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
This preview has intentionally blurred sections. Sign up to view the full version.
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
.
This is the end of the preview. Sign up
to
access the rest of the document.
 Fall '08
 Serra
 Macroeconomics, Supply And Demand, real wage, search market, Search Technology, ﬁrst order condition

Click to edit the document details