HyperByte - How the fee calculated in uniswap-v2?

Download uniswap whitepaper: https://app.uniswap.org/whitepaper.pdf

it’s not very friendly to read, I learned so hard, then this is my summaries.

Origin AMM formula:

Assume the fee is 0.3% (in uniswapv-v2, this is a fixed fee), dx is the token you’re trying to exchange, the fee will be charged on this dx, so we just change the dx of the formula above like list:

this is the dy we correctly will get. And, this formula actually is what the getAmountOut function does, look:

function getAmountOut(...) {
    ...
    uint amountInWithFee = amountIn.mul(997);  //1-0.003=0.997
    ...
}

Once the fee is taken out, it flows into the pool, not to a specific account, so the liquidity will increase as swaps accumulate.

Here is the breakdown of the flow.

Here is pool for USDT/DAI, initially their liquidity are: (100USDT and 100 DAI, they’re woth same)

Liquidity(√k)
   ^
   |   100	   100
   |   ██      ██
   |   ██      ██
   |   ██      ██
   |   ██      ██
   +--+---+---+---+---> token
       USDT    DAI

After some swaps, USDT got 104, DAI got 98.

Liquidity(√k)
   ^
   |   104
   |   ██      98
   |   ██      ██
   |   ██      ██
   |   ██      ██
   +--+---+---+---+---> token
       USDT    DAI

But immediately, they return back a balanced position. Bcz there usually are arbitrage bots are keep detecting whether there are an imbalance that can profit from , which brings the pool back to balance.

Liquidity(√k)
   ^
   |   101     101
   |   ██      ██
   |   ██      ██
   |   ██      ██
   |   ██      ██
   +--+---+---+---+---> token
       USDT    DAI

Can easily see they’ve increased, say again: once the fee is taken out, it flows into the pool, not to a specific account, so the liquidity will increase as swaps accumulate.

pool liquidity
- →
  
  increased liquidity
  
  The growth in liquidity , that's exactly where LPs earn, as the pool growth , so do their profit. Amount of the profit just is equivalent to the `increased liquidity`, equivalent to sum of all gathered fees. But, where does the project team who writes contracts benefit from?
  - in uniswap-v2’s code project team doesn’t charge fee for themselves by default, can open it by giving an account address which receives fee to the feeTo variable in UniswapV2Pair.sol’s factory’:
```
function _mintFee(...) returns (bool feeOn) {
    ...
    address feeTo = IUniswapV2Factory(factory).feeTo();
    feeOn = feeTo != address(0);  // feeTo=0x0, feeOn=false, by default.
    if (feeOn) {
        ...
    }
}
```
  - If we set feeTo to an account address , what will that be?
    - Obviously, according to the _mintFee function above, it will get into the if charges fee to the account.
    - there’s a φ=1/6 , means project team will take away 1/6 of the increased liquidity.

let φ=1/6,

Sm goes to project team, the rest goes to the LPs.