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1998 Quarter 3
An Introduction to the Search Theory
of Unemployment
2
by Terry J. Fitzgerald
What Labor Market Theory Tells Us
about the “New Economy”
16
by Paul Gomme
Unemployment and Economic Welfare
by David Andolfatto and Paul Gomme
FEDERAL RESERVE BANK
OF CLEVELAND
25
Vol. 34, No. 3
http://clevelandfed.org/research/review/
Economic Review 1998 Q3
1998 Quarter 3
An Introduction to the Search Theory
of Unemployment
2
by Terry J. Fitzgerald
What Labor Market Theory Tells Us
about the “New Economy”
16
by Paul Gomme
Unemployment and Economic Welfare
by David Andolfatto and Paul Gomme
25
1
ECONOMIC REVIEW
1998 Quarter 3
Vol. 34, No. 3
An Introduction to the Search
Theory of Unemployment
2
by Terry J. Fitzgerald
The search theory approach to understanding unemployment flourished
during the 1980s and 1990s. It has provided economists with a rich set of
models for analyzing unemployment and labor market issues more generally. Unfortunately, while economists have found modern search theory to
be an invaluable tool, the insights provided by this approach remain largely
unfamiliar to noneconomists. This review is an attempt to reach out to those
readers who are interested in acquiring a modern perspective by providing
an introduction to the search theory of unemployment.
What Labor Market Theory Tells Us
About the “New Economy”
16
by Paul Gomme
“New economy” proponents claim that favorable supply-side shocks have
permanently lowered the nonaccelerating inflation rate of unemployment
(NAIRU). If true, this would explain why inflation has not risen over the past
couple of years, despite unemployment rates that are well below most
NAIRU estimates. What does economic theory have to say about such
claims? This article shows that a simple search model of unemployment
predicts no long-term change in the NAIRU, although favorable supply
shocks may lower the NAIRU over the short term. The key to this result is
that workers change their reservation wage (the lowest wage that they will
accept) in response to favorable developments in the distribution of wages.
The article then considers some extensions that may allow supply shocks to
lower the NAIRU permanently.
Unemployment and Economic Welfare
Economic Review is published
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Editors: Michele Lachman
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Design: Michael Galka
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by David Andolfatto and Paul Gomme
The rates of employment and unemployment are measures of labor market
activity that have long been used as indicators of economic performance
and welfare. Comparisons of these measures across different regions are
typically based on the idea that low levels of employment and high levels
of unemployment are associated with low levels of economic performance
and general well-being. But in the absence of information concerning the
economic circumstances that determine individual labor market choices,
such comparisons are not justified. In this article, the authors develop a
simple model of labor market activity designed to illustrate the tenuous link
that exists between labor market choices and economic well-being.
Opinions stated in Economic
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ISSN 0013-0281
2
An Introduction
to the Search Theory
of Unemployment
by Terry J. Fitzgerald
Introduction
On any given day, during economic busts and
economic booms alike, millions of Americans
are unable to find desirable employment despite their best efforts. Understanding the reasons for this fact is a chief concern for economists and policymakers, since it is necessary for
designing good labor market policies. Unemployment not only creates hardships for those it
encompasses, but it also seems to represent a
vast pool of idle economic resources.
Classical labor theory is not well suited to
thinking about unemployment, for within this
framework the amount of labor that workers
supply is exactly equal to the amount of labor
demanded by firms at the equilibrium wage—
therefore, there is no unemployment. This feature of classical theory has contributed to the
historical interpretation of unemployment, or at
least a portion of unemployment, as a disequilibrium or an involuntary phenomena. While
such terminology has permeated discussions of
unemployment, it has done little to enhance
our understanding of the underlying determinants of unemployment or its behavior through
time and across countries.1
Terry J. Fitzgerald is an economist at the Federal Reserve Bank
of Cleveland. He thanks Paul
Gomme and Randall Wright for
their comments, and Jennifer
Ransom and Jeff Schwarz for
their research assistance.
A different approach to the study of unemployment, which sought to directly explain the
frequency and duration of unemployment
spells, took root during the 1970s. The building
block of this approach is the simple observation that finding a good job (or a good worker,
in the case of a firm) is an uncertain process
which requires both time and financial
resources. This assumption stands in contrast to
the classical model, in which workers and firms
are assumed to have full information at no cost
about job opportunities and workers. The alternative approach, referred to as the search theory of unemployment, seeks to understand
unemployment in the context of a model in
which the optimizing behavior of workers and
firms gives rise to an equilibrium rate of unemployment. Furthermore, it has the potential to
explain the striking fact that while millions of
workers are unemployed, firms are simultaneously looking to fill millions of jobs.
■ 1 See Rogerson (1997) for an excellent discussion of the language
used to discuss unemployment.
3
The search theory approach to understanding unemployment flourished during the 1980s
and 1990s. Incorporating the simple observation that searching is costly into a theory of
labor markets has resulted in a rich set of models which have helped us not only to understand how unemployment responds to various
policies and regulations, but also to gain a better understanding of other labor market issues
including job creation and destruction, business
cycle characteristics, and the effects of labor
market policies on the aggregate economy
more generally.
Unfortunately, while economists have found
modern search theory an invaluable tool for
understanding unemployment (as well as
numerous other issues), the insights provided
by this approach remain largely unfamiliar to
noneconomists. This is partly a reflection of the
old language of unemployment—terminology
such as “full employment’’ and the “natural rate
of unemployment’’—continuing to dominate
discussions of unemployment in the media and
politics. This review is an attempt to reach out
to those readers who are interested in acquiring
a modern perspective on unemployment by
providing an introduction to the search theory
of unemployment.
In this article I present a model of job search
and analyze how an unemployed worker’s
decision environment affects not only her
employment decisions, but also the overall
level of unemployment.2 The model focuses
on an unemployed worker’s decision to accept
an offered job or to continue searching for a
better job. This is one of the earliest search
models used in labor market analysis; its virtue
is that it provides a simple framework capturing
many of the central ideas upon which labor
search theory is based, as well as interesting
economic insights. Far richer models which
capture many additional characteristics of labor
markets have been developed, but these models are much more complex and will not be
discussed here.
I. A Model
of Job Search
Consider an unemployed worker who is searching for a job by visiting area firms, looking
through help wanted ads, etc. Although the
worker likely has many job opportunities, she
has incomplete information as to the location of
her best opportunities. Hence, she must spend
time and resources searching, and she must
hope she has luck finding one of her better
opportunities quickly. In any given week the
worker may receive a job offer at some wage w.
The decision she faces is whether to accept that
offer and forego the possibility of finding a better job, or to continue searching and hope that
she is fortunate enough to get a better offer in
the near future.
This scenario is captured in a model of job
search using the following assumptions. First,
each week the worker receives one wage offer.
In order to capture the uncertainty of job offers,
I assume that this offer is drawn at random
from an urn containing wage offers between w
and w. Draws from this urn are independent
from week to week, so the size of next week’s
offer is not influenced by the size of this week’s
offer. While I will interpret draws as weekly
wage rates, they can be thought of more generally as capturing the total desirability of a job,
which could depend on hours, location, prestige, and so on. For simplicity, assume that all
jobs require the same number of hours and are
of the same overall quality, so that jobs differ
only in terms of the wage.
Each week the unemployed worker must
decide whether to accept the wage offer w, or
to reject the offer and wait for a better one. If
she rejects the offer, the worker receives unemployment income of w u dollars and draws a
new wage offer the following week. For simplicity, wage offers from previous weeks cannot be recalled and accepted, an assumption
which has no impact on the worker’s decision
to accept or reject this week’s offer. While I will
interpret unemployment income w u as being
unemployment compensation, it may also
include factors such as the pecuniary value of
leisure and home production activities less the
cost of searching.
If the worker accepts the wage offer, she
continues to work at that wage until she is
fired (assume that the worker cannot search
for a better job during this time). An employed
worker faces a constant probability α of being
fired at the end of each week. When an employed worker is fired, she becomes unemployed and begins searching for a new job the
following week. Because an employed worker
would never choose to quit her job in this
model, I have omitted that possibility. Workers
in the model are either employed or unemployed and actively searching for employment.
No worker is out of the labor force (that is, not
seeking employment).
■
2 The presentation of the model in this paper was largely drawn
from chapter 2 of Sargent (1987), which provides a more advanced
overview of search theory. Insights into the model were also drawn from
lecture notes provided by Randy Wright.
4
Workers seek to maximize the expected present value of their lifetime wage income, which
is written as
∞
(1) E ∑ β tyt ,
t=0
where β is a discount factor between 0 and 1,
and yt denotes the worker’s income in period
t.3 Note that (yt = w u) if the worker is unemployed, and (yt = w) if the worker is employed
at wage w. The factor β determines the rate at
which workers discount their future earnings
and can also be written as 1/( 1 + r), where r
is a real rate of interest. While workers in the
model have the good fortune of living forever,
this assumption can be thought of as an approximation of the case where workers have
many periods left to live.
Now consider the unemployed worker’s
decision problem in more detail. In evaluating
a wage offer w, her decision will depend on
how the current offer compares to other offers
which she may receive. If the chances of receiving a substantially better offer next period
are good, then the worker may choose to reject
the current offer with the expectation of receiving a better one in the near future. A worker
who rejects an offer foregoes income this week
in the amount of the offer wage w, less the
amount of unemployment compensation w u.
That loss must be balanced against the potential
gain from receiving a higher wage offer next
week, which the worker would receive in all
future weeks until she is fired. In other words,
the worker must compare the expected present
value of her income if she rejects the offer with
the expected present value of her income if
she accepts the wage offer. As we will see,
just how high the wage offer must be for the
worker to accept depends on the exact shape
of the wage offer distribution, the probability
of being fired, the level of unemployment
compensation, and the rate at which the
worker discounts future earnings.
The specific mathematical structure of an
unemployed worker’s decision problem is laid
out in the appendix, along with the description
of a solution strategy. While the formulation of
this problem makes use of mathematical techniques that are likely to be unfamiliar to noneconomists, the underlying intuition of the
problem is relatively straightforward and will
be highlighted here. Recall that in making her
decision, the unemployed worker must compare the expected lifetime incomes of accepting
or rejecting a particular offer. I describe the
unemployed worker’s decision problem using
the following notation. Let v wait (w) be the
expected present value of lifetime income if
she rejects a wage offer w and waits for a better offer; let v accept (w) be the expected present
value of lifetime income if she accepts w; and
let v offer (w) be the expected present value of
lifetime income upon drawing a wage offer w.
Each of these three functions assumes that the
unemployed worker will behave optimally
(that is, makes the best decisions) in future
periods so as to maximize expected lifetime
income as given by (1).
First consider the value of rejecting an offer
and waiting for a better offer:
(2) v wait (w) = w u + β Ev offer,
where Ev offer is the expected value of v offer(w).
The value of waiting includes the unemployment compensation which the worker receives
this week, plus the discounted expected value
of drawing a new wage offer next week. Notice
that v wait (w) is a constant, which I will write as
v wait, since Ev offer does not vary with w. This reflects the fact that next week’s wage offer is independent of this week’s offer, so the value of
rejecting an offer and waiting for a new offer is
the same regardless of this week’s offer.
Next consider the value of accepting a wage
offer w:
(3) v accept (w) =
w + βα Ev offer + β (1 – α )v accept (w).
If the worker accepts a wage offer w, she
receives income w this week. At the end of the
week she is fired with probability α, in which
case she receives the discounted expected
value of receiving a new offer next week,
β Ev offer, or she continues on the job with probability (1 – α ), in which case she receives the
discounted value of accepting the same wage
offer next week, βv accept (w). This equation can
be rewritten
(4) v accept (w) =
w + β α Ev offer .
