## wpiea2026144

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---

### 2.1 Traditional Parallel Markets — characteristics and frictions
- Decentralized market structure: street dealers, informal brokers, bank-adjacent intermediaries, importers and exporters with access to foreign exchange, and firms or households willing to trade foreign-currency cash.
- Price dispersion: "A large bank, a retail cash dealer, and a street market can quote different prices for dollar value on the same day."
- Frictions generating dispersion:
  - Search costs that hinder household comparison of quotes.
  - Relationship-based access leading to different prices for larger firms or connected agents versus retail households.
  - Settlement risk creating wedges between quoted and executable prices.
  - Geographic segmentation of physical cash markets.
  - Regulatory risk leading traders to charge premia for the possibility that a transaction is delayed, blocked, or penalized.
- Implications:
  - Traditional parallel markets are costly access technologies.
  - Prices from these markets are noisy signals of the aggregate state (local inventory, search, and enforcement risk affect quotes).
  - Before digital dollar markets become active, the economy may have parallel prices without a single transparent parallel price.

### 2.2 Stablecoin Exchanges — transformation, margins, and information effect
- Stablecoins create a digital claim to dollar-like value tradable against local currency; example instrument referenced: USDT.
- The local-currency price of USDT is an exchange rate: local currency per dollar-like digital claim (example: USDT/BOB).
- Portfolio and payment dimensions:
  - Households can hold stablecoins as dollar-like stores of value.
  - Stablecoin transfers can settle transactions or obtain foreign-currency exposure outside official allocation channels.
- How stablecoin markets differ from cash markets:
  - More continuous: high-frequency trading across platforms.
  - Easier to observe: posted quotes and platform prices provide a common reference even with incomplete transaction-level data.
  - More scalable: a larger number of agents can access the same digital instrument than a particular physical cash dealer.
- Key economic tension and information effect:
  - Stablecoins reduce effective access frictions and integrate fragmented markets, producing a common public price.
  - When many agents condition on the same price, it can act as a coordination device:
    - Stabilizing effect: reassures agents that dollar scarcity is limited.
    - Destabilizing effect: a large misalignment becomes commonly visible and can induce coordinated runs.
  - Stablecoin markets change both the technology for acquiring foreign-currency exposure and the information structure of the parallel market.

### 2.3 Bolivia case — from dispersed FX premia to a stablecoin reference price
- Pre-stablecoin environment:
  - Fixed exchange rate with declining foreign-exchange liquidity and rationed access produced a widening gap between official rate and private-obtainment prices.
  - Reported shadow price dispersion (press examples):
    - La Paz episode: parallel dollar reportedly climbed to around Bs 15; several exchange houses had no dollars to sell while others quoted sale prices of Bs 13, Bs 12.50, and Bs 12.
    - Santa Cruz report: across eight exchange houses, dollar’s price varied between Bs 10, Bs 10.50, Bs 11, and Bs 11.30.
  - In mid-February 2024 ASFI began limiting banks’ commissions for FX sales to local agents based on transaction size, increasing pricing opacity.
- Post-policy change and stablecoin emergence:
  - Central Bank of Bolivia rescinded restrictions on electronic payment instruments for virtual-asset transactions in June 2024.
  - Virtual-asset activity expanded quickly: BCB reported transactions with these assets in the financial system multiplied twelvefold from July 2024 to May 2025.
  - As USDT trading expanded, the USDT/BOB price became the common reference for the parallel exchange rate:
    - Appeared in public reporting on dollar scarcity.
    - Shaped formal and informal price-setting practices, including retail transactions where shops used USDT prices to set dollar–boliviano conversion rates.
  - The BCB began publishing USDT prices on its website, indicating USDT/BOB's salience as a monitored public reference price outside the official allocation system.

### 3 Model — setup, information, access, and strategic decisions

- Setup (Section 3.1):
  - Static economy with continuum of households i ∈ [0,1]; fixed or heavily managed exchange rate.
  - Degree of exchange-rate misalignment θ ≥ 0.
  - Two access technologies:
    - Cash-based parallel market: opaque, fragmented, relationship-based, relatively costly.
    - Stablecoin market: lower transaction cost and generates an observable market price (public signal).
  - Households form beliefs about θ, choose hedge quantity, and decide whether to join a run; crisis occurs when run-driven demand exceeds regime defense capacity.

- Information (Section 3.2):
  - Prior: θ ∼ N (μ_θ, σ^2_θ). (Equation (1))
  - Private signal: x_i = θ + ε_i,   ε_i ∼ N (0, σ^2_x). (Equation (2))
  - Public signal from stablecoin market: s = θ + ε_s,   ε_s ∼ N (0, σ^2_s). (Equation (3))
  - Public-signal variance endogenized by stablecoin depth:
    - σ^2_s(Q_s) = ̄σ^2_s / (1 + ρ Q_s),   ρ > 0. (Equation (4))
  - Posterior mean:
    - ˆθ_i = ω_0 μ_θ + ω_x x_i + ω_s s. (Equation (5))
    - Weights:
      - ω_0 = σ^{-2}_θ / (σ^{-2}_θ + σ^{-2}_x + σ^{-2}_s(Q_s)). (Equation (6))
      - ω_x = σ^{-2}_x / (σ^{-2}_θ + σ^{-2}_x + σ^{-2}_s(Q_s)). (Equation (7))
      - ω_s = σ^{-2}_s(Q_s) / (σ^{-2}_θ + σ^{-2}_x + σ^{-2}_s(Q_s)). (Equation (8))
    - As stablecoin activity rises, σ^2_s(Q_s) falls and ω_s rises.

