Solutions to Problem Set 1
EC720.01  Math for Economists
Peter Ireland
Boston College, Department of Economics
Fall 2009
Due Thursday, September 17
1. Profit Maximization
Consider a firm that produces output
y
with capital
k
and labor
l
according to the technology
described by
k
a
l
b
≥
y,
(1)
where 0
< a <
1, 0
< b <
1, and 0
< a
+
b <
1. The firm sells each unit of output at the
price
p
, rents each unit of capital at the rate
r
, and hires each unit of labor at the wage
w
.
Hence it chooses
y
,
k
, and
l
to maximize profits
py

rk

wl
subject to the constraint just shown in (1).
a. The Lagrangian for this problem is
L
(
y, k, l, λ
) =
py

rk

wl
+
λ
(
k
a
l
b

y
)
.
b. According to the KuhnTucker theorem, the values
y
*
,
k
*
, and
l
*
that solve the firm’s
problem, together with the associated value
λ
*
for the multiplier, must satisfy the
firstorder conditions
L
1
(
y
*
, k
*
, l
*
, λ
*
) =
p

λ
*
= 0
,
L
2
(
y
*
, k
*
, l
*
, λ
*
) =

r
+
aλ
*
(
k
*
)
a

1
(
l
*
)
b
= 0
,
and
L
3
(
y
*
, k
*
, l
*
, λ
*
) =

w
+
bλ
*
(
k
*
)
a
(
l
*
)
b

1
= 0
,
the constraint
L
4
(
y
*
, k
*
, l
*
, λ
*
) = (
k
*
)
a
(
l
*
)
b

y
*
≥
0
,
the nonnegativity condition
λ
*
≥
0
,
and the complementary slackness condition
λ
*
[(
k
*
)
a
(
l
*
)
b

y
*
] = 0
.
c. The firstorder condition for
y
*
reveals that
λ
*
>
0 whenever
p >
0. Assuming that this
is the case, the complementary slackness condition requires the constraint to hold as
an equality. Hence, the firstorder conditions can be used together with the binding
1
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constraint to solve for
y
*
,
k
*
,
l
*
, and
λ
*
in terms of the model’s parameters
a
,
b
,
p
,
r
,
and
w
:
y
*
=
a
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 Fall '09
 IRELAND
 Economics, Utility, c∗, firstorder conditions, complementary slackness condition

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