For proof of work networks to function securely, the network as a whole must have a much higher computation rate than an attacker.

Obviously capabilities increase the whole time, so the bitcoin network needs to be able to adapt - to do this, it adjusts its 'difficulty' approximately every two weeks. The power usage corresponds directly to the difficulty level set by the network.

If you add more miners, the difficulty will increase, and power consumption will go up, if miners leave, the difficulty will decrease and power consumption will go down.

A certain amount of bitcoin -- the amount determined by a schedule that was defined before the network became operational -- is given as a mining reward every 10 minutes. The incentives of the individual miners is such that the expenses of the miners (collectively) equals the mining reward -- and the major mining expense is electricity.

If the mining reward is cut in half, the electricity consumption of the network is cut in half, too.

In contrast, if the rate of transactions changes or the number of miners change, electricity consumption stays the same. (More precisely, the expenses of running the network equals the mining reward plus any transaction fees, and since the blocks are of fixed size, for more transactions to compete for space in the blocks increases transaction fees, but I am guessing that transaction fees are currently a small fraction of the mining reward.)

> If the mining reward is cut in half, the electricity consumption of the network is cut in half, too.

That's not true it all -- there is no difficulty adjustment when the reward is halved. There may be pressure on some miners to stop mining, but that adjustment is not immediate and can also be compensated for by change in the price of bitcoin.

> In contrast, if the rate of transactions changes or the number of miners change, electricity consumption stays the same.

Given the network is operating at peak transaction rate already, there's not much change here. Neither block size nor the transaction count has a meaningful affect on the computations required to "solve" a block. The merkle tree for the transactions is computed once, but most of computation is finding a nonce, that combined with the rest of the block header, produces a hash with a certain number of leading zeros.

Changing the block size or transaction count doesn't meaningfully change electricity consumption.

>that adjustment . . . can also be compensated for by change in the price of bitcoin.

It can. When I wrote that if the mining reward is cut in half, the electricity consumption of the network is cut in half, too, I assumed that the price remains constant.

A miner must pay for the electricity he or she uses. Where do you think the money comes from to pay for the massive amount of electricity used by the network?

Do you imagine that rich people (i.e., people who can afford to lose money) are buying the electricity for pro-Bitcoin ideological reasons?

I don't: I believe that it comes out of the revenue made (collectively) by the miners. In other words, the payments for the electricity are parts of (individual) plans to make money through mining.

And I believe that if that (collective) revenue were cut in half -- especially if miners and prospective miners knew of the halving in advance (particularly, before they decided what mining hardware if any to buy) -- then money spent on electricity is approximately cut in half, too. (Because otherwise the plans to make money would not work.)

You are correct that the adjustment in the hash rate is not immediate after a halving of the mining reward. Mostly that is because miners who recently bought mining hardware have to continue mining after the halving to continue to pay for their hardware.

If the halving was announced in advance, then the halving will start exerting downward pressure on the mining difficulty months in advance of the actual date of the halving. The major cause of that downward pressure is miners opting not to upgrade their hardware and prospective miners opting not to enter the mining business in the first place (the effect of which is to make it take longer for the remaining miners with their non-upgraded hardware to solve the proof-of-work puzzles, which in turns causes the software to reduce mining difficulty to return the average time between successive blocks back to 10 minutes).

Basically, it takes many months for newly-manufactured mining hardware to pay for itself, and that delay is the main reason the response to a reward-rate halving is slow. But again the response starts months before the actual halving; and the cumulative effect of the halving on the rate -- more precisely the effect the rate has on how much electricity is consumed by the network over the years -- is approximately the same as it would be if the effect of the halving on the rate were instantaneous.

It is the fact that one of the major expenses (namely, hardware) of the miners is "lumpy" (requires an upfront expenditure that is then recouped over many months) that obscures the simple relationship whereby the collective expenses of the mining community approximately equals the collective revenues of that community -- where most of that revenue is from mining rewards, which is equal to the price of bitcoin (or the value of bitcoin if you prefer) times the rate at which the network dispenses bitcoins from miners as rewards. What makes me confident that expenses = revenues is that miners are rational and consequently are capable of taking into account scheduled halvings of the rewards to mining and the "lumpiness" of the cost of mining hardware (and many other factors).

> if the mining reward is cut in half, the electricity consumption of the network is cut in half

This would only happen if half of the miners shut off their equipment as soon as the reward adjustment occurred, which does not happen [1]. Even then, that doesn't change the electricity consumed by a single miner producing a single block, since that is governed by the difficulty parameter, which also did not halve and took multiple weeks to bottom out before increasing again [2].

There is no simple function between block reward and network or block electricity consumption.

1. There was a drop of 120PH/s to 90PH/s within a week after the halvening. https://www.blockchain.com/charts/hash-rate

2. You can see a similar curve of difficulty _following_ the hash rate of network, albeit over multiple weeks. https://www.coinwarz.com/mining/bitcoin/difficulty-chart

> A certain amount of bitcoin -- the amount determined by a schedule that was defined before the network became operational -- is given as a mining reward every 10 minutes. The incentives of the individual miners is such that the expenses of the miners (collectively) equals the mining reward -- and the major mining expense is electricity.

The 10 minute rate is true, but the expense is not collective. I as a miner am in active competition with my fellow miners and am incentivised to find ways to decrease costs (electricity), but only mine. In fact it would be in my best interest to increase the electricity cost of a competing miner in order to force them off the network, and if I win the block it is only my personal power backing that block.

> If the mining reward is cut in half, the electricity consumption of the network is cut in half, too.

Not necessarily. If the reward is cut in half but the price of BTC has risen considerably since starting mining operations then I will remain on the network as the profit is sufficient to continue. This is why the "halving" event that Bitcoin goes through has never had a major impact on either the price of BTC or the scale of mining operations globally.

> In contrast, if the rate of transactions changes or the number of miners change, electricity consumption stays the same. (More precisely, the expenses of running the network equals the mining reward plus any transaction fees, and since the blocks are of fixed size, for more transactions to compete for space in the blocks increases transaction fees, but I am guessing that transaction fees are currently a small fraction of the mining reward.)

This is very wrong. The expense of running the network is equivalent to all the energy burned by the entire network in order to find the next block. This isn't felt by the individual miner in any way other than a potential increase in competition to find the block, but taken as a whole the network is for example now considered to be using the total energy consumption of Argentina.