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

CHAP. 7] COSTS OF PRODUCTION 175
7.27 For a Cobb-Douglas production function of the form Q ¼ AL a K 12a prove that (a) it shows constant
returns to scale and (b) the AP L is a function of K/L only.
(a)
Since a þ 1 2 a ¼ 1, this Cobb-Douglas production function exhibits constant returns to scale and is said to
be homogeneous of degree one or linearly homogeneous. “Constant returns to scale,” “homogeneous of
degree one,” and “linearly homogeneous” all mean the same thing and are used interchangeably.
Fig. 7-26
(b)
AP L ¼ Q L ¼ ALa K 1
L
a
1
¼ AL a 1 K 1 a ¼ A K L
a
Since for any Cobb-Douglas production function, A and a assume fixed values, AP L ¼ f(K/L) only. That
is, the AP L remains the same regardless of amount of L and K used in production, as long as K/L remains
constant (or along any expansion path or isocline). The same is true for MP L ¼ aA(K/L) 12a ¼ f(K/L).
7.28 (a) If the actual estimated a and b are as given in Table. 7.17, what type of returns to scale does each
industry exhibit? (b) By how much does output rise in the American food industry if L increases by 1%?
If K increases by 1%? (c) It has been found that for the United States and other developed nations only
about one-third of the increase in the standard of living over the years was due to the increase in the
physical units of L and K used. What was the remainder due to?
Table 7.17
Industry Country a b
1. Telphone Canada .70 .41
2. Gas France .83 .10
3. Chemicals India .80 .37
4. Electricity India .20 .67
5. Machinery and tools United States .71 .26
6. Food United States .72 .35
7. Communications U.S.S.R .80 .38