Lecture Note 4: Unemployment
Economics 5021
Dana Galizia
Carleton University∗
One of the key criticisms of RBC models that we noted at the end of LN3 was their assumption
that all markets are Walrasian, i.e., that markets are always in a competitive equilibrium. The
conditions required for a market to be in competitive equilibrium are quite stringent, and include
(among other things): fully flexible prices, buyers and sellers that are price-takers, no barriers
to entry, exit or trade, all market participants have perfect information about the goods/services
being traded, and no externalities. In reality, many (perhaps most) markets fail to meet one or
more of these conditions. In some cases, these failures are minimal, so that taking markets to
be Walrasian might be a reasonable first approximation. In other cases, however, the failures are
egregious, making the Walrasian assumption of RBC models wholly inappropriate, and potentially
highly misleading.
One market for which the Walrasian assumption is clearly inappropriate is the labour market.
While for some real-world non-Walrasian markets we need to take a close look at market conditions
in order to determine that it is indeed non-Walrasian (e.g., by checking whether a small number
of sellers dominates the market, or considering how easy it would be for additional firms to enter
the market), with the labour market we can observe the failure of competitive equilibrium to hold
directly: it is an empirical fact that the labour market essentially never clears. In particular, at
any given moment, the quantity of labour actually exchanged is nearly always below the quantity
of labour supplied, with the result being involuntary unemployment.
In this LN, we consider some non-Walrasian models of the labour market that deliver unem-
ployment in equilibrium. We will consider two different models that produce unemployment for
two different reasons. In the first model, unemployment will arise because the equilibrium wage is
above the market-clearing level, so that supply of labour always exceeds demand. In this model,
the key Walrasian condition that will fail to hold is perfect information: sellers of labour (workers)
will have more information about the quality of their labour than buyers (firms).
In the second model, unemployment results from a barrier to trade that prevents the labour
market from actually clearing even if the wage is such that supply equals demand. As a result, at
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the prevailing wage there will generally be workers willing to work and firms willing to hire them,
but because of some impediment—in this case, a search friction that makes it hard for these firms
and workers to find each other—this exchange may not occur.
1 Some Math Preliminaries
1.1 Value Functions
1.1.1 Value Function in a Deterministic Environment
Let ut denote an agent’s period utility at time t,1 and let β ∈ (0, 1) denote the agent’s discount
factor. Thus, for an agent currently at date t, the discounted present value of the period utility
they will receive at date s ≥ t is given by βs−tus.2 The total discounted present value of all current
and future period utility is then given by
Vt ≡
∞∑
s=t
βs−tus . (1)
V here (which is a function of the date t) is referred to as a value function. Notice that, for any
q > t, we can write
Vt =
q−1∑
s=t
βs−tus +
∞∑
s=q
βs−tus
=
q−1∑
s=t
βs−tus +
∞∑
s=q
βs−q+q−tus
=
q−1∑
s=t
βs−tus + βq−t
∞∑
s=q
βs−qus
=
q−1∑
s=t
βs−tus + βq−tVq ,
where line #1 splits up the summation into the component from t to q − 1 and the one from q to
∞, line #2 simply adds and subtracts q in the exponent of β in the second summation, line #3
factors βq−t out of the second summation, and line #4 then uses the definition of Vq (obtained by
replacing t with q in (1)). Thus, the value function at date t can be decomposed into the sum of (i)
the discounted present value of period utility between t and q − 1, and (ii) the “left-over” amount
βq−tVq.
In this context, Vq is sometimes referred to as the “continuation value”, since it summarizes
1Note that, in an typical model, period utility at t could in general depend on choices made by agents in the model
(e.g., we might have ut = u(ct), where ct is consumption at date t, and u is some utility function over consumption).
2Notice that the “discounting” part of this expression, βs−t, is very similar to what we had in the RBC model of LN3,
and indeed its economic meaning is precisely the same.
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the discounted utility experienced by the agent as time “continues on” from date q. Note that the
reference date for the discounting in Vq (that is, the date future-period utilities are discounted back
to) is q. Since Vt is the value function for an agent currently at date t, however, we need to discount
the continuation value Vq back an additional q − t periods, hence the factor βq−t. Thus, we may
interpret βq−tVq as the continuation value for date q, discounted back to date t.
Figure 1 illustrates these two different ways of viewing Vt. Panel (a) corresponds to the definition
of Vt given in (1), for which the period utility at each date s ≥ t is discounted back to date t, as
indicated by all the arrowed curves going from various dates s back to date t. Panel (b), meanwhile,
corresponds to the decomposition of Vt into flow utility received between t and q − 1, plus the
discounted continuation value. In this case, arrowed curves starting from dates s = t, . . . , q − 1
still go directly back to t. For dates s ≥ q, meanwhile, flow utility is first discounted to date q, as
indicated by the arrowed curves starting from various dates s ≥ q and going to date q. These flow
utilities for s ≥ q are then combined together into Vq, and then discounted as a whole from q back
to t, as indicated by the dashed arrowed curve going from q to t.
Figure 1: Two Ways of Viewing V (t)
(a) Vt =
∑∞
s=t β
s−tus
st
(b) Vt =
∑q−1
s=t β
s−tus + βq−tVq
st q
A particularly relevant case of the above decomposition is when we set q = t+ 1, in which case
Vt = ut + βVt+1 . (2)
Expression (2) just says that the value function today is equal to the current period utility, plus
the discounted present value of the value function tomorrow.
1.1.2 Value Function for a Switching Process
Suppose now there are two possible states, A and B, that the agent can be in at any point
in time, and that each period an agent may or may not switch states from the previous period.
Conditional on being in state A at date t, let λt ∈ [0, 1] be the probability that the agent switches
to B next period. For j ∈ {A,B}, let uj,t be the period utility associated with being in state j at
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date t (which we will take as given), and Vj,t the value function for an agent who is in state j at
date t. Thus, for an agent in state j at t, they experience ut = uj,t and Vt = Vj,t. Based on the
above intuition regarding decomposing the current value into the current period utility plus the
discounted value of tomorrow’s value function (see (2)), we may therefore write
VA,t = uA,t + βEt [Vt+1] = uA,t + β [(1− λt)VA,t+1 + λtVB,t+1] . (3)
The term in square brackets in the last expression is simply the expectation of the value the
household will have at t+1: with probability 1−λt, the household will still be in state A tomorrow,
in which case they will receive value VA,t+1, while with probability λt they will be switched, in
which case they will receive value VB,t+1.
1.2 Masses of Agents
It will often be useful to think about agents in our models as existing on a continuum. That
is, rather than indexing agents by integers, we can imagine assigning them real-number indices,
where every real number in some interval—say, [c, d]—is used as an index for exactly one agent.
Importantly, since there is an uncountable infinity3 of numbers in the interval [c, d], and since by
construction we have the same number of agents as numbers in that interval, we also have an
uncountable infinity of agents. Thus, if we want to measure the “total amount” of agents in some
way, we’re going to have to be a little bit more careful than simply counting them all up. One
useful way to do this is simply to use the size of the interval that indexes the agents. So if we have
agents indexed by the real numbers in the interval [c, d], we would say that there is a total mass of
(or measure) d− c of them.
In addition to thinking about the total mass of agents in the economy, in the case where indi-
vidual agents could be in different states at a given time we can also track the masses of agents in
each possible state at each point in time. For example, suppose again that there are two possible
states, A and B, and let At and Bt denote the masses of agents in each of these states at date t,
where we take the initial allocations A0 and B0 as given. Suppose also that agents randomly switch
between the two states, where λt is the probability that an individual agent who is in state A at
t will be switched to B at date t + 1, and similarly µt is the the probability that a B agent at t
switches to A at t + 1. By the law of large numbers, λt is also the total fraction of A agents who
3A set is said to be countable if we can in principle “count them all up” using the natural numbers, i.e., the counting
numbers 1, 2, 3, 4, . . . . Any finite set (i.e., any set with a finite number of elements) is clearly countable in this way.
Some infinite sets (sets with an infinite number of elements) are also countable. For example, the set of all natural
numbers is infinite, but also clearly countable. There are many other such countably infinite sets, including the set
of integers, and the set of rational numbers (i.e., numbers that can be expressed as the ratio of two integers). As
first pointed out by the great German mathematician Georg Cantor, many infinite sets aren’t actually countable,
meaning that, in a particular sense, it is possible for some infinite sets to be “larger” than others. The following video
from the fantastic Numberphile Youtube channel gives a nice explanation of this unintuitive yet highly important
mathematical fact: https://www.youtube.com/watch?v=elvOZm0d4H0.
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get switched to B, while µt is the total fraction of B agents who get switched to A. Thus, we have
At+1 = (1− λt)At + µtBt , (4)
Bt+1 = (1− µt)Bt + λtAt . (5)
2 A Shirking Model of Efficiency Wages
The first model of unemployment we will consider is the “shirking” model of Shapiro & Stiglitz
(1984).4 As we will see, the key departure in this model from a Walrasian environment is the fact
that buyers of labour—i.e., firms—do not have perfect information about the quality of the labour
they’re buying, while sellers of labour—i.e., the workers—have some control over the quality of the
labour they supply.
