Steven Durlauf, Paul Johnson and Jonathan Temple 1089of Summers and Heston (1988, 1991), has meant that it is now possible to study
convergence for a wide range of countries.^2
In this chapter, we first describe various statistical notions of convergence that
have arisen as part of the modern economic growth literature. Our goal is not only
to characterize the range of convergence notions that have been used in empirical
work, but also to give some sense of their links to substantive economic claims.
23.2 β-convergence
The first statistical convergence concept used in the modern growth literature is
based on the relationship between initial income and subsequent growth. Intu-
itively, this convergence notion is simple: two countries exhibit convergence if
the one with lower initial income grows faster than the other and so tends to
“catch up” with the higher-income country. This is the concept used in Abramovitz
(1986), Baumol (1986) and Marris (1982), and which also plays a central role in
Barro (1991), Barro and Sala-i-Martin (1992) and Mankiw, Romer and Weil (1992).^3
One posited driving force behind catching-up is that a position well below the
technological frontier creates the potential for rapid advancement, through the
installation of capital embodying the current frontier technology, for example.
Another convergence mechanism, which is usually associated with the neoclassical
growth model and which has played a greater role in the literature, emphasizes the
role of diminishing returns. It predicts that countries which begin with a relatively
low level of income will grow relatively rapidly, but this growth will slow down
as the economy approaches its balanced growth path and the marginal product of
capital declines towards its steady-state level.^4
23.2.1 Convergence and the neoclassical growth model
Convergence as a property of the neoclassical growth model may be understood in
terms of the behavior of output around the model’s unique and stable steady-state.
LetYi,tdenote output,Li,tthe labor force, andAi,tthe level of (labor-augmenting)
efficiency in economyiat timet. From these, following the standard logic by
which steady-states are constructed in the neoclassical growth model, defineyEi,t=
Yi,t/(Ai,tLi,t)as output per efficiency unit of labor input at any timetandyEi,∞=
limt→∞yiE,tas its associated stable steady-state value. Assuming thatyiE,0>0, a
log-linear approximation around the stable steady-state implies the law of motion:
logyEi,t=( 1 −e−λit)logyEi,∞+e−λitlogyiE,0. (23.1)The parameterλi, which may be shown to be positive, depends on the other param-
eters of the model and characterizes the speed with whichyEi,tadjusts towards its
steady-state value.^5
Given this general law of motion for output per efficiency unit of labor, it
is straightforward to describe the behavior of the observable output per unit of
labor input,yi,t=Yi,t/Li,t. Lettinggibe the (constant) rate of (labor-augmenting)