Consider the following statements about the Stackelberg game from the slides, assuming both firms are identical: • (1) Denote by qº the Cournot equilibrium quantity produced by each firm, and by q°C the competitive quantity defined by P(qPC) = c (price equals marginal cost). Let s2 denote a strategy where firm 2 plays q2 = q° if it %3D %3D observes g 1 = gC, and plays gɔ = a 1 = gC. and plays gɔ = gPC otherwise, Let s 1
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Q5
Cournot Model Outcomes -
Output of each firm = qc
Strategy (s1 , s2 ) => If , q1 = qc , then q2 = qc
If , q1 /= qc , then q2 = qpc
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- Consider the following statements about the Stackelberg game from the slides, assuming both firms are identical: (I) Denote by qC the Cournot equilibrium quantity produced by each firm, and by qPC the competitive quantity defined by P(qPC) = c (price equals marginal cost). Let s2 denote a strategy where firm 2 plays q2 = qC if it observes q 1 = qC, and plays q2 = qPC otherwise. Let s 1 denote a strategy where firm 1 plays q1 = qC. Then, (s1,s2) is a Nash equilibrium of the Stackelberg game, but it’s not a subgame perfect Nash equilibrium. (II) The first firm is allowed to change its quantity after observing firm 2’s quantity chosen at the second stage. (III) Consumers are worse off in the Stackelberg game compared with the Cournot outcome given the same parameters. Group of answer choices: a. Only II is correct b. Only I is correct. c. All options are incorrect. d. Only III is correct e. More than one option is correct.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 = 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…
- Consider a market that only includes two large firms. The (inverse) market demand is P = 100 – Q. 3q2. Firm 1 has a cost function of C, = 2q1, and firm 2 has a cost function of C2 Use a Cournot model to calculate the Nash equilibrium outputs q, and q2 of the two firms. and 92 (a) Give each firm's profit as a function of (b) Compute the Nash equilibrium q, and q2.If firm 1 and firm 2 are the oligopolistic firms in bottled spring water production in Nomansland. The market demand is given by ? = 5000 −20?, Qd is the number of kilolitres demanded per month while P is the price of kilolitres of bottled water. The marginal cost of a kilolitre of bottled water is R10.How do I Find the Cournot equilibrium quantities and price? and how do I Find the Cournot profits and the monopolist profits?1. Consider two firms producing hats, A and B. The demand functions are as follows: qĄ(Pa, PB) = 592 – 8pA + 4pB and qB (PA, PB) = 400 – 8pg + 4pA Firm 1 has constant marginal cost c prices, i.e. the game's joint strategy space is {(pa, PB) E R² | Pa > 0,Pp > 0}. 10 per hat and firm 2 has c2 = 7. The firms decide on 1.1. What is the nature of the relationship between the hats produced by A and the hats produced by B? [No calculations required.] 1.2. Write down payoff functions for the two firms. 1.3. Derive best-response functions for the two firms.
- 1.7. In Section 1.2.B, we analyzed the Bertrand duopoly model with differentiated products. The case of homogeneous products2. 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.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?
- 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 CConsider a Stackelberg duopoly:There are two firms in an industry with demand Q = 1 − Pd.The “leader” chooses a quantity qL to produce. The “follower” observes qL and chooses a quantity qF.Suppose now that the cost function is Ci(qi) = qi2 for i = L, F. (a) Find the subgame perfect equilibrium. (b) Compare the equilibrium you found with the Nash equilibrium if the game was simultaneous (i.e., Cournot competition). Is the Nash equilibrium of the Cournot game also a Nash equilibrium of the sequential game? Why or why not?uppose that the market demand for a certain product is given by P=370−2QP=370−2Q, where QQ is total industry output. There are only two firms F1,F2 that manufacture that product. The two firms have the following marginal costs: c1=24 and c2=26 a) Determine the output levels that will be produced in a Cournot-Nash equilibrium q1= q2= the price level in such an equilibrium P= and the profit levels u1= u2= b) Determine the output levels that will be produced in a Stackelberg-Nash equilibrium (consider F1F1 is the leader and F2F2 is the follower) q1S= q2S= the price level in such an equilibrium PS= and the profit levels uS1= uS2=