**Rafael Pla-Lopez**

Departament de Matemàtica
Aplicada

Universitat de València

Spain

mailto:Rafael.Pla@uv.es

http://www.uv.es/~pla

**Abstract**

Surplus/deficit of the balance of payment of each country is a main problem in the present world. The actions to overcome it repercusses on the whole economical processes of the society, and specifically on the employment.

We worked in a collective study sponsored by *Generalitat
Valenciana* through *Conselleria de Cultura, Educació i Ciència*
inside the framework of *Pla Valencià de Ciència i Tecnologia*
(Project GV97-TI-D5-20) to simulate the evolution of unemployment through
different approaches [Caselles, 1996; Caselles
*et
al.*, 1999a; Caselles *et al.*, 1999b].
I specifically worked in a minimalist approach of successive disaggregation
of variables to build series of models of growing complexity.

Traditional approaches, which don’t consider the evolution of the variables, aren’t accurate to study economical process, like the unemployment, which are essentially dynamic. So, the main lines of our work used System Dynamics [Aracil, 1992; Forrester, 1968; Forrester, 1971].

In the minimalist approach, we basically simulate the
evolution with temporally discrete lineal dynamical systems: we modelize
that a variable X tends toward a variable Y with a
delay t_{A} through the
relation

X(t+1)=X(t)+(Y(t)-X(t))/t_{A}

with t_{A}³1
(naturally, if t_{A}=1,
then X(t+1)=Y(t) ). We express this relation by means of

X | t_{A}®
Y

On the other hand, models of On-Only-Country are insufficient to simulate economical evolutions in a globalized world. Of course, it is usual to consider outputs and inputs from abroad (nobody would think that a model of a closed system were today realistic), but this approach has difficult to integrate the internal dynamics of other countries, plunged in the "environment". Specifically, it is usual to consider constant outputs from this environment. But it is not realistic, especially in relation to the balance of payments: this balance is also altered for the actions from other countries in order to overcome its own surplus/deficit. Thus, the interaction and accumulation of these actions have to be considered.

Global Models modelize the whole world, by considering
different parts of the Earth, which usually aggregate a set of countries
[Meadows *et al.*, 1985; Castellar,
1998]. Nevertheless, to modelize interchanges we have to work with
each country, and can only aggregate a set of countries, like the European
Union, which are economic units for interchange.

Of course, a model with more than a hundred of countries
would be very complex. But, in a first step, we are interested in the general
dynamic of the interchange, and so we will work with abstract models.

**2 Basic Model**

In a first step, **Atur 0** ("Atur" is the Catalan
word for Unemployment), we have initially considered one only Social System
and worked with global means. We used variables as

QP = Quantity
of Production

QC = Quantity
of Consumption

which are not separated by sectors.

The Model doesn’t treat explicitly with means of payment and monetary relations, although it implicitly requires units of measurement to treat homogeneously Production and Consumption of different sectors.

We supposed that Quantity of Production tends toward Quantity
of Consumption with a certain delay,

QP | t_{1}®
QC

In a second step, **Atur 1** disaggregates the System
in a set of m Social System (countries), although Production inside of
each System continues without to be disaggregated.

So, we suppose that Quantity of Production of each System-Country
i, QP(i), tends toward an ideal value QP^{O}(i)
which depends on its demand from different countries through some coefficients
f(i,j), which expresses the proportion of the Consumption of the System-country
j which generates demand from the System-country i.

Hence,

QP(i) | t_{1}®
QP^{O}(i)

QPO(i) = S_{j
}f(i,j)·QC(j)

S_{i
}f(i,j)=1

**3 Changing The Interchange**

Of course, in equilibrium QP(i) = QP^{O}(i)
, and therefore

QP(i) = S_{j
}f(i,j)·QC(j)

But we should also to suppose that equilibrium requires
a balanced interchange for every System-country, QP(i)=QC(i) . In this
case,

S_{j
}f(i,j)·QC(j)=QC(i)
,

and therefore QC(i) has to be a self-vector of the matrix
f(i,j) with self-value equal to one.

Thus, the version **Atur 1.4** of the Model includes
an evolution process of the coefficients of interchange f(i,j) in case
of unbalanced interchange.

Because f(i,j) have to fulfil the condition S_{i
}f(i,j)=1,
the matrix A_{ij}=f(i,j)-d_{ij}will
fulfil S_{i
}A_{ij
}=
0, and therefore, by being its rows lineally dependent, its
Determinant will value zero, which is precisely the condition for that
f(i,j) admit one as a self-value. Therefore, the only problem is that QC(i)
would be the corresponding self-value, by accurately adapting the coefficients
f(i,j) .

We can do it through an iterative process to search a solution of the corresponding system of equations

S_{j}QC(j)·f(i,j)
= QC(i) .

