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Massachusetts
Institute of
Technology
Department ot Econonnics
Working Paper Series
LIQUIDITY
AND INSURANCE FOR
THE
UNEMPLOYED
Robert Shimer
Ivan Werning
Working Paper 05-23
September 22, 2005
Room
E52-251
50 Memorial Drive
Cambridge,
MA 02] 42
This paper can be downloaded without charge from the
Network Paper Collection
http://ssrn.com/abstract=^^HH^
r^
Social Science Research
at
A (^'\'L^
MASSACHUSETTS INSTITUTE
OF TECHNOLOGV
i
I
NOV
1
h 2005
LIBRARIES
Liquidity and Insurance for the
Unemployed*
Ivan Werning
MIT, NBER and UTDT
Robert Shimer
University of Chicago and
NBER
iwerning@mit.edu
shimerOuchi cago.edu
First Draft: July 15, 2003
This Version: September
22,
2005
Abstract
We
study the optimal design of unemployment insurance
opportunities over time.
absolute risk aversion preferences
unemployment,
on the duration of the
from
of
this
benchmark,
more elaborate
roles for policy
to
smooth
spell,
it is
and the
desirability
workers have constant
optimal to use a very simple policy: a constant
a constant tax during
and
When
employment that does not depend
free access to savings using a riskless asset.
Away
for constant relative risk aversion preferences, the welfare gains
policies are minuscule.
toward the unemployed:
their consumption;
and
(b)
(a)
Our
results highlight
two largely distinct
ensuring workers have sufficient liquidity
providing unemployment benefits that serve as
insurance against the uncertain duration of unemployment
1
workers sampling job
We focus on the optimal timing of benefits
of allowing workers to freel}' access a riskless asset.
benefit during
for
spells.
Introduction
There
wide variation
is
(Figure
1).
in
the duration of unemployment benefits across
In Italy, the United
Kingdom, and the United
OECD
countries
States, benefits last for six months.
comments from Daron Acemoglu, George-Marios Angeletos, Hugo Hopenlrayn,
Narayana Kocherlakota, Samuel Kortum, and Ellen McGrattan, and numerous seminar participants on this
paper and an earlier version entitled "Optimal Unemployment Insurance with Sequential Search." Shimer's
research is supported by grants from the National Science Foundation and the Sloan Foundation. Werning
is grateful for the hospitality from the Federal Reserve Bank of Minneapolis during which this paper was
*We
are grateful for
completed.
.
.
CO
O
^^gc&-SS^&
CD
Pi
*^
:3
forever.
b C
o —
S
N
<U
benefits lapse after one year
Which country has
standard argument
G
<U
oj
^
'3
—
CI.
.
C
—
'
S
OT f^
Kh
"3
'i
C
Cj
C3
rd
2;
n°
Ph
M
CO
Duration of unemployment benefits
and Wages 2002, Table 2.2. *Duration
Germany
A
*^
1:
Benefits
In
^^
Qj
ii
CJ
Q
Figure
d a a
3 •"= ^ S
,
in select
is
OECD
countries.
Source:
OECD
unlimited in Belgium.
and France
after five years.
In Belgium they last
the right policy?
for
terminating benefits after a few quarters
that extending
is
the duration of benefits lengthens the duration of jobless spells (Katz and Meyer, 1990).
But benefits
also proAdde insurance
unemployed (Gruber, 1997).
of optimal
Determining which policy
Many
British,
is
best requires a
dynamic model
unemployment insurance. Shavell and Weiss (1979) and Hopenhayn and
(1997) develop such models and
spell.
and help workers maintain smooth consumption while
show that
Nicolini
benefits should optimally decline during a jobless
economists have interpreted these results as broadly supportive of the Italian,
and American version
of
unemployment insurance. But they hinge on some subtle
assumptions, notably a restriction that the unemployed can neither borrow nor save and
so
consume
dual
role:
their benefits in each period.
This means that unemployment benefits play a
they insure workers against uncertainty in the prospect of finding a job and they
provide workers with the ability to smooth consumption while unemployed.
In this paper,
we reexamine the optimal timing
by allowing the worker to borrow and save.
have
of benefits, distinguishing the
Our main conclusion
sufficient liquidity, in either assets or capacity to
is
that
two
roles
when workers
borrow, a constant benefit schedule
of unlimited duration
against
unemployment
temporary drops
Our
in
optimal or nearly optimal.^ The constant benefit schedule insures
is
risk,
while workers' ability to dissave or borrow allows them to avoid
consumption.
two conceptually distinct
results suggest
roles for policy
toward the unemployed.
ensuring workers have sufficient liquidity to smooth their consumption; and sec-
First,
ond, providing constant unemployment benefits that serve as insurance against the uncertain duration of
unemployment
This dichotomy
spells.
Feldstein and Altman's (1998) recent policy proposal for
is
consistent with the spirit of
unemployment insurance savings
accounts (see also Feldstein, 2005).
We represent the unemployed worker's situation using McCall's
(1970) model of sequential
job search. Each period, a risk-averse, infinitely-lived unemployed worker gets a wage offer
from a known distribution.
forever. If she rejects
it,
Our main purpose
is
If
she accepts the
offer,
she keeps the job at a constant wage
she searches again the following period.
to
compare two unemployment insurance
policies.
We
begin by
considering a simple insurance policy, constant benefits, where the worker receives a constant
benefit while she
is
unemployed and pays a constant tax once she
can borrow and lend using a riskless bond.
reservation
wage although her
reservation
wage
is
assets
An
show that the worker adopts a constant
unemployment
benefit
and the employment
duration-dependent consumption
level for the
that depends on the length of the jobless spell.
An
tax, a
insurance agency dictates a
unemployed, funded by an employment tax
The worker has no
access to capital markets
and so must consume her after-tax income
in
wage
the best insurance system possible.
of
The
utility.
then consider optimal unemployment insurance.
off'ers
spell.
insurance agency sets the level of benefits and taxes to minimize
the cost of providing the worker with a given level of
We
employed. The worker
and consumption decline during a jobless
increasing in both the
form of moral hazard.
We
is
or randomization schemes, this
is
each period.
Absent direct monitoring of
The path
unemployment consumption and employment taxes determines the worker's reservation
wage, which the insurance agency cannot directly control.
It
sets this
path to minimize the
cost of providing the worker with a given level of utility.
Our main
result
is
that with Constant Absolute Risk Aversion
(CARA)
preferences and
no lower bound on consumption, constant benefits and optimal unemployment insurance are
^This does not necessarily imply that the Belgian policy
notably in the
maximum
yearly benefit; see
on the optimal duration of
the optimal level.
benefits. In
OECD
is
optimal. Policies differ along other dimensions,
Benefits and
ongoing work
for
Wages
2002, Table 2.2. This paper focuses
a separate paper
we examine
the determinants of
That
equivalent.
the cost of providing the worker with a given level of utility
is,
her reservation wage
the same, and the path of her consumption
is
insurance systems. In both cases consumption
the worker
unemployed, both during and
is
can borrow and save, this
Our
is
constraints in
the same,
the same under both
by a constant amount each period that
after the
unemployment
consistent with a constant benefit
result that the optimal
benefits with borrowing
falls
is
is
and
spell.
When
the worker
tax.
unemployment insurance can do no better than constant
and savings contrasts with a
large literature
on the need
for savings
dynamic moral hazard models.^ Rogerson (1985) considers an environment
which a risk-averse worker must make a hidden
employer's profits.
He
in
effort decision that affects her risk-neutral
proves that optimal insurance
is
characterized by an "inverse Euler
equation,"
'\P{l+r)u'{ct+i),
u'{ct)
where E^
the expectation operator conditional on information available at date
is
particular, since the function "1/x"
is
t.
In
convex,
u'{ct)<EtPil+r)u'{c+,).
An
individual facing this path of consumption would
and hence
consume
less
today and more tomorrow
"savings-constrained" by optimal insurance. In contrast, in our model a worker
is
confronted with the optimal unemployment insurance policy satisfies this Euler condition
with equality.
We
also explore optimal insurance
We
ences.
do
than
CARA
tion.
And
this for
two reasons.
preferences.
We
want
with Constant Relative Risk Aversion
First,
CRRA preferences
are theoretically
(CRRA)
prefer-
more appealing
to explore the robustness of our findings to this
assump-
second, this introduces a nonnegativity constraint on consumption that limits a
worker's debt to the amount that she can repay even in the worst possible state of the world,
Aiyagari's (1994) natural borrowing limit.
We
highlight an interesting interaction
between
a worker's ability to borrow and optimal insurance.
The
breaks
"A
perfect equivalence between optimal
down with
recent example
CRRA
is
preferences, but
unemployment insurance and constant
we
find that our results with
Golosov, Kocherlakota and Tsyvinski (2003),
who emphasize
CARA
benefits
provide an
that capital taxation
may
discourage saving. Allen (1985) and Cole and Kocherlakota (2001) provide a particularh' striking example of the cost of unobserved savings in a dynamic economy with asymmetric information. They prove that
if
a worker privately observes her income and has access to a hidden saving technology, then no insurance
possible.
is
important benchmark. As
CARA
in tlie
case,
optimal unemployment insurance dictates a
declining path of consumption for unemployed workers and an increasing tax
upon reemploy-
ment. However, the implicit subsidy to unemployment, the amount that a worker's expected
from the insurance agency
lifetime transfer
rises if
she stays unemployed for an additional
period, increases very slowly during a jobless spell.
