Proof-of-Stake Dynamics: The Elusive Price Anchor and Endogenous Volatility Harvesting

Mikhail Perepelitsa

REVIEW 3 major objections 5 minor 13 references

Proof-of-Stake Dynamics: The Elusive Price Anchor and Endogenous Volatility Harvesting

T0 review · 3 major / 5 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read A Proof-of-Stake economy has a unique long-run nominal price anchor, but it takes about 46 years to return to it; passive staking centralizes consensus ownership, while speculative buy/sell cycles can transfer tokens from investors back to

desk verdict A mathematically solid PoS macro model whose advertised 'volatility harvesting' is a consequence of the assumed fixed consumption share; the stability result and the long-relaxation insight are worth taking seriously. read the letter →

arxiv 2607.16622 v1 pith:4UVQV7AD submitted 2026-07-18 econ.GN q-fin.ECq-fin.MFq-fin.TR

Mikhail Perepelitsa This is my paper · ORCID

classification econ.GNq-fin.ECq-fin.MFq-fin.TR

keywords proof-of-staketokenpricedynamicssteady-stateequilibriumopen-economymacromodelvolatilityharvestingstakingcentralizationEthereumcalibrationovershooting

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper builds a macroeconomic model of a Proof-of-Stake network with two agent classes: Consumers, who use the token for transactions and staking and have a fixed propensity to consume fiat, and Investors, who stake passively and inject or withdraw fiat. It proves that the Consumer-only economy has exactly one stable steady state, giving a well-defined long-run nominal price anchor; calibrated to current Ethereum parameters, the system's relaxation half-life is about 46 years, so the price can remain persistently displaced from its equilibrium benchmark. Adding a passive Investor class makes the token price rise monotonically while the Consumers' share of staked tokens falls, implying a centralizing force on consensus ownership. Symmetric buy/sell shocks, however, trigger an endogenous contrarian rebalancing by Consumers—they sell tokens as the price rises and buy as it falls—so a zero-net-fiat speculative cycle ends with physical tokens transferred from the Investor to the Consumer. The paper frames this as volatility harvesting in token units rather than in fiat.

What carries the argument

The argument rests on the Consumers' top-level Cobb–Douglas utility U1 = ν log C + (1−ν) log V, which fixes the fiat consumption share at ν and the crypto retention share at 1−ν. Combined with myopic price expectations, market clearing becomes the identity νW_{t+1,c} = Λ_{t+1} + I_{t+1,c}, so any Investor fiat flow is exactly offset by the Consumer's net fiat contribution. This identity is what turns speculative buy/sell cycles into a contrarian constant-value strategy. The stability proof uses the monotone transition map S_{t+1}=F(S_t) from the recurrence in equation (27), with the square-root issuance rule y=c/√S, to show global convergence to the unique steady state and to compute the rel

What would settle it

An empirical test would be to observe a symmetric, fiat-neutral staking inflow/outflow cycle and check whether the non-speculative staking address ends up with a larger token balance and whether the token price remains below its pre-cycle level. A second check: if the model's steady state is correct, the price should slowly mean-revert with a half-life near 46 years; observing much faster mean reversion in major Proof-of-Stake networks would contradict the calibration. A structural test would replace the Cobb–Douglas upper-tier utility with a CES utility of elasticity different from one, which

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Extended reading notes

Core claim

The central claim is that the nominal token price in a Proof-of-Stake economy is anchored by a unique, globally asymptotically stable steady state determined by Consumer fiat inflows and the network's issuance and burn rates. The price is a fast variable that jumps immediately to clear markets, while the physical staked supply adjusts slowly through issuance and burning, producing a relaxation eigenvalue λ≈1−y*/2 and, under Ethereum-like calibration, a 46-year half-life. Consequently, the observed price can remain far from its evolving fundamental benchmark for decades—an inherent property of the network, not just a symptom of speculation. The second claim is that passive institutional staki

Load-bearing premise

The entire mechanism—the 46-year half-life and the Investor-to-Consumer token transfer—rests on the Consumers' fixed fiat-consumption share from a Cobb–Douglas utility and on myopic price expectations; if real users adjust their consumption propensity or form forward-looking expectations, the price anchor and the volatility-harvesting transfer could weaken, reverse, or disappear.

