Probabilistic independence and conditioning in economic mathematical
modelling
Prof. Stelian STANCU PhD
The Bucharest University of Economic Studies, ROMANIA
Department of Economic Informatics and Cybernetics
[email protected]; [email protected]
Alexandra Maria CONSTANTIN MSc. Student
The Bucharest University of Economic Studies, ROMANIA
[email protected]
Abstract: Probabilistic independence and conditioning in mathematical modeling must be associated to the
complexity of phenomenon and interdependent relations between cause and effect type components, at a
phenomenon level. Thus, the current paper has multiple specific objectives: presenting the complexity of
phenomenons, presenting the independence and probabilistic conditioning in mathematical modelling in
economics, presenting a mathematical model to be used when studying economic phenomenon on a microeconomic status and last, presenting the conclusions.
1. Introduction
Often times man has the impression that he knows the ins and outs of the reality we live in.
This is wrong, as each element of reality constitutes, in and of itself, a complex system, which
cannot be fully “characterized”. Each element of reality is part of a, or multiple, phenomenons.
Phenomenons, as defined by Plato1, are “variables spread through time and space, always different
and always transforming”. For this reason, the phenomenon must be analysed through its relations
with other phenomena from the surrounding reality. Phenomena analysis must be based on studying
the degree of dependency or of the conditioning mode with regards to other phenomena, and, at the
same time, the degree of interdependency between it and other phenomena.
Studying the relations between phenomena can be achieved by quantitatively measuring the
relations between the cause and effect components of these phenomena. Mathematical modelling
can be considered the best “instrument” for quantitative study of the interdependencies between the
constituting elements of the studied phenomena. In other words, mathematical modelling implies
formalizing relations between the components being considered, which is to say putting them in
correspondence to one another, thus establishing the cause and effect components. Thus, through
mathematical modelling we try to offer a rigorous image of a hidden reality, behind these
interdependencies.
2. Probabilistic independence and conditioning in mathematical modelling
In order to better understand probabilistic independence and conditioning in mathematical
modelling, we must approach the notion connected with the basic element, which is the variable. As
mentioned before, the variable is the central component of a phenomenon and it highlights the
1
http://ro.wikipedia.org/wiki/Fenomen
change or possibility of quantitative change of the phenomenon, which is to say it is the
“measurable” part of the phenomenon. The existence of a variable implies multiple criteria: the
existence of a phenomenon, the explained phenomenon must have a measurable characteristic and,
also, the quantitative part must have the ability to change or define differences (for example:
between men and women). These criteria, regarding constituting a variable, can be illustrated
through a simple example: measuring unemployment rate across time. This example is a variable, as
there is an (economic) phenomenon being measured – unemployment. The characteristic of the
phenomenon being considered, unemployment, is given by unemployment rate, which can take
different values throughout time. In research more accent is put on the degree of interaction of
variables. For example, one could study which of the multiple social, psychological or economic
variables correlate with a growth of homicide.
To constitute the mathematical model it is important to know which of the variables are
dependent and which are independent, as well as which are causal and which are effective. In this
sense, Kalof, Dan and Dietz propose a strategy in identifying independent from dependent
variables. They say that2: “if a variable describes things that appear before the things described by
another variable happen, then the first variable can thus be usually taken as the independent
variable, and the second variable as a dependent variable”. To illustrate these said by Kalof, Dan
and Dietz we will take the following example: increased interest rates (independent variable)
contribute to growth of suicide and theft (dependent variables). From here we may deduce their
outcome: people become unemployed, fall into depression and commit suicide or steal to survive.
Studying causality is important especially when we analyse the correlation between two or
more variables, which is to say it explains many other phenomena which are responsible for the
existence and the variation of the phenomenon explained. We will illustrate this through an
example: the variation of delinquency can be explained through the invocation of multiple
determinant factors: the economic state of that collectivity, the efficiency of social control and
integration mechanisms, opportunities for delinquency, etc. Each of these factors are a cause of
delinquency: the variation of these factors produces a variety of delinquency. Taking these things
into account, researchers are interested, more and more, by making predictions rather than studying
causality connections.
