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Econ-474 · economics · Economics

Economic role and fragility of crypto, stablecoins and DeFi

LLM status Open unverified

Statement

Atlas ID: F7; original number: 97; field: Finance. Classification: Editorial research formulation (theoretical/empirical); not a certified unsolved theorem.

Definitions and assumptions

Fix a dated protocol and institutional regime over a finite horizon $H$. Users choose feasible payments, collateralized loans and trades; validators or liquidity providers choose activity under stated incentives. Model $\theta$ includes transaction-service utility, settlement reliability, resource costs, cyber losses and an equilibrium selection. For a redeemable stablecoin, let outstanding claims be $B_t$, redemptions due be $D_t\in[0,B_t]$, liquid reserves be $L_t$ and other reserve liquidation proceeds available by the settlement deadline be $Q_t$; redemption promises require an explicit priority and enforceability rule. Algorithmic or overcollateralized tokens require their own collateral and liquidation constraints rather than this reserve identity. All welfare and loss variables are integrable. Claims and reserve values use the same promised redemption currency.

Formal research question

Compare the dated decentralized or stablecoin arrangement $a^1$ with a specified feasible payment-and-credit alternative $a^0$ serving the same population. Identify \[ \Delta_W=\sum_h\omega_h\mathbb E_\theta[U_h(a^1)-U_h(a^0)], \qquad p_{\mathrm{fail}}=\mathbb P_\theta(L_t+Q_t<D_t \text{ at a required redemption date}), \] where $U_h$ includes service benefits, pecuniary outcomes and losses, and $\omega_h\ge0$ are stated weights. Characterize reserve, governance and collateral rules making $\Delta_W>0$ at a given failure-risk bound. Identify exposure across on-chain positions, exchanges, custodians and reserve assets without counting the same underlying collateral multiple times.

Interpretation and scope

Token prices can reflect transaction-service benefits and network effects; valuing every token solely as a discounted cash-flow claim can omit the object being priced. Reserve sufficiency at one date does not guarantee liquidity under simultaneous redemptions, fire-sale discounts or inaccessible custody. A blockchain records addresses and contract states, not necessarily beneficial owners, enforceable off-chain rights or the entire balance sheet. Experimental or quasi-experimental protocol changes must be dated because users, code and regulation adapt. Specify finality and failure criteria, account for fees as transfers where appropriate, and distinguish energy or computing resource costs from token rewards. A causal efficiency claim needs a comparable counterfactual service.

Source audit and status

[S1] is a protocol white paper; [S2] studies cryptocurrency arbitrage; [S3] models token adoption. Atlas link [50], [S4], identifies a 2022 overview. It does not support the atlas’s precise assertion about recent stablecoin flows and short-term government-debt markets; that assertion is left unverified. The editorial welfare-and-fragility question has no certified current open status.

References

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  1. Satoshi Nakamoto (2008). Bitcoin: A Peer-to-Peer Electronic Cash System. Technical white paper. https://bitcoin.org/bitcoin.pdf. Date/version verification: https://www.metzdowd.com/pipermail/cryptography/2008-October/014810.html. Locator and role: White-paper abstract and Sections 2–5; author’s original release email dated 31 October 2008; Abstract and Sections 2–5 describe signatures, timestamping, proof of work and consensus. Protocol source, not evidence that stablecoins or DeFi improve welfare. The original dated announcement verifies the release year.

  2. Igor Makarov and Antoinette Schoar (2020). Trading and Arbitrage in Cryptocurrency Markets. Journal of Financial Economics 135(2), 293–319. https://www.sciencedirect.com/science/article/pii/S0304405X19301746. Locator and role: Publisher or author abstract / bibliographic record; Market segmentation, price differences and cryptocurrency arbitrage. Provides background or a benchmark result; does not establish that the editorial question remains unresolved in 2026.

  3. Lin William Cong, Ye Li, and Neng Wang (2021). Tokenomics: Dynamic Adoption and Valuation. Review of Financial Studies 34(3), 1105–1155. https://doi.org/10.1093/rfs/hhaa089. Locator and role: Publisher or author abstract / bibliographic record; Token transaction demand, network effects and dynamic adoption; online publication 2020, issue year 2021. Provides background or a benchmark result; does not establish that the editorial question remains unresolved in 2026.

  4. Igor Makarov and Antoinette Schoar (2022). Cryptocurrencies and Decentralized Finance (DeFi). NBER Working Paper 30006. https://doi.org/10.3386/w30006. Locator and role: Publisher or author abstract / bibliographic record; Overview of cryptocurrency and DeFi mechanisms and risks. The atlas’s specific assertion about recent stablecoin flows and government-debt markets is not established by the identified overview; a separate dated empirical source is required.

