Transcript Growth 3 Convergence and TP

Prof. PASQUALE TRIDICO
Università Roma Tre
[email protected]


…there is NO significant evidence of
convergence in growth rates of per capita
income between the countries of the world
and even in the long run
However, convergence in growth rates and
income levels is one of the greatest
conclusions in the exogenous growth theory
(neoclassical/Solow)

The growth rate
tends to be equal to
the rate of growth
of the labor supply
plus the growth of
technical progress
gy  n  
These variables n and λ are generally constant, so the
level of income stays along a "path of steady state,"
that is constantly growing, and when it is not on that
path converges towards it





First degree omogeneous equation (constant
return of scale)
Marginal productivity of each factor decreasing
Isoquants of K and L strictly convess towards
the origin and apsyntothic to the axes
Elasticity of substitution constat (CES) and
equal to 1
Shares of wL and rK constant and equal to  e
a 1-.

wL / Y  1  
rK / Y  
Y  ALt
1
1
t
K t

t



K 
L K
1 
K


yA
 ALt K t  A  K t  A    A 
Lt Lt
L
L 
L 
 Akt
k = K/L

Since demand and supply of labour are equal
and correspond to n (pop growth) the rate of
growth of K/L=gK-n:
syt
sYt
It
n
n 
g Kt   n 
kt
Kt
Kt
hence :

sAkt
 1
 n  sAkt  n
g Kt 
kt
The function
sAkt
 1
Is a decreasing function of k as the exponent
is negative, being α <1. This depends of
course on the assumption of diminishing
marginal productivity of the factors in the
neoclassical production function. In the
graphics function approaches asymptotically
to 0.
n, gk
n
sAkt
k*
IN K*
instead:
on the right
on the left
 1
k
gkt  sAkt 1  n  0
gkt  sAkt 1  n  0...k 
 1
gkt  sAkt
 n  0...k 

Remember in the previous case, A is 1 and gA=0

What about if A>1 and gA>0 ?
◦ Long term effect on level of Y/Lwill be affected (higher
Y/L)
◦ No long term effect on growth of Y/L


SS is then given by a level of K / L, and the
Y/L (since y = Ak t  ) which remains constant
through time and is achieved automatically.
SS represents the growth path at a constant
rate of product = n = with Full Employment
At low levels of K, with K / L small the MPK is high, gk is
larger but decreasing. Poor economy.
If the economy is richer, will have more K, the MPK is
lower, gk is smaller.
n, gk
Poor
Richer
Then, if the country was so rich
with abundant K, with MPK very
low, its growth could not keep
the growth of n, then K / L
decrease until reaching K *
n
sAkt
k*
Very rich
k
 1

TP=>0
t
e  TP..Labour  Augmenting
TP  Y as increasing L at a constant rate .
If K/Y is constant, labour productivity grows at the rate
=
The increase of L caused by  + the effective increase
of L = “labour augmented in efficiecy unit LEU”
t
Yt / e Lt ( Yt / e
t
(   n )t
K t / e Lt ( K t / e
L0 )
(   n )t
L0 )
Income per labour in
LEU
Capital per labour in
LEU
Y  ALt
1

Kt
1
t
L




yA
K t  ALt K t  Akt
Lt
t
y
1
(e L )

Kt  e
Lt
t
 t
t (1 ) 
xt  K t / e Lt  e kt
kt  xt e
t
g kt  g xt  
e
t (1 )  1
kt
 xt
 1

L Kt  e
t (1 )  1
x= k per labour in LEU
kt
It
sYt
syt
g Kt   n 
n 
n
Kt
Kt
kt
hence
syt
 1
t (1 )  1
g Kt 
 n  se
kt  n  sxt  n  g xt  
kt
g xt  sxt
 1
n
As usual to the capital accumulation sx one needs
to subtract pop growth and TP  n+ ( labour
growth in LEU terms)