- Access technologies and costs (Section 3.3):
  - Aggregate FX demand Q_FX.
  - Per-unit cost functions:
    - c_c(Q_FX) = k(Q_FX) + κ_c. (Equation (9))
    - c_s(Q_FX) = k(Q_FX) + κ_s. (Equation (10))
    - k′(Q_FX) ≥ 0; κ_j capture instrument-specific access frictions.
  - Stablecoin share:
    - m_s = Λ_h [ν (c_c(Q_FX) − c_s(Q_FX))] = Λ [ν (κ_c − κ_s)], Λ(z) = 1 / (1 + exp(−z)). (Equation (11))
  - Quantities:
    - Q_s = m_s Q_FX,   Q_c = (1 − m_s) Q_FX. (Equation (12))
  - Effective per-unit cost:
    - c_FX = (1 − m_s) c_c(Q_FX) + m_s c_s(Q_FX). (Equation (13))

- Household consumption and utility (Section 3.4):
  - Individual run demand: q^R_i = ̄r r_i. (Equation (14))
  - Total individual FX demand: q^FX_i = q^H_i + q^R_i. (Equation (15))
  - Utility linear in consumption-equivalent payoff: u_i(θ, s) = c_i(θ, s). (Equation (16))
  - Consumption-equivalent payoff:
    - c_i(θ, s) = c_0 + A^H_i(q^H_i; θ) + r_i [B − (1 − I(θ, s)) τ_f (1 − θ) + I(θ, s) λ θ] − c_FX [q^H_i + ̄r r_i] − I(θ, s) λ θ ξ_Q Q_R. (Equation (17))

- Hedge demand (Section 3.5):
  - Hedge-service value:
    - A^H_i(q^H_i; ˆθ_i) = χ_h / η_h ˆθ_i q^H_i − 1/(2 η_h) (q^H_i)^2. (Equation (18))
  - Optimal hedge:
    - q^H_i = max{0, χ_h ˆθ_i − η_h c_FX}. (Equation (21))
  - Aggregate hedge demand:
    - Q_H(θ, s, Q_s, Q_FX) = ∫ max [0, χ_h (ω_0(Q_s) μ_θ + ω_x(Q_s) x + ω_s(Q_s) s) − η_h c_FX(Q_FX, m_s)] dF(x | θ). (Equation (22))

- Run demand (Section 3.6):
  - Threshold strategy: r_i = 1{x_i ≥ x^*}. (Equation (23))
  - Aggregate run demand:
    - Q_R(θ, x^*) = ̄r [1 − Φ((x^* − θ)/σ_x)]. (Equation (24))
  - Regime breaks when:
    - Q_R(θ, x^*) ≥ α − δ θ. (Equation (25))
  - Realized crisis indicator:
    - I(θ, s) = 1{Q_R(θ, x^*(θ, s)) ≥ α − δ θ}. (Equation (26))
  - Marginal-runner expected gain:
    - G(x_i; s, Q_s) = B − c_FX ̄r + E[ I(θ, s) λ ̃θ − (1 − I(θ, s)) τ_f (1 − ̃θ) | x_i, s ]. (Equation (28))
  - Equilibrium cutoff satisfies G(x^*; s, Q_s) = 0. (Equation (30))
  - Public information affects run incentives via posterior beliefs and crisis event; higher precision can compress beliefs and either raise or lower run demand depending on sign of G_{τ_s}.

### 3.7 Equilibrium and Welfare — fixed point, channels, and state dependence
- Equilibrium defined as tuple (Q_FX, Q_s, Q_c, x^*, m_s, Q_H, Q_R) satisfying:
  - Bayesian updating with public-signal variance implied by stablecoin activity.
  - m_s = Λ[ν(κ_c − κ_s)].
  - Q_FX = Q_H + Q_R and Q_s = m_s Q_FX, Q_c = (1 − m_s) Q_FX.
  - σ_s^2(Q_s) = ̄σ_s^2 / (1 + ρ Q_s).
- Welfare (utilitarian) for state (θ,s):
  - W(θ,s) = ∫_0^1 c_i(θ,s) = c_0 + A_H(Q_H; θ) + Q_R ̄r [B − (1 − I(θ,s)) τ_f (1 − θ) + I(θ,s) λ θ] − c_FX Q_FX − I(θ,s) λ θ ξ_Q Q_R. (Equation (31))
- Mechanism and tradeoffs:
  - Access channel: if κ_s < κ_c, stablecoins lower effective access cost → raise ordinary hedging welfare but can increase run risk by lowering cost of run positions.
  - Information channel: stablecoins raise public-signal precision → improve hedge allocation but can synchronize beliefs and increase coordinated runs.
  - State-dependent welfare: stablecoins can be welfare-improving in normal states and welfare-reducing in fragile states where coordination externalities dominate.

### Quantitative analysis — solution, calibration, and key results
- Solution method:
  - Fully nested numerical equilibrium for θ ∈ [0,1] with damped fixed-point iteration and bisection for cutoff; outer loop updates public-signal variance until stable.
- Benchmark calibration (selected parameters):
  - μ_θ 0.46
  - σ_θ 0.48
  - σ_x 0.34
  - ̄σ_s 1.36
  - ρ 4.39
  - α 0.69
  - δ 0.09
  - τ_p 4.97
  - χ_h 2.29
  - η_h 0.69
  - ̄r 0.45
  - B 0.01
  - τ_f 0.03
  - λ 1.63
  - ξ_Q 13.51
  - a 0.13
  - b 0.20
  - κ_c 0.18
  - κ_s 0.04
  - ν 7.77
  - c_0 5.00

- Results — comparison of three economies:
  - Economies: cash-only benchmark; access-only stablecoin (access gains without information precision improvement); full stablecoin (access + endogenous public-signal precision).
  - Crisis exposure (probability run demand overwhelms defense) rises with θ in all three; strongest increase in full stablecoin economy.
  - Average crisis exposure over misalignment interval:
    - Cash-only: 3.9 percent
    - Access-only stablecoin: 4.8 percent
    - Full stablecoin: 7.4 percent
  - At θ = 1:
    - Cash-only: 4.8 percent
    - Access-only: 7.0 percent
    - Full stablecoin: 12.9 percent
  - Interpretation: cheaper access and public price discovery make exit easier and more coordinated.

- Welfare decomposition (log welfare differences; interpreted approximately as percent changes in consumption-equivalent welfare):
  - Define:
    - ∆_access(θ) = log W_access(θ) − log W_cash(θ)
    - ∆_info(θ) = log W_full(θ) − log W_access(θ)
    - ∆_total(θ) = log W_full(θ) − log W_cash(θ)
  - Main findings:
    - Total welfare effect peaks at 1.2 percent of consumption-equivalent welfare, around θ = 0.4, then declines sharply.
    - Reversal threshold θ∗ = inf{θ : ∆total(θ)≤ 0} ≈ 0.59.
    - At θ = 1 total welfare effect reaches −6.3 percent.
    - Access component is mostly positive (cheaper hedging) but can turn slightly negative at highest θ.
    - Information component is mildly positive at low θ, turns negative before ∆_total crosses zero, and reaches about −5.9 percent by high θ.
    - Main mechanism for reversal: information-driven coordination externality dominates access gains above θ∗.

