Empirically correct model of real economic growth (theory)

Let us start by introducing a new concept describing the evolution of real Gross Domestic Product. Our central claim is quite straightforward – the growth rate, g(t), of real GDP per capita, G(t), is driven by the attained level of real GDP per capita and the change in a specific age population, Ns. According to this model, the growth rate of real GDP (for the sake of brevity we often omit “per capita”) in developed countries is primarily characterized by an annual increment, A, which does not change over time; A = const. All fluctuations around this constant increment can be explained by the change in the number of people of the country-specific age:

g(t) = dlnG(t)/dt = A/G(t) + 0.5dlnNs(t)/dt (1.1)

Equation (1.1) is the quantitative model that has been constructed empirically. Notice that unlike in the mainstream economics, no assumptions were made and no theoretical models were formulated at the initial stage of our empirical study. Instead we have attempted to find a model that would best fit observations. The next subsection will be devoted to discussion of the actual model fitting.

We use the relative growth rate, as represented by dG(t)/G(t)=dlnG(t). In order to better understand the processes defining real growth let us decompose the model and consider each term individually. We will for now assume that the second term in (1.1), i.e. the rate of change in the specific age population, is zero. Accordingly, there is no external force acting on the GDP growth rate and the system is in the state of stationary or inertial (in terms of constant speed) growth. Later on we provide a simple analogue from mechanics, which clarifies why we prefer to call A/G(t) “the inertial growth”.

When the population age pyramid is fixed (dNs≡0), real GDP grows as a linear function of time:

g(t) = dlnG(t)/dt(given dNs(t) = 0) = A/G(t)

G(t) = At + C (1.2)

where G (t) is completely equivalent to the inertial growth, Gi(t), i.e. the first component of the overall growth as defined by (1.1). Relationship (1.2) defines the linear trajectory of the GDP per capita, where C=Gi(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. Relationship (1.3) is equivalent to (1.2), but holds for the inertial part of the total growth:

Gi(t) = Gi(t0) + At (1.3)

The relative rate of growth along the inertial linear growth trend, gi(t), is the reciprocal function of Gi or, equivalently, G:

gi(t) = dlnGi/dt = A/Gi = A/G(t) (1.4)

Relationship (1.4) implies that the rate of GDP growth will be asymptotically approaching zero, but the annual increment A will always be constant. 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”. Then the growth, which is observed without the change in the specific age population, can be called the “inertial growth”.

A textbook analogy of inertia at work from mechanics is rotation of a mass on a rope. Rotation around the centre is accompanied by the change in the direction of motion and is driven by the tension force in the rope. If suddenly the rope is ruptured the mass follows up linear progressive motion at a constant speed along the line defined by the velocity vector at the moment when the rope was ruptured. In other words, the mass continues inertial motion with inertia being the property allowing for constant speed and direction. This works only when there is no net force to change the speed and direction. However, in order to retain constant speed and direction in real world (e.g. a plane flight) one needs to supply nonzero forces to compensate all traction forces. When applied to our model of economic growth, the property of inertia implies that if the specific age population, as turns out to be the 9-year-olds in the USA, does not change over time (net exogenous force is zero) the economy grows at a constant speed, as defined by the constant annual increment A.

In physics, inertia is the most fundamental property. In economics, it should also be a fundamental property, taking into account the difference between ideal theoretical equilibrium of space/time and the stationary real behaviour of the society. Mechanical inertia implies that no change in motion occurs in the absence of net exogenous force and without change in internal energy. (As mentioned above, in real case the net force is zero but one should apply extra forces in order to overcome the net traction force and to keep the body moving at a constant speed.) For a society, the net force applied by all economic agents is not zero but counteracts all dissipation processes and creates goods and services in excess of the previous level. The economy does grow with time and its “internal energy” as expressed in monetary units does increase at a constant speed.

We do not consider the economy as a stone flying through space at a constant speed. The economy is a complex system with all internal forces providing constant speed of growth. The stone has no internal forces, which are able to change its speed. In reality, the space is full of dust and electromagnetic fields which can change the speed. The economy has more traction forces, bumps and barriers. That’s why the speed of inertial growth differs between developed countries as we have confirmed empirically. Moreover, not all economies are organized in a way that results in the optimal stationary behaviour and the highest speed of economic growth.

