THE THREE-SECTOR GROWTH HYPOTHESIS AND THE
EULER-MALTHUS ECONOMIC GROWTH MODEL:
APPLICATION TO THE ANALYSIS OF GDP DYNAMICS OF
BRAZIL, 1985-2004-2020
Michael Sonis
Department of Geography, Bar-Ilan University, Israel
Carlos R. Azzoni
University of Sao-Paulo, Department of Economics, Brazil, and Regional Economic
Application Laboratory, University of Illinois at Urbana-Champaign USA
Geoffrey J.D. Hewings
Regional Economic Application Laboratory, University of Illinois at UrbanaChampaign USA
Abstract. The paper describes the interpolation and extrapolation of annual relative
growth of GDP in the three basic economic activities: Primary, Secondary and
Tertiary 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 1985--2004-2020.
I. INTRODUCTION
Simon Kuznets stated in his Nobel lecture that “rapid changes in production
structure are inevitable” (Kuznets, 1973, p. 250). In a recent survey presented by Jens
Kruger, he argues that “the topic of structural change is frequently neglected in
economic research (Kruger, 2008, p. 331). However, many authors devoted energy to
analyzing the changing structure of the economy in the process of growth, as surveyed
by Kruger. One of these is the Three-sector Growth hypothesis, which is an economic
theory which divides economies into three aggregated sectors of activity: agriculture
and extraction of raw materials (AGM), and manufacturing (IND) and services
(SRV). In was initially developed by Clark (see three editions of his book "Conditions
of Economic Progress", 1940, 1951, 1957) and Fourastie (see Fourastie, 1949).
Clark's studies spread widely over the economics, including the agricultural
economics, macroeconomics, demography, economic policy, economic growth and
national income accounting (see Maddison, 2004).He is credited with the introduction
of the concept of Gross National Product (GNP) (at around the same time as Kuznets'
introduction of Gross Domestic Product (GDP) (Kuznets , 1953)).
GNP gives a qualitative measure of total economic activity of a nation assessed
yearly or quarterly. The GNP equals the GDP plus income earned by domestic
residents through foreign investments minus the income earned by foreign investors
in the domestic markets. GDP is calculated from the total value of goods and services
produces in an economy over the specific period of time. Clark has stressed the
dominance of different sectors (Agriculture + Mineral Industry, Manufacturing and
Services) of economy at different stages of its development and modernization.
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Clark introduced the analysis of comparative performance in three main sectors of
economy and initiated the discussion (in The Conditions of Economic Progress, 1940,
p 176) by quoting what he called the Petty's Law (Petty, 1676, see Hull' collection,
1899). "There is more to be gained by manufacture then husbandry, and by
merchandise than manufacture." He stressed the significance of the three-sector
breakdown and structural change in interpreting economic growth.
According to Fourastie the main focus of an economy's activity shifts from the
Agriculture + Mineral Industry (primary activity), through the Industry (secondary
activity) and finally to the Services (tertiary sector). Forastie (1949) 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. Subsequent work by Kuznets (1966)
elaborated upon these ideas, presenting a sectoral transitions perspective on
macroeconomic growth and development.
In continuation of ideas of Petty, Clark, Forastie and Kuznets, in this paper an
attempt will be made to measure the rates of change in agriculture, industry and
services and present the analytical structure of interaction, development and dynamics
of major three economic activities sectors. Let us formalize the Three-sector growth
hypothesis: let  SRV ,  IND ,  AGM be the average multiplicative rate of change for of
these sectors; over time, (see chapter II). The three-sector growth hypothesis,
including the Petty' Law means the following ordering:  SRV   IND  1   AGM .
But obviously the different forms of such ordering exist for different countries in
different time periods. This difference is caused by different forms of economic
evolution in different countries In Brazil, from 1960 to 1980, one observes an increase
in the share of services and parallel decreases in the share of agriculture, while the
participation of industry first increased and then diminished. From 1980 to 1990,
there was s significant grows in the share of services mainly at the expense of
industry. After 1994, there was smaller increase in the share of services and a further
reduction in the share of industry. (Da Fonceca, 2001)
As will be shown below the recent situation in Brazil corresponds to the
inequalities  SRV   AGM   IND >1 . This implies an 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.
In the three next chapters the mathematical structure of dependences between rates
of growth of three sectors and the rate of growth of all economy will be presented in
detail with the help of deterministic Euler-Malthus growth model and its
modifications. Section II presents the analytical structure of Euler-Malthus growth
model and interpolation/extrapolation method of dynamics of sectoral change; the
major part of the section III describes the approximate equation of the dependencies
between the rate of growth of total economy and the rates of growth of the sectoral
rates; section IV.
II. THE THREE-SECTOR DETERMINISTIC EULER-MALTHUS DYNAMIC
GROWTH MODEL
Consider the three deterministic empirical sequences
mtIND , mtAGM , mtSRV ; t  0,1,..., T
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(1)
presenting the dynamics of annual GDP values generated by three major economic
activities industry, agriculture and services during the time intervals t  0,1,..., T . The
three-activity deterministic Euler-Malthus dynamic Growth models (Euler, 1760;
Malthus, 1798, Hoppenstead, Peskin, 1992) can be introduced in the form of the
following iterative dynamic processes:
IND
SRV
mtAGM
  AGM mtAGM , mtIND
, mtSRV
1
1   IND mt
1   SRV mt
(2)
t  0,1,..., T , T  1, T  2,...
where mtAGM , mtIND , mtSRV are the values of annual GDP for agriculture + mineral
industry, industry and services; the growth parameters  AGM ,  IND ,  SRV are the
average annual growth rates and m0AGM ,m0SRV ,m0SRV
is the initial state of the
iteration processes (2).
It is easy to rewrite the three sector Euler-Malthus model (2) in the simple form:
t
t
t
mtAGM  m0AGM  AGM
, mtIND  m0IND IND
, mtSRV  m0SRV  SRV
.
(3)
t  0,1,..., T , T  1, T  2,...
This form of the three-sector Euler-Malthus model presents the
interpolation/extrapolation procedure for the empirical sequences (1) (interpolation
for t  0,1,..., T , 1985  2004; ; extrapolation for t  T  1, T  2,..., 2005  2020.
An approximation procedure for the derivation of the annual growth parameters
 AGM ,  IND ,  SRV will now be proposed. Calculate the empirical annual growth
parameters for each pair of sequential years i, i  1; i  1, 2,..., T
miAGM
miIND
miSRV
(4)
;


