Download Chapter 5 CONVERGENCE IN THE NEOCLASSICAL MODEL

Chapter 5
CONVERGENCE IN THE NEOCLASSICAL
MODEL
Introduction
The concept of economic growth has been examined thus far by looking
at the performance of a single economy. In particular, we have analyzed the
level and the growth rate of per capita income without any reference to
similar variables in other economies. However, an important aspect of
growth theory involves the analysis of variables in relative terms. For
example, it is interesting to examine the path of per capita income with
reference to a benchmark level, which is either given exogenously or
determined by domestic economic policy.
In broad terms, the concept of convergence of a variable is defined as
the path of the variable towards a specific value. In the case of per capita
income mentioned earlier, this value can be either constant (e.g. a specific
income level expressed in constant prices) or change over time (e.g. the USA
or the average EU income). Below, we examine several particular
interpretations of this general definition.
•
Convergence across countries
This form of convergence occurs when countries attain a common
reference value in terms of any economic variable, like the income
level, the growth rate, the level of consumption, the standard of living,
the interest rate, or any other real or nominal variable. This
interpretation is particularly helpful in growth theory as it is used in the
analysis of differences between the income levels recorded across
countries. For example, Figure 5.1 compares the growth patterns of the
USA and Greece since 1980. Despite the trend for convergence
exhibited by these two developed countries, the real income in the USA
is twice as big as the level in Greece. In addition, the differences
between income levels persist over time.
94
P.Kalaitzidakis – S.Kalyvitis
Figure 5.1. Per capita GDP in the USA and Greece (in constant
dollar prices) CHANGE TO EU INCOME
35000
30000
25000
20000
15000
10000
5000
0
1980
1990
Ελλάδα
1999
Η.Π.Α
Source: World Bank, World Development Indicators (2001).
•
It useful to note that, according to this general definition, convergence
between two countries does not necessarily imply that the less-advanced
economy should always experience a rise in income growth. The
reduction in the gap can be equally well driven by a falloff in the
income of the richer country, whereas the income of the poorer country
remains constant or experiences a relatively slower decline. Hence, in
economic policy terms, achieving convergence implies setting a target
for the specific variable (say income) that is relatively ‘higher’ than the
level in the economy under discussion.
Convergence across regions
Examining convergence across different regions in a country requires
again a point of reference (usually the income level). We can find out, for
example, whether there is a significant difference between the income
levels of Epirus area and Athens and if this gap is likely to persist or wear
off over time. Note again that a declining gap is not necessarily equivalent
to ‘improvement’ in absolute terms. At national level, the aggregate
income may decrease (compared to the GNP of another country), but at
the same time, different regions of the country can exhibit convergence
since the high income of the richer region (e.g. Athens) may decline more
than the income of laggards. In conclusion, convergence across regions
may also occur in the absence of growth when the richer region (e.g. the
‘center’) does worse than the poorer one (e.g. the ‘periphery’).
Economic Growth: Theory and Policy
•
95
Convergence across groups of countries
The analysis of convergence across groups of countries requires a
sample of countries that share an identical level of technological
progress as well as similar economic, social, or even ethnic
characteristics. Actually, it will be shown below that economies with
different structural parameters may exhibit differential growth patterns
despite being governed by common economic policies. Hence, our
analysis should involve ‘similar’ economies since it is irrelevant to
examine completely different economies (e.g. a European vs. an
African country). In this sense, economies with similar characteristics
(like those comprising the EU) is for example an interesting case to
study (see Box 5.1). Nevertheless, we should not rule out a comparative
analysis across groups of countries (e.g. between European and African
economies), which is also likely to yield interesting results concerning
growth and income inequality at the global level.
To conclude this introduction, there are three broad dimensions in the
concept of convergence in growth theory and policy: (i) domestic (i.e.
convergence within a country), (ii) national (i.e. convergence across countries)
and (iii) global (i.e. convergence across groups of countries). To grasp the
meaning of convergence, we will examine below both its theoretical
background and the related stylized facts. As a first step, we will extend the
neoclassical growth model in order to introduce several theoretical
convergence issues and analyze its predictions regarding growth and
convergence. Next, we will use available data on growth and income in order to
investigate various ways of examining the associated empirical hypotheses of
the model regarding convergence.
Convergence in the Solow-Swan model
To address the questions about convergence, we will reconsider the
simplest version of the neoclassical model without technological progress.