By
But
if
its
very definition, constant benefits are always at least as costly as optimal insurance.
the worker has enough liquidity so as to have a minimal chance of approaching any
lower bound on assets, the additional cost of constant benefits
weeks
(or
about 0.01 seconds) of income
in
our leading example.
optimal time-varying and time-invariant subsidy
If
the worker
is
is
minuscule,
The
less
difference
than 10~^
between the
also very small.
near her debt limit, the difference between constant benefits and optimal
is
unemployment insurance
is
larger.
This
is
because benefits are forced to play the dual role of
providing insurance and smoothing consumption. However, using benefits to create liquidity
in this indirect
problem
The
way
is
likely to
be
less efficient
than measures designed to address the liquidity
directly.
general message that emerges from our model
should be simple
— a constant benefit and
tax,
is
the following.
CARA
With
that unemployment insurance policy
combined with measures to ensure that workers
have the liquidity to maintain their consumption
for these results
is
level
during a jobless
the optimal
more
time.
unemployment subsidy
risk averse as
consumption
However, this wealth
unemployment
spell
constant.
falls; this
effect
is
explains
is
able to
we note that our use
Nicolini (1997)
CRRA
With
why
risk; as
utility,
a consequence,
the worker becomes
the optimal subsidy increases over
small during a typical, or even relatively prolonged,
provided the worker
Before proceeding,
Hopenhayn and
is
intuition
and consumption that
utility the fall in assets
occurs during an unemployment spell does not affect attitudes toward
Our
spell.
and many
smooth her consumption.
of a sequential search
others,
model departs from
which assumes that there
is
only a job
search effort decision.'^ There are three reasons for this modeling choice. First, our model
produces stark results on optimal policy
trinsically useful.
On
results. Indeed, the
in a straightforward way,
the other hand, the sequential search model
paper most closely related to ours
is
Werning
which we believe
is
in-
not critical for these
(2002), which introduces
hidden borrowing and savings into the Hopenhayn and Nicolini (1997) search
'Shavell and Weiss (1979) allow for both hidden search effort and hidden
is
effort
wage draws. See
model.
also exer-
Ljungqvist and Sargent (2004). However, both of these models assume that employed workers
cannot be taxed and neither examines optimal benefits when workers have access to liquidity.
cise 21.3 in
and some
of his results are analogous to ones
we
constant benefits and taxes are optimal under
monetary.
Despite
this,
and
in contrast to
benefits are not equivalent to optimal
since
preferences
if
the cost of search
is
our results here, in Werning (2002) constant
CARA
utility,
always desirable to exclude the worker from the asset market.
it is
Feldstein and Poterba (1984), a
unemployment
is
number
empirically relevant.
of authors have
jobs.
The
sequential search model
Third, the sequential search model
is
Starting with the work of
documented that an increase
wage and consequently reduces the
benefits raises workers' reservation
which they find
fact.
CARA
unemployment insurance, even with
Second, the sequential search model
at
report here. For example, he proves that
in
rate
a natural one for thinking about this
is
the backbone of most research on equilibrium
unemployment. At the heart of the Lucas and Prescott (1974) equilibrium search model and
of versions of the Pissarides (1985)
sequential search problems.
economy
More
matching model with heterogeneous firms are individual
recently, Ljungqvist
which each individual engages
in
and Sargent (1998) examine a large
in sequential job search
from an exogenous wage
distribution.
This paper proceeds as follows. Section 2 describes the model's environment and the two
we
policies
der
CARA
consider. Section 3 then establishes the equivalence between the
preferences. Section 4 quantitatively evaluates optimal
and optimal constant benefits with
unemployment insurance and
Two
2
begin describing the
two
policies
There
we
unemployment insurance
highlighting the relationship between
hquidity. Section 5 concludes.
Policies for the
We
2.1
CRRA preferences,
two systems un-
common
Unemployed
physical environment of the model.
consider, constant benefits
We
then discuss the
and optimal unemployment insurance.
The Unemployed Worker
is
a single risk averse worker
who maximizes
the expected present value of utility from
consumption,
CO
E_i5;]/3yQ),
t=0
where
/3
function.
<
1
represents the discount factor and u{c)
is
the increasing, concave period utiUty
At the
A
start of each period, a worker can
worker employed at
w
produces
w
the cumulative distribution function
distribution.
The worker observes
she accepts w, she
and
is
of the next period.
If
F.'^
w >
Let
w
units of the consumption
In either case, the worker decides
The worker cannot
The
it
(3-^
The
in the current
unemployed
consume
to
recall past
If
it.
at the start
at the
wage
end of
offers.
is
employed
does not observe the worker's wage, even after she decides
objective of the
unemployment insurance agency
of providing the worker with a given level of utility.
We
assume
is
to minimize the cost
costs are discounted at rate
-1.
Policy
2.2
how much
is
good
assume that an unemployment agency only observes whether the worker
to take a job.''
=
each period; she
in
denote the lower bound of the wage
she rejects w, she produces nothing and
or unemployed. In particular,
r
or unemployed.
the wage and decides whether to accept or reject
the period, after observing the wage draw.
We
good
w
worker receives a single independent wage draw from
employed and produces
future periods.
all
units of the consumption
An unemployed
never leaves her job.
be employed at a wage
policy
we
I:
call
Constant Benefits
constant benefits
constant employment tax
f,
is
defined by a constant
and perfect access
unemployment
benefit
to a riskless asset with net return r
=
j3~^
6,
a
— 1,
subject to a no-Ponzi-game condition.^
Since the worker's problem
is
stationary
we present
it
recursively. Start
by considering a
who is employed at wage w and has assets a with budget constraint a' = (1 + r)a +
w — f ~ c. Since /3(1 + r) = 1, she consumes her after tax-income plus the interest on her
worker
assets Ce{a,w)
=
lifetime utility
is
ra
+w—
f, so that assets are
Ve{a,w;T)
F
=
kept constant,
^—
.
a'
=
a.
This means that her
(1)
finite expectation and that there is some chance of drawing a
some w > 0.
^If the wage were observable, an unemployment insurance agency could tax employed workers 100 percent
and redistribute the proceeds as a lump-sum transfer. Workers would be indifferent about taking a job and
hence would follow any instructions on which wages to accept or reject. This makes it feasible to obtain
the first best, complete insurance with the maximum possible income. Private information is a simple way
to prevent the first best, but other modeling assumptions could also make the first best unattainable, e.g.
''We assume that
positive wage, so
is
F{w) <
continuous and has
1 for
moral hazard among employed workers.
^That is, debt must grow slower than the interest rate, linit-^ao{i + T)~^at > 0, with probability one, where
at denote asset holdings. Together with the sequence of intertemporal budget constraints this is equivalent
to imposing a present-value lifetime budget constraint, with probability one.
Next consider an unemployed worker with assets a and
and
lifetime utility, given policy parameters b
I4(a;
where
If
a'
=
(1
6,
= y°° max jmax
f)
+ r)a + h —
c.
unemployment
benefits
is
6,
r)
denote her expected
"(^"
,
^)
+_^^-
| dF{w).,
c
and saves
a' in
(1).
solution to the Bellman equation defines the worker's
Cu{a), reservation
wage
u)(a),
and next period's assets
Otherwise, she
the current period, and remains
unemployed into the next period, obtaining expected continuation utihty
The
\4(a';6, f).
unemployment consumption
conditional on this period's
a' {a),
Given these objects, the cost of the unemployment insurance system
assets.
(2)
worker chooses whether to accept a job or not.
given by Ve{a,'w:f) in equation
consumes
6,
14 (a;
This must satisfy the Bellman equation
+ /?\4(a^ 6, f ))
An unemployed
she does take the job, her utihty
collects
{u{c)
f.
let
is
defined
recursively by
S{a] b,T)
A
= lb+
worker with assets a
of the
fails
is
F{iu{a))
is
the unemployment benefit b plus the discounted
S{a'). If she finds a job, the present value of her tax
An unemployment
(3)
.
to find a job with probability F{w{a)). In this event, the cost
unemployment insurance system
continuation cost
1
^-^^
payments
"^'"
is
insurance agency chooses 5 and f to maximize the worker's utility
given some available resources and an initial asset level. Equivalently,
we consider the dual
of minimizing the total resource cost of delivering a certain utility for the worker.
optimal constant benefit policy solves ^{vo.a)
Vu{a]b,f)
=
.
=
minj
.^
5(a;
6,
f)
+
(1
+
The
subject to
7-)a
vo.
Since there are no ad hoc constraints on borrowing, a standard Ricardian equivalence
argument implies
cost, so
2.3
it
that:
follows that
Policy
II:
V{a;
C"^(t;o,
b,
a)
f)
is
= V{a
-\-
x]b
— rx,f
independent of
a.
while a worker
rx).
The same
is
true with total
Abusing notation we write
C'^{vq).
Optimal Unemployment Insurance
Under optimal unemployment insurance, a worker who
6t,
-\-
who
finds a job in period
job, for the remainder of her
life.
