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WELLESLEY COLLEGE
DEPARTMENT OF ECONOMICS
ECONOMICS 20103
JOHNSON
Answers to Problem Set #7
1.
P&R, page 550, #6:
Ifwe set up the profit maximization problem directly, we can solve for the
L*(w,P) function directly:
IT=
P·\w.L~
P(12LL!)wL
~:::~
=
12f'.2.V
L

W
oL
~'?
L :
t
2.
f 
vi
L
lt
~w,r)'"
Now, proceed with the following:
Suppose a firm's production function is given
by Q
=
12L . L ,
I
for L
=
0 to 6, where L is
labor
input
per day
and Q
is
output
per day. Derive
and
draw
the firm's demand for
curve
if
the firm's
output sells for $10
in a competitive market. How
many workers will
the
firm
hire when
the wage
rate
is $30
per day? $60 per day? (Hint: The marginal product
of
is 12  2L.)
The demand for labor is given by the marginal revenue product oflabor. This is equal to
the product ofmarginal revenue and the marginal product oflabor:
MRP
=
(MR)(MP).
L
L
In a competitive market, price is equal to marginal revenue, so
MR
=
10. We are given
MP
=
12 
21,
(the slope ofthe production function).
L
Wage
120
100
80
60
40
20
Labor
1.5
3.0
4.5
Figure 14.6
Therefore, the
=
(10)(12 
21,).
The firm's profitmaximizing quantity of labor
L
occurs where
=
w. If
w
=
30, then 30
=
120 
20L
at the optimum. Solving for
L
L
yields 4.5 hours per day. Similarly,
ifw
=
60, solving for
L
yields 3 hours per day.
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The cost minimizing demands for L and K are homogeneous
of degree ZERO in
wand r. Why? Well,
if we double w and r
,let's say, we do NOT change the
slope
of the isocost line and therefore, for a GIVEN q
,we will NOT change
the cost minimizing demands for L and
K.
Same tangency point, given that the
isocost slope does not change.
However, the TC*(w,r,q) function is homogeneous
of degree ONE in
wand
r.
Why? Well, given that L* and K* don't change when, say, w and r double, it IS
true that the TOTAL COST
ofthat bundle ofinputs is now TWICE as
expensive. Thus, even though the isocost slope doesn't change, the TC value (in
$) associated with that same level ofL*, K* and thus q has to double. Think
of
it this way: the original isocost line would shift IN because
of the doubling of
w and
r. SO, to reach the original isoquant, the TC outlay must also double to
shift the isocost curve back out.
As for the unconditional (profit maximizing) functions, L*(w,r,P) and K*(w,r,P)
are homogeneous
of degree ZERO in
w,
r, and P. Why? Well, firms will hire
L and K up to the point where the marginal product
of the factor equals its real
input price (either the real wage or the real rental rate
ofcapital).
If w , r , and P
all double, say, then the real wage and the real rental price
of capital stay the
same, implying that the profit maximizing firm will not alter its demands for L
and K, thus leaving output unchanged as well.
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This note was uploaded on 06/15/2010 for the course ECON 201 taught by Professor Johnson during the Fall '08 term at Wellesley College.
 Fall '08
 JOHNSON
 Economics

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