The slides of this lecture are based on:
Equilibrium Unemployment Theory
The Labor Market
Pissarides, Christopher A. (2000).
Equilibrium Unemployment Theory,
Aleksander Berentsen
2nd ed., Cambridge (MA): MIT Press.
Uni Basel
This book and the so-called Market Search-Models investigate the
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Equilibrium Unemployment Theory
consequences of decentralized labor markets.
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Introduction
Structure of this chapter
1
Introduction
2
Trade in the Labor Market
3
Job Creation
4
Workers
5
Wage Determination
6
Steady-State Equilibrium
7
Out-of-Steady-State Dynamics
8
Capital
9
Concluding remarks
Introduction
Aims of this chapter:
Point out the nature of unemployment in the steady state.
Show how wages and unemployment are jointly determined in an
equilibrium model.
Central concepts:
Matching: The labor market is decentralized, uncoordinated economic
activity.
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Bargaining: Wages are negotiable.
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Trade in the Labor Market
Trade in the Labor Market
Trade in the Labor Market
The matching-function models the frictions mentioned, without
Matching:
The
matching function captures the implications of trade for market
equilibrium: It gives the number of jobs formed at any moment in time
as a function of the number of workers looking for jobs and the
investigating their causes explicitly.
It takes the inputs of the matching process (i.e., vacant jobs and
job-seekers) and calculates the number of new jobs created.
number of rms looking for workers.
The matching function can be likened to a production function which
The labor market is made up of heterogeneities, information
calculates the amount of production given specic inputs, without
imperfections and other frictions.
analyzing the process of production.
Examples of these are: diverse skills, dierent jobs, uncertainty as to
the location and timing of job creation, and availability of suitable
workers.
This prevents the labor market from clearing automatically, contrary to
classical labor market theory.
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A steady state produces unemployment, because existing labor
relationships are terminated before the unemployed can nd new jobs.
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Trade in the Labor Market
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Trade in the Labor Market
The model:
Workers and rms are familiar with the matching function.
Bargaining:
There is no coordination either among workers or rms.
Trade and production are completely separate activities:
Before jobs can be created, rms and workers have to spend resources.
Existing jobs, on the other hand, yield a return.
The return produced by an occupied job is slitted between the rm
Atomistic competition rules.
Workers who have jobs will never go to the labor market to look for
work (no on-the-job search).
and worker.
Similarly, a rm with a job that is occupied will not look for a new
Negotiation: cooperative Nash-bargaining solution.
worker either.
In equilibrium all parties maximize their utility, given the behavior of
all other parties.
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Trade in the Labor Market
Trade in the Labor Market
Notation:
There are
L workers in the labor market.
u is the unemployment rate.
v is the quota of vacant jobs, i.e.
Mathematically, the matching function is
monotonically rising for
the number of vacant jobs as a
concave and
fraction of the labor force (vacancy rate).
There are
uL unemployed workers and vL vacancies.
homogeneous of degree one.
The number of job matches taking place per unit time (the
function) is given by
uL and vL,
matching
mL = m(uL, vL).
(1)
Constant returns to scale (CRS) produce a constant rate of
unemployment.
CRS are plausible since a in a growing economy constant returns
ensure a constant unemployment rate, not a higher one.
It gives the number of new employment relationships. The number of
created jobs per unit of time thus depends on the number of
unemployed workers and the number of job vacancies.
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Trade in the Labor Market
The elasticity is 1
− η(θ) ≥ 0.
faster unemployed workers will nd jobs.
Firms, however, can ll a job more quickly when
is small, in other
available.
This is seen in that
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θ
words, when there are few vacancies relative to the number of workers
We note
|ξ(θ)| = η(θ).
Equilibrium Unemployment Theory
(3)
q (θ)
The more vacant jobs there are, the larger θ = v /u will be, and the
unemployed worker with probability
Aleksander Berentsen (Uni Basel)
v
m(uL, vL)
= m 1,
= θq (θ).
uL
u
The mean duration of unemployment is 1/θ
a vacant job is matched to an
q (θ)∆t .
