## wpiea2025035-print-pdf - Section 3 characterizes the equilibrium path of the economy and show how interest

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### Model overview and environment
- Two-period model (periods indexed by t = 1, 2) with:
  - nominal frictions for firms (sticky prices; nominal wages fixed in period 1, fully flexible in period 2),
  - financial frictions for intermediaries: (i) an incentives-based leverage constraint generating a financial accelerator, and (ii) coordination-driven panic runs.
- Agents and assets:
  - Households, financial intermediaries, final good firms, intermediate good producers, government (including central bank).
  - Assets: equities (capital), long-term government bonds B (exogenous supply), bank short-term deposits (endogenous).
  - Exogenous physical capital K used in production in both periods; capital consumed by owners at end of period 2.
- Timeline (period 1): shocks realized → government announces policies → depositors decide whether to withdraw → run game outcome realized → consumption/labor/production/pricing choices → portfolio choices → period 2 outcomes.

### Key first-order conditions and firm/financial structure
- Households’ preferences (equation (1)):
  - max_{c1,c2,ℓ2} [ log(c1) − v1(ℓ1) + β ( log c2 − v2(ℓ2) ) ]  with v_t(.) increasing and concave.
- Household FOCs (equation (4)):
  - 1 = β R_2 / (1 + π_2) * c_1 / c_2
  - R_2 = (1 + r_L − β_L P_2 b_{h2} / B) / Q_{L1}
  - R_2 = (1 + r_{K2} − β_K P_2 k_{H2} / K) / Q_{K1}
  - where π_2 ≡ P_2/P_1 − 1.
- Intermediate firms:
  - Rotemberg quadratic price adjustment costs in period 1: θ_1 > 0, θ_2 = 0.
  - Pricing condition (equation (8)):
    - (ε_1−1) [ ε/(ε_1−1) MC_{1i} / P_{1i} − 1 ] = θ_1 π_1 (π_1 + 1)
    - P_{2i} = ε_2/(ε_2−1) MC_{2i}
- Financial intermediaries:
  - Incentive-compatible leverage constraint (equation (19)): φ_P N_1 ≥ Q_{L1} B_{F2} + Q_{K1} K_{F2}, φ = min(φ_P, φ_G).
  - Run microfoundation: N_0 ~ log-normal; depositor private signals η_j; run success condition (equation (13)) and equivalent threshold δ > \bar{δ}(N_0).

### Equilibrium characterization and core equilibrium equations
- Decentralized equilibrium summarized by:
  - Phillips Curve (PC),
  - Euler Equation (EE),
  - Balance Sheet Constraint (BSC) when bank equity low,
  - Run Equation (RE) in relevant state space.
- Key mechanism: interest rate tightening increases risk of runs and exacerbates balance sheet constraint, producing two additional price–financial stability trade-offs beyond standard output–inflation trade-off.

### Section 3.1 — The Trade-off Between Output and Inflation
- Period-1 equilibrium split:
  - Determinants of consumption, output and prices (PC and EE).
  - No-arbitrage, run condition, and BSC (determine asset holdings).
- Phillips Curve (PC) (25) (as derived):
  - (ε1 −1) / ε1 * ε1/(ε1−1) * W / ((1+π1)P0) * α [ (Y1/K)^(1−α) / α −1 ] = θ1 π1 (π1 +1)
  - Relates inflation π1 to output Y1 in period 1.
- Euler Equation (EE) (26):
  - Y1 [ 1 − θ2 π2 1 ] = C2 / (β R2).
- Static monetary trade-off:
  - PC and EE jointly determine π1 and Y1 as functions of C2 and R2.
  - Example mechanism: an increase in ε1/(ε1−1) raises π1 for given Y1; raising R2 lowers C1 and thus Y1 via EE — the usual output–inflation trade-off.

