Bonding Curve

A bonding curve is a mathematical function that links the price of a token to the number of tokens in circulation. A smart contract uses the curve to sell newly minted tokens to buyers and, in many designs, to buy them back and burn them. Because the price rises as supply grows, early buyers pay less than later ones, and anyone can trade with the contract at the price the formula gives without needing another person on the other side.

Bonding curves are a form of automated market making, a role otherwise played by a market maker, and are closely related to the formulas used by automated market makers in decentralized exchanges. They have been used for continuous token fundraising, community and creator tokens, and token launch platforms on networks such as Solana, where new tokens trade on a curve before moving to a regular exchange pool. The idea was developed in the Ethereum community around 2017.

Picture a vending machine that sells tickets to a show and sets the price automatically. The first ticket costs $1, the next $1.10, the next $1.20, and so on. The more tickets sold, the more each new one costs. If someone wants to return a ticket, the machine buys it back at the current price, which then drops a little.

A bonding curve is that vending machine for a token. The money buyers pay is kept by the contract, so it can always buy tokens back. Early buyers get cheaper tokens and can sell at a profit if many people buy after them. But if lots of people sell, the price slides back down the curve just as automatically.

A simple linear bonding curve sets the price as a constant times supply:

The cost of buying tokens as supply rises from $S_0$ to $S_1$ is the area under the curve:

• $P(S)$ is the price of the next token when $S$ tokens exist.
• $m$ is the slope, set by the designer.
• $S_0$ and $S_1$ are the supply before and after the purchase.

Suppose $m = 0.01$ and 100 tokens already exist, so the current price is $1. A buyer purchases 100 more, taking supply to 200. The cost is $0.005 \times (200^2 - 100^2) = 0.005 \times 30{,}000 = 150$, or $150, an average of $1.50 per token, and the new price is $0.01 \times 200 = 2$. If the buyer immediately sold the 100 tokens back, they would receive the same $150, less any fees.

Designers choose curve shapes to suit their goals:

• Linear curves. Price rises steadily with supply.
• Exponential or polynomial curves. Price rises slowly at first and then steeply, strongly rewarding early buyers.
• Logarithmic curves. Price rises quickly early and then flattens.
• Sigmoid curves. Price is flat, then rises sharply in the middle, then levels off.
• Separate buy and sell curves. Some designs sell on a higher curve and buy back on a lower one, with the gap funding a project treasury.

The reserve held by the contract determines whether all holders could, in theory, sell back at curve prices.

Bonding curves let projects raise funds continuously rather than through a one-off token pre-sale, and they provide liquidity from the first moment, since the contract is always willing to trade. Token launch platforms, including Pump.fun on Solana, have used bonding curves so that anyone can create a token, with the price rising as people buy until a threshold is reached and the token moves to a standard liquidity pool. Creator and social tokens have used curves to price access to communities, and some research-funding and public-goods schemes have experimented with them.

The structure rewards early buyers at the expense of later ones, which can resemble a Ponzi scheme if a token has no use beyond speculation. Launch platforms built on curves have been dominated by meme coins, many of which collapse quickly, and by automated traders who buy at the start of each curve, sometimes using a sniper bot. Large sales can push the price down the curve sharply, and slippage on big orders can be severe. Smart contract bugs can drain reserves, and creators may control parameters that can be changed. Some token launches also suffer rug pulls after the token leaves the curve. Regulators may treat tokens sold through curves as securities depending on how they are marketed. Like other crypto trading, buying on a bonding curve is highly speculative, and the formula guarantees a price, not a value.