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:

Anchor tokens pertain to the tokens utilized for minting or purchasing project tokens, such as ETH or USDC.

Anchored Token TVL denotes the cumulative value of anchored tokens secured within the Bonding Curve contract.

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

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:

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

Parameter Setting Example:

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