The Aging of the Baby Boomers:
Demographics and Propagation of Tax Shocks∗
Domenico Ferraro
Arizona State University
Giuseppe Fiori
North Carolina State University
September 11, 2016
Abstract
We investigate the consequences of demographic change for the effects of tax cuts in the
United States over the post-WWII period. Using narratively identified tax changes as proxies
for structural shocks, we establish that the responsiveness of unemployment rates to tax changes
largely varies across age groups: the unemployment rate response of the young is nearly twice
as large as that of prime-age workers. Such heterogeneity is the channel through which shifts in
the age composition of the labor force impact the response of the aggregate U.S. unemployment
rate to tax cuts. We find that the aging of the Baby Boomers reduces the effects of tax cuts
on aggregate unemployment by nearly fifty percent.
JEL Classification: E24; E62; J11
Keywords: Taxes; Demographics; Aging; Baby Boomers; Unemployment; Participation
∗
Ferraro: Department of Economics, W. P. Carey School of Business, Arizona State University, PO Box
879801, Tempe, AZ 85287-9801 (e-mail: [email protected]); Fiori: Department of Economics, Poole
College of Management, North Carolina State University, PO Box 8110, Raleigh, NC 27695-8110 (e-mail:
[email protected]). First version: November 8, 2015. We are grateful to Alexander Bick, Carlo Favero,
John Haltiwanger, Berthold Herrendorf, Bart Hobijn, Nir Jaimovich, Albert Marcet, Benjamin Moll, Pietro
Peretto, Edward Prescott, Valerie Ramey, Todd Schoellman, Matthew Shapiro, Pedro Silos, Nora Traum,
Etienne Wasmer, and seminar participants at Arizona State University, the Triangle Dynamic Macro (TDM)
workshop at Duke University, UC Riverside, the OFCE-SKEMA workshop on Economic Growth, Innovation,
and Corporate Governance, the 2016 Annual Meeting of the Society for Economic Dynamics (SED), and the
2016 Annual Congress of the European Economic Association (EEA), for valuable comments and suggestions.
1
Introduction
The post-World War II baby boom and the subsequent aging of the baby boomers resulted
in dramatic shifts in the age composition of the labor force in the United States. In this
paper, we investigate the consequences of such demographic change for the propagation of
tax cuts in the U.S. labor market. Specifically, we quantify how shifts in the age composition
of the labor force affect the response of the aggregate unemployment rate to unanticipated
tax cuts. We argue that the age composition of the labor force constitutes a quantitatively
important channel for the transmission of tax shocks to aggregate unemployment.
The first contribution of the paper is to document that the unemployment responsiveness
to tax shocks largely varies across age groups: the unemployment response of the young is
nearly twice as large as that of prime-age workers. This heterogeneity is the channel through
which shifts in the age composition of the U.S. labor force affect the response of the aggregate
unemployment rate to tax changes. The second contribution is to quantify the impact of an
aging labor force on the propagation of tax shocks. We argue that the aging of the baby
boomers reduces the effects of tax cuts on aggregate unemployment by fifty percent.
The policy debate in the aftermath of the Great Recession of 2007-2009 has led to renewed
interest in the effectiveness of fiscal policy in stimulating economic activity. Not surprisingly,
a growing strand of the empirical literature investigates the effects of government purchases
and taxes (see Romer and Romer, 2010; Barro and Redlick, 2011; Ramey, 2011; Mertens and
Ravn, 2013, among others).1 The literature has invariably considered aggregate macroeconomic variables, which is the natural starting point for analyzing the economic forces that
shape the aggregate response to fiscal shocks. Yet, our understanding of the transmission
mechanisms of fiscal policy remains incomplete. We pursue instead a disaggregated analysis
by considering one specific dimension of heterogeneity, age. This approach sheds light on the
link between microeconomic behavior and macroeconomic effects of tax changes. We stress
that our ultimate goal is to gauge the implications of demographic change for the aggregate
labor-market response to tax cuts. To date, this paper is the first attempt to tackle such a
question.2
Recent work has also studied the implications of demographic change for macroeconomic
analysis. For instance, Shimer (1999) shows that the entry of the baby boomers into the labor
force in the late-1970s, and their aging, accounts for most of the low-frequency movements
in the U.S. unemployment rate since World War II. Jaimovich and Siu (2009) show that
such demographic change accounts for a significant fraction of the decrease in business cycle
volatility observed in the United States since the mid-1980s. This paper shows that the aging
of the baby boomers considerably reduces the effects of taxes on aggregate unemployment.
Furthermore, we argue that assessing the effects of tax changes across different age groups
is also relevant for distinguishing between competing transmission channels of tax shocks at
1
See also Ramey and Shapiro (1998), House and Shapiro (2006, 2008), Pappa (2009), Brückner and
Pappa (2012), Auerbach and Gorodnichenko (2012), Cloyne (2013), Ramey and Zubairy (2014), Nakamura
and Steinsson (2014), Acconcia et al. (2014), Mertens and Ravn (2014), and Mertens (2015).
2
Anderson et al. (2015) document heterogeneous effects of government spending shocks on consumption,
depending on income and age, whereas Wong (2015) investigates the implications of population aging for
the response of consumption to monetary policy shocks.
1
work in the U.S. labor market. Understanding if and the extent to which young, prime-age
and old workers in the labor force display differences in the unemployment responsiveness to
aggregate tax shocks would seem important for understanding why aggregate unemployment
responds to tax cuts as much as it does. Analogously, Rı́os-Rull (1996), Gomme et al. (2005),
Hansen and İmrohoroğlu (2009), and Jaimovich et al. (2013) assess the aggregate implications
of age-specific differences in cyclical movements of hours worked.
To empirically estimate the responses of aggregate and age-specific unemployment rates
to unanticipated, temporary tax changes, we use narrative identification of tax shocks (see
Romer and Romer, 2009, 2010). Specifically, we use narratively identified tax changes as
proxies for structural tax shocks, and structural vector autoregressions (SVARs) to estimate
the dynamic responses to a temporary tax cut, as in Mertens and Ravn (2013). We consider
average marginal tax rates as we are interested in the transmission mechanism of tax changes
that operates through incentive effects on intertemporal substitution rather than disposable
income. We establish that the responsiveness of unemployment rates to tax changes largely
varies across age groups: the unemployment rate response of the young is nearly twice as
large as that of prime-age workers. By contrast, we show that the age-specific labor force
shares are in fact unresponsive to the (narratively identified) changes in average marginal
tax rates. This empirical finding is, perhaps, not surprising as the observed demographic
trends in the age composition of the labor force in the United States are largely determined
by fertility decisions made long before a specific tax shock. Thus, the age composition of the
labor force is largely predetermined at the time of a legislated tax change. In addition, the
post-war baby boom and the aging of the baby boomers in the last thirty years resulted in
large movements in the age composition of the labor force. As a result, time-series variation
in labor force shares by age abounds.
Given these observations, a natural conjecture is that the responsiveness of the aggregate
U.S. unemployment rate to tax changes depends on the age composition of the labor force.
When an economy is characterized by a smaller share of young workers, everything else equal,
one observes a smaller aggregate response to tax cuts. We show that this is indeed the case.
Specifically, we construct an aggregate unemployment response to tax shocks, that accounts
for the observed movements in the age composition of the labor force. The implied response
provides a simple quantitative accounting of how the observed demographic trends in the
United States impact the effectiveness of tax cuts in reducing aggregate unemployment.
We find that the aging of the baby boomers reduces the response of the aggregate U.S.
unemployment rate to tax cuts by roughly fifty percent.
The implications for fiscal policy are far-reaching. In the United States, given the current
fertility and mortality rates, the working-age population is expected to become older. Similar
estimates and projections apply to Japan and most industrialized countries in Europe. The
results in this paper indicate that tax shocks of the size observed in the United States since
World War II are becoming increasingly less effective in stimulating economic activity. Thus,
policies targeted at labor force participation are the natural candidates for future stimulus
policies.
The rest of the paper is organized as follows. In Section 2, we describe the evolution
of population and labor force shares by age group in the United States for 1950-2015. In
Section 3, we present a theoretical framework to interpret the empirical results. In Section 4,
2
we introduce the econometric methodology. In Section 5, we present estimates for the effects
of tax cuts on aggregate unemployment and participation. In Section 6, we present estimates
for unemployment and participation by age group, whereas in Section 7 we quantify the role
of aging for the aggregate labor-market response to tax cuts. Section 8 concludes.
2
Population Aging in the United States
In this section, we provide a bird’s eye view of the aging of the workforce observed in the
United States in the last thirty years: “The aging of the baby boomers.” The post-World War
II baby boom and the subsequent baby bust resulted in dramatic shifts in the age composition
of the U.S. working-age population. To the extent that young, prime-age, and old workers
feature different labor force attachment, turnover rates, and job search intensities, population
aging arguably has important implications for the aggregate labor-market response to tax
changes. In this paper, we show that is indeed the case.
Working-age population. In Appendix A, panel A of Figure A.1 shows the average
age of the U.S. civilian noninstitutional population of 20-64 years old for the period 19502015. We consider five age groups (20-24,
45-54, and 55-64) and calculate the
35-44,
P 25-34,
a+a
P
P
average age of the population as ā ≡ a 2 φa , where a and a are the lower and upper
bounds of the age group a, respectively, and φPa is the age-specific population share, that is,
the ratio of the population in the age group a to the overall population. Two facts emerge.
First, the U.S. population has been steadily aging since the mid-1980s. Second, the average
age of the population declined over the course of twenty-five years from the early-1960s to
the mid-1980s as a result of the sharp increase in birth rates after World War II. This is the
so-called “baby boom.” However, in the early-1960s, birth rates started to decline towards
levels prior to the baby boom. The consequent baby bust led to the sharp inversion in the
average age observed in the late-1980s. The U.S. population has been aging since then as a
result of the aging of the baby boomers.
Notably, these slow-moving trends in the average age of the population result from the
underlying shifts in the age composition of the population induced by the baby boom and the
subsequent baby bust. Specifically, the share of the 20-34 years old in the overall working-age
population has been declining since the mid-1980s, which has tilted the age composition of
the population towards older ages. In Figure A.1, panel B shows the dramatic changes in the
age composition of the population observed in the United States since 1950. The population
share of the 20-34 years old increased by 10 percentage points (from approximately 35 to
45 percent) over the course of twenty years from 1960 to 1980. During the same period,
the population share of the 35-54 years old has instead declined by approximately the same
amount. In addition, the population share of the 55-64 years old has remained approximately
constant over the same period, it starts declining in the early-1980s to reach a trough in the
mid-1990s, when it sharply reverts to steady growth towards an approximately 22 percent
of the overall population in 2015. These sustained changes in the age composition of the
population are the key drivers of the aging of the U.S. workforce.
Labor force. Next, we show that these underlying demographic trends in the U.S.
population are also affecting the age composition and so the average age of the labor force
3
(employed plus unemployed persons of 20-64 years old). We consider again five age groups
(20-24, 25-34,
45-54, and 55-64) and calculate the average age of the labor force
LF
P 35-44,
a+a
φ
as āLF ≡
a , where a and a are the lower and upper bounds of age group a,
a
2
LF
respectively, and φa is the age-specific labor force share, that is, the ratio of the labor force
in the age group a to the overall labor force. In Figure 1, panel A shows that the average
age of the labor force mirrors the movements in the average age of the U.S. population. The
U.S. labor force has been aging since the mid-1980s. In Figure 1, panel B shows that this
aging is indeed driven by the underlying shifts in the age composition of the labor force.
Specifically, the share of the 20-34 years old in the labor force increased by more than 10
percentage points (from approximately 35 to 48 percent) over the course of fifteen years
from the mid-1960s to early-1980s. During the same period, the share of the 35-54 years old
declined by approximately 10 percentage points, whereas the labor force share of the 55-64
years old starts declining in the early-1970, reaches a trough of approximately 10 percent in
the mid-1990s, and it has been steadily increasing since then.
Employment and unemployment. In Appendix A, Figure A.2 shows that the trends
in the age composition of the employment pool in the United States are remarkably similar to
those experienced by the U.S. labor force, as shown in Figure 1. This observation is perhaps
not surprising as employment represent nearly 95 percent of the economy-wide labor force,
for 1950-2015. In Appendix A, Figure A.3 portraits instead the implications of population
aging for the age composition of the unemployment pool. Remarkably, the movements in
the average age of the unemployment pool, and the associated shifts in the unemployment
shares by age, are substantially larger than those observed for employment. Notably, panel
A of Figure A.3 shows that, in the mid-1950s, the average age of an unemployed worker was
39 years old, whereas, in the 1980s, the average age was as low as 33 years old. Yet, as of
2015, an unemployed worker is on average of nearly the same age of one in the unemployment
pool in the years preceding the baby boom. Panel B, in Figure A.3, shows the dramatic
changes in the age composition experienced by the unemployment pool in the United States.
The unemployment share of the 20-34 years old raised by nearly 20 percentage points (from
approximately 45 to 65 percent) from the mid-1960s to the 1980s. Since the 1980s, the share
of the 20-34 years old has been declining at a nearly constant pace. As of 2015, they represent
nearly 49 percent of the economy-wide unemployment pool, which is a level comparable to
that in the years preceding the baby boom. During the same period, the unemployment
share of the 35-54 years old has instead declined and then raised, as a mirror image of the
unemployment share of the 20-34 years old. Finally, the unemployment share of the 55-64
years old has declined from the mid-1950s to the 1990s, and it has been steadily raising
since then. Notably, as of 2015, the 55-64 years old represent 14 percent of the economywide unemployment pool. To summarize, the data point to a substantial reshuffling in
the age composition of unemployed workers. Furthermore, the share of old workers in the
unemployment pool is expected to increase over time.
4
3
Framework for the Analysis
In this section, we introduce a simple theoretical framework to highlight the key equilibrium
feedback effects of tax changes and to provide a guide for discussing the empirical results.
We stress, however, that the econometric methodology we adopt in Section 4 does not rely
on any specific theory or functional form assumptions.
We acknowledge that the literature has not yet settled on a definitive theoretical model to
evaluate the aggregate employment effects of taxes. There are two widely used frameworks:
(i) the frictionless, neoclassical growth model with an endogenous labor-leisure choice, as
in Kydland and Prescott (1982), modified to include indivisible labor (see Hansen, 1985;
Rogerson, 1988); (ii) the Diamond-Mortensen-Pissarides (DMP) model of unemployment,
featuring search-and-matching frictions in the labor market (see Diamond, 1982; Mortensen,
1982; Pissarides, 1985). One can view the former as a model of labor force participation and
the latter as a model of unemployment, conditional on participation. In both frameworks,
taxes affect aggregate employment by distorting the labor-leisure margin, that is, the relative
return of market versus of non-market activity.
As we shall see in Section 5 and 6, we find that aggregate unemployment is indeed quite
responsive to the (narratively identified) tax shocks, whereas labor force participation is not,
and that the responsiveness of unemployment to such tax shocks largely varies by age. To
interpret these empirical findings, we present an overlapping generations (OLG) model of
unemployment in the extended DMP class. The salient feature of the theory is that workers
generate rents (wages in excess of the value of unemployment), or analogously, job surpluses
that vary over the life cycle. We show that this age profile of worker rents translates into an
age profile of tax incidence. And that, as a result, shifts in the age composition of the labor
force can, in principle, substantially change the aggregate impact of tax changes. The model
encompasses finite-lifetimes, as in Chéron et al. (2011, 2013), and uncertain-lifetimes, akin
to a Yaari-Blanchard perpetual youth model (see Yaari, 1965; Blanchard, 1985), in a simple
unified framework. Furthermore, for old workers near to the mandatory retirement age, the
model collapses to a standard model of labor supply, with indivisible labor, which allows
us to study the participation decision of old workers. The model presented in this section
provides a coherent framework to investigate the determinants of the aggregate labor-market
response to unanticipated tax changes.
3.1
Environment
We now turn to the description of the economic environment. We keep the exposition of the
labor market to a minimum as the search-and-matching structure is standard, as described in
Pissarides (2000). We expand instead on the features of the environment that are important
for the transmission mechanism of tax changes. Time is discrete and indexed by t ≥ 0.
Labor market. There is a growing population of workers of measure Nt . Each worker is
endowed with an indivisible unit of time that can be either supplied as labor services when
employed or used in job search when unemployed. We adopt a two-state representation of the
labor market, such that Nt also denotes the labor force (employed plus unemployed workers).
The economy is also populated by a continuum of employers, that is either matched with
5
a worker, and so producing output, or posting job vacancies to hire unemployed workers.
The mass of employers posting vacancies is determined in free-entry equilibrium. We adopt
the standard frictional view of the labor market and postulate the existence of an aggregate
meeting technology that determines the number of worker-employer contacts. As standard
in the literature, the meeting technology displays constant returns to scale, with respect to
the number of posted job vacancies and the number of unemployed workers seeking jobs,
that can be described in terms of the vacancy-to-worker ratio θ (i.e., labor-market tightness).
The meeting technology provides the probability that a worker meets a vacancy, p (θt ), and
the probability that a vacancy meets a worker, q (θt ), as function of labor-market tightness
θt ≡ Vt /Ut , where Vt and Ut denote aggregate job vacancies and aggregate unemployment,
respectively. Notably, in a tight labor market, it is easy for job seekers to find a job—the
job-finding probability is high—and difficult for employers to hire—the job-filling probability
is low. Keeping a job vacancy open requires a per-period cost of k > 0 units of final output;
the expected cost of opening and maintaining a job vacancy is then k/q (θt ).