Editorial extensions

If this is right

If the calibration is representative, Proof-of-Stake token prices can sustain multi-decade deviations from their equilibrium benchmark after fundamental shocks, so short-run prices are a poor guide to long-run network value.
Passive institutional staking, by compressing the staking yield, can monotonically raise the nominal token price while concentrating staked-token ownership away from transactional users—a centralization channel independent of validator technology.
Symmetric fiat-denominated speculative cycles are wealth-neutral in fiat but token-accretive for Consumers, meaning active speculation can shift control of consensus toward utility users.
A higher Consumer fiat inflow raises the price only in the short run; because issuance outstrips burning during adjustment, the long-run equilibrium price is inversely related to the inflow rate.
The network behaves like an open economy with overshooting: fast price response and slow stock adjustment produce persistent disequilibrium even without rational speculation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The 46-year half-life is tied to the square-root issuance rule and the fixed expenditure share; with a flatter issuance curve or an elastic consumer demand, the price anchor would be much stronger. A natural extension is to compute the half-life under alternative issuance schedules.
The 'constant-value strategy' is not an intentional choice by Consumers but a mechanical consequence of the Cobb–Douglas top-level utility. If consumers had non-homothetic or forward-looking preferences, speculative cycles could transfer tokens in the opposite direction, so the decentralization result is sensitive to preference specification.
A testable signature of the volatility-harvesting mechanism is that after a symmetric fiat-neutral buy/sell cycle, the Consumer's staked fraction rises while the token price remains below its pre-cycle level; this could be checked against on-chain staking and exchange-flow data.
Passive and active speculative capital have opposite effects on consensus ownership, implying that policies shaping capital flows (staking derivatives, ETF structures) could be designed to amplify volatility-harvesting and mitigate passive-accumulation centralization.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Dear colleague,

Here's my take on Perepelitsa's PoS model.

The paper is better than the abstract makes it look. The core construction—two classes, Consumers with dual utility and Investors with exogenous fiat shocks—is coherent, and I checked the main algebra: the clearing price equations (12)–(14), the Consumer-only reduction (26), and the steady state (23) all check out. Theorem 1's claim of global asymptotic stability is plausible and the proof via monotone maps is standard. The eigenvalue/relaxation calculation, once you fix the sign error, gives λ ≈ 1 − y*/2, and the resulting half-life of about 46 years is a good illustration of how slowly the physical supply adjusts. The overshooting analogy to Dornbusch is apt. Lemma 1 on passive accumulation is also straightforward and correctly shows a monotone rise in price and fall in Consumer share.

The real soft spot is the "endogenous volatility harvesting" result. It is not emergent; it is the fixed expenditure share from the log-Cobb-Douglas utility (1). Because C = νW_c, the market-clearing condition (14) becomes an identity, and equation (29) forces the Consumer's net fiat flow to exactly offset the Investor's shock. So the contrarian rebalancing—selling high and buying low—is baked into the preference specification. The paper does not explore whether a non-homothetic or intertemporally optimizing Consumer, or non-myopic expectations, would preserve the transfer. The limitations section is candid about many abstractions, but it does not flag this dependence. As a result, the abstract and §3.1.2 overstate the robustness of the decentralization implication; at best it's conditional on the utility structure.

A few smaller issues. Theorem 1(ii) as printed gives λ > 1, which contradicts stability; the approximation on the same line uses the correct λ < 1. The numerical half-life uses the correct value, so it's a typo, but it will confuse readers. The introduction says "approximately 50 years" while the abstract says 46; they should match. The simulation is not reproducible from the text alone—no code, no data, no error bars. And the half-life itself is not an independent estimate; it's a re-parameterization of the calibrated annual yield target, as the paper's own §4 admits.

Bottom line: this is a serious modeling paper with one genuine mathematical contribution (the stability/relaxation theorem) and one overstated interpretive claim. It deserves a careful referee, and the author should be asked to fix the sign error, reconcile the half-life numbers, add a robustness check on the consumption share, and tone down the harvesting language. I'd read it again after those revisions.

It's a yes for peer review, and I'd probably cite the stability theorem, but not the harvesting mechanism without a big caveat.

