Theoretical Economics Letters, 2013, 3, 245-250
http://dx.doi.org/10.4236/tel.[phone] Published Online August 2013 (http://www.scirp.org/journal/tel)
A New Analytical Model for the Analysis of Economic
Processes
Paolo Di Sia
Faculty of Economics and Management and Faculty of Education, Free University of Bozen-Bolzano,
Bolzano, Italy
Received May 29, 2013; revised June 29, 2013; accepted July 16, 2013
Copyright © 2013 Paolo Di Sia. This is an open access article distributed under the Creative Commons Attribution License, which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
ABSTRACT
In this work it is presented a new powerful model, well checked in the last years with the analysis of the dynamics of
nano-bio-structures. The model permits the detailed study of the dynamical evolution of systems also at macro-level,
offering analytical relations concerning the velocity and the distance of an economic variable, so as its diffusion in time;
it permits the analysis of economic processes in general. In this paper it will be focused in particular about economic
cycles.
Keywords: Economic Processes; Mathematical Modelling; Diffusion; Dynamics; Econophysics
1. Introduction
Theories of economic cycles regard capitalist economic
systems and are characterized by a not uniform devel-
opment, but rather intrinsically marked by oscillations.
Fluctuations in pre-capitalist systems were due to exter-
nal factors to the economic sphere, such as the harvest
variability, the consequences of wars or epidemics, and
so on. The expansion of the capitalist market and the
socio-economic and institutional resulted changes had
the effect of a significant change of the oscillation char-
acteristics. The cyclic evolution was determined in in-
creasingly way by endogenous factors and peculiar pro-
pagation mechanisms to the capitalist system; the nature
of the cyclic processes changed so in essential manner
[1].
This new type of oscillation could be due by the de-
velopment of the banking system and by its ability to
create credit. The treatment of models of cyclic devel-
opment requires the use of complex mathematical tech-
niques for analyzing the modifications of structural na-
ture, which characterize the processes of development;
often they occur in slow but continuous changes and
produce variations, that, cumulatively, become increas-
ingly relevant in relation to the crisis. The cyclic trends
are able to exert a strong influence on the short-term dy-
namics, but they can with the same speed change direc-
tion. The integrated analysis of these differing move-
ments is complicated by the need to take into account not
only of the trends of strictly economic variables, but also
of social, political and institutional variables. The estab-
lished inter-relationships between all these variables
could not be neglected when we want to go beyond the
horizon of short-term time periods [2]. The articulated,
but productive way of interaction between economics
and physics reflects the extensive reality of the mutual
influence of various disciplines through the complexity
theory and other trans-disciplinary theories, such as the
non-linear dynamics, the game theory and other mathe-
matically sophisticated approaches [3].
2. Indicators and Cycle Characteristics
One of the most widely used economic indic ato r s is g iv en
by the performance of gross domestic product (GDP) in
real terms, which succinctly measures the fluctuations in
production at different dates. It depends on many vari-
ables, as the final consumptions, the statal spending,
investments, imports, exports, etc [4,5].
Other indicators of partial phenomena are the trend of
production in the different sectors, levels of employment,
investments, monetary aggregates and credit, applica-
tions for specific products, consumption of raw materials,
prices, profits (Figure 1).
The performances of the different indicators are not
identical, but in many cases quite similar. There is a re-
markable agreement of different indicators in signaling
periods of acceleration and slowing of production.
C
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P. DI SIA 246
Figure 1. Gross domestic product of five major economies
of the European community since 1991 (In million Euro (or
ECU), official data from EUROSTAT) [6].
The study of these changes can be crucial to explain
the relationship between cycles and development [7].
The trend of the time series is usually rough, it is not
easy to recognize cyclic regularity. It needs a work of
cleaning the raw data, firstly to eliminate those compo-
nents that have a cyclic explanation as purely seasonal
(for example the decline of industrial production in Au-
gust) or accidental (as monthly variations of production
caused by the different number of days working), and
then also to eliminate patterns requiring a special “ad
hoc” explanation, determined by important but unique
events, as the effects of war, natural disasters [8,9]. The
relationship between cycle and growth is a phenomenon
coexisting in the representation that the empirical data
provide. The majority of econo mists decided to solve the
problem by making a clear separation of the two phe-
nomena. The statistical methods used for this purpose
differ in details, but in essence all consist in the evalua-
tion of a trend, which should give an account of growth,
and in the calculation of variances, that the considered
time series show against the trend. An example, with a
linear trend, is shown in Figure 2.
A substantially always increasing trend can be de-
composed into two components: one showing a uniform
growth for a long period of time and the other that in-
stead oscillates, assuming positive or negative values in
relation on whether of the considered variable overtakes
the long term trend. Normally it is assumed that the trend
represents the performance of system equilibrium and
that cyclical deviations are due to deviations. The trend
calculation is of necessity made on the basis of data of
the past. But the present and the future may have very
different characteristics from those of the past, especially
if there are modifications of structural character, which is
more probable as longer is the period of data observation
[10].
Figure 2. Possible variations of a linear trend.
Looking at the profile of the deviations, the identifi-
cation of the phases, that usually describe the cycle, be-
comes easy: depression, resumption, boom, recession
(Figure 3).
The amplitude of booms and depression of historical
cycles had not always comparable size, so as their dura-
tion was also highly variable. Therefore, some modern
theorists begun to doubt th e very ex isten ce of phenomena,
that can be properly called cyclic [11]. Every historical
cycle has its own characteristics because it takes place in
different and changing socio-economic situations and is
influenced by a different measure extent with endoge-
nous (economic) and external (non-economic) factors
[12-14]. The various indicators begin to move with
greater velocity, the recovery spreads and becomes boom.
It grows the production, sales, employment, real income,
used production capacities, app lications for raw materials
and finished produc ts, the quantity of money, its velocity
of circulation, the amount of credit and loans to economy.
However the boom does not last indefinitely, economic
activity first begins to decelerate and then to contract.
3. Theoretical Explanations of Cycles
The reflection on the causes and mechanisms that gener-
ate cyclic movements has a long history. Many of the
theories of the cycle, that dominate the scene, have still
now their antecedents in formulations which have ap-
peared in the last century or the beginning of it. The fol-
lowing treatments have benefited of a more rigorous
analytical framework and a much larger amount of em-
pirical knowledge.
The first explanation of commercial crises is traced to
the effects of exogenous factors: the variability of crops
wars, exploitation of new gold deposits at the opening or
closing of foreign markets, but also technical innovation
(Lord Overstone in 1837 and Thomas Tooke in 1838)
[15,16]. In the last century, the crises assumed often fi-
nancial characteristics. Already Overstone and Tooke
emphasized how the fluctuations of credit counted for

