A bonding curve represents a mathematical concept integrated into platforms and applications for computing a token's valuation based on its supply. It is a way of defining the relationship between the price and supply of a particular asset. The bonding curve procedure is about establishing the price of each newly minted token based on the existing supply of the token. 
The bonding curve fundamentally solidifies the connection between price and supply. It is a mechanism where tokens are acquired at a price specified on the curve using collateral in the form of fiat currencies or other cryptocurrencies like Bitcoin (BTC), Ether (ETH), or Binance Coin (BNB). The bonding curve calculates the token's estimated value both when individuals purchase tokens, which results in minting, and when they sell tokens, which leads to token burning. As the bonding curve tokens are minted and burned, the supply fluctuates, which is then reflected in the value indicated by the bonding curve. Since the bonding curve defines a predetermined price at each supply level, it acts as an automated market maker (AMM) for the token at every price level. 
Features of Bonding Curves
- Limitless supply: An unlimited number of tokens can be minted.
- Deterministic price: Token prices increase and decrease with the number of tokens minted.
- Continuous price: The price of token
nis lower than that of token
n+1and higher than that of token
- Instant liquidity: Tokens can be instantly purchased or sold at any moment, with the bonding curve serving as an AMM.
Workflow of a Bonding Curve
- A smart contract is generated to enable users to mint or purchase a new token, such as a Bonding Curve Token (BCT), using a designated reserve currency, like DAI.
- The price of BCT is algorithmically determined based on its existing circulating supply and displayed in its reserve currency, DAI.
- Using the reserve currency (DAI), users can acquire the new BCT token via a smart contract. The DAI received from the sale is held within the smart contract as collateral and is not distributed to any individual or team.
- Once the user finalizes their purchase, the token price will adjust along the bonding curve in response to the quantity of supply that the user has just minted. Typically, this movement in price will result in an upward trajectory for future buyers.
- The holder of the BCT, at any point in time, has the option to burn or sell their token back to the curve. If the token price continues to rise after their initial acquisition, the user is likely to sell at a profit, accounting for any gas and fees. Upon approval, the user will receive the reserved DAI from the smart contract following the completion of their sale. 
How Bonding Curve Works
A set number of tokens are initially minted and offered for purchase upon deploying a bonding curve. The quantity of tokens in this initial supply is typically determined by the curve's parameters and functions as the foundation for the token economy. 
The bonding curve equation regulates how the token price fluctuates as supply either increases or decreases. Also, the type of curve employed influences the price increase at different rates. For instance, a linear curve will maintain a constant price, whereas an exponential curve will accelerate the price increase as token supply decreases. 
Types of Bonding Curves
A linear bonding curve is one of the simplest options, but developers might have various objectives, such as incentivizing early investment or discouraging early selling. Since a bonding curve is integrated into the blockchain and remains unalterable, its specific shape will dictate these aspects when individuals engage in buying and selling activities. The bonding curve, therefore, is of four common types: sigmoid curve, quadratic curve, negative exponential curve, and linear (non-increasing) curve.
Linear (Non-increasing) Curve
Linear (non-increasing) bonding curves establish a direct and unchanging connection between token supply and price. It is suitable for projects where participants are motivated to support a project they believe in without the primary goal of making a profit or incurring losses. This bonding curve type maintains a steady and unchanging cost, making it attractive to those who are primarily focused on endorsing a project they have a strong affinity for. 
Objectives of Choosing the Correct Bonding Curve Types
Various bonding curve shapes are chosen based on the type of investment behavior the developer aims to encourage or discourage. The objectives that can be adopted are:
Rewarding individuals who are involved in early trading activities
Developers seeking to reward individuals who are involved with early trading activities often opt for sigmoid or quadratic bonding curves, especially when they anticipate their projects to gain widespread attention. This approach is particularly suitable for projects with broad audience appeal, such as crypto-based gaming platforms (GameFi), NFT (non-fungible token) creation and sales platforms (ECOMI), or audio-sharing platforms like Audius. By implementing a sigmoid curve, developers can maintain lower costs for those individuals while significantly raising prices once the project reaches mainstream adoption, as evidenced by the sharp price increase at the inflection point of the sigmoid curve. Alternatively, a quadratic bonding curve can offer a more gradual yet still advantageous increase in token costs, which remains substantially lower for individuals who are involved with early trading activities compared to latecomers. 
Incentivizing early investment but not disincentivizing subsequent investment
When developers are implementing a bonding curve for a project that requires extended investment, such as a fundraising initiative, they may consider using a negative exponential curve or a linear bonding curve. The choice depends on their specific goals.