Demographics. The economy has overlapping generations (OLG) of workers who live a
maximum of I periods, with age (time spent in the labor market) denoted by i ∈ {1, . . . , I}.
We adopt a stable population structure: workers are born (i.e., they enter the labor market)
at age normalized to one and survive from age i to i + 1 with the age-specific probability
Qi−1
σj .
σi < 1, with σI = 0. The unconditional probability of being alive at age i is σ i = j=1
In the first period, the measure of newly born workers is normalized to one. Working-age
population, Nt , (or, equivalently, labor force), grows at a constant rate λn ≥ 0, such that the
population of age 1 in period t is N1,t = (1 + λn )t , while that of age 2 is N2,t = σ1 (1 + λn )t−1 ,
and so forth, up to NI,t = σ I−1 (1 + λn )t−I+1 workers who survive until the last period of life.
Newly born workers enter the labor marker as unemployed. An alternative interpretation of
this demographic structure is that workers randomly retire with an age-specific probability
1 − σi , with I representing an exogenously set, mandatory “retirement age.” In the simplest
case, workers participate the labor market for I periods (i.e., σi = 1 for i < I), and retire
afterward with certainty (i.e., σI = 0). Working-age population evolves over time according
to λn ≡ (Nt+1 − Nt ) /Nt = b − d ≥ 0, where b and d denote birth rate (births to population
ratio) and death rate (deaths to population ratio), respectively. An increase in b, that
feeds into higher population growth, represents a baby boom; conversely, a reduction in b
represents a baby bust. Specifically, changes in the birth rate b induce shifts in the age
composition of the labor force by changing the age-specific labor shares φni,t ≡ Ni,t /Nt . Note
that the case σi = σ < 1, for all i ≥ 1, corresponds to the demographic structure of the
Yaari-Blanchard perpetual youth model. That is, workers are of different ages, but have the
same expected lifetime.
Preferences. Age i worker’s lifetime utility is given by:
I
X
β i σ i u (ci , ei ; zi ) ,
(1)
i=1
where β < 1 is the pure time discount factor, σ i is the unconditional probability of being
alive at age i, as previously defined, ci ≥ 0 is consumption, ei ≥ 0 is time spent working, and
zi > 0 is an age-specific, exogenously given parameter, that measures the disutility associated
6
with working, or equivalently, the value of leisure. Period utility is linear in consumption
and subject to a labor indivisibility constraint:
u (ci , ei ; zi ) = ci − zi ei ,
with ei ∈ {0, 1} .
(2)
The age-specific parameter zi allows for the possibility that the disutility of work changes
with age.3 The standard interpretation of linear preferences in consumption is risk-neutrality.
An alternative way to justify the linearity assumption without invoking risk-neutrality, is to
consider an economic environment where workers have access to a risk-sharing arrangement
that provides full insurance against idiosyncratic income risks. In that case, one can study
the effects of taxes on income-maximizing behavior abstracting from risk considerations.
Production. The unit of production is a worker-employer match. An age i worker’s unit
of time can be transformed into zi units of leisure or xi efficiency units of labor input: xi > 0
is an age-specific, exogenously given parameter, that measures the productivity on the job.
Specifically, such age-specific parameter xi allows for the possibility that the productivity
on the job raises over the life cycle. Such life-cycle productivity profile is a robust prediction
of theories based on human capital accumulation and/or worker turnover, and it has been
shown to be important for explaining the age profile of earnings observed in the data. Also,
each job is hit by a transitory match-specific productivity shock . A matched worker of
age i produces then yi = xi units of output. When a match of idiosyncratic productivity
suffers a change, the new value 0 is drawn from a fixed cumulative distribution function,
F (), with bounded support [, ].
Idiosyncratic risk. Workers face individual-level risk. Conditional on employment,
wages fluctuate due to the realization of the transitory match-specific shock. In addition,
conditional on a large adverse realization of the match-specific shock, workers may experience
transitory unemployment spells. Here, we abstract from idiosyncratic risk considerations and
posit a simple risk-sharing arrangement: the family. Specifically, we assume that workers
are organized into stand-in families that pool all of their members’ idiosyncratic risks. Risksharing is perfect within the family. Each member of the family maximizes then the present
value of (net-of-tax rate) labor income. This type of risk-sharing arrangement dates at least
back to Merz (1995) and Andolfatto (1996). We do not believe that consumption during
unemployment spells behaves literally according to the model with full insurance. However,
families may provide some degree of partial insurance against unemployment risk. We view
then the fully-insurance case as a good and convenient first-order approximation to the more
complicated real-world behavior.4
Taxes and redistribution. The government runs a progressive tax system. It uses
a nonlinear income tax and transfers to finance public consumption, Gt , which adjusts to
balance the government budget on a period-by-period basis. Government purchases of final
goods are a “pure waste” as they do not affect either marginal utility of private consumption
or production. According to this tax system, net-of-transfer taxes as function of individual
3
Preference heterogeneity has been shown to be important for explaining the cross-sectional joint distribution over wages, hours worked, and consumption (see Heathcote et al., 2014).
4
Blundell et al. (2008) find evidence of substantial insurance of individual workers against transitory
shocks, such as unemployment. Heathcote et al. (2014) find evidence of partial insurance of permanent
income shocks as sixty percent of idiosyncratic permanent wage fluctuations are effectively smoothed.
7
earnings are given by a simple time-invariant tax function:
Tj,t
(w̃j,t /w̄j,t )1−γ
= w̃j,t − (1 − τt )
w̄j,t ,
1−γ
(3)
where w̃j,t ≡ wj,t ej,t denotes worker j’s earnings, wj,t denotes the wage, and ej,t ∈ {0, 1} takes
1
R 1−γ 1−γ
is
on the value of one if worker j is employed and zero if unemployed; w̄j,t =
w̃j,t dj
R
0
an aggregate of the personal income tax base, and τt = 1 − (w̃j,t /w̄j,t ) 1 − Tj,t dj is the
economy-wide average marginal tax rate (AMTR), that is a weighted average of individual
0
marginal tax rates Tj,t
, with weights given by labor income shares. The parameter 0 ≤ γ < 1
indexes the progressivity of the tax system: (i) when γ = 0 all workers face the same marginal
tax rate, the system features a proportional tax rate τt ; (ii) when γ > 0 marginal tax rates
exceed average rates, the tax system is progressive.5 Since we are interested in level shifts
of the tax schedule, as opposed to changes in the progressivity of the tax system, we will
confine our analysis to the case of γ = 0 such that Tj,t = τt w̃j,t .
3.2
Equilibrium
We now turn to discuss the equilibrium of the model. Productivity shocks are independently
and identically distributed across matches, and there is no aggregate uncertainty (i.e., τt = τ
for all t ≥ 0). The focus is on steady-state outcomes. This allows for analytical tractability,
which is important to convey insight into the key properties of the equilibrium. The steadystate analysis can be interpreted as an approximation for the equilibrium dynamics of the
model induced by persistent aggregate shocks. As we shall see in Section 5, we find that,
reassuringly, in the United States unanticipated cuts in AMTR are indeed greatly persistent.
Wage determination. Upon meeting, employers and workers enter bilateral Nashbargaining that determines the wage payment to the worker. The Nash-bargaining solution
implies that worker and employer receive a constant and proportional share of the total
match surplus, which in turn results in separation decisions that are bilaterally efficient.
This result allows us to conveniently and parsimoniously work with the Bellman equations
that describe the total surplus generated by a worker-employer match, rather than with the
pairs of separate Bellman equations for the worker and employer. With this approach, wage
determination is not explicitly treated. Rather, it is implicitly represented in the Bellman
equations via the worker’s bargaining parameter, η ≥ 0. In the background, associated with
each match, there is a corresponding wage that achieves the Nash-bargaining outcome. As a
result, the equilibrium of the model is fully characterized by (a system of) Bellman equations
for the total match surplus, indexed by worker’s age, and a free-entry condition for vacancy
posting that determines the mass of active employers in the labor market.
Moreover, under Nash-bargaining, if taxes are the only distortion, then it is irrelevant
whether taxes are levied on the employer or the worker—all that matters is the tax levied on
the match. In addition, under full insurance, the individual worker maximizes net-of-AMTR
5
The specification of the tax function in (3), originally proposed by Feldstein (1969), has been widely
used and shown to be a remarkably good approximation of the actual tax and transfer schemes in the United
States (see Heathcote et al., 2014, among others).
8
labor income. As a result, the introduction of the tax rate τt mandates only minor changes
to the surplus equations as it effectively creates a wedge between the worker’s disutility of
work, zi , and what we refer to as the effective disutility of work z̃i,t ≡ zi / (1 − τt ).
Surplus equations by age. After the realization of the match-specific shock, workers
and employers decide whether to remain in the match, and produce output, or to destroy the
match, and become idle. Since total match surplus is monotonically increasing in worker’s
productivity, job separation satisfies the reservation property. That is, there exists a unique
age-specific cutoff Ri such that all matches with workers of age i are endogenously destroyed
when hit by an adverse productivity shock < Ri . Hence, large negative shocks induce
job destruction, but the choice of when to separate is optimally chosen by employers and
workers, jointly.
Specifically, total match surplus generated by a worker of age i is:
Z zi
+ β̃i [1 − ηp(θ)]
Si+1 (0 ) dF (0 ),
(4)
Si () = xi −
1−τ
Ri+1
where β̃i ≡ βσi is an age-specific discount factor; future pay-offs are discounted by a factor
that controls for the age-specific probability σi of surviving from age i to age i+1. Intuitively,
young workers with high σi attach a larger weight to future pay-offs than old workers with low
σi . In the last period of life, σI = 0 such that future pay-offs are irrelevant for the current jobseparation decision. Thus, the parameter β̃i captures the horizon effect on job separations.
The other source of heterogeneity comes from the age profile of worker’s net productivity
ỹi ≡ xi −zi / (1 − τ ), that determines the instantaneous pay-off to the match; ỹi captures the
net-productivity effect on job separations. We stress that from the perspective of the model,
whether heterogeneity in worker’s net productivity comes from xi or from zi , or from both,
is inconsequential for job separations, and thereby for the age distribution of unemployment
rates. Notably, changes in the average marginal tax rate, τ , affect worker’s net productivity
by altering the relative return of market versus non-market activity. Specifically, a cut in τ
acts as a reduction in the effective disutility of work effort.
Vacancy posting. In free-entry equilibrium, employers post job vacancies up to the
point where the expected cost equals the expected benefit of opening and maintaining a job
vacancy:
I−1
X
k
= (1 − η)
β̃i φui Si () ,
q (θ)
i=1
(5)
P
where φui ≡ Ui /U , with U ≡ I−1
i=1 Ui , is the age i worker’s share in aggregate unemployment,
with Ui denoting age-specific unemployment (i.e., number of age i unemployed workers).
Equation (5) determines labor-market tightness θ. As in Mortensen and Pissarides (1994), we
assume that all new jobs are created at the upper support of the match-specific productivity
distribution, , such that the surplus of a new match is positive for all workers, irrespective
of their age. This assumption implies that the probability that a worker meets a job vacancy,
p (θt ), equals the probability that a worker finds a job, ft . As a result, job-finding rates do
not vary by age. In the model, age differences in unemployment rates originate solely from
9
age differences in job-separation rates. Such a characterization of life-cycle unemployment
is consistent with the empirical evidence for the United States that the bulk of the declining
age profile of unemployment rates is accounted by the declining age profile of job-separation
rates (see Gervais et al., 2014; Choi et al., 2015; Menzio et al., 2016).
We stress that in this environment the age composition of the unemployment pool affects
vacancy posting. This property of the equilibrium results from the fact that unemployed
workers of different ages enter a single matching function. In this sense, matching is random,
such that employers posting vacancies form expectations about the likelihood of meeting
workers at different stages of their life cycle. One can also construct an equilibrium in which
search is directed. In that scenario, the equilibrium would display endogenous labor-market
segmentation by age, such that vacancy posting would be completely insulated from the age
composition of the unemployment pool (see Menzio et al., 2016).
3.3
Cross-Sectional Implications
We now turn to discuss the cross-sectional implications of the equilibrium. Specifically, we
study the key mechanisms underlying the age profile of match surplus, which is essential for
understanding job separation decisions over the life cycle, job creation, and thereby the age
distribution of unemployment rates. In the model, there are two sources of heterogeneity in
match surplus over the life cycle: (i) the horizon effect, which derives from the fact that the
workers’ distance to retirement necessarily varies by age; and (ii) the net-productivity effect,
which derives from the fact that worker’s productivity on the job and/or disutility of work
(i.e., the opportunity cost of employment) changes over the life cycle.
Unemployment by age. In the steady state, there is a time-invariant distribution of
unemployment levels indexed by age {Ui }Ii=1 :
Ui = Ui−1 + si Ei−1 − fi Ui−1 ,
(6)
where si ≡ F (Ri ) and fi ≡ p (θ) are age-specific, job-separation rates and job-finding rates,
respectively. We stress that job-separation rates differ by age through age differences in the
reservation cutoff, Ri , whereas job-finding rates are the same across all workers, that are
actively seeking jobs. The number of age i workers in the labor force is Ni = Ui + Ei , such
that ui ≡ Ui /Ni is the age-specific unemployment rate. Equation (6) singles out the flow
rates si and fi as the two key determinants of unemployment over the life cycle.
Reservation cutoff. Key to understanding the behavior of job-separation rates is the
determination of the age-specific reservation cutoff. Since employers and workers have the
option to separate at no cost, a match remains active as long as its continuation value is
above zero. As a result, the cutoff Ri is determined by the zero-surplus condition Si (Ri ) = 0,
which yields the following recursive equation:
Z zi / (1 − τ )
− β̃i [1 − ηp (θ)]
[1 − F (0 )] d0 .
(7)
Ri =
xi
Ri+1
Equation (7) identifies two transmission channels of tax changes. First, keeping labor-market
tightness θ constant, a tax cut directly raises current match surplus, which in turn reduces
10
the reservation cutoff Ri across the board. As a result, job-separation rates fall for all age
groups. Second, the equilibrium features a feedback effect that operates through changes
in market tightness. A tax cut affects vacancy posting, and thereby market tightness, by
changing (i) the age composition of the unemployment pool, and (ii) the surplus generated
by new jobs. Specifically, an increase in market tightness raises the reservation cutoff Ri ,
which leans towards more job separation. In this sense, the key feedback effect at play in
the model dampens the equilibrium response to tax cuts. In addition, everything else equal,
such dampening effect decreases as workers age. Notably, in the year before retirement, the
elasticity of the reservation cutoff with respect to the net-of-tax rate is independent of the
equilibrium response of labor-market tightness. Intuitively, for a worker deciding whether to
work the year before retirement or to enjoy the value of leisure, it is irrelevant to know how
tight is the state of the labor market. A non-employed worker of age I is not searching for a
job, as such she is effectively out of the labor force. Hence, the model points to potentially
large effects of tax cuts on the participation decision of old workers close to retirement. These
predictions are consistent with a standard life-cycle model of labor supply (see Rogerson and
Wallenius, 2009, 2013). And, as we shall see in Section 6, they indeed hold in the data.
However, whether the reservation cutoff Ri in (7), and its sensitivity to tax changes,
raises or declines with age is a priori indeterminate, depending on the relative strength of
the horizon and net-productivity effect over the life cycle.
Horizon effect. To isolate the role played by the pure horizon effect, we next assume
that worker’s net productivity remains constant over the life cycle: ỹi = ỹ, for all i ≤ I. In
this case, match surplus differs by age as workers are heterogeneous in terms of their distance
to retirement I − i + 1. Notably, the theory predicts a greater sensitivity of match surplus
to current surplus, and thereby to the net-of-tax rate, for older workers. Specifically, for
old workers in the year before retirement, it is only current worker’s net productivity ỹ that
matters for the separation decision.
The main age-specific implications of the horizon effect are summarized as follows.
PROPOSITION 1 (Horizon effect). Suppose that worker’s survival probability declines
over the life cycle—σi ≥ σi+1 , for all i ≤ I − 1, with σI = 0. If worker’s net productivity is
constant over the life cycle—ỹi = ỹ, for all i ≤ I—then (i) the reservation threshold Ri is
increasing in age; (ii) total match surplus Si () decreases with age; (iii) the job-separation
rate si increases with age; and (iv) the unemployment rate ui increases with age.
PROOF. See Appendix C.
The testable predictions (iii)-(iv) of Proposition 1 are counterfactual. In the data, jobseparation and unemployment rates monotonically decrease with age (see Gervais et al., 2014;
Menzio et al., 2016). Hence, we interpret these results as evidence that the horizon effect
plays a minor role in determining unemployment rates over the life cycle. Yet, as previously
argued, such horizon effect still plays an important role for the participation decisions of old
workers close to retirement.
Net-productivity effect. To isolate the role played by the net-productivity effect, we
next assume that the worker’s survival probability is constant throughout life, as in a YaariBlanchard perpetual youth model, whereas worker’s net productivity raises throughout life:
11
σi = σ < 1, for all i ≥ 1, and ỹi ≤ ỹi+1 . Hence, workers are of different ages, but have the
same expected lifetime. Note that an increasing age profile of net productivity can be due
to an increasing age profile of worker’s productivity on the job (i.e., xi ≤ xi+1 ) and/or to a
decreasing age profile of the disutility of work (i.e., zi ≥ zi+1 ). Yet, from the perspective of
the model, the specific source of age heterogeneity in net productivity is irrelevant. Notably,
the theory predicts a lesser sensitivity of match surplus to current surplus, and thereby to
the net-of-tax rate, for older workers. In this regard, we emphasize that the shape of the
age profile of worker’s net productivity is inconsequential for the results, qualitatively. Yet,
quantitatively, it is likely to be a major determinant of the model’s ability to account for
the age profile of unemployment responsiveness to tax cuts estimated in the data.
