Paul Johnson, Steven Durlauf and Jonathan Temple 112524.3.1 Growth dynamics: basic ideas
Our exposition in this section and the next closely follows Durlaufet al.(2005). At
the heart of the empirical growth literature is a cross-country regression, founded
on the one-sector neoclassical growth model. That model implies that, to a first-
order approximation,yEi,t, output per efficiency unit of labor, evolves according to:
logyEi,t=(
1 −e−λit)
logyEi,∞+e−λitlogyEi,0, (24.1)whereyiE,∞is the steady-state value ofyEi,tand limt→∞yiE,t=yEi,∞. As is standard,
we define output per efficiency unit of labor input asyEi,t=Yi,t/(Ai,tLi,t), where
Yi,t,Li,t, andAi,tdenote, respectively, the level of output, the labor force, and
the level of efficiency in economyiat timet. The labor force is assumed to follow
Li,t=Li,0enit, where the population growth rateniis constant, whileAi,t=Ai,0egit,
wheregiis the (constant) rate of labor-augmenting technical progress. The param-
eterλimeasures the rate of convergence ofyiE,tto its steady-state value and will
typically depend on other parameters in the model. Assuming thatyEi,0>0 and so
eliminating the trivial equilibriumyiE,t= 0 ∀twhen the convergence rateλi>0,
then the value ofyEi,∞is independent ofyEi,0. In this sense, initial conditions do
not matter in the long run.
Equation (24.1) expresses growth dynamics in terms of the unobservableyEi,t.In
order to describe dynamics in terms of the observable variable, output per labor
unityi,t=YLi,t
i,t
, we can useyi,t=yEi,tAi,t=yiE,tAi,0egitto write (24.1) as:
logyi,t−git−logAi,0=(
1 −e−λit)
logyiE,∞+e−λit(
logyi,0−logAi,0)
, (24.2)so that:
logyi,t=git+(
1 −e−λit)
logyEi,∞+(
1 −e−λit)
logAi,0+e−λitlogyi,0. (24.3)Defining the growth rate of output per worker between 0 andt as γi =
t−^1
(
logyi,t−logyi,0)
and subtracting logyi,0from both sides of equation (24.3)
allows it to be written as:
γi=gi+βi(
logyi,0−logyiE,∞−logAi,0)
, (24.4)whereβi=−t−^1
(
1 −e−λit). This expression decomposes the growth rate in coun-
tryiinto two distinct components: the first,gi, measures growth due to technical
progress, while the second,βi
(
logyi,0−logyEi,∞−logAi,0)
, measures growth due
to the gap between initial output per worker and the steady-state value. This sec-
ond source of growth is one aspect of “catching up” and, ast→∞, its importance,
which reflects the role of initial conditions, diminishes to zero.