Common conventions: Source mathematical conventions

Unless an entry states otherwise, agent and firm sets are finite. State and action spaces are measurable, strategies and outcome maps are measurable, probability laws are specified, and expectations exist for the targets under discussion. Infinite-horizon discount factors lie in $(0,1)$ and discounted objectives are finite. Norms, tolerances and complexity input sizes must be specified for computational comparisons. Constants or primitives that appear locally are inputs, not empirical facts asserted by this catalogue.

Identification. A statistical model $m\in\mathcal M$ implies an observable law $P_m$ and target $\theta(m)$. At law $P$, its identified set is \[ \Theta(P;\mathcal M)=\{\theta(m):m\in\mathcal M,\ P_m=P\}. \] Point identification holds when this set is a singleton, or equivalently when $P_{m_1}=P_{m_2}$ implies $\theta(m_1)=\theta(m_2)$ for all compatible models. Sharp bounds mean bounds attained, or approached when the boundary is not attained, by compatible models. Writing this set is a definition, not a solution: the substantive task is to establish informative restrictions and derive or estimate the set. An unrestricted-confounding model may give only trivial bounds.

Inference. Identification, estimability and valid uncertainty quantification are separate questions. Fix $\alpha\in(0,1)$. A confidence procedure $C_n$ over a declared law class $\mathcal P$ has asymptotic uniform coverage at level $1-\alpha$ when \[ \liminf_{n\to\infty}\inf_{P\in\mathcal P}\Pr_P\{\theta(P)\in C_n\}\geq1-\alpha. \] For set-identified targets the coverage event must be defined separately, for example coverage of the full identified set. If an entry uses identification-indexed models instead of uniquely indexed laws, the same coverage convention is applied over those models. Pointwise limits do not establish uniform validity, and asymptotic validity does not guarantee finite-sample precision. Sample sizes, dependence structures and weak-identification sequences are part of each application.

Counterfactuals. $Y_i(a)$ is a potential outcome under a specified intervention, including its timing, implementation and versions. It is not $Y_i$ conditional on voluntarily choosing $a$. A full intervention vector permits interference; shorthand scalar treatment notation does not silently impose no interference. Assignment, selection, attrition, anticipation and measurement must be specified. A claim about an instrument's local effect is not automatically a population policy effect.

Economic equilibrium. A counterfactual model includes best responses, constraints, feasibility, market clearing, state transitions and expectations consistent with the stated information. When several equilibria exist, either a declared selector is an input or the target is set-valued. Comparisons across policy regimes must use the stated selection convention. Reduced-form relationships are not presumed invariant after an equilibrium-changing intervention.

Optimization and welfare. Optimization is over the stated feasible set. If attainment is not established, $\sup$ or $\inf$ and $\varepsilon$-optimal choices replace an asserted maximizer or minimizer. For example, with value $V=\sup_{a\in\mathcal A}W(a)<\infty$, an $\varepsilon$-optimal set is $\{a\in\mathcal A:W(a)\geq V-\varepsilon\}$ for $\varepsilon>0$. Benefits, costs, risk and health or environmental outcomes can be aggregated only using declared units and conversion weights. Interpersonal utility weights, the treatment of fiscal transfers and foreign profits, discounting, and the behavioral welfare criterion are explicit inputs. A comparative-static derivative additionally requires existence and appropriate differentiability of the selected counterfactual.

Sources. Local labels [S1], [S2], and so forth restart in every question. Locators identify the relevant abstract, model, section or result at the level actually checked; a bibliographic or abstract-level verification is not represented as inspection of an entire proof. Working papers are distinguished from later publications and changing versions. Source summaries are paraphrased. All URL checks and editorial formulations refer to the audit date stated above.

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Econ-474-A1 · Exploration · LLM-generated · gpt 6 astra ultra ·

GPT 6 Astra Ultra attempt on Econ-474

Crypto, stablecoins and DeFi A reserve-liquidity frontier with explicit settlement losses

Problem ID: E42-470. Status: Unresolved; restricted results and explicit gaps.

A dated and narrow comparison

The catalogue combines service welfare, redemption failure and exposures across protocols. Makarov and Schoar [ms] discuss stablecoin reserves and systemic interconnections in Sections 3.1 and 4.5; those passages were inspected in the BIS-hosted version. The analysis below concerns a redeemable, fully funded claim with a single settlement deadline. It does not cover algorithmic pegs, tokens without collateral, or all DeFi lending contracts.

