Pons-Novell, Jordi

English

Research Papers in Economics

39
th REGIONAL  SCIENCE ASSOCIATION
 EUROPEAN CONGRESS
DUBLIN, 23-27 AUGUST 1999
CITIES AND CONVERGENCE HYPOTHESIS:Error! Bookmark not defined.
EVIDENCE FROM CATALONIA
Jordi Pons-Novell
Department of Econometrics, Statistics
and Spanish Economy
University of Barcelona
Avda. Diagonal, 690
08034-BARCELONA
SPAIN
Email: jpons@eco.ub.es
Fax: 34-93-4021821Abstract
This paper analyzes the evolution of income per capita of the towns of
Catalonia in the period 1989-1995 and the expended trend for the coming
years. The procedure used to study the convergence in the level of income
per capita among Catalan towns is founded on the proposal of Quah, based
on the statistical concept of the Markov chains. The results obtained
demonstrate a process of reduction in inequality in terms of income per capita.
The prospects for the coming years are for a situation in which the differences
in the level of income per capita between Catalan towns will continue to
decrease, although this process will be very slow.1. Introduction
Recent years have seen a revival in the literature on economic growth and
convergence, basically looking at the contrast of the hypothesis of
convergence of income or of product per capita between countries, regions or
towns. The aim of this paper is to analyze the evolution of the income per
capita of the towns of Catalonia
1 in the period 1989-1995. Despite the fact that
the period in question is not very long, the differences between Catalan towns
as regards income per capita and the expected trend for the coming years are
analyzed.
Catalonia is an autonomous community within the Spanish State, and has its
own charter of autonomy, goverment and parliament. Its geographical
situation in the Mediterranean region has made in the meeting point of
numerous cultures and influences and has at the same time permitted the
development of an economy based on trade. Over the last decade, the
relative distribution of the active population has increased in the service
industries. However, compared to other regions and countries of the EU, there
is still a high proportion employed in the industrial sector in Catalonia.
The current population of Catalonia is 6,090,040, representing 15.5% of the
total population of Spain. Catalonia occupies a surface of 31,895 sq. km,
equivalent to 6,3% of Spain as a whole. Density of population varies over the
territory. The coastal zone is densely-populated, particularly the metropolitan
area of Barcelona, whilst the inland and mountain areas have a low density of
population. Almost four million people live in Barcelona and surrounding area,
and 90% of the population of Catalonia resides in towns and cities of more
than 5,000 inhabitants.
A study of the convergence of levels of income per capita is important both
from a theoretical and from a practical point of view. Indeed, the neoclassical
model of growth (Solow, 1956) predicts the convergence of the economies
analyzed toward a common stationary state, provided that they share similar
basic characteristics. In short, if the hypothesis of convergence is not rejected,
there will be empirical evidence in favour of this kind of model. From a
practical point of view, there is a certain general consensus regarding
accepting an active economic policy aimed at mitigating inter-regional
differences in production through public investment programmes.
Consequently, a rejection of the hypothesis of convergence could call these
policies into question, as they would not be achieving a re-balancing of the
distribution of income per capita in Catalonia.
This paper studies the question from an empirical perspective
2, taking the
series of income per capita of Catalan towns elaborated by the autonomous
government of Catalonia (Arcarons et al., 1998) and, as statistical
methodology, the dynamic analysis of distributions using Markov chains. Thework is organized as follows: the second section very briefly sets forth the
statistical model used, the results obtained are presented in the third section
and, finally, the main conclusions are summarized in the fourth section.
2. The statistical model
The statistical technique used in this paper is based on the dynamic analysis
of the functions of density of the series considered by means of Markov
chains, in accordance with the proposal of Quah (1993a, 1993b and 1996).
This author has presented an alternative to the traditional studies of
convergence based on the concepts of s-convergence and b-convergence
proposed in the works of Barro and Sala-i-Martín (1992 and 1995) and Sala-i-
Martín (1996). Barro and Sala-i-Martín’s approach says that b-convergence
