Exercise 5bis: Collusion with asymmetric costs and Bertrand competition Two firms compete in a market for a homogeneous product. In this market there are N=1 consumers (the normalization N=1 eliminates one parameter); each consumer buys one unit if the price of the product does not exceed 10, and nothing otherwise. Consumers buy from the firm selling at the lowest price. Assume also that firm 1 has lower marginal costs: c1=0

Managerial Economics: A Problem Solving Approach
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Chapter16: Bargaining
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Exercise 5bis: Collusion with asymmetric costs and Bertrand competition
Two firms compete in a market for a homogeneous product. In this market
there are N=1 consumers (the normalization N=1 eliminates one parameter); each
consumer buys one unit if the price of the product does not exceed 10, and nothing
otherwise. Consumers buy from the firm selling at the lowest price. Assume also
that firm 1 has lower marginal costs: c1=0<c2<10.
Assume no capacity constraints.
(1) Find the monopoly price, that is, the price at which a monopolistic firm would
sell (or the joint-profit maximizing price).
(2) Find the (simultaneous) one-shot Bertrand equilibrium, and the associated
profits for each firm.
(3) Now suppose that the game is repeated infinitely. Let d denote the
discount factor, with 0<8<1. Consider trigger strategies for both firms yielding the
monopoly price and equal shares in each period if nobody deviates and reverting to
the Nash equilibrium of the one-shot game following a deviation. Write the incentive
compatibility constraint for both firms and find the minimal discount factor (as a
function of c2) such that both incentive compatibility constraints are satisfied.
Discuss your findings.
(4) Suppose now that the firms agree on the market shares to be allocated in the
collusive equilibrium, with firm 1 covering a fraction 2 of the market and firm 2 the
residual, 1-8. Find the optimal 2 that relaxes the most the incentive compatibility
constraint (that is, that would make collusion the easiest to sustain). Discuss your
results.
Transcribed Image Text:Exercise 5bis: Collusion with asymmetric costs and Bertrand competition Two firms compete in a market for a homogeneous product. In this market there are N=1 consumers (the normalization N=1 eliminates one parameter); each consumer buys one unit if the price of the product does not exceed 10, and nothing otherwise. Consumers buy from the firm selling at the lowest price. Assume also that firm 1 has lower marginal costs: c1=0<c2<10. Assume no capacity constraints. (1) Find the monopoly price, that is, the price at which a monopolistic firm would sell (or the joint-profit maximizing price). (2) Find the (simultaneous) one-shot Bertrand equilibrium, and the associated profits for each firm. (3) Now suppose that the game is repeated infinitely. Let d denote the discount factor, with 0<8<1. Consider trigger strategies for both firms yielding the monopoly price and equal shares in each period if nobody deviates and reverting to the Nash equilibrium of the one-shot game following a deviation. Write the incentive compatibility constraint for both firms and find the minimal discount factor (as a function of c2) such that both incentive compatibility constraints are satisfied. Discuss your findings. (4) Suppose now that the firms agree on the market shares to be allocated in the collusive equilibrium, with firm 1 covering a fraction 2 of the market and firm 2 the residual, 1-8. Find the optimal 2 that relaxes the most the incentive compatibility constraint (that is, that would make collusion the easiest to sustain). Discuss your results.
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