There is a continuum of mass L¯ of workers and a continuum of mass N of firms. For simplicity,
we will normalize L¯ = N = 1.5 Workers may either be employed or unemployed at a given point in
time. If the are employed, they may choose to either exert effort, in which case their labour is more
productive, or shirk, in which case it is less productive. Importantly, firms will not generally be able
to tell the difference between “exerters” and “shirkers”. Let E, S, and U denote the three possible
“types” a worker can be: employed and exerting (E), employed and shirking (S), and unemployed.
A worker has lifetime vNM utility given by
∞∑
t=0
βtut ,
where
ut =

wt − φ , if type E at date t
wt , if type S at date t
0 , if type U at date t
.
Here, wt is the wage received by an employed worker at date t. Note that, since firms can’t tell
them apart, this wage must be the same regardless of whether the worker exerts or shirks. The
parameter φ > 0 here captures the worker’s disutility from exerting effort. As a result, if it thought
it could get away with it, an employed worker would clearly rather shirk than exert.
Within any period, some workers will be hired and some workers will be fired. We assume that
all hiring takes place at the very beginning of the period (before output is produced). Workers
4Shapiro, Carl and Joseph E. Stiglitz (1984). “Equilibrium Unemployment as a Worker Discipline Device.” American
Economic Review, 74(3), 433-444. Note that Shapiro & Stiglitz formulate their model in continuous time, whereas
we formulate ours in discrete time.
5This normalization doesn’t affect any of the important elements in the model, it just gives us two less variables that
we need to carry around.
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can be fired for either of two reasons. First, all employed workers, regardless of whether they are
E or S types, are subject to being randomly and exogenously fired at the end of a period (after
output is produced). We assume that the probability of being fired in this way in any given period
is constant and equal to b. In addition, for S types, while employers can’t always tell whether
employed workers are E or S, if an employer does detect that a worker is shirking we assume that
the worker is immediately fired “for cause”, and is not paid any wages that period.6 Similar to the
random firings, we assume that the probability that a given S worker is detected shirking in a given
period is constant and equal to q. This timing structure is summarized graphically in Figure 2.
Figure 2: Timing in the Shirking Model
hiring
production
and firings
for cause
random
firings
t t+1
date
Let Et, St, and Ut denote the total masses of E, S, and U types, respectively, immediately after
hiring takes place at the beginning of date t. For E workers, note that a fraction b of the mass Et
will be randomly fired at the end of t. For S workers, a fraction q of the mass St will be caught
shirking and fired during t, and of the remaining fraction 1− q, a fraction b will be randomly fired
at the end of t. Thus, the total masses of E and S workers fired during t are given by bEt and rSt,
respectively, where we’ve defined
r ≡ q + (1− q) b = b+ (1− b) q . (6)
Note that r > b, i.e., the probability that an S worker will be fired is higher than for an E worker.
This is a key feature of this model: without it, employed workers would never under any circum-
stances choose to exert.
Next, let U˜t denote the mass of U workers at the beginning of date t before hiring occurs, and
let Ut denote the mass of U workers immediately after hiring occurs. Since U˜t includes all agents
who were still unemployed after hiring took place at the beginning of t − 1, Ut−1, plus bEt−1 E
workers and rSt−1 S workers that were fired during t− 1, we have
U˜t = Ut−1 + bEt−1 + rSt−1 . (7)
6If this seems unfair, as we’ll see below S workers won’t actually do any work in this model, so in a sense they get
paid exactly what they deserve in that period. In any case, we can easily relax this assumption without changing
anything important, though at the cost of complicating the presentation somewhat.
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Since U˜t workers begin t unemployed, while Ut of them remain unemployed after hiring occurs, the
fraction of those U˜t workers that were hired at the beginning of t is at ≡ 1 − Ut/U˜t. Solving this
expression for Ut, and then substituting (7) in for U˜t, we obtain
Ut = (1− at) (Ut−1 + bEt−1 + rSt−1) , (8)
which gives the evolution of Ut over time. Finally, let Lt ≡ Et + St = 1 − Ut be the total mass of
workers who were employed immediately after hiring occurred at date t.
2.1 Value Functions
Let VE,t, VS,t, and VU,t denote a worker’s value functions associated with being in each of the
three possible states during date t (i.e., after hiring occurs at the beginning of t). Note that,
conditional on being employed, a worker will be willing to choose to be an E type at date t if and
only if the value of being an E type is at least as large as the value of being an S type, i.e., if
VE,t ≥ VS,t. To keep things from becoming overly tedious later, we will assume that if an employed
worker is indifferent between E and S—i.e., if VE,t = VS,t—then it will always choose E.
Next, we assume that a U worker’s probability of being hired at the beginning of date t does
not depend on how long they’ve been unemployed. This probability is given simply by the fraction
of pre-existing U workers who are hired that period, i.e., at.
Consider an employed worker who chooses to be an E type at t. During t, this worker simply
receives period utility wt. To determine the continuation value received at t+1, note that this worker
will be unemployed during t+ 1 (i.e., after all hiring is finished at the beginning of t+ 1) if and only
if (i) they are fired at the end of t (which happens with probability b), and (ii) not immediately
re-hired at the beginning of t + 1 (which happens with independent probability 1 − at+1). Thus,
the probability that this worker will be unemployed during t+ 1 is
pE,t+1 ≡ b (1− at+1) .
If they do end up unemployed, they will receive continuation value VU,t+1 next period. On the other
hand, if they end up employed, they will get to choose whichever is larger of VE,t+1 and VS,t+1.
Based on our analysis from (3), the value function for E types must therefore satisfy
VE,t = wt − φ+ β [(1− pE,t+1) max {VE,t+1, VS,t+1}+ pE,t+1VU,t] . (9)
Consider next an employed worker who chooses to be an S type at t. In terms of the continuation
value they receive at t+ 1, the analysis is essentially identical to the E worker case, just with an S
type’s probability of being fired during period t (i.e., r) in place of an E type’s (i.e., b). The one
key difference is that, at period t, S workers only receive the wage payment if they aren’t detected
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shirking, which happens with probability 1− q. Thus, the value function for S types must satisfy
VS,t = (1− q)wt + β [(1− pS,t+1) max {VE,t+1, VS,t+1}+ pS,t+1VU,t] , (10)
where
pS,t+1 ≡ r (1− at+1)
is the probability that the S worker is unemployed during t+ 1.
Lastly, for a worker who is unemployed for the duration of date t, the probability that they will
remain unemployed after hiring is finished at the beginning of t + 1 is simply 1 − at+1. Thus, the
value function for a U type satisfies
VU,t = β [(1− at+1)VU,t+1 + at+1 max {VE,t+1, VS,t+1}] . (11)
Note again that, if this worker is hired at the beginning of t + 1, they will get to choose which
employed type to be, and will therefore receive the maximum of VE,t+1 and VS,t+1.
2.2 Labour Market Structure
The structure of the labour market at each date is assumed to work as follows. All worker types
and all firms meet together in the labour market. Firms bid against each other for workers, and
workers simultaneously bid against each other to get jobs. As a result, just as in a Walrasian market,
there will be a single market wage that applies to all workers that end up with jobs. Importantly,
however, we assume that if and when there is a shortage of jobs (i.e., if demand for workers at the
prevailing market wage is less than the number of workers willing to work at that wage), existing
employed workers get first priority. Thus, assuming existing E and S types are willing to accept a
job at the prevailing wage, they remain employed with certainty.7 Any gap between the quantity of
existing employed workers and the amount workers firms wish to employ is then made up by hiring
randomly from the initial pool of U workers, U˜t.
This labour market structure might seem a bit strange. Perhaps a more sensible way to model
things would be to have firms only competing with other firms for U workers, rather than also for
existing employed workers. Or perhaps, for maximum realism, we should have firms negotiating
wages one-on-one with all employees (whether existing ones, or potential new hires). The reason
we model things as above instead is that it allows us to avoid certain complications that would
considerably muddy the waters. For example, if firms negotiate one-on-one, we would need to
specify exactly how those negotiations work. This is an interesting topic, and one that we will
7It is in principle possible for the shortage of jobs to be so large that there aren’t even enough for the existing E and
S types. That is, it’s possible that, at date t, firms are willing to employ fewer workers than the number of workers
who remained employed at the end of t − 1, which would necessitate further firings by firms at the beginning of t.
We will ignore this possibility, as it turns out to be irrelevant when we focus on the steady state of the model below.
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consider in detail in our search model of unemployment in Section 3 below, but for now we will
stick with the simpler (though maybe less realistic) labour market structure described above.
2.2.1 Labour Supply
Suppose at date t the prevailing wage in the labour market is wt. Consider the problem of a
worker—who could currently be either an E, S, or U type—who has been offered a job at that wage.
