BARRO and SALA-I-MARTIN: JOEG,1997
Production
Ni
Y i A i L i 1 X ij
i 1, 2
j1
Output Expenditures
Ni
Y i C i X ij RD
j1
C i and X ij require one unit of Y i . Invention
of new variety requires i .
Assume N 1 0 N 2 0 , and new
discoveries occur in country
Stochastic Models of Technological Change
Part A
NYU-Stern Mini-course
September, 2011
Robert E. Lucas, Jr.
These notes are based on
Fernando Alvarez, Francisco Buera, and Robert E. Lucas, Jr. Models of
Idea Flows. NBER WP # 14135, July, 2007.
See also
Scale effects
Lucas:
Human capital accumulation:
h 1 u
h
Romer:
Research Sector:
A H a
A
Charles I Jones, "Time Series Tests of
Endogenous Growth Models;" QJE, v. 110,
1995, 495-525
Mankiw, Romer and Weil, QJE 1992,
407-437
SOLOW
Y K AL 1
Y
K
y AL , k AL
In Aghion-Howitt book, page 57:
Poisson Process modeled as:
Cumulative distr. for event before T
FT 1 e T
fT e T Density
gx prob. that x events occurs in the inteval
x e
x!
1
Final product:
y Ax , 1
p t A t x t 1
because intermediate good price is its
Some Models of Technological Change
Part B
NYU-Stern Mini-course
September, 2011
Robert E. Lucas, Jr.
These notes are based mainly on
Robert E. Lucas, Jr. Ideas and Growth. Economica, 76 (2009): 1-19.
1. Introduction
Model economies discussed in Part A
0.1
Lucas 1988 JME-On the Mechanics of Development
max
c(t)
Z
1
c
0
subject to:
1
1
N (t) et dt
A
N c + K = AK N 1 ,
=
A
1
1
c
H =N
+ AK N 1 N c
1
where c is consumption, N is labor force that grows at the rate , K is physical
capital, is a positive dis
The Lucas-Baseline Model Described: Poisson arrivals
What happens in a meeting?
Agent (a draw z from Fz, t meets another, z .
Not symmetric: z is active, learns from meeting, but z is passive.
If z z, nothing happens; if z z agent z adopts z
How do pe
1
ROMER-JPE 1990
x(i) are intermediate goods. Capital is given by:
K=n
ZA
x (i) di
0
Note that the sum of x (i) does not add up to K : there is a proportionality
factor n: Final good production:
Y = Hy L
ZA
x(i)1
di
0
L is labor and set to unity. Human Ca
More on Convergence
Y fK, HL , L 1, H0 ,
Ht e t , y i
y i t
y i 0
Y i t
Ht
Y i 0
y i T
n
y i 0
1 n Y i T
T
Y i 0
Y i t
Ht
K
f H , L fk, L
Y i t
e t
Y i 0
Y i t t
e
Y i 0
Y i T
n
Y i 0
y T
n i
y i 0
T
1
T
But since from convergence
Notes on Control Theory
t
max ft, xt , ut dt
t
1
#
0
x gt, xt , ut
t 0 , t 1 , xt 0 x 0 fixed, t 1 can be .
xt 1 may be free or fixed
The choice variable is a function ut which
is piecewise continuous, that is we are
allowed to choose ut from the set of
Alternative Specification
A gH i cH i max j A j A i t
A
A i t
A m 0e gH m t A i t
gH i cH i
A i t
A i t A i 0 A m 0e gH i cH i t A m 0e gH m
cH i
cH i gH m gH i
A i 0 A m 0 gH i cH i gH m t
e
A m 0
since gH i cH i gH m 0.
Catch up! Asymptotical