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Serge Coulombe, ECO 6120
–Convergence and growth regressions
Conditional convergence
( ) ( )
( )( )
( *) ( )on steady state
*
( ) ( *)*
( )( ) '( )( ) / is decreasing with since
the expression within brackets is the MPL, so:
T
k sf k n x k
k f ks n x
k kf k n xk s
k f k f ks
k k k
f kf k f kkkf k k k d
dk k
he smaller the larger is for a given *k
k kk
The rate of convergence to the steady state
( )k sk n x k
(1 ) (1 ) log( )( ) ( ),kk sk n x se n xk
The speed of convergence
( / )
log
k k
k
On ss (do it yourselfe)
(1 )( )n x
The speed of convergence
• β is the speed at which economies approach their steady state
• A positive β implies that the growth rate of an economy decreases as the economy approaches its steady state
• With α = 1/3, n = 2 %, δ = 5 %, x = 2 %, β is around 6 %
In practice, log-linear estimation on the growth rate of y:
1,
, 1
*log log ii t
i t
yy y
, , 1 ,log log *i t i t i ty y z
log( ) log( )
and
log log* *
y k
kyy k
1
*log logtt
kk k
Investment per unit of effective worker
(n+x+δ)k
sf(k)
kk*k(1)k(0)
Time series considerations
Cross-section considerations: absolute convergence
n+x+δ
sf(k)/k
kk*k(rich)k(poor)
γ(poor)
γ(rich)
Absolute convergence
n(p)+x+δ
sf(k)/k
kk(p)*k(r)k(p)
γ(poor)
γ(rich)
conditional convergence
n(r)+x+δ
k(r)*
Testing the convergence hypothesis and growth regressions
• Absolute convergence hypothesis: Barro and Sala-i-Martin (1992) U.S. states 1880-1990 (B&siM2004, section 11.2.)
• Point estimate of 0.02 for β
• Coulombe and Lee (CJE, 1995), panel data approach to Canadian provinces, 1961-1991
, ,0 0 ,0(1/ ) log( / ) (1 ) / log( )ti t i i i TT Y Y a e T Y
x X Xi t i t Ni
N
i t, , ,lo g ,
1
1
10
, 10 , , ,
(1 )(1/10) log( / ) log( )
10i t i t i t i t
ey y y
Conditional convergence
• The underlying framework is the relationship between the growth rate and the gap between the actual level of Y versus the steady-state level (here assume y = log(y):
• Proxy variables are used for Y*:
• The long-run equilibrium is:
*, , , 1i t i t i ty y y
, , 1 , ,i t i t i t i t i ty y Z u
ˆˆ( * )*
ˆi ì
i
Zy
A wave of growth empirics study
• Absolute convergence hypothesis not rejected for developed countries
• But was rejected for a broad set of countries when less-developed countries were included using new Heston and Summers data bank
• But conditional convergence not rejected (Mankiw, Romer and Veil (1992) and Barro and Sala-i-Martin (1995)