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DEWEV
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Massachusetts Institute of Technology
Departnnent of Econonnics
Working Paper Series
On The Optimal Timing
of Benefits with
Heterogeneous Workers and Human Capital
Depreciation
Robert Shimer
Ivan Werning
Working Paper 06-1
April 25, 2006
Roorri E52-251
50 Memorial Drive
Cambridge,
MA 021 42
This paper can be downloaded without charge from the
Social Science Research
Network Paper Collection
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at
m
1 S 2006
LIBRARIES
On
the Optimal Timing of Benefits
with Heterogeneous Workers and
Human
Capital Depreciation*
Ivan Werning
Robert Shimer
University of Chicago
A-lIT
shimerOuchicago edu
iwerningOmit edu
.
.
This Draft: April 2006
Abstract
This paper studies the optimal timing of unemployment insurance subsidies
search model. Risk-averse workers sequentially sample
random job
in a
McCall
opportunities.
Our
model distinguishes unemployment subsidies from consumption during unemployment
by allowing workers
of
to save
and borrow
freely.
When the insurance
agency faces a group
homogeneous workers solving stationary search problems, the optimal subsidies are
independent of unemployment duration. In contrast, when workers are heterogeneous
or
when human
constant.
We
capital depreciates during the spell, the optimal subsidy
explore the
main determinants
is
no longer
of the shape of the optimal subsidy
schedule, isolating forces for subsidies to optimally rise or
*Shimer's research
is
fall
with duration.
supported by a grant from the National Science Foundation. Werning is grateful for
first draft of this paper was written.
the hospitality of Chicago and Harvard University during the time the
Introduction
1
When
a
homogeneous worker
insurance subsidy
ing, 2005).
is
faces a stationary search problem, the optimal
unemployment
independent of the worker's unemployment duration (Shinier and Wern-
we examine how
In this paper
timing of unemployment subsidies.
relaxing these assumptions affects the optimal
In particular,
we model an unemployment insurance
agency which sets a schedule of unemployment subsidies as a function of unemployment duration
and
faces either:
a group of homogeneous workers whose
(i)
during unemployment; or
(ii)
significant departure
from pre-
unemployment insurance, which focuses on the optimal contract
vious work on optimal
homogeneous workers with stationary search problems
and
capital depreciates
a group of heterogeneous workers.
Both scenarios are empirically relevant and represent a
liayn
human
Our work
Nicolini, 1997).
also differs
(e.g.
Shavell
and Weiss, 1979; Hopen-
from most research on optimal unemployment
insurance by distinguishing unemployment consumption from unemployment subsidies.
allow workers to borrow and save using risk free bonds.
we axgued that
icy:
this
is
In Shinier
crucial for understanding the design of
a constant subsidy
is
for
We
and Werning (2005)
unemployment insurance
pol-
optimal, but unemployed workers choose a declining path for their
consumption. Intuitively, with homogeneous workers facing stationary search problems, constant subsidies are optimal because the tradeoff between insurance and incentives does not
change over time.
This paper shows that the optimal subsidy schedule
the
benchmark with
is
not
flat
when we move away from,
identical workers facing stationary search problems.
opportunities deteriorate during the spell, or
when the pool
of
When
workers' job
unemployed workers
shifts
from one type to another, the tradeoff between insurance and incentives changes during
the spell
— and subsidies change with
it.
We
provide a simple and tractable framework for
exploring the main determinants of the optimal timing of subsidies. In particular, our model
is
well suited for isolating the
impact of human capital depreciation and heterogeneity on
the timing of subsidies, precisely because subsidies are constant in the version of our model
without these features.
Throughout
this paper,
we
focus on workers with constant absolute risk aversion
preferences, since the absence of wealth effects
larly clean results.
The
results in Shinier
(CARA)
and bounds on borrowing leads to particu-
and Werning (2005) suggest that policy towards
the unemployed distinguish efforts to increase liquidity (Feldstein, 2005; Feldstein and Alt-
man, 1998) from unemployment subsidies that provide insurance. Our paper focuses on the
latter,
on the optimal unemployment subsidy schedule once the worker has already been
ensured adequate liquidity. In the benchmark model with neither
(CRRA)
benchmark model
We
is
preferences. Again, the fact that subsides are constant in the
CARA
an important benefit of our
show that the optimal time- varying path
The main
lesson
specification.
of subsidies
and that there are
capital depreciation or heterogeneity,
subsidy schedules.
we
derive
is
depends on the form of human
forces for increasing or decreasing
that optimal subsidies tend to shift over
the spell in the direction suggested by simple comparative statics. For instance,
but lower
M'ith constant
vironment, then when
human
human
unemployment duration.
position of the
if
capital depreciates over the spell, subsidies should
Similarly,
we could
if
first,
workers are heterogeneous, subsidies
shifts
fall if
who would
towards those workers
with
the com-
merit a lower
We
some
interesting particular conclusions
consider two forms of
human
emerge from
capital depreciation.
In the
job opportunities arrive at a constant rate but the wage distribution they are drawn
from deteriorates
in a
steady and parallel fashion. This case captures
close to that in Ljungqvist
skill
depreciation and
That
that workers
is,
search frictions rise during the
their nearest sources for jobs
In our
subsidies.
unemployment
spell.
become increasingly detached from the labor market
form of depreciation
skill
skill
is
and Sargent (1998). The second form of depreciation has workers
sampling from the same distribution of wages, but the arrival rate of opportunities
time.
fall
have, fictitiously, costlessly separated them.
In addition to this general finding,
our numerical explorations.
a worker
if
capital merits a lower constant subsidy in a stationary en-
unemployment pool
constant subsidy
CARA
unemployment duration when workers have constant
preferences and increase slowly with
relative risk aversion
capital deprecia-
are constant with
unemployment insurance subsidies
tion nor heterogeneity, optimal
human
and turn
One
falls
over
interpretation
is
as they initially exhaust
we
to remoter options. For concreteness,
call
the
first
depreciation and the second form search depreciation.
depreciation exercises
we
find decreasing optimal
unemployment insurance
Constant subsidies would give the long-term unemployed a higher replacement
ratio relative to their potential
wages and induce these workers to become overly picky, or
even drop out, as stressed by Ljungqvist and Sargent (1998). In our exercises, the declining
wage opportunities lead the reservation wage
However, subsidies decline even
the reservation wage
is
faster:
to decline steadily during
unemployment.
the ratio of the unemployment insurance subsidy to
also decreasing.
Interestingly,
we
also find that skill depreciation
creates a force for larger initial unemplo3'ment subsidy levels.
Intuitively,
the shocks to
workers' permanent income from remaining unemployed for an additional week, which
we
seek to insure,
is
larger because they are
no longer simply the missed current earnings, but
also include the lower future earnings.
In contrast,
we
find that search depreciation creates a force for rising subsidies.
unfortunate workers
itively,
who remain unemployed
demand more
of offers and, therefore,
the spell.
