Journal of Mathematical Finance, 2011, 1, 125-131
doi:10.4236/jmf.2011.13016 Published Online November 2011 (http://www.SciRP.org/journal/jmf)
Stochastic Convergence in Regional Economic Activity
Fariba Hashemi
Swiss Federal Institute of Technology, Lausanne, Switzerland
Received August 11, 2011; revised September 21, 2011; accepted October 5, 2011
Abstract
A stochastic model is presented, based on a double process of temporal drift and random disturbance, to fit
the evolution of cross-country distribution of income and economic activity . Instead of assumin g a steady state
as is standard practice, a long run stationary equilibrium distribution is hypothesized, around which econo-
mic activity fluctuates. An empirical application comparing dynamics of growth in Asia and Europe tests the
validity of the proposed method. In particular, results point out that the distribution of income and economic
activity is approaching a long run equilibrium at a faster rate in the case of Asia, and that the dispersion of
the distribution is shrinking over time above all in the case of Europ e. Main implications are supportive of the
convergence hypothesis, and suggest that diffusion may be a potential technique for the analysis of growth
dynamics.
Keywords: Dynamics of Growth, Stochastic Processes
1. Introduction
The study is mo tivated by the observa tion that ear ly leader s
in industrial revolution like the UK and the Netherlands
needed over 50 years for income to double from $2000
to $4000, but countries that reached $2000 after 1950
needed 10 - 20 years [1]. It would not be unreasonable to
consider that the shorten ing of time requ ir ed for doubling
per capita income may be attributed to a change in the
velocity of factor mobility. Consistent with this observa-
tion, we hypothesize that income fluctuates around some
long run stationary equilibrium according to a temporal
drift and random disturbance. A drift-diffusion model is
proposed, to expr ess inco me adjustment pro cess with no ise,
where dynamics of income rely on two counteracting
forces: 1) a mean-reversion process along time, driven
by mobility of factors of production and 2) a diffusion
process across regions, driven by search and learning and
trial and error. The present paper is one of a rather small
group of evolutionary studies which eschews simulation
in favour of analytical derivations.
2. Theoretical Framework
Debates on growth theory have contrasted the conver-
gence predictions of the neoclassical growth models of
Swan [2] and Solow [3] with predictions of potential
non-convergence from endogenous technological pro-
gress of Romer [4] and Aghion et al. [5]. In his classic
Contribution to the Theory of Economic Growth [3], So-
low proposed that we study economic growth by assum-
ing a standard neoclassical production function with de-
creasing returns to capital. In Solow’s world, there are
two inputs, capital and labor, which are paid their mar-
ginal products. Countries may differ in technology, re-
source endowments, geography and institut ions. In standard
neoclassical growth theory with diminishing returns, a
country’s growth rate will be inversely related to its ini-
tial income. If all economies are assumed to have the
same steady state, in the absence of shocks, countries at
different stages of economic development are predicted
to experience absolute convergence. It has been argued
that much of the cross-country differences in economic
outcomes can be traced to differing determinants of the
steady state in the Solow growth model. If, more realis-
tically, economies are assumed to have different steady
states contingent on differences in parameters such as
accumulation of human and physical capital and popula-
tio n gr owth , th en n eoclassical theory only predicts condi-
tional convergence [6].
In particular, since the works of Baumol [7], Barro [8]
and Mankiw et al. [9], the existence of convergence in
economic outcomes within groups of similar countries has
been largely recognized. What is less known, however, is
the dynamics in question. The present study helps fill
this gap, by providing a novel methodology to summarize

