Real GDP per capita in Germany: 1871-2011

We have not presented the case of Germany because of the change in real GDP time series in 1989. However, the historical time series does not differ from those in other developed countries. Figure 1 shows that real GDP per capita, G, in Germany also has two branches – before and after 1940. The Total Economy Database (TEDI) gives two estimates for Germany – for West Germany only, which goes back into the 1950s, and for Germany since 1989. We have plotted the West Germany time series since it is different from the historical time series which is identical to Germany after 1989.

As expected, G follows a linear trend in the long run:

G(t) = At + C (1)

Currently, real GDP in Germany is on the trend line. This means no output gap. We also predicted a recession in Germany.

Figure 1. The evolution of real GDP per capita in Germany between 1871 and 2011

Advice needed: the paradox of real GDP growth after the Second World War

We’ve just presented the historical estimates of real GDP per capita developed by Angus Maddison at the Groningen Growth and Development Centre. Figure 1 summarizes all graphs of our previous post and illustrates the inherent mastery of the GDP evolution after WWII. The rate of real economic growth jumped by a factor of 10 between 1940 and 1950. If to extrapolate the linear trend observed after 1950 into the past, the level of real GDP per capita intercepts the time line between 1920 and 1940. Does that mean that there was no or negative real economic growth in the 19^th century or the rate of economic growth after 1950 has been dramatically biased? If to extrapolate the growth rates observed before 1940, the current level of real GDP per capita in developed counties would have been below $15,000 or even $10,000. Does that mean that the rate of price inflation has been highly underestimated since 1940?

This is a major paradox of the historical GDP time series. Can anybody explain it?

Figure 1. The evolution of real GDP per capita in developed countries: Austria, Australia, Belgium, Canada, France, Italy, Japan, Spain, Switzerland, the US, and the UK.

Another 1000 arguments against the Solow growth model

In May 2011, we presented 1000 arguments against the Solow growth model, which states that the rate of change in real GDP per capita must approach some constant level. Here we present new arguments against the Solow model as based on the historical GDP data developed by Angus Maddison at the Groningen Growth and Development Centre. In two previous posts we presented the cases of USA and Austria. We showed that the annual increment of GDP per capita is constant from 1871 to 2010 with an artificial structural break between 1940 and 1950. In other words, the slope of linear trend in real GDP per capita and thus the mean annual increment jumped by a factor of 10 between 1940 and 1950. Here we summarize numerous observations for the developed counties presented in the May’s post and validate our model.   

Under our empirical framework [1,2,3], real GDP per capita in developed countries grows as a linear function of time, we call it inertial growth, when population pyramid does not change much in the long run: 

G(t) = At + C                             (1) 

Relationship (1) defines the linear trajectory of the GDP per capita, where C=G_i(t₀)=G(t₀) and t₀ is the starting time. In the regime of inertial growth, the real GDP per capita increases by the constant value A per time unit. The relative rate of growth along the inertial linear growth trend, g(t), is the reciprocal function of G:

g(t) =  A/G(t)               (2) 

Relationship (2) implies that the rate of GDP growth will be asymptotically approaching zero, but the annual increment A will always be constant. This is different from the Solow model where the rate of growth is a positive (nonzero) value. Moreover, the absolute rate of GDP growth is constant and is equal to A [$/y]. This constant annual increment thus defines the constant “speed” of economic growth in a one-to-one analogy with Newton’s first law. Hence, one can consider the property of constant speed of real economic growth as “inertia of economic growth” or simply “inertia”.   

In Figure 1 (borrowed from the post in May 2011), we present annual increments of real GDP per capita (borrowed from the Conference Board Total Economic database, TEDI) in the biggest developed economies as a function of real GDP per capita in sense of equation (1). These plots validate our empirical finding and reject the Solow model. Overall, there were 19 countries analyzed in the study and no one has any distinct positive trend over the past 60 years, i.e. between 1950 and 2010. So, we had 1000 years supporting the hypothesis of a constant (but country dependent) annual increment.

Figure 2 extends all time series back to 1871 by using the Maddison’s historical estimates of real GDP and population in developed courtiers. All estimates are in 1990 International Geary-Khamis dollars which allow a cross-country comparison. These dollars are different from 2010 EKS dollars in Figure 1. Thus the mean values may not coincide between these Figures. We have also plotted the estimates of real GDP per capita from the Total Economic Database (TEDI) now available through 2011. Essentially, this is the same data set as in the historical database and all curves after 1950 have to coincide, but Japan and Spain show significant discrepancy during the most recent period. For Japan, the TEDI and historical curve started to deviate in 1993.  