### Financial-frictions-implied wedges and policymaker objectives
- Policymakers’ instruments chosen after the shock but before the run game: {K_G, N_G, R_2}.
- Ramsey-optimal allocation maximizes:
  - E[(log C_1 − v(ℓ_1)) + β(log C_2 − v(ℓ_2)) | N̄_0] subject to competitive equilibrium constraints.
  - Expectation E(.) captures uncertainty about run risk; policymakers know mean N̄_0.
- Two wedges introduced by financial frictions:
  - σ: wedge from binding balance-sheet constraint (related to β_K P_2 K_H2 / K and exacerbated by increases in R_2).
  - ξ: wedge from run risk (probability of systemic run).

### Outside the constrained/run zones (ξ = 0, σ = 0)
- No role for credit policy or equity injection: K_G = N_G = 0.
- Social planner problem with v_1(ℓ_1) = χ log ℓ_1 and Y = ℓ^α K^{1−α}:
  - W = max_{Y_1, π_1, K_G, N_G} [ (1−χ/α) log Y_1 + log(1−θ_2 π_2^1) + β log C_2 ] s.t. 0 = PC(Y_1, π_1)
- Lemma 3 (Baseline) — optimal interest rate (equation 31 as presented):
  - R_2 = − PC_Y / PC_π
        θ π_1 (1−χ/α)(1−θ_2 π_2^1)^2 / (β C_2)
- Intuition: steeper Phillips curve or higher C_2 raises optimal R_2.

### Inside the constrained zone (BSC binds)
- When balance-sheet constraint binds, social planner solves with PC and BSC constraints; a wedge σ opens.
- Wedge and formulae:
  - σ tightly related to β_K P_2 K_H2 / K and increases with R_2.
- Lemma 4 (Constrained - optimal interest rate):
  - R_2 = Ω R̄_2  (32)
  - Ω = 1 / (1 + Ω_0 σ (dK_H2/dR_2)) < 1  (33)
  - Ω_0 = 1 + r_K2 P_2 [ 1−θ_2 π_2^1 ] [ 1−χ/α ] / Y_1  (34)
- Interpretation:
  - Ω_0 σ (dK_H2/dR_2) lowers optimal R_2 relative to R̄_2 (the no-frictions optimum).
  - Deviation larger when σ = β_K P_2 K_H2 / K (1 + r_K2) is larger, when household holdings K_H2/K are larger, and when β_K is larger.
- Optimal credit policy (Lemma 5):
  - If β_G > 0:
    - K_G = (β_K / β_G) K_H2 (BSC_{K_G} / BSC_{K_H2})
  - If β_G = 0: interest rate equals baseline (equation 31) and K_G > K_{G2} with K_{G2} given by the long expression in the source.
- Equity injection:
  - If β_N > 0, optimal R_2 same as baseline and:
    - N_G = (β_N / β_G) (K_H2 / K) (BSC_{N_G} / BSC_{K_H2})
  - If β_N = 0, optimal R_2 = baseline and minimum N_G given by exact expression in the source.
- Macroprudential, deposit insurance, and LOLR:
  - Relaxing φ_G can help if φ_G < φ_P; ineffective if incentive-based constraint binds.
  - Deposit insurance / LOLR less effective when distortion is a binding collateral constraint.
- Extreme cases:
  - If β_K → +∞, set K_H2 = 0 and R_2 deviates maximally to preserve intermediation.
  - If β_G = 0, public intermediation at no cost restores first best; interest rate follows baseline.