1 – β (1 – α)
■ 3 The assumption that workers maximize expected lifetime income
can be interpreted in several ways: 1) workers are risk-neutral, so they do
not care about smoothing consumption; 2) workers are able to perfectly
insure themselves against any idiosyncratic income risk, so the worker first
maximizes expected income and then arranges her consumption stream so
as to maximize utility; or 3) c and w can be reinterpreted as being the utility
value of being unemployed and of working at a job with wage w respectively, in which case equation (1) can be reinterpreted as expected, discounted utility.
5
F I G U R E
1
Expected Lifetime Earnings
of w less than w r, v wait is greater than
v accept(w), so the worker is better off rejecting
the offer. For w greater than w r, v wait is less
than v accept (w), so the worker is better off accepting the offer. The function v offer (w) is defined by the maximum of these two functions,
and is illustrated in blue. Notice that w r will
depend on the specific value of v wait and the
function v accept (w), which themselves depend
on v offer (w) through the term Ev offer. Furthermore, Ev offer depends on the value of w r, and it
will be helpful to make this dependence
explicit by writing Ev offer (w r ) .
The wage w r is called the reservation wage
and represents the lowest wage offer that an
unemployed worker will accept. As I will show
in the next section, the exact value of the reservation wage depends on the wage offer distribution, the firing rate α, unemployment compensation w u, and the discount factor β.
Solving for the
Reservation Wage
Notice that v accept (w) increases linearly with w.
The problem for a worker with an offer w in
hand is deciding whether to accept the offer,
which has value v accept, or reject the offer,
which has value v wait. The value of having an
offer w in hand is given by
Next I briefly describe how to solve for the
value of the reservation wage. As shown in figure 1, the reservation wage w r is the value of w
which satisfies
(6) v accept (w r ) = v wait
(5) v offer (w) = max {v accept (w), v wait },
or, using equations (2) and (4),
which takes into account that offers will be
accepted only when accepting is more beneficial than waiting.
A solution to this problem is characterized
by functions v offer (w) and v accept (w), and a constant v wait, that satisfy equations (2), (4), and
(5). Associated with the function v offer (w) is a
decision rule which indicates whether the
worker accepts or rejects each wage offer w
between w and w. Unfortunately, computing
these functions is not as straightforward as it
might first appear. The function v accept (w) and
the constant v wait which define v offer (w)
depend themselves on v offer (w) through the
term Ev offer. None of these elements can be
solved for independently.
To gain insight into the nature of the solution to the unemployed worker’s job decision,
it is helpful to graph v accept (w) and v wait against
the value of the wage offer w (figure 1). The
decision to accept or reject each wage offer w
depends on whether v wait is greater than or less
than v accept (w). The figure shows that this decision takes a particularly simple form. For values
(7)
w r + βα Ev offer(w r )
= w u + βα Ev offer (w r )
1 – β (1 – α )
This expression says that the reservation wage
is the wage at which the value of accepting the
wage offer (the left side) is equal to the value
of rejecting the offer (the right side). That is,
the reservation wage is the wage at which the
worker is just indifferent between accepting or
rejecting the offer. Before we can solve this
equation for w r, we must first provide an
explicit expression for Ev offer (w r ).
Obtaining an expression for Ev offer (w r )
requires that I be explicit about the distribution
of wage offers that are contained in the wage
offer urn. Assume that these offers are uniformly distributed between w and w. This
implies that all wages between w and w are
equally likely to be drawn and makes the computation of Ev offer (w r ) straightforward. It is proportional to the area under the v offer curve between w and w. After doing some algebra, one
finds that
6
(11) w r = w u + ϕ (w r ).
F I G U R E
2
Determination of the
Reservation Wage
(8) Ev offer (w r ) =
1 1 –1 β 2 3w
u
4
+ s ( w – w r )2
2 (w – w )
where s = 1/(1 – β (1 – α)).
Using equation (8) to substitute Ev offer out of
equation (7), one obtains a single equation
containing w r:
β (1 – α)
(w – w r )2
.
(9) w r = w u +
1 – β (1 – α) 2(w – w )
3
4
However, w r appears on both sides of this
equation, so a little more work is needed.
To simplify notation in what follows, define
a new function,
(10) ϕ (w r ) ≡
(w – w ) ,
31 β– (1β (1– α– )α)4 2(w
– w)
r 2
which is the second term on the right side of
equation (9). This function can be interpreted as
the expected benefit of drawing a new wage
when the unemployed worker has an offer w r
in hand. Notice that this function is decreasing
in w r, which indicates that the expected gains
from drawing a new wage diminish as w r increases. If w r is set to w, this function is 0, reflecting the fact there can be no gain from drawing a new offer since w is the highest possible
wage. Equation (9) can be rewritten
I have arrived at a single equation, (11),
which determines the value of the reservation
wage w r given values for all the parameters in
the model. The left side can be regarded as the
benefit of accepting a wage offer at the reservation wage. The more selective a person is (e.g.,
the higher her reservation wage), the higher the
value of accepting a job offer at the reservation
wage. Hence the left side of the equation is
increasing in w r. The right side can be regarded
as the value of rejecting the offer and waiting. It
includes the value of unemployment compensation w u plus the expected gain from drawing a
new age. The expected gain from receiving
additional wage offers again depends on how
selective the person is. The pickier she is, the
lower the chances of getting such an offer and
the lower the value of waiting. Thus the right
side is decreasing in w r. The equilibrium reservation wage is the wage at which the benefit of
accepting is equal to the benefit of rejecting.
The next question to consider is whether a
unique value of w r exists which satisfies equation (11). Figure 2 graphs both sides of this
equation. Denote the left side, w, the “accept
curve,” and the right side, w u + ϕ (w), the
“reject curve.’’ Since the accept curve is increasing in w and the reject curve is decreasing in w,
the intersection of the two curves, if one exists,
will be unique. However, there may not exist
such a value. In this case the solution to the
problem will correspond to a reservation wage
of w (or lower) or to a reservation wage of w
(or higher). In any case, the functions v offer (⋅)
and v accept (⋅) and the constant v wait which
solve the problem are unique, as is the decision
rule for accepting and rejecting wage offers
within the set of possible wage offers. Figure 2
will be useful later when we discuss how
changes in various parameter values impact the
reservation wage.4
It is interesting to note that the reservation
wage behavior of the unemployed worker in
this model is observable in “real world” behavior. Each week many unemployed workers
choose to continue their job searches even
though they could accept low-paying jobs at,
for instance, a local fast food restaurant. They
obviously do so with the expectation that they
will find a better job in the near future.
■ 4 Equation (9) is quadratic in w r, and the quadratic formula can be
used to directly solve for w r.
7
Unemployment
Duration and
Unemployment
Rates
Although this model abstracts from the behavior of firms and the process by which the wage
distribution is determined, unemployment
durations and unemployment rates can be constructed nonetheless. First, assume that there
are many identical workers in the model who
act independently and have independent wage
offer draws when unemployed. Also assume
that the firing of employed workers occurs at
the end of each week. At the beginning of the
next week, each unemployed worker arrives at
a firm, receives a wage offer w, and decides
whether to accept or to reject that offer.
By setting the reservation wage w r relatively
high, the worker is less likely to receive an
acceptable wage offer and will, on average,
spend more time waiting for an acceptable
offer than if she set w r lower. The probability
of accepting a wage offer, called the jobacceptance rate or the hazard rate, is simply
equal to the fraction of offers greater than or
equal to w r. Let ψ denote the job-acceptance
rate. Because the wage offer distribution is uniform, ψ is computed as
(12) ψ =
w – wr
.
w–w
The average number of weeks it takes to
receive an acceptable offer, referred to as the
average waiting time, is given by (1/ψ ). Notice
that if w r is equal to w, the job-acceptance rate
is 0 and the average waiting time is infinity
since there is zero chance of drawing w from
the uniform distribution. If w r is equal to w ,
the job acceptance rate is 1 and the average
waiting time is one week. That is, a job is always accepted in the first week upon becoming unemployed.
Given the assumptions on the transition between employment and unemployment, the
average duration of unemployment is the average waiting time less one week, [(1/ψ ) – 1].
So, for example, if the job-acceptance rate is 1,
then the average duration of unemployment
is 0 weeks since all unemployed workers except a job offer at the beginning of the week.
If the job-acceptance rate is 0.10, or one out of
10, then the average waiting time is 10 weeks
and the average duration of unemployment is
nine weeks.
The path of the unemployment rate through
time can be computed for any given initial
unemployment rate u1 as follows. Let ut be the
fraction of workers who are unemployed during
week t (the unemployment rate), and let L
denote the total population. Total unemployment is thus (Lut ), while total employment is
(L – Lut ). Given ut , we can compute ut + 1 by
keeping track of how many workers enter and
exit unemployment each week. This is expressed as
(13) Lut + 1 = Lut (1 – ψ )
+ [(L – Lut ) α (1 –ψ )].
This equation says that total unemployment
next week (Lut + 1 ) is equal to the number of
unemployed workers this week who do not accept a job at the start of next week [Lut (1 –ψ )],
plus the total number of employed workers this
week who are fired and do not accept a job at
the start of next week [(L – Lut ) α (1 –ψ )]. This
equation can be rewritten by dividing through
by L and rearranging terms to get a simpler
expression for ut + 1, referred to as the law of
motion for ut :
(14) ut +1 = α (1 – ψ ) + [(1 – ψ )(1 – α )]ut .
Given any unemployment rate u1, equation
(14) can be used to compute the path of the
unemployment rate through time. One property of this law of motion for ut is that the unemployment rate converges to the same level for
any given initial unemployment rate u1. The
unemployment rate to which these paths converge can be computed from equation (14) by
setting ut +1 = ut = us. Solving for us produces
(15) us =
α (1 – ψ ) .
α (1 – ψ ) + ψ
The value us is the steady state unemployment rate. It is the point at which the flow of
people into unemployment equals the flow of
workers out of unemployment, so that the unemployment rate remains constant through time.
Notice that steady state unemployment depends only upon the firing rate α and the jobacceptance rate ψ. It is easy to show that higher
firing rates and lower job-acceptance rates each
imply higher unemployment rates, results which
match one’s intuition. Remember that while the
firing rate α was exogenously given (that is, given as a parameter and not part of the solution),
the job-acceptance rate ψ is endogenously determined (that is, not given as a parameter but
part of the solution) and depends on all the
parameters in the model. Thus, through ψ the
steady state unemployment depends on all the
parameters in the model.
8
F I G U R E
3
Unemployment Paths
At this point let me briefly return to two
points raised in the introduction. First, notice
that unemployment in this model arises solely
from incomplete information about wages and
jobs that is costly to acquire; unemployment
here is not a disequilibrium phenomena. Unemployment occurs even though all workers behave optimally and results from the costly but
socially beneficial activity of achieving good
matches between workers and jobs. Second,
the model illustrates that distinctions between
voluntary and involuntary unemployment are
unclear and not useful. Here unemployment is
voluntary in the sense that workers choose to
reject wage offers. But unemployment is involuntary in the sense that any unemployed worker (whom we know has only received wage
offers less than w r ) would prefer to switch
places with any employed worker (who is receiving a wage of w r or larger).
II. A Numerical
Example
In order to make the insights provided by this
model concrete, it is helpful to work with a
numerical example. Consider a distribution of
wage offers that is uniform between 200 and
800. This means that an unemployed worker is
equally likely to receive any wage offer between $200 and $800 per week, and implies an
average wage offer of $500. Let the firing rate
α be 0.005, or 1/2 percent per week, and the
discount rate β equal 0.999, which corresponds
to a 5 percent annual real interest rate. Lastly,
set unemployment compensation w u to $200.
