Involve a profit-maximizing monopolist. Using time-series data, the demand function for the monopolist has been estimated as
$Q _ { d } = 142,000 - 500 P + 6 M - 400 P _ { R }$
where
$Q _ { d }$ is the amount sold, P is price, M is income, and
$P _ { R }$ is the price of a related good. The estimated values for M and
$P _ { R }$ in 2012 are $25,000 and $200, respectively. The short-run marginal cost curve for this firm has been estimated as:
$M C = 200 - 0.024 Q + 0.000006 Q ^ { 2 }$
Total fixed cost is forecast to be $500,000 in 2015.
-What is the optimal price?

Question

Involve a profit-maximizing monopolist. Using time-series data, the demand function for the monopolist has been estimated as
 $Q _ { d } = 142,000 - 500 P + 6 M - 400 P _ { R }$ 
where
 $Q _ { d }$ is the amount sold, P is price, M is income, and
 $P _ { R }$ is the price of a related good. The estimated values for M and
 $P _ { R }$ in 2012 are $25,000 and $200, respectively. The short-run marginal cost curve for this firm has been estimated as:
 $M C = 200 - 0.024 Q + 0.000006 Q ^ { 2 }$ 
Total fixed cost is forecast to be $500,000 in 2015.
-What is the optimal price?

Quizplus · Accepted Answer

To find the optimal price, set marginal revenue (MR) equal to marginal cost (MC). First, derive the inverse demand function and then the MR function. Solve for Q where MR = MC, and use the demand function to find the corresponding price. The calculations show that the optimal price is $408.

Involve a Profit-Maximizing Monopolist Qd=142,000−500P+6M−400PRQ _ { d } = 142,000 - 500 P + 6 M - 400 P _ { R }Qd​=142,000−500P+6M−400PR​

Involve a Profit-Maximizing Monopolist $Q _ { d } = 142,000 - 500 P + 6 M - 400 P _ { R }$