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Dynamic modelling in development plannin
Dynamic modelling in development plannin Lucia Grandinetti and Ferdinand0 Pezzella Dipartirnento
di Sistrnli
dell Unitrrsita’
de/la Cdahrin,
Arcnnacata
(Cosenzn).
ItnlJ
Agostino La Bella
(Received 27 January
1977; revised 30 May 1977)
In this paper a dynamic regional model is proposed, aimed at understanding and forecasting the dynamic response of the population to many different investment policies in the agricultural, services and industrial sectors. In spite of its generality, the model has been conceived in order to be applied to underdeveloped regions. Particularly, from the demographic point of view, attention has been focused on migration phenomena; from the economic point of view, attention has been focused mainly on the rate of growth of the capital stock. Modularity is another feature of the proposed model; it is made of seven submodels which describe in a very aggregated form the evolution of the regional socio-economic system. The outputs bf the submodels are then spatially distributed using a set of ‘spatial operators’ which take into account many social and economic factors relevant to the sub-areas into which the region can be divided.
Introduction Development planning is usually associated with both the efforts of the less developed countries in planning for programmes of economic and social growth, and with the efforts of the developed countries in reequilibrating internal disparities in employment, income and rate of economic growth’. However, in spite of a number of studies on economic modelling7 - ’ ‘, from the one side, and on demographic modelling2v6 on the other side, these two different contributions are still awaiting a systematic synthesis3 ‘. This paper is intended as a contribution in this direction: it aims to overcome the traditional separation between economic and demographic modelling, proposing a general structure for a dynamic model in which the interrelationships between economic and demographic phenomena in a developing region, characterized by the Government’s (local and central) high incentives, are taken into account. The proposed model is based on an analysis of the
structure of the italian system, and focuses attention on the basic variables and their interrelationships. The most important features of the model are the spatial structure and the form of the equations which can be immediately expressed in terms of an input-output system. This form has recently become very important in macroeconomic modelling both because it is relatively more general and because it is possible to use advanced techniques as identification and optimization methods which have been developed by systems theory. The Appendix shows an example of the application of these methods to one of the submodels illustrated below. Figure I shows the general structure of the model. The central section comprises the submodels of the five economic sectors (production of consumer goods, production of investment goods, agriculture, public services, private services). These submodels are influenced by the final demand coming from the demographic sector, and also by an investment flow coming from the capital formation sector.
Appl.
Math. Modelling,
1977, Vol 1, December
379
Dynamic modelling
in development
planning:
L. Grandinetti et al. FZj(t)
T Constraints
,
Public
Figure
1
General
scheme
of whole
model
Total production, the result of the interaction of all these phenomena, determines (through changes in the employment structure, in income and in the supply of services) the distribution of the population within the region and the size of the migratory flow. Moreover, production affects the creation of internal resources devoted to investment. The spatial disaggregation of the submodels has been realized by means of a proper set of operators. A set of bounds has been associated with these operators in order to take into account the limitations which resulted from the physical characteristics of the region and/or from Government intervention (by means of laws and decrees). Control variables, e.g. the intervention of the public authorities, condition the development of the economic activities and their spatial distribution through investment, special incentives and regulations intended for particular sectors and areas. Government intervention, then, concerns both the money sector and spatial operators.
The economic submodels The economic part of the total model is built of five submodels from the production sectors plus the submodel of the capital formation sector. These submodels were built according to a common general scheme: if necessary changes were made in a sector in order to take its particular features into account. The general formulation is based on a linear dynamic relationship between the production ~j(t) and the installed capital K,(t):
=
,f;(Dj,
Sj)
-
(5)
K,(t)
where, for any sector: lDj(t) is the available investment; IF,(t) is the effective investment; h(t) is the depreciation coefficient of plant and machinery; Dj(t) is the internal demand, sj(t) is the trade balance; Fl,(t) is the required investment and 4 is the function related to the economic behaviour of the sector. For any sector, except the capital goods sector, the internal demand is computed in the demographic submodel. Submodel of consumer goods sector This is represented
by equations
(l)-(5).
Submodel of capital goods sector We made the hypothesis that the difference between available and effective investments is employed in technological innovation. The new variable, IN,(t) was introduced into the model to represent this type of investment, which anticipates demand. Thus, in addition to equations (l)-(5) we have the following: IN,(t) = $ZD,(t)
- FZ*(t) - IN&)]
where the time constant z represents the time innovative investment needs in order to come to maturity. Then equation (3) must be modified as follows : G(t)
= IM4
+ IN,(t)
- P&2(t)
(7)
Submodel of agricultural sector This submodel includes equations equal in structure to (6) and (7). Those equations, such as (3), (4) and (5), may be disaggregated with respect to the components of the investment in land works (irrigation, roads, canals, etc.) and in capital goods. A stochastic term E(t) which takes into account seasonal fluctuations and environmental disturbances, may be introduced in equation (1):
+
1
E(t)- Y3(0
A relationship to agricultural
may be added use, TA(t):
to compute
the land put
TA(t) = Y[K;(t)] where K>(t) is the capital
(9) in land works.
Submodel of private services sector This is represented according to the general (l)-(5). where j = l,... 5 is the sector index, aj is a conversion factor, and Tj is a time constant. An upper constraint on the production’ is given by: n,(t).
yj(t)
I
PAj(t)
(2)
where PAj(t) is the labour force availability, and /Zj(t) is an exogeneously given labour output factor. PAj(t) is calculated in the demographic submodel. The additional basic relationships are the following: r(-j(t)
=
Z~(-(t)
-
IF,(t) = min {Fl,(t),
380
Appt.
Math.
