Question

Demand function: P=20-Q

Total Cost function: C=Q22 +8Q+2

1. What output maximizes total profit? What are the corresponding values of price, profit, and total revenue (sales)?

2. What output maximizes sales, and what are the corresponding values of price, profit, and total revenue?

Answer 1

Demand function: P = 20-Q. Total Cost function: C = Q^2 + 8Q + 2

1. Profit = PQ - C

= (20 - Q)Q - Q^2 - 8Q - 2

= 20Q - Q^2 - Q^2 - 8Q - 2

= 12Q - 2Q^2 - 2

Profit is maximum when its derivative is 0

12 - 4Q = 0

Q = 3 units

P = 20 - 3 = $17

Sales = PQ = $51

Cost = 3^2 + 8*3 + 2 = $35

Profit = 51 - 35 = $16

2. Sales = PQ

= 20Q - Q^2

Sales are maximized when its derivative is 0

20 - 2Q = 0

Q = 10

Price = 20 - 10 = $10

Profit = 12*10 - 2*10^2 - 2 = -$82

Sales = 10*10 = $100