Theoretical Economics Letters, 2011, 1, 33-37
doi:10.4236/tel.[phone] Published Online August 2011 (http://www.scirp.org/journal/tel)
Dynamics of Income Distribution—A Diffusion Analysis
Fariba Hashemi
Swiss Federal Institute of Technology, Lausanne, Switzerland
E-mail Fariba.Hashemi@epfl.ch
Received June 14, 2011; revised July 19, 2011; accepted August 18, 2011
Abstract
The study is motivated by the observation that the distribution of income across countries varies as a function of
time. It would not be unreasonable to assume that there exists a statistical equilibrium distribution of income
with a certain mean and variance, towards which the ensemble of countries considered tend to converge, and
there is a speed of adjustment towards this said equilibrium. In order to quantify this process, the evolution
through time of income around its trend is modeled using a classic stochastic differential equation. The model
describes the diffusion of shocks across space, via an income adjustment process with noise. The dynamics rely
on two opposing flows: (i) a factor equalization process, and (ii) a counteracting diffusion process. It is hypothe-
sized that these flows follow simple evolutionary laws that can be described with five parameters—parameters
that can be estimated from historical data with some accuracy. The dynamic behavior of the model is analytically
derived. Both the extent and speed of adjustment of income are analyzed. An empirical application of the pro-
posed model to the evolution of the distribution of income for 25 countries in the European Union tests the va-
lidity of the proposed method and suggests that diffusion may be a preferable technique for the analysis of in-
come dynamics.
Keywords: Cross-sectional Distribution of Income, Diffusion
1. Introduction
While much attention has been devoted in the economics
literature to the explanation of the shape of income distri-
bution at a given point in time by reference to steady state
arguments [1-3], the dynamics in question have been rela-
tively ignored. The main objective of this paper is to help
fill this gap by introducing a diffusion model that allows to
incorporate dynamics which are typically neglected when
looking at convergence of incomes. Our picture of world
development is one where some economic forces push in
the direction of convergence, whilst other forces are diver-
gent. Consistent with this observation, we explore a repre-
sentation for the dynamics in the evolving distributions
where the growth distribution of incomes can be generated
by a single stochastic process in which income follows a
Brownian motion. An empirical application to personal
income for 25 countries in the European Union helps fill a
second gap in the literature, as only few diffusion studies
have employed real statistical data when analyzing income
dynamics.
2. Theoretical Framework
Consider a region consisting of a constant number of
countries with different levels of personal income (wages).
The set of personal incomes forms a distribution which
evolves over time. Responding to regional wage differen-
tials, labor (capital) migrates from low-wage (high-wage)
to high-wage (low-wage) states within the region. As the
capital/labor ratio declines in the high- wage areas, wage
growth decelerates, while as the capital/labor ratio rises in
the low-wage areas, growth of wages accelerates. Richer
states start out with higher capital-to-labor ratios; however,
for given savings rates, diminishing returns to capital set
in faster in these economies. This process is reinforced as
technology and knowledge capital flow from rich to poor
states1 [4]. A counteracting force exists, in the form of
economic, political and institutional blockages and faster
population growth which cause bottlenecks and generate
divergence in the system2 [5,6]. In view of the above, it is
assumed that there exists an equilibrium distribution of
personal income with a certain unknown mean and vari-
1Convergence thus, is a result of adjustment of capital-labor ratios to
common steady-state levels, starting from different initial values.
Globalisation is typically presumed to reinforce factor-price equaliza-
tion and the resulting convergent trend, through international trade,
international migration, the flow of capital towards capital poor econ-
omies and technology transfer.
2It is worth recalling that mobility of labor and capital may be impeded
by the presence of consumption and production externalities.

F. HASHEMI 34
ance, determined by the tension between counteracting
forces of convergence (migration of labor and flow of
ideas) and divergence (bottlenecks to flow of labor, capital
and ideas). Given an exogenous shock, the ensemble of
country incomes considered tend to converge to this
long-run equilibrium3.
3. The Model
A classic attempt to describe the distribution of income is
due to Gibrat [7], where it is proposed that income is
governed by multiplicative random processes, with luck
having a permanent effect4. It would not be unreasonable
however, to assume that due to competition amongst sup-
pliers and demanders of labor, labor which is paid an un-
reasonably low wage would search and find a better-paid
job, and labor who is paid an unreasonably high wage,
would be pressured to accept a lower wage, or risk getting
replaced by someone who is willing to accept the lower
wage. Thus, one might include a drift term in the model,
representing mean reversion In general, one can study a
Markov process generated by a matrix of transitions from
one income to another, where the Markov process can be
treated as income diffusion. Then one can apply the gen-
eral Fokker-Planck equation to describe evolution in time
of personal income. Hence, assuming that personal in-
come behaves like a stochastic process and that it is con-
tinuous and Markovian, we consider the most natural
candidate; a classical linear stochastic differential equa-
tion driven by Gaussian white noise:

2
ttt
dSuS ddB   t  (1)
where St is personal income.

denotes velocity of ad-
justment to stationary equilibrium interpreted as income
adjustment rate5, u denotes the mean of the stationary
equilibrium distribution, 0

 is a constant diffusion
parameter, and Bt is the Brownian motion.
More precisely, for the drift spread, it is assumed that
there exists some equilibrium distribution of wages with a
certain mean and variance, towards which the ensemble
of agents gravitate. Drift is driven by diminishing returns
to capital, which in turn leads to factor price equalization.
For the diffusion spread, noise is generated by diffusion
of knowledge and learning [10], [11], and limited by the
presence of obstacles in the form of trade barriers and
such. Random effects cause a spread of personal income
from high density towards lower density. This happens as
a result of learning and the phenomenon of catch up,
themselves due to economic integration [12]. The income
adjustment process is thus interpreted as depending on
learning speed which in turn is proportional to the mobil-
ity of factors of production and the speed with which di-
minishing returns set in6. This learning process generates
randomness in the system [13].
3.1. Analysis of the Model
The second order partial differential equation associ-
ated with equation (1) can be expressed by:
 
