Growth

 Crown of Creation

Life is change

How it differs from the rocks

I've seen their ways too often for my liking

New worlds to gain

My life is too survive

And be alive for you

Aah

And growth is life!\

The most recent episodes of the podcasts Radio Lab and Planet Money were on grwoth.  I like the name “Malthusian Swerve” that they used but mathematically it already exists as the Cumulative Distribution Function, CDF, of the Exponential Distribution’s PDF, Probability Density Function. On second thought Malthusian Swerve does roll more trippingly off the tongue and is probably better branding. 

There is a way to look at growth as NOT constant and thus uncontrollable ( and after all what is cancer but uncontrolled growth). The phrase “Life is change” might be from the Jefferson Airplane's song Crown of Creation, but the concept goes back to the ancient Greek philosopher Heraclitus. The Malthusian way of looking at constant growth is as a power function, Future  = Present * Growth rate to  the  power  of  Time,                                             

Future =Present * Growth Rate ^Time. 

This is NOT the CDF of an exponential distribution. An exponential distribution would be  Future = Present * exp(time * growth rate)),

  Future= Present * e ^{Growth Rate* Time.}  

The base in this equation is e, Euler’s number. The Natural Logarithm of the exponential of x is  ln(x). 

What was not anticipated is a disruption during the forecast time period. The examples in the podcasts did not include this, but in the 1880s the fear was that New York City would be buried in horse manure by 1930. This was a reasonable trend given the horse power in use at the time. What was not expected is that horse power would be replaced by gasoline engine power. But while history is rife with disruptive and unforeseen examples (what economists would call substitution of products), it would be nice to establish what the growth rate should be absent any unexpected swerves.

The Federal Reserve sets its target for inflation at 2%. This is low enough to almost be unnoticeable. But it is still not ideal. And it is also a Compound (and thus constant) Annual Growth Rate, CAGR. It is said that you can boil a live frog in a pot if you keep the growth in temperature low enough. Two percent is also the rule of thumb rate of growth that I was taught as a young engineer. It is close to 1/ln(2)%, 1.44%, which is the ideal growth rate given the two dimensions, space and time, of reality, using exponential growth. It sounds like exponential growth is even higher growth than compound growth, but words can be deceiving.

Trending, assuming constant growth, is NOT a good idea. Take a young child..  If you trend out his current rate of growth he should be 10 feet tall at the age of 20. But you don’t expect him to be. As Han Solo  said in Star Wars VII, “That is NOT how the Force works". Exponential Distribution’s CDF, a sigmoid curve, for the win!