___________________________________________________________________________________
1
The Three-sector Growth Hypothesis: Application to the analysis of GDP relative
dynamics of Brazil, 1994-2004-2014
Michael Sonis, Department of Geography, Bar-Ilan University, Israel, and Regional Economic
Application Laboratory, University of Illinois at Urbana-Champaign USA, sonism@mail.biu.ac.il.
Carlos R. Azzoni, University of Sao-Paulo, Department of Economics, Brazil, and Regional Economic
Application Laboratory, University of Illinois at Urbana-Champaign USA, cazzoni@usp, br
Geoffrey J.D. Hewings, Regional Economic Application Laboratory, University of Illinois at UrbanaChampaign USA, hewings@uiuc.edu
Abstract: The paper describes the interpolation and extrapolation of annual relative
growth of GDP in Agriculture, Industry and Services economic activities. The annual
regional economic growth is evaluated with the help of the Euler-Malthus dynamic
growth model in three different analytical realizations. As a regional example of annual
economic growth in Brazil represents the results of interpolation and extrapolation of
empirical dynamics of GDP in Brazil in the years 1994-2004-2014.
Key words: Euler-Malthus non-linear dynamics model, Interpolation/Extrapolation
procedure. The-sector growth hypothesis. Log-linear probabilistic chain of economic
growth. GDP dynamics of Brazil.
I. Introduction.
The Three-sector Growth hypothesis is an economic theory which divides economies into
three sectors of activity: Agriculture and extraction of raw materials (primary),
Manufacturing (Industry – secondary) and Services (tertiary). In was developed by C.
Clark and J. Fourastie (see Fourastie, 1949). According to the theory the main focus of
economy's activity shifts from the primary, through the secondary and finally to the
tertiary sector. Fourastie saw the process as essentially positive and writes of the increase
in quality of life, social security, blossoming of education and culture, higher level of
qualifications, humanization of work and avoidance of unemployment.
We will try to measure the rates of change in Agriculture, Industry and Services. The
three-sector GDP Growth hypothesis means that if the relative rate of change  in some
economic sector is lesser than 1 thus the GDP dynamic pointed on decrease; if the relative
rate of change  is bigger than 1 than the corresponding GDP dynamics is increasing.
The situation corresponding to decrease of Agricultural production and to the increase of
Manufacturing and Services (as in the end of forties when the Tree-sector Hypothesis first
appeared) corresponds to the following inequalities:  SRV   IND  1   AGR
.
Historically, from 1960 to 1980, one observes an increase in the share of Services and
parallel decrease in the Agriculture share, while the participation of Industry is increased
and diminished in 1970. From 1980 to 1990, there was s significant grows in the share of
Services mainly on the expense of Industry. After 1994 there was smaller increase of the
share of Services and the reduction in Industry. (Da Fonceca, 2001)
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2
As will be shown below the recent situation in Brazil corresponds to the inequalities
 SRV   AGR   IND >1 . This means the increase of the GDP dynamics in all of three
sectors of activity; the difference in the position of Agriculture is connected with the
extension of the area of sugar-cane for the production of ethanol for car fuel oil and with
the fact that Agriculture outperformed the Industry continuing the pattern during 19801990 (Guilhoto and Hewings, 2001, p.5).
In the next three chapters the analytical treatment of deterministic Euler-Malthus
dynamic growth model and its modification can be represented in detail.
I. The tree-sectors Deterministic Euler-Malthus dynamic growth model.
Consider the three deterministic empirical sequences
mtIND , mtAGR , mtSRV ; t  0,1,..., T ,
(1)
representing the dynamics of annual GDP values generated by three major economic
activities Industry, Agriculture and Services during the time intervals t  0,1,..., T .
It is possible to write that
mtIND  mtAGR  mtSRV  M t ; t  0,1,..., T
(2)
where M t , t  0,1,..., T , are the annual GDP absolute values of economic region.
The tree-activities deterministic Euler-Malthus dynamic Growth models (Euler, 1760;
Malthus, 1798, Hoppenstead, Peskin, 1992} can be introduced in the form of the
following iteration dynamic processes:
AGR
mtAGR
,
1   AGR mt
IND
mtIND
,
1   IND mt
(3)
SRV
mtSRV
.
1   SRV mt
t  0,1,..., T , T  1, T  2,...
where mtAGR , mtIND , mtSRV are the values of annual GDP for Agriculture, Industry and
Services; the growth parameters  AGR ,  IND ,  SRV are the average annual growth rates
and m0AGR ,m0SRV ,m0SRV
is the initial state of the iteration processes (3).
It is easy to rewrite the tree sector Euler-Malthus model (3) in the simple form:
___________________________________________________________________________________
3
t
mtAGR  m0AGR AGR
,
t
mtIND  m0IND IND
,
(4)
t
mtSRV  m0SRV  SRV
.
t  0,1,..., T , T  1, T  2,...
This
form
of
the
tree-sector
Euler-Malthus
model
presents
the
interpolation/extrapolation procedure for the empirical sequences (1) [interpolation
for t  0,1,..., T , ; extrapolation for t  T  1, T  2,... ].
We will propose now the approximation procedure for the derivation of the annual growth
parameters  AGR ,  IND ,  SRV . Let us calculate the empirical annual growth parameters for each
pair of sequential years i, i  1; i  1, 2,..., T
 i , AGR 
miAGR
;
miAGR
1
 i , IND 
miIND
;
miIND
1
(5)
miSRV
miSRV
1
The average annual rates of GDP growth in Agriculture, Industry and Services are given
by averages:
 i , SRV 
1 T
  iAGR ;
T 1
1 T
   iIND ;
T 1
1 T
   iSRV
T 1
 AGR 
 IND
 SRV
The approximated initial state m0AGR , m0IND , m0SRV
(6)
of the Euler-Malthus model is
given by the averages:
AGR
0
m
IND
0
m
1 T miAGR
=  i ,
T i 1  AGR
1 T miAGR
=  i ,
T i 1  AGR
m0SRV =
1 T miAGR
 i
T i 1  AGR
( 7)
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4
The partial sectoral correlation coefficients RIND , RAGR , RSRV representing the goodness
of fit between the empirical GDP dynamics (1) and the model (4) can be calculated with
the help of formulae:
T
RAGR 
m
AGR
t
t 0
mtAGR
1
2