- Sensitivity (selected experiments reported):
  - Crisis gap (average over grid of full-stablecoin crisis exposure minus cash-only) and reversal threshold θ∗ reported:
    - Benchmark: Crisis gap 3.50, θ∗ 0.59
    - ρ +15%: Crisis gap 3.60, θ∗ 0.57
    - λ +15%: Crisis gap 4.00, θ∗ 0.40
    - ξQ +15%: Crisis gap 3.50, θ∗ 0.55
    - κc − κs −0.0203.10.63  (presented in source as an experiment altering κc − κs; reported outcome formatting preserved)
  - Mechanism-consistent patterns:
    - Higher ρ raises crisis gap and lowers θ∗.
    - Higher λ raises crisis gap and lowers θ∗.
    - Higher ξQ raises welfare cost of synchronized exit and lowers θ∗.
    - Reducing stablecoin access-cost advantage (κc − κs) lowers crisis gap and delays reversal.

### Extensions — heterogeneous access, distributional incidence, and quantitative exercise
- Two-type model (types g ∈ {B,U}):
  - Population shares πB = 0.60, πU = 0.40 in calibration.
  - κc,B = 0.15, κc,U = 0.22, κs,B = κs,U = κs; preserves aggregate κc via 0.60(0.15) + 0.40(0.22) = κc.
  - Type-specific stablecoin share: ms,g = Λ{ν [κc,g − κs,g]}.
  - Aggregate public-signal precision uses Qs = Σg πg ms,g QFXg with σ^2_s(Q_s) = ̄σ^2_s / (1 + ρ Q_s).
- Distributional results:
  - Stablecoins deliver larger private access gains to high-friction (unbanked) households because they compress the access-cost gap.
  - Information externality is aggregate and affects all households when any group deepens stablecoin depth.
  - In normal states stablecoins can be progressive (larger gains for high-friction households); in fragile states aggregate information externality reduces welfare for all.
- Quantitative heterogeneous-access exercise (selected table entries preserved):
  - Connected households (θ = 0.1, 0.5, 0.9):
    - Access gain: 0.002, 0.004, -0.005
    - Information effect: 0.005, 0.001, -0.060
    - Net welfare gain: 0.007, 0.005, -0.065
  - High-friction households (θ = 0.1, 0.5, 0.9):
    - Access gain: 0.006, 0.014, 0.010
    - Information effect: 0.005, 0.001, -0.060
    - Net welfare gain: 0.010, 0.015, -0.050

### Policy implications and recommendations (as presented)
- State-contingent approach:
  - Preserve low-cost stablecoin access in normal states to support inclusion and reduce reliance on opaque cash markets.
  - Use temporary, targeted frictions on large, speculative, or run-like flows when misalignment is high to limit the aggregate information externality and reduce coordination risk.
- Trade-offs:
  - Broad restrictions on stablecoins can be regressive by removing low-cost dollar access for households and small firms most constrained by the traditional system.
  - Unrestricted adoption can be fragile because aggregate stablecoin depth improves precision of a public signal that may coordinate exit when misalignment is high.
- Complementarity with macro policies:
  - These measures are not substitutes for macroeconomic adjustment; priority in fragile states remains on restoring credibility, easing FX scarcity, and moving official allocation closer to market-clearing conditions.

*Source: wpiea2026144*

### 2.1    Traditional Parallel Markets

### 2.1    Traditional Parallel Markets

### Characteristics and frictions
- Decentralized market structure: street dealers, informal brokers, bank-adjacent intermediaries, importers and exporters with access to foreign exchange, and firms or households willing to trade foreign-currency cash.
- Price dispersion: "A large bank, a retail cash dealer, and a street market can quote different prices for dollar value on the same day."
- Frictions generating dispersion:
  - Search costs that hinder household comparison of quotes.
  - Relationship-based access leading to different prices for larger firms or connected agents versus retail households.
  - Settlement risk creating wedges between quoted and executable prices.
  - Geographic segmentation of physical cash markets.
  - Regulatory risk leading traders to charge premia for the possibility that a transaction is delayed, blocked, or penalized.
- Implications for the model:
  - Traditional parallel markets are costly access technologies.
  - Prices from these markets are noisy signals of the aggregate state (local inventory, search, and enforcement risk affect quotes).
  - Before digital dollar markets become active, the economy may have parallel prices without a single transparent parallel price.

### 2.2    Stablecoin Exchanges

### Market transformation and economic margins
- Stablecoins create a digital claim to dollar-like value tradable against local currency; example instrument referenced: USDT.
- A household can use local currency to buy a stablecoin on an exchange or through a peer-to-peer desk; the local-currency price of USDT is an exchange rate: local currency per dollar-like digital claim.
- In countries with exchange controls, the stablecoin price becomes linked to the parallel dollar price (example: USDT/BOB as the local-currency price of Tether in bolivianos).
- Portfolio and payment dimensions:
  - Households can hold stablecoins as dollar-like stores of value.
  - Stablecoin transfers can settle transactions or obtain foreign-currency exposure outside official allocation channels.
- Model emphasis: portfolio and access margin, while allowing market depth to affect payments and price discovery.

### How stablecoin markets differ from cash markets
- More continuous: high-frequency trading across platforms.
- Easier to observe: posted quotes and platform prices provide a common reference even with incomplete transaction-level data.
- More scalable: a larger number of agents can access the same digital instrument than a particular physical cash dealer.

### Key economic tension and information effect
- Stablecoins reduce effective access frictions and integrate fragmented markets, producing a common public price.
- When many agents condition on the same price, it can act as a coordination device:
  - Stabilizing effect: reassures agents that dollar scarcity is limited.
  - Destabilizing effect: a large misalignment becomes commonly visible and can induce coordinated runs.
- Stablecoin markets change both the technology for acquiring foreign-currency exposure and the information structure of the parallel market.

### 2.3    Bolivia as a Motivating Case: From Dispersed FX Premia to a Stablecoin Reference Price

### Pre-stablecoin environment
- Bolivia maintained a long-standing fixed exchange rate against the U.S. dollar with declining foreign-exchange liquidity and rationed access to dollars, producing a widening gap between the official exchange rate and private-agent dollar-obtainment prices.
- Shadow price dispersion across seller-specific cash quotes, inventory constraints, bank commissions, transfer thresholds, and authorization frictions.
- Reported price dispersion examples (press reports):
  - La Paz episode: parallel dollar reportedly climbed to around Bs 15; several exchange houses had no dollars to sell while others quoted sale prices of Bs 13, Bs 12.50, and Bs 12.
  - Santa Cruz report: across eight exchange houses, dollar’s price varied between Bs 10, Bs 10.50, Bs 11, and Bs 11.30.
- In formal channels, FX premia appeared as privately quoted variable commissions charged by banks for foreign-exchange operations in offshore accounts.
- In mid-February 2024, Autoridad de Supervisión del Sistema Financiero (ASFI) began limiting banks’ commissions for FX sales to local agents based on transaction size, increasing pricing opacity.