The main age-specific implications of the net-productivity effect are summarized as follows.
PROPOSITION 2 (Net-productivity effect). Suppose that worker’s net productivity
raises throughout life—ỹi ≤ ỹi+1 , for all i ≥ 1. If worker’s survival probability is constant
throughout life—σi = σ, for all i ≥ 1—then (i) the reservation threshold Ri is decreasing in
age; (ii) total match surplus Si () increases with age; (iii) the job-separation rate si decreases
with age; and (iv) the unemployment rate ui decreases with age.
PROOF. See Appendix C.
The testable predictions (iii)-(iv) of Proposition 2 are consistent with data (see Gervais
et al., 2014; Menzio et al., 2016). Hence, we interpret these results as evidence that the
net-productivity effect plays an important role for life-cycle unemployment. Here, we do
not take a stand on the specific theory underlying the increasing age profile of worker’s net
productivity. Theories based on human capital accumulation and/or worker turnover are
natural candidates to generate such a profile as an equilibrium outcome. Yet, we stress that
all these theories would ultimately affect the age distribution of unemployment rates insofar
as they affect the surplus of a job (or worker’s rents) over the life cycle.
3.4
Aggregate Implications
We now turn to discuss the aggregate implications of the equilibrium. First, we detail the
aggregation from micro (i.e., by age) to macro labor-market variables. Second, we study the
implications of tax changes for age-specific and aggregate unemployment rates.
Aggregate unemployment. In
PIour context, aggregation is straightforward. Precisely,
aggregate unemployment is Ut =
i=1 Ui,t , whereas the aggregate unemployment rate is
ut ≡ Ut /Nt . Changes in aggregate unemployment are then driven by the number of employed
workers who lost jobs in the previous period minus the number of unemployed workers who
found jobs:
Ut+1 = Ut + st Et − ft Ut ,
(8)
where st and ft are the aggregate job-separation and job-finding rates, respectively. Note
that aggregate job-separation and job-finding rates are a weighted average of job-separation
and job-finding rates of workers of different ages, respectively:
12
st =
I
X
φei,t si,t ,
and ft =
i=1
I−1
X
φui,t fi,t ,
(9)
i=1
where the weights φei,t ≡ Ei,t /Et and φui,t ≡ Ui,t /Ut are, respectively, age-specific employment
and unemployment shares. Note that, in the model, there are no age differences in job-finding
rates, such that fi,t = ft , for all i < I.
Micro and macro unemployment elasticities. Next, we discuss the effects of tax
changes on the unemployment rates by age (“micro-elasticities”) and the implications for
the aggregate unemployment rate (“macro-elasticity”). Changes in taxes affect age-specific
unemployment rates through (i) changes in the age-specific job-separation rates and (ii)
changes in the aggregate job-finding rate. Micro- and macro-elasticities can, in principle,
greatly differ depending on the age composition of the labor force.
The elasticity of the age-specific job-separation rate with respect to the tax rate is:
f (Ri )
dRi
d ln si
=
×
,
dτ
F (Ri )
dτ
for i ≤ I.
(10)
Equation (10) identifies two channels through which tax changes affect the age-specific jobseparation rate: (i) a change in the tax rate affects the reservation cutoff for endogenous job
separation. This effect is captured by the term dRi /dτ —cutoff-driven tax effect; (ii) the magnitude of the cutoff-driven tax effect depends on a measure of homogeneity of the workforce
at the margin (akin to a hazard rate). This effect is captured by the term f (Ri )/F (Ri )—
hazard-driven tax effect—that is the ratio of the mass of workers at the margin to the mass
of workers on the left of the reservation cutoff.
The elasticities of labor-market tightness and aggregate job-finding rate with respect to
the tax rate are:
I−1
X
1
dφui
d ln θ
u dSi ()
β̃i φi
=
+ Si ()
,
dτ
dτ
dτ
[1 − f (θ)] S̄ i=1
and
d ln θ
d ln f
= f (θ)
,
dτ
dτ
(11)
respectively, where f (θ) ≡ θf 0 (θ)/f (θ) is the elasticity of the aggregate job-finding
rate with
P
u
respect to labor-market tightness (vacancy-unemployment ratio), and S̄ ≡ I−1
β̃
i=1 i φi Si ()
is an age-adjusted measure of aggregate surplus in the economy. Note that, everything else
equal, changes in the age-adjusted surplus S̄ directly affect the elasticity of labor-market
tightness. Since match surplus varies by age, movements in the age composition of the
unemployment pool can in principle affect the aggregate surplus in the economy, and thereby
the sensitivity of vacancy posting to tax changes.
Equation (11) identifies two channels through which changes in taxes affect labor-market
tightness and thereby the aggregate job-finding rate: (i) there is a direct effect on the agespecific match surplus that operates through changes in worker’s net productivity, captured
by the term dSi () /dτ —surplus-driven tax effect; (ii) there is an equilibrium effect on the age
composition of the unemployment pool, that is captured by the term dφui /dτ —unemployment
composition-driven tax effect.
13
Next, we discuss the implications of tax changes for the aggregate unemployment rate. In
a steady state, the inflows into unemployment, st Et , equal the outflows from unemployment,
ft Ut , such that the steady-state unemployment rate equals uss
t = st / (st + ft ), where st and
ft are aggregate job-separation and job-finding rate, respectively. Hall (2005) and Shimer
(2012) argue that the measured magnitudes of the two hazard rates make uss
t a strikingly
good approximation of the actual unemployment rate in the United States, at the quarterly
frequency (and thus at lower frequency as well): that is, uss
t ' ut , for all t ≥ 0. We consider
then the steady-state approximation of the actual unemployment rate as our benchmark for
the analysis of the aggregate effects of tax changes.
The elasticity of the (steady-state approximation of) aggregate unemployment rate with
respect to the tax rate can be decomposed as:
d ln uss
d ln s
= (1 − uss )
−
dτ
{z dτ }
|
separation-driven
tax effect
d ln f
(1 − uss )
.
{z dτ }
|
hiring-driven
tax effect
(12)
Changes in taxes affect the aggregate unemployment rate via (i) changes in the aggregate jobseparation rate—separation-driven tax effect—and (ii) changes in the aggregate job-finding
rate—hiring-driven tax effect.
The elasticity of the aggregate job-separation rate can be decomposed as follows:
I d ln s
1 X e dsi
dφei
=
φi
+ si
.
dτ
s i=1
dτ
dτ
(13)
Equation (13) identifies two channels through which tax changes affect the aggregate jobseparation rate: (i) there is a direct effect on the age-specific job-separation rates, that is
captured by the term dsi /dτ ; and (ii) there is an equilibrium effect on the age composition
of the employment pool, that is captured by the term dφei /dτ .
3.5
Reduced-Form Implications
We now turn to discuss the reduced-form implications of the equilibrium.6 The framework
presented here has two appealing features. First, to completely trace the time-series behavior
of the aggregate unemployment rate, it suffices to know the values of the job-separation and
job-finding rate, jointly with a given initial level of the unemployment rate. This property
implies that the dynamic response of the unemployment rate to any realization of tax shocks
can be reconstructed from the dynamic responses of the two flow rates. Specifically, under
the steady-state approximation of the unemployment rate, one can repeatedly apply equation
(12) to get the equilibrium reduced-form response of the aggregate unemployment rate to
any realized path of tax shocks:
6
Cole and Rogerson (1999) adopt a similar approach to evaluate the ability of the textbook DMP model
to reproduce key business cycle facts.
14
d ln uss
d ln sh
d ln fh
h
= (1 − ūss )
− (1 − ūss )
,
(14)
dτ
dτ
dτ
where d ln xh = ln xh − ln x̄ indicates percent deviations of the generic variable xt from its
long-run value x̄ ≡ limt→∞ xt . Here, h ≥ 1 indexes the “horizon” of the equilibrium reducedform response. Note that, within the model, the tax rate τt can be viewed as an exogenous
driving force. Specifically, one can think of τt as a mean-reverting stochastic process, whose
long-run mean is τ̄ , such that ūss is the long-run unemployment rate associated with τ̄ . Thus,
we associate dτ to the shock we aim to identify in the data. The economic interpretation of
this shock comes then from the structure of the model.
In Section 5.3, we establish that the application of (14) to the estimated responses of st
and ft to an identified tax shock, provides a remarkably good approximation of the dynamic
response of the actual unemployment rate, at the annual frequency. This result will allow
us to quantify the relative importance of separation- and hiring-driven tax effects, which is
key for understanding the transmission mechanism of tax cuts at play.
Second, the model provides a theory-based decomposition of the aggregate unemployment
rate response to tax changes by age. Such a decomposition is additive, such that the sum of
the age-specific shares sum to the overall effect of a tax change. To derive this decomposition,
it is useful to rewrite the aggregate unemployment rate
P as the weighted average of the
unemployment rates of workers of different ages, ut = Ii=1 φni,t ui,t , where the weights are
the age-specific labor force shares φni,t ≡ Ni,t /Nt . Note that age-specific labor force, Ni,t , and
aggregate labor force, Nt , are exogenous driving forces, as such they are, by assumption,
invariant to tax changes. Thus, the equilibrium reduced-form response of the aggregate
unemployment rate can be decomposed as follows:
I
X d ln uss
d ln uss
i,h
h
φ̄ui
=
,
dτ
dτ
i=1
(15)
where φ̄ui denotes the long-run value of the age-specific unemployment share. The age-specific
share of the overall effect of a tax change is then calculated as:
shareih
≡
φ̄ui
ui,h
× u,
h
(16)
PI
i
u
ss
u
ss
such that
i=1 shareh = 1, and i,h ≡ d ln ui,h /dτ and h ≡ d ln uh /dτ are age-specific
(micro-) and aggregate (macro-) elasticities, respectively. As discussed in Section 3.3, the
model predicts that the age-specific elasticities can, in principle, largely differ depending on
the relative strength of the horizon and the net-productivity effect over the life cycle. These
age differences in unemployment elasticities are either amplified or dampened by movements
in age-specific unemployment shares.
What does a baby boom (and a baby bust) do? Demographic change affects the
aggregate unemployment rate via two channels: (i) it directly changes age-specific labor force
shares, φ̄ni ; and (ii) it indirectly affects age-specific unemployment rates. In the model, a
baby boom is easily engineered by raising the birth rate b, which feeds into higher population
growth. Higher population growth in turn tilts the age composition of the labor force
15
towards the young. To the extent that young, prime-age, and old workers feature different
unemployment responsiveness to tax changes, a baby boom can change the aggregate impact
of tax shocks. We stress that key to this argument is the assumption that, in the model, birth
rates and so age-specific labor force shares are exogenous driving forces, whose dynamics is
invariant to tax changes. In Section 6, we establish that the age composition of the labor force
can indeed be viewed as predetermined with respect to the tax shocks we identify in the data.
This result allows us to identify the direct effect of population aging on the aggregate impact
of tax changes. As for the indirect effect of aging on age-specific unemployment elasticities,
we will work under the identifying assumption that unemployment elasticities are constant
over the sample, such that we estimate unconditional age-specific unemployment elasticities.
Structural vs. reduced-form approach. In search-and-matching models, but more
broadly in any dynamic equilibrium model, the exact mapping from structural parameters to
elasticities of outcome variables with respect to policy variables is generally unknown. This
remains true in the model presented here. Specifically, micro- and macro-elasticities depend
on the features of the economic environment, functional form assumptions, and parameter
values for preferences, production, and matching technologies. Yet, several extensions of the
framework would yield the same reduced-form representation of the equilibrium, but change
the mapping from structural parameters to micro- and macro-elasticities. As an example,
introducing a choice of job-search intensity or a different bargaining protocol would require
additional functional form assumptions and parameter values about which we may have little
or no information, nonetheless it would deliver the same reduced-from expressions in (14)
and in (15).
Here, we adopt a reduced-form approach. Precisely, we estimate age-specific responses to
tax changes, by relying on the narrative approach to identification of tax shocks. Importantly,
the estimation takes in account equilibrium feedback effects and expectations about future
changes in taxes. We acknowledge that a structural approach would be required to carry
out counterfactual policy analysis. Yet, we argue that the estimates provided here can serve
as overidentifying restrictions useful in disciplining quantitative analyses of taxation and
unemployment.
4
Econometric Methodology
In this section, we briefly describe the empirical strategy and then introduce the econometric
methodology (identification, data and estimation).
4.1
Empirical Strategy
We now review some key aspects of the empirical strategy that are important for interpreting
the empirical results.
Labor-leisure margin. We focus on changes in average marginal tax rates (AMTRs) as
opposed to changes in average tax rates. Hence, we study the transmission mechanism of tax
changes that operates through incentive effects on intertemporal substitution, rather than
income effects through changes in disposable income. This approach is consistent with the
16
modelling choices underlying the theoretical framework in Section 3, where individual-level
unemployment risk is fully insured within the family, and tax changes represent unanticipated
changes in the opportunity cost of employment.
Recently, Mertens (2015) shows that real economic activity responds primarily to changes
in marginal rather than average tax rates. Specifically, changes in average marginal tax rates
on personal income lead to similar income responses regardless of changes in average tax
rates. In addition, changes in average tax rates reflect changes in marginal tax rates as well
as changes in tax brackets and tax expenditures (e.g., exemptions, deductions, and credits),
which arguably have different effects at the individual and aggregate level. Unanticipated
changes in AMTRs instead have a more straightforward interpretation, akin to a structural
disturbance in our economic theories.
As in Barro and Redlick (2011), AMTR is the average marginal personal income tax
rate, that is calculated as the weighted average of the marginal income tax rates, where the
weighting is based on a broad concept of labor income that includes wages, self-employment,
partnership income, and S-corporation income. The AMTR combines the federal individual
income tax and the social security payroll tax (FICA). Given our focus on labor income, the
constructed AMTRs apply most clearly to the labor-leisure margin.
We consider two indicators of the labor input as key outcome variables of interest: the
unemployment rate (fraction of unemployed workers in the labor force) and the participation
rate (fraction of the working-age population in the labor force) at both the aggregate and
the disaggregated level by age group. Thus, we focus on the “extensive margin” of the labor
market, that is well-known to be the leading driving force of fluctuations in hours worked
over the U.S. business cycle (see Lilien and Hall, 1986; Rogerson and Shimer, 2011). This is
consistent with our focus on unanticipated tax changes, akin to business cycle shocks.
Age heterogeneity. The literature has invariably considered aggregate macroeconomic
variables, which is the natural starting point for analyzing the economic forces that shape the
aggregate response to tax changes. This focus on aggregate time series is arguably due to the
predominance of economic theories based on the representative agent framework. We pursue
instead a disaggregated analysis by considering one dimension of heterogeneity, that can be
measured uncontroversially, age. Specifically, we study whether the effects of unanticipated
cuts in AMTRs systematically differ by age group. Understanding if and the extent to which
young, prime-age, and old respond differently to tax shocks is instrumental in understanding
why the aggregate labor market responds to tax cuts as much as it does.
Equilibrium effects of tax changes. We further stress that the effects of a change in
AMTRs on age-specific labor-market outcomes necessarily incorporate equilibrium feedback
effects. These feedback effects result from the fact that changes in marginal income tax rates
impact the labor-leisure margin of workers in all age groups, rather than just the specific
age group considered. Here, we aim at quantifying the contribution of each age group to the
aggregate labor-market response to tax shocks, akin to a decomposition exercise, after all
the equilibrium feedback effects have played out.
17
4.2
Identification, Data and Estimation
We now turn to the econometric methodology. We use structural vector autoregressions
(SVARs) to gauge the dynamic effects of changes in AMTRs. SVARs have been extensively
used in the macroeconomics literature to evaluate the effects of monetary and fiscal policy
actions as well as other aggregate shocks (e.g., technology, oil price shocks). In our context,
we associate a “tax shock” to a VAR innovation to AMTR that jointly satisfies three criteria:
(i) it is unpredictable, given current and past information; (ii) it is uncorrelated with other
structural shocks; and (iii) it is unanticipated, that is, it is not news about future policy
actions. We refer the reader to Ramey (2015) for a survey of the SVAR literature.
Identification of tax shocks. Based on Mertens and Ravn (2013, 2014), identification
of the structural tax shocks is obtained by the means of SVARs and the use of a proxy for
exogenous variation in tax rates as an external instrument (“proxy SVAR”). We use the
time series of exogenous changes in individual income tax liabilities constructed by Mertens
and Ravn (2013) as the designed proxy. These narratively identified, tax liabilities shocks
are then used as an instrument to sort out the contemporaneous causal relationship between
AMTRs and labor-market variables in an annual sample of fifty-seven years, 1950-2006.
Mertens and Ravn (2013) provide a narrative account of legislated individual income tax
liability changes in the United States, at the federal level, for the quarterly sample 1950:Q12006:Q4. Their approach builds on the seminal work by Romer and Romer (2009, 2010).