(3)
~jKj(t)
(4)
ZDj(t))
Modelling,
1977,
Vol 1, December
equations
Submodel of public services sector In this case the available investment comes solely from Government (central and/or local) intervention. The underlying hypothesis is that a fraction p of the appropriation ug is used to offer the public services at a ‘political price’, which is usually lower than the real cost. The supply of a service at a political price causes an increase of the demand, due both to an increase of the propensity to consume and to the fact that demand expressed in money terms is higher than real demand. Thus, we have:
Dynamic
W(r) = (1 - Pbdr)
(10)
o;(r) = g(W), P.4))
(11)
where D;(r) is the effective demand. Submodel of capital formation sector This submodel is the key to the economic dynamics of the whole system. The investment available for all economic activities, except the public services sector, originates here. The structure of this submodel differs significantly from that of the previous ones, and will therefore be described in detail. The output of each sector yj,j = 1,2,3,4 contributes to the formation of internal reserves for a percentage sj, 0 I sj I 1. Total liquidity M(r) of the monetary system increases by that amount and decreases by total investment 17: given by: IT(r) = W) -PI(r)
+ zz(r) + b(r) -At)
pj(r) = ZDj(r) - IF,(r)
j=
+ W)
1,4
Q(r) = M(r) 1 R,(r) ;m I
Ij(t)
Total
=
investment
=
(17) J
YZ;is a suitable coefficient; difference between marginal productivity cj of and the interest rate r’ paid to banks for the Thus we have: 1 Pj(t)
=
wj(t)
4
4
1 j=
Wj(t)
w;(r) o
=
i
wj
+
1
4
1 j=
r’j.Tj
+
1
if
w>(t) 2 0
if
w>(t) < 0
(18)
Rj
C j=
1
w>(r) = hj(r).Kj(r) + 6;.%(r) + qj(C~ - T;)
(19) (20)
in the sector j is defined
by:
(21)
z.(r) + u.(r) J
[ ’
-
sjo 1 vj
(22)
where ij, Zj and Vj are suitable coefficients. As regards the form of Rj,,j = 1,2,3,4, it should be noted that these terms have been defined so that the outgoing capital would be proportional to the fraction of external capital over total capital installed, and would decrease, for every sector, in proportion to the increase both in the ratio of marginal productivity of capital at external average level and also in the rate at which available. funds are drawn. Thus : R,(r)
where (3) the capital sector.
I ~j
= ~
where J” is the active rate of interest. The use of internal resources is defined in the model by the rate pj at which borrowers draw on available funds. This rate is calculated on the basis of: (1) the estimate %(r) of the development rate of the sector”, which is given by:
,q;.h;(r) + h;(r) = $
a certain degree of of the development rates 14 suitable constants which take the interest rates on
mj(t) = bj:.gj([) J
J
where ys is a suitable coefficient. (2) the estimate of the development rate h;(r) of public intervention uj(r) in the sector. which is given by:
et al.
where mj(r) represents external capital invested in the sector j; mj(r) is calculated on the basis of an estimate gj(r) of internal investment-including public intervention-and of the ratio between the marginal productivity of capital at local level cj and the marginal productivity of capital at average external level Cj. Thus,
4.(r)
(15)
L. Grandinetti
Pj(WWr)
in
where R,(r) represents the rate at which capital relative to the various sector goes outside the region. These terms will be explained later. Thus M(r) becomes:
planning:
ZDj(t) = Zj(t) + nj(t) + mj(t)
(12)
(14)
+ r”.M(r)
in development
where 8: and S; represent confidence in the estimate and Mi and +j and Ij are into account the effect of investment:
(13)
where Zj(r): internal resources devoted to investment sector j. From total liquidity one must also subtract the capital Q(r). which goes outside the region:
A%(r)= t yj(r).sj - IT(r) -m(r) ,= 1
modeling
Cj(
‘j
1 + nj.pj(t)) .
where nj are suitable weights associated with pj It must be stressed how important this submodel is within the framework of the dynamics of the whole system. In fact, virtually all control variables (i.e. public investment uJ and interest rates Y’~and r’j’)affect this submodel (for j # 5) thus contributing to the interaction variables ZOj, j # 5, which intervene in the dynamics of the other submodels.
Demographic and employment sector The model of this sector generates population, employable population, number of employees, the wage levels in the various activities, and final demand in the different sectors. Given the particular orientation of the sector under consideration, we have found it necessary to relate the spatial mobility of the population to variables such as unemployment, availability of public services and income’. Moreover, the process of demand formation
Appl.
Math.
Modelling,
1977,
Vol 1, December
381
Dynamic
modeling
in development
planning:
L. Grandinetti
et al.
