Reorg and cleanup of economics section of docs (#14868) (#14889)

docs/src/inflation/adjusted_staking_yield.md

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+---
+title: Adjusted Staking Yield
+---
+
+
+### Token Dilution
+
+Similarly we can look at the expected *Staked Dilution* (i.e. *Adjusted Staking Yield*) and *Un-staked Dilution* as previously defined. Again, *dilution* in this context is defined as the change in fractional representation (i.e. ownership) of a set of tokens within a larger set. In this sense, dilution can be a positive value: an increase in fractional ownership (staked dilution / *Adjusted Staking Yield*), or a negative value: a decrease in fractional ownership (un-staked dilution).
+
+We are interested in the relative change in ownership of staked vs un-staked tokens as the overall token pool increases with inflation issuance. As discussed, this issuance is distributed only to staked token holders, increasing the staked token fractional representation of the *Total Current Supply*.
+
+Continuing with the same *Inflation Schedule* parameters as above, we see the fraction of staked supply grow as shown below.
+
+![](/img/p_ex_staked_supply_w_range_initial_stake.png)
+
+Due to this relative change in representation, the proportion of stake of any token holder will also change as a function of the *Inflation Schedule* and the proportion of all tokens that are staked.
+
+Of initial interest, however, is the *dilution of **un-staked** tokens*, or $D_{us}$. In the case of un-staked tokens, token dilution is only a function of the *Inflation Schedule* because the amount of un-staked tokens doesn't change over time.
+
+This can be seen by explicitly calculating un-staked dilution as $D_{us}$. The un-staked proportion of the token pool at time $t$ is $P_{us}(t_{N})$ and $I_{t}$ is the incremental inflation rate applied between any two consecutive time points. $SOL_{us}(t)$ and $SOL_{total}(t)$ is the amount of un-staked and total SOL on the network, respectively, at time $t$. Therefore $P_{us}(t) = SOL_{us}(t)/SOL_{total}(t)$.
+
+$$
+\begin{aligned}
+	D_{us} &= \left( \frac{P_{us}(t_{1}) - P_{us}(t_{0})}{P_{us}(t_{0})} \right)\\
+		&= \left( \frac{ \left( \frac{SOL_{us}(t_{2})}{SOL_{total}(t_{2})} \right) - \left( \frac{SOL_{us}(t_{1})}{SOL_{total}(t_{1})} \right)}{ \left( \frac{SOL_{us}(t_{1})}{SOL_{total}(t_{1})} \right) } \right)\\
+
+\end{aligned}
+$$
+
+However, because inflation issuance only increases the total amount and the un-staked supply doesn't change:
+
+$$
+\begin{aligned}
+	SOL_{us}(t_2) &= SOL_{us}(t_1)\\
+	SOL_{total}(t_2) &= SOL_{total}(t_1)\times (1 + I_{t_1})\\
+\end{aligned}
+
+$$
+
+So $D_{us}$ becomes:
+
+$$
+\begin{aligned}
+	D_{us} &= \left( \frac{ \left( \frac{SOL_{us}(t_{1})}{SOL_{total}(t_{1})\times (1 + I_{1})} \right) - \left( \frac{SOL_{us}(t_{1})}{SOL_{total}(t_{1})} \right)}{ \left( \frac{SOL_{us}(t_{1})}{SOL_{total}(t_{1})} \right) } \right)\\
+	D_{us} &= \frac{1}{(1 + I_{1})} - 1\\
+\end{aligned}
+$$
+
+Or generally, dilution for un-staked tokens over any time frame undergoing inflation $I$:
+
+$$
+D_{us} = -\frac{I}{I + 1} \\
+$$
+
+So as guessed, this dilution is independent of the total proportion of staked tokens and only depends on inflation rate. This can be seen with our example *Inflation Schedule* here:
+
+
+![p_ex_unstaked_dilution](/img/p_ex_unstaked_dilution.png)
+
+
+### Estimated Adjusted Staked Yield
+
+We can do a similar calculation to determine the *dilution* of staked token holders, or as we've defined here as the ***Adjusted Staked Yield***, keeping in mind that dilution in this context is an *increase* in proportional ownership over time.  We'll use the terminology *Adjusted Staked Yield* to avoid confusion going forward.
