Consider the model of second degree price discrimination that we introduced in class. (a) Complete the graphical representation of the optimal pricing problem of the monopolist, and describe visually the payments that he will receive from the buyers. (b) Complete the analysis of the optimal quality provision by analyzing the associated transfer payments in equilibrium. Verify that the remaining participation and incentive constraints that we assumed to be slack, i.e. nonbinding, are indeed nonbinding in the optimal solution. 2 (c) Describe in detail the nature of the solution as either becomes small or h l becomes large. Describe the economic intuition behind the solution. (d) Finally suppose that there three di§erent types of customers 0 < l < m < h, with prior probabilities 0 < l ; m; h < 1. Extend the analysis of the optimal second degree price discrimination from two to three types. i. Start with the guess that the only binding constraints is the individual participation constraint of l and that the binding incentive constraints are m to l and h to m. Give an interpretation of the binding constraint and give an argument as to why these might be the only binding constraint. ii. Now compute the optimal solution under the above hypothesis. What can you say about the relative size of ql ; qm; qh in the Örst best case (perfect price discrimination or social welfare maximizing) and the second best (revenue maximizing solution under incomplete information.) iii. Finally, illustrate the revenue that the Örm and the net utility that the agents get in the (x; y) diagram used above, where the xaxis describes the type and the y axis the quantity.
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