SOLUTION MANUAL
All Chapters Included
1
, Solutions Manual for
AUCTION THEORY*
Alexey Kushnir and Jun Xiao
August 2009
Contents
2 Private Value Auctions: A First Look........................................................................ 2
3 The Revenue Equivalence Principle ............................................................................ 8
4 Qualifications and Extensions .....................................................................................11
5 Mechanism Design ....................................................................................................... 17
6 Auctions with Interdependent Values ....................................................................... 25
8 Asymmetries and Other Complications....................................................................34
9 Efficiency and the English Auction........................................................................... 40
10 Mechanism Design with Interdependent Values...................................................... 43
11 Bidding Rings .............................................................................................................. 48
13 Equilibrium and Efficiency with Private Values ......................................................52
15 Sequential Sales ........................................................................................................... 55
16 Nonidential Objects .....................................................................................................60
17 Packages and Positions ............................................................................................... 62
2
,2 Private Value Auctions: A First Look
Problem 2.1 (Pomer distribution) Suppose there are tmo bidders mith private values
that are distributed independently according to the distribution F (x) = xa over [0, 1]
mhere a > 0. Find symmetric equilibrium bidding strategies in a first−price auction.
Solution. Since N = 2, G(x) = F (x) = xa. Thus, using the formula on page 16 of
the text,
β (x) = x — dy = x — dy = x
O G (x) O xa 1+a
∫ x ∫ x
I G (y) ya a
Problem 2.2 (Pareto distribution) Suppose there are tmo bidders mith private values
that are distributed independently according to a Pareto distribution F (x) = 1 —
(x + 1)—2 over [0, ∞). Find symmetric equilibrium bidding strategies in a first−price
auction. Shom by direct computation that the expected revenues in a first− and second−
price auction are the same.
Solution. Again, since N = 2, G (x) = F (x) = 1 — (x + 1)—2. Thus,
∫
x
G (y)
β (x) = x —
I
dy
G (x)
∫O
= x— —
x
1 (y + 1)—2
x dy
O 1 — (x + 1)—2
=
x+2
In the first-price auction, the expected revenue of the seller is
E RI = 2E mI (x)
= 2E G (x) × βI (x)
3
, ∫ ∞ x
= 2 1 — (x + 1)—2 2 (x + 1)—3 dx
O x+2
= 1/3
Let Y2 be the second highest value, and its density is ƒ2 (y) = 2 (1 — F (y)) g (y)
(see Appendix C).
In a second-price auction, the expected revenue of the seller is
E RII = E [Y2]
∫∞
= y2 (y + 1)—2 2 (y + 1)—3 dy
O
= 1/3
Therefore, the expected revenues in the two auctions are the same.
4
All Chapters Included
1
, Solutions Manual for
AUCTION THEORY*
Alexey Kushnir and Jun Xiao
August 2009
Contents
2 Private Value Auctions: A First Look........................................................................ 2
3 The Revenue Equivalence Principle ............................................................................ 8
4 Qualifications and Extensions .....................................................................................11
5 Mechanism Design ....................................................................................................... 17
6 Auctions with Interdependent Values ....................................................................... 25
8 Asymmetries and Other Complications....................................................................34
9 Efficiency and the English Auction........................................................................... 40
10 Mechanism Design with Interdependent Values...................................................... 43
11 Bidding Rings .............................................................................................................. 48
13 Equilibrium and Efficiency with Private Values ......................................................52
15 Sequential Sales ........................................................................................................... 55
16 Nonidential Objects .....................................................................................................60
17 Packages and Positions ............................................................................................... 62
2
,2 Private Value Auctions: A First Look
Problem 2.1 (Pomer distribution) Suppose there are tmo bidders mith private values
that are distributed independently according to the distribution F (x) = xa over [0, 1]
mhere a > 0. Find symmetric equilibrium bidding strategies in a first−price auction.
Solution. Since N = 2, G(x) = F (x) = xa. Thus, using the formula on page 16 of
the text,
β (x) = x — dy = x — dy = x
O G (x) O xa 1+a
∫ x ∫ x
I G (y) ya a
Problem 2.2 (Pareto distribution) Suppose there are tmo bidders mith private values
that are distributed independently according to a Pareto distribution F (x) = 1 —
(x + 1)—2 over [0, ∞). Find symmetric equilibrium bidding strategies in a first−price
auction. Shom by direct computation that the expected revenues in a first− and second−
price auction are the same.
Solution. Again, since N = 2, G (x) = F (x) = 1 — (x + 1)—2. Thus,
∫
x
G (y)
β (x) = x —
I
dy
G (x)
∫O
= x— —
x
1 (y + 1)—2
x dy
O 1 — (x + 1)—2
=
x+2
In the first-price auction, the expected revenue of the seller is
E RI = 2E mI (x)
= 2E G (x) × βI (x)
3
, ∫ ∞ x
= 2 1 — (x + 1)—2 2 (x + 1)—3 dx
O x+2
= 1/3
Let Y2 be the second highest value, and its density is ƒ2 (y) = 2 (1 — F (y)) g (y)
(see Appendix C).
In a second-price auction, the expected revenue of the seller is
E RII = E [Y2]
∫∞
= y2 (y + 1)—2 2 (y + 1)—3 dy
O
= 1/3
Therefore, the expected revenues in the two auctions are the same.
4