exists if a negative relationship is found between the growth rate of the income
per capita and the initial level of income. That is, there is b convergence if the
poor economies tend to grow faster than the rich ones. On the other hand,
there is s convergence if the dispersion of income per capita between different
economies reduces with time.
Over recent years different studies have emerged which have examined those
definitions of convergence based on the works of Barro and Sala-i-Martín with
a critical attitude. In particular, Quah has set out two arguments which could
call these concepts into question. Firstly, he suggests a potential
misinterpretation of the convergence regressions used in this approach and,
secondly, he notes that the difference between convergence and divergence
is very subtle and may not be properly illustrated on studying cross-sectional
data.
The procedure proposed by Quah to study the convergence between a set of
areas is based on studying the dynamic evolution of income per capita over
time. This procedure, based on the statistical concept of Markov chains, is
discussed very briefly below. For a comprehensive treatment of Markov
chains, Anderson and Goodman (1957) and Hamilton (1994) can be
consulted.
jt is the density function of variable yit in period t in a set of n towns. It is
considered that jt evolves over time according to an autoregressive process
of the first order of the type:
This density function is segmented into k intervals and the above expression is
transformed into a Markov process, M being its transition probabilities matrix.
A generic entry (i,j) of this matrix can be interpreted as the probability of
j j t 1 + t    M =  moving from state i at time t to state j at time t+1.
The information contained in the transition matrix synthesizes the internal
movement of the variable analyzed; this makes it possible to quantify its
mobility and, at the same time, to determine the asymptotic behaviour of jt,
that is the distribution of the variable analyzed in the future. The prediction of j
t+h starting from jt can be obtained from the following expression:
It can be demonstrated that the matrix M always has at least a unitary own
value and, consequently, it is established that:
' u M lim
h
h p = ¥ ﬁ
where u is a vector of ones and p represents the autovector of transition
matrix M associated with the unitary autovalue. The characteristics of p, also
known through ergodic distribution, are determinants to analyze the existence
of convergence, as it shows the distribution toward which the variable
analyzed will tend as time passes.
3. Empirical evidence
The variable analyzed, yit, is the income per capita of town i in period t; the
153 towns which make up Catalonia are considered and the period studied is
1989-1995. To avoid temporary trends which could make the estimation and
the interpretation of the model difficult, the observations are expressed in
deviations in relation to the mean of each of the seven years considered.
The final aim of the study is to determine transition matrix M and to do so jt
has previously been segmented into five groups (k=5) and the joint distribution
cutoffs {yit; i=1,...,153; t=1989,...,1995} have been established so that each of
the groups has the same number of elements. Thus, if a value 100 is
assigned to the mean income per capita of the whole of Catalonia for the
seven years considered and the value of the income per capita of each of the
regions is expressed in relation to this mean value, the regions can be split
into five groups depending on their level of income per capita: low, medium-
low, medium, medium-high, and high.
Table 1 presents the estimate of the transition matrix. A generic entry (i,j) of
this matrix can be interpreted as the probability of moving from state i at time t
j j t
h
h + t   M   =  to state j at time t+1. Thus, a city in the low income interval at time t, had a
92.40% probability of staying in the same income interval at time t+1, and a
7.60% probability of moving to medium-low income interval. It can be
observed that the elements of the main diagonal indicate the probability that a
town will remain in the same situation as regards the income per capita over
two consecutive years. Thus, the last column shows the distribution of
balance, that is to say it indicates the distribution toward which the towns will
tend over time in relation to their level of income per capita.
Table 1. Transition matrix
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not
defined.Int
ervals
  Low Mediu
m-Low
Mediu
m
Mediu
m-High
  High Equilibrium
probabilitie
s
Low
92.40
%
 7.60%    0    0    0    12.72%
Medium-
Low
 