One can
t
is
pays a tax
unemployed
Tj,
in period
t
consumes
depending on when she finds a
conceive of more complicated insurance policies
where the agency asks the worker to report her wage draws, advises her on whether to take
the job, and makes payments conditional on the worker's entire history of reports.
is,
one can model unemployment insurance as a revelation mechanism
We
problem.
deterministic
Given
{bt}
Her hfetime
prove in Appendix
mechanism
and
A
that the policy
{rj}, consider a
who
worker
in a principal-agent
we consider here does
as long as absolute risk aversion
is
That
as well as
any
non-increasing.
chooses a sequence of reservation wages
{u't}-
utility is
'f-i
The worker
is
unemployed
at the start of period
a wage below Wt, she rejects
it
and her period
she takes the job and gets utility
u{w —
Now consider an unemployment
with probability Hslo F{ws).
utility
is
u{bt). If she
she draws
draws a wage above
iDj,
each period, forever.
{bt, Tt}
to minimize the cost of providing the worker with
Vq-.
C*{vo)=
mm
f]{l + r)-{]lF{iLi,)](btF{wt)-^-^Tt{l-F{wt))),
subject to two constraints.
ommended
reservation
least as well using the
U[{wt,bt,Tt})
>
First, the worker's utility
wage sequence,
recommended
U(^{iUt,bt,Tt})
t'o
=
U[{wt,bt,Tt}).
reservation
That
is,
must equal
And
wage sequence
as
vq
if
second, she must do at
any other sequence {wt},
the agency recognizes that the worker will choose
problem describes optimal unemployment
It is
useful to express this
solve the
-
problem
(5)
she uses the rec-
her reservation wage sequence {wt} to maximize her utihty given {bt^Tf}.
this
If
insurance agency that sets the sequence of unemployment
consumption and employment taxes
utility
Tt)
t
The
solution to
benefits.
recursively.
The
cost function defined
above must
Bellman equation
C*{v)=
mm
((b +
^f^)F{w)-^-^T{l~F{w)))
(6)
subject to
V
=
{u{b)
+
Pv')F{w)
u(w -
f°°
+
/
\
Jw
«W+'^"' =
Moreover, the optimal sequence
_
t)
'
dF{w)
(7)
P
-'
1^^
{t(;t,6t,Tt}
(8)
must be generated by the BeUman equation's
pohcy functions.
An unemployed
worker starts the period with some promised
chooses consumption for the unemployed
employed
v'
,
in the
6,
the tax r
it
will collect
current period, the worker's continuation utility
and the reservation wage
w
in order to
the reservation wage then the cost
is
minimize
its cost. If
if
utility v.
The agency
on workers who become
she remains unemployed
the worker gets an offer below
the unemployment consumption h plus the discounted
cost of delivering continuation utility v' in the next period.
If
instead the worker finds a
job above the reservation wage then the agency's costs are reduced by the present value of
Equation
taxes.
equation
(8) is
reservation
(7)
imposes that the policy must deliver
wage
at the point of indifference
ance. First, there
and
tion
r^
is
CARA
relative to optimal
utility
unemployment
is
insur-
a restriction on the time path of unemployment benefits and taxes, so
are constant. Second, the planner does not directly control the worker's
and so
Finally,
between accepting and rejecting the wage.
There are two disadvantages to constant benefits
bt
v to the worker.
the incentive constraint, which incorporates the fact that the worker sets her
Equivalence for a Benchmark:
3
utility
consump-
constrained by her savings choices. This can be thought of as an additional
dimension of moral hazard. In general, constant benefits are more costly than optimal un-
employment insurance: C^{v) >
C''{v);
however, in this section
we prove
analytically that
constant benefits achieve the same outcome as optimal unemployment insurance for the case
with
CARA
limit
on the amount of debt that workers can accrue and
towards
preferences, u{c)
lotteries over future
show that these
= -exp(— pc)
with c 6 K.
A
all
key feature
is
that there
good benchmark
10
no
workers have the same attitude
wages, which makes the model particularly tractable.
results provide a
is
We
for other preference specifications.
later
For the results in this section
it is
convenient to define
= u-M
CE(t7))
u{msix{i-u,w})dF{u>)
/
the certainty equivalent for a worker of a lottery offering the
CARA
u{ci
maximum
of
w
and
w~
F.
The
—u{ci)u{c2) for any
Ci
and
Co.
Constant Benefits
3.1
We
(9)
.
has a convenient properties that we exploit throughout this section,
utility fiuiction
+ C2) =
)
characterize constant benefits in two steps.
given unemployment benefits
b,
employment taxes
choose these parameters optimally.
subsidy hj
The
It is
we
First,
f,
and
characterize individual behavior
assets a.
Then we
discuss
how
to
convenient to define the net benefit or unemployment
B = b + f.
first
step follows from solving the Bellman equation (2).
Proposition
1
Assume
CARA
preferences.
The reservation wage, consumption and
utility
of the unemployed satisfy
{l
+ r}w = CE{w) + rB.
w—f
Cu{a) = ra
ujra - f + CE(tD))
j—
K{a) =
(10)
(11)
-{-
Proof.
In
Appendix B.
Equation (10) generalizes a standard eciuation
wage,
e.g.
(12)
for a risk-neutral worker's reservation
equation (6.3.3) in Ljungqvist and Sargent (2004), to an environment with risk
aversion and savings.
It
can be reexpressed as the condition that a worker
between accepting her net wage
benefit today,
w —
f today and rejecting
and then earning the certainty equivalent
id
—
f
-
^
CEfti;)
Equation (10) indicates that the reservation wage
subsidy B. This
is
it,
CE(tt;)
getting her
—
unemployment by
11
unemployment
— r
iD is
raising
indifferent
f thereafter:
increasing in the net
the essence of the moral hazard problem in our model
to protect the worker against
is
unemployment
unemployment
— the more one
benefits
tries
and funding
the benefits by an employment tax, the more selective she becomes.
that a worker's assets a do not affect her reservation wage, so
it is
The equation
also
shows
constant during a spell of
unemployment.
Consumption
in equation (11) has a
ary savings component. Assets
wage
in excess of the
permanent income form with a constant precaution-
by w;—
fall
unemployment
>
J5
subsidy,
as long as there
F{B) <
1.'
is
some chance of getting a
Consumption
CE{w) — w
by
falls
each period that the worker remains unemployed. Unemployed workers face uncertainty: a
wage draw above CE(u))
draw below CE(w)
The next
step
is
is
good news leading to an increase
is
bad news leading
Sia;
6,
wage
consumption.
independent of
-—
1 +
=
r
\
r
—
——
13
.
r [w)
J
Optimal constant benefit policy minimizes S{a;b,f)
a.
Using the
constant, equation (3) becomes
is
r
is
consumption while a wage
to minimize the cost of providing the worker with utility vq.
result that the reservation
which
to a decline in
in
-I-
(1
+ r)a
subject to (10) and (12).
Proposition 2 AssiLme
CARA
independent ojvq. The reservation wage
^
,
,
CE(iD)
b
and f
+
r
- wFCw)
— F{w)
are then determined by equation (10)
C'ivo)
independent of
=
^
the optimal constant benefits policy
is
w* G argmax^j, $(u;) where
satisfies
1
and
Then
preferences.
and
{u-\{l - p)vo)
-
,
(12).
(1
The minimum
+ r)$(ir))
^
cost is
(15)
,
a.
=
Vu{a;
b, f).
result characterizes the worker's allocation given optimal policy.
We
take
Proof. Use equations (10) and (12) to solve
for b
and f
as functions of
w and
v
Substituting into the cost (13) delivers the desired result.
Our next
current
unemployment
utility
and describe the evolution
v as a state variable, express the allocation as a function of
v,
of v.
^Substitute equation (11) into the unemployed worker's budget constraint to get a'
w <
B, condition (10) implies w > CE(7l;). But the definition of the certainty cciuivalent
possible only if F{ui) = 1, a contradiction.
12
—
(9)
a
+B —
w.
implies this
If
is
.
Proposition 3 Assume
CARA
ployed at the beginning of a period. Then
c^{v)
and her
unem-
preferences. Let v denote the utility promised to the
=
agent remains unemployed, she consumes
if the
w* - CE{w*)
+ u-^ {{1 -
P)v)
(16)
utility evolves to
= -u{w* -
v'{v)
wage w, she forever
// she accepts a job at
after
CE{in*))v.
(17)
consumes
cAv,w = w - CE[w*) ^ u-\{l Proof.
I3)v)
(18)
=
This follows directly by changing variables from a to f
tions (11)-(12), Ce{a,w)
=
ra
w —
-\-
This proposition will be useful
f,
and the budget constraint
when comparing constant
y„(a;6, f) using equaa'
=
+
(1
r)a
+
benefits with optimal
b
—
c.
unem-
ployment insurance, which we turn to now.
Optimal Unemployment Insurance
3.2
We characterize optimal unemployment
so, it is
convenient to
first
insurance using the Bellman equation (G)-(8). To do
deduce the shape of the cost function directly from the sequence
problem.
Lemma
Assume
1
CARA
{?!}(*,
6f*
let {zDj*,
+ X, Tj* —
Proof. Let
bf
=
6^*.
t^} denote the
x} with X
bt
+
x and
=
cost function satisfies
= -^u-\{l-P)vo) +
C*{vo)
Moreover,
The
preferences.
