Consider the market of some good with a unique firm in it. In this market there are only two consumers, one of low valuation and one with high valuation. The firm is able to perfectly distinguish between them. Each consumer's gross consumer surplus is given by the following expression: for 1' : 1,2: The monopolist's cost function is given by C(qi) : 2g,2 + 10, for 1' : 1,2: a) (5 points) Suppose the monopolist wishes to supply the market using packages Set up the firm's maxi- mization problem, detailing the objective function and the constraints b) (5 points) Briey explain why, at a solution, both constraints must hold with equality c) (5 points) Assume a : 2, 91 = 10 and 02 = 15. Solve for the firm's optimal packages: (Tf, q?) and (T5, q) d) (5 points) Find the firm's profit under this solution e) (5 point) What is the market's total surplus? (Hint: you do not need to do any calculation to answer this question) f) Solve the monopolist's problem but suppose now that the monopolist uses two-part tariffs: T1 = A1 + p1 ql and T2 = A2 + pzqg. Follow these steps: 2 i) (5 points) Set up the firm's problem. Be very careful when defining the objective function (profits should be a function of prices and parameters) ii) (5 points) Take FOCs With respect to prices to find optimal prices, and then replace these values in the demand functions to find equilibrium quantities. iii) (5 points) Calculate the fixed part of each tariff iv) (4 points) Finally, conclude this exercise by showing that the solution to this problem (under twopart tariffs) is exactly the same as the solution obtained before when the firm was using packages