The full text on this page is automatically extracted from the file linked above and may contain errors and inconsistencies.
Federal Reserve Bank of Chicago
The Effects of Progressive Taxation on
Labor Supply when Hours and Wages
are Jointly Determined
Daniel Aaronson and Eric French
REVISED July, 2004
WP 2002-22
The Effects of Progressive Taxation on Labor
Supply when Hours and Wages are Jointly
Determined
Daniel Aaronson and Eric French∗
Federal Reserve Bank of Chicago
July 22, 2004
Abstract
This paper extends a standard intertemporal labor supply model to account for progressive taxation as well as the joint determination of hourly wages and hours worked.
We show, qualitatively and quantitatively, that these two factors have implications for
estimating the intertemporal elasticity of substitution. Furthermore, we show how to use
the intertemporal elasticity of substitution to interpret the labor supply response to a tax
change. Failure to account for wage-hours ties within a progressive tax system leads to
an hours response to a change in marginal tax rates that may be understated by as much
as 10 percent for men and 17 percent for women.
∗
Comments welcome at efrench@frbchi.org and daaronson@frbchi.org. We thank Jeff Campbell, Jane
Gravelle, Kevin Hasset, Dan Sullivan, James Ziliak, and seminar participants at the Federal Reserve Bank of
Chicago, American Enterprise Institute, and the Econometric Society for helpful comments and Kate Godwin
for excellent research assistance. The views of the authors do not necessarily reflect those of the Federal
Reserve Bank of Chicago or the Federal Reserve System. Recent versions of the paper can be obtained at
http://www.chicagofed.org/economists/EricFrench.cfm/. Author correspondence to Daniel Aaronson or
Eric French, Federal Reserve Bank of Chicago, 230 S. LaSalle St., Chicago, IL 60604. Telephone (312)322-6831,
Fax (312)322-2357.
1
1
Introduction
When evaluating the costs and benefits of modifications to the tax system, as in Altig et
al (2001), a critical elasticity of interest is the intertemporal labor supply elasticity. While
some recent research explicitly studies reactions to specific tax reforms, a more common
approach to approximating these effects is to employ estimates of the labor supply response
to wage changes using the methods of MaCurdy (1981), Altonji (1986), and Browning et al
(1985). Among men, this labor supply elasticity is commonly believed to be low, with most
estimates ranging from 0 to 0.5. For women, the estimate is considerably more uncertain but
believed to be around 1. Yet, some recent studies find larger income responses to specific tax
changes than what would be expected given the estimated labor supply responses to wage
changes.1 This is potentially verification that estimated wage elasticities lead tax analysts to
underpredict the labor supply response to specific tax changes.
In this paper, we emphasize two erroneous simplifying assumptions in standard labor
supply models that could also contribute to different inferences about behavioral responses
to tax changes. First, most labor supply models ignore the joint determination of hours
worked and hourly wages.2 Second, many intertemporal models ignore progressive labor
taxation.
In this paper, we examine how progressive taxation and the joint determination of hours
and wages affects estimates of structural preference parameters. We also consider how to use
estimated preference parameters to predict the likely labor supply responses to tax changes.
We show that failure to account for progressive taxation and the joint determination of hours
and wages leads to a small bias when estimating the intertemporal elasticity of substitution.
However, it is important to consider tied wage-hours offers and progressive taxation when
using this estimated preference parameter to predict the likely labor supply responses to tax
changes.
1
Feldstein (1995) and others attribute this difference to tax avoidance and retiming and reshifting of
transactions, rather than labor supply adjustments. See Slemrod (1998) for a useful nontechnical summary
and discussion of the literature.
2
Aaronson and French (2004) discuss identification and estimation of a causal link from hours worked to
hourly wages - the so-called part-time wage penalty. They identify this relationship using exogenous variation
in hours worked resulting from social security rules. Papers that use other identification strategies, primarily
related to mothers returning to the workforce, include Rosen (1976), Moffitt (1984), Lundberg (1985), Biddle
and Zarkin (1989), Blank (1990), and Ermisch and Wright (1993).
2
Solving a standard life-cycle labor supply model, augmented to include tied wage-hours
offers and progressive labor income taxation, illuminates two fundamental model misspecification problems. First, in a model where the wage is a function of hours worked, an increase in
the post-tax wage resulting from a tax cut potentially leads to an increase in hours worked.
This increase in hours worked leads to an increase in the pre-tax wage through the tied
wage-hours effect, further escalating hours worked. Therefore, there is a larger labor supply
response to a tax change than to an equally sized wage change. Since most models do not
account for tied wage-hours offers, the latter effect (i.e. the effect of increased hours worked
on increasing wages, which should in turn further increase hours worked) is ignored. Therefore, this model misspecification problem causes tax analysts to understate the labor supply
response to a tax change.
However, a tax cut may increase hours and consequently income, which in turn can shift
the individual into a higher tax bracket. This type of “bracket creep” reduces the variation
in the post-tax wage, implying that progressive taxation should dampen the labor supply
response to the tax cut. Consequently, the impact of tied wage-hours offers and progressive
taxation on labor supply tends to offset one another. Nevertheless, since the progressive
taxation effect seems less important than the effect of tied wage-hours offers, tax analysts are
likely to continue to underpredict the labor supply response to tax changes.
We are not the first to observe that the labor supply function must be augmented to
account for the marginal effect of work hours on wages and progressive tax schedules.3 However, we believe that we are the first to show analytically why failure to account for tied
wage-hours offers in both proportional and progressive tax systems will produce labor supply
elasticities that are different than the elasticity of interest to tax analysts.
We consider strategies for consistently identifying the structural preference parameter,
the intertemporal elasticity of substitution, showing that many estimation schemes do not
recover this parameter in the presence of progressive taxation and hours-wage ties. Because
of the criticisms raised against maximum likelihood estimation of labor supply models using
kinked budget constraints (MaCurdy et al. (1990)), we follow the approach of MaCurdy
et al. (1990) and Ziliak and Kniesner (1999) and use smooth approximations to the tax
3
See Rosen (1976), Moffitt (1984), and Lundberg (1985) on tied wage-hours within static labor supply
frameworks. See MaCurdy (1983), Hausman (1985), MaCurdy et al. (1990), Mulligan (1999) and Ziliak and
Kniesner (1999) on progressive taxes.
3
code.4 In particular, we analyze a common instrumental variable strategy in the presence of
progressive taxes and hours-wage offers. We then show how to use the intertemporal elasticity
of substitution to interpret the labor supply response to a change in marginal tax rates. Using
the Panel Study of Income Dynamics, labor supply responses to tax changes that account
for tied wage-hours and progressivity are compared with those that do not and the resulting
difference can be up to 10 percent for men.
Finally, we analytically evaluate the labor supply response to a tax change using a range
of relevant parameter values for the labor supply response to a wage change, the tied wagehours relationship, and the progressivity of the labor income tax schedule. With enough
progressivity, the tied wage-hours and progressivity effects can completely offset each other.
But assuming a level of progressivity observed, on average, in the U.S. over the last 30 years
results in a difference of around 8 percent for men, and potentially up to 17 percent for
women.
2
Dynamic intertemporal labor supply elasticities with tied
wage-hours offers and progressive taxation
2.1
Model
We begin with the canonical intertemporal labor supply model,5 as in MaCurdy (1985),
augmented to account for tied wage-hours offers and a potentially progressive labor income
tax schedule. Preferences take the form:
1
1+
hit σ
β v(cit ) − exp(−εit /σ) ×
U = E0
1 + σ1
t=1
T
t
(1)
where U is the expected discounted present value of lifetime utility, cit is consumption, v(.) is
some increasing concave function, hit is hours worked, and εit is the person and year specific
preference for work. The parameter σ is the intertemporal elasticity of substitution, the usual
4
Alternative approaches to handling these criticisms are in Blundell et al. (1998) and Heim and Meyer
(2003).
5
The key results from this section do not depend on whether the model is static or dynamic. However, the
intertemporal model simplifies the analysis because it allows us to focus more on the substitution effect of a
tax change. In static models and models with liquidity constraints, tax changes cause an additional change in
the marginal utility of wealth. Moreover, if individuals do make forward looking decisions, many measures of
non-labor income that are used in static models are endogenous and inconsistent estimates will result.
4
object of interest in dynamic labor supply studies.
Labor supply models typically assume that a worker receives a fixed wage offer, then
chooses the number of hours to work given that wage. However, firms may not be indifferent
to the number of hours worked. For example, Lewis (1969) and Barzel (1973) argue that the
fixed cost involved in hiring and retaining workers, including the cost of training and aspects
of compensation unrelated to hours worked, can be spread over more hours of work, causing
the wage to be increasing in hours worked.6
Operationally, it is typical in the empirical literature to specify the wage as a linear
function of hours worked:
ln wit = αit + θ ln hit
(2)
where αit represents an individual’s underlying productivity or technology during a specific
year and θ maps hours worked into the wage.
Two aspects of equation (2) are worth highlighting. First, the linearized relationship in
equation (2) provides a good approximation to a structural relationship between the wage
and hours worked, at least in the range of hours to which the majority of workers in our
empirical example are situated. This case is made in detail in appendix A. Second, the
estimate of θ that we use in the analysis is based on samples of workers that do not switch
employers. This is important because virtually all of the estimates in the literature, as well
as the static models of Lewis and Barzel, call into question whether the estimated wage-hours
relationship represents a long-run equilibrium, where hours and wages changes only happen
across jobs. But in Aaronson and French (2004), workers who cut their hours receive wage
reductions even when working for the same employer, consistent with the hypothesis that
employers face fixed costs of work.