That is, changes in total tax liability are classified as exogenous based on the motivation for
the legislative action being either long-run considerations that are unrelated to the current
state of the economy (e.g., the business cycle) or inherited budget deficits. Notably, Mertens
and Ravn decompose the total tax liability changes recorded by Romer and Romer (2009) into
two main sub-components: individual income tax liabilities (including employment taxes)
and corporate income tax liabilities changes.
An additional concern is that the legislated tax changes are often implemented with a
considerable lag, which arguably hints at the presence of anticipation effects. Mertens and
Ravn (2012) indeed provide evidence of aggregate effects of legislated tax changes prior to
their implementation. We instead focus on unanticipated tax changes and so use observations
on individual income tax liability changes legislated and implemented within the year to avoid
anticipation effects, as in Mertens and Ravn (2013). As a result, the time series of individual
income tax liabilities changes used for identification consists of 13 quarterly observations.
Most of these changes were legislated as permanent. Moreover, we scale these tax liability
changes by the personal income tax base in the previous quarter. The personal income tax
base is defined as personal income, less government transfers, plus contributions for social
insurance (see Appendix A for data definitions and sources). This scaling allows us to convert
tax liability changes into the corresponding changes in the average tax rates on individual
income. Henceforth, we refer to these scaled changes in individual income tax liabilities as
“narrative shocks.”
Data. Figure 2 shows AMTRs and the resulting narrative shocks (after subtracting the
mean of non-zero observations). Panel A shows a marked upward trend in AMTRs from
1950 to the early-1980s. Specifically, the AMTR increased by nearly 6 percentage points
(from 20 to 25.9 percent) from 1950 to 1952, it then fluctuates in the 23-27 percent range
18
over the course of approximately twenty years from the early-1950s to the early-1970s. In
the 1970s, AMTRs sharply rise from 25 percent towards the post-war peak of 38 percent in
the early-1980s. Such an acceleration was primarily due to bracket creep effects induced by
rising inflation (the so-called “Great Inflation” of the 1970s). After the 1980s, the sustained
rises in the social security payroll tax have been almost entirely offset by reductions in federal
individual income taxes, which have remained in the 20-25 percent range since then.
Beyond these long-run trends, the time series of AMTRs also points to substantial yearto-year variation. The standard deviation of the first-difference in AMTRs is 1.3 percentage
points for 1950-2006. Most of this year-to-year variation is driven by changes in federal
individual income taxes. Note also that AMTRs do not include state-level taxes. However,
the amount of short-run variation in state-level marginal tax rates is rather small (see Barro
and Redlick, 2011). Hence, the full post-WWII history of U.S. federal income tax policy
includes several large increases and decreases in marginal tax rates, which arguably provides
valuable identifying variation. Yet, the vast majority of the observed legislated changes
in AMTRs result from policy actions aimed at offsetting cyclical downturns. This poses
well-known challenges for the identification of the causal effects of tax changes on economic
activity. Panel B shows three major exogenous changes in individual income tax rates. First,
the Revenue Act of 1964 enacted by president Lyndon B. Johnson on February 26, 1964,
substantially reduced statutory marginal tax rates across the board. According to the time
series of narrative shocks, this specific legislation cut average personal income tax rates by
1.4 percentage points. Second, the Revenue Act of 1978 enacted by president James Carter
on November 6, 1978, lowered individual tax rates. It also widened and reduced the number
of tax brackets such that taxpayers would not be pushed to a higher tax rate due to higher
inflation. This legislation cut average personal income tax rates by 0.8 percentage points.
Third, the Jobs and Growth Tax Relief Reconciliation Act of 2003 (JGTRRA) enacted by
president George W. Bush on May 28, 2003, reduced marginal tax rates on individual income,
capital gains, and dividends. Nearly all of the cuts in JGTRRA were temporary and set to
expire after 2010. This legislation cut average personal income tax rates by approximately 1
percentage point. Furthermore, the Tax Relief, Unemployment Insurance Reauthorization,
and Job Creation Act of 2010 enacted by president Barack Obama on December 17, 2010,
extended the tax cuts in JGTRRA for two years.
SVAR specification. First introduced by Sims (1980), SVARs have been widely used
to study the joint dynamic behavior of multiple aggregate time series by allowing for general
feedback mechanisms. Specifically, SVARs first isolate unpredictable variation in policy and
outcome variables and then sort out the contemporaneous causal relationships by imposing
identifying restrictions. Since the system allows for all possible dynamic causal effects, any
linear (or linearized) dynamic stochastic economic model can be expressed in a state space
form that yields a VAR representation for observables that are available to the econometrician
(see Fernández-Villaverde et al., 2007). In addition, SVARs also identify the expected future
path of policy variables. This is important for interpreting the estimates as expectations
about the persistence of policy actions are arguably key drivers of the behavioral response
to discretionary tax changes.
We consider a sample of annual observations for the period 1950-2006. Our estimates of
the effects of shocks to AMTRs on the U.S. labor market are based on a SVAR with five
19
variables, Yt ≡ [AMTRt , ln (PITBt ) , ln (Gt ) , Xt , ln (DEBTt )]0 , (the superscript “ 0 ” denotes
the transpose operator), where (i) AMTRt is the average marginal personal income tax rate,
as constructed by Barro and Redlick (2011); (ii) PITBt is the personal income tax base
in real per capita terms; (iii) Gt is government purchases of final goods in real per capita
terms; (iv) Xt is the specific labor-market variable of interest: aggregate unemployment and
participation rates, and job-separation and job-finding rates, in Section 5; and unemployment
and participation rates by age in Section 6; (v) DEBTt is federal debt in real per capita terms.
The baseline reduced-form VAR specification is:



  AMTR 
AMTRt
AMTRt−1
et
 ln (PITBt ) 

 ln (PITBt−1 )   ePITB




  t
 ln (Gt )  = dt + B(L)  ln (Gt−1 )  +  eG
,
(17)
t




 
X






Xt
et
Xt−1
eDEBT
ln (DEBTt )
ln (DEBTt−1 )
t
where dt contains deterministic terms and B(L) is a lag polynomial of finite order p − 1. The
lag length in the VAR is set to p = 2, based
on the Akaike information criterion (AIC). The
AMTR PITB G X DEBT 0
contains the reduced-form VAR innovations.
, et , et , et
, et
vector et ≡ et
We acknowledge that the observed trends in the age composition of the workforce might
affect the time-series behavior of the AMTRs, by changing the underlying distribution of
marginal tax rates, that is, both marginal income tax rates and individual income shares of
total income. This possibility arises because (i) age-earnings profiles are hump-shaped; (ii)
unemployment and labor force participation rates largely vary by age; and (iii) because of
retirement decisions of old workers. Changes in the age composition of the workforce indeed
affect the relative importance of these three channels. Yet, such aging-driven movements in
constructed AMTRs would seem relevant for the labor-market dynamics at the low frequency.
However, difficulties may still arise if the Romer and Romer’s narrative shocks, that we use
as an instrument, are caused by the age composition of the workforce itself. We think that
this type of concern does not apply to our exercise. According to the narrative records
of post-war tax policy (see Romer and Romer, 2009), legislated tax changes are driven by
considerations that are unrelated to population aging.
Government debt is an important variable to include in the VAR specification: given the
government’s budget constraint, any change in tax rates must eventually lead to adjustments
in other fiscal instruments, as such we deem appropriate to explicitly allow for the feedback
from debt to taxes and spending.7 Furthermore, since tax changes are often motivated
by concerns about government deficits and so debt accumulation, the inclusion of a set of
contemporaneous and past fiscal variables most likely provides relevant information to isolate
the unanticipated innovations in tax rates.
The personal income tax base (PITB) is an indicator of the business cycle. Specifically,
the contemporaneous correlation of PITB with the real GDP per capita (RGDP) is nearly
unity, for 1950-2006. The inclusion of PITB in the VAR specification in (17) is indeed
important as, otherwise, the fiscal variables might reflect business cycle dynamics. We keep
7
Burnside et al. (2004) argue that the effects of shocks to government purchases may differ depending on
the endogenous response of other fiscal instruments.
20
PITB in the baseline specification, as in Mertens and Ravn (2013) and Mertens (2015), but
we explore the sensitivity of our findings to the inclusion of RGDP.
Several other variables could enter the VAR. However, we note that omitted variables
that are orthogonal to the fiscal variables (once lagged business cycle indicators are included
in the VAR specification) would not bias the estimated effects of changes in AMTRs.
SVAR estimation. If the dynamic system in (17) generates unpredictable innovations
to the vector of observables Yt , then the vector of such reduced-form
innovations is a linear
PITB G X DEBT 0
transformation of the underlying structural shocks t ≡ AMTR
,
, t , t , t
, such
t
t
0
that: (i) E [t ] = 0, (ii) E [t t ] = Σ is a diagonal
0 matrix (we further impose Σ ≡ I,
where I is the identity matrix), and (iii) E t t−j = 0 for j 6= 0. The vector of such
structural shocks consists then of exogenous innovations in tax rates and other observables
that are uncorrelated with each other. In the SVAR literature, the structural shocks t
are treated as latent variables that are estimated based on the prediction errors of the
observables, Y
the informational content in finite distributed lags of Yt ,
t , 0 conditional
on
0
0
that is, Yt ≡ Yt−1 , . . . , Yt−p .
We posit that et = Ht , where H is a matrix of parameters that determines the impact
response of the vector of observables, Yt , to the structural shocks, t , we aim to identify.
Specifically, we are interested in identifying the parameters in the first column of H, that
is, Hi,1 , with i = 1, . . . , dim(Yt ), that determine the impact response of the observables,
. Identification of Hi,1 is achieved by imposing the same
Yt , to the shock to AMTRs, AMTR
t
identifying restrictions in Mertens and Ravn (2013, 2014) and hinges on the availability of a
proxy variable, mt , for the latent
rate, AMTR
, that jointly satisfies
t
structural
shock to the tax
AMTR
i
the identifying assumptions E mt t
6= 0 and E [mt t ] = 0 for i ≥ 2 (the superscript “
i
” denotes the i-th element of the vector). The first orthogonality condition requires the
proxy to be contemporaneously correlated with the underlying shock to the average marginal
tax rate. The second orthogonality condition requires the proxy to be contemporaneously
uncorrelated with all other structural shocks. Based on the proxy SVAR approach of Mertens
and Ravn (2013, 2014), our proxy variable is the series of narrative shocks depicted in panel
B of Figure 2. Once the contemporaneous (or impact response) parameters of interest are
identified and estimated, the effects of an unanticipated exogenous tax shock in subsequent
years can be traced out using the estimated system in (17). The resulting impulse response
functions (IRFs) measure the expected dynamic adjustment of all the endogenous variables
to the initial shock to the average marginal tax rate. IRFs allow for general feedback effects
and so convey insight into the propagation mechanism of tax changes.
5
Aggregate Effects of Tax Changes
In this section, we establish new facts on the dynamic response of the U.S. labor market to
unanticipated changes in taxes. Specifically, we consider two variables that are key indicators
of economic slack: the aggregate unemployment and participation rate.
21
5.1
Aggregate Employment Response to Tax Changes
We now turn to discuss the aggregate employment effects of tax cuts. In order to interpret
the main empirical findings of this section, we make use of the following decomposition of
the employment to population ratio:
employment
=
population
unemployment
employment+unemployment
1−
×
.
employment+unemployment
population
|
{z
} |
{z
}
1 minus the unemployment rate
participation rate
This decomposition shows that employment as a fraction of the population of 16 years and
older is equal to the employment rate (fraction of employed workers in the labor force, one
minus the unemployment rate) times the participation rate. Hence, the response of the
employment to population ratio to tax cuts is accounted by the dynamic response of either
the unemployment rate or participation rate, or both. Next we show that unemployment is
indeed quite responsive to tax cuts, whereas participation is not. These results indicate that
the aggregate response of the employment to population ratio is in fact the mirror image of
the response of the aggregate unemployment rate. Therefore, unemployment, as opposed to
participation, is the key margin for understanding the aggregate response of the U.S. labor
market to unanticipated and temporary changes in AMTRs.
It would seem reasonable to think that participation rates and unemployment rates are
both jointly determined. And that, as a result, any tax policy that affects one margin is
likely to affect the other margin as well. Yet, the empirical findings in this paper do not
support this view. This observation, in turn, suggests that a unified theory of the aggregate
labor market that incorporates employment, unemployment, and non-participation is indeed
the natural framework for understanding our findings.
Unemployment. Figure 3 shows the dynamic response of the aggregate unemployment
rate, for workers of 16 years and older, to a 1 percentage point cut in AMTR. The estimates
point to large aggregate effects of tax changes. Notably, the peak response, that occurs 3
years after the initial shock, implies that a 1 percentage point cut in AMTRs leads to an
approximately 0.7 percentage points decrease in the aggregate unemployment rate. In terms
of the U.S. labor force (employed plus unemployed) in 2007, a decline of 0.7 percentage points
in the unemployment rate amounts to nearly 1.1 million jobs, that are either preserved or
created.
We note that historically a 1 percentage point cut in AMTR is not an unusual event
as the standard deviation of changes in AMTRs is 1.3 percentage points, for the post-war
period 1950-2006. These findings point then to highly significant and quantitatively large
effects of tax changes on aggregate U.S. unemployment. The magnitude of these effects is
somewhat greater than that found by Mertens and Ravn (2013) in response to exogenous
changes in average effective tax rates on personal income. This observation suggests that
unanticipated changes in marginal tax rates arguably have larger effects than changes in
average tax rates as they operate through incentive effects on intertemporal substitution.
Furthermore, we find that tax cuts are long-lasting. The estimated dynamic response of
22
the AMTR to the tax shock is highly persistent. Specifically, AMTRs remain persistently
below average up to 5 years after the shock. This high persistence in the adjustment dynamics
of the AMTRs contrasts with the relatively fast mean reversion observed for average personal
income tax rates, as estimated by Mertens and Ravn (2013), but it is in line with Mertens
and Ravn (2013), where they estimate the response of real GDP per capita to tax cuts using
the same constructed AMTRs.
Participation. Figure 4 shows the dynamic response of the aggregate participation rate,
for workers of 16 years and older, to an equally-sized 1 percentage point cut in AMTRs. In
contrast with the results for the unemployment rate, the response of the participation rate
is both statistically and economically insignificant. This finding is, perhaps, not surprising
since the participation rate has displayed pronounced low-frequency movements over the
post-war period, that are arguably driven by long-run demographic trends, which are hardly
affected by the magnitude of the shocks observed in the period 1950-2006.
However, one cannot a priori dismiss the hypothesis that larger shocks to AMTRs could
generate a substantially different response in labor force participation. In that scenario, the
main concern would be whether linear SVARs remain reliable in recovering the true dynamic
response to such large shocks.
5.2
Robustness
We now turn to discuss a number of robustness checks. Figure D.3 and D.4, in Appendix D,
show that the findings for aggregate unemployment and participation rates are robust to the
inclusion of the real GDP per capita (RGDP) in the SVAR system. Specifically, we estimate
a specification of the VAR where the personal income tax base (PITB) is replaced by RGDP.
The estimated responses with RGDP reproduce both the magnitude and the shape of the
responses with PITB. Thus, they confirm the finding that the aggregate unemployment rate
is responsive to tax cuts, whereas the participation rate is not.
As an additional robustness check, we also consider how the results change when an
indicator of monetary policy and/or credit market conditions is included in the SVAR system.
Specifically, we consider the 3-month treasury bill (T-Bill), and then estimate a VAR with
six variables. Note that the contemporaneous correlation between the 3-month T-Bill and
the federal funds effective rate (FFER) is 0.99, at the annual frequency, for 1955-2006. As
Figure D.5 and D.6 show, the results are very similar to our baseline.
5.3
Inspecting the Propagation Mechanism of Tax Changes
What is the propagation mechanism of tax shocks? To answer this question, we next focus
on the reduced-form implications of the model presented in Section 3. In the model, changes
in aggregate unemployment are driven by the number of employed workers who lost jobs in
the previous period minus the number of unemployed workers who found jobs:
Ut+1 − Ut = st Et − ft Ut ,
(18)
where the variables Ut and Et are the number of unemployed and employed, respectively,
and st and ft are job-separation and job-finding rates, respectively. Hence, equation (18)
23
identifies the job-separation rate (the “separation margin”) and the job-finding rate (the
“hiring margin”) as key driving forces of changes in the aggregate unemployment rate, which
are then also responsible for the response of the aggregate unemployment rate to tax shocks.
In the model of Section 3, and more generally in the extended class of DMP models of
endogenous job separations (see Mortensen and Pissarides, 1994; Ramey et al., 2000; Fujita
and Ramey, 2012), the flow rates st and ft are jointly determined as equilibrium objects.
Yet, we can measure st and ft from actual data under arguably minimal assumptions. Next,
we aim at decomposing the relative contribution of separation- versus hiring-driven effects
to the aggregate unemployment rate response to tax cuts.
Job-separation and job-finding rate. We now turn to the responses of the two flow
rates: st and ft . Based on publicly available data on unemployment and employment from
the Current Population Survey (CPS), time series for st and ft are easily constructed (see
Shimer, 2012).8 Figure 5 shows the responses of such constructed series to a 1 percentage
point cut in AMTRs. In panel A, the job-separation rate drops by −0.14 percentage points
at the trough of the response; in panel B, the job-finding rate raises by approximately 3
percentage points at the peak of the response.
Separation- vs. hiring-driven tax effects. To quantify the relative contribution of
separation and hiring margin, we develop a decomposition of the unemployment rate response
to tax shocks that compares the contribution of job-separation and job-finding rates on an
equal footing. To this aim, we proceed in two steps.
