Three firms produce identical products and compete in a market where the inverse demand function is P(q1, q2, q3) = 62 − q1− q2− q3. Each has a per-unit cost of 10 and zero fixed cost. They simultaneously choose quantities. In scenario (a), find the Nash equilibrium of this game and let A = firm 2's profit in the Nash equilibrium.In scenario (b), assume that the firms form a cartel, i.e., they act as a monopoly and split the profit evenly. If the total quantity produced by the cartel is Q, then the inverse demand is P(Q) = 62 - Q. Let B = firm 2's profit in the cartel. Calculate the value of A - B and enter your answer in the box below. Please round your answer to 3 decimal places (e.g., write 4/3 as 1.333).
Question
Three firms produce identical products and compete in a market where the inverse demand function is P(q1, q2, q3) = 62 − q1− q2− q3. Each has a per-unit cost of 10 and zero fixed cost. They simultaneously choose quantities. In scenario (a), find the Nash equilibrium of this game and let A = firm 2's profit in the Nash equilibrium.In scenario (b), assume that the firms form a cartel, i.e., they act as a monopoly and split the profit evenly. If the total quantity produced by the cartel is Q, then the inverse demand is P(Q) = 62 - Q. Let B = firm 2's profit in the cartel. Calculate the value of A - B and enter your answer in the box below. Please round your answer to 3 decimal places (e.g., write 4/3 as 1.333).
Solution
To solve this problem, we first need to find the Nash equilibrium in scenario (a) and then calculate the profit for firm 2. After that, we will find the profit for firm 2 in scenario (b) where the firms form a cartel. Finally, we will subtract B from A to find the value of A - B.
Step 1: Find the Nash equilibrium in scenario (a) The Nash equilibrium is a situation in which no player can benefit by changing their strategy while the other players keep theirs unchanged. In this case, the firms are choosing quantities, so we need to find the quantities that maximize each firm's profit given the quantities chosen by the other firms.
The profit function for each firm is π = P(q1, q2, q3)q - C(q), where C(q) is the cost function. Given that the per-unit cost is 10 and there are no fixed costs, the cost function is C(q) = 10q. Substituting the inverse demand function and the cost function into the profit function, we get π = (62 - q1 - q2 - q3)q - 10q.
Taking the derivative of the profit function with respect to q (the quantity chosen by the firm) and setting it equal to zero, we get the reaction function for each firm: q = (52 - q_other1 - q_other2) / 2. Since the products are identical and the firms are symmetric, in the Nash equilibrium, each firm will choose the same quantity: q1 = q2 = q3 = q. Substituting q for q_other1 and q_other2 in the reaction function, we get q = (52 - 2q) / 2, which simplifies to q = 17.33.
Step 2: Calculate A, the profit for firm 2 in the Nash equilibrium Substituting q = 17.33 into the profit function, we get A = (62 - 17.33 - 17.33 - 17.33) * 17.33 - 10 * 17.33 = 125.44.
Step 3: Find the profit for firm 2 in scenario (b) where the firms form a cartel In this case, the firms act as a monopoly and split the profit evenly. The profit function for the cartel is π = P(Q)Q - C(Q), where Q is the total quantity produced by the cartel. Substituting the inverse demand function and the cost function into the profit function, we get π = (62 - Q)Q - 10Q.
Taking the derivative of the profit function with respect to Q and setting it equal to zero, we get Q = 26. Substituting Q = 26 into the profit function, we get the total profit for the cartel: π = (62 - 26) * 26 - 10 * 26 = 676. Since the profit is split evenly, the profit for each firm is B = 676 / 3 = 225.33.
Step 4: Calculate A - B Finally, subtracting B from A, we get A - B = 125.44 - 225.33 = -99.89. Rounding to three decimal places, the value of A - B is -99.889.
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