^"'((1
it
=
Tt
optimum for
—
P)vo)
—x
IS
for all
t.
initial
C*(^^y
promised
utility
optimal for any other
Use equation
show that adding a constant x to unemployment consumption
(4)
in
(19)
and
u(0)/(l
initial
— /3). Then
promise
CARA
vq.
preferences to
each period and subtracting
the same constant x from the employment tax simply multiplies lifetime utility by the positive
constant —u{x), that
is
U[{wt,
bt,
tj})
=
—u{x)U[{ijj,
13
6,
f })
.
The
result follows immediately.
The optimal path
for
wage
for the reservation
since promised utility
reservation
wealth effects with
To
is
CARA
lemma
two
while the path
features. Indeed,
implies that the optimal
results are implications of the absence of
preferences.
to eliminate the
(8)
p) then
employment tax
r.
The Bellman
r
1
+
\l-f3))
r
'
^
-^-±L(^^^u-\{l-P){u{b)+(3v'))){\-F{w))\
subject to
The
the cost
-^^ =
-{u{h)
+ [5v')u{CE{w) -
solution to this cost minimization problem
6
+ ti~-'((l — p)v')/r
unemployed. The
first
+
eciuivalent to b
= w —
eliminate b and
v',
and
=
P'v'
u{b)/{l
C'E(tu).
—
subproblem
solve the
+
=
+Pv'
of
minimizing
to those remaining
u{b)
(22)
The promise keeping
/3).
constraint (21)
is
then
Substitute these conditions into the Bellman equation to
—
solve for C*(u(0)/(1
(l
is
(21)
order condition for this problem yields
^Ju{0)\
Cl-^]=where $(tD)
must
(20)
w).
of providing a given level of utility u{b)
{l-p)v'
or equivalently u{b)
(6)
becomes
wAv'W
\1-P)
reflects these
utility,
problem further, we substitute the cost fimction from (19) into
—
u(0)/(l
These
be constant.
and use the incentive constraint
=
with promised
unchanged. The cost function
will
solve the agency's
equation at v
shifts in parallel
a state variable for the problem, the
is
wage path
consumption
p)) to obtain
F{iu)w-CE{w)
^mm^—
^/,
r)2
.
^
=
(1
+
r)^
^^
^max$u)),
^
(23)
defined in by equation (14).
The optimal
reservation
wage w*
is
independent of promised
utility
and hence constant
over time. Substituting equation (23) into equation (19) proves that the cost to the agency
of providing a worker with utility v
is
identical to the cost with constant benefits C'^(u) in
equation (15).
Once we have found the optimal reservation wage
consumption, employment tax and continuation
and
(22) along with
Lemma
1.
The next
u;*,
the associated unemployment
utility fall out using
equations
(7),
(8),
proposition summarizes the main result of this
14
section.
CARA
Proposition 4 Assume
reservation wage
is
Under optimal unemployment insurance,
preferences.
constant over time and maximizes <&(w), given by (14)-
U
the
agent
o-'n
and remains unemployed, she consumes Cu{v) (equation 16) and has
has expected
utility v
continuation
utility v'{v)
(equation 17). If she accepts a job at wage w, she consumes Ce{v,w)
(equation 18) forever. This
is the
the cost is the same, C^(fo)
=
same
allocation as
under an optimal constant
and
benefit
C*{vq).
Thus, when the worker can borrow and lend at the same rate as the agency, a very simple
policy attains the
same
allocation as optimal
unemployment insurance. Of
equivalence implies that the timing of transfers
unemployment.
If
bt
not pinned down, only the net subsidy to
is
a worker takes a job in period
present value terms.
course, Ricardian
she must pay taxes equal to
i,
"'"'"''^'
in
^
she remains unemployed for one more period, she receives a benefit
If
^^
and then pays taxes
in present value terms.
The sum
of these
is
the unemployment
subsidy, a measure of insurance:
B,
,
(,,
+
(1±I)!£ _ !i±l
r
Using
+
((1
bt
=
r)w*
c^{vt)
—
and
Tf
CE(u}))/r.
problem with constant
At the other end
= w —
c'^{vt,u')
of the
extreme
(1979) and
it
calls for
case, benefits
Hopenhayn and
by equation
is
constant and the same as
B
=
in the
(10).
spectrum from Ricardian equivalence, imagine a worker who
can neither borrow nor save and so
this
with eciuations (1G)-(18), we find that Bt
The unemployment subsidy
benefits, given
(24)
r
and taxes
lives
hand-to-mouth consuming current income.
are uniquely pinned
Nicolini (1997).
One
down, as
in Sliavcll
and Weiss
interpretation of this extreme case
decreasing benefits and increasing taxes. However,
it is
In
is
that
equivalent to think of the
insurance agency simultaneously lending to the worker and providing her with a constant
unemployment
subsidy.
Conceptually, even in this case,
between these two components of
4
remains useful to distinguish
policy.
Liquidity and Wealth Effects:
The sharp
it
CRRA
utility
closed form results obtained so far were derived under an assumption of
preferences and in particular allowed consumption to be negative.
15
We now
CARA
turn to workers
with constant relative
consumption.
Let a
risk aversion
>
(CRRA)
preferences with nonnegative constraint on
denote the coefficient of relative risk aversion.
Then the period
=
log(c) corresponding to risk aversion of
again consider our two alternative policies:
optimal unemployment insurance and
utility function
is
= j^^
u{c)
for
cr
with n(c)
7^ 1,
one.
We
constant benefits.
The
erences. Nevertheless,
equivalence between these two policies breaks
we
find little welfare gain in
unemployment insurance. Moreover, we
analytically with
CARA
down with
moving from constant
find that the optimal policy
CARA
case
is
pref-
benefits to optimal
and allocations obtained
provide an excellent approximation for our
For both reasons, we conclude that the
CRRA
CRRA
specifications.
indeed a very useful benchmark.
Optimal Unemployment Insurance
4.1
Optimal unemployment insurance solves the Bellman equation
utility function is different
with
CRRA
preferences than with
(6)-(8).
CARA
The form
of the
preferences and so the
We
analytical expression for the cost function in equation (19) no longer holds.
therefore
use numerical simulations to examine the economy.
To proceed we need
to
of relative risk aversion a,
(1997),
we view
make
choices for the discount factor
and the wage distribution
a period as representing a week
discount factor of 0.949.
We
fix
and
F{iu).
set
/3
so
Q."^
we normalize by
Euler's constant,
With
CRRA
setting z
=
in
+ r)"\
\.
=
the coefficient
Hopenhayn and
0.999, ecjuivalent to
cr
=
Nicolini
an annual
1.5
but later
6.^
= exp{—zw~^)
adopt a Frechet wage distribution, F{w)
parameters z,6 >
As
{1
the coefficient of relative risk aversion at
consider the robustness of our results to a higher value, a
We
=
p —
with support
(0,
00),
and
the parameter z acts as an uninteresting scaling factor,
The mean
and the standard deviation
log
wage draw
of log
wages
is
is
then
^,
where 7
~
0.577
is
-^ ~ ^^.
^Hopenhaj'ii and Nicolini (1997) use the much lower value of ct = 1/2 in their baseline calibration. They
argue that over short horizons a high intertemporal elasticity of substitution may be appropriate. In our
view, this remark resonates introspectively, but
is
regarding consumption and net income paths.
In their
lent;
at the
same time misleading since it confounds attitudes
model consumption and net income were equiva-
but our model allows saving and borrowing, and as a result a worker displays an
infinite elasticity of
substitution with respect to the timing of transfers.
^A
Frechet distribution has
some
desirable properties in this environment.
First,
it
displays positive
n wage draws within a period from a Frechet distribution with
parameters (2, 9), and must decide whether to accept the maximum of these draws. The distribution of the
maximum wage draw is also Frechet with the same d and z = nz. Thus there is no loss of generality in our
assumption that the worker gets one wage draw per period.
skewness. Second, suppose a worker receives
16
40
30--
^
20
'Ed
a
CD
Q
10--
1.00
0.95
1.05
1.10
Wage w
Figure
2:
Wage
=
density: F'{w)
ew-^-^e-"'~\
=
103.56
Following Hopenhayn and Nicolini (1997), we set 9 so that the
unemployment
spell
is
about ten weeks, consistent with evidence
weekly job finding probability of ten percent
6
=
the density function F'{'w).
wage
We
1.2 percent. ^"^
and the expected duration of unemployment.
specification with those obtained from the
coefficient of absolute risk aversion equal to a/c.
functions obtained for a
CARA
case.
CARA
CRRA
worker with coefficient of absolute risk aversion p
we chose
minimum
6 so that a risk-neutral worker without
making use of equation
c
note that
has local
worker with the approximation provided by those of a
begin by discussing our results for the
0.90,
this end,
who consumes
=
a/u~^{{l
the consumption equivalent utility as a proxy for consumption.
•'^Specifically,
F{w) =
To
This suggests comparing the cost or pohcy
can be computed analytically using our results from Section
We
Figure 2 plots
which increases the dispersion
a worker with a constant coefficient of relative risk aversion a
we take
Meyer (1990) on a
be useful to have a way of comparing the cost or policy functions obtained from
CRRA
titious
an
also consider the robustness of our results to changes in the
in wages, raising the option value of job search
our
wages of about
distribution, in particular to a substantial decrease in 6,
It will
in
of
United States. This requires setting
for the
103.56, giving a standard deviation of log
mean duration
(10).
