These results confirm the findings of Bernard and Jones (1996a) who also report lack
of labour-productivity convergence in Manufacturing, weak evidence for convergence in
Agriculture and strong evidence in Services. It is to be noted, however, that in most sectors
that display evidence of convergence, estimates of β are rather small, indicating that lagging
countries catch-up only very slow. Using the estimated values of β, the rate at which the
productivity level is converging to a uniform productivity level can be derived (e.g., Barro
and Sala-i-Martin 1992, Mankiw et al. 1992, Islam 1995). Let y* be the steady state
productivity level and y(t) its actual value at any time t. Approximating around the steady
state, the speed of convergence is given by
∙∙'l°g<'∙ω> = λ [log(y •)_ log(y (f ))] (2)
dt
which implies that:
logG, (t))=(i - e - λ )log (y*)+ e - λ logG, (o)) (3)
where (y(0)) is energy- or labour productivity level at some initial date. Subtracting log (y(0))
from both sides yields
log(y(t )) - log(y (o)) = (1 - e - λt Xl°g(y *)- log(y (0))] (4)
in which -(1-e λ) = β. Hence, the speed of convergence, λ, is given by λ = - [1/T log(β+1)]
with T denoting the time interval under consideration.15 The values of the implied λ are
shown in Table 2. They confirm the finding of a slow rate of convergence: the time t needed
for energy productivity to move halfway its initial level (y(0)) and the steady state y* varies
from 8 years (Textiles) to 225 years (Non-Ferrous metals); the half life for labour
productivity lies in between 16 years (Wood) and 87 years (Manufacturing).16
As previously noted, β-convergence is a necessary but not a sufficient condition for σ-
convergence. Our findings confirm that those sectors showing evidence of σ-convergence
(see section 4) also display evidence of β-convergence: a decreasing cross-country variation
of productivity levels implies by definition that countries with relatively low initial energy-
and labour-productivity levels grow relatively fast. However, the opposite is not necessarily
true, as is illustrated for labour productivity by the sectors Machinery, Non-Metallic Minerals
and Textiles: they pass the test for β-convergence without showing evidence of σ-
convergence (see Figure 3b).
15 Since we use five-year time intervals, T = 5 in our analysis. Note that in Islam (1995) λ = -[(1/T)ln(β)] due to
the fact that he takes ln(y)it instead of [ln(y)it - ln(y)it- 1] as dependent variable.
16 The half life (H) is derived from e-λH = 0.5 « H = ln(2) / λ
12