Econ 467
Professor: John Nachbar
May 2, 2011
Renements of Nash Equilibrium
1
Overview
In game theory, renement refers to the selection of a subset of equilibria, typically
on the grounds that the selected equilibria are, according to some criteria, more
p
Economics 504
Spring 2016
Marcus Berliant
Paulo Natenzon
LECTURE OUTLINE
[Numbers on the right indicate textbook chapters]
0.
Organizational Matters
I.
Introduction (or, The World According to Uncle Marcus)
A.
B.
C.
The Nature of Economic Theory
Why is ma
Q1: Y
T
L
R+
= cfw_0.
Q2: False, since v (cfw_1, 2) = 1 < 2 = v (cfw_1) + v (cfw_2).
Q3:
(1) Since U
L |L=0 = 2 > 1 =
Q = 0, L = 0.
U
Q ,
it's a corner solution. The P.O. allocation is
(2) First we know CE allocations are P.O., so from (1) we know that we
Economics 504
Spring 2015
Marcus Berliant
Brian Rogers
MIDTERM EXAMINATION
March 4, 2015
Please:
1)
Answer all questions.
2)
Be brief. There is a correct answer to each problem. I am not looking
for long essays.
3)
Write your name on all of your answer bo
Economics 504
Spring 2016
Marcus Berliant
Paulo Natenzon
FURTHER READING
This list is clearly biased by my training and research experience. Free
disposal is always an option. The list consists of (older) articles that you should
be able to read after we
VII - The Edgeworth Box: CES Utility
Solution:
A.
M RSS =
M RSK =
21
hS
gS
12
hK
=
gK
US (hS , gS ) /hS
=
US (hS , gS ) /gS
UK (hK , gK ) /hK
UK (hK , gK ) /gK
Letting M RSS = M RSK , we get:
hS
hK
16 hS
=
=
=4
gS
gK
4 gS
i.e.: The contract curve is the
Economics 504
Spring 2016
Marcus Berliant
Paulo Natenzon
MICROECONOMICS II
PART A:
GENERAL EQUILIBRIUM THEORY AND WELFARE
ECONOMICS
COURSE DESCRIPTION
This segment of the Ph.D. course sequence in microeconomic theory will
cover general equilibrium theory
Econ 504A: Problem Set N Answer Key
TA: Wan-Jung Cheng
I. The Utility Possibilities Set
Fasheng and Junnans utility functions are
U (x1 , x2 ) = x1
F
F
F
F
U (x1 , x2 ) = (x2 )2
J
J J J
x1 = U
F
F
x2 =
U
J
J
Endowment: e = (2, 0)
Production Set: Y =
Econ 504A: Problem Set N+1 Answer Key
TA: Wan-Jung Cheng
I. Local Non-Satiation
x Rl+ , > 0, we can find a constant > 0 such that x x + B (x),
where x + = (x1 + , x2 + , ., xl + ).
By strong monotonicity, x x.
Hence we have x Rl+ , > 0, x B (x) such that
Economics 504
Spring 2016
Marcus Berliant
Paulo Natenzon
PROBLEM SET N
DUE: FEBRUARY 17, 2016
Note: As in all of the homework sets for this course, the problems below are
rigged so that the answers turn out to be nice numbers, except where noted.
If you s
Economics 504
Professor: John Nachbar
Spring 2011
Homework 1
Answers
1. Prove: If a strategy i is a best response to i then for any pure strategy si
with i (si ) > 0, si is also a best response.
Proof. By contraposition. Suppose that si is not a best resp
Economics 504
Professor: John Nachbar
Spring 2011
Homework 6
Answers
1. (a) The condition is 4/(1 ) 6 + /(1 ), implying = 2/5.
(b) The condition is 1/(1 ) 0 + 4/(1 ), implying = 1/4.
(c) If there were an SPE involving mutual play of tit-for-tat then, from
Econ 504
Professor: John Nachbar
March 1, 2011
Notes on the Myerson-Satterthwaite Theorem
1
Overview
The Myerson-Satterthwaite Theorem (MS) is an impossibility result on bargaining
with asymmetric information.1 One player, the seller, owns one unit of an
Econ 504
David K. Levine
John Nachbar
May 2, 2007
Econ 504
Final
You have until 12:30PM.
Use good penmanship (or pencilmanship).
Make your answers clear, concise, and complete. Label any diagrams
clearly and explain the diagrams in your text. We will ma
Econ 504
David K. Levine
John Nachbar
February 28, 2007
Econ 504
First Midterm
You have until 11AM.
Use good penmanship (or pencilmanship).
Make your answers clear, concise, and complete. Label any diagrams clearly
and explain the diagrams in your text.
Econ 504
David K. Levine
John Nachbar
April 6, 2007
Econ 504
Second Midterm
You have until 11:30AM.
Use good penmanship (or pencilmanship).
Make your answers clear, concise, and complete. Label any diagrams
clearly and explain the diagrams in your text.
Professor: John Nachbar
Spring 2011
Economics 504
Final
ANSWERS
1. (a) In addition to (A, a), and (B, b), there is a NE in mixed strategies,
32
,
55
,
4x
6x
,
2(5 x) 2(5 x)
.
(b) Changes in x (0, 4) have no aect on the pure strategy equilibria and
leave P
Professor: John Nachbar
Spring 2011
Economics 504
Midterm
You have until 4 PM. You can use either pen or pencil but write legibly and with
good syntax. A correct but unintelligible answer is a wrong answer.
1. Consider the following game
a
b
A 6, 4 x, 0
B
Econ 504
Professor: John Nachbar
Spring 2011
Homework 3
Answers
1. I consider only pure strategy equilibria.
(a) In any pure strategy pooling WPBE, the buyer oers
xa = xb = x cfw_90, 91, . . . , 100.
Note that 90 is the expected value of the object to th
Econ 504
Professor: John Nachbar
Spring 2011
Homework 4
Answers
1. (a) Note rst that in any separating NE, the high cost type produces 3 today.
In a purported NE in which the high cost rm chooses anything other than
3, a deviation to 3 today increase prot
Econ 504
Professor: John Nachbar
Spring 2011
Homework 5
Answers
1. (a) By the revelation principle, it suces to consider truth telling equilibria
of direct revelation auctions. In any such NE, if a player has value vi but
reports vi then his expected payo
Economics 504
Spring 2016
Marcus Berliant
Paulo Natenzon
PROBLEM SET N+1
DUE: MARCH 2, 2016
Note: The answers to problems VI and VII are slightly messy. Its hard to
construct core examples without messy expressions, since they involve utility
levels.
I.
L