Do all economies grow equally fast?
Yaniv Dover1 , Sonia Moulet1 , Sorin Solomon1,2 and Gur Yaari2,1
May 14, 2009
2
1
Racah Institute of Physics, Hebrew University, IL-91904 Jerusalem, Israel
Institute for Scientific Interchange, via S. Severo 65, IT-10113 Turin, Italy
Abstract
The stochastic spatially extended generalized Lokta-Volterra approach
introduced in Solomon [48], Challet et al. [18], Yaari et al. [58], is extended to the study of interactions between economic sectors, countries
and blocks. The theory predicts robustly in a very wide range of conditions systematic regularities in the growth rates evolution of various subsystems. The J-curve phenomenon which was studied in Challet et al. [18]
is revisited and more empirical support is given to the theory. In particular to the connection between the economic minimum and the crossover
of the new emergent leading sector with the old decaying one. We describe the ’Growth Alignment Effect’ (GAE), it’s theoretical basis and
demonstrate it empirically for numerous cases in the inter-national and
intra-national economies. The GAE is the concept that in steady state
the growth rates of the GDP per capita of the various system components
align. We differentiate the GAE predictions from the usual convergence
or divergence conceptual framework. Further investigations of GAE and
subsidiaries are suggested and possible uses are proposed. Due to it’s
simple and robust nature, the method can be used as a tool for economic
decisions and policy making.
1
Introduction
Macro-Economic systems are considered to be highly complicated and very hard
to predict. There has been an extensive amount of literature on the theoretical and numerical modeling of phenomena such as ’Business Cycles’ [52] and a
plethora of Economic Growth patterns [3] etc. The recent advances in the understanding of multi-agent systems [e.g., 19, 30, 24, 35] and the recent abundance
of available ”Economic Data” in the last years have created an opportunity for
an educated search for patterns in the data, using the newly gathered intuition from said advances. Previous work [58, 18] has uncovered some universal
behavior that follows arbitrary changes / shocks in the economic systems. In
1
particular we found the existence of three distinct periods following a shock (the
liberalization of the Polish economy in the particular case of Yaari et al. [58]):
• An initial decay of the previously leading sectors and of the entire economy
with the exception of isolated previously sub-dominant system components
that display accelerated growth.
• A minimum of the economy coinciding with the moment when the old
leading sector crosses in size with the new emerging as the leader after the
shock.
• A return to growth led by those sectors that displayed the accelerated
growth and whose growth rate is gradually approached by the rest of the
system components.
We call the last growth phase, characterized by the fact that all system components approach the same growth rate the Growth Alignment Effect (GAE).
Note that we do claim to find neither in the theory nor in the empirical data
the usual convergence effects. Growth Alignment means only that the growth
rates of various components become equal after a certain time. The mathematical formulation corresponding to such a rate alignment existed for some time in
the shape of the Perron-Frobenius theorem [31], but was used in different contexts (mainly the steady state, zero growth rate case) and was never applied to
growth phenomenology. The importance of the existence of such an effect, also,
in non-stationary systems is unquestionable. Therefore, here, we extend the
patterns found in the Polish post-liberalization economy [58] and show that the
alignment of the growth rates holds generically in complex systems at various
scales. We demonstrate it at the intra-national level (Economic Sectors) and
the inter-national level (countries and aggregations of countries).
2
The convergence issue
The convergence issue has interested economics since a long time. Nobel Laureate Robert Lucas explained that when he examines the wide variation of crosscountry economic rates of growth ”it is hard to think about anything else.” [37].
A key issue in economics is whether inequalities between countries will persist
[10]. Are there forces that make them converge to a steady state in the longrun? This question motivates the debate within the growth theory [3] both on
the theoretical and on the empirical point of view.
2.1
The theoretical economics framework
Let us review briefly the main points under debate.
The premise of the argument of the neo-classical growth theory is that the
growth has exogenous origins and that there is a diminishing returns to capital.
This leads to the conclusion that economies will eventually reach a point at
2
which no new increase in capital will create economic growth. This point is
called a ’steady state’.
This branch of growth theory was initiated by Solow [49] and Swan [54]
in the 1950s. They were the first attempts to model long-run growth analytically. This model assumes that countries use their resources efficiently and that
there are diminishing returns on capital and labor increases. From these two
premises, the neo-classical model makes three important predictions: First, increasing capital relative to labor creates economic growth, since people can be
more productive given more capital. Second, poor countries with less capital per
person will grow faster because each investment in capital will produce a higher
return than rich countries with ample capital. Third, economies will eventually
reach a steady state in the long run.
The key assumption of the neoclassical growth model is that capital is subject to diminishing returns. Given a fixed stock of labor, the impact on output
of the last unit of capital accumulated will always be less than the one before.
Assuming for simplicity no technological progress or labor force growth, diminishing returns implies that at some point the amount of new capital produced
is only just enough to make up for the amount of existing capital lost due to depreciation [49]. Up to this point, because of the assumptions of no technological
progress or labor force growth, the economy ceases to grow. Solow [50] shows
one year after that the same logic applies if we assume technological progress.
