Consider a two-firm industry producing two differentiated products indexed by i = 1,2. For simplicity, we assume that production is costless. Suppose that the demand functions for the two products are given by q1 = 168 2p1 + p2 and q2 = 168 - 2p2 + p1. a Suppose that the two firms set their price independently and simultaneously. Find the Bertrand-Nash equilibrium. b Suppose that the two firms move sequentially rather than simultaneously. In particular, suppose that firm 1 sets its price first. After observing firm 1' selection, firm 2 chooses its price. Find the Subgame Perfect Nash Equilibrium.
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- There are two firms in the market (duopoly). These two firms are competingsimultaneously. The first firm chooses its output level (x) by predicting the second firm’soutput (y). Let c denote the total cost function c(x) = x and c(y) = y. Also, let’s assumethat the inverse demand function is p(Y) = 7 - Y where Y = x + y. (1) Obtain the reactionfunction of the first firm. (2) Find the equilibrium (output and profit of each firm) whentwo firms simultaneously competeLet 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 1.10Two firms produce differentiated products. The demand for each firm’s product is as follows: Demand for Firm 1: q1 = 20 – 2p1 + p2 Demand for Firm 2: q2 = 20 – 2p2 + p1 Both firms have the same cost function: c(q) = 5q. Firms compete by simultaneously and independently choosing their prices and then supplying enough to meet the demand they receive. Please compute the Nash equilibrium prices for these firms.
- Problem 3. Consider the following game with three firms. First, firms 1 and 2 si- multancously choose quantities q1 and q2 respectively. After observing firm 1 and 2's quantities, firm 3 chooses its quantity q3. There is no production cost and the inverse demand function is p= 12 – (91 +2 + 93). (a) Compute the SPNE of this game. (b) Give an example of Nash equilibrium s* with s = 4 and s, = 6 , that is not subgame perfect. game theory questionConsider a duopoly market, where two firms sell differentiated prod- ucts, which are imperfect substitutes. The market can be modelled as a static price competition game, similar to a linear city model. The two firms choose prices pi and p2 simultaneously. The derived demand functions for the two firms are: D1 (p1, P2) and D2 (p1, P2) = + 2, where S > 0 and the parameter t > 0 을 + S + P2-P1 2t 2t measures the degree of product differentiation. Both firms have constant marginal cost c > 0 for production. (a) Derive the Nash equilibrium of this game, including the prices, outputs and profits of the two firms.Assume firms' marginal and average costs are constant and equal to c and that inverse market demand is given by P = a - bQ where a, b > 0. Suppose now the market is served by k firms that choose quantities for their identical products simultaneously. Calculate: i. ii. iii. iv. The Nash equilibrium quantities for the Cournot firms as functions of k. 2 Market output and price as a function of k Firm profit as a function of k Using your answers in i, ii, iii and iv, describe what happen to firm output, market output, market price and firm profit as the number of firms increases.
- Q2. Consider a two-firms Cournot model with constant returns to scale. Assume also that the inverse demand function is P = 100 – 2Q, with marginal cost equal to 20for both firms, where Q = q1 + q2 . b) How do equilibrium outputs and profits vary when firm1’s cost changes. Draw a picture of this outcome.Two firms produce identical products at zero cost, and theycompete by setting prices. If each firm charges a low price,then both firms earn profits of zero. If each firm charges ahigh price, then each firm earns profits of £30. If one firmcharges a high price and the other firm charges a low price,the firm that charges the lower price earns profits of £50, andthe firm charging the higher price earns profits of zero. (a) Which oligopoly model best describes this situation?(b) Write this game in normal form.(c) Suppose the game is infinitely repeated. Can theplayers sustain the "collusive outcome" as a Nashequilibrium if the interest rate is 50 percent? Explain. Please answer the a, b and c parts.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 the market demand for carbonated water be given by Q = 100 - 5P. Let there be twofirms producing carbonated water, each with a constant marginal cost of 2. If both firmschoose their prices simultaneously, what price would each of them charge in the Nashequilibrium?What is the homogeneous-good duopoly Cournot equilibrium if the market demand function is Q= 1,800 - 1,000p. and each firm's marginal cost is $0.28 per unit? The Cournot-Nash equilibrium occurs where q, equals and 92 equals (Enter numenic responses using real numbers rounded to two decimai places.) Furthermore, the equilibrium occurs at a price of $ (Round your answer to the nearest penny.)the set of Fationalizable strategies will be the same for party R.) Explain your reasoning. ve Intel and AMD, the primary producers of computer central processing units (CPUS), compete with one another in the mid-range chip category (among other categories). Assume that global demand for mid-range chips depends on the quantity that the two firms make, so that the price (in dollars) for mid-range chips is given by P = 210 – Q, where Q = qin- || GAMD and where the quantities are measured in millions. Each mid- + 9AMD tel range chip costs Intel $60 to produce. AMD's production process is more streamlined; each chip costs them only $48 to produce. (a) Write the profit function for each firm in terms of qintel and qAMD- Find each firm's best-response rule. (b) Find the Nash equilibrium price, quantity, and profit for each firm. (c) (Optional) Suppose Intel acquires AMD, so that it now has two sep- arate divisions with two different production costs. The merged firm wishes to maximize total…