Debt grows exponentially while physical economy grows logistically

I found this concept described at George Washington’s Blog in The Elephant In The Room: Debt Grows Exponentially, While Economies Only Grow In An S-Curve. During the dot com era the concept was referred to as “trees don’t grow to the sky”. Hubbert of Peak Oil fame demonstrated that given a finite amount of a resource (oil, in his case), production of a resource can temporarily grow exponentially but peters out as the resource is depleted to zero. The graphical representation of this is the logistic curve or “S-Curve”:

The article linked above quotes economist Michael Hudson:

"Every economist who has looked at the mathematics of compound interest has pointed out that in the end, debts cannot be paid. Every rate of interest can be viewed in terms of the time that it takes for a debt to double.

At 5%, a debt doubles in 14½ years; at 7 percent, in 10 years; at 10 percent, in 7 years. As early as 2000 BC in Babylonia, scribal accountants were trained to calculate how loans principal doubled in five years at the then-current equivalent of 20% annually (1/60^th per month for 60 months). “How long does it take a debt to multiply 64 times?” a student exercise asked. The answer is, 30 years – 6 doubling times.

No economy ever has been able to keep on doubling on a steady basis. Debts grow by purely mathematical principles, but “real” economies taper off in S-curves. This too was known in Babylonia, whose economic models calculated the growth of herds, which normally taper off. A major reason why national economic growth slows in today’s economies is that more and more income must be paid to carry the debt burden that mounts up. By leaving less revenue available for direct investment in capital formation and to fuel rising living standards, interest payments end up plunging economies into recession. For the past century or so, it usually has taken 18 years for the typical real estate cycle to run its course."

The logistic curve is a description of exponential increase and decline in growth rates. The inflection point where the trend shifts from increase to decrease is the commonly referred to "peak", reference with respect to oil production, for example.