# PMTP - Description and Examples

Protocol Monetary Trade Policy (PMTP) is a mechanism Sifchain has designed to support the purchasing power of Rowan holders. The mechanism utilizes a gradual percentage increase in purchasing power to reach a desired target, which will be set by governance.

Sifchain’s swap formula, which sets a target purchising power increase for assets in its two-sided liquidity pools, is as follows:

$$y(x)=\frac{xYX}{(x+X)^2}$$

Please see [AMM Specification](https://hackmd.io/6VK2LSYjRTyeNCoHpVt2hg) for more details if needed.

The extension of this formula in the PMTP mechanism includes oracle driven pricing or other forms of unequal pooling:

$$y(x)=Y(1-(\frac{X}{x+X})^\frac{w_x}{w_y}) (1-\frac{x}{x+X})$$

Here, $w_x$ and $w_y$ control the weight that is assumed for each token in the pool, removing the pre-existing assumption that equal quantities in the pool result in equal prices (equivalently the special case where $w_x=w_y=1/2$). Please note that $w_x + w_y = 1$ will always hold.

For the first version, governance can set this ratio and a target block height by which that ratio should be achieved. The protocol will shift the ratio to be one unit closer for every block until it reaches its target.

Let's see how this looks in practice.

Example 1:

Let’s say at time = 0, we start with $w_{xt} = 0.50$ and $w_{yt} = 0.50$. If we use a 2% increase in purchasing power at each time step, after 10 units of time have passed, the weights become $w_{xt} = 0.55$ and $w_{yt} = 0.45$.

Instead of the previous condition of raising to the power of 1.00 in the swap formula, in this case we are raising to the power of 0.55/0.45 = 1.22, increasing the Rowan purchasing power by 22%.

Example 2:

Let’s say we are starting at the end point from the previous example, at the tenth unit of time where $w_{xt} = 0.55$ and $w_{yt} = 0.45$. We want to increase the purchasing power of Rowan only by a small amount, to weights of $w_{xt} = 0.56$ and $w_{yt} = 0.44$, at the 30th unit of time. We can achieve this by using a 0.12% increase in purchasing power at each unit of time in that interval. Then we can derive the weights for each intermediate step between time = 10 and time = 30 to arrive at $w_{xt} = 0.56$ and $w_{yt} = 0.44$.

Example 3:

Same as example 1 but half the liquidity is removed in the 5th epoch of 10 being the total steps targeted.

Renormalization at each epoch into target exponent . The first 5 are the same, in the second 5 the rowan exponent increases by a higher amount of ? per epoch (and the rowan amount decreases by the rate to maintain the sum of exponents being 1 at each step).