Hence, the mean duration of a job is 1/q (θ).
q 0 (θ) ≤ 0
The elasticity of q (θ), is ξ(θ) with 0 ≤ ξ(θ) ≤ −1.
p(θ) ≡
(2)
is the number of vacant jobs per unemployed worker.
∆t ,
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Similarly, the rate at which unemployed workers move into employment is
u m(uL, vL)
q (θ) ≡
=m
,1
vL
v
During a small time interval,
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Trade in the Labor Market
The rate at which vacant jobs are lled is thus:
θ = v /u
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q (θ) is falling in θ.
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Trade in the Labor Market
θ
Trade in the Labor Market
is a measure of labor-market tightness for the rm:
Every job-seeker and every vacant job cause so-called search externalities:
The tightness of the labor market is measured by the relationship of
An additional job-seeker causes a positive externality for the rm but a
available jobs to job-seekers.
negative externality for the other job-seekers.
The higher
θ
is, the tighter the labor market is for the rm.
Each additional job-seeker produces an increased probability of
This expresses the fact that a relatively small number of job-seekers
have a large number of vacant jobs to choose from when
θ
is high.
1
− θq (θ)∆t
that a co-seeker will not nd a vacancy.
At the same time, the addition increases the probability
It is therefore dicult to ll vacant jobs; and so, the market is said to
specic vacancy will be lled.
be tight.
An additional vacancy will have an analogous eect.
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Trade in the Labor Market
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q (θ)∆t
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that a
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Trade in the Labor Market
The ow into unemployment (job destruction):
A shock (reduction in productivity, fall in the relative price of goods
produced, etc.) can make it no longer protable for the rm to oer
the job.
This kind of shock occurs with probability
The ow out of unemployment (job creation):
λ.
In this simple model, every shock leads to immediate job separation
Job creation takes place when a rm and a searching worker meet and
(which in this model is equal to job destruction). Hence, the job
agree to form a match at a negotiated wage.
separation rate is
λ.
The number of job-seekers who nd a job is
This process of job separation is exogenous in this version of the
θq (θ)uL∆t .
model.
(5)
The probability of a worker becoming unemployed in a small time
interval is given by
λ∆t .
Without economic growth (L constant), the workers who enter
unemployment in a short time interval
∆t
is
λ(1 − u )L∆t .
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(4)
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Trade in the Labor Market
Job Creation
Equilibrium unemployment:
The evolution of mean unemployment is given by the dierence of the
Firms: Job Creation
ows into and out of unemployment:
u̇ =
du
dt
= λ(1 − u ) − θq (θ)u
(6)
Job creation takes place when a rm and a worker meet and agree to
an employment contract.
In equilibrium (steady state) the mean rate of unemployment is
constant; i.e.:
For convenience, we assume that rms are very small and employ only
λ(1 − u ) = θq (θ)u
(7)
one worker.
A rm can re a worker any time and will do so when there is a shock.
We derive from this the equilibrium unemployment rate:
u=
A rm starts producing only once it has hired a worker.
λ
λ + θq (θ)
(8)
Products can be sold on the market at a constant price
p > 0.
This
represents the productivity of a worker.
The equilibrium unemployment rate is dependent on both transition
probabilities; i.e., job creation and job destruction.
The higher the rate of job-separations relative to the rate of
job-matchings, the higher unemployment is in equilibrium.
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Job Creation
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Job Creation
Let
J
be the present discounted value of expected prot from an occupied
job and
When a job is vacant, the rm is actively engaged in hiring at a xed
cost
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V
the present discounted value of expected prot from a vacant
job. The value of a job when a rm enters the market is
pc > 0 per unit time.
V
= −δ pc + q (θ)δ J + [1 − q (θ)]δ V .
(9)
The hiring cost is proportional to productivity of the worker, on the
This can be rewritten as
grounds that more productive workers are more costly to hire.
(1 − δ)V = −δ pc + δ q (θ)(J − V )
The number of jobs oered is endogenous and maximizes the prot of
the rm. Any rm is free to open a job vacancy and engage in hiring.