;


i , IND
i , SRV
IND
SRV
miAGM
m
m
1
i 1
i 1
The average annual multiplicative rates of GDP growth in agriculture + mineral
industry, industry and services are:
 i , AGM 
 AGM 
1T
  iAGM ;
T i1
 IND 
1T
1T
  iIND ;  SRV    iSRV
T i1
T i1
(5)
The approximated initial state m0AGM , m0IND , m0SRV of the Euler-Malthus model is
given by the averages:
1 T miAGM
1 T miIND
1 T miSRV
IND
SRV
(6)
m0AGM =
,
m
=
,
m
=
 i
 i
 i
0
0
T  1 i 0  AGM
T  1 i 0  IND
T  1 i 0  SRV
The partial sectoral correlation coefficients RAGM , RIND , RSRV representing the
goodness of fit between the empirical GDP dynamics (1) and the model (3) can be
calculated with the help of equations:
T
RAGM 
T
 mtAGM mtAGM
t 0
1
2

AGM 2  
AGM 2 
  mt     mt  
 t 0
  t 0

T
T
1
2
; RIND 
m
t 0
IND
t
1
2
mtIND

IND 2  
IND 2 
  mt     mt  
 t 0
  t 0

T
T
1
2
;
T
RSRV 
p
t 0
SRV
t
1
ptSRV
1
T
2
2
T
SRV 2  
SRV 2 
m
  t     mt  
 t 0
  t 0

