Dominick Salvatore Schaums Outline of Microeconomics, 4th edition Schaums Outline Series 2006

CHAP. 7] COSTS OF PRODUCTION 177
Since with neutral technological progress the MP L and MP K increase in the same proportion, there is no substitution
of L for K (or K for L) in production at unchanged w/r, so that K/L remains unchanged at K/L ¼ 1 (see
point E 2 in panel A of Fig. 7.27). Since with K-using technological progress MP K increases proportionately more
than MP L , K is substituted for L in production at constant w/r so that K/L rises to K/L ¼ 3 (see point E 3 in panel B).
With L-using technological progress K/L falls to K/L ¼ 1/3 at constant w/r (see point E 3 in panel C).
7.31 (a) What is the relative share of NNP going to L and K and the ratio of the relative shares going to L and
K? (b) How do the different types of technological progress affect relative shares if w/r remains
constant?
(a)
(b)
Let w ¼ average wave rate, r ¼ average return on capital or interest rate, L ¼ total amount of L employed in
the economy, K ¼ total amount of capital, P ¼ general price index, and Q ¼ general quantity index (so that
PQ ¼ NNP). Then, the relative share of NNP going to L is wL/PQ, the relative share going to K ¼ rK/PQ,
and the ratio of the relative share going to L and K ¼ (wL/PQ) 4 (rK/P2) ¼ wL/rK.
Since neutral technological progress leaves K/L unchanged, the ratio of the relative share going to L and K
remains unchanged if w/r remains unchanged. Since K-using technological progress increases K/L (which
means that L/K falls), wL/rK falls. Finally, since L-using technological progress reduces K/L, wL/rK
increases.
COSTS OF PRODUCTION WITH CALCULUS
7.32
7.33
A firm faces the general cost function of C ¼ wL þ rK and production function of Q ¼ f(L, K). Derive
by using calculus the condition to minimize the cost of producing a given level of output (Q ).
Forming function Z 0 , which incorporates the cost function to be minimized to produce output Q , we get
Z 0 ¼ wL þ rK þ l 0 ½Q
f (L, K)Š
where l 0 is the Lagrangian multiplier. Taking the first partial derivative of Z 0 with respect to L and K and
setting them equal to zero, we get
@Z 0
@L ¼ w
Dividing the first equation by the second, we get
@f
l0
@L ¼ 0 and @Z 0
@K ¼ r
w @f =@L
¼
r @f =@K ¼ MP L
¼ MRTS LK
MP K
or
@Z0
l0
@K ¼ 0
MP L
w
¼ MP K
r
Given Q ¼ 100K 0.5 L 0.5 , w ¼ $30, and r ¼ $40. (a) Find the quantity of labor and capital that the firm
should use in order to minimize the cost of producing 1444 units of output. (b) What is this minimum
cost?
(a) Z 0 ¼ $30L þ $40K þ l 0 ½Q 100L 0:5 K 0:5 Š
@Z 0
@L ¼ $30 l0 50L 0:5 K 0:5 ¼ 0
@Z 0
@K ¼ $40 l0 50L 0:5 K 0:5 ¼ 0
Dividing the first partial derivative equation by the second, we have
3
4 ¼ K
so that K ¼ 3 (L)
L
4
By then substituting this value of K into the given production function for 1444 units of output, we get
p
1444 ¼ 100L 0:5 (0:75L) 0:5 so that 1444 ¼ 100L ¼ ffiffiffiffiffiffiffiffiffi
0:75
and L ¼ 1444
86:6 ¼ 16:67