With a iteration of iteration de Jacobi we have

f_{t+1}(i,j)
= (QC(i)-S_{ }_{k¹
j} QC(k)·f_{t}(i,k))/QC(j)
.

Note that the relation S_{i
}f(i,j)=1
is conserved in this iteration. In effect,

S_{i
}f_{t+1}(i,j)
= (S_{i }QC(i)-S_{k¹
j }QC(k)·S_{i
}f_{t}(i,k))/QC(j)
= (S_{i }QC(i)-S_{k¹
j }QC(k))/QC(j) = QC(j)/QC(j) = 1

Now then, this iteration haven't to converge (the corresponding matrix won't fulfil, in general, the condition of convergence of being strictly diagonally dominant by rows).

See you the particular case of m=2 Systems-countries. In this case the relation will be f(0,0)·QC(0)+f(0,1)·QC(1)=QC(0), which, by applying the condition f(0,0)+f(1,0)=1 , becomes f(0,1)·QC(1)=f(1,0)·QC(0) . Hence, if we define A=f(0,1)·QC(1)/( f(1,0)·QC(0)) , the condition for that QC(i) fulfilled the condition of equilibrium with f(i,j) will be A=1 .

Now then, if we apply the indicated iteration we will
have

f_{t+1}(0,1)=
f_{t}(1,0)·QC(0)/QC(1),

f_{t+1}(1,0)=
f_{t}(0,1)·QC(1)/QC(0)
,

and so

A_{t+1}=
f_{ t+1}(0,1)·QC(1)/(
f _{t+1}(1,0)·QC(0))
= f_{t}(1,0)·QC(0)/(f_{t}(0,1)·QC(1))
= 1/A_{t} .

Hence, if A¹ 1 initially we will have a succession (A,1/A,A,1/A,...) which, of course, won't converge.

Observe you the practical meaning of this process: if
each System-country directly adapts its coefficients f(i,j) to equilibrate
its interchange with the other, the result is not this equilibrium, but
a series of periodic oscillations

**4 Slowing The Change**

To avoid these oscillations we could slow the iteration
by introducing a delay tf, so
that, with

f^{O}_{t}(i,j)=
(QC(i)-S_{k¹
j }QC(k)·f_{t}(i,k))/QC(j)
,

the coefficients f(i,j) with j¹
i tended toward this value with a delay tf
,

f_{t}(i,j)
| t_{f}®f^{O}_{t}(i,j)
, and so

f_{t+1}(i,j)
= f_{t}(i,j) + (f^{O}_{t}(i,j)-f_{t}(i,j))/t_{f
}=
f_{t}(i,j)+(QC(i)-S_{k¹
j} QC(k)·f_{t}(i,k)-QC(j)·f_{t}(i,j))/QC(j)/t_{f
}=

= f_{t}(i,j)
+ (QC(i) - S_{k }QC(k)·f_{t
}(i,k))/
QC(j)/t_{f}.

We can verify that the relation Si f(i,j)=1 is conserved:

S_{i
}f_{t+1}(i,j)
= S_{i }f_{t}(i,j)+(S_{i
}QC(i)-S_{k
}QC(k)·S_{i
}f_{t
}(i,k))/QC(j)/t_{f
}=
1 + (S_{i
}QC(i)
- S_{k
}QC(k))/QC(j)/t_{f
}= 1 .

In the case m=2 we will have

f_{t+1}(0,1)
= f_{t}(0,1)(t_{f
}-1)/t_{f }+
f_{t}(1,0)·QC(0)/QC(1)/t_{f}

f_{t+1}(1,0)
= f_{t}(1,0)(t_{f
}-1)/t_{f }+
f_{t}(1,0)·QC(1)/QC(0)/t_{f}

and therefore

A_{t+1 }=
((t_{f }-1)A_{t}+1)/(t_{f
}-1+A_{t})

Naturally, if t_{f }=1
were fulfilled we would newly obtain A_{t+1}=1/A_{t
}. And if t_{f}®
¥, A_{t+1}®A_{t}

If we want that A_{t+1}=1,
then we will have to require (t_{f }-1)A_{t}+1=t_{f
}-1+A_{t
}, which is equivalent to
(t_{f }-2)(A_{t}-1)=0,
which will be fulfilled if A_{t}=1
or t_{f}=2 . So then,
the delay t_{f}=2 always
permits to obtain the coefficients of equilibrium in one only step. We
could say that the Systems-countries have "splitted the difference" in
its search of its equilibrium of interchange. And, in general, if t_{f
}³2 we can anticipate
a convergence of the iteration without oscillations.

We will demonstrate that we can generalise this result
to the general case of m Systems-countries, so that t_{f
}=m carries to obtain the convergence in one
step.

