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OPTIMAL JOB SEARCH IN A CHANGING WORLD
Lones Smith
95-3
Jan. 1995
massachusetts
institute of
technology
50 memorial drive
Cambridge, mass. 02139
OPTIMAL JOB SEARCH IN A CHANGING WORLD
Lones Smith
Jan. 1995
95-3
MASSACHUSETTS INSTITUTE
OF TECHNOLOGY
MAR 2
9 1995
LIBRARIES
INTRODUCTION
1.
This paper explores some theoretical and economic implications of a nice asym-
metry
in
many
classic job search models,
jobs at will, but can retain
To make
more
this
them
whereby individuals are permitted to drop
at their pleasure
definite, consider the
is
assumed to be searching
wage
offers,
when the
While searching, she pays a
for one.
and
fixed flow
positive flow
coming from some nonstationary but deterministically
The
(perhaps discontinuously) evolving distribution.
will
individual does not possess a
unemployment insurance) and enjoys a
cost (possibly negative, such as
arrival rate of
the reverse cannot occur).
problem faced by an individual who must
search for jobs (or 'prizes'). Suppose that
job, she
(i.e.
How
start searching again.
may drop
individual
a job at
then does she behave?
Search theory as a branch of economics has solved a wealth of such problems
in
both continuous and discrete time, but has largely confined
its
attention to the
steady-state theory, where quits do not occur. This paper analyzes the job search
problem by supposing, just as
in
Van Den Berg
(1990), that the
wage distribution
But unlike that paper,
evolves in a nonstationary but deterministic fashion.
I
do
not deny searchers the option of ever dropping a job once taken up. This restriction,
while briefly justified in the context of his model,
—
for instance, in recent analyses of
will
is
inappropriate for
many
contexts
continuum agent two-sided mutual tenant-at-
employment matching models with heterogeneity, such
as
Smith (1994a) and
Shimer and Smith (1994).
As
it
turns out,
Van Den
Berg's assumption that jobs cannot be dropped also
conceals at least two intriguing effects that
I
wish to explore. For one, the identity
between the decision-maker's reservation wage
unemployed value
of motion.
As
it
W
t
fails,
t
and we must independently derive
turns out, casual finance theory intuition
Indeed, the reservation wage
is
an option to renew at the same
local relationship
Another major finding
t
of time, 9 t need not be, but
show that
in
their respective laws
may
then be misleading.
that while
is
W
t
;
W
t
of being
namely, the flow return with
This turns out to be related to the fact that
price!
W
W
between 9 and
is
and her average
not the flow return on the Bellman value
unmatched, but has an even more inviting link to
no simple
(or flow utility)
t
t
exists.
is
necessarily a continuous function
always upper semi-continuous in time.
more general search models, the optimal acceptance
In fact
set is
I
a lower
hemicontinuous correspondence of time. This happens to be a simple example of
a very general principle underlying optimal individual behaviour in the presence of
sticky state variables.
The paper concludes with a simple example
well-known comparative statics
out some
new ones
that
may
fail in
become apparent.
of
how some
a nonstationary environment, and point
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THE MODEL
2.
At any given moment
in time, the individual is either searching (waiting) for
a prize while experiencing a flow search cost
The
w.
c,
individual can hold at most one at a time.
time intervals with parameter p
>
0.
The wage
let
F
t
has support on
[0,
oo) with uniformly
is
1
holding some job with wage
Wage
offers arrive at
Poisson
distribution facing the individual
deterministically evolving, with time-dependent
that
or
c.d.f.
bounded
F.(w) be a piecewise continuous function of time
t
{F
t
(w) \t,w
finite
G
for fixed w.
That
t;
is,
number
distribution can shift discontinuously at possibly a countable
Assume
H£+}.
mean for fixed
is
further
the wage
of distinct
times.
The individual cannot search while holding a job, 2 nor is she permitted to hold
more than one job at a time. But any job may be retained forever if desired, or
may be
freely
dropped at
will.
3
While
drop existing jobs, for definiteness,
I
allow searchers the additional latitude to
I
do not permit them to 'drop out' of the search
market altogether, and thus forego any search
At any time
all
t,
the individual seeks to maximize the expected present value of
future jobs held less search costs borne, namely
s;
where m(s)
is
A
strategy
to
is
and job
wage
w
s - t]
s,
m(s)ds
and the expectation
is
taken with respect
realizations.