Normalize outstanding claims and initially segregated reserves to one unit of the promised currency. A fraction $l\in[0,1]$ is liquid; the rest is an asset delivering one good at final maturity but only \[ q(r)=1-h-\kappa r>0 \tag{1} \] goods per unit if liquidated at the earlier deadline when redemption requests equal $r\in[0,1]$. Assume $h,\kappa\geq0$ and $h+\kappa<1$. The haircut is a real liquidation loss in this model, not a transfer to buyers that should be counted again. Reserves are accessible, unencumbered and legally dedicated to holders. Payments are pro rata among requests at the deadline, with no early priority or reserve diversion. Later claims receive the remaining assets. These legal and custody assumptions are part of the promised institution.

Maximum available redemption resources are $C(l,r)=l+(1-l)q(r)$. Failure means $r>C(l,r)$; equality is successful settlement. Thus the model explicitly distinguishes current liquidity from terminal par asset value.

The exact risk frontier

Theorem (Failure threshold and minimum liquid reserves): Define \[ c(l)=\frac{1-h+hl}{1+\kappa(1-l)}. \tag{2} \] Failure occurs exactly when $r>c(l)$. If $h+\kappa>0$, $c$ is strictly increasing from $c(0)=(1-h)/(1+\kappa)$ to $c(1)=1$. Let the redemption law have CDF $F$ and let $q_\epsilon=\inf\{x\in[0,1]:F(x)\geq1-\epsilon\}$ for $0\leq\epsilon<1$. The exact set of reserve fractions with failure probability at most $\epsilon$ is $[l_\epsilon,1]$, where \[ l_\epsilon=\begin{cases} 0,&q_\epsilon\leq c(0),\\[2pt] \displaystyle\frac{q_\epsilon(1+\kappa)-(1-h)}{h+\kappa q_\epsilon}, &q_\epsilon>c(0). \end{cases} \tag{3} \] If $h=\kappa=0$, failure probability is zero for every $l$.
Proof: The inequality $r\leq l+(1-l)(1-h-\kappa r)$ rearranges to $r[1+\kappa(1-l)]\leq1-h+hl$, proving (2). Its derivative has numerator $h(1+\kappa)+\kappa(1-h)=h+\kappa$, and its denominator is positive, establishing the monotonicity and endpoints. Because $F$ is right continuous and nondecreasing on a compact support, $F(c)\geq1-\epsilon$ is equivalent to $c\geq q_\epsilon$, including atoms at the threshold. Failure probability is $1-F(c(l))$. If the quantile is already below $c(0)$ no liquid reserve is required for this risk tolerance. Otherwise solving $c(l)\geq q_\epsilon$ gives (3); its denominator is then positive and the result lies in $(0,1]$. In the zero-haircut case, $C(l,r)=1$ and $r\leq1$ always. ∎
A high reserve asset's terminal value is not enough: simultaneous demand also lowers its deadline conversion rate through $\kappa$. The law of $r$ in this theorem is a specified intervention-invariant stress law. It may be estimated from a validated model, but is not identified merely by observing reserve balances.
Proposition (What survives arbitrary endogenous redemptions): If $h+\kappa>0$, guaranteeing settlement for every possible redemption choice $r\in[0,1]$ requires and is achieved by $l=1$. More generally, if every allowed equilibrium redemption law satisfies $\Pr(r>x)\leq1-F_0(x)$ for all $x$, choosing $l$ by (3) using $F_0$ guarantees failure probability at most $\epsilon$ in every such equilibrium.
Proof: At any $l<1$, (2) is strictly below one, so the feasible redemption choice $r=1$ causes failure. At $l=1$ every $r\leq1$ is paid. Under the stochastic envelope, evaluate the asserted tail inequality at $x=c(l)$ and apply Theorem 1 to $F_0$. ∎
This is a mechanical robust guarantee, not a proof that a run occurs whenever $l<1$. Endogenous redemption incentives, reputation, public backstops and sequential priority require a separate game and selection rule. A reserve rule calibrated to a historical no-run distribution does not establish the envelope assumption.