The reason
in
time have lower arrival rates
insurance to deal with their heightened duration
Long-term unemployed workers have lower
they become choosier. Indeed,
for a long
exit rates
Intu-
risk.
from unemployment, but not because
our exercises we find declining reservation wages during
for the lower
employment
rate
that they receive fewer job offers.
is
Since the moral hazard problem becomes less severe, but risks loom greater, this leads to
rising insurance subsidies.
we
also find forces for increasing or decreasing sub-
fall if
the pool of unemployed shifts over time towards
In our exercises with heterogeneity
Roughly speaking, subsidies
sidies.
workers that would have required lower constant subsidies had the agency faced them
isolation.
unemployment insurance.
Conversely,
spell.
differ in their
value of leisure. Workers
who
more have a higher reservation wage and longer unemployment duration but
leisure
for
For example, suppose workers
if
workers
In this case, subsidies tend to decline during an
differ in the variance of their
less
in
value
need
unemployment
wage draws, those who
face
more
idiosyncratic uncertainty have a higher option value of search, a higher reservation wage,
longer
unemployment duration, and a
tend to
rise
during an unemployment
In both cases,
we
greater need for
unemployment insurance. Benefits
spell.
find that optimal subsidies tend to overshoot their long-run target. To'
be concrete, suppose individuals
differ in their value of leisure. If
optimal subsidies would be constant for each type and higher
less.
for
types could be separated,
who
workers
value leisure
A common decreasing subsidy schedule partially imitates this desirable but unattainable
property.
The reason
longer for workers
is
that with any
who have a
per-period subsidies
is
common
subsidy schedule, unemployment duration
is
higher value of leisure and so the expected present value of
lower for such workers
when the subsidy schedule
is
decreasing with
unemployment duration.
Put
differently, the
model predicts that optimal unemployment subsidies
for
the long-
term unemployed should be low both because the long-term unemployed mostly have a high
value of leisure and because this provides better a tradeoff between insurance and incentives
for
workers in the early stages of an unemployment
points to a time consistency problem;
agency would
like to raise
if it
unemployment
spell.
The second
piece of the
argument
were possible to reset the subsidy, the insurance
subsidies.
By
constraining
itself
from doing that.
possible to provide better incentives early in the
it is
Throughout
we
this paper,
focus on the optimal timing of
than unemployment benefits and taxes.
This
is
spell.
unemployment
subsidies rather
because of Ricardian equivalence.
many unemployment
workers can borrow and lend,
in
unemployment
benefit
When
and tax schedules are equivalent
the sense that they do not affect workers' budget sets and hence do not affect their
However, the model uniquely determines the optimal timing of unemployment
behavior.
subsidies, the present value of transfers from the
worker remains unemployed
Our model
and the
man
skill loss
unemployment
unemployed workers to explain higher European unemployment.
as stochastic, so their story actually also combines elements of het-
erogeneity. In particular, during 'tranquil' times
unemployment generating unimportant amounts
human
capital depreciates steadily during
of heterogeneity
among
the unemployed. In
contrast, during 'turbulent' times a fraction of workers lose skills immediately at the
off,
the
unemployment insurance. Ljungqvist and Sargent (1998) emphasize hu-
capital depreciation of
they are laid
if
one additional period.
incorporates elements present in various positive models of
effects of
They model
for
unemployment insurance agency
moment
generating significant amounts of heterogeneity.'
Our paper departs
in
important ways from existing normative analyses that focuses
on homogenous workers facing stationary search problems. In these contexts, Shavell and
Weiss (1979) and Hopenhayn and Nicolini (1997) showed thai consumption should decline
during unemployment.
declining
unemployment
Although these results are often interpreted as a prescription
subsidies,
for
Werning (2002) and Shimer and Wcrning (2005) highlight
the importance of distinguishing consumption from unemployment subsidies and suggest a
different interpretation:
workers collect constant, or near constant, unemployment benefits
but choose to have consumption declining as they draw down their assets or borrow, by
the usual permanent-income reasoning, to smooth their consumption.
Pavoni (2003) and
Violante and Pavoni (2005) are two papers that study optimal insurance in environments
with
human
capital depreciation. However, these papers
do not focus on the optimal timing
of subsidies because they do not attempt to distinguish
thej^ focus
on the
latter
consumption from subsidies, and
by assuming that the unemployment insurance agency can control
consumption.
^
Ljungqvist and Sargent (1998, pg. 548) conclude that "during tranquil times, the depreciation of skills
is simplj' too slow to have much effect on the amount of long-term
during spells of unemployment
[.
.
.
]
The primary cause of long-term unemployment in our turbulent times is the instantaneous
loss of skills at layoffs. Our probabilistic specification of this instantaneous loss creates heterogeneity among
laid-off workers having the same past earnings."
unemployed.
The
rest of the
paper
is
organized into four sections.
Section 2 describes the model.
Section 3 characterizes the worker's search problem for any given unemployment insurance
Section 4 describes optimal unemployment insurance policy.
schedule.
homogeneous workers who
face skill
Section 5 studies
and search depreciation during unemployment. Section 6
turns to the case of heterogeneity. Section 7 contains our conclusions.
The Model
2
We adapt the model from Shimer and Werning
(2005, 2006). These papers provide a tractable
version of a McCall (1970) search problem enhanced to incorporate risk-averse workers that
can save and borrow
Time
freely.
maximize expected discounted
is
continuous and
infinite
t
G
Workers seek to
[0,c>o).
utility
/"OO
Eo /
e-P'u{c{t))dt
(1)
Jo
where
c{t) is
We
consumption.
=
functions: u{c)
—e~'^'^
work with Constant Absolute Risk Aversion (CARA)
with c G
IR.
This assumption allows us to solve the model
utility
in closed
form."
Unemployed workers sample job opportunities
tinguished by their wage
forms of
human
w
drawn from a
We
capital depreciation.
Budget Constraints. Workers can
at Poisson arrival rates X{t). Jobs are dis-
distribution
F{w,
assume jobs
t).
This introduces two potential
last forever.
save and borrow at the market interest
Their budget
r.
constraints are
a{t)
where
a{t)
are assets
and
=
ra{t)
+
y{t)-c{t),
(2)
y{t) represents current non-interest income:
it
equals the un-
emploj^ment subsidy, B{t), during unemployment and the wage, w{t), during employment.
Initial assets a(0)
fers,
as
we
=
ao axe given, although there
discuss further below.
>
limt_oo e""*Q(t)
may
In addition workers
be
initial
must
lump-sum taxes and
satisfy the
No-Ponzi condition
0.
Heterogeneity. To capture worker heterogeneity we assume that there are
by n
=
1, 2,
.
,
.
.
A'^
types indexed
A and index the primitives of the worker's search problem by the type,
^Shimer and Werning (2005) verify that
solution with
CRRA
trans-
preferences
is
CARA
very close to the
preferences provide a good benchmark:
CARA
closed form solution.
A„(/:),
the numerical
F„(zi;,t), u„(c), 7n,
and
/)„.