### Post-policy change and stablecoin emergence
- Central Bank of Bolivia rescinded restrictions on electronic payment instruments for virtual-asset transactions in June 2024.
- Virtual-asset activity expanded quickly: BCB reported that transactions with these assets in the financial system multiplied twelvefold from July 2024 to May 2025 (Banco Central de Bolivia, 2025a).
- As USDT trading expanded, the USDT/BOB price became the common reference for the parallel exchange rate across the economy:
  - Appeared in public reporting on dollar scarcity.
  - Shaped formal and informal price-setting practices, including retail transactions where shops used USDT prices to set dollar–boliviano conversion rates.
- The BCB began publishing USDT prices on its website (Banco Central de Bolivia, 2025b), indicating USDT/BOB's salience as a monitored public reference price outside the official allocation system.

### 3    Model

### 3.1    Setup
- Static economy with a continuum of households i ∈ [0, 1]; fixed or heavily managed exchange rate.
- Degree of exchange-rate misalignment denoted θ ≥ 0: higher θ means the official exchange rate is more overvalued relative to the shadow market-clearing value of foreign currency.
- Misalignment effects:
  - Raises private value of holding foreign currency as insurance against future adjustment.
  - Weakens the regime’s ability to satisfy synchronized foreign-currency demand without a disorderly break.
- Two access technologies for foreign-currency exposure:
  - Traditional cash-based parallel market: opaque, fragmented, relationship-based, relatively costly.
  - Stablecoin market: lower transaction cost and generates an observable market price (public signal).
- Households’ three decisions: form beliefs about θ, choose how much foreign currency to acquire for hedging, decide whether to join a run. A crisis occurs when run-driven demand exceeds regime defense capacity. Model is static and captures the run decision at a point in time.

### 3.2    Information
- Households do not observe θ directly; prior:
  - θ ∼ N (μ_θ, σ^2_θ). (Equation (1))
- Private signal for household i:
  - x_i = θ + ε_i,   ε_i ∼ N (0, σ^2_x). (Equation (2))
- Public signal from the stablecoin market:
  - s = θ + ε_s,   ε_s ∼ N (0, σ^2_s). (Equation (3))
- Shocks ε_i and ε_s are independent of each other and of θ.
- Stablecoin activity improves public-signal precision:
  - Let Q_s denote aggregate stablecoin-denominated foreign-currency demand.
  - Public-signal variance: σ^2_s(Q_s) = ̄σ^2_s / (1 + ρ Q_s),   ρ > 0. (Equation (4))
  - Interpretation: thicker stablecoin market aggregates order flow and reveals a more precise public price.
- Cash-only benchmark: households observe a public parallel-market signal with variance fixed at baseline ̄σ^2_s; stablecoin adoption endogenizes public-signal precision.
- Posterior mean for household i:
  - ˆθ_i = ω_0 μ_θ + ω_x x_i + ω_s s. (Equation (5))
  - Weights:
    - ω_0 = σ^{-2}_θ / (σ^{-2}_θ + σ^{-2}_x + σ^{-2}_s(Q_s)). (Equation (6))
    - ω_x = σ^{-2}_x / (σ^{-2}_θ + σ^{-2}_x + σ^{-2}_s(Q_s)). (Equation (7))
    - ω_s = σ^{-2}_s(Q_s) / (σ^{-2}_θ + σ^{-2}_x + σ^{-2}_s(Q_s)). (Equation (8))
  - As stablecoin activity rises, σ^2_s(Q_s) falls and households place more weight on the common public signal.

### 3.3    Access Technologies and Effective-Cost Timing
- Aggregate foreign-currency demand denoted Q_FX.
- Per-unit cost functions for cash and stablecoins:
  - c_c(Q_FX) = k(Q_FX) + κ_c. (Equation (9))
  - c_s(Q_FX) = k(Q_FX) + κ_s. (Equation (10))
  - k′(Q_FX) ≥ 0. Common scale-dependent trading cost k(Q_FX) rises with aggregate FX demand; κ_j capture instrument-specific access frictions (search costs, platform frictions, settlement risk, regulatory costs).
- Stablecoins have an access advantage when κ_s < κ_c.
- Stablecoin share of demand:
  - m_s = Λ_h [ν (c_c(Q_FX) − c_s(Q_FX))] = Λ [ν (κ_c − κ_s)], Λ(z) = 1 / (1 + exp(−z)). (Equation (11))
  - Interpretation: because the common scale-dependent component is shared, instrument choice depends only on κ_c − κ_s. Appendix A.2 derives this from a logit instrument-choice problem with idiosyncratic platform or access costs.
- Quantities intermediated:
  - Q_s = m_s Q_FX,   Q_c = (1 − m_s) Q_FX. (Equation (12))
- Effective per-unit cost of foreign-currency access:
  - c_FX = (1 − m_s) c_c(Q_FX) + m_s c_s(Q_FX). (Equation (13))

### 3.4    Household Consumption and Utility
- Household i chooses ordinary hedge demand q^H_i ≥ 0 and a run decision r_i ∈ {0, 1}.
- Individual run demand: q^R_i = ̄r r_i. (Equation (14))
- Total individual foreign-currency demand: q^FX_i = q^H_i + q^R_i. (Equation (15))
- Utility linear in consumption-equivalent payoff: u_i(θ, s) = c_i(θ, s). (Equation (16))
- Consumption-equivalent payoff:
  - c_i(θ, s) = c_0 + A^H_i(q^H_i; θ) + r_i [B − (1 − I(θ, s)) τ_f (1 − θ) + I(θ, s) λ θ] − c_FX [q^H_i + ̄r r_i] − I(θ, s) λ θ ξ_Q Q_R. (Equation (17))
  - Interpretations:
    - c_0: baseline consumption.
    - I(θ, s): realized crisis indicator.
    - A^H_i(q^H_i; θ): service value of ordinary hedge balances.
    - B: private benefit of exit.
    - τ_f(1 − θ): false-run cost paid when the household runs but the regime survives; decreasing in θ.
    - λ θ: private crisis loss avoided by running when the regime breaks; increasing in θ.
    - c_FX(q^H_i + ̄r r_i): resource cost of acquiring total foreign-currency exposure.
    - ξ_Q Q_R: aggregate disruption from synchronized exit; households are atomistic and take Q_R as given (do not internalize their contribution).

