# Augmented Token Bonding Curves
## Investment Model
This is a way to create an economically self-sustaining organization, by rewarding contributors with:
1) tokens whose value is tied to a good/service, in return for purchase of tokens
2) the development of said good/service, in return for funding of projects.
Anyone can contribute at any time, either as an investor or worker. Investors contribute funds to the organization's
1) Reserve Fund
and
2) Project Fund
with the funds split as follows:
### Goods
For organizations providing goods, investments are split in a $$\lambda:1-\lambda$$ $$\text{Reserve}:\text{Project}$$ ratio set by the organization (for example, $.1:.9$ is fitting for an early-stage startup expecting high returns, while $.5:.5$ would be fitting for a mature startup with positive cash flow).
### Services
For organizations providing services, investors pay a $$1-\lambda\%$$ $$\text{annual service fee}$$ on their purchased token holdings to the Project Fund (for example, $6\%$ annually is reasonable for an index fund).
## A Proof
The market drives the $\text{Reserve}:\text{Project}$ ratio which determines the reserve funds which determines the bonding curve which determines the investment returns, as follows:
Say an investor wants to invest an amount $I$ into the organization, so
$X = \lambda I =$ value of funds incoming to reserve.
Let
$R =$ current value of reserve funds
$C(R) =$ number of tokens in circulation.
**The price of tokens is always "how much you give" divided by "how much you get,"** so
$p(R) =\dfrac{dR}{dC} =$ buy price of token.
**The tokens are always backed by the reserve funds,** so they are redeemable at any time for
$s(R) =\dfrac{R}{C(R)} =$ sell price of token.
Now the investor pays $p(R)$ per token, thus receiving $X/p(R)$ tokens, and
$R+X =$ new value of reserve funds
$C(R+X) =C(R)+X/p(R) =$ new number of tokens in circulation.
What if the sell price immediately after purchase is equal to the buy price? Then we must have
$\begin{align*}
&S(R+X) =p(R)\\
&\frac{R+X}{C(R)+X/p(R)} =p(R)\\
&R+X =C(R)p(R)+X\\
&R/C(R) =p(R)\\
&s(R) =p(R) =s(R+X) =p(R+X)\\
\end{align*}$
So **there is no way to** make the sell price immediately after purchase equal to the buy price, unless the price is always constant.
And **there is no way to** make the sell price immediately after purchase greater than the buy price, because prices must always be backed by the reserve.
**Therefore** the sell price immediately after purchase is always less than than the buy price: $s(R+X)<p(R)$.
**Therefore** $\boxed{R <C(R)p(R)=C(R)\dfrac{dR}{dC}}$.
### Case I.
If $C(R)\dfrac{dR}{dC}=kR,k>1$, then this works:
$R=aC^k$ for any constants $a$ and $k>1$.
### Case II.
Otherwise, it is difficult to find a bonding curve function that works.
## Mathematical Formula
We use the Augmented Token Bonding Curve as proposed in [Bancor Protocol](chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/https://storage.googleapis.com/website-bancor/2018/04/01ba8253-bancor_protocol_whitepaper_en.pdf), described in [Deep Dive: Augmented Bonding Curves](https://medium.com/giveth/deep-dive-augmented-bonding-curves-3f1f7c1fa751), and proven [above](https://hackmd.io/zj__4bvISWSwlx_0XhQPcg?both#A-Proof).
Assume $R = aC^k$ where $k>1$ and $a = \dfrac{1}{V_0}$ are constants. Then the entire result can be derived from token price $p(R) = \dfrac{dR}{dC}$: $$p(R) = akC^{k-1} = \dfrac{kR}{C} = \dfrac{kR}{(RV_0)^{1/k}} = \dfrac{kR^{1-1/k}}{V_0^{1/k}}.$$
The buy price of tokens at the time of purchase is: $$p(R) = \dfrac{kR}{C(R)}.$$
The sell price of tokens immediately after purchase is: $$\begin{multline*}
s(R+X) = \dfrac{R+X}{C(R)+X/p(R)} = \dfrac{R+X}{C(R)+XC(R)/(kR)}\\
= \dfrac{kR(R+X)}{C(R)(kR+X)} = p(R)\dfrac{R+X}{kR+X}.\end{multline*}$$
So $\dfrac{p(R)}{k} =p(R)\dfrac{R+X}{k(R+X)}<s(R+X)<p(R).$
The price immediately after purchase is **always smaller than** the purchase price by a factor between 1 and k, and **the greater your investment the more equal prices you will get**.
## References
* [Bancor Protocol](chrome-extension://oemmndcbldboiebfnladdacbdfmadadm/https://storage.googleapis.com/website-bancor/2018/04/01ba8253-bancor_protocol_whitepaper_en.pdf)
* [Deep Dive: Augmented Bonding Curves](https://medium.com/giveth/deep-dive-augmented-bonding-curves-3f1f7c1fa751)
* [Rewriting the Story of Human Collaboration (or, an Introduction to Token Bonding and Curation Markets)](https://blog.goodaudience.com/rewriting-the-story-of-human-collaboration-c33a8a4cd5b8)
* Special thanks to Matt, James, Jake, Dan, Tom, Peter, Fil, Dan, Peter, and Sam

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