The worker needs to decide whether or not to accept the job (in which case they remain/become
an E or S type), or to reject it (and remain/become a U type). Clearly, it will be optimal to accept
the job if max{VE,t, VS,t} ≥ VU,t, and to reject it if max{VE,t, VS,t} < VU,t. We make the following
claim:
Proposition 1. If wt > 0 for all t (which we henceforth assume), then VS,t > VU,t for all t as well.
Proof. Note that, from the perspective of the worker, there are two differences between being
an S type and a U type:
1. An S type receives the wage w (which is strictly positive by assumption) for as long as
they remain employed, while a U type receives nothing.
2. At any date, an existing S type can choose between remaining an S type, becoming an
E type, or becoming a U type (by rejecting any wage offers). In comparison, in most
periods a U type has no choice but to continue being a U type. The only time they do
have a choice is when their number comes up in the hiring lottery, at which point they
have exactly the same menu of options available to them as S types have had all along.
The first difference is clearly always strictly favourable to S types. Depending on whether
the U type currently has a job offer in hand, the second difference is never any worse for S
types, and often better. Putting this together, it follows that being an S type must always be
strictly preferable; that is, we must have VS,t > VU,t for all t.
A direct implication of Proposition 1 is that we must always have max{VE,t, VS,t} > VU,t, and
thus a worker who is offered a job will always be willing to accept that job regardless of the value
of the wage (as long as it’s positive, which we’ve assumed is always the case). Since all workers
are prepared to accept a job at any positive wage, the supply of workers in this model is simply
1, regardless of the wage. In a typical supply diagram, this would correspond to a vertical labour
supply curve at L = 1.
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2.2.2 Labour Demand
Firms are assumed to produce output according to production function F (H), where H is
the amount of effective labour input, and we assume that F ′ > 0, F ′′ < 0, F (0) = 0, and
limH→0 F ′(H) = ∞. From a firm’s perspective, the main difference between E and S workers
is in how much they contribute to effective labour; namely, an E worker contributes more to H than
an S worker. We can capture this idea via something like H = αEE +αSS, with αE > αS ≥ 0. We
could keep things in this relatively general form, but for simplicity let’s just set αE = 1 and αS = 0,
so that H = E: S types contribute nothing to production.
In terms of its costs, the firm must pay all of its employed workers in a period, except for the
fraction q of S workers that are detected shirking in that period. The firm’s profits at date t are
therefore given by
pit = F (Et)− wt [Et + (1− q)St] . (12)
The key takeaway from this expression is that, while un-detected S types contribute nothing to
production, they still have to be paid. Notice that this wouldn’t be the case if firms could tell E
and S types apart, since in that scenario they would be unwilling to pay S types anything. Thus,
the information friction in this model (i.e., the “friction” that prevents firms from telling E and S
workers apart) shows up quite clearly in equation (12).
The firm’s problem at date t is to maximize its profits (12) taking the wage as given. However,
the firm does not get to choose Et and St directly. Instead, it only gets to choose the total amount
of workers it employs, Lt. Letting σt ≡ Et/Lt be the fraction of a firm’s workers who exert effort,
note that Et = σtLt and St = (1− σt)Lt. Thus, we can re-write (12) as
pit = F (σtLt)− wt [σt + (1− q) (1− σt)]Lt ,
which, for a given σt, the firm wants to maximize by choice of Lt. Note that σt is something that
is known to the firm: based on existing market conditions and the wage this particular firm is
offering to pay its workers, it can deduce what fraction of its workers will choose to exert effort.8
Specifically, if VE,t < VS,t, then all employed workers will shirk, so that σt = 0. On the other hand,
if VE,t ≥ VS,t, then all employed workers will exert,9 in which case σt = 1.
If σt = 0, then clearly a firm will not want to hire any workers at all regardless of the wage, i.e.,
it will choose Lt = 0, in which case its profits would also be zero (since we have assumed F (0) = 0).
On the other hand, if σt = 1, the optimal choice of employment satisfies the usual FOC equating
8Importantly, being able to deduce the total fraction of workers that exert does not imply that the firm knows which
specific workers are exerting.
9Recall our earlier assumption that if workers are indifferent, i.e., if VE,t = VS,t, then they always exert.
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the marginal product of labour to the wage, i.e.,
wt = F ′ (Lt) . (13)
We can interpret (13) as the firm’s labour demand curve for the σt = 1 case, and since we’ve
assumed F ′′ < 0 one can verify that this demand curve is downward-sloping. Note also that, given
our assumption that limH→0 F ′(H) =∞, it follows that the optimal choice for Lt is always strictly
positive in the σt = 1 case,10 and firm profits are also strictly positive.
Before proceeding to discuss the equilibrium of this model, it’s worth noting that we have so far
used the variables Et, St, and Lt to denote both the total masses of E types, S types, and employed
workers, respectively, and also to denote the quantities of E types, S types, and total workers
hired by a single firm. This was not an accident, and indeed, these “aggregate” and “individual
firm” quantities must actually be equal to one another. For example, since there is a continuum of
identical firms, each firm will employ the same amount of E workers as all the others. Put slightly
differently, each individual firm must employ the average number of E workers, which is in turn
equal to the total mass of E workers, Et, divided by the total mass of firms, N . Since we’ve assumed
that the total mass of firms is N = 1, it follows that each individual firm employs exactly Et E
workers. This might seem a bit counter-intuitive, but it’s just a mathematical oddity arising from
the fact that we’re working with continua of firms and workers.
2.3 Equilibrium
We turn now to characterizing the equilibrium of this economy. Because the reasoning is a bit
complicated, we will proceed by stating (and then proving) a number of intermediate results (i.e,
propositions). After stating these intermediate results, we can then put them together to get a
clearer picture of what the equilibrium looks like.
Proposition 2. Holding the path for the hiring probability aτ unchanged for τ ≥ t, VE,t and VS,t
are both increasing in wt.
Proof. Note that such a change in the wage has no effect on either the likelihood of becoming
unemployed (which is controlled by the exogenous values b and r), the period utility a worker
gets if they do become unemployed (which is always zero), or on the likelihood of being re-
employed (which is controlled by the hiring rate aτ , which we are holding constant here). Thus,
10To see this, note that as Lt decreases toward zero, the right-hand side of (13) approaches∞. Thus, for Lt sufficiently
small, the marginal product of labour will necessarily exceed the wage, meaning the firm could increase profits by
increasing Lt.
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the only effect of an increase in wt is that workers receive a higher wage as long as they’re
employed, which clearly has a positive effect on both VE,t and VS,t.
Proposition 3. The increase in VE,t in Proposition 2 is larger than the increase in VS,t.
Proof. Since E types receive the current wage with certainty, while S types only receive it if
they don’t get caught shirking, an increase in the current wage exerts a larger effect on the
value of being an E type than an S type.
The basic content of Propositions 2 and 3 is easiest to see in the case where β = 0 (i.e., HHs care
only about the present period). In this case, (9) and (10) become, respectively, VE,t = wt − φ and
VS,t = (1− q)wt. Figure 3 plots VE,t and VS,t as functions of wt for this case. These functions differ
in two respects. First, for wt = 0, we have VE,t = −φ < 0 = VS,t: when wt = 0, neither type receives
any wages. For S types this means they receive zero value, while E types must bear the exertion
cost φ so they actually receive negative value. Second, ∂VE,t/∂wt = 1 > 1 − q = ∂VS,t/∂wt > 0:
as the wage increases, the value of being either type increases (Proposition 2), but the increase for
E types is greater than for S types (Proposition 3). The reason for this is that, when wt increases,
E types are certain to receive those extra wages. On the other hand, S types only receive them if
they aren’t detected shirking, which happens with probability 1− q. Thus, an increase in wt by one
unit leads to only a (1− q)-unit increase in S types’ expected wages.
Figure 3: VE,t and VS,t as Functions of wt (β = 0 Case)
wt
VE,t
VS,t
wt~
-ϕ
Note that the combination of these two properties—VE starts below VS , but increases faster
with wt—implies that there is some “break-even” value of the wage w˜t > 0 at which VE,t = VS,t
(i.e., the worker is indifferent between shirking and exerting), and beyond which VE,t > VS,t. This
value is indicated for the β = 0 case in Figure 3. As long as the wage is high enough (wt ≥ w˜t), we
see that VE,t ≥ VS,t, in which case all employed workers will choose to be E types.
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While the β > 0 case is a bit more subtle than the β = 0 one, the intuition is fundamentally the
same: All else equal, employed workers would prefer to shirk (since exerting decreases their period
utility by φ). All else is not equal, however, and in particular shirking workers are more likely to
be fired, in which case they (temporarily) lose out on wages. This serves as a deterrent to shirking.
Importantly, higher wages means the cost of being fired for shirking is greater, and therefore the
deterrence effect is stronger. If wages are sufficiently large, the deterrence effect is strong enough
to more than offset the utility benefit workers get from shirking, in which case VE,t ≥ VS,t.
Proposition 4. Let w˜t denote the break-even wage, i.e. the value of wt for which VE,t = VS,t. In
equilibrium, we must always have wt ≥ w˜t.