Unemployment Insurance
of
unemployment insurance
Policy. In this paper we are interested in the optimal timing
subsidies,
and not
in larger,
more comprehensive,
This motivates the policy problem we consider, which
reforms.
unemployment subsidies {B{t)}t>o that
remains unemployed at
t.
is
to select a schedule of
stipulates the subsidy B{t) received
Subsidies can be conditioned only on
The optimal unemployment insurance
policy problem
social welfare
by a worker that
unemployment
we study
is
duration."^
to find the best such
schedule of subsidies. For the case with heterogeneous workers one must, in general, specify
However, to avoid redistributional concerns we allow
a welfare criterion.
between workers types. These transfers are equivalent to redistributions
transfers
initial assets oq.
is
It
turns out that, with
then uniquely pinned down:
schedule by varying the
we do not need
Pareto
initial
all
initial
CARA
Pareto
in
terms of
preferences, the optimal schedule {B(t)}t>o
efficient allocations
lump-sum
lump-sum
can be achieved with the same
between workers. Thus,
transfers (initial assets)
to specify any particular welfare criterion and our analysis characterizes
all
efficient schedules.
Worker Behavior
3
In this section
we
characterize the behavior of a single unemplo^yed type
n worker confronted
with any subsidy schedule {B{t)}t>oFor any job-acceptance policy, workers solve a standard consumption-savings subproblem,
maximizing
utility in
Ponzi condition. As
equation
is
well
(1)
subject to the budget constraint equation (2) and the No-
known, the solution to any consumption-savings problem must
satisfy the usual intertemporal Euler equation
u.'(Q)
where {q}
Un{c)
=
e-(^"-'-)'E,[u„'(Q+,)],
the optimal stochastic process for consumption.
With
CARA
preferences
7„n„(c), so this imphes
^We do
likewise
is
=
not consider, for instance, menus of schedules that may
we do not consider constraining workers access to savings.
self-select
and separate worker types;
It follows
that
/oo
r
Jo
which conveniently
Let Vn{t) and
relates lifetime utility to current consumption.
c{t) represent
respectively, after duration
t;
the lifetime
utility
and consumption
of an
unemployed worker,
note that both are deterministic functions of
/.
Then applying
the argument above implies that
uum^
^
(3)
r
Unemployed workers wait around
offer
if
the wage
is
with current assets
job
for
An unemployed
offers.
worker accepts a job
higher than the wage Wn{t) which makes her indifferent. Since a worker
a{t)
and a job paying
y
/^^
u)„(t) forever
Unirajt)
^
+
+
lUnjt)
consumes
{pn
-
ra{t)
+ iUn{t) + {pn — i^)/lnf,
r)/7„r)
r
Time
differentiate
= — 7„ii„(c)
and use Un{c)
=
Vn{t)
During unemployment,
a{t)
•
=
ra{t)
+
to get
—fnV{t){ra{t)+^n{t)).
B{t)
—
c{t).
(5)
Eciuations (3) and (4) imply:
c{t)
=
r)/7„r,
(6)
a{t)
= B{t)-iIJn{f)-{pn-r)hnr.
(7)
+
ra[t)
Wn{t)
+
(p„
-
Substituting equation (7) into equation (5) yields
V;(i)
With
a reservation
= K(t)(- 7.(^(^(0
wage
-^^V,,(i))
rule, lifetime utility
+
^^'n(i))+Pn-r).
during unemployment
is
(8)
a function of time
and evolves according to
Pr^Vnit)
=
Un{c{t))
+
Vn[t)
+
-n{rait)
X.{t)
l
I
+
u,
+ ip.„-r)M _
^^^^^^
^^^^^^^^^^
J Wn(t)
Use equation
(3) to
eliminate u„(c(i)), equation (8) to ehminate K(i)i ^i^d divide through
7
by Vn{t) as given
in
=
-
i!n{t)
r{w„{t)
equation
(4):
Bit))
r
^n{t)
In
=
Since u„{a)/un{b)
-Un{a -
(^
Un{ra{t)
Ju,At) V
+ w + {pn-r)hnr)
+ w^{t) + [pn - r)hnr)
Un[ra{t)
_
^p,
^.
j
,
equivalent to
b), this is
^,,{t)^Gn{iUn{t),t)-rB{t),
where
Gn{w,t)
—
= rw
In
This
is
{l+Un{w -
/
(10)
In a stationary environment, with constant
and a constant wage distribution F{w), the stationary
subsidies B, a constant arrival rate A,
down
w))dF„{tu,t).
Jul
a crucial equation for our analysis.
solution in equation (9) boils
(9)
to the reservation
wage equations
in Shinier
and Werning
(2005, 2006).
The
relations in equations (3)-(9) are necessary conditions for worker optimality. Indeed,
they completely characterize behavior over any
"transversality" conditions, which pin
equation
(9),
they are also sufficient
for benefits {B{t)},
one solves the
down
for
finite horizon.
the relevant solution to the ordinary differential
In particular, given a path
worker optimality.
ODE
Together with appropriate
(9)
for the reservation
wage path
equation (6)-(7) can be solved for the path of assets a{t) and consumption
unemployment. This then characterizes the
The appropriate
B{t) are constant
paper,
(for
(i)
when
c{t)
during
primitives F{w,t) and X{t) and benefits
=
0.
For the purposes of this
to characterize "transversality" conditions in
more general cases
since our relations completely characterize behavior over
will suffice for
Then
entire allocation.
the constant reservation wage with w{f)
we do not need
two reasons:
they
is
solution to equation (9)
w{t).
any
finite
for
horizon
our theoretical results, derived using dynamic programming arguments
example. Proposition
1
below);
(ii)
economy by assuming that primitives and
our numerical work truncates,
benefit policy are constant for
bj' necessity,
t
>
the
T, after some
long duration T.
Relation between
in
equation
extensively.
(9)
It is
u'-„(t)
and
B(f).
We
shall use the characterization of the relationship
between the reservation wage path
useful to understand
what
{ilin{t)}
this relation
and subsidy schedule {B{t)}
does and does not imply.
Suppose we are
time
in Wr,
This
more
t.
Then, at a steady
it
is
in a stationary
environment so that Gn(w,t)
state, wliere
follows that there
is
iD„,
0,
we have
B=
independent of
is
Gn{wr,)/r. Since
G„
is
increasing
a positive relation between subsidies and the reservation wage.
intuitive since a higher subsidy
attractive without
=
— Gn{w)
makes the option
making employment any more
becomes more picky about what jobs
of waiting for higher
desirable.
As a
wage diaws
result, the
worker
to accept.
However, the steady-state relationship does not imply that along any dynamic path
Wn{t) and B{t) will rise and
both
w-ni't)
and
'Wn[t).
fall
in
Informally,
if
tandem. In equation
the reservation wage
the unemployed worker's lifetime utility
future, so current subsidies
This implies that there
instance, suppose u}n{t)
is
is
is
B{t) are determine
rising sharply,
it
indicates that
doing the same; things are better
must be temporarily
is
(9) subsidies
in the
near
low.
no simple relation between the paths of
monotonically increasing; then,
it
may seem
iUn{i)
and B{t). For
reasonable to expect
subsidies B{t) to also be increasing. This will be the case as long as iUn{t) does not accelerate
too much, so that w-nit) does not
rise
too abruptly; otherwise, equation (19) implies that
subsidies will decrease over a range where iD„ rises quickly.
not imply monotonic subsidies B{t).
B{t)
is
monotonic then
u)n(t) is
The
converse, however,
Thus, a monotonic Wn{t) does
is
true:
if
the subsidy schedule
monotonic.
