Mathematics for Economists
Chapter 1:
Differential Equations
S. Aguey, PhD
African School Of Economics
Chapter 1 Systems of Two First
Order Equations
We introduce systems of two first order equations
In this chapter, we consider only systems of two
first
Mathematical Economics
Chap 4: Functional Equations (ctd3)
S. Aguey, Phd
African School Of Economics
6
Example: optimal growth model
6.1
Characterizing the policy
We introduce the neoclassical optimal growth model and see how to derive
properties of optim
Integrals and Primitives
S. Aguey, PhD
African School Of Economics
S. Aguey
1 / 34
Intuitive notion of the derivative
Suppose that a cyclist rolls from a position A to a position B
Starting from the point A, he looks at his watch and noted the time tA.
At
A Glimpse at Hyperbolic Discounting
An individual lives forever from t = 0,1,_.,oo. Think of the individual
as actually consisting of different personalities, one for each period. Each
personality is a distinct agent (timet agent] with a distinct utility
Mathematical Economics
Chap 4: Functional Equations
S. Aguey, PhD
African School Of Economics
1
Solving the FE
Now we make more assumptions on the primiti ves of the problem:
X is a convex subset of Rl ,
F (x; y) is continuous and bounded,
is continuous a
Mathematical Economics
Chap 4: Functional Equations (ctd1)
S. Aguey, Phd
African School Of Economics
2
Showing that T is a contraction
2.1
Blackwells sufficient conditions
One important step in applying our argument to the map T is to show that T
is a con
Chapitre 0: Do you remember these?
S. Aguey, PhD
African School Of Economics
S. Aguey (ASE)
1 / 37
Natural numbers
Natural numbers are used to count collections of objects indivisible
or designed as such.
"The set of natural numbers" is noted N
N = f0, 1,
Mathematical Economics
Chap 4: Functional Equations (ctd2)
S. Aguey, Phd
African School Of Economics
4
Continuity of the policy function
We are in the case of bounded returns and we are assuming strict concavity of
F and convexity of .
We proved that V is
Mathematical Economics
Chap 3: Optimal Control
S. Aguey, PhD
African School Of Economics
1
A portfolio problem
To set the stage, consider a simple nite horizon problem.
A risk averse agent can invest in two assets:
riskless asset (bond) pays gross return
Chapter 6
A DYNAMIC MODEL OF AGGREGATE
DEMAND AND AGGREGATE SUPPLY
S. Aguey, PhD
Macroeconomics II
African School Of Economics
In this chapter, you will learn:
how to incorporate dynamics into the
AD-AS model we previously studied
how to use the dynamic
Chapter 4
THE OPEN ECONOMY REVISITED:
THE MUNDELL-FLEMING MODEL AND
THE EXCHANGE-RATE REGIME
S. Aguey, PhD
Macroeconomics II
African School Of Economics
In this chapter, you will learn:
the Mundell-Fleming model
(IS-LM for the small open economy)
causes
Solutions Problem Set 2 Macro II (14.452)
Francisco A. Gallego
04/22
We encourage you to work together, as long as you write your own solutions.
1
Intertemporal Labor Supply
Consider the following problem. The consumer problem is:
M ax E0
fCt g;fNt g
t=T
Chapter 3
AGGREGATE DEMAND II:
BUILDING THE IS -LM MODEL ADAPTED
S. Aguey, PhD
Macroeconomics II
African School Of Economics
CONTEXT
Chapter
1 introduced the model of aggregate
demand and supply.
Chapter 2 developed the IS-LM model,
the basis of the agg
Chapter 5
AGGREGATE SUPPLY AND THE
SHORT-RUN TRADEOFF BETWEEN
INFLATION AND UNEMPLOYMENT
S. Aguey, PhD
Macroeconomics II
African School Of Economics
In this chapter, you will learn:
two models of aggregate supply in which
output depends positively on the
Macroeconomics II
Chapter 7: Real Business Cycles
S. Aguey, PhD
African School Of Economics
2
Aims of this lecture
To extend the Ramsey model by endogenizing the labour supply decision of
households
To turn the model into an RBC model by assuming stocha
Please sorry for the typo !
