Lesson 16 - Changes in Liquidity

$p_{x} = \frac{y}{x} = \frac{y_{n e w}}{x_{n e w}} = \frac{y + Δ y}{x + Δ x}$
$p_{x} \cdot (x + Δ x) = y + Δ y$
$\frac{y}{x} \cdot (x + Δ x) = y + Δ y$
$y + \frac{y Δ x}{x} = y + Δ y$
$Δ y = \frac{y Δ x}{x}$

Liquidity changes the invariant:

$x_{n e w} \cdot y_{n e w} = k_{n e w}$
$(x + Δ x) \cdot (y + Δ y) = k_{n e w}$
$x y + x Δ y + y Δ x + Δ x \cdot Δ y = k_{n e w}$
$k + x Δ y + y Δ x + Δ x \cdot Δ y = k_{n e w}$

Minting or burning of pool shares:

For Uniswap 2, this is the formula for the initial pool shares:

$i n i t i a l p o o l s h a r e s = \sqrt{i n i t i a l T o k e n Y \cdot i n i t i a l T o k e n Y}$
$S_{p} = \sqrt{x_{0} \cdot y_{0}}$

Once the pool exists, the change in share pools as liquidity is added:

$\frac{Δ S_{p}}{S_{p}} = \frac{Δ x}{x}$

$S_{n e w} = S_{p} + Δ S_{p} = S_{p} + S_{p} \cdot \frac{Δ x}{x}$

Agent shares then are the additional shares the agent receives of the total Shares, proportional to the amount contributed.

This is the same the amount of additional Pool Shares minted to accomodate the increased liquidity:

$n e w A_{P} = A_{P} + S_{p} \cdot \frac{Δ x}{x}$