This discussion emphasizes the dynamic nature of worker's search problem. The reservation wage
a
is
not only affected by the current subsidy, but also by future subsidies.
result, current subsidies
unemployment
may
As
generally provide a poor measure of the current subsidy to
implicit in the entire schedule.''
Perhaps a better measure
serve the effect on the actual reservation wage, which
is
is
simply to ob-
forward looking and incorporates
the dynamics of future subsidies.
Optimal Policy
4
We
of
imagine an unemployment insurance agency that wishes to maximize a weighted average
unemployed workers'
on average.
If
lifetime utility subject to the constraint that
it
must break even
workers are heterogeneous, unemployment insurance redistributes income
across individuals.
To
focus attention on insurance,
we allow
for the possibility of
lump-
To take an extreme example, suppose that subsidies are negative during the first week of unemployment
but they then jump up to a positive level, much higher than any potential wage. Few would describe the
situation faced by the worker in the first period as providing a tax on unemployment that encourages finding
^
a job.
sum
Given the structure of preferences
transfers between groups of workers at tiipe zero.
in equation (4), tlris implies the
schedule B{t) to maximize the
unemployment insurance agency
sum
of the reservation
wages net
selects a single subsidy
of the discounted cost the
program.
;v
J]K-;„(OK(0)-rC,
n=l
where Wn{0)
is
the time-zero reservation wage of type n workers,
type n workers in the unemployed population at time
0,
and
C
is
//ri(0)
is
the measure of
the expected cost of the
unemployment subsidy system:
Jo
„_i
n=l
where
l^^t)
is
=
exp (^-
the probability of being unemployed at time
//"(iy'\5)
is
H^iiv^s), s)ds^
=
(11)
J^
t
and
A"(s)(l-F"(iD",s))
(12)
the hazard rate of accepting a job.
The agency
n workers
recognizes that for any benefit schedule B{t), the reservation wage of type
solves equation (9)
and the transversality condition, while the measure of unem-
ployed type n workers decreases as these workers find jobs according to equation (11). The
solution to this problem maximizes total surplus, which can then be split, using
transfers, in
any way.
Use equation
(9) to
eliminate B{t) from the objective function
N
5]]ilv(0)/i,,(0)
n=l
lump-sum
N
- rC =
J]-u;„(0)m„(0)
n=l
N
-
/
^0
10
e-^*
J2
„=i
('^"(^)
-
Gn{wn{t),t))ii„{t)dt
(13)
Use integration-by-parts to eliminate the term of the integral involving w{t):
OO
/-OO
e-'*Wn{t)j^n{t)dt
^
-lDn{0)fin{0)
-
=
-u)„(0)/^^„(0)
+
e-'^XVn{t){iln{t)
/
JQ
-
r ^n{t))dt
/OO
where the second Une
e-''wn{t){r
/
Jo
+
H„iw,{t),t))^„{t)dt,
differentiates equation (11) to get
fln{t)
=
-fin{t)Hn{iOn{t),t).
(14)
Substitute this back equation (13) to simphfy the objective function w{0)
must choose a sequence
of reservation
wages
for
— rC. The
planner
each type to maximize
N
/•OO
e-^*Vj4u),,(t),i)/x„(i)dt,
/
(15)
where
JniWnit),t)=Wnit){r
Since there
is
+ Hn{Mt),t))-Gn{Wn{i)^t)-
(16)
a single unemployment subsidy schedule, the evolution of the reservation wages
are linked by equation (9):
An{t)
where A„(i)
type
1
We
=
iD„(t)
—
=
iui{t) is
G„iiv,{t)
+
Ar,{t),t)
-
G,{w,{t),t),
(17)
the difference between the reservation wage of type
n and
workers. Finally, the share of each type evolves according to equation (14).^
can view this as an optimal control problem where the planner chooses a sequence
for the reservation
wages.
wage
of type
1
workers and an
initial
value for other types' reservation
This then determines the evolution of the differences A„(t), n
the unemployment rates
fin{t),
n =
1,2,
.
.
.
,N.
Of
=
2,...,N and
course, the planner also faces a set
transversality conditions.
Appendix
A
discusses our solution
method
for the case of A^
=
2,
when
this reduces to
an optimal control problem with three state variables and one control variable.
how
^
to eliminate one of the state variables and apply Pontrya^gin's
we
some
Formally,
this requires
also require that the the implied worker behavior be optimal.
"transversality" conditions to be met, which
purposes.
11
we
Maximum
We
show
Principle to
As discussed
in Section
.3
fortunately do not need to specify for our
find the solution/'
Human
5
In this section
Capital Depreciation
we consider the case
of a single worker type that faces a non-stationary search
F are changing over time.
problem, with A or
The optimal unemployment insurance problem
simplifies considerably:
oo
max
e
/
-rt
J{w{t),t)fx{t)dt
(18)
{^} Jo
subject to
where J
defined in equation (16) and
—yu(i)i/(u'(i),t),
=
/i(0)
given/
1 is
Constant and Non-Constant Benefits
5.1
An
is
=
/i(t)
interesting property of optimal subsidies that follows from our reformulation
schedule
is
entirely forward looking; only future values of A
immediately from
result then follows
Proposition
1
problem \{t)
=
B{t)
=B
To
(
Shinier- Werning, 2005)
A and F{iu,t)
for some
B>
problem
An
alternative
is
are relevant.
The next
'^
With a single worker type facing a station.ary
F{iu) for
problem we write
all t
>
the optimal subsidy schedule is flat:
t)
= max
it
recursively.
Let
^{/.ij.)
be the value function
This value function solves the Bellman equation
in eciuation (18).
r^ifi,
^
=
F
that the
0.
tackle the general
for the
this observation.
and
is
*"
(j{w,
to formulate the
t)fj,
-
$,,,(/i,
t)//(u^
t)fi
+
$t(/i, t))
dynamic programming Hamilton-Jacobi-Bellman
partial-differential
equation of the problem.
^
Formally,
we
also require that the implied worker behavior be optimal
conditions, as discussed in Section
*
Note that
this
is
true even
— which requires "transversality"
3.
when we
consider explicitly any "transversality" condition required to ensure
worker optimality.
^ This result is proven
in Shinier and Werning (2005) in a discrete time version of the model using a
argument. That paper shows that the planner does not want to distort savings. In contrast, here
we have simply assumed that the policy problem does not consider introducing such distortions.
different
12
Note that the vahie function
is
homogeneous
of degree one in
r0(i)
= max
we have an ordinary
Equivalently,
(j{w,
where the law of motion function
M{(t>,t)
t)
-
(p{t)H[w,
=
M
=
is
<j){t))
min Ur
w
follows that the optimal
w
(19)
+
is
- J {w,t)).
H{iD,t))(t)
(20)
motion function M{(p,
of
and
id
increasing in
(p
is
increasing
(p.
to solve this ordinary
When
differential equation (19), ensuring that the transversality condition holds.
down
t) is
negative in the objective
To characterize optimal unemployment insurance, we simply need
primitives
subsidies that implement this reservation
system
to
system and
satisfies
the transversality conditions.
some time T,
that
(p
=
Fo{w)
at
for
which point they switch forever
alH <
evolves according to
We know
Thus, the
T
and
X{t.)
