1. Under the assumption that each bidders density fi is continuous and strictly
positive and that each vi 1Fi(vi) fi(vi) is strictly increasing.
(a) Show that the optimal selling mechanism entails the seller keeping the
object with strictly positive proba
(Principal-Agent Problem with Hidden Information) Consider a principal-agent
problem with hidden information in which the principal faces the problem of
designing an optimal (i.e., payoff maximising) contract for an agent who will
come to possess private
(Stochastic Dominance) Consider an N-bidder first-price auction with
independent private values. Let b be the symmetric equilibrium bidding
strategy when each bidders value is distributed according to F on [0, c].
Similarly, let b be the equilibrium strat
Consider direct mechanisms that can be derived from the first- and secondprice auctions. (a) Use the equilibrium of the second-price, sealed-bid auction
to construct an incentivecompatible direct selling mechanism in which the ex
post assignment of the ob
In this problem, you shall explore the consequences of risk aversion on the
part of bidders. There are N bidders participating in a first-price auction. Each
bidders value is independently drawn from [0, 1] according to the distribution
function F, having
Consider the second-price, sealed-bid auction. Show that bidding ones value
is weakly dominant for bidders with independent private values (you may
show that either bidding higher or bidding lower than ones value is weakly
dominated in terms of payoffs).
Show that the equilibrium bidding strategy of a first-price auction b(v) = 1 F
N1 (v) Z v 0 xdF N1 (x) (as derived in class) is strictly increasing. Also show
that b(v) < v for any finite number N of bidders and that b(v) converges to v
as N goes to infin
2. (Optimal Auction) There is a single object for sale and there are two
potential buyers. The value assigned by buyer 1 to the object V1 is uniformly
drawn from the interval [0, 1 + k] whereas the value assigned by buyer 2 to
the object V2 is uniformly d
[Dana & Spier (1994), Designing a private industry: Government auctions
with endogenous market structure, Journal of Public Economics] Two firms, j
= 1, 2, compete for the right to produce in a given market. A social planner
designs an optimal auction of
Microeconomics
Midterm Examination
March 24, 2014
1. Consider the following chicken game.
A2
B2
A1
2, 2
0, 3
B1
3, 0
-1, -1
(i) Find the all Nash equilibria (including that in mixed-strategy). (5%)
(ii) Find the minimax payo for both players 1 and 2. (15%