Suppose a monopolist produces Soma which is used as an input by two industries producing two different goods (say Amos and Mosa). In turn, these two firms are monopolists in their respective industries. Assume that both firms convert one unit of Soma into one unit of output. The demand for Amos has a constant elasticity of demand of 3/2, and for Mosa the elasticity of demand is 2. That is, the inverse demand function for Amos is p^AMOS(y) = y^ 2/3 and the inverse demand function for Mosa is p^MOSA(y) = y^1/2 . Finally, assume that the monopolist's constant marginal cost of production is $1. Suppose the monopolist is able to charge different prices for the same good (Soma) to the two firms, an example of third-degree price discrimination. We want to determine the price it will charge to each of them.

3.1 Write the profit of the producer of Amos as a function of the quantity of Amos produced and the price charged by the producer of Soma, p SOMA. (Hint: the cost function of this firm is C(y AMOS) = p^SOMA y^AMOS. Why?) Find the profitmaximizing quantity for this firm. You have just derived the demand function of the producer of Amos for Soma. Why?

3.2 Using the demand you derived in the previous step, compute the optimal quantity the producer of Soma should sell to the producer of Amos and the optimal price it should charge.

3.3 Repeate the previous steps for the producer of Mosa. Which of the two firms is charged a lower price? Why?