Measure Theory and Probability Theory
Stphane Dupraz
In this chapter, we aim at building a theory of probabilities that extends to any set the theory of probability
we have for finite sets (with which you are assumed to be familiar). For a finite set with

Problem Set 2, Suggested Solutions
1. Let A be a square matrix of size n. Assume A has a left-inverse: there exists B such that BA = In .
Consider the homogeneous equation Ax = 0. Pre-multiplying by B, BAx = 0, ie. x = 0. This shows that
Ax = 0 has only t

Problem Set 4
Due Tuesday, August 25, in class.
1. Let (X, d) be a metric space.
(a) Show the reverse triangle inequality: x, y, z X, |d(x, z) d(z, y)| d(x, y).
Let S be a subset of X and x X a point of X. We define the distance between x and S as d(x, S)

Problem Set 5, Suggested Solutions
1. (a) The constraint set is the circle of center 0 and radius
2a: it is bounded, and closed, hence compact.
Since the objective is continuous, there exists a maximum and a minimum by Weierstrass theorem.
The Lagrangian

Problem Set 1, Suggested Solutions
1. (a) Assume f is surjective. We show the equality by showing both inclusions. First f f 1 (T ) T . This
inclusion is always true. Let y f f 1 (T ). By definition of the image, there exists x f 1 (T ) such
that y = f (x

Proving Things
Stphane Dupraz
1
Using and Proving Implications and Equivalences
Let P and Q be two statements.
We say that P implies Q, or if P then Q, and note P Q, if Q is true when P is true.
We say that P is a sufficient condition for Q, and Q is a

Problem Set 3
Due Friday, August 21, in class
1. Let X be a set. Define the discrete metric as:
d(x, x) = 0
d(x, y) = 1 if x 6= y
Show that the discrete metric is a metric.
2. Consider the plane R2 and on it the metrics corresponding to the 1-norm, 2-norm

Problem Set 4, Suggested Solutions
1. (a) From the triangle inequality d(x, z) d(x, y) + d(y, z), we get d(x, z) d(y, z) d(x, y). From the
triangle inequality d(y, z) d(y, x) + d(x, z), we get d(y, z) d(x, z) d(x, y). Hence |d(x, z)
d(y, z)| = max(d(x, z

Economics G6410 - Mathematical Methods for Economists - Fall 2013
Department of Economics
Columbia University
Math Camp Exam
Date: 9.00am - 10.20am, September 5, 2013
Answer all questions - each question has equal weight.
1. Consider the vectors
1
1
0
1

Multivariate Calculus
Stphane Dupraz
1
Differentiation of Functions of One Variable
We aim at generalizing the notion of differentiation defined for real-valued single-variable functions to function
from Rn to Rm multivariate and multi-valued. To do this,

Math Camp 2015
Department of Economics, Columbia University
Monday 8/10 - Thursday 9/3
Instructor: Stphane Dupraz ([email protected]).
TA: Tuo Chen ([email protected]).
Courses website: https:/sites.google.com/site/mathcamp2015cu/.
Schedule: (IAB = In

Linear Algebra
Stphane Dupraz
1
Vectors and Vector Spaces
1.1
Vector Spaces
A vector space is a collection of vectors, an abstract notion to designate pretty much anything: numbers,
functions, sequences, etc. This generality is the benefit of the abstract

Problem Set 5
Due Friday, August 28, in class.
1. Find the maxima and minima of the following functions subject to the following constraints:
(a) f (x, y) = xy st. x2 + y 2 = 2a2 (for some a > 0).
(b) f (x, y) = 1/x + 1/y st. (1/x)2 + (1/y)2 = (1/a)2 .
(c

Static Optimization
Stphane Dupraz
An optimization problem in Rn consists in maximizing or minimizing an objective function f : Rn R
over a constraint set D Rn :
max / minf (x) s.t. x D
xRn
Because minimizing f is equivalent to maximizing f , all results

Convexity (and Homogeneity)
Stphane Dupraz
This chapter deals with both convex sets and convex functions. Convexity and concavity (and quasi-convexity
and quasi-concavity) of functions play an essential role in economics first because they play an essenti

Economics G6410 - Mathematical Methods for Economists - Fall 2014
Math Camp Exam, Solutions
1. (a) (10 points) det(A) = 3 12 = 15 6= 0, so the matrix is invertible. It inverse is A1 =
1
3
1
.
15
4 3
The characteristic polynomial of A is P () = 2 tr(A) +

Economics G6410 - Mathematical Methods for Economists - Fall 2012
Department of Economics
Columbia University
Solutions to Math Camp Exam
1. Consider the system of difference equations xt+1 = Axt + b where xt R2 and
A=
0
1
1
1.5
, b =
1
1
(a) Does the sy

Problem Set 6, Suggested Solutions
1. (a) The objective is concave as the sum of two concave functions, under linear, hence concave, constraints:
the KT conditions are necessary and sufficient for an optimum. Besides, if u is continuous, an
optimum exists

Analysis
Stphane Dupraz
1
Distance and Metric Spaces
In real analysis, you have encountered notions of convergence for real sequences and functions from R to R.
Intuitively, it seems also meaningful to say that a sequence of vectors of Rn tends toward a l

Problem Set 6
Due Wednesday, September 2, in class.
1. Consider the utility maximization problem in n + 1 goods:
max u(x) + z
xRn
+ ,zR
s.t. p0 x + z = w
where u : Rn+ R is strictly concave and pi > 0, w > 0 are parameters. (This utility function is calle

Correspondences
Stphane Dupraz
Simply put, a correspondence is a function that is allowed to take multiple values instead of a single one.
Correspondences are given much emphasis in economics (more than in mathematical analysis classes), due to
two of the

Problem Set 1
Due Friday, August 14, in class.
1. Let X and Y be two sets and f : X Y . Prove that:
(a) f f 1 (T ) = T for all T Y iff f is surjective.
(b) f 1 f (S) = S for all S X iff f is injective.
(c) f (S1 S2 ) = f (S1 ) f (S2 ) for all S1 , S2 X if

Taylor Expansions and (log)linearizing
Stphane Dupraz
1
Mean Value Thoerem
Theorem 1.1. Mean Value Theorem in R
Let f : [a, b] R.
If f is continuous on [a, b] and differentiable on (a, b), then there exists z (a, b) such that:
f (b) f (a) = f 0 (z)(b a)
P

Math Camp Exam Solutions
1. (a) The inverse is
1 5
=
10 + 2
2
A1
1
2
=
5/12 1/12
1/6
1/6
.
(b) The characteristic equation is
= (2 )(5 ) + 2 = 2 7 + 12 = 0
5
2
2
1
which has two roots, 1 = 3 and 2 = 4. These are the eigenvalues of A. To find the eigen

Basic Set Theory
Stphane Dupraz
1
Sets
The notion of set or collection is a primitive notion that we take in its everyday meaning. A set is characterized
by what objects it contains; we call them its elements or members. Two sets that have the same member

Economics G6410 - Mathematical Methods for Economists - Fall 2015
Math Camp Exam
Date: 10:10am - 12pm, September 8, 2015.
Answer all questions.
1. Let A =
2
1.5
1
0.5
.
(a) Is A invertible? If so, give its inverse.
(b) Is A diagonalizable? If so, diagona

Epilogue
My dear Bowley,
I have not been able to lay my hands on any notes as to Mathematico-economics that would be
of any use to you:
and I have very indistinct memories of what I used to think on the subject.
I never read mathematics now:
in fact I hav