Question

Please show work.

The XYZ corporation, a monopolist, receives a report from a consulting firm concluding that the demand function for its product is:

Q = 80 – 1.5P + 2.3M + 0.75A

Where Q is the number of units sold, P the price of its product, M is buyers’ per capita income, and A is the firm’s advertising expenditure. The total cost function is:

C(Q) =50,000 + 42Q – 8Q2 + 1.5Q3

Buyer’s per capita income is estimated to be $10,000 and the firm spends $200,000 on advertising.

a. How much output should the firm produce to maximize profit (minimize loss)?   

b. What price should be charged for the output?

c. How much profit (loss) does the firm make?

Answer 1

The XYZ corporation, a monopolist, receives a report from a consulting firm concluding that the demand function for its product is:

Q = 80 – 1.5P + 2.3M + 0.75A

Buyer’s per capita income is estimated to be $10,(000) and the firm spends $200(,000) on advertising. This implies that demand function is Q = 80 - 1.5P + 2.3*10 + 0.75*200

Q = 253 - 1.5P

Inverse demand is 1.5P = 253 - Q or P = 168.67 - Q/1.5. MR = 168.67 - 2Q/1.5

The total cost function is:

C(Q) =50,000 + 42Q – 8Q2 + 1.5Q3

MC = 42 - 16Q + 4,5Q^2

a. How much output should the firm produce to maximize profit (minimize loss)?   

Use the rule MR = MC

168.67 - 2Q/1.5 = 42 - 16Q + 4.5Q^2

4.5Q^2 - 14.7Q - 126.67 = 0

This gives Q = 7.1846

b. What price should be charged for the output?

Price = 168.67 - 7.1846/1.5 = 163.88

c. How much profit (loss) does the firm make?

Profit = TR - TC = 163.88*7.184 - (50,000 + 42*7.184 – 8*(7.184^2) + 1.5*(7.184^3) = -49267.7