(7)
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In the next two sections the different ramifications of Euler-Malthus growth model
will be presented: the total Euler-Malthus dynamics and the probabilistic chain
dynamics.
III. THE TOTAL GDP EULER-MALTHUS DYNAMICS
The three-sector Euler-Malthus dynamics (2, 3,) yields the simple total GDP
dynamics (2):
M t  mtAGM  mtIND  mtSRV 
t
t
t
 m0AGM  AGM
 m0IND IND
 m0SRV  SRV
=
(8)
t
t
t
 M 0  p0AGM  AGM
 p0IND IND
 p0SRV  SRV

where
m0AGM
m IND
m SRV
; p0IND  0 ; p0SRV  0
M0
M0
M0
is the initial probabilistic sectoral distribution of the GDP.
p0AGM 
Let us consider further the annual multiplicative rate of change  i 
(9)
Mi
M i 1
and the average rate of change
1 T
1 T miAGM  miIND  miSRV
(10)
   i   AGM
SRV
T i 1
T i 1 mi 1  miIND
1  mi 1
Next we replace the Arithmetical mean with the Geometrical mean. Using the
formulae (4,5,) we obtain the approximate equation
1
    AGM   IND   SRV 
(11)
3
which represents the connection between total average rate of growth and the rates
of growth of all sector rates of growth.
IV. THE PROBABILISTIC THREE-SECTOR GDP EULER-MALTHUS
DYNAMICS
The probabilistic three-sector GDP growth dynamics
mtAGM
mtSRV
m1IND
AGM
IND
SRV
pt

; pt 
; pt 
(12)
Mt
Mt
Mt
can be written with the help of equations (2, 3, 8, 9 and 12) in the following
probabilistic chain dynamics:
t
p0AGM  AGM
ptAGM  AGM t
;
t
t
p0  AGM  p0IND IND
 p0SRV  SRV
IND
t
p

t
p0AGM  AGM
t
p0IND IND
;
t
t
 p0IND IND
 p0SRV  SRV
(13)
t
p0SRV  SRV
t
t
t
p0AGM  AGM
 p0IND IND
 p0SRV  SRV
This probabilistic chain dynamics is equivalent to the following logistic growth
probabilistic chain (see Sonis, 1987, 2003):
ptSRV 
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ptAGM

1
ptIND
1 
ptSRV
1 
 AGM ptAGM
 AGR ptAGM
,
  IND ptIND   SRV ptSRV
 AGM ptAGM
 IND ptIND
,
  IND ptIND   SRV ptSRV
 AGM ptAGM
 SRV ptSRV
  IND ptIND   SRV ptSRV
(14)
Let introduce the multiplicative rates of growth in each sector frequency:
 AGM
1 T piAGM
1 T piIND
1 T piSRV
  AGM ;  IND   IND ;  SRV   SRV
T i 1 pi 1
T i 1 pi 1
T i 1 pi 1
(15)
Analogously to the approximate equation (11) it is possible to prove that
 AGM 
 ARM

