FOM: Increasing Net Worth

[Harvey Friedman]

For our purposes, the net worth of individuals worldwide will be measured
in US dollars, rounded up to the nearest whole dollar. People in debt are
considered to have net worth 0.

We assume that there is a constant population of 6 billion individuals in
the world who live forever. Net worth is calculated annually on the last
day of the calendar year, starting this year.

The richest individual (highest net worth) may vary from year to year. We
make the modest assumption that the highest net worth in a given year is at
least 1% of the highest net worth in the succeeding year. The richest
individual at the end of this year has at most 100 billion dollars.

THEOREM 1. In some future year, everybody's net worth will be at least as
great as it was in some common year previous to that year.

How many years do we have to wait for this to happen? Of course, this
depends on how the net worth of individuals changes over time. So the
problem is to give an upper bound to how long we have to wait, regardless
of what the net worths actually become, subject to our assumptions.

ANSWER: More than the 6 billionth level of the Ackerman hierarchy at 100
billion. Less than the 6 billion 1st level of the Ackerman hierarchy at 100
billion.