Economics 201b
Spring 2010
Solutions to Problem Set 7
John Zhu
1a. Suppose there is a portfolio z such that Rz 0 and Rz = 0. Then q z = Rz > 0. If,
however, we only have 0, then it is possible that the nonzero coordinates of and the
nonzero coordinates of
Matthew Rabin
Department of Economics
University of CaliforniaBerkeley
Economics 201A
Fall 2010
Problem Set D
Handed Out: Tuesday, November 23. Optimal perception for when problems are due: tuesday,
December 7. Problems are due: never. (Hence, the usual "
Econ 201A
Fall 2010
Problem Set 4 Suggested Solutions
1. MWG 3.G.15. Consider the utility function
u(x1 , x2 ) = 2 x1 + 4 x2
(a) Find the demand functions for goods 1 and 2 as they depend on prices and wealth.
If you set up the Lagrangian
L(x1 , x2 , ) =
Econ 201A
Fall 2010
Problem Set 2 Suggested Solutions
1. Prove that the lexicographic preference (dened in Example 3.10 in the course notes)
is complete, transitive, and antisymmetric (if x y and y x, then x = y ).
Proof. Completeness Let x, y R2 be given
201B Final Spring 2004
Answer all of the questions below. Be as complete, correct, and concise as possible. There
are 6 questions for a total of 180 points possible. You have 180 minutes to complete the
exam. Use the points as a guide to allocating your t
201b Final Spring 2008
Answer all of the questions below. Be as complete, correct, and concise as possible. There
are 6 questions for a total of 180 points possible. You have 180 minutes to complete the
exam. Use the points as a guide to allocating your t
Economics 201b
Spring 2011
Chris Shannon
Problem Set 3 Due Thursday April 7
1. Consider a production economy with three goods (two outputs denoted good 1 and good 2,
and one input, denoted good 3), two consumers, and two rms. The rms have production
funct
Economics 201b
Spring 2011
Problem Set 4 Solutions
1. A common misperception about the boundary condition on excess demand is to
think that it says that if the price of a good goes to zero, then excess demand
for that good goes to innity. Although intuiti
114.Under IFRS No. 9: which is not a category for accounting for investments?
A.
B.
C.
D.
Fair value through profit and loss.
Fair value through other comprehensive income.
Held-to-maturity.
Amortized cost.
115.Which of the following is NOT true about the
Section 2 : Choice, Preferences, and Utility
ECON 201A, Fall 2010
GSIs: Omar Nayeem and Aniko Oery
These notes were originally prepared by Juan Sebastin Lleras during the Fall 2007 semester and have since been
a
revised by Juan Sebastin and us. We are gra
2
Preference and choice
Denition 2.1. A binary relation
on X is a preference relation if it is a weak order, i.e. complete
and transitive. For any binary relation
, let
x
denote the asymmetric component of
y y
denote the symmetric component of
x;
y and n
Economics 201A: Economic Theory (rst half )
Tu-Th 12:302:00
9 Lewis
1
Description
Economics 201A is the rst semester of the required microeconomic theory sequence for rst year
Ph.D. students in the economics department. The rst half of the fall semester f
Suppose x
C
y . By denition, there exists some set A with x, y A and x C (A). Then
x C (A) because C = C . By denition, x
z for all z A. But y A, so x
y.
Proposition 2.15 is a uniqueness result: the only preference relation that can rationalize C is its
r
Proposition 2.10. If
is a preference relation, then C (A) = whenever A is nite.
Proof. The proof is by induction on the size of |A|. Base step: |A| = 1. Then if x A, x
x by
completeness and the only element of A is x, so C (A) = cfw_x.
Inductive step. Sup
8
The Mixture Space Theorem
Let be a convex subset of Rn , i.e. if , , then + (1 ) for all (0, 1).
on is independent if, for all , , and (0, 1),
Denition 8.1. A binary relation
+ (1 )
Example 8.2. Suppose = R2 and x
+ (1 ).
2
2
y if and only if x2 + x2
1
Binary relations
Denition 1.1. R X Y is a binary relation from X to Y . We write xRy if (x, y ) R and
not xRy if (x, y ) R.