As we will see, this model yields the central predictions on convergence
used by economists.
Let us assume an economy i with a Cobb-Douglas production function:
Yi = AK ia L1i − a
(5.1)
96
P.Kalaitzidakis – S.Kalyvitis
Box 5.1. European Union policies on real convergence
An officially declared policy goal of the European Union is cohesion
across member states. This goal is supported by the implementation of
social and economic measures aiming at real convergence (i.e.
convergence of income and productivity across countries). A relevant
question is therefore whether convergence between richer and poorer
members has indeed taken place and, if this is the case, what is the
degree of convergence between the developed and the less developed
member states. The next figure plots the evolution of income levels in
Greece, Ireland, Spain and Portugal, which are the poorest EU
members, against the average EU level over the period of the 1990s.
EU average GDP versus GDP in the less-developed EU economies
(ΕΕ-15=100)
120
EE-11
110
100
Ireland
90
Spain
80
Greece
70
Portugal
60
50
1989 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999
Source: European Commission (1999).
Obviously, Ireland is the only economy that has attained real
convergence, whereas the other member states remain far from the
EU average GDP. To support the goal of convergence, the European
Union has launched a set of policies -termed Structural Policies- which
aim at eliminating income disparities in a sustainable manner. The
main financial instruments used to achieve these goals are the
Structural Funds and the Cohesion Fund. Financial transfers are
allocated for the development of infrastructure in vital sectors, like
telecommunications, transports and energy, for the improvement of
human capital, and for the support of various structural measures.
Economic Growth: Theory and Policy
97
As was shown in previous chapters, the following relation gives the
steady-state capital-labor ratio in this economy:
1
 sA  1−α
ki = 

n + δ 
(5.2)
Consequently, the expression for the steady-state income per capita is:
yi =
1
1
−
A α
a
 s  1−α
n + δ 


(5.3)
Finally, the relation y i = Ak ia yields the growth rate gy of country i as:
g yi = ag k i
(5.4)
Recall that the main predictions of the neoclassical model without
technological progress are that at the steady state g y = g k = 0 for any
economy, and that the capital-labor ratio k increases at a diminishing rate
while approaching the steady-state point k (provided of course that the
initial capital stock k (0) < k ). Using these results, we can now state the
following proposition concerning convergence across economies.
Proposition 5.1. Consider two economies i=1,2 that have the same
production function given by equation (5.1) and the same values of the
parameters s, n, δ. In the long-run equilibrium, these economies will exhibit
the same level of income per capita, given by equation (5.3).
The conclusion above is one of the most powerful predictions in the
theory of exogenous economic growth. According to Proposition (5.1),
structurally similar economies will converge in the long run to the same
income level.
Recall now that along transition to the steady state, the dynamics of the
growth rate of k can be expressed algebraically as follows:
g k = sAk − (1− a ) − (n + δ )
(5.5)
98
P.Kalaitzidakis – S.Kalyvitis
where the subscript i is omitted for simplicity. Equations (5.4) and (5.5)
imply then that:
g y = asAk − (1− a ) − a (n + δ )
(5.6)
∂g y
< 0 , meaning that the growth rate is
∂k
inversely proportional to the capital-labor ratio along the economy’s
transition to the steady-state. Therefore, the lower the initial capital-labor
ratio, the higher the growth rate will be. We can now sum up these results in
the following Proposition, which is known as the ‘absolute convergence’
hypothesis.
Equation (5.6) yields that
Proposition 5.2. (‘absolute convergence’ hypothesis) In the neoclassical
model of exogenous growth, the economy tends to grow faster for a low level
of initial capital-labor ratio compared to the case of a higher capital-labor
ratio.
Proposition 5.2 reveals again the basic mechanism that drives
economies during their transition to the long-run equilibrium. In particular, if
capital exhibits diminishing returns (α<1), an economy with lower capitallabor ratio exhibits a higher marginal product of capital and thus, grows
faster compared to a similar economy with a higher capital-labor ratio.
Hence, the differences across countries will tend to fade out over time, with
per capita income and its growth rate gradually converging until reaching an
identical long-run equilibrium level for both countries, respectively.
