If we are where I think we are, price should be $50k per coin. The fact that it's not means that at least one of the assumptions is wrong. But which one?
Intro. In a previous post (https://bitcointalksearch.org/topic/how-many-coins-is-a-lot-finally-answer-inside-946938), I discussed the distribution of wealth, the Gini coefficient, and the number of people who have bitcoins and used those quantities to calculate the number of coins that you need in order to have "a lot". That result is independent of the price of coins. It just tells you how many coins is "a lot" relative to other people who have coins.
Here, I use the same principles to calculate the price of coins.
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.
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.
In my previous post, I also tried values of Gini = 0.90 and even 0.95. The reasoning is that the distribution of coins might be even more ruthless than the distribution of fiat wealth.
How many people? rpietila (https://bitcointalksearch.org/topic/distribution-of-bitcoin-wealth-by-owner-316297) still guesstimates that there are currently 1.0 million holders. I have no idea what his methodology is. Based on this, we can see what happens when the number of holders is 1 million, above, and below.
Calculation.
Code:
x = 0.5
p = {2g/(1-g)} x^{2g/(1-g) - 1} / N
x = 0.5 means that we are looking at the median holder. g is the wealth Gini. N is the number of coin holders. p is the proportion of all coins held by our median holder. Right now, there are 14.38M coins. In the "long term", the total number of coins will be 21M. This gives us the number of coins held by the median coin holder.
How much wealth does a median coin holder have? The median coin holder is probably American / Westerner. According to (https://en.wikipedia.org/wiki/Wealth_in_the_United_States), a median family's wealth is $80k. However, bitcoins are still a long way from being mainstream, so the median coin holder and the median person is not the same thing. Maybe once coins are mainstream, we could say that the median coin holder has $80k in wealth, but not now.
Most bitcoiners are probably young. For those under 35, the median wealth is $10k. What percentage of one's wealth does the median holder hold in coins? No idea. Let's say 1%.
To simplify the table, I assume that the dollar value of wealth held in coins is $100 in call cases, which is also $10k * 1%. If you think this values might be different / bigger, multiply the prices accordingly.
The dollar value of wealth held in coins divided by the number of coins is the price per coin. Let's see what we get.
Code:
gini N p coins.now coins.max usd.btc price.coins.now price.coins.max
1 0.95 100,000 0.00000000000000276 0.0000000398 0.0000000581 100 2,520,000,000 1,720,000,000
2 0.9 100,000 0.00000000137 0.0197 0.0288 100 5,060 3,470
3 0.8 100,000 0.000000625 8.99 13.1 100 11.1 7.62
4 0.7 100,000 0.00000367 52.8 77.2 100 1.89 1.3
5 0.95 1,000,000 0.000000000000000276 0.00000000398 0.00000000581 100 25,200,000,000 17,200,000,000
6 0.9 1,000,000 0.000000000137 0.00197 0.00288 100 50,600 34,700
7 0.8 1,000,000 0.0000000625 0.899 1.31 100 111 76.2
8 0.7 1,000,000 0.000000367 5.28 7.72 100 18.9 13
9 0.95 2,000,000 0.000000000000000138 0.00000000199 0.0000000029 100 50,300,000,000 34,400,000,000
10 0.9 2,000,000 0.0000000000687 0.000987 0.00144 100 101,000 69,400
11 0.8 2,000,000 0.0000000312 0.449 0.656 100 223 152
12 0.7 2,000,000 0.000000184 2.64 3.86 100 37.8 25.9
13 0.95 10,000,000 0.0000000000000000276 0.000000000398 0.000000000581 100 252,000,000,000 172,000,000,000
14 0.9 10,000,000 0.0000000000137 0.000197 0.000288 100 506,000 347,000
15 0.8 10,000,000 0.00000000625 0.0899 0.131 100 1,110 762
16 0.7 10,000,000 0.0000000367 0.528 0.772 100 189 130
Here is what the table says. Let's say Gini = 0.9, which is what I would consider to be a ruthless coin distribution. The number of coin holders stays the same at N = 1M. This means that the median holder has this proportion of coins: p = 0.000000000137. This means that he has 0.00197 of a coin. (Tiny!) If he is storing $100 of his wealth in bitcoins, that gives us a price of $50k per coin.
The more unequal the distribution of coins (higher Gini), the fewer coins the median holder has, the higher the price of each coin. Yey for inequality! Put another way, when a few big holders hold most of the coins, there are fewer coins in circulation for the average holders to buy.
The higher the number of coin holders, again, the fewer coins in the hands of a single holder, the higher the price. However, even if the number of holders collapses by a factor of 10 from rpietila's guesstimate, price can still easily go to $5k.
Holding $100 of wealth in coins seems pretty small, but remember, we are talking about the median holder, so maybe it's right. I don't know. As bitcoin goes more mainstream, both the number of holders (N) and the amount of wealth held in coins will increase -- both of these have a positive effect on price.
The huge problem with the table is that it gives an insane range of prices, depending on the scenario. According to the table, price per coin can range all the way from the seemingly impossible $1.89 to the completely laughable $252 BILLION per coin!
If we live in a somewhat "egalitarian" world (gini = 0.7), which I don't think we do, and if the number of holders collapses by a factor of 10, then price could go below $2.
If we live in a very ruthless world (gini = 0.95), and the number of holders goes up 10x, price could really be -- I won't even write it again it's so ridiculous.
If we are at Gini = 0.9, N = 1M, which is where I think we are now, then price could easily be $50k today! The fact that it's not means that one or more of the assumptions is seriously off. Is the distribution of coins a lot less ruthless than I think? Is the number of holders a lot less than rpietila thinks? Does the median holder hold a lot less than $100 worth of coins? Where are the assumptions wrong?