Digitized by the Internet Archive
in
2011 with funding from
Boston Library Consortium
Member
Libraries
http://www.archive.org/details/reservationwagesOOshim
6
DEWEY
HB31
.M415
Massachusetts Institute of Technology
Departnnent of Econonnics
Working Paper Series
RESERVATION WAGES &
UNEMPLOYMENT INSURANCE
Robert Shimer
Ivan Werning
Working Paper 06-1
February 16,2006
Revised: April 28, 2006
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02142
This paper can be downloaded without charge from the
Social Science Research
Network Paper Collection
http://ssrn.com/abstract=902066
at
^^^^^^^^^^M!^^"'''
JUN
2 C 2005
LIBRARIES
Reservation Wages and
Unemployment Insurance*
Robert Shimer
University of Chicago and
Ivan Werning
MIT, NBER and UTDT
NBER
shimerOuchicago edu
iwerningSmit edu
.
.
April 28, 2006
Abstract
This paper argues that a risk-averse worker's after-tax reservation wage encodes
all
the relevant information about her welfare. This insight leads to a novel test for
the optimality of unemployment insurance based on the responsiveness of reservation
wages to unemployment
benefits.
raising the current level of
Some
existing estimates imply significant gains to
unemployment insurance but
research on the determinants of reservation wages.
complements those based on Daily's (1978)
it
uses less of the structure of the model,
separate risk-aversion estimates, and
it is
test.
it is
highlight the need for
Our approach
Some advantages
entirely behavioral
is
intuitive
more
and
of our test are that
and does not require
robust to various extensions including worker
heterogeneity.
supported by a grant from the National Science Foundation. Werning is grateful
Reserve Bank of Minneapolis and Harvard University. We are grateful to
seminar participants at Berkeley, Harvard, and MIT and especially for detailed comments from Raj Chetty.
*Shimer's research
is
for the hospitality of the Federal
Introduction
1
The
goal of this paper
using a minimal
is
amount
to develop a test for the optimal level of
of economic theory and a
by studying a risk-averse worker
unemployment insurance
minimal amount of data.
in a sequential job search setting
We
approach
(McCall, 1970).
this
Our main
— the difference between her
reservation wage and the tax needed to fund the unemployment insurance system —encodes
theoretical insight
is
that the worker's after-tax reservation wage
of the relevant information about her welfare. This
all
and lend
are able to borrow
to
smooth
true regardless of whether workers
is
consumption or whether they must must
their
live
hand-to-mouth.
The
intuition
make a worker
to
clear: the after-tax reservation
is
indifferent
wage
tells
us the take-home pay required
between working and remaining unemployed. Since take-home
pay translates directly into consumption,
the simplicity of the argument,
it
it is
a valid measure of the worker's
utility.
Given
should not be surprising that this insight turns out to be
robust to several variations of our basic model.
To prove
this result,
we develop a formal dynamic model
of job search with risk-aversion.
Workers draw wages from a known distribution and accepted jobs
of time. In order to abstract
risk aversion
(CARA)
by an arbitrary
from wealth
preferences.^
level of
We
unemployment
effects,
first
a fixed
amount
we assume workers have constant absolute
consider
benefits
last for
how workers behave when confronted
and reemployment taxes and show how the
answer depends on whether workers are able to borrow and lend. In both cases we find that
a worker's
If
utility while
unemployed
is
a monotone function of her after-tax reservation wage.
she has no access to capital markets, her unemployment
equivalent units,
is
utility,
equal to her after-tax reservation wage.
If
measured
in
consumption
she can borrow and lend,
it is
equal to her after-tax reservation wage plus the annuity value of her assets. This implies that
optimal unemployment insurance
benefits
—the
policy of an agency which chooses
and reemployment taxes to maximize an unemployed worker's
the expected discounted cost of the unemployment insurance equally zero
maximize workers'
'
desirable whenever
is
In Shimer and VVerning (2005),
relative risk aversion
and
CARA
(CRRA)
preferences.
more general
utility subject to
—simply seeks to
after tax reservation wage.
This insight leads to a novel
benefits
unemployment
preferences.
it
test for the optimality of
raises the after-tax reservation wage.
we show
preferences
Thus we
unemployment insurance:
is
that the behavior
raising
This criteria can be
and insurance needs of a worker with constant
same absolute risk aversion
similar to that of a worker with the
believe that the results
we report here
are quantitatively reasonable for
decomposed
into
two
effects.
unemployed and therefore
wage
is
On
tfie
one liand, higher benefits reduce the cost of remaining
raise the pre-tax reservation wage.
Thus,
if
the pre-tax reservation
very responsive to unemployment benefits, raising unemployment benefits has a
strong positive effect on workers' welfare. However, the increase in benefits must be funded
by an increase
in the
responsive
to
it is
employment
unemployment
The
higher
the unemployment rate or the more
is
benefits, the greater
optimality condition nets out both
While a large
tax.
is
the needed increase in the tax.
Our
effects.
unemployment
literature studies the responsiveness of
duration to unemployment benefits
Meyer, 1990), there
(e.g.,
Two
sponsiveness of reservation wages to benefits.
is
or
unemployment
less research
on the
re-
notable exceptions are Fishe (1982) and
Feldstein and Poterba (1984). Fishe (1982) uses information on actual wages to infer reser-
vation wages, while Feldstein and Poterba (1984) uses direct survey evidence on reservation
wages. Both papers find that a $1 increase in benefits
may
raise pre-tax reservation
by as much as $0.44. Feldstein and Poterba (1984) interpret
this as evidence of the
wages
moral
hazard cost of raising unemployment benefits, but our approach turns this logic around, since
our theory
tells
us that the reservation wage measures the welfare of unemployed workers.
the numbers in Fishe (1982) and Feldstein and Poterba (1984) are correct,
If
we show
that a fully-funded $1 increase in weekly benefits, at a cost of approximately $400 million per
year in the U.S. economy,
is
equivalent to
somehow
giving every employed and unemployed
worker an additional $0.37 of consumption per week,
in additional
consumption.
Of
course, Fishe's (1982)
to creating $2.6 billion per year
i.e.
and Feldstein and Poterba's (1984)
estimates are valid for small policy changes; according to the model, sufficiently high unem-
ployment benefits would eventually eliminate
all
economic
activity.
Moreover, more recent
estimates of the responsiveness of reservation wages to benefits are smaller and imply that
current benefit levels are too high, In our view, the uncertainty around this critical variable
calls for
more
precise estimates of
Within the public finance
ployment insurance
is
it.
literature, the
standard approach to measuring optimal unem-
based on the Daily (1978)
"The optimal unemployment insurance
drop
in
workers (evaluated at the
is
set
when
the proportional
level of
of relative
consumption when unemployed)
equal to the elasticity of the duration of unemployment with respect to balanced
budget increases
While
benefit level
consumption resulting from unemployment, times the degree
risk aversion of
is
test:
this
approach
is
in
UI [unemployment insurance] benefits and
close in spirit to the one
we adopt
here,
we
taxes."
(p.
390)
see several advantages
to our test.
First,
our test
is
entirely behavioral, while the Baily test requires independent
estimates of risk- aversion. Indeed, Chetty (2005) argues within a Baily framework that the
relevant risk-aversion parameter depends on the context
ment
or
risk.
and may be higher
unemploy-
for
In light of such concerns, the fact that our test does not requires selecting this,
any other, parameter
is
particularly convenient.
Second, Chetty (2005) shows that in a dynamic environment, the Baily test requires a
long panel data set with information on total consumption. Unfortunately, no such data set
exists, so the best
known implementation
of the Baily test,
Gruber
(1997), uses panel data
on food expenditure. There are two main limitations to using food expenditure as a proxy
for total
consumption: recent work by Aguiar and Hurst (2005) shows that the link between
food expenditure and food consumption
in
tenuous because of varying amounts of time spent
is
household production; and food consumption
is
likely to react significantly less
than total
consumption to income or wealth shocks.^
Third, our exact test
is
We
robust to a number of extensions.
allow for the possibility
that a worker's costly search effort affects the arrival rate of offers, that jobs
both
in their
but there
is
wage and
basic conclusion that the reservation
wage
is
differ
and that workers are heterogeneous
in their average tenure length,
a single unemployment benefit system.
may
None
of these extensions affects our
a sufficient statistic for the unemployed and
therefore substantially alters our behavioral test for optimal
unemployment insurance.
