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

144 THEORY OF PRODUCTION [CHAP. 6
(c)
Panel C shows decreasing returns to scale. Here, to double output per unit of time, the firm must more than
double the quantity of both inputs used per unit of time. Thus, OS , ST , TZ.
6.26 With respect to the production function in Table 6.14, (a) indicate whether we have increasing, decreasing,
or constant returns to scale. (b) Which of these points are on the same isoquant? (c) Is the law of
diminishing returns operating?
(a)
Table 6.14 indicates that Q ¼ f(L, K). This reads: The quantity of output produced per unit of time is a function
of (depends on) the quantity of labor and capital used per time period. With 1L and 1K, Q ¼ 50; with 2L
and 2K, Q ¼ 100; with 3L and 3K, Q ¼ 150. Thus we have constant returns to scale.
Table 6.14
3K 80 120 150
2K 70 100 120
1K 50 70 80
1L 2L 3L
(b)
(c)
The general equation for an isoquant is given by Q ¼ f(L, K) and refers to the different combinations of labor
and capital needed to produce a given level of output of a good or service. It can be seen from Table 6.14 that
an output of 70 units can be produced with either 1L and 2K or 2L and 1K. These are two points on the isoquant,
representing 70 units of output. Similarly, the firm can produce 80 units of output (and thus remain on
the same isoquant) by using either 1L and 3K or 1K and 3L. Finally, 120 units of output can be produced with
either 2L and 3K or 3L and 2K. These are two points on a higher isoquant.
The law of diminishing returns is a short-run law. In the short run, we look at how the level of output varies,
either by changing labor and keeping capital constant, or vice versa. This can be written in functional form as
Q ¼ f (L, K) orQ ¼ f (L, K). By doing this we get the TP L function and the TP K function, respectively. Note
that we get a different TP L function for each level at which we keep capital constant. (Similarly, by keeping the
amount of labor used constant at different levels, we generate different TP K functions.) If K ¼ 1, and labor
increases from 1 unit to 2 units and then to 3 units, Q increases from 50 units to 70 units and then to 80
units. Since the MP L falls continuously (from 50 to 20 to 10), the law of diminishimg returns is operating continuously.
The same is true for the TP L functions given by row 2 and row 3. The law of diminishing returns also
operates continuously along the TP K functions given by columns (1), (2), and (3). (The implicit assumption we
made in the last three sentences is that f(O, K) ¼ f(L, O) ¼ 0.)
THEORY OF PRODUCTION WITH CALCULUS
6.27
Starting with the general production function Q ¼ f(L, K), which states that output Q is a function of or
depends on the quantity of labor (L) and capital (K) used in production, derive the expression for the
slope of the isoquant using calculus.
Taking the total differential and setting it equal to zero (because output remains unchanged along a given
isoquant) we get
dQ ¼ @f @f
dL þ
@L @K dK ¼ 0
Thus, the expression for the absolute slope of the isoquant is
dK @f =@L
¼
dL @f =@K ¼ MP L
¼ MRTS LK
MP K
6.28
A firm faces the general production function of Q ¼ f(L, K) and given cost outlay of C ¼ wL þ rK,
where w is the wage of labor and r is the rental price of capital. Determine by using calculus the
amount of labor and capital that the firm should use in order to maximize output.