In the short-run the rate of growth slows as diminishing returns take effect and
the economy converges to a constant ’steady-state’ rate of growth without economic growth per-capita. Moreover, including non-zero technological progress,
a new steady state is reached with constant output per worker-hour required for
a unit of output and the per-capita output is growing at the rate of technological
progress in the ’steady-state’ [32].
We do not enter in more details of this literature and advice the reader to
refer to Solow [51] or Ruttan [46] for more reflexion about the current state of
this literature.
Endogenous growth theory [e.g., 45, 43, 5] is based on microeconomic foundations. Households are assumed to maximize utility subject to budget constraints
while firms maximize profits. Crucial importance is usually given to the production of new technologies and human capital.
For the endogenous growth theory, the convergence is not systematic. Endogenous growth theory demonstrates that policy measures can have an impact
on the long-run growth rate of an economy. Countries that save more grow
faster indefinitely and countries need not converge in income per capita, even if
they have the same preferences and technologies.
In the past decade, an important debate in the economics of growth has
been related to the so-called ’convergence’ issue [39]. Several lines of research
have been followed up to now so as to reconcile theory with the empirics of
non-systematic and global convergence in growth rates and per capita income
3
[28].
2.2
The empirical growth convergence
As Durlauf [26] underlines ’few issues in empirical growth economics have received as much attention as the question of whether countries exhibit convergence’.
Within the empirical literature of growth theory, we distinguish two different approaches [21]. The first approach try to identify the immediate source
of growth by measuring the rates of accumulation of productive factors and
weighting them according to their share in national income [e.g., 11]. The
second approach use statistical techniques to quantify the impact of different
variables on growth.
Durlauf and Quah [27] provides one overview of the current state of macroeconomist’s knowledge on cross-country growth. Until now, several concepts
of convergence (e.g, β-convergence1 , σ converge2 ) have been proposed (or even
compared [47, 33]) and different econometric methodologies have been employed.
Cross-sections [e.g., 8], time series [e.g., 15, 16], clustering and classification
[e.g., 6], and panel data [e.g., 34] have been exploited in the empirical analysis.
Most of the papers in this field tend to interpret the convergence or divergence
as a confirmation or a falsification of the validity of the neoclassical economic
growth theory [20]. Since there is an important debate around the validity and
the added values of those methods, we do not enter in details [see 41, for more
details].
Nevertheless, from Madison’s work, a common desire persist: quantify economics and the relationships between countries. Once more, we do not enter in
details but rather we give some extreme example that seem to be representative
to the variety of results we can obtain by looking at different set of data with
different tool.
There are some ambiguous definitions that put back into questions some
empirical conclusion. For example, using a time series data base of productivity
levels for 16 Western nations, Abramovitz [2] finds support for the catch-up hypothesis. According to Abramovitz [2], a country’s ability to catch-up to richer
countries is determined by the country’s ’social capacity’ to absorb new technology. Baumol [11] also using the Madison data, finds evidence of convergence
amongst the developed countries but DeLong [22] points out that Baumol’s definition of developed countries suffers an ex post problem, and when an ex ante
definition of ’developed’ is used the convergence disappears.
1 According to Sala-i-Martin [47], there is β-convergence if there is a negative correlation
between the average growth rate and the initial per capita income.
2 There is σ-convergence if the dispersion of real per capita income across a group of
economies falls over time
4
Also, there are strong assumptions from the literature that are subject to
validation of the Solow Model. This is the case of the ’conditional convergence’.
The neoclassical growth model assumes convergence conditional on all countries
having the same steady-state, i.e. the same technology, same savings rate and
same population growth rate. Using the Summers and Heston [53] data, Mankiw
et al. [40] attempt to test for ’conditional convergence’. They show that an augmented Solow model that includes accumulation of human as well as physical
capital provides an excellent description of the cross-country data. They also
examine the implications of the Solow model for whether poor countries tend to
grow faster than rich countries. The evidence indicates that, holding population
growth and capital accumulation constant, countries converge at about the rate
the augmented Solow model predicts.
Looking at the empirical literature, we remark that studies tend to give
contrary results when applying different methodologies. On one hand, studies
based on cross-section tests generally reject the ’no converge null’ hypothesis in
the cases of advanced industrialized economies [11, 25] and US regions [9, 10] as
well as in large international cross-sections after controlling for variables such as
population growth or savings rates [7, 39]. But, on the other hand, time series
tests have generally accepted the ’no convergence null’ for a range of data sets,
as shown by Quah [42], Bernard [14] and Bernard and Durlauf [15].
When comparing developed and developing nations, the issue of openness
of markets becomes a central issue. Romer [44] emphasizes, the implications of
openness3 (e.g. openness to international trade, foreign direct investment, and
the flow of knowledge and ideas) can differ significantly across different growth
models. Empirically however, openness may be a driving force in convergence
[13, e.g.,]. Another difficult issue to address the level of economic ”freedom”
that exists in an economy. Economic freedom is an important variable in long
run economic growth and may play an important role in level of economic convergence. However, cross-country time series data on measurement of economic
freedoms is relatively scarce. More generally, Ben-David [12] finds convergence
among the world’s wealthiest countries and also finds convergence among the
world’s very poorest countries.
