Suppose a firm's total cost curve is given by the equation $T C = Q ^ { 2 } + 2 Q + 100$. The firm's marginal cost is $M C = 2 Q + 2$. At what level of $Q$ does the firm's average cost curve reach a minimum?
A) 100
B) 2
C) 10
D) 20

Question

Quizplus · Accepted Answer

To find the level of Q at which the firm's average cost is minimized, we need to find the derivative of the firm's total cost function and set it equal to zero.

\begin{aligned} A C &= \frac { T C } { Q } \ &= \frac { Q ^ { 2 } + 2 Q + 100 } { Q } \ &= Q + 2 + \frac { 100 } { Q } \ \end{aligned}

Now we take the derivative of the average cost and set it equal to zero to find the minimum point:

\begin{aligned} \frac { d A C } { d Q } = 1 - \frac { 100 } { Q ^ { 2 } } &= 0 \ Q ^ { 2 } &= 100 \ Q &= \sqrt { 100 } = 10 \end{aligned}

Therefore, the firm's average cost is minimized at Q = 10. The average cost is calculated by dividing total cost by quantity. Therefore, the average cost equation is:
$$
A C = \frac{T C}{Q} = \frac{Q ^ { 2 } + 2 Q + 100}{Q} = Q + 2 + \frac{100}{Q}
$$
To find the minimum of the average cost, we need to take the derivative of it and set it to zero:
$$
A C ^ { \prime } = 1 - \frac{100}{Q ^ { 2 }} = 0
$$
Solving for Q, we get Q=10. Therefore, the firm's average cost curve reaches a minimum at Q=10.

Suppose a Firm's Total Cost Curve Is Given by the Equation