给MBA课程当TA，批作业的时候发现挺好玩的。下面这道题目，有兴趣的来做做吧！

Consider a market characterized by two segments. A customer's preference for a product in this market with a quality index Q is given by qQ. The parameter q captures the intensity of preference for quality. A customer with high q gets higher utility from consuming a product of quality Q than another customer with low q. A customer buys that quality that maximizes his net utility = qQ - P(Q) where P(Q) is the price the firm charges for the quality Q. A customer desires only one unit of any quality. The two segments are different only in their q values and all consumers within a segment have the same q value. This means that all consumers from a given segment either buy none or the same product. A monopoly firm in this market has developed two distinct qualities Q₁ and Q₂. The marginal cost of production of quality Q is a constant given by aQ² where a is a parameter >0. That is to produce a higher quality product it costs much more than a lower quality product but, the tenth unit of producing the same quality product costs no more than the first unit. We can write down the profit function for the firm as

P = S1 (P(Q₁)- aQ₁²) + S2 (P(Q₂)- aQ₂²)

where S1 and S2 are the sales in units of product with qualities Q₁ and Q₂ respectively. Of course, if the firm markets only one product then one of the terms in the above equation would be zero.

Assume that the two qualities are Q₁=1 and Q₂=2 and a= 0.5. The segment sizes and their q values are as follows.

Segment Size (# of consumers) q value

I            500                            2

II           300                            4

Answer the following questions

a. If the firm wants to market only one product out of the two that are available which one should it choose, what would be the optimal price and profit, and which segment(s) is (are) served?

b. If it markets both qualities but can perfectly discriminate between the two segments (that is, segments cannot cross over and buy the other product not assigned to them), what prices should it choose for the two products, what would be the profits, and which segment buys what? Note that we are ruling out selling the same product at different prices to the two segments.

c. Now consider the case when consumers are smart and perfect discrimination is not possible. If two products are offered, a consumer compares the prices of both and buys the one that yields the highest net utility. In this case, what would be the prices the firm should set and what would be the profit. Compare this case with (b).

d. Illustrate how your answer in (c) changes as the segment sizes change and as the ratio of the q values changes. That is, fix the q value for one segment and increase the q value for the other segment. Similarly, fix the size of segment 1, and increase the size of segment 2. How would the solutions change? That is, what would be the changes, if any, in the breadth of the product line or in the assignment of segments to products?