Let us now consider the second growth component – the relative rate of change in the number of “s”-year-olds. As a rule, in Western Europe the integral change in the specific age population during the last 60 years is negligible, and thus, the cumulative input of the population component is close to zero. In the USA, the overall increase in the specific age population is responsible for about 20% of the total growth in real GDP per capita since 1960. In (1.1), the term 0.5dlnNs(t)/dt is the halved rate of growth in the number of s-year-olds at time t. The factor of 1/2 is common for developed countries. The only exception we have found so far and report later in this Section is Japan, where this factor is 2/3 as obtained from the rate of growth.

Figure 1.1 depicts an arbitrary GDP evolution curve, lnG(t), which exhibits episodes of rapid growth (t1) and recession (t2). It is easier to illustrate the performance of the model on extreme cases and then to proceed by showing how GDP growth relates to the two defining components at t1 and t2.

Figure 1.1. Illustration of the growth model for real GDP per capita

The growth rate is nothing but the first derivative of the function lnG(t). So, we are interested in how the tangent to the curve behaves. Let’s first consider the case of the rapid growth in t1. The overall growth rate g(t1) is the tangent to the curve at point t1. Please notice that if the age specific population is fixed (dNs(t1) = 0), the inertial growth rate gi would be the tangent to the lnG(t). Nevertheless, we observe that dlnNs(t) > 0 what results in a rise in the GDP above the inertial level of growth.

The second case is similar, but differs in the direction of the overall growth. Please notice that gi(t1) > gi(t2) as the attained level of real GDP per capita is higher at t2 and A/G(t1)>A/G(t2). Furthermore, the rate of change of age specific population is negative, which leads to the overall negative growth as the GDP declines.

The two component growth model is quite intuitive. Relationship (1.1) can furthermore be reversed in order to define the evolution of the number of s-year-olds as a function of real economic growth:

d(lnNs(t)) = 2.0(g(t) - A/G(t))dt (1.5)

However, equation (1.5) is only implicit and does not represent the correct causal direction. Instead of integrating (1.5) analytically, we use relevant annual estimates for all involved variables and rewrite (1.5) in a discrete form:

Ns(t+Δt) = Ns(t)[1 + 2Δt(g(t) - A/G(t))] (1.6)

where Δt is the time step, fixed at one year in our study. Equation (1.6) uses a simple discrete representation of time derivative of the population estimates, where the derivative is approximated by its estimate at point t. Since we use actual data at each point in time, this crude approximation of the derivative does not damage the overall quantitative description.

Both time series g(t) (equivalently, G(t)) and Ns(t) are measured independently. In order to obtain the best prediction of Ns(t) using (1.6) one has to vary coefficient A and (only in the range of uncertainty of the corresponding population estimates) the initial population level – Ns(t0). The best-fit parameters can be obtained by standard LSQ techniques minimizing the difference between predicted and measured series. At this stage we will use only visual fit between these curves. As a result, our models might not provide the lowermost standard deviation.

The final rearrangement of the model is presented below and describes the divergence of the observed growth from its inertial path:

g(t) – A/G(t) = 0.5[Ns(t+Δt)/ Ns(t)-1] (1.7)

Equation (1.7) can be interpreted in the following way - the deviation between the measured growth rate of GDP per capita and the rate of inertial growth is completely defined as a half of the change rate of the number of s-year-olds. This deviation has nothing to do with the well-know production gap as introduced in the mainstream economics. The difference between the overall and inertial growth can not be treated as over- or underperformance of a given economy.

We would like to stress that the reversed interpretation of (1.7) is hardly correct - the number of people of some specific age cannot be completely, or even in any significant part, defined by the contemporary real economic growth. Specifically, the causality principle prohibits the present to influence the birth rate nine years ago. Econometrically, the number of s-year-olds has to be a weakly exogenous variable relative to real economic growth.

Availability of high quality data is a mandatory condition for successful modelling. However, the quality of GDP and population estimates in developed countries is inferior to those measurements, which are usually obtained for variables in physics. We would like to emphasise that the main measurement problem likely consists in numerous revisions to definitions of GDP. Essentially, GDP has been measured in randomly varying units since the very creation of the notion. Unfortunately, there is no procedure to correct the past measurements because necessary information is missing and statistical agencies openly declare the non-compatibility of the data over time. Furthermore, the number of s-year-olds is significantly biased by the balancing procedure among adjacent age groups. We consider several important issues associated with the accuracy of the population estimates in Section 1.4.