Ilj + lj.~j(t)
Gj(t) =
C(“j + lj.rrj(f)
(30)
where Uj and lj are suitable coefficients introduced with the purpose of taking into account the effects that any change in the wage level might have on the structure of the potentially working population. The unemployment rate in this sector can be obtained as follows : PA(t).Gj(t) - E,(t)
DLj(t) = Figure
PA(t).Gj(t)
I
I
2
DemographIc
and employment
submodel
takes into account the prices of public services which will probably be ‘political’ prices. Figure 2 illustrates the population section of the submodel under consideration. Total population P(t) is calculated on the basis of birth rate 72v, and death rate TPI, ‘and on the basis of the net migratory flow FNM: P(t) = 7N.P(t) + FMN(t)
- TM.P(t)
(23)
FMN(t) = A(t).P(t)
(24)
where A(t) is a coefficient of attraction of the region. The following contribute to the formation of A(t): the comparison between availability of public services per cupitu at local level and availability of public services per capitu external to the region under consideration; the comparison between the local unemployment rate DL and the external unemployment rate DN; the comparison between regional and external income prr capitu. We have obviously assumed that the population responds to the variations in these factors of relative attraction with some delay. This delay varies from one factor to the other. Thus: A(t)
=
hJ(K(%
ff2(%
(25)
ff3(0)
- DN(t) - H,(t)]
i?&(t) = ;[DL(t) 6
(26)
where Ej represents employment in thejth sector and PA the potentially active population. We can obtain the wage level aj(t) in the sector j by comparing DLj with the corresponding variable external to the region by means of a function f,: 6j(t) = ~[S_(DLj(t)
where Ts, T,, Ts are suitable time constants; PN, represents national population; SN, public services available at national level; RPN, external income per capita at national level; f6, is a suitable function of the variables HI, H2, H3 and RPC is the regional income
(32)
RPCj can be calculated once we have taken into account the non-wage factors NWj and the rate of taxation on the incomes of the sector TRj. Thus we have : RPC,(t)
=
I
fyt) =
[aj(t).Ej(t) + NY(f)l.[l -TRj(t)l (33) Q(t) PAj( t) FPA
PA,(t) = Gj(t).PA(t)
(35)
where Pj(t) represents the part of population who have an income produced by the sector j. Final demand can be obtained by means of the vector-valued function F, which generates, on the basis of the per capitu income, the vector bj whose components represent the expenditure coefficients for goods and services which may be found on the market by the fraction of population Pj. So we have:
(28)
- MN(t)
- DNj(t)) - oj(t)]
9
(27) MT(t)
(31)
b,(t) = ~(RPCj(t), pj(t)>$3)
(36)
?b,(t)1
(37)
h;‘(r) = [hlJ(O>~~
Note that the control variable y; has been introduced into the function 4 with the purpose of taking into consideration the effects that the ‘political’ prices of public services might have. Moreover, as there is clearly no final demand for investment goods, h,,(t) must be equal to zero for any j. Hence final demand may be obtained as follows: D(t) = B(t).j(t)
per capita.
The regional follows: DL(t) =
unemployment
rate is calculated
where:
as
D”(r) = [DZ,(t), . . , DlJt)] FPA.P(t)
- E(t)
(29)
FPA.P(t)
Appl.
Math.
Modelling,
= vJ,(t)l. ” lpJ,@)l
(38)
j'(t) = [(I- s,)y,(t),... ,(l - S&)51
where FPA is a coefficient which provides the fraction of the potentially active population and E represents total employment in the region. The potentially active population has been disaggregated according to activities by using the vector G(t) whose jth element is given by:
382
et)
1977,
Vol 1, December
Spatial
disaggregation
It is important to give spatial dimension to the model under consideration in order to understand the
Dynamic
dynamics of the economic and demographic phenomena internal to the region. However, a model becomes very complex both from the point of view of its structure and from the calculations involved when one attempts to spatially disaggregate it. To avoid all these difficulties, we have decided to distribute the outputs of the various economic sectors over the region by means of conveniently defined spatial operators. The regional structure of the demographic sector can thus be obtained dynamically on the basis of the economic variables obtained for each separate zone. This procedure is based on the following hypotheses : (1) Each zone is large enough to ensure that workers will live and work in the same area. Consequently, commuting has not been taken into account. (2) The development rates of the various zones are proportional to the capital installed. (3) Government investment is the main incentive to economic development. (4) Services are set up in those areas where there are already nuclei of economic activities. (5) All the control variables of the model are specified in their spatial dimension. According to this hypothesis, we can assume that the capital installed K:(t) in thejth sector of the ith zone may be obtained as follows: qto)e-%“-‘“’
K~(t) =
+
~j(t)
K,(t)
modelling
in development
Total employment
planning:
L. Grandinetti
for every single zone
E’(t) = c Ej(t) enters the spatial
section
of the
demogrAphic model together with public services availability and wages. In fact, the spatial distribution of the population within the region is based on the calculation of the coefficients of attraction A’ of every single zone. These coefficients are determined, employing suitable functions, by the comparison between the local levels of wages, unemployment rate and availability of public services per capita and the regional levels of wages, unemployment rate and availability of public services per capita.
We have also introduced suitable time constants T,,, Ti 1, T12 between changes in the above variables and in the coefficients of attraction to take into account the way in which the population perceives changes in the economic structure of the region. Thus
- RS(t) - I/AR;(t)
VAR’,(t) = ;[DL’(t) I/k&(t)
1 x:
1
- DN(t) - VAR;(t)]
= -1 y’;(t) ~ - Y&) __ - I/AR;(t) T P’(t) P(t)
DL’(t) =
FPA.P’(t)
I/AR;(t).
I/AR;(t))
i cc
-q(t)
i
j
where:
1, j=k4
6.=
J
i
0, j-1,2,3
of, = public investment in the jth sector of the ith zone. Ef = employment in the jth sector of the ith zone tij = is such that
C Kj(t) = Kj(t)
Then, spatial distribution of capital from production distribution:
can be obtained
(44)
(46) (47)
RS(t) = Coj(t).&(t)/P(t)
.
(43)
(45)
A’(t)?(t) P’(t) = ~ c A’(t)
(39)
(42)
- Ei(t)
FPA.P’(t)
A’(t) = f’(VAR’,(t).
cj Ej(t)
et al.
(48)
where DL’ represents the unemployment rate and Pi the population of the ith zone. In conclusion: the spatial distribution of activities and population is controlled by the size of public investment in each zone, by {u:} and {A;}. The number of integrators of the model increases by 8 x n, where n represents the number of zones into which the region has been divided. This proved to be less complex than if n interacting models had been built, as the spatial disaggregation we have suggested is mainly algebraic. We have not dealt with the system of constraints, illustrated in Figure I, because this system is closely linked with the characteristics of the area under consideration and with the type of subdivision we have adopted.