The hypothesis of absolute convergence provides the main building
block for empirical tests on the fit of the model when confronted with data
on groups of economies. According to this hypothesis, if all economies have
similar characteristics described by equations (5.1) to (5.6), the ‘poor’
(‘rich’) countries with low (high) initial capital k0, and, thus, low (high)
initial income y0, will display a higher (lower) growth rate. That is, in
practice these two variables are expected to be negatively correlated.
Later on in this chapter we will examine whether this hypothesis is
supported by empirical observations. At the current stage, we will attempt to
answer the following question: is it always true that two economies with
different levels of initial income will exhibit different (and inversely related)
growth rates? In other words, is it possible for a typical African country with
low capital and income levels to grow faster than Switzerland?
Economic Growth: Theory and Policy
99
The answer is of course ‘no’. The reason is that the hypothesis of
absolute convergence applies exclusively to economies with identical
structural parameters, which are will display the same level of per capita
income in the long run. So, for example, if the population growth rate n is
higher in a country than in another one, then according to equations (5.3) and
(5.6), the former will exhibit lower steady-state income and growth rate
along transition to equilibrium than the latter. We can therefore conclude
that, according to our model, convergence should only be anticipated across
structurally ‘similar’ economies (i.e. with identical parameter values).10
This new insight introduces the concept of convergence known as
‘conditional convergence’.
Proposition 5.3. (‘conditional convergence’ hypothesis) In the neoclassical
model of exogenous growth, economies tend to converge faster to their own
steady state the further away they are from it.
According to Proposition 5.3, simply looking at the initial equilibrium
point and the growth rate cannot yield accurate predictions about
convergence. We should first allow for possible differences between the
countries’ steady states. So, an economy is likely to exhibit both a high
initial income and fast growth simply because its steady-state income is
simultaneously high and relatively far from its current income. For instance,
we know that large values of the technology constant A result in high steadystate capital-labor ratio and income (see equation 5.3). A straightforward
implication is that a simple comparison with an economy having the same
initial capital stock and income level without looking at parameter A, would
lead to inaccurate conclusions regarding its convergence pattern.
In the following sections, we will use Proposition 5.3 in order to
introduce the theoretical background for empirical tests on the convergence
hypotheses. Obviously, while conducting these tests we should allow for the
differential determinants of the steady state (which in our neoclassical
setting are given by parameters Α, α, n, s, δ) and then examine the transition
towards this point.
10
It is worth noting that according to the theory two economies with identical parameters will
achieve the same steady state. However, the opposite does not hold: as can be readily seen in
the model, two economies may coincidentally exhibit the same steady-state income level
despite having different parameter values. Therefore, two economies sharing the steady-state
income level cannot be viewed as ‘similar’ without further knowledge of their structural
charactheristics.
100
P.Kalaitzidakis – S.Kalyvitis
The theoretical background of tests on convergence
As we have seen so far, to establish the presence of convergence we
have to test the specific hypothesis according to which there is an inverse
relation between the growth rate and the initial income of a country.
Algebraically, this hypothesis can be expressed as follows:
g yi , t , t + T = f [log( yi ,t )]
(5.7)
where g yi , t , t + T = log( yi ,t +T / yi ,t ) / T is the average growth rate of economy i
during the period from t to t+T, and log( yi ,t ) denotes the logarithm of
income of economy i in period t.11
For the Cobb-Douglas production function, the capital growth rate,
which determines the income growth rate, is given by equation (5.5). The
linear approximation of this equation around the steady-state yields:12
gk =
d log( K )
k
≅ −(1 − a)(n + δ ) log( )
dt
k
(5.8)
y
k
Using equation (5.4) and the relationship log( ) = a log( ) , we can also
y
k
approximate the income growth rate by:
The use of logarithms will allow us to express the growth rates as first differences of the
logarithm of income.
12
The capital growth rate can be written as:
11
gk =
d log(k )
= sAk −(1− a ) − (n + δ )
dt
and its linear approximation around the steady state – given by equation (5.2) – takes the
following form:
g k ≅ sA(k ) −(1− a ) − (n + δ ) − (1 − a) A(k ) a − 2
Substituting above the expression for
d log(k )
k
log( )
dk
k
k yields relation (5.8).
Economic Growth: Theory and Policy
y
g y ≅ −(1 − a)(n + δ ) log( )
y
101
(5.9)
The general form of the above equation is:
y
g y = − β log( )
y
(5.10)
where β=(1-α)(n+δ). It can be easily seen that convergence is determined by
coefficient β. In particular, for β>0, if the initial income is higher (lower)
than the steady-state income, the economy will grow faster (more slowly).
