(1) Two firms produce goods that are imperfect substitutes. If firm 1 charges price pi and firm 2 charges price p2, then their respective demands are q1 = 12 – 2p1 +p2 and 42 = 12 + P1 – 2p2. So this is like Bertrand competition, except that when p1 > P2, firm 1 still gets a positive demand for its product. Regulation does not allow either firm to charge a price higher than 20. Both firms have a constant marginal cost c= 4. (a) Construct the best reply function BR1(p2) for firm 1. That is, pi = the optimal price for firm 1 if it is known that firm 2 charges a price P2. Construct a Nash equilibrium in pure strategies for this game. Are there any Nash equilibria in mixed strategies? If yes, construct one; if no provide a justification. BR1 (p2) is

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Chapter1: Making Economics Decisions
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(1) Two firms produce goods that are imperfect substitutes. If firm 1 charges price pi and
firm 2 charges price p2, then their respective demands are
q1 = 12 – 2p1 + p2 and q2 = 12 + p1 – 2p2.
So this is like Bertrand competition, except that when p1 > P2, firm 1 still gets a positive
demand for its product. Regulation does not allow either firm to charge a price higher
than 20. Both firms have a constant marginal cost c=4.
(a) Construct the best reply function BR1 (p2) for firm 1. That is, pi =
the optimal price for firm 1 if it is known that firm 2 charges a price p2. Construct a
Nash equilibrium in pure strategies for this game. Are there any Nash equilibria in mixed
strategies? If yes, construct one; if no provide a justification.
BR1 (P2) is
(b) Notice that for any given price p1, firm 1's demand increases with p2, so firm 1 is better
off when firm 2 charges a high price p2. What is the best reply to p2
best reply to p2 = 0?
20? What is the
(c) What prices for firm 1 are not strictly dominated? What prices would survive two
rounds of strict dominance? Provide a reason for each strategy that you eliminate.
(d) Challenge question: If you continue the iterative elimination of strictly dominated
strategies, what strategies will survive?
Transcribed Image Text:(1) Two firms produce goods that are imperfect substitutes. If firm 1 charges price pi and firm 2 charges price p2, then their respective demands are q1 = 12 – 2p1 + p2 and q2 = 12 + p1 – 2p2. So this is like Bertrand competition, except that when p1 > P2, firm 1 still gets a positive demand for its product. Regulation does not allow either firm to charge a price higher than 20. Both firms have a constant marginal cost c=4. (a) Construct the best reply function BR1 (p2) for firm 1. That is, pi = the optimal price for firm 1 if it is known that firm 2 charges a price p2. Construct a Nash equilibrium in pure strategies for this game. Are there any Nash equilibria in mixed strategies? If yes, construct one; if no provide a justification. BR1 (P2) is (b) Notice that for any given price p1, firm 1's demand increases with p2, so firm 1 is better off when firm 2 charges a high price p2. What is the best reply to p2 best reply to p2 = 0? 20? What is the (c) What prices for firm 1 are not strictly dominated? What prices would survive two rounds of strict dominance? Provide a reason for each strategy that you eliminate. (d) Challenge question: If you continue the iterative elimination of strictly dominated strategies, what strategies will survive?
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