9.18%
80.29
%
10.53
%
   0    0    27.26%
Medium    0 11.69
%
73.25
%
15.06
%
   0    36.42%
Medium-
High
   0    0  9.20% 79.61
%
11.19
%
   15.70%
High    0     0  2.88% 14.57
%
82.55
%
    7.90%
The main conclusions which can be drawn from this table are:
-A model of significant persistence is observed, given the high values of the
elements of the main diagonal. Thus, it is especially difficult for a town with a
low level of income per capita to escape from this situation, as the probability
that it will remain in this situation on studying two consecutive years is 92.40%.
-The mobility of the towns is not very high. In addition, it is characterized by
greater movements toward the groups with a higher level of income per
capita. Thus, for example, the probability that a region with an medium level of
income per capita will move to the medium-high level over two consecutive
years is 15.06%, while that of experiencing a downward movement to the
medium-low level is 11.69%. In short, the elements which are below the main
diagonal of transition matrix M are smaller than those in the lower part, which
indicates that the towns present a tendency to move toward higher relativelevels of income per capita.
-The ergodic distribution would appear to indicate that in the future the
differences between Catalan towns will reduce in relation to income per capita,
as out of all 153 towns, 12.72% will tend to have a low level of income per
capita, 27.26% medium-low, 36.42% medium, 15.70% medium-high and
7.90% high.  However, these values should only be considered as indicative
of the expected trend of the future distribution of income per capita in Catalan
towns. Beyond the mere figures, these results reveal that a large number of
towns will in the future be positioned around the average income for the whole
of Catalonia. In short, the future prospects are of a situation in which few
towns will have a value of income per capita greatly above or greatly below
the Catalan average, and most will be positioned below this average, that is to
say a situation of reduction of the inequalities in the distribution of income in
Catalonia.
An especially important issue is to determine the speed with which the ergodic
situation will be obtained, and therefore to estimate the degree of mobility
associated with matrix M. To measure this, Quah (1996) proposed the
following index:
l being the second largest own value of matrix M. If m(M)=0 it can be
concluded that the mobility is nil and, therefore, the solution of balance is
never achieved. On the other hand, if m(M)=1, the system tends toward
balance instantly. Consequently, as this index comes closer to zero the
transition toward balance will be slower, and the more it approaches one the
faster it will be. In our case m(M) is equal to 0.0912, which indicates that it
tends toward a situation of balance in a much slower manner or, in other
words, the reduction in the inequalities in the level of income per capita in the
towns of Catalonia will take place very slowly over the coming years.
4. Conclusion
The Catalan towns demonstrate a process of reduction in equality in terms of
income per capita, although this process of convergence will take place very
slowly. The prospects for the coming years are for a situation in which few
towns will tend to have a value of income per capita greatly above or greatly
below the Catalan average, while the majority will be positioned around the
average for the whole of Catalonia.
These results on convergence are especially important on referring to one
l m - =1 (M)figure - income per capita - which is determined subsequent to when the
redistribution mechanisms come into operation. A reduction is therefore
observed in the inequalities, which justifies the active economic policies
carried out by the different public authorities aimed at lessening the inter-
regional differences in production by means of public investment programmes.
Consequently, accepting the hypothesis of convergence validates these
policies, as they are achieving a re-balancing of the distribution of income in
Catalan towns.
Notes
1. A town has been defined as an urban area with a population of over 5,000
inhabitants.
2. A similar analysis was carried out by Eaton and Eckstein (1997) for the
towns of France and Japan.
5. References
ANDERSON, T.W. and GOODMAN, L.A. (1957) Statistical inference about
Markov chains, Annals of Mathematical Statistics, 28, pp. 89-110.
ARCARONS, J.; LURIA, J.; TARRACH, A. and POVEDA, C. (1998):
Estimació de l'indicador de Renda Familiar Disponible de les comarques i els
municipis de Catalunya 1989-1995. Barcelona: Generalitat de Catalunya.
BARRO, R.J. and SALA-I-MARTÍN, X. (1992) Convergence, Journal of
Political Economy, 100, pp. 223-251.
BARRO, R.J. and SALA-I-MARTÍN, X. (1995) Economic Growth. Boston:
McGraw Hill.
EATON, J. and ECKSTEIN, Z. (1997) Cities and growth: Theory and evidence
from France and Japan, Regional Science and Urban Economics, pp. 443-
474.
HAMILTON, J.D. (1994) Time series analysis. New Jersey: Princeton
University Press. 
MANKIW, N.G.; ROMER, D. and WEIL, D.N. (1992) A contribution to the
empirics of economic growth, Quarterly Journal of Economics, 107, pp. 407-
437.
QUAH, D. (1993a) Empirical cross-section dynamics in economic growth,European Economic Review, 37, pp. 426-434.
QUAH, D. (1993b) Galton's fallacy and tests of the convergence hypothesis,
Scandinavian Journal of Economics, 95, pp. 427-443.
QUAH, D. (1996) Convergence empirics across economies with (some)
capital mobility, Journal of Economic Growth, 1, pp. 95-124.
SALA-I-MARTÍN, X. (1996) Regional cohesion: Evidence and theories of
regional growth and convergence, European Economic Review, 40, pp. 1325-
1352.
SOLOW, R.M. (1956) A contribution to the theory of economic growth,
Quarterly Journal of Economics, 70, pp. 65-94.

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