Proof. Suppose to the contrary that there is an equilibrium where wt < w˜t. By Propositions
2 and 3, this implies that VE,t < VS,t, and thus all employed workers choose to shirk. In that
case, σt = 0, and as noted in Section 2.2.2, firms wouldn’t hire anybody and would all earn
zero profits. But if that were the case, a single individual firm could offer to hire workers at the
wage w˜t. Since all workers would currently be unemployed, and since all workers are willing to
accept a job regardless of the wage (see Section 2.2.1), this firm could hire as many workers as it
would like at this wage. Further, by definition of w˜t, upon being employed at this wage workers
would choose to exert, and therefore the fraction of that particular firm’s workers who exert
would be σ = 1. As noted in Section 2.2.2, this firm would therefore want to hire a positive
amount of workers, and would earn strictly positive profits. Clearly this is better than earning
zero profits, so this firm would choose to do this (i.e., hire some workers at wage w˜t). But this
contradicts our initial supposition that the market was in equilibrium. We therefore conclude
that we cannot have an equilibrium where wt < w˜t.11
Note that Proposition 4 implies that the equilibrium of our model must feature VE,t ≥ VS,t, and
therefore all workers exert (i.e., σt = 1), and thus St = 0 and Et = Lt > 0.
Proposition 5. If Lt < 1 in equilibrium, then we must have wt ≤ w˜t.
Proof. Suppose to the contrary that the equilibrium features Lt < 1 but wt > w˜t. From
Propositions 2 and 3, this implies that VE,t > VS,t. Suppose an individual firm decides to offer
to hire workers at wage w˜t instead of wt. Since Lt < 1, there are some unemployed workers, and
since all such workers are willing to accept a job regardless of the wage (see Section 2.2.1), this
11This type of proof is referred to as a “proof by contradiction”. The basic logic for such proofs is as follows. We’d like
to prove that A implies B (i.e., that if A is true, then B must be true as well), where A and B are two statements.
Suppose to the contrary that A is true but B is actually false. If this leads to a logical contradiction, then we
conclude that we can’t have both A true and B false. This in turn means that if A is true then B must also be true,
which is exactly what we were trying to prove. In the above example, A would be the statement “the economy is
in equilibrium”, while B would be the statement “wt ≥ w˜t”. The proof then supposed that A was true but B was
false, and ended up concluding that A must be false, which is clearly a logical contradiction.
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firm will definitely be able to hire workers at this wage. Furthermore, by definition of w˜t, these
workers would choose to exert, and therefore that particular firm would face a σ = 1. Thus,
by offering w˜t instead of the prevailing wage wt > w˜t, the firm can still hire workers and still
have them all exert, but it doesn’t have to pay them as much, and therefore it can earn higher
profits. This is clearly beneficial to this firm, so it would choose to do so. But this contradicts
the initial supposition that the economy was in equilibrium. We therefore conclude that, in
equilibrium, we cannot have both Lt < 1 and wt > w˜t.
Note that, importantly, the logic of the proof of Proposition 5 only holds when Lt < 1. If instead
Lt = 1, then a firm wouldn’t actually be able to find any workers to hire at a wage less than the
prevailing market wage. Thus, Proposition 5 does not apply in that case, and therefore we cannot
rule out the possibility that wt > w˜t when Lt = 1.
Combining Propositions 4 (which says that wt ≥ w˜t) and 5 (which say that if Lt < 1, we have
wt ≤ w˜t), we can summarize the possible types of equilibria as follows.
• Type 1 equilibrium: Lt < 1 and wt = w˜t, in which case, by definition of w˜t, we have
VE,t = VS,t , (14)
and no workers shirk (σt = 1).
• Type 2 equilibrium: Lt = 1 and wt ≥ w˜t, in which case VE,t ≥ VS,t, and again σt = 1.
2.4 Steady State Equilibrium
It turns out that analyzing the above model in its current general formulation any more than
we already have is very difficult. Luckily, we can get a good grasp of the important elements of
the model by focusing on its steady state (SS), that is, on the case where none of the variables
are changing over time any more. Typically we interpret the SS as telling us about the long-run
behaviour of the model economy.
Since SS quantities don’t change over time, we can drop all dependence on t from all of our
notation. Let us first solve for the SS value of the break-even wage w˜ in the case where the
equilibrium features w = w˜.12 Note that this implies that VE = VS . In steady state, equations (9),
(10), and (11) then become, respectively,
VE = w˜ − φ+ β [(1− pE)VE + pEVU ] , (15)
VE = (1− q) w˜ + β [(1− pS)VE + pSVU ] , (16)
VU = β [(1− a)VU + aVE ] , (17)
12Recall that if L < 1 then we must have w = w˜, while if L = 1 we could have w = w˜.
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where we have replaced all occurrences of VS with VE , and of w with w˜, and you’ll recall that
pE = b(1 − a) and pS = r(1 − a). Since in equilibrium E = L, S = 0, and U = 1 − L, in steady
state, equation (8) becomes
1− L = (1− a) (1− L+ bL) . (18)
Equations (15)-(18) give us four equations in five endogenous variables (VE , VU , w˜, a, and L),
so we cannot solve them uniquely for all five of these variables.13 We can, however, fix one of these
variables at some arbitrary level, and then solve uniquely for the remaining four in terms of that
one fixed variable. To that end, let’s fix L, and do some work to try to solve uniquely for w˜ in terms
of L.
First, solve (17) to obtain
VE =
1− β (1− a)
aβ
VU . (19)
Substituting this into (16) to eliminate VE , and then solving for VU yields (after some algebra, and
using pS = r(1− a))
VU =
aβ (1− q)
(1− β) [1− β (1− a) (1− r)] w˜ , (20)
and therefore from (19) we have
VE =
[1− β (1− a)] (1− q)
(1− β) [1− β (1− a) (1− r)] w˜ . (21)
Next, substituting (20) and (21) into (15) to eliminate VU and VE , after some tedious algebra, and
using pE = b(1 − a), and (6) to eliminate r, we can solve for w˜ as a function of a and exogenous
variables only as
w˜ =
{
1 + 1− q
q
[1− β (1− b) (1− a)]
}
φ . (22)
Next, we can solve (18) for a to get
a = bL1− (1− b)L . (23)
Substituting this into (22) and then simplifying and re-arranging gives us w˜ as a function of L, i.e.,
w˜ =
{
1 + 1− q
q
[
1− β + βb1− (1− b)L
]}
φ , (24)
which is precisely what we were after.
Note that, in the limit as L→ 1, (24) implies w˜ → φ/q. While (24) was obtained supposing we
were in the L < 1 case, it’s nonetheless straightforward to check that if L = 1 then the break-even
wage is w˜ = φ/q.14 Thus, we can characterize a Type-2 SS equilibrium simply as a case where
L = 1 and w ≥ φ/q.
13To do that, we’ll need a fifth equation, which we’ll identify later.
14This would be a good practice exercise to try on your own.
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Summarizing, then, we conclude that a Type-1 SS equilibrium features L < 1 and w = w˜, where
w˜ is given in (24), while in a Type-2 equilibrium we have L = 1 and w ∈ [φ/q,∞). We can combine
this information to write down a relationship between the wage w and employment L that must
hold in any equilibrium:w =
{
1 + 1−qq
[
1− β + βb1−(1−b)L
]}
φ , if L < 1
w ∈ [φ/q,∞) , if L = 1
. (25)
Relationship (25) is a key condition for this model. Since it implicitly gives us a condition relating
w and L that is required to ensure that VE ≥ VS and therefore no workers shirk, we will call (25)
the “no-shirking condition” (NSC).
As noted above, in order to actually solve uniquely for the SS equilibrium of this model, we need
one more equation. The one key equation we haven’t used yet is the SS version of the firm’s FOC
(13), i.e.,
w = F ′ (L) , (26)
which implicitly gives the firm’s labour demand (LD). The NSC and LD conditions give us two
relationships between the two variables w and L that must both hold in equilibrium, and therefore
together they will uniquely pin down the SS equilibrium.
2.4.1 Benchmark: Commitment
Before analyzing the equilibrium of this model, as a benchmark let us briefly consider what
would happen in a hypothetical version of this model where workers could credibly promise—or
“commit”—to exert if hired. Clearly, in this case firms would only hire workers who made such a
commitment, and therefore all employed workers would exert effort.
Recall that we argued in Section 2.2.1 that all workers are willing to accept a job at any positive
wage. The basis of that argument was that an employed worker could never do worse than being an
S type, and being an S type means receiving a positive wage, which is always better than being a U
type and receiving nothing. When employed workers are forced to exert, however, this argument no
longer necessarily follows. In particular, employed workers must now necessarily be E types, which
requires them to incur an exertion cost φ. If w < φ, then workers would not find it beneficial to be
employed, and would therefore choose not to work, so that labour supply would be L = 0. On the
other hand, if w > φ, then arguments similar to those made in Section 2.2.1 imply that all workers
would choose to be employed if given the option, so that labour supply would be L = 1. Finally, if
w = φ, workers are exactly indifferent between working and being unemployed, so that workers are
willing to supply any amount of labour in the interval [0, 1]. We may refer to this labour supply
curve as the “no-shirking labour supply” (NSLS) curve, since it is the labour supply curve for the
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case where no employed workers shirk. The NSLS is shown in (L,w)-space (i.e., on a graph with
L on the horizontal axis and w on the vertical) in panel (a) of Figure 4. The curve is vertical at
L = 0 for w < φ, horizontal at w = φ for L ∈ [0, 1], and vertical at L = 1 for w > φ.
