EC744 Lecture Notes: Projection Method
Jianjun Miao
Solve a functional equation problem
T f = 0;
or a xed point problem
T f = f:
References
Adda, J. and R. Cooper, 2003, Dynamic Economics, MIT Press.
Judd, K., 1992, Projection Methods for Solving Aggregat
EC744 Lecture Note 1
Overview
Prof. Jianjun Miao
1
Deterministic Growth Model
max
X
tU (Ct) subject to
t=0
Ct + It F (Kt, Nt) , K0 > 0 given,
Kt+1 = (1 ) Kt + It, 0 Nt 1.
Assumptions:
1. F : R2 R+ is homogeneous of degree one, strictly increasing, and
+
EC744 Lecture Note 2
Deterministic Models:
Mathematical Preliminaries
Prof. Jianjun Miao
Functional equation:
v (k) =
max
0y f (k)
U (f (k) y ) + v (y ) .
Method of successive approximations:
v1 (k) =
vn+1 (k) =
Question: vn v ?
max
0y f (k)
max
U (f (k)
EC744 Lecture Notes:
Economic Dynamics
Prof. Jianjun Miao
1
Deterministic Dynamic System
State vector xt 2 Rn
State transition function xt = g x0; t;
; x0 = x0; parameter
A parametrized dynamic system: (X; g ) where X
Rp ! X
Rn and g : X
Dierence equation
Dynare
Wouter J. Den Haan
University of Amsterdam
July 26, 2010
Introduction
Do it yourself
Tricks
IRFs & Simulations
Introduction
What is the objective of perturbation?
Peculiarities of Dynare
Some examples
Pruning
Practical
Introduction
Do it yourself
T
FINAL EXAM STUDY SHEET
Google
SEO: Search Engine Optimization. The process of
improving the volume and quality of traffic to a
web site from search engines via "organic" search
results for targeted keywords.
SEM: Search Engine Marketing. The practice of
d
Friedman:
"SocialResponsibilitiesofBusiness"
Theindividualswhoaretoberesponsiblearebusinessmen,whichmeans
individualproprietorsorcorporateexecutives
o Acorporateexecutiveisanemployeeoftheownersofthebusiness.
o Hehasdirectresponsibilitytohisemployers.Thatr
Explain how Singers effective altruism follows from his commitment to utilitarianism.
What is the utilitarian principle which forms the basis for effective altruism?
Singer-does it really matter that you dont have to walk around these children as you walk
FINAL EXAM STUDY SHEET
Google
SEO: Search Engine Optimization. The process of
improving the volume and quality of traffic to a
web site from search engines via "organic" search
results for targeted keywords.
SEM: Search Engine Marketing. The practice of
First day of notes
Pauls argument is that the gentiles dont need to follow mosaic law such as circumcision,
Passover and the day of atonement because these were replaced by the sacrament
o So baptism replaced circumcision brecause it welcomes you into the
QUIZ 2 STUDY SHEET
What makes Zara (Inditex) different?
Be able to identify and use many (not necessarily)
all of these differences to explain Zaras strategic
advantage. Remember that Inditex is parent
company.
Vertically Integrated (v. contract manufact
QUIZ 2 STUDY SHEET
What makes Zara (Inditex) different?
Be able to identify and use many (not necessarily)
all of these differences to explain Zaras strategic
advantage. Remember that Inditex is parent
company.
Vertically Integrated (v. contract
ma
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NOCTI
All rights reserved
2
Sample Study Guide Information
This document provides information about NOCTIs Study Guides which are available for all Job Ready
and Pathway assessments. To learn more about using Study Guides as a helpful prepa
EC744 Lecture Note 3
Dynamic Programming under Certainty
Prof. Jianjun Miao
Sequence Problem (SP):
v (x0) =
sup
X
cfw_xt+1 t=0
t=0
tF (xt, xt+1)
subject to
xt+1 (xt) , t = 0, 1, 2, ., x0 X given.
Functional Equation (FE):
v (x) = max F (x, y ) + v (y )
EC744 Lecture Note 4
Deterministic DP with Unbounded Returns
Prof. Jianjun Miao
Unbounded Returns without the Boundedness Condition
Theorem: Let X, , F, and satisfy Assumptions 1-2. Suppose there is a
b
b
b
b
function v : X R such that (i) T v v, (ii) lim
EC744 Lecture Note 5
Applications of Deterministic DP
Prof. Jianjun Miao
1
Consumption-Savings Problems
max
X
tU (ct)
t=0
subject to
ct + at+1 = (1 + r) at + yt, a0 0 given,
or
ct + xt+1/R = xt + yt, x0 0 given,
where R = 1 + r and x = (1 + r) a is nanci
EC744 Lecture Note 6
Stochastic Models:
Mathematical Preliminaries
Prof. Jianjun Miao
Motivation
Optimal growth model
v (x, z ) =
max
0y f (x)z
U (f (x) z y ) + E v y, z 0
for IID shock z
How to dene E []?
Finite many values of z or continuous values?
Mea
EC744 Lecture Note 7
Stochastic Dynamic Programming
Prof. Jianjun Miao
Principle of Optimality
Measurable spaces (X, X ), (Z, Z )
Transition function Q : Z Z [0, 1]
Return function F : X X Z R
Constraint correspondence : X Z X.
Let the state space be S =
EC744 Lecture Note 8
Applications of Stochastic DP
Prof. Jianjun Miao
1
Consumption-Savings with Exponential Utility
ec . The income follows an AR(1) process
Let U (c) =
yt = yt1 + y + t,
where t is iid with a standard normal distribution. The budget con
EC744 Lecture Note 9
Convergence of Markov Processes
Prof. Jianjun Miao
Policy and Transition Functions.
The policy function g induces a Markov process st = (xt, zt) on the product
space with transition function P.
Theorem. Let (X, X ) and (Z, Z ) be meas
EC744 Lecture Notes:
Discrete State Space Method
Prof. Jianjun Miao
Discretization of AR(1) Shock
Method 1. Tauchen (1986)
2.
where ut is iid N 0,
yt = yt1 + ut
q
The standard deviation of yt is y = / 1 2.
Step 1. Choose N points discrete state space.
ym
EC 744 Lecture Notes:
Incomplete Markets and Bewley Models
Jianjun Miao
Spring 2009
1
Measurement of Inequality
Data Sources Olivetti, Silverman and Hong (2002)
Measures of Interest
Earnings: labor income before taxes
Income: household income before taxes
DYNARE for Macroeconomic Analysis
CAMA Lecture
Philip Liu
[email protected]
Centre for Applied Macroeconomic Analysis (CAMA)
Research School of Pacic and Asian Studies
The Australian National University
March 28, 2006
L EX - DYNARE for Macroeconomic A