In
contrast, although Chetty (2005) shows that extensions of the consumption-based Baily
test are possible, in
our view they
may be
implement because they require an
difficult to
empirically challenging comparison of the average marginal utility of consumption during
employment with that during unemployment over the worker's
of
consumption data not analyzed by Gruber (1997),
for
can also deliver easily implementable consumption-based
derivation uses the full structure of the model,
here,
evidence on reservation wages
is
usefulness as a welfare statistic,
^
less
— a moment
example. Nevertheless, our model
tests,
but we point out that their
robust than the
new
test
we propose
and requires unexplored consumption measures from panel data.
As mentioned above, one challenge
much
is
entire lifetime
to implementing our behavioral test
scarce.
may
Our hope
is
lead to greater interest in reservation
Indeed, Chetty (2005) extends the consumption test so that
food, instead of risk aversion.
a parameter
for the
that empirical
that this paper, by underscoring
as Baily's (1978) theoretical contribution led to empirical research
nately, the test then requires setting
is
it
its
wage evidence,
on how much con-
applies to food consumption. Unfortu-
curvature of the utility function with respect to
sumption declines when workers
complementary.
Both
lose their job (Gruber, 1997). Ultimately, the
assess the optimality of
unemployment insurance, but
two
tests are
exploit very
data sources.
different
Macroeconomists have generally taken a
approach to optimal unemployment
different
insurance, calibrating a stochastic general equilibrium model and then performing policy
experiments within the model (Hansen and Imrohoroglu, 1992; Acemoglu and Shimer, 2000;
Alvarez and Veracierto, 2001).
we
is
that
it
can address issues
do that, these papers rely heavily on the entire structure of the model and
in order to
calibration,
advantage to this approach
impact of unemployment insurance policy on capital accumulation.
neglect, such as the
But
An
which sometimes obscures the economic mechanisms at work and their empirical
validity.
This approach also makes evaluating the robustness of the results expensive.
contrast,
by focusing on the worker's
general equilibrium models
partial equilibrium
—we are able to highlight,
that seem important for understanding optimal
how
its
problem
In
— a component
in richer
main
tradeoffs
in a tractable way, the
unemployment insurance and
to point out
the relevant forces can be measured.
A
third strand of the literature focuses on the timing of benefits,
whether unemployment benefits should
1979;
Hopenhayn and
Nicolini, 1997).
fall
and
in particular,
on
during an unemployment spell (Shavell and Weiss,
This paper emphasizes the optimal
level of benefits
but assumes that benefits and taxes are constant over time. In Shimer and Werning (2005)
we argue
that, provided workers are given
earnings,'^ constant benefits
enough
and taxes are optimal, or nearly
emphasis, there are two modeling differences. The
time rather than
portantly, here
spells.
in discrete time, a superficial
we allow
is
important
for
first is
so.
Besides this difference in
that here
we work
in
continuous
change that simplifies the algebra. More im-
for separations, so that
This generalization
borrow against future
liquidity to easily
workers experience multiple unemployment
any quantitative exercise focusing on the
level of
benefits.
The remainder
of the paper proceeds as follows:
of sequential search. Section 3 analyzes
unemployment
benefits
and constant
The next
section presents our
model
how workers behave when confronted with constant
taxes.
We
consider two financial regimes. In the
first,
workers have unlimited access to borrowing and lending at a constant interest rate, subject
only to a no Ponzi-game condition. In the second, workers must
suming
their
choosing the
Such
income
in
level of
liquidity
five
hand-to-mouth, con-
each period. Section 4 describes the problem of an insurance agency
unemployment insurance subject
to a budget constraint.
might be provided by unemployment insurance savings accounts (Feldstein,
Section 5
200.5)
describes our
new
test for
optimal unemployment insurance and discusses the available em-
on the relevant parameters
pirical evidence that bears
a
number
of generalizations to our
Section 6 considers
of that test.
model and shows that our
test
is
unaffected by those
Section 7 derives a version of the Baily (1978) test for our model, showing that
changes.
the exact test depends on
behavioral
test.
We
all
the details of the model and hence
conclude in Section
robust than our
is less
S.
Unemployment and Sequential Search
2
There
is
a single risk-averse worker
who maximizes
the expected present value of utility from
consumption,
J^oo
'
where p >
e-'"u{c{t))dt,
CARA,
out the body of the paper that the utility function exhibits
coefficient of absolute risk aversion
At any moment
7 >
becomes unemployed.
An unemployed
of job opportunities.
The worker
distribution function
F
An employed
A.'^
When
observes the wage and decides whether to accept or reject
There
is
lasts for exactly
T <
b
When
*
If
it.
t
periods
w
units
the job ends, she
for the arrival
recall past
is
offer,
she
she accepts, employment
00 periods.'^
an unemployment insurance agency whose objective
If
she rejects, she
wage
offers.
With
not binding.
is
to
b
maximize an unem-
and constant employ-
expected cost of the unemployment insurance
T,^ subject to the constraint that the
if
with
a worker gets a wage
not optimal, so this last assumption
Section 6.4 shows that our results are robust
w
and waits
ployed worker's utility by choosing a constant unemployment benefit
ment tax
with
wage draw from a cumulative
produces nothing and remains unemployed. The worker cannot
is
r.
worker receives a benefit
with Poisson arrival rate
preferences recall
—e~'''^
worker produces a flow of
receives an independent
commences immediately and the job
=
0.
consumption good and pays an employment tax
of the single
assume through-
u{c)
time a worker can be employed, at some wage
in
remaining in the job, or unemployed.
CARA
We
represents the subjective discount rate in continuous time.
a worker's search effort affects the arrival rate of job
offers.
^
Section 6.2 shows that our main results are robust
Section 6.3 shows they are robust
if
if
the duration of a job
the worker draws both a wage and a job duration.
is
uncertain.
and Werning (2005) we show that this simple unemployment insurance system is optimal
with no job separations when the worker can borrow and lend at interest rate r. With job separations, as
we allow here, this simple policy may not be fully optimal, but it remains an important benchmark.
^
In Shinier
system
is
zero
when discounted
sum
subsidy to unemployment, the
B =
+
t denote the net
of the benefit a worker receives while
unemployed and
We
the employment tax she avoids paying.
only on the net unemployment subsidy.
We
= p?
at the interest rate r
Let
b
show below that a worker's behavior depends
•
consider two financial environments.
In the
first,
the worker has access to finan-
markets, namely a riskless borrowing and savings technology, facing only the budget
cial
constraint
a{t)
=
ra{t)
+
y{t)-c{t),
and the usual no Ponzi-game condition.^ Here
a{t)
assets, c{t)
is
represents current income, equal to the current after-tax
employed, or benefits
b,
otherwise.
The
rate of return r
unemployment insurance agency and equal
environment, the worker
=-0
a(t)
We
for all
t,
lives
is
is
consumption, and y{t)
wage w{t) — r
the worker
if
is
the same for the worker and the
to the discount rate p for simplicity. In the second
She has no access to a savings technology,
hand-to-mouth.
and so must consume her income
in
each period,
c{t)
=
y{t).
study these two extremes because they span the spectrum of financial environments
and because both cases are analytically
in closed
tractable.
The intermediate
cases cannot be in solved
form but could be studied numerically to see whether our two cases provide a good
benchmark; doing so goes beyond the scope of
this paper.
Finally, define
at
=
1
_
g-'"*
/*
=
r
This
The
is
/
e ''^ds.
Jo
the present value of receiving an additional unit of income for the next
present value of income from a
new job with wage
w
is
arw. Note that
if
r
=
t
periods.
0,
a^
=
t.
Worker Behavior
3
We
start
system
by characterizing how a worker behaves when confronted with any constant benefit
(6, t).
We
first
access to borrowing
consider a worker with no liquidity problems, that
and lending
at rate
r.
We then
is,
a worker with
turn to the opposite end of the spectrum
and consider a hand-to-mouth worker who must consume her current income.
^
Section 6.1 shows that our main results are robust
*
The no-Ponzi condition
states that debt
if
the discount rate and interest rate are not equal.
must grow slower than the
with probability one. Together with the budget constraints d(i)
imposing a single present-value constraint, with probability one.
=
ra{t)
interest rate, limj^oo e~''*a(i)
+y{t) —
c{t), this is
>
0,
equivalent to
Workers with Liquidity
3.1
A worker who
can borrow and lend at the interest rate r
=
p keeps her consumption constant
during an employment spell since she faces no uncertainty.
She saves, however, gradually
accumulating assets while on the job. In contrast, consumption steadily declines during un-
employment, because remaining unemployed represents a negative permanent income shock.
This
accompanied by dissavings, as assets are run down during unemployment
is
spells.
Consumption jumps up when an unemployed worker becomes employed, because finding a
job
wage
is
a discrete positive shock.
is
policy, accepting jobs
When
unemployed, the worker uses a constant reservation
above some threshold w. Finally, the after-tax reservation wage
a sufficient statistic for the welfare of the unemployed.
