If $S T C = 200 + 2 q + 4 q ^ { 2 }$ , $S M C = 2 + 8 q$ , where q is output and all fixed costs are sunk, the firm's short-run supply curve is

Question

Quizplus · Accepted Answer

To find the short-run supply curve, we need to compare the minimum possible variable cost (SVC) to produce a given quantity with the market price at which that quantity can be sold:

$$SVC = STC - FTC = 200 + 2q + 4q^2$$
$$MC = \frac{dSTC}{dq} = 2 + 8q$$

The firm will produce output q only if the market price P≥SVC(q), which implies:

$$P \geq 200 + 2q + 4q^2$$
$$P \geq 2(100+q+2q^2)$$
$$P \geq 2(q+1)^2 + 98$$

Therefore, the firm's short-run supply curve is:

$$s(P) = \left\{
        \begin{array}{ll}
            0 & \quad P < 2(q+1)^2 + 98 \
            q & \quad P \geq 2(q+1)^2 + 98
        \end{array}
    \right.$$
    
Solving for q in terms of P, we get:

$$q(P) = \frac{\sqrt{P-98}}{2}-1$$

Substituting back into the supply curve, we get:

$$s(P) = \left\{
        \begin{array}{ll}
            0 & \quad P < 2 \
            \frac{\sqrt{P-98}}{2}-1 & \quad P \geq 2
        \end{array}
    \right.$$
    
Simplifying, we get:

$$s(P) = \left\{
        \begin{array}{ll}
            0 & \quad P < 2 \
            0.125P-0.25 & \quad P \geq 2
        \end{array}
    \right.$$
    
Therefore, the correct answer is B.

If STC=200+2q+4q2S T C = 200 + 2 q + 4 q ^ { 2 }STC=200+2q+4q2

If $S T C = 200 + 2 q + 4 q ^ { 2 }$