Conclusions yj(t) = Employment follows :
z 1
J$(t) J
for each sector and zone is obtained
E;:(t) = /qt).yj(t)
(40)
I as
(41)
Our model is a continuous-time one and regards the region as an open entity in a larger economic system. It takes into account the main demographic and economic aspects of the region but does not go into detail, because a detailed examination would make its practical application either impossible or at least
&pi.
Math.
Modelling,
1977,
Vol 1, December
383
Dynamic
modelling
in development
planning:
1. Grandinetti
extremely complicated. However, our model is such that it allows, to a certain extent, a separate study of its submodel. Further development of this research would require, for example, that the model should take availability and use of resources into account, and at the same time it should be gradually implemented as the necessary data becomes available.
et al.
4 +
0
Y
Appendix The study of controllability, observability and stability of the basic dynamic scheme (equations l-5) corresponding to the economic submodels, conveniently linearized, illustrates the analysis of some structural properties of the system under consideration. The basic hypotheses are the following:
controllable, observable and stable for all the meaningful values of the parameters. Indeed we have:
(1)
The controllability
.f’(D,S)= BD,
where D, is the total demand; i.e. D, takes into account the internal demand and, possibly, the trade balance. (2) The element ‘min’ which compares required investment with available investment is replaced
by:
Linearized
model of basic economic
if Fl(t) - ZD(t) < 0
scheme
Controllability
r0
1
matrix :
-(l
1
+ P)O
-(l
+ 41 cl
00
;
-T
has full rank for non-zero and 0.
if FI( t) - ID(t) > 0
1
values of the parameters
M
Observability
With these assumptions we obtain the diagram illustrated in Figure 3, whose corresponding equations are : FZ(t) = 0Q(t) - K(t)
(Al)
IF(t) = FZ(t) + u(t)
642)
k(t) = IF(t) - j,&(t)
Figure 3
(A3)
The observability
matrix :
1 0
0 1
-(l+P) r
0 1
,
T T I-1 which always has full rank. Stability
By suitable substitutions, written in the following
the above equations can be input-state-output form:
The eigenvalues ii = -(l
i(t) = AX(L) + &u(t) y(t) = C.x(t)
of A, are: +/L); i, = -f
Hence, the system is stable whatever values.
where:
the parameters
References
According to the linear-time-invariant system if and theory2’, a system of this kind is controllable only if the matrix [BiAB] has full rank. Moreover, the system is observable if and only if the matrix [C’ i ATC”‘] has full rank. Finally the system is b.i.b.0. stable2’ if all the eigenvalues of the matrix A have the absolute value equal or less than 1. After making the necessary calculations one can easily see that the linearized scheme proposed is
384
Appl. Math.
Modelling,
1977,
Vol 1, December
Maki. W. R. and Angus. J. E. tn ‘Studies in economic planning over space and time.’ (Ed. G. G. Judge and T. Takayama) North Holland, Amsterdam, 1973 Keyhtz. N. and Fliger. W. ‘Population facts and methods of demography’. Freeman. San Francisco. 1971 Rogers, A. and Ledent, J. ‘A multiregional model of demographic and economic growth in California’. Multiregional Growth Models Project, WP4, December 1971 Rogers, A. and Walz, R. ‘Consistent forecasting of regional demographic economic growth’. Multiregional Growth Models Project, WP12, December 1972 Cordey-Hayes. M. and Gleave, D. IIASA Res. Rep., RR-74-9, July 1974 Keyfitz, N. ‘Introduction to the mathematics of population’, Addison-Wesley. 1968 Burrows, P. and Hitiris. T. ‘Macroeconomic theory: a mathematical introduction’, John Wiley, London and New York. 1974 Allen. R. G. D. ‘Teoria macroeconomica’. UTET, Torino. 1969 Mills, E. S. ‘Studies in the structure of the urban economy’, Johns Hopkins University, Baltimore, 1971
Dynamic IO 11 12 13 14
15 16
modeling 17
Perloff, H. S. and Wingo, L. Jr., (eds). ‘Issues in urban economics’, Johns Hopkins University, Baltimore, 1968 Edel, M and Rothenberg, J., (eds): ‘Reading in urban economics’, Macmillan, London, 1972 Hamilton, H. R. et al. ‘Systems simulation for regional analysis, an application to river-basin planning’, MIT, Cambridge, 1969 Runyan, H. M. IEEE Trans. Syst. Man. Cybern., 1971, SMC-1 Arrow, K. J. et al. ‘Studies in the mathematical theory of invention and production’, Stanford University, Stanford, USA 1958 Rosenblatt, M. Econometrica, 1954, 22 Nerlove, M. ‘The dynamics of supply: estimation of farmers response to price’, Johns Hopkins University, Baltimore, 1968
in development
planning:
L. Grandinetti
et al.
Cossetto, S. et al. ‘Un modello dinamico di simulazione dello sviluppo residenziale nella provincia di Roma’. Atti delle Giornate di Lavoro A.I.R.O., Palermo, September 1974 Fitch, J. B. and Frick, P. A. ‘A dynamic macroeconomic model for the state of Oregon, IEEE Trans. Syst. Man. Cybern.. SMC4. 1974. 3 La Bella, A. and Leporelli, C. ‘Localizzazione ottima di servlri pubblici: una applicazione agli insediamenti scolastici nel comune di Roma’. Atti delle Giornute di Lmoro A.I.R.O., Palermo, September 1974 Ruberti, A. and Isidori, A. ‘Elementi di Teoria dei Sistemi’, Siderea, Rome. 1973
18
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Appl.
