3. Use the best-response curves to find all Nash equilibria (in both pure and mixed strate-gies) of the Battle of the Sexes game.4. Find the Nash equilibrium of the following modified Rock-Paper-Scissors game:• When rock (R) beats scissors (S), the winner's payoff is 10 and the loser's payoffis –10.• When paper (P) beats rock, the winner's payoff is 5 and the loser's payoff is –5.• When scissors beats paper, the winner's payoff is 2 and the loser's payoff is -2.• In case of ties, both players receive 0 payoff. 1. Consider the following Cournot game:• There aren>1 firms.• Firm i,i1, 2,· , n, chooses an output q; E [0, 0)...• The market inverse demand is P = a – bq, where a, b > 0 and q = E=1 9i.• Firm i's cost function is ci(qi)cqi, where c > 0.||(a) Solve the monopoly problem, in which there is only one firm in the market.(b) Find the Nash equilibrium outputs with n firms. Find the price in the equilibrium.(c) Show that the Nash equilibrium of the above game covers cases of market outcomesarranging from monopoly (n = 1) to perfect competition (n →).

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3. Use the best-response curves to find all Nash equilibria (in both pure and mixed strate- gies) of the Battle of the Sexes game.

Microeconomic Theory

12th Edition

ISBN:9781337517942

Author:NICHOLSON

Publisher:NICHOLSON

Chapter15: Imperfect Competition

Section: Chapter Questions

Problem 15.4P

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1. Two firms (A and B) play a competition game (i.e. Cournot) in which they can choose any Qi from 0 to ¥. The firms have the same cost functions C(Qi) = 10Qi + 0.5Qi2, and thus MCi = 10 + Qi. They face a market demand curve of P = 220 – (QA + QB). a. Assume firm A chooses quantity first. Frim B observes this choice and then chooses its own quantity. What is Frim B's profit as a function of QA and QB? b. Firm B has MRB = 220 – 2QB – QA. What is firm B’s best response to an arbitrary QA selected by firm A? c. Given that firm A expects firm B’s best response, what is firm A’s profit as a function of QA? (Hint: the only unknown variable in the profit function should be QA) d. Firm A has MRA = 150 – 4QA/3. What are the equilibrium QA and QB selected in this game? e. What is the equilibrium price, and how much profit does each firm collect?
1. The market (inverse) demand function for a homogeneous good is P(Q) = 10 - Q. There are two firms: firm 1 has a constant marginal cost of 2 for producing each unit of the good, and firm 2 has a constant marginal cost of 1. The two firms compete by setting their quantities of production, and the price of the good is determined by the market demand function given the total quantity. a. Calculate the Nash equilibrium in this game and the corresponding market price when firms simultaneously choose quantities. b. Now suppose firml moves earlier than firm 2 and firm 2 observes firm 1 quantity choice before choosing its quantity find optimal choices of firm 1 and firm 2.
4. In 2056, there are two mining firms operating on the moon, extracting Helium 3. Once both firms have entered the market, they compete a la Cournot. The market inverse demand function is given by P(Q) = 8 - Q. Assume that both firms have the total cost functions C(q) =2+2q. Let the star superscript* denote equilibrium quantities/prices/profits. Which of the following statements is true? (a) q₁ =q2 = 4 (b) qt > 92 (c) p* = 6 (d) π₁ < π₂ (e) T₁ = π = 2 the C
Q4. Consider two firms competing in a Cournot fashion. Each firm has MC=10 and the market demand is given by P=100-Q, where Q is the total market output. What is firm 1's Response function? a. q1=45-.5q2 b. q1=30 c. q2=45-.5q1 d. firm 1 will set p=MC
1. Best responses in a Cournot Oligopoly Firm A and Firm B sell identical goods Total market demand for the good is: The inverse demand function is therefore 1 P(QM) = 780 -Q=780 -0.02222QM 45 QM is total market production (i.e., combined production of firm's A and B. That is: Q(P) = 35, 100- 45P 2M = A +QB As a result, the inverse demand curve for each firm is: P(QA, QB) = 780- -1/32₁-752 45 Unlike the example in class, the two firms have different costs. = 4000A TCA (QA) TCB (QB) = 260QB = 780 -0.022220A -0.02222QB a. Using the demand function and the cost functions above, what is firm A's profit function. b. Using the profit function above and assuming that firm B produces Qg, calculate what firm A's best response is to firm B’s decision to produce QB- Note: Firm A's best response should be a function of B
2. A homogenous good industry consists of two firms (firm 1 and firm 2). Their cost functions are cq and cq2, respectively, where c<2. The market demand function is p=10-Q, where Q=q₁+q₂. (a) Assume that the two firms play the Bertrand price game. Find the firms' choices in the Bertrand-Nash equilibrium. (b) Assume that the two firms play the Cournot quantity game. Find the firms' choices in the Cournot-Nash equilibrium. (c) Assume the two firms play the Stackelberg game with firm 1 as the leader. Find the firms' equilibrium choices in the Stackelberg equilibrium.
