Nothing is proportional

by Admin
Updated: June 17, 2018

Direct proportion is intuitive and expected but nothing is proportional in the real world

The concept of direct proportionality philosophically intuitive: If you do something to get a result then doing twice as much gets twice the result. However, most beneficial information comes from the ways in which direct proportion doesn’t apply or is broken.

Direct proportion is mathematically simple: When the relationship of one variable to another is plotted on a graph, it produces a nice straight line which is easy to understand.

Out in the real world, wherever direct proportionality is expected, it proves to be too simplistic and using calculations that depend on it will at best result in missed opportunities and at worst could prove to be disastrous.

How direct proportion gets messed up

There are three main ways that direct proportionality is ruined:

  1. Drag effects
  2. Slow start (exponential growth)
  3. A sudden change (a “catastrophe”)

This can be illustrated graphically as follows:

  • The straight black line shows direct proportionality (y ∝ x, or y = a.x where a is a constant, e.g. 125)
  • The solid blue line shows the effect of drag (e.g. y = a.x - b.xn, where b=1 and n=2)
  • The solid red line shows the effect of a slow start (e.g. y = b.xn, where b=0.5 and n=3)
  • The dashed lines represent a departure (a “catastrophe”)

Drag effects

One example of drag in business is the law of diminishing returns.

Another is the effect of your own activity in a market, e.g. if you are a large enough player, your buying activity pushes the unit price up reducing your margin for profit.

Another buying large quantities at a discount which translates to a drag effect for the supplier.

Exponential growth

The law of increasing returns - economies of scale - can lead to exponential growth which can generate great wealth.

A concept I call “the ordeal of exponential growth” a.k.a. the deception of linear vs exponential growth refers to surviving the early part of the curve while returns are low.

Digital businesses typically need to plan for exponential growth. Going viral on social me can lead to massive overnight growth - growth that comes with zero tolerance for failure!

Catastrophe theory

Catastrophe theory is an area of mathematics that studies how small changes in circumstances can produce sudden shifts in behavior. For example, an elastic band is stretchy up to a point beyond which it breaks suddenly.

Catastrophes under this theory are departures from what seemed to be predictable behavior - they are not necessarily bad, e.g. a novelist suddenly getting published after years of writing.

Unfortunately, for most practical situations we encounter as individuals, the equations aren’t much help.

More info

mathsisfun.com has a good intro to the mathematics of direct (and inverse) proportionality and some useful everyday examples can be found at futurelearn.com.

ING’s eZonomics.com has a nice description of Fiscal drag and the FT has an interesting explanation of the drag effects of volatility on leveraged funds can be found at FT.com.

The law of increasing returns is defined and explained in some depth on Economicsconcepts.com and a couple of fun interactive demos can be found at mathwarehouse.com.

A qualitative (i.e. no equations) primer for catastrophe theory is at Encyclopedia.com and Dangerous Intersection by Steven Strogatz (nytimes.com, 2012) offers a more in-depth, yet still accessible and entertaining analysis with applications to finance and economics.

Internal links

Diminishing returns The graph of success Should I play the lottery? Go to Articles
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