# Appendix

### Mathematical Equations

#### Equation 1: Reserve Ratio Equation

Let:
$P$
- the price of one 1G$in Supported Currency; $R$ - the amount of Supported Currency in the reserve; $S$ - total G$ coins in circulation.
Then
$r$
- the reserve ratio - is defined as:
$r=\frac{R}{P \cdot S}$
In the beginning,
$r=1$
nd will be reduced daily according to the Expansion Rate. As can be seen from, given
$R$
and
$r$
the system has two degrees of freedom,
$P$
and
$S$
that will be calculated as described below.

#### Equation 2: Buy/Sell Function & Current Price(Bancor Formula for Automated Market Making)

Current Price function:

#### ​$Price = \frac{Reserve~Balance}{Smart~Token's~Total~Supply~ \cdot ~Reserve~Ratio}$

$Tokens~Issued = Supply \cdot ((1+\frac{Connected ~Tokens~Paid}{Balance}) ^ {CW} -1)$
Sell function:
$Connected~Tokens~Paid~Out = Balance \cdot (\sqrt[CW]{(1+\frac{Tokens~Destroyed}{Supply})} -1)$

#### Equation 3: Expansion Rate Formula

$q=\frac{r_t}{r_{t-1}}$

#### Equation 4: Mint G$from Declining Reserve Ratio - Once a day the reserve ratio is reduced while all other parameters are unchanged. The GoodDollar Reserve contract mints an amount of G$ to satisfy Equation 1
$P=\frac{R}{r \cdot S}$
. If P and R do not change and the daily change in r=𝚫r and the corresponding change in S is 𝚫S, then
$r s=(r+\Delta r)(S+\Delta S)$
and from here we can see that:
$\Delta S=S \cdot \frac{-\Delta r}{r+\Delta r}=S \cdot \frac{r_{t-1}-r_{t}}{r_{t}}$
From Equation 2 we get
$q=\frac{r_t}{r_{t-1}}$
and thus:
$S \frac{r_{t-1}-r_{t}}{r_{t}}=S \frac{1-q}{q}$

#### Equation 5: Mint G$from Deposit Formula - The number of new tokens based on the daily interest deposited in the Reserve. Z - Value deposited into the reserve E - Newly minted tokens The formula does not change the price or reserve ratio: $E = \frac {\frac{R + Z}{Rr} - P \cdot S}{P}$ ### Monetary Example This is an illustrative example with no relation to real parameters. • 10 Supporters have staked a total value of US$10 million in supported crypto to a third-party protocol
• Protocol annual interest rate of 10%
• Daily interest of US$2,736 • G$ Price = US$1 • GoodReserve = US$1,000,000
• Supply of G$= 1,250,000 • G$ Market Capitalization = US$1,250,000 • G$ Reserve Ratio = 80%
• Daily Expansion Rate = 1%
Every day, G$minting occurs via two methods: the decline in reserve ratio and from the deposit of interest to the GoodReserve. 1. 1. Minting derived from deposite of daily interest to the GoodReserve • Deposit of US$2,736 to GoodReserve
• Calculated as follows:
1. 1.
$S \cdot P=\frac{R}{Rr}$
2. 2.
(1,250,000 +X)*1=(1,000,000 + 2736)/0.8
3. 3.
1,250,000 + X = 1,002,736/0.8
4. 4.
1,250,000 + X = 1,253,420
5. 5.
X = 3,420
• 3,420 G$are minted • 2,736 G$ are allocated back to the 10 supporters
• 684 G$are distributed as basic income 2. 2. Mint from Reserve Ratio Decline 1. 1. Lower the Reserve Ratio from 80% to 79% 2. 2. Calculated as follows: 1. 1. $S \cdot P=\frac{R}{Rr}$ 2. 2. (1,253,420+X)*1= 1,002,736/0.79 3. 3. 1,253,420 + X = 1,269,286 4. 4. X = 15,866 3. 3. 15,866 G$ are distributed as UBI
In total, on that day 16,550 G\$ were distributed as basic income to Claimers.