From this follows that
There is free market entry with a zero prot condition.
rV
Note that
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(10)
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= −pc + q (θ)(J − V ).
r = (1 − δ)/δ and r
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(11)
is exogenous.
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Job Creation
Job Creation
Equation (11) states that the return on the asset, a vacancy
equal in size to the capital costs
rV .
V , is
The asset value of an occupied job,
The net return (return minus hiring fees) of a job to an employer
equals
J − V.
J = δ(p − w ) + (1 − λ)δJ
In equilibrium rents from vacancies are zero owing to free market entry.
Therefore,
V
=0
(13)
which is implying
J=
Filled jobs thus yield a return, i.e.
As 1/
J , satises a value equation similar to
the one for vacant jobs:
pc
q (θ)
A net return of
(12)
p−w
is earned, where
w
is the cost of labor. In
addition, the job runs the exogenous risk of an adverse shock (job
destruction).
J > 0.
V
Note that
q (θ) is the expected duration of a vacancy.
is not a part of this equation because it is assumed that a
rm aected by a shock disappears from the market. It will not
reappear later to oer a job (complete irreversibility).
Equation (12) in equilibrium, market tightness is such that the
expected prot from a new job is equal to the expected cost of hiring
a worker.
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Job Creation
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Job Creation
Equation (15) produces a negative relationship between
wage
w
in the
θ,
w
θ = v /u
and the
space.
The asset value of a lled job satises a value equation, similar to the one
for vacancies:
If
rJ = p − w − λJ
(14)
J = pc /q (θ) is substituted in equation (14) we get
p−w −
(r + λ)pc
=0
q (θ)
(15)
This equation gives the marginal condition for labor demand. If the rm
had no hiring costs, i.e.
condition
p=w
Figure: The job creation curve (JC)
c = 0, then the standard marginal productivity
would result.
The downward sloping labor demand curve is also called the
job creation
condition (JC).
In order to determine equilibrium, the supply side of the market has to be
considered. We therefore now turn to workers.
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Workers
Workers
Workers
Let U denote the present-discounted (cash) value of the expected
income stream of an unemployed worker.
In this model the labor force supply
L is constant.
W is the present-discounted (cash) value of the expected income
stream of an employed worker.
Moreover, each worker's search intensity is xed.
p.
A worker earns w when employed, and z when searching for a job.
The expected income steam of an unemployed worker is thus
Workers all have the same productivity
U = δz + θq (θ)δW + [1 − θq (θ)]δU .
Every worker is either employed or searching for employment.
z
covers unemployment insurance benets or some return from
Reformulated, this gives
self-employment.
z
rU = z + θq (θ)(W − U ).
includes the imputed real return from unpaid leisure activities.
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Workers
λ.
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W
equations (11) and (14).
The asset that is valued is the unemployed worker's human capital.
This gives
W − U.
is the minimum compensation that an unemployed worker requires
to give up search, or the reservation wage.
w
and lose their jobs at the exogenous rate
Hence, the expected income of a worker is
Equation (17) has the same interpretation as the rm's asset
rU
(17)
Workers
Employed workers earn a wage
The worker's net return is
(16)
rW
= δ w + λδ U + (1 − λ)δ W .
rW
(18)
= w + λ(U − W ).
is not equivalent to the wage
(19)
w , because it reects the risk of
unemployment.
Workers stay in their jobs for as long as
W
≥ U.
The necessary and sucient condition for this is when
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w ≥ z.
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Workers
Workers
Substituting equations (17) and (19) in each other's equations gives
(r + λ)z + θq (θ)w
,
r + λ + θq (θ)
λz + [r + θq (θ)]w
=
r + λ + θq (θ)
Since
rU =
rW
(20)
W
≥ U ).
Without discounting (
r = 0), unemployed workers are not worse o
than employed workers.
Reason: Job allocation is random, and every worker is employed
w −z
W −U =
r + λ + θq (θ)
Equilibrium Unemployment Theory
employed workers have higher permanent incomes than unemployed
workers (so
(21)
TIP: First calculate the dierence
Aleksander Berentsen (Uni Basel)
w ≥ z , it follows from (20) and (21) that with discounting,
sometime (in an innite time horizon).
