Mathematical Tools in Economy: Changes in National Debt and Public
Expenditure in Spain before European Convergence. A Simulation Study
with Techniques from Systems Dynamics.
José Manuel González Rodríguez
Department of Applied Economy
La Laguna University
Campus de Guajara s/n, La Laguna 38075
SPAIN
Abstract: In this paper we analyse a Model that simulates conveniently the interrelations between Inflation, Interest
rates, Public Expenditure and national Debt in Spain before the Maastrich' s scenario. We use techniques of Dynamic
Simulation and Growth Curves Theory as mathematical tools in Economic analysis research.
Key-Words: Macroecomic scenario, Dynamic simulation, Growth Curves, European convergence
1. Introduction
Spain and the rest of the countries of the European
Union were confronted with the difficult challenge of
harmonizing their economies in search of the long
awaited monetary and commercial union. We know
that this convergence of interests was only attainable
when each country could commit itself through its
economic policies to reducing four basic indicators:
Inflation, Interest Rates, Public Expenditure and
National Debt.
In this paper we face up to this challenge by
analyzing a Model that simulates conveniently the
interrelations between the above variables. Taking up
the proposal of Prof. Ramón Tamames, we can use the
techniques of Dynamic Simulation with the aim of
resolving this question with the modesty required on
using quantitative methods.
2. Problem Formulation
A large number of variables are recognised in the
problem of interrelating the indicators set out in the
Maastricht treaty, our first objective being to describe
a mathematical model that permits us to obtain
information about them. We have chosen one of the
simulation type due to the following necessities:
1. The model must accommodate the
fundamental interest in analyzing the
internal logic of the system, which
accurately
determines
the
structural
relationships governing its behaviour.
2. Its formation must reflect the opinions of
experts in the areas it covers.
3. The model will attempt analysis of complex
problems, where numerous variables and
relations have their influence.
4. It will permit testing behaviour patterns
other than those observed in reality, so that
through a sensitive analysis the most
meaningful behaviour may be identified.
A recent technique that has come to be used in
such cases: Systems Dynamics (SD Techniques), has a
great variety of areas of competence and a wide
spectrum of positive results This technique was
chosen not only for its perfect suitability to the
problems under study, but also, by the high level of
mathematical formality it offers given the author´s
professional training.
As instruments used in the mathematical modelling
of regional systems, SD techniques attempt to discern
in these the characteristic methods of servomechanism
design, joining these to a behaviour analysis of certain
types of nonlinear differential equations. Essentially,
the system modelling fundamental problem is directed
to of explaining why and how the variables of state
vary with time. The ultimate consequence of this if
certain stability conditions of the parameters are
verified, a procedure is obtained to know (or simulate)
how the system will behave in the future, faced with
diverse circumstances, alternative hypotheses or
scenarios.
In practice, our model attempts to reproduce the
convergence conditions agreed on by the European
Union member countries. For this we state a
fundamental initial hypothesis:
“The convergence conditions cause an approach of
the variables: Interest Rate, Inflation and Deficit
towards asymptotic tendencies proper to logistic or
sigmoidal growth (describing lengthened S-shaped
curves)”.
We limit our model the causal relationships
between those variables, which act over the feed-back
relations that satisfy that hypothesis. According to
this, we must take in account a diagram
interconnecting the variables of the model in the
following positive feedback loops:
Retail Price Index Objectives for Convergence
Optimum Objective in Social Welfare
Expenditure as Percentage of GNP
Welfare Payments
 Variation in Interest Rates, IR; an
instrumental variable delimiting changes
during each annual period.
 Percentage Social Expenditure, SE; valued
as % GNP, an instrumental variable
modelling the real tendency in the Spanish
economy towards reaching the levels of
social protection in the more economically
developed EU countries.
 Percentage of Public Expenditure, PE,
permitted by convergence rules and the
Inflation Objective, RPIOBJ
The flow variables selected are:
 Budgeted Public Deficit, BPD; permitted
 Deviation from Public Deficit, DVPD;
annual negative balance between budgeted
and real deficits.
 Expenditure Deviation, ED in percentage og
GNP
Inflation RPI appears in a first simulation test as an
exogenous variable, alien to the model’s internal
dynamics.
The above may be depicted as follows:
Expenditure Deviation
GNP
Budgeted Deficit
ED
PE
National Debt
Inflation Objective
Deviation of Debt Repayment
DVPD
ND
BPD
Annual Excess in Debt Repayment
SE
RPI
Interest Rates
Gross National Product
In this diagram both natural causal relationships
(those which by definition delimit the variation in
fundamental level variables: National Debt, GNP and
Interest Rates), and those which are only operative as
they include level and flow variables reproducing the
objective pursued in the model: a logistic approach to
the convergence conditions of the European Monetary
Union. Therefore, the specific classification of the set
of variables will respond to this criterion and will thus
be level variables:
 National Debt, ND; a stock variable
accumulated each year.
