Job Search: The Choice of Intensity
Jess Benhabib and Clive Bull
New York University
We integrate the optimal sequential and the optimal sample-size
search strategies. In contrast to the existing search literature in
which individuals specialize in either search or work, the model in
this paper allows for an optimal choice of search intensity that results
in individuals' choosing to work and search simultaneously. The
existing sequential search strategy literature follows optimal stop-
ping theory in treating work as an absorbing state. We do not, how-
ever, require the searcher to stay in a job once it has been accepted
but derive this behavior as a result of the searcher's optimization
problem.
I. Introduction
Search theory has shown that some of the unemployment we observe
can be explained by the search behavior of unemployed workers.
However, the literature has concentrated on the sequential nature of
an optimal search strategy, rather than the intensity of search, by
imposing the constraint that unemployed searchers receive at most
one wage offer per period of search.' In contrast, the seminal paper
We would like to thank an anonymous referee for his comments and Pfizer, Inc. and
the New York University Challenge Fund for funding this research.
' See, e.g., Mortensen (1970) and Lippman and McCall (1976a) for a survey of the
literature. Lippman and McCall (1976a, pp. 164-66) have treated the number of offers
received by a searcher as a random variable rather than a fixed number. However, the
distribution of the number of offers received is not treated as a control variable of the
searcher. Lippman and McCall (1976b, pp. 376-78) allow the searcher to choose a
sample size, but the sample size affects only the probability of receiving an offer and
not, as sampling theory would suggest, the distribution of the best offer. For further
results, in the context of different models concerning the optimal choice of sample size,
see Gal, Landsberger, and Levykson (1981) and Morgan (in press).
[Journal of Political Economy, 1983, vol. 91, no. 5]
? 1983 by The University of Chicago. All rights reserved. 0022-3808/83/9105-0001$01.50
747
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748 JOURNAL OF POLITICAL ECONOMY
in this field by Stigler (1961) concentrated entirely on the intensity of
search, that is, the sample size of wage offers selected by the searcher
given that he or she must take a job at the end of one period. This
paper will combine these two approaches by allowing workers to
search sequentially and, in each period, to choose optimally that pe-
riod's sample size, that is, the intensity with which they search.
Most of the search literature has followed optimal stopping theory
by treating work as an absorbing state; that is, once a searcher accepts
a job, he or she is not allowed to quit it.2 This appears to be a very
strong assumption, and it would be of interest to see under what
conditions it could be derived as a result. If, for instance, the reserva-
tion wage was nonincreasing with the passage of time, then an accept-
able wage offer would remain acceptable throughout the worker's
working life and we could conclude that he would not quit his job,
ceteris paribus.
There are simple proofs in the literature (see, e.g., Lippman and
McCall 1976a, p. 167) showing that in a finite-horizon model in which
quits are not allowed the reservation wage does not increase with the
passage of time. However, to show as a result that quits will not take
place, instead of simply assuming it, one has to show that the reserva-
tion wage is a nonincreasing function of time in a model where the
worker is given the option to quit. This problem was posed by Lipp-
man and McCall (1976b, p. 382) but left unresolved. We show in
Section II that the reservation wage will not rise over time and so an
individual with a finite horizon will not quit a job once it has been
accepted.
Our generalization of the sequential search model enables us to
show that the optimal sample size when search takes place may be
greater than one but will be less than that chosen in a Stigler-type
optimal consequential search model. We then examine how the op-
timal sample size changes with the passage of time, the costs of sam-
pling, and the level of nonlabor income during periods of search.
In Section III, we modify the model of Section II to allow for
simultaneous work and search, thereby allowing us to tackle the issue
of the existence of simultaneous work and search by workers. An
unfortunate outcome of standard sequential search strategy models is
that a worker will specialize either entirely in work or entirely in
2 See the references in Lippman and McCall (1976a) and esp. Kohn and Shavell
(1974, p. 96). Lippman and McCall (1976b) allow workers to quit but only in response
to an exogenous change in the distribution of wage offers. Wilde (1979) and Lippman
and McCall (1981) also allow for quits but only in response to postacceptance revelation
of the nonwage characteristics of the job. Burdett (1978) has dealt with endogenous
quits, though only in the context of a model that allows simultaneous work and search.
See Flinn and Heckman (1982, app. B) for a further development of this approach and
possible empirical tests.
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JOB SEARCH 749
search; that is, it is not optimal to work and search simultaneously.
Moreover, the result holds even where the individual has a concave
utility function. The intuitive explanation for this result is that the
reservation wage in these models, which is the marginal opportunity
cost of work, is not a function of the intensity of search within the
period. Thus, if the reservation wage is above one's current wage, it
pays to quit and just search, and vice versa if the reservation wage is
below one's current wage. To remove the dichotomous nature of the
worker's optimal program it is necessary to make the marginal oppor-
tunity cost of work fall as the proportion of time in the period used
for search rises. Seater (1979) achieves this by assuming that the
"technology of search" is such that the time taken to generate one
further wage offer in a given period rises with the number of offers
already obtained. He motivates this assumption by a spatial view of
search. The model in this paper provides a direct explanation of
simultaneous work and search. Here the marginal cost of search rises
with the intensity of search simply because of the concavity with re-
spect to the sample size of the expected value of the best offer.3 This is
precisely the same property that Stigler relied on in his original analy-
sis of nonsequential search strategies.