This form of convergence is known in the literature as β-convergence.
The latter also introduces the concept of convergence speed, which has
already been discussed in Chapter 3. In particular, according to equation
(5.8), coefficient β shows how fast the economies approach their long-run
equilibrium: higher (lower) β implies faster (lower) convergence. It is
obvious that coefficient β can shed light in the analysis of economic growth:
if β is large and convergence is fast, the economy is expected to approach its
long-run equilibrium in a relatively short time period and, therefore, we can
use steady-state analysis to examine the growth pattern. However, in the case
of a low β, the economy will be far from its long-run equilibrium and its
growth path can be better approximated by the transitional dynamics.
Now, solving equation (5.10) yields:
log( y t ) = (1 − e − βt ) log( y ) + e − βt log( y 0 )
(5.11)
Income at any time t depends on the constant value of the initial income
y0 and of the steady-state income y . In addition, for higher values of β,
income yt will be closer from its steady-state level. At t=T, equation (5.11)
becomes:
yT
)
y0
y
(1 − e − βT )
=
log( )
T
T
y0
log(
(5.12)
Equation (5.12) is a simplified form of equation (5.7) and shows that
the average growth rate after T periods is inversely related to the levels of
the initial and the steady-state income. Therefore, just as in the case of the
102
P.Kalaitzidakis – S.Kalyvitis
absolute and conditional convergence, β-convergence depends not only on
initial income level, but also on its steady-state level. We can now state the
following Proposition about the different forms of β-convergence.
Proposition 5.4. (absolute and ‘conditional’ β-convergence) In the
neoclassical model of exogenous growth, absolute β-convergence is
measured in terms of initial income of the economy, whereas ‘conditional’
β-convergence is given by the deviation of initial income from the its steadystate level.
Just as in the case of absolute and ‘conditional’ convergence discussed
earlier, the above forms of β-convergence have similar properties. Therefore,
in order to establish whether the conditions supporting convergence are
satisfied, we should look at the steady states of the economies.
We can now examine how the above results apply to a collection of data
from different economies, i.e. when cross section data are used. In particular,
we raise the following question: if each of these economies tends to
converge to its steady-state income level, will then convergence emerge at
the global level, i.e. will the inequality of income distribution be worldwide
diminished?
At first glance, there should be a positive correlation between the two
forms of convergence: for a small income gap across economies worldwide,
individual economies should tend to converge to equilibrium income per
capita. But does the opposite hold, namely does β-convergence lead to small
income dispersion? This time the answer is ‘no’ and the reason is that this
income gap can stay constant (or increase) despite the presence of βconvergence. For example, a poor country may grow much faster than a rich
one so that the income of the former may turn out to exceed the level of the
latter and thus the income gap will increases while there is β-convergence in
both countries under consideration.13
13
This phenomenon is known in the scientific literature as Galton’s Fallacy. Back in the 19th
century, Sir Francis Galton observed that children having fathers that were taller than the
average tended to have heights close to the average of the population heights and not close to
the average of their family members’ heights (which was higher than the population average).
This tendency of heights to converge to the average level (called mean reversion) was
regarded by Galton as a proof of the fact that the dispersion of all individuals’ heights should
diminish over time. However, this hypothesis does not hold in reality. For more details on the
relation between Galton’s Fallacy and the empirical studies on growth and convergence, see
Quah (1993) and Bliss (1999).
Economic Growth: Theory and Policy
103
Now, what are the implications in terms of the relationships developed
previously? In the simplest case of successive time periods (Τ=1) between t1 and t, equation (5.12) can be rewritten as follows:
log( y t / y t −1 ) = (1 − e − β ) log( y ) − (1 − e − β ) log( y t −1 )
⇔ log( y t ) = e − β log( y t −1 ) + c
(5.13)
where c = (1 − e − β ) log y . Income dispersion is then given by the variance:14
Var[log( y t )] = e −2 β Var[log( y t −1 )]
(5.14)
Defining the variance as Var[log( y t )] ≡ σ t2 and substituting above implies:
σ t2 = e −2 β σ t2−1
(5.15)
Hence, in order for income dispersion to be reduced intertemporally we must
have that β>0 (β-convergence), in which case:
σ t2 < σ t2−1
(5.16)
Inequality (5.16) is known in the growth literature as σ-convergence and
states the reduction of income dispersion (captured by its variance) over
time. As can be readily seen by equation (5.15), a necessary condition for σconvergence to hold is that β>0. On the other hand, the reverse is not true,
i.e. we may get σ t2 > σ t2−1 even when β>0, provided that the initial value of
the variance is lower than the one prevailing in the steady state (see Barro
and Sala-I-Martin, 1995, chapter 11). In turn, the following Proposition can
be stated on the relationship between β-convergence and σ-convergence.