The Japanese economy

As mentioned in our previous post on Japan, it has no good prospective in the long run. This post was initially pubslished in November 2010. It is instructive to update it and demonstrate again that Japan has no stellar future in sense of the lowered rate of real growth, slightly elevated unemployment and very extended (decades) period of deflation. This is an excerpt from our monograph “mecђanomics” or “Economics as Classical Mechanics”

Japan is a country with a modern statistical service. The Statistics Bureau (JSB, 2006) of the Ministry of Internal Affairs and Communications provides information on various economic and demographic variables. In this Section, we are specifically interested in real GDP per capita and the estimated and enumerated distributions of the Japanese population by distinct years of age. The OECD (2010) and the Conference Board (2010) provide additional population data and estimates of GDP per capita, as converted at various PPPs.

The peculiarity of real economic growth in Japan can be characterized by the difference in annual increment of real GDP per capita with the mean value of $494 between 1960 and 1991 and only of $168 between 1992 and 2003. Here we use real GDP published by the Conference Board (CB, 2006) as expressed in 1990 US dollars (i.e. converted at Geary Khamis PPPs). Excluding the smaller economies of Norway and Ireland, no other developed country has experienced a GDP per capita decline as significant as $326 (in mean value) between periods of strong growth and depression.

Obviously, there were periods of poor performance as well as prosperous years in many developed countries. But the duration and the amplitude of these phenomena in Japan need an explanation far beyond the current understanding provided by the mainstream economics. Kydland and Prescott (1982) developed a theory (RBC) explaining business cycle by exogenous shocks to productivity. Being a relatively useful tool for formal description of stochasticity during stationary periods, the RBC fails to predict the essential jump in the Japanese time series, if no extraordinary assumptions are used (Hayashi and Prescott, 2002). In addition, their approach does not suggest any solution for recovery from the current state.

There were several practical attempts to revitalize the Japanese economy based on various economic theories and assumptions. All failed as one can conclude from the figures of economic growth and inflation during the past thirty years. Almost all economic problems in Japan have been aggravating over time, mirrored by deflationary period that started in 1999 as well as frequent recessions. Only in 2005, some indications of potential recovery from deflation were discussed. Chapter 2 of this book will however shows that price deflation will likely extend into 2050. After the past twenty years of unsuccessful attempts to produce a model that would consistently explain the unprecedented and very special case of the Japanese economy, a new insight is necessary for the explanation of the poor economic performance.

Here, we apply the model presented in Section 1.1.1 to describe real economic growth as related to only one source - population. This growth is defined as a sum of two components: inertial growth and fluctuations. Inertia is associated with a constant annual increment in real GDP per capita. In the USA, the relative amplitude of fluctuations around the trend is equal to a half of the relative change in the number of nine-year-olds; this defining age may vary across developed countries.

The population-based economic concept is straightforward and parsimonious. It involves only one defining parameter and is accompanied by the advantage that any desirable accuracy is attainable provided precise enumeration of population is available. Before the true population is counted, any improvements in methodology and practice of this enumeration would result in more accurate predictions of economic growth.

For Japan, via using the trial and error approach for the estimation of coefficients in equation (1.1), we have originally revealed a stronger dependence on the change in population. Therefore, the relationship for the growth rate has to be re-written in a more general form:

dG(t)/G(t)dt = A/G(t) + BdNs(t)/Ns(t)dt (1.8)

where A and B are empirically determined coefficients, Ns(t) is the number of people of the defining age. For Japan, the defining age of eighteen years has been found.

Relationship (1.8) implies that the growth rate of GDP depends explicitly and entirely on the attained level of real GDP per capita and the population change. If to gather relevant terms on both sides of the equation, this relationship can be simplified in the following form:

d[G(t) - (At + C)]/G(t) = BdNs(t)/Ns(t) (1.9)

where C is the constant of integration, i.e. the initial condition of the initial value problem.

