Avalanche Economic Model Specification

Overview

The Avalanche network is a complex economic system with multiple interacting subsystems: Staking Dynamics, Token Supply, Fee Dynamics, L1 Ecosystem, and Governance. This specification defines the mathematical framework for modeling these interactions using a differential specification approach.

State

Name	Symbol	Definition	Initial Value
Total Supply	$S$	Total AVAX tokens in existence	457,300,000
Circulating Supply	$S_c$	$S-S_s-S_l$	218,100,000
Staked Supply	$S_s$	$S_v+S_d$	239,200,000
Validator Stake	$S_v$	Total stake by validators	202,800,000
Delegator Stake	$S_d$	Total stake by delegators	36,300,000
Locked Supply	$S_l$	Tokens locked in vesting	36,669,600
Cumulative Burned	$B$	Total fees burned	2,730,000
Active Validators	$V$	Number of active validators	1,956
Primary Validators	$V_p$	Validators on Primary Network	1,491
L1 Validators	$V_{l1}$	Validators on L1s	465
Active L1s	$L$	Number of active L1s	66
Modern L1s	$L_m$	L1s using ACP-77	27
Legacy L1s	$L_l$	L1s using old model	39
Gas Price	$P_g$	Current gas price (nAVAX)	25
Excess Gas	$X_g$	Excess gas consumption	0
Daily Issuance	$I_d$	Daily token issuance	25,885
Daily Burning	$B_d$	Daily token burning	750
Staking APR	$r_s$	Average staking return	0.0613
Avg Stake Duration	$D_s$	Average staking period (days)	180
Delegation Utilization	$U_d$	Delegation capacity used	0.356
L1 Continuous Fees	$F_{l1}$	Daily L1 fees (AVAX)	20.57

Parameters

Name	Symbol	Definition	Initial Value
Max Supply Cap	$S_{max}$	Maximum AVAX supply	720,000,000
Min Validator Stake	$s_{v,min}$	Minimum validator stake	2,000
Max Validator Stake	$s_{v,max}$	Maximum validator stake	3,000,000
Min Delegator Stake	$s_{d,min}$	Minimum delegator stake	25
Min Stake Duration	$D_{min}$	Minimum staking period	14 days
Max Stake Duration	$D_{max}$	Maximum staking period	365 days
Max Delegation Weight	$w_{max}$	Max delegation per validator	5×
Min Consumption Rate	$c_{min}$	Minimum reward rate	0.10
Max Consumption Rate	$c_{max}$	Maximum reward rate	0.12
Minting Period	$T_m$	Annual minting period	365 days
Block Time	$\tau$	Time between blocks	2 seconds
Min Gas Price	$P_{g,min}$	Minimum gas price	1 nAVAX
Gas Update Constant	$K_g$	Gas price update constant	97,000
Target Gas/Second	$G_{target}$	Target gas consumption	15,000,000
L1 Base Fee Rate	$F_{base}$	Base L1 fee rate	512 nAVAX/s
L1 Target Capacity	$C_{l1}$	Target L1 validators	10,000
L1 Max Capacity	$C_{l1,max}$	Maximum L1 validators	20,000
Validator Restaking Rate	${\rho }_v$	Validator reward restaking	0.70
Delegator Restaking Rate	${\rho }_d$	Delegator reward restaking	0.50
Uptime Requirement	$u_{min}$	Minimum validator uptime	0.80
Delegation Fee	$f_d$	Average delegation fee	0.071

Mechanisms

Staking Subsystem

Stake

A token holder

$i$ can stake tokens by locking

$s_i\ge s_{min}$ for duration

$D_i\in [D_{min},D_{max}]$:

For validators:

$$S_v^{+}=S_v+s_i\text{ where }{s}_i\ge s_{v,min}$$

$$V^{+}=V+1$$

For delegators choosing validator

$j$:

$$S_d^{+}=S_d+s_i\text{ where }{s}_i\ge s_{d,min}$$

$$\text{subject to: }\sum _{k\in {\mathcal{D}}_j}{s}_k\le w_{max}\cdot s_j$$

where

${\mathcal{D}}_j$ is the set of delegators to validator

$j$.

