Graduate Macroeconomics 2
Problem set 8. - Solutions
Question 1 – Cyclical change in the matching model
1. See Lecture 9 for the derivation of the equilibrium of the endogenous job
destruction model (NJD). The equilibrium market tightness θ and reservation
productivity R jointly satisfy the job creation condition
(1 − β)
1−R
r+λ
=
c
q (θ)
(JC)
and the job destruction condition
λ
R+
r+λ
Z
1
(z − R) dG (z) =
R
b
β
+
cθ.
p 1−β
(JD)
To summarize the steps:
(a) For any job with productivity x, write down the flow value equations
for J (x) and W (x) ,
Z
rJ(x)
1
= px − w(x) + λ
J(z)dG(z) − λJ(x)
R
Z
rW (x)
1
W (z)dG(z) + λG(R)[U − λW (x)]
= w(x) + λ
R
then solve the Nash Bargaining problem for this job to determine the
wage w (x) (taking market tightness θ and the value of the unemployed
U and value of vacancy V as given). The flow value equations imply
that
−
dJ (x)
1
dW (x)
=
=
dw (x)
r+λ
dw (x)
so the Nash Bargaining splits a fixed surplus W (x)+J(X)−U −V . This
implies that w (x) satisfies:
β (J (x) − V ) = (1 − β) (W (x) − U )
(b) Now write down the flow value equations for U and V,
rV
= −pc + q(θ)(J(1) − V )
rU
= b + θq(θ)(W (1) − U )
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The free entry condition V = 0 implies
J (1) =
pc
q (θ)
(1)
Subtract (1 − β)rW (x) from βrJ(x), using the above given expressions
for rJ(x) and rW (x), use that βJ(x) − (1 − β)W (x) = βV − (1 − β)U for
all x ∈ [0, 1], use that V = 0 and simplify to the the wage for a job with
productivity x:
w (x) = (1 − β) b + βp (x + cθ) .
(c) Substitute w(x) into flow equation of J(x) to derive
Z
1
(r + λ) J (x) = (1 − β) (px − b) − βpcθ + λ
J (z) dG (z)
(2)
R
a relationship between value of job J (x) and the equilibrium variables
θ and R.
(d) Substitue x = R in (2) and subtract it from the generic version of (2),
finally use J(R) = 0 (which holds by definition of R) to obtain an expression for the value of a job J(x):
J(x) =
(1 − β)p
(x − R)
r+λ
(3)
Combining with (1) and (2), we obtain the JC condition.
(e) Substitute x = 1 into (3) to get the value of a newly created job:
J(1) =
(1 − β)p
(1 − R).
r+λ
Combine the above with (1) to get the job creation condition
(1 − β)p
pc
(1 − R) =
.
r+λ
q(θ)
(JC)
(f) Substitute x = R into (2) and use the expression for a generic J(z) from
(3) in the integral to get the job destruction condition:
R+
λ
r+λ
Z
1
(z − R) dG (z) =
R
2
b
β
+
cθ.
p 1−β
(JD)
higher p v
wage curve
w
JC’
JC
w∗∗
w∗
higher p
BC
JC
θ∗θ∗∗
u
θ
Figure 1: Equilibrium θ, w, u and v in the XJD model
The steady state unemployment rate is given by the Beveridge curve
u=
λG (R)
λG (R) + θq (θ)
(BC)
2. & 3. In the exogenous job destruction model (XJD), we have shown that
changes in p only shift the job creation curve (θ changes) but not the Beveridge curve (the term G(R) in BC is equal to 1 for the XJD model). An
increase in p increases θ and shifts the job creation curve upward in the u − v
space. As a result the equilibrium u decreases (movement along the Beveridge curve).
In the NJD model, where there are non-trivial productivity differences, i.e.
existing jobs have different idiosyncratic x, and changes in p shift both the
job creation curve and the Beveridge curve. For example, an increase in p
increases θ and reduces R, so in the u − v space, the job creation curve rotates
upwards and the Beveridge curve shifts inwards. When comparing with the
XJD model, the key is that the BC shifts in the NJD model. If the shift in the
BC (↔ the fall in R) is sufficiently large, then u is more volatile and v is less
volatile in the NJD. (See Beveridge curve and JC curve diagram).
The magnitude of change in θ is also different in the two models, and this
also plays a role in the effect of a change in p in the two cases. Below we
explicitly derive the change in θ.
Under the XJD model, equilibrium θ satisfies:
(1 − β) (p − b) = βpcθ +
3
(r + λ) pc
q (θ)
(4)
v
R
JC’
JD
JC
higher p
R∗
R∗∗
BC
JC
BC’
θ∗ θ∗∗
u
θ
Figure 2: Equilibrium θ, R, u and v in the NJD model
Total differentiation with respect to p implies
(r + λ) c
(1 − β) = βcθ +
+
q (θ)
βpc −
(r + λ) pcq 0 (θ)
!
2
[q (θ)]
dθ
dp
Re-arranging:
(1 − β) − βcθ − (r+λ)c
dθ
q(θ)
=
(r+λ)pcη
dp
βpcθ + q(θ)
Using the equilibrium condition (4) to substitute for the numerator, we have
the elasticity of θ with respect to p for the XJD model:
−1
dθ p (r + λ) pcη
= (1 − β) b βpcθ +
dp θ
q (θ)
η=
−q 0 (θ) θ
q (θ)
(5)
Note that this is a bit different than the elasticity derived at the end of Lecture
9, as in the lecture notes it was assumed that vacancy cost is c rather than pc.
