Charlie Plopp sells used construction equipment in a quiet Oklahoma town. He has run short of cash and needs to raise money quickly by selling an old bulldozer. If he doesn’t sell his bulldozer to a customer today, he will have to sell it to a wholesaler for $1,000. Two kinds of people are interested in buying bulldozers. These are professional bulldozer operators and people who use bulldozers only for recreational purposes on weekends. Charlie knows that a professional bulldozer operator would be willing to pay $6,000 for his bulldozer but no more, while a weekend recreational user would be willing to pay $4, 500 but no more. Charlie puts a sign in his window. “Bulldozer Sale Today.” Charlie is disappointed to discover that only two potential buyers have come to his auction. These two buyers evidently don’t know each other. Charlie believes that the probability that either is a professional bulldozer operator is independent of the other’s type and he believes that each of them has a probability of 1/2 of being a professional bulldozer operator and a probability of 1/2 of being a recreational user. Charlie considers the following three ways of selling the bulldozer:
Method 1. Post a price of $6,000, and if nobody takes the bulldozer at that price, sell it to the wholesaler.
Method 2. Post a price equal to a recreational bulldozer user’s buyer value and sell it to anyone who offers that price.
Method 3. Run a sealed-bid auction and sell the bulldozer to the high bidder at the second highest bid (if there is a tie, choose one of the high bidders at random and sell the bulldozer to this bidder at the price bid by both bidders.)
(a) What is the probability that both potential buyers are professional bulldozer operators? ________. What is the probability that both are recreational bulldozer users? ________. What is the probability that one of them is of each type? ________.
(b) If Charlie sells by method 1, what is the probability that he will be able to sell the bulldozer to one of the two buyers? ________. What is the probability that he will have to sell the bulldozer to the wholesaler? ________. What is his expected revenue? ________.
(c) If Charlie sells by method 2, how much will he receive for his bulldozer? ________.
(d) Suppose that Charlie sells by method 3 and that both potential buyers bid rationally. If both bidders are professional bulldozer operators, how much will each bid? ________. How much will Charlie receive for his bulldozer? ________. If one bidder is a professional bulldozer operator and one is a recreational user, what bids will
Charlie receive? ________. ________. Who will get the bulldozer? ________. How much money will Charlie get for his bulldozer? ________. If both bidders are recreational bulldozer users, how much will each bid? ________. How much will Charlie receive for his bulldozer? ________. What will be Charlie’s expected revenue from selling the bulldozer by method 3? ________.
(e) Which of the three methods will give Charlie the highest expected revenue?