Answer: the profit-maximizing price is 60. Option c. 50 is incorrect
Step-by-step explanation:
o answer the questions, we need to analyze the Cournot duopoly model using the given demand curve and marginal cost.
Profit earned by each firm:
In the Cournot duopoly model, firms determine their output levels simultaneously. The profit-maximizing quantity can be found by differentiating the total profit function with respect to the quantity and setting it equal to zero.
Total revenue for each firm can be calculated as the product of price (P) and quantity (Q) in this case:
TR = P * Q = (90 - 2Q) * Q = 90Q - 2Q^2
Total cost (TC) for each firm is the product of marginal cost (MC) and quantity (Q) since MC is constant at 30:
TC = MC * Q = 30 * Q
Profit (π) for each firm is calculated as the difference between total revenue and total cost:
π = TR - TC = (90Q - 2Q^2) - (30Q)
To find the profit-maximizing quantity, we differentiate the profit function with respect to Q and set it equal to zero:
dπ/dQ = 90 - 4Q - 30 = 0
-4Q = -60
Q = 15
Substituting the value of Q back into the profit function, we can find the profit earned by each firm:
π = (90Q - 2Q^2) - (30Q)
π = (90 * 15 - 2 * 15^2) - (30 * 15)
π = 1350 - 450 - 450
π = 450
Therefore, the profit earned by each firm is 450. Option c. 500 is the closest answer, but the correct answer is 450.
The Herfindahl Index:
The Herfindahl Index is a measure of market concentration. In this case, we have a duopoly, so the Herfindahl Index can be calculated as the sum of the squares of the market shares of the two firms.
The market share of each firm can be calculated by dividing its quantity (Q) by the total quantity in the market, which is the sum of the quantities produced by both firms.
Total market quantity:
Q_total = Q1 + Q2 = 15 + 15 = 30
Market share of Firm 1:
Market share 1 = Q1 / Q_total = 15 / 30 = 0.5
Market share of Firm 2:
Market share 2 = Q2 / Q_total = 15 / 30 = 0.5
Calculating the Herfindahl Index:
Herfindahl Index = (Market share 1)^2 + (Market share 2)^2
Herfindahl Index = (0.5)^2 + (0.5)^2
Herfindahl Index = 0.25 + 0.25
Herfindahl Index = 0.5
Therefore, the Herfindahl Index is 0.5. Option d. 1250 is incorrect.
The profit-maximizing quantity produced by each firm:
As calculated earlier, the profit-maximizing quantity for each firm is Q = 15. Option a. 10 is incorrect.
The profit-maximizing price:
To find the profit-maximizing price, we substitute the profit-maximizing quantity (Q = 15) into the demand curve equation:
P = 90 - 2Q
P = 90 - 2 * 15
P = 90 - 30
P = 60