Graduate Macroeconomics 2
Problem set 7. - Solutions
Question 1
1. See Lecture 8 for the derivation of the equilibrium of the search and matching
model with exogenous job destruction. The equilibrium market tightness θ
and w jointly satisfy the job creation condition
p−w−
(r + λ)pc
=0
q(θ)
(JC)
and the wage curve
w = (1 − β)b + βp(1 + cθ)
(1)
2. An increase in general productivity, p, shifts the wage curve and the job creation curve upwards. Intuitively, a rise in p increases the size of the pie which
the worker and the employer split between them according to the sharing
rule (from Nash bargaining). Consequently, the worker receives a higher
wage and the employer receives higher profits. The higher wage for any θ
implies that the wage curve shifts up, see effect of p in (1). The higher productivity implies that the JC curve shifts up as well: for a given wage the
expected profits increase more than the expected vacancy costs, therefore the
zero-profit market tightness has to increase. The effect on equilibrium wages
higher p v
wage curve
w
JC’
JC
w∗∗
w∗
higher p
BC
JC
θ∗θ∗∗
θ
Figure 1: The effect of a higher p in the equilibrium
is clearly positive, as the shift of both curves implies this. The rightward shift
of the JC curve implies a higher θ for any w, which via the higher arrival rate
of jobs to workers reinforces the ’direct effect’ of higher p on wages (captured
1
u
by the upward shift of the wage curve). The effect on equilibrium market
tightness might appear to be ambiguous. The ’direct effect’ of a higher p on
θ is captured by the rightward shift of the JC curve, and thus a higher θ for
any w. The upward shift of the wage curve partially offsets this: the higher
wages drives the firms to diminish the number of vacant jobs.
However, it becomes clear that value of θ increases after an increase in productivity if we use the job creation condition and the wage curve to characterize the equilibrium value of θ. To see this plug (1) into the (JC), divide by
p (and rearrange a bit) to get an implicit equation for θ:
b
(r + λ)c
(1 − β) 1 −
= cβθ +
p
q(θ)
Finally, let’s look at the steady-state value of unemployment in the u - v space.
The Beveridge curve is defined by:
u=
λ
,
λ + θq(θ)
which gives a steady state u for any level of v (or equivalently θ). We can see
that the Beveridge curve is not affected by a change in p. As the equilibrium
value of θ increases, the JC curve rotates upwards, and the equilibrium value
of unemployment decreases (a movement along the Beveridge curve).
It is important to note that the individual productivity effects depend strongly
on the hypothesis that the gains of unemployed persons b do not hinge on labor productivity. There are good reasons to believe that this parameter (like
the cost of vacancy posting pc) is dependent on individual productivity in
the long run: unemployment benefits are most often defined as a fraction of
past wages - which is the same as linking them to labor productivity. In such
a case, it is easy to verify by using the wage curve and the JC that the level
of productivity would no longer have any influence on equilibrium market
tightness,it would only increase equilibrium w. This result implies that the
unemployment rate is likely affected by the level of productivity in the short
to medium run, but it is independent in the very long run.
A note on Shimer’s critique (2005): The effect of a productivity shock (as
seen in the Figure 1.) leads to a big increase in the real wages and a marginal
increase in the labor market tightness. In the u - v space, this entails a small
increase in the slope of the job creation curve which decreases the unemploy-
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ment rate and increases the vacancy rate, albeit weakly. This is the unemployment volatility puzzle raised by Shimer which highlights that the canonical
search and matching model is unable to replicate the large unemployment
fluctuations observed in the US. Shimer argues that this failure results from
the high flexibility of the real wage stemming from Nash bargaining. As
seen from the wage curve, productivity shocks have a direct effect on the
real wage. It also has an indirect effect through the impact of p on θ. Hence,
the real wage rises sharply following a positive productivity shock. This implies that the profit of the firm increases but only marginally after a positive
productivity shock. Refer to Lecture 9 for more on this.