Relationship (1.9) demonstrates that the evolution of GDP depends only on the population change term with constants A, B, and C to be determined by calibration and initial conditions. It is worth noting that the number of people of defining age is an exogenous parameter because it does not depend on the history of GDP per capita. There is a menu of tools to control such demographic characteristics as birth rate, mortality rate, and net immigration in addition to the level of GDP per capita. Besides, many real forces influencing general demographic processes are out of control. However, there is correlation between birth rate and the speed of economic growth, which potentially introduces a slightly coherent interference.

We use two estimates of real GDP per capita provided by the OECD (2000 US$) and the Conference Board (1990 US$). These values are obtained as the overall real GDP divided by total population. As discussed in Section 1.1.2, GDP per capita should be related to working age population. So, both GDP series are corrected for the working age to total population ratio, which is displayed in Figure 1.17. The bump around 2000 is likely of artificial character and is associated with a sudden increase (after 2000 census) in the total population without any response in the working age population.

Figure 1.18 presents two GDP series: the OECD’s one, which is equivalent to the JSB’s time series, and the one from the Conference Board. Both time series practically coincide except for the decade between 1980 and 1990, where the OECD estimates are slightly higher. In 2009, the growth rate of real GDP per capita was -5.3% per year. This is due to the overall fall in real GDP and also due to the decrease in the working age population.

Figure 1.17. The ratio of total and working age population in Japan.

Figure 1.18. Growth rate of real GDP per capita (corrected for working age population) as reported by the Conference Board and OECD. Notice the difference between 1980 and 1990. In 2009, the CB estimate is at the level of -5.3% per year. Solid line represents the trend as obtained from term A/G, where A=$600 (2000 US$).

Annual single-year-of-age population estimates are available from 1920 to 2009 (JSB, 2010). The accuracy of these estimates is apparently decaying back in the past. The population estimates between censuses are usually based on current information related to birth rate, age and sex dependent mortality, and net migration. In Japan, censuses are conducted every five years, i.e. twice as often as in the USA. The most recent census with the data available for analysis was conducted in October 2005. The intercensal estimates, relevant surveys, statistics and methodology are tested by the census data.

In practice, censuses are considered as a more reliable and accurate source of population related information than that associated with the intercensal estimates. In Japan, for example, it is obligatory to answer the census questions. It happens very often that the population estimated at the end of an intercensal period does not coincide with that enumerated in the later census. This effect is known as the “error of the closure” and sometimes reaches several per cent in such developed countries as the USA and the UK.

In order to match the enumerated figures, the estimated population is adjusted for the error of the closure. This correction is usually age dependent and may significantly differ even for neighbouring ages. Figure 1.19 illustrates the magnitude and timing of relevant corrections. The relative increment in the number of people of age i, [(Ni+1(t)-Ni(t-1)]/Ni(t-1), per one year is plotted for the number of 17- and 18-year-olds. One can easily find the census years in this Figure: sharp and high amplitude adjustments are very typical for statistical and census agencies over the world.

For the purposes of our study, strong disadvantage of these step corrections consists in the difference of their amplitudes as applied for adjacent years. For example, in 1995, the number of 18-year-olds was corrected by about 0.4% compared to the mean annual increment of 0.03% during the previous four years. At the same time, the correction applied to the number of 17-year-olds is very small. Thus, for 1994, it is apparent that the number of 18-year-olds is biased. In particular, the difference of 18-year-olds for 1995 is biased by 0.4%. The difference for 1996 is less biased because it involves two corrected values.

In 2000, the corrections in Figure 1.19 are opposite in sign, which indicates even larger measurement errors in the intercensal estimation procedure. In 1970, the corrections were as large as 2%. So, one has to be careful when using population estimates in economic analysis. Of course, the inherent uncertainty of population surveys and macroeconomic measurements cannot be avoided and, in quantitative analysis, one may only rely on larger population differences. Any discrepancy in amplitude between predicted and observed value, which is comparable to the inherent uncertainty in population, inflation or GDP measurements, might be neglected. Measurement errors may be uncorrelated over time and can be smoothed out with a zero residual by a long period filter or in cumulative representation.

Figure 1.19. Relative growth rate of a single year of age population per one year: [(Ni+1(t)-Ni(t-1)]/Ni(t-1).