2.3
Our position with respect to the literature
As we saw until now, there is no absolute way to define the empirical growth
convergence, to measure it or to explain it. Durlauf [26] writes ’ ... the empirical growth literature has failed to develop a coherent approach to assessing
convergence as an economic phenomenon’. Economists and econometricians try
to predict the dynamics of economic growth, by using several techniques and
sometimes making strong assumptions. It is somewhat very hard to extract
empirical predictions or real conclusions on what is going on and hence, it is
3 The
openness is considerate as the ratio between imports and GDP
5
quite hard to apply policies that should have an impact on the growth [17, 4].
Consequently, we propose an orthogonal way to look at the growth dynamics
which can be traced back to the ideas of Romer [44]. We study the system both
empirically and theoretically. The theoretical model has stochastic background
and hence the strong predictions have statistical meaning: the growth rates
of strongly inter-connected countries is expected to align asymptotically. Note
that this prediction refers to the rate and not to the absolute value of GDP
per capita. This feature to the extent of our knowledge was not discussed so
far in the growth literature. We do not look at the per capital real income but
rather at the growth in the GDP of a country. Not to confuse with the classical
definition of convergence [15, 16], we use the term ’alignment’ and consider that
two countries are ’aligned’ if their GDPs growth rates approach one another.
On a semilogaritmic plot the two curves become parallel (thus the term ”alignement”) while their distance may stay constant and large. We start by showing
the theoretical considerations leading to this prediction, then we demonstrate
it’s empirical evidences in various scales: clusters of countries, countries and
sectors within a country.
We wish by this paper to add yet one more point of view to the important field of economic growth, and by this to catalyze scientists to put back
into question the definitions of the various types of dynamics existing in real
economic systems. We believe that the form and strength of the international
trades influence significantly the dynamics of growth rates. In particular the
frictions inherent in getting capital, production skils, technological environment
and products across geographical, political, social, educational and economic obstacles may affect crucially the convergence scenario of Solow Model. Our long
run ambition is to study the impact of commercial exchanges on the growth of
the GDP in terms of the IO matrix representing the world trade and economic
exchanges.
3
3.1
The Model
Generalized Lotka Volterra
The traditional starting point for the investigations of any system in which there
is some sort of consumption of a finite amount of resources is the Logistic model
[55, 56, 57]:
dw
= r · w(t) − c · w(t)2 .
(1)
dt
This single equation represents the aggregated dynamical model for a population
that proliferates at a constant rate, r relative to its current size w and has a
maximal saturation value of wmax = rc where c is the competition constant.
Later, the above 1 was generalized into a theoretical framework where the subpopulations were modeled by a set of coupled differential equations [23, 36]. The
differential equations paradigm was shown to be less than adequate when it came
6
to reproducing realistic results and then effects of discreteness and stochasticity
were introduced into the modeling scheme [48]. This change of paradigm have
brought about some success in describing realistic systems and we intend to
continue and make use of it here. The so called Generalized Lotka Volterra
model [48, 38] is a natural generalization of the aggregated version in the Eq(1):
N
N
N
X
X
X
dwi
= ri · wi (t) +
aij · wj (t) −
aji · wi (t) −
cij · wi (t) · wj (t)
dt
j=1
j=1
j=1
wi (t + 1) = wi (t),
(2)
j = 1, .., N, i 6= j
The above is an analytical description of an interacting system composed of
N agents. The first term on the right hand side of Eq(2) is the self-growth
term which can be a stochastic random variable, per agent. The second term
expresses the inflow from other agents to agent i. The third can be interpreted
as the outflow and the fourth term is the competition term. In passing, we note
that although the Logistic and Lotka-Volterra exist for a considerable time in
the literature, only recently the effects of discreteness and stochasticity, which
modify completely their behavior, were studied in detail. As a result they
were shown to produce unexpected results such as the emergence of adaptive
collective objects which insure the sustainability of the system in conditions
which naively should lead to total collapse [48, 18, 58]. Therefore, the reinforced
realism of this model makes it a worthy candidate when investigating complex
systems.
For our purposes, a further simplifying assumption will be made, namely a
sort of ”mean field” assumption of the competition term:
N
N
X
X
dwi
= ri · wi (t) +
aij · wj (t) −
aji · wi (t) − c(w̄, t) · wi (t).
dt
j=1
j=1
(3)
This assumption means that we assume that the competition is between the
agent and the system as a whole. We note that there is no assumption regarding
its time dependence - so, we are still including arbitrary changes in the nonlinear
competition term.
3.2
A Prediction: The Coupling of Growth Rates
Looking into the model represented in (3) we can find that it has an interesting
property. Writing the temporal transformation in a matrix form:
dw
~
= M(t, w̄(t)) · w(t)
~
(4)
dt
where w
~ denotes the vector of all N agents (while w̄ denotes the average) and
the, Markovian like, transformation matrix is defined, using (3):
Mii (t, w̄(t)) ≡ ri −
N
X
j=1
7
aji − c(w̄(t), t)
Mij (t, w̄(t)) ≡ aij .