Having shown the possibility of simultaneous work and search we
then extend the results of Section II to the case of simultaneous work
and search. As this model is both formally and economically of a
character different from that of Section II, some of the results differ
across the models; in particular, the optimal sample size is a nonin-
creasing function of time in the case of simultaneous work and search,
whereas when work and search are mutually exclusive it is a nonde-
creasing function of time.
Section IV contains some concluding remarks.
II. The Choice of Sample Size in a Sequential
Search Strategy
In sequential optimal search strategy models the sample size of offers
one receives if one searches is fixed, typically at unity. However, in
many actual search situations, especially where there is a significant
delay between making an application and receiving an offer, the size
of the sample is a choice variable. Consider the fate of a new econom-
ics ABD. Having sent out his or her first application to a university,
the candidate faces a long wait for a reply. During this time the
3 Recent empirical research by Chirinko (1982) supports the hypothesis of dimin-
ishing marginal returns to search.
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750 JOURNAL OF POLITICAL ECONOMY
candidate can either work on his dissertation or submit more applica-
tions, give more seminars, etc.4 Thus the ABD is faced not only with
the decision whether or not to search in a specific year but also how
intensively to search. The model below characterizes this decision
problem.
The worker/searcher is assumed to be an expected income max-
imizer whose working life terminates at T. He knows that there is a
distribution of wage offers, w E [0, oc), across firms but he cannot tell
which firm will make which wage offer. Thus if he searches he will
contact firms randomly. Let the cumulative probability and probabil-
ity density functions of the wage offers be denoted by F(w) and f(w),
respectively. The cost of making contact with a firm is a positive
constant c. We assume that all firms contacted within the period make
their offers at the end of the period for work to begin next period.5
Moreover, the wage, if accepted, will remain constant while the
worker stays in the job and he has tenure in that job, though he can
quit if he wishes to. Finally, we assume that if the worker/searcher
decides not to work in any period he receives income s - O.6
In general the worker/searcher has three options available: to work,
to search, or to remain idle. Now consider his position at the end of
period t - 1 (< T).7 If he has neither worked nor searched that
period then his alternatives in period t are either to search or to
remain idle. If, on the other hand, he had searched in period t - 1,
he would have the additional opportunity in period t of working at a
wage equal to the best wage offer he had received in t - 1. Finally, if
he had worked in t - 1 he would have the alternatives of searching,
being idle, or continuing to work at the same wage as in t - 1. Thus
the expected income or value function of the worker/searcher, given
that he has the opportunity of earning w 0 in period t, is given by
Vt(w) = max w + 3Vt+ I(w), s -cnt
(1)
+ 0Vt+ l (w)-y(w; n*8)dwo, s + P3Vt+ I sl
4 In Sec. III we will allow him or her both to submit more applications and to work on
the thesis.
5 These offers cannot be stored across periods; i.e., he samples without recall. Mor-
gan (in press) deals with the variable sample size sequential search strategy with recall.
He does not, however, allow the worker the opportunity to quit.
6 The variable s could be interpreted as unemployment insurance benefits.
7 We assume that his postretirement income is independent of his search strategy and
that he cannot recall offers made in previous periods.
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JOB SEARCH 751
where 0 < ,3 ' 1 is the discount factor and y(w; n) is the probability
density function of the largest wage offer (the first order statistic) W
for a sample of size n ? 1.8
The nt in equation (1) is the worker/searcher's optimal choice of
sample size should he search, that is, the value of n maximizing the
middle term of (1), the value of search, over the positive integers.
Carrying out this maximization over the real line for n - 1 yields the
first-order condition
C = i { d V?I (w)dwo. (2)
The n satisfying this condition, ni, will be the optimal sample size since
we show in the Appendix that the value of search, the middle term on
the right-hand side of (1), is concave in n. Although n1 may not be an
integer, it follows from concavity that the optimal choice of n, n*,
when n is restricted to be a positive integer, will be either n1 if n1
happens to be an integer or one of the two integers immediately
adjacent to n.9 Thus when n is not an integer, the searcher will have to
compare the values of search at the adjacent integers.
Having calculated his optimal sample size if he searches, the
worker/searcher now chooses the maximum of (1). If he works he will
receive w and the discounted value of the opportunity to continue
working at w in t + 1. This is the first term on the right-hand side of
(1). Alternatively, he could choose to search optimally, in which case
he will receive s, incur search costs cn*, and gain the discounted ex-
pected value of the opportunity to work at the highest offer he re-
ceives. This is the middle term of (1). If he neither works nor searches
he receives the third term in (1), which represents his nonlabor in-
come s and the discounted value of the opportunity to receive s next
period.