We now
state these results formally:
Proposition
1
Assume
the lifetime utility of
a worker has access to financial markets. For a given policy {b,T),
an unemployed worker with assets a
Vu{a)
= -u{ra + w —
is
t).
(1)
r
The consumption of an unemployed worker with assets a and of an employed worker with
assets a,
t
periods remaining on the job, and a wage
The reservation wage
w
is
Cu{a)
=
ra
c{a, t,w)
=
r(^a
+
u)
+
—
w
are respectively
T,
at{w
(2)
—
u;))
+w~
t.
(3)
constant and solves
A
j{w-B) = '
f°°
{l
+ u{raT{w~w)))dF(w).
(4)
J id
For the purposes of this paper, the most important part of this proposition
To
get
some
intuition for this result, suppose a worker could accept a job at
lasts forever, so her after-tax
income would he
w—t
in all future periods.
rate equal to the interest rate, a worker with a concave utility function
consumption constant and so would consume
ra.
That
is,
is
this
With
equation
wage
w
(1).
that
the discount
u would keep her
income plus the annuity value on her
assets,
= ra + w — r, her assets would be constant, d = 0,
-u{ra + w — t). Now define the reservation wage w so that
she would consume c{a, oo, w)
and her lifetime
utility
would be
an unemployed worker without a job
offer is indifferent
between remaining unemployed and
working forever at
This logic
catch
a wage for a
may
Vu{a)
=
^u{ra
so simple that
+w—
r), giving
equation
(1).
might seem to extend beyond our
it
the notion of a reservation wage. In general, a worker
lies in
worker
is
lu,
finite
amount
of time but unwilling to take the
wage
specific model.
may be
The
willing to accept
For example, a
forever.
take a low wage for a while, accumulate assets, and eventually quit to search for
a higher wage. Acemoglu and Shimer (1999) explore this possibility in an environment with
We
decreasing absolute risk aversion.
CARA
A
that this cannot happen with
preferences since a worker's attitude towards risk and hence her reservation wage
independent of
3.2
prove in Appendix
is
assets.
Hand-to-Mouth Workers
We now consider worker behavior under an extreme alternative, financial autarky, so a worker
must consume her income in each period: c„"* = b and cl"^{w) = w — t. Under financial
autarky, a worker's consumption will typically
she leaves her job. Although this
is
jump up when she
qualitatively different than
to financial markets, one critical property
is
and down when
finds a job
when
the worker has access
unchanged, the worker's lifetime
utility
depends
only on her after-tax reservation wage:
Proposition 2 Assume a worker must consume her income. For a given
lifetime utility of
unemployment
policy {b,T), the
is
Vr' = -uiw^^'-T),
where
u)""' is the
(5)
reservation wage, the solution to
/oo
u(u)""'
This result
is
To prove
utility of
r)
=
u{b)
+ arX
/
{u{w
-
independent of the form of the period
this result,
we use a
r)
-
u(u)°"*
- T))dF{w).
(6)
utility function u.
pair of recursive equations. Let \4^"' denote the expected
an unemployed worker living under autarky and
let V'/"'(ti),T)
denote the corre-
,
spending value
for
a newly-employed worker at a wage w. These solve
/CO
pF-t ^
^^
^^^^
max {K,""*(u;, T) -
/
V;""',
0}dF(Ti;)
JO
K/"'(u;,
T)=
f e-f"u{w - T)dt
+
e'^'^V^''^
Jo
The
flow value of an
rate A she gets a
or reject.
unemployed worker comes from her current
wage draw
An employed
has continuation value
w
which she
utility of
r,
for a
t) for the next
T
V^^"S
periods and then
newly-employed worker implies
so the reservation
an unemployed worker
wage
is
^r = ^Mw - r) - p^r
solves u{w'^^^
~
given by equation
equation for an unemployed worker gives equation
It is
—
—
V^^"'.
Vr\w, T) -
=
accept, giving capital gain V^'^^{w,T)
worker in a new job earns u{w
The Bellman equation
since p
may
utility u{b). In addition, at
t)
(5).
—
)
P^u"^- Equivalently, the lifetime
Substituting this into the Bellman
(6) for the reservation
wage.
worth noting that, since the reservation wage summarizes a worker's
utility
both un-
der perfect liquidity and financial autarky, the difference in the reservation wage summarizes
the value of access to financial markets.
More
Proposition 3 A hand-to-mouth worker has
precisely.
a lower reservation
wage then a worker with
access to capital markets.
Moreover, the difference in their reservation wages
gain from access
markets, measured in units of per-period consumption.
The proof
4
is
to capital
in
the utility
Appendix B.
Optimal Unemployment Insurance
We now
turn to the problem of an unemployment insurance agency which chooses the unem-
ployment benefit
The agency
in the
is
b
and the employment tax r to maximize an unemployed worker's
recognizes that the worker chooses her reservation
wage
utility.
optimally, as described
previous section. Thus benefits and taxes affect the expected discounted net cost of
the unemployment insurance agency;
we
require that this
is
equal to zero, which turns out
to be equivalent to
The
hand
left
side
is
the expected cost of
unemployment
benefits during one
unemployment
spell,
the value of benefits divided by the hazard rate of finding an acceptable job.
right
hand
side
the product of the reemployment tax and the factor ar, the present
is
value of a unit of income for the duration of a job.
unemployment insurance problem
w
is
equivalently
w —
utility given
The balanced budget
taxes r, and a reservation
by equation
(1) or
equation
wage
or
(5),
(7).
constraint seems natural in a large
independent across workers.
If all
workers are
optimal unemployment insurance.
If
initially
now and
economy where wage draws
unemployed,
some workers
interests are not perfectly aligned with those of
workers pay taxes
b,
subject to the reservation wage equation (4) or equation (6) and the
t,
budget balance equation
call this
Putting this together, the optimal
to choose benefits
maximize the unemployed worker's
to
The
it
should be clear
are
why we
start off employed, however, their
unemployed workers since initially-employed
only receive benefits
later.
That
is, if
we
start with
some work-
employed and some unemployed, optimal benefit policy has elements of both insurance
ers
and
To focus on insurance, we
redistribution.
implicitly
assume that the unemployment
insurance agency does not start taxing workers until they begin their
we assume
Equivalently,
spell.
a worker's
initial
employment
realigns the interests of
tic, it
the agency has access to
status.
lump-sum
Although we do not view
first
unemployment
transfers conditional
this
assumption as
employed and unemployed workers and allows us
on
realis-
to focus
on
insurance rather than redistribution.
5
A
Behavioral Test
Optimal unemployment benefits maximize a worker's after-tax reservation wage
the tax
is
we need
set to balance the
to
know
is
budget in equation
how a balanced-budget
after-tax reservation wage. It
is
(7).
To
w — r when
see whether this condition holds,
all
increase in taxes and benefits affects a worker's
not necessary to
make any assumptions about
risk- aversion,
discount rates, the speed of finding a job, the duration of a job, the distribution of wage offers,
or about
utility
is
While
whether workers have liquidity or must consume hand-to-mouth since workers'
a monotone function of the after-tax reservation wage
this result
is
theoretically appealing,
10
it
may be
w—
t.
difficult to
implement because
it
may be hard
to discern
how much
principle this question might be
left
taxes must rise to balance an increase in benefits.
to a budgetary authority like the Congressional
but such an organization would
Office,
in benefits raises
unemployment
still
unemployment duration.
Budget
need to understand how much the increase
Instead,
we show that
benefits affect the pre-tax reservation wage, then
if
we can observe how
we can
use information on
the elasticity of unemployment duration with respect to benefits to characterize
must change and hence to characterize optimal
5.1
In
how
taxes
policy.
Theory
Equation
benefits
(4) or ecjuation (6) implies
and
taxes, w{b,T).
It
that the reservation wage depends on unemployment
follows that the resource constraint (7) defines taxes as a
function of benefits,
Dib,T{b))b
where D{b,T)
=
1/A(l
—
=
aTr{b),
(8)
the expected duration of an unemployment spell.
F(iZ)(6, r))) is
Differentiate this with respect to b to get
^
^^^
where subscripts denote partial
or
bD,{b,T{b))
D{b,rib))
'
aT-bD,{b,r{b))
derivatives.
