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 Author Topic: ❢❢❢ How many coins is a lot? (Finally, answer inside!)  (Read 4082 times)
Joe200
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 February 05, 2015, 09:05:58 PM

How many coins do you have to have to have "a lot" of coins in the long term? This question is asked periodically. Here is the calculation.

tl;dr. Depending on assumptions about the long term future, anywhere from eight(!) coins in a somewhat egalitarian world that sees massive bitcoin adoption and you are well off but not rich to 8,000 coins in a very ruthless world that sees a massive shrinkage of the bitcoin economy and you are the top dog.

If the bitcoin economy shrinks (fewer people using coins), you will actually need more coins to be rich within that economy -- to have what is considered "a lot" of coins. I am not talking about the fiat price of coins. That's a separate issue. (If the fiat price of coins drops, then those riches won't be worth much in the fiat world.)

Under the most conservatively realistic scenario in my view, to be the top dog, you need 400 800 coins. What do you think about that? Comment below.

Major caveat #1. All this is figuring out how much is a lot of bitcoins relative to the bitcoin economy. Having a lot of coins  absolutely does not mean that you will be rich in fiat terms. If the fiat price of bitcoin goes to shit, then you will own a large chunk of shit. That's all.

Major caveat #2. I assume a particular distribution of wealth. This distribution is thought to fairly accurately describe reality. To the extent the theoretical distribution deviates from reality, the calculations will be wrong. This is especially an issue at the tails of the distribution, since that's where the biggest deviation from reality is likely to be. Well, in this post, that is precisely what I am doing -- looking at the tails. I am even looking at the 100th percentile for crying out loud!

What this means is that the calculated 95% level ("well off but not rich") is probably within the ballpark of reality. However, the 100% level ("top dog") is highly "theoretical", shall we say.

The Gini coefficient (https://en.wikipedia.org/wiki/Gini_coefficient) is a single number that measures the inequality of income or wealth. Gini = 0 represents "perfect equality", where everyone has the same amount. Gini = 1 means that a single person has everything and the rest have nothing.

Let x be a proportion of people, ordered by the wealth that they have, from lowest to highest. Let F(x) be the proportion of wealth that x owns. Then, let's say that F(x) is

Code:
F(x) = x^{2g/(1-g)}

Then, for this F(x), the Gini coefficient is g.

Values of wealth Gini. Do not confuse Gini coefficients for income and wealth. We are interested in the wealth Gini. (https://en.wikipedia.org/wiki/List_of_countries_by_distribution_of_wealth) is a table of wealth Gini by country.

Wealth Gini varies from the more "egalitarian" 0.55 for Japan and China, with Spain and South Korea close by, to the very unequal 0.85 for Namibia and Zimbabwe, with Denmark, Switzerland, and United States close by, all at 0.80+. For the world as a whole, wealth Gini is 0.80.

One argument is that, long term, the wealth Gini for coins will resemble the wealth Gini for the world, since people from all over the world will just transfer some portion of their wealth into coins.

Another argument is that the world of coins is more "laissez faire" than the fiat world, with less wealth transfer -- this would lead to a higher wealth Gini.

Both of these arguments point to high long term values of wealth Gini for coins. With wealth Gini = 0.8, we have the top 20% of coin holders holding 83% of the coins. This is the famous 80/20 rule.

A single person. F(x) tells us the proportion of coins the bottom X% or the top 1 - X% have. It doesn't tell us the proportion of coins for a single individual. Here is the proportion of coins owned by a person who is the richest among x. This is the person on the threshold between the x poorest and 1-x richest.

Code:
p = dF(x)/dx * dx
p = {2g/(1-g)} x^{2g/(1-g) - 1} dx
p = {2g/(1-g)} x^{2g/(1-g) - 1} / N

dx is a small change in the proportion of individuals, such that the change in individuals = 1. N is the total number of individuals within the bitcoin economy / who hold bitcoins. N * dx = 1, so dx = 1/N.

How many people? How many people will be in the bitcoin economy? This is a very uncertain number that has a huge influence on p. If there are few people in the bitcoin economy, everyone has a lot of coins, so you have to have really a lot to be "rich".

Assuming that the bitcoin economy will become as large as a country, we can go back to (https://en.wikipedia.org/wiki/List_of_countries_by_distribution_of_wealth) and look at the number of adults in each country. This ranges from 24k in Saint Kitts and Nevis to 842M in China, all the way to 3.7B in the world as a whole.