Given (1), the agent's decision to work will depend on his compari-
son of the first two terms, that is, the values of work and search.
Setting these terms equal and solving for w yields the reservation
wage we of the individual provided we is greater than or equal to s. 10
If we is less than s the agent will choose to be idle in period t rather
than work or search, and so s will constitute the reservation wage.
8 r(o; n) = nFn- I (w)f (a). We assume that f ' wf (w)dw exists. This ensures that the
expected value of w exists for all sample sizes greater than zero (see Gumbel 1958, pp.
42-51).
9 We define -y(w; n) to be zero for all X when n = 0.
10 This depends on the value function increasing monotonically in its argument,
which can be shown to be the case by an argument essentially identical to that of
Lippman and McCall (1976b, theorem 1, pp. 372-73).
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752 JOURNAL OF POLITICAL ECONOMY
Setting the first two terms on the right-hand side of (1) equal and
solving gives the reservation wage,
w* = s - cn* + V,,1(w)yw; n*)dw - Vtl(w*)1. (3)
The time path of this reservation wage is of interest because it
determines whether the individual, having accepted a job, will ever
decide to quit it at a later date. As we noted in Section I, most authors,
borrowing from the optimal stopping literature, assume that employ-
ment is an absorbing state, that is, that workers will never quit. But
this assumption will be restrictive unless the reservation wage does not
increase over time. If it were to rise over time, then there could be an
incentive for workers to quit. As the only existing results on the time
path of the reservation wage in finite-horizon search models are based
on the assumption of no quits, it is important to investigate the prop-
erties of the time path in a model that specifically allows quits to take
place. In such a model if one could show that the reservation wage is
nonincreasing over time, then one would obtain the result that quits
will not occur.11 We now prove this result.
PROPOSITION 1: In a finite-horizon, sequential job search strategy
where both quits and optimal sampling are allowed, the reservation
wage will be a nonincreasing function of time and so quits will not
occur.
PROOF: Consider the expressions for the reservation wage, similar
to (3), in two adjacent periods t and t + 1:
w* + IVt I(w*) = s - cnt + I Vt 1+I(w) -y(w(; n*)dw, (4)
wt+1 + IVt+2(w?+1) = s - cnt+1 + 13 Vt+2(W))Y(W; n* 1)dw. (5)
Now proceed by contradiction. Let w* 1 be the correct reservation
wage at t + 1 and assume that we = = w*. Then the value of
work at the reservation wage, that is, the left-hand side of (4), must
have dropped by exactly the same amount as the fall in the value of
search at the reservation wage, that is, the right-hand side of (4). This
we can check.
Taking the value of work first, substituting for V,? +I() in the left-
hand side of (4), and subtracting the left-hand side of (5) yields
Ow* + ( - I)Vt+2(w*). (6)
" Lippman and McCall (1976b, p. 382) raised the issue but did not solve for the
properties of the time path of the reservation wage.
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JOB SEARCH 753
Now consider the value of search. As n* is the integer value of n, that
maximizes the right-hand side of (4), substituting in nO.,1 will not raise
its value. Making this substitution and subtracting the right-hand side
of (5) from that of (4) yields
13{[Vt+I(W) - Vt+2(W)]Y(W; n* 1)do, (7)
a number less than or equal to the fall in the value of search. Now
Vt+I(W) = X + IVt+2(W) if W 2 W8+1 = w*, and Vt+?(w) = we +
I3Vt+2(w*) if X w8+ I = w*. Thus (7) equals
I 13{ + (a - 1)Vt+2(w)]Y(w; n* I)dw
W* (8)
+ 13{ [w* + P3Vt+2(w*) - Vt+2(O)]w(W; n* 1)dw.
The first part of expression (8) can be made smaller if we replace
the first w under the first integral sign by its lowest value, w*. Simi-
larly, the second part of expression (8) can be made smaller if we
replace f3Vt+2(w*) under the second integral sign with Vt +2(w), since
Vt+2(W) is increasing in w. This gives the expression
3[w* + (13 - 1)Vt+2(W)]^y(W; n* 1)dw
? 13{ [w* + (1 - I)Vt+2(W)]Y(W; n* I)dw,
which is smaller than (8) and simplifies to
w* {y(w; n* 1)dw + (13 - 1)Vt+2(W))^y(W; n* 1)dw,
which in turn is equal to
13w* + 13(13 - 1){ Vt+2(W))^y(W; n* 1)dw. (9)
Because (9) was obtained by reducing both of the integrals that make
up expression (8), it represents an expression that is strictly less than
the fall in the value of search. Comparing (9) to the fall in the value of
work given by (6), we see that the fall in the value of search exceeds
the fall in the value of work only if
.Vt+ 2((-t))-((); n*+ 1)d(.) ' Vt+2(W*). (1 0)
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754 JOURNAL OF POLITICAL ECONOMY
Rearranging (5) and remembering that we set wt1 = * = w*, we
have
w* - s + cn*1+ = Vt+2(W)y(W; n* 1)dw - Vt+2(w*)1. (11)
From (1), it is obvious that w~t+ 1 = w ?s and therefore that both sides
of equation ( 11) are nonnegative. This implies that (10) holds and that
the fall in the value of search at the reservation wage is strictly greater
than the fall in the value of work, contradicting the equality of the
reservation wages in the two periods, that is, of w* = w* I = w*. It
follows that for both (4) and (5) to hold w* must be raised such that w*
> wt+1. Q.E.D.