The circulating supply decreases:

$$S_c^{+}=S_c-s_i$$

Claim Rewards

At the end of staking period, staker

$i$ receives rewards:

$$r_i=\frac{(S_{max}-S)\cdot s_i}{S}\cdot \frac{D_i}{T_m}\cdot ECR(D_i)$$

where the Effective Consumption Rate is:

$$ECR(D_i)=c_{min}+(c_{max}-c_{min})\cdot \frac{D_i}{D_{max}}$$

For delegators, the net reward after commission:

$$r_{d,i}=r_i\cdot (1-f_{d,j})$$

where

$f_{d,j}$ is the commission rate of validator

$j$.

Unstake

When stake period ends, tokens return to circulation:

$$S_s^{+}=S_s-s_i$$

$$S_c^{+}=S_c+s_i+r_i$$

If restaking:

$$S_s^{+}=S_s-s_i+\rho \cdot r_i$$

$$S_c^{+}=S_c+s_i+(1-\rho )\cdot r_i$$

where

$\rho \in \{{\rho }_v,{\rho }_d\}$ depending on staker type.

Token Supply Subsystem

Issuance (Synthetic State Change)

Per epoch (day), new tokens are minted according to staking participation:

$$I_d=\frac{(S_{max}-S)\cdot S_s}{S}\cdot \frac{1}{T_m}\cdot \overline{ECR}$$

where

$\overline{ECR}$ is the stake-weighted average ECR:

$$\overline{ECR}=\frac{\sum _{i\in \mathcal{S}}{s}_i\cdot ECR(D_i)}{\sum _{i\in \mathcal{S}}{s}_i}$$

The total supply increases:

$$S^{+}=S+I_d$$

This is allocated proportionally to all stakers but not realized until claim.

Burning

All transaction fees are burned permanently:

$$B^{+}=B+B_d$$

$$S^{+}=S-B_d$$

$$S_c^{+}=S_c-B_d$$

where daily burning is:

$$B_d=\sum _{t\in \mathcal{T}}fee(t)$$

Fee Dynamics Subsystem

Gas Price Update

The gas price adjusts based on network congestion:

$$P_g^{+}=P_{g,min}\cdot e^{X_g/K_g}$$

where excess gas consumption evolves as:

$$X_g^{+}=max(0,X_g+G_{actual}-G_{target})$$

Transaction Fee

For a transaction

$t$ consuming resources:

$$fee(t)=P_g\cdot (b_t+1000\cdot r_t+1000\cdot w_t+4\cdot c_t)$$

where:

• $b_t$: bandwidth (bytes)
• $r_t$: read operations
• $w_t$: write operations
• $c_t$: compute (microseconds)

L1 Ecosystem Subsystem

Create L1

An operator can create a new L1 by recruiting validators:

$$L^{+}=L+1$$

$$V_{l1}^{+}=V_{l1}+n_{l1}$$

where

$n_{l1}$ is the number of validators for the new L1.

L1 Continuous Fees

Each L1 validator pays continuous fees:

$$F_{l1,i}=F_{base}\cdot e^{(V_{l1}-C_{l1})/K_{l1}}$$

Total daily L1 fees burned:

$$F_{l1}=\sum _{i\in \mathcal{L}}{n}_i\cdot F_{l1,i}\cdot 86400$$

where

$n_i$ is the validator count for L1

$i$.

Abandon L1

If an L1 cannot pay fees:

$$L^{+}=L-1$$

$$V_{l1}^{+}=V_{l1}-n_{l1}$$

Validators return to available pool or leave system.

