20 Rule of Thumb: the Economic half-life Time

Already in the measured values for the investment banking industry a certain half-life can be seen. So we ask ourselves what such dynamic effects are a result of and whether there is a way to estimate such times, at least roughly. Thus we consider the following highly simplified model of a closed economy: An economy has the economic performance given to a start time. Whereas a lot of money is delivered. is slightly larger than , so that all things can be paid for with money in circulation , as well as a general reserve asset can be accumulated: (20.1)

With the growth of the economy now is growing the capital stock. Thus we can write  (20.2)

It is considered that arises more money with the economic growth41 , and the inflation rate becomes effective and, finally, the deposits with the banks earn of interest. We compute, as always, on an annual basis , therefore the time factor is the number of years. We are now following the question: At what juncture is the capital stock greater than the sum of tradable goods? So mathematically, the inequality: (20.3) ?

Now, the relative share of capital stock at all levels of the economy, goods and assets, results from the ratio of capital stock to total national wealth (20.4)

and to reduce the fracture results in (20.5). This gives a dimensionless value that is called the "significance number" of capital in an economy. When this value gets greater than 0.5 = 50%, so there will be a transition, after which the meaning of production falls behind the financial sector . This equation can now be easily resolved to the critical time , as in the case of equality holds (20.6)

and thus after a short algebra we get (20.7).

As is , one can also see that the critical time goes to infinite for or , which corresponds to a stable system. Typical values for the savings rate and interest rate are about 10% and 5% in the FRG. This gives a critical time of years.

For moderate values, about 7% and 3%, the corresponding time is years,

which is already more than two generations. The critical time of a monetary economy depends so much on the return on investment: With every percentage point it drops considerably. Somewhat less strongly is the influence of the savings rate. With only 1% for each and , the critical time is more than 400 years. An increase in interest rates to 10% results in a lifetime of only a few decades. An increase in the savings rate alone to 10% reduced but this time not much, it is then still almost 300 years. At moderate values of and the critical period is therefore from years. The rule of thumb for the critical period can be reduced to a simple rule because of the shallow curve of the natural logarithm. As and are significantly smaller than one and commute to values of about 5%=0.05, one can roughly approximate: and (20.8).

So that we can remember as a "raw rule of thumb" for the average lifetimes of capital driven economies: (20.9).

With average interest rate on all assets of about 5%, we obtain for example years     (20.10)

magnitude as for the estimation of the saturation level.