(5)
Looking at the normalized (perhaps ”deflated”) space, we can see an even simpler form of the transformation 34. Changing variables to w(t)
w̄(t) (as was shown
in [58] and [48]) the matrix becomes:
Mii (t, w̄(t)) ≡ r̃i (t) −
N
X
aji
j=1
Mij (t, w̄(t)) ≡ aij
(6)
where r̃i (t) is the fluctuating part of the stochastic self-growth term in (3). In
this version (6) the ”temporal nonlinear” term, c(w̄, t) · wi (t) does not appear
and the time dependence arises only endogenously. On time scales larger than
the one of the fluctuations, the steady state solution of (4) in the case of (6) is
the eigenvector of M corresponding to its largest eigenvalue [58]. This means
that the growth steady state solution is:
w(t)
~
= W0 · w
~ max (t) · eλmax ·t
(7)
where W0 is a diagonal matrix representing the initial conditions and w
~ max (t)
is the eigenvector of M corresponding to the highest eigenvalue λmax . This
means that after some mixing time which is approximately −1/ ln(λ0 ) (λ0 being
the second highest eigenvalue) the whole system is driven toward a steady state
growth where the growth rates of all agents are equal. There is a small caveat
that fortunately applies only to a very small amount of countries: if there is
complete decoupling between a system component (or a group) and all the
others, then of course there is no equalization of the growth rates (this might
happen mainly for dictatorial isolationist regimes or for regimes under universal
embargo).
3.3
The J-curve prediction. Universal transient behavior
after shocks.
The present framework has often unexpected powerful implications in situations
in which there is almost no a priori knowledge on the details of the conditions
in which the system is placed.
Suppose a policy maker comes and say: I am going to make some reforms
that completely reshuffle the present national priorities. What is going to happen next?
The first reaction could be ” well, it depends of what changes you make”. As
it turns out this is in general not true: ”a complete reshuffle ” means essential
by definition that most likely the leading components of the economy are going
to change. Mathematically, this means that the interaction matrix is going to
change and in particular its highest eigen vector. Thus most likely the largest
8
component, which by the Frobenius-Perron theorem had until now the highest
intrinsic growth rate, is going to be substituted by another component that was
sub-dominated until now. From the moment that the old dominant is going
down and until the moment that the new dominant component becomes comparable in size with it, the entire system will have a downward tendency. This
will change only after the new fast developing component becomes large enough.
Thus, without knowing the details of the change , just based on the interaction
matrix concept and its eigenvector analysis one obtains that following a generic
shock the economy will follow a j-shape: first down and then stronger up. The
existence of a J-shape is not related with the desirability of the change or with
the way it is implemented: it is a generic fact that has to be taken into account.
One may in certain conditions optimize by taxation and subsidies the transition
period but it is virtually impossible to eliminate it. This is just one simple
example[18] in which the new approach allows one to inform politicians of nontrivial yet generic facts that might strongly affect the outcome of their policies.
Of course more knowledge of the system will allow more detailed analysis and
advice.
4
4.1
Results
Global Scale
As a first example, we present evidence for our claims at the global level of
aggregates of entire countries4 . In Fig. 1 one can see the different aggregates
and how many converge with time. Notice the clear convergence of rates of
the US, Australia and the Western Europe countries. Also oe can see clearly
from Fig. 1 the divergence triggered (according to the above mechanism) by
an exogenous shock: the fall of the Iron curtain towards the end of the 1990’s.
A clear J-shape is evident in the growth of the eastern bloc (and it’s effect on
the whole world). It is relatively easy to recognize cliques from the graphs,
not only by looking at the absolute value of growth (σ-convergence) but also
of the exponential rate of growth (i.e. the slope, GAE) which, as we explained
above, marks strong economic coupling. Another interesting and approximately
universal phenomenon we observe is the relatively high growth rate for countries
with regimes that are less liberal than what is common in ”Western countries”.
Such examples include China in Fig. 1 and Greece, Spain and Portugal in Fig.
2 before 1973 (i.e. before the regime change). At some point, though, such a
4 For all countries the series are expressed in 1990 US dollars converted at ”Geary-Khamis”
purchasing power parities (PPPs). The 1990 US dollar estimates are in almost all cases derived
from Maddison, A. (2007), Historical Statistics for the World Economy: 1 − 2003 AD with a
downward adjustment of 22.6 percent made to China’s GK PPP-converted GDP level in U.S.
dollars, reflecting a partial adjustment to recent GK PPP estimates by the World Bank for
2005 to better represent urban price levels. To extend the GDP series to the period before
1950, these series can be linked to Maddisons historical series, which often go back to the 19th
century and in some cases even further back. Series on real gross domestic product in both
constant 1990 and current 2008 US dollars are presently available for 123 countries. Series
are expressed in market prices and cover the period 1950 − 2008 for all countries. (2009)
9
country is bound to couple to the external economy due to its size and extent of
activity. This will eventually align its growth as well as its much of its economic
functioning with the rest of the world. This has happened with Greece, Spain
and Portugal (see the next section) and may happen to China in the future.
