Consider the price discrimination example we discussed in the class. A firm produces two goods, with one being a downgraded (worse) version of the other. Suppose that each good has zero cost of production. There are N (even) consumers. Half of the consumers have high valuations: their valuation of good 1 is 100, and of good 2 is vH < 100. The other half are low valuation consumers: they value good 1 for 90, and good 2 for vL < min {90, vH}. So the valuation of consumers in the first group of each good is higher than the corresponding valuation of consumers in the second group. Also note that good 2 is considered worse by both groups of consumers each group is willing to pay less for it than for the other good. The firm cannot charge different prices for the same good to different consumers.

(a) In the class showed that when (vH, vL ) = (94, 89) the firm increases its profits by selling both versions of the good. Clearly it is not true in general. Characterize all pairs (vH, vL ) such that selling both versions is more profitable for the firm than offering only a better version (that is mathematically describe a set of such pairs). Carefully describe all the steps of your analysis. If you make some additional assumptions, you must explicitly introduce them.

(b) Prove that for each (vH, vL ) such that selling both versions is more profitable for the firm than offering only a better version, the social welfare defined as the sum of consumer and producer surplus is lower when a firm can sell both versions, than when it can sell only the better version of the good.

(c) The above observation is not as intuitive as it might seem. In fact it is possible that the social welfare goes up when a firm can introduce the downgraded version of a product. Find an example that demonstrating this. You can assume that there are still two groups of consumers. Find the number of consumers in each group, and their valuations of each good that yields you a desired conclusion.