Overall the time series before 1940 and after 1950 are both well approximated by linear time trends with slopes suddenly increasing by approximately a factor of 10.  Not considering the reasons for this break in the tie series we just conclude that the annual increment is constant before 1940 and after 1950, as stated by our model of real GDP growth. This validates our model and rejects the possibility for the Solow model to be right.  

There are important implications of the constant annual increment for economic policy in developed countries. Economists and economic authorities (like FRB and CBO) are waiting for a significant increase in the rate of real economic growth to close so called output gap, i.e. the difference between the measured level of real GDP and that expected form exponential extrapolation of the trend observed before 2007. In reality, there is not output gap, as Figure 1 and 2 demonstrate.  Only Italy and Japan are far below the linear trend in real GDP per capita and Australia is above the expected level. France is also slightly below its long term linear trend. These countries might expect a recovery to the trend in the long run, depending on the evolution of their age pyramids. 

The US and UK have been returning to the trend during the recent crisis and should not wait for any elevated rate of real economic growth.  The expectation of a growth rate of 3.5% per year (in terms of GDP per capita) which has been explicitly articulated by the FRB and CBO in their economic outlooks is a naïve extrapolation of exponential growth related to the exponential population growth.   

Figure 1. Dependence of annual GDP increment on GDP (both real per capita) for select developed countries. Original GDP data are extended by those corrected for the ratio of total and working age population (the latter must be used in GDP per capita calculations). For both time series linear regression lines and equations are shown with corresponding slopes. For the biggest countries these slopes are very close to zero but can be positive or negative. A zero slope corresponds to constant annual increment.



Figure 2. The evolution of real GDP per capita in developed countries between 1871 and 2011.  

Real GDP per capita in Japan: 1871-2008

We continue presenting historical estimates of real GDP per capita in developed countries. The USA and Austria have demonstrated that the period between 1871 and 1940 supports our model of constant annual increment of real GDP per capita. We have also found that there was a transition period between 1940 and 1950 when the mean annual increment rose by a factor of 5. It might be the result of a new measurement procedure introduced during this period after the concept of GDP had been developed by Simon Kuznets. The next country we would like to present is Japan. For the period from 1871 to 1940, Japan also has a constant annual inctement.

All measurements in developed countries do support the concept of constant annual increment in real GDP per capita, G(t), which can be described by a simple model: in the long run, the GDP growth as a linear function of time:

G(t-t₀)=G₀+A(t-t₀) (1)

where G₀ is the initial level of GDP per capita at time t₀ in a given country, A is the country dependent annual increment measured in PPP dollars. Unlike in the Solow model and its successors, the rate of growth of real GDP per capita, dG/Gdt, has a decelerating nonlinear trend. Differentiating with respect to time and dividing both sides of (1) by G(t), one obtains

dG/Gdt = A/G (2)

This model has given excellent statistical results and explained the evolution of real GDP per capita in developed countries [1,2] since 1950. Figure 1 presents the case of Japan: annual increment in real GDP per capita is plotted against the level of real GDP per capita. (Equation (2) uses time implicitly.) It demonstrates the accuracy of our concepts. Since the increment is assumed to be constant, the mean value of the annual GDP increment should coincide (at least should be very close to) with its linear trend. The linear regression line for Japan is very close to the constant level. Actually, it slightly oscillates around the mean value over time, as the cases for 2007 (upper panel) and 2009 (lower panel) demonstrate. The hypothesis of the constant increment looks sound.

Figure 1. Annual increment of real GDP per capita (2007 and 2009 US$) vs. real GDP per capita in Japan for the period between 1950 and 2007 (upper panel) and between 1950 and 2009 (lower panel). Two sets are presented - the original (open circles) and that corrected for population (filled diamonds). Subsequent values of the latter set are connected by a solid line for illustration of the evolution in time. Bold lines represent the mean value of $605 (2007 US$) and $596 (2009 US$) for the population corrected sets. Two solid lines show linear regressions lines. Corresponding linear relationships are displayed, the lower relationship being associated with the original data set.

Figure 2 presents the whole period between 1871 and 2008. There is a spurious time trend which can be split into to segments and one transition period. Before 1945, the mean annual increment was $32 (1990 International Geary-Khamis dollars) as Figure 3 shows and practically no increase in the annual increment with time (no linear time trend). After 1950, the mean increment jumped to $341 also without linear trend (see Figure 1).

All in all the case of Japan validates our model of inertial economic growth by its 130-year history of measurements. One should not expect the rate of growth observed in the second part of the 20^th century to extent into the 21^st century. 

Figure 2. Annual increment of real GDP per capita (historical data) between 1871 and 2008. There is no linear trend.

Figure 3. Annual increment of real GDP per capita (historical data) between 1875 and 1944. There is no linear trend.