Under the NJD model, equilibrium θ and R satisfy (JC) and (JD). Total differentiation of both with respect to p gives
1−β
r+λ
−dR
dp
Z 1
λ
dR
1−
dG (z)
r+λ R
dp
=
=
−cq 0 (θ)
2
[q (θ)]
dθ
dp
−b
βc dθ
+
.
2
p
1 − β dp
Together they imply that the elasticity of θ wrt p in the NJD model:
−1
Z 1
dθ p (r + λ) pcη
λ
1−
dG (z)
= (1 − β) b βpcθ +
dp θ
q (θ)
r+λ R
4
(6)
Comparing (5) and (6), the key difference is
λ
1−
r+λ
Z
1
dG (z) < 1,
R
so, the elasticity is larger under the NJD model. Therefore, volatility in θ is
higher in the NJD model. But by how much? That depends on 1 − R which
is pinned down by the JC condition,
1−R=
c (r + λ)
,
q (θ) (1 − β)
which is usually a small number in calibration. Therefore, the change in θ is
higher in NJD but not by much, which implies that the shift in JC (in the u − v
diagram) for both models are similar.
Question 2 – Search externalities
1. The social planner maximizes the sum of leisure value and net output (output
minus vacancy costs), subject to a matching technology which describes the
dynamics of the unemployment rate. The matching function implies
q (θ) =
v −η
m
q (θ)
=µ
= µθ−η =⇒ q 0 (θ) = −η
v
u
θ
which is decreasing in θ and has elasticity η. The job arrival rate is θq (θ) =
µθ1−η .
2. The Hamiltonian is:
H = e−rt [(1 − u) p + uz − pcθu] + ψ (λ (1 − u) − q (θ) θu)
the first order conditions are:
∂H
∂θ
∂H
∂u
=
0 ⇔ −e−rt pcu + ψ (−q (θ) u − q 0 (θ) θu) = 0
(7)
=
−ψ̇ ⇔ e−rt [−p + z − pcθ] + ψ (−λ − q (θ) θ) = −ψ̇
(8)
3. Use the matching function in (7) to get
e−rt pc = ψ (η − 1) q (θ) .
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Differentiate with respect to time and divide by the above equation to get:
−r =
ψ̇ q 0 (θ)
ψ̇
θ̇
+
θ̇ = − η
ψ
q (θ)
ψ
θ
(9)
and (8) implies
ψ̇
[p − z + pcθ] (η − 1) q (θ)
=
+ λ + q (θ) θ
ψ
pc
From (9),
θ̇ = 0 ⇔ r + λ + q (θ) θ +
[p − z + pcθ] (η − 1) q (θ)
=0
pc
(10)
∗
which implies that the optimal θSP
is independent of u. Also
u̇ = 0 ⇔ λ (1 − u) − q (θ) θu = 0 =⇒ u =
λ
λ
=
λ + q (θ) θ
λ + µθ1−η
so u decreases in θ. The Beveridge diagram is obtained by substituting θ ≡ uv .
∗
solves (10) which can be simplified to
4. The θSP
r+λ
pc + θpc − [p − z + pcθ] (1 − η)
q (θ)
r+λ
pc + ηθpc − (1 − η) (p − z)
q (θ)
=
0⇔
=
0.
(11)
Recall that in the decentralized equilibrium:
JC : w = p −
r+λ
pc;
q (θ)
W C : w = (1 − β) z + βp (1 + cθ)
and that θ∗ solves:
r+λ
q (θ)
pc + βθpc − (1 − β) (p − z) = 0.
∗
Therefore θSP
= θ∗ if and only if β = η. Intuitively, if β > (<) η, then the
wage determined in the Nash Bargaining process between the firm and the
worker may be too high (too low). The parameter β measures the fraction
of the surplus going to the worker. On the other hand, η measures the job
searcher’s share in the matching production function. In a perfectly competitive market, one should expect η to be worker’s income share. When β = η,
the Nash Bargaining works just like a perfectly competitive world, rewarding
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the worker with the ’fair’ value, and the search externality is internalized.
5. Since θ∗ solves
(r + λ) pc + ηθq (θ) pc − (1 − η) (p − z) q (θ) = 0
(12)
Given θ∗ , we can plot JC : v = θ∗ u, which is a straight line with slope θ∗
in the u − v space. Together with the BC curve (u̇ = 0), we have a dynamic
system for u and v. Moreover the saddle path (the optimal decision rule) is
just the JC curve which describes the optimal v to post for a given u.
Now consider a productivity shock, p increases. Differentiate equation (12)
to get:
(ηpc[q(θ∗ ) + θ∗ q 0 (θ∗ )] − (1 − η)(p − z)q 0 (θ∗ )) dθ∗ = ((1 − η)q(θ∗ ) − (r + λ)c − ηθ∗ q(θ∗ )c)dp
dθ∗
(1 − η)q(θ∗ ) − (r + λ)c − ηθ∗ q(θ∗ )c
=
dp
ηpc(q(θ∗ ) + θ∗ q 0 (θ∗ )) − (1 − η)(p − z)q 0 (θ∗ )
The numerator is positive, since from (12) we know that (1 − η)q(θ∗ ) − (r +
λ)c−ηθ∗ q(θ∗ )c = p1 (1−η)zq(θ∗ ) > 0 for θ∗ . The denominator is positive, since
a0 (θ) > 0 and q 0 (θ) < 0. Hence an increase in p increases θ∗ , so the JC curve
rotates counter-clockwise. The BC remains the same. The adjustment path
is to start at the original intersection point, jump vertically, i.e. stays constant
u, to the new JC curve, then down to the new intersection. It’s counterclockwise but not as smooth as the data. If the shock is anticipated, then the
adjustment takes place before the shock happens, and the adjustment path is
smoother.
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