### Inside the run zone (ξ > 0)
- Policymaker maximizes expected welfare over good and run equilibria:
  - W = max_{...} (1−ξ)[ good-state welfare ] + ξ[ run-state welfare ] s.t. PC(Y_1, π_1) = PC(Y_1^*, π_1^*)
- Run introduces wedge ξ from coordination failure; policies chosen before run realization.
- Lemma 7 (Run - optimal interest rate policy) (equation (35)):
  - R_2 = (1 − ξ Ω_1) R̄_2 + ξ Ω_1 R_2 − ξ′ Ω_2 log(C_2^* / C_2)
  - with
    - Ω_1 = χ/α [ 1 − (1 − χ/α) R_2 / R_2 ]^{-1}
    - Ω_2 = α / χ (1 + β) (1 − χ/α)^{-1} Y_1^* Ω_1
    - ξ′ = ξ_Y + ξ_π − PC_Y / PC_π
- Interpretation:
  - Optimal R_2 set below R̄_2 to account for run risk via:
    1) ξ Ω_1 (R̄_2 − R_2): central bank trades off higher inflation in good state to avoid deeper recession in run state; acts like a weighted average of shadow rates.
    2) − ξ′ Ω_2 log(C_2^* / C_2): because ξ′ < 0 and log(C_2^* / C_2) < 0, this term further lowers R_2; captures R_2’s effect on run probability and welfare losses from a run.
- Lemma 8 (Run - other tools):
  - If β_G, β_N > 0:
    - N_G = (ξ_{N_G} / ξ_{K_H}) (β_H / β_N) (K_H / K)
    - K_G = (ξ_{K_G} / ξ_{K_H}) (β_K / β_G) K_H2
  - Intuition: equity injections and asset purchases used in proportion to costs 1/β_N, 1/β_G and efficacy at reducing ξ.
- Separation limits:
  - If β_N → +∞, N_G = 0 and central bank must moderate tightening.
  - If β_N = β_G = 0 (no-cost tools), full separation achievable: interest rate equals baseline (equation 31) and government injects equity/purchases assets until ξ = 0.
- Deposit insurance and LOLR:
  - Act as state-contingent asset purchases; lower effective β_N and improve separation.
- Historical example referenced: 2023 U.S. regional banking turmoil — FDIC systemic risk exception; U.S. Treasury allocated $25 billion to support a new Fed lending facility.

### Robustness (Section 5.5)
- Strict inflation targeting:
  - All qualitative results hold under strict inflation targeting.
  - When other tools are costly, central bank should adopt a less aggressive policy stance; optimal interest rate is lower in "constrained" zone and strictly decreasing in K_H2 and B_H2.
  - When tools are not costly, governments use them to remove frictions; equilibrium: households hold no assets, no spread between policy rate and asset returns, and C2 = Y2 + K; policy rate follows outside-zone path.
- Runs and policy under run risk:
  - Policies set before run realization; policymakers minimize expected squared deviations from target.
  - Appendix C shows optimal policy rate strictly lower than outside run zone; central bank tolerates some inflation in no-run path to avoid run-period disinflation — a risk-management approach.
  - Credit policy, equity injection, deposit insurance used to decrease ξ and raise C_2^*, allowing higher R_2.
- Large financial crisis (ξ = 1):
  - Banks lose all net worth N1 = 0 absent interventions; households intermediate all capital → drop in C2 and C1.
  - Central bank trade-off reduces to inflation stabilization vs preserving output; optimal R2 given by expression in source.
  - With β_G = 0, government should intermediate all assets: K_G = K.

### Empirical evidence (Appendix D) — monetary policy and financial instability
- Baseline LP specification (equation (49)):
  - C_{i,t+h} = α_{i,h} + β_h × Δr_{i,t} + Σ_{l=0}^L Γ_{h,l} X_{i,t−l} + ε_{i,t+h}.
  - Δr_{i,t} = change in nominal short-term interest rates (three-month yields/money market rates).
  - IV: Trilemma instrument z_{i,t} from Jordà et al. (2017); estimation via LP-IV.
- Supply vs demand shock decomposition:
  - Aggregate demand: ỹ_t = −α π_t + d_t.
  - Aggregate supply: π_t = β ỹ_t + η_t.
  - SVAR mapping: Az_t = Σ A_j z_{t−j} + ε_t (equation (50)).
  - Data: Jordà-Schularick-Taylor Macrohistory Database, annual data for 18 advanced economies, 1870–2016.
- Empirical findings:
  - Monetary policy tightening can exacerbate financial stability risks, particularly following supply shocks.
  - Unconditional effect: financial-crisis risk peaks at 2.5 percent one year after a one percentage point increase in short-term nominal rates.
  - Average unconditional annual probabilities: bank equity crash = 3.5 percent; banking panics = 3.4 percent.
  - Rate hikes increase probability of banking panics by 1 percentage point in the same year; bank equity crashes become significant three years after initial rate hike.
  - Channels: rate hikes lead to declines in real stock prices, real house prices, real bank loans; effects on credit and housing prices persist up to five years.
- Interpretation: both intermediary capacity constraints and depositor-run risks matter for how rate hikes affect financial instability; declines in asset prices and lending capacity are key transmission channels.