P. DI SIA 247
Figure 3. The various levels of a displacement by a trend.
much in exalting the phases of expansion and in aggra-
vating that of contraction.
Later, John Stuart Mill pointed out the influence of
speculation and the forecast errors of the operators that
caused irrational fluctuations in the demand for credit,
which, consequently, had the effect of enhancing the
cyclic phenomena. At the beginning of the last century,
the different theories explained the crisis in terms of un-
der-consumption and savings excess [17].
It has seen that, in the case of damped os cillations, the
permanence of the cycle time can be ensured by a suc-
cession of irregular, random nature shocks. Theories
based on the interaction between multiplier and accel-
erator did not give exclusive importance on this fact [18].
On the other hand, the empirical investigations seem to
suggest that the values of the parameters of the model are
such as to give rise to a solu tion of “explosive” type, i.e.
to oscillations or amplified in exponential growth. The
most well-known theories have recognized this pecu-
liarity, but have introduced upper and lower limits, which
block the divergent trends of the system bouncing in the
opposite directions [19].
Explosives trends occur at high values of the accelera-
tor. With this increase in demand (exog enous), it follows
a significant increase in investment and thus a rapid
growth. The boom is self-fueled and the system is pushed
towards the upper limit b y full capacity utilization (espe-
cially in sectors producing capital goods) or by the full
employment of labor (with the required characteristics).
The introduction of lower and upper limits allowed for a
greater realism of the theory and has eliminated some
important deficiencies. Richard M. Goodwin introduced
the ideas of ceilings and floors [20], John R. Hicks an
exogenous trend of growth for individual investments
[21]. The introduction of lower and upper limits made
non-linear the original model; the accelerator does not
work as before when limits are reached. This is the ad-
vantage of the model, but also a limit, because at linear
level cycle disappears or becomes explosive.
4. A New Derived Model
A recent theoretical analytical formulation showed to fit
very well experimental scientific data and offers interest-
ing new predictions of various peculiarities in nanos-
tructures [22-24].
The new model was utilized at today at scale length of
order of nanometers. But the de Broglie thermal wave-
length refers to the dual nature of reality. Associating th e
wavelength to the impulse through the de Broglie re-
lation, it is possible to define a thermal wavelength t

for every object at macrolevel via the relation:
3
t
hmkT  B (1)
In Equation (1), h is the Planck constant, kB the Bolt-
zmann’s constant, m and T mass and temperature of the
considered system respectively. With this gauge factor it
is so possible to study the dynamics of reality processes
presenting oscillations in time, so as characteristics of
diffusivity in time.
The model offers the analytical formulation of the
most important quantities related to the dynamics of a
generic proces s , i.e.:
a) The velocities correlation function:




0T tvv (2)
at temperature T, from which the velocity v(t) is obtain-
able;
b) The mean squared deviation of position:
 
