Question 1 (40 points) Consider a homogeneous duopoly market where two firms compete in prices. Demand is given by D 8-2p, where p is price. Marginal cost of production is 2. a)lf the individual capacity of both firms is 2, is there an equilibrium in pure strategies? If so, what are the equilibrium prices? If not, provide a proof. b) Consider then that a third firm enters the market and that all three firms have a capacity of 2. Does an equilibrium in pure strategies exist? If so, what are the equilibrium prices? If not, why not? c) Does you answer under b) change if the firms' capacities are respectively equal to 1, 2 and 3?.
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Question 1 (40 points) Consider a homogeneous duopoly market where two firms compete in prices. Demand is given by D 8-2p, where p is price. Marginal cost of production is 2. a)lf the individual capacity of both firms is 2, is there an equilibrium in pure strategies? If so, what are the
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- Question 1 (40 points) Consider a homogeneous duopoly market where two firms compete in prices. Demand is given by D 8-2p, where p is price. Marginal cost of production is 2. a)lf the individual capacity of both firms is 2, is there an equilibrium in pure strategies? If so, what are the equilibrium prices? If not, provide a proof. b) Consider then that a third firm enters the market and that all three firms have a capacity of 2. Does an equilibrium in pure strategies exist? If so, what are the equilibrium prices? If not, why not? c) Does you answer under b) change if the firms' capacities are respectively equal to 1, 2 and 3?.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.QUESTION 13 Consider a market where two firms (1 and 2) produce differentiated goods and compete in prices. The demand for firm 1 is given by D₁(P₁, P2) = 140 - 2p1 + P2 and demand for firm 2's product is D2 (P1, P2) 140 - 2p2 + P1 Both firms have a constant marginal cost of 20. What is the Nash equilibrium price of firm 1? (Only give a full number; if necessary, round to the lower integer; no dollar sign.)
- 2. Consider a Cournot competition environment with one good and two firms, Firm 1 and Firm 2. The (pure) strategy space of Firm 1 is S₁ = [0, 1], and the strategy s₁ of Firm 1 corresponds to the amount of the good they produce. Similarly, the (pure) strategy space of Firm 2 is S2 = [0, 1], and the strategy s2 of Firm 2 corresponds to the amount of the good they produce. If Firm 1 were to produce quantity s₁ and Firm 2 were to produce quantity s2, the prevailing price in the good market would be 1 — 81 — 82, the utility of Firm 1 would be their profit, u₁ (81, 82) = (1-81-82 - c)81, and the utility of Firm 2 would be their profit, u2($1, $2) = (1-81-82-c)s2, where 0 ≤ c< 1 is the marginal cost of production for both firms. (a) Find the pure-strategy Nash equilibria of this game. (b) Are there other Nash equilibria in this game.Suppose that Toyota and GM are considering entering a market for electric cars and that their profits from entering or not entering the market are given in the table below. (a) Does GM have a dominant strategy? If so, what is it? (b) Does Toyota have a dominant strategy? If so, what is it? (c) What is the Nash Equilibrium? (d) Is there any incentive for the two firms to have a collusive outcome different from the Nash equilibrium? Please explain.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?
- QUESTION 10 Suppose there are two firms that produce an identical product. The demand curve for the product is given by P = 62 - Q where Q is the total quantity produced by the two firms. Both firms choose their individual quantities qı20 and q22 0 simultaneously. Each firm has a marginal cost of 37. What is the market price when both firms produce the quantities in the unique Nash equilibrium? Give your answer as a number to two decimal places.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…13) Two identical firms are engaged in Cournot competition, with cost functionsTCA(QA) = 30 QA and TCB(QB) = 30 QB. The market demand is given by P = 480 –3Q.a) Plot the best response functions and report the Cournot-Nash equilibrium quantities, price and profits.b) What are the prices, quantities, and profits for the firms if they decide to collude and share profits equally?c) Show that firms have an incentive the deviate from the collusive outcome.d) Find the Stackelberg equilibrium if A leads and B follows.e) Show the equilibria in the previous parts on the inverse demand function. Calculate and identify consumersurplus and deadweight loss in each equilibrium. If you can only answer a limited amount of questions, please answer d and e :)
- 2. An industry contains two firms that have identical cost functions C(q)=10+2q. The inverse demand function for the market is P=50-2Q where Q is the total industry output. Assuming the firms compete in quantities: Find the firms' best response functions. b. Solve for the Cournot Nash Equilibrium of the game. What is the total industry output in equilibrium? What is the equilibrium price? с. i. If both firms could collude, what would the industry output and price be? Suppose they decide that each firm produces half of the industry output found in part (i). Is this agreement self-enforcing? Explain. ii. a.2) Suppose the Market Demand function is Q = 60,700 – 520P. The two firms in the market compete under Bertrand Competition. The Marginal Cost (MC) for the two firms is MC = $39.90. What is the market price at the Nash Equilibrium?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