In order to construct a mathematical model that is concise and rigorous we must take into
account the problem of probabilistic conditioning of the random variables being considered (two or
more), discreet or continuous, in order to explain in probabilistic terms the appearance or evolution
of a phenomenon that is conditioned by the existence of another phenomenon or phenomena. In the
Theory of Probability, the connection between two or more random variables is studied through the
correlation coefficient. Next, we will briefly approach the relation, the link (the conditioning),
between random variables.
Taking into account that random variables can be discreet or continuous, we will attempt to
differentiate between the two.
The random variable is a magnitude, that in the event a process is achieved, can take a value
from a well-defined set, named the set of possible variables. If this set is discreet, then we are
looking at a discreet variable, and if the set is an interval (finite or infinite) the random variable is
continuous.
2
Kalof, L., Dan, A., Dietz, T., Essentials of social research, Berkshire, England: Open University, 2008, pag. 36.
In order to calculate the correlation coefficient, we must calculate3 the mean of the product
of the two variables, the mean for each variable and the dispersion for each variable. The formula of
the
correlation
coefficient
for
the
two
random
variables
is:
(X ,Y ) 
cov( X , Y )
D( X ) D(Y )

M ( X  Y )  M ( X )  M (Y )
D( X f (t )) D(Y f (t ))
, where M ( X f (t )Y f (t )) is the mean of the product,
M ( X f (t )) and M ( X f (t )) are the means of the variables X f (t ) and Y f (t ) respectively, and
D( X f (t )) and D(Y f (t )) are the dispersions of the variables X f (t ) and Y f (t ) respectively.
If  ( X , Y )  0 then the variables are not correlated, which means that the change in status of
one phenomenon does not affect the other. Still, we cannot say if the two random variables X f (t )
and Y f (t ) , associated with the two phenomena, are or not independent. Thus, in order to study their
independence it is necessary that we check if p X Y  p X  pY , where p X and p Y are the
f
f
f
f
f
f
probabilities corresponding to the considered variables, X f (t ) and Y f (t ) . In this case we may say
that the two random variables are independent, and if not, that they are dependent.
3. Example of a mathematical model for the study of economic phenomena on a
microeconomic level
Starting from the complexity of the phenomenon to studying it in rapport with other
phenomena, we can build a multitude of mathematical models taking into account the independence
and probabilistic conditioning of phenomena.
Of course, we can create mathematical models in the economic field, both on a micro and
macro level. Henceforth, we will give an example of mathematical modelling in studying economic
phenomena on a micro level, in an enterprise.
Thus, taking into account the process of bread fabrication, we will develop a mathematical
model to determine production costs based on production factors frequently used in the fabrication
process, at different points in time. Developing the model implies developing the production cost
random variable. Thus, taking into account the process of bread fabrication, we will develop a
mathematical model to determine production costs based on production factors frequently used in
the fabrication process, at different points in time. Developing the model implies developing the
production cost random variable.
Varying components of production costs are the actions undertaken in production: mixing
the dough, separating and fermenting, baking the bread, etc. In this way, the enterprise, through
production, transforms4 production factors (water, flower, yeast, electric energy, machines, etc.)
into the economic good: bread.
When modelling production costs we will introduce time intervals, noted with I (t ), t  1, T ,
considering T finite. For the duration of a fixed time interval, I (t ) , f (t ) take place, production
acts, where, f  {1,2,3,..., M } , M is finite, and t  1, T .
Events that we will “annex” are: production factor, quantity and price of the production
factor.
3
4
For ease of approach we will take into consideration two random variables.
In this context, the enterprise may be viewed as a cybernetic system.
The production process implies using n production factors, D ni (t ) , for each production act
f (t ) , where: f  {1,2,3,..., M } , M is finite, t  1, T is the time frame analysed and n  1,2,..., N are
the production factors used in the production act f (t ) .
The production factor thus depends on5 quantity and price, or Dnf (t )  X nf t , Ynf (t ) . Also
worth mentioning is that X nf (t )  is the random variable associated to price, named price, and Ynf (t )
is the random variable associated to quantity, named from here on quantity.
The quantities of production factors depend on the market demand for the produced good
and on regulations or decisions regarding production factors. In the time interval analysed, I (t ) , in
the production act, f (t ) , t  1, T we assume the undertaking of a singular production factor (n=1),
which is why the discreet random variable Ynf (t ) can also be written as Y f (i )
Also, the production factor consumed in the interval I (t ) of the production act f (t ) is bought
at different prices, which are, at the same time, the prices with which the factor enters the
production process. Price repoartition is considered known.