Note that we do not say that China’s GDP needs to reach the western world
level before its growth rate will align itself with the western world level - but it
is enough that the upper Chinese class (say, the people that live in big cities)
will have comparable income to the western world that they will start buying
more intensively from the rest of the world and by this will join the block of
highly connected economies. Joining this block means aligning the growth rate
with rest of the world.
4.2
International Scale
One of the striking examples of the coupling of growth rates is the coupling of
Spain, Portugal and Greece to the 12 European Countries 5 . This, of course,
is evidence for the inter-national effect. Looking at Figure 2, one can see that
before the year 1973, Spain, Greece and Portugal (denoted by lines with symbols) had a long-scale growth rate (Logarithm slope) that is distinctly different
than that of the main 12 countries (denoted by solid lines - no symbols). After
the practical joining, in 1973, the Logarithm slopes of all countries aligns on
scales of decades. One can also see that the slope of the 12 European countries
for the pre-1973 period, also changed as a result of the transition. Mathematically speaking, this is expected when one considers that the eigenvalue of
the original growth matrix (or block) of the 12 European countries is attached
to the growth matrix of Spain, Greece and Portugal. This means that the
new matrix will have a different maximal eigenvalue than the original matrix.
Since the 12 Countries are the majority, obviously the change of their maximal eigenvalue is less prominent than that of the more dramatic change of the
growth rate of Spain, Greece and Portugal. This can also be seen in Figure 2.
One can note that the actual official joining of the three countries to the
European Union did happen up to 8−10 years later - Spain(1986), Greece(1981)
and Portugal (1986). Obviously, a proper recognition of this kind of effect
can assist policy makers in analyzing and identifying of underlying couplings
between Economic entities. This, of course, could also have crucial influence on
the creation of Monetary and Economic policies.
All three countries have been through some kind of a transition of regimes
around 1973−1974. In this work we only analyze the empirical evidence without
going into the complexities of the political situation, which will be done elsewhere. In order to accumulate evidence to verify the nature of the effect seen
in Figure 2, we’ve also analyzed Trade-Volume data [29]. For demonstration
purposes, we’ve analyzed the total inflow of trade into the the three countries,
5 The 12 European countries are: UK, France, Germany, Norway, Sweden, Italy, Germany,
Netherlands, Finland, Denmark, Belgium and Austria
10
Spain, Greece and Portugal. We do not argue that this is the only or the best
way of measuring connectivity, but we do argue that this demonstrates the validity of our claims. The detailed discussion in methods of measuring connectivity
and the comparison of the coupling of growth rates effect to the results of these
methods, will be dealt with elsewhere.
As we can see in Figure 3, the normalized inflow of trade (for details see
caption) has a step-like behavior around the year 1973. The normalized trade
volume increase in a factor of 2 − 3 within the time interval 1970-1980, which is
unusually high in relation to the preceding years. We interpret it as suggesting
a jump in the amount of trading activity. Even though the documented trade is
only a fraction of other activities that may connect Economies of countries, it
should be proportional to the aggregated inter-activity. This is indeed only a
simple starting point, but we suggest that this step-like behavior is an indicator
for an establishment of a connection of the the above three countries to the
12 European countries, as was seen, independently, through the growth rates
(Figure 2).
Examples for the same growth rate coupling effect on a higher level of international aggregation can be seen in Figure 1.
One can see in Figure 1 that all throughout the time range, Australia and the
US share a similar growth rate (Logarithm slope) on scales of more than a few
years. We interpret that as their being connected. Mathematically speaking,
this means that an interaction exists between them - i.e. that in M (5) one
of the non-diagonal elements is not zero. In case of two or more sub-groups
of connected Economic entities, we expect that M will be composed of blocks.
One can also see in Figure 1 the convergence of Western Europe to the US and
Australia growth rates at around 1975 (which is also, curiously, more or less
coincident with the transition year observed in Figure 2). Prior to 1975, we can
see that Western Europe’s growth rate seems to be similar to that of the Eastern
Bloc. This is an interesting observation that awaits clarification. One can clearly
see that at the point of liberalization (around 1989) there is a clear divergence
of growth rates, suggesting that the system is composed of underlying, isolated
blocks. One can use the fact that during shocks to the system, the divergence
contains information regarding the structure of the interaction matrix M and
reconstruct it. Another apparent observation in Figure 1 is the coupling between
the East European countries and the USSR, up until the liberalization point.
Last but not least, one can spot the convergence of the East European countries
towards the ”Western” growth rate (West Europe, US and Australia). It is
tempting to suggest that in the near future, the Russian growth rate might also
converge with the western rate, but we do not have enough numerical evidence
at present to make anything but an intuitive suggestion.
The exercise we demonstrated above, of course, can be done for all international aggregates and for a variety of historical eras. We have taken the above
example as a strong indicator of our claims to the existence of the GAE on the
international level.
