ECON217 | Microeconomics in Economics - Pennsylvania State University

1. Some online auctions are run as ascending auctions with a “Buy It Now” option. That is, at any point during the auction, any bidder can immediately claim the object at a predetermined price. Model the ascending portion of the auction as a first price auction: the price rises continuously from 0 until either all but one bidder has dropped out or someone has decided to pay the “Buy It Now” price. Suppose there are two bidders, with independent private values drawn from the uniform distribution on [0, 100]. Suppose the “Buy It Now” price is 50, and that there exists a symmetric equilibrium where both bidders play the following strategy: if si < 50, remain active until the price reaches si and then drop out. If si > 50, remain active until the price reaches g(si ) and then end the auction by bidding the “Buy It Now” price, where g : [50, 100] → [0, 50] is some function.

(a) Show that if g is strictly decreasing, the highest value agent gets the object in equilibrium. (b) Assuming that such a g exists, show that, conditional on winning, a bidder’s expected payment must be the same in this auction as it would be in a standard ascending auction with no “Buy It Now” price. 1 (c) (YOU DON’T HAVE TO, BUT IF YOU’RE CURIOUS) You can use this to calculate the function g, that is, the price at which each type of bidder with value above 50 jumps to a bid of 50.