^ = Mq{iP)
=
for
t
independent of
is
reservation
wage and subsidies
1.
primitives are constant up
after.
That
is,
=
Fi(w)
<T, and
then p
=
we have
for all
t
>
M-i{ip) for
t
X{t)
initial
T—
>
it
=
and
Aq
T. This imphes
>T.
that the optimal solution must reach the steady-state point 0* of
which point
indeed, as
The
1, is
and F{w,t)
Ai
l\Ii
at
t
=
T.
value p{0) must start somewhere to the right of point p^ and increase
accelerating with the explosive dynamics of
at
ODE
solves the
1
when
simple non-stationary case, illustrated in Figure
F{w,t)
the
eciuation (9) for B(t).
the stationary solution in Proposition
implicit in this solution are constant, as in Proposition
A
And
= M{ip). Since M{il>) is increasing there exists a unique steady-state value
= M{ip*). Then note that the stationary solution 'ip{t) — ip* solves the
satisfying
ODE
how
wage are found by inverting
equation (19). Since primitives are constant the law of motion
M{ip,t)
time:
•0*
instructive to verify
in
The
one can solve backwards from the long-run steady-state.
in the long-run
optimal reservation wage solves the right hand side of equation (20) at each date.
It is
=
given by
Moreover, since the cross-partial derivative of
it
+
t)
M(0(t),t)
Note that the envelope condition implies that the law
settle
(p{t)
differential equation for 0(i):
0(t)
function,
and so we can define
solving
<J>(/i.,t)/yU
in 0.
/i
remains constant there.
Mq — until
The
oo then p{0) limits to pQ.
13
larger
it
reaches ^j exactly at time
is
T
t
=
T,
the closer p{0) must be to Pq;
Figure
The
Law
1:
of
Motion
for
4>.
implications for subsidies B{t) are immediate translations of those derived for 0(t).
Let B* denote the optimal constant subsidies for the stationary problem with
Then
subsidies B{t) converge to the optimal constant subsidy
and they
start
somewhere near Bq. This
we imagine time extending
that
fined limits limf^_ooA(i)
=
=
on both
indefinitely
Proposition 2 Suppose
result
we have
\{t)
is
in the long
and F^{w).
run as
t
—
>
oo
generalized in the next proposition, where
sides.
and F{iv,t) defined for
Aq and \imt^-oaF{iu,t)
Then B{t)
B^
Aj
=
all t
E
M.
Fo{w) and limj^oo
such that limj^^nc B{t)
= Bq
with well de-
A(/:)
—
Xj
and limt_co B{t)
=
and
B\,
limf_oo F{iu,
t)
where B*
defined as the optimal constant subsidy levels for the economies with constant
is
Fi{tu).
is
primitives at A^ and F^{w).
An
Aside:
model
Q-Theory Analogy. Our model
can be
of investment with constant returns to scale
mapped
into the adjustment cost
which Hayashi (1982) used to related
investment to "Tobin's Q".
In the case of certainty the investment
model can be formulated
profits
f
TT{l{t),t)K{t)dt
Jo
14
as
maximizing discounted
subject to K{t)
=
i{t)K{t),
where
=
i{t)
I{t)/K{t)
is
the mvestment rate. There are constant
returns to scale in the net profit function, which equals revenues net of investment costs,
and constant returns
that the value function
No assumption
investment.
in
is
average value of capital,
homogeneous
q,
of concavity
of degree one,
V{K) = qK
One can show
required.
is
and
so that the marginal
,
often referred to as "Tobin's Q", solves
rq
=
max{7r(i,
t)
+
iq}
+
q.
i
The important
future
result for this theory
that the investment rate
is
is
a function of
the role of
w
q,
playing the role of investment
ji
playing the role of
and n given by w{r
i,
+
We
This section describes the outcome of two numerical experiments.
The purpose
depreciation, then search depreciation.
Our
definite quantitative conclusions.
model and perhaps get a tentative
Our
baseline parameterization
=
r
1,
=
p
.001
and A
interest rate of 5.3%.
set e
t))
=
The
=
1
goal
is
playing
,
— G{w, t).
is
consider
skill
not to obtain
to understand the qualitative workings of the
feel for their
quantitative importance.
close to that in
is
first
of these explorations
Shimer and Werning (2005).
We
set
and interpret a period to be a week, with an implied annual
distribution
is
assumed to be Frechet: F{w)
=
exp(— u;~^).
We
103.5.
This baseline calibration has wages concentrated near
of around 10 weeks, which
optimal constant subsidy
to insure
is
is
in line
in this
1
and
delivers
with unemployment durations
economy turns out
in
to be very low,
an expected duration
the United States.
around
0.01.
The
The
desire
small enough, while the moral-hazard problem severe enough, that low subsidies
result at the
is
H{iD,
/\
Numerical Explorations
5.2
=
the entire
captured by this forward looking variable.
is
This model maps directly into our framework, with
7
g;
optimum. As discussed by Shimer and Werning (2005),
liquidity, in contrast,
important: workers are able to smooth their shocks, spreading their impact over time, by
dissaving or borrowing.
5.2.1
Skill
In this
wages
first
shifts
Depreciation
exercise
we keep
downwards
\{t)
=
1
constant and instead assume that the distribution of
in a parallel fashion.
Our
15
specification
is
inspired by the depreciation
Benefit/Reservation-wage
Optimal Benefits
ratio
0,16
0.35
0.14
0.3
g
S 0.12
<u
0.25
en
n]
S
c
0.1
O
m 0.08
0.2
£
r
0)
0.15
(I
0.1
0,06
0.04
I
0.05
0.02
400
400
800
600
Figure
Figure
Optima.l Benefits with Skill De-
2:
3:
Ratio of subsidies to the reserva-
tion wage: B{t)/w{t).
preciation.
process used in Ljungqvist a.nd Sa.rgent (1998). Specifically
where F{w)
distribution
shifts
is
is
F{w,
t)
= F{w -
exp(-(5F
F{w,
t)
= F{w -
exp{-SF T))
t))
•
we
let
t<T
t>T
Thus, at
the baseline Frechet distribution defined above.
simply a rightward
continuously to the
of convergence to 6p
Our approach
steady state for
We
1000
600
time (weelis)
lime (weeks)
t
then compute
=
left,
=
the wage
Over time the distribution
shift of the baseline distribution.
converging back to the baseline distribution.
We set
the speed
0.01.
is
to solve the
>
T.
u!{t)
t
We
ODE
system
then solve the
and solve eciuation
equation (19).
in
ODE
backwards up
(9) for
B{t)
=
We first solve the system's
to t = 0. This gives us ^(i).
^{G{iv{t),t)
—
w{t))}^
Figure 2 shows the outcome of this exercise for the optimal schedule of subsidies.
schedule
and
is
decreasing with unemployment duration, starting at subsidies just above 0.30
falling to the steady-state value of
0.01— equal to the value of subsidies
Figure 4 shows that these subsidies induce the reservation wage to
ment
spell.
The
The
rate at which the reservation
during the unemploy-
wage drops, however, does not keep up with
the rate of decline in the distribution of wages.
