# Appendix

### Mathematical Equations

**Equation 1: Reserve Ratio Equation **

**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.

Current Price function:

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

Buy function:

$Tokens~Issued = Supply \cdot ((1+\frac{Connected ~Tokens~Paid}{Balance}) ^ {CW} -1)$** **

Sell function:

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

#### Equation 3: *Expansion Rate* Formula

*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. **

**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. **

**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.**

Minting derived from deposite of daily interest to the GoodReserve

Deposit of US$2,736 to GoodReserve

Calculated as follows:

$S \cdot P=\frac{R}{Rr}$

(1,250,000 +X)*1=(1,000,000 + 2736)/0.8

1,250,000 + X = 1,002,736/0.8

1,250,000 + X = 1,253,420

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

Mint from Reserve Ratio Decline

Lower the Reserve Ratio from 80% to 79%

Calculated as follows:

$S \cdot P=\frac{R}{Rr}$

(1,253,420+X)*1= 1,002,736/0.79

1,253,420 + X = 1,269,286

X = 15,866

15,866 G$ are distributed as UBI

In total, on that day 16,550 G$ were distributed as basic income to Claimers.

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