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Appendix

Mathematical Equations

Equation 1: Reserve Ratio Equation

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

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

Current Price function:

​
Price=Reserve BalanceSmart Token′s Total Supply ⋅ Reserve RatioPrice = \frac{Reserve~Balance}{Smart~Token's~Total~Supply~ \cdot ~Reserve~Ratio}

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

​

Equation 3: Expansion Rate Formula

​
q=rtrt−1q=\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=Rr⋅SP=\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
rs=(r+Δr)(S+ΔS)r s=(r+\Delta r)(S+\Delta S)
 and from here we can see that:
​
ΔS=S⋅−Δrr+Δr=S⋅rt−1−rtrt\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=rtrt−1q=\frac{r_t}{r_{t-1}}
 and thus:
Srt−1−rtrt=S1−qqS \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=R+ZRr−P⋅SPE = \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$2,736 to GoodReserve
    • Calculated as follows:
      1. 1.
        ​
        S⋅P=RRrS \cdot P=\frac{R}{Rr}
      2. 2.
        (1,250,000 +X)*1=(1,000,000 + 2736)/0.8
      3. 3.
        1,250,000 + X = 1,002,736/0.8
      4. 4.
        1,250,000 + X = 1,253,420
      5. 5.
        X = 3,420
    • 3,420 G$ 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⋅P=RRrS \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.
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