5. Power Law Growth (L-curves)
Power laws, or L-curves, are another useful growth curve, as they tell us how learning or performance increases in closed systems, temporary environments of fixed complexity. Power laws have similarities to the saturation phase of S-curves, though they are likely each due to different physical mechanisms. In either case, we see declining marginal learning over time.
With a power-law curve, each time cumulative output doubles, learning improves by a fixed percentage (for example, 20%). For an individual doing practice trials to learn some new skill or method, or organization or industry of a fixed size learning production, if each of these are capable of a standard output each year, learning or performance will start out very rapid, as it is very easy to double a small batch size at first, but it will increasingly slow. Over time, there’s less new efficiency to be discovered, and less learning left to be done to improve performance.
Many manufactured technologies improve their performance over time based on manufacturing experience curves, a type of industry learning curve. As these are power law curves, each time cumulative output volume doubles, value added costs fall by a constant percentage (for example, 10%, 20%, or 30%, as in the figure at right), a rate which will vary by industry and technology. When plotted on a non-log scale (picture left), we see the “L” shape of the L-curve, with declining learning/efficiency improvement over time with each new unit produced.
Notice that in the special case where output doubles every year or less, say in a young industry rapidly growing to serve a large market, power-law growth can look like exponential growth. That’s why startups are such dynamic environments. Many things look exponential at first, even those that ultimately aren’t. Some people confuse the two, but exponential growth continues its steady annual pace, while power-law growth increasingly peters out. But if the industry is young, if new technologies and processes are continually involved, or if more people with different skills keep entering the mix, power-law learning keeps returning to the beginning of its learning curve. In that case, performance gains can look exponential for long periods, and even briefly superexponential if powerful new technologies or processes enter the mix.
There is also a power-law distribution, another L-curve (picture below), which is not a growth curve but a distribution of features often seen in collectives of natural systems. Power law L-curves have “fat heads” and “long tails” (see picture right), and are the source of the 20/80 rule, or Pareto Principle, the reality that 20% of the cities, people, products, and activities often control 80% of the popularity, wealth, and power, while the rest represent the “long tail” of evolutionary diversity. This also gives us the observation that 20% of our effort will often get us 80% of our desired results. Thus strategy, focus and satisficing (versus perfectionism) are key! Good books on using power law distributions in business and life include Chris Anderson’s The Long Tail (2008), and Richard Koch’s The 80/20 Principle, 1999. Knowing the basics of both power law distributions (L-curves) and normal distributions (Bell curves) is important for general foresight.