This is great analysis! The near-constant exponential rate of growth for the ETH-DAI liquidity token value -- in spite of a 40x increase in the size of the pool -- suggests that the ETH-DAI Uniswap pool is still in the small regime (compared to the global market) and has much more room to grow. The 11% APR appears to be an emergent property determined by:

  1. The natural price variability of the ETH-USD global market, measured on the timescale of the Ethereum block time.
  2. The Uniswap trading fee of 0.3%.
  3. (And to a smaller extent, probably) The Ethereum gas price.

Three events drive trading volume on the pool:

  1. A non-arbitraging user trades with the pool for their own reasons, possibly because it provides the least price slippage for their order size.
  2. Sometimes, due to this trade, the pool price moves enough that an arbitrage opportunity exists, after fees, between the global market and the pool. An arbitrageur then submits a trade to the pool to move its price back toward the global market.
  3. Or independently, the global price of ETH-USD moves enough away from the pool price that an arbitrage opportunity exists and is corrected in the same way.

The volume of event #1 is limited by the size of the pool, because users will not submit orders that overwhelm the pool and move the price significantly.

The volume of event #2 is limited even more so, since only larger orders that cause a price imbalance, or a succession of smaller orders in the same direction, will drive #2 trades.

The volume of event #3 is both limited by the pool size but also driven by it. This is key! When the pool is large and the global price moves, a large volume of trades is required to correct the price of the pool. So long as the size of the pool does not dominate the global market, a larger pool will force higher trading volume by arbitrageurs who refuse to leave money on the table, even in the complete absence of natural trading in the pool.

I believe that most of the trading we have seen so far is category #3, and thats why it has scaled up as the pool has grown. But it could also be that the demand for #1 also far exceeds the pool capacity. To decide this, we would need to analyze what portion of trades occurred when the global price provided an arb and what portion when there is no arb motivation.

It is interesting to consider the effect of the trading fee. If the fee were smaller, then smaller fluctuations in the global price would trigger #3-type trades that otherwise would not be made. I think its an open question how the total fee collection would vary as a function of the fee rate. If the ETH-USD price series (at the 15-second scale, say) naturally fluctuates 0.1% 10 times as often as it fluctuates 0.3%, it could be that a lower fee would yield higher fee collection.

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