A world renowned monopoly sells flacons of their most exquisite perfume. The monopoly has two types of consumers in equal proportions: high type and low type, denoted by H and L. Each consumer would like at most two flacons. Their valuations of the good are as follows. The H type would be willing to pay 35 + 10X to get one flacon, and 75 + 10X to get two. The L type would be willing to pay 30 + 10X to get one unit, and 40 + 10X to get two. You should substitute X for the last number of your student ID number. Getting no flacon is worth zero.

Let T(q) denote the price charged when consuming q units, where T(0) = 0. The monopolist chooses T(q) to maximize her profit. We assume that the cost of production is zero.

From now on, assume now that each consumer's type (H or L) is private information.

c-) Assume that the monopolist wants to sell two units to the high type and one unit to the low type. Write the participation constraints and the incentive constraints for both types and find the optimal price scheme T(1) and T(2). Explain carefully your results.

d-)Explain if the scheme found in part (c) maximizes the profits or whether there is an alternative scheme that generates higher profits. Does your answer change if the willingness to pay of the L type for two units changes to 55 + 10X (keeping everything else fixed)?