Answer ALL of the following questions.
1. Let the demand function D (p) be such that the price elasticity of demand ∈(p) = -dD(p)/dp.p/D(p) ≥ 1 everywhere. Denote by P (q) the inverse demand function D-1 (q). The two firms have capacity constraints, so firm i‘s output must satisfy qi ≤ q-i for i = 1,2. The marginal cost of production is zero up to q-i and ∞ after qi-. Show that it is a pure strategy Nash equilibrium that both firms charge the market clearing price p* = P (q¯i + q¯j ) and earn profits Πi (q¯i + q¯j ) = q¯iP (q¯i ‡ q¯j ) for the following two cases:
(a) The efficient (parallel) rationing rule is used, i.e., the residual demand is given by Di(pi, pj) = D(pi) - q¯j for pi ≥ pj = p*.
(b) The proportional (randomised) rationing rule is used, i.e., the residual demand is given by Di(pi, pj) = D(pi).(D(pj -q¯j)/D(pj) for pi ≥ pj = p*.
2. Consider an n-firm supergame framework. The firms have constant marginal cost c. The demand function at date t is qt = µtD (pt), where µδ < 1 (δ is the discount factor). Derive the set of discount factors such that full collusion (i.e., monopoly solution) is sustainable as an equilibrium of the supergame. What would this model predict about the relative ease of sustaining collusion in expanding and declining industries?