39 The General Quantity Equation
First of all we have to define the full and thus General Quantity Equation:
(39.1)
The norm of this matrices and vectors are defined now by the following summation-norm:
(39.2)
Sum up into the two main parts of capital driven economy:
(39.3)
which gives by virtue of the sum-norm
:
which can be sorted to:
(39.4)
Now assuming symmetry and/or small valued mixed coefficients
(39.5)
follows:
(39.6)
Now we define
(39.7)
which leads to:
(39.8)
which can be interpreted as:
loans to real economy + banks own business + interests flow from R to I=
production by loans + financial products + real econ. interest drain from R to I
Now we invest Euler-Lagrange relations, and thus we define in a first step
(39.9),
although this is not entirely correct, as
the units 1/y and cur./buy in standard quantity equation
disagree, which should not happen for the variational
calculus as units [] should be
equal. Thus we
denote this badly defined quantities with a b index.
Now we do
with
(39.10)
which results in
gives 
(39.11 a)
gives 
(39.11 b)
gives 
(39.11 c).
The last equation also means 
or
. Now we get by
adding this three equations
(39.12)
and by subtracting them we get also
(39.13)
Subtracting and adding this two equations from results in
(39.14)
which means by some redefinitions
(39.15),
as 
can
be interpreted as the savings of the economy to banks, and 
as the net result
of the loans and interests from the GDP. The constants d,e,c are just factors due to our badly defined
units of the standard Quantity Equation. So we get the principle form of the
basic model derived in the first chapters of this book. The second equation is
just trivial, as
can be written as 
which then can be interpreted as
(39.16)
with f,g again
some constant factors resulting from the bad definition of the standard. As we
see is seemingly
and
and also we may
set 
by definition. The
term
is the sum-effect
of giving loans to the real economy on one hand, and to get interests in return
on the other hand. The net effect now can be summarized in our
net-business-rate by some sorting and redefinition:
(39.17)
This brings then back our the basic equations from the first part of the book
and 
,
which just had to be shown.