AGR 2  
AGR 2 
  mt     mt  
 t 0
  t 0

T
T
1
2
;
T
RIND 
m
t 0
IND
t
1
2
mtIND

IND 2  
IND 2 
  mt     mt  
 t 0
  t 0

T
T
1
2
;
(8)
T
RSRV 
p
t 0
SRV
t
1
ptSRV
1
T
2
2
T
SRV 2  
SRV 2 
  mt     mt  
 t 0
  t 0

II. The total GDP Euler-Malthus dynamics.
The tree-sector Euler-Malthus dynamics (3, 4) gives the simple total GDP dynamics (2):
M t  mtAGR  mtIND  mtSRV 
t
t
t
 m0AGR AGR
 m0IND IND
 m0SRV  SRV
=
(9)
t
t
t
 M 0  p0AGR AGR
 p0IND IND
 p0SRV  SRV

where
p0AGR 
m0AGR
;
M0
p0IND 
m0IND
;
M0
m0SRV
p 
M0
is the initial probabilistic distribution of GDP dynamics.
SRV
0
III. The probabilistic tree-sector GDP Euler-Malthus dynamics.
The probabilistic tree-sector GDP growth dynamics
(10)
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5
ptAGR 
mtAGR
;
Mt
ptIND 
m1IND
;
Mt
(11)
mtSRV
Mt
can be written with the help of formulae (3, 4, 9, 10) in the following probabilistic chain
dynamics:
t
m AGR
p0AGR AGR
ptAGR  t
 AGR t
;
t
t
Mt
p0  AGR  p0IND IND
 p0SRV  SRV
ptSRV 
IND
t
p
t
mtIND
p0IND IND

 AGR t
;
t
t
Mt
p0  AGR  p0IND IND
 p0SRV  SRV
(12)
t
mtSRV
p0SRV  SRV
 AGR t
t
t
Mt
p0  AGR  p0IND IND
 p0SRV  SRV
This probabilistic chain dynamics is equivalent to the following Logistic growth probabilistic chain
(see Sonis, 1987, 2003):
 AGR ptAGR
ptAGR

,
1
 AGR ptAGR   IND ptIND   SRV ptSRV
ptSRV 
ptIND
1 
SRV
t 1
p
 IND ptIND
,
 AGR ptAGR   IND ptIND   SRV ptSRV
(13)
 SRV ptSRV

