# Bonding Curves for L1 Token Economies

## Introduction

Bonding curves are fundamental to DeFi, primarily shaping the field through Automated Market Makers (AMMs). When leveraged as [Primary Issuance Markets](https://mirror.xyz/0x8fF6Fe58b468B1F18d2C54e2B0870b4e847C730d/1Pxl_fbIPifIQ4_y0xoJGZGEk70qfOM3Gi9nWycm-8k), these curves define the active relationship between a token's price and its supply, serving as a crucial mechanism for price discovery and supply management.

The strategic integration of a token within a project's economic and business framework is critical for achieving optimal coherence and effectiveness. This integration becomes vital for projects developing Proof-of-Stake (PoS) blockchains, commonly called Layer 1 (L1) networks. Such projects face the intricate challenge of determining an appropriate token issuance strategy without precise knowledge of future demands—essentially, they must make educated guesses about the total number of tokens needed throughout the project's lifecycle before launch.

Moreover, PoS L1 networks hinge on robust cryptoeconomic security to safeguard against potential Sybil attacks, where a single entity could disrupt the network by controlling a disproportionate number of tokens. Initially, these networks often rely on a genesis stake from trusted participants. Over time, the goal is to achieve a level of decentralized security, or "maturity," at which point the network is sufficiently robust and no longer vulnerable to such attacks.

This article explores a mathematical approach: A modified sigmoid bonding curve tailored to address the unique economic conditions of growing L1 blockchain projects. By employing adaptive parameters, this model optimally positions the bonding curve at the heart of the network’s maturity and scalability strategy, effectively using the sigmoid's inflection point as a marker for the network's transition into maturity.

## The Modified Sigmoid Bonding Curve

### Basic Concept
A sigmoid bonding curve is typically characterized by its S-shaped growth, representing a slow initial price increase followed by rapid acceleration and tapering off as it approaches a maximum cap. Our modified model enhances this framework by incorporating dynamic elements that allow the curve to adapt over time, more accurately reflecting the project's growth and market conditions.

### Mathematical Formulation
The modified sigmoid function for the token price $𝑃(𝑆)$ in terms of the supply
$𝑆$ is given by:

$$P(S) = \frac{L(S)}{1+e^{-k(S-S_0)}}+C$$ where

- $L(S)=L_0 + \alpha⋅S^\beta$ modulates the progressively increasing price cap, which continues to rise as the supply expands, controlled by the parameters of $L(S)$ function.
- $L_0$ is the initial price cap.
- $\alpha$ and $\beta$ control how the cap increases with supply.
- $k$ determines the steepness of the curve.
- $S_0$ is the midpoint of the sigmoid, where the inflection occurs.
- $C$ is a constant added to elevate the entire price curve, setting a minimum price level.

### Variable Constraints
To ensure that $𝑃(𝑆)$ is always positive and non-decreasing, the following constraints must be maintained:

- **Positive Parameters**: $𝐿_0$, $\alpha$, $k$, and $C$ must be positive. This ensures that the function does not return negative values, which would be economically nonsensical.
- **Non-negative Supply**: $S$ must be non-negative, as negative supply is not meaningful in any economic context.
- **Positive Slope Requirement**: $\beta$ must be chosen such that the derivative of $𝑃(𝑆)$ with respect to $S$ is non-negative across the entire domain. $\beta>0$ typically ensures that $L(S)$ is an increasing function.
- **Adequate Steepness**: $k$ must also be sufficiently large to ensure that the increase driven by the sigmoid function component overcomes any potential decrease from other components at any point.

### Optimizing Reserve Stability
Optimizing the reserve function $R(S)$ is crucial to ensuring a blockchain network's long-term sustainability and economic stability. The reserve represents the total amount of value (or collateral) held to back the tokens in circulation and reflects the overall liquidity of the token.

Here, we detail a methodological approach to achieve this using numerical integration and parameter optimization techniques.

#### Numerical Integration of the Reserve Function
The reserve function $R(S)$ is defined as the integral of the price function $P(S)$ from $0$ to $S$:

$$R(S) = \int_0^S P(S') dS'$$

where $P(S)$ is defined by our modified sigmoid curve. Given the complexity of the price function's integral, which involves parameters leading to a non-linear relationship within the sigmoid curve, an analytical solution is not feasible. Therefore, to compute
$R(S)$, we can employ numerical integration techniques that allow us to accurately evaluate the area under the curve, providing a real-time estimation of the total reserve for any given supply.

The $R(S)$ calculation offers a macroeconomic perspective, which indicates the overall health and robustness of the economic system.

#### Parameter Optimization for Stability
One could use a grid search method (or other optimization methods) to find the optimal set of parameters that minimize fluctuations in the reserve relative to the supply and thereby enhance the system's stability. This method systematically explores a range of values for parameters such as $𝐿_0$, $\alpha$, $\beta$, and $k$, computing the variance of the Reserve x Supply product for each combination. The objective is to identify the parameter set that results in the lowest variance, indicating a stable Reserve x Supply relationship, crucial for reducing economic volatility and ensuring predictable market behavior.

