13 Classical Growth Models
It is to justify the considerable mathematical complexity, which requires full field theory of macroeconomics, we add at this point a chapter which deals with the criticism of classical mathematical models of growth of the economy.
It is undisputed that there is an interaction
between financial capital stock
and gross
domestic product
of an economy.
However, the correct connection is currently modeled unsatisfactory.
The empirical data shows for the FRG, the picture Fig.10 as already known:
Tangibly increases the capital stock28 in the long term significantly
faster than GDP. Contrary, to be seen even in all the developed countries, a
tendency to stagnation of GDP growth arises. This also manifests itself in the
figure 11 of the total capital coefficient K/Y. This is by no means
constant, but grows steadily in favor of the capital
side.
In detail, one also sees that this ratio, from an initial 0.38 in 1950, already in the middle of the 1960-year exceeded the value of unity 1. The small "hump" around 1990 resulted from the integration of the GDR into the FRG, the huge jump of a factor of 2 to 3 times in the 1990s, arised from the dotcom bubble. The jump was due to the dotcom speculative exaggeration in the context of the first internet hype. It reflects the fact that this wave of innovation of the 90s left distinct traces in the total of assets and liabilities, but barely detectable traces left in the GDP.
This fact contradicts the usual assumption that
technological innovations would cause increases in GDP. As seen in the details
of the GDP curve (see Fig. 23) of the FRG, innovation waves like cell phones or
PC left no significant increases in (the natural slope of) GDP. The effect of
technological thrusts for GDP is so amazingly low. In contrast to this the
integration of the GDR into the FRG can be seen clearly again, where the sudden
population increase
is received
immediately. In particular, the real data show the remarkable fact that the
capital productivity29 or “marginal productivity of
debt”
, i.e. the ratio
of the two long-term rates of change, decreases with time and finally goes to
zero or even negative. This clearly observable effect is a worldwide effect and
is contrary to general assumptions of economic models currently still in force.
The growth model still used by the IMF (IWF 2005) can be written in the original approach30 as:
(13.1)
with production(GDP), capital stock(assets
or liabilities of credit institutions) and 
(investment to the real economy). 
is the growth rate on an ad hoc basis equal for
all three functions. Then
is the current and 
the previous year. These functions can be
mathematically more meaningful represented as a system of differential
equations:
(13.2)
These three equations have of course a trivial solution and since each equation is only self-referential are completely independent from each other, namely:
(13.3)
Accordingly, all the three economic key
functions would have to grow in unison exponentially indefinitely. Then the
capital coefficient 
would remain
constant, and also the capital productivity, which gives
, as is easily
verified. However, this contradicts the experience and the official
measurements. One can try to rescue the relationship by replacing the same
growth rateby more
realistically uneven growth rates
.
However, the singularity of the equations does not change, because they can
continue to operate independently of one another and be integrated separately.
The capital coefficient is thus
(13.4)
an exponential function of the ratio with the
newly defined constants
and
.
As an empiric rule
applies, and
this would give a weak exponentially rising function for the capital
coefficient.
(13.5)
and the capital productivity should be
(13.6)
according to a slightly falling curve, which however, could never be negative. A comparison with the figure 13 shows, that the real data do not hold this behavior.