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Background
The Poisson process
The Shapiro-Stiglitz model
Macroeconomics 4 - Lecture 1
The Labor Market
Daniel Cohen
Master APE
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
U S
nemployment rate in the U and France
12
10
8
%
6 US
France
4
2
0
Q1 1950
Q3 1953
Q1 1957
Q3 1960
Q1 1964
Q3 1967
Q1 1971
Q3 1974
Q1 1978
Q3 1981
Q1 1985
Q3 1988
Q1 1992
Q3 1995
Q1 1999
Q1 2006
Q3 2002
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
Insiders / Outsiders
Ascar Lindbeck and Denis Snower.
I The wage is not determined on a market, but by the insiders.
Those who have a job try to keep it, so it is impossible for the
economy to go back to an equilibrium after a shock.
I Unemployed workers are not considered as a stock but as a
‡ow.
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
I Doc 1 : Survival rates of employed workers :
How many people from a group of workers still have the same job
x months later ?
x US France
6 months 95% 98.1%
1 year 88.3% 94.5%
2 years 81.2% 92.5%
1 year later, 88.3% of the cohort never experienced an
unemployment spell in the US.
Twice more people move into unemployment in the US.
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
I Doc 2 : How many people from a cohort of unemployed were
never employed x months later ? (a job for more than two
weeks is considered as employment)
x US France
1 month 92.3% 97.2%
3 months 70.8% 91.5%
6 months 57.6% 85%
1 year 43.1% 64.9%
2 years 37.1% 56%
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
I Doc 3 : Cohort of unemployed workers who have just lost their
job. How many of them were never employed x years after ?
x US France
1 month 83.3% 87.1%
3 months 46.2% 67.4%
6 months 25.2% 51.5%
1 year 12.5% 26.9%
2 years 5.6% 14.7%
In France it is twice longer to reach the same percentage.
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
The Poisson process
Probability of an event to happen before a certain date.
ex : the probability that an earthquake happens between today t0
and T : Z T
λ (s t 0 ) λ (T t0 )
λe ds = 1 e
t0
The survival rate is : e λ(T t0 ) . It is not linear, except when t0 and
T are close, and than it can be approximated by a linear process.
NB : If T tends to in…nity, we are certain that the event would
happen.
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
1st model
s : separation rate.
h: hiring rate
u: the unemployed
1 u : the labor force (we suppose there is 1 unit of people)
dut = ut dt
dut = (1 ut )sdt hut dt
ut = s (1 ut ) hut
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
The steady state is given by u = 0.
s
u∞ =
s +h
ut = (s + h)(u∞ ut )
s
u∞ depends only on the ratio h :
s
h
u∞ = s
1+ h
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
Introduction of value functions
A …rm gets a ‡ow of incomes : fxt gt 0
The net present value (NPV) is :
Z +∞
rt
J0 = e xt dt
0
Z +∞
r (s t )
Jt = e xs ds
t
Z +∞
= e rt e rs
xs ds
t
Jt = rJt xt
rJt = Jt + xt
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
Consider an asset Ja (t ) that is associated with a ‡ow of dividends
fxta gt 0 but that switches to Jb (t ) with a Poisson probability of
parameter λa .
Z +∞ Z T
λa (T t) r (s t ) a r (T t)
Ja ( t ) = λa e dT e xs ds +e Jb ( t )
t t
Z +∞
(r +λ)(T t ) a
= e [xT + λJb (T )] dT
t
Ja ( t ) = ( r + λ ) J a ( t ) [xta + λJb (t )]
rJa (t ) = (xta + Ja (t ) + λ(Jb Ja )(t )
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
Using these tools to consider the unemployment problem :
Let JE the value function of being employed, and JU the value
function of being unemployed.
rJE = w + s (JU J E ) + JE
rJU = b + h(JE JU ) + JU
where b represents the unemployment bene…ts.
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
If s and h are given,
w + sJU
JE =
r +s
b + hJE
JU =
r +h
w b
JE JU =
r +s +h
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
The Shapiro-Stiglitz model
I A worker can produce the e¤ort required, or shirk.
JE = max fJS ; JNS g
I S : shirk. Corresponding wage : w
I NS : not shirk. Corresponding wage : w e (e is the
monetary equivalence for the e¤ort).
I There is a probability q for a shirker to be discovered.
rJNS = w e + s (JU JNS )
rJS = w + s ( JU J S ) + q ( JU JNS )
rJU = b + h ( J E JU )
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
e + q (JNS JU )
JNS JS =
r +s +q
I e
We need JNS JS > 0, i.e. JNS JU > q .
I Working and produce an e¤ort must lead to a gain higher than
e
being unemployed, and the di¤erence must be higher than q .
I The bigger q, the less important the di¤erence needed.
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
Unvolontary unemployement : high wages are the result of the
s
…rm’ willing because of the incompleteness of contracts. There
will never be full employment and it is not due to wages resistance.
w e b e
JE JU = JNS JU = >
r +s +h q
e
w > e + (r + s + h ) + b
q
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
contract
curve
w
e+b
marg prod of labour
(1)
f’
û u*
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
I A new technology that allows to watch better the employed
lowers w and u.
I When s increases, there is less incentive to work. The …rm
must raise w to o¤set this e¤et, and u increases.
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
Discussion
If there are two segments on the labour market characterized by
two di¤erent s but with equally productive workers. Ex : men and
women. s is higher for women. The model predicts that women
have higher wage and higher unemployment rate.
In reality, high turnover goes with low w .
D. Cohen Macro 4 - 1
Background
The Poisson process
The Shapiro-Stiglitz model
Extensions
I Suppose each worker makes a deposit when she is hired, and
lose it either if she is caught shirking, or if she loses her job.
In the …rst case, the bene…t from shirking is now w qd.
There is a level of deposit that for sure creates an incentive to
work (ex. China).
I Suppose now that the deposit is replaced by the fact that a
worker is underpaid at the beginning of his career and
overpaid later on. According to Schleifer and Summers, this
was the implicit wage contract before the …nancial revolution.
The …nancial revolution consisted in the repudiation of the
debt to old workers.
D. Cohen Macro 4 - 1
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