The JSB’s population estimates are used for the prediction of the growth rate of real GDP per capita. According to (1.8), the relative change dN18/N18 defines all fluctuations in real economic growth around the inertial growth as determined by constant annual increment A. Figure 1.18 depicts the growth rate of measured GDP per capita, as obtained from the OECD and the Conference Board. Inertial growth, defined as a reciprocal function of GDP per capita with a constant increment A=$600 (2000 US dollars), is also shown in the Figure. The inertial component is not smooth because we use actual readings of GDP per capita. Currently, the inertial growth is above the average growth rate over the last 20 years. This is due to the negative input of the falling number of 18-year-olds.

Coefficients A and B in (1.8) have been determined in a calibration procedure aimed at matching the observed and predicted values of growth rate. By varying A and B one can reach the best visual resemblance between the curves. Figure 1.20 shows a model with A=$600 and B=2/3, as obtained with the OECD data available in 2010. All values of GDP per capita are expressed in 2000 US dollars. We found factor B to be somewhat larger than 0.5 for Japan. This finding might imply that the economic growth fluctuations in Japan are more sensitive to the change in the specific age population.

For Japan, the principal feature to be modelled is the sharp fall in growth rate that started in 1991. This is a critical point for any theoretical description of the Japanese economic evolution. Our model links this drop to the dramatic change in the number of 18-year-olds. Figure 1.21 displays the evolution of population for several adjacent ages. The specific age of 18 years has been chosen because this age is characterized by a fast decay starting in 1991. When extrapolated from N10, as shown in Figure 1.21, N18(t) (=N10(t-8)) approaches the level of 1,200,000 in 2010 and does not fall further in the 2010s.

Figure 1.20. Modelling the observed evolution of growth rate of GDP per capita using relationship (1.7). The most important feature is the fall in the growth rate in 1991.

There is a discretization problem associated with timing of the GDP and the population readings. By definition, GDP per capita values are given for the last day of corresponding years. The population estimates are published for the first day of October. So, formally these variables are separated by one quarter. Then the number of 17-year-olds should be used if to judge by the start of decrease demonstrated in Figure 1.21. One has to bear in mind, however, that for N18 the mid-term point is April 1. This date divides N18 in approximately equal portions. Thus, we consider the estimate of N18 (April 1, 1991) as the closest to the end of 1990 and use this age population as the defining one. We have to shift the predicted curve by a quarter back (from April 1, 1991 to January 1, 1991) in order to synchronize these curves. The procedure has brought an excellent match in the most important period between 1990 and 1993. One can also use N17 with a one year shift or any other younger age with relevant time shift.

Figure 1.21. The evolution of single-year-of-age populations. Shown is the number of 10-, 17-, 18- and 19-year-olds. The number of 18-year-olds starts to decrease in 1991.

Figure 1.22 forecasts real economic growth for the next ten years. We use two projections of N18: the one extrapolated from the estimated number of 8-year-olds in 2009 and that from the 2005 (census) age pyramid. Supposedly, both projections are relatively good approximations for the future demographic development in Japan.

Figure 1.22. Modelling the observed and future evolution of growth rate of GDP per capita. The prediction till 2020 is given from the number of 8-year-olds (N8) and the 2005 population age distribution extrapolated into the number of 18-year-olds. Both approximations give close predictions.

The difference between the measured and predicted dG/G in Figures 1.20 and 1.22 is less than 1% between 1985 and 2003. In the second half of the 2000s, the actual growth rate is higher than the predicted one. Still the difference is within the tolerance range as related to the measurement errors. So, it is instructive to use (1.8) and predict N18 from GDP.

Figure 1.23 depicts the predicted time series and two enumerated ones: the estimated N18 and that projected from the 2005 age pyramid. Both actual curves coincide in 2005 and the adjacent years, but the projected curve is below the enumerated one in the past. This is opposite to the effect observed in the United States (see Figures 1.12 through 1.14). Obviously, the difference consists in the rate of the overall population growth. In Japan, the population shrinks and the US population grows. However, the predicted curve fits the number of 18-year-olds between 1975 and 2005. The deviation between 2005 and 2010 is likely to be compensated by the 5.3% fall in 2009.

