Transaction Fee

On every transaction, there is a small fee that is captured by the liquidity pool. This serves two purposes: 
1.Revenue - This is seen as an aggregate system revenue based on Gross Merchandise Volume (GMV)
2.Stability - Given the system is built on an automated market maker, the system holds the liquidity but over time 
Stability Analysis
The problem is that for constant product function AMMs is that the smaller the K-value, the more the price moves. This means that since each transaction incrementally decreases the K-value, over a long time, the internal economic model gets more and more unstable. See a set of transactions below where the K-value delta between transactions is negative.
Transaction Impact Analysis without Fee
Looking at the exact same liquidity pool and set of transactions but now with a small 0.05 gold fee that feeds into the pool. This over time makes the system more stable because the K-value gets slightly larger.
Transaction Impact Analysis with Fee
This makes even more sense from the perspective that as more transactions occur at a higher velocity, every incremental transaction has an infinitesimally smaller and smaller impact on price movements.
At the launch of alpha the gold fee has been set to 0.05 zGOLD on every transaction and this is not felt by the user because it looks appears closer to a small amount of slippage on the transaction.

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Stability Analysis