Thus we have shown that a worker/searcher will indeed not quit a
job once it has been accepted, ceteris paribus,12 and so the assumption
in the literature that such quits cannot take place is not restrictive.
We now turn to the optimal sample size n*, or intensity of search. In
contrast to the standard sequential search models, the agent here will,
if he chooses to search, in general pick a sample size greater than one.
The question then arises as to the relationship between n* and the
optimal sample size in a single-period optimal size strategy. We first
consider the optimal sample size in this latter case.
The searcher is the same as the one considered previously except
that during t - 1 he neither worked nor searched and that he is
allowed to search only in period t. At the end of period t he must
accept a job, which he will keep until T. His objective is to maximize
with respect to the sample size mt ? 0 his expected income Wt, where
Wt is given by
T-t
Wt= S - Cqt +l ? s wy(w; -q)dw. (12)
The first-order condition for this maximization is given by
T-tx
C = a dw. (13)
Let the value of -q that satisfies (13) be denoted by Fi. We will now
prove the following proposition.
PROPOSITION 2: The optimal sample size chosen in a sequential job
search strategy will be less than or equal to that chosen in a single-
period optimal sample size strategy.
12 Quits can of course take place if the state of the world, e.g., the distribution of wage
offers (Lippman and McCall 1976b), a nonwage characteristic of the job (Wilde 1979),
suddenly changes, but such quits are not endogenous to the individual's search strat-
egy.
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JOB SEARCH 755
PROOF: Consider value function (1) and assume that in period t - 1
the individual neither worked nor searched, so that the value function
was Vt(s). Now consider the optimal choice of nt under condition (2).
If we imposed upon the individual the choice of n, = ft and found
that c was greater than the right-hand side of (2) we could argue by
concavity that n't < t. From (2) and (13) we see that this is the same as
showing that
T-t
i( Vt +1 ) -I E an (t) d(. < ?. (14)
The maximum over a set of strategies, one of which is to accept the
best job offer and work until T, is V? 1(w). Thus for w E [0, co),
E3Vt+ 1 (w) ? EA_ Wl. For X < w* 1 this inequality must hold strictly as
the individual is not choosing to accept the job. Moreover, the lower is
X in this range the greater will be [L3Vt+ 1(w) - n PT (].
Now consider the value of [IVt+ 1(w) - T-tITW] for w ? +w;
AVT+ 1(W) will differ from IT-1 PTdw by exactly the value of the option
to quit in the future. However, proposition 1 implies that the individ-
ual will never wish to quit, so the value of this option will be zero.
Putting the results for X < w* 1 and X - w* 1 together, we see that
[I3Vt+ 1(W) _JT--t AdW] is a nonincreasing function of w for w E [0, c).
To show that condition (14) holds we note from the Appendix that
there exists an X E (0, xc) such that (ay/an)(w; f1) c 0 according to W c
C. Splitting the integral in (14) at X and noting the nonincreasing
nature of the integrand in w we get the following inequalities:
fOWL ~~T-t __T-t1{
it+ I (@) - XITS] b dw L iVt+ l(W) - E T y dw
fL~~vt~i~ TZt LT-t __dw
a- t+ I(@) - E TS]) A P Vt i+ I (@) - E T(1) dA d
Note that yQ; ) is a density function and so f (a8y/an)dw = 0; combin-
ing the inequalities above yields
TP
{4 vt +I -LP A) dw?LIVt?1(w I Z13N01f0 = 0.
Jo [ T~~T= I ]an [T= I ] o an
Q.E.D.
Note that if we assume further that the distribution y(; ) has some
mass below w* 1, then the expression above holds with strict inequal-
ity, thereby confirming (14). Hence nit < t, and so n*< K* except
where "rounding" to the appropriate integer causes them to be equal.
This result tells us that the sequential problem is a true dynamic
programming problem rather than a series of independent one-
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756 JOURNAL OF POLITICAL ECONOMY
period optimal sample size problems. The intuition for the decreased
optimal sample size is as follows. In the single-period problem raising
one's sample size has conceptually two marginal benefits. Obviously it
raises the expected value of the best offer received. However, it also
provides increased insurance that one will not be stuck with a low
wage all the way to T. The sample size is increased until the sum of
these marginal benefits equals the marginal cost c. But when we allow
sequential search, that is the possibility of refusing all the offers and
searching again next period, the insurance benefit of raising the sam-
ple size is reduced. As the marginal cost of increasing the sample size
is the same in both cases, the reduction in one of the components of
the marginal benefits requires the sample size to be reduced if mar-
ginal cost is to equal marginal benefits as required by the first-order
condition (2).
