Economics/Mathematics 103
Fall 2014
Chris Shannon
Problem Set 4 Due Thursday September 25
1. Consider a marriage market allowing for indierences, in which M = cfw_m1 , m2 , m3 , W =
cfw_w1 , w2 , w3 , and preferences given by
m1 : w1 m1 w3 m1 m1 w2
w 1 :
Econ. C103, 2003
Daniel McFadden
MECHANISM DESIGN, DIRECT SELLING MECHANISMS, EFFICIENT AUCTIONS
The theory of mechanism design provides some general insights into the construction
of resource allocation mechanisms that achieve specified objectives. One o
Analysis and Linear Algebra
Lectures 1-3 on the mathematical
tools that will be used in C103
Set Notation
A,B
AcB
A1B
A\B
N.
AfB
. Ac
a0A
aA
sets
union
intersection
the set of objects in A that are not in B
Empty set
inclusion (A is contained in B)
the co
7. This mechanism is clearly not incentive compatible.
To see this, let us denote by cfw_v1 ,., vN +1 the true valuations of the good for the N+1
consumers, where vn represents the nth highest valuation. Clearly, the maximum amount
that a consumer would b
C103, Fall 20003, Problem Set 4 (due October 2)
1. Suppose a consumer has a concave utility function U(x), where x = (x1,.,xn) is a vector of n
goods and services, and maximizes utility subject to x $ 0 and a budget constraint p@x # y, where
p > 0 is a ve
PROBLEM SET II
1.
This question refers to the notation and equations of Appendix A.1 of
D. McFadden Definite Quadratic Forms Subject to Constraints in M. Fuss and D. McFadden Production
Economics, Vol. 1.
Prove that if Gx ( x ) in (SND) does not have maxi
C103, Fall 20003, Problem Set 3 (due September 25)
1. In a Robinson Crusoe economy where the goods are leisure (H) and yams (Y), the feasible
resource allocations lie on or below the curve Y = [6(24-H)]. Robinson has preferences that can
be described by a
Econ C103, 2003
McFadden
Existence of Walrasian Equilibrium
Theorem (Grandmont-McFadden, 1972)
Define the closed unit simplex U* = cfw_p0m | p $ 0 and 1@p = 1 and the open unit
simplex U0 = cfw_p0U* | p>0. Suppose there exists a set U with U0 f U f U* and
Optimization Theory
Lectures 4-6
Unconstrained Maximization
Problem Maxim afunctionf: 6withinaset A
:
ize
fn.
n
Typically, Ais , or thenon-negativeorthant
n
cfw_x0 |x$0
n
Existence of a maximum:
Theorem. If A is compact (i.e., closed and bounded)
and f is
Econ C103, 2003
Daniel McFadden
THE THEORY OF FIRST-PRICE, SEALED-BID AUCTIONS
1. Within the class of first-price, sealed-bid auctions, there are a number of
possible variations in environment, information, and rules:
(1) The number of potential bidders i
Consumer Theory: The Mathematical Core
Dan McFadden, C103
Suppose an individual has a utility function U(x) which is a function of non-negative
commodity vectors x = (x1,x2,.,xN), and seeks to maximize this utility function subject to the budget
constrain
Economics/Mathematics 103
Fall 2014
Chris Shannon
Problem Set 1 Suggested Solutions
1. Prove that there is no greatest even integer. (Hint: prove this by contradiction)
Solution: Suppose, by way of contradiction, that M is the greatest even integer. Thus
Economics/Mathematics 103
Fall 2014
Chris Shannon
Problem Set 3 Due Thursday September 18
1. Consider the marriage market example from class and problem set 2 #3, with M = cfw_m1 , m2 , m3 ,
W = cfw_w1 , w2 , w3 , and preferences given by
m1 : w2 m1 w1 m1
Economics/Mathematics 103
Fall 2014
Chris Shannon
Problem Set 1 Due Thursday September 4
1. Prove that there is no greatest even integer. (Hint: prove this by contradiction)
2. Prove that if r is a rational number and x is an irrational number, then r x i
Economics/Mathematics 103
Fall 2014
Chris Shannon
Problem Set 2 Due Thursday September 11
1. Let
be a preference relation on a set X. For each x X, dene
I(x) := cfw_y X : y x
So I(x) is the set of elements in X that are ranked indierent to x according to
MATH/ECON C103 Problem Set 1 Comments
General Notes
1. PLEASE WRITE LEGIBLY
Please do not write write very small, and please leave adequate
space between your text so that it is easier for your graders to
read. Please also write as legibly and do not writ
Proof by Deduction
Proof by Deduction: A list of statements, the last of which is
the statement to be proven. Each statement in the list is either
an axiom: a fundamental assumption about mathematics, or
part of denition of the object under study; or
a
Economics/Mathematics C103
Introduction to Mathematical Economics
Fall 2014
T-Th 8-9:30
3106 Etcheverry
Professor Chris Shannon
1
Description
C103 is an interdisciplinary topics course in mathematical economics, focusing this semester on
matching and mark
Econ. C103, 2003
Daniel McFadden
Problem Set 10. Example Final Exam Questions
(For practice, not to be handed in)
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
Econ C103, 2003
Daniel McFadden
THE WINNERS CURSE
Consider an auction for a single item whose value to a buyer is not known with certainty,
but must be estimated. Each players bid will be based on his estimate, which in turn will
be based on his own infor