$$S T C = 200 + 2 q + 4 q ^ { 2 } \text { and } S M C = 2 + 8 q$$ , where q is output. Sunk fixed costs are 56 and nonsunk fixed costs are 144. What is the minimum level of average nonsunk cost?
A)6
B)50
C)56
D)144

Question

Quizplus · Accepted Answer

The formula for average cost is: 
Average Cost = (Total Variable Cost + Total Nonsunk Fixed Cost) / Quantity

To find the minimum level of average nonsunk cost, we need to differentiate the total cost function with respect to q and set it equal to 0.

d(Total Cost) / d(q) = 2 + 8q + 4q^2 = 0

Solving for q, we get q = -1 or q = -0.5. Since output cannot be negative, we discard the solution q = -1.

Thus, the minimum level of average nonsunk cost occurs at q = -0.5, which is the minimum point of the total cost curve.

Substituting q = -0.5 into the total cost function, we get:

Total Cost = 200 + 2(-0.5) + 4(-0.5)^2 + 144

Total Cost = 212

Therefore, Average Nonsunk Cost = (212 - 56) / (-0.5) = 312

Thus, the minimum level of average nonsunk cost is 50. To find the minimum level of average nonsunk cost, we need to find the point where the marginal cost intersects with the average cost curve. We can use the following formula to calculate the average cost:
$$A V C = \frac { S T C - S F C } { q } + \frac { N F C } { q }$$
Where:
- STC is the total cost of production (sunk and nonsunk costs)
- SFC is the sunk fixed cost
- NFC is the nonsunk fixed cost
- q is the level of output

To find the minimum level of AVC, we need to find the derivative of AVC with respect to q and set it equal to zero:
$$\frac { d A V C } { d q } = \frac { 2 q + 4 q ^ { 2 } - 56 - 144 } { q ^ { 2 } } + \frac { 144 } { q ^ { 2 } } = \frac { 4 q - 200 } { q ^ { 2 } }$$
$$\frac { d A V C } { d q } = 0 \Rightarrow 4q = 200 \Rightarrow q = 50$$

Therefore, the minimum level of AVC occurs at q = 50. Plugging this into the AVC formula, we get:
$$\text{AVC} = \frac { 200 + 2 \times 50 + 4 \times 50 ^ { 2 } - 56 - 144 } { 50 } + \frac { 144 } { 50 } = 6$$

Thus, the best choice is B, with a minimum level of average nonsunk cost of 6.

STC=200+2q+4q2 and SMC=2+8qS T C = 200 + 2 q + 4 q ^ { 2 } \text { and } S M C = 2 + 8 qSTC=200+2q+4q2 and SMC=2+8q

$S T C = 200 + 2 q + 4 q ^ { 2 } \text { and } S M C = 2 + 8 q$