 Gross National Product, GNP; measured in
its entirety, accumulated annually.
IR
RPI
RPIOBJ
3 Problem Solution
The differential equations governing the dynamics
of the model reproduce the internal structure of the
Forrester diagram as follows:
The equation defining the annual increase in the
National Debt will be a classical level equation given
by:
(1) NDt+1 = NDt + EDt,t+1 + BPDt,t+1 + DVPDt,t+1
with flows determined by the identities:
(1.1) BPDt,t+1 = GNPt  PEt
(1.2) EDt,t+1 = SEt  GNPt  V1/100
(1.3) DVPDt,t+1 = NDt  V2
The auxiliary variables V1 and V2 verify by
definition the following equations:
(1.4) V1 = 0, if RPI  RPIOBJ or RPI - RPIOBJ in
other case
(1.5) V2 = Table Function(IR, RPI - RPIOBJ)
where the Table Function is to be understood as a law
of interpolation that establishes the IR (Interest Rates)
variation exogenously in accordance with that of RPI RPIOBJ. In the language DYNAMO used in
numerical resolution of equation system a subroutine
exists to perform naturally the calculations necessary
to determine the value of the dependent variable
according to the values known for the independent
variable with this subroutine, the intrinsic difficulty of
modelling Interest Rate variations due to sudden
changes in RPI, is avoided. What is more in any case
after a first approach to the equation defining (DVPD)
we have assumed IR variation to be endogenous to the
model i.e. it only depends on historical developments
in IR’s, allowing to be expressed in the form:
(1.6) DVPDt,t+1 = NDt  1800  [IRt+1 -IRt]/228,915
The parameters 1800 and 228,915 were obtained
from data from 1995 on the interests that should be
paid at the present Debt owing level.
Lastly, the differential equations governing
behaviour of level variables we qualify as
instrumental have been chosen according to the
following argument.
It is known that the variables subject to a tendential
objective are governed by a logistic or sigmoidal
curve (lengthened S). These growth curves are
characterized by presenting in their evolution regime
two types of growth: A moderately expansive initial
one followed by another tempered similarly to the
“S”-curve asymptotic objective or carrying capacity
that should be reached when the time parameter tends
towards infinity. Given that the sigmoidal growth of a
variable X(t) can be simulated by resolving one (or
various) equations of type:
(1.7) Xt+1 = Xt + a  Xt [1 - Xt/OB]
modelling a time series X1(t) with the aid of these
curves essentially consists of finding those values of
parameters a and OB (plus any other that appears in
other auxiliary equations) that best reproduce changes
in the variable called X1 in the simulated variable X.
Therefore, from the graph of a data series X1 and
from an analysis of those mathematical elements that
best characterize its mode of variation, we must select
those components (binding conditions) that
determine parameter values (degrees of freedom) for
the Differential Equation. The most important are:
 Carrying capacity OB, the tendency of the
variable X when t tends towards infinity.
 Initial value X1(0), first datum of the series.
 Inflection point t, moment in time when the
curve changes from concave to convex,
coinciding with the time at which the slope
of curve X1(t) reaches its greatest value, i.e.
the derivative of X1 in it, will be maximum.
 Value of X1' s derivative in it, i.e. expression
of maximum slope.
These elements being known, characterizing the
life cycle duration of the variable, its rapid growth
intensity and the exact payment delay period, the
instrumental variables of the model may be modelled
as follow:
The Inflation Objective will decrease with time to
a carrying capacity around 3% annual increase, a
tendential objective corresponding to the retail price
index of the EU countries which had lower inflation
rates in 1995, increasing the latter by two points
(convergence criterion). In this way the logistic
equation verifying this variable will be:
(2) RPIOBJt+1 = RPIOBJt, + a  RPIOBJt 
[1 - ,RPIOBJt/3]
parameter “a” being the maximum growth intensity
of RPIOBJ, equal to 0.0747. With this simulation it is
found that the approximation between simulated and
real variables does not exceed of 3.65% and error in
the year 1994 (see figure 1):
14
12
10
8
6
2.5
5
7.5
10
12.5
15
Figure 1. Simulation of RPIOBJ evolution (19791994)
The percentage of Social spending rose quickly at
the end of the eighties (Figure 2)
17
16.5
16
15.5
15
14.5
2
4
6
8
10
12
14
Figure 2. Growth of Social Spending in % of GNP
This boom aided the approach of this variable to its
optimal objective: 21% of GNP (average for the more
developed EU countries). As consequence the
modelling of variable SE can be obtained applying
this modified logistic equation:
(3) SEt+1 = SEt + 0.619  [SEt + 77.1748] 
[1 - (SEt + 77.1748)/106.017]
that offers a fit between real tendency and simulated
variables with a mean relative error less than 2.3%
(figure 3). This equation remember the approach to
diffusion and expansion of a new product, in such way
as was formulated for Bhargava (1991) when its life
cycle its modified by prices considerations.