With
CARA
utility
and either perfect
liquidity
hand-to-mouth consumption, the reservation wage and hence unemployment duration
depends only on the sum of benefits and taxes
so
+
Db
=
Dr. Then letting Sofi
=
(see
equation 4 and equation
6,
respectively),
bDi,{b,T)/D{b,T) be the the elasticity of unemployment
duration with respect to unemployment benefits, we can write the previous equation as
_
D{b,T{b)){l
+
en,)
^'-
}'-- c.T--D{KT{b))eo^,
Next, since unemployment benefits should maximize
for
optimal benefits
'w{b, T{b))
T{b),
a necessary condition
= Wr
under
is
W,{b,T{b))+Wr{b,T{b)y{b)
where as usual subscripts denote
and so combining
—
this equation
partial derivatives.
with equation
11
=
T'{b),
Again, Wb
(9) gives
CARA
our test for optimal benefits:
utility,
Proposition 4
//
unemployment
benefits are optimal,
If
—^(l+EAfc)-
=
^h
the left-hand-side of equation (10)
is
benefits has a big effect on the reservation
tax cost, and so a small increase
Roughly speaking, the
employed.
r
—
>
Qt
0,
larger than the right-hand-side, an increase in
wage and hence on workers'
^-^
represents the fraction of time that a worker
suppose we pay a worker 1/r each period she
precisely,
the worker starts off unemployed, the expected cost
If
—
*
r, so this
unemployment
rate u
is
= y^-
At an optimum, a unit increase
employment duration with respect
to
unemployment
than the unemployment
slightly larger
is
^
^p
rate,
but
unemployment
in
1
i.e.
un-
the
benefits
plus the elasticity of un-
benefits. If there is discounting,
in practice
is
In the limit as
.
just the fraction of time the worker spends unemployed,
should raise the reservation wage by the unemployment rate times
is
utility relative to the
welfare-improving.
is
coefficient
More
spends unemployed.
(10)
the difference
is
^-^
quantitatively
small.
5.2
Measurement
To implement
the test proposed in Proposition
=
set the interest rate at r
We set expected
T = 165 weeks,
0.001, equivalent to an annual interest rate of 5.1 percent.
unemployment duration
=
at Z>
According to Meyer (1990,
of finding a job with respect to benefits
10 weeks and the duration of a job at
unemployment
consistent with a 5.7 percent
the U.S. since 1948.
think of the time unit as a week and
4,
p.
779), the elasticity of the hazard rate
—0.88; since the hazard rate
is
=
expected unemployment duration, this implies eo^b
larger than 0.5,
summary"
which Krueger and Meyer (2002,
of the literature,
unemployment
benefits.''
rate, the average value in
p.
0.88.
2351)
call
£D,b
=
0.88, the right
hand
the inverse of
This estimate
is
somewhat
"not an unreasonable TOugh
and so provides a conservative bound
With
is
for the cost of raising
side of equation (10) evaluates
^ The elasticity
£D,b is partial, holding taxes constant, not the elasticity of duration with respect to an
increase in benefits and a balanced-budget increase in taxes. Most of the theoretical literature has focused on
the latter concept, but our reading of the empirical literature suggests that
and so we define the
elasticity
is
ela.sticity
that
way
again the conservative choice.
unemployment
benefits
and employment
with respect to benefits alone
is
here.
If
it
measures the partial
elasticity
In any case, interpreting Meyer's estimates as a partial
they
the impact of a balanced-budget change in
one can show that the elasticity of duration
in fact give
taxes, iD,b
slightly smaller, £D,b
=
=
12
0.88,
£D,bCtT/{{^
+
£D,b)D
+
Qt)
=. 0.78.
to 0.116.
Reasonable parameter changes do not much
unemployment duration
T=
330, so the
is
twice as long,
unemployment
rate
is
D =
affect this
number. For example,
but job duration
20,
is
if
also twice as long,
unchanged, the right hand side increases slightly to
0.125.
There are several studies that estimate the responsiveness
unemployment
benefits. -^^ In our view,
none of these calculations
different answers they provide point to the
ness of reservation wages to benefits.
History
files for
Florida, a 5 percent
need
for
more
wage
of the reservation
is definitive.
to
Instead, the
precise estimates of the responsive-
Fishe (1982) uses the Continuous
Wage and
Benefit
sample of state residents from 1971 to 1974. He
infers
the reservation wage from information on actual wages. His Table 2 shows that a $1 increase
in potential
If this
A
weekly benefits raises the (unobserved) reservation wage by $0.44.
estimate
is
correct, there
$1 balanced-budget increase in
is
a substantial gain from raising unemployment benefits.
unemployment
benefits raises the after-tax reservation
wage
by
Wb{l
+
T
(6))
number
or $0.37 using Fishe's (1982)
this raises the welfare of all
-r{b) =
is
Measuring utiUty
tt);,.
all
per year. Of course, even
if
consumption-equivalent
study a supplement to the
^°
[0.13,0.42].
The
year. This
is
equivalent to
these estimates are correct, they are only correct locally. Raising
yield $2.6 trilhon per year in additional
Feldstein and Poterba"( 1984)
"self-fepofted TeservaticTn wages."
May
1976 Current Population Survey (CPS) that includes such
1
percentage point increase in the ratio
wage wq
raises the ratio of the reservation
wage wq by somewhere between
0.13 and 0.42 percentage points, so
benefits b to the previous
to the previous
Raising unemployment
workers by $0.37 per week, at a cost of $2.6 billion
In their Table 4, they report that a
unemployment
G
them 37
utility.
Another approach uses
W\,
all
week would probably not
benefits by $1000 per
information.
in time.
week would cost approximately $400 million per
(somehow) raising the consumption of
w
as giving
revenue neutral. Put differently, there are about 135 million workers in the U.S.
benefits by $1 per
wage
consumption,
dates in the future, but the increase in unemployment
economy, with about 7.7 million unemployed at any point
of
in units of
unemployed workers by the same amount
cents of additional consumption at
benefits
for
,
lowest slope estimate
for job losers
is
on layoff and the highest
is
for
Early but indirect evidence that the reservation wage responds to unemployment benefits comes from
Ehrenberg and Oaxaca (1976), who
wage jobs.
find that workers
who
13
receive higher
unemployment
benefits get higher
other job losers; the slope estimate for job leavers
substantial gains from increasing
unemployment
is
0.29.
This study also therefore suggests
benefits. Curiously, Feldstein
and Poterba
(1984) interpret their estimates of the responsiveness of reservation wages to benefits as an
argument
shows
for
that,
benefits
On
lowering unemployment benefits because of the moral hazard costs.
on the contrary,
must be serving
if
the reservation
their purpose,
wage
is
Our model
sufficiently responsive to benefits,
then
improving the welfare of unemployed workers.
the other hand, some more recent estimates of Wb from other countries are smaller.
For example, a recent study by Bloemen and Stancanelli (2001) uses self-reported reservation
wages
They
for
unemployed workers
in the
Dutch socio-economic panel from 1987
report in their Table 4 that a 1000 Florin increase in
to 1990.
unemployment income
raises
the reservation wage of household heads by 4.4 percent and of spouses by 9.0 percent,
though the
wage
is
latter figure
is
not statistically different from zero. Since the
mean
is
0.07 for both groups.
Taking
permanent
Of
5 cent increase in consumption.
this calculation since
compares estimates of the
it
course, there are
elasticity of
the U.S. with estimates of the slope of the reservation
is
it
equivalent
some problems with
unemployment duration from
Still, it clarifies
the need for more
of Wb-
Extensions
6
We
and up-to-date estimates
the
wage function from the Netherlands,
a country with relatively high unemployment benefits.
precise
1),
this small estimate at face value,
suggests that current benefit levels are too high and that reducing benefits by $1
to a
reservation
1521 Florin for household heads and 828 Florin for spouses (see their Table
estimated value of Wb
al-
think the most attractive feature of the behavioral test for optimal unemployment insur-
ance
is
that, while
of the model.
it is
theoretically well-grounded,
it
does not rely on
much
of the structure
For example, we have already shown that we do not need to know whether
workers have easy access to financial markets or no access at
all.
In this section,
we
dis-
cuss several modifications of and extensions to our basic framework in order to establish
the robustness of our approach. Each of these modifications alters the formula for
reservation
wage
reacts to benefits, but none of
test in Proposition 4.
To
them
simplify the presentation
we
and keep the mathematical formalities to a minimum.
14
how
the
substantially changes the behavioral
discuss each
new element
separately
To
and Discount Rates
Different Interest
6.1
simplify the exposition
discount rate.
we have assumed throughout that the
affects
consumption,
level-shift in
is
it
is
equal to the
While the relationship between
p.
easy to show that with
CARA
preferences the effect
is
and p
r
simply a
consumption:
=
Cuia)
where 7
is
unemployment insurance does
Fortunately, our characterization of optimal
not depend on the relationship between r and
interest rate
+w—
ra
T
-\
,
r7
the coefficient of absolute risk aversion Therefore the objective of the unemploy-
ment insurance agency
budget constraint
to maximizing the after-tax reservation
is still
in equation (7)
and so the characterization
in
wage subject
equation (10)
is
to the
unchanged."'^
Heterogeneity in Job Length
6.2
we assumed
In our baseline model,
that
We now
only in the wage opportunity.
jobs last for
all
T
periods and are heterogeneous
prove that our results easily extend to the case
when
jobs differ both in terms of their wage offer and in terms of their duration.