17
—
f3)v),
fic-
where
The approximations
3.
cost of providuig a worker with
unemployment insurance would have exactly
0.015
0.010
0.005
CD
O
a
V
Sh
0-^
CD
an
Q
-0.005 --
-0.010
-0.015
0.4
0.2
Utility
Figure
a given level of
measured
CRRA
utility.
we
1.0
(consumption equivalent) u^^{{l
Difference between actual values and
specification,
CRRA
0.8
CARA
Recall that with
find that the cost
we do not graph the
is
utility, this
function of the worker's promised
shows how
Uo
=
is
we compare the
utility,
when
utility
same
utility
is
For our
slope. For
cost obtained from
CARA exercise. The solid
and this CARA approximation
exceeds a certainty equivalent of
policy, the left panel in Figure 4
unemployment consumption
and r evolve over an unemployment
b
— /?). Although
initially
spell starting
slightly negative, after a sufficiently long
together, a worker's expected utility
Vt
with
unemployment consumption
falls
0.3.
shows that, as a
b is increasing (solid
line).
initial
The
right panel
promised value
high and the employ-
is
unemployment
tax rises to a high level and unemployment consumption
it is
when
see equation (19).
while employment taxes r are decreasing (dashed orange
^i(l.l)/(l
ment tax
linear
is
specification with an approximation provided by the
Turning to the optimal allocation and
line)
approximations.
nearly hnear with almost the
cost function. Instead,
small, less than 0.0001 in absolute value
brown
P)v)
cost
^,
black line in Figure 3 shows that the difference between C*{v)
is
—
CARA
consumption equivalent units with a slope of
in
this reason,
the
3:
0.6
spell the
employment
to nearly zero. Putting these
declines over time. This line
is
not graphed because
only slightly higher than, and would be scarcely distinguishable from, unemployment
consumption
We also
bt-
look at the subsidy to unemployment, the additional resources that a worker gets
18
1.25
0.2
0.6
0.4
0.8
200
1.0
u-\{l - p)v)
Figure 4: Optimal unemployment consumption
subsidy Bt = bt + ^^^^ - ^^.
by remaining unemployed
for
400
800
600
1000
Time (weeks)
Utility
employment taxes
r^,
and unemployment
one more period, as previously defined
in
equation (24).
bt,
dash-dot blue line in Figure 4 shows that this unemployment subsidy
is
small
when
The
utility
is
high at the start of an unemployment spell and then increases gradually as promised utility
falls
for
and the
Bt of the
spell continues.
CARA
The dash-dot
approximation with p
The unemployment subsidy Bt
blue line in Figure 3 illustrates the high accuracy
=
a/ir^{{l
lar.
The
picture for
Werning
finds that
it
(2(J02)
is
bt
and
r^
(5)v).
paints a very different picture of optimal unemploy-
ment insurance than do unemployment consumption
tion.
—
in
Hopenhayn and
bt
or
employment taxes
Nicolini
(1997)
in isola-
qualitatively simi-
is
computes the net subsidy to unemployment from
r^
their allocation
and
nearly constant, starting quite low and rising very slowly. This distinction
between unemployment consumption and subsidies
is
crucial in understanding the differ-
ence between the results of this paper on the one hand, and Shavell and Weiss (1979) and
Hopenhayn and Nicohni
Finally, Figure 5
(1997) on the other.
shows the probability that an unemployed worker accepts a
job,
1
—
F{wt), under optimal unemployment insurance. This starts just above ten percent per week
when promised
utility is high,
then
initially rises before falling
low. This non-monotonicity illustrates
two opposing
19
when promised
utility is
very
forces at play as the worker gets poorer:
0.1005-
^
I
tT 0.1000
0.0995
;S
<^
O
"^
0.0990
bC
^
0.0985
r
0.0980
1
1
0.6
0.4
0.2
Utility
Figure
5:
0.8
u~\{l -
400
200
1.0
\
1
600
800
Time
f3)v)
Optimal Job Finding Probability
1
—
1000
(weeks)
F{wt).
the increase in absolute risk aversion and the increase in unemployment subsidies.
effect
encourages the worker to accept more jobs, while the second
an endogenous response to the
first,
effect,
which
encourages her to become more selective.
red line in Figure 3 shows the high accuracy of the
CARA
approximation.
It
The
is
first
partly
The dashed
follows that
the non-monotonicity of the job finding rate with respect to promised utihty v found in our
CRRA
case.
specification reflects a non-monotonicity with respect to risk aversion p in the
The next subsection
policy with
CRRA
Before closing,
ability that a
10"^.
explores this notion further by studying the constant benefits
utility.
it is
important to emphasize that in this
worker remains unemployed
for 100
CRRA
weeks or more
is
specification, the prob-
remote, on the order of
Over the relevant time period, the unemployment subsidy and the job finding prob-
ability are virtually constant.
benchmark
4.2
CARA
for the
CRRA
In this sense, our results with
CARA
provide an excellent
specification.
Constant Benefits
The Bellman equation
a constant benefit
b
(2) describes
the problem of an unemploj-ed worker with assets a facing
and a constant employment tax
20
f.
In addition, since
consumption
is
nonnegative assets
cannot
a!
A
natural borrowing limit.
below some, possibly negative,
fall
level a, Aiyagari's (1994)
worker can borrow as long as she can pay the interest on her
debt following any sequence of wage draws. The natural borrowing limit
w—
max{6,
~
is
max{5, w} — f
f]
r
r
Thus, the details of the natural borrowing limit depend on whether the smallest possible
wage, w,
is
In the
bigger or smaller than the net
first case,
w>B,
unemployment
and the natural borrowing
limit
is
This implies that a change
offer.
B = b + f.
—2^^^, since a worker with
consumption and pay the
assets above this level could always have positive
debt by taking the next job
benefit,
interest
on her
wage
in the left tail of the
distribution can substantially affect a worker's debt limit, and potentially her behavior, even
if
she
is
extremely unlikely ever to accept a wage from this part of the distribution,^^
w <
li
B, the natural borrowing limit
unemployment income
unemployment
effects.
job (insurance) and
(liquidity).
there
is
The
it
for Vu{a;b-,f)
by an increase
is
increase in the net
and a budget balance change
in
in f,
which the worker does not
absent from the model with
CARA
is
then
find a
unemployed
preferences because
limit.
method
(3)
+
/)a
(1
S{a]b,f).
+
S{a;
6,
we begin with a
affect the
Bellman equation
of solving the worker's
It is
then simple to choose
b
/3
=
specification
0.999,
where
(2)
and f to minimize
f) of providing a worker with utility v
parameterize the economy as before;
where benefits
= — -. An
ability to use her
allows the worker to go further into debt while she
and equation
103.56. Thus,
in b
income to states
transfers
discusses an efficient
the total resource cost
We
=
It
liquidity effect
no borrowing
Appendix C
.
to pay the interest on her debt, so a
benefit, obtained
has two distinct
determined by a worker's
is
=
Vu(a;
= 1.5, F{iu) = exp(— to^"),
w = < B so that we are in the
a
b,
f).
and
case
borrowing constraint. In the next subsection we turn to the other
case.
To
start,
we examine the
oretically this
is
showing the cost
cost of providing the worker with a given level of utility. The-
higher than the cost under optimal unemployment insurance. Rather than
directly, the solid
purple hne in Figure 6 plots the additional cost of con-
stant benefits on a logarithmic scale. At small values of utihty, the cost of constant benefits
is
reasonably large, equal to a few weeks consumption.
^^With
is
CARA
preferences, Proposition
1
irrelevant for equiUbrium behavior. This
show that the
is
at high levels of utility, near
distribution of wages below the reservation
because there
21
But
is
no borrowing
limit
with
CARA
wage
preferences.
I
10^
m
10-2
tuO
O
X
O
\
10-
w=
«- =
S'
O
0.95
Ad Hoc
10
6
__
.1—
Q
10-
+
+
0.2
0.4
+
0.6
0.8
1.0
Utility (consumption equivalent) ^"^((1
Figure
6:
—
Additional cost of constant benefits.
those corresponding to a net resource cost of zero, around v
using a constant benefits system
On
for
is
is
1-/3
very small, about 0.01 weeks of consumption.
who
typically
starts
much
an unemployment
spell
demand
higher than the optimal time-varying
unemployment subsidy Bt
line).
Workers with lower
because they are closer to
utility
is
more important
them
to
it.
Liquidity
goal of this section
this, first
is
to isolate the insurance role of
unemployment
observe that under the Prechet wage distribution F{w)
probability that a single
wage draw
is
less
than 0.95
is
=
> w =
0.95
and
F(ti;)
=
otherwise,
i.e.
To do
benefits.
exp{—w~^°^'^^)
,
the
minuscule, approximately 10"^^.
Consider an economy very similar to this one but with the wage distribution F{io)
if lu
for
higher benefits for two reasons. First, they have higher absolute risk aversion, and
value insurance more. Second, relaxing the borrowing constraint
The
B
with a given level of utility (solid purple
a worker with the same level of utility (dash-dot blue
4.3
the additional cost of
^
the other hand, Figure 7 shows that the optimal constant unemployment subsidy
a worker
line)
P)v)
with a (small) mass point at 0.95.
= F{w)
It
turns
out that the optimal reservation wage always exceeds 0.95 and this change has no effect on
22
icq
a
CO
CD
0.4
0.2
Figure
0.6
=
Net Unemployment Subsidy B^
7:
optimal unemployment insurance policy.