After solving the model for these parameter values, I will discuss how the reservation wage,
the average duration of unemployment, and the
unemployment rate respond to changes in
these values.
Take a guess at what the reservation wage is
for this example. Will the worker hold out for a
wage greater than $500, the average wage offer?
The answer is yes. In fact, the reservation wage
w r is $737.62. If you think this number is surprisingly large, consider the fact that with a firing rate α of 0.005, the average length of employment, which is given by 1/α , is 200 weeks
or almost four years. This means that once a
wage offer is accepted, the worker expects to
receive that wage for the next four years—thus
providing an incentive to hold out for a relatively high wage. Of course, every week an offer is rejected is a week with foregone wage income, so the worker doesn’t hold out for $800.
The job-acceptance rate for this example is
equal to 0.104. This says that each week there
is a 10.4 percent chance of receiving an acceptable wage offer—which is any wage greater
than or equal to $737.62. This job-acceptance
rate implies an average duration of unemployment of 8.6 weeks.
Finally, the steady state unemployment rate
is 0.041, or 4.1 percent. Figure 3 illustrates the
time paths for two different initial unemployment rates, 7.1 percent and 1.1 percent. Each of
these paths converges to the steady state rate of
0.041. As discussed in the previous section, this
convergence to steady state occurs for any initial unemployment rate.
What causes the unemployment rate to be
above or below the steady rate in the first
period? Loosely speaking, one could imagine
that a one-time unexpected shock hits the
economy which changes the unemployment
rate. For example, this could reflect a temporary increase (decrease) in the wage offer distribution, perhaps due to a productivity shock,
which implies that more (fewer) wage offers
are above (below) the reservation wage and
thereby lowering (increasing) unemployment.
After this temporary shift, the wage offer distribution returns to its original form, and the
unemployment rate steadily returns to its steady
state value. Throughout the remainder of this
paper, I will focus on the determinants of the
steady state unemployment rate.
9
F I G U R E
4A
Effect of Higher Real
Interest Rate
The Effects of
Changes in the
Environment
While the model presented here is relatively
stark and simple, it nonetheless provides interesting insights on how elements of the economic environment influence the unemployment rate. In the following subsections I
explore how changes in the discount rate, firing
rate, wage offer distribution, and unemployment compensation each influence the solution
to the numerical example. For each of these
elements I first examine the effect on the reservation wage, then trace the effects on the average duration of unemployment and the steady
state unemployment rate.
Changes in the
Discount Rate
F I G U R E
Effect of Changes in the
Real Interest Rate
u
4B
First consider the effect of an increase in the
real interest rate, which implies a lower value for the discount factor β (recall that β =
1/(1 + r )). It is not immediately obvious how
this change will effect the reservation wage or
the unemployment rate. However, intuition
suggests that because a higher interest rate implies discounting future earnings more rapidly,
an increase in the real interest rate lowers the
benefits of waiting for a higher wage. This suggests that the reservation wage will decrease.
Indeed, figure 4a shows that a higher interest
rate causes the reject curve to shift inward, resulting in a lower reservation wage.
The lower reservation wage implies a higher
job-acceptance rate, lower unemployment duration, and lower steady state unemployment.
That is, higher real interest rates lead to lower
steady state unemployment. It is informative to
consider extreme cases to gain insight into the
underlying logic of the model. For example,
consider setting the real interest rate infinitely
large, which corresponds to setting β to 0. In
this case the worker completely discounts future
earnings. Thus, she sets her reservation wage to
$200, accepts any job offer, and the unemployment rate is 0. Unemployment in the model is
due, in part, to workers’ willingness to wait for
a high wage offer.
Figure 4b shows how the reservation wage
and unemployment rate vary with the real
interest rate. Both are steadily decreasing in the
interest rate. As the interest rate approaches 0
(the case in which workers do not discount the
future at all), the reservation wage and the
unemployment rate increase to $743 and 4.5
percent, respectively.
10
F I G U R E
5A
Effect of Higher Firing Rate
F I G U R E
5B
Effect of Changes
in the Firing Rate
α
Changes in
the Firing Rate
Next suppose there is an exogenous increase
in the firing rate α, perhaps resulting from a
change in government regulations. While the
effect on the reservation wage may not be
apparent in this case, it seems obvious that an
increase in the firing rate must result in an
increase in the unemployment rate. Figure 5a
illustrates that an increase in α causes the reject
curve to shift inward toward 0. This results in a
decrease in the reservation wage and a corresponding increase in the job-acceptance rate.
This finding is not particularly surprising since,
all things being equal, an increase in the firing
rate reduces the expected length of time at a
given job, and thus reduces the benefit of waiting for a relatively high wage offer. For example, if you are likely to hold the same job for
only a few months, then it is not worth spending a long time searching for a high wage job.
Surprisingly, the effect of an increase in the
firing rate on the unemployment rate is
ambiguous and depends on the size of the
increase. Figure 5b shows that the unemployment rate increases steadily as the firing rate
rises to roughly 0.30, but then declines for
higher firing rates. To understand why this
occurs, note that the reservation wage falls as
the firing rate rises. This implies that the jobacceptance rate is increasing with α and the
average duration of unemployment is falling.
Thus there are two competing effects on
the steady state unemployment rate.The increase in the firing rate raises unemployment,
while the increase in the job-acceptance rate
lowers unemployment. Which effect dominates depends upon the specific numerical
values used in the example and the magnitude
of the increase in the firing rate. In the extreme case where all workers are fired every
period (α = 1), unemployed workers accept all
wage offers (w r = 200) and the unemployment
rate is 0. The average weekly wage that workers receive falls from $737.62 when the unemployment rate is 4.1 percent, to $500 when the
unemployment rate is 0; meanwhile, the expected lifetime earnings, Ev offer, of an unemployed worker fall from $740,086 to $500,000.
This example makes clear that policies which
reduce unemployment do not necessarily
benefit workers.
The potential for such surprising effects is
one reason that it is important to rigorously
model economic behavior. While intuition is
certainly useful as a guide, relying on intuition
alone often provides an incomplete picture,
and is sometimes just plain wrong.
11
F I G U R E
where w r′ is the reservation wage given the
new wage offer distribution. With a little bit of
algebra, it is straightforward to show that
6
Effect of Changes in the
Mean of Wage Distribution
w r ′ = (1 + λ)w r .
λ
That is, the reservation wage increases by the
same percentage as the wage offers.
Next consider what happens to the jobacceptance rate, ψ ′, which is determined by
(17) ψ ′ =
(1 + λ) w – (1 + λ) w r w – w r
=
(1 + λ) w – (1 + λ) w w – w = ψ .
λ
λ
Changes in the Wage
Offer Distribution
Next I address the impact of changes in the
wage offer distribution. More specifically, I
examine the effect of a permanent upward shift
in the entire wage offer distribution, perhaps
resulting from a permanent increase in productivity, and the effect of an increase in the “riskiness’’ of the wage offer distribution.
An Upward Shift in
the Distribution
Suppose that the distribution of wages were to
increase by the fraction λ , or λ times 100 percent, as the result of a permanent, across-theboard increase in productivity. This implies that
the uniform distribution of wage offers shifts
from [200,800] to [200(1 + λ) , 800(1 + λ)].
Consider two cases. First, suppose that
unemployment compensation, w u, increases
by the same percentage as all the wage offers.
Equation (9), which determines the reservation
wage, could then be rewritten
(16) w r ′ = (1 + λ)w u +
3
β (1 – α)
31 – β (1 – α) 4
((1 + λ ) w – w r ′) 2
,
2(( 1 + λ) w – ( 1 + λ) w )
The job-acceptance rate is unaffected by the
shift in the wage offer distribution. This implies
that the average duration of unemployment and
the unemployment rate are also unchanged!
While this result is certainly a striking one, it
is perhaps not so surprising. It essentially says
that if the costs and benefits of searching for a
job all go up by the same proportion, then the
reservation wage will increase by the same proportion and unemployment will be unaffected.
Consider an example where we simply measure
the wage offer distribution and unemployment
compensation in cents instead of in dollars.
Clearly we would expect the reservation wage
to increase from $737.62 to 73,762 cents, with
unemployment duration and rates unaffected.
Now consider a second case. Suppose that
unemployment compensation does not increase
with the wage distribution. Figure 6 shows that
in this case the reservation wage increases less
than proportionally with the wage offer distribution (w r ′/(1 + λ) < w r ). This implies that the
job-acceptance rate increases, unemployment
duration falls, and the unemployment rate declines. The relative cost of searching increases
since unemployment compensation, which
serves as a subsidy to searching, does not increase with the wage distribution.
As an example, consider a 5 percent increase in the wage distribution, so that λ equals
0.05. For this case the reservation wage increases by 4.9 percent to $773.96, and the
unemployment rate falls slightly, from 4.13 percent to 4.09 percent. For this particular example, the 5 percent increase in the wage offer
distribution has little impact on unemployment.
12
F I G U R E
7A
Effect of Changes in the
Variance of Wage Distribution
δ
F I G U R E
7B
Effect of Changes in Wage Riskiness
on Expected Lifetime Earnings
δ
Changes in the
Riskiness of the
Distribution
Next I examine the effect of a change in the
“riskiness’’ of the wage offer distribution. To do
this, I must first clarify what I mean by riskiness.
I define riskiness as the difference between
the highest and lowest possible wage offers,
(w – w ). An increase (decrease) in riskiness will
be defined as an increase (decrease) in this
spread which does not affect the mean. Define
the lower and upper bounds on wage offers
to be 500 – δ /2 and 500 + δ /2, where 0 ≤ δ ≤
$1000. Thus δ is the measure of riskiness since
(δ = w – w ), and was set equal to 600 in the
baseline numerical example. Notice that the
mean of the distribution is 500 regardless of the
value of δ. When δ is set to zero, there is no
riskiness in wage offers in the sense that all
wage offers are exactly $500.
What happens to the reservation wage and
the unemployment rate as the riskiness of the
distribution changes? Figure 7a shows that the
reservation wage increases with wage offer riskiness. This is not too surprising, given that the
spread of the distribution is increasing. Furthermore, the job-acceptance rate decreases as riskiness increases, which implies that the unemployment rate rises. This seems to bear out the
intuition that riskiness is bad for workers.
Before reaching that conclusion, however,
consider the case in which there is no riskiness
(δ = 0). Since there is no uncertainty in wage
offers, there is no reason to search. Each job
pays $500, and workers who are unemployed
at the beginning of the week always accept the
offer. The steady state unemployment rate in
this case is 0. But is an unemployed worker
better off?
Let’s compare the expected discounted lifetime earnings, Ev offer, for an unemployed
worker first in the model with no riskiness, and
then in the baseline numerical example with
riskiness (δ = 600). In the case with no riskiness, Ev offer is slightly less than $500,000, the
present value of $500 per week forever (with
no unemployment spells). But in the case with
riskiness, Ev offer is $740,086.
At first blush it may seem surprising that an
unemployed worker in the model with wage
riskiness and higher unemployment has substantially higher expected lifetime earnings than
an unemployed worker in the model with no
riskiness and no unemployment. But it should
not be. Given that the average duration of a job
is almost four years, an unemployed worker
would be much better off spending more time
searching for a relatively high-paying job than
she would be in a world where all jobs paid the
average wage. Recall that the reservation wage
in our numerical example was $737.62, which
is almost 50 percent higher than the average
wage offer of $500. Figure 7b shows that expected lifetime earnings steadily increase as the
spread in the wage distribution increases. Here,
13
F I G U R E
8A
Effect of Higher Unemployment
Compensation
riskiness is good. Again, note that versions of
the model with low unemployment are not
necessarily the environments which benefit the
workers most.
