Consider a duopoly market, where two firms sell differentiated products, which are imperfect substitutes. The market can be modelled as a static price competition game. The two firms choose prices p1 and p2 simultaneously. The demand functions for the two firms are: D1 (p1; p2) = S (1/2 + p2p1/2t) and D2 (p1; p2) = S(1/2 + p1p2/2t) , where S > 0 and the parameter t > 0 measures the degree of product differentiation. Both firms have constant marginal cost c > 0 for production.

(a) Derive the Nash equilibrium of this game, including the prices, outputs and prices of the two firms.

(b) From the demand functions, qi = Di (pi ; pj) = S(1/2 + pjpi/2t) , derive the residual inverse demand functions: pi = Pi(qi ; pj) (work out Pi(qi ; pj)). Show that for t > 0, Pi(qi ; pj) is downward sloping, i.e., dPi (qi ;pj )/dqi < 0. Argue that, taking pj > 0 as given, firm i is like a monopolist facing a residual inverse demand, and the optimal qi (which equates marginal revenue and marginal cost) or pi makes Pi(qi ; pj) = pi > c, i.e., firm i has market power.

(c) Calculate the limits of the equilibrium prices and prices as t tends to 0. Argue that the Bertrand Paradox (i.e., the prediction of the static Bertrand duopoly model, where p 1* = p 2* = c) holds only in the extreme case of t = 0, i.e., the firms selling homogeneous products.

(d) In the field of competition policy there has been a view that as long as a market is not a monopoly (or an effective monopoly through collusion), it can be treated as perfectly competitive. On the basis of your analyses and arguments given in parts (a) to (c) above, (briefly) relate this view to the Bertrand Paradox (for support), and critically comment on the validity (of such support)