Econ/Math C103 - Problem Set 1 Solutions
Question 1
Let N = cfw_1, . . . , n be the set of people. Foreach i N , let ki cfw_0, 1, . . . , n
be the number of people whose hands person i shakes (so if person i shakes
5 peoples hands, ki = 5). Let h denotePt
LECTURE 1
Preferences
Preferences
Our economic agent will soon be advancing to the stage of economic
models. Which of his characteristics will we be specifying in order to
get him ready? We might have thought name, age and gender, personal history, brain
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
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
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 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
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
Econ/Math C103 - Midterm Solutions
10/15/2015
1. Let X = R+ be nonnegative monetary prizes and consider an agent with vNM
utility function u(x) = x. Would her preferences over simple lotteries be the
same if she instead had the vNM utility function: ua (x
Econ/Math C103 (2016) - Problem Set 2
Due 09/22/2016
1. Let X = [0, 1] [0, 1] = cfw_x = (x1 , x2 )|x1 , x2 [0, 1]. Let the preference relation
R over X be defined by:
(x1 , x2 )R(y1 , y2 ) [(x1 > y1 ) or (x1 = y1 and x2 y2 )],
for all x = (x1 , x2 ), y =
Econ/Math C103 - Problem Set 3 Solutions
Question 1
The Tournament SWF satisfies Rationality and Unanimity but not IIA.
1. Rationality: Rationality follows because the tournament SWF at
any preference profile R is representable by the real-valued function
Chpt1
Denition 2 Let B be a binary relation on X. Then
B is weak-order if it is complete and transitive.
B is a linear/strict order if it is complete, transitive, and antisymmetric.
Denition 3 A preference R over alternatives is rational if it is comple
Econ/Math C103 (2016) - Problem Set 1
Due 09/6/2016
1. A group of people met and some of them shook each other hands. Prove that the
number of people who shook others hands an odd number of times is, in fact,
even.
2. Let X be a nonempty set, and let 2X d
Econ/Math C103 (2016) - Problem Set 3
Due 10/6/2016
1. Read the definition of the tournament SWF from the lecture notes. Which of
the three properties: Rationality, IIA, and Unanimity, does the tournament SWF
satisfy?
2. For the case when |X| = 2 and n 2,
Econ/Math C103 - Midterm
10/15/2015
Instructions: This is a closed book exam. You have 75 minutes. Each question is
worth 20 points. Write clearly, explain your answers, and be concise. You may use any
result from class. Good luck!
1. Let X = R+ be nonneg
Math C103 Problem Set #1
Siyuan Tao 24150524
1. Once two people, say A and B shake hands, 2 will be added to the total number of
hands shakes M. So the total number of hands shake M must be an even number.
Suppose: M(n) := the amount of hands shakes made
Econ/Math C103 - Problem Set 2 Solutions
Question 1
To show that R is rational, it is sufficient to show that R is complete and
transitive.
R is complete
Given (x1 , x2 ) and (y1 , y2 ) such that x1 , x2 , y1 , y2 [0, 1] there are three
possibilities:
1.
LECTURE 3
Choice
Choice Functions
Until now we have avoided any reference to behavior. We have talked
about preferences as a summary of the decision makers mental attitude
toward a set of alternatives. But economics is about action, and therefore
we now m