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 Urbana-Champaign 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.
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 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:.
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 countriesIn 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. 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
(1)
presenting the dynamics of annual GDP values generated by three major economic activities industry, agriculture and services during the time intervals. 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:
(2)
where are the values of annual GDP for agriculture + mineral industry, industry and services; the growth parameters are the average annual growth rates and 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:
(3)
This form of the three-sector Euler-Malthus model presents the interpolation/extrapolation procedure for the empirical sequences (1) (interpolation for; extrapolation for
An approximation procedure for the derivation of the annual growth parameters will now be proposed. Calculate the empirical annual growth parameters for each pair of sequential years
(4)
The average annual multiplicative rates of GDP growth in agriculture + mineral industry, industry and services are:
(5)
The approximated initial state of the Euler-Malthus model is given by the averages:
(6)
The partial sectoral correlation coefficients representing the goodness of fit between the empirical GDP dynamics (1) and the model (3) can be calculated with the help of equations:
(7)
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):
(8)
where
(9)
is the initial probabilistic sectoral distribution of the GDP.
Let us consider further the annual multiplicative rate of change
and the average rate of change
(10)
Next we replace the Arithmetical mean with the Geometrical mean. Using the formulae (4,5,) we obtain the approximate equation
(11)
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
(12)
can be written with the help of equations (2, 3, 8, 9 and 12) in the following probabilistic chain dynamics:
(13)
This probabilistic chain dynamics is equivalent to the following logistic growth probabilistic chain (see Sonis, 1987, 2003):
(14)
Let introduce the multiplicative rates of growth in each sector frequency:
(15)
Analogously to the approximate equation (11) it is possible to prove that
(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 2004[1].
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.
Table 1. Brazilian sectoral GDP growth dynamics: Empirical data and Interpolation 1985-2004; Extrapolation 2005-2020, (billion R$).
Years / Total / Total / AGM / AGM / IND / IND / SRV / SRVmodel / model / model / model
1985 / 1083 / 1051.885 / 157 / 111.6944 / 737 / 733.0998 / 189 / 207.091
1986 / 1137 / 1082.831 / 162 / 115.2315 / 753 / 747.6841 / 222 / 219.9155
1987 / 1156 / 1114.973 / 134 / 118.8807 / 795 / 762.5585 / 227 / 233.5342
1988 / 1183 / 1148.371 / 130 / 122.6454 / 831 / 777.7289 / 222 / 247.9963
1989 / 1206 / 1183.084 / 123 / 126.5293 / 860 / 793.201 / 223 / 263.354
1990 / 1257 / 1219.18 / 114 / 130.5363 / 844 / 808.9809 / 299 / 279.6627
1991 / 1359 / 1256.726 / 111 / 134.6701 / 886 / 825.0748 / 362 / 296.9814
1992 / 1291 / 1295.796 / 101 / 138.9348 / 902 / 841.4888 / 288 / 315.3725
1993 / 1231 / 1336.467 / 108 / 143.3346 / 856 / 858.2294 / 267 / 334.9026
1994 / 1322 / 1378.819 / 153 / 147.8737 / 838 / 875.303 / 331 / 355.6421
1995 / 1495 / 1422.939 / 136 / 152.5566 / 905 / 892.7163 / 454 / 377.666
1996 / 1503 / 1468.917 / 137 / 157.3878 / 856 / 910.476 / 510 / 401.0537
1997 / 1585 / 1516.851 / 138 / 162.3719 / 902 / 928.589 / 545 / 425.8897