First, we rely on a theoretical approximation argument, that hinges on the irrelevance of
turnover (or adjustment) dynamics for empirically relevant values of st and ft . In the absence
of any type of disturbance, the unemployment rate converges to the theoretical steady-state
value uss
t = st /(st + ft ), which only depends on contemporaneous values of job-separation
and job-finding rates. Hall (2005) and Shimer (2012) show that, at the quarterly frequency,
such steady-state approximation generates values that are nearly indistinguishable from the
actual unemployment rates observed in U.S. data: that is, uss
t ' ut at the quarterly frequency
(and thus at lower frequencies as well). This happens because, in the data, st + ft is typically
close to 0.5 on a monthly basis, such that the half-life of a deviation from the steady-state
unemployment rate is approximately one month. In our sample, monthly job-separation and
job-finding rates are 3.4 and 45.3 percent on average, respectively.9
Second, based on Elsby et al. (2009), Fujita and Ramey (2009), and Pissarides (2009), we
decompose the changes in the (steady-state approximation of) actual unemployment rates
into the contribution due to changes in the job-separation rate and the contribution due to
changes in the job-finding rate:
ss
ss
duss
t ≈ ū (1 − ū )
dft
dst
− ūss (1 − ūss ) ¯ ,
s̄
f
(19)
where dxt = xt − x̄ indicates deviations of the generic variable xt from its sample average
8
Job-separation and job-finding rates are constructed under two assumptions: (1) workers do not transit
in and out of the labor force; and (2) workers are homogeneous with respect to job-finding and job-separation
probabilities. We refer the reader to Shimer (2012) for further details.
9
Figure D.7 in Appendix D confirms that uss
t is indeed a strikingly good approximation of the actual U.S.
unemployment rate for the post-war period 1950-2006.
24
x̄, and ūss ≡ s̄/ s̄ + f¯ , where s̄ and f¯ are sample averages of job-separation and jobfinding rates, respectively. Equation (19) provides an additive decomposition in which the
contributions of job-separation and job-finding rates are comparable on an equal footing
with respect to their impact on the observed changes in the unemployment rate. Note that
equation (19) holds as equality only for infinitesimal changes. For discrete changes, instead,
it is only an approximation. However, Fujita and Ramey (2009)’s regression-based estimation
of the decomposition in (19) verifies that unconditionally it works well for discrete changes.
We next show that such decomposition works well also for the conditional response of the
U.S. unemployment rate to narrative tax shocks. Specifically, the counterfactual response
implied by equation (19) is
b
b
c ss ≡ ūss (1 − ūss ) dsh − ūss (1 − ūss ) df h ,
du
h
s̄ }
f¯
|
{z
|
{z
}
jsr
jfr
c
c
du
du
h
h
(20)
b h and df
b h are the estimated responses
where h is the number of years after the shock, and ds
of job-separation and job-finding rates, respectively, as shown in Figure 5.
In Figure 6, panel A shows that the counterfactual response of the unemployment rate
c ss , (dashed line with diamonds), is remarkably
under the steady-state approximation, du
h
close to the estimated response of the actual U.S. unemployment rate (full line with circles).
Thus, at the annual frequency, turnover dynamics is in fact negligible for understanding the
aggregate response of the actual U.S. unemployment rate to narrative tax shocks. This novel
empirical finding bears important implications for theoretical modelling. To the extent that
we are interested in the year-to-year dynamic response to “small” tax shocks, theories based
on the flow approach to labor markets can safely abstract from adjustment dynamics as
steady-state comparisons capture the bulk of the observed aggregate effects of tax cuts.
This irrelevance of turnover dynamics for studying the effects of tax cuts can be explained
as follows. First, the U.S. labor market is extremely fluid; month-to-month worker flows are
enormous. In the United States, over the period 1996-2003, approximately 15 million workers
changed their labor market status from one month to the next (see Fallick and Fleischman,
2004; Davis et al., 2006). These large workers’ flows translate then into the relatively high,
monthly job-separation and job-finding rates observed at the aggregate level, which are in
turn responsible for the extremely fast adjustment dynamics featured by the aggregate U.S.
unemployment rate. Second, the tax shocks we identified in the data are, on the contrary,
extremely persistent. As shown in panel A of Figure 3, the initial 1 percentage point cut in
the AMTR persists for nearly 5 years. In a nutshell, unanticipated changes in AMTRs can
be viewed as nearly permanent from the perspective of the average unemployed worker.
In Table 1, we use the decomposition in (20) to quantify the relative contribution of jobseparation and job-finding rates to the response of the actual U.S. unemployment rate to tax
shocks. The results show that the job-separation rate accounts for approximately 60 percent
of the overall unemployment rate response in the first year after the shock, whereas the split
between job-separation and job-finding rates is inverted from the second year after the shock
onward. These results indicate that to fully account for the effects of tax shocks, a successful
25
theory has to reproduce the joint dynamic behavior of job-separation and job-finding rates.
We stress that these novel empirical findings complement those in previous studies on the
relative importance of job separation and hiring for unemployment fluctuations over the U.S.
business cycle (see Davis et al., 2006; Fujita and Ramey, 2009; Elsby et al., 2009). In this
regard, we emphasize that the results in Table 1 derive from the estimated response of the
unemployment rate conditional on a specific, narratively identified shock to AMTRs. Since
shocks to AMTRs are identified as orthogonal to business cycle shocks, the decomposition in
Table 1 provides additional, independent evidence on the propagation mechanism of shocks
at work in the U.S. labor market.
6
Age-Specific Effects of Tax Changes
In this section, we detail the demographics of the U.S. labor market response to tax shocks.
To this aim, we study if and the extent to which the dynamic responses of unemployment
and participation rates to tax cuts vary by age. In doing so, it is imperative to keep in mind
that the effects of a shock to AMTRs on age-specific labor-market outcomes incorporate
equilibrium feedback effects that result from the fact that changes in average marginal tax
rates impact workers in all age groups, rather than just the specific age group considered.
Ultimately, we are interested in decomposing the response of the aggregate unemployment
rate into the relative contribution of each age group, after all the equilibrium feedback effects
have played out. This will allow us to quantify the relative importance of the young, primeage, and old in shaping the aggregate response of the U.S. unemployment rate to tax cuts.
6.1
Age-Specific Unemployment Response to Tax Changes
We now turn to analyze the role of age-specific unemployment rates and labor force shares
for the response of the aggregate unemployment rate. Towards this aim, we consider the
following decomposition of the aggregate unemployment rate:
unemployment X
=
labor force
a∈A
labor forcea
|labor{zforce}
age-specific
labor force share
×
unemploymenta
,
labor forcea
|
{z
}
age-specific
unemployment rate
(21)
where “ a ” indicates an age group in the list A = {16-19, 20-24, 25-34, 35-44, 45-54, 55+},
and the labor force is defined as employed plus unemployed workers of 16 years and older, in
accord with the definition used by the Bureau of Labor Statistics (BLS). The decomposition
in (21) shows that the response of the aggregate unemployment rate to tax cuts is accounted
by the response of either age-specific labor force shares or age-specific unemployment rates,
or both. Next, we show that age-specific unemployment rates are indeed responsive to tax
cuts, whereas age-specific labor force shares are not.
Labor force shares vs. unemployment rates by age. To disentangle the relative
contribution of age-specific labor force shares from that of age-specific unemployment rates,
we construct a counterfactual series of the aggregate unemployment rate, uFLFS
, in which
t
26
age-specific labor force shares are fixed at their sample averages, φ̄LF
a , for 1950-2006, whereas
we let age-specific unemployment rates, ua,t , vary over time:
≡
uFLFS
t
X
φ̄LF
a × ua,t ,
(22)
a∈A
P
LF
a∈A φ̄a
where
= 1. We then re-estimate the proxy SVAR by replacing the actual U.S.
unemployment rate with the counterfactual unemployment rate with fixed labor force shares,
uFLFS
, in (22). In Figure 6, panel B shows that the impulse response to the AMTR shock
t
of the counterfactual unemployment rate (dashed line with diamonds) is nearly indistinguishable from the impulse response of the actual U.S. unemployment rate (full line with
circles). Thus, we conclude that age-specific unemployment rates are indeed quite responsive
to tax cuts, whereas age-specific labor force shares are not. This observation corroborates
the empirical finding we discussed earlier in this paper that labor force participation, at the
aggregate level, seems to be unresponsive to the exogenous changes in AMTRs we identified
in the data.
As argued in Section 2, labor force shares by age display marked low-frequency movements
in the post-war period. However, such low-frequency movements are due to the underlying
demographic trends that pervade the entire U.S. economy, which are unlikely to be affected
by temporary changes in marginal tax rates. The composition of the workforce is largely
pre-determined by fertility decisions made prior to the observed changes in tax rates.
These findings are important for the scope of this paper as they provide an empiricallyvalidated restriction, akin to an orthogonality condition, that will enable us to quantify
the role of an aging labor force in shaping the aggregate unemployment response tax cuts.
Specifically, we can view the impulse response function (IRF) of the aggregate unemployment
rate as a weighted average of the IRFs of the age-specific unemployment rates, where the
weights are (sample averages of) the age-specific labor force shares:
X
ch ≈
c
du
φ̄LF
(23)
a × dua,h ,
a∈A
c h and du
c a,h indicate the IRFs of the aggregate unemployment rate and age-specific
where du
unemployment rates, respectively, and h is the number of years after the initial shock to the
c FLFS ,
AMTR. Note that the impulse response of the counterfactual unemployment rate, du
h
shown in panel B of Figure 6, guarantees that the right-hand side of (23) is indeed a strikingly
c h . With the
good approximation of the impulse response of the actual unemployment rate, du
IRF’s approximation in (23) at hand, we can decompose the contribution of each age group
to the aggregate response to tax cuts. We note that if there was no heterogeneity
in the IRFs
P
across age groups, then the labor force shares would become irrelevant as a∈A φ̄LF
a = 1, by
construction. Therefore, changes in the age composition of the labor force affect the response
of the aggregate unemployment rate insofar as the unemployment rate responses to tax cuts
differ by age.
Unemployment rates by age. We next establish that the response to tax cuts of the
aggregate U.S. unemployment rate in Figure 3 in fact masks substantial heterogeneity by
27
age. Specifically, the unemployment rate response of the young is nearly twice as large as
that of the prime-age and old workers in the labor force. This age-specific heterogeneity in
the responsiveness to tax cuts is the channel through which shifts in the age composition of
the U.S. labor force affect the response of the aggregate unemployment rate to tax cuts.
Figure 7 shows the dynamic responses of age-specific unemployment rates to an equallysized 1 percentage point cut in the AMTRs. The estimates display stark differences in the
responses of the young (20-34 age group), prime-age (35-54 age group), and old workers (55+
age group). In panels A and B, the unemployment rates for the 16-19 and 20-24 years old
fall by 1 percentage point at the peak of the response, that occurs 3 years after the initial
shock. The magnitude of these responses is somewhat larger than the 0.7 percentage points
fall of the aggregate U.S. unemployment rate in response to a shock of the same size. For the
25-34 years old, instead, the unemployment rate falls by nearly 0.8 percentage points, which
is then more in line with the response of the aggregate unemployment rate of 0.7 percentage
points. However, the responses of the unemployment rates for the age groups 35-44, 45-54,
and 55 years and older, are nearly half as large as those of the 16-19 and 20-24 years old.
Figure D.1 and D.2, in Appendix D, show that the differences in unemployment responses
between the young and the prime-age and old are statistically significant. This heterogeneity
is the prerequisite for a quantitatively important role of age composition in determining the
aggregate unemployment response to tax cuts.
6.2
Age-Specific Participation Response to Tax Changes
We now turn to analyze the role of age-specific participation rates and population shares for
the response of the aggregate participation rate. In Section 5, we argued that participation
does not respond to tax shocks. Yet, this lack of responsiveness may mask heterogeneity
by age. To address this concern, we consider the following decomposition of the aggregate
participation rate:
populationa
labor forcea
,
×
population
populationa
|
{z
}
|
{z
}
a∈A
age-specific
age-specific
population share participation rate
labor force X
=
population
(24)
where “ a ” indicates an age group in the list A = {16-19, 20-24, 25-34, 35-44, 45-54, 55+}.
The decomposition in (24) shows that the response of the aggregate participation rate to tax
cuts is accounted by the response of either age-specific population shares or participation
rates, or both. Next, we show that age-specific population shares are in fact unimportant in
accounting for the response of the aggregate participation rate. And that the participation
rates for all age groups, but the old (55 years of age and older), are indeed unresponsive
to the identified shocks to AMTRs. Notably, such age profile of participation responses is
consistent with the theory presented in Section 3, and more generally with standard life-cycle
models of labor supply (see Rogerson and Wallenius, 2009, 2013).
Population shares vs. participation rates by age. To disentangle the relative
contribution of age-specific population shares from that of age-specific participation rates,
28
we construct a counterfactual series of the aggregate participation rate, nFPS
, in which aget
P
specific population shares are fixed at their sample averages, φ̄a , for 1950-2006, whereas we
let age-specific participation rates, na,t , vary over time:
nFPS
≡
t
X
φ̄Pa × na,t ,
(25)
a∈A
where a∈A φ̄Pa = 1. We then re-estimate the proxy SVAR by replacing the actual U.S.
participation rate with the counterfactual participation rate with fixed population shares,
, in (25). In Figure 6, panel C shows that the impulse response of the counterfactual
nFPS
t
participation rate (dashed line with diamonds) is nearly indistinguishable from the impulse
response of the actual participation rate (full line with circles), as shown in Figure 4. Thus,
we conclude that the age-specific population shares are indeed irrelevant for the response of
the aggregate participation rate to tax shocks.
Participation rates by age. We next address whether the lack of economically and
statistically significant effects of tax cuts on the aggregate participation rate, as documented
in Figure 4, masks heterogeneity by age group. Figure 8 shows the dynamic responses of
the age-specific participation rates to an equally-sized 1 percentage point cut in AMTR. The
responses of young and prime-age (16-19, 20-24, 25-34, 35-44, and 45-54) are statistically and
economically insignificant. On the contrary, tax cuts have persistent positive effects on the
participation rate of the 55 years of age and older. Specifically, the participation rate of the
old raises by 0.5 percentage points. In terms of the U.S. population in the 55+ age group in
2007, an increase of 0.5 percentage points in the participation rate amounts to nearly 343.8
thousand workers reentering the labor force.
P
7
Demographic Change and Tax Policy
In this section, we quantify the role of demographic change for the effects of tax cuts on
the aggregate U.S. unemployment rate. We adopt a quantitative accounting approach. The
approach consists of two steps. First, we quantify the relative contribution of each age group
to the average response of the aggregate unemployment rate to tax cuts. Second, we quantify
by how much the aggregate impact of a tax cut changes when we account for the shifts in
the age composition of the labor force observed in the data.
7.1
Quantifying the Role of Age Composition
We now turn to quantify the contribution of each age group to the response of the aggregate
U.S. unemployment rate to tax shocks. Specifically, we decompose the impulse response of
the unemployment rate of 16 years and older, as shown in Figure 3, into additive shares that
measure the relative contribution of each group to the aggregate unemployment response.
Next, we show that the young (20-34 years old) and the prime-age (35-54 years old) account
for more than 80 percent of the aggregate unemployment response to tax cuts.
Contribution to the P
aggregate response by age. We use the IRF’s approximation
65+
LF
LF
ch ≈
c
in (23) such that du
are the sample averages of the
a=16 φ̄a × dua,h , where φ̄a
29
c a,h are the estimated IRFs of actual
age-specific labor force shares, for 1950-2006, and du
age-specific unemployment rates, as shown in Figure 7. Hence, each age group accounts for
shareah of the response of the aggregate unemployment rate:
shareah ≡ φ̄LF
a ×
c a,h
du
,
ch
du
(26)
P
a
with 65+
a=16 shareh = 1, at any horizon h = 1, . . . , 5 (years after the shock). As defined in
(26), the share of each group in the aggregate unemployment response is calculated as the
ratio of impulse responses (IRF-ratio), weighted by age-specific labor force shares.
Table 2 shows the results of the age decomposition. Two extended age groups account for
the bulk of the aggregate unemployment response to tax cuts. Specifically, the age groups
20-34 and 35-54 account for nearly 88 percent of the aggregate response in the first year after
the shock, and for nearly 80 percent of the response from the second year after the shock
onward. The age groups 16-19 and 55-64, combined, account for most of the remaining share
of the response of the aggregate unemployment rate, whereas the age group of the 65 years
and older is in fact negligible for nearly all years after the initial tax shock.
