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Consider the Bertrand game (oligopoly where firms select prices) discussed in class where N = 2, ci = 1 for i ∈ {1, 2} and α = 15 so D(P) = 15 − MIN{p1, p2}. Show (10, 10) is not a NE, that is show that both players charging 10 is not a Nash Equilibrium of the game

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Casey Durgan
Casey DurganLv2
27 Aug 2019

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Related questions

Consider the following Bertrand competition model. Two firms choose their prices simultaneously and the market demand is Q(p). A firm with the lowest price serves the entire market. They split the demand equally if prices are equal. Each firm i = 1, 2 has constant unit of production (i.e., marginal cost) ci > 0 where c1 > c2,and so the firms marginal costs are different. a) Show that p1 = p2 = c1 is not a Nash equilibrium. b) Show that there is no Nash equilibrium where the firms pick the same price (i.e., p1 = p2). c) Is there a Nash equilibrium of this game? d) Suppose now that the prices are discrete and so firm is price must be a multiple of a cent: That is, pi - {0.01, 0.02, 0.03, ..., 0.01n, ...} where n is some positive integer. What is the Nash equilibrium now?

As what we learned so far, if all firms have the same marginal cost and average cost (denoted by C), the pure-strategy Nash equilibrium under Bertrand model (price competition) is . Now consider a special case that only two firms exist in the market. But firm 1's marginal cost and average cost C1 is greater than firm 2's marginal cost and average cost C2, i.e, C1 greater than C2

1. Is p1 = p2 = C1 or p1 = p2 = c2 a pure-strategy Nash equilibrium? Why or why not?

2. Is p1 = c1 > p2 = c2 a pure-strategy Nash equilibrium? Why or why not?

Consider the following market game: An incumbent firm, called firm 3, is already in an industry. Two potential entrants, called firms 1 and 2, can each enter the industry by paying the entry cost of 10. First, firm 1 decides whether to enter or not. Then, after observing firm 1's choice, firm 2 decides whether to enter or not. Then, after observing firm 1's choice, firm 2 decides whether to enter or not. Every firm, including firm 3, observes the choices of firms 1 and 2. After this, all of the firms in the industry (including firm 3) compete in a Cournot oligopoly, where they simultaneously and independently select quantities. The price is determined by the inverse demand curve p = 12 − Q, where Q is the total quantity produced in the industry. Assume that the firms produce at no cost in this Cournot game. Thus, if firm i is in the industry and produces qi, then it earns a gross profit of (12 − Q)qi in the Cournot phase. (Remember that firms 1 and 2 have to pay the fixed cost of 10 to enter).

(a) Compute the subgame perfect equilibrium of this market game. Do so by first finding the equilibrium quantities and profit in the Cournot subgames. Show your answer by designating optimal action on the tree and writing the complete subgame perfect equilibrium strategy profile. [Hint: In an n-firm Cournot oligopoly with demand p = 12 − Q and 0 costs, the Nash equilibrium entails each firm producing the quantity q = 12/(n + 1).]

(b) In the subgame perfect equilibrium, which firms (if any) enter the industry?

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