Finally, the individual faces the dynamic budget constraint:
Ait+1 = (1 + rt (1 − τA ))(Ait + wit (log hit )hit + yit − τit − cit )
(3)
where Ait are time t assets, rt the interest rate, τA is the tax rate on capital income, yit is
6
Barzel also contends that exhaustion eventually causes marginal productivity (and thus the wage) to
declines once the workday reaches a certain threshold.
5
spousal income, and τit denotes labor income taxes:7
τit = τ (wit (log hit )hit + yit )
(4)
Maximization of (1) subject to equations (2) and the dynamic budget constraint (3) yields
the labor supply function:
log hit = σ log(1 − τ (.)) + log wit + log(1 + θ) + σ log λit + εit .
(5)
The term in square brackets is the logarithm of the opportunity cost of time. The first
part of this term reflects the cost of taxation that arises from additional working hours and
is sometimes referred to as the log of the “net of tax price”. Note that τit is the marginal tax
rate and thus 1 − τit is the share of labor income that the individual keeps at the margin.
The second part, the wage, arises because income increases with hours worked, holding the
wage fixed. The third part occurs because the worker is paid a higher hourly wage when she
works more hours, if hours and wages are tied. If changes in hours of work impact neither
the wage (i.e. θ = 0) nor the amount of taxes paid (i.e. τ (.) = 0), equation (5) becomes the
standard estimating equation in intertemporal labor supply models. The term λit ≡ v (cit )
represents the marginal utility of wealth.
To estimate σ, we first difference equation (5):
∆ log hit = σ ∆ log(1 − τit (.)) + ∆ log wit + σ∆ log λit + ∆εit .
(6)
¿From equation (6), it is clear that obtaining consistent estimates of σ requires valid
controls for changes in marginal tax rates, preferences, and the marginal utility of wealth.
For the latter, we follow MaCurdy (1985) and derive an estimating equation that controls for
changes in the marginal utility of wealth:8
7
This analysis looks at anticipated changes in tax rates. If a tax change is unanticipated, we must consider
both movements along and ”parametric shifts” (e.g. MaCurdy, 1985) in the lifecycle wage profile. Furthermore,
we assume that capital income does not affect labor income tax rates, which simplifies the analysis (Blomquist,
1985) but is problematic in that interest and dividends are taxed like ordinary income. Capital gains were
taxed like ordinary income prior to 1997 and are still taxed that way for investments held less than one year.
For long-term investments, there are currently two marginal rates. However, if capital gains are primarily
concentrated among higher income households (see Burman and Ricoy (1997) for evidence), these rates could
be considered significantly more proportional in practice than labor income. For tractability and due to
limitations in the data, we therefore ignore these aspects of the progressive tax schedule.
8
He shows that the marginal utility of wealth, and in approximation its log, follows a random walk with
6
β(1 + rt−1 (1 − τA ))it
∆ log hit = σ ∆ log(1 − τ (.)) + ∆ log wit − σ log β(1 + rt−1 (1 − τA )) + σ
+ ∆εit .
λit−1
(7)
where it is the innovation to the marginal utility of wealth.
The remainder of this paper examines two general questions: how to obtain consistent
estimates of σ and how to use σ to infer the labor supply response to a tax change. Sections
2.2 and 2.3 consider, in turn, the roles of tied wage-hours offers and progressive taxation for
these issues.
2.2
The case of proportional taxes
When taxation is progressive, analyzing the effects of taxes on labor supply becomes a bit
complicated. In this section, we consider proportional taxation in order to develop intuition
about the effect of tax changes on labor supply in the presence of tied wage-hours offers.
Proportional taxes imply that a constant share of labor income is taxed and therefore the
marginal tax rate is a constant:
τit (.) = τ .
(8)
In this case, marginal tax rates disappear from equation (7).
First, consider the problem of identifying σ. Note from equations (2) and (5) that changes
in εit will affect hours, which will in turn affect the wage. Therefore, log wit is correlated with
εit . This is the simultaneous equations bias problem. In addition, wage changes are likely
correlated with the marginal utility of wealth. Consequently, a good instrument needs to be
correlated with ∆ ln wit but uncorrelated with rt , it , and ∆εit . If such an instrument, Zit ,
can be found, then the instrumental variables estimator converges in probability to
∗
=
σIV
E[Zit ∆ log hit ]
=σ
E[Zit ∆ log wit ]
(9)
∗ is a consistent estimator of σ.9
and thus σIV
drift. See appendix B for a derivation of equation (7).
9
This result relies on the assumption that the log wage increases linearly in log hours. However, Barzel
7
However, the parameter σ is no longer sufficient for understanding the labor supply response to taxation if wages are tied to hours. In particular, tax analysts are interested in the
effect of taxes on labor supply,
∆ log hit
∆ log(1−τ )
:
∆ log hit
∆ log λit
∆ log hit
=σ 1+θ
+
.
∆ log(1 − τ )
∆ log(1 − τ ) ∆ log(1 − τ )
(10)
There are three pieces on the right hand side of equation (10), reflecting different labor
supply incentives arising from a tax change. The first term reflects changes in the post-tax
wage, holding the pre-tax wage fixed. A reduction in taxes causes an increase in the post-tax
wage, which in turn affects labor supply. This is the usual object of interest in intertemporal
labor supply studies. The second term arises from the effect of hours worked upon the wage.
If σ > 0, reductions in taxes cause increases in hours worked, which in turn increases the
pre-tax wage (because of tied wage-hours offers). Because the pre-tax wage increases, hours
worked increase further. The final term is the effect of the tax change on the marginal utility
of wealth. Increases in (1 − τ ) (i.e., decreases in marginal tax rates) tend to increase lifetime
log λit
wealth and thus decrease its marginal utility, ∆∆log(1−τ
) ≤ 0. Nevertheless, the labor supply
response to tax changes, holding the marginal utility of wealth constant, is an important
object since it is used to calibrate many of the important models used for tax analysis (Altig
et al. (2001)) and it is a measure of the deadweight loss associated with tax changes (Ziliak
and Kniesner, 1999). Therefore, we assume
d log λit
d log(1−τ )
= 0 and rearrange equation (10) as10
σ
∆ log hit
.
=
∆ log(1 − τ ) λit
1 − σθ
(11)
Equations (9) and (11) demonstrate that the labor supply response to a one percent
increase in 1 − τ is larger than the labor supply response to a one percent wage increase,
holding the marginal utility of wealth constant. Therefore, the strategy used to identify the
labor supply elasticity can be critical. The magnitude of this difference, and identification
strategies used to uncover it, are discussed further below.
(1973) speculates that at very long work weeks, an increase in hours might lower wages as exhaustion reduces
productivity, so w (log hit ) < 0. Nevertheless, the existence of tied wage-hours offers need not necessarily
lead to inconsistent estimates of σ. It is non-linearity in the wage-hours relationship that causes inconsistent
estimates of σ. See appendix A for more discussion of this issue.
10
If θ > 0 then the budget set is not convex. However, equation (11) still represents an equilibrium condition
so long as σθ < 1. This condition is satisfied for reasonable parameter values.
8
2.3
The case of progressive taxes
The above analysis provides an assessment of the importance of model mis-specification
introduced by wage-hours ties. In this section, we discuss a further complication, allowing for
the possibility that increased hours of work push households into a higher tax bracket. This
type of bracket creep reduces the variation in the post-tax wage, implying that progressive
taxation should dampen the labor supply response to a pre-tax wage and tax change.11
Ignoring progressive taxation leads to a downward biased estimate of σ and an upward biased
estimate of the labor supply response to a tax change for a given σ. It is the latter effect that
is more important, however. An increase in the marginal tax rate causes a decrease in work
hours, naturally decreasing labor income and potentially lowering the marginal labor tax
rate that the worker faces. Therefore, progressive taxation attenuates the effect of the initial
increase in the marginal tax rate. Consequently, the impact of tied wage-hours offers and
progressive taxation on labor supply tends to offset one another.
In order to capture a potentially progressive (or regressive through, for example, the
Earned Income Tax Credit) tax schedule, we let the marginal tax rate depend on a polynomial
in log(wit hit + yit ) :12
log(1 − τ (wit hit + yit )) =
K
k
γk log(wit hit + yit )
(12)
k=0
which can be approximated using a first order Taylor’s series approximation:
K
k=0
K
k
γk log(wit hit + yit ) =
γk log((wit hit )(1 +
k=0
K
yit k
yit k
) ≈
γk log(wit ) + log(hit ) +
wit hit
wit hit
k=0
(13)
∗ (the probability limit of the IV
Recall that our interest is in the relationship between σIV
estimator using the pre-tax wage) , the structural parameter σ, and the labor supply response
11
Of course, the extent of this effect depends on the distribution of taxpayers on the tax schedule. If most
are far from the kinks, the effect will be small.
12
This approach follows MaCurdy et al. (1990) and Ziliak and Kniesner (1999). In practice, we use a third
order polynomial in log income. We also tried higher order polynomials, although this adjustment did not
affect our results. A differentiable tax function makes the evaluation of the labor supply response to tax
changes more straightforward, as in equation (14).
