What a bonding curve is, and what happens when it ends
A bonding curve is a pricing rule written into a contract: the more tokens exist, the more the next one costs. This page shows how the rule works, how it differs from an exchange pool, and what becomes of it at the end.
A bonding curve is a formula in a smart contract that sets the price of a token from the number of tokens already issued. You pay the contract, and it creates new tokens for you at the price on the curve; this is called minting. Every purchase moves the price up for the next buyer. The curve sets the mint price. It does not set the price on an exchange.
How a bonding curve works
An ordinary token sale has a seller and a price list. A bonding curve has neither. The contract itself is the seller, and the price list is a formula.
The formula takes one input: the supply, the number of tokens issued so far. You send the contract the currency it accepts: a stablecoin, or the coin of the network it runs on. What the contract collects is its reserve. It reads the supply, works out the price and mints new tokens for you. The supply is now larger, so the next buyer starts from a higher price.
Many curves also work in reverse. You hand tokens back, the contract destroys them — burns them — and pays you from the reserve at the price on the curve, and the price steps down. Such a curve is two-way. A one-way curve only mints. Holders leave through a market, or, if the contract has a separate rule for it, through redemption: handing tokens back for a share of the reserve.
The formula, with numbers
The simplest curve is a straight line. The price starts at a base and grows by a fixed step with every token:
Take a base of $0.10 and a slope of $0.001. With 1,000 tokens issued, the next token costs $0.10 + $0.001 × 1,000 = $1.10.
A purchase of several tokens does not pay one price. Each token is a little dearer than the one before it. So the cost of the whole purchase is the area under the curve, between the old supply and the new one. For 100 tokens bought at a supply of 1,000:
That is $1.15 a token on average. The buyer who comes next starts at $1.20.
An exponential curve replaces the fixed step with a fixed percentage: price = base × e^(k × supply). It stays low over a long stretch of supply and then climbs steeply. The arithmetic of a purchase is the same: the area under the curve.
Linear, exponential and other shapes
| Shape | Formula | What it does |
|---|---|---|
| Linear | base + slope × supply | The price grows by the same step with every token. Early and late buyers pay different prices, but the gap widens slowly. |
| Exponential | base × e^(k × supply) | The price grows by the same percentage with every token. The gap between early and late buyers widens fast. |
| Power | a × supply^n | With n = 1 it is a straight line from zero. A larger n bends the curve upwards; an n below 1 flattens it. |
| S-curve | a logistic function | Slow, then fast, then flat: the price levels off as it nears a ceiling. |
The shape is a decision about who pays what. On a steep curve early and late buyers pay very different prices. On a flat one they pay almost the same. No shape makes a token worth its price: that depends on what stands behind the token.
Bonding curve versus liquidity pool
An exchange pool, the heart of an automated market maker, also prices by formula, so the two are easy to confuse. They answer different questions.
| Question | Bonding curve | Liquidity pool (AMM) |
|---|---|---|
| Who is on the other side | the contract that issues the token | a pool of two tokens, put in by liquidity providers |
| What a purchase does | mints new tokens: the supply grows | moves existing tokens: the supply stays the same |
| What the price depends on | the number of tokens issued | the ratio of the two balances in the pool |
| Can the price fall | only if the curve buys tokens back | yes, with every sale |
The two often work side by side. The curve issues tokens, and a pool lets holders trade them with each other. Such a token has two prices at once: the mint price, set by the curve, and the market price, set by trades in the pool. The two need not be equal. While minting is open, the market price cannot stay above the mint price for long: anyone can mint and sell. Nothing holds the market price from below unless the contract takes tokens back, and then only near what the contract pays.
What happens when a bonding curve is complete
A curve can be written to stop. It ends when a set number of tokens has been issued or sold, or when the reserve reaches a set size. What follows depends on the design. Three outcomes are worth knowing.
- 01
The curve hands over to a pool
The usual outcome on token launch platforms. There the whole supply is usually created in advance, and the curve sells a set part of it: it sells from stock instead of minting, and its price follows the number of tokens sold. Once that part is sold, the curve closes, and the reserve it collected goes into an exchange pool together with the tokens held back for the pool. From then on the pool sets the price, and that price can fall.
- 02
The curve never ends
A continuous token has no last token. The contract mints and buys back along the same curve for as long as it exists.
- 03
The rule changes
The contract switches to a second rule for the mint price: for example, growth with time instead of growth with supply. Minting goes on under the new rule.
If you hold a token on a curve, find out which of the three applies before the curve gets there. Look for it in the contract.
What to check before you use one
- Can you sell back to the curve? If not, your way out is the market, at the market’s price, unless the contract has a separate rule for redemption.
- What is in the reserve, and who can take it out? A curve is only as good as the reserve behind it. See what “backed” means.
- Can anyone change the formula? A curve in an upgradeable contract is a curve until someone replaces it. See how to tell whether a team can still change the contract.
- What happens at the end? One of the three outcomes above, and the contract should say which.
A bonding curve is a pricing rule, not a promise. It tells you what minting costs. It says nothing about what the token will fetch on a market.
Common questions
Is a bonding curve the same as a liquidity pool?
No. A bonding curve mints new tokens at a price set by the supply. A liquidity pool trades existing tokens at a price set by the balances in the pool. One token can have both.
Can the price on a bonding curve go down?
On a two-way curve, yes: when holders sell tokens back, the supply shrinks and the price steps down the curve. On a one-way curve the mint price never goes down, but the market price of the token still can.
What happens to my tokens when the curve completes?
Nothing happens to the tokens themselves. What changes is where the price comes from. On most launch platforms it moves from the curve to an exchange pool.
Is a bonding curve a Ponzi scheme?
Not by itself: a curve is arithmetic. What matters is where the money goes: into a reserve that holders can redeem, or to earlier buyers. Five tests tell the two apart.
How Assetrix does it
Assetrix is a protocol on Arbitrum One. It issues ASTRX, a token backed by collateral, whose mint price is set by the contract and never falls. In Phase 1 the mint price of ASTRX follows a published curve: $0.001 for the first token, $200 for the hundred-millionth. The curve counts every token ever created. Redemption does not reduce the count, so the mint price does not fall when holders leave.
The contract does not set the market price of ASTRX. It sets two numbers, and the market price sits between them: the backing per token below and the mint price above. Inside that corridor the market decides, and the price can fall.
Read next
Phase 2: what happens when the curve runs out
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Can a token only go up?
A design question with a negative answer, and what the answer leaves you.
Why a rising issue price is not a pyramid
It is a fair suspicion and it deserves a direct answer rather than an offended one.