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Wage Determination
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Wage Determination
Wage Determination
In equilibrium, occupied jobs yield a total return that is strictly greater
that the sum of the expected returns of a searching rm and a
For a given wage rate
satises
searching worker.
A lled job yields a pure economic rent that is equal to the sum of
For the worker, it is
expected search costs of a searching rm and a searching worker.
It is assumed that the monopoly rent is shared according to the Nash
w , the rm's expected return from the job, J ,
rJ = p − w − λJ .
rW
(22)
= w − λ(W − U ).
solution to a bargaining problem.
The net return from a job match contract is
This rent is divided by xing the wage rate.
for the worker.
J −V
(23)
for the rm and
W −U
Since all workers and all jobs are identical in this model, a uniform
wage
w
is established.
This is an atomistic market, i.e. no individual participant is able to
inuence the market.
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Wage Determination
Wage Determination
The Nash bargaining solution identies a value for
w
that maximizes
the weighted product of the worker's and rm's net returns from the
job match.
In order to form the job match, the worker gives up
rm gives up
V
for
J.
U
for
W
In equation (24),
and the
U
and
V
are called thread points.
If the two parties are unable to agree, the worker will remain
unemployed and the job vacant (which will not happen in this model
Therefore the wage rate for the job satises
given the assumptions on productivity and the arrival process of
idiosyncratic shocks).
w = arg max(W − U )β (J − V )1−β .
In symmetric Nash bargaining solutions
β=
(24)
1
2 . A dierent
β
implies
dierent measures of bargaining strength or rates of impatience.
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Wage Determination
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Wage Determination
The FOC from equation (24) satises
rU
W − U = β(J + W − V − U )
(25)
in the equilibrium solution (26) is not particularly interesting.
Another
method for deriving the wage equation results by following the subsequent
steps:
In equilibrium, equation (20) holds. Consequently,
β
is labor's share of the total surplus that an occupied job creates
q (θ) =
(called the sharing rule).
In order to obtain equilibrium,
W
and
J
are substituted from (23) and
(22) into (25), and the equilibrium condition
V
=0
is imposed. The
Workers receive their reservation wage
rU
J=
(26)
and a fraction
β
of the net
(27)
The FOC, equation (25), can be rewritten as
wage equation is thus
w = rU + β(p − rU )
pc
.
J
Note that in equilibrium
V
1
−β
(W − U ).
β
(28)
= 0.
surplus that they create by accepting the job.
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Wage Determination
Wage Determination
Equations (27) and (28) are inserted into
rU = z + θq (θ)(W − U )
(29)
pc θ is the average hiring cost for each unemployed worker (since
pc θ = pcv /u and pcv is total hiring cost in the economy).
Workers are rewarded for the saving of hiring costs that the
representative rm enjoys when a job is formed.
This gives
rU = z +
β
pc θ.
1−β
(30)
θ
indicates the tightness of the labor market. With a high
θ
there is a
large number of jobs relative to number of workers.
The resulting equation for the reservation wage can now be
In this situation, the workers have a strong bargaining position which
substituted back into the original wage equation (26):
has a positive eect on their wages.
This produces an upward-sloping relationship between
w = (1 − β)z + β p(1 + c θ)
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(31)
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w
and
θ
This is the case in spite of a xed labor force size.
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Steady-State Equilibrium
Steady-State Equilibrium
The upward-sloping relation is expressed by the wage setting function
(subsequently called wage curve).
Equation (31) replaces the labor supply curve of Walrasian models.
We have a triple
(u , v , w )
that satises the ow equilibrium condition (8)
u=
λ
,
λ + θq (θ)
(32)
(r + λ)pc
= 0,
q (θ)
(33)
the job creation condition (15)
p−w −
and the wage equation (31)
w = (1 − β)z + β p(1 + c θ).
Figure: The wage curve (WC)
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(34)
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Steady-State Equilibrium
Steady-State Equilibrium
v consecutively.
If u and θ are known, the number of lled jobs, (1 − u )L, and the
number of vacancies, θ uL, are also known.