Note that the age-specific share of the aggregate unemployment response, as defined in
(26), depends on both the average of the age-specific labor force share, φ̄LF
a , and the IRFc
c
ratio, dua,h /duh . In principle, then, age differences along both dimensions jointly determine
the quantitative importance of the different age groups. How much of the age differences in
shareah , as shown in Table 2, is due to the differences in labor force shares by age as opposed
to the ratio of impulse responses? To answer this question, we construct a counterfactual
age-specific share, sharea,FLFS
, in which average labor force shares are fixed at the same
h
value across age groups:
c a,h
du
1
×
,
(27)
ch
na
du
where na = 5 is the number of age groups considered. In Appendix D.1, Table D.1 reports the
results for the counterfactual decomposition in (27). Note that any age-specific heterogeneity
in sharea,FLFS
now comes exclusively from the heterogeneity in the unemployment rate
h
responses to tax cuts across different age groups. Hence, the difference between the agespecific shares with and without fixed labor force shares can be entirely attributed to the
age composition of the labor force. The results point in fact to a quantitatively important
role of age composition. Specifically, the two extended age groups 20-34 and 35-54 now
account for 76 percent of the aggregate unemployment response in the first year after the
shock, and for approximately 63 percent from the second year after the shock onward. These
figures are considerably smaller than those in Table 2. Age composition alone is responsible
for a decrease of 12 percentage points for the first year after the shock, and 17 percentage
points for the second year after the shock onward. Most of these differences are due to the
changes in the relative shares of the 35-54 years old. This is because the 35-54 years old are
relatively unresponsive to tax cuts (see panel D and E, in Figure 7), but they represent on
average 42 percent of the U.S. labor force, for the period 1950-2006. By contrast, the shares
of the 16-19 years old raise by 7 percentage points for the first year after the shock and by
sharea,FLFS
≡
h
30
approximately 11 percentage points from the second year after the shock onward. This is
because the 16-19 years old are responsive to tax cuts (see panel A, in Figure 7), but they
represent on average only 7 percent of the labor force. See Table A.1, in Appendix A, for
average labor force shares by age group, for the period 1950-2006.
Unemployment elasticities by age. We now investigate the extent to which the agespecific heterogeneity in the unemployment rate responsiveness to tax shocks, as measured
by the IRF-ratio in (26), can be attributed to differences in unemployment elasticities across
age groups as opposed to differences in average unemployment rates. To this aim, we use a
slightly modified version of the expression in (26):
shareah ≡ φ̄LF
a ×
ūa ˆua,h
× u ,
ū
ˆh
(28)
where ūa and ū are averages of age-specific and aggregate unemployment rates, respectively,
c h /ū are estimated elasticities of age-specific and aggregate
c a,h /ūa and ˆu ≡ du
and ˆua,h ≡ du
h
unemployment rates, respectively. As re-written in (28), the share of each group equals the
ratio of the age-specific to the aggregate unemployment elasticity (elasticity-ratio), weighted
by the product of (i) the age-specific labor force share and (ii) the ratio of the age-specific
to the aggregate average unemployment rate (UNR-ratio).
In Appendix A, Table A.1 shows UNR-ratios by age group, for the period 1950-2006.
Unemployment rates decrease monotonically with age. This age profile of unemployment
rates is a stylized fact that theories of worker turnover and life-cycle unemployment have
successfully explained (see Jovanovic, 1979; Chéron et al., 2013; Esteban-Pretel and Fujimoto, 2014; Papageorgiou, 2014; Gervais et al., 2014; Menzio et al., 2016). Note that these
differences are quantitatively large. Specifically, the average unemployment rate of 16 years
and older is 5.64 percent. The unemployment rate of the 16-19 years old is 15.63 percent,
that is nearly 2.8 times higher than the average of the aggregate unemployment rate, whereas
the unemployment rate of the 20-24 years old is 9 percent, that is 1.6 times higher than that
of the 16 years and older. For the 25-34 years old, instead, the unemployment rate averages
5.29 percent, whereas the unemployment rates for the remaining age groups (35-44, 45-54,
55-64, and 65 years and older) are nearly 30 percent lower than the average of the aggregate
unemployment rate.
Next, we establish new facts on the life-cycle profile of unemployment elasticities to tax
shocks. We re-estimate the proxy SVAR using the unemployment rate in logs, separately
for each each group, such that we recover unemployment elasticities by age. Table 3 reports
the unemployment elasticities estimates by age group for the first year (“impact elasticity”)
and the second year after the shock (“lagged elasticity”). The estimates provide evidence
of an hump-shaped life-cycle profile of unemployment elasticities. The least elastic are the
16-19 years old and the 65 years and older, whereas the most elastic are the 20-34 years
old (the “young”) and the 35-54 years old (the “prime-age”). Same life-cycle patterns hold
for unemployment elasticities up to five years after the initial tax shock (see Table D.2, in
Appendix D).
These age-specific differences in unemployment elasticities are new life-cycle facts that
can be instrumental in disciplining theories of life-cycle unemployment, as they provide
overidentifying restrictions for quantitative analysis of taxes and unemployment. We further
31
stress that these empirical results also complement the well-known observation that business
cycle volatility in labor-market outcomes declines with age (see Clark and Summers, 1981;
Gomme et al., 2005; Jaimovich and Siu, 2009; Jaimovich et al., 2013).
7.2
Quantifying the Role of Demographic Change
We now turn to evaluate the quantitative implications of the aging of the baby boomers for
the propagation of tax cuts in the United States. Our approach resembles that in Shimer
(1999) and Jaimovich and Siu (2009), where the authors quantify the role of age composition
of the U.S. workforce for low-frequency movements in the aggregate unemployment rate and
cyclical volatility in hours worked, respectively. The results in Shimer (1999) indicate that
the observed trends in the age composition of the labor force have an impact on the level of
the aggregate U.S. unemployment rate. The entry of the baby boomers in the labor force
in the late-1970s, and their aging, accounts for a substantial fraction of the rise and fall in
unemployment rates observed in past 50 years. Jaimovich and Siu (2009) argue that the age
composition of the labor force has a causal impact on the volatility of hours worked over
the business cycle. Since young workers feature less volatile hours worked than prime-age,
the aging of the labor force accounts for a significant fraction of the decrease in business
cycle volatility observed since the mid-1980s in the United States over the so-called Great
Moderation. Next, we show that the demographic change experienced by the U.S. labor
market also has quantitatively important implications for the effectiveness of tax policy.
We find that the aging of the baby boomers considerably reduces the effects of tax cuts on
aggregate unemployment.
To establish this result, we implement a quantitative accounting exercise. Specifically,
we re-construct the U.S. history of aggregate unemployment responses to a tax cut, by using
age-specific labor force shares and unemployment rates observed at a specific point in time,
and unemployment elasticities estimated over the entire sample period 1950-2006, as shown
in Table 3. The implied unemployment responses provide a quantitative accounting of how
the trends in the age composition of the labor force affect the response of the aggregate U.S.
unemployment rate to tax cuts.
We construct an aggregate unemployment response to tax cuts, that is adjusted for age
composition (AC-adj), as follows:
c AC-adj ≡
du
h,t
65+
X
φ̄LF
ˆua,h ,
a,t × ūa,t × (29)
a=16
c AC ,
where the time subscripts indicate that the AC-adj aggregate unemployment response, du
h,t
is allowed to vary over time based on the observed demographic trends, that drive changes
in age-specific labor force shares. We use equation (29) to generate unemployment responses
to an equally-sized tax cut, at 5-year intervals, from 1950 to 2005, that are adjusted for the
movements in the age composition of the labor force.
Is this exercise informative of the transmission mechanisms of tax changes? Note that
this exercise assumes that the age-specific unemployment elasticities are independent of the
age composition of the labor force. Specifically, it assumes the absence of indirect effects of a
32
changing age composition of the labor force on the aggregate unemployment response, that
operate through changes in age-specific unemployment elasticities. The responses in (29)
tell then what would have been the response of the aggregate unemployment rate to a tax
cut, if the age composition was that of a specific year in the sample. Hence, to the extent
that the age composition of the U.S. labor force has dramatically changed over the post-war
period, one would expect the aggregate unemployment response to tax cuts to greatly differ
over time. Next, we show that this is indeed the case.
Figure 9 shows the AC-adjusted responses of the aggregate U.S. unemployment rate, as
constructed in (29), for the first year (“impact response”) and the second year after the
shock (“lagged response”). Same patterns hold up to five years after the shock. Note that,
in Figure 9, larger negative values indicate that an equally-sized 1 percentage point cut in
AMTRs reduces aggregate unemployment by more. The results indicate large changes in the
response of the aggregate unemployment rate to tax cuts. The largest impact and lagged
responses occur in the mid-1970s. The peak impact response to a 1 percentage point cut in
AMTRs is 0.42 percentage points, that is nearly 50 percent larger than the response of 0.27
percentage points in 2005, whereas the peak lagged response is 0.81 percentage points, that
is nearly 56 percent higher than the 0.52 percentage points in 2005. Overall, the magnitude
of the counterfactual responses is nearly constant from the mid-1950s to the early-1970s, it
markedly increases (in absolute value) during the mid-1970s, and starts to steadily decrease
since then.
Figure 10 shows that these changes are indeed associated with the marked shifts in the age
composition of the labor force. Specifically, the aggregate unemployment responses closely
track the share of the 35-54 years old in the labor force. The entry of the baby boomers
in the labor force in the 1970s caused a fall in the share of the 35-54 years old of nearly 10
percentage points. At the same time, the share of the 20-34 years old increased by nearly the
same amount. Since the young are more responsive to tax cuts than prime-age workers, the
responsiveness of the aggregate unemployment rate dramatically increased over the period.
However, as the aging of the baby boomers unfolds, the effects of tax cuts on the aggregate
unemployment rate are reduced to a level comparable to that of early-1950s.
8
Conclusion
In this paper, we investigate the consequences of population aging for the transmission of
tax changes to the aggregate labor market in the United States. After isolating exogenous
variation in average marginal tax rates in SVARs, using a narrative identification approach,
we document that the responsiveness of unemployment rates to tax changes largely varies
across age groups: the unemployment rate response of the young is nearly twice as large as
that of prime-age and old. This heterogeneity is the channel through which shifts in the age
composition of the labor force impact the response of the aggregate U.S. unemployment rate
to tax changes. We find that the aging of the baby boomers considerably reduces the effects
of tax cuts on aggregate unemployment.
These results indicate that the age composition of the workforce is a potentially important
transmission channel of tax changes, that deserves further attention in fiscal policy analysis.
33
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37
Tables and Figures
Table 1: Separation- versus Hiring-driven Tax Effects
h (years after shock):
1
2
3
4
5
c jsr as % of du
c ss
du
h
h
0.58
0.40
0.36
0.35
0.42
c jfr as % of du
c ss
du
h
h
0.42
0.60
0.65
0.58
Notes: See equation (20) for definitions of
ss
c ,
du
h
0.64
jsr
c ,
du
h
and
jfr
c .
du
h
Table 2: Shares of Aggregate Unemployment Response by Age
h (years after shock):
1
2
3
4
5
share16-19
h
5.54
11.19
11.56
10.69
7.62
share20-34
h
51.96
46.53
44.46
43.33
42.96
share35-54
h
35.80
33.60
34.04
35.27
38.66
share55-64
h
6.16
7.42
8.53
9.42
10.20
share65+
h
0.54
1.26
1.41
1.29
0.56
shareah .
Notes: See equation (26) for the definition of
Age-specific shares are reported
P65+
a
in percent such that a=16 shareh = 100 (column-wise summation).
38
Table 3: Unemployment Elasticities by Age
Age group:
16+
16-19
20-24
25-34
35-44
45-54
55-64
65+
Impact elas.
1.52
0.25
1.27
2.31
1.99
2.11
1.36
0.15
1
0.16
0.84
1.52
1.31
1.39
0.89
0.10
2.92
1.05
3.11
3.47
3.56
3.67
3.23
1.58
1
0.36
1.07
1.19
1.22
1.26
1.11
0.54
Impact elas.-ratio
Lagged elas.
Lagged elas.-ratio
Notes: “Impact elas.” refers to the unemployment rate elasticity of a specific age group at
horizon h = 1 (one year after the shock); “Lagged elas.” refers to the unemployment rate
elasticity of a specific age group at horizon h = 2 (two years after the shock). Each of the
impact and lagged elasticities estimates are based on a separate SVAR system that includes
the log of the unemployment rate of a specific age group and a common set of regressors,
as specified in (17), for 1950-2006. Elasticities are with respect to the average marginal tax
rate (AMTR), and are reported in percent. “Impact elas.-ratio” refers to the unemployment
rate impact elasticity by age, relative that of 16+ years old. “Lagged elas.-ratio” refers to the
unemployment rate lagged elasticity by age, relative that of 16+ years old. See Table D.2, in
Appendix D, for elasticities estimates up to 5 years after the shock.
39
B. Labor Force Shares by Age
A. Average Age of Labor Force
42
55
25
20−34
35−54
55−64
41.5
41
50
20
45
39.5
39
15
Percent
40
Percent
Average age
40.5
40
38.5
10
38
35
37.5
37
1950 1960 1970 1980 1990 2000 2010
Year
30
5
1950 1960 1970 1980 1990 2000 2010
Year
Figure 1: Trends in the Age Composition of the U.S. Labor Force, 1950-2015
Notes: Panel A shows the average age of the U.S. labor force (employed plusunemployed
workers of 20-64
P
a+a
LF
LF
years old). The average age of the labor force is calculated as ā ≡ a∈A 2 φa , where a and a are
respectively lower and upper bounds of the age group a ∈ A, with A = {20-24, 25-34, 35-44, 45-54, 55-64},
and φLF
a is the age-specific labor force share (the ratio of the labor force in the age group a to total labor
force). Panel B shows the labor force shares by three age groups: (i) full line with circles (left axis) shows
LF
LF
LF
φLF
20-24 + φ25-34 ; (ii) dashed line with squares (left axis) shows φ35-44 + φ45-54 ; and (iii) dashed-dotted line
LF
with diamonds (right axis) shows φ55-64 . See Appendix A for data definitions and sources.
40
A. Average Marginal Tax Rate
40
Percent
35
30
25
20
1950
1955
1960
1965
1970
1975
1980
Year
1985
1990
1995
2000
2005
1990
1995
2000
2005
B. Narrative Shocks
1
Percent
0.5
0
−0.5
−1
−1.5
1950
1955
1960
1965
1970
1975
1980
Year
1985
Figure 2: Average Marginal Tax Rates and Narrative Shocks, 1950-2006
Notes: Panel A shows the average marginal tax rate (AMTR) constructed by Barro and Redlick (2011).
AMTR is the average marginal personal income tax rate weighted by a concept of income that is close to
labor income: wages, self-employment, partnership income, and S-corporation income. AMTR consists
of two components: federal individual income tax and social security payroll tax (FICA). Panel B shows
the exogenous changes in individual income tax liabilities divided by the personal income tax base in
the previous quarter (personal income less government transfers plus contributions for government social
insurance), as constructed by Mertens and Ravn (2013). See Appendix A for data definitions and sources.
41
B. Unemployment Rate 16+
0.5
0
0
Percentage points
Percentage points
A. Average Marginal Tax Rate
0.5
−0.5
−1
−1.5
−0.5
−1
1
2
3
4
Years after shock
−1.5
5
1
2
3
4
Years after shock
5
Figure 3: Unemployment Rate Response to a Tax Cut
Notes: The figure shows the response to a 1 percentage point cut in the average marginal personal income
tax rate (AMTR). Full lines with circles are point estimates; dashed lines are 95 percent confidence bands.
See Appendix A for data definitions and sources.
42
A. Average Marginal Tax Rate
B. Participation Rate 16+
1
0.5
0.5
Percentage points
Percentage points
0
0
−0.5
−0.5
−1
−1
−1.5
1
2
3
4
Years after shock
−1.5
5
1
2
3
4
Years after shock
5
Figure 4: Participation Rate Response to a Tax Cut
Notes: The figure shows the response to a 1 percentage point cut in the average marginal personal income
tax rate (AMTR). Full lines with circles are point estimates; dashed lines are 95 percent confidence bands.
See Appendix A for data definitions and sources.
43
A. Job−Separation Rate 16+
B. Job−Finding Rate 16+
0.2
5
4
0.1
Percentage points
Percentage points
3
0
2
1
0
−0.1
−1
−0.2
1
2
3
4
Years after shock
−2
5
1
2
3
4
Years after shock
5
Figure 5: Job-Separation and Job-Finding Rate Response to a Tax Cut
Notes: The figure shows the response to a 1 percentage point cut in the average marginal personal income
tax rate (AMTR). Full lines with circles are point estimates; dashed lines are 95 percent confidence bands.
Data for job-separation and job-finding rates was constructed by Robert Shimer (see Shimer, 2012, for
details). See Appendix A for data definitions and sources.
44
A. Unemployment Rate 16+
B. Unemployment Rate 16+
0.5
Percentage points
Percentage points
0.5
0
−0.5
−1
0
−0.5
−1
Actual data
Steady−state approx.
−1.5
1
2
3
4
Years after shock
Actual data
Fixed labor force shares
−1.5
5
1
2
3
4
Years after shock
5
C. Participation Rate 16+
Percentage points
0.5
0
−0.5
−1
Actual data
Fixed population shares
−1.5
1
2
3
4
Years after shock
5
Figure 6: Unemployment and Participation Rate Response To a Tax Cut — Actual Data
vs. Counterfactuals
Notes: In all panels, full lines with circles are point estimates for the response to a 1 percentage point
cut in the average marginal personal income tax rate (AMTR); dashed lines are 95 percent confidence
bands. In panel A, dashed line with diamonds shows the steady-state approximation of the response
for the actual unemployment rate, as implied by equation (20). In panel B, dashed line with diamonds
shows the response of the counterfactual unemployment rate with fixed age-specific labor force shares.
In panel C, dashed line with diamonds shows the response of the counterfactual participation rate with
fixed age-specific population shares. See Appendix A for data definitions and sources.
