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Suggested Solutions to Problem Set 12
Econ 702, Spring 2004
Prepared by Ahu Gemici and Omer Kagan Parmaksiz
May 3, 2004
Solution 1
Suppose we have a world described by the following
Γ
matrix, in
which probability of ﬁnding a job and loosing a job is .1. Find the stationary
distrubution of this economy and calculate the average duration of unemploy
ment.
Γ =
±
0
.
9
0
.
1
0
.
1
0
.
9
²
Stationary distribution
x
*
is the solution to the following equation,
Γ
T
x
*
=
x
*
(1)
0
.
90
e
+ 0
.
10
u
=
e
(2)
0
.
10
e
+ 0
.
90
u
=
u
(3)
both of which implie
e
=
u
and with normalization
e
+
u
= 1
(4)
the stationary distribution for employment states is
x
*
=
±
1
2
,
1
2
²
(5)
also the duration of unemployment is given by the inverse of probability of
transiting from unemployment to employment which is given by,
D
u
=
1
.
1
= 10
.
Solution 2
Consider the optimal unemployment insurance problem we covered
in class,
c
(
V
) = min
c,a,V
u
{
c
+
β
[1

p
(
a
)]
c
(
V
u
)
}
(6)
s.t
V
=
u
(
c
)

a
+
β
³
p
(
a
)
V
E
+ (1

p
(
a
))
V
u
´
(7)
Derive the envelope condition explicitly and show that
c
(
V
)
is convex.
1
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View Full Document To solve the problem, construct Lagragian function
L
=
c
+ [1

p
(
a
)]
βc
(
V
u
) +
θ
±
V

u
(
c
) +
a

β
±
p
(
a
)
V
E
+ (1

p
(
a
))
V
u
²²
FOC: (c)
θ
=
1
u
c
(8)
(a)
c
(
V
u
) =
θ
³
1
βp
0
(
a
)

(
V
E

V
u
)
´
(9)
(V
u
)
c
0
(
V
u
) =
θ
(10)
To derive EC, we write
c
(
V
)
=
min
c,a,V
u
c
+ [1

p
(
a
)]
βc
(
V
u
)

θ
±
V

u
(
c
) +
a

β
±
p
(
a
)
V
E
+ (1

p
(
a
))
V
u
²²
Take ﬁrst derivative of
c
(
V
) with respect to
V
when
c, a, V
u
take optimal value,
we get
c
0
(
V
)
=
∂c
*
∂V

p
0
(
a
)
∂a
*
∂V
βc
(
V
u
*
) + [1

p
(
a
*
)]
βc
0
(
V
u
*
)
∂V
u
*
∂V
+
θ
³
1

u
0
(
c
)
∂c
∂V
+
∂a
∂V

p
0
(
a
)
∂a
∂V
β
(
V
E

V
u
)

[1

p
(
a
)]
β
∂V
u
∂V
´
Substitute (9) and (10),
c
0
(
V
)
=
∂c
*
∂V

p
0
(
a
)
∂a
*
∂V
βθ
³
1
βp
0
(
a
)

(
V
E

V
u
)
´
+ [1

p
(
a
*
)]
βθ
∂V
u
*
∂V
+
θ
³
1

u
0
(
c
)
∂c
∂V
+
∂a
∂V

p
0
(
a
)
∂a
∂V
β
(
V
E

V
u
)

[1

p
(
a
)]
β
∂V
u
∂V
´
(11)
=
∂c
*
∂V

∂a
*
∂V
θp
0
(
a
)
∂a
*
∂V
βθ
(
V
E

V
u
)
+ [1

p
(
a
*
)]
βθ
∂V
u
*
∂V
(12)
+
θ
³
1

u
0
(
c
)
∂c
∂V
+
∂a
∂V

p
0
(
a
)
∂a
∂V
β
(
V
E

V
u
)

[1

p
(
a
)]
β
∂V
u
∂V
´
(13)
Combine (8), we can show envelope condition as
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This note was uploaded on 11/12/2010 for the course ECON 8108 taught by Professor Staff during the Spring '08 term at Minnesota.
 Spring '08
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