In our previous examples of bonding curves, we have assumed that no base tokens enter the reserves and no target tokens are burned without taking from the reserves. What if we want to set a relationship between the target supply and reserves that always holds, even in these situations?
This situation is called a Sponsored Burn, which you can read more about here
Let's take an example bonding curve with price, dictated by:
We can define , the amount of
Base tokens stored in the reserves as the area under the curve up to the current supply:
Therefore, taking the indefinite integral, we can say the reserves are
Now, what happens when we inject
Base tokens into the Reserves?
This causes all holders to see an increase in value of the token. Our formula above does not account for this possibility in pricing. So how do we support this model?
We say that the
Reserve is always proportional to the
Supply in a relationship that should never change.
Holding this invariant proportion, if we increase the reserve by some amount , the increase in supply should maintain this formula:
We will not walk through solving the following equations, as we solved them with the aid of computers
Solving for , we can say that given some amount of the base token of the curve , the target tokens we get back should be
Likewise, solving for R, we can say our price for some amount of target tokens S is
Worth noting is that both of these equations fail for R = 0 or S = 0. When this is true, we revert to the base pricing formula of
Where c is a constant that will help set the slope of the initial curve, and thus the initial price.
The above math has one interesting property. We can define a curve in terms of a price relation, current
Reserve, and current
Supply. This means that, without changing tokens, we can change the shape of our curve. The new invariant will be satisfied for the new curve, and any future purchases or sales will need to follow the new invariant.
Royalties that go back to Holders
A powerful model that comes from this feature is having a way to reward all token holders. A simple way to see this at work is to put a 5% buy and sell royalty in terms of the
Base token that goes back into the
This setup allows holders to improve their position via transaction fees, encouraging circulation of the token.