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tags: tokenomics
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# Discounting the Token Distribution

Every new protocol starts with a pie chart which breaks the token supply between investors, team and users. The distribution reflects a balance between the need of rewarding the directe contributors of the application, and accelerating adoption by incentivizing usage. The genesis token allocation makes transparent to all parts (funders, team, users, ...) the financial allocation and distribution through time, algorithmically enforced by the immutability of the token smart contract.

## Distribution schedule

There are multiple ways to look at the supply distribution. One way is to focus on the circulating supply that excludes reserves and time-locked tokens. Another is to adopt a protocol's life-cycle view and look at which rate the tokens are progressively emitted and to whom. In the very long-term, all tokens are distributed and only the maximum supply and fully diluted value matter.

The emission schedule is a key information beyond the allocation pie chart. Consider for instance the following two distributions:

| | Distribution 1 | Distribution 2 |
| -------- | --------: | --------: |
| Funders/team 1y locked | 100m | 100m |
| Community 1y locked | 100m | 0 |
| Community 2y locked | 0 | 100m |
| Community share | 1/2 | 1/2 |

The two distributions allocate *the same* quantity of tokens to the team/investors and community, yet the second distribution is less favorable to the latter as funders and team members are allowed to sell their allocation before users can. <!--The token market has one-way trades the first year, with the first group selling their allocation the second one. -->

## Discounted distribution

It is a simple financial principle that receiving an asset in the future is less valuable than receiving it in the present due to:
- the opportunity cost of not being able to obtain a return from investing the asset
- risk of selling at a lower price in the case the application does not deliver on its promise
- losing the option of selling early rather than lately.

Denoting $r>0$ the period discount rate and $\gamma = 1/(1+r) \in(0,1)$ the discount factor, \$1 obtained in one period is worth \$$\gamma<1$ now.

In previous example, the team/funders allocation is worth $\gamma 100$ whereas it is $\gamma 100$ for the community in distribution 1 and $\gamma^2 100$. It follows that distribution 2 actually gives the community less than one half of the total supply in discounted terms.

In a more general setting, $a_t$ is the allocation unlocked at date $t$ to a particular group (investors, team, users, ...), $s_t$ is all unlocked tokens composing the circulating supply . The discounted share of tokens distributed to this group is:
$$
\dfrac{\sum_{t=0}^T \gamma^t a_t}{\sum_{t=0}^T \gamma^t s_t}
$$

with $T$ the number of periods after which all tokens are distributed.

### Examples of discounted distribution

As an illustration, we present the token distributions of two financial applications, Euler Finance, a lending protocol and Curve an AMM protocol, both with and without flow discounting.


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The share distributed to the community (liquidity providers in Curve, borrowers in Euler) is higer in Curve than in Euler and the share given to investors is higher in Euler than in Curve. In both applications, the share given to the team and investors is larger in discounted termes due to shorter vesting periods compared to the period over which incentives are distributed.

This difference can be observed in the emission schedules of Euler (right-hand side figure) and Curve (left-hand side figure) in which distibution to investors is gradually unlocked before the distribution to the team and users:


*I thank Benoït Laurent for computing Curve's and Euler's discounted distributions and making the graphs.*

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Consider this second example

| | Distribution 1 | Distribution 2 |
| -------- | -------- | -------- |
|Investors/team <1y | 100m | 100m |
| Community <2y | 50m | 50m |
| Community >2y | 0 | 50m |
| Community share | 1/3 | 1/2 |