;  IND 
 IND

;  SRV 
 SRV

(17)
In the next sections we will present the analysis of the tendencies of sectoral
growth in the economic GDP dynamics in Brazil, 1985-2004-2020.
V. BRAZILIAN RELATIVE GDP GROWTH DYNAMICS, 1985-2004
Brazil is a good case to apply the above methodology, since it is a large country,
and has been under intensive productive changes, notably after 1990. Starting from a
typical Third World agricultural economy previously to World War II, in which
coffee exports accounted for most of its foreign-oriented activities, the country
followed an import substitution strategy until the mid-80s, but remained quite closed
to foreign trade. Starting in 1990, an opening-up process took place, and nowadays,
although still relatively closed, its economy is more open. Coinciding with this
opening process, a strong movement of increased agricultural production took place,
together with an impressive investment in energy-related crops, of which its ethanol
program is emblematic. As a result, Brazil is a major international player nowadays in
beef, grains, cotton, coffee, and, of course, energy-related products. These changes
took place in a context of a fast-growing internal market, since its population
increased from 41 millions in 1940 to 187 millions in 2008. These productive changes
had consequences in the shares of the three main productive sectors in total output. In
the decade 1985-1995 the shares of primary and secondary activities were almost the
same, around 20% - 25%, with the tertiary sector accounting for the remaining share.
From mid-1985s to the2004, the share of the secondary activities changes between
56% and 71%, its highest share ever. This came about from a decrease in the share of
primary activities, reaching around 9%- 10% in late 1990s, and even the tertiary
activities, with around 35% in that period. Opening the economy to foreign trade
cased important changes in these shares. Primary activities, in spite of the success of
Brazilian exports in the above mentioned products, stabilized their share around 10%
in 2000.and growing to 14% in 20041.
Given the changes mentioned, we will deal in this paper with a period, 1985-2004.
Table 1 represents the distribution of Brazilian GDP during two decades, 1985-2004.
1
See Contas Regionalis de Brazil, 1985-2004, and Guilhoto and Hewings (2001) and Baer (2007) for
a detailed description of the evolution of Brazilian economy.
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Table 1. Brazilian sectoral GDP growth dynamics: Empirical data and Interpolation
1985-2004; Extrapolation 2005-2020, (billion R$).
SRV
SRV
model
IND
IND
model
AGM
AGM
model
Total
Total
Years
model
207.091
189
733.0998
737
111.6944
157
1051.885
1083
1985
219.9155
222
747.6841
753
115.2315
162
1082.831
1137
1986
233.5342
227
762.5585
795
118.8807
134
1114.973
1156
1987
247.9963
222
777.7289
831
122.6454
130
1148.371
1183
1988
263.354
223
793.201
860
126.5293
123
1183.084
1206
1989
279.6627
299
808.9809
844
130.5363
114
1219.18
1257
1990
296.9814
362
825.0748
886
134.6701
111
1256.726
1359
1991
315.3725
288
841.4888
902
138.9348
101
1295.796
1291
1992
334.9026
267
858.2294
856
143.3346
108
1336.467
1231
1993
355.6421
331
875.303
838
147.8737
153
1378.819
1322
1994
377.666
454
892.7163
905
152.5566
136
1422.939
1495
1995
401.0537
510
910.476
856
157.3878
137
1468.917
1503
1996
425.8897
545
928.589
902
162.3719
138
1516.851
1585
1997
452.2638
558
947.0624
886
167.5139
138
1566.84
1582
1998
480.2712
544
965.9032
893
172.8187
154
1618.993
1591
1999
510.0129
512
985.1189
869
178.2916
162
1673.423
1543
2000
541.5965
504
1004.717
897
183.9377
182
1730.251
1583
2001
575.1359
483
1024.705
895
189.7626
215
1789.603
1593
2002
610.7524
470
1045.09
949
195.772
240
1851.615
1659
2003
648.5744
500
1065.881
1018
201.9718
249
1916.427
1767
2004
688.7387
1087.086
208.3678
1984.192
2005
731.3902
1108.712
214.9664
2055.069
2006
776.683
1130.769
221.7739
2129.226
2007
824.7807
1153.265
228.7971
2206.842
2008
875.8569
1176.208
236.0426
2288.107
2009
930.0961
1199.607
243.5176
2373.221
2010
987.6941
1223.472
251.2293
2462.396
2011
1048.859
1247.812
259.1853
2555.856
2012
1113.812
1272.636
267.3931
2653.841
2013
1182.787
1297.954
275.861
2756.601
2014
1256.033
1323.775
284.5969
2864.405
2015
1333.816
1350.11
293.6095
2977.535
2016
1416.415
1376.969
302.9076
3096.292
2017
1504.129
1404.363
312.5
3220.992
2018
1597.275
1432.301
322.3963
3351.973
2019
1696.19
1460.795
332.6059
3489.591
2020
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Table 2. Interpolation and extrapolation relative GDP dynamics of Primary,
Secondary and Tertiary economic activities, Brazil, 1985-2004-2020 (billion R$).
Tertiary
Tertiary
