Growth II

 

What Do I Know?

No university, no degree, but lord knows
Everybody's talking 'bout exponential growth
And the stock market crashing in their portfolios
While I'll be sitting here with a song I wrote
Sing, love could change the world in a moment
But what do I know?

You probably know more than you think!

Any exponential growth function can be expressed as a compound growth function. Compound growth functions have a constant and continuous growth. Exponential growth functions have a continuous but not constant growth. However any compound growth rate can be expressed as an exponential function.

If the growth rate, in demand, is 10% per period, then the compound growth formula would be

Future value=Present value*(100%+10%)^x

where x is the number of time periods from the present to the future.

At infinity, this will have a value of infinity. But this assumes that the time periods are also infinite. It is possible for the Future value to consume all of available supply before an infinite time unless the growth in supply is also 10% per year. If it is less than 10% then the future value in demand will exceed the future value in supply long before an infinite number of time periods have been reached. In other words, in compound growth you will approach infinity in a finite period. In exponential growth you will only approach infinity only after an infinite period.

The rich have a disparaging comment about the poor that they are poor because they live off principal instead of living off interest. Doh, if the present value, principal, is zero then it makes no difference what the interest rate is, the future value will always be zero. The ideal is to have enough principal that you CAN live off the interest. But if your living expenses are higher than the interest on your principal then your principal gets reduced and the next Future value will be smaller because the Present value in that next time period is also smaller.

An exponential growth can be made from this compound growth

Present value*(100%+10%)^x=Present value*exp(λx)

 if

λ=ln (110%)

In fact any compound growth rate, r%, can be made into an exponential growth if

λ=ln (100%+r%)

If λ is negative, less than zero, this is said to be a decay function, for example a radioactive element decay function, and λ can be converted into a half-life. Long half-lives are stable, good, and short half-lives are radioactive, unstable, bad. When λ is positive, greater than zero, this is said to be a growth function. But λ can still be converted into a time, say a doubling period. But while this will be the inverse, the qualifications are still valid. A long doubling period is good and a short doubling period is bad. Short doubling periods have lots of energy but expire soon. Long doubling periods have less energy but they last longer.

Given that, what should the best growth rate be? The question should be what is the growth rate of a market? The best is arguably an example of the Lake Wobegon effect ( “all the children are above average”) or the Yogi Bear effect (“smarter than the average bear”). If the population is normal, then the maximum, "best", of the crowd should be twice the average of the crowd ( e.g. the Z-score of 99.9% is approximately 2.)

Sustainable growth should thus be twice the growth of the market ( yes a  firm could grow by 10% if the market only grows by 2%, but eventually that firm will have 100% market share.)   The maximum sustainable growth  should not exceed twice the growth of a market. Twice is only sustainable if the average  IS the maximum ( which is only true when the standard deviation is zero, but then that is no longer normal.)

Should that firm enter other markets? Absolutely, but that growth still should not exceed 2 times the growth in its combined markets ( the original market plus the new market). If the best growth rate is compared to the revenue from the original market, the growth in  revenue will appear to be greater than the growth in the original market. But this is only because of the choice of the dominator in that growth rate.

This also defines the size of the new market. If a firm wishes a growth rate, r, that is more than twice the growth of its market, it should pursue new markets whose value, V, is larger than its current market according to.

Proposed Growth Rate=

2*(V_{Old Market}*r_{Old Market} +V_{New Market}*r_{New Market} )/(V_{Old Market)} +V_{New Market})

This also explains why you should pursue new markets.  If your old market is buggy whips and you don't go after any new markets, then the best growth rate you can expect have is twice the growth rate of the old market. Having 100% of the buggy whip market will still mean a growth rate of 0% since the buggy whip market is not growing.