Equations (33) and (34) determine the wage rate w and the tightness
For convenience, we will work with
of the labor market
θ
and instead of
θ.
The unemployment rate
u can be calculated from equation (32).
Equilibrium is unique (which is illustrated with the help of the
following two diagrams).
Figure: Equilibrium wages and market tightness.
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Aleksander Berentsen (Uni Basel)
Steady-State Equilibrium
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Steady-State Equilibrium
The Beveridge diagram (gure on slide 46):
The job creation curve, equation (33), says that a higher wage rate
leads to reduced job vacancies and thus lowers the equilibrium ratio of
jobs to workers.
Equation for this
The wage curve, equation (34), says that a tighter labor market
negotiated.
(θ, w )
θ
(1 − β)(p − z ) −
is at the intersection ot the two curves and it is
r + λ + βθq (θ)
pc = 0.
q (θ)
JC is a line through the origin, with slope
Equilibrium Unemployment Theory
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is independent of
can be explicitly derived by substituting wages
unique.
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θ
from (34) into (33), to get the job creation line (JC)
increases the bargaining strength of workers and a higher wage is
Equilibrium
The gure on slide 42 shows that the equilibrium
unemployment.
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(35)
θ.
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Steady-State Equilibrium
Steady-State Equilibrium
The steady-state condition for unemployment, equation (32), is the
Beveridge curve (BC).
The Beveridge curve is convex to the origin by the properties of the
matching technology: When there are more vacancies, unemployment
is lower because the unemployed nd jobs more easily. Diminishing
returns to scale to individual inputs in matching imply the convex
shape.
Equilibrium vacancies and unemployment are at the unique
intersection of the job creation line and the Beveridge curve.
Figure: Equilibrium vacancies and unemployment
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Steady-State Equilibrium
Comparative Statics
Productivity
p:
Higher productivity
pc
p leads to higher wages and lower unemployment (with
held constant).
Since
β < 1,
JC in gure on slide 42 shifts by more, so both
w
and
θ
increase (see also equation (35))
In gure on slide 46 this rotates the job creation line anticlockwise,
increasing vacancies and reducing unemployment.
Remark: This result cannot be maintained if
in
p.
z
increases proportionally
Figure: Eects of a higher productivity
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Steady-State Equilibrium
Unemployment income
z:
Higher unemployment income
unemployment (a higher
β
z
Steady-State Equilibrium
leads to higher wages and higher
produces similar eects):
Workers claim a higher wage because their reservation wage increases
with a higher
z.
Firms nd it less attractive to create jobs.
It is important that the disincentive eects are ignored here, i.e. in
spite of higher
z , the unemployed continue to seek employment.
Figure: Eects of a higher unemployment income
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Steady-State Equilibrium
Interest rate
r:
A higher interest rate
r
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Steady-State Equilibrium
leads to lower wages and higher unemployment.
The reason is to be found in the heavy discounting of future revenues
in the short-term horizon.
We have shown that workers are indierent between being unemployed
and employed with
r = 0.
r
The less patient workers are (
increases), the more important it is for
them to nd a job. Hence, wages drop.
Figure: Eects of a higher interest rate
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Steady-State Equilibrium
Steady-State Equilibrium
Beveridge curve (BC):
An exogenous fall in the matching function can cause the BC to shift
out. This may be caused by frictions in the labor market.
A worsening in the matching function leads to lower wages and higher
unemployment.
Another cause of changes that can shift the Beveridge curve is an increase
in exogenously occurring shocks
A higher
λ
λ:
shifts the Beveridge curve out because at a given
unemployment rate
u a higher λ implies a bigger ow into
unemployment than out of it. Unemployment needs to increase to
bring the ow out of unemployment into equality with the higher
inow.
Figure: Eects of a shift in the Beveridge curve
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Out-of-Steady-State Dynamics
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Out-of-Steady-State Dynamics
How does the economy react to an increase in productivity
Out-of-Steady-State Dynamics
Immediate reaction:
w
and
θ
p?
jump instantly to their new values. There is
no adjustment dynamic.
The model's dynamics:
Changes in the parameters have an immediate eect on the wage
w
(through the assumption that wages can be renegotiated any time).
