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
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)
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
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
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
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 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 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 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 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 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 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 )
DYNARE for Macroeconomic Analysis
CAMA Lecture
Philip Liu
philip.liu@anu.edu.au
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