We use two datasets to study the convergence process across European regions. Relying on Quah (1966a,1997), we examine the dynamics of income distribution and find evidence of polarization whatever the time horizon considered. Regions whose incomes were close together at an initial period transit subsequently to widely different income levels.distribution dynamics

Jalal El Ouardighi

Phu Nguyen Van

Theophile Azomahou

Thi Kim Cuong Pham

English

Research Papers in Economics

Income distribution dynamics across European regions 
Theophile Azomahou Jalal El Ouardighi
BETA, Universite Louis Pasteur BETA, Universite Louis Pasteur
Phu Nguyen Van Thi Kim Cuong Pham
THEMA BETA, Universite Louis Pasteur
Abstract
We use two datasets to study the convergence process across European regions. Relying on
Quah (1966a,1997), we examine the dynamics of income distribution and find evidence of
polarization whatever the time horizon considered. Regions whose incomes were close
together at an initial period transit subsequently to widely different income levels.
We thank Prof. Ping Wang, François Laisney, Raouf Boucekkine and an anonymous referee for valuable comments. All
remaining errors are our own.
Citation: Azomahou, Theophile, Jalal El Ouardighi, Phu Nguyen Van, and Thi Kim Cuong Pham, (2005) "Income distribution
dynamics across European regions." Economics Bulletin, Vol. 15, No. 12 pp. 1−9
Submitted: January 18, 2005.  Accepted: March 4, 2005.
URL: http://www.economicsbulletin.com/2005/volume15/EB−05O40001A.pdf1I n t r o d u c t i o n
Issues on economic convergence have important implications to policy makers with regard
to regional development and economic integration in the European Union (EU). Indeed, the
European economic integration is generally considered as bene￿cial for its regions in terms
of higher allocation eﬃciency and increased competition. Moreover, the convergence of the
economies is one of the requirements for the new countries eager to integrate the EU. Yet to
prevent an increase of inequalities between regions, an important part (0.51% of the GDP of
the 15 countries in the European community) of the European structural funds is devoted to
regional development; see e.g. European Commission (1996, 1999) for further details. Even if
eﬀorts have been made to increase the attractiveness of European regions, inequalities seem
persistent. According to Neven and Gouyette (1995), a process of divergence would be even
in hand in the European regions. All occurs like if, in less prosperous countries, only the
richest regions bene￿ted from the convergence process with the rich countries.
Research over the last decade has considerably improved our understanding on conver-
gence and has also raised several con￿icting views. The convergence hypothesis has then
induced a huge literature; see e.g., the 1996 Economic Journal symposium. Beginning with
Baumol (1986), the convergence hypothesis has been extensively tested empirically. Sala-
￿-Martin (1996) analyses the regional β−convergence hypothesis on a data of 90 European
regions for the 1950-1990 period and ￿nd that these regions converge at the 2% rate per year.
The author also ￿nds that the cross-regional income dispersion was reduced over time, which
is in favour of the σ−convergence hypothesis.1 However, as pointed out by Quah (1996, 1997),
traditional approaches to convergence hypothesis focus on the average behaviour rather than
the dynamics of income distribution. Therefore a particular attention has to be paid to
analysing and interpreting convergence empirics which account for the dynamic structure of
it, rather than a static average behaviour. According to Quah (1996), β−convergence or more
generally, the coeﬃcients in a cross-section regression is uninformative for a distribution￿s dy-
namics. Cross-section regressions can inform only about the average behaviour, not about
the behaviour of the entire distribution. Checking for σ−convergence would not get round
this diﬃculty. Quah (1996, 1997) underlines an ￿emerging twin peaks￿ or polarization in the
income distribution in a cross-section of countries. Polarization means that starting from
period t, two peaks or modes emerge in the horizon t+τ income distribution. At the regional