The comparative statics of the optimal sample size are as one would
expect and are summarized in the following proposition.
PROPOSITION 3: The optimally chosen sample size in the finite-
horizon sequential job search strategy is a nondecreasing function of
s, the payments received when not working, and a nonincreasing
function of c, the cost per firm contacted.
PROOF:
i) In (2), V,+ 1(w) is nondecreasing in s and so, by concavity, dn*lds
0.
ii) Raising c raises the left-hand side of (2) but does not raise the
right-hand side. This latter point can be shown by a simple induction
argument. From (1) we see that, given V,, 1(w), V,(w) is nonincreasing
in c. By writing out (1) for periods T and T - 1 we see that VT(w) and
VT- l(w) are also nonincreasing in c. Thus dV,? I/dc < 0. As the left-
hand side of (2) has been raised while the right-hand side has not, by
the concavity argument used previously, dn*ldc ' 0. Q.E.D.
Finally, we turn to the time path of the optimally chosen sample
sizes in the sequential search strategy. We shall prove the following
proposition.
PROPOSITION 4: The optimally chosen sample size in the finite-
horizon sequential job search strategy is a nondecreasing function of
time.
Consider the first-order condition shown in (2) for the choice of the
optimum sample size in two consecutive periods t and t + 1:
c = { a Y Vt ,?I(w)dw, (15)
c = a| j' - Vt 2(w)dw. (16)
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JOB SEARCH 757
Let the value of n, that solves (15) be nt and set ntt+ 1 = nt. Now if we
subtract (16) from (15) we get
0 = o Y Vt+ 1(( - Vt+2(oj) ]d(o. (17)
a t~
Using the notation r(y; n) = fY -y(w; n)dw = F" (y), we can rewrite the
right-hand side of (17) as
P[W*+ I + PVt +2 (W~t+ 1) -a (w*t+,; Mt + P~f Vt+2((A)) ando
8n 8nn A
- [W*+2 + P3Vt+3(wt+2)] (w~t+2; nt) - P Vt+3(OJ) ddo
an an
(18)
which in turn can be written, since 1impe arJ(; n)/an = 0, as
13* [Vt+2((O) - 3W1St+l - PVt+2(Wtl+ )] aw do
- i [Vt+3(W) - W* - I2Vt+3(w~t+2)] dw. (19)
Integrating (19) by parts and noting that, since for all n iim,7 r1(; n)
= 1, we obtain
-1 n ('w; nt) d~t+3dw + oW* d1 (@; nt)dVt+2dw
{W+ 2a8n t0 t)V?d t+ 2a
XC d (20)
+ 4t W an (@; nt))dVt+2dw.
Since for values of the wage above the reservation wage the value
function is equal to the value of work, dVt+2(C) = 1 ? dVt+3(W) =
= 0 PtI. So (20) can be written as
00 ~~T-t-1I T-t-1I
| r 1t+ I 8r
817* (CO;i) n ITdw?1 JW0 * 8n (W; ni ) P Wdw < 0, (2 1)
since ar(w; nt)/an < 0. Thus (21) and therefore the right-hand side of
(17) are negative, and if we are to restore the equality in (17), by
concavity of the value of search, we must have nt < nt+ l and so nt ?
no 13QE
'3 The weak inequality stems from the integer nature of the problem. If the sample
size were allowed to be a nonnegative real number the optimal sample size would
strictly increase with time. The result also holds if the cost of search is concave in the
sample size.
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758 JOURNAL OF POLITICAL ECONOMY
III. Simultaneous Work and Search
The analysis of Section II followed the standard search literature in
not allowing the individual both to work and to search during the
period. However, it is unlikely that voluntary specialization in search
is at all common. Rather, one would expect an allocation of time
between search and work within the period in such a way that the
marginal benefit of time spent on each was equal; that is, one would
not expect a corner solution. It is unfortunate, therefore, that when
the standard models are generalized to allow simultaneous work and
search they yield, as an optimal choice, complete specialization in
either work or search. The reason for this result is that in these
models the marginal benefit of search time does not decline as the
amount of time spent on search rises, even when the individual's
utility function is concave in income and leisure. We show in this
section that when one allows the individual to choose an optimal
sample size in an intertemporal problem, the marginal benefit of
search declines with the intensity of search and thus, in general. he
will not specialize completely in search or work. This provides a direct
explanation of the simultaneous work and search "problem" raised by
Seater (197 7, 1979). Seater assumes, on the basis of spatial considera-
tions, a technology of search such that there is a rising marginal cost
to obtaining job offers. The solution put forward here requires no
such special assumptions about "search technology."
