ECON 2500 Fall 2008 Dr. Tufte
Homework 23
Due: Monday, October 27, at the start of class (by 8:30 AM if you want to get it back in class on Monday)
If possible, answer questions in the space provided.
Suppose a firm is producing output by hiring labor, N, at the wage, w, and
renting capital, K, at the price, r. Their costs are thus
C=wN =rK
. The
production of output, y, they obtain is governed by a Cobb-Douglas
production function, with
y=N
α
K
1−α
.
1) What is the firm’s demand for labor that maximizes their production
subject to a constraint that they can’t pay more than M for labor and
capital combined? Note that you can, and should, set this up as a
problem to maximize the natural log of the production function.
max L=αln
(
N
)
+
(
1−α
)
ln (K) + λ(C−wN −rK)
∂ L
∂ N =α
N−λw=0
∂ L
∂ K =1−α
K−λr=0
α
N
(1−α)
K
=w
r
αrk=
(
1−α
)
wN
C=wn+
(
1−α
)
wN /α
N=αC
w
2) Can you show that you will get the same answer if you minimize costs
subject to the constraint that output be greater than some number Z?
Hint: once again, do yourself a favor and express the constraint so
that ln(Z) minus the log of the production function.
min L=wN+rK +λ[ln (Z)−αln
(
N
)
−
(
1−α
)
ln
(
K
)
]
∂ L
∂ N =w−λα
N=0
∂ L
∂ K =r−λ1−α
K=0
