Short Version : The electrical power consumed by the Bitcoin network - in megawatts - is equal to the price of one bitcoin - in dollars.
(“Law” valid up to the next reward halving under the hypothesis described below)

Long Version :

I wanted to know how much electricity is consumed by the Bitcoin network. Unfortunately most of the available resources make assumptions on the efficiency of the miners and the mining difficulty.

I state that the difficulty does not matter, nor the efficiency of the miners, and that the only factors that have an influence on the electrical power of the network are :
- the price of electricity paid by miners
- the content of the block reward

The fun conclusion is that with the current block reward of 25 BTC per block and a mean price of electricity of $0.15 per kWh, the electric consumption of the network in MW (megawatts) is exactly the price of one bitcoin in dollars.

Hypothesis :
1. Miners are rational actors. Therefore once they have bought a mining rig, they will not stop it unless the cost of running it is higher than the price of the mined bitcoins. However if the price drops or if the difficulty grows too high they should stop mining.
2. Miners pay their electricity bill. There is no such thing as free electricity. Some miners may not pay the real price and I will discuss the effect of that later. I will take a price of $0.15 per kWh.
3. The network is in a stable state : 25 BTC per block = 3600 BTC per day (I know this is not the case nowadays). The fees are negligible.

So a rational miner :
- has already a mining rig, which consumes electric power if running.
- Every day he has to make a simple choice : either he runs its rig or not.
- The price at which he bought the rig is irrelevant to this decision. It’s a “sunk cost”.

If the marginal cost of running the rig can turn the electricity into a profit (price of electricity >= price of the bitcoins mined) then the rational miner should run it. If not he should turn if off.

Now let’s suppose that the bitcoin network in only made of rational miners : the equilibrium is reached when the price of all newly mined bitcoin is strictly equal to the price paid as electricity by the miners.

A few calculations :

Miners mine 3600 bitcoins a day.
Miners collectively spend the same amount of dollars in electricity (equilibrium point of the equation that follow rational miners).

3600 BTC = amount spent daily in electricity

P = power of the network in MW (megawatts)
X = price of one bitcoin
Price of electricity = 0.15 USD/kWh = 150 USD/MWh

So we have 3600*X = P*24*150 or P=X.

Therefore : given the current reward, in a stable bitcoin network run by only rational miners, the electric power consumed by the network in megawatts equals the price of one bitcoin in US dollars.

At the current price ($600 per BTC) it means that the network consumes something like 600MW. That is one third of the electric capacity of the plant of Fessenheim that made the news today (in France).

I just wanted to share that with you…

(originally posted in “Economics” but feel free to move it elsewhere if needed)