Price's law and the 80/20 rule

Recently I got thinking about Price's Law and Pareto Principle and if they are compatible or saying the same thing.

I've known about 80/20 for some years, back as far as university. The 50 per cent and square root I'd head of more recently, in the last few months. I was wondering how to reconcile them.

under Price's Law it states

50% of the work is done by the square root of the total number of people who participate in the work.

Now under the 80/20 Rule, also known as the Pareto Principle it can be stated as

80% of the work is done by 20% of the total number of people who participate in the work.

I will use these definitions as cited. Though I've found things a bit strange because on some sites I've heard the term Pareto used with square root and half.

So there is overlap between them. The 80% of work under Pareto 80/20 is included in the 50% of work under Price. The Price contributors are a subset of the Pareto contributors.

I got a bit confused thinking about it. I was thinking about cutoff points and in terms of small worker sets. At small sizes it can seem a bit confusing and even contradictory between Pareto and Price.

They overlap twice. Let X be the number of workers. solving for  

sqrt(X) = X/5

0 = X² - 25X

with solutions of X = 0 and X = 25

so X = 0 makes sense. no workers do no work. It was X = 25 that I got stuck on a bit.

At X = 25 under Price, the top 5 people do 50% of the work. However under Pareto 80/20 the top 5 do 80% of the work. So it seems at low sample numbers Price and Pareto don't work quite so well.

The trick is to use larger samples. At larger sizes X/5 dominates sqrt(X), the numbers stabilize and it becomes clearer.

Let's say that the number of units of work done is equal to the number of people. This would match up to say an outsourcing contract where each worker generates 8 billable hours each day. So to the client with 100 contractors from the outsourcing firm they purchase total 100 units of work a day. Now who within the 100 people gets what actual useful work completed is somewhat opaque to the client.

So if X = 100; then under Pareto the top 20 people do 80 units of work, and under Price the top 10 people do 50 units of work. So the 10 Price people average 5 units of work each, and the (20 - 10 = 10) Pareto people do (80 - 50 = 30) units of work, or 3 units of work each. The remaining 80 people do 20 units of work or 0.25 units each.

Now if X=10,000; then under Pareto the top 2,000 people do 8,000 units of work, and under Price the top 100 people do 5,000 units of work. At this 10,000 number (about the size of a company on the NYSE, a phone company, or power utility) the ratio of Pareto to Price people is 1,900:100, or 19:1. The Price people now accomplish 50 units of work each and the Pareto people are still solid contributors at about 1.6 units of work each. The remaining 8,000 are alas invariant at 0.25 units of work each.

So as the population scales, if we accept Price as invariant that sqrt(X) will do 50%, the exceptionals really soar. Still the "solid contributors", the non-Price Pareto people will do 30% of the work, while being essentially 20% of the population, as X/5 - sqrt(X) approaches to X/5 for larger X. So the non-Price Pareto people will approach 1.5 units of work each.

Which is pretty good for the solid contributors. The "remaining 80%", the fungibles will get laid off first. At a ratio of 1.5 units per person to 0.25, the average solid contributor gets 6 times as much done than the average remaining 80 person. So you want to know who the solids are and keep them around.