level, this method was used by Quah (1996) and Johnson (2000) to study the convergence
across the US states over the 1948-1989 period and over the 1929-1993 period, respectively.
However, they ￿nd no evidence of polarization in the cross-state income distribution. This
￿nding suggests that there is a convergence between the US states over the periods of the
study.
In this note, we use the method proposed by Quah (1996, 1997) to examine the conver-
gence process across European regions. This method relies on two approaches : (i) one based
on discrete transition probability matrix, and (ii) the other based on conditional density
estimation. The latter provides us with a nonparametric approach to convergence empirics
that captures the behaviour of the entire income distribution as if it evolves over time. The
main empirical result of this study is the emerging of a polarization in the EU. We ￿nd that
regions whose incomes were close together at an initial period transit subsequently to widely
diﬀerent income levels.
1σ-convergence in a cross-section of economies means that the cross-section dispersion of income per capita
falls over time whereas β-convergence means that poor economies tend to grow faster than wealthy economies.
12D a t a
T w od a t a s e t sw e r eu s e di nt h i ss t u d y .T h e￿rst sample is the same as in Sala-￿-Martin (1996)
and Att￿eld et al. (2000).2 The data contain series on real GDP per capita (in US $) in 1950
and 1990 of 90 regions from Germany (11 regions), United Kingdom (11), Italy (20), France
(21), The Netherlands (4), Belgium (3), Denmark (3), and Spain (17). More details on this
data sample can be found in Sala-￿-Martin (1996).
The second is a new and extensive dataset of 445 European regions observed in 1990 and
1996. It is provided by the Observatory of Sciences and Technology (OST). This dataset
contains series on real GDP (measured in European Common Currency (ECU)3,i n1990
prices) and the level of population for the years 1990 and 1996. For reasons of homogeneity
of the regions of the Nomenclature of Territorial Units for Statistics (NUTS)4,t h eO S Tu s e s
regions on level NUTS-0 for Luxembourg (1 region), level NUTS-2 for Germany (38), Austria
(9), Belgium (11), Greece (13), Finland (6), The Netherlands (12), Portugal (7) and Sweden
(8); and on level NUTS-3 for Denmark (15), France (97), Ireland (8), Italy (103), Spain (52)
and United Kingdom (65). To the best of our knowledge, it is the most extensive database
available at the regional scale in Europe.5 Note that our databases include both prosperous
regions (for example from West Germany, France, United-Kingdom, Belgium, Luxembourg,
Denmark, and The Netherlands) as well as less prosperous regions (from Spain, Portugal,
and Greece).
The data present disparities between regions around the European average that are strong.
Indeed, the highest level of GDP per capita ￿ in Sweden (in 1990), Belgium-Luxembourg and
Denmark (in 1996), is approximately twice the lowest level ￿ in Portugal and Greece. The
disparities grow even more marked if one observes the intra-country GDP per capita. The
regional range is particularly high in France and Germany, but remains weak in Greece,
Sweden and in Portugal. The evolution of GDP per capita between 1990 and 1996 shows
a national average growth of 2.35%. The rise is particularly important in the Portuguese,
Greek and Irish regions with an average growth rate of about 4%. The Finnish and Swedish
regions are characterized by a lowest increase, respectively 0.1% and 0.36% on average. More
details on this database can be found in OST (1999).
3 Methodology and empirical results
To study the dynamics of income distribution over time, we compute the transition probability
matrix, say, M,f r o mp e r i o dt to t + τ. Diagonal entries in M traduce the persistence, that
is to say, some regions rich at t + τ had already being rich at time t and similarly, others
poor at t+ τ had already been poor at t.N o t et h a tw es t a n d a r d i z et h ed a t a{xt,i,x t+τ,i}
n
i=1
by subtracting the sample mean and premultiplying by the inverse of the transpose of the
Cholesky factor for the estimated sample covariance matrix to have zero mean and identity