 AGR ptAGR   IND ptIND   SRV ptSRV
The system of difference equations (13) equivalent to the following system of difference equations:
  IND  IND   SRV  SRV 
AGR
ptAGR
 ptAGR
 pt  1 
 pt 
1  pt
1  1 
  AGR 
  AGR 

  AGR  AGR   SRV  SRV 
IND
ptIND
 ptIND
 pt  1 
 pt 
1  pt
1  1 
  IND 
  IND 

(14)
  AGR  AGR   IND  IND 
SRV
ptSRV
 ptSRV
 pt  1 
 pt 
1  pt
1  1 
  SRV 
  SRV 

AGR
AGR
p
p
The expression t 1 AGR t
presents the relative increment of the share ptAGR
pt 1
combination of the shares ptIND , ptSRV
as a linear
. The similar interpretation of the second and third row in (14)
exists for relative increments of the shares ptIND , ptSRV
the co-influence matrix
.This gives the possibility of introduction of
___________________________________________________________________________________
6


 
1  IND 1  SRV 
 0
 ARG
 ARG 

 
 
(15)
  1  AGR
0
1  SRV 
 IND 
  IND
 


0 
1  AGR 1  IND
 SRV
  SRV

representing the indirect influence of the shares of growth of different sectors of activity on the
relative increments of the shares of other sectors.
IV. Brazilian relative GDP growth dynamics, 1994-2004.
Table 1 represents the relative distribution of Brazilian GDP during the decade, 19942004.
t
Year
Total  M t 
mtAGR
mtIND
mtSRV
0
1994
1387
135
921
331
1
1995
1429
122
853
454
2
1996
1503
122
871
510
3
1997
1585
122
918
545
4
1998
1582
125
899
558
5
1999
1591
125
922
544
6
2000
1543
116
915
512
7
2001
1583
152
947
504
8
2002
1593
153
957
483
9
2003
1660
172
1018
470
T  10
2004
1766
168
1098
500
Table 1. Brazilian sectoral GDP growth dynamics, 1994-2004 (billion R$).
These empirical statistical data can be used for the evaluation of parameters of EulerMalthus models (3, 4). Formulae (6) give the aggregate relative growth
rates  AGR  1.025424;  IND  1.018638 :  SRV  1.048314 ; i.e. the aggregate relative
change in Services is bigger then in Agriculture and Industry. The initial states of GDP
dynamics calculated with the help of formulae (7) are:
___________________________________________________________________________________
7
m0AGR  117.9, m0IND  854, m0SRV  390
.
The interpolation and extrapolation GDP dynamics give the forecast of GDP relative
dynamics for the Brazilian GDP, 1994-2004-2014 (see tables 2, 3, 4):
Empirical GDP
Agricaltial Dynamics
Interpolation
Extrapolation
mtIND
mtIND
t  0,1,...T
mtIND
t  T  1, T  2,...
1994
1995
1996
1997
135
122
122
117.9
120.9
124.0
122
127.2




4
5
6
7
8
1998
1999
125
125
116
2001
2002
132
130.4
133.7
137.1
140.6
153
144.2
9
T  10
11
12
2003
2004
172
168
147.8
151.6
13
2005
2006
2007






14
15
16
17
18
2008
2009
2010
2011
2012










167.6
171.8
176.2
180.7
185.3
19
20
2013
2014




190.0
194.9
t
Year
Initial State
t 0
1
2
3
2000







155.4
159.4
163.4
Table 2. Interpolation and extrapolation relative Agricultural GDP dynamics,
Brazil, 1994-2004-2014 (billion R$).
The goodness of fit between empirical Agricultural GDP relative dynamics and the
model can be measured with the help of partial correlation coefficient (8), which gives
us the following value: RAGR  0.748443 .
___________________________________________________________________________________
8
Empirical GDP
t
Year
Interpolation
Extrapolation
mtIND
m
t  0,1,...T
mtIND
t  T  1, T  2,...
Industrial Dynamics
IND
t
Initial State
t 0
1
1994
1995
921
853
854
870


2
3
1996
1997
871
918
806.1
902.6


4
5
6
7
8
1998
1999
2000
2001
2002
899
922
915
947
957
915.4
936.6
954
971.8
989.9