2
2
f
f
usf
ts
s




  (2)
where f denotes probability density and s denotes per-
sonal income7.
The time-development of the distribution can be ex-
pressed by:


2
2
2
,t
s
ut
ta
fst Nee
a





 (3)
where
2
0
2
a



21
2
t
e









0
1tt
t
utEfueu e

 


2
222
01t
t
tee



 



0
urepresents the initial mean of personal income distri-
bution and 2
0

represents the initial variance of the dis-
tribution and N is the normalization constant8.
4. Empirical Application
4.1. Data and Descriptive Statistics
Our data consists of labor compensation per hour of
6Convergence in the context of our model would mean collapsing o
f
the cross-sectional distribution. Divergence would mean that the
cross-section distribution is not collapsing, but replicates itself because
for example it happens to be the stationary distribution for many inde-
p
endent and identically-distributed country incomes.
7The process derived from the diffusion model evolves according to an
Ornstein-Uhlenbeck, but with a transition, such that the mean tends to
u , instead of 0. This type of model has been widely used in Biomathe-
matics [14], [15], [16], [17], [18] and [19].
8By considerations of analytic tractability, the initial distribution is
approximated to be normal.
3Equilibrium in this paper refers to a statistical equili
b
rium, which is
characterized by a stationary probability distribution of personal in-
comes. This equilibrium can also be associated with level of personal
income which is in line with potential incomes and outputwhich in
themselves are due to natural, technological and institutional con-
straints.
4See [8] and [9] for a review.
5For simplicity we assume this rate to be constant.

F. HASHEMI35
work ($US PPP adjusted)9, for 25 countries in the Euro-
pean Union, recorded from 1976 up to 2007. Data was
compiled from Eurostat. The countries are: Austria, Bel-
gium, Bulgaria, Cyprus, Czech Republic, Denmark, Es-
tonia, Finland, France, Germany, Greece, Hungary, Ire-
land, Italy, Latvia, Lithuania, Luxembourg, Netherlands,
Poland, Portugal, Slovakia, Slovenia, Spain, Sweden,
United Kingdom. Table 1 shows the descriptive of
wages per hour of work for the N = 32 years of data from
1976-2007.
4.2. Estimation
A second order partial differential equation model has
been proposed to express the dynamics of personal in-
come. The model has five parameters: , u,
0
u

, 2
0

,
and

. denotes the initial mean of the distribution,
and u denotes where the initial mean is heading.
0
u
0

is
the standard deviation at time zero,

represents the
diffusion parameter, and

determines the income ad-
justment rate. To examine the behavior of the model as
t→∞, one observes that

,

f
st →

f
s
where



2
2
2
,
su
a
f
sttNefs





 
 (4)
and that


lim t
tEf u  
The model has been applied to the wage distribution of
the population as a function of time. Tables 2 reports
estimates for the five model parameters , u,
0
u

, 2
0

,
and

, along with the standard errors and t-values, us-
ing the first and second moments of the distribution.
To ascertain whether the parameter estimates conform
to the real data presented in the descriptive analysis, Fig-
ure 1 illustrates the actual versus predicted time plot that
can be generated from the real and estimated data.
Table 1. Descriptive analysis.
N Minimum Maximum Mean Std
Deviation
Wage 32 9.326 34.138 22.416 6.711
Table 2. Parameter estimates.
Parameter Value Std Error t-value
λ 1.46 1.12 0.35
u 1.27 0.02 37.22
u0 1.06 0.02 15.04
σ0 0.86 0.03 12.47
ε 0.45 0.03 11.32
Figure 1. Actual vs. predicted distribution.
Figure 1 graphically illustrates the evolution of the
density over time, superimposed on histograms which
describe the time evolution of the distribution in the data
(for selected years 1976, 1991, and 2007). The solid and
dotted curves in this Figure illustrate the distributions as
predicted by the model and data respectively. The verti-
cal axes denote frequency, and the horizontal axes meas-
ure wages in logarithms.
The following observations can be made concerning
our results:
1). The mean and variance of the distribution is clearly
9Controlling for inflation could produce different results.

F. HASHEMI 36
evolving, corresponding to our theoretical predictions.
2). The value for the income adjustment rate

is
positive as expected.
3). The value for the diffusion parameter

is small
and positive as expected.
4). The diffusive limit, i.e., the limit as of the
variance is:
t
2
lim t
t

 
 The results predict that if we
start with a normal distribution and let the model drive the
distribution, the distribution variance will tend toward a
constant

 and concentrated around a mean u.
5. Final Remarks
A methodology has been proposed which is a more
transparent way to quantify the dynamics of income, as it
avoids the complications associated with dynamic infer-
ence and statistical regression fallacy inherent in stan-
dard cross-section tests [20-22]. The present study is in
the spirit of probabilistic models of Krugman [23] who
studies city sizes, Axtell [24] and Hashemi [25,26] who
study firm sizes, and Hashemi [27,28] who studies in-
come and rate of unemployment. Our suggestion is that
these models could provide interesting insights as well, if
applied to spatial dynamics of personal income.
One fruitful extension of the present study would be to
relax the homogeneity assumptions of the model pa-
rameters. Another interesting extension would be to ex-
amine if the driving forces of convergence and diver-
gence are the same for other components of income such
as profit, interest and rent. We hope the present study
illustrates that the endeavor is promising.
6. Acknowledgments
The author has benefited from discussions with Minimax
consulting on the statistical analysis in this paper.
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