Question

Exercise 6. Consider a firm with monopoly power that faces the demand curve P= 100 – 3Q +4A 1/2 and has the total cost function C = 4Q+ 10Q + A where A is the level of advertising expenditures, and P and Q are price and output a. Find the values of A, Q and P that maximizes the firm's profit. b. Find the maximum level of profit.

Answer 1

Answer 6

(a)

Profit = TR - C

where TR = Total Revenue = P*Q = (100 - 3Q + 4A^1/2)Q

and C = Total Cost = 4Q² + 10Q + A

Thus, Profit(Pr) = TR - C = (100 - 3Q + 4A^1/2)Q - (4Q² + 10Q + A) ------------------(1)

First Order condition :

$\frac{\partial Pr}{\partial Q} = 0 =>100 - 6Q + 4A^{1/2} - 8Q - 10 = 0 => 90 - 14Q + 4A^{1/2} = 0$

$\frac{\partial Pr}{\partial A} = 0 =>((1/2)4/A^{1/2})Q -1 = 0 => A^{1/2} = 2Q$

From above we have 90 - 14Q + 4A^1/2 = 0 => 90 - 14Q + 4*2Q = 0 => 6Q = 90 => Q = 15

=> A = (2Q)² = (2*15)² = 900

P = 100 - 3*15 + 4*900^1/2 = 175

Hence, Profit maximizing level of A = 300, Q = 15 and P = 175

(b)

Putting these values in (1) we get :

Profit(Pr) = (100 - 3*15 + 4*900^1/2)15 - (4*15² + 10*15 + 900) = 675

Hence, Maximum level of profit = 675