model
Secondary
Secondary
model
Primary
Primary
Years
model
0.196876
0.174515
0.696939
0.680517
0.106185
0.144968
1985
0.203093
0.195251
0.69049
0.662269
0.106417
0.14248
1986
0.209453
0.196367
0.683925
0.687716
0.106622
0.115917
1987
0.215955
0.187658
0.677246
0.702451
0.1068
0.10989
1988
0.222599
0.184909
0.670452
0.713101
0.106949
0.10199
1989
0.229386
0.237868
0.663545
0.67144
0.107069
0.090692
1990
0.236313
0.266372
0.656527
0.65195
0.107159
0.081678
1991
0.243381
0.223083
0.649399
0.698683
0.10722
0.078234
1992
0.250588
0.216897
0.642163
0.69537
0.107249
0.087734
1993
0.257932
0.250378
0.634821
0.633888
0.107247
0.115734
1994
0.265413
0.303679
0.627375
0.605351
0.107212
0.09097
1995
0.273027
0.339321
0.619828
0.569528
0.107145
0.091151
1996
0.280772
0.343849
0.612182
0.569085
0.107045
0.087066
1997
0.288647
0.352718
0.604441
0.560051
0.106912
0.087231
1998
0.296648
0.341923
0.596607
0.561282
0.106745
0.096794
1999
0.304772
0.331821
0.588685
0.563189
0.106543
0.10499
2000
0.313016
0.318383
0.580677
0.566646
0.106307
0.114972
2001
0.321376
0.303202
0.572588
0.561833
0.106036
0.134965
2002
0.329849
0.283303
0.564421
0.572031
0.10573
0.144665
2003
0.338429
0.282965
0.556181
0.576118
0.10539
0.140917
2004
0.347113
0.547873
0.105014
2005
0.355896
0.539501
0.104603
2006
0.364772
0.53107
0.104157
2007
0.373738
0.522586
0.103676
2008
0.382787
0.514053
0.103161
2009
0.391913
0.505476
0.102611
2010
0.401111
0.496863
0.102026
2011
0.410375
0.488217
0.101408
2012
0.419698
0.479545
0.100757
2013
0.429074
0.470853
0.100073
2014
0.438497
0.462147
0.099356
2015
0.44796
0.453432
0.098608
2016
0.457455
0.444716
0.097829
2017
0.466977
0.436003
0.09702
2018
0.476518
0.427301
0.096181
2019
0.486071
0.418615
0.095314
2020
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These empirical statistical data can be used for the evaluation of parameters of
Euler-Malthus models (2, 3). Equation (5) provides the aggregate relative growth
rates,  SRV  1.0619 :  AGM  1.0317;  IND  1.0179 . The type of the GDP growth in
Brazil has a form:
(18)
 SRV >  AGM   IND  1
with average total rate of growth
1
  ( SRV +  AGM   IND )  1.0372   Total =1.0279
(19)
3
This means that the aggregate relative change in services is larger than in
agriculture and industry. Because each sectorial rate of growth is bigger then 1 the
GDP is increasing in each economic sector. Agriculture presents the important
unusual behavior. The initial states of GDP dynamics calculated with equation (6) are:
m0AGM  111.7, m0IND  333.1, m0SRV  189.2 .
The interpolation and extrapolation GDP dynamics provide the interpolation and
extrapolation forecast of Brazilian GDP relative dynamics for 1985-2004-2020 (see
table 2):
Inte rpolation/Extrapolation Brazilian GDP
Dynamics: 1985-2004-2020.
4000
Extrapolation
Interpolation
3500
3000
Total
Total Model
2000
1500
R$ billions
2500
AGM
AGMmodel
IND
gIND mo
SRV
SRVmodel
1000
500
2020
2004
34 31 28 25 22 19 16 13 10
7
4
0
1985
1
YEARS
Fig. 1. Interpolation and Extrapolation of Brazil GDP dynamics in different sectors
and total annual GDP dynamics.
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The goodness of fit between empirical agricultural GDP relative dynamics and the
model can be measured with the help of the partial correlation coefficient (8), which
generates the following value RAGR  0.6963 . Using the same equation, the
goodness of fit between empirical industrial GDP relative dynamics is RIND  0.8324 .
For services the value is RSRV  0.9833 .
The low value of the correlation coefficient for Agriculture + mineral Industry can
be explained by jump in change in 2001 in Primary sector during the decade 19942004 (see table1).
Figure 1 presents the geometrical presentation of the GDP empirical and model
dynamics for the three macro sectors in Brazil including interpolation for 1985-2004
and extrapolation for 2005 through 2020 (cf. tables 1-2). We also include in this
figure the total relative growth of GDP in Brazil calculated by equations (8, 9).
VI THE PROBABILISTIC CHAIN OF SECTORAL GROWTH IN BRAZIL,
1985-2004-2020
Probabilistic Chain of basic e conomic activ itie s
in Brazil, 1985-2004-2020
0.8
0.7
Extrapolation
0.6
Primary
0.5
Primary model
0.4
%
Interpolation
0.3
Secondary
Secondary model
Tertiary
Tertiary model
0.2
0.1
2020
2004
1985
34 31 28 25 22 19 16 13 10
7
4
0
1
YEARS
Fig. 2. Probabilistic Chain of Main Sectors in Brazil.
Equation (16) presents the chain of probabilistic dynamics of relative growth GDP
dynamics of sectors of main economic activities in Brazil. The relative rates of growth
in frequencies of GDP growth in different sectors of Brazilian economy calculated
with the help of formula (19) are:
 AGM 
 ARM