We
fall
at the baseline.
This
is
shown
in
Figure 5 where we plot
employed Matlab's ode45 routine to solve the ODE, along
We computed 4>(t) and then backed out the implied
reservation wage w(t). We then fit a piecewise cubic shape-preserving spline through w{t) (using Matlab's
interpl routine with the pchip option) to obtain the derivative il'{t) from it.
^°
The numerical
details are as followed.
with the fminbnd routine
for
the required optimization.
16
Reservation
Wage
400
Acceptance Probability
600
400
200
time (weeks)
Figure
4:
Reservation Wage.
Figure
the acceptance probability of job opportunities,
1
—
5:
of out
unemployment,
the ratio B{t)/w{t),
As Figure
=
in this case, since A(i)
A(/:)(l
it is
—
Benefits, however,
F{uj{t))).
all
even
hazard rate out of unemployment
their skills
which increases the dispersion
of
is
greatest.
Figure 3 plots
faster:
is
high, especially at the
Workers become more willing to accept
from depreciating.
We
computed the
solution for ^
=
15
wages making search more attractive. Figure 6 plots optimal
Once again we
the hazard rate out of
from aroun(d 80% to
the acceptance probability equals the hazard rate
1
beginning of the spell where depreciation
subsidies for this case.
offer falls
in(duces
decreasing.
5 illustrates the
bad matches to prevent
Acceptance Probability.
The optimal schedule
F{w{t)).
workers to become pickier: the probability of accepting a job
10%. Note that
600
time (weeks)
see that subsidies are decreasing. Figure 7
unemployment
much
is
shows that
lower now, and close to the hazard at the
benchmark without depreciation.
5.2.2
Search Depreciation
In this second exercise
specification,
A(0
and
we assume that the wage
but that the arrival rate
= Aoexp(— 5a t) +
Aj = .1, so that the
•
Ai for
t
falls
< r and
distribution
is
fixed at the baseline's Frechet
continuously over time.
Specifically,
constant thereafter; we set 8\
arrival rate starts at our
benchmark
=
we
0.01, Aq
week and
of one offer a
=
let
.9
falls
continuously towards one offer every 10 weeks.
Figure 8 shows the results of this exercise for optimal subsidies.
schedule, which
risk of
is
in line
with Proposition
unemployment prompting higher
2,
We
find
an increasing
since a lower arrival rate increases the duration
subsidies.
17
The
level of subsidies is quite
low
in this
400
600
400
Figure
6:
Optimal Benefits with 9
=
Figure
15.
case throughout, so the increase in subsidies
even though subsidies
600
lime (weeks)
time (weeks)
rise the
is
become
Hazarid rate.
Figure 9 shows that
not very spectacular.
acceptance probabihty (blue
opportunities become rarer. Workers
7:
less
line) rises
with duration as job
picky and lower their reservation wages.
However, the resulting hazard rate out of unemployment X{t){l — F{w{t))), also shown (green
line),
comes out
We
to be slightly declining.
have found that the limiting long-run subsidy
Ai for low
enough
values,
and become quite large
shows optimal subsidies when
6 years
Ai
=
—similar to what we found
.01.
in
close to zero, subsidies rise sharply
is
for Ai
quite sensitive to the long-run value
near zero. To illustrate this Figure 10
Note that subsidies
rise
only modeiately for about
Figure S for higher Aj. However,
and asymptote to a high
level,
with these parameters only an insignificant fraction of workers make
unemployment. Nevertheless,
this illustrates that the
when
around
it
X{t) gets ver)^
0.73.
Of
course,
this far into long-term
magnitude of the increase
in subsidies
depends on parameters of the problem.
As explained
in Section 3,
where subsidies axe decreasing.
for the
when
the reservation
We found that
this occurs over
an intermediate region of time
extreme parameter value of Ai =0. Figure 12 shows that subsidies are decreasing
over an intermediate region. Indeed, they
low and close to zero
initially.
become
slightly negative there
The nonmonotonicity occurs
that an insignificant fraction of workers will reach.
made
wage accelerates there may be regions
earlier:
that subsidies
may be
after
because they were
around 6 years, a duration
However, this
illustrates the point
a poor measure of the subsidy to
unemployment
workers are forward looking and incorporates the d3'namics of future subsidies.
IS
we
since
optimal Benefits
400
200
600
800
time (weeks)
Figure
8:
Optimal Benefits with Search De-
Figure
9:
Acceptance Probability and Haz-
ard Rate with Search Depreciation.
preciation.
Heterogeneity
6
When
workers are heterogeneous,
for
any given common subsidy schedule, the pool of un-
employed worker types typically varies over time. That
lower hazard rates
affects the
by n
=
become more prevalent
optimal subsidy schedule.
1. 2.
To bring out the
arrival rate of offers A„
We
is,
worker types that tend to have
as time passes. This section focuses on
how
this
study the case with two types of workers indexed
role of heterogeneity
we suppose that
for
each worker type the
and the distribution of wages Fn does not vary with time
t.
Numerical Explorations
6.1
As with the depreciation
case, the
purpose of our explorations
is
not to obtain definite
quantitative conclusions but to understand the qualitative workings of the model.
In all our exercises
jj,2
=
1/2-
We
we
start with equal population fractions for
begin with heterogeneity
in
to
B„
.
The
values of
Heterogeneity
specification:
5„
in 0.
Fn{w)
it
=
/ii
=
the distribution functions of potential wages. All
the other parameters are set at our baseline, as described in Section
separate workers, then
both worker types
5.
If
we could
costlessly
would be optimal to give worker n some constant subsidies equal
will
be useful references
In this experiment
we
exp(— tu"^"). Workers
in the exercises below.
set 6i
of type
=
1
50 and 02
=
100, in our Frechet
tend to have a lower hazard rate
out of unemployment because they sample from a distribution with thicker
a higher option value of waiting for
offers.
As a consequence,
19
tails
and have
their fractions rises in the
400
400
600
Figure
10:
Optimal Benefits with Search
Depreciation
when
Ai
=
Figure
Acceptance Probabihty
11:
Hazard Rates when Ai
.01.
unemployment pool over
time.
The impUed constant optimal
workers, but slightly higher for type-1 workers:
=
J5i
rise
=
an(J
.01.
subsidies are small for both
0.024 and §2
Figure 14 shows the optimal subsidy schedule for this case.
around B2
600
time (weeks)
time (weeks)
=
0.012.
Overall subsidies start
with duration and limit to 5], as type-1 workers prevail in the unemployment
and
pool. Indeed, subsidies rise
initially
overshoot Bi and then come
Figure 15 shows the evolution of the fraction of type
hazard rate Hi{t)/H2{t). In
side the relative
1
workers
example the
this
iii{t)
down back towards
{n.i{t)
relative
+
fi2{t))
hazard rate
along-
is
quite
constant, hoovering around 1/2, and the fraction of type-1 workers rises steadily towards
the hazard rates are also quite constant and around Hi
«
.48
and H2
~
it.
1;
.95.
In this example, given the chosen parameters, the overall subsidy levels are quite low.
