The European Union fined Sotheby's auction house more than €20 million for operating (along with rival auction house Christie's) a price-fixing cartel (see "The Art of Price Fixing" in MyEconLab, Chapter Resources, Chapter 14). The two auction houses were jointly setting the commission rates sellers must pay. Let r denote the jointly set auction commission rate, Di(r) represent the demand for auction house i's services by sellers of auctioned items, p denote the average price of auctioned items, F represent an auction house's fixed cost, and v denote its average variable cost of auctioning an object. At the agreed-upon commission rate r, the profit of an auction house i is (i = r(Di(r) - [F + vDi(r)].
a. What is the sum of the profits of auction houses i and j?
b. Characterize the commission rate that maximizes the sum of profits. That is, show that the commission rate that maximizes the sum of profits satisfies an equation that looks something like the monopoly's Lerner Index profit-maximizing condition, Equation 11.11.
c. Do the auction houses have an incentive to cheat on their agreement? If Christie's does so while Sotheby's continues to charge r, what will happen to their individual and collective profits?