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Two firms produce a homogeneous product, but firm 1 has lower marginal cost of production than firm 2

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Two firms produce a homogeneous product, but firm 1 has lower marginal cost of production than firm 2. The demand for the product is given by p = 18 − q, where q is aggregate output. Firm 1 has constant marginal cost of $ 2. Firm 2 has a constant marginal cost of $ 4.

a) Suppose first that each firm i = 1, 2 chooses its output qi , taking the output of the other firm as given [Cournot Competition].

i) Determine the best-response functions q br 1 (q2) and q br 2 (q1). Note that since marginal cost are different, these functions will not be symmetric.

ii) Calculate the Nash equilibrium quantities and the market price

b) Now suppose firm i = 1, 2 chooses its price pi , taking the price of the other firm as given [Bertrand Competition]. Assuming that prices are quoted in dollars and cents (the smallest unit of measurement is a cent), determine Nash equilibrium prices and quantities

c) Now suppose the two firms collude, i.e., they choose either q1 and q2 (under Cournot competition) or p1 and p2 (under Bertrand competition) so as to maximize joint profit. Determine the new equilibrium price and quantities. (Note: you have to be very careful about how the cartel will allocate production across ‘plants’ because marginal cost of production differ). Under which type of competition is the incentive to “cheat” (to break the cartel agreement) highest?