Exercise sessions 12
Course: Mathematical methods in Economics
Lecturer: Dmytro Vikhrov
Date: May 14, 2013.
Problem 1
An object is to be sold on an auction. There are two risk-neutral bidders with independent
private values of the object, v1 and v2, distributed according to a continuous distribution
function F(·). A bidder with the highest bid gets the object.
1. Find agent’s expected utility from bidding r when the agent’s private value is v1.
2. Find the bidding function b∗(v).
3. Show that the bidding function is strictly increasing in v.
4. Obtain the biding functions when 1. v ∼ U[0, 1] and 2. v ∼ exp(λ). Plot them on the
graph in (v, b(v)) locus.
Problem 2
Consider a modification to the setup of Problem 1. The highest bid wins the object and
every bidder pays the seller the amount of his bid. Find the bidding function and compare the
bidding behavior to Problem 1.
Problem 3
Consider the following modification of a first-price auction. The highest bidder pays his bid
but receives the object only if the outcome of the toss of a fair coin is heads. If the outcome
is tails, the seller keeps the object and the high bidder’s bid. Find the bidding function and
compare it to b∗(v) from Problem 1.
Problem 4
In the first-price sealed bid auction with two bidders find the seller’s revenue.
Problem 5
Suppose that there are N risk-neutral bidders with independent private valuations. Rederive
b∗(v) and seller’s revenue. Do the comparative statics exercise.
Problem 6
Introduce the second-price sealed bid auction. Show that bidders bid their true value,
bsp(v) = v. Compare bsp(v) to b∗(v).
Problem 7
Show the equivalence of expected revenues in the first and second-price sealed-bid auctions.
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