Referee Report

3 major / 5 minor

Summary. The paper builds a two-class open-economy model of a Proof-of-Stake network. In a Consumer-only economy with a log-Cobb-Douglas utility over fiat consumption and crypto holdings, the authors prove global asymptotic stability of a unique steady state (Theorem 1), derive a closed-form price anchor, and calibrate a relaxation half-life of about 46 years. They then introduce an Investor class. Lemma 1 describes a regime in which passive staking raises the token price and shifts staked-token ownership away from Consumers. In §3.1, a simulation of symmetric buy/sell shocks shows that the Consumer's fixed fiat-consumption propensity generates a contrarian rebalancing that transfers tokens from Investor to Consumer while leaving the Consumer's fiat-denominated wealth unchanged, which the paper labels 'volatility harvesting.'

Significance. If the stability and slow-relaxation results hold, they constitute a valuable contribution to the macroeconomics of PoS networks: the 46-year half-life is a striking quantitative claim with practical implications for understanding price deviations from fundamentals. The model is tractable, with explicit market-clearing conditions and a closed-form steady state; the proof of global stability is a genuine theoretical result. However, the second main finding—volatility harvesting and the associated token transfer—is largely a direct consequence of the fixed expenditure share imposed by utility (1). This reduces the novelty and robustness of that claim relative to the presentation in the abstract and introduction. The paper is transparent about many limitations but does not analyze whether the mechanism survives a more general utility specification.

major comments (3)

[§3.1, eqs. (1), (14), (29)] The 'endogenous constant-value strategy' and the Investor-to-Consumer token transfer are hard-wired by the fixed expenditure share in utility (1). The market-clearing identity νW_{t+1,c}=Λ_{t+1}+I_{t+1,c} (14) and its corollary I_c^t−νW_c^t=−Λ_t (29) follow immediately from C=νW_c and the accounting condition; they do not represent an emergent market mechanism. The paper should state this explicitly in the main text and, ideally, test robustness to a non-constant consumption propensity (e.g., Stone-Geary or CRRA). Without such a check, the abstract's claim of an 'endogenous constant-value strategy' is overstated and potentially circular.
[Theorem 1(ii), eq. (28)] The displayed expression for λ appears to read λ = 1 + (1/2)y*/(1 + y* + (1/2)(y*)^2), which exceeds 1 and contradicts the global stability proved in part (i). The intended formula is presumably the fraction (1 + (1/2)y*)/(1 + y* + (1/2)(y*)^2), which is approximately 1 − (1/2)y*. The numerical half-life uses the correct value, so this is a typesetting/notation error, but it should be corrected because the theorem as printed is internally inconsistent.
[Introduction, Abstract, and Lemma 1] The paper claims that passive institutional staking 'compresses the endogenous staking yield below the Consumer-only equilibrium level.' However, Lemma 1 does not prove this; it assumes 0<y_1<y* as a sufficient condition. While a large Investor stake likely makes S_1>S* and hence y_1<y*, this is not derived from the model's primitives. The result should be stated conditionally, or a proof of yield compression from the Investor's presence should be supplied.

minor comments (5)

[Comment 3, §2.2] The statement that increased fiat inflow lowers the long-run equilibrium price is counterintuitive and deserves a more explicit explanation of the supply-side channel (higher inflow → more staking → higher issuance → lower price).
[§2.2, eq. (25)] The 'asset-market clearing locus' is stated without derivation. It appears to follow from (21)-(22), but the intermediate steps would help readability.
[General] Several references are to the author's own unpublished arXiv preprints ([10], [11]). If published versions exist, citing them would improve the paper's verifiability.
[§2.0.4, eq. (19)] Equation (19) defines S_{t+1} implicitly through y_{t+1} on both sides. The text should note this explicitly to avoid confusion about the nature of the transition map.
[§3.1, Figure 2] The post-shock horizon in the simulation is finite; the paper should clarify whether the two-class economy converges to a new steady state or remains displaced indefinitely. The persistent displacement claim would be stronger with a longer-run analysis.

