Appendix

Mathematical Equations

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:

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

$r=1$

$R$

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

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

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%

- 1.Minting derived from deposite of daily interest to the GoodReserve
- Deposit of US$2,736 to GoodReserve
- Calculated as follows:
- 1.$S \cdot P=\frac{R}{Rr}$
- 2.(1,250,000 +X)*1=(1,000,000 + 2736)/0.8
- 3.1,250,000 + X = 1,002,736/0.8
- 4.1,250,000 + X = 1,253,420
- 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.Mint from Reserve Ratio Decline
- 1.Lower the Reserve Ratio from 80% to 79%
- 2.Calculated as follows:
- 1.$S \cdot P=\frac{R}{Rr}$
- 2.(1,253,420+X)*1= 1,002,736/0.79
- 3.1,253,420 + X = 1,269,286
- 4.X = 15,866

- 3.15,866 G$ are distributed as UBI

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

Last modified 15d ago

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