Suppose that workers sample jobs distinguished by
some
joint distribution function
reservation
wage
accepting
rule,
F{w,T).
all
a
about how long the job
indifferent
is
consuming ra
lasts.
straightforward to prove that workers use a
is
indifferent
about accepting the job and therefore
unemployed worker with
In particular, an
assets
about accepting a job offering her reservation wage forever, and therefore
+w —
t
forever.
This pins the value of unemployment, unchanged from
eciuation (1) in the case with liquidity
both
wage-duration pair {w,T) from
jobs that pay at least w, independent of T. Intuitively, a
worker employed at her reservation wage
indifferent
It is
their
cases, a worker's utility
is still
and equation
(5) in
the case of financial autarky. In
increasing in the after-tax reservation
wage
w—
t.
Optimal unemployment insurance maximizes the after-tax reservation-wage- subject the
resource constraint, a slight generalization of equation
(7):
u
E(q;7-|u;
>
'w)t,
X{l-Fiw))
where E{aT\w > w)
reservation wage.
^'
If
is
w
the expected value of
and
T
ar
are independent,
conditional on a
E^arlw >
iD)
wage draw exceeding the
= Ear,
the unconditional
This argument ignores any possible general equilibrium effects of unemployment benefits on interest
channel that we think is unlikely to be quantitatively important.
rates, a
15
expected value of or, and so our behavioral characterization of optimal unemployment
surance
is
virtually
unchanged from equation
In general, however, the expected value of
benefits
and
if
=
(10):
ar depends on
the reservation
wage and hence on
taxes. This leads to the following generalization of equation (10):
=
^b
where a
in-
K{aT\w > w) and
(H)
n (l+g£',b~^a,6),
a+V
,
ea,b is
the elasticity of
a with respect
to benefits. For example,
higher wage jobs last longer, an increase in benefits raises both employment duration so
>
£a,b
0.
This equation
employment
is
E(T\w >
spell,
ployment rate
is
easy to interpret
e(ti J>z^Vf d
w).
'
if
r
=
a measures the average duration
so
Since the unemployment rate
^^^ difference in elasticities, eo^b
the unemployment-employment ratio with respect to benefits.
last for longer, the increase in
—
If
unemployment duration, reducing the
tiveness of
unemployment
£&,b,
unemployment insurance has neglected
is
^™'
the elasticity of
higher wage jobs tend to
relevant elasticity
To our knowledge, the
insurance.
^"^^ ^^^
E(r|w>tu)+£>
employment duration from an increase
increase in
6.3
is
of an
in benefits offsets the
and raising the attrac-
existing literature on optimal
this possibility.
Job Loss Risk
To focus on the
risk of
unemployment duration we abstracted from job
that the duration of a job
is
known
as soon as the job
is
accepted.
loss risk
by assuming
In reality, of course,
workers dp _facejuncertainty_ regarding job length, _and. would value insurance against the jisk
of early separations.
If all
it is
job losses are exogenous
optimal to
—that
is, if
there
fully insure against these shocks.
is
no form of moral-hazard involved
The
— then
right instrument to address this
would
not be unemployment insurance, which pays some benefit per period remaining unemployed,
but a lump-sum severance payment at the time of dismissal. The fact that unemployment
insurance
from job
is
not the obvious instrument
loss risk in
for this risk
was part of our motivation
our baseline model. However, even
to understand the determinants of
in this
case
it
may
for abstracting
still
be of interest
unemployment insurance when such severance payments
16
are ruled out.
Once
again, our behavioral test
To be concrete, suppose
s.
all
jobs end according to a Poisson process with arrival rate
Since a worker earning her reservation wage
=
is
about when her job ends, she
indifferent
no uncertainty and therefore keeps her consumption and assets constant:
effectively faces
c^{a,w)
virtually unaffected.
is
=
Cu(a)
ra
+w—
T.
This pins down the value of unemployment, an increasing
function of the after-tax reservation wage with both financial market structures.
The
when job duration
resource constraint changes slightly
uncertain, so equation (7)
is
becomes
r
X{l-F{w))
r
+
-T.
s
Note that l/(r + s) represents the expected present value of a unit of income
analogous to ax in the case of
until a job ends,
This modification carries through the algebra until
finite jobs.
ecjuation (10), yielding the optimality condition
Wh
= ^--—{l + £D,b)-
(12)
r+s
Setting r
=
0.001,
D=
10, s
=
1/T
=
1/165, and £D,b
to 0.124, slightly larger than the 0.116 obtained
=
when
0.88, the right
all
hand
side evaluates
Indeed, the only difference between these numbers comes from discounting.
and yj^Td ^'^^ both equal to the unemployment rate.
Of course, we can also examine what happens when the hazard of job
jobs.
w
If
and
s are
independent, the expected value of l/(r
of eciuation (12). If they are correlated, the
wage exceeding
w
and the relevant
with respect to benefits, exactly as
elasticity
in the
is
in the
+
s)
If
loss varies across
enters the denominator
denominator must condition on the
that of the unemployment-employment ratio
model without job
loss risk.
Costly Search
6^4
We
term
T periods.
r = 0, ^;^
jobs last for exactly
have so
far
focused on a worker's choice of which jobs to accept as the source for the
moral-hazard problem.
An
alternative approach models workers as
effort choice that affects the arrival rate of a
making a
costly search
homogeneous job opportunities. Reality
likely
combines both elements; fortunately, so can our model.
To maintain the
choices,
for
tractability of our
we assume that the search
some
CARA
effort is
disutility of effort function v{e),
specification with
monetary so that the
where
17
e
is effort.
no wealth
effects
utility function
is
on job
u{c — v{e))
Effort improves the arrival of job
opportunities A(e).
With
workers optimally choose some constant level of
this specification,
pendent of their wealth
reduces unemployment income by ^(e*). While
level. Effectively this
this naturally alters the reservation
wage equation
(4), it
does not alter the value of an un-
employed worker conditional on her reservation wage, which
and equation
Similarly, the
(5).
effort e*, inde-
budget constraint equation
is
unchanged from equation
(7) is
unchanged by introduc-
ing search effort, although one must recognize that the arrival rate of job offers
wage
reservation
unaffected by this modification since
ticity of
all
unemployment duration with respect
4
why
to benefits, not the reason
is
and the
our main result
line is that
that matters for deriving equation (10)
unemployment duration. Thus Proposition
is
the elas-
is
benefits affect
unchanged by a monetary cost of search.
Worker Heterogeneity
6.5
Up
The bottom
are both affected by policy.
(1)
to this point
a single worker.
we have considered the problem
Obviously, this problem
The
identical workers.
analysis
is
neous workers, and the agency can
of worker.
We now
fractions
tt".
risk aversion
rion,
finitely
We
also of
immediate relevance
immediately applicable
tailor the
if
there are
if
many
there are
unemployment insurance design
many
heteroge-
to each type
pursue a generalization that allows worker heterogeneity but assumes
that there can be only one
There are
also
is
of an insurance agency confronted with
unemployment insurance
many
policy that applies to
types of workers denoted by n
=
1,2,...
A''
all
worker types.
with population
allow the distribution of wages F'^{w), the duration of jobs T", and the
parameter 7" to depend on the worker type.
we assume worker types
are observable
To motivate our
welfare crite-
and that lump-sum transfers are
feasible.
We
introduce lump-sum transfers to focus the problem on insurance rather than redistribution.
If
lump-sum
transfers were infeasible,
redistributive role,
much
like in an- economy
some
are initially unemployed.
ment
status, the objective
which using equation
unemployment
(1) or
is
benefits have both an insurance
where some workers are
With lump-sum
initially
transfers across types
and
and a
employed and
initial
employ-
simply to maximize average consumption-equivalent welfare,
equation
(5) is
yv
J2^V-T,
(13)
n=l
where w^ represents the reservation wage used by a type
18
n.