1998 / 1582 / 1566.84 / 138 / 167.5139 / 886 / 947.0624 / 558 / 452.2638
1999 / 1591 / 1618.993 / 154 / 172.8187 / 893 / 965.9032 / 544 / 480.2712
2000 / 1543 / 1673.423 / 162 / 178.2916 / 869 / 985.1189 / 512 / 510.0129
2001 / 1583 / 1730.251 / 182 / 183.9377 / 897 / 1004.717 / 504 / 541.5965
2002 / 1593 / 1789.603 / 215 / 189.7626 / 895 / 1024.705 / 483 / 575.1359
2003 / 1659 / 1851.615 / 240 / 195.772 / 949 / 1045.09 / 470 / 610.7524
2004 / 1767 / 1916.427 / 249 / 201.9718 / 1018 / 1065.881 / 500 / 648.5744
2005 / 1984.192 / 208.3678 / 1087.086 / 688.7387
2006 / 2055.069 / 214.9664 / 1108.712 / 731.3902
2007 / 2129.226 / 221.7739 / 1130.769 / 776.683
2008 / 2206.842 / 228.7971 / 1153.265 / 824.7807
2009 / 2288.107 / 236.0426 / 1176.208 / 875.8569
2010 / 2373.221 / 243.5176 / 1199.607 / 930.0961
2011 / 2462.396 / 251.2293 / 1223.472 / 987.6941
2012 / 2555.856 / 259.1853 / 1247.812 / 1048.859
2013 / 2653.841 / 267.3931 / 1272.636 / 1113.812
2014 / 2756.601 / 275.861 / 1297.954 / 1182.787
2015 / 2864.405 / 284.5969 / 1323.775 / 1256.033
2016 / 2977.535 / 293.6095 / 1350.11 / 1333.816
2017 / 3096.292 / 302.9076 / 1376.969 / 1416.415
2018 / 3220.992 / 312.5 / 1404.363 / 1504.129
2019 / 3351.973 / 322.3963 / 1432.301 / 1597.275
2020 / 3489.591 / 332.6059 / 1460.795 / 1696.19
Table 2. Interpolation and extrapolation relative GDP dynamics of Primary, Secondary and Tertiary economic activities, Brazil, 1985-2004-2020 (billion R$).
Years / Primary / Primary / Secondary / Secondary / Tertiary / Tertiarymodel / model / model
1985 / 0.144968 / 0.106185 / 0.680517 / 0.696939 / 0.174515 / 0.196876
1986 / 0.14248 / 0.106417 / 0.662269 / 0.69049 / 0.195251 / 0.203093
1987 / 0.115917 / 0.106622 / 0.687716 / 0.683925 / 0.196367 / 0.209453
1988 / 0.10989 / 0.1068 / 0.702451 / 0.677246 / 0.187658 / 0.215955
1989 / 0.10199 / 0.106949 / 0.713101 / 0.670452 / 0.184909 / 0.222599
1990 / 0.090692 / 0.107069 / 0.67144 / 0.663545 / 0.237868 / 0.229386
1991 / 0.081678 / 0.107159 / 0.65195 / 0.656527 / 0.266372 / 0.236313
1992 / 0.078234 / 0.10722 / 0.698683 / 0.649399 / 0.223083 / 0.243381
1993 / 0.087734 / 0.107249 / 0.69537 / 0.642163 / 0.216897 / 0.250588
1994 / 0.115734 / 0.107247 / 0.633888 / 0.634821 / 0.250378 / 0.257932
1995 / 0.09097 / 0.107212 / 0.605351 / 0.627375 / 0.303679 / 0.265413
1996 / 0.091151 / 0.107145 / 0.569528 / 0.619828 / 0.339321 / 0.273027
1997 / 0.087066 / 0.107045 / 0.569085 / 0.612182 / 0.343849 / 0.280772
1998 / 0.087231 / 0.106912 / 0.560051 / 0.604441 / 0.352718 / 0.288647
1999 / 0.096794 / 0.106745 / 0.561282 / 0.596607 / 0.341923 / 0.296648
2000 / 0.10499 / 0.106543 / 0.563189 / 0.588685 / 0.331821 / 0.304772
2001 / 0.114972 / 0.106307 / 0.566646 / 0.580677 / 0.318383 / 0.313016
2002 / 0.134965 / 0.106036 / 0.561833 / 0.572588 / 0.303202 / 0.321376
2003 / 0.144665 / 0.10573 / 0.572031 / 0.564421 / 0.283303 / 0.329849
2004 / 0.140917 / 0.10539 / 0.576118 / 0.556181 / 0.282965 / 0.338429
2005 / 0.105014 / 0.547873 / 0.347113
2006 / 0.104603 / 0.539501 / 0.355896
2007 / 0.104157 / 0.53107 / 0.364772
2008 / 0.103676 / 0.522586 / 0.373738
2009 / 0.103161 / 0.514053 / 0.382787
2010 / 0.102611 / 0.505476 / 0.391913
2011 / 0.102026 / 0.496863 / 0.401111
2012 / 0.101408 / 0.488217 / 0.410375
2013 / 0.100757 / 0.479545 / 0.419698
2014 / 0.100073 / 0.470853 / 0.429074
2015 / 0.099356 / 0.462147 / 0.438497
2016 / 0.098608 / 0.453432 / 0.44796
2017 / 0.097829 / 0.444716 / 0.457455
2018 / 0.09702 / 0.436003 / 0.466977
2019 / 0.096181 / 0.427301 / 0.476518
2020 / 0.095314 / 0.418615 / 0.486071
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,. The type of the GDP growth in Brazil has a form:
(18)
with average total rate of growth
(19)
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: .