- Negative Exponential Curve: This curve encourages individuals by allowing them to acquire tokens at a low initial cost, potentially resulting in a profit on their investment. As the project gains more attention and attracts additional investment, the curve gradually flattens, resulting in a more modest rate of increase. This approach is beneficial for projects that aim to incentivize early participation and provide an opportunity to profit from their early involvement.
- Linear Bonding Curve: In contrast, a linear bonding curve involves a steady, linear increase in token costs as more people participate in the project. While it can still be profitable for early participants, the difference in costs between early and late participants is not as pronounced as with sigmoid and quadratic curves. This approach may be suitable for projects where the goal is to maintain a more consistent and predictable rate of increase in token value.
Ultimately, the choice between these bonding curve shapes depends on the developer's strategy, project goals, and the type of investment behavior they wish to encourage throughout the project's lifecycle. 
Keeping costs constant
A linear (non-increasing) bonding curve is a suitable choice for projects where people are not primarily seeking to generate profits or incur losses from their investment. In this scenario, the token cost remains constant, meaning individuals neither gain nor lose value over time. This type of bonding curve can be particularly well-suited to those who are primarily interested in supporting a project they believe in or contributing to its success rather than pursuing financial gains. It aligns with the idea of community or philanthropic involvement in a project rather than speculative investment. 
Benefits of Bonding Curves
Bonding curves serve various purposes, including enhancing valuation methodologies, pre-determining the rate at which token value will rise or fall, eliminating the need for exchanges, and facilitating the coexistence of multiple tokens within a single ecosystem. 
Enhancing valuation methodologies
Bonding curves exhibit transparency since they are integrated into blockchains and offer predictability and precision due to their mathematical foundation. Moreover, bonding curves represent a dynamic method for computing cryptocurrency values because they account for ecosystem expansion. A bonding curve acknowledges that as an ecosystem grows, the quantity of its associated tokens increases, leading to a proportional rise in its value. 
Pre-determining the rate at which the token value will rise or fall
A bonding curve establishes that token and coin prices are contingent on their supply, leading to either a decrease or an increase in value, thereby forming a continuous token model. If, for instance, a developer wants to influence this feature significantly, they can select a particular bonding curve shape, which in turn dictates the extent to which a token's value will increase in response to changes in supply. 
Eliminating the need for exchanges
Functioning as a fully automated market maker (AMM), bonding curves not only facilitate the computation of a token's price but also enable seamless transactions. The mathematical algorithm determines the token's cost and presents it to an individual, who can then conveniently buy or sell their tokens directly through this mechanism. This capability is particularly thrilling as it contributes to the decentralization of cryptocurrency, reducing reliance on centralized intermediaries. 
Facilitating the coexistence of multiple tokens within a single ecosystem
The ability to enable the creation of multiple tokens within a single ecosystem is another crucial role of a bonding curve. Developers can incorporate multiple bonding curves into the ecosystem, granting them the flexibility to use various tokens for distinct projects based on their intended functionality. This versatility allows for the utilization of different tokens across various blockchains, contingent on the specific use cases and the interconnections established through smart contracts. 
- Automated Market Makers (AMM)
- Continuous Organizations
- Continuous token-curated registries
Automated market makers (AMM)
On a typical cryptocurrency exchange, whether centralized or decentralized, market makers create buy or sell orders for a specific token trading pair. Since both buy and sell orders come with attached prices, market fluctuations can lead to a situation where some orders may take a significant amount of time to execute, or in some cases, they may not get accomplished at all. However, automated market makers (AMMs) employ bonding curves to enable tokens to be instantly and seamlessly converted into one another, eliminating the necessity of matching buyers with sellers. With AMMs, users can consistently buy and sell directly against the AMM contract. An example of an AMM utilizing bonding curves is the Bancor Protocol.
Bancor Protocol, launched in 2017, is an AMM that aims to make crypto asset markets more liquid by offering incentives for individuals to create and maintain pools of assets. It facilitates automatic price determination and an autonomous liquidity mechanism for tokens and continuously recalculates prices to maintain equilibrium between buy and sell volumes. 
Bancor's AMM service is automated as users deposit assets into pools to receive a new token in return. Each pool is made up of a pair of tokens and a reserve of BNT, its native cryptocurrency. Bancor Protocol has the following features: 
- It performs instant, automated token trades.
- It proposes a token for whitelisting.
- It issues flash loans.
- It provides liquidity with single-sided interest-earning.
- It integrates its on-chain trading and yielding features into any application.
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