# Consider the following three firm version of Bertrand pricing game

that we discussed in class. Four firms each simultaneously choose a price p1,p2,p3 ≥ 0. There is one consumer whose demand function is given by:

q(p) = 120 − p.

Since all three firms produce the same product, the consumer buys goods only from the lowest price firm. If the lowest price is offered by more than one firm, then these firms equally share the quantity demanded. Suppose that firms 1 and 2 have marginal cost of production equal to 10 while firm 3 has marginal cost of production equal to 20.

a. Write down the best response functions of each firm.

b. Find one Nash equilibrium of this game. Justify why it is a Nash equilibrium.

c. What are all of the Nash equilibria of this game?

d. Is there a Nash equilibrium that is Pareto efficient (from the perspective of just the firms) in this game? Justify your answer.

e. Consider the 10 firm version of the above game where firms 1 − 5 have marginal cost of 10 and firms 6 − 10 have marginal cost of 20. In this case, what are all of the Nash equilibria?

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