By the assumption that the expected prot from the creation of a new
job vacancy is zero, rms have to be able to adjust their vacancies
immediately, i.e.
v
and thus
θ
are both jump variables.
However, the unemployment rate
u does not jump because it is tied to
the matching function that matches unemployed workers with
vacancies over time.
Figure: Eects of an increase in productivity
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Out-of-Steady-State Dynamics
Capital
This direct eect immediately rotates the JC in the Beveridge diagram.
Vacancies jump immediately to their new value. The unemployment rate
Capital
however moves along the JC to the new equilibrium point C.
Assumptions:
There is a perfect second-hand capital market.
The interest rate is exogenous.
The rm can buy and sell capital at the price of output.
Since capital is costly, vacancies do not own capital.
Figure: Eects of an increase in productivity
Vacancies have the tendency to overshoot. Firms create a lot of new jobs
which are then closed in the matching process over a period of time until
equilibrium is re-established.
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Aleksander Berentsen (Uni Basel)
Equilibrium Unemployment Theory
Capital
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Capital
Asset value of a vacant job:
Productivity
Still given by equation (11),
p is reinterpreted as a labor-augmenting productivity
rV
parameter that measures the eciency units of labor.
K is aggregate capital.
N is aggregate employment.
F (K , pN ) is an aggregate production function with positive but
= −pc + q (θ)(J − V ).
Asset value value of an occupied job:
J + pk .
The real capital cost of the job is r (J + pk ).
The job yields the net return pf (k ) − δ pk − w .
The asset value of a job is now given by
diminishing marginal products and constant returns to scale.
k is the ratio K /pN .
f (k ) = F (K /pN , 1) is the output per eciency unit of labor.
f (k ) satises f 0 (k ) > 0 and f 00 (k ) < 0.
The job runs a risk of an adverse shock
J
λ,
which leads to a loss of
is hence determined by the condition
r (J + pk ) = pf (k ) − δpk − w − λJ .
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Equilibrium Unemployment Theory
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J.
Aleksander Berentsen (Uni Basel)
Equilibrium Unemployment Theory
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(36)
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Capital
Capital
A rearrangement of equation (36) gives
rJ = p[f (k ) − (r + δ)k ] − w − λJ ,
(37)
which generalizes equation (14). It can be seen that only the job product
p
is aected by the introduction of capital.
Therefore the model can be solved as before but with the generalization
that product
p is multiplied by [f (k ) − (r + δ)k ]:
(38)
p[f (k ) − (r + δ)k ] − w −
(39)
(r + λ)pc
= 0,
q (θ)
w = (1 − β)z + β p[f (k ) − (r + δ)k + c θ],
λ
u=
.
λ + θq (θ)
Equilibrium Unemployment Theory
r , equation (38) gives the capital-labor ratio.
With knowledge of r and k , equations (39) and (40) give wages and
With knowledge of
market tightness.
With knowledge of
f 0 (k ) = r + δ,
Aleksander Berentsen (Uni Basel)
This equilibrium system is recursive:
Spring term 2009
θ,
equation (41) determines unemployment.
Notice: The essential features of the unemployment model remain
unaltered and the capital decision is unaected by the existence of
(40)
matching frictions. Hence, the eects described with the gures on
slides 42 and 46 stay the same with capital.
(41)
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Concluding remarks
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Concluding remarks
Concluding remarks
This model demonstrates how both unemployment and vacancies can
exist concurrently in labor market equilibrium.
Existing frictions in the labor market are
formed by the matching
function.
The job creation rate is the same as the job destruction rate in the
The key reason for this is that the searching activities of the
steady state. If there is a deviation from it, there will be a dynamic
unemployed and job-seekers bear frictions that prevent the labor
readjustment that equalizes both rates.
market from clearing automatically.
Search externalities play a role in the derivation of the results. They
are the reason that price is not the sole allocation mechanism. For
Since the job destruction rate is given as exogenous with
λ, q (θ)
adapts.
every price, there is always a positive probability that a vacancy will
not be lled or an unemployed worker will not nd a job.
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