Proposition 5.5. (relation between β-convergence and σ-convergence) In the
neoclassical model of exogenous growth β-convergence is a necessary but
insufficient condition for σ-convergence.
14
The variance of a variable Χ around its mean Ε(Χ) is given by the formula Var(X)=E(XE(X))2 and satisfies the following property: Var(kX)=k2Var(X) for k=constant.
104
P.Kalaitzidakis – S.Kalyvitis
As we will see, this theoretical relationship is of little use in the
empirical context, as the two forms of convergence usually coexist.
However, for reasons of strictness we should clarifying once again the exact
meanings of the two concepts: β-convergence characterizes the path of the
economy’s income towards its steady-state level, whereas σ-convergence
refers to the intertemporal evolution of global dispersion of income. Hence,
both concepts are crucial in the analysis of convergence to the steady-state
income per capita.
So far we have presented the most important theoretical issues on
convergence. As we will see, these concepts also support the empirical tests
of convergence hypotheses. In the next section we will look at empirical
measures of convergence and discuss the results of many empirical studies
attempting to determine different measures of convergence.
Convergence tests: empirical studies
During the last decades, and especially after the publication of Solow’s
exogenous growth model, applied macroeconomists turned their attention to
empirical studies testing convergence across countries. The following
reasons account for the increased interest in this kind of tests:
• The model of exogenous growth yielded the first specific and, more
importantly, measurable results concerning the convergence process.
• Apart from its importance for economics, empirical measurements of
convergence also shed further light upon the social and political aspects
of the issue of global income distribution.
• A larger variety of statistical data became easily available through
several economics data banks maintained consistently by various
economists and research teams.
• Improvement in theoretical econometric methods involving, among
other things, endogeneity and heterogeneity issues, along with the
development of user-friendly computer-based estimation offered more
accurate answers to many quantitative issues involving the nature of
data on economic growth.
A straightforward test regarding the presence of convergence can be
carried out by plotting countries’ income per capita over a long time horizon.
To gain a first picture of the long-run performance in various economies we
plot in the figure below the log of income per capita in three selected
economies (USA, UK, Portugal) between 1870 and 1988. These economies
are chosen based on data availability and on their substantial disparities in
the level of income.
Economic Growth: Theory and Policy
105
Figure 5.2. Log of per capita GDP in 3 countries, 1870 – 1988
10
9
8
7
6
1870
1890
1910
USA
1930
U n ite d K in g d o m
1950
1970
P o rtu g a l
Source: Easterly and Rebelo (1993).
-
-
Examining the figure above we can draw the following conclusions:
Despite starting from the same income level, the two more developed
economies (USA and UK) diverged significantly during this period.
Obviously, if this path persists, per capita GDP levels in these countries
will continue to diverge in absolute terms. Therefore, a question that
arises naturally is whether one of these countries diverges along an
unstable transition path during the period under consideration (implying
a rejection of the Solow model) or if their income levels simply
converge to a new long-run equilibrium level, which is not the same for
both countries.
Income in Portugal, which is the less developed country, displays a
similar evolution. However, there is a large gap between Portugal’s per
capita income and that in the two developed countries. Moreover, this
difference persists over time. The relevant question in this case is
therefore whether less developed economies can possibly converge to
the developed ones, or if less developed countries have a different
(lower) long-run equilibrium income.
However, we should always keep in mind that the simple graphical
illustration of per capita income suffers from several shortcomings:
• The conclusions drawn above concern only certain developed
economies for which long-run data is available. No predictions can be
drawn for the majority of developed economies.
106
•
•
•
P.Kalaitzidakis – S.Kalyvitis
Usually, the sample of countries analyzed is not chosen irrespective of
the likelihood of convergence. The economies for which data can be
found are typically those that have converged to a high level of income,
which is actually the reason for which statistical data is available.