With A, k is no longer constant in the long run

A
 g  A  A0 e gt
A
and : Y  K  ( AL)1  k  A1



y
k
A
   (1   )
y
k
A

A
g
A
g y  gk  g

The new SS is K/AL  k/A which is constant along the balanced growth because
gk=gA=g
...
k  K / AL  k / A : ratio : kapital / techno log y
...
...
y  k
...
y  Y / AL  y / A : ratio : output / techno log y
...
rewriting : Kapital _ accumulation _ in _ terms _ of _ k :

...



k K A L
  
k K A L

...
...
...
k  s y  (n  g  d ) k
...

...
...
...
k  s y  (n  g  d ) k
Now, line n+ is higher than n line (convergence is obtained at higher rate of
growth)
x* indicates a level of K/L
in LEU which remains
constant through time
and it correponds to a
level > of K/L in normal
unit of labour because it
is augmented now by TP
n,g(x)
n+
n
x*
sxt
X
 1
k*  x * e
t



If n=0, that is L is not growing, but it only
grows in terms of efficiency, driven by TP. In
this case y must grow at the rate  otherwise,
if it grew less,  U, since it would take less L
with TP now in order to produce Y.
In other words as if TP increased number of L
at rate .
In this model (with constantly increasing
TP>0) y, per capital GDP, grows also in the
long run, through a balanced path of growth
TP labour Augmenting (=K/Y):
1. The growth rate of the product per worker
is g (y) = g (Y) -n  g (Y) = n + 
2. The growth rate is lower the higher the level
of income per worker
3. The level of income per worker tends
towards the path of SS associated with
x*e
t
k*
These 3 propositions are the basis of the
neoclassical theory of convergence
n, gk
s2 Akt
n
 1
K*
K**
K
s1 Akt
 1



Both economies converge toward the same
growth rate BUT NOT towards the same path
of SS. The economy 2, with s2 < s1, has a
path of SS with K / L (and also with Y / L)
lower than that of Economy 1.
=g of SS,  path of SS. The Economy 2 with
s2 <s1 has a path of SS lower, and income
per worker also lower
The richer and poorer economies converge
towards the same level of Y / L only if they
have the same s (conditional convergence)


The convergence in income levels per
employee, in the neoclassical model, is
therefore conditional to the propensity to
save s.
The absolute convergence of growth rates of
income per employee is instead always an
obtained result in the neoclassical model.
PT=free good, no costs, accessable to all
countries

MP of K e L decreasing
Otherwhise no:

TP= is a result of knowledge, competences,
effort, invenstment, institutional framework,
rules, state, etc

Specific national factors, institutions and models:
….. different path of growth: endogenous
growth therry

1. leaders
2. Late comers  Followers
The Gap of technology is a delay but also an advantage.
When a leader replaces old K with the new, the old K is
already quite modern.
This difference among followers is instead bigger.
However, the greater the gap the higher is the growth
potential of Y and productivity among followers
(see f.i USA and Europe after 1945)



Otherwise with an institutional system
inefficient, lack of "social capability", and
processing capacity and imitation, the gap
is rooted  accumulation delayed of TP
Researches have shown that catching up is
difficult, and the gap more than a potential
advantage can be a trap. A part of society
resists changes.
Socio-institutional factors Are needed ... etc
(Kuznets1966; Rosowski and Ohkawa 1973;
Gerschenkron)


1879-1950 Consolidation advantage
- Fordism, productivity growth and w
- Accumulation of knowledge, technology and
skills, virtuous process
- The imitation was made difficult by the fact that
even to imitate countries had to have certain
organizations and institutions.
-Hence the delay in catching-up. Also massive
invest. In R & D in private and public sector
(especially in the military sector) have widened the
gap.
Since 1950, instead, the followers improved
institutional and social capacity  EU and Japan
have recovered



If there is some convergence between AE, this
is not true for LDE
One can only speak of convergence among a
groups "Club Convergence" and during the
"Golden Age" (1950-73).
On average, poor countries do not grow
faster than rich countries