126 F. HASHEMI
growth dynamics. A model is proposed to describe the
fluctuations over time in the density of cross-sectional
distribution of income and economic activity. It is hy-
pothesized that economic activity fluctuates around its
long run stationary equilibrium according to a temporal
drift and random disturbance. These flows follow simple
stochastic laws that can be described with a few parame-
ters; parameters which can be estimated from historical
data with some accuracy.
3. Model
Consider a region consisting of a constant number of
countries with different levels of income and output1. The
set of incomes forms a distribution which evolves over
time. Consider the basic conservation law, with flux inter-
preted as the number of countries entering and number of
countries exiting an income/output interval. Assume that
flux is made of two different parts: a mean reversion proc-
ess—call it drift, and a random process—call it diffusion.
The counteracting forces of drift and diffusion result in a
long-run stationary equilibrium distribution of economic
activity. The equilibrium is a result of tension between
counteracting forces of convergence and divergence.
More precisely, for the drift spread, it is assumed that
there exists some equilibrium distribution of economic
activity with a certain mean and variance, towards which
the ensemble of countries gravitates, and the process is
governed by a velocity of convergence. Convergence is a
result of adjustment of capital-labor ratios to common
steady-state lev els star ting fro m different initia l values [1].
This adjustment is driven by diminishing returns [2,3].
For the diffusion spread, a search and learning process
generates randomness in the system [10-13]. Bottlenecks
in the flow of labor and capital and random effects cause
a spread of income/output from high density towards
lower density. Noise is generated by diffusion of knowl-
edge and learning, and limited by the presence of obsta-
cles in the form of trade barriers and the like.
In order to interpret and quantify this relationship, con-
sider a classical linear stochastic differential equation in
which the flux of probability co nsists of a drift and some
diffusion. The history of a country’s economic activity is
governed by an ordinary differential equation

dd= ,
X
tgXt
where

,

g
xt is the drift. Letting

,

X
xt be the so-
lution such that


,0 =
X
xx, we obtain:

 
d,d= ,,,
X
xttg Xxtt


,0=
X
xx (1)
Assume that and that the solution >0x


,
X
xt
remains positive (see Appendix A).
From the point of view of diffusion processes, con-
sider a stochastic differential equation of the Ito type:



 
d,=d ,d  ,
X
xtWtgXxtt


,0 =
X
xx (2)
where

is a small positive constant, and g is the drift.
W is a standard Wiener process (see Appendix B).
Consider the form


=

g
xx  u where
and >0,u
>0

.

denotes the adjustment rate, which for
simplicity is assumed constant. u denotes the mean of the
stationary equilibrium distribution. In this case, each point
moves toward the position u, but never reaches it. In ge-
neral, one can study a Markov process generated by a
matrix of transitions from one level of income/output to
another, where the Markov process can be treated as in-
come/output diffusion. Then one can apply the general
Fokker-Planck equation to describe evolution in time of
economic activity. Hence, assuming that income/output
behaves like a stochastic process and that it is continuous
and Markovian, we consider the most natural candidate;
a classical linear stochastic differential equation driven
by a standard Wiener process2. Our picture of world de-
velopment is thus one where convergence is counterbal-
anced by divergence. Convergence in the context of the
present model would mean collapsing of the cross-section
distribution. Divergence would mean that the cross-section
distribution replicates itself because for example it hap-
pens to be the stationary distribution for many inde-
pendent and identically-distributed country outputs.
4. Empirical Analysis
4.1. Data
We use data on GDP per capita for nine Asian econo-
mi e s between th e years of 1980 to 2007. The countries are:
China, Hong Kong, Indonesia, Japan, Malaysia, South
Korea, Singapore, Taiwan and Thailand. Figure 1 illus-
trates the evolution of the distribution of average GDP
per capita. We observe a steady growth with the East Asian
Financial crisis of 1997 standing out. Table 1 reports the
descriptive of GDP per capita for this population.
1In this study, “income” and “output” will be used interchangeably to
represent economic activity.
2The process evolves according to an Ornstein-Uhlenbeck. This type o
f
model has been widely used in Biomathematics [15,16]. Ref [17-19]
develops and provides a full analysis of this model albeit in a different
context.
For comparison, con sider GDP per capita for 24 coun-
tries in the European Union: Austria, Belgium, Bulgaria,
Cyprus, Czech Republic, Denmark, Estonia, Finland,
C
opyright © 2011 SciRes. JMF

127 F. HASHEMI
Figure 1. Evolution of GDP Asia.
rance, Germany, Greece, Hungary, Iceland, Italy, Lat-
.2. Estimation
he expectation of the distribution representing the time
F
via, Lithuania, Luxembourg, Netherlands, Poland, Slo-
vakia, Slovenia, Spain, Sweden, and United Kingdom.
Data were recorded from 1991 to 2007. Figure 2 illus-
trates the evolution of the distribution of average GDP
per capita. Table 2 reports the descriptive of GDP per
capita for this population. All data has been collected
from Datastream.
4
T
development of the differential equation expressing the
growth dynamics is:
t
uu