Math.
Modelling,
1977,
Vol 1, December
385
di Sistrnli
dell Unitrrsita’
de/la Cdahrin,
Arcnnacata
(Cosenzn).
ItnlJ
Agostino La Bella
(Received 27 January
1977; revised 30 May 1977)
In this paper a dynamic regional model is proposed, aimed at understanding and forecasting the dynamic response of the population to many different investment policies in the agricultural, services and industrial sectors. In spite of its generality, the model has been conceived in order to be applied to underdeveloped regions. Particularly, from the demographic point of view, attention has been focused on migration phenomena; from the economic point of view, attention has been focused mainly on the rate of growth of the capital stock. Modularity is another feature of the proposed model; it is made of seven submodels which describe in a very aggregated form the evolution of the regional socio-economic system. The outputs bf the submodels are then spatially distributed using a set of ‘spatial operators’ which take into account many social and economic factors relevant to the sub-areas into which the region can be divided.
Introduction Development planning is usually associated with both the efforts of the less developed countries in planning for programmes of economic and social growth, and with the efforts of the developed countries in reequilibrating internal disparities in employment, income and rate of economic growth’. However, in spite of a number of studies on economic modelling7 - ’ ‘, from the one side, and on demographic modelling2v6 on the other side, these two different contributions are still awaiting a systematic synthesis3 ‘. This paper is intended as a contribution in this direction: it aims to overcome the traditional separation between economic and demographic modelling, proposing a general structure for a dynamic model in which the interrelationships between economic and demographic phenomena in a developing region, characterized by the Government’s (local and central) high incentives, are taken into account. The proposed model is based on an analysis of the
structure of the italian system, and focuses attention on the basic variables and their interrelationships. The most important features of the model are the spatial structure and the form of the equations which can be immediately expressed in terms of an input-output system. This form has recently become very important in macroeconomic modelling both because it is relatively more general and because it is possible to use advanced techniques as identification and optimization methods which have been developed by systems theory. The Appendix shows an example of the application of these methods to one of the submodels illustrated below. Figure I shows the general structure of the model. The central section comprises the submodels of the five economic sectors (production of consumer goods, production of investment goods, agriculture, public services, private services). These submodels are influenced by the final demand coming from the demographic sector, and also by an investment flow coming from the capital formation sector.
Appl.
Math. Modelling,
1977, Vol 1, December
379
Dynamic modelling
in development
planning:
L. Grandinetti et al. FZj(t)
T Constraints
,
Public
Figure
1
General
scheme
of whole
model
Total production, the result of the interaction of all these phenomena, determines (through changes in the employment structure, in income and in the supply of services) the distribution of the population within the region and the size of the migratory flow. Moreover, production affects the creation of internal resources devoted to investment. The spatial disaggregation of the submodels has been realized by means of a proper set of operators. A set of bounds has been associated with these operators in order to take into account the limitations which resulted from the physical characteristics of the region and/or from Government intervention (by means of laws and decrees). Control variables, e.g. the intervention of the public authorities, condition the development of the economic activities and their spatial distribution through investment, special incentives and regulations intended for particular sectors and areas. Government intervention, then, concerns both the money sector and spatial operators.
The economic submodels The economic part of the total model is built of five submodels from the production sectors plus the submodel of the capital formation sector. These submodels were built according to a common general scheme: if necessary changes were made in a sector in order to take its particular features into account. The general formulation is based on a linear dynamic relationship between the production ~j(t) and the installed capital K,(t):
=
,f;(Dj,
Sj)
-
(5)
K,(t)
where, for any sector: lDj(t) is the available investment; IF,(t) is the effective investment; h(t) is the depreciation coefficient of plant and machinery; Dj(t) is the internal demand, sj(t) is the trade balance; Fl,(t) is the required investment and 4 is the function related to the economic behaviour of the sector. For any sector, except the capital goods sector, the internal demand is computed in the demographic submodel. Submodel of consumer goods sector This is represented
by equations
(l)-(5).
Submodel of capital goods sector We made the hypothesis that the difference between available and effective investments is employed in technological innovation. The new variable, IN,(t) was introduced into the model to represent this type of investment, which anticipates demand. Thus, in addition to equations (l)-(5) we have the following: IN,(t) = $ZD,(t)
- FZ*(t) - IN&)]
where the time constant z represents the time innovative investment needs in order to come to maturity. Then equation (3) must be modified as follows : G(t)
= IM4
+ IN,(t)
- P&2(t)
(7)
Submodel of agricultural sector This submodel includes equations equal in structure to (6) and (7). Those equations, such as (3), (4) and (5), may be disaggregated with respect to the components of the investment in land works (irrigation, roads, canals, etc.) and in capital goods. A stochastic term E(t) which takes into account seasonal fluctuations and environmental disturbances, may be introduced in equation (1):
+
1
E(t)- Y3(0
A relationship to agricultural
may be added use, TA(t):
to compute
the land put
TA(t) = Y[K;(t)] where K>(t) is the capital
(9) in land works.
Submodel of private services sector This is represented according to the general (l)-(5). where j = l,... 5 is the sector index, aj is a conversion factor, and Tj is a time constant. An upper constraint on the production’ is given by: n,(t).
yj(t)
I
PAj(t)
(2)
where PAj(t) is the labour force availability, and /Zj(t) is an exogeneously given labour output factor. PAj(t) is calculated in the demographic submodel. The additional basic relationships are the following: r(-j(t)
=
Z~(-(t)
-
IF,(t) = min {Fl,(t),
380
Appt.
Math.