Despite the high quality of available data, cross-country comparisons
are not always feasible due to different statistical measurements used by
national statistical offices and the often measurement errors, especially
during the earlier years of the sample.
In addition, despite the fact that the long-time horizon (over 100 years)
is relevant for tracing the long-run growth tendency of economies, it
may still not be enough to reveal important dimensions of convergence,
such as β-convergence and σ-convergence, or the long-run equilibrium
income in these economies.
These limitations were largely overcome by the publication of the Penn
World Tables (a description was given in Chapter 4). The immediate step
was to use this data in order to test the basic prediction about convergence of
the neoclassical model, namely that initial income is inversely related to the
average growth rate or, equivalently, that richer (poorer) economies exhibit
lower (higher) growth rates – i.e. they display absolute convergence.
Following this approach, Figure 5.3 plots per capita income during the
1960s against the average growth rate between 1960 and 1989 for 119
countries.
Figure 5.3.
Growth rate versus initial per capita income (119 countries)
8
Per capita income (in logs), 1960
7
6
5
4
3
2
1
0
-2
-1
0
1
2
3
4
G ro w th ra te , 1 96 0 -1 9 8 9
Source: Levine and Renelt (1992).
5
6
7
8
Economic Growth: Theory and Policy
107
As it can be readily seen, there is no sign of correlation between the two
variables plotted above. More exactly, a fitted linear trend shows a positive
correlation between per capita income and the growth rate implying that
richer (poorer) economies tend to have higher (lower) growth rates! This
conclusion challenges one of the main predictions of the neoclassical model
since it questions the existence of β-convergence, which is also the necessary
condition for σ-convergence.
But is this indeed what the neoclassical model predicts about
convergence? Recall that in the previous section this form of convergence
has been termed ‘absolute’ β-convergence. In fact, the neoclassical model
bases its main results on variables like economy’s initial income and its own
long-run equilibrium level. Therefore, we shouldn’t test for convergence
economies that are likely to have different long-run equilibrium income.
Since there is no reliable method for estimation of a country’s steady-state
income, a general criterion should be adopted in order to classify economies
into homogenous categories. Only after this classification takes place can
any convergence test be applied; as expected, this recalls us the prerequisites
of ‘conditional’ β-convergence.
Following this idea, Figure 5.4 illustrates the two variables for a sample
of 24 OECD countries. This is a group of economies that consistently exhibit
high per capita income over many decades. So, we expect that they fall into
a category of economies with similar per capita income and also many
common features from the standpoint of their structure and behavior during
the period analyzed.
Figure 5.4.
Growth rate versus initial per capita income (24 ΟECD countries)
6
6
Per capita income (in logs), 1960
5
5
4
4
3
3
2
2
1
1
2
3
4
5
6
G ro w th ra te , 1 9 6 0 -1 9 8 9
Source: Levine and Renelt (1992).
7
8
108
P.Kalaitzidakis – S.Kalyvitis
Figure 5.4 reveals a completely different image from Figure 5.3. Here,
initial income and growth are inversely related; moreover, this inverse
relation is statistically significant.15 Indeed, many empirical studies have
supported the accuracy of the figure above on the basis of data that confirm
the existence of ‘conditional’ β-convergence across similar economies. SalaI-Martin (1996) examined the relation given in equation (5.12) for a sample
of 110 countries and found that the coefficient β fluctuates around 0.02,
meaning that annual convergence amounts to 2% of the distance between
actual and steady-state income. That is, in general economies converge to
their long-run equilibrium income, with their growth slowing down as they
approach their equilibrium income. However, this does not mean that the
long-run per capita income is the same across countries.
It should be noticed that the tests mentioned thus far do not apply
exclusively to countries. In fact, an extensive literature has been developed
around the issue of convergence across regions within a country. Since it is
expected that regions display relatively smaller differences in their per-capita
income than countries, ‘conditional’ β-convergence should provide a suited
hypothesis to test. Barro and Sala-I-Martin (1995) studied the implications
of equation (5.12) in the U.S. states, for which there is available data for the
period 1880-1990. They estimated that β is not far from 0.02, which is close
to the result found by similar studies of regions for other developed
countries.
Coefficient β also denotes the speed of convergence. In particular, a
value of 2% means that it will take 35 years for economies to cover half of
the gap between initial and equilibrium income. Thus, it turns out that
convergence is a rather slow phenomenon, even when we allow for the
determinants of steady-state income.
