In problem 3(c), you found the optimal nondiscriminatory fixed fee-per unit charge pricing scheme for the manufacturer to employ. Now consider general nondiscriminatory schemes. Imagine that the manufacturer is able to publish any sort of price scheme she wants. Being formal about it, the manufacturer picks a function P : (0, oo) - [O, oo] with the following interpretation: A retailer can purchase quantity x from the manufacturer at a total cost of P(x). We allow P( x) = oo for some quantities x ,with the interpretation that the manufacturer is unwilling to sell that quantity. The manufacturer is able to control resale of this good, so each retailer must live with the scheme P ; given P , each retailer chooses that lvel x to buy such that its revenues selling x less its costs P( x) are as large as possible. But the manufacturer is unable to discriminate among the retailers; the manufacturer must set a single cost schedule P and allow the different retailers each to choose his own preferred point. (a) Prove that the best the manufacturer can do with such a general nondis criminatory scheme is strictly worse than what she can do if she is allowed to discriminate (and use general schemes). (b) To what extent does your proof in part (a) depend on the exact num bers and parameterization in problem 3? Specifically, suppose that we looked at an example in which there were a number of different retail ers, all of whom faced demand with the same B but with different Ai . Would your proof work in this case? And what if both the Ai and Bi varied? Would your proof still work in all cases? Would it only work in some? If the latter, can you characterize precisely those circumstances in which a nondiscriminatory scheme can do just as well as the optimal discriminatory scheme?
114. In problem 3(c), you found the optimal nondiscriminatory fixed fee-per unit charge pricing scheme for the manufacturer to employ. Now consider general nondiscriminatory schemes. Imagine that the manufacturer is able to publish any sort of price scheme she wants. Being formal about it, the manufacturer picks a function P : (0, oo) - [O, oo] with the following interpretation: A retailer can purchase quantity x from the manufacturer at a total cost of P(x). We allow P( x) = oo for some quantities x ,with the interpretation that the manufacturer is unwilling to sell that quantity. The manufacturer is able to control resale of this good, so each retailer must live with the scheme P ; given P , each retailer chooses that lvel x to buy such that its revenues selling x less its costs P( x) are as large as possible. But the manufacturer is unable to discriminate among the retailers; the manufacturer must set a single cost schedule P and allow the different retailers each to choose his own preferred point.
(a) Prove that the best the manufacturer can do with such a general nondis criminatory scheme is strictly worse than what she can do if she is allowed to discriminate (and use general schemes).
(b) To what extent does your proof in part (a) depend on the exact num bers and parameterization in problem 3? Specifically, suppose that we looked at an example in which there were a number of different retail ers, all of whom faced demand with the same B but with different
Would your proof work in this case? And what if both the Ai and Bi varied? Would your proof still work in all cases? Would it only work in some? If the latter, can you characterize precisely those circumstances in which a nondiscriminatory scheme can do just as well as the optimal discriminatory scheme?
(c) Go back to the parameterization of problem 3. What is the best general nondiscriminatory scheme from the point of view of the manufacturer? (This is difficult. You will be in better shape to tackle this part of this problem after finishing chapter 18.)
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