A monopoly has two types of consumers: {L, H}, with L = 3 and H = 4. (The parameter reflects how much they appreciate the good). The proportion of low demand consumers is = 1 2 . The utility of the consumers is given by ln q p if they consume, and 0 otherwise, where q and p represent the quantity consumed and the price paid respectively. The monopoly wants to send two bundles: a cheap one with a low quantity, (qL, pL), and a more expensive with a higher quantity, (qH, pH). The monopolist has a cost of production equal to c(q) = q and, therefore, the total profits are given by = 1 2 (pL qL) + 1 2 (pH qH). (a) Find the two optimal bundles (qL, pL) and (qH, pH) when the monopolist can recognize the different types of consumers and can perfectly discriminate. [6 marks] From now on, assume now that each consumer's type (H or L) is private information. Assume, furthermore, that the monopolist wants to optimally serve both types. [Note: in the next questions, assume that ln qi > 0 for both types.] (b) Write the participation constraints and the incentive constraints for both types. [8 marks] (c) Write the maximization problem of the monopolist as an unconstrained maximization problem and find the optimal prices and quantities. [6 marks] (d) Discuss how the results in part (c) differ from part (a). Which of these features are typically found in optimal non-linear pricing?