+
+To see the functional form, we calculate, $Y_{adj}$, or the *Adjusted Staked Yield* (to be compared to *D_{us}* the dilution of un-staked tokens above), where $P_{s}(t)$ is the staked proportion of token pool at time $t$ and $I_{t}$ is the incremental inflation rate applied between any two consecutive time points. The definition of $Y_{adj}$ is therefore:
+
+$$
+	Y_{adj} = \frac{P_s(t_2) - P_s(t_1)}{P_s(t_1)}\\
+$$
+
+As seen in the plot above, the proportion of staked tokens increases with inflation issuance. Letting $SOL_s(t)$ and $SOL_{\text{total}}(t)$ represent the amount of staked and total SOL at time $t$ respectively:
+
+$$
+	P_s(t_2) = \frac{SOL_s(t_1) + SOL_{\text{total}}(t_1)\times I(t_1)}{SOL_{\text{total}}(t_1)\times (1 + I(t_1))}\\
+$$
+
+Where $SOL_{\text{total}}(t_1)\times I(t_1)$ is the additional inflation issuance added to the staked token pool. Now we can write $Y_{adj}$ in common terms $t_1 = t$:
+
+$$
+\begin{aligned}
+Y_{adj} &= \frac{\frac{SOL_s(t) + SOL_{\text{total}}(t)\times I(t)}{SOL_{\text{total}}(t)\times (1 + I(t))} - \frac{SOL_s(t)}{SOL_{\text{total}}(t)} }{ \frac{SOL_s(t)}{SOL_{\text{total}}(t)} }  \\
+	&= \frac{ SOL_{\text{total}}(t)\times (SOL_s(t) + SOL_{\text{total}}(t)\times I(t)) }{ SOL_s(t)\times SOL_{\text{total}}\times (1 + I(t)) } -1 \\
+\end{aligned}
+$$
+
+which simplifies to:
+
+$$
+Y_{adj} =  \frac{ 1 + I(t)/P_s(t) }{ 1 + I(t) } - 1\\
+$$
+
+So we see that the *Adjusted Staked Yield* a function of the inflation rate and the percent of staked tokens on the network. We can see this plotted for various staking fractions here:
+
+
+![p_ex_adjusted_staked_yields](/img/p_ex_adjusted_staked_yields.png)
+
+
+It is also clear that in all cases, dilution of un-staked tokens $>$ adjusted staked yield (i.e. dilution of staked tokens). Explicitly we can look at the *relative dilution of un-staked tokens to staked tokens:* $D_{us}/Y_{adj}$. Here the relationship to inflation drops out and the relative dilution, i.e. the impact of staking tokens vs not staking tokens, is purely a function of the % of the total token supply staked. From above
+
+$$
+\begin{aligned}
+Y_{adj} &=  \frac{ 1 + I/P_s }{ 1 + I } - 1,~\text{and}\\
+D_{us} &= -\frac{I}{I + 1},~\text{so} \\
+\frac{D_{us}}{Y_{adj}} &= \frac{ \frac{I}{I + 1} }{ \frac{ 1 + I/P_s }{ 1 + I } - 1 } \\
+\end{aligned}
+$$
+
+which simplifies as,
+
+$$
+	\begin{aligned}
+	\frac{D_{us}}{Y_{adj}} &= \frac{ I }{ 1 + \frac{I}{P_s} - (1 + I)}\\
+	&= \frac{ I }{ \frac{I}{P_s} - I}\\
+	\frac{D_{us}}{Y_{adj}}&= \frac{ P_s }{ 1 - P_s}\\
+	\end{aligned}
+$$
+
+Where we can see a primary dependence of the relative dilution of un-staked tokens to staked tokens is on the function of the proportion of total tokens staked. As shown above, the proportion of total tokens staked changes over time (i.e. $P_s = P_s(t)$ due to the re-staking of inflation issuance thus we see relative dilution grow over time as:
+
+![p_ex_relative_dilution](/img/p_ex_relative_dilution.png)
+
+As might be intuitive, as the total fraction of staked tokens increases the relative dilution of un-staked tokens grows dramatically. E.g. with $80\%$ of the network tokens staked, an un-staked token holder will experience ~$400\%$ more dilution than a staked holder.
+
+Again, this represents the change in fractional change in ownership of staked tokens and illustrates the built-in incentive for token holder to stake their tokens to earn *Staked Yield* and avoid *Un-staked Dilution*.