That
belongs to
A
is,
4
t
measure zero equivalence,
.
the individual
Note that
A
t
is
A=
Q K+
1
€
willing to accept or retain a job
x
a time-dependent sequence of acceptance sets
oo)) for each time.
iff its
Pe-^
the income flow at time
to arrival times
[0,
costs.
(A
t
1
can only possibly be well-defined up
in the sense that
any At that contains At on a set of
times of measure zero yields the same payoff. Although a strategy might possibly
also
depend on whether an individual currently holds a job, an optimal strategy
A* cannot. This must be
true, since
retaining a current job, or accepting a
1
2
new
one, are identical.
more generally, one could modify the model to suppose that the
number of prizes she can simultaneously enjoy.
Or,
in the
both the costs and outside options of either
individual
is
constrained
Less stringently, one could suppose that one experiences a lower flow arrival rate of job offers
while matched than while unmatched.
3
lump-sum
quitting costs, then reservation wages will no longer exist; that is, an
have a lower 'reservation wage' than will her unemployed counterpart.
This is somewhat analogous to the effect of switching costs in the multi-armed bandit, where
Gittin's indices no longer exist. See Banks and Sundaram (1994).
4
This implicit restriction to pure strategies is without loss of generality, given a mild generalization of a result due to Dvoretsky, Wald and Wolfowitz (1951).
If
there are
employed individual
will
THE MAIN RESULTS
3.
Optimal Strategies
3.1
Let
W(
be the time-f (Bellman) optimal average value of being unemployed, and
W (w) the analogous value with wage w > 6 hand. While would be nice to say
an option,
that w € A exactly when W (w) > W since quitting 'immediately'
necessarily always true
So say that a wage w
the inequality W (w) > W
agreeable
time
W (w) > W Then wage w called agreeable and
w G A exactly when W {w) > W or W (w) = W but an -optimal strategy
in
t
t
t
t
at
that.
if
t
t
t
t
t
entails retaining that
w=
agreeable wage
wage
liirin^oo
is
,
is
t
strictly
t is
t
t
it
is
.
if
,
t
t
for
a positive period of time.
wn
for
some sequence
follow that any
(It will
of strictly agreeable wages
wn
.)
Notice further that an optimal strategy entails monotonic preferences at every
instant in time:
so
is
must w' 6
closed in
Lemma
I
If
w
€
.4* for all s
.4* for w'
R
for all
t.
>
As
w.
[t, t
+ A],
then at least in the same time interval
assume that
all
agreeable wages are accepted, .4*
Hence,
(Monotonicity)
1
I
€
The optimal strategy
is
A*
=
at time
[9 t ,oo)
shall refer to 9 t as the individual's reservation wage; in the
t.
more general model
considered in subsection 3.4, a more neutral term might be the (lower) threshold.
I
shall
soon argue that this
unmatched
at time
(W ),
Given
is
conceptually distinct from her flow value of being
t.
the determination of the threshold time path (0 t )
t
in optimality, while the calculation in the reverse direction
accounting. Suppose that (6 t )
next time after
define
O^iw) =
t
that 9 t will assume (or exceed,
inf{s
>
WtH =
t
/
1
Finally,
(W
t
Wt; =
t
The
must
)
J°°
given. In order to
is
\
>
6S
is
if it
t
jumps) the
in
must know the
I
level w.
That
is,
w}. Then
w + e " WrlW "
t)
W-M-«')
W
if
-^<
(1)
else
t
solve the following recursive accounting equation:
pe-^-V (-c(l - e-M*-*) + e-^ s -
t]
f°°
W {w)dF (w)) ds
s
s
(2)
Van Den Berg (1990),
to be dropped. On the other hand, Van Den Berg's
function uniquely exists, by means of a standard
functional form of (2) fundamentally differs from that in
on account of
my
allowing jobs
proof that a continuous value
contraction-mapping argument, applies here with few changes, 5 and
5
an exercise
a simple exercise
(W ),
compute
is
is
thus omitted.
His proof, for instance, assumed stationarity after some finite time T; however, a continuous
Lemma
(The Value Function)
2
to (2), that is
continuous and
There exists a unique continuous solution
1
t
we have
1
(»)-*)
J Of
(Ww) _ w) _ Wt
)
dFt ( w )
(
3)
Average Values and Thresholds
3.2
I
are
now must characterize optimal behaviour, and compute (8 ) given (W ). There
many approaches to this problem. For instance, one could treat (9 ) as an
t
t
t
infinite-horizon control,
route
much more
is
and use the calculus of
level.
