1. Consider a price competition with differentiated products with N = {D, H} and pi € S₁ = [0, +∞) for each i N. Let firm i's demand be D; (pi, Pj) 100 - 2p; + p; and firm i's cost function be C; (qi) = c;q; such that cp = 10 and CH = 30. Find a Nash equilibrium of this game. =
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- 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 2.1. 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. 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?
- 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.5**Practice** Consider a market with three firms. Each firm has a different cost function. We assume C1(q1) = 7q1, C2(q2) = q2 + 1, and C3(q3) = 3q3.
- 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.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.)⁴
- 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.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.Let ci be the constant marginal and average cost for firm i (so that firms may have different marginal costs). Suppose demand is given by P=1-Q. Calculate the Nash equilibrium quantities assuming there are two firms in a Cournot market. Also compute market output, market price, firm profits, industry prof- its, consumer surplus, and total welfare. Represent the Nash equilibrium on a best-response function diagram. Show how a reduction in firm 1’s cost would change the equilibrium. Draw a representative isoprofit for firm 1.