The best fit model in Figure 1.23 is characterized by A=$550 (2000 US$), but B=1/2 that is different from the previously estimated value of 2/3. This discrepancy is associated with the poor resolution of the dG/G prediction. Essentially, we have fit only the drop in 1991 and neglected the long-term behaviour. The prediction of N18 uses the level of GDP per capita instead of its first difference. As a result, the short-term fluctuations in the dG/G curve are cancelled out and the predicted N18 curve fits observations much better. It is interesting that the deep and sharp trough in N18 observed in 1984 is expressed by a wider but shallower depression between 1983 and 1989. This is the effect to be investigated in detail. Otherwise, our model shows a reasonable level of accuracy for data between 1970 and 2009. If the number of 18-year-olds will follow up the predicted curve in Figure 1.21, one may expect the rate of growth between 1% and 2% per year in the 2010s. Essentially, the growth will follow up the inertial component, A/G, since the defining age population will be constant.

Figure 1.23. Enumerated and predicted number of 18-year-olds.

There is almost no migration and the Japanese population structure is very stable with a well-predictable death rate. Hence, it is possible to predict the GDP growth rate with a high reliability. Having the forecast and knowing the principal mechanism driving real economic growth one can propose a new migration strategy, however, in order to speed up the economy. Any means to accelerate the birth rate will give results only in 18 years. It is obviously too long a wait for such means to be incorporated in the current socio-economic policy. On the other hand, the Japanese have paid fifteen years of low performance for the ignorance of the importance of demographic processes. Reoccurrence of such a depressive economic period should ideally be avoided in the future.

The long term trend in Austrian GDP

We continue testing our model of the real economic growth by presenting more developed countries. The next example is Austria. Originally, 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. (No mainstream macroeconomic model has ever been tested by data according to strict statistical procedures.)

Figure1 presents the case of Austria: annual increment in real GDP per capita is plotted against the level of real GDP per capita. (Equation (1) uses time implicitly.) 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. One can see this effect of the example of Ireland.

In 2009, the trend is almost 0 and the hypothesis of the constant increment looks sound. The next case is Belgium.

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

References

1. Kitov, I., (2006). Real GDP per capita in developed countries, MPRA Paper 2738, University Library of Munich, Germany, http://ideas.repec.org/p/pra/mprapa/2738.html

2. Kitov, I., (2009). The Evolution of Real GDP Per Capita in Developed Countries, Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. IV(1(8)_ Summ), pp. 221-234.

Real GDP per capita in Japan

Couple months ago we presented the concept of constant annual increment in real GDP per capita, G(t), as observed in developed countries. The concept 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. This is an empirical model and is based only on observations of real GDP in developed countries. This is in striking contrast to the mainstream macroeconomic models based on axioms; not empirically proved axioms.

Unlike in the Solow model and its successors, the rate of growth of real GDP per capita, dG/G, has a decelerating nonlinear trend. Differentiating with respect to time and dividing both sides of (1) by G(t), one obtains

dG/G = A/G (2)

This model gives excellent statistical results and explains the evolution of real GDP per capita in developed countries [1,2] since 1950. This year is considered as the year of relatively accurate measurements of GDP. We are using the data base provided by the Conference Board.

In the post related to labor productivity in Turkey, we introduced a model explaining the evolution of productivity as based on the deviation from constant annual increment of real GDP per capita. Therefore, model (1) provides an empirical framework for the productivity model and we need to illustrate the predictive power of (1).

Figure1 presents a very important case of Japan: annual increment in real GDP per capita is plotted against the level of real GDP per capita. (Equation (1) 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.

Both original linear regression line is practically parallel to the x-axis. The corrected line is characterized by a slightly negative trend. There were two periods of very quick growth between $12000 and $20000 and between $28000 and $33000. Both ended in periods of low (sometimes - negative) growth rates. This effect might be expected in any country which demonstrates very fast growth during an extended period of time. A good example is Ireland. A candidate is China, but its growth is supported by the army of unemployed with very low salaries. Therefore, China may grow mainly due to extensive factors and real GDP per capita do not grow so fast as the overall GDP.

Following the general rule of the constant increment, one may expect a slow recovery of the Japanese economy over decades. However, this recovery is unlikely because the Japanese population is on long-term decline [3].

References
1. Kitov, I., (2006). Real GDP per capita in developed countries, MPRA Paper 2738, University Library of Munich, Germany, http://ideas.repec.org/p/pra/mprapa/2738.html

2. Kitov, I., (2009). The Evolution of Real GDP Per Capita in Developed Countries, Journal of Applied Economic Sciences, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. IV(1(8)_ Summ), pp. 221-234.