Circularity Check

0 steps flagged · score 0.0 of 10

No load-bearing circularity; headline results are derived from stated assumptions and a self-contained stability proof.

full rationale

The paper's derivation chain is self-contained. The Consumer-only steady state and the global stability claim are proven in Section 5 from the model equations (26)–(31), with no reliance on prior results. Equation (23) is described as obtainable from the preceding equations, so the citation to [11] is a pointer, not the source of the load-bearing claim. The 46-year half-life is a closed-form consequence of the calibrated yield: Theorem 1(ii) gives λ ≈ 1 − y*/2, and Table 2 fixes the monthly yield y* = 0.0025 to a 3%-annualized external target; the half-life is then computed, not fitted. This is ordinary calibration, not a fitted input renamed as a prediction. The 'endogenous constant-value strategy' in §3.1.1 is explicitly derived from the fixed expenditure share in utility (1) and the market-clearing identity (14)→(29); the paper labels the mechanism 'endogenous' and does not present it as an externally observed fact. The directness of this implication is a modeling feature, not circularity, because the direction is assumption → theorem. Self-citations [10] and [11] introduce the Dual-Consumption model and steady-state algebra, but they are not load-bearing: the substantive theorems and simulations are derived in this paper. Section 4 candidly lists the model's abstractions and the illustrative nature of the calibration, which are limitations and robustness concerns, not circular reasoning.

Assumptions & free parameters 7 free parameters · 7 assumptions · 0 invented entities

The model is closed-form and parameter-free relative to its own assumptions: the price anchor, half-life, and transfer magnitudes are functions of the assumed utility shares, the exogenous fiat inflow, the issuance curve, and the calibration targets (y* ≈ 0.0025/month, S* = 40M ETH). No external data are fitted and no new empirical entity is introduced. The "endogenous constant-value strategy" is a property of the fixed expenditure share in U1 (eq. 1), not an emergent equilibrium object; the paper's own limitations section (§4) concedes the calibration is illustrative and the model stylized.

free parameters (7)

ν (fiat consumption propensity) = 0.01
Cobb-Douglas share in U1 (eq. 1). Chosen, not estimated. Sets price level, equilibrium yield y* = γν/(ξI_c), and the scale of the Consumer's contrarian flows.
I_c (steady fiat inflow) = $350M/month
Exogenous steady inflow; "representative" baseline (Table 2), not estimated. Anchors the steady-state price and supply via eq. (23).
γ (gas utility parameter) = 86.625×10^6
Set so that y* = γν/(ξI_c) = 0.25%/month. Not estimated from any data.
c (square-root issuance coefficient) = 15.81
Set so that y* = c/√S* with S* = 40M ETH. Not estimated from any data.
y* (calibrated monthly yield target) = 0.25% (3% annualized)
Effective calibration target. By eq. (28), λ ≈ 1 − y*/2, so the 46-year half-life is t1/2 = ln2·2/y* — a restatement of this assumed input at leading order.
Λ_t (speculative shock schedule) = ±$300M/month, months 20–30 and 40–50
Symmetric buy/sell cycle chosen for the simulation in §3.1; magnitudes and timing not justified from data.
Initial conditions (S_0, S_c0, S_i0, p_0) = 40M, 10M, 30M, ≈$3,456
Initial stake split (10M Consumer / 30M Investor) chosen to represent a large passive Investor; determines the scale of the post-shock token transfer.

assumptions (7)

domain assumption Square-root issuance curve y = c/√S
Table 1; eqs. (17)–(19); used in the proof of Theorem 1 (monotonicity of R, H) and in the eigenvalue. Mimics an Ethereum-style issuance schedule but is assumed, not derived.
domain assumption Myopic price expectations E[p_{t+1}] = p_t
§2, immediately before eq. (3). Underpins the explicit price equation (13) and all stability/overshooting results; rational or adaptive expectations are not analyzed.
domain assumption Exogenous, constant fiat inflow I_c in the steady-state analysis
§2.2. All steady-state formulas (23) and the price-anchor claim are conditional on constant I_c; the "moving benchmark" discussion in §2.2.1 requires I_c to fluctuate on timescales shorter than the 46-y relaxation, but no stochastic I_c process is modeled.
domain assumption Frictionless markets; no taxes, slippage, staking delays, leverage, or validator heterogeneity
Stated in §4. The volatility-harvesting transfer and the Investor's zero-fiat-net-loss result depend on frictionless price clearing.
domain assumption Investor never uses block space and stakes the entire balance (eq. 6)
§2, eq. (6). The centralization result (Lemma 1) and the transfer asymmetry require the Investor to be purely yield-driven.
standard math 0 < y* < 1 (standard macroeconomic conditions)
Theorem 1 statement. Needed for 1 + y − y* > 0 in the proof; holds for the ETH calibration (y* = 0.0025).
domain assumption Consumer fiat inflow dominates gas fees: I_c ≫ γ/(1+y)
Comment 1 (§2.0.3); used to assert S_{t+1,c} > ξS_{t,c} regardless of Investor buying.