Changes in
Unemployment
Compensation
F I G U R E
Effect of Changes in
Unemployment Compensation
8B
Finally, consider what happens when unemployment compensation is increased. Figure 8a
shows that an increase in unemployment compensation causes the reject curve to shift outward, implying an increase in the reservation
wage. This is not surprising: Since unemployment compensation acts as a subsidy to searching, the worker is willing to wait longer for a
high-paying job and thus increases her reservation wage.
The higher reservation wage implies a
lower job-acceptance rate, an increase in the
average duration of unemployment, and an
increase in the unemployment rate. Figure 8b
shows that the reservation wage and the unemployment rate increase steadily with increases in unemployment compensation. In
the extreme case where w u is set to 800, it is
clear that unemployment will be 100 percent
since no job pays better than collecting unemployment compensation.
Consider an increase in w u from $200 to
$300 per week. In this case the reservation
wage increases from $737.62 to $743.35, average unemployment duration increases from 8.6
weeks to 9.6 weeks, and the unemployment
rate increases from 4.1 percent to 4.6 percent.
There is a great deal of empirical evidence
which supports the finding that increases in
unemployment compensation result in higher
unemployment. This does not imply that unemployment insurance necessarily makes workers
worse off in the real world. The findings do
suggest, though, that a tension exists between
maintaining low unemployment rates and providing insurance for the unemployed.
III. Concluding
Remarks
Search models of unemployment provide a
valuable tool for understanding the factors
which determine the unemployment rate and
the impact of labor market policies and regulations on unemployment. Furthermore, search
theory provides an alternative perspective to
the view that unemployment represents idle
resources. In this theory unemployed workers
14
are not idle, but instead are engaging in the
socially beneficial activity of finding a productive job match. The simple version presented
in this paper illustrates how search models can
be used to examine the influence of elements
of the economic environment on the unemployment rate.
The search model discussed here is often
referred to as a one-sided search model because it focuses solely on the job decisions of
unemployed workers and abstracts from the
search decisions of firms. More complex twosided search models examine the optimizing
decisions of workers and firms simultaneously.
In addition, these models have incorporated a
variety of other considerations which are
abstracted from in the simple model, and they
have proven to be capable of explaining many
features of unemployment data within and
across countries. This process of building better
theories is perhaps the most important step in
designing good economic policies, and search
theory is playing a critical role in that process.
w r + βα Ev offer(w r ) = w u + β Ev offer (w r ),
1 – β (1 – α)
which can be rewritten
(A3)
(A4) w r = c [1 – β (1 – α)]
+ [ β (1 – β )(1 – α)Ev offer (w r )].
Assuming a uniform distribution for F makes
it possible to obtain a closed form solution for
the integral expression that defines Ev offer. This
integral can be rewritten
(A5) Ev offer = ∫ wv offer (w ′)dF (w ′)
w
1
= w–w
w
∫w v offer (w ′)dw′ =
Appendix
v wait (w – w)
Solving the Model
+
In this appendix I lay out the basic mathematical structure of the model and describe a strategy for solving it. The wage offers in each period
are drawn from the same wage distribution
F (w), where F denotes the cumulative distribu∧
∧
tion function. That is, F (w)
= prob(w ≤ w).
The definition of Ev offer, the expected value of
the v offer value function, is
(A1) Ev offer = ∫wwv offer (w ′)dF (w ′).
The Bellman functional equation for v offer is
written
(A2)
w
using the fact that the density function for a
uniform distribution on [ w , w ] is 1/(w – w ).
This latter integral is simply the area under the
v offer curve, whose shape is illustrated in figure
1. This integral can be written
(A6)
v offer (w)
∫ w v offer (w ′)dw ′
1
(w – w r )s (w – w r )
2
where s = 1/(1 – β (1 – α )) is the slope of
v accept. The first term is the area of the rectangle
with width (w – w ) and height v wait, and the
second term is the area of the triangle with
width (w – w r ) and height s (w – w r ). Note,
however, that this expression is still a function
of Ev offer since v wait equals (wu + β Ev offer ).
Substituting equation (A6) and the definition
of v wait into equation (A5), one obtains
1
(A7) Ev offer (w r ) =
w–w
1
2
3
3 (w u + β Ev offer (w r ))(w –w )
=
4
5
1
+ 2 (w – w r )s(w – w r )
w + βα∫ wwv offer(w ′)dF (w ′) ,
1 – β (1 – α )
= w u +βEv offer (w r )
max w u + β∫wwv offer (w ′)dF (w ′),
6
where the first term is the value of waiting and
the second term is the value of accepting the
wage offer. The equation determining the reservation wage w r is
+
s (w – w r )2
.
2(w – w)
This expression can be rewritten to obtain
Ev offer as a function of w r
(A8) Ev offer (w r ) =
11 –1β 2 3w
u
r 2
4
+ s (w – w ) .
2(w – w)
15
Finally, this expression for Ev offer can be
combined with (A4) to obtain an equation in
w r alone:
(A9) w r = w u +
β (1 – α)
11 – β (1 – α)21
s(w – w r )2
2(w – w ) .
2
This is a quadratic equation in w r. It can be
shown that the smaller of the two roots for this
expression is the equilibrium reservation wage
if the solution is interior (w < w < w ). Given
w r, equation (A8) can be used to obtain Ev offer,
which in turn can be used to obtain v wait,
v accept, and v offer using equations (2), (4 ), and
(5 ) in the text.
References
Rogerson, Richard. “Theory Ahead of Language in the Economics of Unemployment,’’
Journal of Economic Perspectives, vol. 11,
no. 1 (Winter 1997), pp. 73 –92.
Sargent, Thomas J. Dynamic Macroeconomic
Theory. Cambridge, Mass.: Harvard University Press, 1987.
16
What Labor Market
Theory Tells Us about
the “New Economy”
by Paul Gomme
Introduction
The average unemployment rate for 1997 was
4.9 percent, well below most estimates of the
nonaccelerating inflation rate of unemployment
(NAIRU).1 One would therefore have expected
to see an increase in inflation in 1997; yet, as
measured by the CPI, inflation fell from 3.3 percent to 1.7 percent (December to December).
This phenomenon of low unemployment accompanied by falling inflation has prompted
some observers to claim that the economy is
now operating under a new set of rules. The
explanation is often couched in terms of a
favorable technology shock which has permanently lowered the NAIRU.
This article asks whether economic theory
supports the claim that a technology shock can
change the natural rate of unemployment. This
term is preferred to NAIRU in the context of the
theory used below, which is silent on the determination of nominal magnitudes like the price
level and inflation.2 Rather, the theory speaks
to the determination of real as opposed to
nominal wages (that is, in terms of goods
rather than dollars). Consequently, changes in
the natural rate of unemployment need not
have any repercussions for inflation.
Paul Gomme is an economic advisor at the Federal Reserve Bank of
Cleveland. The author thanks
David Andolfatto for his helpful
suggestions.
Proponents of the view that a technology
shock can change the natural rate of unemployment often rely, at least informally, on
neoclassical labor demand and supply. A positive improvement in technology shifts labor
supply to the right, since firms find all workers
more productive. In equilibrium, total hours
worked and output rise without contributing to
inflation, since improved technology raises the
real wage rate. However, as shown below, the
neoclassical model cannot explain unemployment per se. Any individual who does not
work has chosen not to work and so cannot be
described as unemployed.
Next, a search model of unemployment is
developed. This environment is characterized
by imperfect information: Workers do not
know the locations of well-paying jobs, and
firms do not know the identities of highly productive workers. Consequently, workers must
seek out firms in order to receive wage offers,
■ 1 The Economic Report of the President for 1998 estimated a
NAIRU of 5.4 percent, revised down from 5.5 percent in the 1997 report.
■ 2 That is, money is neutral: A once-and-for-all change in the level of
the money supply will have a proportional effect on the price level but will
leave all real magnitudes unchanged. In fact, here money will be superneutral: Changes in the time path of the money stock will have no real effect.
17
just as firms evaluate potential employees.
Workers choose a reservation wage above
which they accept employment (since the costs
of continued search outweigh the expected
benefits), and below which they reject job
offers (since the opposite is true). In the basic
search unemployment model, a permanent,
positive technology shock will shift the distribution of wages to the right. That is, each
worker is more productive at all potential jobs
and so will receive higher wage offers from any
employer he contacts. Suppose that the costs of
search rise in proportion to productivity. This
will be true if, for example, the only search
costs are forgone wage income and the delay
in receiving a new wage offer. In that case, an
individual’s reservation wage will also rise in
proportion to productivity and the improved
technology will have no effect on the unemployment rate.
If individuals are initially unaware of the shift
in the wage distribution, they will not change
their reservation wages. As a result, the unemployment rate may fall in the short run, since
individuals find a greater proportion of wage
offers meeting their reservation wage. Over
time, as individuals learn of the shift in the
wage distribution, they will revise their reservation wage upward, and the unemployment rate
will be unchanged. The analysis thus far casts
doubt on a fall in the natural rate of unemployment that is driven by technology shocks.
Alternatively, if, following a technological
improvement, search costs rise more than
benefits, then the unemployment rate may fall.
Two plausible reasons for this scenario are:
1) a cap on unemployment insurance benefits,
and 2) unchanged benefits of leisure or home
production opportunities enjoyed during a spell
of unemployment. Both reasons operate by
reducing the effective subsidy rate to search,
thus raising search costs relative to benefits.
The basic search model can be extended to
incorporate search effort. Consider, first, the
problem faced by someone who is unemployed. In choosing his search intensity, he
must make a conjecture about the level of
recruiting by firms which affects his likelihood
of successfully meeting up with a firm. A good
time to be looking for a job is when plenty of
firms are trying to hire. Next, notice that firms
must likewise form a conjecture regarding the
level of search by the unemployed: Posting lots
of job vacancies does not do much good if
there are few unemployed people looking for
work. Owing to these conjectures—or expectations—regarding the behavior of agents on the
other side of the job market, there may be multiple equilibria with self-fulfilling expectations.
High and low unemployment equilibria can
exist in an economy with identical fundamentals: The difference is in the expectations of
firms and the unemployed. If the economy
starts in a high unemployment equilibrium, a
positive technology shock may move the economy to the low unemployment equilibrium.
Firms raise their recruiting efforts since the
value of filling jobs has increased, and the
unemployed increase their search effort as a
consequence. Firms then recruit more, and so
on. The externality to search—for example, that
the unemployed benefit from increased recruiting by firms—-leads to the reinforcing effects of
search effort on both sides of the market. The
net result is an increase in the number of
matches between firms and the unemployed,
hence a lower unemployment rate.
Under the multiple equilibrium story, the
technology shock need not be permanent in
order for the unemployment rate effect to be
permanent. By permanently changing expectations regarding search effort, even a temporary
technology shock may permanently lower
unemployment. Notice, as well, that old-fashioned Keynesian “pump priming” would have
the same effect. For example, government
could hire people into temporary jobs, increasing the returns to search by the unemployed.
Observing that more individuals are looking for
work, firms will find that the returns to recruiting are higher and so increase their hiring.
Thus, another chain of events is put into
motion which can move the economy from a
high to a low unemployment equilibrium.
A final variant of the search model looks at a
matching function. This model postulates that
the number of successful matches depends on
the number of unemployed persons and on the
number of vacancies posted by firms. Rather
than affecting the productivity of jobs/workers,
suppose that the technological improvement
operates on the matching function: For the
same number of vacancies and unemployed,
more matches are consummated. While this
technological improvement will lower the
unemployment rate permanently, the mechanics are far different from those typically invoked
by the advocates of the “new economics.” Of
course, improvements in the matching function
may be positively correlated with aggregate
productivity gains. For example, computer
technologies are generally credited with much
of the aggregate productivity gains in recent
years, and also make it easier for firms and the
unemployed to contact each other.