3. Kitov, I., 2006. "The Japanese economy," MPRA Paper 2737, University Library of Munich, Germany

Does crude drive the price index of steel and iron?

This update includes the readings of the producer price indexes of crude oil and iron&steel for November 2010.

In September 2009, we reported that the price index of crude oil had been likely evolving in sync with that of iron and steel, but with a lag of two months [1].  In order to present both indexes in a comparable form, the difference between a given index, iPPI, and the overall PPI was normalized to the PPI: (iPPI(t)-PPI(t))/PPI(t). The 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 since June 2010.  Simple visual inspection reveals the following feature: the (normalized deviation from the PPI of the) index of iron and steel lags by 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 2010

(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 invaluable 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

Between 2006 and 2010, 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.  

References

1. Kitov, I., Kitov, O., (2009). Sustainable trends in producer price indices, Journal of Applied Research in Finance, Spiru Haret University, Faculty of Financial Management and Accounting Craiova, vol. I(1(1)_ Summ), pp. 43-51

Labor productivity in Austria: further fall in 2010 and 2011

As in the previous post, Figure 1 is borrowed from our paper on productivity [1] (see Figure 4 in the paper). It presents the case of Austria. This is a less difficult example with the rate of productivity growth, dP/P, on a steady descent since the 1970s.  Between 1975 and 2005, the rate of productivity growth is oscillating around the level of 0.015y^-1. Notice the excellent prediction of the severe drop in the productivity after 1970.  This fall was induced by an increase in the growth rate of real GDP per capita relative to its inertial level, as Figure 2 depicts. The elevated rate of real growth induced a higher increase in the rate of participation, and thus, the drop in productivity.  It is worth stressing again that there was no shock to productivity or a structural break, as the mainstream economists would suggest. Our model presumes that labor productivity in Austria has been following the only driving force – real GDP per capita.

Figure 1. Observed and predicted (from real GDP pee capita) change rate of productivity in Austria.  The observed curve is represented by MA(5) of original version. Model parameters are as follows: A₂=$335, N(1959)=100000, B=-500000, C=0.243, T=3 year.

 Lets return to the deviation from the inertial growth, which is unambiguously determined by constant annual increment of real GDP per capita. Figure 2 shows that the rate of inertial growth is decreasing with the increasing level of GDP as a reciprocal function of GDP.  Coefficient A2 has to be determined empirically for each developed country. For Austria, the initial estimate was A₂=$335 (1990 U.S. dollars at GK PPPs as presented by the Conference Board). The current economic and financial crisis manifests itself in a severe drop in GDP, with dGDP/GDP=-0.045 y^-1 in 2009. 

A significant feature of the model is the presence of a delay between the change in real GDP and the reaction of P. This effect is similar to the delay of thunder relative to lightning. Any economic system needs some time to adjust to the exogenous change. In Austria, productivity lags by 3 years behind GDP, as caption of Figure 1 indicates. For details of the model see [2]. For the purpose of this blog, the three year lag means that the current drop in real GDP per capita will result in a hike in labor productivity three years later. Also, the currently observed decline in the rate of productivity growth is actually induced by several years of intensive real economic growth observed before 2009.

Figure 2. Comparison of the growth rate of real GDP per capita, dGDP/GDP, with the rate of inertial growth defined as A₂/GDP.

Finally, Figure 3 tests the model by adding two new data points to Figure 1. These new measurements are borrowed from the Conference Board database [3]. One can conclude that the model gave an excellent prediction for 2008 and 2009. The period of the productivity decline will continue in Austria for another couple years. Since 2012, the rate of productivity growth will show high positive values in response to the current drop in GDP and labor force particiaption. This will be a striking upturn which is always a challenge to any productivity model or concept. We will revisit the case of Austria for further validation of the model. In 2010 and 2011 the rate of labor productivity in Austria will be falling.

Figure 3. Same as in Figure 1 with two new points – 2008 and 2009. The original and MA(5) productivity series are shown. One can expect positive rate of productivity growth in 2012.