Question 2 – Unemployment insurance sanctions
1. Write down the flow equation for the value of vacancies
rV = −pc + q (θ) (J − V )
(2)
rU = b + a (θ) (W − U )
(3)
for unemployed workers
The value for a firm of job with wage wi :
rJ (wi ) = p − wi − λJ (wi )
(4)
rW (wi ) = wi − λ (W (wi ) − U )
(5)
and worker:
where wage wi is determined by Nash Bargaining.
1−β
wi = arg max (J (wi ) − V )
β
(W (wi ) − U ) .
Rewrite the maximization as
max [(1 − β) ln (J (wi ) − V ) + β ln (W (wi ) − U )]
wi
The first order condition is
1−β
J (wi ) − V
0
J (wi ) +
3
β
W (wi ) − U
W 0 (wi ) = 0
Note that U and V are independent of wi and from (4) and (5) we get that,
J (wi )
=
W (wi )
=
p − wi
−1
=⇒ J 0 (wi ) =
r+λ
r+λ
wi + λU
1
=⇒ W 0 (wi ) =
r+λ
r+λ
which implies that the total surplus is independent of the wage rate. Thus
the optimal solution implies that β fraction of the surplus goes to the worker:
W (wi ) − U = β (J (wi ) − V + W (wi ) − U )
(6)
Use (3) and (5) to obtain
W −U =
w−b
.
r + λ + θq(θ)
The free entry condition implies V = 0, and using (4) and (2) we obtain:
J=
pc
p−w
=
.
r+λ
q (θ)
(7)
Substitute W − U and J − V into (6), to get:
(1 − β)
w−b
r + λ + θq(θ)
pc
⇔
q (θ)
pc
= β
(r + λ) + βpcθ
q (θ)
=
(1 − β) (w − b)
β
and finally use (7) to get the wage equation:
w = (1 − β) b + β (1 + cθ) p.
(8)
2. Equilibrium w and θ jointly satisfy (8) and (7). We can compute ∂w/∂b and
∂θ/∂b jointly from (8) and (7). Graphically, in the θ − w space, the wage curve
(8) is upward sloping and the job creation curve (7) is downward sloping.
Higher b shift the wage curve upward, so w increases and θ decreases. Intuitively, higher b increases the value of unemployment (the threat point) of the
worker in the wage bargain. The decrease in θ is due to the movement along
the job creation curve: higher wage reduces surplus of a job and so reduces
vacancy for given unemployment, i.e. θ falls.
3. The equations for V, U, W (wi ) and J (wi ) are the same as before, but during
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bargaining, the worker’s threat point becomes U0 , where
rU0 = θq(θ) (W − U0 )
(9)
because he does not receive any unemployment insurance if he rejects the
wage offer. Since W (wi ) and J(wi ) are the same as before, the solution to the
Nash Bargaining satisfies
W (wi ) − U0 = β (J (wi ) − V + W (wi ) − U0 ) ⇔ W (wi ) − U0 =
β
J (wi )
1−β
(10)
Use (3) and (9) to obtain U − U0 ,
U − U0 =
b
r + θq(θ)
Finally use (5) and (9) to obtain
r (W − U0 ) = w − λ (W − U ) − θq(θ) (W − U0 )
which implies
(r + λ + θq(θ)) (W − U0 ) = w − λ (U0 − U )
substitute J from (7), Wi − U0 and U − U0 from above, we get
(r + λ + θq(θ)) β
pc
q (θ)
β (r + λ) pc
+ βpcθ
q (θ)
=
=
(1 − β) w + λ
b
r + θq (θ)
λ (1 − β)
(1 − β) w +
b
r + θq (θ)
⇔
again use (7), and we obtain the wage equation
w=−
λ (1 − β)
b + β (1 + cθ) p.
r + θq (θ)
Now higher b shifts the wage curve downwards, and so w decreases and θ
increases.
4. Under the tougher rules, when a firm and a worker meet, the worker loses
UI qualification. To regain it, he has to accept the job, so higher b now makes
job acceptance more attractive to the worker, reducing the bargained wage.
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