22 Conformity to Quantity Equation
For this purpose we first consider the
numerical integration of the simplified model. So we will first check whether
our model actually withstands the requirements of the quantity theory. Then
follows a theoretical foundation of the relationship. We consider again the
basic model and we calculate the money supply. This consists on the one hand
from the GDP of money in circulation
,
i.e. the amount of money not saved. In addition there are contributions that
are deducted from interest rates on savings, if one assumes that most market
participants are interested42 to preserve their nominal assets.
So we get
(22.1).
The product
is of course the GDP
in
our model. Due to the quantity theory 
must
now obtain trivially to our basic model. Thus in our numerically integrated
model should rule
(22.2).
We can now easily calculate these value over
time. In fact the result (Fig. 22) shows, that the quantity equation is indeed
respected. Over the entire non-critical area of development over time, it
remains closely at the value 1, except for small deviations from numerical
integration. Alike can also be calculated the velocity of money, which is so
important. This is
, with our total
assets
constitute a
superset of what the money supply
is
only one component. In a first approximation
can be assumed constant. Thus, the monetary velocity is computed as
(22.3)
In the next graph, the corresponding values
from the numerically integrated K(t) and Y(t) were obtained. From
the considerations of the previous chapter one can justify
which gives
indeed numerical values of
as
it is to be expected. In Figure 22 we see that from about 2007, the velocity
begins to leave the liquidity area. Nonlinearities are beginning to be felt
after 2010. At the latest by 2020, a simple linear approximation is no longer
justified. Then a computation must be based on the general field theory, which
we treat later.
One can see clearly only after the crisis years in the vicinity of the collapse, the quantity equation in our simplified model loses its generality. Because then begins the regime where a linear approximation is no longer present. Especially since external sources play in crisis years an increasing role. About the occurring nonlinearities, we will also need to make more considerations later in this book. Significant is also the course of the resulting velocity. They tend to behave exactly as it is known empirically from official data43.
The ratio KV/HP=1 is obviously a constant of the economy. A deeper foundation receives this more than 400 years old empirical fact by comparison with physical problems. Systems with conserved quantities always satisfy a corresponding equation of continuity. These are basically on the structure
(22.4)
(21.4) is a normal continuity equation as we
know from physics. Here it means that the charge
is a conserved value and it is the source of a current
. One can write
such an equation for short as a four-vector:
(22.5)
The four-divergence is defined as
(22.6)
where
is
the unit-speed. In economics, there is now trivially
(22.7)
Using this simple description one can now see
at the heart of the classical quantity theory. Because 
is a speed and
the frequency of
transactions, which we can regard as a time derivative. We can therefore write
in one-dimensional analogy:
(22.8)
The quantity theory is obviously just the necessary continuity equation, as with any well defined theory must exist such conservation values. It says:
The prices or price levels are the sources of funds currents.
If there are changes
to a source this
results always automatically in resulting currents 
. And vice versa
rules:
No currents can exist without an associated change in the sources.