(3)
~jKj(t)
(4)
ZDj(t))
Modelling,
1977,
Vol 1, December
equations
Submodel of public services sector In this case the available investment comes solely from Government (central and/or local) intervention. The underlying hypothesis is that a fraction p of the appropriation ug is used to offer the public services at a ‘political price’, which is usually lower than the real cost. The supply of a service at a political price causes an increase of the demand, due both to an increase of the propensity to consume and to the fact that demand expressed in money terms is higher than real demand. Thus, we have:
Dynamic
W(r) = (1 - Pbdr)
(10)
o;(r) = g(W), P.4))
(11)
where D;(r) is the effective demand. Submodel of capital formation sector This submodel is the key to the economic dynamics of the whole system. The investment available for all economic activities, except the public services sector, originates here. The structure of this submodel differs significantly from that of the previous ones, and will therefore be described in detail. The output of each sector yj,j = 1,2,3,4 contributes to the formation of internal reserves for a percentage sj, 0 I sj I 1. Total liquidity M(r) of the monetary system increases by that amount and decreases by total investment 17: given by: IT(r) = W) -PI(r)
+ zz(r) + b(r) -At)
pj(r) = ZDj(r) - IF,(r)
j=
+ W)
1,4
Q(r) = M(r) 1 R,(r) ;m I
Ij(t)
Total
=
investment
=
(17) J
YZ;is a suitable coefficient; difference between marginal productivity cj of and the interest rate r’ paid to banks for the Thus we have: 1 Pj(t)
=
wj(t)
4
4
1 j=
Wj(t)
w;(r) o
=
i
wj
+
1
4
1 j=
r’j.Tj
+
1
if
w>(t) 2 0
if
w>(t) < 0
(18)
Rj
C j=
1
w>(r) = hj(r).Kj(r) + 6;.%(r) + qj(C~ - T;)
(19) (20)
in the sector j is defined
by:
(21)
z.(r) + u.(r) J
[ ’
-
sjo 1 vj
(22)
where ij, Zj and Vj are suitable coefficients. As regards the form of Rj,,j = 1,2,3,4, it should be noted that these terms have been defined so that the outgoing capital would be proportional to the fraction of external capital over total capital installed, and would decrease, for every sector, in proportion to the increase both in the ratio of marginal productivity of capital at external average level and also in the rate at which available. funds are drawn. Thus : R,(r)
where (3) the capital sector.
I ~j
= ~
where J” is the active rate of interest. The use of internal resources is defined in the model by the rate pj at which borrowers draw on available funds. This rate is calculated on the basis of: (1) the estimate %(r) of the development rate of the sector”, which is given by:
,q;.h;(r) + h;(r) = $
a certain degree of of the development rates 14 suitable constants which take the interest rates on
mj(t) = bj:.gj([) J
J
where ys is a suitable coefficient. (2) the estimate of the development rate h;(r) of public intervention uj(r) in the sector. which is given by:
et al.
where mj(r) represents external capital invested in the sector j; mj(r) is calculated on the basis of an estimate gj(r) of internal investment-including public intervention-and of the ratio between the marginal productivity of capital at local level cj and the marginal productivity of capital at average external level Cj. Thus,
4.(r)
(15)
L. Grandinetti
Pj(WWr)
in
where R,(r) represents the rate at which capital relative to the various sector goes outside the region. These terms will be explained later. Thus M(r) becomes:
planning:
ZDj(t) = Zj(t) + nj(t) + mj(t)
(12)
(14)
+ r”.M(r)
in development
where 8: and S; represent confidence in the estimate and Mi and +j and Ij are into account the effect of investment:
(13)
where Zj(r): internal resources devoted to investment sector j. From total liquidity one must also subtract the capital Q(r). which goes outside the region:
A%(r)= t yj(r).sj - IT(r) -m(r) ,= 1
modeling
Cj(
‘j
1 + nj.pj(t)) .
where nj are suitable weights associated with pj It must be stressed how important this submodel is within the framework of the dynamics of the whole system. In fact, virtually all control variables (i.e. public investment uJ and interest rates Y’~and r’j’)affect this submodel (for j # 5) thus contributing to the interaction variables ZOj, j # 5, which intervene in the dynamics of the other submodels.
Demographic and employment sector The model of this sector generates population, employable population, number of employees, the wage levels in the various activities, and final demand in the different sectors. Given the particular orientation of the sector under consideration, we have found it necessary to relate the spatial mobility of the population to variables such as unemployment, availability of public services and income’. Moreover, the process of demand formation
Appl.
Math.
Modelling,
1977,
Vol 1, December
381
Dynamic
modeling
in development
planning:
L. Grandinetti
et al.