This
now claim
the
is
moment
will
(W ),
For any c
the reservation
t
wage
(W - e-^ -^Ws )/{l - e"^"')),
'
w
>
0,
8^
its
current
1
(6 t ),
t
.
I
and given time path of aver-
0-
-
-Wt/P
Wt--e-W w- (w*- I(t) -o
t
t
1
=
()
{
,
The
result for #
=T
6
(i,
-1
oo).
0-
•
t)
w
0-
t
inequality
dropped at time T.
-1
=
t
W
<
= oo
follows from (1)
t
<
W
for allt. Moreover,
t
or equivalent^
f
fl
The
(£)
be at least as high as
satisfies 6 1
s
lim,_> -i (t)
0~ 1 (t)
ambitious
less
-1
that a maximizing individual will drop a job paying
(Reservation Wages)
1
age value functions
Proof:
But a
that optimal behaviour yields
Theorem
=
variations.
6
For ease of notation, define #
transparent.
namely the next time that the reservation wage
6t
regardless
(2),
a continuous and differentiable function of
is
r (w + e-^'"
= (3W + (5c-p
t
W
or F.,
time. In fact, together with (1),
W
t
a.e. differentiable.
Observe that by the Fundamental Theorem of Calculus applied to
of the functional form of Or
(W
Ht)
Ht)
l
(t)
=t
>t
= CO
and the identity
W =W
t
t
(0 t ).
Van Den Berg (1990). Next, suppose
Then optimal behaviour demands that the job paying
be
(£)
is
essentially
t
W
Hence,
t
=
t
+
e"^T_t
'[}Vr
—
t ],
yielding the result for
l
Finally, suppose that 0~ {t)
= t. By monotonicity, if w >
then
it must be optimal to hold the job paying w for some positive period of time, i.e.
W < w + e~^s ~^(W8 — w) for some s > 0; conversely, if w < then it is strictly
(£)
>
t.
t
t
t
better to stay unemployed,
and the reverse inequality holds
= linw(W - e-^ -')Ws )/(l s
t
t
e^
8 "*)).
for all s
In that case,
t
=
>
Thus,
0.
W -W
t
t
/P, by
l'Hopital's rule.
function on
[0,
T]
<0>
is
uniformly continuous, and so a simple limiting exercise suffices to complete
the argument.
Van Den
Berg's 'no-quitting' assumption
the law of motion for (6 t ) and
(W ). The
t
meant that he did not have
to independently derive
analysis that follows thus did not arise in his paper.
)
In words,
we
the reservation wage
unmatched value
to the
as
when
shall see);
>
for all s
when
weakly monotonically decreasing on
Some
in
comes
Shimer and Smith (1994),
W
implicitly through
wage
relationship
t, its
[t,
oo), then 6 S
s
and
falling,
it is
W
=
into play.
Financial Intuition
As done
(ipt)
=
But when
again a purely local relationship arises.
t,
later to rise higher, global optimality
3.3
so that 0~ x {t)
purely local (and has a natural financial interpretation,
is
it is
is rising,
=
t
°°
/
Pe~^
f
s
I
can define the spot market wage system
~^i)} s ds.
at time t in a hypothetical spot market.
Interpret
An
as the reservation
ip t
individual
is
always indifferent
between remaining unemployed and accepting her spot market reservation wage
an arbitrarily short period of time, during which she cannot search.
formula
explicit
flow return and
t
W
t
t
t
the associated instantaneous capital gain of the asset 'being
t
unemployed', whose present value
But Theorem
is
establishes that
1
W
t /f3.
ip t
^
8 t in general.
flow return with an option to renew at the
same
rate.
It
turns out that 9 t
for as long as she wishes at the
same wage. That
is,
When
is
that
to
=
t,
it is
is
>
to) for
fixed (for as long as she wishes). If the option
then 6 t
=
tp t .
and so 6
t
=
W
t
-
If
_1
(£)
=
oo,
A
I
it
the flow return
never exercised,
i.e.
>
ip t
then the unemployed asset
W
the renewal option
is
future date, then no purely local relationship between 6 t and
3.4
in
indifferent
Otherwise, the renewal option has value, and so 6
forever exercised, in the case #
effectively sold,
is
is
the
from
may remain
at time to, she
about exchanging the variable return on the asset (Wt,t
w=
is
Intuitively, this follows
the fact that an individual taking a job does so knowing that she
Q~ l (t)
The more
= W — Wt/P follows by the Fundamental Theorem of Calculus.