Governance Subsystem

Parameter Update

Governance can update parameters within bounds:

For parameter

$\theta$ with proposed change

$\mathrm{\Delta }\theta$:

If time since last change

$t_{last}>t_{min}$:

$${\theta }^{+}=\theta +min(\mathrm{\Delta }\theta ,h(\theta )\cdot \theta )$$

where

$h(\theta )$ is the hysteresis function limiting change magnitude:

$$h(\theta )=\{\begin{array}{ll}0.20& \text{if }{t}_{last}>180\text{ days}\\ 0.10& \text{if }{t}_{last}>90\text{ days}\\ 0.05& \text{if }{t}_{last}>30\text{ days}\\ 0& \text{otherwise}\end{array}$$

System Dynamics

The continuous-time evolution of the system follows:

Supply Dynamics

$$\frac{dS}{dt}={\mathrm{\Psi }}_i(S_s,S,D_s)-{\mathrm{\Psi }}_b(F)$$

where:

• ${\mathrm{\Psi }}_i$: Issuance rate function
• ${\mathrm{\Psi }}_b$: Burning rate function

Staking Dynamics

$$\frac{d{S}_s}{dt}={\varphi }_s(S_c,r_s)-{\psi }_u(S_s,D_s)+{\eta }_r(I_d,\rho )$$

where:

• ${\varphi }_s$: Staking inflow rate
• ${\psi }_u$: Unstaking outflow rate
• ${\eta }_r$: Restaking rate

Fee Market Dynamics

$$\frac{d{P}_g}{dt}=\mu \cdot P_g\cdot [\exp \left(\frac{X_g}{K_g}\right)-1]$$

$$\frac{d{X}_g}{dt}=G(t)-\delta \cdot X_g$$

where:

• $G(t)$: Gas consumption rate
• $\delta$: Decay parameter
• $\mu$: Adjustment speed

L1 Ecosystem Dynamics

$$\frac{dL}{dt}={\lambda }_{create}(P_g,F_{l1})-{\lambda }_{abandon}(L,F_{l1})$$

$$\frac{d{V}_{l1}}{dt}=n_{avg}\cdot \frac{dL}{dt}+{\theta }_{convert}(V_p,V_{l1})$$

where:

• ${\lambda }_{create}$: L1 creation rate
• ${\lambda }_{abandon}$: L1 abandonment rate
• ${\theta }_{convert}$: Validator conversion rate

Equilibrium Analysis

The system reaches equilibrium when all derivatives equal zero:

Staking Equilibrium

At equilibrium, staking inflows equal outflows plus restaking:

$${\varphi }_s(S_c^{\ast },r_s^{\ast })={\psi }_u(S_s^{\ast },D_s^{\ast })-{\eta }_r(I_d^{\ast },\rho )$$

Empirically observed:

$S_s^{\ast }/S^{\ast }\approx 0.523$

Supply Equilibrium

Issuance equals burning at equilibrium:

$${\mathrm{\Psi }}_i(S_s^{\ast },S^{\ast },D_s^{\ast })={\mathrm{\Psi }}_b(F^{\ast })$$

Fee Market Equilibrium

Gas price stabilizes when:

$$X_g^{\ast }=K_g\cdot \ln \left(\frac{P_g^{\ast }}{P_{g,min}}\right)$$

Stability Conditions

The system is locally stable if the Jacobian eigenvalues have negative real parts:

$$J=\left[\begin{array}{ccc}\frac{\partial \dot{S}}{\partial S}& \frac{\partial \dot{S}}{\partial S_s}& \cdots \\ \frac{\partial \dot{S_s}}{\partial S}& \frac{\partial \dot{S_s}}{\partial S_s}& \cdots \\ ⋮& ⋮& \ddots \end{array}\right]$$

Key stability requirements:

1. $\frac{\partial {\mathrm{\Psi }}_i}{\partial S}<0$ (issuance decreases with supply)
2. $\frac{\partial {\varphi }_s}{\partial r_s}>0$ (staking increases with APR)
3. $\frac{\partial {\mathrm{\Psi }}_b}{\partial P_g}>0$ (burning increases with gas price)

Implementation Notes

1. Time Steps: Use daily epochs for staking calculations, block-level for fees
2. Numerical Method: 4th-order Runge-Kutta for differential equations
3. Conservation: Ensure
   $S=S_c+S_s+S_l$ at all times
4. Bounds: Enforce all parameter constraints and non-negativity

References

• Avalanche Consensus Whitepaper
• Avalanche Platform Specification
• ACP-77: Reinventing Subnets
• ACP-103: Dynamic Fees
• cadCAD Documentation