But
this slight
borrowing
The
limit,
change
i.e.
1.0
in the
wage
b^
+
{l
—
+ r)Tt
r
in the original
if
then the natural borrowing limit was —b/r while
Thus,
0.8
(consumption equivalent) u~^{{l
Utility
in the
may
distribution
p)v)
^^
r
and
B
b
+
economy we had
modified economy
T.
B <
—{w —
it is
in the availability of liquidity.
green dashed lines in Figure 6 and Figure 7 show
unemployment subsidy turns
=
0.
high.
into a pure insurance
In fact, the optimal subsidy
unemployment subsidy
is
at the
For example, at v
same
—
is
how
When w >
this matters.
level of utility
i^J
>
0.95 but 0.540
when w
is
high,
subsidy (Figure
A
w =
0.
and hence
There
little
collecting
is
much
lower than with
(dash-dot blue
line)
,
the optimal time- varying subsidy
spell.
is little
when
at least
is
0.0278, rising to
The optimal constant subsidy
need
for time- varying
utility
is
0.0288
if
unemployment subsidies
additional cost of providing utility through a constant
6).
slight modification in policy
Suppose that
and
ii
mechanism and
The
only slightly higher than the optimal time- varying
0.0283 during a ten week unemployment
w=
f)/r.
lead to a significant increase in the
B, unemployment subsidies no longer play a role in increasing a worker's liquidity.
u)
.95,
after the
w—f
has an effect that
is
similar to this change in technology.
worker draws a wage, she has the option of exiting the labor market
thereafter.
She accepts
this option
23
if
her reservation wage
w
falls
below
^
ui}'-^
When
this
(^+'')("'~'^)
happens, the cost
Like the lower
since
^
bound on the wage
must be paid
it;
in all future periods.
distribution, this option can substantially affect a
worker's debt limit and her behavior even
The only
(^)
is
if
she
is
extremely unlikely ever to exercise
it.
is
that the policy involves a cost to the planner, while the alternative
distribution does not.
However, since the odds of getting a single wage draw below 0.95
difference
are negligible, the odds of the worker ever accepting
w=
benefits are indistinguishable with
summary, when the distribution
lems, optimal
unemployment insurance
0.95 or with
of
is
wages
is
cost of the policy are
to ensure that workers are able to
smooth
w=
0.95.
such that workers have liquidity prob-
well-mimicked by a two part policy: a subsidy to
unemployment, which insures workers against the
of
and hence the
This means that the cost and optimal unemployment subsidy under constant
infinitesimal.
In
w
their
failure to find a
good
job;
and measures
consumption over unusually long sequences
bad wage draws. Insuring workers against the small probability of a very bad shock pro-
vides liquidity.
Together the two policies mimic optimal unemployment insurance, which
unemployment subsidy
involves a nearly constant
for
a long period of time, followed by a
sharp increase in the subsidy when workers are sufficiently poor (Figure
An open
question
is
how
wage distribution from
even
if
F(0.95)
~
to interpret the finding that raising the lower
10~^^. In our view,
model as a
4.4
we view the
it is
a shortcoming of exogenous incomplete markets
simplicity
and transparency
can easily circumvent
the
market
Constraints
anj^ tighter
ad hoc
limit.
prohibited but that the the natural borrowing limit
worker a lump-sum transfer
unemployment
benefit to b
— (1
+
-l-
limit,
To be
is
it is
useful to note that pol-
concrete, suppose borrowing
negative, a
r)a at the start of the
<
an ideal instrument
for
is
is
Consider giving the
0.
initial period,
ra and raising her employment tax to f
changes the timing of payments, and
is
On
virtue.
Ad Hoc Borrowing
loan, but
borrowing.
affect
of the exogenous incomplete
Although our analysis focuses on the natural borrowing
icy
bound on the
to 0.95 can have a significant effect on the constant benefit policy
models that vanishingly small probability events can significantly
other hand,
4).
while lowering her
—
ra.
This simply
equivalent to providing the worker with a risk- free
circumventing any ad hoc borrowing constraint.
^"In our numerical examples, a worker only accepts
w
on her debt.
24
if
doing so
is
the only
way
she can pay
tiie
interest
Optimal unemployment insurance, taken
mouth consumer,
taxes
much
(as in
Tj
is
literally as
a policy geared towards a hand-to-
also a loan. It pays out the duration-dependent sequence
Hopeuhayn and
Nicolini, 1997). Figure 4
and
collects
shows that the net subsidy Bt may be
lower than consumption while unemployed, reflecting the accumulating employment
tax liability over the jobless
spell.
Thus, an important component of the agency's gross
transfers are not net present value transfers; the agency pays out early
much
b^
on and
collects later,
as a loan.
For completeness,
we
also consider briefly the case where, for
some unspecified and
ar-
bitrary reason, the insurance agency does not circumvent the ad hoc borrowing constraint.
This
may
significantly affect
both the cost and
level of
an extreme case, suppose a worker has no assets and no
sume her
benefit each period she
and taxes
for this case.
is
unemployed.
The black dotted
line in
We
optimal constant benefits. To take
ability to borrow, so she
must con-
compute the optimal constant
benefit
Figure 6 shows that this raises the cost of
providing the worker with a particular level of utility by about three to six weeks income,
a substantial amount given that unemployment spells
to provide both insurance
much
subsidy
4.5
last for
The need
only ten weeks.
and consumption smoothing makes the optimal unemployment
higher, in excess of 0.5 over the usual range of utility (Figure
7).
Robustness
This section asks the extent to which our results depend on the wage distribution,
in partic-
ular on the assumption that a worker finds a job in ten weeks on average. There are a few
reasons to explore this assumption. Fust, our results indicate that constant unemployment
benefits
and constant employment taxes do almost
insurance policy.
last longer
notably
It
could be that this result would go away
and therefore presented a bigger
much
as well as a fully optimal
of Europe,
unemployment
risk to individuals.
unemployment duration
at least in part a response to
if
unemployment
is
Second, in
unemployment
spells
tended to
many
countries,
substantially longer, although this
benefits that are high
is
compared to workers'
income prospects (Ljungqvist and Sargent, 1998; Blanchard and Wblfers, 2000). And
third,
workers typically experience multiple spells of unemployment before locating a long-term job
(Hall, 1995).
raising
Although modehng
this explicitly
would go beyond the scope of
unemployment duration may capture some aspect
To explore
this possibility,
we choose
9
=
this paper,
of this longer job search process.
21.084 so that the weekly job finding probability
25
is
about
1
— F{w) =
0.020, one-fifth of the earUer level.
standard deviation of wages, by a factor of
We
of job search.
revisit
five,
to 0.027, which increases the option value
our main conclusions under this alternative parameterization:
Under optimal unemployment insurance,
•
This raises the unconditional
''^
a worker with utility equal to a constant consumption of
0.080 and rises to 0.145 during the
time her
first
10.75 years of
1.2.
The optimal job finding rate changes
slowly.
The optimal subsidy
0.6.
same experiment,
In the
is
unemployment, during which
a consumption equivalent of
utility falls in half, to
Suppose we start
the subsidy Bt rises slowly.
2.005 percent per week to 2.010 percent per week during the
first
it
rises
10.75 years of
from
unem-
ployment.
The
•
CARA
CARA
case provide a good approximation.
ployment subsidy of 0.084,
rising to 0.156
would suggest an
when absolute
initial
unem-
risk aversion doubles.
The
approximate and exact job finding probabilities are indistinguishable.
• If the lowest
wage
is
high, here
w=
1.03, the
optimal constant subsidy
similar to the
is
optimal time-varying unemployment subsidy, 0.087 at the start of the unemployment
spell
and 0.175 once the worker's
benefits
is
utility
has fallen to
small, approximately 0.0004 at the start
0.6.
Moreover, the cost of constant
and 0.017
for a
worker with utihty
0.6.
• If lowest
wage
zero, the
is
at the start of the
The
This
optimal constant unemployment subsidy
unemployment
cost of constant benefits
last
number
still
is
spell
and 0.698
also higher, 0.745
for
a worker with
and 5.56
at these
higher, 0.240
is
utility of 0.6.
two
utility levels.
only represents about a one percent increase in the cost of the
unemployment insurance system.
We
cr
=
6.
have also examined the robustness of our results to higher
Optimal unemployment benefits are higher than the benchmark with a
CARA
approximations would also suggest. For example,
u(l)/(l
— /?),
the optimal
Otherwise this change
^^Specifically,
1
risk aversion
- F{w) =
we
unemployment subsidy
in preferences
set 6 to
has
rises
little effect
for
=
by setting
1.5, as
the
a worker with utility equal to
by a factor of four from 1.4% to 5.5%.
on our
results.
ensure that a risk-neutral worker without unemployment insurance would have
0.020.
26
Conclusion
5
This paper characterizes optimal unemployment insurance
search model.
with
Our main
result
is
borrowing and lending of a
free access to
preferences, constant benefits coupled
riskless asset
optimality of constant benefits breaks down.
rises very slowly over time.
is
optimal.
CRRA
With
inefficient to distort the worker's savings behavior.
is
if
CARA
that with
McCall (1970) sequential
in the
In particular,
it
preferences, the exact
We find that the optimal unemployment subsidy
However, we find
to a constant
little loss
unemployment subsidy
workers are given free access to enough liquidity. This quantitative result
is
robust to the
key parameters of the model.
