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.

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