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. All fixed costs are sunk. What is the equation of the shortrun market supply curve?

Question

Quizplus · Accepted Answer

In perfect competition, the short-run market supply curve is derived by horizontally summing the individual firm’s supply curves. The short-run market supply curve equation is the sum of the individual firms' supply curves. Therefore, we need to derive the equation of an individual firm's supply curve.

To derive an individual firm's supply curve, we need to find its profit-maximizing level of output. At the profit-maximizing level of output, the firm's marginal cost (MC) equals its marginal revenue (MR), which is equal to the market price (P) in perfect competition.

The marginal cost curve for an individual firm is given as:
$S M C = 100 + 200 q$
Equating this to the market price:
$P = S M C = 100 + 200 q$
Solving this gives us:
$q = \frac{P - 100}{200}$

The short-run market supply is the sum of the individual firms' supply curves. Therefore, 
$Q ^ { s } = N q$
where N is the number of firms which is 1,000 in this case.

Thus, the equation of the short-run market supply curve is:
$Q ^ { s } = 1,000 \times \frac{P - 100}{200}$ 
which simplifies to:
$Q ^ { s } = 5 P - 500$ for $P \geq 100$, and $Q ^ { s } = 0$ otherwise.

Therefore, the best choice is A.

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 $S T C = 100 + 100 q + 100 q ^ { 2 }$