answer:To find the profit-maximizing output for firm B, we first need to find the reaction function for firm B. The reaction function shows the optimal output for a firm given the output of its competitor. Let qA be the output of firm A and qB be the output of firm B. The total output in the market is Q = qA + qB. The demand function is given by P(Q) = 96 - 4Q. The marginal cost for both firms is 10 per unit. First, we need to find the revenue for firm B, which is given by the product of price and quantity: R_B = P(Q) * qB. Substitute the demand function into the revenue function: R_B = (96 - 4Q) * qB Now, substitute Q = qA + qB into the revenue function: R_B = (96 - 4(qA + qB)) * qB Next, we need to find the cost function for firm B, which is given by the product of marginal cost and quantity: C_B = 10 * qB. Now, we can find the profit function for firm B, which is the difference between revenue and cost: π_B = R_B - C_B. Substitute the revenue and cost functions into the profit function: π_B = (96 - 4(qA + qB)) * qB - 10 * qB Now, to find the profit-maximizing output for firm B, we need to take the first derivative of the profit function with respect to qB and set it equal to zero: dπ_B/dqB = 0 Take the derivative: dπ_B/dqB = -4qA - 8qB + 86 Set the derivative equal to zero and solve for qB: -4qA - 8qB + 86 = 0 Given that firm A produces 20 units, we can substitute qA = 20 into the equation: -4(20) - 8qB + 86 = 0 Solve for qB: -80 - 8qB + 86 = 0 -8qB = -6 qB = 6/8 = 3/4 So, to maximize its profit, firm B should produce 3/4 units of output. However, since output must be a whole number, firm B should produce either 0 or 1 unit of output to maximize its profit.

answer:To determine the quantity of the product that firm A should produce to maximize its profits, we first need to find the reaction function for firm A. The reaction function shows the optimal quantity for firm A to produce given the quantity produced by firm B. First, let's find the inverse demand function. The demand function is given by Q = 200 - P. We can rewrite this as P = 200 - Q. Now, let's denote the quantity produced by firm A as qA and the quantity produced by firm B as qB. The total quantity in the market is Q = qA + qB. We can now substitute the inverse demand function into the total revenue function for firm A: Total Revenue (TR) = P * qA = (200 - (qA + qB)) * qA = 200qA - qA^2 - qA * qB The marginal cost (MC) for each firm is 10, so the total cost function for firm A is: Total Cost (TC) = 10 * qA Now we can find the profit function for firm A: Profit (π) = TR - TC = (200qA - qA^2 - qA * qB) - 10qA To maximize profits, we need to find the first-order condition (FOC) by taking the derivative of the profit function with respect to qA and setting it equal to 0: d(π)/dqA = 200 - 2qA - qB - 10 = 0 Now we can solve for qA in terms of qB: 2qA = 190 - qB qA = (190 - qB) / 2 This is the reaction function for firm A. Given that firm A believes firm B will produce 75 units, we can plug this value into the reaction function to find the optimal quantity for firm A to produce: qA = (190 - 75) / 2 = 115 / 2 = 57.5 So, firm A should produce 57.5 units of the product to maximize its profits, given that it believes firm B will produce 75 units.

answer:To find the equilibrium level of output for each firm and the corresponding market price using the Cournot model, we need to follow these steps: 1. Find the reaction functions for both firms. 2. Solve for the equilibrium quantities for both firms. 3. Calculate the total quantity produced and the market price. Step 1: Find the reaction functions for both firms. Let q1 and q2 be the quantities produced by firm 1 and firm 2, respectively. The total quantity produced in the market is Q = q1 + q2. The inverse demand function is P = 20 - Q. Firm 1's revenue is R1 = P * q1 = (20 - Q) * q1 = (20 - (q1 + q2)) * q1. Firm 2's revenue is R2 = P * q2 = (20 - Q) * q2 = (20 - (q1 + q2)) * q2. Since both firms have the same constant marginal cost of 4 per unit, their cost functions are C1 = 4 * q1 and C2 = 4 * q2. Firm 1's profit is π1 = R1 - C1 = (20 - (q1 + q2)) * q1 - 4 * q1. Firm 2's profit is π2 = R2 - C2 = (20 - (q1 + q2)) * q2 - 4 * q2. To find the reaction functions, we need to find the first-order conditions for profit maximization for both firms: ∂π1/∂q1 = 20 - 2 * q1 - q2 - 4 = 0 ∂π2/∂q2 = 20 - 2 * q2 - q1 - 4 = 0 Step 2: Solve for the equilibrium quantities for both firms. We can solve the system of equations to find the equilibrium quantities for both firms: q1 = 8 - 0.5 * q2 q2 = 8 - 0.5 * q1 Substituting the first equation into the second equation, we get: q2 = 8 - 0.5 * (8 - 0.5 * q2) q2 = 8 - 4 + 0.25 * q2 0.75 * q2 = 4 q2 = 16/3 ≈ 5.33 Now, we can find q1: q1 = 8 - 0.5 * q2 q1 = 8 - 0.5 * (16/3) q1 = 8 - 8/3 q1 = 16/3 ≈ 5.33 Step 3: Calculate the total quantity produced and the market price. The total quantity produced in the market is Q = q1 + q2 = 16/3 + 16/3 = 32/3 ≈ 10.67. The market price can be found using the inverse demand function: P = 20 - Q P = 20 - 32/3 P = 28/3 ≈ 9.33 So, the equilibrium level of output for each firm is approximately 5.33 units, and the corresponding market price is approximately 9.33.

answer:To find the Cournot equilibrium output levels and prices for both firms, we first need to find the reaction functions for both firms. The reaction function shows the best response of one firm to the output level chosen by the other firm. Let's start with firm A. Firm A's profit function is given by: πA(qa, qb) = P(Q) * qa - CA(qa) = (240 - (qa + qb)) * qa - 40qa To find the best response of firm A, we need to maximize its profit function with respect to qa. To do this, we take the first-order derivative of the profit function with respect to qa and set it equal to zero: dπA(qa, qb) / dqa = (240 - 2qa - qb) - 40 = 0 Solving for qa, we get firm A's reaction function: qa = (200 - qb) / 2 Now, let's find the reaction function for firm B. Firm B's profit function is given by: πB(qa, qb) = P(Q) * qb - CB(qb) = (240 - (qa + qb)) * qb - 60qb To find the best response of firm B, we need to maximize its profit function with respect to qb. To do this, we take the first-order derivative of the profit function with respect to qb and set it equal to zero: dπB(qa, qb) / dqb = (240 - qa - 2qb) - 60 = 0 Solving for qb, we get firm B's reaction function: qb = (180 - qa) / 2 Now, we need to find the Cournot equilibrium output levels by solving the system of reaction functions: qa = (200 - qb) / 2 qb = (180 - qa) / 2 Substituting the second equation into the first equation, we get: qa = (200 - ((180 - qa) / 2)) / 2 Solving for qa, we get: qa = 80 Now, substituting the value of qa back into the second equation, we get: qb = (180 - 80) / 2 qb = 50 So, the Cournot equilibrium output levels are qa = 80 and qb = 50. Now, let's find the equilibrium price. We know that the market demand curve is given by P(Q) = 240 - Q, where Q = qa + qb. Plugging in the equilibrium output levels, we get: P(Q) = 240 - (80 + 50) = 240 - 130 = 110 So, the Cournot equilibrium price is 110. In conclusion, the Cournot equilibrium output levels for firms A and B are 80 and 50, respectively, and the equilibrium price is 110.