/
When X = Y and R X X , we write R is a binary relation on X .
Exercise 1.2. Suppose R, Q are two binary relations on X . Prove
complete and transitive preferences over cfw_a, b, c as follows:
a
1
b
1
c
b
2
a
2
c.
c
3
a
3
b
Now suppose we let
C (A) = cfw_x A : |cfw_i : x C i (A)| |cfw_i : y C i (A)|, for all y A.
In words, x C (A) if there is no alternative which would be chosen b
1
Binary relations
Denition 1.1. R X Y is a binary relation from X to Y . We write xRy if (x, y ) R and
not xRy if (x, y ) R.
/
When X = Y and R X X , we write R is a binary relation on X .
Exercise 1.2. Suppose R, Q are two binary relations on X . Prove
1. C meets Sens and ;
2. C meets Houthakkers axiom;
3. C is rationalizable.
Proof. We will show (1) implies (2) implies (3) implies (1).
Step 1: Sens and imply Houthakkers Axiom. Suppose C meets Sens and .
Assume x, y A B , x C (A), and y C (B ). Applying
Section 6 : Production & vNM utility
ECON 201A, Fall 2011
GSIs: Aniko Oery and Mich`ele M
uller
These notes were originally prepared by Juan Sebasti
an Lleras during the Fall 2007 semester and have since been
revised by Juan Sebasti
an, Omar Nayeem and us
Section 2 : Choice, Preferences, and Utility
ECON 201A, Fall 2011
GSIs: Aniko Oery and Mich`ele M
uller
These notes were originally prepared by Juan Sebasti
an Lleras during the Fall 2007 semester and have since been
revised by Juan Sebasti
an, Omar Nayee
Section 5 : Afriats Theorem
ECON 201A, Fall 2011
GSIs: Aniko Oery and Mich`ele M
uller
These notes were originally prepared by Juan Sebasti
an Lleras during the Fall 2007 semester and have since been
revised by Juan Sebasti
an, Omar Nayeem and us. We are
Section 1 : Preference and Choice
ECON 201A, Fall 2011
GSIs: Aniko Oery and Mich`ele M
uller
These notes were originally prepared by Juan Sebasti
an Lleras during the Fall 2007 semester and have since been
revised by Juan Sebasti
an, Omar Nayeem and us. W
11
Basics of Savage expected utility
This entire section is totally optional.
In both the von NeumannMorgenstern and AnscombeAumann models, we assumed the existence of
some objective randomizing device. Ideally, all uncertainty in the model would be subje
7
Producer behavior
Definition 7.1. A production set is a subset Y Rn .
Definition 7.2. Y satisfies:
no free lunch if Y Rn+ cfw_0n ;
possibility of inaction if 0n Y ;
free disposal if y Y implies y 0 Y for all y 0 y;
irreversibility if y Y and y 6= 0n
Econ 201A (2011)
Microeconomic Theory I
Haluk Ergin
Game Theory - Basics I
Normal Form Games
Game Theory
How may a group of self-interested
individuals behave if each of them
is affected by the others' actions?
We need to specify
Who are the players?
Wh
Econ 201A (2011)
Microeconomic Theory I
Haluk Ergin
Game Theory - Basics II
General Extensive Form Games and
Sequential Equilibrium
An extensive form game
l
(5,-2)
t
1
c
a
r
2
m
y
z
x
b
y
2
y
(2,0)
x
(0,2)
(3,0)
2
d
(2,0)
(2,1)
x
1
(-1,2)
(1,2)
fc(a|(l)=f
Econ 201A (2011)
Microeconomic Theory I
Haluk Ergin
Repeated Games with Observable
Actions (Perfect Monitoring)
Repeated Games with Observable Actions
Let G = (N, A, u) be a finite normal form game. Let G(T )
denote the extensive form game where:
At each
Econ 201A (2011)
Microeconomic Theory I
Haluk Ergin
Reputation Formation
Reputation & Equilibrium Selection
in Games with a Patient Player
Fudenberg & Levine (1989)
Repeated Games with one Long-Run and
many Short-Run Players: The Complete
Information Benc