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It seems to me that the Pareto principle says that for any $n$-many people that produced $m$-many goods, $\sqrt[2]{n}$-many people would produce $\frac{m}{2}$ many goods out of the total $m$ many goods (until all the goods are produced, then the left out people would be useless).

Question 1: Is my understanding above, correct?

Now, I also read that the principle is also named the $80/20$ rule, which I think means that $80\%$ of the $m$-many goods are produced by $20\%$ of the $n$-many people. But I cannot see how the Pareto principle leads to the numbers $80$ and $20$.

As Wiki says:

It is an axiom of business management that "80% of sales come from 20% of clients".[4]

Since Wiki didn't mention anything about the total sales volume, or the total number of clients, I assume their total/absolute values are not relevant.

Question 2: But, then, how does the Pareto principle eventually give the numbers $80$ and $20$? I tried to get this, but I failed.

My attempt at Question 2:

So I simulated this in Python:

import math
POPULATION = 100
PRODUCTION = 10000000

people_left = POPULATION
resource_left = PRODUCTION

total_successful_people = 0
total_resource_harvested = 0

while True:
    total_successful_people += math.sqrt(people_left)
    people_left             -= math.sqrt(people_left)

    total_resource_harvested += resource_left / 2
    resource_left            -= resource_left / 2

    rat_successful_people = total_successful_people/POPULATION*100
    rat_resource_harvested = total_resource_harvested/PRODUCTION*100

    print('{}% of people produced {}% of goods.'.format(
        rat_successful_people,
        rat_resource_harvested,
    ))

    if rat_resource_harvested > 80:
        break

It seems that the PRODUCTION parameter is asymptotically stable, so doesn't hurt setting it to a very big number.

But I only seem to get something close to the $80/20$ numbers if I set POPULATION to $100$ (as in code above) as follows:

10.0% of people produced 50.0% of goods.
19.486832980505138% of people produced 75.0% of goods.
28.45974594227485% of people produced 87.5% of goods.

But if I enlarge POPULATION into something like, say, $1000$, then I get:

3.162277660168379% of people produced 50.0% of goods.
6.274153659301887% of people produced 75.0% of goods.
9.335621384767862% of people produced 87.5% of goods.

So what's going on here? Is the Wiki quote wrong? Or what exactly?

caveman
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    The simple answer is that it is a very rough approximation. – Tim Apr 21 '19 at 13:42
  • So rough that it's totally wrong when number of resource gatherers is not 100? – caveman Apr 21 '19 at 15:28
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    People use this name to denote any event where minority of some population is responsible by majority of some events no matter of the actual numbers. If you look at the observed proportions in many of the examples quoted in Wikipedia, you'll see that the actual numbers are not exactly 80/20. – Tim Apr 21 '19 at 15:54
  • Yeah, but look at GDP distribution table in Wiki (for example). It shows that the top 20% of the population have about 80-ish percentage of the income. Yes, not exactly 80%, but rather 82-something%, but at least it's an 80-ish number. However, if you look at my code, and my simulation outcome, I get WAY OFF and instead of getting the 80-ish number with 20%, I get it with 6-9-ish!!! I AM WAY OFF. – caveman Apr 21 '19 at 18:48
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    Results of any simulation depend on the assumptions about the stochastic process that you made. I'm sure that you'd agree that your code does not reflect how does economy works in real world. Also, this is a "principle" not a theorem. It does not state any assumptions and obviously it is not the case that no matter what and how you simulate, you'd get the 80/20 split in your results. It is a set of anecdotes with an extra layer of post factum interpretations. – Tim Apr 21 '19 at 19:34
  • I think my simulation is based on $\sqrt[2]{n}$ many people will generate $m/2$ of the total $m$ revenue. I don't see how this is approximately related to $20%$ of people make the $80%$ revenue. My simulation shows that it depends on the population (only when 100 people are around I get 20-80-ish numbers). Is my simulation wrong? – caveman Apr 21 '19 at 20:48
  • I suspect that my simulation is wrong. I'm basically applying $\sqrt[2]{n}$ and $m/2$ recursively on the remaining population and production. Not sure if this is what Pareto said. – caveman Apr 21 '19 at 20:51

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