Consider the worker/searcher of the previous section. For simplic-
ity we will assume that he supplies inelastically one unit of nonleisure
time per period. Given the amount of leisure he consumes, his utility
is a linear function of consumption. He can both work and search if
he wishes, and the cost of making contact with a potential employer is
simply the time forgone, which is 1 units, 0 < 1 ' 1. The individual
maximizes his expected utility subject to the constraint of spending no
more than one unit of time on work and search together. His value
function in period t, V(w, t), is then
V(u', t) = max PV(w, t + 1) y(w; n)dw + PV(w, t + l)y(w; n)dw
?I (22)
+ P(1 - ln)w + X(1 - In),
wher-e UP is the wage he has in period t and X is the Lagrangian
multiplier on his time constraint. If this latter constraint is not bind-
ing, then (22) says that his expected utility equals the discounted value
of his utility next period given his current wage times the probability
that he does not find a better wage in t + 1, plus the probability-
weighted discounted expected utility next period given that he does
not find a wage above w, plus his discounted earnings in t + 1.
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JOB SEARCH 759
Again we assume that n lies on the real line n ?-1 and solve for the
optimum n, n'. As the value function in (22) can be shown to be
concave, the first-order conditions will be sufficient and the optimal
integer n will be n, if n2 happens to be an integer, or one of the
immediately adjacent integers. Differentiating (22) yields the first-
order conditions
13V(w, t + I Y dw + { V(w, t + 1) do
-lw-lK?O,1-ln?O. ~~~~(23) - Iw - IA c< 0, I - In -O0. 3
When the time constraint is not binding, In < 1, K = 0, and so
lw = V(w, t + 1){ a'Y dw + P3V(W, t + 1) dw. (24)
an a~~~~~~n
Note that the n2 which solves (22) may often not require complete
specialization in search or work.
We now extend the propositions of the previous section to the case
where simultaneous work and search is allowed. We will start with
proposition 2, which compares the optimal sample size in a sequential
search strategy with that of a single-period search strategy. The sin-
gle-period search problem is as follows. Assume the worker has a
current wage of wt and will search for only one more period. His
problem is to maximize, by his choice of -rt,
Wt = (1 - llq)w t + f 1 -y((,); -qt)dw + wt Z T ET (t?; -qt)d(w,
subject to -t ? 0, ant < 1. The first-order conditions, which are
sufficient conditions for a maximum here, are
T-t 0 t)T-t W -~;'t
lot '_ IA + XP",T| (I) ) dw + wt X PT dw,
(25)
Iqt 1 l, K(1 - I&t) = 0.
PROPOSITION 2': For the case where simultaneous work and search
is allowed, the optimal sample size chosen under a sequential search
strategy will be less than or equal to that chosen under a single-period
search strategy.
PROOF:
i) Consider first the case where both the optimal choice of nt, nt, and
the optimal choice of -rt, 't, are interior, and K = 0. Then (25) can be
written as
1W E T E t d dX+ IE V TfY(w t) I (
= a-q dw?+ Wt >' J dw. (26)
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760 JOURNAL OF POLITICAL ECONOMY
Using the same argument as in the proof of proposition 2 we can set n,
= It and consider the difference between the right-hand sides of (24)
and (26). If
00 ~~~T-t
P V((,@ t + 1) - E A '( di
{Lvt~ r3 J
T-t (27)
?[P v(W~t? 1)- WtZ 3j PT a'y~w~ dw?O< ,
then, by the concavity of V(wt, t) in nt, we can conclude that nit < ft and
so n* ? a Condition (27) can be shown to hold by an argument
identical to that used to prove proposition 2.
ii) To complete the proof we must rule out cases where nit = 1/1 and
It < 1/1 and where 't = 0 and nt > 0. Let us deal with the first of these
cases. If nit = 1/1, equation (24) will include a term - 1 on the right-
hand side where A - 0, and so
lw ' V(w, t + 1){ aY dw + { V(w, t + 1) dw.
Now consider lowering nit to Ft. By concavity this must raise the right-
hand side of the equation immediately above thereby making it
strictly greater than 1w. But this term is therefore greater than the
right-hand side of (26), which is equal to 1w. However, weak inequality
(27) must hold at the optimally chosen nt and -qt; thus we have con-
tradicted our initial assumption that nii = 1/1 and
argument it can be shown that it cannot be the case that nit > 0 and
t = 0, thereby completing the proof. Q.E.D.
Consider now the time path of the optimal sample size. As the cost
of search in the simultaneous work and search case is the opportunity
cost of the working time used for search, the optimal sample size will
be a function of the wage rate currently received. Thus we wish to ask,
What will be the time path of the optimal sample size given a certain,
constant wage rate? This is answered in the following proposition.
PROPOSITION 4': For the case of simultaneous work and search, for
any given wage, w, the optimal sample size is a nonincreasing function
of time.
PROOF:
i) First assume that at the given w both nit and nt+ I are interior, and
so the first-order conditions are given by, respectively, (24) and (24)
updated by one period. If nit = nt+ 1, then subtracting the updated
equation from (24) and dividing by 3 yields
[V(w, t + 1) - V(w, t + 2)]7 d = d
+ |[V((x, t + 1) -V((x, t + 2)] n^ tdxi = .
