Question

Suppose that GNV Pool owns the only community pool in​ Gainesville, Florida. The market demand for annual passes to this pool​ is:

Qd = 600 - 2P

where Qd is the quantity of annual passes demanded and P is the​ per-pass price. GNV Pool knows that its total cost of providing annual passes​ is:

TC (Q ) = 25,000 + 0.25Q ^2.

The​ profit-maximizing price is ​$ ________. The​ profit-maximizing quantity is ________ annual passes. The amount of economic profit​ (or loss) is ​$ _________. ​(Enter a negative sign if your answer is a loss​.)

Answer 1

As GNV pool owns the only community pool so it has a monopoly and thus determines its profit maximizing quantity according to the rule MR = MC.

Qd = 600 - 2P
So, 2P = 600 - Q
So, P = (600/2) - (Q/2)
So, P = 300 - 0.5Q
Total Revenue, TR = P*Q = (300 - 0.5Q)*Q = 300Q - 0.5Q²
Marginal Revenue, MR = d(TR)/dQ = 300 - 2(0.5Q) = 300 - Q

Marginal Cost, MC = d(TC)/dQ = 2(0.25Q) = 0.5Q

Now, MR = MC gives,
300 - Q = 0.5Q
So, 0.5Q + Q = 1.5Q = 300
So, Q = 300/1.5
So, Q = 200

P = 300 - 0.5Q = 300 - 0.5(200) = 300 - 100 = 200

Economic profit = TR - TC = P*Q - (25,000 + 0.25Q^2) = (200)*(200) - 25,000 - 0.25(200)² = 40,000 - 25,000 - 10,000 = 5,000

The​ profit-maximizing price is ​$ ___200_____. The​ profit-maximizing quantity is ____200____ annual passes. The amount of economic profit​ is ​$ ____5,000_____.