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
Expenditure Deviation Budgeted Deficit
National Debt
Inflation Objective
Deviation of Debt Repayment
Annual Excess in Debt Repayment
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:
Þ National Debt, ND; a stock variable accumulated each year.
Þ Gross National Product, GNP; measured in its entirety, accumulated annually.
Þ Variation in Interest Rates, IR; an instrumental variable delimiting changes during each annual period.
Þ 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.
Þ Percentage of Public Expenditure, PE, permitted by convergence rules and the Inflation Objective, RPIOBJ
The flow variables selected are:
Þ Budgeted Public Deficit, BPD; permitted
Þ Deviation from Public Deficit, DVPD; annual negative balance between budgeted and real deficits.
Þ 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:
GNP
ED
PE
DVPD ND BPD
SE
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:
Þ Carrying capacity OB, the tendency of the variable X when t tends towards infinity.
Þ Initial value X1(0), first datum of the series.
Þ 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.
Þ 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):
Figure 1. Simulation of RPIOBJ evolution (1979-1994)
The percentage of Social spending rose quickly at the end of the eighties (Figure 2)
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.
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:
(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).
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).