Issuance Equilibrium Explorer

Three properties of this model

• The crossing sets the steady state. Two curves that cross R(d) at the same point have the same steady state, whatever their peaks look like. Their shape away from the crossing decides how far that steady state moves when demand moves (the range row).
• Stakers' dilution-adjusted all-in yield is the same for every curve, by assumption. The model has one representative investor whose required return is fixed, so this yield equals 1/β̃ − 1 whatever the curve (Jermann's note, pp.3 and 6). Curves change how much ETH is staked and how fast supply grows. With no taxes, costs or staker types, the model cannot say how a particular staker fares.
• A zero point bounds staking only under a condition. It caps the steady state when the required nominal CL yield there is positive, that is, when fee income alone does not meet stakers' required return: at these settings, a zero point above . Below that, a curve can settle past its zero point. A floor removes the bound only if the yield it pays at the zero point sits above the required nominal CL yield there; for an issuance floor, that yield is the floor divided by the zero point.

What this page cannot show

• Who stakes. The model has one representative staker, so it says nothing about solo stakers versus pools, a frequent objection in the EIP-8363 discussion. Work that separates staker types disagrees. Beccuti et al.'s model predicts that 'a reduction in issuance is likely to crowd out solo stakers' (p.1); EIP-8363 argues that today's curve forces them out (p.2).
• The path. Steady state only: no transition period, no exit queue and, as in Jermann's note, no L2 and no uncertainty (note p.2). The page holds the effective discount factor β̃ = βγ (Jermann's notation) fixed across curves.
• Security beyond one proxy. Staked ETH, which sets the cost of attack for an adversary who must buy stake, is the only security measure in ETH on the page; the value-of-staked-ETH row gives its model-implied counterpart in real terms.
• Two channels EIP-8363 argues from. Its case also rests on concentration of stake and on liquid staking tokens displacing ETH (EIP-8363, pp.2–3). The model has neither. It shows the curve's effect on staked ETH and on dilution, which the EIP also cites (p.3), but not those two.
• Staking past a 50% zero point. At every setting this page allows, stakers' required nominal CL yield at 50% staked is positive, so without a floor that pays more than that yield at 50%, a zero point at 50% always keeps staking below 50%. Outside the model that depends on how much fee and MEV income stakers keep after any burn.
• Curves set in ETH. The page converts curves defined in ETH staked (Elowsson's k; EIP-8363's saturation balance of 60.25M ETH) at a 120.5M ETH supply, so 60.25M is a 50% ratio. Like Jermann's note, the page assumes such a curve is re-set from time to time to keep that ratio (note p.2).

Model and sources

R(d) = (1−β̃)·(c·d − φθ) / (d·(m − d))
c = (1−θ) + φθ,  m = θ + β̃(1−θ)
money growth = (d − θ)(1−β̃)/(m − d)
x = (m − d)/(1−β̃) (pp.6–7); value of staked ETH ∝ x·d/(1−d), of all ETH ∝ x/(1−d): the page's step from p̂M^U = x and M^U = (1−d)M (p.3); the note does not report these values

Your curve: y(d) = g·K0/√d·T(d), K0 = 0.025·√0.33
No taper: T = 1
Taper to zero: T = (1 − (d/dz)^p)^q below dz (p set by the peak position)
Taper, positive tail: T = 1/(1 + (d/dk)^a), peak at d_p = dk·(2a−1)^−1/a
Cosine taper: the positive-tail taper up to its peak (d_p, I_p), then issuance d·y = (I_p/2)·(1 + cos(π·(d − d_p)/(dz − d_p))) down to zero at dz
Tail: zero issuance from dz; issuance floor: d·y ≥ I_f past the peak, so issuance holds at I_f beyond dz; yield floor: y ≥ y_f; negative: y = −n·(d − dz)/0.1 past dz

A particular staker, with eff = effectiveness, the share of ideal rewards earned: y_s = eff·y; with a deduction sized on ideal rewards, y_s = y·(eff − b)/(1 − b), b = 1 − y/y_today
net = (y_s + fee and MEV yield)·(1 − pool fee), less tax if positive, less fixed cost

• U. Jermann, A Macro Finance Model for Proof-of-Stake Ethereum (2025): the model. U. Jermann, EIP 8363: Tapered Issuance Burn in an Equilibrium Model, note of 19 Aug 2026: its steady state without L2 or uncertainty, the calibration and Table 1. Today's curve is his ratio form, 2.5% nominal CL at 33% (p.2). The page solves his steady-state equations itself for every curve; Table 1 serves only as a check.
• Today's staking ratio: 36.1% of ETH supply, from actual balances in the finalized beacon state of 18 Oct 2026 (35.0% counting effective balances).
• The page draws every curve with a published level at Jermann's level. The protocol's own curve, cF/√D with c ≈ 2.6 and F = 64 (Elowsson, Properties of issuance level, p.3), is 5.6% higher at 120.5M ETH, so each curve here is 0.947 of its published level. Captions give the published figures.
• EIP-8363 in ratio form: burn fraction (d/0.5)^3/2, zero issuance from 50%.
• A. Elowsson, Reward curve with tempered issuance, 1 Apr 2024: today's curve divided by 1 + D/k (k = 2²⁶ near term, 2²⁵ final, p.1); Option C cuts F to 32 (p.11).
• Elowsson, cubed: A. Elowsson, Practical endgame on issuance policy, 23 Oct 2024, Fig. 3 (pp.8–9): today's curve divided by 1 + (D/k)^a for exponents 1 to 3.5, with k set so each curve peaks at 0.5% issuance. The exponent-2.5 curve is his red 'cubed' curve, k around 35.4M ETH (p.9). At 120.5M ETH the same peak gives k ≈ 35.7M.
• Croissant: Justin Drake's curve. V. Buterin, Possible futures of the Ethereum protocol, part 3: The Scourge, 20 Oct 2024, draws it with APR = sqrt[(½−S)/S] and issuance = sqrt[(½−S)×S]. Drake's 'croissant issuance' graphic (tweet, 5 Feb 2025) puts peak issuance at 1%/year at 25% staked and a 'soft cap' at 50%. The page combines the two as issuance = 0.04·√(d(½ − d)) at the published level (0.0379 here), deriving the 0.04 from the graphic's 1% peak at 25%, and sets issuance to zero past 50%, where neither source defines the curve.
• Frankenstein: an unpublished construction. A rise shaped like Elowsson's cubed curve but with exponent 1.5, to an issuance peak of 0.25% of supply a year near 7.4M ETH, then a cosine taper of issuance to zero at 30M ETH, at the protocol's level. The page applies the same 0.947 factor as for the other curves.
• J. Beccuti, T. Chantramonklasri, M. Hafner and N. Oderbolz, Towards a Formal Framework of the Ethereum Staking Market, 22 Jul 2025 (arXiv:2503.14385v2), p.1.