*Source: wpiea2025035-print-pdf - Section 3 described the financial-frictions-implied wedges(σ,ξ)and the trade-offs faced by central banks*

### Section 3 characterizes the equilibrium path of the economy and show how interest

### wpiea2025035-print-pdf - Section 3 characterizes the equilibrium path of the economy and show how interest

### Model overview and purpose
- Introduces a two-period model (periods indexed by t = 1, 2) with:
  - nominal frictions for firms (sticky prices; nominal wages fixed in period 1, fully flexible in period 2),
  - financial frictions for intermediaries: (i) an incentives-based leverage constraint generating a financial accelerator, and (ii) coordination-driven panic runs.
- Goal: analyze optimal conduct of interest rate policy and other tools when financial fragilities exist.
- Builds on Gertler and Kiyotaki (2015) and Gertler et al. (2020) and global-game literature on bank runs.

### Environment and timeline (Section 2.1)
- Agents: households, financial intermediaries, final good firms, intermediate good producers, government (including central bank).
- Assets: equities (capital), long-term government bonds B (exogenous supply), bank short-term deposits (endogenous).
- Exogenous physical capital K used in production in both periods; capital consumed by owners at end of period 2.
- Timeline (Figure 1 summary):
  - beginning of period 1: shocks realized → government announces policies → depositors decide whether to withdraw → run game outcome realized → consumers choose consumption and labor, firms choose production and prices → households and banks choose portfolios → period 2 production and consumption.

### Households: preferences, portfolio and first-order conditions (Section 2.2)
- Preferences:
  - max_{c1,c2,ℓ2} [ log(c1) − v1(ℓ1) + β ( log c2 − v2(ℓ2) ) ]  (equation (1)), with v_t(.) increasing and concave.
- Period-1 holdings and returns:
  - start with b_{h1}, k_{H1}, d_1; returns: Q_{L1}+r_L / Q_{L0}, Q_{K1}+r_{K1} / Q_{K0}, and R_1 on deposits (returns depend on run outcome).
  - If no successful run: return on deposits = \bar{R}_1; if successful and banks liquidated: R_1 = R^*_1 (random, depends on fraction withdrawing and bank asset values).
- Period-1 budget constraint (equation (2)):
  - P_1 c_1 + d_2 + Q_{L1} b_{h2} + Q_{K1} k_{H2} = R_1 d_1 + (Q_{L1}+r_L) b_{h1} + (Q_{K1}+r_{k1}) k_{H1} + T_1 + W_1 ℓ_1
- Period-2 budget constraint with quadratic portfolio-holding costs (equation (3)):
  - P_2 c_2 = R_2 d_2 + [1 + r_L − β_L 2 P_2 b_{h2} / B] b_{h2} + [1 + r_{k2} − β_K 2 P_2 k_{H2} / K] k_{H2} + T_2 + W_2 ℓ_2
- Key assumptions:
  - households less efficient at holding capital and bonds than intermediaries; quadratic costs parameterized as β_K 2 and β_L 2.
  - labor supply: elastic in period 1; v(ℓ2)=0 for ℓ2 < \bar{ℓ} and v(\bar{ℓ}) = +∞ ⇒ ℓ2 = \bar{ℓ}.
- First-order conditions (equation (4)):
  - 1 = β R_2 / (1 + π_2) * c_1 / c_2
  - R_2 = (1 + r_L − β_L P_2 b_{h2} / B) / Q_{L1}
  - R_2 = (1 + r_{K2} − β_K P_2 k_{H2} / K) / Q_{K1}
  - where π_2 ≡ P_2/P_1 − 1.