Cross country

estimates of the
country's income (y)
i ,t
i at time t
y
 (log
y i ,t
yi ,0
)/t
Average growth rate between 0 and t that estimates the HP of absolute
convergence

y i,t  a - logy i,0  u i,0,t
= parameter of convergence. If  is significantly ≠
0, then the average growth rate of poor countries
is higher of the richer countries
But usually  is not significantly ≠ 0.
 absolute convergence unconfirmed



The existence of an economic convergence in which the
poorest regions grow at rates greater than those initially richer
(the so-called beta-convergence) is thus NOT confirmed.
In the long run this process should lead to equality in the
levels of per capita wealth between the various economic
systems.
The mechanism behind this process of "absolute” convergence,
given the restrictive assumptions of the model, is found in the
lower initial endowment capital of the poorest countries that
guarantees them higher returns and growth over time.


The -Convergence implies an inverse
relationship between the rate of change of
output per capita and the initial level of output
per capita: this results in a greater expectation
of growth in poorer countries than in rich ones.
If there is no beta convergence can not be
sigma convergence, necessary but not
sufficient condition. Since the variance may
increase or decrease depending on whether the
country is above or below the steady state.
Paradoxically: “countries can diverge while
converging”

Parametrically, in addition to  another
parameter that must be tested in a regression of
convergence is . It indicates the dispersion
within a group of countries, of their income
level, measured for example by the standard
deviation of the logarithm of GDP per capita in a
group of states or regions. If this diminishes
over time there is -convergence
  E[ X i  E ( X )]




There is -convergence when the standard
deviation of output per capita of the
countries at the time t2 decreases with
respect to time t1.
The -convergence implies a decline in the
dispersion of the variable considered in all
countries over time
This type of convergence is easily
influenced by the presence of outliers from
average.
The scattering cross-section can be
measured as the variance of the logarithm
of GDP per capita
1.
2.
3.
4.
The TP does not allows for convergence 
Endogenous growth theory
TP  Technological Gap theory a
convergence/Cacthing up is possible among
leader ad followers
Absolute convergence non confirmed ()
Conditional convergence (?)

Unlike the absolute
n, gk
convergence the
conditional one
suggests the
possibility for poorer
economies to stay on a
SS path < because of a
<s and/or 
sxt
 1
can be <because
of a  s and 
s 2 xt
X0
 1
g xt
X1,2
n+
X1,1
X
s1 xt
 1

C1 and C2 have the same K / Y in LEU
initial X0 at time 0.
C1: s1> s2.
C1 will have a growth rate greater than n
+ , even when C2 has reached the growth
rate of SS in X1,2.
C1 converges at a higher speed towards
the growth rate of SS for its path of SS
with K / L = X1,1> X1,2
g xt
n+,
g(x)t
g xt
n+
sxt
sxt
xt0
xt1,2
xt1,1
X
 (1) 1
 ( 2 ) 1

In reality it is possible that poorer countries
are simultaneously with s and  parameters
different.

y i,t  a - logy i,0  u i,0,t    j S j ,i ,t  u i,0,t
j
S j ,i ,t  Indicates one of the independent variables intended
to capture the structural and institutional differences
and  the corresponding factor.
If  is significantly ≠ 0 ( the relationship between
initial income and growth is negative) there is
conditional convergence effect of the variable S. The
higher the growth rate is the lower its initial level,
given that s is the same
Many critical and controversial results.
However, empirical studies when
testing the convergence with
neoclassical assumptions, it remain
linked not only to certain periods and
groups of countries, but also to
specific variables such as
Initial income (-)






I (+)
R&D (+)
Capital Accumulation (+)
Education and Human Capital (+)
Openess, Export and international trade (+)
INSTITUTIONS(+)
Countries with similar technology and human capital
Countries with different level of technology and human capital