18
F I G U R E
1
Neoclassical Labor
Market Equilibrium
F I G U R E
2
A Shift in Labor Demand
number of hours individuals are willing to
work at that wage.
Consider the effects of a permanent improvement in technology. Since firms find each
and every worker more productive, they are
willing to offer a higher wage to each one, and
labor demand shifts to the right (see figure 2).
In the new equilibrium, both the labor input
and the wage rate are higher. Since the technological improvement is permanent, the increase
in the labor input is also permanent.
If firms and workers are fully informed about
all prices in the economy, then it is irrelevant
whether the wage rate discussed above is
expressed in nominal terms (in dollars) or in
real terms (in terms of goods). In the classical
model, money is said to be neutral: The level of
the money supply determines the general price
level, but has no influence on real quantities
like the level of employment. Consequently, the
change in employment owing to an improved
technology need not be inflationary.
Notice that nothing has yet been said of
unemployment. According to the neoclassical
model, there is no unemployment, since anyone not working at the prevailing wage rate
has chosen not to work; presumably, they have
better things to do with their time. As a consequence, the neoclassical model cannot explain
the current situation of low unemployment and
low inflation.
II. A Basic
Search Model
Perhaps the most important reason why individ-
I. The Neoclassical
View of the Labor
Market
In the neoclassical model, the labor market is
like any other. That is, the labor market is
treated as a continuous auction, with equilibrium given by the intersection of labor
demand with labor supply (see figure 1). At
the equilibrium wage rate, w *, the quantity of
labor required by firms is just equal to the
uals are unemployed is that they do not know
which firms will offer high wages. Likewise,
firms post vacancies because they are ignorant
of the identities of highly productive workers.3
Each individual in the economy is endowed
with a unit of time. For now, assume that people receive no utility from leisure. Thus, when
employed, an individual will supply the entire
unit of time; when unemployed, he will use the
entire unit of time looking for a job.4 Suppose
that each period (for example, a week), an
unemployed individual contacts exactly one
firm. Once contact has been made, both the
firm and the individual learn the individual’s
productivity at that firm. That is, each match
■ 3 For a more comprehensive treatment of the model, see Sargent
(1987), Jovanovic (1979), and Lucas and Prescott (1974).
■ 4 Equivalently, suppose that the utility cost of working is equal to
that of searching. Then these utility costs wash out of the analysis.
19
F I G U R E
3
Wage-Offer Distribution
F I G U R E
4
A Shift in the WageOffer Distribution
has an idiosyncratic component that depends
on both the firm and the individual. The outcome of a bargaining process between the firm
and unemployed person will be a wage offer.5
Once a firm and worker have agreed to a
wage, they are assumed to enter a long-term
relationship in which the worker continues
supplying labor to the firm at the agreed wage.6
Now, consider the decision process of an
unemployed individual. This person is assumed
to know the distribution of wages which he
will receive; when he contacts some firm, he
knows the probability of receiving a wage offer
of, say, w. One such distribution is given in figure 3. This individual must decide whether to
accept a wage offer, w. Suppose that this offer
is quite low, as it would be if his productivity at
a particular firm was also very low. Since deciding to work for a firm means entering into a
long-term relationship with it, agreeing to such
a wage would imply accepting a low wage for
several years. An individual who rejects such an
offer is hoping to receive a higher wage offer
from some other firm in the future. Figure 3
shows that the probability of receiving such an
offer, given by the area under the wage distribution curve to the right of w, is quite high. Of
course, there is some possibility of receiving an
even lower wage offer, but receiving a higher
wage offer is more likely.
A particularly high wage offer will almost
certainly be accepted, since the chances of
receiving an even higher one are remote. This
means that an individual who rejects a very
high offer in the hope of an even higher one
will have a long wait.
The outcome to the individual’s decision
problem can be summarized by a reservation
wage, w r: The individual will reject all wage
offers below w r and accept all other offers. The
reservation wage balances the costs of continued search against the benefits. In this model,
the cost of prolonging a search is the wages
lost while the individual waits for a new offer.
The benefit of a longer search is the expectation of being matched with a firm offering a
higher wage. At w r, the (expected) benefits of
continued search just equal the costs.
Figure 3 shows the probability of receiving a
wage at least equal to the reservation wage.
This is also the probability of an individual
leaving the pool of unemployed. Individuals
choose to be unemployed (in the sense that
they are rejecting wage offers) because it is
rational to do so. A spell of unemployment can
be thought of as an investment in finding a
well-paying job. Since individuals are allocating
themselves to relatively more productive jobs in
the economy, search unemployment is both
privately and socially desirable.
■ 5 Assume that the firm always offers a wage rate at which it would
actually be willing to hire the worker. That is, the firm does not lose money
by hiring the worker at the wage offered.
■ 6 On-the-job search is ruled out solely in the interests of parsimony.
20
F I G U R E
5
An Unanticipated Shift in
the Wage-Offer Distribution
An Unanticipated
Aggregate
Productivity
Increase
Now, suppose that while all firms know that the
A Permanent
Technology Shock
productivity of all jobs has increased 10 percent,
individual workers are initially unaware of this
improvement in technology. Then the unemployed will have no reason to alter their reservation wage, and as a group they will receive
more acceptable job offers. Equivalently, the
likelihood that an unemployed individual will
receive an acceptable wage offer increases
(compare the shaded areas in figure 5). Under
this scenario, the unemployment rate will fall,
since the job-finding rate has increased.
One would anticipate that, over time, workers will learn about this shift in productivity. As
a result, reservation wages will gradually creep
up until they have risen 10 percent. Once all
workers have found out about the 10-percent
improvement in wage offers, the analysis proceeds as above when workers were fully informed of the increase in wage offers. That is,
the unemployment rate will fall only in the short
term;7 in the long term, it will be unchanged.
Suppose that the productivity of all jobs increases. To start, suppose that both firms and
workers are aware of this improvement in productivity, and that firms are now willing to pay
10-percent-higher wages. In this case, the distribution of wages faced by an unemployed
individual will shift to the right, as depicted in
figure 4.
How should an unemployed individual’s
behavior change? It turns out that since all
wage offers have risen 10 percent, his reservation wage should also rise 10 percent. Increasing the reservation wage in this manner will
imply that the search costs (at this new reservation wage) will increase 10 percent, as will the
benefits, since all wage offers have increased
10 percent. The original reservation wage
equated the costs and benefits of search, so a
10 percent increase in the reservation wage will
continue to equate the costs and benefits of
search. Given this increase in the reservation
wage, the likelihood that an unemployed individual will receive an acceptable offer is
unchanged: All wage offers have risen 10 percent, as has the reservation wage. Consequently, there will be no effect on the unemployment rate. These points are developed in
more detail in the appendix.
Possibilities for
Long-Run Effects?
Thus far, the analysis has relied on the fact that
search costs increase in proportion to the benefits. Consider two factors which may contribute
to a larger increase in the search costs. First,
most states impose a maximum on unemployment insurance (UI) benefits. In this context, it
is useful to think of UI benefits as a subsidy to
search. For individuals whose UI benefits are
capped, the 10-percent increase in the wage
offer distribution implies that the subsidy to
unemployment has fallen relative to wages,
effectively increasing search costs. Such individuals should increase their reservation wage by
less than 10 percent in order to reduce the
■ 7 The view that the current low unemployment rate is a temporary
phenomenon has been expressed by Federal Reserve Governor Lawrence
H. Meyer, among others. In a recent speech, Meyer (1997) said: “The consensus estimates of NAIRU as this expansion began—about 6 percent—
did not prepare us for the recent surprisingly favorable performance.... my
response is to update my estimate of NAIRU and add other explanations
consistent with this framework, but not to abandon this concept. One possible explanation is that one or more transitory factors, for the moment, are
yielding a more favorable than usual outcome. A coincidence of favorable
supply shocks is clearly, in my judgment, an important part [of] the answer
to the puzzle.”
21
costs of search (since a lower reservation wage
implies a shorter duration of unemployment).
Thus, the aggregate unemployment rate should
fall. However, to the extent that legislation
maintains a link between the cap on UI benefits and average wages, this channel for reducing the unemployment rate will likely be of limited duration.8
Second, the unemployed may have more
opportunities to pursue leisure and home production activities than those who are engaged
in full-time work. These alternative uses of time
also act as a subsidy to unemployment. Suppose, for example, that the value of leisure is
unchanged following a shift in the wage distribution, then as with maximum UI benefits, the
unemployed should increase their reservation
wage by less than 10 percent (that is, the reservation wage should decrease relative to the
average wage), and the unemployment rate
should fall.
III. Search Intensity
The unemployed can vary their search intensity
by sending out more resumés, filling out more
job applications, calling more employers, or
pursuing prospective jobs more aggressively.
Likewise, employers can alter their search
intensity by posting more job vacancies, using
larger advertisements, assigning more employees to recruiting, and sending recruiters to
more places where potential employees are
concentrated, such as university campuses. Of
course, these activities are costly to firms.
An individual who increases his search
intensity will reduce the (expected) length of
time between job offers, or equivalently will
increase the number of job offers per unit of
time. The payoff to increased search intensity
by the unemployed will, in turn, depend on the
search intensity of firms: It does little good to
look hard for a job if firms simply are not hiring. Consequently, if the unemployed believe
that firms are recruiting intensively, then the
unemployed will do the same, since they are
more likely to encounter a firm offering an
acceptable wage.
Likewise, if firms believe that the unemployed are searching intensively, then they will
want to do likewise: A good time to be seeking
employees is when lots of people are looking
for jobs. These beliefs of firms and the unemployed are self-fulfilling in the sense that intensive search by the jobless is justified by vigorous
firm recruiting, and vice versa. With high search
intensity on both sides of the job market, unem-
ployment will be low, since many unemployed
individuals are finding suitable jobs.
Of course, a high unemployment equilibrium is also possible. In this case, the unemployed do not look very hard for jobs, since
they believe that firms are not engaged in
much recruiting; firms do not recruit heavily
because they believe that the unemployed are
not searching very hard. Again, these beliefs
are self-fulfilling.
Starting from a high unemployment equilibrium, consider the effect of a positive technology shock that shifts the wage-offer distribution.
Suppose that the unemployed initially increase
their reservation wage in proportion to the shift
in the wage-offer distribution, and do not
change their search effort. Prior to the technological improvement, firms chose the number of
vacancies to be posted in such a way that the
marginal cost of posting another vacancy just
equaled the (expected) marginal benefit due to
sharing in the surplus created by a successful
match with an unemployed person. For simplicity, assume that the cost of posting a job vacancy is unchanged. Then firms will wish to recruit
more heavily, since the expected marginal benefits of posting a vacancy now exceed the marginal cost. In response, the unemployed will
find it optimal to increase their search intensity.
Firms, in turn, will want to intensify their
recruiting efforts further, and so on. The net
result is an increase in search effort by both
firms and the unemployed (with the final
increase being larger than the impact effect),
leading to more matches and so to a lower
unemployment rate. That is, a positive technology shock may move the economy from a high
to a low unemployment equilibrium. Furthermore, the fall in the unemployment rate will be
permanent, regardless of whether the productivity shock is permanent or temporary.
Naturally, the shock effecting the move
from a high to a low unemployment equilibrium need not be technological. For example,
the government could temporarily hire people
to perform socially useless activities (for example, digging holes on even-numbered days,
and filling them in on odd-numbered days). By
■ 8 In light of this analysis, one might well ask why the state subsidizes unemployment (through the UI system), since the outcome will necessarily be a higher unemployment rate. Two obvious answers come to
mind. First, some workers are simply unlucky in that they lose their jobs
through no fault of their own. UI benefits allow such individuals to better
smooth their consumption over time. That is, UI is an insurance against
some forms of consumption risk. Second, not all unemployment is necessarily bad. As pointed out above, unemployment can be thought of as an
investment in well-paying jobs. It probably would not be desirable to
require an unemployed person to accept the first job that is offered.
