Three firms produce identical products and compete in a market where the inverse demand function is P(q1, q2, q3) = 87 − q1− q2− q3. Each has a per-unit cost of 6 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) = 87 - 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) = 87 − q1− q2− q3. Each has a per-unit cost of 6 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) = 87 - 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.
(a) In the Nash equilibrium, each firm maximizes its profit given the quantities chosen by the other firms. The profit function for each firm i is:
πi = (P - c)qi = (87 - q1 - q2 - q3 - 6)qi
Taking the derivative with respect to qi and setting it equal to zero gives the reaction function for each firm:
∂πi/∂qi = 87 - 2qi - qj - qk - 6 = 0
where j and k are the other two firms. Since the products are identical and the firms are symmetric, in the Nash equilibrium, we have q1 = q2 = q3. Substituting this into the reaction function gives:
81 - 3qi = 0
Solving for qi gives qi = 27 for i = 1, 2, 3. Substituting qi = 27 into the profit function gives the profit for each firm:
πi = (87 - 27 - 27 - 27 - 6)27 = 27^2 = 729
So, A = 729.
(b) In the cartel, the firms maximize their joint profit. The total quantity Q is the sum of the quantities produced by the three firms, so Q = q1 + q2 + q3. The profit function for the cartel is:
Π = (P - c)Q = (87 - Q - 6)Q
Taking the derivative with respect to Q and setting it equal to zero gives:
∂Π/∂Q = 87 - 2Q - 6 = 0
Solving for Q gives Q = 40.5. Since the profit is split evenly, the quantity produced by each firm is Q/3 = 40.5/3 = 13.5. Substituting Q = 40.5 into the profit function gives the total profit:
Π = (87 - 40.5 - 6)40.5 = 40.5^2 = 1640.25
The profit for each firm is Π/3 = 1640.25/3 = 546.75. So, B = 546.75.
Finally, A - B = 729 - 546.75 = 182.25. Rounded to three decimal places, this is 182.250.
Similar Questions
The market for aircraft features only two firms: Boeing (firm 1) and Airbus(firm 2). The inverse demand in the market is given byP = 18 − Q,where Q is the aggregate quantity. Both firms have a total cost function given by C(qi) =3qi, i ∈ {1, 2}.(a) Find the Nash equilibrium (q∗1 , q∗2 ) when firms do not collude, and compute theprice and profits of each firm in this equilibrium.(b) Suppose that Boeing and Airbus collude to act as a monopolist and split profitsevenly. Compute the new equilibrium quantity, price and profits.(c) Now, suppose the total cost functions are C(q1) = 3q1 and C(q2) = 6q2. Find thenew Nash equilibrium and compute the price and profits of each firm in this newequilibrium.1
Question 1 Suppose there are two firms in the market of good X. The cost functionof firm 1 is C1(q) = q2, whereas the cost function of firm 2 is C2(q) = 3q2. Suppose thefirms operate in a perfectly competitive market and face a perfectly elastic demand atP = 30. Find the equilibrium quantity produced in this market. (Hint: Obtain eachfirm’s inverse supply curve and then add them horizontally.)
Two firms (Firm 1 and Firm 2) compete in selling identical products. They choose their outputlevels simultaneously and face the inverse market demand curve given by P = 100 – Q, whereQ = Q1 + Q2 . Q is the total market quantity produced, while Q1 and Q2 represent the outputsproduced by the two firms, respectively. The total cost functions for firm 1 and firm 2 are givenby TC(Q1 ) = 40Q1 and TC(Q2 ) = 40Q2 , respectively.2.1 Determine the Cournot-Nash equilibrium in this market. [9]2.2 Calculate the profit (loss) for each firm.
Two firms X and Y produce the samecommodity. Due to the productionconstraints, each firm is able to producepackages of 1, 3 and 5 units of the product.The cost of producing qx units for firm X is< [6 +2xq – 2qx + 5], and firm Y has theidentical cost function < [6 +2yq – 2qy + 5]for producing qy units. p is the price of oneunit for firm X. We assume that the marketis in equilibrium. The outcomes are theprofits of the firm shown in the form of amatrix A = [aij] (pay-off matrix). Write(i) a11, (ii) a22, (iii) a21, if the demandfunction D(p) is given by D(p) = 50 – p.
Consider a Cournot duopoly model we discussed in the lecture where the inverse demand function is P(Q)=5 - Q and the marginal cost for each firm is 8. Which of the following is the Nash equilibrium of this game? A. q1 = q2 = (1,1) B. q1 = q2 = (3,3) C. q1 = q2 = (0,0)
Upgrade your grade with Knowee
Get personalized homework help. Review tough concepts in more detail, or go deeper into your topic by exploring other relevant questions.