The research inspired by Barro (1991) has shown how the prediction of

convergence in the neoclassical model needs considerable quali¬cation. If all

economies had identical savings rates, population growth rates and unlimited

access to the same technology, then relative capital intensities would deter-

mine output per capita differentials between countries. Poor countries with

low capital intensities are predicted to grow faster than rich countries in the

period of transitional dynamics en route to the common steady state equilib-

rium. In this situation there will be unconditional or absolute convergence.

Clearly, given the restrictive requirements, this outcome is only likely to be

observed among a group of relatively homogeneous countries or regions that

share similar characteristics, such as the OECD economies and US states. In

reality, many economies differ considerably with respect to key variables

(such as saving propensities, government policies and population growth)

and are moving towards different steady states. Therefore the general conver-

gence property of the Solow model is conditional. ˜Each economy converges

to its own steady state, which in turn is determined by its saving and popula-

tion growth rates™ (Mankiw, 1995). This property of conditional convergence

implies that growth rates will be rapid during transitional dynamics if a

country™s initial output per capita is low relative to its long-run steady state

value. When countries reach their respective steady states, growth rates will

then equalize in line with the rate of technological progress. Clearly, if rich

countries have higher steady state values of k* than poor countries, there will

be no possibility of convergence in an absolute sense. As Barro (1997) notes,

˜a poor country that also has a low long-term position, possibly because its

public policies are harmful or its saving rate is low, would not tend to grow

rapidly™. Conditional convergence therefore allows for the possibility that

rich countries may grow faster than poor countries, leading to income per

capita divergence! Since countries do not have the same steady state per

capita income, each country will have a tendency to grow more rapidly the

bigger the gap between its initial level of income per capita and its own long-

run steady state per capita income.

This can be illustrated as follows. Abstracting from technological progress,

we have the intensive form of the production function written as (11.34):

618 Modern macroeconomics

y = k± (11.34)

Expressing (11.34) in terms of growth rates gives (11.35):

y y = ±k/k

™

™/ (11.35)

Dividing both sides of Solow™s fundamental equation (11.26) by k gives

equation (11.36):

k/k = sf ( k )/k ’ (n + δ )

™ (11.36)

Therefore, substituting (11.35) into (11.36), we derive an expression for the

growth rate of output per worker given by equation (11.37):

y y = ±[sf ( k )/k ’ (n + δ )]

™/ (11.37)

™

In Figure 11.6 the growth rate of the capital“labour ratio ( k/k ) is shown by

the vertical distance between the sf(k)/k function and the effective deprecia-

tion line, n + δ (see Jones, 2001a; Barro and Sala-i-Martin, 2003). The

intersection of the savings curve and effective depreciation line determines

the steady state capital per worker, k*. In Figure 11.7 we compare a rich

sf(k)/k

·

k /k

n+δ

sf(k)/k

k* k

Figure 11.6 Transition dynamics

The renaissance of economic growth research 619

sf(k)/k

c

a

SP

b e (n + δ)P

SR

d

(n + δ)R

sR f(k)/k

sP f(k)/k

kP k*P kR k*R k

Figure 11.7 Conditional convergence

developed country with a poor developing country. Here we assume (realisti-

cally) that the developing country has a higher rate of population growth than

the developed country, that is, (n + δ)P > (n + δ)R, and also that the developed

country has a higher savings rate than the developing country. The steady

state for the developing country is indicated by point SP, with a steady state

*

capital“labour ratio of k P . Similarly, the steady state for the developed coun-

*

try is indicated by points SR and k R . Suppose the current location of these

economies is given by kP and kR. It is clear that the developed economy will

be growing faster than the developing country because the rate of growth of

the capital“labour ratio is greater in the developed economy (distance c“d)

than the developing country (a“b). Figure 11.7 also shows that even if the

developed country had the same population growth rate as the developing

country it would still have a faster rate of growth since the gap between the

savings curve and the effective depreciation line is still greater than that for

the developing country, that is, a“b < c“e.

Robert Lucas (2000b) has recently presented a numerical simulation of

world income dynamics in a model which captures certain features of the

diffusion of the Industrial Revolution across the world™s economies (see

Snowdon, 2002a). In discussing prospects for the twenty-¬rst century Lucas

concludes from his simulation exercise that ˜the restoration of inter-society

income equality will be one of the major economic events of the century to

come™. In the twenty-¬rst century we will witness ˜Convergence, Big Time™!

In short, we will witness an ever-growing ˜convergence club™ as sooner or

later ˜everyone will join the Industrial Revolution™.

620 Modern macroeconomics

In Lucas™s model the followers grow faster than the leader and will eventu-

ally converge on the income per capita level of the leader, ˜but will never

surpass the leader™s level™. As followers catch up the leader Lucas assumes

that their growth rates converge towards that of the leader, that is, 2 per cent.

The probability that a pre-industrial country will begin to grow is positively

related to the level of production in the rest of the world which in turn re¬‚ects

past growth experienced. There are several possible sources of the diffusion

of the Industrial Revolution from leaders to followers, for example:

1. diffusion via spillovers due to human capital externalities (Tamura, 1996),

the idea that ˜knowledge produced anywhere bene¬ts people everywhere™;

2. diffusion via adopting the policies and institutions of the successful

countries thus removing the barriers to growth (Olson, 1996; Parente and

Prescott, 1999, 2000);