As a consequence,
so
is
the rise in subsidies. This need not be the case: Figure 16 and 17
and 15 but using
are comparable to Figure 14
subsidy levels are
is
more
now Bi =
now
significant
In the previous
0.267 and B2
when
example inverts
=
10 and 6
0.073. Figure 16
as subsidies rise from a value near Bi
20.
The
associated constant
shows that the
rise in subsidies
and asymptote
to B2.
unemployment
subsidy, in the sense that Bi
>
policy cannot separate workers subsidies tend to rise over time.
B2.
As a
Our next
this logic.
Parallel Shifts.
We now
assume that workers
of type
shifted to the left in a parallel fashion, so that Fi{w)
One
=
two examples workers with lower hazard rates were also those workers
that merited a higher constant
consequence,
=
6*
interpretation for the source of heterogeneity
20
1
sample from a distribution that
= F2{w +
is
is
ui).
that the wage workers sampled
is
to
0-8
0.6
0,4
0.2
400
600
400
time (weeks)
Figure
12:
Optimal Benefits with Search
Depreciation
when
Ai
=
Figure
be interpreted broadly as a wage net of work
can be interpreted as "lazy" relative to type
is
that type
capital, lowering their productivity
model
upon being
is
Acceptance ProbabiHty and
1
effort cost
2, in
on the
Ai
=
.01.
Then workers
job.
workers have suffered a discrete
on the job by a constant amount. This
laid off, the rest are spared.
of type
1
that they have a higher cost of working.
and Sargent (1998), where a fraction of workers
in Ljungqvist
in skills
13:
Hazard Rates when
.01.
Another interpretation
600
time (weeks)
loss in
is
human
similar to the
an immediate
suffer
loss
This introduces a form of heterogeneity that
very similar to that explored here. Their model also includes a steady rate of depreciation
during the unemployment
spell,
that can be roughl}' captured by our previous analysis,''
Thus, we can captures both elements, heterogeneity and depreciation,
in
Ljungqvist and
Sargent's (1998) specification.
We
solve the
model with F2
These choices imply Bi
Figure
19, Benefits fall
=
set to the Frechet specification
0.185 and Bo
=
0.242.
from around 0.225, not too
The
far
with 9
results are
=
shown
10 and
in
uj
—
1/2.
Figure 18 and
from B2, and asymptote to the lower
value of Bi
In
all
these examples subsidies start
verge to Bi. This
is
somewhere between B2 and §1 and eventually con-
intuitive since workers of type
the long-run. Note, however, that in
all
Our
^^
One
human
is
unemployment pool
in
in
our explorations with homogeneous workers
capital depreciation.
intuition for this finding
difference
prevail in the
these examples subsidies actually overshoot the Bo.
Note that such overshooting never occurred
suffering from
1
tliat
is
as follows.
The planner would wish
they model the steady depreciation during the
deterministic, partly for numerical reasons.
21
to separate workers
spell as stochastic instead of
^—
1
0.9
0.8
/
'
07
0.6
0)
a.
/
;
;
1
:
/
=i0
i
\
,/
r
i
;
:
5
gLO.4
0.3
0.2
i
0.1
150
100
50
)
time (weeks)
Figure
Optimal Benefits with Hetero-
14:
geneity in
=
50, ^i
=
offer
them
different
9; 9i
by types and
cannot do so there
when a worker
is
100.
wage
unemployment,
all
effective subsidy for
schedule.
i.e.
Bi and Bo. Although he
to continue searching
1
note that
and collecting subsidies, the relevant
an average of the unemployment subsidies over the
is
is
found. Since workers of type
this subsidy reflects
things the same, a
type
this,
1
have lower hazard
a longer average of subsidies.
downward
sloping subsidy schedule implies a lower
workers than for type 2 workers; the reverse
is
true of an increasing
Thus, tithng the schedule helps imitate desired discriminatory policy.
titling that
is
long-run level
Note
levels,
rates.
something that imperfectly mimics separation. To see
rejects a
follows that,
Fraction of type-1 wor leers and
hazard
unemployment subsidy
span of time until the next suitable job
It
15:
relative
per-period subsidy in her calculation
rates out of
Figure
responsible for the overshooting: subsidies rise or
in
fall
It is this
past the desired ex-post
order to provide better incentives ex ante.
that, by the
same
logic, this
phenomena
is
symptomatic of a time-consistency
issue:
ex post the overshot subsidies are not optimal, they were provided to affect incentives ex
ante.
Formally, as
we discussed
briefly
above (and
in detail in
Appendix A) the planning
problem can be formulated with a two-dimensional state vector using the fraction of type
workers
/j,i{t)/{i^ii{t.)
However, only the
+ i^oit))
first
is
and the promised difference
(i.e.
variable, that
is is
to keep track of
general, the promised differences in reservation
would wish
wages W2{t)
a physically unalterable state variable, the second
sometimes termed, a 'pseudo' state
planning problem recursive
in reservation
to reoptimize over W2{t)
—
U!i{t),
what
"promise-keeping" constraints).
will
be
inefficient
is
In
ex post: the planner
ignoring the inherited promised value.
22
— Wi{t).
introduced to render the commitment
all
wages
is,
1
p
0.9
0.8
•|
0.7
/^
i
i
1
:
^0.6
1
/
1-
]
\
£0.5
-4
1-0.4
•
0.3
'
'
7
"
0.2
-
:;::::
0.1
200
400
600
800
1000
1200
1400
200
400
time (weeks)
Figure
16:
geneity in
Optimal Benefits with Hetero9; Ox
=
50, 9i
=
Figure
100.
Recall that, in contrast,
re(iuced so that at
7
800
1000
1200
Fraction of type-1 workers ancd
relative hazarcd rates.
when workers
any point
17:
in
time
it
are
is
homogeneous the planning problem can be
entirely forwarcd-looking.
variables except for calendar time (and none whatsoever
It
600
time (weeks)
does not, therefore, feature a time inconsistency
when
There are no state
the problem
is
stationary).
issue.
Conclusions
studying the optimal timing of unemployment
This paper provided a tractable framework
for
subsidies over the unemploj'ment spell.
Subsidies should be constant
homogeneous and
face a stationary search problem. In contrast,
human
when workers
are
capital depreciation
and worker heterogeneity can lead to increasing or decreasing schedules, depending on the
precise nature of the depreciation or heterogeneity;
A
simple heuristic
is
we provided some
useful dichotomies.
suggested by our finding that optimal subsidies generally shift
direction suggested by a comparative-static analysis.
23
in the
1
'
'
0.9
0.8
0:7
|0.6
/
-J
...i
I--
;
£0.5
g;0 4
0,19
...-.
0,3
j
0.2
0,1
1000
500
2000
1500
2500
1000
time (weeks)
Figure
18:
geneity in
1500
time (weeks)
Optimal Benefits
support of F{w).
Hetero-
witli
Figure
relative
19:
Fraction of type-1 workers an(d
hazard
rates.
Appendix
A
Optimal Control with Heterogeneous Agents
we describe
In this appeniiix
its
in detail the
associated Hamiltonian, the resulting optimality conditions, and our numerical strategy
for solving
it.
We
being with some definitions.
Definitions. Let the difTerence
of
optimal control problem with two worker types,
motion
for this difference
in reservation
1
u'2(i)
-
= G2{m+A)-G,iwr),
wiit).