9
to a tax change. However, with progressive taxation, it is impossible to know the relationship
∗ and σ without knowing the distribution of preference and productivity shocks,
between σIV
αit and εit , as the higher order moments include covariances between income and αit and εit .
Unfortunately, no evidence exists on these parameters because it is difficult to distinguish
variation in αit and εit from variation in hours and wages induced by measurement error.
Nevertheless, it is still possible to obtain consistent estimates of σ using instrumental
variables procedures. Instead of using the relationship between the pre-tax wage and labor
supply, it is necessary to use the relationship between the post-tax wage and labor supply.
Next, we describe the association between σ and a tax change, γ0 . Note that a one
percentage point change in γ0 increases the after tax wage by one percentage point, holding
pre-tax income constant. Assuming
yit
it hit
dγ0
dw
= 013 and combining equations (12), (13), and
(5), it can be shown that the elasticity of hours worked with respect to γ0 is14
d log hit
=
dγ0 λit
1 − σ θ + (1 + θ)
σ
K
k=1 kγk log(wit ) +
log(hit ) +
yit k−1
wit hit
(14)
.
K
Relative to equation (11), this derivative has an extra term, σ(1 + θ)
k=1 kγk log(wit ) +
k−1
. The first part of this term, (1 + θ), represents the percent increase in
log(hit ) + wityithit
own labor income due to a one percent increase in hours. The second term depicts the percent
change in the quantity 1−τit caused by shifting own and spouse’s labor income by one percent.
Therefore, the entire term is roughly the percent change in 1 − τit caused by changing hours
by one percent. Intuitively, this term captures the result that when γ0 increases (in other
words, as marginal tax rates fall), individuals supply more hours to the market. However, this
initial effect is dampened by progressive taxation since increased income pushes the worker
into a higher marginal tax rate, thus attenuating the effect of γ0 .15
Equations (11) and (14) differ only in that individuals are aware that changes in labor
supply cause changes in the marginal tax rate in the latter equation. Equation (11) em13
This assumption implies that changes in the marginal tax rate will equally impact husband’s and wife’s
labor supply, leaving the ratio of the wife’s to husband’s income unchanged.
14
Note that the elasticity of interest is most likely with respect to a vertical shift in the marginal
tax
d log hit
rate schedule. The connection between this elasticity and the one in equation (14) is d log M T R
=
λit
M T R d log hit
M T R−1
dγ0
.
λit
k−1
yit
15
Recall that progressive taxation implies that K
< 0.
k=1 kγk log(wit ) + log(hit ) + wit hit
10
phasizes only tied wage-hours and how failure to account for this relationship leads to an
understatement of the importance of tax changes. Failure to account for progressive taxation, on the other hand, causes the researcher to overstate the importance of tax changes.
Therefore, the two effects tend to offset.
Although the relationship between σ,
∗ ,
σIV
and
d log hit
dγ0
λit
is complicated, it is still straight-
∗ given the approaches we have discussed. Equation (14) and
forward to estimate σ and σIV
d log hit
K
estimates of {γk }k=1 also allow us to predict dγ0 . We present such estimates in section
λit
5.
Moreover, if log(1− τ (.)) is linear in log labor income (i.e., γk = 0 for k > 1), it is possible
to obtain simple analytic solutions to help give our results some intuition. First, it is possible
∗ < σ. In particular, appendix C illustrates that
to qualitatively show that σIV
∗
σIV
=
σ(1 + γ1 )
.
1 − σγ1
(15)
Intuitively, σ measures the labor supply response to a change in the post-tax wage, whereas
∗ measures the labor supply response to a change in the pre-tax wage. Note that a 1
σIV
percent increase in the pre-tax wage causes less than a 1 percent change in the post-tax
wage. Therefore, an anticipated 1 percent change in the post-tax wage causes a σ percent
change in hours worked. However, a 1 percent change in the pre-tax wage will lead to less
than a 1 percent change in the post-tax wage and thus less than a σ percent change in hours
worked.
Finally, the relationship between
∗
σIV
equations (14) and (15). Again assuming
hit
and d log
can
dγ0
λit
that log(1 − τ (.)) is
be derived analytically using
linear in log labor income and
contemporaneous and lagged preference changes are uncorrelated, we can show that:
∗
σIV
d log hit
=
.
∗ θ 1+γ
dγ0 λit
(1 + γ1 ) − σIV
1
(16)
After describing the estimation strategy and data in the next two sections, section 5
provides estimates of σ and the tax function directly. Section 6 uses plausible ranges of γ1
∗ to calibrate d log hit .
and σIV
dγ0
λit
11
3
Estimation Strategy
In Section 2, we pointed out problems with inferring the labor supply response to a
tax change using the intertemporal elasticity of substitution. However, failure to account
for progressive taxation also leads to inconsistent estimates of the intertemporal elasticity
of substitution. Moreover, failure to account for tied wage-hours offers sometimes leads
to inconsistent estimates, depending on the instrument set. These points are somewhat
technical, so we derive the asymptotic properties of different estimators in Appendix C.
Our strategy for analyzing the importance of jointly determined hours and wages in a
progressive tax world is to directly estimate σ, accounting explicitly for jointly determined
hours and wages and progressive taxes. We compare estimates that account for wage-hours
ties and progressive taxes with those that ignore both factors. This allows us to assess the
bias described in the previous section when data and other methodological choices are fixed.
There are five terms on the right hand side of our estimating equation (7). The first term,
changes in the marginal tax rate, are explicitly simulated for each individual using the NBER’s
Taxsim program, augmented with payroll tax rates obtained from the Tax Policy Center at
the Urban Institute.16 The third term, log β(1 + rt−1 (1 − τA )) is accounted for by including
year dummies and education controls. The year dummies account for changes in the interest
rate over time. The education group controls account for variation in subjective discount rates
across education groups.17 Health status change regressors capture the observed component
of preference shifters, the fifth term, with the remaining portion of that term assumed to be
white noise.
However, an important problem emerges with regard to the first, second and fourth terms
of equation (7). First, the marginal tax rate is endogenous because hours choices affect this
rate. Consequently, E[(∆ log(1 − τ (.)))(∆εit )] = 0. Second, the wage change is potentially
correlated with the innovation to the marginal utility of wealth if the wage change is unan
ticipated, and thus E (∆ log wit )it = 0. Therefore, we need anticipated sources of post-tax
wage variation that are uncorrelated with preferences to identify σ.
One common strategy to solve this problem is to exploit the life cycle wage profile and
assume that workers are able to anticipate future post-tax wage growth based on their age,
16
See www.nber.org/taxsim/ for more details. Marginal rates are computed relative to the next $1,000 in
wage income. The data section describes the computations in more detail.
17
See Mulligan (1999) for a discussion of the cross-sectional evidence.
12
as in MaCurdy (1981) and Browning et al. (1985), among many others. The age profile will
give consistent estimates of σ so long as age-specific variation in preferences is fully accounted
for using health status and an age trend.18 Appendix C contains a more thorough discussion
of the identification difficulties of standard instrumental variables strategies in a setting with
tied wage-hours. It shows that using age as an instrument will yield consistent estimates of
σ. One important point of this discussion is that just as the effects of tied wage-hours offers
and progressive taxation tend to offset when estimating the labor supply response to a tax
change for a given σ, the effects of these two factors are likely to offset when computing the
bias in the estimate value of σ.
4
Data
Similar to many previous studies of taxes and labor supply, we use the PSID to estimate
σ. Our sample consists of male household heads aged 25 to 60 between 1977 and 1989. We
drop the self-employed because their capital and labor income (as well as taxes) is difficult
to distinguish. We also drop those workers with fewer than 300 or more than 4,500 hours, as
well as those who earn less than $3 or more than $100 per hour. Our selection criterion leads
to a sample of 2,393 working men encompassing 15,989 person-years observations.
Two variables require further elaboration. First, we use a common measure of the hourly
wage, annual earnings divided by annual hours. However, such a measure introduces a nonstandard measurement error problem called “division bias” by allowing measurement error
in hours to enter both the left hand and right hand side of the estimating equation (7). This
can drive estimates of the wage elasticity to negative values.19
18
An alternative strategy is to assume workers can anticipate future wage growth based on their current wage
and thus use lagged wages or wage changes as instruments, as in Altonji (1986), Holtz-Eakin et al. (1988), and
Ziliak and Kniesner (1999), among others. However, in the presence of tied wage-hours offers, changes in hours
worked caused by changes in preferences will impact the wage. This violates the orthogonality assumptions
of the life cycle labor supply model. Because lagged wages depend on lagged hours, lagged wages will only
be a valid instrument for the current wage if E[∆εit εit−k ] = 0 for wages lagged k periods. It is possible to
show that a slightly modified version of the lagged wage instrument that adjusts lagged wages by θ log hit can
potentially eliminate this feedback effect. Results are available upon request. But it appears to us that the
age profile is clearly a cleaner instrument in a setting with tied wage-hours offers.
19
One potential solution we have tried is to instrument for the current wage change using twice lagged wages.