11
4.3
Intranational Scale
In Figure 6 we plot the Added-Value of a ten-sector partition of the Brazilian
Economy over the last 30 − 40 years. In other words, we plot the production
growth estimate of a certain sectoral partition of a one-country Economy. For
political or economical reasons that await explanations, we see convergence of
growth rates from the mid 1990’s and onwards. Also, in the case of Bolivia’s
ten-sector partition, there is also a convergence of most of the sectors’ growth
rate as is seen in Figure 6 from the late 1990s onward. We have plotted the
three sectors deviating from this pattern in weaker dotted lines as to enhance
the distinction. We interpret this as the Bolivian economic structure composed
of one big 7-sector (matricial) block (i.e. 7 sectors that are connected) and the
three other isolated sectors, which might be connected to each other to form a
second block, or not. What we see here is a measure for the substructure of the
system. It is quite clear, though, that within these 7-sector sub-economy there
is a convergence of growth rate.
Another example for the convergence effect, we can see in Figure 6 where
a 6-sector partition of the Romanian economy is plotted (Added Value). Here,
also in the late 1990s, we see a clear convergence of all sectors. But, we see
also from the year 1982 to around 1988 a clear convergence of growth rates.
This convergence breaks at the liberalization point and is only fully regained,
as we said above, in the late 1990s. This scrambling of growth rates, at the
liberalization point is crucial in the understanding the underlying connectivity
of the sub systems, M (5).
4.4
Reconstruction of Economic Networks
Using our model and the assumption of growth rate convergence, we can also
use available data to uncover hidden interactions and coupling networks. The
straightforward way of doing so, is to calculate the ”distance” between exponential growth rates of interacting components (countries, sectors or aggregates).
The distance was defined to be:
d(t)i,j =
(β(t)i − β(t)j )2
σ(t)2
(8)
Meaning, the distance between components i and j at time t equal the distance
between exponential growth rates (β(t)) normalized by the standard deviation
of the exponential growth rate of that year among components. In order to take
out the local noise, different smoothing lengths were used for the exponential
growth rates. Using our model, one can say that d(t)i,j should be proportional
to the coupling between components i and j. One case study was calculated for
Argentina for the period of 20 years (1960 − 1980), which was an eventful one
as can be seen in its economic growth during these years (Fig. 5a and b ). The
result is presented in the form of a matrix of coupling strengths in Fig. 5c for
these years. It is easy to identify in this form cliques and coupling within the
matrix. Using the growth data of the sectors alone, one can infer the microscopic
12
coupling of the sectors and by that identify clusters and their dynamics. In the
case of Argentina, one can see that a ”shock” around the year 1965, has brought
to light the decoupling of two cliques (Seen by two separated bright squares).
Following the decoupling stage with time, as we explained above and after some
reshuffling of interactions we see more uniformity in the matrix suggesting a
convergence of the most of the sectors, as is also seen in Fig. 5a and b. One has
to note that in this mode of analysis the resulting matrix is symmetrical while in
the economic reality the inter-country influence is not necessarily symmetrical.
So the coupling strengths measured here is a measure of the overall interaction
and not the directional one. Note that the reshoufling of the interaction matrix
is contemporary with the aftershock J shape minimum and with the crossing
between the decaying ”old” leading component and the growing new one.
5
Discussion
The numerical evidence we have shown above, indicates that economic systems
at various scales display generically the growth rate alignment effect(GAE).
This is a much weaker claim than the usual convergence hypothesis: we only
claim that in steady state the growth rates of the GDP per capita of the various
system components align. We do not address the issue of the equalization of
the GDP per capita itself. Our hypothesis is that for the equalization of rates
it is sufficient to have free capital flow between the components while for the
equalization of the absolute GDP/capita values one needs also free man-power
flow. Contrary to the neoclassical assumptions underlying the Solow models it
is difficult to achieve manpower interchangeability not only between countries
but also between sectors and regions in the sam country: one cannot expect the
workers of the mining sector to start producing high-tech software as soon as
.com sector becomes profitable (see for example the alignement at very different
absolute values of the GDP of Polish counties [58]).
The rate convergence and the ratio between the absolute GDP/capita values
depend on the matrix that contains as the diagonal terms the intrinsic growth
rates of the components and as the off diagonal terms the flow between the
components. In a sense one can consider this matrix as an extension of the
Leontief matrix. Thus with some abuse of language one may claim that GAE
is the Frobenius-Perron theorem applied to the Leontief matrix. According to
our model, one can extend the Perron-Frobeniun theorem to nonlinear, nonstationary systems too. This is confirmed empirically by evidence at various
inter- and intra- national scales. We do not claim that the analytical model
based on the exponentiation of the generalization of the Leontieff matrix is
the most general or ultimate model for multi-component economic systems ,
but we do note that not only it exhibits stylized facts seen in real life data
(even in conditions where other analytical methods are impracticable) but it
also has some quantitative relative success in describing complex phenomena
[48, 18, 58]. This work is only a starting point. The numerical method awaits
further specifications and the abundant relevant data is still to be confronted.
13
One should also try to understand the ramifications of the success of such models
and investigate the details of the dynamics leading to alignment and out of it.