45
−1
−1.5
−2
1
2
3
4
Years after shock
Percentage points
Percentage points
D. Unemployment Rate 35−44
0.5
−0.5
−1
−1.5
−2
1
2
3
4
Years after shock
5
−0.5
−1
−1.5
−2
5
0
0
Percentage points
−0.5
B. Unemployment Rate 20−24
0.5
1
2
3
4
Years after shock
E. Unemployment Rate 45−54
0.5
0
−0.5
−1
−1.5
−2
1
2
3
4
Years after shock
5
C. Unemployment Rate 25−34
0.5
0
−0.5
−1
−1.5
−2
5
Percentage points
0
Percentage points
Percentage points
A. Unemployment Rate 16−19
0.5
1
2
3
4
Years after shock
5
F. Unemployment Rate 55+
0.5
0
−0.5
−1
−1.5
−2
1
2
3
4
Years after shock
5
Figure 7: Unemployment Rate Response to a Tax Cut by Age
Notes: The figure shows the response to a 1 percentage point cut in the average marginal personal income
tax rate (AMTR). Full lines with circles are point estimates; dashed lines are 95 percent confidence bands.
See Appendix A for data definitions and sources.
46
0
−0.5
−1
−1.5
1
2
3
4
Years after shock
−1
−2
5
0.5
0
−0.5
−1
−1.5
1
2
3
4
Years after shock
2
3
4
Years after shock
5
0.5
0
−0.5
−1
−2
0
−0.5
−1
−2
5
−1.5
1
0.5
−1.5
E. Participation Rate 45−54
1
Percentage points
Percentage points
−0.5
−1.5
D. Participation Rate 35−44
1
−2
0
1
2
3
4
Years after shock
5
F. Participation Rate 55+
1
Percentage points
−2
0.5
C. Participation Rate 25−34
1
Percentage points
0.5
B. Participation Rate 20−24
1
Percentage points
Percentage points
A. Participation Rate 16−19
1
0.5
0
−0.5
−1
−1.5
1
2
3
4
Years after shock
5
−2
1
2
3
4
Years after shock
5
Figure 8: Participation Rate Response to a Tax Cut by Age
Notes: The figure shows the response to a 1 percentage point cut in the average marginal personal income
tax rate (AMTR). Full lines with circles are point estimates; dashed lines are 95 percent confidence bands.
See Appendix A for data definitions and sources.
47
Unemployment Rate 16+: AC−adj Impact and Lagged Response to a Tax Cut
−0.15
0
−0.2
−0.2
−0.3
−0.4
−0.35
−0.6
−0.4
Percentage points
Percentage points
−0.25
−0.45
−0.8
−0.5
Impact response
Lagged response
−0.55
1950
1955
1960
1965
1970
1975
1980
Year
1985
1990
1995
2000
2005
−1
Figure 9: AC-adjusted Unemployment Rate Response to a Tax Cut
Notes: The figure shows the AC-adjusted responses of the unemployment rate of 16 years and older to a 1
percentage point cut in the average marginal personal income tax rate (AMTR). “Impact response,” full
line with circles (left axis), shows the AC-adjusted response of the unemployment rate at horizon h = 1
(one year after the shock); “Lagged response,” dashed line with diamonds (right axis), shows the ACadjusted response of the unemployment rate at horizon h = 2 (two years after the shock). AC-adjusted
impact and lagged responses are constructed using equation (29). See Appendix A for data definitions
and sources.
48
A. Unemployment Rate 16+: AC−adj Impact Response to a Tax Cut
50
−0.2
45
−0.25
−0.3
40
Percent
Percentage points
−0.15
−0.35
35
Impact response
Labor force share 35−54
−0.4
−0.45
1950
1955
1960
1965
1970
1975
1980
Year
1985
1990
1995
2000
2005
30
B. Unemployment Rate 16+: AC−adj Lagged Response to a Tax Cut
50
−0.4
45
−0.5
−0.6
40
Percent
Percentage points
−0.3
−0.7
35
Lagged response
Labor force share 35−54
−0.8
−0.9
1950
1955
1960
1965
1970
1975
1980
Year
1985
1990
1995
2000
2005
30
Figure 10: Demographic Change and Unemployment Rate Response to a Tax Cut
Notes: In panel A, full line with circles (left axis) shows the AC-adjusted response of the unemployment
rate at horizon h = 1 (one year after the shock, “impact response”). In panel B, dashed line with
diamonds (left axis) shows the AC-adjusted response of the unemployment rate at horizon h = 2 (two
years after the shock, “lagged response”). AC-adjusted impact and lagged responses to a 1 percentage
point cut in the average marginal personal income tax rate are constructed using equation (29). In panels
A and B, dashed-dotted lines with squares (right axis) show the labor force share of the 35-54 age group.
See Appendix A for data definitions and sources.
49
Appendix — For Online Publication
This appendix is meant for online publication only.
A
A.1
Data
Definitions and Sources
The labor force is the number of employed plus unemployed persons. The unemployment rate is the number unemployed divided by labor force. The participation rate
is labor force divided by population. Population is the civilian noninstitutional population of 16 years of age and older. See the Bureau of Labor Statistics (BLS) glossary at
http://www.bls.gov/bls/glossary.htm for further details on data definitions and sources.
Data for the labor force, unemployment rate, participation rate, numbers of employed and
unemployed, and population for the total economy and by age groups are obtained from the
Current Population Survey (CPS), published by the BLS and available at the CPS home
page at http://www.bls.gov/cps/. Job-separation rates and job-finding rates are
constructed based on the methodology in Shimer (2012) and available at Robert Shimer’s
website at https://sites.google.com/site/robertshimer/research/flows. The average marginal tax rate (AMTR) is the average marginal personal income tax rate
weighted by a broad concept of labor income: wages, self-employment, partnership income,
and S-corporation income. AMTR combines the federal individual income tax and social
security payroll tax (FICA). Data for AMTRs has been tabulated by Robert Barro and
Charles Redlick in Barro and Redlick (2011). Output is real GDP, obtained from the National Income and Product Accounts (NIPA), published by the Bureau of Economic Analysis
(NIPA Table 1.1.3 line 1) divided by population. Government spending is real federal
government consumption expenditures and gross investment (NIPA Table 1.1.3 line 22) divided by population. The personal income tax base is personal income (NIPA Table
2.1 line 1) less government transfers (NIPA Table 2.1 line 17) plus contributions for government social insurance (NIPA Table 3.2 line 11). Debt is federal debt held by the public,
obtained from Favero and Giavazzi (2012) (DEBTHP), divided by the GDP deflator and
population. The narrative shocks are exogenous changes in individual income tax liabilities divided by the personal income tax base in the previous quarter. Data for personal
income tax base, real GDP, government spending, and federal debt (all in real per capita
terms), and narrative shocks are obtained from Mertens and Ravn (2013) and available
at Karel Mertens’s website at https://mertens.economics.cornell.edu/research.htm.
T-Bill is the 3-month treasury bill (FRED series: TB3MS), obtained from Federal Reserve
Economic Data (FRED), published by the Federal Reserve Bank of St. Louis and available
at https://research.stlouisfed.org/fred2/series/TB3MS. FFER is the federal funds
effective rate, obtained from the Board of Governors of the Federal Reserve System and
available at http://www.federalreserve.gov/releases/h15/data.htm.
50
A.2
Additional Facts by Age Group
Table A.1: Average Unemployment Rates and Labor Force Shares by Age
Age group:
16+
16-19
20-24
25-34
35-44
45-54
55-64
65+
Avg. UNR
5.64
15.63
9.01
5.29
4.04
3.64
3.62
3.51
1
2.77
1.60
0.94
0.72
0.65
0.64
0.62
100
7.03
11.57
23.96
22.96
19.18
11.75
3.55
Avg. UNR-ratio
Avg. LFS
Notes: Average unemployment rate (UNR) and labor force share (LFS), for 1950-2006, are
reported in percent. The second row indicates average unemployment rates by age, relative
to that of 16+ years old (UNR-ratio). See Appendix A.1 for data definitions and sources.
51
B. Population Shares by Age
A. Average Age of Population
42
55
25
20−34
35−54
55−64
41.5
50
20
45
15
Percent
40.5
Percent
Average age
41
40
40
39.5
10
35
39
38.5
1950 1960 1970 1980 1990 2000 2010
Year
30
5
1950 1960 1970 1980 1990 2000 2010
Year
Figure A.1: Trends in the Age Composition of the U.S. Population, 1950-2015
Notes: Panel A shows the average age of the U.S. civilian noninstitutional
population (20-64 years old).
P
a+a
P
P
The average age of the population is calculated as ā ≡ a∈A 2 φa , where a and a are respectively
lower and upper bounds of the age group a ∈ A, with A = {20-24, 25-34, 35-44, 45-54, 55-64}, and φP
a is
the age-specific population share (the ratio of the population in the age group a to total population).
Panel B shows the population shares by three age groups: (i) full line with circles (left axis) shows
P
P
P
φP
20-24 + φ25-34 ; (ii) dashed line with squares (left axis) shows φ35-44 + φ45-54 ; and (iii) dashed-dotted line
P
with diamonds (right axis) shows φ55-64 . See Appendix A.1 for data definitions and sources.
52
B. Employment Shares by Age
A. Average Age of Employment
42
55
25
20−34
35−54
55−64
41.5
41
50
20
45
39.5
39
15
Percent
40
Percent
Average age
40.5
40
38.5
10
38
35
37.5
37
1950 1960 1970 1980 1990 2000 2010
Year
30
5
1950 1960 1970 1980 1990 2000 2010
Year
Figure A.2: Trends in the Age Composition of U.S. Employment, 1950-2015
Notes: Panel A shows the average age of the U.S.employment pool (20-64 years old). The average age
P
of employment is calculated as āE ≡ a∈A a+a
φE
a , where a and a are respectively lower and upper
2
bounds of the age group a ∈ A, with A = {20-24, 25-34, 35-44, 45-54, 55-64}, and φE
a is the age-specific
employment share (the ratio of employed in the age group a to total employment). Panel B shows
E
employment shares by three age groups: (i) full line with circles (left axis) shows φE
20-24 + φ25-34 ; (ii)
E
dashed line with squares (left axis) shows φE
35-44 + φ45-54 ; and (iii) dashed-dotted line with diamonds
E
(right axis) shows φ55-64 . See Appendix A.1 for data definitions and sources.
53
B. Unemployment Shares by Age
A. Average Age of Unemployment
40
70
65
39
25
20−34
35−54
55−64
60
38
20
55
36
45
15
Percent
50
Percent
Average age
37
40
35
35
34
10
30
33
25
32
1950 1960 1970 1980 1990 2000 2010
Year
20
5
1950 1960 1970 1980 1990 2000 2010
Year
Figure A.3: Trends in the Age Composition of U.S. Unemployment, 1950-2015
Notes: Panel A shows the average age of the U.S.
pool (20-64 years old). The average age
unemployment
P
a+a
U
U
of unemployment is calculated as ā ≡ a∈A 2 φa , where a and a are respectively lower and upper
bounds of the age group a ∈ A, with A = {20-24, 25-34, 35-44, 45-54, 55-64}, and φU
a is the age-specific
unemployment share (the ratio of unemployed in the age group a to total unemployment). Panel B shows
U
unemployment shares by three age groups: (i) full line with circles (left axis) shows φU
20-24 + φ25-34 ; (ii)
U
U
dashed line with squares (left axis) shows φ35-44 + φ45-54 ; and (iii) dashed-dotted line with diamonds
(right axis) shows φU
55-64 . See Appendix A.1 for data definitions and sources.
54
B
B.1
Model
Full Insurance and the Family’s Budget Constraint
Here we first detail key steps for the derivation of the budget constraint of the family, then we
discuss the implications of full risk-sharing for individual-level income-maximizing behavior.
The family pools net-of-taxes earnings of all workers in the labor force. Thus, the family’s
budget constraint is:
Z
Z
Tj,t dj ,
(B.1)
Ct ≤
wj,t ej,t dj −
j∈Nt
j∈Nt
{z
} | {z }
|
total earnings total taxes
R
where Ct = j∈Nt cj,t dj is family’s total consumption. With full insurance, the family allocates
consumption independently of employment status, a result that follows immediately from
the assumption of separability of consumption and disutility of work in the utility function.
In the budget constraint (B.1), the tax function Tj,t takes the following form:
(w̃j,t /w̄j,t )1−γ
w̄j,t ,
(B.2)
1−γ
1
1−γ
R
1−γ
dj
where w̃j,t ≡ wj,t ej,t denotes individual j’s earnings, w̄t ≡ j∈Nt w̃j,t
, and τt is the
average marginal tax rate (AMTR). Note that the economy is also populated
by a continuum
R
of risk-neutral employers, who own claims on total profits Πt ≡ Yt − j∈Nt w̃j,t dj − kVt , where
Yt and Vt denote aggregate output and job vacancies, respectively.
The government runs a
R
balanced budget on a period-by-period basis, such that Gt = j∈Nt Tj,t dj. Next, using the
tax function in (B.2), the budget constraint in (B.1) can be rewritten as:
Z
1−γ
1 − τt
w̃j,t
Ct ≤
w̄t
dj.
(B.3)
1−γ
w̄t
j∈Nt
Tj,t = w̃j,t − (1 − τt )
Next, using the expression for w̄t , the budget constraint in (B.3) can be rewritten as:
Ct ≤
1 − τt
1−γ
Z
1−γ
w̃j,t
dj
1
1−γ
.
(B.4)
j∈Nt
Note that if γ = 0 then the budget constraint in (B.4) collapses to:
Z
Ct ≤
(1 − τt ) w̃j,t dj,
(B.5)
j∈Nt
where (1 − τt )w̃j,t is net-of-AMTR individual-level earnings.
In our setting, the family maximizes individual-level utilities, taking the AMTR as given.
Specifically, workers enter bilateral Nash-bargaining with employers and solve an individual
optimization problem, taking the net-of-AMTR wage (1 − τt )wj,t as the instantaneous return
to working.
55
B.2
Bellman Equations
The equilibrium of the model is characterized by a collection of Bellman equations, indexed
by age i ∈ {1, . . . , I}: (i) the value of working JiW for the worker; (ii) the value of being
unemployed JiU for the worker; (iii) the value of a filled job JiF for the employer; (iv) the
value of an unfilled vacancy J V for the employer. Note that J V is not indexed by age, as
the model features random search, in the sense that workers and employers have no ability
to seek or direct their search towards age-specific types of jobs. In what follows, we impose
that the reservation productivity rule yields the same threshold Ri for the worker as for the
employer. This is without loss of generality. Under Nash-bargaining, job separations are
bilateral efficient. As a result, workers and employers agree on the reservation threshold Ri .
Worker’s Bellman equations. The value of working for a worker of age i ≤ I is:
Z 0
0
U
W
W
(B.6)
Ji () = (1 − τ ) wi + β̃i
Ji+1 ( ) dF ( ) + F (Ri+1 ) Ji+1 ,
Ri+1
where (1 − τ ) wi is the net-of-tax rate wage. The wage rate wi is determined via bilateral
Nash-bargaining (see Appendix B.3 for details on the determination of the wage rate, wi ).
The parameter β̃i ≡ βσi < 1 is the age-specific discount factor, and Ri is the reservation
cutoff determined by the zero-surplus condition Si (Ri ) = 0, such that if < Ri the match
is endogenously destroyed (see Appendix B.4 for details on the derivation of the reservation
cutoff, Ri ). The value of being unemployed for a worker of age i ≤ I is:
W
U
JiU = zi + p (θ) β̃i Ji+1
() + [1 − p (θ)] β̃i Ji+1
,
(B.7)
where zi > 0 is the worker’s disutility of work, and p (θ) ∈ (0, 1) is the probability that a
worker meets an employer. Note that, in the value equation in (B.7), we assumed that all
new jobs are created at the upper support of the distribution of match-specific productivity
shocks, , such that the surplus of a new match is guaranteed to be positive for workers of
all ages. This assumption implies that the worker-employer meeting probability equals the
aggregate job-finding rate, such that f = p(θ), for all i ≤ I − 1.
Employer’s Bellman equations. The value of a filled job for an employer matched
with a worker of age i ≤ I is:
Z F
0
0
V
F
Ji+1 ( ) dF ( ) + F (Ri+1 ) J ,
(B.8)
Ji () = yi − wi + β̃i
Ri+1
where yi = xi is output of the worker-employer match, the parameter xi is the age-specific
worker’s efficiency units of labor, and is a match-specific productivity shock. In freeentry equilibrium, the value of an unfilled job vacancy is J V = 0, for all t ≥ 0, such that
labor-market tightness θ is determined by a free-entry condition:
k = q(θ)
I−1
X
β̃i φui JiF () ,
(B.9)
i=1
where k > 0 is the per-period cost of posting and maintaining a job vacancy, q (θ) is the
56
job-filling probability, such that k/q (θ)P
is the expected cost of posting and maintaining a
u
job vacancy, and φi ≡ Ui /U , with U = I−1
i=1 Ui , is the age-specific unemployment share.
B.3
Nash-Bargaining Equilibrium
The wage rate wi is determined via bilateral Nash-bargaining:
η
wi = arg max JiW () − JiU × JiF ()1−η ,
(B.10)
where η ≥ 0 is the worker’s bargaining weight. Bellman equations for the worker’s value
of working, JiW (), value of unemployment, JiU , and employer’s value of a filled job, JiF (),
are defined in equation (B.6), (B.7), and (B.8), respectively. The problem in (B.10) yields a
tax-adjusted sharing rule:
(1 − η) JiW () − JiU = η (1 − τ ) JiF () .