The interpolation and extrapolation GDP dynamics provide the interpolation and extrapolation forecast of Brazilian GDP relative dynamics for 1985-2004-2020 (see table 2):
Fig. 1. Interpolation and Extrapolation of Brazil GDP dynamics in different sectors and total annual GDP dynamics.
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. Using the same equation, the goodness of fit between empirical industrial GDP relative dynamics is. For services the value is .
The low value of the correlation coefficient for Agriculture + mineral Industry can be explained byjump inchangein 2001in Primary sector during the decade 1994-2004 (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
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:
(20)
The type of the rates of growth in frequencies of GDP growth in different economic sectors
(21)
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.
References
Baer, W., 2007. Brazilian Economy: Growth and Development, Lynne Rienner Publishers, 6th Revised Edition.
Clark C., 1940, 1951, 1957. "Conditions of Economic Progress", Macmillan, London.
Contas Regionalis de Brazil, 1985-2004.
Da Foncseca M.A.R, 2001. “Analysis of Brazil's Macroeconomic Trends.” In J.J.M. Guilhoto and G.J.D. Hewings (eds), 2001. Structure and structural change in the Brazilian Economy, Aldershot, Ashgate, pp.9-31.
Fourastie J, 1949. Le Grand Espoir du XXe Siecle. Progress Technique, Progress Economique, Progress Social. Paris, Presses Universitaires de France, 224 p. (Reprinted in 1989 collection, Tel Gallimard.)
Hoppenstead F.C. and Peskin C.S., 1992. Mathematics in Medicine and Life Sciences. Berlin, Heidelberg, New York, Springer Verlag.
Guilhoto JJM and GJD Hewings (eds), 2001. “Structure and structural change in the Brazilian Economy, Aldershot, Ashgate, pp. 5.
Hull C.H. (ed), 1899, The Economic Writings of Sir William Petty, 2 vols, Cambridge University Press.
Kuznets S, 1953. Economic Change, Nortin, New York.
Krüger, J. J. (2008) Productivity and structural change: a review of the literature, Journal of Economic Surveys, Vol. 22, No. 2, pp. 330-363
Sonis, M., 1987. "Spatio-temporal Stability and Potentials of Migration Flows: a Logistic Growth Model." Paper presented on Fifth European Colloquium on Quantitative and Theoretical Geography, Bardonecchia, Italy, September 9-12, 1987.
Sonis, M, G.J.D. Hewings, 1990. "The Matrioshka Principle in the Hierarchical Decomposition of Multiregional Social Accounting Systems". Chapter 7, Part I.3 in L. Anselin and M. Madden (eds) New Directions in Regional Analysis. Multiregional Approaches, Pinter, London, pp.101-111.
Sonis M, GJD Hewings and Y Okuyama, 2001. “Feedback Loop Analysis of Japanese Interregional Trade, 1980-85-90.” Journal of Economic Geography, 1, pp. 341-362.
Sonis, M., 2003. ”Discrete Non-Linear Probabilistic Chains”, In M. Drachlin and E. Litsyn (eds) Functional-Differential Equations, Ariel, Israel, 10:445-487.
7-1
[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.