19 Discussion of the Analytical Solution
The piece-wise solutions of Chapter 9 of the
simplest model equations give a lot of approaches for a comprehensive
discussion of the principle behavior of a credit-driven economy. First, we
compare the properly integrated piecewise solution to the IMF 2005 formula.
Exemplary of the GDP at the beginning of the development (
) is:
(19.1)
(19.2)
We see that the singular IMF 2005 model (18.1) promises a constant growth rate g. The IMF model sets for g the general growth of the capital stock, but in truth, it is of course only the share of invested capital into the GDP. In fact (18.2), it is a combination of exponential and harmonic growth. The hyperbolic functions can be written as exponential function, because it holds
(19.3).
For vanishingly small
holds
and thus for the
smallalso holds 
. We can
therefore write as an approximation for small arguments
:
(19.4)
The argument is
(19.5).
and thus applies to the beginning of the development
(19.6).
Hence we actually start early in the economy with approximately
(19.7)
Analogously, this applies also for
, as is easily
verified. At the beginning there is an exponential slope not only of the
capital stock, but also of the GDP. However, not globally but only locally, and
also are there not the same values 
for
the slope rates, but
for the GDP and
for the capital
stock. Since in the beginning almost all capital growth gains from direct
investment in GDP, also these values are initially very similar. No wonder then
that singular interpolating models may initially give the impression of a global
scope.
Over time, however, the terms
and
develop a very
different behavior as they change to
and
terms. Since they
are harmonic functions the argumentis
of particular importance. The argument of a time-dependent harmonic function
has the structure
, with
the so-called
angular frequency. There are the familiar relationships
with
the frequency.
Wavelength
and frequency are
linked by the wave velocity
. The
characteristic time
of course, is the
reciprocal of the frequency
.So now we can
determine these values closer. The characteristic frequency is given by
(19.8)
and further, the characteristic time to
(19.9).
The timecan
be calculated if one uses the official figures of the statistics of the FRG. At
the beginning of development
was approximately
-17.9% and
about 4.2%. So it
follows immediately
years (19.9)
as the characteristic time. This characteristic
time is, in fact, be of the same magnitude as the critical time
, which we will
calculate in the next chapter of this book from a simple rule of thumb.
However, it has a slightly different meaning. The meaning ofis basically the
answer to the question: In which time it would be possible with the given
initial conditions, to increase GDP by a factor of
? With time-dependent increasing of the capital coefficient but this
gets always more difficult, and the characteristic time therefore gets larger.
We can see
these characteristic values in the following figures, calculated according to
the official figures of the FRG: In the beginning the characteristic timerepresents the
critical periodgiven by the compound
interest rule of thumb next chapter. For the range of initial conditions
we get about 
years.
But after
thatgrows strongly
and ultimately turns sharply to negative values. This inflection point is given
by the year 2000. For the growth path of the economy thus the parameter
is crucial. It must be kept negative to stay in
the growth-regime of theand thus does not
enter into the realm of the down-sloping .
Since 2001, but this value is positive in the FRG. The share of domestic
lending business has declined since 1950 by almost 73% to less than 40%. To
assess the growth parameters we reflect again on the importance of individual
variables.
It is the condition
(19.10)
to comply, which in turn gives the conditions
forgreater or less
zero:
:
(19.11)
:
(19.12)
As we see it is in each of the four cases in
turn critical if the relative share of investment in the real economy
is larger or less
than 50%, which is the same as
is greater than
or less than zero.
In the next
graph we can see, calculated according to official figures from the Bundesbank,
the growth frequency(scaled40 by the factor 50) and the share of
real investment
(scaled by 4).
One can clearly see how the crisis situation since 2000 is associated with the
decline of real credit share below 50% and also the drop in growth rate below
zero.
Potential strategy for how you can still stay on a positive growth path, we must therefore continue to consider the four different cases F1 to F4:
(F1)
(19.13)
(F2)
(19.14)
(F3)
(19.15)
(F4)
(19.16)
Until 2000 the case F2 was present in
the FRG. We see here on the official figures, the FRG typical high savings
rate. The relevant limit value
is
maintained until the year 2000.
Afterwards,
however, the value of
changes its sign,
because since then the value of the share of investment to the real economy
fell below 50%.
Thus, we switch to F1 case with the condition
for the growth
path. Currentlyis only about 40%
at a nominal interest rate of less than 2%, and thus
would have to
apply. This means that the savings rate would have to be less than <0.1%,
thus best negative, in order to remain on a growth path. Alternatively, would
also be a negative nominal interest rate
of bank assets imaginable. One possibility would then be the case F3.
That would mean that with a high proportion of investment banking, with a
savings rate of
, e.g. 
,
the growth condition would also continue to be
met with a positive savings rate.
Remains finally the case F4: So likewise, a negative nominal interest
ratesof assets, but at
reduced investment share, and a savings rate of condition of
, e.g.
.
The last example also shows the growth
condition, that meets the growth path only with negative savings rate. One can
clearly see the importance of the relative ratio of investment in the real
economyin the next
graphic. Because of positive nominal interest rate always
is to demand.
The latter
value does, however, with the fall ofbelow
the 50% level, and thus the preponderance of bank equity business, a sign
change. So unattainable with the FRG typical high savings rate is above
condition. To return to growth path thus the possibilities are limited. The
result would call into question, to reduce the savings rate to well below zero.