# Pricing Formula

### Virtual Liquidity Pool

The trading of Power adopts AMM's constant product formula. Here, we introduce the concept of a 'virtual liquidity pool'. This liquidity pool doesn't actually exist; we use this concept to explain the calculation of the price of Power.

Suppose $Liquidity(ETH)$ and $Liquidity(Power)$ are the ETH amount and the Power amount in the Virtual Liquidity Pool respectively. We adopt the AMM's constant product formula: $Liquidity(ETH) \times Liquidity(Power) = k$ to determine the price of Power and the ETH cost.

### Total Supply, Initial Liquidity and $k$

The Topic Power price** **starts from **0.001 ETH**. The Subtopic Power price starts from **1 Topic Power**.

When a Topic is launching, its **Total Supply** is determined by the popularity of the Topic. Typically, it ranges between 5,000 and 50,000, with 10,000 being the most common.

Initially, the total supply of Power is all in the virtual liquidity pool, $Liquidity_{initial}(Power) = TotalSupply$

From this, we can derive that $Liquidity_{initial}(ETH)=0.001\times TotalSupply$

$k=Liquidity_{initial}(ETH)\times Liquidity_{initial}(Power)=0.001\times TotalSupply^2$

The price of Topic Power is $0.001 \times \left(\frac{TotalSupply}{Liquidity(Power)}\right)^2 \, \text{ETH}$

We set a Subtopic's total power supply to be 1/20 of its Topic's total power supply. For example, if the total supply of a topic power is 10,000, each of its Subtopic will have a 500 total supply.

Thus, the price of Subtopic Power is $0.0025 \times \left(\frac{TotalSupply}{Liquidity(Power)}\right)^2 \, \text{Topic Power}$

### Buying Process

Assuming a user purchases $x$ Power with $y$ ETH (or $x$ Subtopic Power with y Topic Power), based on the AMM's constant product formula $(Liquidity(ETH)+ y)\times (Liquidity(Power)-x) = k$, we'd get

$x=Liquidity(Power) - \frac{k}{Liquidity(ETH) + y}$, and the average price of this transaction would be $\frac{y}{x}$

### Selling Process

Similarly, if a user sell $x$ Power, and would get $y$ ETH (or sell $x$ Subtopic Power and get y Topic Power),

$y=Liquidity(ETH) - \frac{k}{Liquidity(Power) + x}$, and the average price of this transaction would be $\frac{y}{x}$

The above calculation doesn't include trading fee. For details on these fees, please refer to Trading Fee.

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