Ilj + lj.~j(t)
Gj(t) =
C(“j + lj.rrj(f)
(30)
where Uj and lj are suitable coefficients introduced with the purpose of taking into account the effects that any change in the wage level might have on the structure of the potentially working population. The unemployment rate in this sector can be obtained as follows : PA(t).Gj(t) - E,(t)
DLj(t) = Figure
PA(t).Gj(t)
I
I
2
DemographIc
and employment
submodel
takes into account the prices of public services which will probably be ‘political’ prices. Figure 2 illustrates the population section of the submodel under consideration. Total population P(t) is calculated on the basis of birth rate 72v, and death rate TPI, ‘and on the basis of the net migratory flow FNM: P(t) = 7N.P(t) + FMN(t)
- TM.P(t)
(23)
FMN(t) = A(t).P(t)
(24)
where A(t) is a coefficient of attraction of the region. The following contribute to the formation of A(t): the comparison between availability of public services per cupitu at local level and availability of public services per capitu external to the region under consideration; the comparison between the local unemployment rate DL and the external unemployment rate DN; the comparison between regional and external income prr capitu. We have obviously assumed that the population responds to the variations in these factors of relative attraction with some delay. This delay varies from one factor to the other. Thus: A(t)
=
hJ(K(%
ff2(%
(25)
ff3(0)
- DN(t) - H,(t)]
i?&(t) = ;[DL(t) 6
(26)
where Ej represents employment in thejth sector and PA the potentially active population. We can obtain the wage level aj(t) in the sector j by comparing DLj with the corresponding variable external to the region by means of a function f,: 6j(t) = ~[S_(DLj(t)
where Ts, T,, Ts are suitable time constants; PN, represents national population; SN, public services available at national level; RPN, external income per capita at national level; f6, is a suitable function of the variables HI, H2, H3 and RPC is the regional income
(32)
RPCj can be calculated once we have taken into account the non-wage factors NWj and the rate of taxation on the incomes of the sector TRj. Thus we have : RPC,(t)
=
I
fyt) =
[aj(t).Ej(t) + NY(f)l.[l -TRj(t)l (33) Q(t) PAj( t) FPA
PA,(t) = Gj(t).PA(t)
(35)
where Pj(t) represents the part of population who have an income produced by the sector j. Final demand can be obtained by means of the vector-valued function F, which generates, on the basis of the per capitu income, the vector bj whose components represent the expenditure coefficients for goods and services which may be found on the market by the fraction of population Pj. So we have:
(28)
- MN(t)
- DNj(t)) - oj(t)]
9
(27) MT(t)
(31)
b,(t) = ~(RPCj(t), pj(t)>$3)
(36)
?b,(t)1
(37)
h;‘(r) = [hlJ(O>~~
Note that the control variable y; has been introduced into the function 4 with the purpose of taking into consideration the effects that the ‘political’ prices of public services might have. Moreover, as there is clearly no final demand for investment goods, h,,(t) must be equal to zero for any j. Hence final demand may be obtained as follows: D(t) = B(t).j(t)
per capita.
The regional follows: DL(t) =
unemployment
rate is calculated
where:
as
D”(r) = [DZ,(t), . . , DlJt)] FPA.P(t)
- E(t)
(29)
FPA.P(t)
Appl.
Math.
Modelling,
= vJ,(t)l. ” lpJ,@)l
(38)
j'(t) = [(I- s,)y,(t),... ,(l - S&)51
where FPA is a coefficient which provides the fraction of the potentially active population and E represents total employment in the region. The potentially active population has been disaggregated according to activities by using the vector G(t) whose jth element is given by:
382
et)
1977,
Vol 1, December
Spatial
disaggregation
It is important to give spatial dimension to the model under consideration in order to understand the
Dynamic
dynamics of the economic and demographic phenomena internal to the region. However, a model becomes very complex both from the point of view of its structure and from the calculations involved when one attempts to spatially disaggregate it. To avoid all these difficulties, we have decided to distribute the outputs of the various economic sectors over the region by means of conveniently defined spatial operators. The regional structure of the demographic sector can thus be obtained dynamically on the basis of the economic variables obtained for each separate zone. This procedure is based on the following hypotheses : (1) Each zone is large enough to ensure that workers will live and work in the same area. Consequently, commuting has not been taken into account. (2) The development rates of the various zones are proportional to the capital installed. (3) Government investment is the main incentive to economic development. (4) Services are set up in those areas where there are already nuclei of economic activities. (5) All the control variables of the model are specified in their spatial dimension. According to this hypothesis, we can assume that the capital installed K:(t) in thejth sector of the ith zone may be obtained as follows: qto)e-%“-‘“’
K~(t) =
+
~j(t)
K,(t)
modelling
in development
Total employment
planning:
L. Grandinetti
for every single zone
E’(t) = c Ej(t) enters the spatial
section
of the
demogrAphic model together with public services availability and wages. In fact, the spatial distribution of the population within the region is based on the calculation of the coefficients of attraction A’ of every single zone. These coefficients are determined, employing suitable functions, by the comparison between the local levels of wages, unemployment rate and availability of public services per capita and the regional levels of wages, unemployment rate and availability of public services per capita.
We have also introduced suitable time constants T,,, Ti 1, T12 between changes in the above variables and in the coefficients of attraction to take into account the way in which the population perceives changes in the economic structure of the region. Thus
- RS(t) - I/AR;(t)
VAR’,(t) = ;[DL’(t) I/k&(t)
1 x:
1
- DN(t) - VAR;(t)]
= -1 y’;(t) ~ - Y&) __ - I/AR;(t) T P’(t) P(t)
DL’(t) =
FPA.P’(t)
I/AR;(t).
I/AR;(t))
i cc
-q(t)
i
j
where:
1, j=k4
6.=
J
i
0, j-1,2,3
of, = public investment in the jth sector of the ith zone. Ef = employment in the jth sector of the ith zone tij = is such that
C Kj(t) = Kj(t)
Then, spatial distribution of capital from production distribution:
can be obtained
(44)
(46) (47)
RS(t) = Coj(t).&(t)/P(t)
.
(43)
(45)
A’(t)?(t) P’(t) = ~ c A’(t)
(39)
(42)
- Ei(t)
FPA.P’(t)
A’(t) = f’(VAR’,(t).
cj Ej(t)
et al.
(48)
where DL’ represents the unemployment rate and Pi the population of the ith zone. In conclusion: the spatial distribution of activities and population is controlled by the size of public investment in each zone, by {u:} and {A;}. The number of integrators of the model increases by 8 x n, where n represents the number of zones into which the region has been divided. This proved to be less complex than if n interacting models had been built, as the spatial disaggregation we have suggested is mainly algebraic. We have not dealt with the system of constraints, illustrated in Figure I, because this system is closely linked with the characteristics of the area under consideration and with the type of subdivision we have adopted.
