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)