22
increasing the economy’s recruiting, the government provides the unemployed with an
incentive to increase their search effort. In
turn, firms will increase their recruiting efforts,
since more unemployed people are looking for
work. Again, a sequence of events is put in
motion which will move the economy to a low
unemployment equilibrium. Once this new
equilibrium is reached, the government can
terminate its hiring activities.
IV. A Matching
Technology
To simplify the analysis somewhat, suppose
that in a given period, an unemployed individual either will or will not receive a wage offer.
If he receives a wage offer, it is some constant,
w. Likewise, a firm with a vacancy either has a
job applicant or not. All applicants are assumed
to be equally productive, so the firm hires any
applicant, paying the wage rate, w.
The matching technology works as follows:
The number of matches in the economy (that
is, the number of jobless people who successfully find work) depends on the number of
unemployed and on the number of job vacancies posted. Of course, the number of matches
cannot exceed the number of unemployed
individuals, nor can it exceed the number of
posted job vacancies. Assume that each unemployed person is equally likely to receive a
wage offer. Likewise, suppose that all firms
posting vacancies are equally likely to have a
job applicant in any given period. The number
of matches in a given period is increasing in
the number of unemployed people looking for
jobs and in the number of vacancies posted by
firms. Finally, the matching technology is typically assumed to exhibit constant returns to
scale: Doubling the number of unemployed
and the number of vacancies doubles the number of matches consummated.
Instead of an aggregate technology shock
which affects the productivity of workers on
the job, suppose that the shock affects the rate
at which matches occur. That is, for a given
number of unemployed and vacancies, there
are simply more matches. By way of example,
the Internet has made it easier for employers to
post vacancies and for the unemployed to
search for jobs (particularly in faraway places).
Since the number of matches has increased, the
unemployment rate must fall.
Of course, improvements in the matching
function and in overall economic productivity
may move in tandem. The computer example
is a particularly apt one, since productivity
gains in recent years have been largely attributed to the adoption and spread of computer
technologies.
V. Conclusions
This article uses economic theory to assess
recent claims that the economy has a new
“speed limit”—that the economy can operate at
a lower unemployment rate without exerting
upward pressure on the inflation rate due to an
improvement in technology. The models analyzed above embody the classical dichotomy
between the real and nominal sides of the
economy. As a consequence, there need not be
any relationship between inflation and unemployment in these models. The key question is
whether a technological improvement will permanently lower the unemployment rate.
In the neoclassical view, the labor market
operates as a continuous auction market. An
implication of this model is that there is no
unemployment; individuals without jobs have
chosen not to work at the equilibrium wage
rate. This observation prompts a look at search
unemployment models.
In the basic search unemployment model,
the outcome of the decision problem faced by
the unemployed is a reservation wage: Offers
below this wage are rejected, while all others
are accepted. In this model, a permanent
improvement in productivity of all jobs which
increases wage offers will, at best, lower the
unemployment rate only temporarily. Once
workers are fully aware of the shift in the
wage-offer distribution, they will increase their
reservation wage so that the fraction of acceptable wage offers is the same as it was before
the productivity change.
The no-change-in-unemployment result in
the basic search model relies on the assumption that search costs increase by the same proportion as search benefits. Should the costs of
search increase by more than the benefits—
perhaps due to caps on unemployment insurance benefits or the unchanged value of leisure
and home production opportunities which may
be enjoyed in greater abundance when an individual is unemployed—then the unemployment rate may fall.
In an extended search model, both firms and
the unemployed are permitted to vary their effort. This model can be characterized by multiple rational expectations equilibria. That is,
there can be high and low unemployment
23
equilibria whose only difference lies in expectations. Specifically, a low unemployment equilibrium will result if the unemployed search
hard because they believe that firms are recruiting heavily, and firms recruit energetically
because they believe the unemployed are
searching intensively. Conversely, a high unemployment equilibrium will result if neither side
of the market searches vigorously because each
believes that the other is not searching very
hard. Now, even a temporary technology shock
may move the economy from a high to a low
unemployment equilibrium by initiating a chain
of events that intensifies search efforts by both
firms and the unemployed.
The final model is characterized by a matching technology which depends on the number
of unemployed and the number of vacancies.
Here, a permanent improvement in the matching technology will lead to a lower unemployment rate.
Most advocates of the “new economy” paradigm seem to view recent events as an improvement in worker productivity, not in the
matching technology. No doubt, many would
be uncomfortable with the multiple equilibria
explanation of events—if only because traditional Keynesian tools could also move the
economy between equilibria. This leaves the
basic search model, which predicts a permanent
fall in unemployment only if the costs of search
rise by more than the benefits (a scenario that
could result from a cap on unemployment
insurance benefits) or if the technology shock
does not change the value of alternative uses of
time while an individual is unemployed.
Appendix
The Basic Search
Model: The Worker’s
Problem
The typical worker seeks to maximize expected
lifetime utility, given by
∞
^ b tct ,
t=0
where
5
wt if employed and
ct = 0 otherwise.
Notice that workers are assumed to be riskneutral (utility is linear in consumption).
Wage offers are distributed according
to
_
g(w), which is defined over [w
,
w
].
The
asso_
ciated
cumulative density function is G(w)
_
w
≡ e g(w)dw. Let
w
_
5
wt + 1 = 0 with probability p
wt with probability 1 – p,
where p is the exogenous separation rate.
The worker’s problem can be cast using the
tools of dynamic programming. The value of
working at a particular wage, w, is given by
V w (w) = w + b [pV u + (1 – p)V w (w)].
Similarly, the value of being unemployed is
given by
_
w
Vu
=
G(w r)V u
+ eV w (w)g(w)dw,
wr
where, as above, w r is the reservation wage
rate. G(w r) is the probability of rejecting a
wage offer, given the reservation wage. Notice
that the reservation wage will have the property that
V w (w r ) = V u .
The shift in the wage distribution owing to
an improvement in technology should be
thought of as a “stretching out” of the wageoffer distribution. That is,
G (w ) = G~(w /l) for all w and for all l,
where G~ is the new wage-offer distribution.
The claim that a technology shock resulting
in a shift in the wage distribution by a factor l
will increase the reservation wage by the same
factor l is now relatively straightforward to see.
In particular, provided the reservation wage
does increase by the factor l, all the quantities
describing the value functions V w (w ) and V u
will also rise by the same factor l.
24
References
Economic Report of the President, February
1998. Washington, D.C.: U.S. Government
Printing Office, 1998.
Jovanovic, Boyan. “Job Matching and the
Theory of Turnover,” Journal of Political
Economy, vol. 87, no. 5, part 1 (October
1979), pp. 972–90.
Lucas, Robert E., Jr., and Edward C.
Prescott. “Equilibrium Search and Unemployment,” Journal of Economic Theory,
vol. 7, no. 2 (February 1974), pp. 188–209.
Meyer, Laurence H. Remarks before the 1998
Global Economic and Investment Outlook
Conference, Carnegie–Mellon University,
Pittsburgh, Penn., September 17, 1997.
Sargent, Thomas J. Dynamic Macroeconomic
Theory. Cambridge, Mass.: Harvard University Press, 1987.
25
Unemployment and
Economic Welfare
by David Andolfatto and Paul Gomme
David Andolfatto is a professor
of economics at the University of
Western Ontario. Paul Gomme is
an economic advisor at the Federal Reserve Bank of Cleveland.
Both authors are members of the
Centre for Research on Economic Fluctuations and Employment in Montreal, Canada. They
thank Lutz-Alexander Busch,
Gordon Myers, and Ed Nosal for
helpful comments and criticisms.
This research is part of a larger
project for which Andolfatto
acknowledges the financial support of SSHRCC.
Introduction
Statistics that measure labor market activity,
such as employment and unemployment, are
often interpreted in the press and by politicians
as measures of economic performance and
social well-being. Discussions that focus on
cross-country comparisons of unemployment,
for example, seem to be based without exception on the premise that unemployment represents a social and economic ill, so that less of it
is generally to be preferred. The purpose of this
note is to demonstrate that some care should
be exercised when constructing a map between
labor market behavior and economic welfare
and that, generally speaking, such interpretations are not justified in the absence of information concerning the economic circumstances
that determine individual labor market choices.
I. Some Labor
Market Facts
Each month, the Current Population Survey
(CPS) assigns the noninstitutional civilian population of the United States to one of three mutually exclusive groups: Employment, Unemploy-
ment, or Nonparticipation. The survey begins
by determining whether a person is employed,
which is defined roughly as having allocated
any time at all toward paid work in the previous week. Those that are not employed in this
sense are defined as nonemployed. The survey
then asks all nonemployed individuals a series
of questions designed to detect some minimum
level of active job search. Those nonemployed
individuals that report themselves as having
engaged in some minimum level of active job
search over the previous week are classified as
unemployed.1 The remaining group of nonemployed individuals are classified residually as
nonparticipants.
Of course, these three classifications are
extremely crude. We know, for example, that
there is a tremendous amount of variation in
hours worked per month across employed
individuals. While it is unclear how much time
is typically devoted to job search activities, we
can safely assume it varies from a few hours
per month browsing over help-wanted ads
■ 1 Individuals who report themselves on temporary layoff are also
classified as unemployed whether or not they report any search activity. It
should be pointed out, however, that these individuals form only a small
fraction of the total number of unemployed.
26
to many hours per month seeking out job opportunities that are well-matched to an individual’s attributes.
Concerning the curious category of unemployment, it seems apparent that the CPS (and,
indeed, most people) implicitly attaches a great
deal of weight to the time used for job search
by the nonemployed. Despite the popularity of
the unemployment measure among commentators and policymakers, many economists
question the usefulness of the concept of
unemployment, preferring instead to focus on
employment (or nonemployment) and on the
allocation of time across other activities, for
example, on-the-job search, learning, and
household production. It is also interesting to
note that many extensive time-use studies,
such as those surveyed in Juster and Stafford
(1991), do not even include a category for job
search, let alone job search conducted by nonemployed individuals. Ultimately, the justification for isolating job search as a crucial activity
distinct from the many other competing uses of
time among the nonemployed has to be based
on the theoretical and empirical relevance of
the concept.
Abstracting from seasonal variation, the CPS
reveals that net monthly changes in employment and unemployment tend to be relatively
small. However, the stability displayed by these
stocks masks the very high degree of turnover
that exists in the labor market: Each month, literally millions of individuals make transitions
between different labor market states. Historically, about 2 percent of the adult workforce in
the United States flows into and out of employment every month.2 Based on current population estimates, this represents approximately
4 million workers either losing or leaving their
jobs, and roughly the same number acquiring
jobs, resulting in a total turnover of about 8 million workers per month.3
A second striking feature of the CPS flows
data concerns the degree of mobility displayed
by the group of individuals labeled “nonparticipants.” Contrary to what one might expect,
fully half of the flows into and out of employment are accounted for by individuals making
transitions to and from nonparticipation. While
nonparticipants are, by definition, not “actively”
seeking employment opportunities, this apparently does not preclude the possibility of being
available for employment (for example, if
called on by a former employer). This feature
of labor market behavior calls into question the
usefulness of attempting to make a distinction
between unemployment and nonparticipation.