 0.9946;  IND 
 IND

 0.9814;  SRV 
7-161
 SRV

=1.0238
(20)
The type of the rates of growth in frequencies of GDP growth in different
economic sectors
(21)
 SRV >1>  AGM   IND
This mean that frequency of GDP in Tertiary sector (Services) is increasing; the
frequency of Primary and Secondary sectors (Agriculture+ Mineral Industry and
Industry) are decreasing in such a way that decrease in Secondary sector bigger then
in Primary sector.
The chains of the probabilistic distribution of GDP for the aggregated sectors in
Brazil (see figure 2) reveal the growth of the relative portion (%) of GDP in services,
the stability of the relative portion (%) of GDP in Primary activities and the decrease
in relative portion (%) of GDP in industry.
VII. CONCLUSIONS
The theoretical part of this paper introduced 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 dynamic growth forms provide the basis for 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 the three-sector growth of GDP in Brazil, based on empirical data of
annual GDP in Primary, Secondary and Tertiary activities for the period 1985-2004,
interpolation of this data using average annual rates of GDP growth and the
extrapolation of Brazilian economic dynamics for 2004-2020. It is possible to stress
that the analytical scheme of GDP dynamics and especially the formula (11) presented
in this paper can be expanded to the analysis of GDP of multi-sectoral and
multuregional dynamics in Brazil and in this way to classify types of regional growth
of GDP in Brazil in different time periods. Here the notions of so called Matrioska
principle (Matrioshka principle represent the nesting hierarchy of economic regions)
(cp Sonis and Hewings, 1990; Sonis, Hewings and Okuyama, 2001) could be used.
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