
June 19, 2011, 08:37:23 PM 

Like many here, I'm trying to get a handle around the bitcoin economy, and where it is going. One of the important factors driving price is the difficulty level. This is an indication of how long it will take a computer or pool to find a block of bitcoins. The difficulty level is set every 2016 blocks to set the rate of finding this amount to be 2 weeks. Thus if there is more computing power coming online, the difficulty rate will increase.
This gives another comparison point: the cost of entry. Many people are seeing that bitcoin mining is profitable and are purchasing their own rigs. In fact, with a modest investment, someone can suddenly get a passive income of a few hundred dollars/month. However, it's actually not the case that you get dollars/month and instead you get btc/month.
Add into this that you can, through an exchange like Mt. Gox, instead purchase btc, there is an interesting equilibrium that can be found, showing the relationship between bitcoin trade price, MH/s price, and difficulty level.
I'm going to pick specific values, because it illustrates this much better. Let's assume that Cost of a bitcoin = 17 USD Cost of 1000 MH/s = 900 USD
Therefore 1000 MH/s is about 53 btc
At the current difficulty, 1000 MH/s gives about 1.3759203893 btc. So now what we try to figure out is at what difficulty change rate will this pay back 53btc? This isn't easy (I don't know the closed form for it), but it can be found by finding how much one would earn in this difficulty period, and then the next period, and so on and so forth. One key aspect is that you have to adjust the size of the period to the assumed difficulty change. Thus if you assume the difficulty change is 25%, then you have to adjust the period to be 11.2 days. When putting this into a spreadsheet and extending out several periods, it is easy to find the bounds of this.
In this case, we end up with an average change of around 36.2%. At that rate of change, purchasing the 1000 MH/s system will yield about 53 bitcoins over its lifetime.
Now this does simplify a lot out. For example, it doesn't take into consideration the cost of electricity, space, infrastructure, etc. For another, it assumes a perfectly barrierless entry into bitmining which is close to true, but there are still capital requirements and risk management that can act as barriers.
However, 36.2% represents almost an upper bound when assessing the difficulty change (given the exchange rate stays the same).
So what can we draw from this?
If the rate of difficulty change goes above this amount and all other things stay equal, then it will stop making sense to purchase systems and instead it makes more sense to buy bitcoins. This will drive up the price of bitcoins, lowering the btc per MH/s cost for a mining rig. It is possible the difficulty could fall if people come offline, but this is doubtful as it will still make sense to run an existing system.
If the rate of difficulty change goes below this point and all other things stay equal, then it will stop making sense to buy bitcoins and instead it makes more sense to buy a system. With more people coming online, this will push up the difficulty level.
If the exchange rate of usd to btc goes up and all other things stay equal, then the btc cost of MH/s goes down, and it makes more sense to mine than buy. With more people coming online, this will push up the difficulty level.
If the exchange rate of usd to btc goes down and all other things stay equal, then the btc cost of MH/s goes down, and it makes more sense to buy than mine. This will slow the difficulty rate.
If the dollar cost of a mining rig goes down, it makes more sense to mine than buy. This could have the effect of lowering the btc exchange rate or increasing difficulty or both.
This is just the basic analysis for now. There are some more interesting things that come out of this equilibrium relationship that I'll post on if there's any interest.