### 3.5    Hedge Demand
- Hedge-service value:
  - A^H_i(q^H_i; ˆθ_i) = χ_h / η_h ˆθ_i q^H_i − 1/(2 η_h) (q^H_i)^2. (Equation (18))
  - Service value increases in misalignment and decreases in quantity of hedge balances.
- Household chooses q^H_i ≥ 0 to maximize A^H_i(q^H_i; ˆθ_i) − c_FX q^H_i. (Equation (19))
- First-order condition (interior):
  - χ_h / η_h ˆθ_i − 1/η_h q^H_i − c_FX = 0. (Equation (20))
- Non-negativity constraint gives:
  - q^H_i = max{0, χ_h ˆθ_i − η_h c_FX}. (Equation (21))
- Aggregate hedge demand in state (θ, s):
  - Q_H(θ, s, Q_s, Q_FX) = ∫ max [0, χ_h (ω_0(Q_s) μ_θ + ω_x(Q_s) x + ω_s(Q_s) s) − η_h c_FX(Q_FX, m_s)] dF(x | θ). (Equation (22))
  - Q_s affects hedge demand through public-signal precision via posterior weights; Q_FX affects it through access costs.
- Hedge component captures fundamentals-driven foreign-currency demand (non-strategic).

### 3.6    Run Demand
- Strategic motive: households may acquire foreign currency for exit; monotone strategies assumed with a private-signal cutoff x^* such that r_i = 1{x_i ≥ x^*}. (Equation (23))
- Aggregate run demand in state θ:
  - Q_R(θ, x^*) = ̄r [1 − Φ((x^* − θ)/σ_x)]. (Equation (24))
  - ̄r > 0 scales run intensity; Φ is the standard normal CDF.
- Regime breaks when run-driven demand exceeds effective defense capacity:
  - Q_R(θ, x^*) ≥ α − δ θ. (Equation (25))
  - α: baseline reserves or policy capacity; δ > 0 captures erosion of capacity as misalignment rises.
- Realized crisis indicator:
  - I(θ, s) = 1{Q_R(θ, x^*(θ, s)) ≥ α − δ θ}. (Equation (26))
- Household runs if expected consumption-equivalent gain from running is non-negative:
  - E[c_i(1) − c_i(0) | x_i, s] ≥ 0. (Equation (27))
- For the marginal household, expected gain:
  - G(x_i; s, Q_s) = B − c_FX ̄r + E[ I(θ, s) λ ̃θ − (1 − I(θ, s)) τ_f (1 − ̃θ) | x_i, s ]. (Equation (28))
  - Running insures against losses if regime breaks; premature exit entails false-run cost when crisis risk is low.
- Run-demand evaluation for candidate fundamental ̃θ:
  - Q_R(̃θ, x^*) = ̄r [1 − Φ((x^* − ̃θ)/σ_x)]. (Equation (29))
  - Marginal runner treats crisis risk as dependent on synchronized exit.
- Equilibrium cutoff satisfies:
  - G(x^*; s, Q_s) = 0. (Equation (30))
  - Boundary solutions arise when running is strictly dominant or strictly dominated over the relevant signal range.
- Public information affects run incentives via posterior beliefs and the crisis event: more precise public signals can align beliefs around adverse fundamentals, lower the cutoff, and raise aggregate run demand in fragile states or after bad public signals. 

*Italicized source: wpiea2026144 - 2.1    Traditional Parallel Markets*

### 3.7    Equilibrium and Welfare

### 3.7    Equilibrium and Welfare

### Equilibrium definition and fixed point
- An equilibrium for a given state (θ,s) is a tuple (Q_FX, Q_s, Q_c, x^*, m_s, Q_H, Q_R) satisfying:
  - Households update beliefs according to Bayes’ rule, taking the public-signal variance implied by stablecoin activity as given.
  - Stablecoin share: m_s = Λ[ν(κ_c − κ_s)].
  - Hedge demand Q_H aggregates individual hedge demand over the cross-sectional distribution of private signals.
  - Run cutoff x^*_s satisfies the marginal household’s indifference condition (boundary solutions allowed).
  - Run demand Q_R is generated by the cutoff strategy.
  - Total foreign-currency demand: Q_FX = Q_H + Q_R.
  - Access clearing: Q_s = m_s Q_FX, Q_c = (1 − m_s) Q_FX.
  - Public-signal precision consistent with stablecoin depth:
    σ_s^2(Q_s) = ̄σ_s^2 / (1 + ρ Q_s).
- Proposition 1: existence and local uniqueness under standard regularity conditions (see Appendix B.1).
- Feedback captured by the fixed point:
  - Stablecoin demand → public-signal precision → posterior beliefs → run cutoff → run demand → crisis exposure and total foreign-currency demand → common scale-dependent access costs → instrument choice via κ_c − κ_s.  
  - Information channel feeds back into stablecoin activity through Q_R and Q_FX.

### Social welfare
- Utilitarian welfare for state (θ,s):
  W(θ,s) = ∫_0^1 c_i(θ,s) = c_0 + A_H(Q_H; θ) + Q_R ̄r [B − (1 − I(θ,s)) τ_f (1 − θ) + I(θ,s) λ θ] − c_FX Q_FX − I(θ,s) λ θ ξ_Q Q_R.  (Equation (31))
- Welfare components reflect ordinary hedging benefits, run-related payoffs, access costs, and crisis-loss amplification by run demand.

### Mechanism: access and information channels
- Access channel:
  - If κ_s < κ_c, stablecoins raise the stablecoin share and lower effective per-unit foreign-currency access cost.
  - Lower access costs make hedges easier, raising ordinary hedging welfare (hedge-margin effect).
  - However, lower access costs also reduce the cost of run-related foreign-currency positions, increasing run risk in fragile states.
  - Net effect: positive on ordinary hedging margin; ambiguous when interacting with run margin.
- Information channel:
  - Stablecoin markets create a public price aggregating dispersed information about exchange-rate misalignment.
  - Improves hedge allocation (allocative benefit) by reducing mismatch between hedge demand and underlying state (Proposition 3).
  - But public information can coordinate runs: higher signal level raises perceived misalignment; higher precision makes beliefs more common and actions more correlated (Proposition 4).
  - As stablecoin depth rises, precision increases and belief dispersion compresses; coordination externality can raise crisis exposure even if average posterior beliefs fall.
- State-dependent welfare tradeoff:
  - In normal states: lower access costs + better hedge allocation dominate → stablecoins can raise welfare even if crisis exposure rises.
  - In fragile states: coordination externality can dominate → stablecoin depth may increase synchronized foreign-currency demand and crisis losses.
  - Therefore, stablecoins have a state-dependent welfare effect (not uniformly beneficial or harmful).

