The Bonding Curve

How to make it go brrr?

Bonding curves establish a clear relationship between token supply and price. As more tokens are bought or sold in the bonding curve, the price per token adjusts according to a predefined mathematical function. This provides clarity and predictability for the user.

Printr’s chain-agnostic bonding curve establishes a clear relationship between supply and price, independent of the specific blockchain.

Curve Expression

The expression for the curve is:

$$f(x) = \frac{MP(x + a)}{M - (x + a)}$$

where:

• M = Maximum token supply

• P = Base price factor

• x = New supply added

• a = Virtual reserve (used to smooth pricing)

The function applies within this range:

$$0 \leq x < M - a$$

How It Works

Price Calculation

• The numerator MP(x + a) represents the virtual reserve multiplied by the new supply.

• The denominator M - (x + a) prevents the price function from exceeding the total supply limit.

• The result is a dynamically increasing price curve that rises as supply nears its maximum.

Properties of the Curve

• Supply-Driven Price Increases: Prices rise as users mint more tokens, following the supply curve.

• Liquidity Smoothing: The virtual reserve a reduces sudden price jumps when users add new tokens.

• Supply Constraints: The function only works within valid supply limits, maintaining ecosystem stability.

The expression provides us a reliable pricing model for printed tokens while providing stability through the token's lifecycle.

Implications

The implications of using this bonding curve expression are:

• Once the specified token supply is sold, the contract graduates the token and moves it to the DEX by creating a liquidity pool.

• Each chain operates independently, meaning tokens may graduate at different times.

• If a token is deployed across multiple chains, its supply is split evenly.

• The price volatility is higher based on demand due to the reduced per-chain supply.

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Common Questions

Since the bonding is chain-agnostic, what happens when a token graduates on a chain faster than others?

• Nothing exceptional really happens. The token on that chain becomes tradable on DEXs but is still tradable through our contract as it routes trades to DEXs under the hood.

• Graduation on one chain might trigger graduation on the other as a chain reaction. If a token is graduated, it means its price is higher. So it's profitable to buy it where it hasn't graduated yet, bridge, and sell where it's graduated already.

• This model's effect is cross-chain liquidity flow for the token. Price consistency is an important aspect of liquidity unification, and our chain-agnostic bonding curve model would help achieve that.