Then
the law
(21)
workers as
a{t)^
\\'ith
=
is
A{t)
Define the fraction of type
wages be A{t)
f^iit)
Mi) + Mt)'
the law of motion
a{t)
=
a{t){l
-
a{t))
[H^iMt) +
24
A(t))
-
H^{w,it))
(22)
The
objective function
is
then
fOO
-Ttl
l^a{t)Ji{m{t))
+
- a{t))J2{m{t) +
{1
A{t))y.{t)dt,
(23)
JO
where
J„(iD)
=
w„[r
+
—
//n(u'„))
equation (23) over Wi{t), A(i), a{t) and
=
A(i)
It is
fi{t)
-,jit)ia{t)H,{wAt))
useful to think of iu\{f) as the control
Truncated Control Version.
is
Thus, the planning problem
Gn{'il>„).
+
wages of
A
T
-
(1
we
a{t))H2{iMt)
+
//(t)
as state variables.
>
t
T, for
T
Let
large.
1
where policy
^['(q.
with a constant subsidy that implements a difference
and with the fraction of type
maximizing
A(i))).
solve a version of the problem
restricted to offering constant subsidies for
continuation value at
to
subject to equation (21), equation (22) and
and A{t), a{i) and
In practice,
is
A) be the
in reservation
workers equal to a. The problem can be written
as
/
—
with q(0)
Q'o
e-''j{a{t),iD,{t),A{t),f)fj.it)dt
and
//(O)
=
+
e-'-'^'^{a{T),A{T),T)^i{T).
(24)
^q are given, where
=
J{a,Wi,A,t)
Hamiltonian. The Hamiltonian
aJi(^iDi,t)
for this
+
-
(l
problem
is
a)J2{'Wi
H-
A,
t)
(omitting the arguments in functions
to save on notation)
where
= -{aHi{wi) + {I -a)H2{iui + A))
M"{a,Wi,A,t) = a{l - a){H2{wi + A) - Hii'w^))
M^(iDi,A,0 = G2{w, + A,t) - G,{wi,t)
M^(Q;,u)i,A,f)
The
II
functions
M'^
is
M°
and M'^ are the laws of motion
the law of motion for
/i
and that
M^
of motion.
25
for
itself
a and
A
derived above.
Note that
can be thought as a "normalized" law
Maximum
Principle. The reservation wage solves
msxH
And
(25)
the co-states evolve according to
= ru^-{J^^u^NP)
(26)
t'A
=
ruJ^,ll
- Ha
(27)
Va
=
rVajl
- Ha
(28)
v,,
Equations (25)-(28), together with the law of motion equations
ODE
system
for the
6-dimensional vector
Normalized System.
It
(/x,
a,
A,
for
^u,
a and A, comprise an
v^, Ua, i^a)-
proves convenient to work with the normalized Hamiltonian and
co-states:
H = J + k^M"" + kaM" + k.AM^
,.
Kn
The
_
—
,
"'a
l^Li
benefit of this transformation
5-dimensional one by dropping
Since
fi
is
that
_
—
I'a
,
"-A
_
—
I'A
we can reduce the 6-dimensional problem
fi..
can be factored out, the optimality condition (25)
is
eciuivalent to
maxH
and that using
(26)-(28)'
to a
we can obtain the laws
(29)
of
motion
for the
transformed co-states
becomes
k^
= rk^-{J+k^M^)
kA
= kA{r-M^)-HA
ka
^ Kir - A4n - na
Equations (29)-(32), together with the law of motion ecjuations
ODE
system
for the
5-dimensional vector {a, A,
26
t'^, !/„, i^a)-
(30)
.
"
for
(31)
(32)
a and A, comprise an
Terminal Conditions.
Since A(0)
a free variable
is
fcA(O)
Ai=T
costate should be initially zero
=
•
(33)
we must meet the terminal conditions that
^v(^)
= *(a(T),A(T),^)
(34)
fcA(T)
= *A(a(T),A(r),T)
(35)
UT) =
Algorithm. The idea
the states are
role in the
(2,
A
and
T
to start at
is
it
to solve this
is
T
k^ and
/c^,
As
ka-
before,
is
//
-uii
and
plays no
/ca, ka.
/c^,
One can then
a{T) and A(r).
This
using equations (34)-(3r)).
and «(0). One can then search
is
and work backwards. The control variable
to guess the values of
the S5'stem backwards for Wi, a, A,
Qq, that
(36)
can be ignored.
the values of the co-states at
and a{0) =
^M{T),A{T),T)
a; along with their co-states
dynamics and
One way
/ca(0)
its
is
then sufficient to solve
One can then compute
the implied value of
and A(r) that has
for the initial guess o:{T)
obtain
to satisfy the optimality condition equation (33)
/ca(0)
and the given
=
initial
condition for a.
Useful Expressions.
We now
collect
some expressions that
are needed. First,
some
atives of the Hamiltonian function:
We
'Ha
= Ji-J2-
Hi,
= [l-
a)
(
{H,
J^
-
- Ho)
Av/Za)
((1
+
-
2a) k^
kAG'2
+
+
k,,)
k^a{l
- a)H^
also need the derivatives:
J'{iL')
=
r
+ H{w) + wH'{w) -
G'{iu)
/oo
G'{w)
=
r
-
Xit)
/
n„(u;
- iD)dF{w, t) =
+
r
j{G{iv)
-
riu)
JW
We
define the value from a steady state policy as
*(«, A)
=
m_ax{QSi{iui)
27
+
(1
- a)S2{wi + A))
+
A(l
- F{w))
deriv-,
subject
to,
= M^{wi, A),
where we define
for
convenience the steady state surplus function:
r
It
follows immediately that 'i/{-,A)
As
for the
for
A
always differentiable and
=
other partial derivative, note that the constraint
as a function of Wi,
maximization
is trivial.
maximization
is
However,
is
+ H„{w)
which we write as A*(m;i).
In general this function
nontrivial.
This
may
at points of differentiability
^A(a-,
A)
If
M^{tvi, A) can be solved
this function is invertible then the
may be non-monotone,
so that the above
also introduce kinks in the value function ^(a,-).
we must have
=
iaS[{w,)
+
(1
-
=
{aS[{i-u,)
+
(1
- a)S',{m + A))
a)S',{wi
+
A))
—1— +
- M^
—^
+
''''(ill
28
(1
(1
-
q)5^(iZ;i
+ A)
- Q)S',{m +
A).
References
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2005, 95 {1), 1-24. 1
_ and Daniel Altman, "Unemployment
ing Paper 6S60.
Insurance Savings Accounts," 1998.
NBER Work-
1
Hayashi, Fumio, "Tobin's Marginal q and Average
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(1),
q:
A
Neoclassical Interpretation,"
213-24. 14
Hopenhayn, Hugo and Juan Pablo
Journal o} Political Economy, 1997, 105
Nicolini,
(2),
"Optimal Unemployment Insurance,"
412-438.
1,
4
Ljungqvist, Lars and Thomas Sargent, "The European Unemployment Dilemma,"
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McCall, John, "Economics
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Human
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4
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Staff
_ and Ivan Werning,
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8
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4
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