### 4    Quantitative Analysis

### Solution method
- Fully nested equilibrium computed numerically for θ ∈ [0,1].
- Inner loop: damped fixed-point iteration on z = (Q_s, Q_c, x^*); update posterior weights, integrate hedge demand over private signals, solve marginal-runner cutoff by bisection, update Q_R, clear demand across cash and stablecoin technologies.
- Outer loop: update aggregate public-signal variance implied by average stablecoin demand and iterate until variance is stable.
- Numerical checks: single-crossing condition verified on simulated grid; local-contraction checked via finite-difference Jacobian spectral radius < 1.

### Benchmark calibration (selected parameters reported in Table 1)
- μ_θ 0.46 (prior mean of misalignment)
- σ_θ 0.48 (prior standard deviation)
- σ_x 0.34 (private-signal standard deviation)
- ̄σ_s 1.36 (baseline public-signal standard deviation)
- ρ 4.39 (stablecoin-depth information elasticity)
- α 0.69 (baseline defense capacity)
- δ 0.09 (defense-capacity erosion with misalignment)
- τ_p 4.97 (smooth crisis-probability slope)
- χ_h 2.29 (hedge-demand sensitivity to beliefs)
- η_h 0.69 (hedge-demand sensitivity to costs)
- ̄r 0.45 (run-demand scale)
- B 0.01 (private benefit of early exit)
- τ_f 0.03 (false-run cost scale)
- λ 1.63 (crisis-loss severity)
- ξ_Q 13.51 (crisis-loss amplification by run demand)
- a 0.13 (common access-cost intercept)
- b 0.20 (common scale-cost slope)
- κ_c 0.18 (cash-market access friction)
- κ_s 0.04 (stablecoin access friction)
- ν 7.77 (cost sensitivity in instrument choice)
- c_0 5.00 (baseline normal-state consumption)

### Results — comparison of three economies
- Economies compared:
  - Cash-only benchmark: stablecoin access shut down, public-signal precision fixed at baseline.
  - Access-only stablecoin: stablecoins reduce access costs and intermediate foreign-currency demand, but stablecoin depth does not improve public-signal precision.
  - Full stablecoin: stablecoins reduce access costs and endogenously increase public-signal precision.
- Foreign-currency demand:
  - Total foreign-currency demand increases with θ.
  - Stablecoin demand rises with θ because stablecoins intermediate a cost-driven share of total demand; rising total demand increases stablecoin depth and public-signal informativeness.
- Crisis exposure (probability that run-driven foreign-currency demand overwhelms effective defense capacity):
  - Crisis exposure rises with θ in all three economies; increase is strongest in the full stablecoin economy.
  - Average crisis exposure over the misalignment interval:
    - Cash-only: 3.9 percent
    - Access-only stablecoin: 4.8 percent
    - Full stablecoin: 7.4 percent
  - At θ = 1:
    - Cash-only: 4.8 percent
    - Access-only: 7.0 percent
    - Full stablecoin: 12.9 percent
  - Interpretation: these are gross risk margins (not total welfare effects).
  - Mechanism: cheaper access makes exit easier; public price discovery makes exit easier to coordinate via higher precision and focality.
- Welfare decomposition (log welfare differences; interpreted approximately as percent changes in consumption-equivalent welfare):
  - Define:
    - ∆_access(θ) = log W_access(θ) − log W_cash(θ)
    - ∆_info(θ) = log W_full(θ) − log W_access(θ)
    - ∆_total(θ) = log W_full(θ) − log W_cash(θ)
  - Findings:
    - Total welfare effect is positive at low misalignment and peaks at intermediate misalignment (figure illustrates the decomposition into access and information channels).
    - Access channel tends to raise ordinary hedging welfare; information channel raises hedge allocation but can increase coordination risk and crisis losses in fragile states.

*Source: wpiea2026144 - 3.7    Equilibrium and Welfare.*

### 1.2 percent of consumption-equivalent welfare, around θ = 0.4, and then declines sharply.  It

### wpiea2026144 - 1.2 percent of consumption-equivalent welfare, around θ = 0.4, and then declines sharply.  It

### Welfare effects of stablecoins (state dependence)
- Total welfare effect:
  - Peaks at 1.2 percent of consumption-equivalent welfare, around θ = 0.4, and then declines sharply.
  - Crosses zero at the reversal threshold θ∗ = inf{θ : ∆total(θ)≤ 0} ≈ 0.59.
  - Reaches −6.3 percent at θ = 1.
- Interpretation:
  - Stablecoins can raise welfare in normal states (low misalignment) but reduce welfare once misalignment is sufficiently high.
  - Stablecoins raise crisis exposure throughout the displayed range, but at low misalignment access and hedge-allocation gains dominate the small increase in crisis exposure; at high misalignment coordination externalities overturn those gains.

### Channels: access component and information component
- Access component (lower-cost foreign-currency access, holding public-signal precision fixed):
  - Primarily reflects benefit of cheaper access to ordinary foreign-currency hedges.
  - Rises at low levels of misalignment because cheaper access expands valuable hedging demand when crisis risk is small.
  - Remains positive through most of the displayed range.
  - Turns slightly negative at the highest level of misalignment because cheaper access also facilitates run-related foreign-currency demand.
- Information component (endogenous public-signal precision):
  - Better information can improve ordinary hedge allocation.
  - The same public signal can create a coordination externality by synchronizing exit.
  - Mildly positive at low misalignment, turns negative before the total welfare effect crosses zero, then falls sharply as the economy becomes fragile.
  - Main force behind the welfare reversal: above the reversal threshold, information loss dominates access gain.
  - By high misalignment the information component reaches about −5.9 percent of consumption-equivalent welfare.

### Sensitivity (Table 2: sensitivity of crisis exposure and the welfare reversal)
- Notes: The crisis gap is the average, over the displayed misalignment grid, of full-stablecoin crisis exposure minus cash-only crisis exposure, reported in percentage points. The reversal threshold θ∗ is the first value of misalignment at which the total welfare effect of stablecoins is weakly negative.
- Reported experiments and outcomes (as presented):
  - Benchmark: Crisis gap 3.50, θ∗ 0.59
  - ρ +15%: Crisis gap 3.60, θ∗ 0.57
  - λ +15%: Crisis gap 4.00, θ∗ 0.40
  - ξQ +15%: Crisis gap 3.50, θ∗ 0.55
  - κc − κs −0.0203.10.63
- Mechanism-consistent patterns:
  - Higher information elasticity ρ raises the crisis gap and lowers the welfare-reversal threshold because stablecoin depth makes the public signal more precise.
  - Higher private crisis-loss parameter λ raises private running incentives (increasing crisis exposure) and increases welfare cost of crisis states; lowers reversal threshold.
  - Higher social run-amplification parameter ξQ mainly raises welfare cost of synchronized exit rather than private probability of crisis; lowers reversal threshold.
  - Reducing the stablecoin access-cost advantage (κc − κs) lowers the crisis gap and delays the reversal.

