10 December 2017

The Correct Price of Bitcoin

I noticed the crazy Bitcoin bubble, and tried to think about what its price should be. I think I've come up with a model for it, based on the optimal inventory of Bitcoin for a Bitcoin user. This idea follows from the optimal inventory of cash.

The basis for this model is the idea is that we need a supply of cash for purchases over the year, perhaps our demand is d = $500/year. To take cash out of the bank, we lose the interest i, perhaps 1%,  on that money. So we are incentivized to take out only small amounts. But taking cash out of the bank has a fixed cost f, including any withdrawal fee (as from a banking machine) and the hassle of getting to the bank, perhaps $2/transaction. The fixed cost of withdrawal incentivizes us to take out large amounts.

So the two competing costs have an optimum, which we can calculate using the classic Wilson economic order quantity formula. For more background on the optimal inventory of cash, see this, which references Baumol. The economic order quantity formula is Q = (2fd/i)^0.5, where Q is the optimal withdrawal amount.  Assuming we use the cash evenly over time, we will have on average $Q/2 dollars in our pocket.

We can apply this model to Bitcoin to estimate its correct price. Assume:

  • a 3% interest rate, and 
  • a $2/transaction withdrawal fee to get Bitcoin (e.g., to convert it from $US),
  • users want to spend about $500 worth of Bitcoin per year.
Using the formula above, we find that each user would hold on average about $129 worth of Bitcoin.

Assume 20,000,000 Bitcoin users. Together, they need $129*20,000,000 = $2.59 billion Bitcoin.

With 21,000,000 Bitcoin in existence, the Bitcoin should cost $2.59 billion/21,000,000 = $123/Bitcoin.

I tried this on the amount of U.S. currency, and was off by a factor of approximately 2, which could be due to people hoarding cash. So let's multiply by 2 for hoarding. We get a price of Bitcoin of only $246.

Below is some sensitivity analysis. I think $15,000 is really high!

Low Geometric mean High
Max number bitcoin 21,000,000 15,874,508 12,000,000
Number of users 20,000,000 44,721,360 100,000,000
Interest rate/year 3% 1.7% 1%
Transaction cost $2 $2.83 $4
Demand/year/user $500 $1,414 $4,000
Optimal holding/user $129 $340 $894
Total cash needed $2,581,988,897 $15,196,713,713 $89,442,719,100
$/bitcoin $123 $957 $7,454
With hoarding, x2 $246 $1,915 $14,907

I put the calculated fields in bold.

The world has few substitutes for US dollars, because many things can be purchased only in US dollars. The world has a lot of substitutes for Bitcoin - the only people who accept only Bitcoin are thieves. And Ethereum looks like a much better cryptocurrency, with its smart contracts and higher trading rate.

And what do you get when you buy Bitcoin? You get a secret number. That's pretty much it.

This Bitcoin bubble is a zero-sum game. When you take money home, someone else is likely to lose it. A true pyramid scheme, with the Winkelvoss twins at the top. My advice? Get out while you can.

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