Assuming lower numbers is more "conservative". First, though we all love bitcoin, as far as the world is concerned, coins have still to prove themselves. Second, a lower number of people in the bitcoin economy means that you have to have more coins to reach the same percentile / level or richness. It sets your sights higher.

rpietila (https://bitcointalk.org/index.php?topic=316297.0) guesstimates that there are currently 1.0 million holders. I have no idea what his methodology is. The country with a million adults is Estonia. Seems like a fairly small country, so why not. Chile has 10 million adults. Belize has 120k.

Here is an interesting multiplier effect. If the bitcoin economy grows, then presumably the price of each coin will grow too, and also the number of people in the economy will grow. This means that you will have more wealth in fiat terms even for a fixed number of coins, and, separately from that, you will need fewer coins to be bitcoin rich.

If the bitcoin economy goes to shit, both the price and the number of people in the economy plummet. You will have less wealth in fiat terms for a fixed number of coins, and you will need more and more coins just to stay bitcoin rich in the shitty bitcoin economy. Hope that makes sense.

The results. Let's try wealth Gini = 0.95 (very ruthless), 0.9 (ruthless), 0.8 (US, the world), and 0.7 (Russia, Syria). , and 0.65 (Netherlands, New Zealand).

Long term number of people in the economy = 0.1M (decline relative to where rpietila thinks we are right now, Belize), 1M (where rpietila thinks we are right now, Estonia), 2M (growth, Lebanon), 10M (major growth, Chile).

What percentile is rich? Richest among 95% (well off but not rich) and 100% (top dog).

Assuming we are in the "long term", so the number of coins is 21M.

If you'd like me to try different parameter values, please comment below.

Code:
gini          N rich           p   coins
1  0.95    100,000 0.95    0.000057   1,200
2   0.9    100,000 0.95   0.0000753   1,580
3   0.8    100,000 0.95   0.0000559   1,170
4   0.7    100,000 0.95   0.0000387     812
5  0.95  1,000,000 0.95   0.0000057     120
6   0.9  1,000,000 0.95  0.00000753     158
7   0.8  1,000,000 0.95  0.00000559     117
8   0.7  1,000,000 0.95  0.00000387    81.2
9  0.95  2,000,000 0.95  0.00000285    59.8
10  0.9  2,000,000 0.95  0.00000376      79
11  0.8  2,000,000 0.95  0.00000279    58.7
12  0.7  2,000,000 0.95  0.00000193    40.6
13 0.95 10,000,000 0.95  0.00000057      12
14  0.9 10,000,000 0.95 0.000000753    15.8
15  0.8 10,000,000 0.95 0.000000559    11.7
16  0.7 10,000,000 0.95 0.000000387    8.12
17 0.95    100,000    1     0.00038   7,980
18  0.9    100,000    1     0.00018   3,780
19  0.8    100,000    1     0.00008   1,680
20  0.7    100,000    1   0.0000467     980
21 0.95  1,000,000    1    0.000038     798
22  0.9  1,000,000    1    0.000018     378
23  0.8  1,000,000    1    0.000008     168
24  0.7  1,000,000    1  0.00000467      98
25 0.95  2,000,000    1    0.000019     399
26  0.9  2,000,000    1    0.000009     189
27  0.8  2,000,000    1    0.000004      84
28  0.7  2,000,000    1  0.00000233      49
29 0.95 10,000,000    1   0.0000038    79.8
30  0.9 10,000,000    1   0.0000018    37.8
31  0.8 10,000,000    1   0.0000008    16.8
32  0.7 10,000,000    1 0.000000467     9.8

If we will see a very ruthless distribution of coins (gini = 0.95) and if the number of people using coins decreases by a factor of 10 to 100k, then you need 7,980 to be the top dog.

At today's prices, you need about \$2M to buy that. There is a problem here. Why would the number of people using coins decrease? The only scenario I see is if the price goes to shit. Which means you actually won't want those 8k coins anyway. Or, if you do, you can wait until the collapse to buy them for a lot less than \$2M.

Say the distribution of coins is somewhat "egalitarian" (gini = 0.7). This is like Russia or Syria, not exactly paradises of equality. But, interestingly enough, this is much more equal than the US. I don't think this will ever happen. Maybe with some altcoin that prioritizes "fairness", but not with bitcoins. Say also that the number of coin holders explodes 10x to 10M people. Compared to the world population, that's still a tiny number. But it's 10 times more than what we've got. And say you just want to be well off, not rich (x = 0.95). In that case, all you need is 8.12 coins.