covariance matrix. The standardized data, de￿ned in the Cartesian product set [−2,2] ￿
[−2,2], allow us to analyse the regional income per capita relative to the EU average.
We discretize the set of possible values into intervals at {< −1, [−1,0), [0,1), ≥ 1},
denoted group 1, 2, 3, and 4, respectively. Then, M is a 4 ￿ 4 matrix with the (j,k) entry
2The data is extracted from Eurostat. We are grateful to Att￿eld et al. (2000) for providing the dataset.
3The former European Curency Unit is replaced by the Euro since 1999.
4The nomenclature in European regions is a hierarchical classi￿cation established by Eurostat. This clas-
si￿cation starts from country level information (NUTS-0) to community level (NUTS-5).
5The list of these 445 regions is available from the authors upon request.
2Table 1: Transition probability matrices for the 1950-1990 data (90 regions) and for the
1990-1996 data (445 regions).
GDP 1990
#r e g i o n s G D P1950 < −1[ −1,0) [0,1) ≥ 1
18 < −1 0.222 0.389 0.333 0.056
20 [−1,0) 0.050 0.100 0.550 0.300
38 [0,1) 0.211 0.289 0.342 0.158
14 ≥ 1 0.357 0.429 0.071 0.143
GDP 1996
#r e g i o n s G D P1990 < −1[ −1,0) [0,1) ≥ 1
59 < −1 0.169 0.305 0.356 0.169
141 [−1,0) 0.156 0.546 0.255 0.043
194 [0,1) 0.072 0.485 0.366 0.077
51 ≥ 1 0.020 0.078 0.647 0.255
Note: Bold ￿gures represent the highest transition probabilities conditional on starting state.
being the probability that a region in income group j at time t transits to income group
k at time t + τ. These probabilities are computed as the empirical frequencies, that is the
proportion of regions falling into a given group. Table 1 tabulates matrix M for the ￿rst and
the second dataset, respectively. The ￿rst row (corresponding to group 1 in 1950) in Table 1
shows that 22% of regions in group 1 do not move at all from 1950 to 1990, 39% transit from
group 1 to group 2; 33% transit from group 1 to group 3 and only 6%, transit from group 1
to group 4. It follows from this transition matrix that regions in groups 1 and 2 have a higher
probability to see their income increased whereas there is a persistence for group 3 and a
decrease for group 4. Table 1 also reports transition patterns for the 1990-1996 data sample.
We observe that regions in group 1 transit to higher income group, group 2 is persistent, and
groups 3 and 4 transit to lower income group. However, two income groups are formed at the
arrival period. In light of these results, it seems that a phenomenon of polarization emerges.
Nevertheless, the transition matrix M is a discretized version of the dynamics of income
distribution. We now investigate the polarization underlined above in a continuous income
space. Following the notations in Johnson (2000), assume that the cross-region income dis-
tribution (in logarithm) at time t can be described by a density function ϕt (xt), where xt
denotes the income at period t.
As in general the income distribution evolves over time, we assume that the density
prevailing at period t+τ,f o rτ > 0,i sϕt+τ (xt+τ). We suppose that the process describing the
evolution of the distribution is time-invariant and ￿rst-order, then the relationship between
the two distributions can be summarized as
ϕt+τ (xt+τ)=
Z ∞
0
fτ (xt+τ|xt)ϕt (xt)dxt, (1)
where fτ (xt+τ|xt) denotes the conditional density of xt+τ, τ periods latter. Compared to
the above discrete approach, fτ (xt+τ|xt) represents the continuous analogue of the transition
matrix. Let us denote gτ (x) by the joint density, where x =[ xt,x t+τ].W ee s t i m a t egτ (x) as
3gτ (x)=
1
nh2
n X
i=1
K
￿
1
h
(x − xi)
¶
, (2)
where K (.) is a kernel function, and h is the bandwidth or smoothing parameter. We use
a bivariate Gaussian kernel, K (x)= 1
2π exp
¡
−1
2x0x
¢
and the optimal bandwidth, see e.g.,
Silverman (1986).6 We estimate fτ (xt+τ|xt) by gτ (x)/gt (xt), where gt(xt) is the marginal
density of xt, which is obtained by integrating gτ (x) over xt+τ.W e c a l c u l a t e t h e k e r n e l
density estimates from the standardized data on a grid of evaluation points de￿n e do nt h e
Cartesian product set [−2,2] ￿ [−2,2].7
Figure 1 and 2 display the surface plot of the joint distribution of log of real GDP per
capita for 1950 and 1990 for the ￿rst sample, and for 1990 and 1996 for the second sample,