### Extension: heterogeneous access (two-type model)
- Setup:
  - Two household types g ∈ {B,U}: B = banked/connected, U = unbanked/high-friction.
  - Population shares πg with πB + πU = 1.
  - Access cost functions: cc,g(QFX) = k(QFX) + κc,g, cs,g(QFX) = k(QFX) + κs,g.
  - High-friction type has worse traditional access: κc,U > κc,B.
  - Stablecoins are inclusion-enhancing when κc,U − κs,U > κc,B − κs,B (stablecoins reduce relative access-cost disadvantage of high-friction type).
- Type-specific choices and aggregates:
  - Type g stablecoin share: ms,g = Λ{ν [κc,g − κs,g]}.
  - Effective per-unit cost: cFXg = k(QFX) + (1− ms,g)κc,g + ms,g κs,g.
  - Ordinary hedge demand: qHig = max{0, χh ˆθi − ηh cFXg}.
  - Type-specific foreign-currency demand: QFXg = QHg + QR.
  - Aggregate foreign-currency demand: QFX = Σg πg QFXg.
  - Aggregate stablecoin demand: Qs = Σg πg ms,g QFXg.
  - Public signal variance: σ̄s^2/(1 + ρ Qs) (presented as σ^2s(Qs) = ̄σ^2s 1 + ρQs).
  - Information externality: aggregate stablecoin depth raises public-signal precision observed by all households.
- Type-specific consumption-equivalent utility and welfare:
  - Consumption-equivalent utility: cig(θ,s) = c0 + AHi(qHig; ˆθi) + ri[B − (1− I(θ,s))τf(1− θ) + I(θ,s)λθ] − cFXg[qHig + ̄rri] − I(θ,s)λθ ξQ QR.
  - Type-specific welfare: Wg(θ,s) = ∫ cig(θ,s) di.

### Distributional incidence (qualitative results)
- Stablecoins deliver larger private access gains to households facing higher traditional FX frictions (type U) because stablecoins compress the access-cost gap between connected and less-connected users.
- The information externality is aggregate: adoption by any group deepens the stablecoin market, improves public-signal precision, and affects the run environment for all households.
- Net effect:
  - In normal states: access channel can make stablecoins progressive by benefiting high-friction households more.
  - In fragile states: aggregate information externality reduces welfare for all households.

### Quantitative heterogeneous-access exercise (Table 3 and calibration)
- Calibration used for the two-type exercise:
  - πB = 0.60, πU = 0.40, κc,B = 0.15, κc,U = 0.22, and κs,B = κs,U = κs.
  - These values preserve the benchmark aggregate traditional access friction: 0.60(0.15) + 0.40(0.22) = κc.
- Table 3 (log welfare differences; Access gain, Information effect, Net welfare gain as defined in Section 4):
  - Connected (θ = 0.1, θ = 0.5, θ = 0.9):
    - Access gain: 0.002, 0.004, -0.005
    - Information effect: 0.005, 0.001, -0.060
    - Net welfare gain: 0.007, 0.005, -0.065
  - High-friction (θ = 0.1, θ = 0.5, θ = 0.9):
    - Access gain: 0.006, 0.014, 0.010
    - Information effect: 0.005, 0.001, -0.060
    - Net welfare gain: 0.010, 0.015, -0.050
- Quantitative patterns:
  - At low misalignment stablecoins raise welfare for both types; larger gains for high-friction households.
  - At high misalignment the common information externality dominates and net welfare turns negative for both types.
  - Loss is larger for connected households because their access benefit is small and can turn negative once cheaper access facilitates run-related demand, while they remain exposed to the same aggregate crisis-risk externality. High-friction households also lose, but larger access gains partially cushion the decline.

### Policy implications and recommendations
- State-contingent approach:
  - Preserve low-cost stablecoin access in normal states to support inclusion and reduce reliance on opaque cash markets.
  - Use temporary, targeted frictions on large, speculative, or run-like flows when misalignment is high to limit the aggregate information externality and reduce coordination risk.
- Trade-offs:
  - Broad restrictions on stablecoins can be regressive in access terms because they remove low-cost dollar-access technology from households and small firms most constrained by the traditional system.
  - Unrestricted adoption can be fragile because aggregate stablecoin depth improves precision of a public signal that may coordinate exit when misalignment is high.
- Complementarity with macro policies:
  - These measures are not substitutes for macroeconomic adjustment. In fragile states priority remains on restoring credibility, easing foreign-exchange scarcity, and moving the official allocation system closer to market-clearing conditions.

### Conclusion (summary)
- Stablecoins function both as an access technology and an information technology: easier dollar-like claims and a visible, high-frequency public price that aggregates dispersed information about exchange-rate misalignment.
- Welfare consequences depend on interaction of access benefits and information-driven coordination externalities:
  - Low misalignment: access and hedge-allocation gains dominate; stablecoins can be welfare-improving.
  - High misalignment: precise public price compresses belief dispersion, strengthens coordinated exit incentives, and raises crisis-state losses; stablecoins can be welfare-reducing.
- Central message: in fragile exchange-rate regimes, financial innovations that improve access and price discovery can also change the coordination environment; preserve inclusion while limiting coordination risk.