We
unemployment insurance agency's budget balances when averaged
require that the
across types:
bD =
Ta,
(14)
where
—
U=
are, loosely speaking, the
to
—
and
:
r,
a
=
average duration of unemployment and employment spells, weighted
downplay workers who experience fewer unemployment
spells, either
ployment duration Z?" or their employment duration T" and hence a"
=
In the special case of r
D'^/{D^
+ T")
of their
0,
is
=
Since type
T".
ratio
when
longer.
n workers spend a
=
r
To summarize, optimal
1
E:=i(1-^^"K"
fraction u^
=
u
Thus D/a measures the employment-
the population unemployment rate.
unemployment
is
unemployed, these expressions simphfy further:
life
^
where u
a"
because their unem-
0.
policy consists of a choice of benefits and taxes which maximizes
the average after-tax reservation wage in equation (13) subject to the budget constraint in
equation (14). This gives the following necessary condition for optimal policy, analogous to
equation (11):
TV
J]<7r" =
where w^
is
benefit level
The
D
^(1
+ £^_, -
(15)
£^,fc),
the derivative of the reservation wage of type
n workers with
respect to the
h.
left-hand-side of this equation uses the population weights
tt"
and thus corresponds
to studies like Fishe (1982),
who
Florida's population. This
not necessarily equal to the average value of ui"
is
infers reservation
wages from a representative sample of
among unem-
ployed workers, the quantity that Feldstein and Poterba (1984) and Bloemen and Stancanelli
(2001) measure using self-reported reservation wages.
When
the
there
is
no discounting, the right-hand-side of ecjuation (15) only requires data on
unemployment
rate
and
its
b
responsiveness to benefits:
,
'^
a+
D
^ ^bb~
^OL,b)
—
19
"(I
+
£u,6
—
ei-u,b))
where
e^^b
and
ei_u,6 are the elasticity of the
unemployment
rate
and the employment rate
with respect to benefits. With a quantitatively reasonable amount of discounting, this ap-
proximation
A
7
The
is
likely to
be
close.
Consumption-Response Test
goal of this section
is
model with existing
to link our
tests for optimal
unemployment
insurance which are based on the response of consumption to becoming unemployed (Baily,
1978; Gruber, 1997; Chetty, 2005).
and show how we can use the
full
To do
this,
we return
to the
benchmark model
of Section 2
structure to derive a test linking the decline in consumption
during an unemployment spell to risk aversion and the elasticity of unemployment duration
Our exact
with respect to benefits.
depends on whether workers have
test
liquidity.
they do, our test looks at the average drop in consumption during an unemployment
In the
hand-to-mouth model, our
test
examines the
unemployed worker and a worker employed
diflference in
If
spell.
consumption between an
at her reservation wage. It should
be clear that
each of the extensions analyzed in Section 6 would potentially introduce further modifications
to our consumption-response tests since, in contrast to our behavioral test, these tests build
on the
full
structure of the model including the determinants of consumption and reservation
wages.
possible to derive other consumption-based tests that do not rely heavily on the
It is
structure of the model, including ones which are identical in the hand-to-mouth and liquidity
cases (Chetty, 2005).
marginal
we
utility of
Unfortunately, such tests
a worker
so that
we have
to
tests
we
on the
make some
we can extrapolate the
consumption
us to
compare the average
lifetime
when employed and when unemployed. To implement such
either need a very rich data set
individuals or
tell
lifetime
path of consumption
implicit assumption about the
desired
moments from a
for a large
tests,
panel of
economic environment
limited data set.
In contrast, the
derive here use the full structure of an explicit model, including whether
workers have access to
liquidity, to derive expressions
with modest data requirements. Both
of these approaches highlight the need for implicit or explicit assumptions
on the structure of
the model, especially workers' financial environment, which our behavior test largely avoids.
20
Workers with Liquidity
7.1
We
start with the case
when workers have
access to financial markets.
In this case, our
consumption-response test relates the speed of decline in consumption to the elasticity of
unemployment duration with respect
unemployment
to
Proposition 5 Assume workers have
benefits:
access to financial markets. If
are chosen optimally, the expected absolute decline in
unemployment
benefits
consumption during an unemployment
spell is
Iot +
D
eD,b
,-,„.
(16)
.
ar
7
1
Alternatively, the expected percentage decline in
should be
aT
1
a
where a
+D
+ ^D,b
consumption during an unemployment
spell
Sdm
ar
1
'
+ eD,b
the coefficient of relative risk aversion evaluated at the consumption level at the
is
start of the
unemployment
The proof
is
=
a
spell,
mostly algebraic.
7Cu(ao).
First, take the partial derivative
with respect to
b of
both
sides of equation (4), holding fixed the tax rate r:
"f{wh
—
1)
= — iDfcQxA
u'{raT{u>
/
—
u)))dF{w).
Jw
= —^u{c), we
5 = 6 -H r gives
Since u'{c)
sion for
can eliminate the integral using equation
I
Second, note that while a worker
is
B
r,
consumption
falls linearly
ar
unemployed, assets
where the second equality uses equation
by
I
f
(4).
Solving this expres-
+D
fall
at rate a
=
ra + b — Cu{a)
= B — w,
Since a unit decrease in assets reduces Cu{a)
(2).
during an unemployment
spell, c^
= r{B —
w). Substitute
from the previous equation.
1
arj
f
ar +
\
\ujb
D
D
This holds for any tax and benefit policy. At the optimal policy, we can eliminate Wb using
equation (10) to get
Dc^
=
7
Ot
21
l+£D.b'
Finally,
if
an unemployment
spell lasts for
t
periods, the drop in consumption
density of the duration of an unemployment spell
sumption during an unemployment
spell
in
most consumption-response
e~^^^/D, so the expected drop
in
The
con-
is
Combining these equations gives the condition
As
is
is Cut.
Proposition
in
5.
optimality condition relates the average
tests, this
decline in consumption to the elasticity of duration with respect to benefits, but there are
some important
differences:
(i)
we use the
unemployment duration with
partial elasticity of
respect to benefits holding taxes fixed, ^d^, whereas previous studies have considered the
effect of
a balanced budget increase in benefits and taxes;
average decline in consumption during an unemployment
pression
is
somewhat
These points need
(ii)
the expression describes the
spell;
and
(iii)
the elasticity ex-
different than in previous work.
we use the
First
clarification.
partial elasticity holding taxes fixed
because we believe this corresponds to the empirical evidence on the responsiveness of un-
employment duration
ments, workers
who
to
unemployment
receive higher
benefits.
For example,
unemployment
response to policy experi-
in
benefits are typically not expected to
higher subsequent taxes. Similarly, in cross-sectional data, workers
employment
who
pay
receive higher un-
benefits do not typically pay proportionately higher taxes. In contrast, existing
studies measure the elasticity of
unemployment duration with respect
+
b{D,{b,T)
to benefits as
Dr{b,r)T'{b))
Dib,T)
where
r'(6) is the
change
in taxes
required to keep the budget balanced.
analysis fully incorporates the balanced-budget requirement.
the elasticity
is
is
(ii),
Baily's (1978) original analysis
simply a question of how
and Gruber's (1997) subsequent
based on a static analysis. These papers focus on the discrete drop
between employment and unemployment. For example,
Baily's (1978) test,
for
a worker
course, our
defined.
Turning now to point
work
It is
Of
who
is
Gruber (1997) uses PSID data
employed
in year
t
to look at the
and unemployed
Chetty (2005) develops a version of Baily's
in his empirical
in
year
f
-I-
in
consumption
implementation of
drop in food consumption
1.
In his
dynamic
analysis,
test that suggests looking at the difference in the
average marginal utility between employment and unemployment over the worker's entire
22
lifetime. Indeed,
a similar condition can be derived for our model. Unfortunately, measuring
the required difference in marginal utilities
That
empirically impractical.
is
is,
in general
it
does not equal the consumption drop used in Gruber (1997), nor the average consumption
drop during unemployment required by our
Point
now
(iii) is
easily explained. In these papers the optimality condition equates
measure from consumption data to the
elasticity of duration. Instead,
involving the elasticity, but not equal to
differ,
so
test.
it.
find
an expression
As explained above, the consumption measures
should not be surprising that the optimality conditions
it
we
some
call for
equating these to
different expressions involving the elasticity.
To implement
this test,
we plug the usual
values r
0.88 into equation (16). In addition,
assume that the
the start of the unemployment spell
is
c
=
2.
Then
=
0.001,
optimal.
is
unemployment
We know
If
of
would
spell,
is
eD,b
=
the unemployment benefit
is
smaller, a decrease in
raise welfare.
but there
is
some
indirect evidence based
expenditure. Gruber (1997) reports that food expenditures
a worker
if
no direct evidence on the magnitude of the decline
an unemployment
and
10,
the model predicts that consumption
instead the observed decline in consumption
benefits
D=
165,
coefficient of relative risk aversion at
should decline by 25 percent during an unemployment spell
level
T=
fall
in
consumption during
on food consumption and
by about
6.8 percent
employed one year and unemployed the next. Aguiar and Hurst (2005)
when
find that
the unemployed spend 19 percent less on food than do the employed using cross-sectional
data; however, because of an increase in time spent on shopping
translates into only a 5 percent drop in food consumption.
elasticity of
food consumption
is less
than
1,
it
consumption of other goods declines more than
addition, even
if
seems
and food preparation,
Of
course, since the
likely that the
this during
this
income
expenditure on and
an unemployment
spell.
