(d) Suppose that Charlie sells by method 3 and that both potential buyers bid rationally. If both bidders are professional bulldozer oper- ators, how much will each bid? $6 , 000. How much will Cher- lie receive for his bulldozer? $6 , 000 . If one bidder is a profes- sional bulldozer operator and one is a. recreational user, What bids will Charlie receive? Professional bids $6,000. Recreational user bids $4,500. Who will get NAME 215 the bulldozer? The professional. How much money will Charlie get for his bulldozer? $4: , 500 . If both bidders are recre- ational bulldozer users, howr much will each bid? $4 , 500 . How much will Charlie receive for his bulldozer? $4 , 500. What will be Charlie's expected revenue from selling the bulldozer by method 3? $(1/4) x $6, 000 + (3/4) x $4, 500 = $4, 875. (3) Which of the three methods will give Charlie the highest expected revenue? Method 3 . 17.2 (2) 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 214 AUCTIONS (Ch. 1?} 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: I Method 1. Post a price of $6,000 and if nobody takes the bulldozer at that price, sell it to the wholesaler. a Method 2. Post a price equal to a recreational bulldozer user's buyer value and sell it to anyone who offers that price. a 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) 1What is the probability that both potential buyers are professional bulldozer operators? 1/4 . What is the probability that both are recre- ational bulldozer users? 1 /4 . 1W'hat is the probability that one of them is of each type? 1/2 . (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? 3/ 4. \"That is the probability that he will have to sell the bulldozer to the wholesaler? 1/4. \"That is his expected revenue? $(3/4) X $6, 000 + (1/4) x $1,000 = $4, 750. (c) If Charlie sells by method 2, how much will he receive for his bulldozer? $4 , 500 . 17.3 (2) We revisit our nancially aficted friend, Charlie Plopp. This time we will look at a slightly generalized version of the same problem. All else is as before, but the willingness to pay of recreational bulldozers is an amount 0 C whenever C $(3/4)6, 000 + (1/4) 1, 000 = 4, 759. 0') For what values of C is Charlie better oif selling by method 1 than by method 3? This happens when 4,750 > 1,500+ 1, 40 which is the case whenever C