Real GDP per capita in Austria: from 1871 to 2010.

In our previous post, we have started to present historical GDP data which were estimated by Angus Maddison from Groningen University. We have demonstrated that there was no linear time trend in the annual increment of real GDP per capita in the USA between 1871 and 1940. The year of 1940 was selected to introduce a short transition period between two long intervals of constant annual increment. The Second World War and the development of the GDP concept clearly cut the period from 1871 and 2011 in two pieces. One can consider the transition between 1940 and 1950 as a structural break or change in measurement units.  

Here we continue testing our model of the real economic growth by presenting the case of Austria. The model is based on an extensive set of observations in developed countries which show that real GDP per capita, G(t), evolves along a linear time trend with all fluctuations related to the change in age pyramid. In our article, we calculated the inertial term A in

G(t-t₀)= G₀+A(t-t₀)   (1)

where G(t) is real GDP per capita as observed in developed countries; G₀ is the initial level of GDP per capita at time t₀ in a given country; and  A is the country dependent annual increment measured in PPP dollars. Since the empirical model and is based only on observations of real GDP in developed countries its predictive power depends on how well it fits observations.   

Several years ago we presented a model for Austria. Figure 1 depicts annual increment in real GDP per capita as a function of the level of real GDP per capita instead of time. Since the increment is assumed to be constant, the mean value of the annual GDP increment should coincide (at least should be very close to) with its linear trend. In 2002, the linear regression line for Austria shows a distinct positive trend of +0.0041. According to (1) such deviations must be compensates in the long-run by negative rates of growth. However, the years after 2002 have been demonstrating increasing positive trend. This deviation has been compensated by a severe decline in 2009. Therefore, the inertia of real economic growth has won again. Any deviation creates a returning force likely proportional to the size of the deviation. In 2009, the trend is almost 0 and the hypothesis of the constant increment looks sound.

 

Figure 1. Annual increment of real GDP per capita (2002 and 2009 US$) vs. real GDP per capita in Austria for the period between 1950 and 2002 (upper panel) and between 1950 and 2009 (lower panel). Two sets are presented - the original (open circles) and that corrected for population (filled diamonds). Subsequent values of the latter set are connected by a solid line for illustration of the evolution in time. Bold lines represent the mean value of $548 (2002 US$) and $700 (2009 US$) for the population corrected sets. Two solid lines show linear regressions lines  

In Figure 2, we have plotted all historical GDP data (annual increments) from 1871 to 2008 (138 years). All historical data are estimated in 1990 International Geary-Khamis dollars which are different from 2002 US$ and 2009 US$ in Figure 1. One might suggest that there is a linear trend in the time series, but we have to remove the transition period between 1940 and 1950 and plot only the estimates between 1871 and 1950. Figure 3 evidences that there was not positive trend in this time series. Hence, the overall period has two lengthy periods of constant annual increment and a short transition period,  During the transition, the mean increment jumped from $30 (from 1871 to 1940) to $353 (from 1951 to 2008). In other words, there two shelves and a ladder.

The case of Austria validates our model of inertial economic growth with a constant annual increment. This is equivalent to the rate of real GDP per capita grows decays inversely proportionally to the attained level:

dG/dt = A  or dG/Gdt=A/G. 

Therefore, the rate of real economic growth should asymptotically approach the zero line. It should also be noted that the gap in real GDP per capita between developed countries can hardly be closed and the lag of developing courtiers is forever. In relative terms, convergence is possible when the rate of growth in all countries approaches zero. Empirically speaking, the Solow model is wrong.

Figure 2. Annual increment in real GDPper  capitafrom 1871 to 2008.

Figure 3. Annual increment in real GDP per capita from 1871 to 1940.

A 130-year long argument against the Solow growth model

We have already presented strong quantitative arguments against the Solow growth model, which presumes that the rate of change in real GDP per capita must approach some constant level. In developed countries, the annual increments of real GDP per capita have been rather oscillating around constant level since 1955. We use the estimates of real GDP per capita published by the Conference Board’s total economy database.  Since the late 1990s this database has been developed and maintained in conjunction with the Groningen Growth and Development Centre (University of Groningen, The Netherlands). As of the summer of 2007 the database has been transferred from the University of Groningen to The Conference Board and is maintained there. The GGDC also provides historical estimates of real GDP developed by Angus Maddison.  It is instructive to use these historical estimates in order to reject the Solow model by empirical data.  