If measurement error is white noise, twice lagged wages (or wage changes) will be uncorrelated with the current
wage change. However, French (2004a) and Ziliak and Kneiser (1999) provide evidence that the measurement
error in earnings and hours is autocorrelated and thus cannot solve inconsistency problems associated with σ.
We have also tried using the reported wage of hourly workers. Its advantage is that it overcomes the division
bias problem since measurement error in the reported hourly wage is likely to be uncorrelated with both
13
Second, effective marginal rates are computed for each household using the NBER’s
Taxsim program. We augment these rates with payroll tax schedules obtained from the
Tax Policy Center at the Urban Institute. For the state and federal calculations, we assume
that all married households file jointly and use the standard deduction. We also assume that
income is provided solely through the head and spouse’s wages and salaries. The number of
dependents, including those who qualify for the age 65 exemption, are provided by the PSID
and accounted for in the computations.
Figure 1 displays marginal tax rates for individuals in our sample.20 Circles represent
single filers, squares represent heads of household, and triangles represent joint filers. There is
variation within income level due to cross-sectional differences in state tax law, variation over
time in federal and state tax law, differences in the number of dependents across households,
and filing status across households. Nevertheless, the dominant source of variation in marginal
tax rates is from labor income. A simple regression of log(1 − τit ) on log income has an R2
of 0.49. A third order income polynomial, as we use, yields an R2 of 0.52.
hours and earnings. However, there are two distinct disadvantages. First, only hourly employees are included,
which limits the sample size substantially and introduces potentially important nonrandomness to the sample.
Second, overtime pay and bonuses are excluded. The latter concern is critical since overtime and bonuses are
an important source of wage variation.
20
To account for substantial changes in the tax code introduced by the 1986 law changes, we show the rates
separately pre- and post-reform. It is also important to note that there are few households facing negative
marginal tax rates because we include payroll taxes and limit the sample to those households headed by men
with at least $5,000 in annual income. However, the EITC is accounted for in the calculations.
14
single
headofhousehold
married
.6
marginal tax rates
.5
.4
.3
.2
.1
0
10000
20000
40000
income on log scale
80000
160000
Marginal tax rates, 1977−1986
single
headofhousehold
married
.6
marginal tax rates
.5
.4
.3
.2
.1
0
10000
20000
40000
income on log scale
80000
Marginal tax rates, 1987−1989
Figure 1: Marginal Tax Rates
15
160000
5
Results
Table 1 reports our estimates of the various labor supply elasticities. The first two columns
report findings when the contemporaneous wage change is defined as annual earnings divided
by annual hours and the parameter θ, the wage-hours tie, is set to 0 in column 1 and 0.4
in column 2. The 0.4 estimate is in the middle to upper end of the estimates in the tied
wage-hours offer literature.21 It implies that cutting weekly work hours from 40 to 20 leads
to a 24 percent reduction in the offered hourly wage. A θ = 0 assumes that the hourly wage
is not a function of hours worked. In both columns, the findings are based on specifications
that use a third order age polynomial as a means of exploiting the life cycle profile of wages.
The top panel displays the F − statistic and R2 from the first-stage regressions to show
the power of this instrument. The instruments seems to be strongly associated with contemporaneous wage changes, with the F − statistic exceeding standard thresholds.
∗ , σ, and
The bottom panel reports the size of the four key labor supply parameters: σIV
d log hit
hit
and d log
the objects of interest to tax analysts, d log(1−τ
)
dγ
. These elasticities are
0
it
λit ,εit
described in equations (11) and (14).22
21
λit
See Aaronson and French (2004), Blank (1990), Ermisch and Wright (1993), and Rosen (1976). Biddle
and Zarkin (1989) estimate values in excess of 3.
d log hit
22
Recall that d log(1−τ
is somewhat difficult to interpret because the marginal tax rate is a function
it )
λit
d log hit
of hours worked. However, for many cases, tax analysts are interested in d log(1−τ
, which can still be
)
it
λit
d log hit
dγ0
λit
, or the percent increase in labor supply given a change in γ0 that is sufficiently
interpreted as
d log(1−τ )
it
dγ0
λit
large to increase log(1 − τit ) by 1 percent.
16
Dependent variable Hourly wage Hourly wage Annual earnings
θ=
0
0.4
0
First Stage Estimates, Dependent Variable is ∆ log wit
Annual earnings
0.4
F − statistic
5.4
5.4
18.6
18.6
R2
0.016
0.016
0.025
0.025
N
15,989
15,989
15,989
15,989
Second Stage Estimates, Dependent Variable is ∆ log hit
17
∗
σIV
0.62
(0.16)
0.62
(0.16)
0.81
(0.06)
0.81
(0.06)
σ
0.64
(0.22)
0.64
(0.22)
1.13
(0.35)
1.13
(0.35)
0.64
0.86
1.13
2.06
(0.22)
(0.40)
(0.35)
(1.16)
0.57
0.69
0.92
1.31
d log hit
)
d log(1−τit
λit
d log hit
dγ0
λit
(0.17)
(0.26)
(0.23)
(0.47)
Life cycle instrument set is a third order age polynomial.
Other right hand side variables are year dummies, health status change, and education.
Table 1: Estimated Labor Supply Elasticities, PSID 1977-1989
∗ and σ are 0.62 (standard error of .16)23 and 0.64 (0.22).24 Note
We that find that σIV
that, as argued in appendix C, failure to account for progressive taxation does lead to a
downward biased estimate of σ (i.e. 0.64 versus 0.62). However, this effect is small. Allowing
wage-hours ties (i.e., setting θ = 0.4) increases the hours response to a change in (1 − τit ) by
34 percent, to 0.86, relative to σ. That is, a 1 percent increase in (1 − τit ) has an initial effect
of increasing the after tax wage by 1 percent, which in turn increases hours by 0.64 percent.
However, the longer workweek further increases the hourly wage, due to the wage-hours tie.
This leads to a further increase in hours worked. Thus, the initial 1 percent increase in
(1 − τit ) increases hours by 0.86 percent.
But this is not the end of the story. When we introduce progressive taxation, the tax
hit
elasticity of interest, d log
dγ0 , falls to 0.69, only 8 percent higher than σ and 11 percent
λit
∗ .25 This result arises from higher income leading to a higher marginal tax
higher than σIV
rate, which dampens the labor supply response to the original tax change. As it turns out, in
this case, the effect of progressivity offsets much, but not all, of the tied wage-hours effect.26
In the data section, we noted that division bias, in combination with small samples, leads
to estimates that are biased downward. To minimize this problem, we respecify the labor
supply function in terms of log earnings rather than log wages.27 It can be easily shown that
this modification results in σ being biased to zero rather than -1 from measurement error.
However, Ghez and Becker (1975) point out that omitted variables potentially lead to an
23
Standard errors are computed using the multivariate delta method and correct for arbitrary forms of
heteroskedasticity and serial correlation.
24
These estimates are at the high end of the literature for men, although consistent with the findings of Lee
(2001) who uses a similar sample and instrument set. Lee finds that using unbalanced data and a parsimonious
instrument set overcomes small sample bias, and thus leads to higher estimates of the intertemporal elasticity
of substitution.
25
The results are similar when we restrict our sample to those 12,533 workers with lagged earnings and hours,
d log hit
as in the lagged wage instrument regressions reported in columns 3 and 4. Here, σ = 0.70, d log(1−τ
= 0.98
it )
λit
hit
and d log
= 0.76.
dγ0
26
λit
When there is no wage-hours tie, ignoring progressivity leads to a 8 percent reduction (from 0.62 vs. 0.57)
in the labor supply response to a one percent change in marginal rates. This is in contrast to Mulligan (1999),
who finds that progressivity biases downward labor supply responses.
Mulligan emphasizes the difference
d log hit
∗
between σIV and σ, but not the difference between σ and dγ0 . Our results show that the latter effect
λit
is more important.
27
The estimating equation becomes
β(1 + rt−1 (1 − τA ))it
1
εit
+∆
∆ log hit = σ̃ ∆ log(1 − τ (.)) + ∆ log Eit − σ̃ log β(1 + rt−1 (1 − τA )) + σ̃
λit−1
1+σ
(17)
18
upward bias using this specification. Results are in columns 3 and 4 of table 1. Using
the age
d log hit
to 0.80,
polynomial instruments, substituting log earnings for log wages drives d log(1−τ
it )
λ
it
d log hit
σ to 1.13, and dγ0 to 1.30 when θ = 0.4.
λit
We also estimated equation (7) on men in the outgoing rotation files of the Current Population Survey (CPS). The key advantage of the CPS, particularly the outgoing rotation files,
is large samples. Using similar sample selection criterion as those in our PSID sample, almost
700,000 men between 1979 to 1999 can be used in the estimation. Although the questions
are more limited than the PSID, we can recreate the PSID specification, less information on
health status. The drawback is that only two observations per person are available. Our
estimates, based on the age polynomial instruments, are smaller than the PSID. We get esti∗ of just below 0.20, which is inelastic enough that the bias that arises from tied
mates of σIV
wage-hours and progressive taxation is hard to detect.
