Mathematical Analysis of the Historical Economic Growth

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Historical economic growth is analysed using the method of reciprocal values. Included in the analysis is the world and regional economic growth. The analysis demonstrates that the natural tendency for the historical economic growth was to increase hyperbolically.
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  1 Mathematical Analysis of the Historical Economic Growth Ron W Nielsen 1  Environmental Futures Research Institute, Gold Coast Campus, Griffith University, Qld, 4222, Australia September, 2015 Historical economic growth is analysed using the method of reciprocal values. Included in the analysis is the world and regional economic growth. The analysis demonstrates that the natural tendency for the historical economic growth was to increase hyperbolically. Introduction Historical economic growth is defined as the growth before it started to be diverted to mathematically distinctly different trajectories, which, in contrast, could be described as representing modern economic growth. This distinction will become clear when we analyse the economic growth data (Maddison, 2010). We shall see that the transitions from the historical to modern growth occurred between around the end of the 1800s and the mid-1900s. The analysis of the modern growth will be discussed in separated publications. The primary objective of the analysis presented here is to find the  simplest   mathematical representations of the historical economic growth data. It is not   a curve-fitting exercise. It is not an attempt to fit the historical economic growth data by the incomprehensibly complicated mathematical formulae just to reproduce their trajectories. Our aim is to find the simplest and if possible the uniform mathematical representation of the economic growth data  because our ultimate aim is not   just to   describe  the historical economic growth but to   explain  it. We shall be looking for a common pattern  of growth. In a separate publication we shall  propose mechanism to explain this common pattern. Complicated mathematical distributions might be impressive if they fit the data perfectly well but it might be also difficult or maybe even impossible to find a convincing explanation of the mechanism of growth represented by such complicated mathematical distributions. Our analysis will be based on using the method of reciprocal values (Nielsen, 2014). This method is based on the observation (von Foerster, Mora & Amiot, 1960) that the growth of human population during the AD era was increasing hyperbolically. Recent but limited analysis (Nielsen, 2014) indicated that historical economic growth was also increasing hyperbolically. Hyperbolic distribution describing  growth  is represented by a reciprocal   of a linear function: 1 ( ) S t a kt  , (1) where ( ) S t  is the size of the growing entity, while a  and k   are  positive  constants. 1 AKA Jan Nurzynski, r.nielsen@griffith.edu.au; ronwnielsen@gmail.com;  http://home.iprimus.com.au/nielsens/ronnielsen.html  2 The reciprocal of such hyperbolic growth,  1 ( ) S t  , is represented by a decreasing   linear function: 1 ( ) S t a kt  . (2) Hyperbolic distributions  should not be confused with hyperbolic  functions  ( sinh( ) t  , cosh( ) t  , etc). Furthermore, in our notation, 1 ( ) S t   does not represent the inverse  function of ( ) S t   but its reciprocal  , i.e. 1 ( ) 1/ ( ) S t S t  . Reciprocal values help in an easy and generally unique identification of hyperbolic growth  because in this representation hyperbolic growth is given by a decreasing straight line. Apart from serving as an alternative way to analyse data, reciprocal values allow also for the investigation of even small deviations from hyperbolic distributions because deviations from a straight line can be easily noticed. Reciprocal values allow also for an easy identification of different components of growth. This property can be used, for instance, in the investigation of the validity of the Unified Growth Theory (Galor, 2005, 2011), which is based on the postulate of different regimes of growth. When comparing mathematically-calculated distributions with the reciprocal values of data, we have to remember that the sensitivity of the reciprocal values to small deviations increases with the decreasing size S   of the growing entity. Suppose we have two values of S   at a given time: 1 ( ) S t  and 2 ( ) S t  , representing, for instance, the empirical and the calculated values. It is clear that 121 2 S S S S S S  , (3) where S   is either 1 S  or 2 S  . For a given S  , 1 S   increases rapidly with the decreasing S  . The separation of small values of data from calculated distributions are magnified. It should be also noted that the decreasing   reciprocal values describe  growth , while a deviation to larger   reciprocal values describes decline . Consequently, a diversion to a  faster   trajectory will be indicated by a downward   bending of a trajectory of the reciprocal values, away from an earlier observed trajectory, while the diversion to a  slower   trajectory will be indicated by an upward   bending. The data describing the historical economic growth (Maddison, 2001, 2010) do not allow for a detailed analysis below AD 1500 because there are two large gaps in the data: between AD 1 and 1000 and between AD 1000 and 1500. The best sets of data are from AD 1500. However, the compilation prepared by Magnuson appears to be the best source of the historical growth data. Throughout the analysis, the values of the Gross Domestic Product (GDP) will be expressed in billions of the 1990 International Geary-Khamis dollars. Furthermore, in order to examine the quality of fitted distributions to the small values of data, we shall display the GDP values and their corresponding calculated curves using semilogarithmic scales of reference. All diagrams are presented in the Appendix.  