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/ 2 +p2,p1/ 2t and D2 (p1,p2) = S/ 2 + p1,p2 /2t , where S > 0, and the parameter t > 0 measures the degree of product differentiation. Both firms have constant marginal cost c > 0 of production.

1:Derive the Nash equilibrium of this game, including the prices, outputs and profits of the two firms.

2:From the demand functions, qi = Di (pi,pj) = S /2 + pj,pi /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., @Pi(qi;pj) @qi < 0. Argue that, taking pj 0 as given, rm 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., rm i has market power.

3:Calculate the limits of the equilibrium prices and profits as t ! 0. What is Pi(qi,pj) as t= 0? Is it downward sloping? 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.