6
Calibration
The estimation results suggest that progressive taxation offsets much but not all of the
impact of wage-hours ties. We
generalize this result in table 2 by describing calibrations of
hit
∗
the key tax derivative, d log
dγ0 , when plausible ranges of the underlying parameters, θ, σIV ,
λit
and γ1 are introduced. For θ, we allow the wage-hours relationship to vary from 0 to 0.60,
∗ to
which seems to cover the range of estimates in the literature. Most studies measure σIV
be between 0 and 0.5 for continuously employed men but are often greater than 1 for women
(e.g. Heckman and MaCurdy (1980)). Therefore, we allow this parameter to vary between 0
and 1.5 to account for the vast majority of estimates in the literature.
Finally, we allow γ1 to take on four values: 0, -0.10, -0.18, and -0.28. Zero represents
a proportional tax schedule. Larger negative values of γ1 characterize more progressive tax
systems. In the U.S., we estimate γ1 to be, on average, -0.18 for the 1977-1989 period.28
where
σ=
σ̃
.
1 − σ̃
28
(18)
This is based on a regression of the PSID respondents’ effective marginal tax rate on log income. Adding
a more complicated log income polynomial has only a marginal impact on the progressivity parameters as well
as the general fit of the regression.
19
∗
Panel A displays the proportional tax case. When σIV
= 0.5 and θ = 0.4, the bias
∗
= 1, a
introduced by tied wage-hours offers is 26 percent (0.63 versus 0.50). With σIV
relevant case for women, the bias introduced by θ = 0.4 is 67 percent. However, inelastic
labor supply or a small wage-hours tie results in a smaller bias.
Panel B introduces progressive taxes but at a level almost half that of the U.S. The
offsetting effect of progressivity is readily apparent. Rather than a 26 percent bias when
∗ = 0.5 and θ = 0.4, we see a 14 percent difference (0.57 versus 0.50). For σ ∗ = 1 the bias
σIV
IV
drops from 67 to 35 percent. With no tied wage-hours relationship, ignoring progressivity
∗ when it is between 0.5 to 1.0.
leads to a 4 to 9 percent overstatement σIV
When progressivity is assumed to be at the average level in the U.S. during the 1977 to
1989 period (panel C), the bias introduced by θ = 0.4 falls to 8 to 17 percent, for values of
∗ between 0.5 and 1.0. This is consistent with the empirical exercise of the last section.
σIV
Finally, only when tax progressivity is almost 50 percent higher than what we have seen in
the U.S. (i.e. γ1 = −0.28) or when θ = 0.2, roughly half of what is found in Aaronson and
French (2004), does progressive taxation completely offset the impact of hours-wage ties.
7
Conclusions
There are two important caveats to our analysis. First, we consider the decision of how
many hours to work (the “intensive margin”), not the decision of whether to work (the
“extensive margin”).29 Heckman (1993) contends that most of the variability in labor supply
is at the extensive margin. Furthermore, French (2004b) argues that a large fixed cost of
work is necessary to reconcile a high labor supply elasticity at the extensive margin, but a
low labor supply elasticity at the intensive margin. It is not clear to what extent the results
in this paper extend to a model with a labor force participation decision when there are fixed
costs of work.
The second concern is that we focus only on the substitution effect associated with tax
wage changes. Understanding the substitution effects is arguably sufficient for understanding
the labor supply response to short-term tax adjustments. However, to understand the im29
See Kimmel and Kniesner (1998) for a decomposition of labor supply elasticities into the intensive and
extensive margins).
20
A. γ1 = 0
θ
d log hit
d log wit
0
0
0
0.5
0.50
1
1.00
1.5
1.50
B. γ1 = −0.10
.4
0
0.63
1.67
3.75
.6
0
0.71
2.50
15.0
.4
0
0.57
1.35
2.46
.6
0
0.64
1.79
4.41
.4
0
0.54
1.17
1.93
.6
0
0.59
1.45
2.82
θ
d log hit
d log wit
0
0
0
0.5
0.48
1
0.91
1.5
1.30
C. γ1 = −0.18
.2
0
0.52
1.09
1.70
θ
d log hit
d log wit
0
0
0
0.5
0.46
1
0.85
1.5
1.18
D. γ1 = −0.28
d log hit
d log wit
.2
0
0.56
1.25
2.14
.2
0
0.50
0.98
1.46
θ
0
0
0.44
0.78
1.06
.2
0
0.47
0.88
1.25
.4
0
0.50
1.01
1.52
.6
0
0
0.5
0.54
1
1.18
1.5
1.94
d log hit
Table 2: Value of dγ0
λit
portance of fundamental tax reform, it is necessary to recognize the wealth effects associated
with tax changes.
Nevertheless, we believe that we have shown, both qualitatively and quantitatively, that
augmenting a standard intertemporal labor supply model to account for tied wage-hours offers
and progressive taxation affects estimates of the intertemporal elasticity of substitution and
the labor supply response to tax changes. Using common methods to estimate men’s labor
supply functions, we find that the hours response to a change in marginal tax rates may be
biased by as much as 10 percent, relative to many of the estimates in the literature, when
not accounting for these features of the data. The bias could be up to 20 percent or so for
21
populations with more elastic labor supply, such as women. Therefore, tax analysts inferring
the extent of behavioral responses to tax changes should consider the source of variation used
for identification.
Appendix A: The specification of tied wage-hours offers
To formally capture the link between hours worked and the offered wage, we first note
that, in equilibrium, perfectly competitive firms cover their fixed costs so that total output
equals the wage bill plus the fixed cost of work:
pit hit = wit hit + φ
(19)
where φ is the fixed cost per employee, pit is productivity of worker i at time t, hit is hours
worked, and wit is the offered hourly wage. By rewriting equation (19) as
wit = pit −
φ
,
hit
(20)
it is obvious that the offered hourly wage is rising in hours worked. This relationship implies
that at points in the life cycle or tax cycle that hours worked are high, the offered wage
should also be high.
Empirical research typically estimates a linearized version of the hours-wage relationship
as in equation (2) in the text. For example, Aaronson and French (2004) estimate θ = 0.4,
a result that appears to be well within the bounds found in the literature. The only papers
that we are aware of that test for the existence of a nonlinearity in ln hit are Moffitt (1984)
and Biddle and Zarkin (1989). While both papers find that equation (2) is misspecified, we
have been unable to find any evidence of nonlinearities in either the Panel Study of Income
Dynamics (PSID) or Current Population Survey (CPS).30
Regardless, it is straightforward to compute the approximation bias assumed in equation
(2) at different hours levels. The left panel in figure 2 plots the estimated relationship
between hours worked and the offered hourly wage, using equation (2), and an estimate of
θ = 0.4 derived from Aaronson and French (2004). It also presents the structural relationship
30
Furthermore, at least in the case of Biddle and Zarkin, even their smallest estimates of the elasticity of
wages with respect to hours worked appear implausibly large. As we show in section 6, their implied estimates
would suggest huge biases to the estimation of intertemporal labor supply elasticities, in cases where this
elasticity is sufficiently large.
22
between hours worked and the offered hourly wage using equation (20), again fitted to match
Aaronson and French’s estimate of θ. The right hand panel plots the elasticity of the wage
with respect to hours worked implied by equations (20) and (2).31 Between 1,700 hours and
2,500 hours, encompassing 68 percent of our sample, the implied elasticity from equation (20)
is 0.48 to 0.28, versus the constant elasticity implied by equation (2). Therefore, we conclude
the linearized relationship in equation (2) provides a good approximation to the structural
equation (20).
Moreover, the estimated value of θ seems to provide a plausible estimate of the fixed cost
of work. We find φ = $13,450 and pit = $23.30, implying that 28 percent of firm’s labor costs
13,450
13,450+17.26∗1,941
are fixed. This accords reasonably well with the studies on recruitment and
training costs cited in Malcomson (1999).
Figure 2: Offered Hourly Wage as a Function of Hours
31
We use our estimate of θ = 0.4, and pick αit to match the average work year length (1,941 hours) and
wage ($17.26, in 1996 dollars) from the sample of older PSID (age 50 to 70) males for equation (2). We pick
pit and φ to match the average wage and an elasticity of 0.4 at 1,941 hours of work for our fitted equation
(20).
23
Appendix B: Controlling for changes in the marginal utility of wealth
This appendix describes our approach for dealing with changes in the marginal utility
of wealth in order to derive equation (7) from the first differenced labor supply function
illustrated in equation (6). The discussion follows MaCurdy (1985), in which the marginal
utility of wealth and, in approximation, the log of the marginal utility of wealth are shown
to follow a random walk with drift. This result falls out of the Euler equation of the model
described in section 2.1. In particular, the Euler equation indicates that individuals equate
expected marginal utility across time according to:
λit−1 = β(1 + rt−1 (1 − τA ))Et−1 λit
(21)
where rational expectations32 implies that innovations to the marginal utility of wealth, denoted it , should be uncorrelated with lagged values of the marginal utility of wealth:
λit = Et−1 λit + it
(22)
Equations (21) and (22) can be rewritten as
β(1 + rt−1 (1 − τA ))λit
=
λit−1
β(1 + rt−1 (1 − τA ))it
1+
λit−1
Taking logarithms of both sides of (23) and approximating log(1 +
(23)
β(1+rt−1 (1−τA ))it
)
λit−1
β(1 + rt−1 (1 − τA ))it
log λit − log λit−1 + log β(1 + rt−1 (1 − τA )) = log 1 +
λit−1
≈
yields
β(1 + rt−1 (1 − τA ))it
λit−1
(24)
We assume that the approximation in (24) holds with equality, a valid assumption as
innovations in the marginal utility of wealth become arbitrarily small.
