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Homework answers / question archive / For the cubic total cost function TC = a + bQ + cQ^2 + dQ^3 , where a = 0, b = 25, c = -10, and d = 1, the firm's average variable cost is minimized at an output of: a
For the cubic total cost function TC = a + bQ + cQ^2 + dQ^3 , where a = 0, b = 25, c = -10, and d = 1, the firm's average variable cost is minimized at an output of:
a. ten.
b. five.
c. two.
d. one.
The correct answer is: b. five.
Given the total cost curve:
TC=a+bQ+cQ2+dQ3TC=a+bQ+cQ2+dQ3
Where:
The firm's total variable cost will be:
VC=25Q−10Q2+Q3VC=25Q−10Q2+Q3
The average variable cost curve will be given by:
AVC=VCQ=25−10Q+Q2AVC=VCQ=25−10Q+Q2
The average variable cost curve is minimized when the first-order derivative of the average variable cost curve with respect to Q is equal to zero.
Minimizing the average total cost curve:
ΔAVCΔQ=−10+2Q=0ΔAVCΔQ=−10+2Q=0
−10+2Q=0−10+2Q=0
Solving for Q:
2Q=102Q=10
Q=102=5Q=102=5 units.
Thus, the firm's average variable cost curve is minimized at Q = 5 units.
