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Exercise 4.2" Two firms (1' = 1.2) produce one good each, at marginal cost r. (i = 1. 2]. Each rm has a monopo- ly power in the production of its good. The goods are perfect complements. The demand curve is :7 = Dip). where p E p1 + p; is the price of the composite good and p,- is the price of good 1' (i = 1. 2'). Let r E ct + :2. (i) Reinterpret the variables to show that the case of a single good produced by a manufacturer and distributed by a retailer fits into this model. Assume that the elasticity of demand. a = D'p/D. is constant in order to simplify computations. (ii) What is the optimal p For the horizontally inte grated structure? (iii) Consider the nonintegrated structure. Suppose that Firm 1 chooses its price rst and takes into account the eFtect of its choice on rm 2's price. Show that the Lerner index is higher than under integration. More precisely. show that p = MI I/rziz. (iv) Suppose now that the two Firms choose their prices simultaneously. Assume that each firm maximizes its pro Fit given the other rm's pricethat is. the choices are simultaneous: a firm does not try to influence the other Firm's price (see chapter 5). Show that the Lerner index is even higher than in the case of sequential choice of prices. More precisely. Show that p = r/(I 2m}. Interpret this result. (Hint: Start from the simultaneous choice equilibrium and change rm 1's price slightly.)