Figure 4: Steady State Equilibrium of Shirking Model
(a) Commitment Possible (b) Commitment Not Possible
L
ϕ
w
1
LD1w
NSLS
LD2
L2* L
ϕ
w
1
LD
NSC
w
w*
L*
ϕ/q
The firm’s labour demand is unchanged relative to the case where there is no commitment, and
is thus still given by (26). Since F ′′ < 0, this curve is downward-sloping in (L,w)-space. The
intersection of this LD curve with the NSLS curve then gives us the equilibrium values of L and w
for this “commitment” case.
Panel (a) of Figure 4 illustrates the two possible configurations for such an equilibrium. In the
first case, LD is given by the curve labeled “LD1”, which intersects the NSLS curve on the part of
it that’s vertical at L = 1. In this case, the equilibrium features full employment (L = 1), with a
wage given by w¯ ≡ F ′(1) > φ, i.e., w¯ is equal to the marginal product of labour when L = 1. In this
case, firm productivity is sufficiently high that, even at full employment, the marginal product of
labour exceeds the exertion cost φ, and thus firms are willing to pay workers more than is necessary
for them to be willing to exert.
In the second configuration, LD is given by the curve labeled “LD2”, which intersects the NSLS
curve on its horizontal part. In this case, F ′(1) < φ, and therefore in order for firms to be willing to
hire all the workers (i.e., to have L = 1), the wage would have to be below the exertion cost φ, which
can’t happen in equilibrium. As a result, the equilibrium wage is simply given by φ (i.e., workers
are just indifferent between exerting and not working), and employment by the value L = L∗2 that
solves (26) with w = φ.
While either of these cases is possible in principle,15 for concreteness let us assume henceforth
15Recall that, since we have assumed limL→0 F ′(L) = ∞, it is not possible to have an equilibrium with L = 0. This
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that, if workers could commit to exerting, the equilibrium would feature full employment. That is,
let us assume henceforth that w¯ > φ. Rest assured, however, that none of the important conclusions
we’ll draw from the shirking model will rely on this assumption.
2.4.2 Allowing Shirking
Returning to the case where workers can’t commit to exerting, note from (24) that
dw˜
dL
= 1− q
q
βb (1− b)
[1− (1− b)L]2φ > 0 ,
d2w˜
dL2
= 21− q
q
βb (1− b)2
[1− (1− b)L]3φ > 0 ,
w˜
∣∣
L=0 =
{
1 + 1− q
q
[1− β (1− b)]
}
φ > φ , w˜
∣∣
L=1 =
φ
q
.
Thus, if we draw the NSC (25) in (L,w)-space, for the part of it where L < 1 it will be upward-
sloping, convex, with w greater than φ at L = 0, and w reaching a maximum of φ/q at L = 1. For
the part where L = 1, meanwhile, any wage w ≥ φ/q is consistent with no-shirking, and therefore
the NSC will be vertical over the range of wages w ≥ φ/q. This curve is illustrated as “NSC” in
panel (b) of Figure 4.
The labour demand curve LD, meanwhile, has the same shape as in the benchmark case. Since
both the NSC and the LD conditions must be satisfied in equilibrium, the steady state equilibrium
values of L and w for the shirking model are given simply by the intersection of these two curves.
The exact nature of this equilibrium depends on the precise parameters of the model. In par-
ticular, our earlier assumption that w¯ > φ ensures that the LD curve intersects L = 1 at a wage
above φ. Depending on the value of q, however, this intersection could occur at a wage either above
or below φ/q; that is, w¯ could be greater than or less than φ/q. In panel (b) of Figure 4, we have
drawn LD for the case where q < φ/w¯, so that φ < w¯ < φ/q. The equilibrium is then at (L∗, w∗).
By arguments identical to those made in the benchmark case of Section 2.4.1, the amount of
labour that would be supplied in this model if all employed workers chose to exert is given by
L = 0 when w < φ, L = 1 when w > φ, and any value in [0, 1] when w = φ; that is, the
“true” labour supply is given by the NSLS curve from panel (a). Thus, the market-clearing level of
employment (i.e., the level of employment where LD would intersect the NSLS) is clearly L = 1.
Furthermore, the associated market-clearing wage is given by F ′(1) = w¯ > φ. We therefore see that
the market-clearing (i.e., Walrasian) outcome for this model is precisely the equilibrium outcome
for our benchmark “commitment” case where shirking is not allowed: (L,w) = (1, w¯).
The most important thing to notice about the actual equilibrium shown in panel (b) of Figure
4—which occurs where LD intersects the NSC, rather than the NSLS—is that w∗ > w¯ and L∗ < 1.
That is, as long as w¯ isn’t too large (i.e., w¯ < φ/q), the equilibrium wage in the shirking model is
rules out the possibility that the LD curve intersects the remaining part of the NSLS curve, i.e., the part that’s
vertical at L = 0.
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above its Walrasian level, while employment is below its Walrasian level; that is, the equilibrium
features unemployment.
The intuition for this result is as follows. The chance of getting caught and punished by being
sent to the U state is ultimately what deters workers from shirking. The strength of this punish-
ment is affected by two endogenously determined factors: (i) the expected length of time spent
unemployed, and (ii) the amount of wages lost per period while unemployed. Since, in any equi-
librium with L < 1 workers must be just indifferent between shirking and not shirking (VE = VS),
if the strength of one of these factors decreases, then the strength of the other must increase to
compensate. As we now show, an increase in L decreases the expected length of the unemployment
spell, so that w must increase in order to compensate.
To see this, recall that, since there is no shirking in equilibrium, there is a mass bL of existing
workers that gets fired each period. To keep L constant (as it must be in steady state), these workers
must immediately be replaced at the beginning of the next period from the pool of unemployed
workers at that point, of which there is a mass U + bL = 1 − (1− b)L (i.e., the U unemployed at
the beginning of the period, plus the additional bL E workers who were randomly fired during the
period). Thus, we need the hiring probability a to satisfy bL = a[1 − (1− b)L]: bL workers were
fired during the period, a[1− (1− b)L] were hired, and in SS these must be equal. For a larger SS
value of L, the mass of fired workers bL is larger, while the mass of unemployed workers to be hired
from the next period, 1 − (1− b)L, is smaller. Thus, when L is higher, more fired workers need
to be replaced each period, but from a smaller pool of unemployed workers. This implies that the
hiring probability a has to be much higher. This in turn implies that when L is higher, a U worker
expects to be hired more quickly, i.e., to spend less time unemployed. As noted above, this requires
a compensating increase in w in order to keep VS from rising above VE . This explains why, along
the NSC, w goes up when L goes up.
Recall that, for L = 1, the minimum wage required to ensure no-shirking is w˜ = φ/q. Assuming
that w¯ < φ/q (as we did in panel (b) of Figure 4), w˜ will exceed firms’ marginal product of labour
when L = 1, and therefore firms would not be willing to employ all workers at a wage that would
prevent shirking. Thus, we must have L < 1 in equilibrium, which in turn implies that w > w¯.
This equilibrium wage is an example of what is sometimes referred to as an efficiency wage, i.e., a
wage that is set above the market-clearing level in order to improve worker productivity. We may
then refer to w − w¯ as the wage premium.
Before proceeding, it is worth briefly highlighting the alternative case where w¯ > φ/q. In that
case, the LD curve would intersect L = 1 at a wage above φ/q, and thus the equilibrium will feature
L = 1 and w = w¯, i.e., we’ll have the market-clearing (Walrasian) equilibrium. The intuition for this
result is as follows. Note that, even if L = 1, in which case a = 1 so that workers who get fired in the
current period are definitely re-hired next period, workers still face a potential cost from shirking,
which stems from the fact that if they’re caught they get fired immediately and don’t receive the
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current period’s wage. If the current wage is high enough, then even if the worker knows they’ll
be re-hired for sure next period, the possibility of losing today’s wages is enough to disincentivize
them from shirking. Mathematically, φ/q is precisely the minimum wage that’s enough to ensure
no-shirking even when L = a = 1. As a result, any w ≥ φ/q is consistent with no-shirking when
L = 1 (which is precisely why the NSC is vertical at L = 1 for w ≥ φ/q). When w¯ > φ/q, since LD
is downward-sloping we see that the equilibrium w must be at least φ/q, and therefore we have a
no-shirking equilibrium with L = 1 and w = w¯.
2.4.3 Comparative Statics
Suppose the detection probability q increases from q1 to q2. From (25), we see that this shifts
down the NSC, with no effect on LD. This is illustrated in Figure 5 for the case where w¯ < φ/q2.