Starting from notions in the Theory of Probability and taking into account a production
T
factor with the two specific elements, quantity and time in a given time frame I, where I   I (t ) ,
t 1
defines 6 the production cost random variable.
The product between price and quantity, which is to say between X f t  and Y f t  , of a
production factor utilized in the act of production f t , f  {12,3,..., M }, t  1, T , in the time frame
I t  is called production costs, written C f t  and is defined: C f t   X f t   Y f t  .
Taking , K, P  a field of probability and Y f t  , X f t  , f  {1,2,..., M }, t  1, T , the quantity
variables, respectively price X f t  , Xx f t , Y f t  :   R . Thus, X f t   Yf t  is a random variable.
Henceforth,
the
repartition
of
the
production
cost
random
variable
is:
 x sf (t )  y qf (t ) 

,where p rqf (t )  P{ | X f (t )( )  xrf (t )  Y f (t )  y qf },
C f t  : 
 p (t )

 sqf
 ( s , q )(1,..., S )(1,...,Q )
f  {1,2,..., M }, t  1, T .
The probabilities prqf (t ) in the repartition are known, in the Theory of Probabilities, as
marginal probabilities with the help if which we can deteremine dependency or indepedency of the
marginal variables, Y f t  and X f t  , t  1, T .
For the model to be viable we must analyse the link or conditioning between the two
variables
with
the
help
of
the
correlation
coefficient.
Thus,
we
get:
 ( X f (t ), Y f (t )) 
cov( X f (t ), Y f (t ))
D( X f (t )) D(Y f (t ))

M ( X f (t )Y f (t ))  M ( X f (t ))  M (Y f (t ))
D( X f (t )) D(Y f (t ))
, where M ( X f (t )Y f (t ))
is the mean of the product, M ( X f (t )) and M ( X f (t )) are the means of the variables X f (t ) and
Y f (t ) , and D( X f (t )) and D(Y f (t )) are the dispersion of variables X f (t ) , and Y f (t ) .
5
http://www.economicswebinstitute.org/glossary/costs.htm
Damian, D., Studiul Analizei variaţiei şi modele statistice în studiul fenomenelor aleatoare, Teză de doctorat,
Universitatea "Transilvania", Braşov 2010.
6
When
prqf (t )  P{ | X f (t )( )  xrf (t )  Y f (t )  y qf }, f  {1,2,..., M }, t  1, T
and
 ( X f (t ), Y f (t ))  0 , the variables are uncorrelated and the variation of price does not depend on the
required amount, and thus the two variables are independent. Otherwise, there can be positive or
negative correlation.
4. Conclusions
In conclusion, studying the probabilistic conditioning and interdependency between
analysed phenomena plays a major role in developing a complex mathematical mode, that mirrors
as concisely as possible the links between hidden elements, existent outside human consciousness
and independent of it.
Also, the complexity of the model must take into account the equilibrium between simplicity
and its presentation. In this sense, Occam7 says: “between the models that have the same
representation and predictive power, it is recommended that one choose the simplest one”. From
here we must understand that the model must be simple but not simplistic, and as we increase the
complexity of the model we must take into account several aspects: tracking the growth of its
representation, its ability to expose reality in a realistic way.
On the other hand, we must not ignore other aspects of mathematical modelling, such as:
difficulties in model analysis, software problems regarding the size of the model developed, as well
as conditioning between component variables.
Bibliography
1. Kalof, L., Dan, A., Dietz, T., Essentials of social research, Berkshire, England: Open University,
2008, pag. 36.
2. Damian, D., Studiul Analizei variaţiei şi modele statistice în studiul fenomenelor aleatoare, Teză
de doctorat, Universitatea "Transilvania", Braşov 2010.
3. Teilhard de Chardin, P., Christinity and Evolution, Harvest/HBJ 2002.
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4. http://ro.wikipedia.org/wiki/Fenomen
5. http://www.economicswebinstitute.org/glossary/costs.htm
6. http://en.wikipedia.org/wiki/Occam's_razor
7
http://en.wikipedia.org/wiki/Occam's_razor