In
food consumption could proxy for total consumption, these measures do
not generally represent the average decline during a
spell.
We
conclude that, after viewing
the available evidence through the lens of our consumption-response
whether current benefits are much too high, much too low, or just
test,
right,
we
even
if
are unsure
we
are sure
workers have access to liquidity.
7.2
Hand-to-Mouth Workers
We now
turn to hand-to-mouth workers.
In this case, our test relates the difference in
consumption between a worker at the reservation wage,
23
iu^^^
— t, and
an unemployed worker.
b,
to the elasticity of
unemployment duration:
Proposition 6 Assume workers must consume
ment
income
in each period.
If
unemploy-
chosen optimally, the difference between the consumption of an employed
benefits are
worker
their
at the reservation
wage and the consumption of an unemployed worker
is
-log{l +eD,b)
(17)
7
when a worker
Equivalently, the percentage drop in consumption
reservation
loses a job
paying her
wage should be
-log(l +£D,b),
a
where a
is
the coefficient of relative risk aversion evaluated at the consumption level of a
worker earning the reservation wage,
Again, the proof
u'iw^"'
r)
is
+ arX
iv^^^
—
r.
algebraic. Totally differentiate equation (6):
H
u'(u)^"*
- T)dF{w)] «"* +
u;^"V'(6)
-
r'(6))
noo
=
The
left-hand-side
Then
is
zero
if
u'{h)
- qtA
benefits are chosen optimally, to
u'{w
/
maximize
-
T)dF{w)T'{b).
'u)^"'(6,r(6))
—
r(6).
use equation (9) to eliminate r'{b) from the right-hand-side:
Q;r(H-e£>,6)
u'{h)
E(u'(u;-r)|u;
>
ar-Depfi
iZ)^"')
where the denominator on the left-hand-side
is
the expectation of the marginal utility of
consumption conditional on the wage drawn from
Under
CARA
utility, this simpfiffes lurther
F
exceeding
u)^"*.
since the ratio of marginal utility
as the ratio of utility,
u{b)
E{u{w — t)\w >
ctrjl
tD'^"*)
+
gD,fe)
ar — DeD,b
Since equation (6) implies
ar
u[b)
E{u{w - t)\w >
{D + ar r^'^Zr^
w^^')
24
-^
is
the
same
the previous two equations give
u{b)
=
UiW
Since u{c)
Once
=
—e~'^'^,
+ eD,b-
1
Proposition 6 follows immediately.
again, there are three important differences between our condition
formulas based on the response of consumption to unemployment:
elasticity
(h)
eo.b',
we use the
we use the
(i)
between the lowest acceptable
difference
and most existing
level of
partial
consumption
while employed and consumption while unemployed, rather than the average difference; and
the final expression
(iii)
than
is
slightly different
in previous work,
with log(l
+
e)
rather
e.
Given the usual values of
mouth model
is
than
eD,b
=
0.88 and
a
=
2,
the critical question in the hand-to-
whether the consumption of a worker employed
is
32 percent more than the consumption of unemployed workers.
need to know both the drop
reservation wage.
in
To measure
this,
we
consumption following unemployment and the worker's
Data on food expenditures and consumption from Gruber (1997) and
Aguiar and Hurst (2005) suggest that many workers
raise their
wage
at her reservation
consumption by
less
may be
willing to take jobs
which
than 32 percent, which suggests that workers are currently
over-insured. However, this conclusion depends strongly on the hand-to-mouth hypothesis;
Proposition
1
consumption
shows that
is
if
a worker with liquidity takes a job at her reservation wage, her
unchanged.
In our view, there are three drawbacks to the consumption-response tests
sented here.
at
The
first is
that the
depend on the structure of
of reliable, high frequency
moments
of the
financial markets.
consumption data
for
we have
pre-
consumption data that we should look
The second drawback
the unavailability
is
goods other than food.
In contrast, the
behavioral test requires data on the responsiveness of reservation wages to unemployment
benefits.
This can either be measured using self-reported reservation wages or inferred from
the observed pattern of accepted wages.
tions like the predictability of job loss
modifications
is
likely to further
is
robust to assump-
of heterogeneity.
Introducing these
Finally, the behavioral test
and the extent
change the consumption-response
25
tests.
Conclusions
8
This paper argues that the after-tax reservation wage measures the well-being of unemployed
workers.
Any
policy that raises the average after-tax reservation
and the benefit can be measured by the average increase
While we have applied
insight
is
more
this
the unemployed
liquidity.
—examples
Going beyond
therefore beneficial,
in the after-tax reservation
wage.
shows that the after-tax reservation wage
this paper,
when
evaluating any policy towards
include severance payments, reemployment bonuses, training
and job search centers
subsidies,
is
mainly to thinking about optimal unemployment insurance, the
general. For example. Proposition 3
encodes the value of
wage
—the key question
is
whether the policy
raises the after-tax
reservation wage.
We have assumed CARA preferences throughout
is
the body of this paper. This assumption
convenient but probably not essential. Proposition 2 shows that the after-tax reservation
wage measures a hand-to-mouth worker's welfare regardless
of her preferences. Moreover, in
our companion paper Shimer and Werning (2005), we argue that the behavior of a worker
with constant relative risk aversion
a worker with
CARA
have access to
liquidity.
this
true:
is
of an
preferences and the
preferences
same
is
quantitatively similar to that of
coefficient of risk aversion
Indeed, our intuition for the proof of Proposition
if
1
both workers
explains
why
the only reason the after-tax reservation wage would not measure the welfare
unemployed worker
nently.
(CRRA)
While
this
is
is if
workers are willing to take jobs temporarily but not perma-
a theoretical possibility,
we doubt that the phenomenon
is
quantitatively
important.
Finally, our
tion
wage
to
paper implies that a key empirical issue
unemployment
benefits or other labor
is
market
the responsiveness of the reservapolicies.
Some
existing estimates
suggest that reservation wages are very responsive, implying huge gains from increasing
unemployment
benefit levels.
-levels^are tocrhigh.
precise estimates of
Other estimates are much smaller and imply current benefit
An important-goal^rfuture
how
empirical research should be to obtain
labor market policies affect reservation wages.
26
more"
Appendix
A
A
Proof of Proposition
Convenient
CARA
Vu{a)
Property.
=
We
1
start
and
-u{cu{a))
by proving that
=
V{a,t,w)
r
where V{a,t,w)
is
We
With
u '(c,)
CARA,
u'{c)
walk with
is
= — 7u(c),
=
V5'
taken using
>
s,
the information available
all
when
Cg is
chosen.
so the Euler equation implies implies per-period utility
is
With
random
a
=
e('-''«^'-^)E,u(c30.
(19)
consider the lifetime utility Vg at time s of a worker facing some stochastic future
all
future dates
s':
poo
Vs=
pCG
e-''^''-'^^,u{c,,)ds'
Js
=
/
J
The second equation
-|
e-P^''-'^e-^'-P'^^''-'^u{c,)ds'
ra'
-u{c,).
'
uses equation (19) while the third equation solves the integral.
value functions follow immediately from equation (18).
consume
=
s
Shape of the Consumption and Value Functions. The shapes
to
periods remaining on
drift:
consumption path at
a'
t
we have an Euler equation
e(^-^)(^'-^)E,T.'(cy)
u(c,)
Now
a,
prove this for the general case where r and p are not necessarily
additively separable utility,
where the expectation
(18)
r
the value of an employed worker with assets
the job, and a wage w.
equal.
-u{c{a,t,w)),
more than a worker with
assets
of the
It is feasible for
and
vice- versa,
consumption and
a worker with assets
assuming the two have
the same employment duration and wage. This implies
c{a,t,w)
=
ra
+
c{Q,t,w).
Next, consider two employed workers, one at a wage
has
t
w
and another
(20)
at a
wage w'
.
If
each
periods remaining in his job, the present value (as of the end of the previous period)
27
.
of the difference in earnings
is
{w
—
w')
e~^^ds
/
=
—
at{w
w').
