1090 The Econometrics of Convergence
technological progress, so thatAi,t=Ai,0egit, equation (23.1) may be rewritten as:
logyi,t−git−logAi,0=( 1 −e−λit)logyiE,∞+e−λit(logyi,0−logAi,0) (23.2)so that:
logyi,t=git+( 1 −e−λit)logyiE,∞+( 1 −e−λit)logAi,0+e−λitlogyi,0.^6 (23.3)Lettingγi=t−^1
(
logyi,t−logyi,0)
denote the growth rate ofyi,tbetween 0 andt
and subtracting logyi,0from both sides of (23.3), division bytyields:
γi=gi+βi(logyi,0−logyiE,∞−logAi,0), (23.4)whereβi=−t−^1 ( 1 −e−λit).
Equation (23.4) decomposes the growth rate ofyi,tinto two parts. The first,
gi, is growth due to technological progress, and the second,βi(logyi,0−logyiE,∞
−logAi,0)is growth due to the closing of the initial gap between output per
worker and the steady-state value – “catching up.” Becauseβi→0ast→∞,
the importance of this term, and hence the role of initial conditions in determin-
ing contemporaneous output, diminishes to zero. The long-run rate of growth of
the economy isgi, the rate of technological progress.
While equation (23.4) provides a characterization of the sources of economic
growth, it is not yet in a form that can be estimated. One reason is that the
parametersλiandgiare country-specific. The empirical growth literature often
assumes that these parameters are identical across countries, so that we have
λi=λandgi=g,∀i. Under these assumptions, (23.4) simplifies to:
γi=g−βlogyiE,∞−βlogAi,0+βlogyi,0. (23.5)Equation (23.5) implies,ceteris paribus, a negative relationship between average
rates of growth and initial levels of output per capita, over any time period, when
estimated for a cross-section of countries. Those countries with low income are
further below their balanced growth path and will grow relatively quickly: their
low income implies that the capital-output ratio is lower, and the marginal prod-
uct of capital higher, than in countries starting with a higher level of income.
This mechanism leads to a period of relatively fast growth, so that the countries
initially behind will catch up with other countries that have the same levels of
steady-state output per effective worker and initial efficiency. Similarly, countries
that begin above their balanced growth path, perhaps because some determinants
of steady-state income have deteriorated over time, must grow relatively slowly.
In these economies, the capital-output ratio is high and the marginal product of
capital relatively low, leading to a period of growth at below the rate of technical
progress. This movement towards a balanced growth path is the economic notion
of convergence implied by the neoclassical model.