answer:To find the Stackelberg equilibrium quantity produced by both firms, we need to first find the reaction function of the follower firm and then find the profit-maximizing quantity for the leader firm. 1. Find the reaction function of the follower firm: The follower firm's profit function is given by: π_F = (P - MC_F) * q_F = (100 - Q - MC_F) * q_F = (100 - (q_L + q_F) - 30) * q_F = (70 - q_L - q_F) * q_F To maximize its profit, the follower firm will choose q_F such that its marginal profit is equal to its marginal cost: d(π_F) / dq_F = 70 - q_L - 2 * q_F = 30 Solving for q_F, we get the reaction function of the follower firm: q_F = (70 - q_L) / 2 2. Find the profit-maximizing quantity for the leader firm: The leader firm's profit function is given by: π_L = (P - MC_L) * q_L = (100 - Q - MC_L) * q_L = (100 - (q_L + q_F) - 20) * q_L = (80 - q_L - q_F) * q_L Substitute the reaction function of the follower firm into the leader firm's profit function: π_L = (80 - q_L - ((70 - q_L) / 2)) * q_L = (90 - (3/2) * q_L) * q_L To maximize its profit, the leader firm will choose q_L such that its marginal profit is equal to its marginal cost: d(π_L) / dq_L = 90 - 3 * q_L = 20 Solving for q_L, we get the profit-maximizing quantity for the leader firm: q_L = (90 - 20) / 3 = 70 / 3 3. Find the equilibrium quantity produced by both firms: Substitute the profit-maximizing quantity for the leader firm into the reaction function of the follower firm: q_F = (70 - (70 / 3)) / 2 = (70 * 2 / 3) / 2 = 70 / 3 The Stackelberg equilibrium quantity produced by both firms is: Q = q_L + q_F = 70 / 3 + 70 / 3 = 140 / 3 ≈ 46.67 units

answer:To find the profit-maximizing output levels for both firms in this Stackelberg competition, we need to follow these steps: 1. Find the inverse demand function (P as a function of Q). 2. Calculate the reaction function for Firm 2 (Q2 as a function of Q1). 3. Calculate the profit function for Firm 1 (π1 as a function of Q1 and Q2). 4. Substitute the reaction function of Firm 2 into the profit function of Firm 1. 5. Maximize the profit function of Firm 1 with respect to Q1. 6. Find the optimal output level for Firm 2 using the reaction function. Step 1: Inverse demand function Q = 100 - P P = 100 - Q Step 2: Reaction function for Firm 2 Firm 2's profit function: π2 = P * Q2 - C2 π2 = (100 - Q) * Q2 - 8Q2 π2 = 100Q2 - Q * Q2 - 8Q2 To maximize its profit, Firm 2 will set the derivative of its profit function with respect to Q2 equal to zero: dπ2/dQ2 = 100 - 2Q - 8 = 0 Q2 = (100 - 8) / 2 = 46 - Q1/2 (reaction function) Step 3: Profit function for Firm 1 π1 = P * Q1 - C1 π1 = (100 - Q) * Q1 - 10Q1 π1 = 100Q1 - Q * Q1 - 10Q1 Step 4: Substitute the reaction function of Firm 2 into the profit function of Firm 1 π1 = 100Q1 - (Q1 + Q2) * Q1 - 10Q1 π1 = 100Q1 - (Q1 + (46 - Q1/2)) * Q1 - 10Q1 π1 = 100Q1 - (Q1 + 46 - Q1/2) * Q1 - 10Q1 Step 5: Maximize the profit function of Firm 1 with respect to Q1 dπ1/dQ1 = 100 - 2Q1 - 46 + Q1/2 - 10 = 0 Solving for Q1: (3/2)Q1 = 44 Q1 = 44 * (2/3) Q1 = 88/3 ≈ 29.33 Step 6: Find the optimal output level for Firm 2 using the reaction function Q2 = 46 - Q1/2 Q2 = 46 - (88/3) * (1/2) Q2 = 46 - 88/6 Q2 = 188/6 ≈ 31.33 The profit-maximizing output levels for Firm 1 and Firm 2 in this Stackelberg competition are approximately 29.33 and 31.33 units, respectively.

