Suppose the demand curve for a monopolist is

QD = 500 − P, and the marginal revenue function
is MR = 500 − 2Q. The monopolist has a constant
marginal and average total cost of $50 per unit.
a.Find the monopolist’s profit-maximizing output
and price.
b. Calculate the monopolist’s profit.
c. What is the Lerner Index for this industry?

I am not sure about the answer but I think that for profit maximizing output . MR = MC , therefore 500-2Q=50 ( MC = 50 per unit ) , after calculating , we get the output of 275 and the price of 225. The profit is 48,125.

This is what I got for the lender index:

L = (P- MC) / P
L = (225 - 50) /225
L = (175) / 225
L = 0.77

These are my final responses:

a) Find the monopolist’s profit-maximizing output and price.
Output is 225 and price is 50
MR = 500 − 2Q
50 =500 – 2Q
-450 = -2Q
225 = Q (Profit maximizing quantity)
P = 500 − 2Q
P = 500 – 2(225)
P = 500 – 450
(Profit Price) P = 50
b) Calculate the monopolist’s profit.
TR = P *Q
TR = 50 * 225
TR = 11,250
TC = 500 + (50*225)
TC = 500+11250
TC = 11,750
Profit = TR – TC
Profit = 11250 – 11750
Profit = -500
c) What is the Lerner Index for this industry?
L = (P- MC) / P
L = (50 - 50) /50
L = (0) / 50
L = 0
I saw a Youtube video by Marginal Revolution University called "Office Hours: Calculating Monopoly Profit"

To find the monopolist's profit-maximizing output and price, we need to use the marginal revenue and marginal cost approach.

a. To maximize profit, the monopolist should produce at the quantity where marginal revenue (MR) equals marginal cost (MC).
In this case, MR = 500 - 2Q, and we know that MC is a constant $50 per unit.

Setting MR equal to MC:
500 - 2Q = 50

Solving for Q:
2Q = 500 - 50
2Q = 450
Q = 225

So the monopolist's profit-maximizing output is 225 units.

b. To find the price corresponding to the profit-maximizing output, we can substitute the quantity (Q) into the demand function.
QD = 500 - P

500 - P = 225
P = 500 - 225
P = 275

Therefore, the monopolist's profit-maximizing price is $275.

c. The Lerner Index measures the degree of market power a firm has by calculating the ratio of the difference between price and marginal cost to price.
Lerner Index = (P - MC) / P

In this case:
P = $275 (the price we found in part b)
MC = $50 (given in the question)

Lerner Index = (275 - 50) / 275
Lerner Index = 225 / 275
Lerner Index ≈ 0.818

Therefore, the Lerner Index for this industry is approximately 0.818.