17
16.5
GNPt+1 = GNPt + 0.1541  GNPt 
[1 - GNPt/421012]
would follow, and the solution to this problem of
initial values will adjust itself more or less adequately
to the real GNP evolution in accordance with a value
for the free parameter c = 0.1541 that minimizes the
quadratic error of the fit.
The Spanish GNP valued in constant 1986 pesetas,
will be perfectly simulated with the aid of an (5)-type
equation with an error of 1.2% for 1994, in such a way
that its foreseeable historical variation will not exceed
42 billion constant pesetas in a nearby horizon of 30
years. It seems this modelling method not only adapts
well to the real scenario under study, but, in turn,
proposes a reasonable prospective future horizon in
accordance with feasible economic considerations
(Figure 4).
(5)
16
3600
15.5
3400
15
3200
14.5
3000
2
4
6
8
10
12
14
2
Figure 3. Approach to SE growth tendency
The Public Deficit percentage is also found to be
limited by the convergence criteria. Its asymptotic
objective must no exceed 3% of GNP and may be
simulated in the form:
(4) BPDt+1 = BPDt + 0.07541  BPDt 
[1 - BPDt/0.03]
On simulating GNP growth, one should consider
the well known succession of characteristic cycles in
economic phenomena, reflecting periods of recession
then expansion. Modelling these crests and troughs is
no easy task as they follow one another with diverse
amplitudes and frequencies, difficult to regulate under
deterministic laws. A possible evolution of the GNP
may thus be conceived of according to the moderate
exponential growth that follows from the solution of
the Differential Equation:
(1.7) GNP'(t) = C  GNP(t)
where C must be chosen as mean constant rate of
variation. This is the way most demand projection
studies usually model GNP evolution.
In this simulation the added national value grows
without limit and according to parameter C, reaches
explosive growth rates difficult to reproduce in reality.
Therefore, the hypothesis might follow that such
variable will approach a certain asymptotic horizon or
carrying capacity OB, attainable but not exceedable in
a foreseeable period of time. Thus a sigmoidal o
logistic evolution of the type:
4
6
8
10
12
14
Figure 4. Simulation of Spanish GNP (1979-1994)
Lastly, changes in Interest rates may be modelled
whenever we know that of their annual rates of
variation. As the monetarist theories of inflation often
make use of regression equations where these rates are
interrelated, we have resorted to Friedman’s classical
formulation who considers the evolution of IRt to be
given by:
(1.8) IRt = Cte. + yt*
*
being yt the expected nominal growth rate (H. Frisch,
1983, p.128).
This new rate can be evaluated by following a
monetarist model with rational expectations
(monetarism Mark II) in the form:
(1.9) yt* = RIPt + xt
where RIP represents the current inflation rate and xt
denotes the relative annual growth rate of the GNP. In
this way, developments in the Interest Rate can be
modelled with the equation:
(6) IRt+1 = IRt + IRt  [0.022 + RIPt +
(GNPt+1 - GNPt)/GNPt]
whose solution provides a fit with the real IR data
with an mean relative error of 3.2 % in 1994 (Figure
5).
16
14
12
10
2
4
6
8
10
12
14
Figure 5. IR simulated showing qualitative accuracy
4 Conclusion
In summary, a system of equations: (1)-(6) has
been obtained that determines developments in the
simulated variables. Their solution contributes a
simulation of the main level variable: National Debt,
which is perfectly adjusted to its real variation (Figure
6).
22500
20000
17500
15000
12500
10000
7500
2
4
6
8
10
12
14
Figure 6. Final Result of National Debt simulation
In particular, simulation of ND reflects both
quantitative changes and qualitative variation in the
variable tabulated by the Spanish State Accounting
Bureau. It only remains to ask about its modellization
reproduced the future of that level in the convergence
scenario: 1995-2000.