The NSC shifts down from the solid to the dashed curve. As can be seen in the diagram, the
equilibrium w falls from w∗1 to w∗2, and the equilibrium L rises from L∗1 to L∗2. Intuitively, since
shirking is now more likely to be detected, the deterrent to shirking associated with any given wage
w is stronger. As a result, firms don’t have to pay as high a wage premium to prevent shirking, and
therefore w will fall. This in turn makes firms willing to hire more workers, causing L to increase.16
Figure 5: Shirking Model: An Increase in q or Fall in b
L
w
1
LD
NSC
w*
L*
ϕ/q1
1
1
ϕ/q2
L*2
w*2
Note that, given our maintained assumption that w¯ > φ, for q close enough to one we’ll have
w¯ > φ/q, i.e., the equilibrium will be as in the case discussed above where L = 1 and w = w¯. In
other words, as long as shirking is detected with a high enough probability, the wage premium in
this model disappears, and we’re back to the Walrasian outcome.
Next, suppose instead that the random firing probability b decreases. Qualitatively, this has a
similar effect on the equilibrium as a rise in q: it shifts the NSC down, so that, assuming initially
16Note that, by reducing the expected length of time workers are unemployed, the resulting increase in L partially
mitigates the degree to which w falls in response to the increase in q. Thus, in the end, the wage falls by less than
would be required if L remained at L∗1.
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L < 1, w will fall and L will rise (just as shown in Figure 5). The mechanism for this change is a
bit more subtle than for the change in q, however. There are two distinct effects at play here: a
direct one, and an indirect one.
• Direct effect: The slower rate of random firing tends to increase both VE and VS , but the
effect on VE is larger. Intuitively, the fall in b means that both E and S workers get to be
employed (and therefore earn wages) for longer on average, which raises the value of being in
either of those states. However, while an E worker only ever gets fired for random reasons, S
workers can also get fired for shirking. An S worker will only be randomly fired if they aren’t
detected shirking first, and therefore a change in b is less likely to be relevant to an S worker
than it is to an E worker. The upshot of this is that VE will increase by more than VS does
in response to a fall in b. This in turn implies that VE − VS increases, making shirking less
appealing.
• Indirect effect: The lower firing rate b implies that, in order to maintain a given L, firms don’t
need to hire as many new workers each instant, and therefore the hiring rate a falls (you can
verify this by differentiating (23) with respect to b and noting that ∂a/∂b > 0). As a result, if
a worker is fired for shirking, they expect to remain unemployed for longer, thereby reducing
the incentive to shirk.
Since both of these effects work to increase the disincentive to shirking (for any given combination
of w and L), firms don’t need to pay as high of a wage premium for any given L, and therefore the
NSC shifts down. As a result, in equilibrium w falls and L rises.
3 A Search Model
In the shirking model of Section 2, the inability to perfectly monitor shirking behaviour meant
that firms were only willing to hire workers if there was a positive level of unemployment, since this
was the only way workers would actually care about the possibility of being fired if caught shirking.
As a result, in equilibrium, firms were unwilling to hire all of the workers that were willing to work
at the prevailing wage. In the current section, we consider an alternative modeling environment
where unemployment arises in equilibrium even though firms would be happy to hire more workers
at the prevailing market wage if they could. As we’ll see shortly, something (a “friction”) prevents
such hires from being made.
As in the shirking model, there is a mass one of workers. Unlike the shirking model, however,
we will now imagine there is an unlimited (i.e., infinite) mass of potential firms. As we’ll see, the
actual mass of active firms will be endogenous (and finite in equilibrium).
There are two possible states a worker can be in: employed (E) or unemployed (U). The masses
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of such workers at date t are given by Et and Ut, respectively. Workers get lifetime utility
∞∑
t=0
βtut ,
where
ut =
wt , if type Eb , if type U
is period utility at t. Here, wt is the wage, and b > 0 is an exogenous amount that we can interpret
as an unemployment insurance benefit (or as value derived from leisure).
Firms, meanwhile, can be in one of three states. In the first state, which we’ll call dormant
(D), firms do nothing and neither pay nor receive anything. Alternatively, a firm can be in one of
two active states. In order to be active, a firm must pay the exogenous operating cost c > 0 per
period. Each active firm has a single “job” available, which may either be filled by a worker (in
which case the firm is type F), or vacant (in which case the firm is type V). Let Vt denote the mass
of V firms, and note that, since there is exactly one F firm for each E worker, and vice versa, the
mass of F firms must always equal the mass of E workers (i.e., Et). The discounted present value
of firm profits is given by
∞∑
t=0
βtpit ,
where
pit =

y − wt − c , if type F
−c , if type V
0 , if type D
is profits at date t. Here, y is the (exogenous) level of output produced by a filled job. We assume
that y − c > b, which ensures that, after deducting the firm’s operating cost c, there is enough
output left over such that it’s possible to pay workers a wage that exceeds their unemployment
benefit b. If this weren’t the case, there could never be any circumstances under which a worker
would accept a job.
Similar to the shirking model, we will assume that existing employed workers are randomly fired
from their jobs at the end of each period with probability λ ∈ (0, 1]. In the more conventional
terminology of search models, we may alternatively say that existing E/F worker-firm pairs are
exogenously separated with probability λ. Regardless of which terminology we use, as a result of
this separation a mass λEt−1 of workers who were employed during t− 1 will be unemployed at the
beginning of t. By the same token, a mass λEt−1 of firms who had filled jobs during t − 1 will be
either V or D types at the beginning of t.
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3.1 Search and Matching
In the shirking model of Section 2 and in the RBC model of LN3, the labour market essentially
had the feature where all workers and firms meet together in a single central location and then bid
against each other for jobs/employees. In the present model, on the other hand, there is no such
place. Instead, individual U workers and V firms need to search to find each other, and depending
on a number of factors, they may not actually find a counterpart. Further, if and when they do find
a counterpart, there aren’t any other workers or firms around to bid against. Instead, negotiations
over the wage take place one-on-one between that firm and that worker.
The process by which U and V types find each other is assumed to be independent of how long
they have been U or V types (e.g., a worker who has been unemployed for many periods is not any
more or less likely to be hired than a worker who has only been unemployed for one period). The
mechanism by which this occurs is often referred to as random matching, since we may imagine that,
at the beginning of the period, some “matching technology” randomly picks out an equal number
of U and V types and puts them together into a collection of “matched” worker-firm pairs. We
assume that the total mass of new matches µt at time t depends on the existing masses of U and V
workers according to
µt = M (Ut, Vt) ,
where the matching function M is assumed to be strictly increasing in both arguments and strictly
concave. We also require that M(Ut, Vt) ≤ min {Ut, Vt}: there can’t be more matches than there
are U workers or V firms.
As is typically done, for convenience we will also assume that the matching function exhibits
constant returns to scale (CRS).17 Recall that this implies
aM (U, V ) = M (aU, aV ) , for a ≥ 0 .
Setting a = 1/U in this expression, we may obtain that
M (U, V )
U
= M (1, θ) ≡ m (θ) ,
where θ ≡ V/U . The ratio θ is often referred to as tightness, since a larger θ implies that there are
more searching V firms relative to searching U workers (i.e., the labour market is “tighter”).
Next, the probability that a U worker finds a job is equal to the mass of new matches per U
worker, i.e., M(U, V )/U . From above, we see that this job-finding rate is simply m(θ). Meanwhile,
the probability that a V firm finds a worker—the vacancy-filling rate—is equal to the mass of of
17Empirical explorations suggest that the CRS assumption is reasonable. Nonetheless, we can in principle relax this
assumption, though at the cost of making the analysis more complicated.
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new matches per V firm, i.e.,
M (U, V )
V
= M (U, V ) /U
V/U
= M (1, θ)
θ
= m (θ)
θ
≡ q (θ) ,
where we have again used the CRS property. Note that
m′ (θ) = MV (1, θ) > 0 .
Also, since we can alternatively write q(θ) = M(θ−1, 1),18
q′ (θ) = − 1
θ2
MU
(
θ−1, 1
)
< 0 .
Thus, the job-finding rate is increasing in θ, while the vacancy-filling rate is decreasing in θ. This
should make sense: if there are more V firms searching for a given pool of U workers, then it should
be easier for U workers and harder for V firms to find matches.
Timing-wise, we assume that workers/firms that are not matched at the beginning of t (i.e., U
workers, and V/D firms) can search during t, but if they are matched during t, that match won’t
become productive until t + 1. This implies, for example, that a worker who is a U type at the
beginning of t can’t become an E type until at least t + 1, with a similar property holding for V
and D firms
3.2 Value Functions and Wage Negotiation
3.2.1 Match Surplus
At the beginning of every date t, each matched worker-firm pair (whether it’s an existing E/F
pair, or a new pair that was first matched at t − 1) must either come to an agreement on a wage,
in which case they produce y, or else they separate and become U/V types.