However, the absolute size of the flows between
employment and unemployment are as large as
those that occur between employment and nonparticipation. This, together with the fact that
the unemployment stock is much smaller than
the stock of nonparticipants, implies that the
average probability that an unemployed person
makes a transition to employment is much
higher than the corresponding probability for a
nonparticipant. This feature of the data is consistent with the notion that unemployment is a
labor market state that facilitates the job-finding
process, an interpretation that conflicts with the
common textbook perception that “unemployment represents wasted resources.”4
The remainder of this paper is concerned
with developing a simple theoretical framework
that might be used to interpret the labor market
behavior described above; this interpretive
device is then used to determine under what
conditions changes in employment and unemployment can be associated with changes in
economic welfare. The analysis proceeds in
two steps. First, a basic model of employment–
nonemployment is developed and analyzed.
This model is then extended to incorporate the
phenomenon of unemployment.
II. A Simple Model
of Worker Turnover
Consider an economy consisting of a fixed number of individuals. Each person has preferences
given at each point in time by U = ln(c) + z,
where c represents the consumption of market
goods and services and z represents the consumption of services produced in the nonmarket sector.5 Notice that according to this specification of preferences, individuals find it very
painful to subsist at very low levels of market
consumption; i.e., U → – ∞ as c → 0.
Each person is endowed with an indivisible
unit of discretionary time, which may be utilized either in the production of market goods
or services (employment) or in some other
activity (nonemployment). People generally differ in how their time is valued across alternative uses. Below, we interpret this heterogeneity as emanating from differences in individual
■ 2 See Blanchard (1997, p. 295).
■ 3 These figures actually underestimate the degree of turnover, as
they abstract from job-to-job transitions.
■ 4 Mankiw (1994, p. 137).
■ 5 The structure of the economy will be such that myopic decisionmaking is optimal.
27
F I G U R E
1
The Work Decision
F I G U R E
2
The Effect of Nonlabor Income
Generally, we shall think of each of these parameters as differing across individuals at any
given point in time as well as changing periodically over time for any one person.6
We are interested in modeling an environment where individual labor market transitions
are associated with changes in economic wellbeing, as is likely the case in reality. For this to
be true, financial markets must to some extent
be incomplete, since otherwise individuals
could insure themselves perfectly against any
idiosyncratic labor market risk. For simplicity,
we assume an extreme form of incompleteness
and abstract from financial markets entirely.
In the absence of financial markets, each
person faces a simple set of period budget constraints: c ≤ wn + a and z ≤ v(1 – n), where n
∈ {0,1} represents the time allocation decision.
An individual facing economic circumstances
(w, a, v) must choose how to best allocate time
between employment and nonemployment.
The utility payoff associated with employment
(n = 1) is given by ln(w + a), while the utility
payoff associated with nonemployment (n = 0)
is given by ln(a) + v. Clearly, the individual
should choose the action that yields the highest
utility payoff.
For a given configuration of (a, v), one can
define a reservation wage wR such that any person with an employment opportunity w ≥ wR
will choose to work, while any person with an
employment opportunity w < wR will choose
some nonmarket activity. The reservation wage
is defined to be that wage for which an individual is just indifferent between working or not;
i.e., wR satisfies:
(1)
economic circumstances as summarized by the
triplet (w, a, v). Here, w represents the market
value of an individual’s particular skill (real
wage) or, equivalently, the amount of output
that can be produced with one unit of labor
(productivity). The parameter a represents an
individual’s nonlabor income, for example,
interest income on property, income from a
spouse, unemployment insurance, or welfare,
charity, and so on. The parameter v represents
the value of time allocated to nonmarket activities, for example, home production or leisure.
ln(wR + a) = ln(a) + v ,
which can be solved explicitly as wR = (e v – 1)a.
Figure 1 plots the reservation wage as a function of v , holding fixed the level of nonlabor
income a.
The reservation wage has a very useful economic interpretation. In particular, it can be
thought of as representing a person’s level of
“choosiness” over available job opportunities: A
higher reservation wage means that a person is
more discriminating. Theory sensibly suggests
that a person’s level of job-choosiness should
depend positively on the level of nonlabor
income and on the quality of opportunities in
the nonmarket sector. People are more discriminating when they can afford to be. Figure 2
■ 6 In order to maintain the optimality of myopic decision-making,
assume that (w, a, v ) are identically and independently distributed random
variables.
28
F I G U R E
(4) N = ∑n (w, a, v)g (w, a, v).
3
w,a,v
Possible Changes in the Value
of Leisure and the Wage Rate
plots the reservation wage function for two different levels of nonlabor income aH > aL.
With the reservation wage so defined, the
optimal time allocation decision is given by
(see figure 1):
(2) n (w, a, v) =
510 ifif ww ≥< (e(e
v
v
– 1)a ;
– 1)a,
with maximum utility given by:
(3) W (w, a, v) = max{ln(w + a), ln(a) + v }.
Theory suggests that an individual is more
likely to be employed when w (the return to
working) is high, and less likely to be employed
when either a or v are high (as these latter variables increase the reservation wage). The welfare function W (also referred to as the indirect
utility function) tells us that individual wellbeing is an increasing function of nonlabor
income and a nondecreasing function of both
the real wage and the value of time in the nonmarket sector. All of this makes perfect sense.
It also implies that there is no necessary correlation between employment status and economic
well-being.
This assertion holds true at the aggregate
level as well. For a theory of aggregate employment, one must describe how the economic
attributes (w, a, v) are distributed over individuals. Let g(w, a, v) denote the fraction of the
population with attributes (w, a, v). Then
aggregate employment is given by:
Improvements in aggregate economic conditions can be modeled as changes in the distribution function g such that more individuals
are concentrated over higher values of w, a,
or v. In the first case (higher values of w), aggregate employment can be expected to rise,
while in the second and third cases, employment can be expected to fall. In each case, any
reasonable measure of social welfare can be
expected to increase.
Note that, even in the absence of aggregate
uncertainty (i.e., a stationary distribution function g), the equilibrium of this economy will in
general feature flows of workers into and out of
employment (recall that individuals begin each
period by independently drawing a new realization of (w, a, v) from the distribution g). Examples of such transitions are plotted in figure 3.
Keep in mind that, because financial markets
are absent, these transitions are typically associated with significant changes in personal living
standards. Two points deserve to be made here.
First, note that one cannot infer any change in
personal well-being simply on the basis of an
observed change in labor market status. Consider, for example, a person who begins the
period at point A in figure 3. Suppose that at the
end of the period, we observe that the individual exits employment. Whether this person is
better or worse off clearly depends on the
change in economic circumstances that triggered the transition. For example, a deterioration in the value of market time (point B) or an
improvement in the value of nonmarket activities (point C ) may both trigger such a transition.
Second, note that these transitions are not the
direct cause of any change in living standards;
rather they represent the “rational” behavior of
individuals in response to exogenous changes
in economic circumstances. The following example will illustrate this latter point.
Imagine that individuals in the economy
described above differ only with respect to their
employment opportunities w and that w > 0,
so that everyone always has the option of working at a job that produces positive output (note:
w may be arbitrarily close to zero so that the
opportunity may not be particularly attractive).
Let F (w) denote the fraction of workers with a
job with a wage no better than w and assume
that workers independently draw a new wage
every period from the distribution F. Individuals
value nonmarket activities identically according
to v > 0, and we assume that each person has
zero nonlabor income; i.e., a = 0.
29
Recall that the reservation wage is given by
wR = (e v – 1)a, so that in this example, wR = 0
since a = 0 (people cannot afford to be very
choosy here). Since w > 0 by assumption, it follows that everyone chooses to work in this
economy, and that, consequently, transitions
into and out of employment are absent. Judging by these aggregate labor market statistics,
the economy appears tranquil (low turnover)
and robust (high employment).
However, these statistics hide the fluctuations in individual well-being that occur as people find the return to their labor changing over
time. Some individuals may experience precipitous wage declines as the demand for their
labor all but disappears (perhaps owing to the
arrival of a new technology that is not wellmatched to their skills). These unfortunate people refuse to exit from employment (as they
arguably should in order to pursue relatively
more valuable nonmarket activities such as
retraining), since they must work in order to
eat; as such, they become a part of the “working poor.”
This equilibrium is inefficient relative to one
in which insurance markets (or some equivalent institution) operated to alleviate individual
income risk. Recall that, at the beginning of
each period, an individual draws a new wage
according to the distribution F; the expected
utility payoff for the representative individual in
this world is given by:
(5) EU A = ∫0 ln(w)dF (w).
In addition, note that per capita output is
given by y A = ∫0 wdF (w) with an employment
level N A = 1.
Consider now the allocation that would be
chosen by a social planner wishing to maximize the expected utility of the representative
individual (the same allocation would result in
a world with a perfectly functioning insurance
market). The social planner must choose a
reservation wage wR that determines who
works and who does not, along with a feasible
set of consumption levels for the employed y e
and nonemployed y n. Conditional on these
choices, the representative individual has an
expected utility payoff given by:
(6) EU = [1 – F (wR )]ln(y e ) + F(wR )[ln(y n) + v],
where F (wR ) represents the probability of nonemployment. Assume that the planner chooses
(wR , y e, y n) in order to maximize EU subject to
the feasibility constraint (total consumption
cannot exceed total output):
(7) [1 – F(wR )]y e + F(wR )y n ≤
∫wR wdF(w).
The reader can verify that the solution to
this problem entails a reservation wage that is
strictly positive, wR* > 0, together with equal
consumption across labor market states, y e
= y n = y * = ∫wR* wdF (w). The expected utility delivered to the representative individual is EU *
= ln(y* ) + F (wR* )v. It can be easily demonstrated
that N * = 1 – F (wR* ) < N A (employment is lower
under the planner), y * < y A (output is lower under the planner), and that EU * = ln(y * ) + F (wR* )v
> EU A (people are better off under the planner).
In addition, as time unfolds, note that individuals will generally experience transitions into
and out of employment under the allocation
chosen by the planner.
The availability of consumption insurance
means that people who temporarily find their
earnings capabilities severely diminished need
not waste valuable time engaged in very low
productivity tasks; time can instead be reallocated to more productive nonmarket applications. Employment and market incomes in such
an environment are necessarily lower (relative
to a situation where everyone is compelled to
work), but this does not necessarily imply that
economic well-being is lower.
III. Unemployment
Recall that the CPS definition of an unemployed
person is someone who is both nonemployed
and actively searching for employment. Why
are there people in the economy whose economic circumstances are such that they are
compelled to spend precious time looking for
buyers of labor willing to pay an acceptable
price for their particular job skill? It must be the
case that people have incomplete information
concerning the location of their best job opportunity, and that the job search activity generates
information whose expected return exceeds the
value of this foregone time spent in alternative
activities. Incomplete information of this sort is
likely to be a natural feature of any dynamic
economy in which changes in the structure of
technology and tastes randomly create, destroy,
and reallocate employment opportunities across
different sectors.
There are several ways in which one might
model the job search activity of nonemployed
workers. Here, we shall take a particularly
simple approach that is in keeping with the
30
analysis developed earlier.7 Following the
setup above, assume that all individuals have
access to some employment opportunity in the
market sector. While some readers may view
such an assumption as a gross violation of reality, our view is quite the opposite. In particular, note that we do not place any restriction
on the quality of potential employment opportunities, so that our setup does allow for the
possibility that there is a scarcity of what might
be considered to be “good” jobs.
As with the earlier analysis, assume that individuals are distributed in some exogenous
manner over the space (w, a, v). In that analysis, it was implicitly assumed that individuals
had complete information about the location of
their best job opportunity (w), so that the job
search (and hence unemployment) in that environment proved unnecessary. However, suppose now that while individuals are endowed
with a job opportunity w at the beginning of
the period, they are generally aware that better
(and worse) prospects exist elsewhere. Assume
that these prospects p are distributed according
to a known distribution Q(x) = Pr[p ≤ x], where
Q ′ > 0. Job search is modeled as a random
draw from this distribution.