To tackle this problem we must first model the
annual inflation rate RPI, a variable considered to be
exogenous to the model but susceptible to modelling
in an endogenous manner. In fact, monetarist theories
of inflation tend to evaluate this variable in terms of
the GNP growth rate, unemployment rate and interest
rate variation (see Frisch, 1988). So according to
Okum’s law and Philips’ curve, the RPI is explained
as a regression equation in the form:
(7) RPIt = RPIt* + a  (xt -x*) - b  (ut-1 - u*) + t
RPIT* being the expected inflation rate,
(xt - x*) the unforeseen acceleration of annual
GNP growth,
ut-1 will be the unemployment rate affected by
a retard, and
u*, the natural unemployment rate.
Moreover, as Okum’s law relates the rates u
and xt, we have the equation:
(7) RPITt = RPIt* + A  (xt - x*) + t*
In such a way that under the equilibrium
hypotheses in the long-term:
(4.1) x* = 0 and RPIt+1* = RPIt +   (RPIt - RPIt*)
i.e. the expected inflation rate will vary as a constant
proportion () of the error that would have been
predicted (RPIt - RPIt*). With these two equations we
can model the evolution of the variable RPI in the
scenario 1979-1994, being that the values A = -0.298
and  = 0.95 give us a good fit for this simulation
Taking the above into account, adapting these
equations to the real development of inflation in
Spain, it was found that the system:
(8) RPIOBJt+1 = RPIOBJt, + a  RPIOBJt 
[1 - ,RPIOBJt/3]
(9) SEt+1 = SEt + 0.619  [SEt + 77.1748] 
[1 - (SEt + 77.1748)/106.017]
(10) BPDt+1 = BPDt + 0.07541  BPDt 
[1 - BPDt/0.03]
(11) GNPt+1 = GNPt + 0.1541  GNPt 
[1 - GNPt/421012]
(12)
RPIt+1* = RPIt* - 0.95 0.298  xt =
= -0.95  0.298  (lnGNPt)'
(13)
IRt+1 = IRt + IRt  [0.022 + RIPt +
(GNPt+1 - GNPt)/GNPt]
(14)
NDt+1 = NDt + EDt,t+1 + BPDt,t+1 + DVPDt,t+1
reproduce the future evolution of variable RPI with a
high degree of accuracy, and the constants 0.95 and
0.298 are those best fit the real case.
In summary, having incorporated the variable RPI
into our model the set of levels, we obtain the
previous differential equation system, that, under the
hypothesis of maintaining the causal relations in the
original diagram, it will be necessary to simulate the
joint evolution of Deficit, Debt, GNP, IR and RPI for
the past scenario in the year 2000.
The behaviour of the state variable of the system
(8)-(14) will depend only on their equilibrium points
and its stability, such that analyzing the points where
the second parts of equations cancel out, it will be
necessary to work out the future behaviour of all the
variables. Making all these equations equal to zero
gives the following equilibrium values for the first
four equations:
RPIOBJ = 0 or 3
SE = -77.1748 or 28.84 % of GNP
GNP = 0 or 42  1012
OBJPD = 0 or GNP's 33%
As the value of RPI in the equilibrium
coincides with this, according to the previous
discussion with
, the only equilibrium points
with real meaning will be:
RPIOBJ = 3
SE = 28.84 % of GNP
GNP = 42  1012
OBJPD = GNP's 33%
The stability of this equilibrium point depend on
the eigenvalues of the Jacobian matrix for the system,
the characteristic equation of which having 7 rows and
7 columns can be worked out more simply by applying
the Variety of Centres theorem. According to this one
can replace the study of coordinates with an
asymptotically stable equilibrium and reduce the study
to those where such stability is not guaranteed (see
J.M. González, 1995 and E. Freire, Gramero and E.
Ponce, 1990).
Then, if we assume the most favourable and real
situation, where the simulated RPI never exceeds 3%
near its equilibrium point, the system approaches the
equilibrium point at which the Inflation Objective
reaches 3% per year, a tendency also associated with
the foreseen inflation, the Public Spending represents
28.84% of GNP, which in turn reaches 42 billion
constant pesetas and the Public Deficit disappears.
The variations in National Debt and Interest Rates
prevent all the variables converging at a single reliable
attractor.
This situation, which reflects the rational future
expectations for these variables with greater accuracy,
is not repeated likewise in the other equilibria
appearing from nonreal scenarios. For example, when
the equilibrium admits the hypothesis according
RPIOBJ and OBJPD equals to zero, the National Debt
allows growth to any percentage of GNP, moreover
the variable RPI* must be -3.
It is possible that the previous situation in the
equilibrium could change in the last seven years, but
the stability analysis allows us to accept that all
inflationary scenario must converge to the results of
our Model.
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