(Vertical product differentiation duopoly) Consider an industry where there are two firms 1 and 2, and where consumers have preferences summarised by Ui = Qui - pi, 0 being a parameter which describes the taste for quality of consumers, u; being the quality of good i = 1, 2 and p; being its price. Consumer tastes are distributed uniformly on [0, 0] with unit density. Each consumer can buy at most one unit of the good. (a) Both firms have zero costs of production. Suppose u1 2 u2 2 0 and find the equilibrium prices of the simultaneous moves game where each firm sets prices. Then derive equilibrium quantities and profits. (b) Now set 0 = 10 and consider the following two-stage game. In the first stage, firms simultaneously choose the quality of their product, with u = {0,5,20} (that is, only three quality levels are available.) The higher the quality, the higher the fixed cost a firm has to pay: F(ui) = u? /2. (You can think that you need a larger investment, such as R&D or advertising, to raise the quality of your product.) In the second stage, once chosen the qualities (that everyone can observe), the firms simultaneous choose prices (that is, this stage coincides with the game analysed in the previous point.) Find the qualities that each firm would set at equilibrium. Do you find a result of maximal product differentiation? Please comment