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Cite this review

Pith. "Pith review of Proof-of-Stake Dynamics: The Elusive Price Anchor and Endogenous Volatility Harvesting." pith.science (2026). https://pith.science/paper/4UVQV7AD

@misc{pith2026260716622,
  author       = {Pith},
  title        = {Pith review of: Proof-of-Stake Dynamics: The Elusive Price Anchor and Endogenous Volatility Harvesting},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4UVQV7AD}},
  note         = {Machine review of arXiv:2607.16622}
}

read the original abstract

In this paper, we develop an open-economy macroeconomic model of a Proof-of-Stake network to analyze nominal token-price dynamics and the systemic effects of speculative capital. We first consider a network populated solely by active utility users, who finance network activity through a steady exogenous inflow of fiat currency. We prove the existence of a unique, globally asymptotically stable steady-state equilibrium with a well-defined nominal token price and derive a closed-form expression for the network's relaxation time. Calibrating the model using parameters representative of the current Ethereum network, we estimate a relaxation half-life of approximately 46 years. This extreme macroeconomic inertia implies that the token price may remain persistently displaced from its evolving steady-state benchmark, producing sustained price overshooting as the network adjusts to changing fundamentals. We then introduce an Investor class to examine the effects of passive and active speculative capital. We show that passive institutional staking compresses the native staking yield and creates a structural imbalance that systematically raises the nominal token price while shifting consensus ownership away from active utility users. Active speculative capital has a qualitatively different effect. In response to capital shocks, the Consumer class's rigid preference for fiat-denominated consumption generates an endogenous constant-value strategy. This mechanism shifts staked-token ownership from the Investor class toward active utility users, with potentially favorable implications for consensus decentralization.

Figures

Figures reproduced from arXiv: 2607.16622 by the authors.

**Figure 2.** Simulated dynamics of a Buy-Sell cycle. The Investor is pas [PITH_FULL_IMAGE:figures/full_fig_p017_2.png] view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

13 extracted references · 2 linked inside Pith

[1]

Tokenomics: Dynamic adoption and valuation.The Review of Financial Studies, 34(3):1105–1155, 2021

Lin William Cong, Ye Li, and Neng Wang. Tokenomics: Dynamic adoption and valuation.The Review of Financial Studies, 34(3):1105–1155, 2021

2021
[2]

Expectations and exchange rate dynamics.Journal of Polit- ical Economy, 84(6):1161–1176, 1976

Rudiger Dornbusch. Expectations and exchange rate dynamics.Journal of Polit- ical Economy, 84(6):1161–1176, 1976

1976
[3]

Circulating supply equilibrium for ethereum and minimum viable issuance during the proof-of-stake era

Anders Elowsson. Circulating supply equilibrium for ethereum and minimum viable issuance during the proof-of-stake era. Ethereum Research Forum, 2021. Accessed: 2026-07-16

2021
[4]

Compounding of wealth in Proof-of-Stake cryptocurrencies

Giulia Fanti, Leonid Kogan, Sewoong Oh, Kathleen Ruan, Pramod Viswanath, and Gerui Wang. Compounding of wealth in Proof-of-Stake cryptocurrencies. In Ian Goldberg and Tyler Moore, editors,Financial Cryptography and Data Secu- rity, volume 11598 ofLecture Notes in Computer Science, pages 42–61. Springer, 2019

2019
[5]

Ethereum Proof- of-Stake Consensus Layer: Participation and decentralization

Dominic Grandjean, Lioba Heimbach, and Roger Wattenhofer. Ethereum Proof- of-Stake Consensus Layer: Participation and decentralization. InFinancial Cryp- tography and Data Security: FC 2024 International Workshops, pages 253–280. Springer, 2024

2024
[6]