The most conservatively realistic scenario in my view is that gini = 0.95 and the number of users stays at 1M. If you want to be the top dog here, you need 798 coins. If you are realistically optimistic and think that N will go to 2M, then to be the top dog you need 399 coins.
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unamis76
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 February 05, 2015, 09:15:28 PM

Well... Although I'm still fan of the theory that "says" that you never have a lot, independently of the coins you have, this was still a nice read. There were some concepts I didn't realize that they even existed, such as Gini.

I think you do have a valid point, or at least something going on here...

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jbrnt
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 February 05, 2015, 09:31:40 PM

OP has a lot of maths there. I think OP's approach is owning a "percentage of all coins". It doesn't take into account of the demand for bitcoin and price of bitcoin. If bitcoin is worth \$300, 400btc is a lot, whereas bitcoin is \$50, 400btc is not.
HarmonLi
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 February 05, 2015, 09:36:31 PM

Very interesting approach! I always keep thinking that you actually only need 1 BTC, "just in case". I also think that a lot of Bitcoin's classic adversaries have that one coin, "just in case it does come through". If we achieve a global adoption and really represent, say, 1% of one of the big Ms, 1 BTC will be all you need.

kwukduck
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 February 05, 2015, 09:50:41 PM

Appreciate the approach here

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DieJohnny
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 February 05, 2015, 10:30:51 PM

awesome post that gets me thinking.....

gotta get me some more coins

Those who hold and those who are without property have ever formed distinct interests in society
fr4nkthetank
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cryptominingtalk.com

 February 05, 2015, 10:35:04 PM

too bad bitcoin is worse than the worst country on earth for wealth distribution in my opinion....

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redsn0w
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# Free market

 February 05, 2015, 10:39:15 PM

Thanks for this analysis/theory, I think also .10 btc is a lot or also 10 euros ( you can be rich also with 20 dollars).
Hyena
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 February 05, 2015, 10:47:10 PM

That makes me hold even harder

Ibian
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 February 05, 2015, 11:25:12 PM

too bad bitcoin is worse than the worst country on earth for wealth distribution in my opinion....
So you are a communist or you just don't understand how the fiat economy works?

There is no bubble.
Dafar
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dafar consulting

 February 06, 2015, 03:00:46 AM

What if bitcoin reaches a semi-decent adoption (used by 200M or so people), will someone be well-off with 40btc?

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Wexlike
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 February 06, 2015, 10:32:12 AM

What if bitcoin reaches a semi-decent adoption (used by 200M or so people), will someone be well-off with 40btc?

Just do the math! In your scenario an average user has 0,105 bitcoins.
zimmah
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 February 06, 2015, 11:33:45 AM

What if bitcoin reaches a semi-decent adoption (used by 200M or so people), will someone be well-off with 40btc?

Oh yes, certainly.

21 million coins, divided by 200m means everyone will have just about 0.1 coins in average.

40 bitcoins would be quite a large sum then.
Joe200
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 February 06, 2015, 02:46:24 PM

too bad bitcoin is worse than the worst country on earth for wealth distribution in my opinion....

I agree. I think Gini = 0.9 might be too low. I'll adjust my calculations.
Joe200
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 February 06, 2015, 03:34:02 PM

Thank you all who agreed with the calculations for mathematical reasons in the OP.

Posted an update. Bumped Gini to the very ruthless level of 0.95. Added another caveat. Enjoy!

Ibian
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 February 06, 2015, 04:35:09 PM

Those who liked? Not those who agreed with the calculations for mathematical reasons?

There is no bubble.
Joe200
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 February 06, 2015, 04:53:42 PM

Corrected. It's the middle of the night here. I'm exhausted.
sinner
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 February 06, 2015, 07:16:00 PM

https://bitcointalk.org/index.php?topic=98239.0
rpietila
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 February 11, 2015, 09:14:34 AM

Following. Nice methodology. Sometimes when you cannot observe the stuff directly, it is indeed valuable to construct the distribution solely based on theory, maths and bounds. Otherwise you have nothing to play with, and obviously something is better than nothing, if you know the limits of the something.

My current understanding of the number of bitcoin owners is in the ballpark of 2-5 million (many services report that many wallets), but when we put this into the Gini perspective, the results show that only about 100k own a meaningful amount.

In Satoshi Roundtable, my closing comment was that Bitcoin early adopter period (usually defined as the first 0.1%-2.5% of the eventual adoption of the technology) is hopefully about to begin in 2015 (innovator stage of the 0-0.1% being soon fulfilled )
Q7
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 February 11, 2015, 11:39:13 AM

Wow...I couldn't grasp all the calculation despite having read through until the end but hey, that's a nice figure. For me, even if I have 1000, I will still yearn for more. Nothing is a lot of me.

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