Suppose the The demand curve for a monopolist is Q_D = 47,000 -50 P, and the marginal revenue function is MR = 940 - 0.04Q. The monopolist's Marginal Cost = 40 + 0.02Q and its

Total Cost =250,000 + 40Q + 0.01Q^2.

a. Find the monopolist's profit-maximizing output and price.

b. calculate the monopolist's profit/losses, if any.

c. What is the Lerner Index for this industry at the monopolist's profit-maximizing output and price.

I need to make sure that I am working this problem correctly in particular solving for P....

a. Profit maximizes where MR = MC

940 - 0.04Q = 40+0.02Q

Solving for Q:

900 = 0.06Q

Q = 15,000

Substituting Q in the demand curve, we can obtain P:

15,000 = 47,000 - 50P

P = 1,240

b. Re-arrange the demand curve equation to make P the subject

P = 940 - (Q/50)

TR = P * Q

TR = 940Q - ((Q^2)/50)

Substituting the Q found in part a.:

TR = (940*15,000) - (15000^2)/50)

= 14,100,000 - 4,500,000

= 9,600,000

TC = 250,000 + 40Q + 0.01Q^2

= 250,000 + (40*15000) + (0.01*15000^2)

= 250,000 + 600,000 + 2,250,000

= 3,100,000

Therefore profit = 9,600,000 - 3,100,000 = 6,500