Question

1. You are the manager of a monopoly and your cost function is C(Q) =2Q. You need to determine the optimal level of output for your firm, but the demand for your firm’s product will depend on whether or not a new tax law is passed. If passed, the new tax law will reduce income taxes and increase consumers’ disposable income. Politicians have determined that there is a 70% chance that the tax law will be passed and a 30% chance that will not. If the tax law is passed, the demand for your firm’s product will be:  

Q = 100 – 2P

If the tax law is not passed, the demand for your firm’s product will be:

Q = 75 – 3P

a. How much output should you produce to maximize expected profits?

b. What is the expected price for your product?

c. What are your anticipated profits?

Answer 1

Expected demand = 70%(Q when law is passed) + 30%(Q when law is not passed.)
So, expected demand, Q = 0.7(100 – 2P) + 0.3(75 - 3P) = 70 - 1.4P + 22.5 - 0.9P = 92.5 - 2.3P
Thus, Q = 92.5 - 2.3P
So, 2.3P = 92.5 - Q
So, P = (92.5/2.3) - (Q/2.3)

Profit is maximized when MR = MC.
TR = P*Q = [(92.5/2.3) - (Q/2.3)]*Q = (92.5Q/2.3) - (Q²/2.3)
So, MR = d(TR)/dQ = (92.5/2.3) - (2Q/2.3)

MC = dC/dQ = 2

So, MR = MC gives,
(92.5/2.3) - (2Q/2.3) = 2
So, 2Q/2.3 = (92.5/2.3) - 2 = (92.5 - 2*2.3)/2.3 = (92.5 - 4.6)/2.3 = 87.9/2.3
So, Q = 87.9/2
So, Q = 43.95

b. P = (92.5/2.3) - (Q/2.3) = (92.5 - 43.95)/2.3 = 48.55/2.3 = 21.11

c. Profit = TR - TC = P*Q - 2Q = (P-2)Q = (21.11-2)*43.95 = 839.88