Problem Set 3
EC720.01  Math for Economists
Peter Ireland
Boston College, Department of Economics
Fall 2009
Due Thursday, October 1
Many famous results from microeconomic theory are now understood to be special cases
of the envelope theorem.
This problem set will ask you to invoke the envelope theorem
repeatedly to “prove” some of these results.
1. Hotelling’s Lemma
Consider a firm that produces output
y
with capital
k
and labor
l
according to the technology
described by
f
(
k, l
)
≥
y.
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 technological constraint just shown above.
a. Set up the Lagrangian for this problem, letting
λ
denote the multiplier on the constraint.
b. Next, write down the conditions that, according to the KuhnTucker theorem, must be
satisfied by the values
y
*
,
k
*
, and
l
*
that solve the firm’s problem, together with the
associated value
λ
*
for the multiplier.
c. Assume that the output and input prices
p
,
r
, and
w
and the production function
f
are
such that it is possible to solve uniquely for the values of
y
*
,
k
*
,
l
*
, and
λ
*
in terms of
the parameters
p
,
r
, and
w
. Then the function
y
*
(
p, r, w
) describing the optimal level
of output represents the firm’s supply function, and functions
k
*
(
p, r, w
) and
l
*
(
p, r, w
)
describing the optimal inputs are the firm’s factor demand curves. Along with these
functions, define the firm’s profit function as
π
(
p, r, w
) = max
y,k,l
py

rk

wl
subject to
f
(
k, l
)
≥
y.