32
If workers have rational expectations then at time t they know their state variables αit , θ, rt , εit , τit the
Markov process that determines the evolution of the state variables, and optimize accordingly.
24
Combining (24) and (6) results in
β(1 + rt−1 (1 − τA ))it
+ ∆εit .
∆ log hit = σ ∆ log(1 − τ (.)) + ∆ log wit − σ log β(1 + rt−1 (1 − τA )) + σ
λit−1
(25)
Because the innovation to the marginal utility of wealth is potentially correlated with wage
changes if the wage change is unanticipated, the wage must be instrumented. See section 3
for a discussion on instrument selection.
Appendix C: Bias from failure to control for tied wage-hours offers and
progressive taxation when estimating the intertemporal elasticity of substitution
In this appendix we consider the likely biases caused by failure to control for tied wagehours offers and progressive taxation when estimating the intertemporal elasticity of substitution parameter σ. We show that disregarding progressive taxation leads to a downward
biased estimate of σ, as the econometrician overstates the amount of post-tax wage variability
that the individual faces. The intuition for this result is straightforward. An anticipated 1
percent change in the post-tax wage causes a σ percent change in hours worked. However, a
1 percent change in the pre-tax wage will lead to less than a 1 percent change in the post-tax
wage and thus less than a σ percent change in hours worked.
We also show that overlooking tied wage-hours offers potentially leads to inconsistent
estimates of σ. The fundamental problem that the econometrician must face when estimating
the labor supply response to a wage change is the simultaneous equations bias. Because
hours and wages are jointly determined, the econometrician must be careful that that he is
estimating a labor supply function (where hours are a function of the wage) rather than a
labor demand function (where wages are a function of hours worked). Failure to properly
control for the simultaneous equations bias likely leads to an upward bias in σ, as we show
below.
Therefore, just as the effects of tied wage-hours offers and progressive taxation tend to
offset when predicting the labor supply response to a tax change for a given σ, the effects of
tied wage-hours offers and progressive taxation tend to offset when computing the bias in the
estimated value of σ.
25
In order to simplify the analysis, consider the case where log(1 − τit ()) is linear in the log
of labor income, and that the marginal tax rate is unaffected by spousal income:
log(1 − τ (wit hit + yit )) = γ0 + γ1 log(wit ) + log(hit ) .
(26)
Further, ignore the importance of variable interest rates and observable preference shifters.33
Therefore, equation (7) can be rewritten as:
∆ log hit = σ ∆ log(1 − τ (.)) + ∆ log wit + ∆uit
(27)
(1−τA ))it
+ ∆εit . Combining equations (??), (26), and (7) yields the
where ∆uit = σ β(1+rt−1
λit−1
reduced form equations of the system:
σ (1 + γ1 )∆αit + ∆uit
∆ log hit =
1 − σ(γ1 (1 + θ) + θ)
(28)
(1 − σγ1 )∆αit + θ∆uit + ∆uit
.
∆ log wit =
1 − σ(γ1 (1 + θ) + θ)
(29)
Typically, instrumental variables procedures are used to estimate σ within the misspecified
model
∆ log hit = σ ∗ ∆ log wit + ∆uit
(30)
where σ ∗ is the wage coefficient on the misspecified model.
∗
using our instrumental
Next, we show derivations of the estimated coefficient σ ∗ , σIV
variables procedure. Consider the case where Cov(∆uit , Zit ) = 0 (i.e. the instrument is
uncorrelated with preferences and the marginal utility of wealth)and and Cov(log wit , Zit ) =
σZ2 = 0 (i.e., it is correlated with the productivity parameter ∆αit ). For example, arguably,
the life-cycle wage profile of men measures changes in life cycle productivity but not changes
33
In other words, consider a model where both the log post-tax wage and post-tax hours worked are the
residuals from regressions of the log post tax wage and log hours worked on year dummies and observable
preference shifters. Using the Frisch-Waugh-Lovell Theorem (Davidson and MacKinnon, 1993), it is straightforward to show that using this approach will still yield a consistent estimate of σ.
26
in life cycle preferences. In this case, we can consider the correlation caused by Zit .34 then
∗
=
σIV
σ(1 + γ1 )(1 − σγ1 )σZ2
σ(1 + γ1 )
=
2
2
1 − σγ1
(1 − σγ1 ) σZ
(31)
∗ is the probability limit of the estimate. Recall that γ < 0, so the estimated labor
where σIV
1
∗ = σ. Therefore, many
supply elasticity is biased downwards. However, if γ1 = 0, then σIV
common instrumental variables strategies overcome problems generated by tied wage-hours
offers. However, these strategies will not overcome the model misspecification problem of
using the pre-tax wage rather than the post-tax wage.
Note that in this simplified version of the labor supply model, we can analytically show the
∗ and d log hit . Combining equations (14) and (31), and assuming
relationship between σIV
dγ0
λit
γ2 = γ3 = ... = γK = 0, the relationship is
∗
σIV
d log hit
=
.
∗ θ 1+γ
dγ0 λit
(1 + γ1 ) − σIV
1
(32)
Lastly, we note that instrumental variables estimation of equation (27) does yield consistent estimates of σ. Using equations (26), (27), (28) and (29), the estimate of σ using E(∆αit )
as the instrument for ∆ log(1 − τ (.)) + ∆ log wit will converge to σE(∆αit ) :
σIV =
σ(1 + γ1 )
(1 + γ1 )σZ2 = σ.
σZ2
(33)
By the Frisch-Waugh-Lovell Theorem, by using dummy variables for the interest rate, the
procedure will provide consistent estimates of σ in equation (7) also.
34
More precisely, we can think of an individual’s age-specific productivity as being the sum of two orthogonal
components, or αit = αt + ψit where αt is the age-specific component of wages and ψit is the idiosyncratic
component of wages, and E[αt ψit ] = 0. In this case using αt as the instrument (which is another way of saying
∗
1 )(1−σγ1 )Cov(∆αit ,∆αt )
1)
that we use the average age-specific wage) yields σIV
= σ(1+γ
= σ(1+γ
1−σγ1
(1−σγ1 )2 Cov(∆α ,∆αt )
it
27
References
[1] Aaronson, Daniel and Eric French. “The Effect of Part-Time Work on Wages: Evidence
from the Social Security Rules.” Journal of Labor Economics, 22 (April 2004): 329-352.
[2] Altig, David, Alan Auerbach, Laurence Kotlikoff, Kent Smetters, and Jan Walliser. “Simulating Fundamental Tax Reform in the United States.” American Economic Review
91 (June 2001): 574-595.
[3] Altonji, Joseph. “Intertemporal Substitution in Labor Supply: Evidence from Microdata.” Journal of Political Economy 94 (June 1986): S176-S215.
[4] Barzel, Yoram. “The Determination of Daily Hours and Wages.” Quarterly Journal of
Economics 87 (May 1973): 220-238.
[5] Biddle, Jeff and Gary Zarkin. “Choice Among Wage-Hours Packages: An Empirical
Investigation of Male Labor Supply.” Journal of Labor Economics 7 (October 1989):
415-437.
[6] Blank, Rebecca. “Are Part-Time Jobs Bad Jobs?” In A Future of Lousy Jobs? The
Changing Structure of U.S. Wages, edited by Gary Burtless. Washington, DC: Brookings
Institution, 1990.
[7] Blomquist, N.. “Labour Supply in a Two-Period Model: The Effect of a Nonlinear
Progressive Income Tax.” Review of Economic Studies 52 (July 1985): 515-524.
[8] Blundell, Richard, Alan Duncan, and Costas Meghir. “Estimating Labor Supply Responses Using Tax Reforms.” Econometrica 66 (July 1998): 827-861.
[9] Browning, Martin, Angus Deaton, and Margaret Irish. “A Profitable Approach to Labor
Supply and Commodity Demands Over the Life-Cycle.” Econometrica 53 (May 1985):
503-543.
[10] Burman, Leonard and Peter Ricoy. “Capital Gains and the People Who Realize Them.”
National Tax Journal 50 (September 1997): 427-452.
[11] Davidson, Russell and James MacKinnon. Estimation and Inference in Econometrics.
Oxford: Oxford University Press, 1993.
28
[12] Ermisch, John and Robert Wright. “Wage Offers and Full-Time and Part-Time Employment by British Women.” Journal of Human Resources 28 (Winter 1993): 111-133.
[13] Feldstein, Martin. “The Effect of Marginal Tax Rates on Taxable Income: A Panel Study
of the 1986 Tax Reform Act.” Journal of Political Economy 103 (June 1995): 551-572.
[14] French, Eric. “The Labor Supply Response to Predictable (but Mismeasured) Wage
Changes.” Review of Economics and Statistics , 2004.
[15] French, Eric. “The Effects of Health, Wealth, and Wages on Labor Supply and Retirement Behavior.” Review of Economic Studies , 2004, forthcoming.
[16] Ghez, Gilbert, and Gary Becker. The Allocation of Time and Goods Over the Life Cycle,
NBER, 1975.
[17] Hausman, Jerry . “Taxes and Labor Supply” In Handbook of Public Economics, Volume
1 , edited by A. Auerbach and M. Feldstein. New York: North-Holland, 1985.
