INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
Economics/Mathematics C103:
Introduction to Mathematical Economics
Problem Set 5
Due 4 October 2012
1. Consider the following random allocation mechanism with 2 bidders. Each bidder has a value
xi uniformly distribution on [0, 1]. She can bid any positive
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2013
Econ103Fall03
Prepared by: Theo Diasakos
PROBLEM SET I
(Suggested Solutions)
1.
a) Consider the following:
x1
x=
x2
1 0
A=
2 1
The quadratic form xT Ax is the required one in matrix form.
Similarly, for the following parts:
0
x
5
b) x = 1
A=
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
Economics/Mathematics C103:
Introduction to Mathematical Economics
2012 Midterm #2
YOUR NAME:
Please sign below to conrm that you agree with the following statement:
On my honor, I will not engage in any form of academic dishonesty during
this examination
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
ECON/MATH C103
SPRING 2012
SOLUTIONS TO MIDTERM 1
(1) (a) Suppose (b , b ) is a Nash equilibrium. Then we must have
1 2
b
b
b arg max
1
b + b
b[0,1]
2
and
b arg max
2
b[0,1]
b
b.
b + b
1
Applying the rstorder conditions gives
b =
1
b b
2
2
(1)
and
(2)
b
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
Economics/Mathematics C103:
Introduction to Mathematical Economics
2009 Midterm
NAME:
Please sign below to conrm that you agree with the following statement:
On my honor, I will not engage in any form of academic dishonesty during
this examination, as art
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
Economics/Mathematics C103:
Introduction to Mathematical Economics
2012 Midterm #1
YOUR NAME:
Please sign below to conrm that you agree with the following statement:
On my honor, I will not engage in any form of academic dishonesty during
this examination
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
Economics/Mathematics C103:
Introduction to Mathematical Economics
2010 Midterm
YOUR NAME:
Please sign below to conrm that you agree with the following statement:
On my honor, I will not engage in any form of academic dishonesty during
this examination, a
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
Economics/Mathematics C103:
Introduction to Mathematical Economics
2010 Midterm
YOUR NAME:
Please sign below to conrm that you agree with the following statement:
On my honor, I will not engage in any form of academic dishonesty during
this examination, a
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
Economics/Mathematics C103:
Introduction to Mathematical Economics
2010 Final
YOUR NAME:
Please sign below to conrm that you agree with the following statement:
On my honor, I will not engage in any form of academic dishonesty during this examination, as
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
Economics/Mathematics C103:
Introduction to Mathematical Economics
Spring 2012 Final
YOUR NAME:
Please sign below to conrm that you agree with the following statement:
On my honor, I will not engage in any form of academic dishonesty during this examinati
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
ECON/MATH C103
Fall 2012
Problem Set 8 Suggested Solutions
1. Consider the example in the lecture notes with three players with uniform signals and
where V (x1 , x2 , x3 ) = 3 xi /3. What is the expected revenue to the seller in the
i=1
secondprice aucti
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
Econ C103
Fall 2012
Problem Set 1 Suggested Solutions
1. Suppose that there are three rms. The market demand is p(q1 , q2 , q3 ) = 120 q1
q2 q3 . Firm is prot is
2
ui (q1 , q2 , q3 ) = qi p(q1 , q2 , q3 ) qi .
Compute the Nash equilibrium quantity for ea
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
ECON/MATH C103
Fall 2012
Problem Set 3 Suggested Solutions
1. Consider the exponential distribution F (x) = 1 exp(x). Assume = .
(a) Compute the probability density function.
Solution:
The probability density function can be computed as the rst derivative
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
ECON/MATH C103
Fall 2012
Problem Set 2 Suggested Solutions
1. Suppose that there are N bidders in an allpay auction. In other words, all bidders
must pay their bids to the auctioneer; the highest bid wins the object. Their values
for the object are v1 >
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
Economics/Mathematics C103:
Introduction to Mathematical Economics
2010 Midterm
YOUR NAME:
Please sign below to conrm that you agree with the following statement:
On my honor, I will not engage in any form of academic dishonesty during
this examination, a
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2012
Economics/Mathematics C103:
Introduction to Mathematical Economics
2010 Final
YOUR NAME:
Please sign below to conrm that you agree with the following statement:
On my honor, I will not engage in any form of academic dishonesty during this examination, as
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2013
Econ103Fall03
Prepared by: Theo Diasakos
Problem Set 4
Suggested Solutions
Problem 1
(A) The market demand function is the solution to the following utilitymaximization
problem (UMP):
1
max U ( x1 , x2 , x3 ) = ( x1 x2 ) 3 + x3
( x1 , x2 , x3 )
s.t.
p1
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2013
Econ103 Fall 03
Prepared by: Theo Diasakos
Problem Set #3
Suggested Solutions 1 (A) The given problem is: H max u (Y , H ) = ln Y + (Y , H ) 12 s.t. Y = (6(24 H ) 2
1
(I)
Using the equality constraint we can eliminate Y from the objective. 1 H max u (H )
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2013
Econ103Fall03
Prepared by: Theo Diasakos
Problem Set 2
Suggested Solutions 1. Following the notation in Appendix A.2 of D. McFadden Definite Quadratic Forms Subject to Constraints in M. Fuss and D. McFadden Production Economics, Vol. 1, we have for the (
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2013
Econ103Fall 03
Prepared by: Theo Diasakos
Problem Set #5
Suggested Solutions
1. Consider the utility contour of agent h i.e. the set of consumption bundles
x h = ( a h , b h ) that give the same utility level u . This will be given:
cfw_x
h
: u h ( xh )
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2013
Econ103Fall03
Prepared by: Theo Diasakos
Problem Set 6
Suggested Solutions
N
p z ( p) = 0
1. Walras Law:
i i
i =1
Proof:
Assume that each agent chooses optimally to consume a bundle that lies
on his budget constrain. We have:
N
N
N
i =1
( )
i =1
( ( ) )
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2013
Econ. C103, 2003
Daniel McFadden
Problem Set 10. Example Final Exam Questions
(With Solutions)
1. There are J firms in an industry. Each can try to convince Congress to give the industry
a subsidy. Let Hj denote the hours of effort put in by firm j, and l
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2013
Econ. C103, 2003
Daniel McFadden
Final Exam Solutions
1. Consider the following simultaneousmove strategicform game:
Player 2
A
Player 1
B
C
U
(1,1)
(1,0)
(1,1)
D
(1,2)
(1,0)
(1,1)
Find all the Nash equilibria (pure and mixed) of this game.
Let u b
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2013
Econ103Fall03
Prepared by: Theo Diasakos
Problem Set #9
Suggested Solutions
1. (JR #9.5)
One could actually repeat verbatim the argument given in JR (pp. 379) regarding the
optimal strategy in a secondprice auction. The catchphrase there is regardless o
INTRODUCTION TO MATHEMATICAL ECONOMICS CONTRACT THEORY
ECON 103C

Fall 2013
Econ103Fall03
Prepared by: Theo Diasakos
Problem Set #7
Suggested Solutions
1. Consider that John is playing ( O, F ) according to the mixed strategy
( p,1 p ) : p [0,1] whereas Mary is playing according to the mixed strategy
( q,1 q ) : q [ 0,1] .
The e