard federal income tax schedule is utilized, and it is assumed that those
who report not itemizing deductions take the standard deduction. For those who report
having itemized deductions, an approximation to their deductions is imputed by adding
their reported deductible medical expenses, charitable contributions, and interest paid on
a home mortgage, subject to this sum being at least as large as the standard deduction.
Everyone is assumed to takes exemptions of $2800 per family member. Finally, state income tax schedules are incorporated, as is the employee’s share of the payroll tax.32
B. Functional Forms
For the specification of the couple’s preferences, a quadratic utility specification is
used,33 in which
28. See Borjas (1980) and Eklof and Sacklen (2000).
29. Wage observations and predications are also truncated at the federal minimum wage of $5.15 per hour.
30. This variable still isnÔt ideal, however, since people may earn money from side jobs and investments
that either arenÕt taxable or aren’t reported. The amount of error this induces, however, should be small,
and likely is much smaller than the error in other available measures.
31. See, for example, Hausman (1981), who had to define nonlabor income as an 8 percent return on home
values.
32. The major parts of the tax system that are not incorporated are the Earned Income Tax Credit and the
Alternative Minimum Tax. The AMT is omitted because whether a couple was required to pay the AMT is
not observed. The EITC is omitted for two reasons. First, only 5.7 percent of couples in the sample have
earnings that make them eligible for the EITC, so the bias that results from omitting it is likely negligible.
Second, incorporating the EITC requires altering the hours-finding algorithm to deal with the nonconvex
budget constraints that result. These issues are addressed in Heim (2008), which estimates the model used
here on a sample of married couples with wives that have less than a high school education, and so are more
likely to qualify for the EITC.
33. For good descriptions of the properties of the quadratic utility function, see Stern (1986) and
Goldberger (1987).
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The Journal of Human Resources
Table 1
Sample Statistics
Mean
Standard
Deviation Minimum Maximum
Annual hours of work of husband
2,244.18
556.02
0
Annual hours of work of wife
1,436.69
872.38
0
Husband annual hours positive?
0.99
0.11
0
Wife annual hours positive?
0.81
0.39
0
Wage of husband
25.25
24.47
5.15
Wage of wife
16.02
13.22
5.15
Taxable income of husband and wife 84,745.38 87,965.10
0
Deductions
9,788.62 6,318.26 7,350
Labor income of husband
56,183.89 65,783.17
0
Labor income of wife
23,856.02 22,403.30
0
Nonlabor income of couple
4,705.48 33,636.09
0
Age of husband
41.45
7.96
25
Age of wife
39.74
8.12
20
Number of children
1.34
1.16
0
Presence of children younger than 6
0.28
0.45
0
Years of education of husband
13.52
2.41
3
Years of education of wife
13.57
2.30
2
Husband nonwhite
0.29
0.45
0
Wife nonwhite
0.28
0.45
0
ð15Þ
3,920
3,920
1
1
375
250
2,112,300
134,250
1,156,700
250,000
890,600
55
62
7
1
17
17
1
1
Uðhh ; hw ; CÞ ¼ bhh h2h + bww h2w + bCC C2
+ bhw hh hw + bhC hh C + bwC hw C
+ bh hh + bw hw + bC C
and where bC is normalized to 1. The quadratic utility function can be thought of as a
second order Taylor approximation to an arbitrary utility function. It has the desirable
features that labor supplies can be backward bending, and hours endowments do not
have to be specified or separately estimated. However, nothing in this utility function
constrains it from displaying a positive marginal utility of working at low hours levels.
This functional form also easily allows for the incorporation of heterogeneity due to both
observable and unobservable factors by specifying the coefficients on the linear terms as
ð16Þ
bh ¼ X#h ah + vh
bw ¼ X#w aw + vw
where the vectors Xh and Xw contain observable preference shifters,34 and vh and vw
denote unobservable heterogeneity in taste for work.
34. This method of incorporating observable preference shifters is analogous to the demographic translating in Pollak and Wales (1992).