In particular, assume that an individual may
divert some given fraction of the period time
endowment 0 < (1 – ξ ) < 1 toward job search.
(for simplicity, assume that such an action
necessitates the abandonment of the beginningof-period job opportunity). Following this exertion of job search effort, the individual realizes
a new job opportunity p from the distribution
Q and may at this stage choose to devote any
remaining time ξ toward employment or home
production activities.
Let us now determine the expected utility
payoff associated with the job search decision.
Once the new job opportunity is realized, the
individual faces a standard employment–
nonemployment decision and chooses a reservation wage pR (a, v) that satisfies:
(8) ln(ξ pR + a) = ln(a) + ξ v.
If the new employment opportunity offers a
wage p < pR , the individual will find it optimal
to spend any remaining time at home. With pR
so determined, the expected utility of undertaking the search activity is given by:
(9) λ(a, v) = Q (pR )[ ln(a) + ξv]
+ ∫pR ln(ξ p + a)dQ ( p).
Here, Q (pR ) is the probability that the new job
prospect is of an unacceptably low quality, in
which case the person earns a utility payoff
[ ln(a) + ξ v]. The term dQ (p) can be interpreted as the probability of locating a job with
wage p, which earns utility payoff ln(ξ p + a);
the second term in the right-hand side of the
expression above simply adds up the utility
payoff associated with each acceptable job
weighted by the probability of finding a job of
that particular quality.
In the earlier analysis, which abstracted from
unemployment, a reservation wage wR was
determined that partitioned the population into
employment and nonemployment; these two
groups we shall now refer to as “type-A” and
“type-B” individuals, respectively. Think of
type-A individuals as those who (given current
economic circumstances) prefer work to leisure
(i.e., w > wR ), while type-B individuals are
those who prefer leisure to work (i.e., w < wR ).
With the option of job search available,
some type-A individuals may now choose to
abandon their current employment opportunity
in pursuit of a new (and hopefully better) one.
The return to work is given by ln(w + a), while
the return to search is given by λ (a, v). Clearly,
the optimal strategy is to form a reservation
wage w′R satisfying:
(10) ln(w′R + a) = λ (a, v),
such that all type-A individuals with w > w′R
should choose to work full-time, while those
with w < w′R should abandon their current
employment opportunity in search of another.
It can be demonstrated that w′R ≥ wR for typeA individuals; i.e., the option of a search activity makes these people even more choosy
about their beginning-of-period employment
opportunity.
Likewise, a group of type-B individuals may
now choose to sacrifice some of their leisure
time to look for work (i.e., for a wage that dominates their current employment opportunity).
As the return to leisure is given by ln(a) + v, the
optimal strategy for type-B individuals is to set a
reservation “leisure wage” of vR satisfying:
(11) ln(a) + vR = λ (a, vR ),
■ 7 The model that follows is closely related to that developed by
Burdett, Kiefer, Mortensen, and Neumann (1984); and Andolfatto and
Gomme (1996).
such that all type-B persons with v > vR should
choose full-time leisure, while those with v < vR
should devote some time to active job search.
31
F I G U R E
4
The Search Decision
F I G U R E
5
The Unemployment Decision
Figure 4 plots the reservation wage functions
wR , w ′R and vR for a given level of nonlabor
income.
In order to calculate the equilibrium level of
employment and unemployment, let us assume
that the CPS is undertaken at the end of each
period. To begin, it is clear that all type-A individuals with w > w′R would be classified as
employed, as these individuals work throughout the period. As well, all type-B individuals
with v > vR would be classified as nonparticipants, as these individuals engage in nonmar-
ket activities throughout the period. All remaining individuals allocate at least some time to
search. However, not all of these individuals
would be classified as unemployed by the CPS.
In particular, all searchers who are successful at
finding a suitable job within the reference
period of the survey and work any amount of
positive hours would be classified as employed.
The unemployed are those who search for
work but are unsuccessful at obtaining a suitable job within the reference period of the
labor force survey. In terms of figure 5, the
unemployed would be those who find themselves in the shaded region of the parameter
space at the end of the period.
How are the economic attributes (w, a, v)
related to individual labor market choices?
Recall the earlier analysis of employment and
nonemployment. In that model, conditional on
the level of nonlabor income a, the employed
tended to be those people with high (w/v)
ratios; i.e., those individuals whose productivity
in the labor market dominated their productivity in the home sector. In that model, the employment decision is a poor indicator of economic well-being, as it depends (conditional on
a) primarily on the ratio (w/v), while economic
welfare depends on the levels of w and v.
What general inferences can be made about
unemployment and individual well-being?
According to the model of unemployment
developed above, there is a sense in which the
unemployed tend to be relatively disadvantaged (conditioning on the level of nonlabor
income). Being measured as unemployed for
the period indicates that, at some time in the
recent past, the available job opportunity w was
of poor quality and that the value of time spent
in alternative uses v was also of poor quality.
(Individuals with good qualityw’s tend to be
employed, while individuals with good
qualityv’s tend to be nonparticipants.) As economic well-being depends (indirectly) on the
levels of w and v, it follows that choosing to
search (a prerequisite for being unemployed) is
associated with a low level of welfare.
Having said this, note that there may be
many individuals in the model who are employed and yet experience even lower levels of
welfare than the unemployed (even holding
equal the level of nonlabor income). Recall that
after a job search yields an employment opportunity that pays p, an individual is free to work
for wage p or spend the time at home earning
the “leisure wage” v. If the latter choice is
made, then the job search is deemed unsuccessful and the person is classified as unemployed. If the former choice is made, then the
32
person is classified as employed for the period.
Note that this work–leisure choice depends, as
before, primarily on the ratio (p/v), and so
whether the person chooses employment or
unemployment at this stage reveals very little
about the levels of p or v (and hence the level
of welfare).
For example, consider two individuals with
identical p’s and a’s, but with different v’s. It
is conceivable that the person with the poor
home opportunity will at this stage choose
employment, while the person with the relatively good home opportunity will choose not
to work (and therefore be measured as unemployed). In this example, the unemployed person is clearly better off than the employed
person, while both persons are worse off compared to most other members of the population
who did not feel the need to search.
What about the relationship between the
level of nonhuman wealth a and unemployment? Consider two societies that are identical
in every respect except that one society generates all of its income from labor, while the
other is also endowed with a source of nonlabor income a > 0. What can be said about the
equilibrium level of unemployment and level
of welfare in these two economies?
From the earlier analysis of employment
and nonemployment, we know that there is
no nonemployment (and hence no unemployment) in the economy with zero nonhuman
wealth. Individuals may still choose to search
and generate new job opportunities p, but
when a = 0 it turns out that pR = 0, so that individuals will choose to work at whatever new
prospect makes itself available (as long as p >
0). Thus, we observe paradoxically that the
wealthier economy will exhibit a higher measured rate of unemployment. There is a sense
here in which unemployment represents a “luxury” that only very rich countries can afford.
Citizens of poor countries are compelled to
work (either in the market or at home) or die;
in either case, they are unlikely to be recorded
as being “unemployed” by the CPS.8
More generally, the model suggests an
ambiguous relationship between unemployment and wealth. The reason for this is as follows. First, from the condition determining vR ,
one can demonstrate that vR is a decreasing
function of a. In other words, higher levels of
wealth have the effect of making leisure more
affordable; this effect leads to higher nonparticipation and hence less search activity (and
hence lower unemployment). Second, from the
condition that determines w R′ , it appears that
w ′R may either increase or decrease with higher
levels of wealth. On the one hand, a higher
level of nonlabor income may make an individual more willing to forego the hassles associated with job search. On the other hand, a
higher level of nonlabor income means that an
individual can better afford to engage in job
search activities. Which effect dominates depends on the precise form of preferences, the
level of nonlabor income, and the distribution
of available job opportunities. If w R′ is decreasing in a, then people are less willing to search,
so that unemployment falls, reinforcing the participation effect. In this case, unemployment
unambiguously declines as nonlabor income
rises. If w′R is increasing in a, then people are
more willing to search, leading to an increase
in unemployment, offsetting the participation
effect. The overall effect on unemployment
then depends on the relative strength of these
two effects.
Finally, a remark on the optimal level of
unemployment. With incomplete consumption
insurance markets, the equilibrium level of
unemployment will likely be too low. The reason for this is similar to before: Individuals who
find themselves temporarily in dire straits are
compelled to work rather than search and/or
engage in other nonmarket activities. This basic
result calls into question the conventional wisdom which views unemployment as “idle” or
“wasted” resources.
IV. Conclusions
Economic theory asserts that living standards
(utility) ought to depend primarily on the level
of broadly defined consumption (including
leisure). The simple, yet in many ways plausible, model developed above demonstrated the
tenuous link between labor market choices and
economic well-being. Economic welfare was
shown to be linked indirectly to the level of
human capital in the market and at home, and
to sources of nonlabor income. These parameters determine the individual’s ability to generate high consumption levels.
Labor market choices concerning whether to
be employed or nonemployed, however, in
general reflect the relative returns to engaging
in alternative activities, and hence are poor
indicators of the level of welfare.
■ 8 One might also point out that there is likely very little reason for job
search activity in poor, stagnant economies (aside from migration to cities).
In such environments, the set of available employment opportunities is likely
very limited so that information concerning their location is readily available;
individuals face a standard employment–nonemployment decision.
33
However, the decision to undertake labor
market search was shown to be correlated with
poor opportunities in the market and at home.
Since the extent of search activity and the level
of unemployment are obviously linked, there is
reason to believe that unemployed workers are
generally worse off in welfare terms relative to
that set of the population that appears content
with current market/home opportunities. However, it would be a mistake to infer that the unemployed are the least well-off members of the
workforce. In particular, individuals who are
endowed with very poor human capital and no
outside source of income may be compelled to
work at jobs that others can afford to eschew;
these poorly endowed individuals comprise the
“working poor.”
Thus, while the unemployed tend to be disadvantaged relative to perhaps the majority of
the population, it does not necessarily follow
that they should constitute the primary target of
social policy (should redistribution policy be
deemed desirable). Furthermore, it does not
necessarily follow that the elimination of unemployment would lead to an improvement in
their economic well-being. Whether a reduction
in unemployment is associated with an improvement in welfare would depend on the
particular change in economic circumstances
that altered the return to job search activities.
Unemployment may fall because of any number
of diverse reasons, for example: (1) an improvement in the quality of labor market opportunities; (2) a deterioration in the distribution of
new job opportunities; (3) cutbacks in public
unemployment insurance (reductions in a); or
(4) the arrival of an oppressive regime that im-
poses “work camps” and bans job search activity.9 Clearly, not all of these examples would be
unambiguously associated with improvements
in overall social welfare.
The undue focus on unemployment as a
measure of economic performance and welfare
has contributed much mischief to discussions
concerning the design and implementation of
labor market (and monetary) policy. Throughout the 1980s, for example, the Canadian
unemployment rate averaged about four percentage points higher than the United States,
after decades of close correspondence.10 This
event was widely portrayed as reflecting some
underlying malaise in the Canadian economy, a
belief that seemed to persist despite the fact
that real per capita income growth and employment rates in the two countries remained similar. Indeed, the Canadian economy even managed to maintain a stable after-tax distribution
of income over this high-growth period, while
in the United States the income distribution
widened. Clearly, one must look deeper than
simple measures of labor market activity before
making definitive statements about economic
performance and well-being.11
■ 9 See Eason (1957), who quotes from Pravda (January 31, 1954):
“In 1953, as in preceding years, there has been no unemployment [in the
Soviet Union].’’
■ 10 See Andolfatto, Gomme, and Storer (1997); and Burtless (1997).
■ 11 Rogerson (1997) makes a similar point concerning European
unemployment.
34
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