2
20Rt t




RR (3)
from which the position vector at the time t is
obtainable;

Rt
c) The time-dependent diffusion D(t) of the system:




2
12 ddDtR tt (4)
Equations (2)-(4) are fundamental in deducing the
most important characteristics concerning the transport
phenomena of processes.
The roots of the model lie in the linear response theory
and follow the standard time-dependent approach; the
conductivity



 is in general a complex function of
the frequency

and can be deduced from linear re-
sponse theory. The model is based on a complete Fourier
transform of the frequency-dependent complex conduc-
tivity



 of the system; this is in general a complex
function of the frequency

, which can be deduced
from linear response theory (Green-Kubo formula):




2
00
de d0
it
eV tti

 

 



vv (5)
By inversion of Equation (5), it is possible to find the

P. DI SIA 248
velocities correlation function




0T inside the
integral. The presence of an integration from 0 to  is
however a problem for the analytical inversion, but it can
be overcame evaluating the integral on the entire time
axis (, +). Considering the real part of the complex
conductivity in Equation (5), the extension to the entire
time axis is possible and a complete Fourier transform
can be performed, obtaining directly real velocities. The
integral can be resolved in the complex plane considering
a Cauchy integration; as a result, the velocities correla-
tion function can be evaluated exactly by the residue
theorem [25]. After this step, it is possible to obtain the
analytical form of the mean square deviation of position
(Equation (3)) and of the diffusion coefficient D(t)
(Equation (4)).
tvv
The deduced results are, with
x
t

:
 
   

0exp
cos2 1sin 2
RRR
tKTm x2
x
x 
 

vv (6)


  

  

22
0
21sin2exp
cos2exp2 1
RR
R
Rt KTmxx
xx




2
(7)
 

  

2
24sin 2exp2
RR
DtKT mxx   


(8)
Equations (6)-(8) are related to a first model constant
22
0
41
R
 ;
R

is a real non-negative number. In
the same way the model contains three analytical rela-
tions related to the second model constant
22
0
14
I

  ;


0,1
I
  and real. One of the most
interesting peculiarities of the model is the “time do-
main” used approach, not previously found in such a
contest, contrarily to the existing theoretical approaches
of literature, which are “frequency domain” treatments
and/or numerical methods [22-24].
The model was tested with the most utilized nanoma-
terials at today and describes transport properties of
nano- and bio-materials [26-33], including previous im-
portant models [34] and offering new interesting peculi-
arities, still not experimentally found.
The analytical form of previous cited relations is com-
posed by a superposition of exponentials and/or products
among exponentials and sinusoidal (cosinusoidal) func-
tions, giving rise to trends as indicated in Figures 4-6.
The model is so related to variables as the temperature,
the energy and the mass, the weight of the various states
(at quantum level), which, “mutatis mutandis”, are very
important in a lot of economic processes too, from eco-
nomic geography and gravity models for economy, to
economic behaviours treated with game theory [35], eco-
Figure 4. Velocities correlation function vs t

for two
values of the parameter of the model R

22
0
41. We
note clear damped oscillations in time [22-24]. R

10

→
red line; αR = 30 → green line.
Figure 5. Positive and negative deviation by a linear trend
(represented in this case by t-axis) for some values of the
parameter of the model I

 
22
0
14 [22-24]. αI = 0.1
→ red line; αI = 0.5 → blue line; αI = 0.9 → green line.
Figure 6. Behaviour of the diffusion D for I

varying in
its definition interval [22-24], with τ = 10−12 s. Dots with
error bars represent experimental data. αI = 0.25 → red line;
αI = 0.75 → blue line; αI = 0.85 → green line; αI = 0.95 →
violet line.
nomic analogies to thermodynamics and chaos in busi-
ness cycle theories [36,37].
The relations between physical and economic vari-
ables have been in the past always of great interest; as
above introduced, we can think to gravity models of
trade in international economics, so as in social science,
where the physical variables are related to correspondent
economic variables, as F to the trade flow, M to the eco-
nomic mass of each country, R to the distance. Gravity

P. DI SIA 249
models have also been used in international relations for
the evaluation of the impact of treaties and alliances on
trade, so as for testing the effectiveness of trade agree-
ments and organizations.
For all these reasons, the indicated model can effec-
tively be an interesting technical tool for the implemen-
tation of economic data and the investigation of new
economic peculiarities.
5. Conclusion
The strength of the new derived model consists of its
ability to fit very well experimental data, accommodating
not completely understood behaviours and including pre-
vious models, like the Smith model. The possibility to
wor k a t macrolevel through the use of the gauge factor ( 1 )
allows interesting applications in any sector in which
velocities, distancies, oscillations and diffusion are in-
volved. The started tests with economic data are giving
favorable evidence of an interesting assistance of this
model in the interpretation and comprehension of many
phenomena at economic level.
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