Jo
If
the present value difference happens to equal the difference in the two workers' asset levels,
they have the same resources and
=
c{a,t,w)
Combining with equation
for
behave the same:
will
c{a
+
—
at{w
w'),t,w')
(20) gives
c{a, t,w)
=
finished,
t
r{a
at{w - w'))
+
+
c(0,
w')
t,
(21)
any w'
Note that
c{a,0,w') for
of a worker
if
the job
w
all
who
is
and
w'.
and a^
=
0,
the worker
=
convenient to define Cu{a)
It is
starts a period
=
unemployed and
V-u{a)
=
V{a,T,vj) and so she takes the job
equivalent to c{a,T,w)
>
c^{a),
V{a,T,w) >
if
is
unemployed so
c{a, 0,
w)
=
c{a,0,w) as the consumption
V{a,0,w) be her value function.
Reservation Wage. Consider a worker who accepts a job
is
.
at
wage w. Her value function
Using ecjuation
14(a).
(18), this
which by equation (20) implies a reservation wage
is
rule,
independent of assets, satisfying
c{0,T,w)
Combine equation
sion for the
(22)
=
w, to get a convenient expres-
consumption of a newly employed worker:
T,w)
=
r{a
+ ariw -
Behavior of the Employed. A worker who
That
c^iO).
(22) with equation (21), evaluated at w'
c{a,
in her
=
w))
+
c„(0).
(23)
starts a period with
i
>
periods remaining
job faces no uncertainty until the job ends and therefore keeps consumption constant.
is,
for
any
t
>
0,
dc{a{t),t,w)
_
dt
where
d{t)
=
ra
+w—
t
—
Ca{a,
c{a{t),t,
t,
w) {ra
w)
is
the rate of increase in assets. Differentiating gives
+w—
r
—
28
c{a,
t,
w)^
=
Ct{a,
t,
w),
.
where subscripts denote
partial derivatives.
Note from equation
a differential equation for c as a function of
this
is
The
solution
(21) that Ca{a,t,w)
=
with terminal condition equation
t
r,
so
(23).
is
c{a,
t,
w)
= ra-{w-
r) (e'^^"*)
This provides an alternate expression
Simplifying this equality pins
down
"
l)
+
e"^^"*^
[rariw -
u))
which we know
for c{a,0,vj),
+
is
c„(0))
(24)
equal ra
+
Cu(0).
the constant in the consumption function,
c„(0)
= w-T.
(25)
Substituting equation (25) into equation (23) yields the consumption functions for unem-
ployed and employed workers found in equation (2) and equation
(3),
while substituting
these into equation (IS) gives the value of an unemployed worker in equation
remains
is
(1).
All that
to determine the worker's reservation wage.
Behavior of the Unemployed. Expected marginal
utility for
an unemployed worker
is
a
Martingale. This implies
/oo
ii"(c„(a))<(a)d
where a
c'^{a)
=
=
r,
ra
+b—
+
A
= B—w
Cu{a)
we can rewrite
{u'{c{a,T,w))
/
-
u'{c^{a)))dF{w)
using equation (25). Since u"{c)
—
=
0,
—^u'[c)
=
^~u[c) and
this as
/•oo
-fru{c^{a)){B
- w) =
X
[u{c{a,T,w))
-
u{cu{a)))
dF [w)
Jw
Next, use u{ci)/u[c2)
=
—u{ci
—
C2)
and u{0)
= —1
to get
/•oo
^r{w
-B) =
\
{u{c{a,T,w)
I
Simplify using equation (23) yields equation
worker behavior
in
Proposition
(4).
1.
29
-
c^{a)))
+
l)dF{w).
This completes the characterization of
B
Proof of Proposition 3
We
start with
two
inequalities.
At any
—u{B — w) —
1
The
first
1
=
w>
w,
exp(7(iy
+ u{raT{w —
—
—
B))
> ^{w —
1
> raril + u{w —
vj))
equahty uses the definition of u and the
w)).
inequality uses convexity of the
first
exponential function. To prove the second inequality, note that
a:
>
and y e
When
[0, 1].
x
=
right-hand-side with respect to x
hence e~^
<
any positive
0, this is trivially true.
is
y{e~^
e~^^, so the right-hand-side
x.
If
a;
= ^{w —
w) >
—
is
and y
e~^^).
= rax
G
x.
[0, 1],
—
1
y
>
e~^y
—
ye~^ when
Moreover, the derivative of the
Since y G
decreasing in
B),
and x >
[0, 1]
0,
x
> xy and
Hence the inequality holds
this
is
for
equivalent to the desired
inequality.
Now
suppose
u{B -w) -
w
I
The previous
solves ecjuation (4).
inequalities imply
> j{w - B) =
A
noo
'°°
/
roo
{1
+ u{raT{w —
iv)))dF{w)
> axX
r J U)
It is
[l
+ u{w —
u)))dF{w).
J ill
easy to confirm that the
first
expression
is
decreasing in
w
and the
last expression is
increasing, so the solution to
roo
-u{B -
requires w^^^
<
w. Under
u)^"')
-l^arX
CARA
the reservation wage ecjuation
(l
utihty, u[ci
+
C2)
+ u{w =
w^'''))dF{
—u[ci)u{c2)
,
so this
is
equivalent to
(6).
FfiTatlj^TTmder'finanaal^airt^afky7 equation (^5")^hows"that"an"uriemployed" worker "sTTtitity is"
u{w^^^
— t)/ p. With
The worker
is
access to financial markets, equation (1) shows that
indifferent to the scenarios
the worker's consumption by ra
=
w^"^
if
a
—w
=
(iD^"'
— u))/r,
a reduction
in every future period.
30
it is
u{ra + w — T)/p.
in assets that lowers
References
Acemoglu, Daron and Robert Shimer,
of Political
_ and _
,
Economy, 1999, 107
{5),
"Efficient
Unemployment
Insurance," Journal
893-928.
"Productivity gains from unemployment insurance," European Economic Review,
2000, 44 (7), 1195-1224.
Aguiar, Mark and Erik Hurst, "Consumption versus Expenditure," Journal
Economy, 2005, 113 (5), 919-948.
Alvarez, Fernando and Marcelo Veracierto, "Severance payments
frictions," Journal of Monetary Economics, 2001, ^7(3), 477-498.
Baily,
Martin
Neil, "Some Aspects of Optimal
Public Economics, 1978, 10
(3),
Unemployment
in
of Political
an economy with
Insurance,"
Journal of
379-402.
Bloemen, Hans G. and Elena G.
F. Stancanelli, "Individual Wealth, Reservation
Wages, and Transitions into Employment," Journal of Labor Economics, 2001, 19
(2),
400-439.
Chetty, Raj, "A General Formula
uscript, UC-Berkeley,
September.
Optimal Level of Social Insurance," 2005. Man-
for the
.
Ehrenberg, Ronald G and Ronald L Ocixaca, "Unemployment Insurance, Duration of
Unemployment, and Subsequent Wage Gain," American Economic Review, 1976, 66 (5),
754-66.
Feldstein, Martin, "Rethinking Social Insurance," American Economic Review, March
2005, S5(l), 1-24.
_ and James Poterba, "Unemployment
Insurance and Reservation Wages," Journal of
Public Economics, February-March 1984, 23 (1-2), 141-167.
Fishe,
—
Raymond
P. H.,
"Unemployment Insurance and the Reservation Wage
of the
Unemployed^'-The-Revieuj-of-Econamics-and-Statisiics,- lQ§2—64^-{i)-, -t2=-l-7,
Gruber, Jonathan, "The Consumption Smoothing Benefits
American Economic Review, March 1997, 57(1), 192-205.
of
Unemployment Insurance,"
Hansen, Gary D and Ayse Imrohoroglu, "The Role of Unemployment Insurance in an
Economy with Liquidity Constraints and Moral Hazard," Journal of Political Economy,
1992, 100 (1), 118-42.
Hopenhayn, Hugo and Juan Pablo
Journal of Political Economy, 1997, 105
Nicolini, "Optimal
(2),
31
412-438.
Unemployment Insurance,"
Krueger, Alan and Bruce Meyer, "Labor Supply Effects of Social Insurance," in Alan
Auerbach and Martin Feldstein, eds., Handbook of Public Economics, Elsevier Science
B.V., 2002, chapter 33, pp. 2327-2392.
McCall, John, "Economics
nomics, 1970, 84
(1),
of Information
and Job Search," Quarterly Journal of Eco-
113-126.
Meyer, Bruce, "Unemployment Insurance and Unemployment
Spells," Econometrica, July
1990, 58 (4), 757-782.
Shavell, Steven
and Laurence Weiss, "Optimal Payment
of
Unemployment-Insurance
Benefits over Time," Journal of Political Economy, 1979, 87 (6), 1347-1362.
Shimer, Robert and Ivan Werning,
Staff Report 366, Federal Reserve Bank
600^
"Liquidity
^^
kS
and Insurance for the Unemployed,"
December 2005.
of Minneapolis