IS-LM Curve
How and when the IS and the LM curve will shift?
IS curve
The investment saving curve is the relationship between the interest rate and the income in
the good and services market. Keynes argues that there is a negat
Macroeconomics II
Chapter 8: RBC to NK DSGE
S. Aguey, PhD
African School Of Economics
Criticism 1: Non-neutrality
RBC model cannot replicate evidence of non-neutrality
of money
An increase in money supply
1. Prolonged, but not immediate, positive effect
Master in Mathematics, Economics and Statistics (MMES)
Academic year : 2014-2015
Course
: Macroeconomics II
Date
: February 24th, 2015
Semester : 2nd
Duration : 2H45
Instructor : Juste Som
Midterm Exam
Exercise 1 [60%]
Consider a Real Business Cycle econo
African School of Economics
MMES - Econometrics
Practice Exercises for Final Exam
Exercise 1
Find whether the following AR models are stationary or not. If the model is stationary,
t ;Yt+h )
nd E (yt ) ; V ar (yt ) ; h = Cov (Yt ; Yt+h ) and h = Cov(Y
, f
African School of Economics
MMES - Econometrics
Midterm Practice: Solution
QuestionP1
a) x = N1 N
xi is the observed proportion of "wins."
i=1
1 PN
b) pb = x = N i=1 xi = 0:45 is the estimated probability of winning.
c) V ar (b
p) = p(1100p) , which is es
African School of Economics
Spring Semester
MMES - Econometrics
Instructor: Rachidi Kotchoni
Home Assignment#2: Solution
Question 1
Use the enclosed matlab code to Homework2_1.m to answer question 1. Here is an
example of output:
Y
2:0018
0:0018
0:0890
0:
African School of Economics
Spring Semester
MMES - Econometrics
Instructor: Rachidi Kotchoni
Home Assignment#1
Today: 2015-01-26
Due: 2015-02-12
Note: May be done in groups of 3 or 4 students.
Exercice 1 (6 points/20)
Consider the following linear model:
African School of Economics
MMES - Econometrics
Midterm Practice
Question 1
A money machine works as follows: you insert one dollar and press a button; the machine
may return two dollars (meaning that you win), or nothing (meaning that you loose). You dec
African School of Economics
MMES - Econometrics II
Instructor: Rachidi Kotchoni
Home Assignment #1
Today: March 1st, 2016
Due: March 31st, 2016
Note: May be done in groups of 2 or 3 students. Use your preferred software to answer the questions.
Your submi
African School of Economics
MMES - Econometrics II
Instructor: Rachidi Kotchoni
Home Assignment#2
Today: March 8, 2016
Due: April 8, 2016
Note: May be done in groups of 2 or 3 students.
Question 1 (8 points)
Consider the MA(1) model given by:
y t = c + "t
Violation of the Classical Assumptions in Linear
Regression Models
Rachidi Kotchoni
African School of Economics
January 11, 2016
R. Kotchoni (African School of Economics)
Violation of the Classical Assumptions
January 11, 2016
1 / 32
The Linear Regression
Maximum Likelihood Estimation
Rachidi Kotchoni
A.S.E.
January 11, 2016
R. Kotchoni (A.S.E. )
Maximum Likelihood
January 11, 2016
1 / 32
Principle of Maximum Likelihood
Consider a random variable Y
N , 2
The density of Y is given by:
1
f (y ) = p
exp
2
(y
Generalized Method of Moment
Rachidi Kotchoni
A.S.E.
Jan-May 2015
R. Kotchoni (A.S.E. )
Generalized Method of Moment
Jan-May 2015
1 / 32
Principle of Asset Pricing
Consider an asset that promises non-random cash ows dt ,
t = 1, 2, ., .
Suppose the interes
Review of Bivariate Analysis
Rachidi Kotchoni
African School of Economics
January 11, 2016
R. Kotchoni (A.S.E)
Bivariate Analysis
January 11, 2016
1 / 30
1. Joint distributions
Discrete Case
Consider rolling two dices, labelled "die 1" and "die 2".
Let X