Consider a given matched worker-firm pair. Let’s call this pair j. Since all workers and firms are
identical, and since, if wage negotiations break down, this particular worker and firm have effectively
a zero probability of ever running into each other again, the consequences of failing to agree to a
wage are in no way specific to this particular pair. Thus, if this worker and firm fail to agree to a
wage, they simply receive the value functions for a U worker and a V firm, respectively. Let VU,t
and VV,t denote these value functions.
In contrast, if worker-firm pair j does agree on a wage—say, wjt—the precise level of this wage
will generally impact the values received by this worker and firm. The total value received by the
worker and firm together, however, is independent of this wage. To see this, let ujt and pi
j
t denote
the flow utility and flow profits for worker-firm pair j if they come to an agreement, and V jE,t and
18i.e., q(θ) =M(1, θ)/θ =M(θ−1, 1), where the second equality uses the CRS property with a = 1/θ.
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V jF,t the corresponding value functions. We have
ujt + pi
j
t = w
j
t + y − wjt − c = y − c ,
which clearly doesn’t depend on wjt . The basic intuition here is that increasing the wage simply
re-allocates resources one-for-one from the firm to the worker, without changing the total amount
of resources shared between them. Now, letting T ≥ t denote the date at which worker-firm pair j
ends up being exogenously separated. Since the date at which this will happen is not known with
certainty, T is a random variable. We then have
V jE,t + V
j
F,t = E
[
T∑
s=t
βs−t
(
ujs + pijs
)
+ βT−t (VU,T + VV,T )
]
= E
[
T∑
s=t
βs−t (y − c) + βT−t (VU,T + VV,T )
]
≡ VM,t ,
where the expectations are taken with respect to the random variable T . Since j doesn’t appear
anywhere on the last line, we see that the total value V jE,t+V
j
F,t in match j doesn’t actually depend
on j in any way (and in particular, doesn’t depend on wjt ).
Next, worker j’s “surplus” σjE,t from coming to an agreement is the difference between the value
of agreeing and the value of not agreeing, i.e., σjE,t = V
j
E,t − VU,t. Similarly, firm j’s surplus σjF,t
from agreeing on a wage is σjF,t = V
j
F,t − VV,t. The total surplus of the worker and firm is then
σjE,t + σ
j
F,t = VM,t − VU,t − VV,t ≡ σM,t .
Thus, similar to the case for the values received by the worker and firm in match j if they agree, the
individual surpluses σjE,t and σ
j
F,t will depend on the wage w
j
t , but the total surplus σ
j
E,t+σ
j
F,t = σM,t
does not.
It is straightforward to reason that, assuming employment is strictly positive in equilibrium (i.e.,
Et > 0), the total match surplus σM,t must always be strictly positive. If this weren’t the case, then
it would be impossible for workers and firms to both be better off agreeing than not agreeing, in
which case all negotiations would break down, and there would be zero employment (contradicting
the initial supposition that Et > 0). We can therefore view negotiations over the wage in match
j as the process of choosing how to split a given (positive) total surplus σM,t between the worker
and the firm, where an increase in wjt means more surplus is allocated to the worker and less to the
firm. Since the total surplus is positive, there is clearly always a way to split it so that σjE,t and
σjF,t are both positive, and, since they don’t get any surplus at all if negotiations break down, the
worker and firm will always have an incentive to come to some such agreement.
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3.2.2 Bargaining Protocol
It should be clear that there are many (an infinite number of) ways to split the total surplus σM,t
in a mutually beneficial way. To this point, we haven’t specified any mechanism that will determine
which of these possible splits will actually occur. Such a process is referred to as a wage-bargaining
protocol. There are many different protocols that people have put into search models of this type,
and in principle we could use any of them (or even come up with our own). To make our lives easy,
however, we will stick with the most widely used protocol: Nash bargaining. According to the Nash
bargaining protocol, the worker is assumed to always receive an exogenous fraction φ ∈ (0, 1) of the
total surplus, with the firm receiving the complementary fraction 1− φ. That is, we require
σjE,t = φσM,t , (27)
σjF,t = (1− φ)σM,t .
Since the right-hand sides of these expressions don’t depend on j, the left-hand sides can’t either,
so that we can write σjE,t = σE,t and σ
j
F,t = σF,t. From the definitions of the surpluses and the fact
that VU and VV don’t depend on j, V jE,t and V
j
F,t cannot depend on j either, so that we can write
V jE,t = VE,t and V
j
F,t = VF,t. Thus, substituting the definitions of the σ’s and VM,t = VE,t +VF,t into
(27), and dropping all dependence on j, we may obtain
VE,t − VU,t = φ (VE,t − VU,t + VF,t − VV,t) ,
which can be simplified to
VE,t − VU,t = φ1− φ (VF,t − VV,t) . (28)
Lastly, note that, since all worker-firm pairs are identical, they will all end up choosing the same
wage, so that we can also write wjt = wt, and since the worker and firm surpluses are both strictly
positive (since they each get a positive fraction of a positive surplus), all matched worker-firm pairs
will agree in equilibrium.
3.3 Equilibrium
Since all matched worker-firm pairs agree, the mass Et of E workers/F firms at t will be given
by the level Et−1 from t − 1, less the mass of exogenous separations λEt−1, plus the mass µt−1 of
newly created matches in t− 1; that is,
Et = (1− λ)Et−1 + µt−1 . (29)
Next, note that firms can freely switch between being D types and V types whenever they like.
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Letting VD,t denote the value function for a D firm, this implies that if VD,t > VV,t then all V firms
would switch to being D firms, while if VD,t < VV,t then all D firms would switch to being V firms.
Thus, assuming that both D and V types exist simultaneously in equilibrium (equivalently, that
0 < Vt <∞, where Vt is the mass of vacancies), then we must have VV,t = VD,t.
It turns out that we do not currently have a way to actually pin down the path of VD,t in this
model. However, if we impose a “no bubbles” condition similar to the one encountered in the asset
pricing section of LN2, i.e.,
lim
t→∞β
tVD,t = 0 ,
then the only possible path for VD,t is simply VD,t = 0 for all t,19 which in turn implies VV,t = 0 for
all t as well.
Next, using the principles we used to derive (3), we have20
VV,t = −c+ β {[1− q (θt)]VV,t+1 + q (θt)VF,t+1} .
Here, V types get profits −c at t. If the firm is matched at t, which happens with probability q(θt),
then at t + 1 it will receive value VF,t+1, while with complementary probability 1 − q(θt) the firm
remains a V type at t+ 1 and receives VV,t+1. Since VV,t = VV,t+1 = 0, this in turn implies that
VF,t+1 =
c
βq (θt)
. (30)
Since it’s ultimately derived from the fact that D firms are free to “enter” and become V firms, (30)
is often referred to as the free-entry condition for this model. Note that V types incur a loss of c
each period they remain a V type. The only reason a firm would be willing to incur this loss (rather
than just remaining dormant) is because of the prospect of eventually being matched with a worker
and becoming an F type. Intuitively, given current tightness θ, condition (30) tells us what the
prospective value of becoming an F type has to be in order for firms to be just indifferent between
being a D type and a V type: if VF were any higher, no firms would choose to be D types, while if
VF were any lower, no firms would choose to be V types. Note that the required level of VF is:
• Increasing in c: if the flow loss of being a V type is higher, the eventual payoff when it becomes
an F type must also be higher in order for firms to still be willing to be V types.
• Increasing in θ: when θ is higher, the vacancy-filling rate q is lower (since q′ < 0), which
means firms expect to have to wait longer—and therefore incur more flow losses—before
19For completeness, this is shown formally in Appendix A. While I won’t ask you to prove this on an exam, I
nonetheless encourage you to have a read-through of the proof, as it’s not very long or difficult, and may help
improve your intuition for other results.
20Technically, we need to account here for the fact that V firms could switch back to being D firms. However, since
VV,t = VD,t in equilibrium, if a switch from V to D happens at some point, the value the firm receives is exactly
equal to what it would’ve gotten if it had instead just remained a V type. So for the purposes of finding VV,t, we
can just pretend switches from V to D don’t happen.
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being matched. Thus, again, in order to continue being willing to be V types, the eventual
payoff when it becomes an F type must be higher in order to compensate.
Next, the analogous expressions for the remaining value functions are
VE,t = wt + β [(1− λ)VE,t+1 + λVU,t+1] , (31)
VU,t = b+ β {[1−m (θt)]VU,t+1 +m (θt)VE,t+1} , (32)
VF,t = y − wt − c+ β [(1− λ)VF,t+1 + λVV,t+1] . (33)
Here, we’ve used the fact that E types get period utility wt at t, switch to being a U type at t+ 1
with probability λ, and remain an E type at t+ 1 with probability 1−λ. U types get period utility
b at t, get switched to an E type at t + 1 with the job-finding probability m(θt), and remain a U
type at t+ 1 with probability 1−m(θt). Finally, F types get period profits y−wt− c at t, switched
to being a V type at t+ 1 with probability λ, and remain an F type at t+ 1 with probability 1− λ.