References

1. Kitov, I., Kitov, O., (2009). Modelling and predicting labor force productivity, MPRA Paper 15152, University Library of Munich, Germany, http://mpra.ub.uni-muenchen.de/15152/01/MPRA_paper_15152.pdf

2. Kitov, I., Kitov, O., (2008). The driving force of labor productivity, MPRA Paper 9069, University Library of Munich, Germany, http://ideas.repec.org/p/pra/mprapa/9069.html

http://mpra.ub.uni-muenchen.de/9069/01/MPRA_paper_9069.pdf

3. Conference Board. (2010). Total Economy Database, January 2010. http://www.conference-board.org/data/economydatabase/

On further decline in labor productivity in Turkey

Figure 1 is borrowed from our paper on productivity [1]. It presents the case of Turkey. This is a difficult example with the rate of productivity growth oscillating since 1980. Since the measured time series is smoothed with MA(3), actual oscillation is even more prominent. Such a behavior is a nightmare for the mainstream models based on capital, labor and multifactor productivity. As a rule, the multifactor productivity has to resemble observations and severe “shocks” to productivity are introduced.  This is a lucky hour for an economist – millions of factors to explain these shocks. In reality, the number of explanations is steadily approaching the number of economists involved. At the end of the day, all mainstream models are able to explain only “stylised facts”. This is a euphemism of “failure”.

Our model uses only one variable – real GDP per capita. The intuition behind the model is almost banal.

1. A developed economy is characterized by a constant speed of real economics growth, which we call “economic inertia” in line with mechanical sense of inertia. In other words, the economy would be growing with constant increment per year, i.e. at constant speed, if no change in the population age structure is observed. 
2. Any deviation from the inertial growth results in the change in labor force participation. Obviously, a higher speed of growth may attract more people into the labor.
3. The number of people who are able to join the labor force in response to a given growth above the inertial one is proportional to the relevant deviation.
4. The value added by any newcomer must depend on his/her overall professional capabilities. It is obvious that this characteristic (capability or productivity) is distributed (we claim that this distribution is exponential and personal income distribution is) over the working age population and people with efficiency between 50% and 51 % should bring more value added to the economy than those between 75% and 76%.  In other words, one per cent of “extra” (above the inertial level) economic growth may allow to join the labor, say, 1% of population, when this labor force grows from 50% to 51% , or 5% of population, when the labor grows from 70% to 75%.  These portion must give the same extra input into the real GDP.
5. The extra growth in real GDP has to be reflected in productivity, which is defined as a ratio of real GDP and the level of labor force. As suggested in point 4, the extra labor force depends on the current participation rate. Therefore, the growth in productivity depends on the current rate of participation in labor force for a given increase in real GDP.  As an example, the rate of participation in Italy and Canada is quite different and 1% extra growth in real GDP per capita results in absolutely different change in labor productivity.
6. Mathematical formulation of this simple consideration is given in [2].

Finally, Figure 1 (and the example of Canada ) demonstrate the predictive power of our simple and parsimonious model. 

Figure 1. Observed and predicted (from real GDP pee capita) change rate of productivity in Turkey.  The observed curve is represented by MA(3) of the original version. Model parameters are as follows: A₂=$105, N(1959)=1450000, B=-6000000, C=0.24, T=2 year.

Figure 2 tests the model by adding two new data points to Figure1. These new measurements are borrowed from the Conference Board database [3]. One can conclude that the model gave an excellent prediction for 2008 and 2009. The period of the productivity decline will continue in Turkey for another couple years, and then it will start to grow again. This turn is a challenge for any productivity model or concept. We will revisit the case of Turkey for further validation of the model. Meanwhile, we would not expect good news about labor productivity from Turkey.

Figure 2. Same as in Figure 1 with two new points – 2008 and 2009.

References

1. Kitov, I., Kitov, O., (2009). Modelling and predicting labor force productivity, MPRA Paper 15152, University Library of Munich, Germany, http://mpra.ub.uni-muenchen.de/15152/01/MPRA_paper_15152.pdf

2. Kitov, I., Kitov, O., (2008). The driving force of labor productivity, MPRA Paper 9069, University Library of Munich, Germany, http://ideas.repec.org/p/pra/mprapa/9069.html

http://mpra.ub.uni-muenchen.de/9069/01/MPRA_paper_9069.pdf

3. Conference Board. (2010). Total Economy Database, January 2010. http://www.conference-board.org/data/economydatabase/