The market for sweet potatoes consists of 1,000 identical firms. Each firm has a short-run total cost curve of $S T C = 100 + 100 q + 100 q ^ { 2 }$, and a short-run marginal cost curve of $S M C = 100 + 200 q$ where $q$ is output. What is the minimum level of average variable costs?

Question

Quizplus · Accepted Answer

To find the minimum level of average variable costs, we need to first find the short-run average variable cost (SAVC) curve:

SAVC = AVC = STC/q = 100/q + 100 + 100q

Then, we take the derivative of AVC with respect to q and set it equal to 0 to find the minimum point:

dAVC/dq = -100/q^2 + 100 = 0

Solving for q, we get q = 1. Taking this value of q back into the AVC equation, we get:

AVC = 100/1 + 100 + 100(1) = 300

Therefore, the minimum level of average variable costs is AVC = 300/1,000 = 0.3 or $0.30. The closest choice is C, AVC = $100. In the short run, each firm will produce where marginal cost equals marginal revenue, as long as the price is greater than or equal to average variable costs. Thus, in this case, the firm will produce where $S M C = M R$ as long as $P \geq A V C$. 
Setting $S M C = M R$, we get 
$$100 + 200 q = P $$ 
Solving for $q$, we get 
$$q = \frac{P - 100}{200}$$
The short-run profit-maximizing level of output will be where marginal cost equals marginal revenue, or 
$$100 + 200q = P.$$
Substituting for q:
$$100 + 200 \times \frac{P-100}{200} = P.$$
This simplifies to 
$$P = 300.$$
Therefore, the minimum level of average variable cost is 
$$A V C = \frac{100+200q}{q} = \frac{100 + 200 \times \frac{P-100}{200}}{\frac{P-100}{200}} = \frac{100 + P -100}{\frac{P-100}{200}} = 200.$$ 
Thus, the answer is (C).

The Market for Sweet Potatoes Consists of 1,000 Identical Firms STC=100+100q+100q2S T C = 100 + 100 q + 100 q ^ { 2 }STC=100+100q+100q2

The Market for Sweet Potatoes Consists of 1,000 Identical Firms $S T C = 100 + 100 q + 100 q ^ { 2 }$