Information on the system should also be extracted from the data before the
alignment takes place at the stage of decay after the shock. This may allow the
prediction of the crisis depth and duration and most importantly identify the
components of the system likely to lead the rebound out of it. Consequently one
can suggest the optimal policy to accelerate their take-over of the economy and
eliminate the after-shock crisis completely [58]). In general, one should further
calibrate, tune, and test the model and its parameters. In a wider perspective,
the effects described here, could also be exploited in reconstruction methods for
networks (only where the analogy of the agents’ growth rate is clear). Thus,
we believe that this might be a good anchor point in the understanding of the
nature of interactivity, growth and dynamics of complex systems in general.
Due to its simple and robust nature, we also suggest it as an analytical tool for
economic and politcal decision making and policy testing.
6
Conclusions
We have studied the effects of interactions on the growth rates of the components of various economic systems. We have shown that even in non-stationary
conditions and even under arbitrary shock conditions, one can make definite
non-trivial generic predictions about the evolution of the system. The crucial object turns out to be the matrix characterizing the coupling between the
system components: regions, sectors, countries, economic blocks. The matrix
can be thought as a generalization of the I/O , Leontief matrix in as far as
its entries quantify the transfer of economic activity, production, capital between the system components. The diagonal elements of the matrix are the
intrinsic growth rates of each component. The dynamics of the system can be
then regarded as an exponentiation of the interaction matrix. Thus, applying to the interaction matrix the Perron-Frobenius theorem one predicts that
the various components will reach asymptotically a common growth rate equal
to the highest eigenvalue of the interaction matrix. We call this phenomenon
Growth Alignment Effect(GAE) and document its presence at various scales
in the intra- and inter-national economic scales. Note that according to the
Perron-Frobenius theorem the absolute value of the (per capita) production of
each system component approaches a value proportional to its representation
in the eigenvector corresponding to the highest eigenvalue of the interaction
matrix. Thus the present framework does not predict the previously defined
β or σ convergence. In fact the empirical data support our theoretical prediction: that while the growth rates approach a common value, the ratio between
the production level (per capita) of the various components approach non-trivial
(different from 1) constants. Moreover if due to endogenous or exogenous shocks
the interaction matrix is dramatically affected, our model predicts a cross-over
between the production (per capita) of different system components: the weight
of the old dominant ones shrinks while the components enjoying new large in14
trinsic growth values take-over. Eventually all the components re-align to a
common growth rate but in the intermediate time range after the shock the
model predicts a set of generic stylized facts: the various system components
associated with the ”old” and ”new” leading components follow very different
paths and growth rates: usually the parts of the system strongly connected to
the ”old” dominating component display immediately after the shock of matrix
reshufling negative growth rates while the ”new” components develop rapidly.
This intermediate time period is also characterized by a J-shape evolution of the
entire system with a minimum located at the time when the ”old” and ”new”
components sizes cross [18]
On could see in the example of Argentina 5 this split of the economy in
clusters during transition periods and the return to economic growth rate uniformity across the sectors in the steady periods. Moreover one could see in the
examples of Portugal and China (graphs 6 and 7) the coincidence between the
times when the old dominating and sub-dominating economic sectors cross and
the time of minimum GDP during the post-shock J recovery (called in some
literature ”adaptation”) process.
Another clue given by the present analysis is about the causes of growth
rate alignment taking place and the causes for the apparent absence of GDP
per capita equalization in absolute values.
Indeed, one see in graph 2 that among all countries joining the bulk of EU,
only Ireland has equalized the GDP with the former EU members while all the
others new-comers have equalized only the GDP growth rate. The answer may
lie in the fact that most of the Irish population speaks English and thus there
were no barriers in the Irish manpower to take over any jobs held by previous
EU citizens. For the other countries that joined EU (except East Germany
where the same phenomenon as with Ireland held), it was only the capital that
could now flow freely between them and EU. This might turn out to be the clue
to the puzzle:
• capital / investment exploits, gains and pursue relative growth : investing
in shares of a very productive company with stable equity price is not
producing significant gain.
• As opposed to it, people live on and pursue present absolute value of
wages (proportional to the production per capita), Getting a good job in
a company with stable production is all one wishes.
Thus free movement / interchangeability of workforce insures convergence of
GDP per capita (as in the case of Ireland) while free movement of capital insures
only the alignment of growth rates (GAE, as in the case of Spain and Portugal,
or in the comparison of US, EU and Australia 1). Note that these effects are
somewhat complementary but congenial to the neoclassical analysis of Solow [49]
where the capital flow between countries with various productivity and capital /
manpower ratios is the mechanism for convergence. However, in the neoclassical
analysis the inequivalence between the services and needs characterizing widely
different countries and economies were not included. Such an analysis could fit
15
the convergence of counties within the same state in US [33] but not culturally
or geographical segregated societies (or very different sectors corresponding to
widely separated social groups and interests).
In our case the result of the analysis is rather the alignment of growth
rates(GAE) which is much easier to recognize and study in the empirical structures and data. It is remarkable that so much can be said within the present
framework before even entering the details of the external changes and conditions. It is also hoped that introduction of new concepts such as the interaction
matrix and the Growth Alignment Effect might help to a better capture of the
stylized facts dominating the phenomenology and might allow a better understanding, policy and control of particular empirical situations: crises, social and
technological change, political reforms etc.