[18] Heckman, James. “What Has Been Learned About Labor Supply in the Past Twenty
Years?” American Economic Review 83 (May 1993): 116-121.
[19] Heckman, James and Thomas MaCurdy . “A Life Cycle Model of Female Labour Supply.” Review of Economic Studies 47 (January 1980): 47-74.
[20] Heim, Bradley and Bruce Meyer . “Structural Labor Supply Models when Budget Constraints are Nonlinear.” Working paper, Northwestern University, 2003.
[21] Holtz-Eakin, Douglas, Whitney Newey, and Harvey Rosen. “Estimating Vector Autoregressions with Panel Data Econometrica 56 (November 1988): 1371:1395.
[22] Kimmel, Jean and Thomas Kniesner. “New Evidence on Labor Supply: Employment
versus Hours Elasticities by Sex and Marital Status.” Journal of Monetary Economics
42 (October 1998): 289-301.
[23] Lee, Chul-In. “Finite Sample Bias in IV Estimation of Intertemporal Labor Supply Models: Is the Intertemporal Substitution Elasticity Really Small?” Review of Economics
and Statistics 83 (November 2001): 638-646.
29
[24] Lewis, H.G . ”Employer Interests in Employee Hours of Work.” working paper, University
of Chicago, 1969.
[25] Lundberg, Shelly. “Tied Wage-Hours Offers and the Endogeneity of Wages.” Review of
Economics and Statistics 67 (August 1985): 405-10.
[26] MaCurdy, Thomas. “An Empirical Model of Labor Supply in a Life-Cycle Setting.”
Journal of Political Economy 89 (December 1981): 1059-85.
[27] MaCurdy, Thomas. “A Simple Scheme for Estimating an Intertemporal Model of Labor
Supply and Consumption in the Presence of Taxes and Uncertainty.” International
Economic Review 24 (June 1983): 265-289.
[28] MaCurdy, Thomas. “Interpreting Empirical Models of Labor Supply in an Intertemporal
Framework with Uncertainty.” In Longitudinal Analysis of Labor Market Data , edited
by J. Heckman and B. Singer. Cambridge: Cambridge University Press, 1985.
[29] MaCurdy, Thomas, David Green, and Harry Paarsch. “Assessing Empirical Approaches
for Analyzing Taxes and Labor Supply.” Journal of Human Resources 25 (summer
1990): 415-490.
[30] Malcomson, James. “Individual Employment Contracts.” Handbook of Labor Economics,
Volume 3B , edited by O. Ashenfelter and D. Card. New York: North-Holland, 1999.
[31] Moffitt, Robert. “The Estimation of a Joint Wages-Hours Labor Supply Model.” Journal
of Labor Economics 2 (October 1984): 550-566.
[32] Mulligan, Casey. “Substitution over Time: Another Look at Life-Cycle Labor Supply.”
NBER macroeconomics annual 1998 13 (1999): 75-134.
[33] Rosen, Harvey. “Taxes in a Labor Supply Model with Joint Wages-Hours Determination.” Econometrica 44 (May 1976): 485-507.
[34] Slemrod, Joel. “Methodological Issues in Measuring and Interpreting Taxable Income
Elasticities.” National Tax Journal 51 (December 1998): 773-788.
[35] Ziliak, James and Thomas Kniesner. “Estimating Life Cycle Labor Supply Tax Effects.”
Journal of Political Economy 107 (April 1999): 326-359.
30
Working Paper Series
A series of research studies on regional economic issues relating to the Seventh Federal
Reserve District, and on financial and economic topics.
Extracting Market Expectations from Option Prices:
Case Studies in Japanese Option Markets
Hisashi Nakamura and Shigenori Shiratsuka
WP-99-1
Measurement Errors in Japanese Consumer Price Index
Shigenori Shiratsuka
WP-99-2
Taylor Rules in a Limited Participation Model
Lawrence J. Christiano and Christopher J. Gust
WP-99-3
Maximum Likelihood in the Frequency Domain: A Time to Build Example
Lawrence J.Christiano and Robert J. Vigfusson
WP-99-4
Unskilled Workers in an Economy with Skill-Biased Technology
Shouyong Shi
WP-99-5
Product Mix and Earnings Volatility at Commercial Banks:
Evidence from a Degree of Leverage Model
Robert DeYoung and Karin P. Roland
WP-99-6
School Choice Through Relocation: Evidence from the Washington D.C. Area
Lisa Barrow
WP-99-7
Banking Market Structure, Financial Dependence and Growth:
International Evidence from Industry Data
Nicola Cetorelli and Michele Gambera
WP-99-8
Asset Price Fluctuation and Price Indices
Shigenori Shiratsuka
WP-99-9
Labor Market Policies in an Equilibrium Search Model
Fernando Alvarez and Marcelo Veracierto
WP-99-10
Hedging and Financial Fragility in Fixed Exchange Rate Regimes
Craig Burnside, Martin Eichenbaum and Sergio Rebelo
WP-99-11
Banking and Currency Crises and Systemic Risk: A Taxonomy and Review
George G. Kaufman
WP-99-12
Wealth Inequality, Intergenerational Links and Estate Taxation
Mariacristina De Nardi
WP-99-13
Habit Persistence, Asset Returns and the Business Cycle
Michele Boldrin, Lawrence J. Christiano, and Jonas D.M Fisher
WP-99-14
Does Commodity Money Eliminate the Indeterminacy of Equilibria?
Ruilin Zhou
WP-99-15
A Theory of Merchant Credit Card Acceptance
Sujit Chakravorti and Ted To
WP-99-16
1
Working Paper Series (continued)
Who’s Minding the Store? Motivating and Monitoring Hired Managers at
Small, Closely Held Firms: The Case of Commercial Banks
Robert DeYoung, Kenneth Spong and Richard J. Sullivan
WP-99-17
Assessing the Effects of Fiscal Shocks
Craig Burnside, Martin Eichenbaum and Jonas D.M. Fisher
WP-99-18
Fiscal Shocks in an Efficiency Wage Model
Craig Burnside, Martin Eichenbaum and Jonas D.M. Fisher
WP-99-19
Thoughts on Financial Derivatives, Systematic Risk, and Central
Banking: A Review of Some Recent Developments
William C. Hunter and David Marshall
WP-99-20
Testing the Stability of Implied Probability Density Functions
Robert R. Bliss and Nikolaos Panigirtzoglou
WP-99-21
Is There Evidence of the New Economy in the Data?
Michael A. Kouparitsas
WP-99-22
A Note on the Benefits of Homeownership
Daniel Aaronson
WP-99-23
The Earned Income Credit and Durable Goods Purchases
Lisa Barrow and Leslie McGranahan
WP-99-24
Globalization of Financial Institutions: Evidence from Cross-Border
Banking Performance
Allen N. Berger, Robert DeYoung, Hesna Genay and Gregory F. Udell
WP-99-25
Intrinsic Bubbles: The Case of Stock Prices A Comment
Lucy F. Ackert and William C. Hunter
WP-99-26
Deregulation and Efficiency: The Case of Private Korean Banks
Jonathan Hao, William C. Hunter and Won Keun Yang
WP-99-27
Measures of Program Performance and the Training Choices of Displaced Workers
Louis Jacobson, Robert LaLonde and Daniel Sullivan
WP-99-28
The Value of Relationships Between Small Firms and Their Lenders
Paula R. Worthington
WP-99-29
Worker Insecurity and Aggregate Wage Growth
Daniel Aaronson and Daniel G. Sullivan
WP-99-30
Does The Japanese Stock Market Price Bank Risk? Evidence from Financial
Firm Failures
Elijah Brewer III, Hesna Genay, William Curt Hunter and George G. Kaufman
WP-99-31
Bank Competition and Regulatory Reform: The Case of the Italian Banking Industry
Paolo Angelini and Nicola Cetorelli
WP-99-32
2
Working Paper Series (continued)
Dynamic Monetary Equilibrium in a Random-Matching Economy
Edward J. Green and Ruilin Zhou
WP-00-1
The Effects of Health, Wealth, and Wages on Labor Supply and Retirement Behavior
Eric French
WP-00-2
Market Discipline in the Governance of U.S. Bank Holding Companies:
Monitoring vs. Influencing
Robert R. Bliss and Mark J. Flannery
WP-00-3
Using Market Valuation to Assess the Importance and Efficiency
of Public School Spending
Lisa Barrow and Cecilia Elena Rouse
Employment Flows, Capital Mobility, and Policy Analysis
Marcelo Veracierto
Does the Community Reinvestment Act Influence Lending? An Analysis
of Changes in Bank Low-Income Mortgage Activity
Drew Dahl, Douglas D. Evanoff and Michael F. Spivey
WP-00-4
WP-00-5
WP-00-6
Subordinated Debt and Bank Capital Reform
Douglas D. Evanoff and Larry D. Wall
WP-00-7
The Labor Supply Response To (Mismeasured But) Predictable Wage Changes
Eric French
WP-00-8
For How Long Are Newly Chartered Banks Financially Fragile?
Robert DeYoung
WP-00-9
Bank Capital Regulation With and Without State-Contingent Penalties
David A. Marshall and Edward S. Prescott
WP-00-10
Why Is Productivity Procyclical? Why Do We Care?