There are important advantages to simple
with free access to markets that our
policies
model does not capture. Free choice of savings decisions may be
intrinsically valuable for
philosophical reasons (Friedman, 1962; Feldstein, 2005). Such policies
on practical grounds because they are
likely to
may
also be valuable
be more robust to the numerous real-world
considerations that are not included in our model.
This paper has not focused on the optimal
on
their optimal timing
market.
and on the
who computed
to be optimal for the United States.
optimal unemployment subsidy
determinants of the
is
They
Still,
much
benefits
is
it
to
are not, however, out of line
somewhere between 0-10%
of
wages
possible to construct examples where the
higher.^' In ongoing separate work,
we
focus on the
level of benefits.
have deliberately written a stark model of job search
relatively simple
and focus on the
the model lends
itself to
forces that
we
believe are
in order to
keep the analysis
most important. Nevertheless,
a number of extensions, some of which we mention here.
to keep our analysis comparable to Shavell
(1997),
subsides, but rather
unemployment subsidy turns out
utility is also quite low.
with results in Grubor (1997),
unemployment
desirability of allowing workers free access to the asset
In the examples in this paper, the optimal
be low unless the worker's
We
level of
we have assumed that
all
and Weiss (1979) and Hopenhayn and
jobs last forever.
First,
Nicolini
Relaxing this assumption permits an
examination of how optimal unemployment subsidies depend on a worker's entire labor
market
experience.^-'
^''Suppose the wage
Second, we have focused on deterministic unemployment insurance
draw can take on two
values,
Then
Wi
<
u'2.
Moreover, suppose the
first-best features
the
unemployment subsidy to W2 has the desired
effect and moreover fully insures the unemployed; it attains the first-best and is thus optimal.
'^Wang and Williamson (1996) and Zhao (2000) have explored optimal unemployment insurance, without
borrowing and saving, in an economy with repeated spells of unemployment.
worker
onlj'
accepting the high wage
offer.
setting the
27
—
There are situations
mechanisms.
of optimal
in
which an employment lottery can reduce the cost
unemployment insurance even
if
workers have
CARA
or
CRRA
preferences.-^''
Future research should explore the potential gains from using employment lotteries and
their interpretation.
By
Third,
we have assumed that wage draws
are independent over time.
introducing some serial correlation, this model could potentially capture the idea that
some people
are
much more
early on that a high
extensions,
it
will
wage
is
likely to
obtain a high wage job quickly while others learn
an unlikely event. Our results suggest that
for these
and other
be important to evaluate the relative efficiency of simple benefit policies
coupled with free access to the asset market and to distinguish between insuring workers
against uncertainty in the duration of a jobless spell and ensuring their ability to
smooth
consumption while unemployed.
Appendix
A
General Mechanisms
This section uses the revelation principle to set up the most general deterministic mechanism
that an unemployment insurance agency might contemplate given the assumed
of information.
We
allow the worker to
make
reports on the privately observed
allow taxes to vary during an employment spell.
is
useful:
We
show that neither
asymmetry
wage and we
of these capabilities
the planner does just as well by offering unemployment benefits that depends on
the duration of unemployment, and setting employment taxes that depend on the duration
of the previous
unemployment
spell,
not on employment tenure.
The Recursive Mechanism
A.l
For notational convenience, we present the general mechanism directly in
this can
be
justified along the lines of
its
recursive form
Spear and Srivastava (1987). Our general mechanism
involves the following steps:
1.
The unemployed worker
starts the period with
some promise
utility V.
16
We
are grateful to
Daron Acemoglu
for pointing out this possibility.
28
for
expected Ufetime
The worker then
2.
id to
3.
If
receives a
offer
w
from the distribution F{vj) and makes a report
the planner.
w <
the worker report
and
wage
is
w, she rejects the job, receives unemployment benefit b{w),
promised continuation
utility v'{w), starting
the next period in step
1,
described
above, with this value.
4.
If
subsequent period n
=
w, she accepts the job and pays a tax T[vj,n) in each
1,2,....
The Planner's Problem
A. 2
The
w >
the worker reports
full
planner's problem
may be
expressed recursively as follows:
subject to the promise keeping constraint
v=
{u{h[w))
+ l3v'{w))dF{iu) +
^^
and a
^/?"ii(iy
Vn=0
/
-^"^
-
dF{w)
t{w, n))
/
set of truth teUing constraints for all w, w:
oo
oo
y^ /?"u(t(;
-
r(ty, n))
>
^
/3"n(w
-
r('«:',
W^W
n))
>w
(25)
n=0
71=0
oo
^
/3"^i(ii;
-
t{w, n))
> u (6 (w)) + pv'
(w)
w> w>w
(26)
w >w>w
(27)
w > w,w
(28)
n=0
oo
(6 (tu))
+ pv' (w) >
^
/3"u(w
-
t{iu, n))
n=0
u
We
{b
{w))
+
pv' {w)
>u{b{w))+ pv'
(w)
proceed to simphfy the planner's problem.
Lemma
2
(a)
Suppose an optimum has the schedules
that replaces these with a constant schedule b
abuse of notation),
is
also optimal,
(b)
—
b{iu)
b{w) and
v'
and v'{w), then the mechanism
=
v'{w) for any
w
(with a slight
The incentive constraints (25)-(28) can
29
be replaced
with the single equality condition
oo
i{b)+pv'
=
^P''u{iu-T{iv,n)),
(29)
n=0
and constraint
Proof,
X,
(25).
Condition (28) implies that u{h(w))
(a)
From the
independent of w.
{b{w)
,
v' (w))
+
G argmaxb,t,'{6
one can select a solution that
(b)
For constant b and
side of constraint (27)
is
v'
pv'{w)
—
majCw<u,{u{b{w))
planner's objective function
C{v')} subject to u{b)
is
+
Pv'
=
x
we
is
+
pv'{w))
see that given
optimal.
=
x any
Consequently,
independent of w.
the constraint (28)
,
+
increasing in w,
it is
is
trivially satisfied.
Since the right hand
equivalent to
oo
u{b)
for all
so
it
w>
+ Pv' >
w. Constraint (25) implies
J^/?"u(t(}
n=0
-
t{w, n))
w = w maximizes the right hand side of this inequahty,
reduces to
CX5
L{b)
+
yY^P^'uiw-T^w^n)).
pv'
(30)
n=0
Next note that inequality
(26)
is
now
equivalent for
all
w>w
oo
^/?"ti(w - T{w,n)) > u{b)+pv'.
n=0
li
w>
w, then
oo
where the
oo
oo
y^ P'^u{w - t{w, ?^)) >
^
n=0
n=0
first
inequality uses (25)
Therefore the preceding inequality
-
P'^u{w
t{w, n))
> Y^ p'^u['w -
t{w, n)),
n=0
and the second uses monotonicity of the
is
tightest
when w = w,
utility function.
so inequality (26)
is
equivalent
to
oo
Y^ P^'uiiu -
r(u), n))
>
u{b)
+
pv'.
n=
Inequalities (30)
and
(31) hold
if
and only
if
equation (29) holds, completing the proof.
30
(31)
Constant Absolute Risk Aversion
A. 3
So
we have not made any assumptions about the period
far
utility function
u except that
it is
increasing. This section examines the imphcations of having constant absolute risk aversion
preferences.
Lemma
being
CARA utility,
independent of w and n.
3
With exponential
Proof.
an optimum must feature the tax on the employed T{w,n)
With
the
utility
lo
on both sides of
(25) cancels, implying that the
remaining term X^^0'^"^(~'''(^' ^)) i^^st be some value independent of w. Let x denote
this value. It follows that
an optimum must solve the subproblem
HP'-n=0
oo
subject to X
=
2^/3"^( ~
'Ti:w,n)').
n=0
The
first
(w, n).
order condition for this problem reveals that an t{w, n) must be independent of
B
Lemmas
2
tion prevents
CARA
and
3 allow us to rewrite the Planning
"employment insurance," so the tax rate r
preferences and jobs that last forever, the
plicative taste shock. This ensures that all
transfer schemes, which
makes
it
With non-CARA
cases,
it
utility,
may be
wage
is
in (6)-(8). Private informa-
independent of the wage. With
effectively acts as
a permanent multi-
employed workers have the same preferences over
impossible to separate workers according to their actual
wages. Since workers have concave
some
problem as
utility,
introducing variability in taxes
is
not
efficient.
workers with different wages rank tax schedules differently. In
possible to exploit these differences in rankings to separate workers
according to their wage; see Prcscott and Townscnd (1984) for an example.
decreasing absolute risk aversion
(DARA),
including
wages are more reluctant to accept intertemporal
CRRA
If
workers have
preferences, those earning lower
variability in taxes.
One can
therefore
induce these workers to reveal their wage by giving them a choice between a time-varying
employment tax with a low discounted
wage workers would opt
for the
cost
and a constant tax with a high
time-varying schedule.
income from low wage to high wage workers.
31
It is
High
This does not, however, reduce
the planner's cost of providing an unemployed worker with a given level of
transfers
cost.
utility,
therefore not optimal.
since
it
B
Proof of Proposition
The
u{ra
1
worker's sequence problem implies that the value function must have the form Vu{a)
—
f
+
—
ki)/{l
some constant
P), for
the solution along with
We
ki.