docs/src/inflation/inflation_schedule.md

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+---
+title: Solana's Proposed Inflation Schedule
+---
+
+As mentioned above, the network's *Inflation Schedule* is uniquely described by three parameters: *Initial Inflation Rate*, *Dis-inflation Rate* and *Long-term Inflation Rate*. When considering these numbers, there are many factors to take into account:
+
+- A large portion of the SOL issued via inflation will be distributed to stake-holders in proportion to the SOL they have staked. We want to ensure that the *Inflation Schedule* design results in reasonable *Staking Yields* for token holders who delegate SOL and for validation service providers (via commissions taken from *Staking Yields*).
+- The primary driver of *Staked Yield* is the amount of SOL staked divided by the total amount of SOL (% of total SOL staked). Therefore the distribution and delegation of tokens across validators are important factors to understand when determining initial inflation parameters.
+- [Yield throttling](https://forums.solana.com/t/validator-yield-throttling-proposal-discussion/855/5) is a mechanism currently under consideration that would impact *staking-yields*. **This is not taken into consideration in the discussion here or the modeling below.**
+- Overall token issuance - i.e. what do we expect the Current Total Supply to be in 10 years, or 20 years?
+- Long-term, steady-state inflation is an important consideration not only for sustainable support for the validator ecosystem and the Solana Foundation grant programs, but also should be tuned in consideration with expected token losses and burning over time.
+- The rate at which we expect network usage to grow, as a consideration to the dis-inflationary rate. Over time, we plan for inflation to drop and expect that usage will grow.
+
+Based on these considerations and the community discussions following the initial [design](https://forums.solana.com/t/solana-inflation-design-overview/920), the Solana Foundation proposes the following Inflation Schedule parameters:
+
+> Initial Inflation Rate: $8\%$
+> Dis-inflation Rate: $-15\%$
+> Long-term Inflation Rate: $1.5\%$
+
+These parameters define the proposed *Inflation Schedule*. Below we show implications of these parameters. These plots only show the impact of inflation issuances given the Inflation Schedule as parameterized above. They *do not account* for other factors that may impact the Total Supply such as fee/rent burning, slashing or other unforeseen future token destruction events. Therefore, what is presented here is an **upper limit** on the amount of SOL issued via inflation.
+
+![](/img/p_inflation_schedule.png)
+
+In the above graph we see the annual inflation rate [$\%$] over time, given the inflation parameters proposed above.
+
+![](/img/p_total_supply.png)
+
+Similarly, here we see the *Total Current Supply* of SOL [MM] over time, assuming an initial *Total Current Supply* of `488,587,349 SOL` (i.e. for this example, taking the *Total Current Supply* as of `2020-01-25` and simulating inflation starting from that day).
+
+The expected Staking Yield and Adjusted Staking Yield metrics are then primarily a function of the % of total SOL staked on the network. Therefore we can we can estimate the *Staking Yield*, if we introduce an additional parameter *% of Staked SOL*:
+
+$$
+\%~\text{SOL Staked} = \frac{\text{Total SOL Staked}}{\text{Total Current Supply}}
+$$
+
+This parameter must be estimated because it is a dynamic property of the token holders and staking incentives. The values of *% of Staked SOL* presented here range from $60\% - 90\%$, which we feel covers the likely range we expect to observe, based on feedback from the investor and validator communities as well as what is observed on comparable Proof-of-Stake protocols.
+
+![](/img/p_ex_staked_yields.png)
+
+Again, the above shows an example *Staked Yield* that a staker might expect over time on the Solana network with the *Inflation Schedule* as specified. This is an idealized *Staked Yield* as it neglects validator uptime impact on rewards, validator commissions, potential yield throttling  and potential slashing incidents. It additionally ignores that *% of Staked SOL* is dynamic by design - the economic incentives set up by this *Inflation Schedule* are more clearly seen when *Token Dilution* is taken into account (see the **Adjusted Staking Yield** section below).

docs/src/inflation/terminology.md

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+---
+title: Terminology
+---
+
+
+Many terms are thrown around when discussing inflation and the related components (e.g. rewards/yield/interest), we try to define and clarify some commonly used concept here:
+
+### Total Current Supply [SOL]
+The total amount of tokens (locked or unlocked) that have been generated (via genesis block or protocol inflation) minus any tokens that have been burnt (via transaction fees or other mechanism) or slashed. At network launch, 500,000,000 SOL were instantiated in the genesis block. Since then the Total Current Supply has been reduced by the burning of transaction fees and a planned token reduction event. Solana’s *Total Current Supply* can be found at https://explorer.solana.com/supply
+
+
+### Inflation Rate [%]
+The Solana protocol will automatically create new tokens on a predetermined inflation schedule (discussed below). The *Inflation Rate [%]* is the annualized growth rate of the *Total Current Supply* at any point in time.