A macro finance model for proof-of-stake Ethereum

Urban Jermann. A macro finance model for proof-of-stake Ethereum. Working Paper, 2025

2025
[7]

Optimal issuance for proof-of-stake blockchains

Urban Jermann. Optimal issuance for proof-of-stake blockchains. Working Paper, University of Pennsylvania, 2025

2025
[8]

Rivera, and Fahad Saleh

Kose John, Thomas J. Rivera, and Fahad Saleh. Equilibrium staking levels in a proof-of-stake blockchain. Working Paper, SSRN 3965599, 2021

2021

Show all 13 references

[9]

Rogoff.Foundations of International Macroe- conomics

Maurice Obstfeld and Kenneth S. Rogoff.Foundations of International Macroe- conomics. MIT Press, Cambridge, MA, 1996

1996
[10]

Bubbles vs

Mikhail Perepelitsa. Bubbles vs. baselines: Token valuation and institutional capital in pos networks under eip-1559.arXiv:2606.07445, 2026. 19

2026 arXiv
[11]

The financialization of proof-of-stake: Asymptotic central- ization under exogenous risk premiums.arXiv preprint: 2604.26076, 2026

Mikhail Perepelitsa. The financialization of proof-of-stake: Asymptotic central- ization under exogenous risk premiums.arXiv preprint: 2604.26076, 2026

2026 arXiv
[12]

Evolution of shares in a proof-of-stake cryptocur- rency.Management Science, 67(2):661–672, 2021

Ioanid Ro¸ su and Fahad Saleh. Evolution of shares in a proof-of-stake cryptocur- rency.Management Science, 67(2):661–672, 2021

2021
[13]

A model of cryptocurrencies.Management Sci- ence, 69(11):6684–6707, 2023

Michael Sockin and Wei Xiong. A model of cryptocurrencies.Management Sci- ence, 69(11):6684–6707, 2023. 20

2023

Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.

[1] [1]

Tokenomics: Dynamic adoption and valuation.The Review of Financial Studies, 34(3):1105–1155, 2021

Lin William Cong, Ye Li, and Neng Wang. Tokenomics: Dynamic adoption and valuation.The Review of Financial Studies, 34(3):1105–1155, 2021

2021

[2] [2]

Expectations and exchange rate dynamics.Journal of Polit- ical Economy, 84(6):1161–1176, 1976

Rudiger Dornbusch. Expectations and exchange rate dynamics.Journal of Polit- ical Economy, 84(6):1161–1176, 1976

1976

[3] [3]

Circulating supply equilibrium for ethereum and minimum viable issuance during the proof-of-stake era

Anders Elowsson. Circulating supply equilibrium for ethereum and minimum viable issuance during the proof-of-stake era. Ethereum Research Forum, 2021. Accessed: 2026-07-16

2021

[4] [4]

Compounding of wealth in Proof-of-Stake cryptocurrencies

Giulia Fanti, Leonid Kogan, Sewoong Oh, Kathleen Ruan, Pramod Viswanath, and Gerui Wang. Compounding of wealth in Proof-of-Stake cryptocurrencies. In Ian Goldberg and Tyler Moore, editors,Financial Cryptography and Data Secu- rity, volume 11598 ofLecture Notes in Computer Science, pages 42–61. Springer, 2019

2019

[5] [5]

Ethereum Proof- of-Stake Consensus Layer: Participation and decentralization

Dominic Grandjean, Lioba Heimbach, and Roger Wattenhofer. Ethereum Proof- of-Stake Consensus Layer: Participation and decentralization. InFinancial Cryp- tography and Data Security: FC 2024 International Workshops, pages 253–280. Springer, 2024

2024

[6] [6]

A macro finance model for proof-of-stake Ethereum

Urban Jermann. A macro finance model for proof-of-stake Ethereum. Working Paper, 2025

2025

[7] [7]

Optimal issuance for proof-of-stake blockchains

Urban Jermann. Optimal issuance for proof-of-stake blockchains. Working Paper, University of Pennsylvania, 2025

2025

[8] [8]

Rivera, and Fahad Saleh

Kose John, Thomas J. Rivera, and Fahad Saleh. Equilibrium staking levels in a proof-of-stake blockchain. Working Paper, SSRN 3965599, 2021

2021