0
=1 tt e ue




 (3)
and the variance of the distribution:

222t  2
0
=1
t
tee







(4)
The model has been applied to log GDP per capita
distribution for th e two population s as a function of time,
using non-linear least squares estimation3. A two-step
procedure has been employed to estimate the model pa-
rameters, where 0
u denotes the initial mean of the dis-
tribution, and u denotes where the mean is heading. 0

represents the initial standard deviation,

represents
diffusion parameter, and the

represents th velocity of in-
come/output convergence
Tables 3 and 4 report
e
. estimates for the five model
pa
e the evolution of the distribu-
tio
Table 1. Descriptive of GDP Asia.
Descriptive of GDP (Asia) from 1980 to 2007
rameters, along with the standard errors and t-values
for the two populations.
Figures 3 and 4 illustrat
n of GD P per ca pita for the two populations over time,
superimposed on histograms which describe the time
NMinimumMaximum Mean Std.
Deviation
GDP283624.87 22790.29 11317.03 5473.38
Figure 2. Evolution of GDP EU.
Table 2. Descriptive of GDP EU.
Descriptive of GDP (EU) from 1991 to 2007
NMinimumMaximum Mean Std.
Deviation
GDP 1714549.1931223.53 21089.82 5284.32
Table 3. Parameter estimates for Asia.
GDP Parameter estimates for Asia
Parameter Value Std Error t-value
λ 1.88 1.01 0.31
3The expression representing the time-development of the distribution is:

 

2
22
0
0
2
21
,=
t
t
sue uu
e
ta
fstNee
a










where N is the normalization constant, 2
0
=2 a

and

2
=1
2
t
e




u 1.57 0.02 32.54
u0 1.37 0.02 13.15
σ0 1.06 0.03 10.90
ε 0.58 0.03 9.90

F. HASHEMI
128
Table 4. Paeter estimats for EU. rameevolution of the distribution in the data. The solid and
dotted curves in these figures illustrate the distribution as
predicted by the model and the data respectively.
GDP Parameter estimates for EU
As expected, the mean of the distribution is clearly
evolving for both European and Asian populations, cor-
responding to our theoretical predictions. The variance of
the distribution is likewise evolving for both populations.
Results point out that the distributions are approaching a
long-run equilibrium (at a faster rate in the case of our
Asian economies) and that the dispersion of the distribu-
tions are shrink ing over time (above all in the case of our
European economies). One might speculate that this pres-
sure towards reduction of disparities across European
Parameter t-value Value Std Error
λ 1.79 0.97 0.29
u 1.49 0.02 30.26
u0 1.44 0.02 12.23
σ0 1.11 0.03 10.14
ε 0.61 0.03 9.21
Figure 3. Asia for selected years 1980, 1994, 2007.

129 F. HASHEMI
Figure 4. Europe for selected years 1991, 1999, 2007.
countries is attributed to the establishment of the Maas-
tricht Treaty in 1993. Furthermore, the velocity parame-
ter

e
op
eter
is positive for both regions as expected. This pa-
ramter is stronger for our Asian population than for our
Eurean one. Finally, the value for the diffusion pa-
ram

is small and positive for both regions, like-
wise conforming to our theoretical predictions. The dif-
fusive limit suggests that the distribution variance will
tend toward a constant and concentrated around a mean u,
which is larger for our Asian population than for our
European one.
By considerations of analytical tractability, the model
developed in this paper is simplified. An extension which
would significantly enrich the analysis would be to in-
clude exogenous control variables that shift the condi-
tional mean (e.g., business cycle effects). We also ob-

