Question

Profit Maximization: Equations

21st Century Insurance offers mail-order automobile insurance to preferred-risk drivers in the Los Angeles area. The company is the low-cost provider of insurance in this market but doesn't believe its annual premium of $1,500 can be raised for competitive reasons. Rates are expected to remain stable during coming periods; hence, P = MR = $1,500 (this is a hint which might not be given in future problems). Total cost relation for the company is as follows
TC = $41,000,000 + $500Q + $0.005Q2
Calculate the profit-maximizing activity level (in terms of Q).
Calculate the company's optimal profit (p), and optimal profit as a percentage of sales revenue (i.e. profit margin).

Answer 1

• The profit-maximizing activity level (Q) for 21st Century Insurance is 100,00
• The company's optimal profit (p) is $108,900,000 and the company's optimal profit as a percentage of sales revenue is 72.6%.

Step-by-step explanation:

To calculate the profit-maximizing activity level (Q) for 21st Century Insurance, we need to find the quantity that maximizes profit. This can be determined by finding the level of output where marginal revenue (MR) equals marginal cost (MC).

1. Calculate the marginal cost (MC) by taking the derivative of the total cost (TC) function with respect to quantity (Q):

MC = dTC/dQ

MC = $500 + $0.01Q

2. Set marginal cost (MC) equal to the constant price (P) to find the profit-maximizing quantity:

MC = P

$500 + $0.01Q = $1,500

$0.01Q = $1,000

Q = $1,000 / $0.01

Q = 100,000

Therefore, the profit-maximizing activity level (Q) for 21st Century Insurance is 100,000.

3. Calculate the company's optimal profit (p) by subtracting the total cost (TC) from the total revenue (TR):

TR = P * Q

TR = $1,500 * 100,000

TR = $150,000,000

p = TR - TC

p = $150,000,000 - ($41,000,000 + $500Q + $0.005Q^2)

p = $150,000,000 - ($41,000,000 + $500 * 100,000 + $0.005 * 100,000^2)

p = $150,000,000 - ($41,000,000 + $50,000 + $50,000)

p = $150,000,000 - $41,100,000

p = $108,900,000

Therefore, the company's optimal profit (p) is $108,900,000.

4. Calculate the company's optimal profit margin as a percentage of sales revenue:

Profit margin = (Optimal profit / Sales revenue) * 100

Profit margin = ($108,900,000 / $150,000,000) * 100

Profit margin = 72.6%

Therefore, the company's optimal profit as a percentage of sales revenue is 72.6%.