(B.11)
The tax-adjusted sharing rule in (B.11) implies that worker’s net value of working Hi () ≡
JiW () − JiU is proportional to the net-of-tax rate value of a job J˜iF () ≡ (1 − τ ) JiF ().
Next, let S̃i () ≡ (1 − τ ) Si () = Hi () + J˜iF () denote the net-of-tax rate total match
surplus. The Nash-bargaining solution implies that worker and employer receive a constant
and proportional share of the total match surplus Si (), such that Hi () = ηSi () and
JiF () = (1 − η) Si (). Bellman equations in (B.6)-(B.9), and the Nash-bargaining outcome
in (B.11), yield the age-specific surplus equations in (4) and the free-entry condition for
vacancy posting in (5).
B.4
Reservation Productivity Rule
We next detail key steps for the derivation of the reservation cutoff Ri in equation (7). Since
total match surplus is monotonically increasing in match-specific productivity, a reservation
productivity rule is optimal. Using equation (4), the reservation cutoff on the match-specific
shock, Ri , is determined by the zero-surplus condition Si (Ri ) = 0, that yields:
Z zi
xi Ri =
− β̃i [1 − ηp(θ)]
Si+1 (0 ) dF (0 ).
(B.12)
1−τ
Ri+1
Next, applying integration by parts to the integral on the right-hand side of (B.12) yields:
Z
0
Z
0
Si+1 ( ) dF ( ) = Si+1 () F ()−Si+1 (Ri+1 ) F (Ri+1 )−
Ri+1
0
Si+1
(0 ) F (0 ) d0 . (B.13)
Ri+1
Using F () = 1, the zero-surplus condition for the determination of the productivity cutoff
0
Si+1 (Ri+1 ) = 0, and Si+1
(0 ) ≡ dSi+1 (0 ) /d0 = xi , equation (B.13) yields:
Z Z 0
0
Si+1 ( ) dF ( ) = Si+1 () −
xi F (0 ) d0 .
(B.14)
Ri+1
Ri+1
57
Next, using Si+1 () =
R
Ri+1
Z
xi d0 , equation (B.14) yields:
Z 0
0
Si+1 ( ) dF ( ) = xi
[1 − F (0 )] d0 .
Ri+1
(B.15)
Ri+1
Replacing equation (B.15) into (B.12), and rearranging terms, yields the recursive equation
for the reservation cutoff in (7).
58
C
C.1
Proofs
Proof of Proposition 1
We next detail the key steps for the proof of Proposition 1. To isolate the role of the horizon
effect, let us assume that xi = x and zi = z, for all i ≤ I, such that worker’s net productivity
is constant over the life cycle: ỹi = ỹ, for all i ≤ I. Worker’s survival probability declines
over the life cycle, such that σi ≥ σi+1 , for i ≤ I − 1, with σI = 0.
Part (i). Next, we show that the reservation cutoff Ri is increasing in age: Ri < Ri+1 ,
for all i ≤ I − 1. The result follows immediately from Proposition 2 in Chéron et al. (2011).
Consider the recursive equation for the reservation cutoff in (7):
Z z̃
[1 − F (0 )] d0 , for i ≤ I,
(C.1)
Ri = − β [1 − ηp (θ)] σi
x
Ri+1
with z̃ ≡ z/(1 − τ ). Note that for a worker of age i = I (i.e., in the year before mandatory
retirement age), σI = 0, such that RI = z̃/x. By equation (C.1), it is straightforward that
Ri < RI , for all i ≤ I − 1:
Z Ri − RI = −β [1 − ηp (θ)] σi
[1 − F (0 )] d0 < 0.
(C.2)
Ri+1
Note that, in (C.2), 1 − ηp(θ) > 0, such that the value of labor hoarding is higher than the
value of search activity. Next, we generalize the expression in (C.2) as follows:
Z
0
Ri − Ri+1 = −β [1 − ηp (θ)] σi
0
Z
[1 − F ( )] d − σi+1
Ri+1
0
[1 − F ( )] d
0
< 0. (C.3)
Ri+2
A sufficient condition for the inequality in (C.3) to hold is σi ≥ σi+1 . To see this, consider
the expression in (C.3) for two successive ages I − 2 and I − 1:
(
Z
RI−2 −RI−1 = −β [1 − ηp (θ)] σI−2
[1 − F (0 )] d0 − σI−1
RI−1
Z
)
[1 − F (0 )] d0
. (C.4)
RI
R
R
From (C.2), we know that RI−1 < RI , such that RI−1 [1 − F (0 )] d0 > RI [1 − F (0 )] d0 ,
which implies that RI−2 < RI−1 as long as σI−2 ≥ σI−1 . Applying this argument recursively,
it is straightforward that the inequality in (C.3) holds for all i ≤ I − 1.
Part (ii). Next, we show that match surplus Si () decreases with age: Si () > Si+1 (),
for all i ≤ I − 1. We show that Ri < Ri+1 , for i ≤ I − 1, implies that Si () > Si+1 (). The
inequality in (C.3) implies that:
Z Z 0
0
σi
[1 − F ( )] d > σi+1
[1 − F (0 )] d0 .
(C.5)
Ri+1
Ri+2
59
Note that, using (B.15), and xi = x, for all i ≤ I, we can rewrite the inequality in (C.5) as
follows:
Z Z 0
0
σi
Si+1 ( ) dF ( ) > σi+1
Si+2 (0 ) dF (0 ).
(C.6)
Ri+1
Ri+2
Then, using equation (4), the inequality condition in (C.6) implies that Si () > Si+1 ().
Parts (iii) and (iv). Next, we show that the job-separation rate and the unemployment
rate increase with age. The job-separation rate is si = F (Ri ), such that si < si+1 as long as
Ri < Ri+1 . Given si < si+1 , and fi = f , for all i ≤ I − 1, it is straightforward that ui < ui+1 .
C.2
Proof of Proposition 2
We next detail the key steps for the proof of Proposition 2. To isolate the role of the netproductivity effect, let us assume that worker’s survival probability is constant throughout
life, akin to a Yaari-Blanchard perpetual youth model: σi = σ < 1, for all i ≥ 1. Worker’s
net-productivity raises throughout life, such that ỹi ≤ ỹi+1 (xi ≤ xi+1 and/or zi ≥ zi+1 ).
Parts (i) and (ii). Next, we show that total match surplus Si () increases with age, and
accordingly the reservation cutoff Ri is decreasing in age: Si () < Si+1 () and Ri > Ri+1 , for
all i ≥ 1. Since the probability of surviving from age i to age i + 1 is constant throughout
life, workers are of different ages, but have the same expected lifetime. Specifically, workers
solve an infinite horizon problem, but discount future payoffs with the effective discount rate
βσ < 1. Without loss of generality, we proceed under the assumption that the instantaneous
returns to the match are bounded. That is, worker’s net productivity, ỹi ≡ xi −zi /(1−τ ), is
bounded from above by an upper bound on the worker’s efficiency units of labor, xi ≤ x < ∞.
¯
Next, let I¯ denote the cutoff age at which xi = xI¯ ≤ x, such that Ri = , for all i ≥ I.
Thus, matches with workers of age i ≥ I¯ remain active for any realization of the matchspecific shock, whereas matches with workers of age i < I¯ are endogenously destroyed when
< Ri , with < Ri < . In a steady state, then, there is a time-invariant age distribution of
¯
¯ and UI¯ = 0, for all i ≥ I.
¯ Note that
unemployment {Ui }Ii=1 , such that Ui > 0, for all i < I,
¯
the existence of the unique cutoff age I is guaranteed as total match surplus is monotonically
increasing in worker’s productivity yi = xi . The free-entry condition in (5) now reads:
I¯
X
k
φui Si () .
(C.7)
= (1 − η) β̃
q (θ)
i=1
PI¯ u
On the right-hand side of (C.7),
i=1 φi Si () < ∞, which is required to have a welldefined steady state with q(θ) ∈ (0, 1). Since total match surplus is monotonically increasing
in worker’s productivity, the reservation productivity rules between cutoff age I¯ and age
I¯ − 1, SI¯ (RI¯) = 0, and SI−1
(RI−1
> RI¯. Applying this argument
¯
¯ ) = 0, imply that RI−1
¯
recursively, it is straightforward that Ri > Ri+1 , for all 1 ≤ i < I¯ − 1.
Parts (iii) and (iv). Next, we show that the job-separation rate and the unemployment
rate decrease with age. The job-separation rate is si = F (Ri ), such that si > si+1 as long as
Ri > Ri+1 . Given si > si+1 , and fi = f , for all i ≥ 1, it is straightforward that ui > ui+1 .
60
Empirics
Additional Results by Age Group
0.5
0
−0.5
−1
1
2
3
4
5
Years after shock
D. UNR16−19−UNR45−54
Percentage points
−1.5
0.5
0
−0.5
−1
−1.5
1
2
3
4
5
Years after shock
G. UNR20−24−UNR35−44
0.5
0
−0.5
−1
−1.5
1
2
3
4
Years after shock
5
C. UNR16−19−UNR35−44
Percentage points
Percentage points
B. UNR16−19−UNR25−34
Percentage points
Percentage points
Percentage points
Percentage points
A. UNR16−19−UNR20−24
0.5
0
−0.5
−1
−1.5
1
2
3
4
5
Years after shock
E. UNR16−19−UNR55+
Percentage points
D.1
0.5
0
−0.5
−1
−1.5
1
2
3
4
5
Years after shock
H. UNR20−24−UNR45−54
Percentage points
D
0.5
0
−0.5
−1
−1.5
1
2
3
4
Years after shock
5
0.5
0
−0.5
−1
−1.5
1
2
3
4
5
Years after shock
F. UNR20−24−UNR25−34
0.5
0
−0.5
−1
−1.5
1
2
3
4
Years after shock
I. UNR20−24−UNR55+
5
1
2
3
4
Years after shock
5
0.5
0
−0.5
−1
−1.5
Figure D.1: Difference in Unemployment Rate Response Between Age Groups
Notes: The figure shows the differential response to a 1 percentage point cut in the average marginal
personal income tax rate (AMTR) of the unemployment rate (UNR) between age groups. Full lines with
circles are point estimates; dashed lines are 95 percent confidence bands. The estimates are based on the
baseline VAR specification in (17), that includes the difference between the unemployment rates of two
age groups as the variable of interest, whose impulse response function is plotted.
61
A. UNR25−34−UNR35−44
B. UNR25−34−UNR45−54
−0.5
−1
1
2
3
4
Years after shock
0
−0.5
−1
−1.5
5
0.5
Percentage points
0
−1.5
1
D. UNR35−44−UNR45−54
−0.5
−1
2
3
4
Years after shock
−1
1
5
5
0.5
0
−0.5
−1
−1.5
2
3
4
Years after shock
F. UNR45−54−UNR55+
Percentage points
0
1
−0.5
−1.5
5
0.5
Percentage points
Percentage points
2
3
4
Years after shock
0
E. UNR35−44−UNR55+
0.5
−1.5
C. UNR25−34−UNR55+
0.5
Percentage points
Percentage points
0.5
1
2
3
4
Years after shock
5
0
−0.5
−1
−1.5
1
2
3
4
Years after shock
5
Figure D.2: Difference in Unemployment Rate Response Between Age Groups
Notes: The figure shows the differential response to a 1 percentage point cut in the average marginal
personal income tax rate (AMTR) of the unemployment rate (UNR) between age groups. Full lines with
circles are point estimates; dashed lines are 95 percent confidence bands. The estimates are based on the
baseline VAR specification in (17), that includes the difference between the unemployment rates of two
age groups as the variable of interest, whose impulse response function is plotted.
62
Table D.1: Counterfactual Shares of Aggregate Unemployment Response by Age
h (years after shock):
1
2
3
4
5
share16-19,FLFS
h
12.79
22.86
23.45
22.14
17.09
share20-34,FLFS
h
48.56
40.20
37.57
36.57
37.82
share35-54,FLFS
h
27.65
22.78
22.96
24.32
28.90
share55-64,FLFS
h
8.51
9.07
10.36
11.68
13.71
share65+,FLFS
h
2.49
5.09
5.66
5.29
2.48
sharea,FLFS
.
h
Notes: See equation (27) for the definition of
Age-specific shares with
P65+
fixed labor force shares are reported in percent such that a=16 sharea,FLFS
= 100
h
(column-wise summation).
Table D.2: Unemployment Elasticities by Age (up to 5 years after the shock)
Age group:
16+
16+FLFS
16-19
20-24
25-34
35-44
45-54
55-64
65+
Elas. at year 1
1.52
1.50
0.25
1.27
2.31
1.99
2.11
1.36
0.15
Elas. at year 2
2.92
2.93
1.05
3.11
3.47
3.56
3.67
3.23
1.58
Elas. at year 3
3.38
3.44
1.15
3.30
4.22
4.13
4.26
4.05
2.05
Elas. at year 4
2.53
2.62
0.74
2.32
3.26
3.22
3.36
3.29
1.44
Elas. at year 5
1.51
1.62
0.29
1.28
2.10
2.16
2.26
2.17
0.53
Notes: “Elas.” refers to the unemployment rate elasticity of a specific age group at horizon h = 1, . . . , 5
(years after the shock). Each of the elasticities estimates are based on a separate SVAR system that
includes the log of the unemployment rate of a specific age group and a common set of regressors, as
specified in (17), for 1950-2006. Elasticities are with respect to the average marginal tax rate (AMTR),
and are reported in percent. The column labeled “16+FLFS ” refers to the counterfactual unemployment
rate of 16 years and older with fixed labor force shares as in (22).
63
D.2
Additional Results on the Aggregate Unemployment Rate
and Participation Rate
B. Unemployment Rate 16+
0.5
0
0
Percentage points
Percentage points
A. Average Marginal Tax Rate
0.5
−0.5
−1
−1.5
−0.5
−1
1
2
3
4
Years after shock
−1.5
5
1
2
3
4
Years after shock
5
Figure D.3: Unemployment Rate Response to a Tax Cut — Real GDP per Capita Included
Notes: The figure shows the response to a 1 percentage point cut in the average marginal personal income
tax rate (AMTR). Full lines with circles are point estimates; dashed lines are 95 percent confidence bands.
The estimates are based on a SVAR system that includes the real GDP per capita (RGDP), as a business
cycle indicator, that replaces the personal income tax base (PITB); other regressors remain the same as
in the baseline VAR specification in (17). See Appendix A.1 for data definitions and sources.
64
A. Average Marginal Tax Rate
B. Participation Rate 16+
1
0.5
0.5
Percentage points
Percentage points
0
0
−0.5
−0.5
−1
−1
−1.5
1
2
3
4
Years after shock
−1.5
5
1
2
3
4
Years after shock
5
Figure D.4: Participation Rate Response to a Tax Cut — Real GDP per Capita Included
Notes: The figure shows the response to a 1 percentage point cut in the average marginal personal income
tax rate (AMTR). Full lines with circles are point estimates; dashed lines are 95 percent confidence bands.
The estimates are based on a SVAR system that includes the real GDP per capita (RGDP), as a business
cycle indicator, that replaces the personal income tax base (PITB); other regressors remain the same as
in the baseline VAR specification in (17). See Appendix A.1 for data definitions and sources.
65
B. Unemployment Rate 16+
0.5
0
0
Percentage points
Percentage points
A. Average Marginal Tax Rate
0.5
−0.5
−1
−1.5
−0.5
−1
1
2
3
4
Years after shock
−1.5
5
1
2
3
4
Years after shock
5
Figure D.5: Unemployment Rate Response to a Tax Cut — 3-Month T-Bill Included
Notes: The figure shows the response to a 1 percentage point cut in the average marginal personal income
tax rate (AMTR). Full lines with circles are point estimates; dashed lines are 95 percent confidence bands.
The estimates are based on a SVAR system that includes the real GDP per capita (RGDP), as a business
cycle indicator, that replaces the personal income tax base (PITB), and the 3-month treasury bill (TBill); other regressors remain the same as in the baseline VAR specification in (17). See Appendix A.1
for data definitions and sources.
66
A. Average Marginal Tax Rate
B. Participation Rate 16+
1
0.5
0.5
Percentage points
Percentage points
0
0
−0.5
−0.5
−1
−1
−1.5
1
2
3
4
Years after shock
−1.5
5
1
2
3
4
Years after shock
5
Figure D.6: Participation Rate Response to a Tax Cut — 3-Month T-Bill Included
Notes: The figure shows the response to a 1 percentage point cut in the average marginal personal income
tax rate (AMTR). Full lines with circles are point estimates; dashed lines are 95 percent confidence bands.
The estimates are based on a SVAR system that includes the real GDP per capita (RGDP), as a business
cycle indicator, that replaces the personal income tax base (PITB), and the 3-month treasury bill (TBill); other regressors remain the same as in the baseline VAR specification in (17). See Appendix A.1
for data definitions and sources.
67
Unemployment Rate 16+
12
Actual data
Steady−state approx.
11
10
9
Percent
8
7
6
5
4
3
2
1950
1955
1960
1965
1970
1975
1980
Year
1985
1990
1995
2000
2005
Figure D.7: Unemployment Rate — Actual Data vs. Steady-State Approximation
Notes: The figure shows the unemployment rate of 16 years and older (full line) and the counterfactual
unemployment rate under the steady-state approximation (dashed line). The steady-state approximation
of the actual unemployment rate is constructed as uss
t = st /(st + ft ). Data for job-separation, st , and
job-finding rates, ft , was constructed by Robert Shimer (see Shimer, 2012, for details). See Appendix
A.1 for data definitions and sources.
68