### Final and intermediate good firms (Sections 2.3–2.4)
- Final good firms:
  - CES technology with elasticity ε; take P_t and P_{ti} as given.
  - Demand for variety i: Y_{ti} = (P_{ti}/P_t)^{-ε} Y_t (equation (5)).
  - Free entry ⇒ zero profits; final output Y_t from goods market clearing.
- Intermediate good firms:
  - monopolistic competition; combine labor and capital, face Rotemberg quadratic price adjustment costs in period 1: θ_1 > 0, θ_2 = 0.
  - production: Y_{ti} = ℓ_{ti}^α K_{ti}^{1−α}; price demand relation (equation (7)).
  - Pricing conditions (equation (8)):
    - (ε_1−1) [ ε/(ε_1−1) MC_{1i} / P_{1i} − 1 ] = θ_1 π_1 (π_1 + 1)
    - P_{2i} = ε_2/(ε_2−1) MC_{2i}
  - Marginal cost MC_{ti} given by equation (9).
  - Transfers: (i) cost of price changes transferred to government; (ii) workers receive per-unit transfer so post-transfer labor earnings W_t ℓ_t = α P_{ti} Y_{ti} (equation (10)).
  - Dividends per unit equity and returns (equation (11)):
    - r_{kt} = (1−α) P_{ti} Y_{ti} / K_{ti}
    - R_{kt} = (1−α) P_{ti} Y_{ti} / K_{ti} + Q_{Kt+1} / Q_{Kt}

### Financial intermediaries and run microfoundation (Section 2.5)
- Intermediaries start period 1 with B_{F1}, K_{F1}, owe deposits D_1.
- Run game:
  - Depositors know own deposit holdings d_{1j} and overall bank balance sheet size, but not composition between deposits D_1 and equity N_0.
  - N_0 ~ log-normal: log N_0 ∼ N( log \bar{N}_0, σ_N ). Private signals η_j: log η ∼ N( log N_0, σ_η ). Posterior log N_0 ∼_j N( μ_{N0}(η_j, \bar{N}_0), σ_N^2 ) (equation (12)); denote density p(n|η_j, \bar{N}_0).
- Condition for successful run (equation (13)):
  - R^*_{k1} / Q_{K0} (K − K_{H1}) + R^*_{L1} / Q_{L0} (L − L_{H1}) < \bar{R}_1 D_1 δ
  - In the run equilibrium K^*_F2 = B^*_F2 = 0 (all assets held by households or government) ⇒ asset prices drop ex post.
- Asset prices in run equilibrium (equations (14)–(15)):
  - R^*_{k1} = r^*_{K1} + Q^*_{K1} / Q_{K0}
  - R^*_{L1} = r_L + Q^*_{L1} / Q_{L0}
  - Q^*_{K1} = 1 + r_{K2} − β_K P_2 (K − K_G)/K * R_2
  - Q^*_{L1} = 1 + r_L − β_L R_2
- Equivalent threshold for run share:
  - A run successful ⇔ δ > \bar{δ}(N_0) with
    - \bar{δ}(N_0) = [ R^*_{k1} / Q_{K0} (K − K_{H1}) + R^*_{L1} / Q_{L0} (L − L_{H1}) ] / [ \bar{R}_1 ( Q_{K0}(K−K_{H1}) + Q_{L0}(L−L_{H1}) − N_0 ) ]
- Depositor payoffs and equilibrium trigger strategy:
  - Running yields share of liquidation proportional to d_{1j} but incurs exogenous utility cost ζ.
  - Depositor cutoff signal \bar{η}; equilibrium run share δ^*(N_0) = F(\bar{η}|N_0).
  - Indifference condition for cutoff (equation (16)) (integral condition equated to ζ), and definition of \bar{N} (level where even universal run is unsuccessful):
    - \bar{N} = [(\bar{R}_1 − R^*_{k1}) Q_{K0} (K − K_{H1}) + (\bar{R}_1 − R^*_{L1}) Q_{L0} (L − L_{H1})] / \bar{R}_1.

### Intermediaries if no run (Section 2.6) and balance sheet constraint
- Period-1 equity if no successful run (equation (17)):
  - N_1 = \bar{R}_1 N_0 + (R_{k1} − \bar{R}_1) Q_{K0} K_{F1} + (R_{L1} − \bar{R}_1) Q_{L0} B_{F1} + N_G
  - N_G denotes government equity injection.
- Unconstrained arbitrage implication at end of period 1 (equation (18)):
  - R_2 = R_{K2} = R_{L2}
- Incentive-compatible leverage constraint (equation (19)):
  - φ_P N_1 ≥ Q_{L1} B_{F2} + Q_{K1} K_{F2}
  - φ_P > 0 exogenous; macroprudential policy can implement equity-based constraint φ_G; the effective leverage cap φ = min(φ_P, φ_G).
- When constraint binds, returns on bonds and capital exceed deposit rate: R_{K2} = R_{B2} > R_2 due to households being marginal buyers.

