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Problem 3 – Auctions & Reserve Prices
Please consider the following auction problem
You have a single unit of an indivisible good to sell. You have two potential buyers, and decide to auction the good. You do not know the buyers’ values (i.e. their maximum willingness to pay), but you do know that in each case these are uniformly distributed on [0, 1]. In other words, all values between $0 and $1 are equally likely (the valuations of the two buyers are independent). If it is unsold, the good is worth nothing to you (e.g. because it is perishable). As the auctioneer, you raise the price in very small increments starting from $0. The buyers are instructed to keep their hands up so long as they are willing to buy at the current price (i.e., so long as their value exceeds the price). When the first bidder drops out (puts his hand down), the other bidder gets the good at the price just called.
a) Set up a simple Excel model that models this scenario. Designate a cell that calculates your profit from the auction.
What is your expected profit from this auction?
Include an Exhibit to support your argument as Exhibit 3-A.
You wonder whether profits can be increased by setting a reserve price. Everything else remains the same, but you are now allowed to announce a reserve price R. This is the lowest price at which you are willing to sell – so you start raising the price from $R instead of $0. The risk is that both people drop out at $R itself, in which case you make nothing. If one person drops out right away, the good is sold at $R to the remaining bidder. If both are still in the bidding, the auction proceeds as before with the price being increased until one person remains in the bidding.
b) What is expected profit if you set a reserve price of $0.4? [Attach an image of the @Risk model as Exhibit 3-B. Use Ctrl-~ to display formulae.]
c) Examine the following options for R: 0.3, 0.35, 0.4, 0.45, 0.5, 0.55, 0.6, 0.65, 0.7, and 0.75, using a RiskSimTable.
What is the best reserve price?
At that reserve price, what is the expected profit?
At that reserve price, what is the standard deviation of profit?