> For the complete documentation index, see [llms.txt](https://printr.gitbook.io/printr-docs/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://printr.gitbook.io/printr-docs/how-printr-works/the-bonding-curve.md).

# The Bonding Curve

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.**

### Printr Bonding Curve Formula

The formula for the curve is:

$$
Constant Product: k = (v + r) × t
$$

where:

* v = virtual reserve = (T₀ × P₀) / PRECISION where P₀ = initial price
* r = actual reserve (ETH/WETH raised)
* t = token reserve (unsold tokens)
* T₀ = initial token reserve for curve
* PRECISION = 10^18

**Price: P** = (PRECISION × v × T₀) / t² (quadratic increase)

**Buy Cost: cost** = k/(t - Δt) - v - r Key: Initial price sets virtual reserve size, which anchors the entire curve. Higher P₀ → higher v → higher starting prices throughout.

#### 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.

<figure><img src="/files/O4V0sGD6I9gepFs7X0iJ" alt=""><figcaption></figcaption></figure>

***

### 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.