P(W /(3) = ip + VVt//? makes it clear that ^ is flow value or the
ip t
Rewriting this as
for
t
.
f
not exercised at some
W
t
can
exist.
General Continuity Result
temporarily abstract away from the model discussed above, in order to derive a
somewhat more penetrating
u(x) to the owner, where
search problem, there
a;
result.
is
Treat the jobs as prizes x providing flow utility
a vector of prize attributes. For instance, in the job
may be a
multidimensional job characteristics space, with
wage not the only consideration. Suppose that there
iCR"
is
given a Borel space of prizes
whose distribution follows a known deterministic evolution. As above, one
either possesses a prize or
must search
for one.
Observe the following simple property of the graph of the optimal acceptance
sets
(A* ) over time.
t
Theorem
2 (Lower Hemicontinuity)
// the utility function
continuous, and satisfies local non-satiation, then
A*
u
nonnegative,
is
a lower hemi-continuous
is
correspondence of time.
Proof:
Assume
c
=
0, for
below.
Note that
it
is
=
A*
>
R", since u
obvious that
We
.
t
For definiteness,
n.
0, for that strategy
must show that
—Y x
a sequence of prizes x n
is
^
.4*
suffices to
it
for all sequences of times
W
t
W
=
(x)
W
So suppose that
definition.
t
{x)
W
t
W
s
W
(x)
>
>
W
(x)
s
t
W
So
.
and
s
W
s
enough to
for s close
s
>
optimality considerations. Suppose that
=
<
For s
x.
t,
W
let
onto prize x until time
for s close to
and
t,
enough to
soi£ A1n
by
t,
Hence, x € A*
tn
\.
W
t
W
t
>
W
t
,
tB
(y)
W
-
tn
(x)
(x)
for all large
W
=
definition).
t
enough
W
s
(x)
x n -> x yielding
W
>
s
even though x £ A* (so that x E
>
u.
0.
By
utilities
—>
the right side must vanish as n
for y).
s
>
t
close
u(x n )
>
u(x) and such that
W
all
>
0,
6 to
>
8 t —e on
average unmatched value
how
By
>
t
local
enough to
t
(by
tn
—
{x n )
must
W
tn
exist prizes
{x)
>
0, i.e.
In the job search problem, the reservation
an upper semicontinuous function of
e
for s
<0>
time.
This follows from Theorem 2 and the definition of what an
that for
s
Thus, for large enough n, we have
continuity and local nonsatiation, there
Corollary (Upper Semicontinuity)
is
W
oo, while the inner
,
t
n.
u(x). Write
x n G A*tn as required.
wage
>
(x)
A*
Here must appeal to the properties of
x and by monotonicity
— yVt n (x) >
yVt n (y)
enough
n.
one has positive liminf because both x,y G A*s for
for
t
= [Wtn (y) - Wtn (y)] + [Wtn (y) - Wtn (x)} + [Wtn (x) - Wtn (x)]
Then the outer terms on
assumption
>
and consider the sequence
nonsatiation, there exists a prize y close to x yielding utilities u(y)
W
(u(x))
Here we must appeal to simple
t.
(x)
Due
x.
in time, yielding
for all large
t
and then behaving optimally. Then
suppose that
Finally,
close
t
t.
=
1
be the not necessarily optimal average value of holding
(x)
s
both right continuous
(x) are
t,
some time 6^
until
4- 1-
enough n by
large
all
employ the sequence x n
shall
x must be held
Next, consider the second case, with
xn
I
.
t
with t n
first case,
then x € A*n for
,
W
>
to the accounting identity (1), prize
if
t,
argue that this limit separately holds for both
even though x E A*t
t
—>
tn
such that x n £ A*tn for all large enough
increasing and decreasing sequences of times. Consider the
If
dominated by
is
for all utilities.
Suppose that x € A*
there
positive search costs will not invalidate the arguments
W
t
is
u.s.c.
function
some small enough neighbourhood of t
.
a continuous function of time, Theorem
the reservation wage can discontinuously
namely,
So while the
1 illustrates
jump up when \V jumps.
t
is;
acceptance set
Intuitively, the
may suddenly
'implode' but can never 'explode',
and the reservation wage can 'jump up' but can never 'jump down'.
derscore
how
general
is
should un-
the underlying idea here. Indeed, Smith (1994b) considers
a class of models where a heterogeneous stock variable
property (hard to build up, but easy to destroy).
tia'
I
an 'upward
satisfies
iner-
preferences over the stock
If
composition have a deterministic evolution, then the optimal policy entails flows
that are lower hemi-continuous in time:
One might
well suddenly wish to discretely
destroy a discrete portion of the stock, but one only continuously builds
model, think of the stock of capital in set
this search
w€
x € A, or a job with wage
prize
A] the flow
is
A
up. In
it
as the probability one has a
the acceptance set at any time.
