certain production process employs two inputs--labor (L) and raw materials (R).
ID: 1177599 • Letter: C
Question
certain production process employs two inputs--labor (L) and raw materials (R). Output (Q) is a function of these two inputs and is given by the following relationship:
Q = 6L2 R2 - 0.10L3 R3
Assume that raw materials (input R) are fixed at 10 units.
(a) Find the number of units of input L that maximizes the total product function.
(b) Find the number of units of input L that maximizes the marginal product function.
(c) Find the number of units of input L that maximizes the average product function.
(d) Determine the boundaries for the three stages of production.
Explanation / Answer
a) Letting R = 10, we have Q(L) = 6(L^2)(10^2) - 0.10(L^3)(10^3) = 600L^2 - 100L^3.
for maximization dQ/dL=0
1200L-300L^2=0
L=4
b) MPL(L) = dQ/DL = 1200L - 300L^2.
Same As Above,
1200=600L
L=2
c) APL(L) = Q(L)/L = [600(L^2) - 100(L^3)]/L = 600L - 100L^2.
for maximum d(APL)/dL=0
600=200L
L=3
d) Depends on your definition of the three stages...
Increasing output per unit: 0 < L < 3
APL and MPL declining, TPL still increasing: 3 < L < 4
TPL Decreasing: 4 < L
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