#### Analysis of Economic Stability
To assess the long-term stability of our token economy, we analyzed how the reserve $R$, the product $R \times S$, and the ratio $\frac{R}{S}$ evolve with an increasing supply of tokens $S$. This analysis is crucial in understanding the macroeconomic implications of our bonding curve and ensuring sustainable growth.

#### Reserve Growth
Our findings show that the reserve increases alongside the supply, a behavior that aligns with our economic model’s intent to enhance the collateral backing for each token issued. This proportional growth suggests a stable reserve-to-token issuance strategy, which is critical for maintaining investor confidence in the token's inherent value.

- **Total Value Analysis** ($R \times S$ vs. Supply)
The product of the reserve and supply, $R \times S$, indicative of the system’s total value, also shows an upward trend. As it scales, this is a positive indicator of value accrual in our ecosystem. However, the unbounded nature of this increase poses questions about potential speculative behavior and the model’s incentives for early versus late adopters.

- **Per-Token Reserve Ratio** ($\frac{R}{S}$ vs. Supply)
An upward trajectory in the ratio of the reserve to the supply, $\frac{R}{S}$, suggests that each additional unit of supply is backed by an increasingly larger reserve. While this indicates strong backing for the token, it also raises concerns about its scalability and accessibility. As token acquisition costs rise, it may deter new market participants, potentially reducing market liquidity and velocity.

Depending on a project's specific stability goals, additional metrics such as price elasticity, liquidity depth, volatility, and Gini coefficient might also be considered.

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## Implications of the Model

- **Flexibility**: The model adapts to the project's lifecycle, supporting initial growth with a lower entry price and sustaining investor interest as the project matures.
- **Incentive Alignment**: Early adopters are rewarded, but late entrants are also encouraged to participate due to the gradual increase in the price floor. In addition, various locking and vesting mechanisms can be built on top of a bonding curve, ensuring no unnecessary sell pressure and well-aligned interests.
- **Market Stability**: Adding $C$ provides a safety net against price crashes, ensuring a baseline return on investment.
- **Fee Capture Mechanism**: The bonding curve mechanism is created not only to manage the token supply but also to capture transaction fees when there is an arbitrage opportunity between the primary and secondary markets. These fees provide an additional source of revenue, making the platform financially sustainable. Optimized for trading, the bonding curve mechanism can result in Protocol-Owned Automated Market Makers ([poAMMs](https://medium.com/@demirelo/protocol-owned-automated-market-makers-0cfdb110f5a3)).
- **Security and Decentralization**: The gradual increase and decrease in token price promote the equitable distribution of tokens, which is necessary for a decentralized holder base and essential for the security of PoS networks. It prevents early adopter dominance, which is common in traditional bonding curves, and supports fair and widespread network validation.
- **Adaptability to Network Growth**: The modified sigmoid curve is designed to reflect the network's journey toward maturity. During the early phase of the curve, prices are set lower to encourage adoption and stake accumulation, which are essential for ensuring network security. As the network matures and the inflection point is crossed, the rate of price growth slows down. This aligns with a more robust and secure network that can handle an increased transactional throughput and complexity.
- **Dynamic Parameter Adjustment**: Unlike traditional bonding curves that may remain static after deployment, the parameters of the sigmoid model ($\alpha$, $\beta$, $C$, $k$, and $S_0$) are adjustable. This allows for post-deployment fine-tuning to accommodate changes in network usage, economic conditions, or community governance decisions.

## Interactive Visualization
To better understand the dynamics of the modified sigmoid bonding curve and explore the effects of varying its parameters, you can interact with a [live model](https://www.desmos.com/calculator/hedvwp3pmc) hosted on Desmos. This visualization allows you to adjust parameters in real time, providing a hands-on experience of how each parameter influences the shape and progression of the curve.

## Conclusion

Introducing a modified sigmoid bonding curve framework offers a practical approach to managing token issuance and economic design dynamics in new PoS L1 blockchain projects. By incorporating adaptive parameters that reflect the growth stages of these networks, this model facilitates a more controlled and predictable development of token economies. As a foundational approach, it can serve as a baseline for exploring further mathematical models tailored to the specific needs of blockchain ecosystems. This method represents a step toward refining token engineering practices, potentially aiding blockchain projects in achieving a balance between security, scalability, and decentralization.

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Special thanks to [Jeff Emmett ](https://twitter.com/jeffemmett)for his invaluable feedback and discussion.

You can find me on [Twitter](https://twitter.com/demirelo) and [Medium](https://medium.com/@demirelo), where I write about crypto-related topics.

*Disclaimer: This article is for informational purposes only and should not be considered financial, investment, or technical advice. Always do your own research and consult with a professional before making any financial decisions.*