For a firm, let total cost be $T C ( Q ) = 160 + 10 Q ^ { 2 }$ and marginal cost be $M C ( Q ) = 20 Q$. What is the minimum efficient scale for this firm?
A) 0
B) 2
C) 4
D) indeterminate

Question

Quizplus · Accepted Answer

The minimum efficient scale occurs where average total cost is at its minimum, which is where marginal cost equals average total cost. So we set $MC(Q) = ATC(Q)$. 
$$\begin{aligned} 
ATC(Q) &= \frac{TC(Q)}{Q}\
&= \frac{160+10Q^2}{Q}\
\end{aligned}$$
Setting $MC(Q) = ATC(Q)$ gives us: 
$$20Q = \frac{160+10Q^2}{Q}$$
$$20Q^2 = 160 + 10Q^2$$
$$10Q^2 = 160$$
$$Q^* = 4$$
Thus, the minimum efficient scale for this firm is 4. To find the minimum efficient scale for the firm, we need to find the quantity level where the average cost of producing each unit of output is at its minimum. The formula for average cost is:
AC(Q) = TC(Q)/Q

Substituting the given total cost function, we get:
AC(Q) = (160 + 10Q^2)/Q

To find the minimum point of this function, we take the derivative with respect to Q and set it equal to zero:
d/dQ (160/Q + 10Q) = 160/Q^2 + 10 = 0
160 = -10Q^2
Q^2 = -16

Since a squared quantity cannot be negative, this equation has no real solution. Therefore, the minimum efficient scale cannot be found and the answer is (D) indeterminate.

Note: This result may seem counterintuitive, but it is because we have assumed a cost function that does not follow the typical U-shape. In reality, most cost functions do have a minimum point where average cost is at its lowest, and this is where the firm should aim to produce in order to be most efficient.

For a Firm, Let Total Cost Be TC(Q)=160+10Q2T C ( Q ) = 160 + 10 Q ^ { 2 }TC(Q)=160+10Q2

For a Firm, Let Total Cost Be $T C ( Q ) = 160 + 10 Q ^ { 2 }$