The reason gas prices have been declining for the past 4 quarters despite having over 100% gas usage

The Enigma

One interesting curiosity I've noticed is that Ethereum gas prices have been falling 20-40% each quarter since EIP-1559 initiated DESPITE gas usage being over the 100% 15M target gas usage.

This is surprising because normally under EIP-1559, if each block stays exactly at the 100% gas target, gas price will also stay exactly the same.

So why is this happening?

This counter-intuitive phenomenon can be explained by simple math. After EIP-1559, gas price changes whenever the previous block's gas usage deviates from 15M gas. It increases by 12.5% at 30M gas, and decreases by 12.5% at 0M gas. Any value between the 2 limits affects price proportionally up to 12.5%.

So what happens if a 30M-gas block were followed by a 0M-gas block?

(x+a) * (x-a) = x^2 - a^2 (1 + 0.125) * (1 - 0.125) = 98.4% => 2.6% price decrease 

Thus if a 30M-gas block were followed by a 0M-gas block, the gas price would still fall 2.6% even though the average gas usage stays at 15M gas. The larger the volatility from the 15M-gas target, the larger the negative price pressure.

In fact, in an extreme theoretical case where there were one full day of 30M-gas blocks followed by another full day of 0M-gas blocks, the gas price on the 3rd day would be 10⁵⁰ times smaller despite that the average usage between the 2 days was still 15M gas. Obviously, validators will not magically be able to process transactions 10⁵⁰ times cheaper, so this is completely theoretical.

Nevertheless, the effect is there, and we have been experiencing a 20-40% decrease in gas prices quarterly since EIP-1559 was introduced in Aug 2021. Though the decrease has slowed down since The Merge.

A mix of these 2 cases will continue happening until a dynamic market equilibrium is reached:

1. Gas price keeps falling every quarter
2. Daily gas usage rises noticeably above its current 102% average. (Edit: I did some simulations afterwards. Gas usage actually remains around 102-103% of 15M when the price remains stable. So it's actually not as noticeable as I thought.)

Edit: I thought about it more, and it does seem to either be a bug or a purposeful design flaw.

Let's say you have 3 fee changes of a, b, and c where the sum of all 3 is zero. You have to find a fee formula f(x) where the product, f(a) * f(b) * f(c) = 1.

So I figured out the correct formula is to use any exponential function. For example:

a + b = -c e^a * e^b * e^c = e^a * e^b * e^(-a-b) = 1 

I was going to report it, but looks like someone else already mentioned it a long time ago: https://ethresear.ch/t/make-eip-1559-more-like-an-amm-curve/9082/2

Guess they never bothered fixing it.

Edit 2: Oh wait, that "someone else" was vbuterin.