=
determine this constant, and the rest of
it.
The maximization with
respect to consumption in equation (2) delivers
=
Cu{a)
+
ra
+ r)-\rb +
{1
-
ki
f)
(32)
Substituting this back into the value fimction (2) gives
T
/
1
=
^
Ktlfi)
/
This implies that the worker accepts
hy
w=
{rB
+
ki)/{\
+
r),
—u{ci)u{c2) to write (33)
.
7
^
K(a;6,f)
= -
j
wages
all
B =
where again
+ ki-f\
rb
\
(
™3^ <u\ra-\
f°°
n
b
lu
+
,
,
u[ra
+w—
-xl
t)
>
,^.
exceeding a reservation wage
f.
Use
and the identity
this
u(ra-f) f^
\_^
,, _,
u{ma.x{w,w})dF{w) =
,
,
,
w
u{ci
defined
+
C2)
J
Substituting
A;i
=
u(ra - f + CE{w))
^
^_^
^
CE(u;) into equation (32), and
^"i)/(l
+
C
Computing the Worker's Value Function
We
are interested in solving equation (2). Ricardian equivalence implies that
delivers equations (10)
employment tax f
(33)
=
as:
establishing equation (12).
?')
V
dF[w).
and
impossible to work with the natural borrowing limit a
——
.
^
(34)
w = {rB +
(11).
to zero without loss in generality. First
ad hoc borrowing constraint, a >
,
^^
We
assume
= ——
,
B>
w.
we can
It is
numerical^
and so instead we impose an
consider values of a arbitrarily close to
test the sensitivity of the results to the exact
borrowing
limit.
set the
The
results
we report
——
to
in the
paper are not sensitive to this choice.
Take a worker with
consume enough
assets a very slightly greater than ag
=
a.
A
worker at this point
to reach the borrowing limit. This implies
K(a) = max ( (u((l + r)a +
B~ ao)
+ f3K,{ao)) F {w) + /" !^ir^^dF(u:
32
will
We
can evaluate this directly at a
u{rao
K (ao
In addition,
we can
=
= max
ao:
+ B)F{wo) +
jhg
JZ ""(^^o + w)dF{
.
I
tuo
—
fit [wo)
differentiate with respect to a using the envelope condition
and evaluate
at a to get
=
Vl{ao)
where we use
1
—
Next, for any
level of a
Equation
/J
(1
= r^
n >
+
U'(rao
+
5)F(u;o)
+ /
+
w'(rao
«Orf^(ti')]
,
to simplify the expression.
we
1,
r)
define recursively a„, Ki(an),
and
V„'(a„).
Let a„ be the highest
such that a worker with assets a this period wants to have assets a„_i next period.
(2) implies
u'((l+r)a„
which uniquely defines a„ since
reservation
wage
+ 5-a„_i)
=/?yj(a„_i),
decreasing and \4'(a„„i)
u' is
is
known.
It
follows that the
solves
w((l
+
r)a„
+5
-a„_i) +/?K(an-i)
=
u{;ran
+ Wn)
^Z
n
the value function solves
14(a„)= (u((l
+
r)a„
+ 5-a„_i)+/?K(a„_i))F(a)„)+
and, using the envelope theorem again,
K(aj =
(1
+
r)
U'((l
+
r)an
its
H
''^''^^^''^
dF{w),
derivative solves
+ B-
a„_i)F(tD.„)
+ /
u'{ran
+
aOrfF(t^)J
.
For each level of assets a„, we can also compute the expected discounted cost of the
unemployment insurance system. At
So
Oq,
So \
= f=
S + -——
l
+ rj
F{wo)
33
-
{l
+ r)BF{wo)
l+r-F{vj)
-
while thereafter
Sn= [B + -~The
]
F{Wn-l).
cost of providing the worker with utihty Kt(a„)
compute the
cost at arbitrary utility levels
is (1
+ r)an + 5„. We
and choose unemployment
interpolate this to
benefits
B
to minimize
cost.
Next consider
cannot
fall
w>B,so
below — =
r
.
accepting any job
In fact,
'
if
a worker
is
the worker's best option.
who ends one
do
=
a
slightly
,
a violation of the borrowing constraint.
= — ^ilf)
Below
changed by
this constraint:
this point a
,,,
,
K(ao)
worker's assets
period
with assets a
^
remains unemployed, her assets the following period are no higher than
5 < —=
A
a'
=
< — 4tt-t
r(H-r)
(1
Thus the natural borrowing
+
r)a
+
limit
is
worker must accept any job. K,(ao) and VJ(ao) are
/^ u(rao + u;)fl!F(ii;)
= -=
^—^
,
and
V^iao)
=
(1
+
r)
u'irao
/
+ w)dF{w).
J w_
The
cost of a worker at the borrowing limit
initial conditions, a„, Ki(a„), V^(a.n))
is
zero, since she accepts
any
job.
Given these
and 5„ are defined using the same inductive formulae
as before.
References
Aiyagari, S. Rao, "Uninsured Idiosyncratic Risk and Aggregate Saving," Quarterly Journal
of Economics, 1994, 109 (3), 659-684. 4, 21
Allen, Franklin, "Repeated Principal-Agent Relationships with Lending and Borrowing,"
Economic
Letters, 1985, 17, 23-31. 4
Blanchard, Olivier and Justin Wolfers, "The Role of Shocks and Institutions in the
Rise of European Unemployment: the Aggregate Evidence," Economic Jownal, March
2000, 110 (462), C1-C33. 25
Cole, Harold
and Ncirayana Kocherlakota,
"Efficient Allocations
with Hidden Income
and Hidden Storage," Review of Economic Studies, 2001, 68, 523-542. 4
34
Feldstein, Martin, "Rethinking Social Insurance," American Economic Review, March
2005, 95 (1), 1-24.
3,
27
_ and Daniel Altman, "Unemployment
Insurance Savings Accounts," 1998.
NBER Work-
ing Paper 6860. 3
_ and James Poterba, "Unemployment
Insurance and Reservation Wages," Journal of
Public Economics, February- March 1984, 23 (1-2), 141-167. 6
Friedman, Milton, Capitalism and Freedom, University
of Chicago Press, 1962. 27
Golosov, Mikhail, Narayana Kocherlakota, and Aleh Tsyvinski, "Optimal
and Capital Taxation," Review of Economic Studies, 2003, 70 (3), 569-587. 4
Gruber, Jonathan, "The Consumption Smoothing Benefits
American Economic Review, March 1997, 57(1), 192-205.
Hall,
Robert
E., "Lost Jobs," Brookings Papers on
of
Unemployment
2,
27
Economic
Indirect
Insurance,"
Activity, 1995, 1 (1),
221-
273. 25
Hopenhayn, Hugo and Juan Pablo
Nicolini, "Optimal
Journal of Political Economy, 1997, 105
(2),
412-438.
Unemployment
Insurance,"
2, 5, 15, 16, 17, 19, 25,
27
Katz, Lawrence and Bruce Meyer, "The Impact of the Potential Duration of Unemployment Benefits on the Duration of Unemployment," Journal of Public Economics, February
1990, 41 (1), 45-72. 2
^-
^
.
Ljungqvist, Lars and Thomas Sargent, "The European Unemployment Dilemma,"
Journal of Political Economy, June 1998, 106 (3), 514-550. 6, 25
_ and _
,
Recursive Macroeconomic Theory, second
ed.,
MIT
Press, 2004.
5,
11
Lucas, Robert E. Jr. and Edward C. Prescott, "Eciuilibrium Search and Unemployment," Journal of Economic Theory, February 1974, 7(2), 188-209. 6
McCall, John, "Economics
nomics, 1970, 5^
(1),
of Information
113-126.
3,
and Job Search," Quarterly Journal of Eco-
27
Meyer, Bruce, "Unemployment Insurance and Unemployment
Spells," Econometrica, July
1990, 58 (4), 757-782. 17
Pissarides, Christopher A., "Short-Run Ecjuilibrium Dynamics of Unemployment, Vacancies,
and Real Wages," The American Economic Review, September 1985, 75
(4),
676-690. 6
.
Prescott,
Edward
C. and Robert Townsend, "Pareto Optima and Competitive Equi-
libria with Adverse Selection and Moral Hazard," Econometrica, 1984, 52
35
(1),
21-45. 31
Rogerson, William, "Repeated Moral Hazard," Econometrica,
Shavell, Steven
and Laurence Weiss, "Optimal Payment
of
1985, 53 (1), 69-76. 4
Unemployment-Insurance
Benefits over Time," Journal of Political Economy, 1979, 57(6), 1347-1362.
2, 5, 15, 19,
27
Spear, Stephen and Sanjay Srivastava, "On Repeated Moral Hazard with Discounting,"
Review of Economic Studies, 1987, 54 (4), 599-617. 28
Wang, Cheng and Stephen Williamson, "Unemployment
in a
U,
Dynamic Economy," Carnegie- Rochester Conference
Insurance with Moral Hazard
Series on Public Policy, 1996,
1-41. 27
Warning, Ivan, "Optimal Unemployment Insurance with Unobservable
Mimeo.
5, 6,
Savings," 2002.
19
Zhao, Rui, "The Optimal Unemployment Insurance Contract:
2000. Manuscript, University of Chicago, July
''284
^1.
O
36
8.
27
Why a Replacement Ratio?,"
MIT LIBRARIES
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