Endogenous Token Supply Stability in Algorithmic Blockchain Protocols

Three flows govern token supply in blockchain protocols: issuance, burning, and staking lock-ups. We study this supply adjustment mechanism in isolation from price dynamics, treating market returns and on-chain activity as predetermined observables. This design permits analytical examination of supply-side feedbacks without modeling price endogeneity. A representative staker's utility maximization problem yields an equilibrium staking function with a linearly decreasing dependence on market return. From this microeconomic foundation we derive a first-order ordinary differential equation (ODE) for circulating supply with endogenous feedbacks from all three supply channels. The direct Lyapunov method applied to the linearized system yields a closed-form sufficient condition for local asymptotic stability. The condition requires that deflationary absorption (burning plus staking outflow) grows faster with circulating supply than issuance does. The model is then extended to a stochastic differential equation (SDE) via Ito's Lemma, yielding a critical diffusion threshold crit σ below which supply trajectories converge to equilibrium in mean-square. Calibration draws on 1,295 daily Ethereum observations (post-Merge, EIP-1559, 2022–2026). Four sequential OLS regressions are estimated, covering the issuance scale, burning sensitivity, equilibrium staking share, and mean-reversion speed, each with Newey–West HAC correction. The stability parameter min λ is computed analytically from the estimated partial derivatives, and its 95% confidence interval is obtained via block bootstrap (B = 2,000 iterations, block length = 30 days). The stability condition is satisfied empirically: min λ = 0.00274 day-1 (95% CI: [0.00175, 0.00373]; P( min λ > 0)= 100% by block bootstrap), with a characteristic supply half-life of approximately 253 days and a 13-fold stochastic safety margin ( / 0.077 emp crit σ σ ≈ ). Sub-period analysis confirms a structural break at the Shanghai upgrade (April 2023), after which min λ rises to 0.00492 day-1, consistent with the bidirectional mean-reversion mechanism. The stability parameter min λ and the critical diffusion threshold crit σ are proposed as standardized risk metrics for regulatory disclosure under EU Regulation 2023/1114 on Markets in Crypto-Assets (MiCA).