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JOB SEARCH 761
However, V(w, t) is decreasing in t and V(w, t + 1) - V(w, t + 2) is
increasing in w. Thus the condition above is strictly positive and can
be fulfilled only by raising n't or lowering n't I or both. Therefore, nt >
nt + 1, so n*t' n ,*+ 1-
ii) To complete the proof we must rule out the case where n*< 1/1
and n*+1 = 1/1. Note that if n* 1 = 1/1 then A - 0, so the expression
above would include a term IA - 0 on the left-hand side. Clearly, this
simply makes the expression larger and so violates the condition
above even more strongly. Q.E.D.
Interestingly, direct extension of proposition 1 to the case of simul-
taneous work and search is not possible, as there does not exist a
reservation wage in the sense defined in the previous section. Now the
individual will have instead two switch points. At some low wage, and
certainly at a zero wage, the worker/searcher will devote all his time to
search, that is, choose n*(O, t) = 1/1. Similarly, for some high wage he
will choose to specialize in work and so pick a sample size of zero. In
view of this, let us define wt such that n*(wt, t) = 1/1 (he specializes in
search) and n*(w, t) < 1/1 for all w > wt, since it is easily shown from
(23) that dn*ldwt < 0 for n* < 1/1. Similarly, we define an upper switch
point zvt such that n*(wt, t) = 0 (he specializes in work) and n*(w, t) > 0
for all w < wvt. Thus, in the context of simultaneous work and search
we must consider the time paths of these switch points rather than
that of the now nonexistent reservation wage.
PROPOSITION 1': In a simultaneous work and search model the max-
imum wage which leads to specialization in search, wt, and the
minimum wage which leads to specialization in work, zt, are nonin-
creasing functions of time.
PROOF: Consider the first-order condition (23). As w approaches _t
from above, n*(w1, t) approaches 1/1. At wt (19) still holds with equal-
ity. At time t + 1, however, n*(wt, t + 1) ' n*(wt, t) = 1/1 by proposi-
tion 4'. Thus the maximum w at t + 1 for which n*(w, t + 1) = 1/1
must be less than or equal to wt., since dn*+ I/dwt+ 1 < 0. By essentially
the same argument one can show that zt+ 1 ? w-t. Q.E.D.
Finally, as in proposition 3, we can ask, What are the responses of
the optimal sample size with respect to the cost of search, which here
is wl? The answers are provided in the following proposition and are
proved by the arguments used immediately above proposition 1'.
PROPOSITION 3': The optimal sample size chosen at time t, given
that simultaneous work and search is allowed, is a nonincreasing func-
tion of the wage in the current job and the time required to contact a
potential employer.
IV. Conclusion
The major result of this paper is that generalizing the sequential
search strategy literature to allow for an optimal choice of sample size
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762 JOURNAL OF POLITICAL ECONOMY
in each period will inevitably make the expected marginal returns to
search within each period decline with the sample size. From this it
follows naturally that the individual will not always wish to specialize
in search or in work, as is the case in the previous literature, but may
choose to do both simultaneously if this is technically possible. If it is
not possible for some reason-that is, he is constrained to specialize in
either work or search, and he is allowed to quit his job-we have
shown that once he accepts a job he will not quit it at a later date. This
result has been an assumption in the previous literature, with the
exception of Burdett (1978).
Although the models in this paper are described as models of job
search there is nothing specific to the labor market in them, and so the
results are applicable to, for instance, search across sellers for the
cheapest source of a product.
Appendix
Concavity of the Value of Search
The value of search at t, St, is given by
St(n) = s - cnt - A| Vt+I(w)^y(w; nt)dw. (Al)
We wish to show that (Al) is concave in n, n 2 1. We will show this by showing
that 82S/8n2 < 0.
Note first that y(w; n) is a density function and so satisfies f y(w; n)dw = 1,
V n 2 1. Therefore, f (aylan)dw = f (a2 ylan2)dw = 0. By the definition of a
first order statistic,
wy(w; n) =nF (w))n- if(w),
and so
a'Y = f(w)F n'- I()[n In F (o) + 1].
an
For w 0 X ' o?, In F (w) is a continuous function mapping w E [0, o) into
(-o, 0]. Thus for every n 2 1 o = o I [n in F (X) + 1] = 0, so that
( $ Co [n ln F(w) + 1] Z O
and
d = f (w)F '() In F(w)[n In F(w) + 2].
For 0 ? X ' oo, in F(u) is a continuous function mapping w into (-co, 0].
Thus, for every n 2 1,
3w = C I [n In F (w) + 2] = 0.
