# Parameters for Exponential Curve

If exponential curve is chosen to establish the relationship between the price of tokens and the supply of tokens, then:

$$
p=ae^\frac{s}{b}
$$

| Parameter | Meaning      |
| --------- | ------------ |
| p         | Token Price  |
| s         | Token Supply |
| a         | Parameter 1  |
| b         | Parameter 2  |

The function provided allows for the calculation of the correlation between *Anchor Token TVL* and the *Token Supply*:

{% hint style="info" %}
Anchor tokens pertain to the tokens utilized for minting or purchasing project tokens, such as ETH or USDC.
{% endhint %}

{% hint style="info" %}
Anchored Token TVL denotes the cumulative value of anchored tokens secured within the Bonding Curve contract.
{% endhint %}

$$
TVL=\int\_{0}^{s}ae^\frac{s}{b}=abe^\frac{s}{b}-ab
$$

In the context of defining the curve within Burve protocol, specifying the values of **a** and **b** is essential.

When the supply is 0，it follows that **p=a**. In this scenario, the value of **a** represents the initial price of the token：

$$
a=p\_0
$$

The parameter **b** in the equation $$p=ae^s$$ signifies the scaling on the **X**-axis for the function, which often lacks practical significance. To address this, the Burve Protocol introduces the parameter **A** as a replacement.

The **A**-value is defined as the Anchor Token TVL divided by the increase in token price over the initial price.

The specific value of the Anchor Token TVL is already known:

$$
TVL=abe^\frac{s}{b}-ab
$$

The increase in token price from the initial price is a multiple of:

$$
\frac{p}{p\_0}=\frac{ae^\frac{s}{b}}{a}=e^\frac{s}{b}
$$

Thus **A**-value is：

$$
A=ab-\frac{ab}{e^\frac{s}{b}}=ab-\frac{ab}{\frac{TVL}{ab}+1}
$$

When TVL is relatively large:

$$
A\approx{ab}
$$

Then **b**-value is calculated with the Parameter **A**:

$$
b=\frac{A}{a}
$$

<mark style="color:purple;">**Parameter Setting Example:**</mark>

If the project aims for an Initial Token Price of 0.01USDC and plans for the token price to surge by 100 times when the TVL reaches 1,000,000USDC, the following settings can be implemented:

* Initial Token Price: 0.01 USDC
* A-value: 100,000 (derived from A = 1,000,000 / 100)

In this configuration, the relationship between the supply of tokens, price, and TVL would adhere to the specified conditions:

| Supply    | Price(USD)    | TVL(USD)      |
| --------- | ------------- | ------------- |
| 100,000   | 0.01105159459 | 1,051.594593  |
| 200,000   | 0.0122137743  | 2,213.774304  |
| 300,000   | 0.0134981682  | 3,498.168205  |
| 400,000   | 0.01491762827 | 4,917.628274  |
| 500,000   | 0.016486358   | 6,486.357997  |
| 600,000   | 0.01822005449 | 8,220.054489  |
| 700,000   | 0.02013606557 | 10,136.06557  |
| 800,000   | 0.02225356333 | 12,253.56333  |
| 900,000   | 0.02459373602 | 14,593.73602  |
| 1,000,000 | 0.02718       | 17,180        |
| 1,100,000 | 0.0300382341  | 20,038.2341   |
| 1,200,000 | 0.03319703856 | 23,197.03856  |
| 1,300,000 | 0.03668802118 | 26,688.02118  |
| 1,400,000 | 0.04054611365 | 30,546.11365  |
| 1,500,000 | 0.04480992104 | 34,809.92104  |
| 1,600,000 | 0.0495221081  | 39,522.1081   |
| 1,700,000 | 0.05472982621 | 44,729.82621  |
| 1,800,000 | 0.06048518514 | 50,485.18514  |
| 1,900,000 | 0.0668457745  | 56,845.7745   |
| 2,000,000 | 0.07387524    | 63,875.24     |
| 2,100,000 | 0.08164392029 | 71,643.92029  |
| 2,200,000 | 0.0902295508  | 80,229.5508   |
| 2,300,000 | 0.09971804157 | 89,718.04157  |
| 2,400,000 | 0.1102043369  | 100,204.3369  |
| 2,500,000 | 0.1217933654  | 111,793.3654  |
| 2,600,000 | 0.1346010898  | 124,601.0898  |
| 2,700,000 | 0.1487556676  | 138,755.6676  |
| 2,800,000 | 0.1643987332  | 154,398.7332  |
| 2,900,000 | 0.1816868151  | 171,686.8151  |
| 3,000,000 | 0.2007929023  | 190,792.9023  |
| 3,100,000 | 0.2219081753  | 211,908.1753  |
| 3,200,000 | 0.2452439191  | 235,243.9191  |
| 3,300,000 | 0.271033637   | 261,033.637   |
| 3,400,000 | 0.2995353877  | 289,535.3877  |
| 3,500,000 | 0.3310343671  | 321,034.3671  |
| 3,600,000 | 0.3658457621  | 355,845.7621  |
| 3,700,000 | 0.4043179046  | 394,317.9046  |
| 3,800,000 | 0.4468357569  | 436,835.7569  |
| 3,900,000 | 0.4938247634  | 483,824.7634  |
| 4,000,000 | 0.5457551085  | 535,755.1085  |
| 4,100,000 | 0.6031464206  | 593,146.4206  |
| 4,200,000 | 0.666572972   | 656,572.972   |
| 4,300,000 | 0.7366694253  | 726,669.4253  |
| 4,400,000 | 0.8141371837  | 804,137.1837  |
| 4,500,000 | 0.8997514097  | 889,751.4097  |
| 4,600,000 | 0.9943687815  | 984,368.7815  |
| 4,700,000 | 1.098936065   | 1,088,936.065 |
| 4,800,000 | 1.214499587   | 1,204,499.587 |
| 4,900,000 | 1.342215707   | 1,332,215.707 |
| 5,000,000 | 1.483362385   | 1,473,362.385 |
| 5,100,000 | 1.639351971   | 1,629,351.971 |
| 5,200,000 | 1.811745338   | 1,801,745.338 |