+
+
+### Inflation Schedule
+A deterministic description of token issuance over time. The Solana Foundation is proposing a dis-inflationary *Inflation Schedule*. I.e. Inflation starts at its highest value, the rate reduces over time until stabilizing at a predetermined long-term inflation rate (see discussion below). This schedule is completely and uniquely parameterized by three numbers:
+
+- **Initial Inflation Rate [%]**: The starting *Inflation Rate* for when inflation is first enabled. Token issuance rate can only decrease from this point.
+- **Dis-inflation Rate [%]**: The rate at which the *Inflation Rate* is reduced.
+- **Long-term Inflation Rate [%]**: The stable, long-term *Inflation Rate* to be expected.
+
+
+### Effective Inflation Rate [%]
+
+The inflation rate actually observed on the Solana network after accounting for other factors that might decrease the *Total Current Supply*. Note that it is not possible for tokens to be created outside of what is described by the *Inflation Schedule*.
+
+- While the *Inflation Schedule* determines how the protocol issues SOL, this neglects the concurrent elimination of tokens in the ecosystem due to various factors. The primary token burning mechanism is the burning of a portion of each transaction fee. While $100\%$ of each transaction fee is currently being destroyed, it is planned on reducing this burn rate to $50\%$ of each transaction fee, with the remaining fee to be retained by the validator that processes the transaction.
+- Additional factors such as loss of private keys and slashing events should also be considered in a holistic analysis of the *Effective Inflation Rate*. For example, it’s estimated that $10-20\%$ of all BTC have been lost and are unrecoverable and that networks may experience similar yearly losses at the rate of $1-2\%$.
+
+### Staking Yield [%]
+The rate of return (aka *interest*) earned on SOL staked on the network. It is often quoted as an annualized rate (e.g. "the network *staking yield* is currently $10\%$ per year").
+- *Staking yield* is of great interest to validators and token-holders holders who wish to delegate their tokens to avoid token dilution due to inflation (the extent of which is discussed below).
+- $100\%$ of inflationary issuances are to be distributed to staked token-holders in proportion to the SOL they have staked with validators.
+    - There may be future consideration for an additional split of inflation issuance with the introduction of *Archivers* into the economy. *Archivers* are network participants who provide a decentralized storage service and should also be incentivized with token distribution from inflation issuances for this service.
+	- Similarly, early designs specified a fixed percentage of inflationary issuance to be delivered to the Foundation treasury for operational expenses and future grants. However, inflation will be launching without any portion allocated to the Foundation.
+- *Staking yield* can be calculated from the *Inflation Schedule* along with the fraction of the *Total Current Supply* that is staked at any given time. Since inflation interest is split between staked-token holders, the Foundation pool and, potentially, *Archivers* - only the portion delivered to validators should be considered in the calculation of *Staking Yield*. Currently, the only split discussed is the $5\%$ to the Foundation, which is likely to be directly staked upon receipt, so not impacting the yield calculation. Below, this is specified by setting the 'Fraction to Validators' to $100\%$. The explicit relationship is given by:
+
+$$
+\begin{aligned}
+\text{Staking Yield} &= \text{Inflation Rate}\times \text{Fraction to Validators}\times \left( \frac{1}{\%~\text{SOL Staked}} \right) \\
+\text{where:}\\
+\%~\text{SOL Staked} &= \frac{\text{Total SOL Staked}}{\text{Total Current Supply}}
+\end{aligned}
+$$
+
+
+### Token Dilution [%]
+
+Dilution is defined here as the change in proportional representation of a set of tokens within a larger set due to the introduction of new tokens. In practical terms, we discuss the dilution of staked or un-staked tokens due to the introduction and distribution of inflation issuance across the network. As will be shown below, while dilution impacts every token holder, the *relative* dilution between staked and un-staked tokens should be the primary concern to un-staked token holders. Staking tokens, which will receive their proportional distribution of inflation issuance, should assuage any dilution concerns for staked token holders. I.e. dilution from 'inflation' is offset by the distribution of new tokens to staked token holders, nullifying the 'dilutive' effects of the inflation for that group.
+
+### Adjusted Staking Yield [%]
+
+A complete appraisal of earning potential from staking tokens should take into account staked *Token Dilution* and its impact on the *Staking Yield*. For this, we define the *Adjusted Staking Yield* as the change in fractional token supply ownership of staked tokens due to the distribution of inflation issuance. I.e. the positive dilutive effects of inflation.