I think you’ve done a great job in modeling the basic formulas and dynamics around the fee switch. I’m surprised there aren’t more comments on your work, perhaps you’ve received them outside the forum.
Reading your paper, everything ultimately depends on the values of sticky volumes s1, s2, and the LPs acceptable difference d. These are unknowns for Uniswap before trying out the fee switch in the open. Additionally, I consider d > 0 a fair assumption for practically all pools, since 1) Uniswap is the leader in market share 2) the contracts have been battletested with the most DEX volume.
In terms of describing the real world, the weakness of the model is that it’s fully static in terms of the total trade volume V. You have clearly stated this in the assumptions, but I’d like to emphasize the significance of this on how the model behaves. It ultimately leads to the majority of conclusions from the model to be against the fee switch. The model basically tells how much volume Uniswap would lose by introducing take rates. What if the total volume would start to explode upwards?
Particularly, I believe that activating the fee switch could supercharge Uniswap adoption by generating revenue to own UNI and making everyone more interested in the project.
We are still in the early stages of growing what is possible “the orderbook of the world”.
Therefore, the model could be improved by adding a growth factor g > 0 that depends on the revenue generated to UNI holders. The factor g would affect the total trading volume V, creating a positive feedback loop effect. Then, in addition to the constant cash flow, we could add the future expected growth of the cash flow to the analysis and compare that against the costs now presented in the model (loss of volume/liquidity).
Obviously, this would make the model more complicated but arguably a better one for real world decision making. But once more, really high quality work and a good starting point for further analysis.