### Government, central bank tools, and costs (Section 2.7)
- Central bank controls deposit rate R_2.
- Government instruments: issue short-term deposits D_{G2}, purchase equities K_G, inject equity N_G, set transfers T_1, T_2; pays interest on long-term bonds and repays principal in period 2.
- Government budget constraints (equations (20)–(21)):
  - D_{G2} = T_1 + θ_{t2} π_2 / 1 P_t Y_t + r_L B + Q_{K1} K_G + N_G  (equation (20))
  - [1 + r_{k2} − β_G 2 P_2 K_G / K] K_G − β_N P_2^2 N_2^G = T_2 + (1 + r_L) L + (1 + R_2) D_{G2}  (equation (21))
- Use of tools (equity injections, credit policy, deposit insurance) is costly; modeled as quadratic pecuniary losses in government budget to capture fiscal costs, moral hazard, mispricing/addiction risks.

### Market clearing (Section 2.8)
- Goods market period 1 (equation (22)):
  - Y_1 [ 1 − θ_2 π_2^1 ] = C_1  where C_1 = ∫ c_{1j} dj
- Normalization and assumption for period 2 (equation (23)):
  - π_2 = 0, set P_2 = P_1 and let W_2 adjust so real wage clears labor market ⇒ abstracts from goods-price inflation between period 1 and 2.
- Asset markets clear (equation (24)):
  - K = K_{H2} + K_{F2} + K_G
  - B = B_{H2} + B_{F2}
  - D_2 = D_{F2} + D_{G2}
  - where D_2 = ∫ d_{2j} dj, B_{H2} = ∫ b_{H2j} dj, K_{H2} = ∫ k_{H2j} dj.

### Characterization of equilibrium and trade-offs (start of Section 3)
- Decentralized equilibrium summarized by a small set of equations:
  - Phillips Curve (PC),
  - Euler Equation (EE),
  - Balance Sheet Constraint (BSC) when bank equity low,
  - Run Equation (RE) in relevant state space.
- Key mechanisms and trade-offs:
  - Interest rate tightening increases risk of runs and exacerbates balance sheet constraint.
  - Two price–financial stability trade-offs emerge in addition to the standard output–inflation trade-off:
    - (i) Interest rate changes affect inflation via Phillips Curve and consumption via Euler Equation.
    - (ii) Interest rate tightening can raise the likelihood of panic-driven runs (RE) and tighten intermediary leverage (BSC), amplifying financial fragility.
- The paper highlights the wedges produced by the two financial frictions (leverage constraint and run coordination) and analyzes implications for optimal policy (further examined in Section 5).

*Source: wpiea2025035-print-pdf - Section 3 characterizes the equilibrium path of the economy and show how interest*

### 3.1    The Trade-off Between Output and Inflation

### 3.1    The Trade-off Between Output and Inflation

### Equilibrium partition and focus
- Period 1 equilibrium conditions split into two subsets:
  - Determinants of consumption, output and prices.
  - No-arbitrage across assets, the run condition, and the balance sheet constraint (determine asset holdings by intermediaries and households).
- The analysis in this section focuses on the first subset to highlight the trade-off between output and inflation.

### Phillips Curve (PC)
- Derived by combining:
  - Intermediary firm’s optimal pricing condition (8),
  - Final goods firms’ optimality condition (5),
  - Symmetry of intermediary firms and production function Y1i = Y1,
  - Definition of marginal cost (9),
  - Wages fixed in period 1.
- Resulting Phillips Curve (PC)(25):
  - (ε1 −1) / ε1 * ε1/(ε1−1) * W / ((1+π1)P0) * α [ (Y1/K)^(1−α) / α −1 ] = θ1 π1 (π1 +1)
  - The Phillips Curve relates inflation π1 to output Y1 in period 1.

