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optimal protocol fee

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tim
2026-01-06 15:06:30 -04:00
parent c0f0916969
commit b1245f27fd
1 changed files with 18 additions and 21 deletions
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doc/optimal_protocol_fee.md
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@@ -12,7 +12,7 @@ Let:
- $v$ = volume (exogenous, constant) - $v$ = volume (exogenous, constant)
- $F$ = total LP fee rate (exogenous, fixed) - $F$ = total LP fee rate (exogenous, fixed)
- $f$ = LP fee share (fraction of $F$ retained by LPs) - $f$ = LP fee share (fraction of $F$ retained by LPs)
- $g$ = protocol fee share (fraction of $F$ retained by protocol) - $g$ = protocol fee share (fraction of $F$ retained by the protocol)
- $r$ = short-term market rate (exogenous, includes risk premium) - $r$ = short-term market rate (exogenous, includes risk premium)
- $L$ = total value locked (endogenous) - $L$ = total value locked (endogenous)
@@ -25,6 +25,7 @@ In equilibrium, LP earnings equal the opportunity cost:
$$v f = r L$$ $$v f = r L$$
Thus: Thus:
$$L = \frac{v f}{r}$$ $$L = \frac{v f}{r}$$
The TVL adjusts until LPs are indifferent between deploying capital here versus the outside market rate $r$. The TVL adjusts until LPs are indifferent between deploying capital here versus the outside market rate $r$.
@@ -50,12 +51,12 @@ Setting $h'(g) = 0$ yields critical points at $g = 1$ and $g = \frac{1}{3}$.
Evaluating: Evaluating:
- $h(0) = 0$ - $h(0) = 0$
- $h(1) = 0$ - $h(1) = 0$
- $h(1/3) = \frac{1}{3} \cdot \left(\frac{2}{3}\right)^2 = \frac{4}{27}$ - $h\!\left(\frac{1}{3}\right) = \frac{1}{3} \cdot \left(\frac{2}{3}\right)^2 = \frac{4}{27}$
The maximum occurs at **$g^* = \frac{1}{3}$**, giving: The maximum occurs at **$g^* = \frac{1}{3}$**, giving:
| Quantity | Value | | Quantity | Value |
|----------|-------| |-----------------------|---------------------------|
| Protocol fee share | $g^* = \frac{1}{3}$ | | Protocol fee share | $g^* = \frac{1}{3}$ |
| LP fee share | $f^* = \frac{2}{3}$ | | LP fee share | $f^* = \frac{2}{3}$ |
| Equilibrium TVL | $L^* = \frac{2 v}{3 r}$ | | Equilibrium TVL | $L^* = \frac{2 v}{3 r}$ |
@@ -73,17 +74,13 @@ Under the assumptions:
the protocol’s optimal fee policy is: the protocol’s optimal fee policy is:
- **Protocol share of the fee pool:** - **Protocol share of the fee pool:**
\[ $\phi_P^* = \frac{1}{3}$
\phi_P^* = \frac{1}{3}
\]
- **LP share of the fee pool:** - **LP share of the fee pool:**
\[ $\phi_L^* = \frac{2}{3}$
\phi_L^* = \frac{2}{3}.
\]
Equivalently, in absolute terms: Equivalently, in absolute terms:
- **Optimal protocol fee rate:** \(F/3\), - **Optimal protocol fee rate:** $F/3$,
- **Optimal LP fee rate:** \(2F/3\). - **Optimal LP fee rate:** $2F/3$.
This fee split balances extracting revenue from trades against maintaining sufficiently attractive LP returns to support a large equilibrium TVL. This fee split balances extracting revenue from trades against maintaining sufficiently attractive LP returns to support a large equilibrium TVL.