*Source: wpiea2026144 (PDF chapter/section).*

### References

### wpiea2026144 - References

### A Model Microfoundations
- Purpose: Provide microfoundations for the tractable blocks used in the main text so that the asset-share and information objects can be derived from standard primitives.
- Endogenous public-signal precision:
  - Each order-flow observation j contains a signal z_j = θ + ν_j,  ν_j ∼ N (0,σ^2_ν). (33)
  - Order flow proportional to that signal: o_j = κ z_j. (34)
  - Normalized order flow:
    - Ō_s = 1/(κ n_s) ∫_0^{n_s} o_j dj = θ + 1/n_s ∫_0^{n_s} ν_j dj. (35)
  - Noise variance in normalized order flow = σ^2_ν / n_s, so stablecoin price reveals public signal s = θ + ε_s,  ε_s ∼ N (0, σ^2_ν / n_s). (36)
  - If effective order-flow depth increases in stablecoin usage, n_s = ̄n + ρ Q_s, (37) then
    - σ^2_s(Q_s) = σ^2_ν / (̄n + ρ Q_s). (38)
    - Setting ̄σ^2_s = σ^2_ν/̄n and rescaling ρ gives σ^2_s(Q_s) = ̄σ^2_s / (1 + ρ Q_s). (39)
- Stablecoin share (logit choice with type-I extreme-value shocks):
  - Indirect utilities: V_ic = −c_c(Q_c) + ε_ic, V_is = −c_s(Q_s) + ε_is. (40)-(41)
  - Stablecoin share:
    - m_s = exp{−ν c_s(Q_s)} / [exp{−ν c_c(Q_c)} + exp{−ν c_s(Q_s)}]. (42)
    - Equivalent share rule: m_s = Λ[ν( c_c(Q_c) − c_s(Q_s) )]. (43)

### B Characterization
- Equilibrium objects collected into z = (Q_s, Q_c, x^*). An equilibrium satisfies z = T(z; θ, s, ψ). (44)
- Cash-only benchmark: set stablecoin share = 0 and fix public-signal variance at baseline ̄σ^2_s. Stablecoin economy allows m_s > 0 and σ^2_s(Q_s) = ̄σ^2_s / (1 + ρ Q_s).
- Regularity conditions:
  - Assumption 1 (Single crossing): For any public signal s and public-signal precision τ_s, the marginal gain from running G(x^*; s, τ_s) is continuously differentiable and strictly increasing in private signal: G_x(x^*; s, τ_s) > 0.
  - Assumption 2 (Local contraction): In a neighborhood of the equilibrium, the equilibrium map T is continuously differentiable and its Jacobian with respect to z has spectral radius smaller than one: ρ(D_z T(z; θ, s, ψ)) < 1.
- Existence and local uniqueness (Proposition 1):
  - If Assumptions 1 and 2 hold, then for each state (θ, s) there exists an equilibrium (Q_s, Q_c, x^*, m_s, Q_H, Q_R).
  - If the equilibrium is interior, it is locally unique and differentiable in (θ, s, ψ).
- Access channel (Proposition 2):
  - Holding public-signal precision and the run cutoff fixed, a lower stablecoin access friction κ_s:
    - Raises the stablecoin share: ∂m_s / ∂κ_s < 0.
    - Lowers the effective per-unit cost of foreign-currency access: ∂c_FX / ∂κ_s |_{Q_FX} > 0 (implying c_FX falls when κ_s declines).
    - Weakly raises ordinary hedge demand: ∂q^H_i / ∂c_FX ≤ 0.
  - Stablecoins raise the hedge-demand component of consumption-equivalent welfare by lowering the cost of foreign-currency insurance; however, allowing the run cutoff to adjust can make the total effect state dependent because lower access costs also reduce private cost of acquiring run-related positions.
  - Stablecoin share expressed as m_s = Λ[ν(κ_c − κ_s)]. (Proof details provided.)
- Information channel:
  - Define public-signal precision τ_s(Q_s) ≡ σ^{-2}_s(Q_s) = ̄σ^{-2}_s (1 + ρ Q_s).
  - Posterior mean:
    - ˆθ_i = ω_0 μ_θ + ω_x x_i + ω_s s, with ω_s = τ_s / (σ^{-2}_θ + σ^{-2}_x + τ_s). (45)
  - Two parts of the information channel:
    - Level effect: The realized public signal s moves posterior beliefs up or down (∂ˆθ_i / ∂s = ω_s > 0).
    - Precision effect: Greater public-signal precision raises the posterior weight on the public signal (∂ω_s / ∂τ_s > 0).
  - Proposition 3 (Information, posterior beliefs, and hedge allocation):
    - Holding access costs and run cutoff fixed, public information improves allocation of ordinary foreign-currency hedges by making posterior beliefs more informative.
    - Holding public-signal precision fixed, a higher public signal raises each household’s posterior mean: ∂ˆθ_i / ∂s = ω_s > 0.
    - Holding realized signal fixed, greater precision raises posterior weight on s: ∂ω_s / ∂τ_s > 0.
    - Ordinary hedge demand q^H_i = max{0, χ_h ˆθ_i − η_h c_FX} is weakly increasing in posterior misalignment, so more informative public signals improve match between hedges and underlying state and raise expected demand-hedging welfare.
    - The posterior-level effect also affects exit incentives: higher public signal raises perceived misalignment and increases private incentive to acquire foreign-currency exposure.
  - Proposition 4 (Public-signal precision and coordination):
    - Stablecoin depth raises public-signal precision: ∂τ_s / ∂Q_s = ρ ̄σ^{-2}_s > 0.
    - Higher precision raises posterior weight on the common public signal: ∂ω_s / ∂τ_s > 0.
    - Around an interior cutoff equilibrium, the cutoff response is
      - ∂x^* / ∂τ_s = − G_{τ_s}(x^*; s, τ_s) / G_x(x^*; s, τ_s). (46)
    - Under single crossing (G_x > 0):
      - If G_{τ_s} < 0 (higher precision lowers marginal household’s perceived gain from running), then cutoff rises and run demand falls.
      - If G_{τ_s} > 0 (higher precision raises marginal household’s perceived gain from running), then cutoff falls and run demand rises.
    - Greater precision compresses cross-sectional belief dispersion and can increase crisis exposure through synchronization even if the public signal is individually reassuring in level terms.
  - Aggregate run demand:
    - Q_R(θ, x^*) = ̄r [1 − Φ( (x^* − θ) / σ_x )], and ∂Q_R / ∂x^* = − ̄r / σ_x φ( (x^* − θ) / σ_x ) < 0 (higher cutoff reduces run demand).
- Mathematical identities and key expressions preserved as in source:
  - σ^2_s(Q_s) = ̄σ^2_s / (1 + ρ Q_s).
  - τ_s(Q_s) = σ^{-2}_s(Q_s) = ̄σ^{-2}_s (1 + ρ Q_s).
  - Posterior weight on public signal: ω_s = τ_s / (σ^{-2}_θ + σ^{-2}_x + τ_s).
  - Stablecoin share: m_s = Λ[ν(κ_c − κ_s)].
  - Ordinary hedge demand: q^H_i = max{0, χ_h ˆθ_i − η_h c_FX}.

*Source: Excerpt from "References" and Appendices A and B in the supplied content unit.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026144.pdf_