The mathematical relationship between Bitcoin's usage and price

Why does this matter?
Until the block reward runs out, the exchange rate is a critical component of Bitcoin's security. It's important to understand exactly how usage effects security.

The equation of exchange
We can use the equation of exchange to see the precise relationship between Bitcoin's usage and price.

The equation of exchange states that M * V = P * T, where:

M = the money supply V = the velocity of money P = the average price per transaction T = the number of transactions

So,

(number of bitcoins) * (velocity of bitcoins) = (average amount of bitcoins per transaction) * (number of transactions)

Or in other words:

(number of bitcoins) * (velocity of bitcoins) = (total value of Bitcoin spending denominated in Bitcoin)

Multiply the right side by 1 = (Bitcoin price)/(Bitcoin price):

... = (total value of Bitcoin spending denominated in Bitcoin)(Bitcoin price in USD)/(Bitcoin price in USD)

Simplifying the right side:

... = (total value of Bitcoin spending denominated in USD)/(Bitcoin price in USD)

Solve for the Bitcoin price in USD:

(Bitcoin price in USD) = (total value of Bitcoin spending denominated in USD)/((number of bitcoins) * (velocity of bitcoins))

Suppose there are 15 million bitcoins in existence at some point in time, the Bitcoin economy contains 60 billion USD per year of economic activity, and the velocity of bitcoins is 5 (about 3.3 times as high as current M2 velocity -- see here). Then the value of a bitcoin would be:

(60 billion)/((15 million) * 5)) = $800

The effect of savings and speculation
The above analysis doesn't explicitly address demand for Bitcoin stemming from a desire to save or speculate on the future price. When someone obtains bitcoins and doesn't use them in the economy, you can account for them in the above math in one of two ways:


 * Reduce the bitcoin supply to only include bitcoins that are being used for spending, and also define the velocity only in terms of these bitcoins.
 * Keep bitcoins used for savings/speculation in the definition of bitcoin supply, and compute velocity over all bitcoins.

As long as you compute velocity over whatever you've defined supply as, the math will work out.

Putting a lower bound on Bitcoin's price
You may think from the above "great, but we don't know what percentage of bitcoins will be used for speculation or savings, so how useful is this?"

We can just assume that all bitcoins are used in commerce to get a lower bound on the future price given a certain level of bitcoin activity and velocity. This is essentially what we did above when we calculated a hypothetical price of $800. Our big assumption was that velocity among coins used in commerce was 5.

How can we know what velocity will be?
Some argue that velocity is just a free variable in the equation of exchange, so the equation is meaningless. However it has a real interpretation: velocity is the amount of times per year each bitcoin is spent. It can't have an arbitrary value -- it needs to correspond to actual economic activity. Velocity of US dollars has stayed between 1.5 and 2.3 since at least 1960. Because Bitcoin is a digital currency without a lot of the friction involved in the traditional banking system (no waiting 3 days for a bank deposit), Bitcoin velocity may be higher than USD velocity. Still, because we have a lot of historical data about the velocity of different forms of money in the past, we can make an educated guess.