0 and the parameter t > 0measures the degree of product differentiation. Both firms haveconstant marginal cost c > 0 for production.SP2-P12tS(a) Derive the Nash equilibrium of this game,...

Extracted text: Consider a duopoly market, where two firms sell differentiated prod- ucts, which are imperfect substitutes. The market can be modelled as a static price competition game, similar to a linear city model. The two firms choose prices p1 and p2 simultaneously. The derived demand functions for the two firms are: D1 (P1, P2) = ; + and D2 (P1, P2) =+ 2, where S > 0 and the parameter t > 0 measures the degree of product differentiation. Both firms have constant marginal cost c > 0 for production. S P2-P1 2t S (a) Derive the Nash equilibrium of this game, including the prices, outputs and profits of the two firms. Pj-Pi derive (b) From the demand functions, q; = D; (pi, Pj) = the residual inverse demand functions: p; = P;(qi, Pi) (work out P:(qi, Pi)). Show that for t > 0, P:(q;, P;) is downward-sloping, aP:(gi-Pj) + 2t i.e., < 0.="" argue="" that,="" taking="" p;=""> 0 as given, firm i is like a monopolist facing a residual inverse demand, and the optimal q; (which equates marginal revenue and marginal cost) or pi makes P;(qi, P¡) = Pi > c, i.e., firm i has market power.