➰What are Bonding Curves?

Bonding curve definition & library of articles

Bonding curves are a mathematical encoding of the relationship between two or more tokenized assets in smart contracts. Their adoption has become mainstream with decentralized exchanges like Uniswap and Balancer, who built automated market maker protocols leveraging custom bonding curve implementations to revolutionize the way token liquidity is approached in blockchain token design. Bonding curves are a ‘supply discovery’ mechanism that allows new tokens to be minted (or burned) by the deposit (or withdrawal) of reserve assets when the demand for new tokens is greater (or less) than the available supply. There are different types of bonding curves employing various mathematical functions that express the encoded relationship and dictate the shape of the curve e.g. constant product, constant function, concentrated liquidity, variable product, or time-weighted. Launched via smart contracts running on top of a blockchain, the first and most basic bonding curve - constant product - allow those assets to be traded against one another, with the bonding curve defining their exchange rate. A popular example of a bonding curve equation is $x \cdot y = K$, which has an ‘Invariant $K$’ that defines the exchange price between token $x$ and token $y$. The ‘curve’ defines how that price changes as the amount of either token increases or decreases in supply. As we’ll see, bonding curves can be applied in different contexts and configurations to provide key infrastructure for projects deploying a token economy.

A graph of two tokens related via different bonding curves. Different ‘shapes’ of bonding curves can result in different properties of the mechanism, which can be helpful for different contexts and use cases.

Since bonding curves are not much more than a mathematical function, it can be difficult to grasp how they could have such a huge impact on token ecosystems. But when these mathematical relationships are encoded into smart contracts, they lay the economic foundations for new tools that can address some of the major challenges of distributed economic systems, such as bootstrapping small economies, providing necessary exchange liquidity, and facilitating demand-responsive dynamic token supply. By embedding bonding curves in smart contracts, we can create novel and intentional market structures with customizable design spaces.

Bonding curves have also been ‘augmented’ with the addition of common pool treasuries or even prediction markets, in order to improve the collective signaling capacity of these new tools for social and ecosystemic benefit.

This is just the beginning of their utility, and current tools have only taken the first steps to explore this design space. Bonding Curves are a gateway for engineered control systems to facilitate and improve the economic management and value accrual systems of Web3. Read and explore more in our library of articles on bonding curves:

BCRG Bonding Curve Library | NotionCuriousRabbit.eth on Notion