So
w .\,Co#> [n n F (w) + ] 91O na 'Y 'on 0. (M
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JOB SEARCH 763
Now, a2 Slan2 = - fo Vt+ l(o)(a2 y/an2)dw. Consider breaking this term into two
parts,
I_ _ _ _ 2 a2 y
1 ads - FV | ( )Vt+ ? I do + { Vt+ I () d dw.
r~an2a2a2
By (A2),
|Vt+I((JL)) 2 dw- Vt+I(CO) a2 dY dw;
an2 an 2
{Vt+I(w) a2+ dw Vt+ a((n) a dw.
Therefore,
1 a2S CVt+(I)(CO) dw = 0;
P an 2 ~ian2~
therefore S is concave in n, strictly concave if f(w) is not degenerate.
The Value Function Is a Decreasing Function of Time
We prove the following proposition by induction.
PROPOSITION Al: For a finite-horizon sequential job search strategy the
value function of the worker/searcher is a decreasing function of time.
PROOF:
i) First we prove that if Vt+ l(w) < Vt(w), then Vt(w) < Vt- l(w). We do this by
contradiction; that is, we assume that Vt(w) - Vt 1 (w). Using (1) we write the
worker/searcher's value functions for the first two of these periods:
Vti(w) = max [w + PVt(w), s - cn*I + V Vt(o)y(w; n* I)dw], (A3)
Vt(w) = max [w + PVt+I(w), s - cn* + Vt+i(w)-y(; nt)dco]. (A4)
As Vt+ 1(w) < Vt(w), the value of work in period t, the first term on the right-
hand side of (A4), is smaller than the value of work in period t - 1, the first
term on the right-hand side of (A3). Thus for Vt(w) ' Vt -(w) to hold, the
value of search in period t, the second term on the right-hand side of (A4),
must be greater than or equal to the value of search in period t - 1, the
second term on the right-hand side of (A3); that is, subtracting the latter from
the former must yield a nonnegative number. Now evaluate the value of
search in period t - 1 at n* rather than n* 1. This cannot increase the value of
search in period t - 1 as n*1 was the (integer) value of nti - that maximized
the value of search that period. Now we subtract the resulting value of search
in period t - 1 from that in period t, which gives
Ho [Vt+,(w) - Vt(w)]y(w; r4)dw. (A5)
But by assumption Vt+ l(w) < Vt(w) for all values of w, so (A5) is negative,
thereby contradicting Vt- l(w) c Vt(w). Thus, if Vt+ l(w) < Vt(w), then Vt(w) <
Vt - 1 (w), for all w.
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764 JOURNAL OF POLITICAL ECONOMY
ii) To complete the proof we show that VT- 1(W) > VT(W) for all w E (0, co).
Using (1) we see that VT(W) = w and so
VT-1(W) = max [w - w, s - cn* -I + { wy(w; n* - )dw.
Thus, VT- 1(W) ' W + AW > VT(W) = w for all w C (0, oo). Q.E.D.
References
Burdett, Kenneth. "A Theory of Employee Job Search and Quit Rates."
A.E.R. 68 (March 1978): 212-20.
Chirinko, Robert S. "An Empirical Investigation of the Returns to Job
Search." A.E.R. 72 (June 1982): 498-501.
Flinn, Christopher J., and Heckman, James J. "New Methods for Analyzing
Structural Models of Labor Force Dynamics." J. Econometrics 18 (January
1982): 115-68.
Gal, Shmuel; Landsberger, Michael; and Levykson, Benny. "A Compound
Strategy for Search in the Labor Market." Internat. Econ. Rev. 22 (October
1981): 597-608.
Gumbel, Emil J. Statistics of Extremes. New York: Columbia Univ. Press, 1958.
Kohn, Meir G., and Shavell, Steven. "The Theory of Search."J. Econ. Theory 9
(October 1974): 93-123.
Lippman, Steven A., and McCall, John J. "The Economics of Job Search: A
Survey: Part I." Econ. Inquiry 14 (June 1976): 155-89. (a)
. "Job Search in a Dynamic Economy."J. Econ. Theory 12 (June 1976):
365-90. (b)
. "The Economics of Belated Information." Internat. Econ. Rev. 22 (Feb-
ruary 1981): 135-46.
Morgan, Peter. "Search and Optimal Sample Sizes." Rev. Econ. Studies (in
press).
Mortensen, Dale T. "A Theory of Wage and Employment Dynamics." In
Microeconomic Foundations of Employment and Inflation Theory, edited by Ed-
mund S. Phelps et al. New York: Norton, 1970.
Seater, John J. "A Unified Model of Consumption, Labor Supply, and Job
Search."J. Econ. Theory 14 (April 1977): 349-79.
"Job Search and Vacancy Contacts." A.E.R. 69 (June 1979): 411-19.
Stigler, George J. "The Economics of Information." J.P.E. 69 (June 1961):
213-25.
Wilde, Louis L. "An Information-Theoretic Approach to Job Quits." In Stud-
ies in the Economics of Search, edited by Steven A. Lippman and John J.
McCall. Amsterdam: North-Holland, 1979.
This content downloaded from 216.165.95.69 on Mon, 29 Feb 2016 21:26:50 UTC All use subject to JSTOR Terms and Conditions
