Basic Economics of Bitcoin Mining

Arvind wrote yesterday about the availability of chips that do super-fast Bitcoin mining. I want to follow up by unpacking the economics of Bitcoin mining, to see what the effect of the new chips will be, and more generally what the future of Bitcoin mining looks like.

For those unfamiliar with Bitcoin, here is a one-paragraph summary: Bitcoin is a digital currency that operates without any centralized issuer. It’s a “real” currency, in the sense that it can be converted readily into traditional currencies such as dollars. Logically, a Bitcoin has a serial number. You can give a coin to someone by adding an entry to the global Bitcoin log, saying that you gave them that coin. The clever part of Bitcoin’s design is how the global log works. The log consists of a chain of “blocks,” each memorializing a bunch of transactions. To create a new block, you have to solve a difficult cryptographic puzzle which can only be done by randomly trying guesses until you find one that works. If you are the first person to solve a puzzle, you can create a new block in the log–and you are allowed to add to the block a special event that gives you a newly minted coin. Once somebody solves a puzzle and creates a block, a new puzzle-solving race begins. The activity of trying to solve puzzles, in order to create new blocks, in order to get the new coins that go to block creators, is called “mining”. Mining is optional: you can receive and spend Bitcoins without doing any mining. Increasingly, mining is a specialized activity.

The economics of mining are interesting. Suppose you are thinking about devoting resources to mining. Let’s assume it will cost you C dollars per second to mine (for equipment and electricity), and you will be able to try P puzzle guesses per second. Every time you solve a puzzle you will get a Bitcoin whose value in dollars is V. Is mining profitable?

The answer depends on how many guesses it takes to solve a puzzle. If it takes G guesses to solve a puzzle on average, then you will solve P/G puzzles per second, earning PV/G dollars per second. If this is greater than your cost C, then you will mine. Solving for G, we see that mining is profitable if G < PV/C.

But what is G in real life? How hard are the puzzles? It turns out that Bitcoin automatically adjusts the difficulty of the puzzles, so that the total rate at which puzzles are solved, by all of the miners in the world put together, is some constant value R. So if there are N active miners, and each is solving P/G puzzles per second like our would-be miner, then the puzzle difficulty will be adjusted so that NP/G = R, which means that the puzzle difficulty G will be G = NP/R.

What does this mean for our would-be miner? Plugging our calculated value for G back in to the profitability calculation, we get that mining is profitable if NP/R < PV/C, or equivalently if NC < RV. In other words, mining is profitable if the worldwide total being spent on mining (NC) is less than the worldwide total revenue being captured by miners (RV). As long as this is true, new miners will join, thereby increasing the left side of the inequality. The influx of miners will stop when the two sides are equal, NC=RV, so that in equilibrium the value of the resources spent on mining is exactly equal to the value of the Bitcoins it generates.

What's interesting about this equilibrium is that it doesn't depend on how efficiently puzzles can be solved. So the arrival of new super-fast, super-efficient Bitcoin mining chips won't fundamentally change the economics of Bitcoin mining. Instead of devoting RV worth of resources to mining with one type of equipment, miners will switch to devoting RV worth of resources to mining with a different type of equipment. The switch might be a shock to some miners–especially those who have sunk-cost investments in previous generations of mining technology, but the dynamics of Bitcoin mining, and Bitcoins generally, won't change much.

Or that's what a simple model tells us. If we make our model a bit more complicated–and arguably a bit more realistic–some interesting things start to happen. But that's a topic for a future post.