In effect,

å_{j
}f_{t+1}(i,j)·QC(j)
= å_{j
}f_{t}(i,j)·QC(j)
+(QC(i) - S_{k}
QC(k)·f_{t}(i,k))·m/t_{f
}= å_{j
}f_{t}(i,j)·QC(j)·(1-m/t_{f})
+ QC(i)·m/t_{f }.

Hence, if t_{f }=m
the condition of equilibrium å_{j
}f_{t+1}(i,j)·QC(j)
= QC(i) will be fulfilled.

In general, the residue will value

R_{t+1}=
å_{j
}f_{t+1}(i,j)·QC(j)
- QC(i) = (å_{j
}f_{t}(i,j)·QC(j)
- QC(i))·(t_{f }-m)/t_{f
}= R_{t}·(t_{f
}-m)/t_{f
}.

Therefore, R_{t+n}
= R_{t}·((t_{f
}-m)/t_{f})^{n
}, and the condition of convergence of the iteration,

limn®¥
Rt+n = 0 , will be |(t_{f
}-m)/t_{f }|
< 1 .

So that, if t_{f }³m
the iteration will converge without oscillations. If t_{f
}<
m , the condition of convergence will be m-t_{f}<t_{f
}, that is to say, t_{f
}> m/2 : if m/2 < t_{f
}< m , the residue will change of sign in
each iteration by being approximate to zero.

If t_{f}=m/2
(like in the case m=2, t_{f}=1),
R_{t+1} = - R_{t
}, and the iteration will produce periodic oscillations
without converging. Si t_{f }<m/2
, lim_{n® ¥}|R_{t+n}|
=
¥ , and the iteration will diverge: the
residue will oscilate between plus and minus infinite.

Observe that, in the limit, if t_{f
}®¥ , then
R_{t+1 }®
R_{t} , to be annulled
the evolution (f_{t+1}(i,j)®
f_{t}(i,j)) .

In the Model **Atur 1.4** we have took t_{f
}=mt_{1
}, the evolution toward the equilibrium of interchange
were gave by

f^{O}_{t}(i,j)
= (QC(i) - S_{k¹
j }QC(k)·f_{t
}(i,k))/QC(j),

and the coefficients f_{t }(i,j)
tended toward f^{O}_{t}(i,j)
with a delay t_{f }=mt_{1
}, and so

Df(i,j) = (QC(i) - S_{k
}QC(k)·f_{t
}(i,k))/QC(j)/t_{f
}.

You can see the full Model in http://www.uv.es/~pla/java/atur1.html and test it graphically in http://www.uv.es/~pla/java/Atur1gra.html through an interactive Java program (in Netscape you have to fill the first window, put OK, recharge the page, fill the second window, put OK and pick on the page).

**5 Conclussion**

According to our model, a hard adjustment carry to increase unbalanced economical state, and a soft adjustment is more efficient.

Our model could so explain the financial and social convulsions which have been generated by policies of hard adjustment of the balance of payment, and suggests the convenience of smooth policies, by searching little and slow changes of this balance, especially in a global world with many countries in interaction.

Of course, these policies will only be effective if its application is generalised in the world. Therefore, the intercommunication in international economical forums becomes essential in order to achieve a balanced interchange between the different countries.

Particularly, in the conditions of our model, and if we
consider annual cycles, the better results would be obtained if each country
programmed its adjustment with a delay of so many years as the number of
countries which interchange: in this way, every them would be able to arrive
to a balanced interchange in **one** year.

**References**

[Aracil, J., 1992] Introducción a la Dinámica de Sistemas, pp.262 ss. Alianza Editorial. Madrid.

[Caselles, A., 1996] A systemic
Approach to Unemployment Control, *Third European Congress on Systems
Sciencie*, 487-491. Edizioni Kappa. Roma.

[Caselles, A., Ferrer, L., Martínez de Lejarza, I., Pla, R., Temre, R., 1999a] Control del Desempleo por Simulación. Universitat de València. València.

[Caselles, A., Ferrer, L., Pla, R., Temre, R., Martínez de Lejarza, I., 1999b] Simulation for Unemployment Control. Diputación de Valéncia, Àrea de Informàtica. València.

[Castellar Busó, V., 1998] Un model sistèmic d'evolució social sostenible, pp.3-30, 169-194. Departament de Matemàtica Aplicada Universitat de València. València. http://www.uv.es/~buso/tesi/intro.zip , http://www.uv.es/~buso/tesi/ap_i.zip

[Forrester, J.W., 1968] Principles of Systems. Wright-Allen Press. Cambridge.

[Forrester, J.W., 1971] World Dynamics. Wright-Allen Press. Cambridge.

[Meadows, D.H., Robinson, J.M.,
1985] The Electronic Oracle: Computer Models and Social Decisions. Wiley.
Chichester.