Conclusions yj(t) = Employment follows :
z 1
J$(t) J
for each sector and zone is obtained
E;:(t) = /qt).yj(t)
(40)
I as
(41)
Our model is a continuous-time one and regards the region as an open entity in a larger economic system. It takes into account the main demographic and economic aspects of the region but does not go into detail, because a detailed examination would make its practical application either impossible or at least
&pi.
Math.
Modelling,
1977,
Vol 1, December
383
Dynamic
modelling
in development
planning:
1. Grandinetti
extremely complicated. However, our model is such that it allows, to a certain extent, a separate study of its submodel. Further development of this research would require, for example, that the model should take availability and use of resources into account, and at the same time it should be gradually implemented as the necessary data becomes available.
et al.
4 +
0
Y
Appendix The study of controllability, observability and stability of the basic dynamic scheme (equations l-5) corresponding to the economic submodels, conveniently linearized, illustrates the analysis of some structural properties of the system under consideration. The basic hypotheses are the following:
controllable, observable and stable for all the meaningful values of the parameters. Indeed we have:
(1)
The controllability
.f’(D,S)= BD,
where D, is the total demand; i.e. D, takes into account the internal demand and, possibly, the trade balance. (2) The element ‘min’ which compares required investment with available investment is replaced
by:
Linearized
model of basic economic
if Fl(t) - ZD(t) < 0
scheme
Controllability
r0
1
matrix :
-(l
1
+ P)O
-(l
+ 41 cl
00
;
-T
has full rank for non-zero and 0.
if FI( t) - ID(t) > 0
1
values of the parameters
M
Observability
With these assumptions we obtain the diagram illustrated in Figure 3, whose corresponding equations are : FZ(t) = 0Q(t) - K(t)
(Al)
IF(t) = FZ(t) + u(t)
642)
k(t) = IF(t) - j,&(t)
Figure 3
(A3)
The observability
matrix :
1 0
0 1
-(l+P) r
0 1
,
T T I-1 which always has full rank. Stability
By suitable substitutions, written in the following
the above equations can be input-state-output form:
The eigenvalues ii = -(l
i(t) = AX(L) + &u(t) y(t) = C.x(t)
of A, are: +/L); i, = -f
Hence, the system is stable whatever values.
where:
the parameters
References
According to the linear-time-invariant system if and theory2’, a system of this kind is controllable only if the matrix [BiAB] has full rank. Moreover, the system is observable if and only if the matrix [C’ i ATC”‘] has full rank. Finally the system is b.i.b.0. stable2’ if all the eigenvalues of the matrix A have the absolute value equal or less than 1. After making the necessary calculations one can easily see that the linearized scheme proposed is
384
Appl. Math.
Modelling,
1977,
Vol 1, December
Maki. W. R. and Angus. J. E. tn ‘Studies in economic planning over space and time.’ (Ed. G. G. Judge and T. Takayama) North Holland, Amsterdam, 1973 Keyhtz. N. and Fliger. W. ‘Population facts and methods of demography’. Freeman. San Francisco. 1971 Rogers, A. and Ledent, J. ‘A multiregional model of demographic and economic growth in California’. Multiregional Growth Models Project, WP4, December 1971 Rogers, A. and Walz, R. ‘Consistent forecasting of regional demographic economic growth’. Multiregional Growth Models Project, WP12, December 1972 Cordey-Hayes. M. and Gleave, D. IIASA Res. Rep., RR-74-9, July 1974 Keyfitz, N. ‘Introduction to the mathematics of population’, Addison-Wesley. 1968 Burrows, P. and Hitiris. T. ‘Macroeconomic theory: a mathematical introduction’, John Wiley, London and New York. 1974 Allen. R. G. D. ‘Teoria macroeconomica’. UTET, Torino. 1969 Mills, E. S. ‘Studies in the structure of the urban economy’, Johns Hopkins University, Baltimore, 1971
Dynamic IO 11 12 13 14
15 16
modeling 17
Perloff, H. S. and Wingo, L. Jr., (eds). ‘Issues in urban economics’, Johns Hopkins University, Baltimore, 1968 Edel, M and Rothenberg, J., (eds): ‘Reading in urban economics’, Macmillan, London, 1972 Hamilton, H. R. et al. ‘Systems simulation for regional analysis, an application to river-basin planning’, MIT, Cambridge, 1969 Runyan, H. M. IEEE Trans. Syst. Man. Cybern., 1971, SMC-1 Arrow, K. J. et al. ‘Studies in the mathematical theory of invention and production’, Stanford University, Stanford, USA 1958 Rosenblatt, M. Econometrica, 1954, 22 Nerlove, M. ‘The dynamics of supply: estimation of farmers response to price’, Johns Hopkins University, Baltimore, 1968
in development
planning:
L. Grandinetti
et al.
Cossetto, S. et al. ‘Un modello dinamico di simulazione dello sviluppo residenziale nella provincia di Roma’. Atti delle Giornate di Lavoro A.I.R.O., Palermo, September 1974 Fitch, J. B. and Frick, P. A. ‘A dynamic macroeconomic model for the state of Oregon, IEEE Trans. Syst. Man. Cybern.. SMC4. 1974. 3 La Bella, A. and Leporelli, C. ‘Localizzazione ottima di servlri pubblici: una applicazione agli insediamenti scolastici nel comune di Roma’. Atti delle Giornute di Lmoro A.I.R.O., Palermo, September 1974 Ruberti, A. and Isidori, A. ‘Elementi di Teoria dei Sistemi’, Siderea, Rome. 1973
18
19
20
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