3.4 Steady State
As was the case in the shirking model, getting a good understanding of the important elements
of this search model is easiest if we focus on the steady state equilibrium where all variables are
constant. In particular, equations (28), (30), (31), (32), and (33) become, respectively,
VE − VU = φ1− φVF , (34)
VF =
c
βq (θ) , (35)
VE = w + β [(1− λ)VE + λVU ] , (36)
VU = b+ β {[1− θq (θ)]VU + θq (θ)VE} , (37)
VF = y − w − c+ β (1− λ)VF , (38)
where we’ve used the facts that VV = 0 and m(θ) = θq(θ). This is five equations in five endogenous
variables VE , VU , VF , w, and θ, which can in principle be solved to obtain the unique steady state
equilibrium.
To that end, solving (38) for VF yields
VF =
y − w − c
1− β (1− λ) , (39)
which gives the value of being an F firm as a function of the wage w. Recall that the free-entry
condition (35) tells us what VF has to be in order for firms to be indifferent between trying to
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hire workers (i.e., being V types) and not trying (i.e., being D types). By substituting VF out of
(35) using equation (39) and re-arranging, we can express the free-entry condition in terms of the
required wage w as
w = y − c− [1− β (1− λ)] c
βq (θ) . (40)
Given θ, this tells us what w has to be in order for firms to be just willing to try to hire workers.
As such, (40) is often referred to as the job-creation condition (JCC). Note that, since q′ < 0 and
β(1 − λ) < 1, if θ increases then the JCC wage decreases: a rise in tightness means V firms have
to incur the loss c longer on average before they get matched, which they’ll only be willing to do if
the payoff once they are matched increases, and this in turn requires the wage to fall.
Next, subtracting equation (37) from (36), we can solve for the worker’s surplus VE − VU in
terms of w and θ as
VE − VU = w − b1− β (1− λ) + βθq (θ) . (41)
Substituting this into the left-hand side of the Nash bargaining equation (34), then using (35) to
replace q(θ) with c/(βVF ), and then rearranging, we get
w − b = φ1− φ {[1− β (1− λ)]VF + cθ} .
Finally, using (39) to replace VF , we can solve to get
w = b+ φ (y − c− b+ cθ) . (42)
Given θ, (42) essentially tells us what the Nash bargaining wage must be, and for that reason it is
often referred to simply as the wage equation (WE). Note that, given our earlier assumption that
y − c > b, (42) necessarily implies that w > b, i.e., that E workers earn more than the UI benefit
received by U workers, as we might have expected. Note also that the Nash wage w is increasing in
θ:
• When θ is higher, so is the job-finding rate m(θ), so that U workers get hired more quickly.
• This tends to increase both VU and VE , since all workers, regardless of whether they’re cur-
rently employed, expect to spend less of their lifetimes unemployed. U types are affected
more directly, however, since they’re the ones that are actually currently looking for a job,
and therefore VU increases by more than VE . As a result, the increase in θ causes the worker
surplus VE − VU to fall (we can actually see this directly in (41) by noting that θq(θ) = m(θ)
is increasing in θ).
• The firm surplus, meanwhile, hasn’t changed: VV always equals zero, while VF is given by
(39), which doesn’t depend on θ. Thus, the total surplus falls by the same amount as the
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worker’s surplus.
• With Nash bargaining, this can’t happen: the worker’s surplus should only fall by a fraction
φ < 1 of the fall in the total surplus, whereas so far it’s fallen by the full amount. By extension,
this implies that the worker is now receiving less than the fraction φ of the total surplus, and
the firm is receiving more than fraction 1 − φ. In order to bring these individual surpluses
back to their required shares of the total surplus, the wage must adjust in order to transfer
some of the firm’s surplus over to the worker, i.e., the wage must rise.
The JCC (40) and the WE (42) give us two equations in two endogenous variables, θ and w.
The unique solution (θ∗, w∗) to this system gives us the equilibrium values. We can visualize this
equilibrium by drawing the JCC and WE in (θ, w)-space. Our discussion above implies that the
JCC is downward-sloping and generally non-linear, while the WE is upward-sloping and linear, with
vertical intercept w0 ≡ b+ φ(y − c− b). Figure 6 illustrates the determination of this equilibrium.
Figure 6: Search Model Equilibrium
θ
w0
w
θ*
JCC
WE
w*
Given a solution for θ and w, we can consider how two other important variables in the model
are determined: the unemployment rate U , and the number of vacancies (i.e., the mass of V firms)
V . Since E = 1− U , and µ = m(θ)U , equation (29) can be solved for U to get
U = λ
λ+m (θ) . (43)
Thus, as long as the separation rate λ > 0, there will be unemployment in this model. Essentially,
because workers are constantly being fired, and it takes time for them to find a new job, there will
always be some workers who are unemployed. Further, since m′ > 0, we see from (43) that, for a
given λ, if equilibrium tightness θ is higher, then the unemployment rate will be lower: U workers
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are finding jobs faster, and therefore the pool of such workers is smaller.
Next, since θ = V/U , the mass of vacancies is given by V = θU . An increase in θ has two
opposing effects on V . First, for a given U , higher tightness directly requires that there be more
vacancies. Second, as noted above, as long as λ hasn’t changed, an increase in θ tends to lower U .
To see which effect is larger, using (43) we have that
V = θ λ
λ+m (θ) =
[1
θ
+ m (θ)
λθ
]−1
=
[1
θ
+ q (θ)
λ
]−1
, (44)
where the last equality uses the fact that m(θ) = θq(θ). Since q′ < 0, both terms inside the square
brackets are decreasing in θ, and therefore V must be increasing in θ, i.e., the first effect described
above is larger.
In combination, (43) and (44) imply that, for a given λ, U and V can only change if θ changes,
and further that U and V must move in opposite directions in response to such a change. Thus,
if we believe that the separation rate λ is more or less constant over time, the model predicts that
the relationship between the unemployment rate and the number of vacancies—which, for reasons
that are not entirely clear to me, is referred to as the Beveridge curve after the English economist,
politician, and prominent eugenicist William Beveridge—should be downward-sloping, a prediction
that seems to be (mostly) borne out by the data.
3.4.1 Comparative Statics
Comparative statics in the steady state of the search model are fairly straightforward. As such,
we’ll just work through one here, leaving additional ones for exercises/problem set questions. In
particular, consider what happens if an increase in productivity causes the amount of output y
produced in a match to increase by an amount ∆y. From (40), we see that, for any given θ,
the JCC wage is higher by ∆y; that is, the JCC curve shifts up by an amount ∆y. From (42),
meanwhile, we see that the WE shifts up by the amount φ∆y < ∆y. These shifts are illustrated in
Figure 7, where the JCC has shifted upward from JCC1 to JCC2, the WE has shifted up from WE1
to WE2, and the magnitude of the shift in the JCC is larger than that for the WE. As a result,
we see clearly that the new equilibrium features both a higher wage and a higher level of tightness,
which in turn implies that the associated equilibrium unemployment rate U must be lower (and V
higher).
Intuitively the increase in y has the initial direct effect of increasing the firm’s surplus, with no
effect on the worker’s surplus. Thus, the total surplus and firm’s surplus initially rise by the same
amount. Because of the Nash bargaining, however, this can’t happen: a fraction φ of the increase
in the total surplus must go to the worker, and as a result the wage must rise. This isn’t the end of
the story, however: even if the firm only received fraction 1− φ of the increase in the total surplus,
this would imply a rise in VF , which would in turn increase VV above its equilibrium level of zero.
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Figure 7: Effect of a Change in y
θ
w
θ*
JCC1
WE1
w*1
2
JCC2
WE2
θ*1
w*2
Δy
ϕΔy
As a result, additional D firms would decide to become active, increasing V and, in turn, θ. As θ
increases, the vacancy-filling rate q(θ) falls, thereby lowering VV . This process must continue until
VV falls all the way back to zero. Thus, in the end, both w and θ will have increased.
APPENDIX
This appendix works through the details of some of the mathematical derivations in the main
text.
A Proof That VD,t = 0
Consider a firm who is a D type at date t. Since VV,t = VD,t in equilibrium, this firm must be
indifferent between remaining a D type at least until t + 1, and switching to a V type at t. If it
switches to a V type immediately, it receives value VV,t = VD,t. If it instead remains a D type for at
least one more period, it receives a period payoff in t of 0, and value max{VD,t+1, VV,t+1} at t+ 1.
Since VV,t+1 = VD,t+1 , the total value of this option as at date t is thus 0+βmax{VD,t+1, VV,t+1} =
βVD,t+1. Since the D firm is indifferent between these two options, we must have VD,t = βVD,t+1
or, re-arranging slightly,
VD,t+1 = β−1VD,t .
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Since this recursive relationship holds for every t, we may write
VD,t = β−tVD,0 . (45)
Thus,
lim
t→∞β
tVD,t = lim
t→∞β
tβ−tVD,0 = VD,0 .
We therefore conclude that the no-bubbles condition is satisfied if and only if VD,0 = 0. From (45),
this in turn implies that VD,t = 0 for all t.
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