2) Assume that there are 2 firms producing an identical product. Both firms have the same total cost function TC(q) =q2. The inverse demand function for the firms' output is p=120 Q, where Q is the total output and p is price. 1. What are the equilibrium price, output and profits of each firm if they are competing with each other? (Hint: Consider the equilibrium in a Cournot game.) 2. What happens if they form a cartel? Calculate the equilibrium price, output, and profits for the cartel? 3. If a single firm cheated, what would its output and profits be, assuming the other firm maintains the cartel price? Calculate the new outputs and profits for bath firms: 4. Discuss why it is hard to enforce a cartel. Explain using words. DO NOT do any calculations. DO NOT draw any graphs. 5. What can the cartel members do to enforce the cartel agreement? Propose a method. Describe the method using words. DO NOT do any calculations. DO NOT draw any graphs.
Please solve D and E part only. Thankyou. Consider three firms, each with cost function C(qi)=4qi, currently competing Cournot. Market demand is P = 20 – Q. a. a. Find the quantities, price, and profits of each firm in equilibrium. b. Imagine firms1 and 2 merged to become one, so after the merger there are now two firms left in the market, firm 1&2 and firm 3. Assume there are no cost efficiencies expected and assume that the two firms play Cournot as before. Find the quantities, price, and profits of each firm in equilibrium c. Was it profitable for firms 1 and 2 to merge in the first place? d. Continuing with b., imagine that firms did not play Cournot after the merger, but rather that the merged firm 1&2 became a Stackelberg leader after the merger and firm 3 became the Stackelberg follower. If this were the case, find the quantities, price, and profits of each firm in equilibrium. e. Was it profitable for firms 1 and 2 to merge in the first place? Did price…
4. In 2056, there are two mining firms operating on the moon, extracting Helium 3. Once both firms have entered the market, they compete a la Cournot. The market inverse demand function is given by P(Q) = 8 – Q. Assume that both firms have the total cost functions = 2+ 2q. Let the star superscript* denote equilibrium quantities/prices/profits. Which C(q): of the following statements is true? (a) qi = 4 = 4 (b) qi > qž (c) p* = 6 (d) nj < T (e) Tỉ = = 2 5. Assume the same demand and cost structures as in question 4, but now firm 1 enters the market first and firm 2 follows, as in the Stackelberg model from lecture (both firms are guar- anteed to enter; the only choice is quantities produced). Which of the following statements regarding the equilibrium outcome is FALSE? (a) The first mover produces a greater quantity than the second mover (b) Total market output is Q* = 4.5 (c) The second mover will receive a negative profit (d) The first mover will receive a greater profit than the…
12. Two firms with differentiated products are competing in price. Firm A and B face the following demand curves: QA = 90 – 2PĄ + Pg and QB = 140 – 2Pg + PA respectively. Assume production is costless. a. Give equations for and graph each firm's reaction curve. b. If both firms set their prices at the same time, what is the Nash equilibrium price, quantity, and profit for each firm? c. Suppose A sets its price first and then B responds. What price and quantity does each firm set now? Is it advantageous to move first? d. Compare the profits from part b and c. Which firm benefits more from the sequential price choosing?
1. Two firms (A and B) play a competition game (i.e. Cournot) in which they can choose any Qi from 0 to ¥. The firms have the same cost functions C(Qi) = 10Qi + 0.5Qi2, and thus MCi = 10 + Qi. They face a market demand curve of P = 220 – (QA + QB). Now assume firm A chooses quantity first. Firm B observes this choice and then chooses its own quantity. d)Firm A has MRA = 150 – 4QA/3. What are the equilibrium QA and QB selected in this game? e)What is the equilibrium price, and how much profit does each firm collect?
a) Suppose that the two firms engage in Cournot competition. Find the equilibrium price PNE in the industry, the equilibrium outputs QANE and QBNE, as well as the profits πANE and πBNE, for each firm. b) Suppose the marginal cost for firm B increases from $20 to $140, while everything else remains unchanged. Find the new equilibrium price PNE in the industry, the new equilibrium outputs QANE and QBNE, as well as the new profits πANE and πBNE for each firm. c) Suppose that, in addition to the marginal cost increase from $20 to $140 from sub question b), firm B also has a fixed cost of $2500, out of which $2100 may be recouped if it shuts down; everything else remains unchanged. In this case, what will firm B’s optimal output be? (Justify your answer.) What will firm A’s profit be?