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 equation of an individual firm's short-run supply curve in this market?

Question

Quizplus · Accepted Answer

An individual firm's supply curve is determined by comparing the market price to the firm's marginal cost curve. In this case, the firm's marginal cost curve is given by $S M C = 100 +$ $200 q$. Setting this equal to the market price, $P$, gives $P = 100 +$ $200 q$. Solving for $q$, we get $q =$ $(P - 100) / 200$. However, if $q$ is negative, then the firm shuts down and produces nothing. Therefore, the correct equation for the supply curve is
$$q = \begin{cases} (P-100)/200-0.5, &\mbox{if } P \geq 100 \ 0, &\mbox{otherwise} \ \end{cases}$$ In a perfectly competitive market, each firm's short-run supply curve is determined by comparing the price (P) with the firm's short-run marginal cost (S MC). If P is greater than or equal to the minimum value of S MC, then the firm will produce the quantity where S MC equals P. If P is less than the minimum value of S MC, the firm will produce zero output.

So, first, we need to find the minimum value of S MC:
$S M C = 100 + 200 q$
To find the minimum value of S MC, we take the derivative and set it equal to zero:
$\frac { dS M C } { dq } = 200 = 0$
So, the minimum value of S MC is 200.

Now we can set up the firm's short-run supply curve:
For P ≥ 200, SMC = P, so:
$P = 100 + 200 q$
Solving for q, we get:
$q = \frac { P - 100 } { 200 }$

For P < 200, SMC > P, so the firm produces zero output.

Therefore, the firm's short-run supply curve is: 
$q = \begin{cases} \frac { P - 100 } { 200 } , & P \geq 200 \ 0, & P < 200 \end{cases}$

Simplifying, we get:
Answer: D
 The firm's short-run supply curve is: 
$q = \frac { P - 100 } { 200 }$ for $P \geq 200$, and $q = 0$ otherwise.

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