Acknowledgements
We enjoyed during the thinking on the issues related to this project discussions with David Bree, Simona Cantono, Marco Lamieri, Moshe Levy, Yoram
Louzoun, Lev Muchnik, Andrzej Nowak, Dietrich Stauffer and Leanne Ussher.
The research was supported in part by the CO3 and Daphnet STREP grants
from ICT and NEST, EU FP6.
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W.Europe
E.Europe
Russia (F. USSR)
China
Latin America
Africa
U.S.
10000
Australia
Real GDP per capita
India
1000
1920
1940
1960
1980
2000
Time[years]
Figure 1: Real per capita GDPs (in units of 1990 US dollars) of countries
aggregates around the world. Data taken from GGDC[1]
20
25000
Real per capita GDP
20000
15000
10000
Netherland
Norway
Austria
Belgium
Denmark
Finland
5000
France
Germany
Sweden
UK
Ireland
Greece
Portugal
Spain
Italy
1970
1980
1990
2000
Time [years]
Figure 2: Real GDPs (in units of 1990 US dollars) for the years 1965 − 2003, of
the 12 European Countries with the Real GDPs of Spain, Greece and Portugal
[1].
21
18
Normalized incoming Trade
16
Portugal
14
Spain
Greece
12
10
8
6
4
2
0
1960
1965
1970
1975
1980
1985
1990
1995
2000
2005
Time [years]
Figure 3: The sum of total incoming trade [29] coming in from the 12 European
Countries (US$). The data is deflated using the US CPI and normalized using
the importing country’s real GDP per capita.
22
Value Added [M. Pesos]
Value Added [M. Reales]
100000
10000
Agric. For. Fish.
Transp. Comm.
Mining Quar.
Finance Insur. Rl. Est.
Manufac.
1975
1980
100
Serv.
PublicUtil
1970
1000
Agric. For. Fish.
Trade Rest. Hot.
Mining Quar.
Transp. Comm.
Manufac.
Finance Insur.
Real Est.
PublicUtil
Dwellings
Construct.
Construct.
Wh. Sale. Ret. Trad.
WholeSale. Ret.
1985
1990
1995
2000
2005
1980
1990
Time [years]
Serv.
Dwellings
2000
Time [years]
(b) Bolivia
Value Added [Const US$]
(a) Brazil
1E10
1E9
Agric.
Whole Sale
Manuf.
Transp. Com.
Construc.
1985
1990
Other
1995
2000
2005
Time [years]
(c) Romania
Figure 4: Value Added (equivalent to sectorial GDP) of three countries. The
list of sectors is given in initials in the legend.
23
Figure 5: The case of Argentina (Data taken from GGDC [1]): (a) One example
of the J-shape of Growth Rate of Economic sectors within a country. Argentina
in the last 60 years which, in this case, is partitioned into 9 sectors (which
include: Agriculture, Mining, Manufacturing, Construction, Finance etc.) [1].
(b)One can see that the J-shape in the real GDP per capita of Argentina during
the 1980s and 1990s result from a switch between dominating groups of sectors.
Data taken from GGDC [1] (c)The di,j matrices of every year in the range of
1961 − 1980 for the Argentina sectoral growth. The color base was chosen to be
inversely proportional to the exponential growth distance (Eq. 8), the lower the
distance is the brighter the color of the cell is (i.e. the brighter it is the stronger
the coupling). The data used for calculation is the data shown in a. Cell i, j
in the matrix, represents the coupling strength between sector i and sector j.
One can see how the different growth periods reflect in the matrix in the form
of different partitioning to blocks.
24
22000
20000
500
18000
400
16000
Group B
14000
12000
300
10000
8000
200
Real GDP per capita
Value Added [M. of Euros]
Group A
GDP
6000
100
1970
1975
1980
1985
Time [years]
Figure 6: The case of Portugal (Data taken from GGDC [1]): (a) Another
example of the J-shape of Growth Rate of Economic sectors within a country.
Portugal during the 1970’s and 1980’s. The two solid lines represent Value
Added (equivalent to sectorial GDP) of the aggregations of the sectors into two
groups, group A contains Construction, Retail, Transport, Communications and
Finance and Group B contains Hotels, Restaurants, Services, Health, Education
and Public administration. The crossing described in the text (switch between
dominating groups of sectors) happens at around 1975 as is also evident in
Portugal’s real GDP per capita(blue line).
25
200
Agriculture
PublicUtil
Industry
Bank. Insur.
160
Construction
Real Estate
Transp. Comm.
Sci. Edu. Cul. Health
140
Commerce
Gov. Party Soci. Org.
Soc. Services
1000
%GDP
120
GDP
100
80
60
Real GDP per capita
180
40
20
0
1960
1970
1980
1990
Time [years]
Figure 7: The case of China (Data taken from GGDC [1]): (a) Another example
of the J-shape of Growth Rate of Economic sectors within a country. China
during the 1950’s until the 1980’s: the graph exhibits percentage of each sector
out of the total Value Added (equivalent to sectorial GDP). In this case the
dominant sectors are Agriculture and Industry. The interplay and crossing
between them around 1960 and around 1970 causes the J-shapes in the real
GDP per capita (blue line).
26