Answer to Question #294128 in Macroeconomics for Jotes@3

Question

A firm has the following total revenue and total cost functions: TR= 21Q-Q^2, Q^3/3-3Q^2+9Q+6 At what level of output does the firm maximize total revenue? Define the firm’s total profit as p = TR - TC. At what level of output does the firm maximize total profit? How much is the firm’s total profit at its maximum?

Answer 1

Solution:

TR = 21Q – Q²

TC = 1/3Q³ – 3Q² + 9Q + 6                                 

 

a.). Output maximizing revenue is where MR = MC:

Derive MR:

MR = $\frac{\partial TR} {\partial Q}$ = 21 – 2Q

 

Derive MC:

MC = $\frac{\partial TC} {\partial Q}$ = Q² – 6Q + 9

 

Set MR = MC:

21 – 2Q = Q² – 6Q + 9

Q = 6

Output maximizing revenue = 6 units.

 

b.). Profit is maximized where MR = MC:

Profit maximizing output = 6

Profit = TR – TC

P = (21Q – Q²) – (1/3Q³ – 3Q² + 9Q + 6)

P = ((21*6) – 6²) – (1/3(6³) – 3(6²) + 9(6) + 6)

P = (126 – 36) – (72 – 108 + 54 + 6)

P = 90 – 24 = 66

Profit = 66