Consider the Cournot model of oligopoly. N firms, indexed by i, compete in a market for a homogeneous good, where they simultaneously choose output quantities, q_i. The total market quantity in the market is denoted by Q = i=1N q_i

1. The inverse demand function is given by P(Q) = a - Q, unless a < Q, in which case P = 0. The firms have no fixed costs and marginal cost equal to

c_i. The firms' payoffs are their profits, i.e., firm i's payoff is i = P(Q)q_i - c_iq_i.

a)Suppose there are two firms in the market, i.e., N = 2, and 012<2a . Derive the unique Nash equilibrium quantities for the two firms as well as the resulting market price and the profits of the two firms.

b)Maintaining the duopoly-assumption from part a), suppose that the cost structure is such that 0122>a+c₁ . Again, derive the unique Nash equilibrium quantities, the market price and the profits of the two firms.

c)Suppose now that there is an arbitrary number of firms, N, in the market, and that these firms all have the same marginal cost, c. Derive the symmetric cooperative cartel solution.

d)Keeping the symmetric oligopoly setup from part c), but assuming that the firms behave non-cooperatively, derive the unique Nash equilibrium quantity of each firm as well as the resulting market price and profits of the firms. Finally, analyze what happens to the Nash equilibrium as N and comment.