Question

Longhorn Insurance offers mail-order automobile insurance to preferred-risk drivers in the state of Texas. The company is the low-cost provider of insurance in this market with fixed costs of $18 million per year, plus variable costs of $750 for each driver insured on an annual basis. The demand curve for the company is: P = $1,500  $0.005Q

A. Calculate price and quantity at the profit maximizing level of output.

B. Calculate price and quantity at the revenue maximizing level of output.

C. Calculate the profits and return-on-sales levels at the profit maximizing level of output.

Answer 1

We are given demand equation as:-

P = 1500 - 0.005Q

Total revenue = P * Q = 1500Q - 0.005Q²

Total cost = Total variavble cost + Total fixed cost = 750 + 18000000 = 18000750

MR = dTR/dQ = 1500 - 2(0.005)Q = 1500 - 0.01Q

MC = 0

(1) Profit maximisation level:-

750 = 1500 - 0.01Q

0.01Q = 750

Q = 75000

Putting the value of quantity in demand curve we get price:-

P = 1500 - 0.005Q

P = 1500 - 0.005(75000) = 1500 - 375 = 1125

(2) Revenue maximisation point is the point when the extra revenue from selling the lastmarginal unit becomes 0

So equting MR = 0

1500 - 0.01Q = 0

1500/0.01 = Q

150000

Putting this value in the demand function we get the price:-

P = 1500 - 0.005Q

P = 1500 - 0.005(150000)

P = 750

(3) Now for profits we have:-

TC= 18000000 + 750(75000) = 74,250,000

TR = 1500(75000) - 0.005(75000)² = 112500000 - 28125000 = 84375000

profit = TR - TC = 84375000 - 74250000 = 10125000

sales margin = P / TR = 10125000 / 84375000 = 0.12