9
T  10
11
12
13
2003
2004
2005
2006
2007
1018
1098



1008.4
1027.2





1046.4
1065.8
1085.7
14
15
16
17
18
2008
2009
2010
2011
2012










1105.9
1126.5
1147.5
1168.5
1190.7
19
20
2013
2014




1212.9
1235.5
Table 3. Interpolation and extrapolation Industrial GDP dynamics, Brazil, 19942004-2014 (billion R$).
The goodness of fit between empirical Industrial GDP relative dynamics can be
measured with the help of correlation coefficient (8), which gives us the following
value: RIND  0.99927 . This high value of fitness is contrasted with a low value of
fitness of Agriculture.
___________________________________________________________________________________
9
Empirical GDP
t
Year
Dynamics of Services
SRV
t
m
Interpolation
SRV
t
Extrapolation
m
t  0,1,...T
mtSRV
t  T  1, T  2,...
Initial State
t0
1
2
3
1994
1995
1996
1997
331
454
510
545
390
409
428.7
449.4




4
5
6
7
8
1998
1999
2000
2001
2002
558
544
512
504
483
471.2
493.9
517.8
542.8
569





9
T  10
11
12
2003
2004
2005
2006
470
500


596.5
625.3




655.6
687.2
13
2007


720.4
14
15
16
17
18
2008
2009
2010
2011
2012










755.2
791.7
830
870.1
912.1
19
20
2013
2014




956.2
1002.4
Table 4. Interpolation and extrapolation relative Services GDP dynamics, Brazil,
1994-2004-2014 (billion R$).
The goodness of fit between empirical Services GDP relative dynamics and the model
can be measured with the help of correlation coefficient (8), which gives us the
following value: RSRV  0.987104
.
In the next Figure 1 we present the graph giving the geometrical presentation of the
GDP empirical and model dynamics of Agriculture, Industry and Services in Brazil
including interpolation for 1994-2004 and Extrapolation for 2005 till 2014 (cf. tables
1-4). We include into this figure also the total relative growth of GDP in Brazil
calculated by formulae (9, 10).
___________________________________________________________________________________
10
Interpolation/Extrapolation of Brazilian GDP
dynamics: 1994-2004-2014
3000
2500
AGR
2000
1000
R$ billon
1500
AGR model
IND
IND model
SRV
SRV model
Total
Total model
500
0
21 19 17 15 13 11 9
7
5
3
1
Years
Figure 1. Interpolation and Extrapolation of Brazil GDP dynamics in different sectors
and total annual GDP dynamics.
V. The probabilistic chain of growth in different sectors of economic activities in
Brazil, 1994-2004-2014.
Formulae (12) present the chain of probabilistic dynamics of relative growth GDP
dynamics of sectors of main economic activities in.
The co-influence matrix (15) based on the parameters of GDP growth of economic
activities  AGR  1.025424;  IND  1.018638 :  SRV  1.048314 has the following form
___________________________________________________________________________________
11

 0

 
  1  AGR
  IND
 
1  AGR
  SRV
 IND

1  SRV
 ARG
 ARG

0
1  SRV
 IND

1  IND
0
 SRV
1


 
0
0.006618 0.022322 
 

0
0.029133 
   0.006662
  0.021835 0.028308

0

 


(16)
This co-influence matrix generates the chain of probabilistic distribution of GDP in the
main sectors of economics in Brazil (see figure 2). This chain represent the growth of
the relative portion (%) of GDP in Services, the stability of the relative portion (%) of
GDP in Agriculture and the decrease of relative portion (%) of GDP in Industry.
Probabilistic chain of main sectors in Brazil,
1994-2004-2014
0.7
0.6
0.5
pAGR
pAGRmodel
%
0.4
0.3
0.2
pIND
pIND model
pSRV
pSRVmod
0.1
0
21 19 17 15 13 11
9
7
5
3
1
Years
VI. Conclusion.
The theoretical part of this paper includes the consideration of the Euler-Malthus
dynamic growth model in three different forms: the three economic activities growth,
the total economic growth and the log-linear probabilistic growth realizations. All these
___________________________________________________________________________________
12
dynamic growth forms results in the interpolation/extrapolation procedure of relative,
total and probabilistic non-linear growth dynamics.
The second part of the paper presents the application of the interpolation/extrapolation
technique to the analysis of three-sector growth of GDP in Brazil, based on empirical
data of annual GDP in Agriculture, Industry and Services in the period of 1994-2004,
interpolation of this data with the help of average annual rates of GDP growth and the
extrapolation of Brazilian economic dynamics for 2004-2014.
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