3 World economic growth Results of mathematical analysis of the world economic growth are presented in Figures 1-3. Reciprocal values of historical data can be fitted using straight line (representing hyperbolic growth) between AD 1000 and 1955. From around 1955, the world economic growth started to be diverted to a slower trajectory as indicated by the upward bending of the reciprocal values, away from the earlier straight line. This part of the data is shown in Figure 2. Hyperbolic fit to the world GDP data (Maddison, 2010) is shown in Figure 3. The fit to the data is remarkably good. The point at AD 1 is only 77% away from the fitted curve. We would need more data between AD 1 and 1000 to decide whether such a difference is of any significance. Hyperbolic economic growth of the historical GDP has been uniquely identified by the straight-line fitting the reciprocal values of data. However, for comparison, we are also showing the best fit using exponential function for precisely the same data  over the same time . While the hyperbolic growth can be easily identified as a straight line in the display of the reciprocal values of data, exponential growth can be easily identified as a straight line in the direct display of data using the semilogarithmic scales of reference. Furthermore, while the reciprocal values for the hyperbolic growth decrease linearly, the reciprocal values for the exponential growth decrease exponentially. Even without trying to fit the data with exponential distributions we can easily see in all the figures presented in this study that the historical growth of the GDP, global or regional, was not exponential. Parameters describing hyperbolic trajectory fitting the data between AD 1000 and 1955 are: 2 1.684 10 a and 6 8.539 10 k  . The point of singularity for this fit is at 1972 t  . From around 1955, the world economic growth started to be diverted to a slower trajectory, as indicated by the upward bending of the reciprocal values shown in Figure 2. The diversion to a slower trajectory bypassed the singularity by 17 years (see Table 1). Western Europe The growth of the GDP in Western Europe is shown in Figures 4-6. It is the total for 30 countries: Austria, Belgium, Denmark, Finland, France, Germany, Italy, The Netherlands,  Norway, Sweden, Switzerland, United Kingdom, Greece, Portugal, Spain and for 14 small,  but unspecified countries. Ireland is the missing country in this list but it was included from 1921. The best hyperbolic fit is between AD 1500 and 1900. The parameters of the hyperbolic distribution are 2 9.859 10 a and 5 5.112 10 k  . The point of singularity for this fit is at 1929 t  . Between 1900 and 1910, economic growth started to be diverted to a slower, but still fast-increasing, trajectory. To demonstrate that it is possible to improve the fit to the historical economic growth by using more complicated mathematical expressions than the first-order hyperbolic distributions defined by the eqn (1), we are showing, in Figure 6, the best fit to the data using the fourth-order hyperbolic distribution: 140 ( )  iii S t at   (4)  4 Parameters fitting the data are: 20  6.860 10 a , 41  6.898 10 a ,  62  1.250 10 a , , 103  7.228 10 a  and 134  1.400 10 a . The calculated distribution reproduces the data  below AD 1500 but is unappealing. We would need more data in this region but even then such a complicated description would be still unattractive. The most complete set of data for Western Europe are for four countries: Denmark, France,  Netherlands and Sweden. These are the only countries that have yearly data from 1820. They are analysed separately and results are presented in Figures 7 and 8. The quality of the hyperbolic fit to the data is virtually the same as for the total of the 30 countries but now the fitted curve passes also through the AD 1 point. However, it still does not reproduce the point at AD 1000. The historical growth of the GDP in Western Europe is consistent with the hyperbolic growth. The diversion to a slower trajectory for these four countries appears to have occurred earlier than for the whole Western Europe, around 1875 rather than around 1900, or 48 years rather than 29 years before the time of the singularity. Parameters describing the historical hyperbolic growth of the GDP in Denmark, France,  Netherlands and Sweden are: 1 3.821 10 a and 4 1.986 10 k  . The point of singularity for this fit is at 1923 t  . From around 1875 economic growth in Denmark, France, Netherlands and Sweden was diverted to a slower trajectory. Eastern Europe Systematic data for Eastern Europe are available only for seven countries: Albania, Bulgaria, Czechoslovakia, Hungry, Poland, Rumania and Yugoslavia. For other countries there are no data until 1990. The analysis of the historical data for Eastern Europe are summarised in Figures 9  –   11. The best hyperbolic fit to the data is between AD 1000 and 1890. Hyperbolic parameters are: 1 7.749 10 a and 4 4.048 10 k  . The point of singularity for this fit is at 1915 t  . From around 1890, the economic growth in Eastern Europe was diverted to a slower trajectory,  bypassing the singularity by 25 years. Former USSR The analysis of the data for countries of Former USSR is presented in Figures 12-14. The  best hyperbolic fit is between AD 1 and 1870. Parameters fitting the data are: 1 6.547 10 a and 4 3.452 10 k  . The point of singularity for this fit is at 1897 t  . From around 1870, or maybe even a little earlier, the economic growth in the Former USSR was diverted to a slower trajectory, bypassing the singularity by at least 27 years. Asia The analysis of the historical economic growth in Asia is summarised in Figures 15, 16 and 18. The best hyperbolic fit is between AD 1000 and 1950. Parameters fitting the data are: 2 2.303 10 a   and 5 1.129 10 k  . The point of singularity for this fit is at 2040 t  .
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