Other typical illustrations of this idea include a firm's behaviour with respect to
diverse capital stock, or individual behaviour with respect to one's reputation.
its
Smith (1994b) explores the general proof of
work on nonstationary job search
some
some a >
when 7 >
0.
(0, 1) until
how this nontrivially
by Van Den Berg (1990). I
generalizes previous
Case
-
and 7 > 0. This corresponds to a foreseen wage distribution 'spike'
0, and a foreseen one-shot upward shift in the wage distribution when
>
have an arrival rate p
offers
>
0,
the interest rate
is (3
>
and the
0,
0.
>T
Since the prospects are deteriorating over time after T,
job once accepted will never be dropped. Thus, equations
= _c +
Since
1,
1
Job
1: t
show how
shall also
that wages
~
and thereafter uniform on (0,ae~"(( t T ^), for
T>
time
(inescapable) search costs are c
,
applications.
basic comparative statics insights change. Suppose, as in figure
are uniform on
w
its
conclude with an example showing
I
=
and
AN ILLUSTRATIVE EXAMPLE
4.
7
this property
r
h
Theorem
„e
1
-<* + ,>,-«,
[c
L
+
asserts that 6 t
f
\
=
6t
=
>
0,
(1)
and
W
(2)
t
—
W
for
t
+
s~
T
t
(
/
2
G
[T, 00),
we have
2
-7(-T)
>
a e
0t,
and a
reduce to
«-*-"+». _*
-^'-M
+
ae-^
ae-^
pe -(/Hp)(s-0[ c
Thus, so long as 6 t
we have
s~
T
WJ(b
>
+ 02 e 7(-T))/2a](te
we have
-(p/2a)[a
2
e-^- T + V<'- T +
)
t
>]
(p
+ 0)6 + /3c
t
(4)
a-
Ot=W
t
Figure
The
Optimal Behaviour given a Foreseen Wage Distribution Spike.
1:
wage (thicker lines) and unmatched value are depicted for a wage
is known to be uniformly distributed at any moment in time (on
slices) on the shaded region. Depending on the parameters p and /?,
reservation
distribution that
the vertical
either the solid or the dashed paths obtain.
For reasons of tractability,
and
>
c
then the problem
0,
6t
is
I
shall confine attention to
stationary,
= max ^0, a(p + P)/p-
two
and so optimally 9
yf^Tp
+ (3) 2 /p 2 -
(a 2
t
-
cases. First,
=
0.
if
7
=
Hence,
2ac/?/p))
>T. Observe that 6 > (and thus the equations are well-defined) precisely
when c < ap/2(3, i.e. when flow search costs are not too high.
= e~ l(t ~ T ^6 T where
Second, if 7 >
and c = 0, then (4) admits the solution
for all t
t
t
9t
Case
2:
If
6T
= max (0, a(p + (5 + 7 )/p -
<T
> 1, then
<
1,
+ P + 7 )yp*-a^
t
the individual will drop any job agreed to in [0,T); thus, she
problem on [0,T), subject to 8t- = 0. Otherwise, if
we can act as if the problem is finite horizon with a terminal job max(0, w —
solves a de facto finite horizon
9t
y/a*(p
,
if
6t)
wage
w
is
held at time T. In general, the Bellman equation
is
7
T
=
[l-e-W-'tyt
pe -^-^(-c{l-e-^-^)
J
+
_
e -^-')(l
e
^
_ ^ )(1 + 9s)/2 +
T
p(l - 6 T )e~ f>
-/»(T-.)
)[(1
+ max(0,e- /3(T Hl -9T
)/2J
t
For #r
>
the last term
1,
**=
As might be
ds)
KV g
"«r + 1 )/
W r- +p^ g t-p(
" 2 + l_ e -/3(T-t)
l_ e--/J(T-t)
c
rt
,
'
'
'
guessed, this differential equation admits no neat closed form solution,
<
available for 0?
=
(5)
and so
is zero,
By
but can be numerically evaluated.