F. HASHEMI
130
serve that the rate of approach to the final equilibrium as
well as the relaxation time in our model only depend on
Vol. 2, No. 1, 1997, p. 126.
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
characterizing the drift term. This feature is entirely
inherent to the linearity of the dynamics considered, and
turns out to be a limitation in the modeling capability
offered by Ornstein-Uhlenbeck. For nonlinear drifts, this
feature does not occur anymore and the noise strength
strongly affects the transient behavior of the probability
density.
5. Conclusions
Over the past two centuries, the interplay of geography
and policies has produced disparities in economic outcomes
greater than any seen before. Tod
istory, a great majority of the world’s population is
m
[1]
[9] G. Mankiw, D. Romer and D. Weil, “A Contribution to
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conomics,” International Review of
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ay, for the first time in
h
bound together in a glob al system. As a result, large parts
of the developing world are narrowing the income gap
between themselves and richer nations [20]. This has
promoted worldwide prosperity and a long-cherished
hope for convergence between nations at different levels
of econoic development. Monitoring this momentous
development requires appropriate techniques for quanti-
fication and interpretation. This paper proposes one such
technique. The results suggest that diffusion is a potential
method to monitor regional growth dynamics.
6. Acknowledgments
Part of this paper has been built on previous work of the
author [21].
7. References
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[20] J. Sachs, “Nature, Nurture and
12 June 1997. http://www.economist.com/node/91003
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tionary Perspective,” Journal
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doi:10.4236/jssm.2011.43033
C
opyright © 2011 SciRes. JMF

F. HASHEMI 131
ng at x
ty density in the ordinary sense
Appendix A
This is a degenerate diffusion with no Brownian “noise”.
Thus the position at time t of income/output starti
does not have a probabili
but is deterministic. It is important that a differential
equation such as (1) defines a flow on the whole interval

. In other words, a set in

0,0,, for example an
interval

,ab, is transformed into another set, the
interval
 
,,, ,
X
atXbt at time t. Assuming


,
g
xt
is a “reasonable” function, the paths from two distinct
points, namely

,tXat and

,tXbt will never
meet, so an interval remains an interval under this flow.
The flow transforms any initial measure on the interval

0, into a different measure. Suppose the initial mea-
sure is given by a density function

0fx, where we
0
can normalize by taking

0
0d=1

fxx . Then the trans-
formed measure attaches to the interval

,ab the mea-
sure

d



f
xx, where

and

(which depend on t)
re the inverse images of a and b:

t
a
X
b  ,




t
X
a.
Assuming that g is not very
rmed measure also has a density
ensity
peculiar and th
, call the us the trans- transformed fo
d

t
f
y. One can interpret

t
f
y
nk of
for a single co
as the mass
)(
0xf as
untry income/
density at time t; alternatively, if we thi
the initial probability density
output, then

t
f
x is the probability density at time t.
For the mass density interpretation, we may think of

d
t
abfyy as the fraction of all country incomes that
are in the interval

,ab at time t. Our picture of the
world thus follows the lines of the derivation used for
gorov’s forward differentiaation. Taking the
probability interpretation, with the initial density 0(),
Kolmol equ
f
x
let t
Y bX-process, using 0 e the rand omized
f
fo
density of the initial point x. Let

wy be a smooth
r the
function vanis


0,

.
hing off some finite subinterval of
The expected value




Et wY can be expressed as:
  


0,,d=0
 

wyfyttgyfyty y
Since this expression is 0 for all functions


wy of
the type described above, we can conclude that the
bracket {} is identically 0.
Appendix B
Under mild conditions on g, Equatio
have a unique solution. Moreoven (2) is kn r, for each own to
>0 and t
each x,


,
X
xt does have a probability density


,,
f
xyt , and


,,
f
xyt satisfies Kolmogorov’s for-
ward equation:





22
,, =12,,fxyt tfxyty
,, ,
g
yt f xyty
  



where x is fixed, indicating the starting point.


,,0
f
xy
entrated at x,
r >0t
e. Fo
is a “Dirac” function, with all the mass conc
and so is not a density in the ordinary sens,
one has Prob d
  
a, ,,
b
a
X
xtbfx yty
ose x is initially random.
. We may
again supp Multiplying this
equation by 0(),
f
x integrating over

0, and letting
 
0
,,, d=
f
ytf xytfxx

 ,
we get:






 
22
,=12 ,
,,
fyt tfyty
g
yt fy



 

yt



To show that the solution of this equation with =0

is the limit of the solution as 0

, see Ref. [14,
Chapter 4].