answer:To find the Stackelberg equilibrium quantities and prices, we first need to find the reaction functions for both firms. The reaction function shows the best response of one firm to the output of the other firm. Step 1: Find the reaction function for Firm B To find the reaction function for Firm B, we need to find the profit-maximizing output for Firm B given the output of Firm A. Profit for Firm B: πB = P * Qb - C(Qb) Substitute the demand function and cost function: πB = (100 - Q) * Qb - 20Qb Simplify: πB = 100Qb - Q * Qb - 20Qb Since Q = Qa + Qb: πB = 100Qb - (Qa + Qb) * Qb - 20Qb Expand: πB = 100Qb - Qa * Qb - Qb^2 - 20Qb Combine like terms: πB = 80Qb - Qa * Qb - Qb^2 To find the best response, we need to maximize πB with respect to Qb. To do this, we take the first-order derivative with respect to Qb and set it equal to zero: d(πB)/dQb = 80 - Qa - 2Qb = 0 Solve for Qb to get Firm B's reaction function: Qb = (80 - Qa) / 2 Step 2: Find the optimal output for Firm A To find the optimal output for Firm A, we need to find the profit-maximizing output for Firm A given the reaction function of Firm B. Profit for Firm A: πA = P * Qa - C(Qa) Substitute the demand function and cost function: πA = (100 - Q) * Qa - 10Qa Simplify: πA = 100Qa - Q * Qa - 10Qa Since Q = Qa + Qb: πA = 100Qa - (Qa + Qb) * Qa - 10Qa Expand: πA = 100Qa - Qa^2 - Qa * Qb - 10Qa Combine like terms: πA = 90Qa - Qa^2 - Qa * Qb Now, substitute Firm B's reaction function into Firm A's profit function: πA = 90Qa - Qa^2 - Qa * ((80 - Qa) / 2) Expand and simplify: πA = 90Qa - Qa^2 - 40Qa + (1/2)Qa^2 Combine like terms: πA = 50Qa - (1/2)Qa^2 To find the best response, we need to maximize πA with respect to Qa. To do this, we take the first-order derivative with respect to Qa and set it equal to zero: d(πA)/dQa = 50 - Qa = 0 Solve for Qa: Qa = 50 Step 3: Find the optimal output for Firm B Now that we have the optimal output for Firm A, we can find the optimal output for Firm B using its reaction function: Qb = (80 - Qa) / 2 Qb = (80 - 50) / 2 Qb = 30 / 2 Qb = 15 Step 4: Find the equilibrium price Now that we have the equilibrium quantities for both firms, we can find the equilibrium price using the demand function: P = 100 - Q P = 100 - (Qa + Qb) P = 100 - (50 + 15) P = 100 - 65 P = 35 The Stackelberg equilibrium quantities and prices are Qa = 50, Qb = 15, and P = 35.

answer:To find the equilibrium quantity and price for firms A and B in a Stackelberg duopoly model, we need to follow these steps: 1. Find the reaction function of firm B (the follower). 2. Find the profit function of firm A (the leader) and maximize it by taking the first-order condition. 3. Substitute the reaction function of firm B into the maximized profit function of firm A to find the equilibrium quantity for firm A. 4. Use the reaction function of firm B to find the equilibrium quantity for firm B. 5. Use the market demand function to find the equilibrium price. Step 1: Find the reaction function of firm B To find the reaction function of firm B, we first need to find its profit function. The profit function of firm B is given by: ΠB = P * QB - CB(QB) Since Q = QA + QB and P = 100 - Q, we can rewrite the profit function as: ΠB = (100 - QA - QB) * QB - 30QB Now, we need to maximize the profit function of firm B with respect to QB. To do this, we take the first-order condition: dΠB/dQB = 100 - QA - 2QB - 30 = 0 Solving for QB, we get the reaction function of firm B: QB = (70 - QA) / 2 Step 2: Find the profit function of firm A and maximize it The profit function of firm A is given by: ΠA = P * QA - CA(QA) Using the same substitution as before, we can rewrite the profit function as: ΠA = (100 - QA - QB) * QA - 20QA Now, we substitute the reaction function of firm B (QB = (70 - QA) / 2) into the profit function of firm A: ΠA = (100 - QA - (70 - QA) / 2) * QA - 20QA Simplifying the expression, we get: ΠA = (130 - 3QA / 2) * QA - 20QA Now, we need to maximize the profit function of firm A with respect to QA. To do this, we take the first-order condition: dΠA/dQA = 130 - 3QA - 20 = 0 Solving for QA, we get the equilibrium quantity for firm A: QA = 50 / 3 Step 3: Use the reaction function of firm B to find the equilibrium quantity for firm B Substituting the equilibrium quantity for firm A (QA = 50 / 3) into the reaction function of firm B: QB = (70 - 50 / 3) / 2 QB = 40 / 3 Step 4: Use the market demand function to find the equilibrium price Now that we have the equilibrium quantities for both firms, we can use the market demand function to find the equilibrium price: Q = QA + QB = 50 / 3 + 40 / 3 = 90 / 3 = 30 P = 100 - Q = 100 - 30 = 70 So, the equilibrium quantities and price for firms A and B are: QA = 50 / 3 QB = 40 / 3 P = 70