T-
ds
- 6r)dr
{1
limtfT Ot
=
1;
the
same
token, only a numerical solution
however, we can apply l'Hopital's rule to
P(l
— 9T
2
)
/2P
>
Thus, the greater
0.
and deduce that
(5),
<
6T
is
1,
is
the lower
T --
is
Comparative Statics
This simple example
1.
offers
FSD
[Improved Wages:
at time
fails:
true that the
many
Shifts] First, the standard reservation
=
Consider 7
T < T, then
useful explanatory power.
improvements with respect to
tonicity with respect to
inance (FSD)
some
shifts
In that case,
0.
down on [V -
(W
unmatched value
t)
if
wage mono-
order stochastic
first
dom-
the wage spike occurs earlier
£,T"), for
some e
>
0.
But
it is still
uniformly increases on [0,T), and in fact
of our stylized steady-state intuitions are best interpreted as extending to
average rather than flow values.
2.
[Increased Risk:
SSD
Shifts] While the example cannot illustrate this,
also true that increases in risk
increase (9 t ). This
is
intuitive
(i.e.
if
it is
a mean-preserving spread) need not uniformly
one imagines just as above that the increase
is
concentrated starting at some time.
3.
[Policy Duration]
A
'more permanent' change
and can increase reservation wage
9t
<
1,
smaller 7) cannot decrease
around the regime
shift.
Indeed,
if
7 pushes up 6T and hence down Ot-. I think this is an
addendum to macro arguments on the consequences of a permanent
then a
interesting
volatility
(i.e.
fall in
,
versus a transitory government policy shift.
By means
were
7
of a different example,
cyclical, say
uniform on
[0, 1]
I
could also show that
at times
[0, 1)
U
[2,
3)
U
•
if
•
the wage distribution
and uniform on
The non-zero maximand consists of the average terminal value if the wage that
exceeds 6? times (the integral) the chance that the first accepted wage exceeds 0t-
is
[0, a]
accepted
MIT LIBRARIES
3 9080 00922 5266
at times
[1,
2)
U
[3,
U
4)
•
then an optimizing individual will also employ a cyclical
•,
a stochastic number of voluntary unemploy-
reservation
wage
ment
and eventual 'permanent' position with a
spells
rule.
This
will lead to
sufficiently high
wage.
shall
I
abstain from providing the mathematics, but the intuition should be clear.
5.
CONCLUSION
There are two modelling assumptions
there
no distributional uncertainty
is
here:
it
I
paper worth commenting on.
First,
assume that the wage distribution
known ex ante. It is not clear how to
would certainly entail much messier analysis, in which
follows a deterministic evolution that
relax this assumption, but
in this
is
one might obtain declining reservation wages as in Burdett and Vishwanath (1988).
Second,
I
have assumed that
it is
the outside environment, and not the current job
characteristics, that are variable. For
it
if
one's
own wage were
subject to change, then
might well be true that no current reservation wage rule alone could optimally
serve as a quitting yardstick (see Shimer
and Smith
(1994)). In that case, or
if
one
were slowly learning one's preferences about the job, then there might exist some
index that would serve the same role as the wage in this paper.
(eg. Gittins)
References
Banks, Jeffrey
S.
and Rangarajan K. Sundaram,
"Switching Costs and the
Gittins Index," Econometrica, 1994, 62 (3), 687-694.
Berg, Gerard J.
Economic
Van Den,
"Nonstationarity in Job Search Theory," Review of
Studies, 1990, 57, 255-277.
Burdett, Kenneth and Tara Vishwanath, "Declining Reservation Wages and
Learning," Review of Economic Studies, 1988, 55, 655-665.
Dvoretsky, A., A. Wald, and
in Certain Statistical Decision
Annals of Mathematical
Shinier,
1994.
J.
Wolfowitz, "Elimination of Randomization
Procedures and Zero-Sum Two- Person Games,"
Statistics, 1951, 22, 1-21.
Robert and Lones Smith, "Matching,
MIT mimeo.
Smith, Lones, "Cross Sectional Dynamics
1994.
—
,
Search, and Heterogeneity,"
MIT
in
a Two-Sided Matching Model,"
mimeo.
"The Optimal Policy Correspondence with Sticky State Variables," 1994.
Work
in progress,
Nancy
MIT.
and Robert E. Lucas, Recursive Methods
Dynamics, Cambridge, MA: Harvard University Press, 1989.
Stokey,
L.
10
in
Economic
90 6
Date Due
m
Lib-26-67