respectively. The two graphs show an almost uni-modal distribution with a probability mass
concentrated around the sample mean values. This means that the two samples contain a
high proportion of middle-income regions.
Figures 3, 4 and Figures 5, 6 display the surface plot of the estimates of fτ (xt+τ|xt) and
the corresponding contour plots for the two samples, respectively. The conditional density
estimates (Figures 3 and 5) show how the cross-sectional distribution at t evolves into t+ τ.
The results for the two samples show a similar bi-modal distribution (a peak corresponding to
weak income in t +τ, 1990 for the ￿rst sample and 1996 for the second sample, and another
corresponding to high income in t + τ). However in the 1990-1996 sample, the bi-modal
pattern is clearly marked and the peak corresponding to high income in 1996 is higher.
Contour plot in Figures 4 and 6 makes this clear. They show a peak below the 45◦ line
and a peak above the 45◦ line. For example, in the second sample, the contour peak values
are located around 0.3, which is equivalent to a value of GDP per capita of 13,448 ECUs
(approximately the mean of the EU).8 T h e n ,f o re a c hs a m p l e ,w eo b s e r v ed i ﬀerent behaviours
for regions having approximately the same income. In the ￿rst sample, a part of low income
regions tends to have decreasing income over the 41-years horizon whereas the other tends to
have increasing income. In other words, some of the low income regions tend to grow quickly
whereas the others do not. We observe a similar dynamic pattern in the second sample (note
that now we have 7-years horizon).
These ￿gures do exhibit some ￿peaks￿ which appear diﬀerent from that observed by
(Quah, 1997, Fig. 5.1) in a cross-country data and by (Johnson, 2000, Fig.1)i nt h ec a s eo f
the US states. Indeed, Quah (1997) ￿nd a polarization. However, the peaks are located on
the main diagonal. The peaks in Johnson (2000) suggest that there is a convergence between
the US states. Our estimate does not imply the same behaviour in the long-run. The key
message from Figures 4 and 6 is that regions whose incomes were close together at an initial
period transit subsequently to widely diﬀerent income levels.
6In calculating the optimal bandwidth, we need the value of the integral, over 2-dimensional real space, of
the squared kernel function. The value of this integral equals 0.0796. Note that other bandwidth selection
methods might be implemented, such as cross-validation, etc.
7All computations in this paper are performed in Gauss.
8Recall that data (in log) were standardized before estimation. Then, to retrieve the value for 1990, 13,448
ECUs, we compute exp(aσ
2+m),w i t ha =0 .3 (standardized income), σ
2 =0 .0833 (variance of unstandardized
income), m =2 .5738 (mean of unstandardized income).
4References
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5Figure 1: Surface plot of the joint distribution of log-real GDP per capita in 1950 and 1990,
g(x) where x =[ x50,x 90], using a bivariate Gaussian kernel and the optimal bandwidth,
h =0 .473.
Figure 2: Surface plot of the joint distribution of log-real GDP per capita in 1990 and 1996,
g(x) where x =[ x90,x 96], using a bivariate Gaussian kernel and the optimal bandwidth,
h =0 .362.
6Figure 3: Surface plot of the conditional density of log-real GDP per capita in 1990 conditional
on that in 1950, f (x90|x50), using a bivariate Gaussian kernel and the optimal bandwidth,
h =0 .473.
Figure 4: Contour plot of the conditional distribution f (x90|x50), where x90 and x50 are log-
GDP per capita in 1990 and 1950, respectively. The straight line denotes the 45◦ diagonal.
7Figure 5: Surface plot of the conditional density estimate of f (x96|x90) using a bivariate
Gaussian kernel with the optimal bandwidth h =0 .36, x96 and x90 are log-GDP per capita
in 1996 and 1990, respectively.
Figure 6: Contour plot of the conditional density estimate of f (x96|x90), where x96 and x90
are log-GDP per capita in 1996 and 1990, respectively. The straight line denotes the 45◦
diagonal.
8

Income distribution dynamics across European regions

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Income distribution dynamics across European regions

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