Heim
Table 2
Auxiliary Log Wage Regressions
Women
Log Wage
Equation
Age
0.0204
(0.0156)
20.0001
(0.0002)
0.1250
(0.0605)
0.3041
(0.0617)
0.7086
(0.0621)
20.0309
(0.0371)
20.0370
(0.0587)
Age squared
High school graduate
Some college
College graduate plus
Black
Other
Presence of children
younger than 6
Number of children
Constant
r
s
l
Log likelihood
N
1.7089
(0.3028)
20.3059
(0.1362)
0.5046
(0.0128)
20.1544
(0.0712)
21,530.32
1,501
Men
Selection
Equation
20.0109
(0.0452)
20.0013
(0.0006)
0.3893
(0.1335)
0.6351
(0.1369)
0.6862
(0.1399)
0.1185
(0.1075)
20.1957
(0.1391)
20.6095
(0.1051)
20.1370
(0.0389)
1.5015
(0.8748)
Log Wage
Equation
Selection
Equation
0.0901
(0.0196)
20.0010
(0.0002)
0.2459
(0.0533)
0.4325
(0.0550)
0.8434
(0.0536)
20.2573
(0.0393)
20.1984
(0.0576)
0.0321
(0.1391)
20.0006
(0.0017)
0.1312
(0.9123)
0.1940
(0.3046)
0.8108
(0.4345)
20.6978
(0.2494)
20.5477
(0.3434)
20.1194
(0.2611)
0.6249
(0.3895)
20.5511
(0.1455)
0.5844
(0.0111)
20.3220
(0.0869)
21,372.85
1,501
2.0911
(2.8728)
Note: Asymptotic standard errors in parentheses.
V. Results
Table 3 presents the results from three specifications of the model.
In Column 1, heterogeneity in tastes for work is accounted for, but these tastes do
not depend on the demographic characteristics of the couple. The first panel of the
table contains the estimated coefficients on the squared terms in the utility function.
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The Journal of Human Resources
Table 3
Estimation Results: 2001 PSID Sample
(1)
Squared and interacted terms
210.994
bhhx102
(0.775)
bwwx102
20.926
(0.052)
0.032
bCCx105
(0.015)
20.854
bhwx104
(0.148)
bhCx104
20.259
(0.133)
20.828
bwCx104
(0.059)
Heterogeneity nodes
Weights
0.033
p1
(0.005)
0.701
p2
(0.015)
0.161
p3
(0.011)
0.029
p4
(0.007)
p5
0.076
Husband nodes
84.271
bh1
(11.828)
470.410
bh2
(32.062)
472.470
bh3
(33.370)
bh4
679.764
(49.689)
694.488
bh5
(47.036)
Wife nodes
17.704
bw1
(2.034)
29.984
bw2
(2.118)
(2)
(3)
25.5441
(0.486)
20.780
(0.050)
20.046
(0.014)
20.801
(0.183)
20.019
(0.142)
20.463
(0.059)
25.613
(0.518)
20.799
(0.057)
20.048
(0.014)
20.796
(0.194)
20.048
(0.152)
20.497
(0.062)
0.031
(0.005)
0.711
(0.015)
0.166
(0.011)
0.022
(0.005)
0.070
0.031
(0.005)
0.712
(0.015)
0.163
(0.011)
0.023
(0.005)
0.072
32.1089
(5.899)
230.5252
(20.978)
230.6359
(21.531)
343.6533
(31.882)
344.5299
(30.801)
35.096
(9.436)
232.781
(23.987)
233.052
(24.518)
345.730
(35.574)
347.737
(34.550)
17.8059
(1.880)
24.9652
(1.910)
19.077
(3.695)
25.977
(3.490)
(continued )
Heim
Table 3 (continued)
bw3
bw4
bw5
Stochastic elements
sh
sw
r
Demographic characteristics
ah
Husband’s education
Wife’s education
Husband’s age
Wife’s age
Husband’s age squared/1,000
Wife’s age squared/1,000
Number of children
Presence of children younger than 6
Nonwhite
aw
Husband’s education
Wife’s education
Husband’s age
(1)
(2)
(3)
212.267
(5.122)
2119.885
(8.737)
31.292
(2.470)
28.377
(3.277)
2121.284
(148.822)
29.125
(2.293)
211.441
(6.048)
225.362
(216.701)
29.772
(3.846)
351.96
(7.99)
571.53
(10.57)
0.134
(0.039)
338.45
(7.19)
557.69
(10.35)
0.111
(0.036)
338.82
(7.29)
557.26
(10.18)
0.109
(0.037)
0.577
(0.193)
20.397
(0.197)
20.268
(0.143)
20.137
(0.145)
0.275
(0.185)
20.214
(0.180)
0.044
(0.205)
0.218
(0.311)
23.504
(0.979)
0.736
(0.196)
20.0