Economics 302
4 Week Summer Session 2006
Answers to Homework #3
Due Monday, June 12, 2006
5/30/06
Homework will be graded for content as well as neatness.
Sloppy or illegible work will
not receive full credit. This homework requires the use of Excel which is a spreadsheet
program.
For this homework please round to two places past the decimal.
1.
(5 points) Suppose you are given the production function
Y = F(K, L) = AK
1/4
L
3/4
Where Y is real GDP, K is capital, L is labor, and A is a parameter expressing
the level of technology available in this economy.
a.
Suppose K and L are both tripled in this economy.
Does this production
function exhibit constant returns to scale? Provide a proof of your answer.
If this production function exhibits constant returns to scale then F(zK, zL)
= zF(K, L) = zY.
So, we look at the question, what happens to Y if K and
L are increased by the same proportion? Let’s define a new level of K, K’
= 3K, and a new level of L, L’ = 3L and consider what happens to our
level of output, Y’.
Y’ =
AK’
1/4
L’
3/4
= A(3K)
1/4
(3L)
3/4
Y’ = A
K
1/4
(3)
1/4
L
3/4
(3)
3/4
Y’ = A(3)
K
1/4
L
3/4
Y’ = 3Y, since Y = AK
1/4
L
3/4
b.
Rewrite the above production function in terms of output per worker, y.
That is, normalize the function so that it can be written as y = f(k) where k
is capital per worker.
Provide a verbal interpretation of your mathematical
result.
Y = F(K, L) = AK
1/4
L
3/4
Y/L = y
y = (1/L)( AK
1/4
L
3/4
) = AK
1/4
L
-1/4
= A(K/L)
1/4
= Ak
1/4
= f(k)
Output per worker, y, depends upon the amount of capital per worker (k)
and the level of technology (A).
c.
Using the expression you found in part (b) and assuming labor is equal to
16 units, fill in the following table. Recall that k, the capital to labor ratio
(K/L), measures capital intensity: that is, it provides a measure of how
much capital each unit of labor has to work with in the economy. Assume
1