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Econ 102  Answer key assignment 8
Josef Schroth, November 16, 2008
Question 1
1. The social planner problem is maximizing consumption.
max
C
s
.
t
.
C
≤
Y
Y
≤
3
X
X
≤
L

V
L
≤
H
1/2
V
1/2
H
≤
1
All these constraints will bind in optimum. Hence we can rewrite the planning problem as
max 3
(
L

L
2
)
.
The ﬁrst order condition is
1

2
L
=
0
⇒
L
=
1
2
.
Then
V
=
1/4,
X
=
1/4, and
C
=
Y
=
3/4.
2. The ﬁrm maximizes its proﬁts. It uses
X
units of labor input to produce the consumption
good. It also uses
V
units of labor to recruit its workforce
L
, where
L
=
X
+
V
. Note that
recruiters recruit themselves in this model. The ﬁrm takes as given the prices for labor,
w
,
and the consumption good,
p
. It also takes as given the productivity
q
of recruiters. Note
that
q
will be determined by an aggregate search technology, but
w
/
p
is a free parameter
1
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View Full Documentso far. The ﬁrm problem is then
max
pY

wL
s
.
t
.
Y
≤
3
X
X
≤
L

V
L
≤
qV
3. All these constraints will bind in optimum. Hence we can rewrite the ﬁrm problem as
max
p
3
(
qV

V
)

wqV
.
The ﬁrst order condition is
p
3
(
q

1
)

wq
=
0
⇒
w
p
=
3
±
1

1
q
²
.
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 Fall '08
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 Macroeconomics

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