Susanto Basu and John Fernald
WP-00-11
Oligopoly Banking and Capital Accumulation
Nicola Cetorelli and Pietro F. Peretto
WP-00-12
Puzzles in the Chinese Stock Market
John Fernald and John H. Rogers
WP-00-13
The Effects of Geographic Expansion on Bank Efficiency
Allen N. Berger and Robert DeYoung
WP-00-14
Idiosyncratic Risk and Aggregate Employment Dynamics
Jeffrey R. Campbell and Jonas D.M. Fisher
WP-00-15
Post-Resolution Treatment of Depositors at Failed Banks: Implications for the Severity
of Banking Crises, Systemic Risk, and Too-Big-To-Fail
George G. Kaufman and Steven A. Seelig
WP-00-16
3
Working Paper Series (continued)
The Double Play: Simultaneous Speculative Attacks on Currency and Equity Markets
Sujit Chakravorti and Subir Lall
WP-00-17
Capital Requirements and Competition in the Banking Industry
Peter J.G. Vlaar
WP-00-18
Financial-Intermediation Regime and Efficiency in a Boyd-Prescott Economy
Yeong-Yuh Chiang and Edward J. Green
WP-00-19
How Do Retail Prices React to Minimum Wage Increases?
James M. MacDonald and Daniel Aaronson
WP-00-20
Financial Signal Processing: A Self Calibrating Model
Robert J. Elliott, William C. Hunter and Barbara M. Jamieson
WP-00-21
An Empirical Examination of the Price-Dividend Relation with Dividend Management
Lucy F. Ackert and William C. Hunter
WP-00-22
Savings of Young Parents
Annamaria Lusardi, Ricardo Cossa, and Erin L. Krupka
WP-00-23
The Pitfalls in Inferring Risk from Financial Market Data
Robert R. Bliss
WP-00-24
What Can Account for Fluctuations in the Terms of Trade?
Marianne Baxter and Michael A. Kouparitsas
WP-00-25
Data Revisions and the Identification of Monetary Policy Shocks
Dean Croushore and Charles L. Evans
WP-00-26
Recent Evidence on the Relationship Between Unemployment and Wage Growth
Daniel Aaronson and Daniel Sullivan
WP-00-27
Supplier Relationships and Small Business Use of Trade Credit
Daniel Aaronson, Raphael Bostic, Paul Huck and Robert Townsend
WP-00-28
What are the Short-Run Effects of Increasing Labor Market Flexibility?
Marcelo Veracierto
WP-00-29
Equilibrium Lending Mechanism and Aggregate Activity
Cheng Wang and Ruilin Zhou
WP-00-30
Impact of Independent Directors and the Regulatory Environment on Bank Merger Prices:
Evidence from Takeover Activity in the 1990s
Elijah Brewer III, William E. Jackson III, and Julapa A. Jagtiani
WP-00-31
Does Bank Concentration Lead to Concentration in Industrial Sectors?
Nicola Cetorelli
WP-01-01
On the Fiscal Implications of Twin Crises
Craig Burnside, Martin Eichenbaum and Sergio Rebelo
WP-01-02
4
Working Paper Series (continued)
Sub-Debt Yield Spreads as Bank Risk Measures
Douglas D. Evanoff and Larry D. Wall
WP-01-03
Productivity Growth in the 1990s: Technology, Utilization, or Adjustment?
Susanto Basu, John G. Fernald and Matthew D. Shapiro
WP-01-04
Do Regulators Search for the Quiet Life? The Relationship Between Regulators and
The Regulated in Banking
Richard J. Rosen
Learning-by-Doing, Scale Efficiencies, and Financial Performance at Internet-Only Banks
Robert DeYoung
The Role of Real Wages, Productivity, and Fiscal Policy in Germany’s
Great Depression 1928-37
Jonas D. M. Fisher and Andreas Hornstein
WP-01-05
WP-01-06
WP-01-07
Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy
Lawrence J. Christiano, Martin Eichenbaum and Charles L. Evans
WP-01-08
Outsourcing Business Service and the Scope of Local Markets
Yukako Ono
WP-01-09
The Effect of Market Size Structure on Competition: The Case of Small Business Lending
Allen N. Berger, Richard J. Rosen and Gregory F. Udell
WP-01-10
Deregulation, the Internet, and the Competitive Viability of Large Banks and Community Banks WP-01-11
Robert DeYoung and William C. Hunter
Price Ceilings as Focal Points for Tacit Collusion: Evidence from Credit Cards
Christopher R. Knittel and Victor Stango
WP-01-12
Gaps and Triangles
Bernardino Adão, Isabel Correia and Pedro Teles
WP-01-13
A Real Explanation for Heterogeneous Investment Dynamics
Jonas D.M. Fisher
WP-01-14
Recovering Risk Aversion from Options
Robert R. Bliss and Nikolaos Panigirtzoglou
WP-01-15
Economic Determinants of the Nominal Treasury Yield Curve
Charles L. Evans and David Marshall
WP-01-16
Price Level Uniformity in a Random Matching Model with Perfectly Patient Traders
Edward J. Green and Ruilin Zhou
WP-01-17
Earnings Mobility in the US: A New Look at Intergenerational Inequality
Bhashkar Mazumder
WP-01-18
The Effects of Health Insurance and Self-Insurance on Retirement Behavior
Eric French and John Bailey Jones
WP-01-19
5
Working Paper Series (continued)
The Effect of Part-Time Work on Wages: Evidence from the Social Security Rules
Daniel Aaronson and Eric French
WP-01-20
Antidumping Policy Under Imperfect Competition
Meredith A. Crowley
WP-01-21
Is the United States an Optimum Currency Area?
An Empirical Analysis of Regional Business Cycles
Michael A. Kouparitsas
WP-01-22
A Note on the Estimation of Linear Regression Models with Heteroskedastic
Measurement Errors
Daniel G. Sullivan
WP-01-23
The Mis-Measurement of Permanent Earnings: New Evidence from Social
Security Earnings Data
Bhashkar Mazumder
WP-01-24
Pricing IPOs of Mutual Thrift Conversions: The Joint Effect of Regulation
and Market Discipline
Elijah Brewer III, Douglas D. Evanoff and Jacky So
WP-01-25
Opportunity Cost and Prudentiality: An Analysis of Collateral Decisions in
Bilateral and Multilateral Settings
Herbert L. Baer, Virginia G. France and James T. Moser
WP-01-26
Outsourcing Business Services and the Role of Central Administrative Offices
Yukako Ono
WP-02-01
Strategic Responses to Regulatory Threat in the Credit Card Market*
Victor Stango
WP-02-02
The Optimal Mix of Taxes on Money, Consumption and Income
Fiorella De Fiore and Pedro Teles
WP-02-03
Expectation Traps and Monetary Policy
Stefania Albanesi, V. V. Chari and Lawrence J. Christiano
WP-02-04
Monetary Policy in a Financial Crisis
Lawrence J. Christiano, Christopher Gust and Jorge Roldos
WP-02-05
Regulatory Incentives and Consolidation: The Case of Commercial Bank Mergers
and the Community Reinvestment Act
Raphael Bostic, Hamid Mehran, Anna Paulson and Marc Saidenberg
WP-02-06
Technological Progress and the Geographic Expansion of the Banking Industry
Allen N. Berger and Robert DeYoung
WP-02-07
Choosing the Right Parents: Changes in the Intergenerational Transmission
of Inequality Between 1980 and the Early 1990s
David I. Levine and Bhashkar Mazumder
WP-02-08
6
Working Paper Series (continued)
The Immediacy Implications of Exchange Organization
James T. Moser
WP-02-09
Maternal Employment and Overweight Children
Patricia M. Anderson, Kristin F. Butcher and Phillip B. Levine
WP-02-10
The Costs and Benefits of Moral Suasion: Evidence from the Rescue of
Long-Term Capital Management
Craig Furfine
WP-02-11
On the Cyclical Behavior of Employment, Unemployment and Labor Force Participation
Marcelo Veracierto
WP-02-12
Do Safeguard Tariffs and Antidumping Duties Open or Close Technology Gaps?
Meredith A. Crowley
WP-02-13
Technology Shocks Matter
Jonas D. M. Fisher
WP-02-14
Money as a Mechanism in a Bewley Economy
Edward J. Green and Ruilin Zhou
WP-02-15
Optimal Fiscal and Monetary Policy: Equivalence Results
Isabel Correia, Juan Pablo Nicolini and Pedro Teles
WP-02-16
Real Exchange Rate Fluctuations and the Dynamics of Retail Trade Industries
on the U.S.-Canada Border
Jeffrey R. Campbell and Beverly Lapham
WP-02-17
Bank Procyclicality, Credit Crunches, and Asymmetric Monetary Policy Effects:
A Unifying Model
Robert R. Bliss and George G. Kaufman
WP-02-18
Location of Headquarter Growth During the 90s
Thomas H. Klier
WP-02-19
The Value of Banking Relationships During a Financial Crisis:
Evidence from Failures of Japanese Banks
Elijah Brewer III, Hesna Genay, William Curt Hunter and George G. Kaufman
WP-02-20
On the Distribution and Dynamics of Health Costs
Eric French and John Bailey Jones
WP-02-21
The Effects of Progressive Taxation on Labor Supply when Hours and Wages are
Jointly Determined
Daniel Aaronson and Eric French
WP-02-22
7
@FedFRASER