A welfare calculation with real costs rather than token rewards

Include the issuer, holders and service providers in the same quasilinear welfare population, with equal monetary weights and a common numeraire. Protocol fees and redemption payments are internal transfers. Relative to a feasible alternative serving this same population, let $b$ be the net service gain after baseline operating and cyber costs, and let $cl$ be an additional real cost of liquid custody, $c\geq0$. Required liquidation of the illiquid reserve is \[ x(l,r)=\min\{1-l,(r-l)_+/q(r)\}. \tag{4} \] If resources are insufficient, all illiquid reserves are liquidated; the unpaid amount is a transfer loss to holders rather than a second destruction of the same reserve. Let $\chi\geq0$ be an additional real settlement service loss upon failure. Social gain is defined by \[ W(l)=b-cl-{\mathbb{E}}[(1-q(r))x(l,r)]-\chi\Pr\{r>c(l)\}. \tag{5} \] The liquidation term records exactly the goods destroyed by early conversion. This declared benchmark omits endogenous adoption and heterogeneous incidence; (5) cannot be inferred from token prices.
Theorem (An exact finite-scenario design algorithm): Suppose $r$ takes finitely many known values $r_j$ with probabilities $p_j>0$ summing to one. Under $h+\kappa>0$, define \[ t_j=\max\left\{0, \frac{r_j(1+\kappa)-(1-h)}{h+\kappa r_j}\right\} \tag{6} \] whenever the numerator is positive, and set $t_j=0$ otherwise. The maximum of (5) subject to failure probability at most $\epsilon$ is attained at a member of the finite set \[ \{l_\epsilon,1\}\ \cup\ \{r_j:r_j\geq l_\epsilon\}\ \cup\ \{t_j:t_j\geq l_\epsilon\}. \tag{7} \] Consequently a reserve rule with positive welfare at that risk bound exists exactly when the largest value of (5) over (7) is positive.
Proof: For fixed $r_j$, the liquidation fraction (4) is continuous and piecewise affine in $l$. Its only possible breakpoints are $l=r_j$, where no liquidation is needed, and $l=t_j$, where full-reserve liquidation gives way to meeting requests with a partial sale. The latter follows by solving $r_j=l+(1-l)q(r_j)$; when its solution is negative there is no positive breakpoint. Since $q(r_j)>0$, a positive solution lies at most at $r_j$ and at most at one. At that point the two formulas for (4) agree. The failure indicator equals one for $l<t_j$ and zero for $l\geq t_j$ when $t_j>0$; it is identically zero if $t_j=0$.

Thus the total cost $b-W(l)$ is affine between consecutive breakpoints in (7), and at a failure threshold it has a weak downward jump, with the threshold value equal to its right-hand limit. The risk-feasible set is exactly $[l_\epsilon,1]$ by Theorem 1. On each interval, an affine cost is minimized at an endpoint: if its slope is nonnegative use the left endpoint; if negative use the right endpoint, whose possible downward jump only lowers cost. Constant intervals admit either endpoint. The finite candidate set therefore contains a global minimum, proving both attainment and the positive-welfare criterion. ∎

The result accommodates any finite number of correlated stress scenarios; it is not limited to a two-state example. Numerical probabilities and service benefits must come from outside the theorem. No calibration or empirical welfare improvement is claimed.

Consolidating protocol exposures

Suppose tokens and wrappers ultimately claim a finite collection of external assets. Let $v_a\geq0$ be the deadline goods obtainable from asset $a$, and let promised holder payments be $d_i\geq0$. Feasible settlement requires an allocation $x_{ai}\geq0$ satisfying \[ \sum_i x_{ai}\leq v_a\quad\text{for every asset }a, \qquad \sum_a x_{ai}\geq d_i\quad\text{for every holder }i, \tag{8} \] with $x_{ai}=0$ where an enforceable claim is absent. Summing (8) proves the necessary aggregate restriction $\sum_i d_i\leq\sum_a v_a$. A wrapper transferring an existing claim does not add a new $v_a$; counting both the wrapper and its reserve as external backing violates that restriction. Aggregate sufficiency need not follow, because accessibility or seniority can prevent the needed allocation. This elementary certificate explains which off-chain rights must supplement a ledger's address balances.

The full target still requires beneficial-owner links, feasible alternative services, endogenous protocol participation and a run equilibrium. The reserve frontier and finite design theorem answer a well-defined portion of that target under explicit enforceability and stress-law assumptions.

Completion Estimate

55%

This is a post hoc, heuristic estimate for the full catalogue target. The redeemable reserve model yields an exact failure risk frontier and a finite scenario algorithm finding welfare maximizing liquid reserves, with collateral consolidation constraints. Endogenous runs and participation, validated stress laws and service benefits, beneficial ownership, custody rights, and broader protocol or credit equilibria remain unresolved.

References

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