Consider the model of conformity and vanity of Section 10.2. Suppose now that there are 300 product A oriented consumers and only 100 product B oriented consumers. Let pA denote the price of product A, and pB the price of product B. The utility function of a product i oriented consumer, i = A, B, exhibits vanity and is given by Ui def =  −qi/2 − pi if he buys product i −qj/2 − 100 − pj if he buys product j = i, where i, j = A, B and i = j. Firms do not bear any production cost. Answer the following questions.

(a) Suppose that all the brand A oriented consumers buy brand A and all brand B oriented consumers buy brand B. Given pB, find the price p
A in which firm A undercuts the price of firm B and grabs all its 100 consumers. Then, given pA, find the price p
B in which firm B undercuts the price of firm A and grabs all its 300 consumers. Hint: Notice that in the case of vanity consumers’ utility decline withan increase in the number of consumers buying the same product. Therefore, when a firm undercuts the other, it must “compensate” the consumers who switch from the other firm for the increase in the network size.

(b) Solve for the UPE prices. Conclude which firm charges a higher price and explain why.

(c) Calculate the profit level of each firm and conclude which firm makes a higher profit.