Under our empirical framework [1,2,3], real GDP per capita in developed countries grows as a linear function of time, we call it inertial growth, when population pyramid does not change much in the long run: 

G(t) = At + C           (1) 

Relationship (1) defines the linear trajectory of the GDP per capita, where C=G_i(t₀)=G(t₀) and t₀ is the starting time. In the regime of inertial growth, the real GDP per capita increases by the constant value A per time unit. The relative rate of growth along the inertial linear growth trend, g(t), is the reciprocal function of G: 

g(t) =  A/G(t)                     (2) 

Relationship (2) implies that the rate of GDP growth will be asymptotically approaching zero, but the annual increment A will always be constant. This is different from the Solow model where the rate of growth is a positive (nonzero) value. Moreover, the absolute rate of GDP growth is constant and is equal to A [$/y]. This constant annual increment thus defines the constant “speed” of economic growth in a one-to-one analogy with Newton’s first law. Hence, one can consider the property of constant speed of real economic growth as “inertia of economic growth” or simply “inertia”.   

In Figure 1, we present the estimates of annual increments of real GDP per capita in the USA since 1870. At first glance, these data support the Solow model, i.e. annual increments increase with time. However, we know that since 1955 the increment has been oscillating around constant level of $423 (2011 US dollars) and the accuracy of GDP measurements (actually obtained by reconstruction) before 1950 can hardly be characterized as a high one.  In Figure 2, we display the estimates of annual increment since 1955 as measured by the Bureau of Economic Analysis. There is no linear time trend in the curve and thus the Solow model does not work well.  

The period between 1940 and 1955 is characterized by extremely large oscillations. This period corresponds to the Second World War and the development of the concept of Gross Domestic Product (Simon Kuznets introduced this idea in 1934). Therefore, the war and the development of measurement procedure may introduce significant structural breaks in the time series and we remove the years between 1940 and 1955 from our consideration as a mixture of an artificial step in measurements (say, the transition from mph to km/h) and the effects of noneconomic factors in economic evolution.  Figure 3 shows the period between 1871 and 1940 (70 years). Not surprisingly, there is no linear time trend and the overall length of the period when the Solow model is not applicable is now ~130 years.  

All in all, the Solow model of economic growth (and all its branches and versions) contradicts hundred and fifty years of observations.   We are going to report the evolution of real GDP per capita in other developed countries.

Figure 1. The estimates of annual increment of real GDP per capita in the USA since 1970.

Figure 2.  Annual increments since 1955 as reported by the BEA. There is practically no linear trend.  

Figure 3. The annual increment between 1871 and 1940. The mean value is $65.

Crude and steel - an unbreakable pair

We have been reporting on the trade-off between the producer price index of crude oil (domestic production) and the PPI of iron&steel since 2009. It has been always a linear and lagged link between them. Our previous update included PPI data through July 2011. Here we present an annual wrap-up.

We reported that the PPI of crude oil had been likely evolving in sync with that of iron and steel, but with a lag of two months in September 2009. In order to present both indices in a comparable form, the difference between a given index, iPPI (i.e. iron&steel and crude), and the overall PPI was normalized to the PPI: (iPPI(t)-PPI(t))/PPI(t). These normalized differences represent the evolution of the rate of deviation from the PPI over years.

Figure 1 depicts the corresponding time histories of the normalized deviations from the PPI, including the most recent period through December 2011. Even a simple visual inspection reveals the following feature: the (normalized deviation from the PPI of the) index of iron and steel lags by approximately two months behind the (normalized) index of crude oil.

Figure 1. The deviation of the iron and steel price index and the index of crude oil from the PPI, normalized to the PPI.

In order to reduce both deviations to the same scale we additionally normalized the curves in Figure 1 to their peak values between 2005 and 2011.

(iPPI(t)-PPI(t))/[PPI(t)*max{iPPI-PPI)}]

This scaling allows a direct comparison of corresponding shapes. In Figure 2, we display the normalized index of iron and steel shifted by two months ahead to synchronize its peak with that observed in the normalized index for crude petroleum. The scaled index of crude demonstrates just short-term deviations from the index of iron and steel in the overall shape and timing of the peak and trough. Simple smoothing with MA(3) makes the curves resemblance even better. As an extra benefit of the resemblance, one can use the two-month lag to predict the future of the iron and steel price index.

Figure 2. Deviation of the iron and steel price index from the PPI, normalized to the PPI and the peak value after 2005 as compared to the deviations of the index for crude petroleum normalized in the same way. The normalized index for iron and steel is shifted two months ahead.

Conclusion

The link between oil and iron seems to be unbreakable. Between 2006 and 2012, the deviation of the price index of iron and steel from the PPI in the USA repeats the trajectory of the deviation of the index of crude petroleum (domestic production) with a two-month lag. Therefore, the prediction of iron and steel price for at this horizon is a straightforward one.