A corrected footnote

[Trond Andresen <trond.andresen@xxxxxxxxxxx>]

Things went a bit too fast for me last night. There were two errors in the
included footnote. It should say "consols" where it said "bonds" in the
first paragraph, and the last paragraph was completely wrong. I include the
corrected footnote anew below. Sorry!

Trond Andresen

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Footnote: The danger of perpetuities
==========================

If a certain share of a capitalist's expenditure is on the condition that it
shall yield a future stream of dividends, and these in the next round are
not spent but financially reinvested in the same proportion, etc., then
simple maths indicate that for certain parameter values the aggregate of all
financial assets (mirrored by corresponding debts) will grow exponentially.
(In the following we consider stocks on a par with consols, i.e. as implying
a permanent claim to a future income stream):

Let -

aggregate net non-money financal assets (debt)   = A(t) [$]
initial aggregate assets at t = 0 is       A(0) = A0
average interest (dividend) rate   = i  [% / year]
the propensity to save out of financial income   = s  [  ]
average loan repayment rate (incl. perpetuities)   = d  [% / year]

Then accumulation will occur following the simplest
possible linear homogeneous differential equation there is:

dA/dt = ( -d + s (i +d) ) A(t)                                        (1)

or, with the net asset growth factor defined as

g = -d + s (i +d) ,                                                       (2)

dA/dt = g A(t)                                                            (3)

which has the solution

A(t) = A0 exp(gt)                                                      (4)

A(t) will grow, and we have asset/debt polarization, if g > 0.
In other words, this can be avoided with g <= 0.
This condition  may be reformulated in three equivalent ways,

            s <=  d / (i+d)                                              (5)

or        (1-s) >=  i / (i+d)                                          (6)

or         is <=  (1-s) d                                               (7)

For low nominal interest rates, savings rates, short average repayment time
(= 1/d) on loans - the system will not experience growing debt burdens.

Note especially, following (7), that a relative increase in s has a stronger
impact towards polarizaton than a corresponding increase in the interest
rate i.