As we have seen earlier, β-convergence is a necessary but insufficient
condition for σ-convergence (see Proposition 5.5). A straightforward
implication is then that σ-convergence cannot occur across economies
worldwide (since there is no sign of β-convergence in these economies), but
it is possible to arise across ΟECD countries. Figure 5.5 illustrates
graphically the dispersion of real income per capita in both cases.
15
The value of the correlation coefficient between the two variables is –0.68, which is
considered high enough for cross-sectional data. Note also that similar conclusions are drawn
if we examine clusters of countries with an approximately equal income level (e.g. lessdeveloped economies, countries with common geographical characteristics etc).
Economic Growth: Theory and Policy
109
Figure 5.5. Intertemporal dispersion of real income per capita
worldwide and across ΟECD countries
1 ,2 0
1 ,0 0
0 ,8 0
0 ,6 0
0 ,4 0
0 ,2 0
1950
1955
1960
1965
1970
W o rld
1975
1980
1985
1990
OECD
Source: Sala-I-Martin (1996).
Figure 5.5 shows that the dispersion of income at global level does not
decline, but on the contrary, increases over time. This confirms the finding
that rich countries tend to become richer, whereas the relative position of
poor countries worsens. In other words, the unequal distribution of global
income is intensified at the expense of poor countries. On the other hand, the
isolated analysis of OECD countries reveals the existence of convergence
(except for the period 1975-1985). Consequently, income inequalities across
OECD countries decrease significantly over time. These findings (i.e. σconvergence except for 1975-1985) also hold when regions of developed
countries are examined.
Most of the available surveys tend to agree that convergence occurs
across similar economies at a speed of 2%. But what are the implications of
this convergence speed according to the neoclassical model? Recall that
relation (5.10) shows that coefficient β is equal to the product between the
labor share and the sum of the population growth rate and the depreciation
rate. Allowing for β to equal approximately 0.02 and assuming that n and δ
0.01 (i.e. 1% average annual rate of population growth) and 0.05 (i.e. 5%
average annual rate of depreciation) respectively, we get that the fraction of
capital should be 0.67 (=1-0.33). Note though that a series of empirical
studies (see Chapter 4) has found that this fraction equals 0.30, which is
much lower than the prediction above.16
16
The model examined here does not allow for technological progress. If productivity grows
at a constant rate, we should include this rate in the sum (n+δ). Assuming that the average
productivity growth rate equals 0.02 (2% yearly), the fraction of capital should be even higher
and roughly equal to 0.75.
110
P.Kalaitzidakis – S.Kalyvitis
This apparent contradiction arising from initial estimations of
coefficient β was resolved by Mankiw, Romer and Weil (1992), whose work
was discussed extensively in Chapter 4. Their empirical methodology
assumes that aggregate capital incorporates both physical and human
components and finds that the fraction of capital approximates 0.7. Hence,
the analysis of Mankiw, Romer and Weil does not only that reveal a new
dimension of the neoclassical model, but also supports its predictions about
convergence.
(1 − e − βT )
Finally, it is worth emphasizing that the coefficient
in
T
equation (5.12) is inversely related to the number of time periods, T. Hence,
when we look at the linear relation between income and growth and assume
more time periods over which convergence is examined, we should expect to
find lower values for this coefficient. The reason is that the economy
approaches the long-run equilibrium at a decreasing rate, which affects the
average growth rate over the period under discussion. It is clear then that the
period size constitutes an important factor of the empirical estimation.17
Conclusions
The issue of convergence is one of the central aspects of the
neoclassical model of exogenous growth. It has drawn the interests of
economists studying the development of countries and regions and, despite
its simplicity, this model remains an important tool in the growth theory.
Its central predictions, which have been confirmed by many empirical
studies, can be summarized as follows:
Each economy converges to its own long-run equilibrium, which is
determined by a series of exogenous parameters.
Convergence to long-run equilibrium income does not lead automatically
to leas income inequality.
Economies worldwide do not show a consistent convergence pattern,
whereas homogenous groups of countries and their regions tend to
converge to their steady-state income.
The annual rate of convergence is very low and equals roughly 2%.
One para here
17
Note that for Τ approaching infinity (i.e. sample size increases), the coefficient tends to
zero, whereas for T approaching zero, (i.e. sample size decreases), the coefficient tends to β.