### Euler Equation (EE)
- Household intertemporal optimality condition (4) and goods market clearing (22) yield (EE)(26):
  - Y1 [ 1 − θ2 π2 1 ] = C2 / (β R2).
- The Euler Equation links period-1 output Y1 (through consumption C1 via market clearing) to C2 and the policy rate R2.

### Static trade-off faced by monetary policy
- The Phillips Curve and Euler Equation jointly determine π1 and Y1 as functions of C2 and R2.
- Mechanism when a markup shock increases (an increase in ε1/(ε1−1)):
  - From the Phillips Curve, inflation π1 increases for a given Y1.
  - If the central bank raises R2, the Euler Equation implies C1 and thus Y1 decrease.
- Conclusion: central banks face the well-known static trade-off between output and inflation when setting R2 following a markup shock.

*Source: wpiea2025035-print-pdf - 3.1    The Trade-off Between Output and Inflation*

### Section 3 described the financial-frictions-implied wedges(σ,ξ)and the trade-offs

### Section 3 described the financial-frictions-implied wedges(σ,ξ)and the trade-offs faced by central banks

### Policymakers’ objectives and instruments
- Baseline objective: maximize expected households’ welfare, a weighted sum of (log) consumption in the current and future period and inflation, subject to the restriction that the allocation is a competitive equilibrium.
- Instruments chosen after the shock but before the run game: {K_G, N_G, R_2}.
- Definition 1 (Optimal Policies): A Ramsey-optimal allocation is a set of quantities {Y_1, C_1, Y_2, C_2, ℓ_1, ℓ_2, K_F2, K_H2, L_F2, L_H2}, policies {K_G, N_G, R_2} and prices/returns {W_1, W_2, π_1, π_2, Q_k1, Q_L1, r_K1, r_K2} that solve the policymakers’ maximization of
  E[(log C_1 − v(ℓ_1)) + β(log C_2 − v(ℓ_2)) | N̄_0]
  subject to competitive equilibrium constraints.
- Expectation E(.) captures only uncertainty about the risk of a run; policymakers know the mean N̄_0 of the distribution.
- Alternative objective: central bank minimizes deviation of inflation from target π̄ while government chooses other tools to maximize welfare; qualitative results robust (discussed in section 5.5).

### Outside of the run and the constrained zones (ξ = 0, σ = 0)
- Regime description: occurs when markup shock is small or banks are well-capitalized (section 3.2); no role for credit policy or equity injection (K_G = N_G = 0).
- Social planner simplifies to a maximization problem subject to the Phillips curve and Euler equation relating R_2 and C_1.
- With labor disutility v_1(ℓ_1) = χ log ℓ_1 (30) and production Y = ℓ^α K^{1−α}, social planner problem:
  W = max_{Y_1, π_1, K_G, N_G} [ (1−χ/α) log Y_1 + log(1−θ_2 π_2^1) + β log C_2 ] s.t. 0 = PC(Y_1, π_1)
- First-order condition (MRS = MRT): (1−χ/α)(1−θ_2 π_2^1) θ π_1 / Y_1 = − PC_Y / PC_π.
- Lemma 3 (Baseline): optimal interest rate
  R_2 = − PC_Y / PC_π
        θ π_1 (1−χ/α)(1−θ_2 π_2^1)^2 / (β C_2)
  (equation 31 as presented)
- Intuition: steeper Phillips curve (e.g., larger cost-push shock ε_1/(ε_1−1)), higher future consumption C_2, and other parameters raise the optimal R_2.

### Inside the constrained zone (balance sheet constraint binds)
- Regime description: balance sheet constraint binds when inflation shock requires larger rate hike or banks are less well-capitalized (section 3.2); economy characterized by Phillips curve, Euler equation, and Balance Sheet Constraint (BSC).
- Social planner problem:
  W = max_{Y_1, π_1, K_G, N_G, K_H2} [ (1−χ/α) log Y_1 + log(1−θ_2 π_2^1) + β C_2(K_H2, B_H2, K_G, N_G) ] s.t. 0 = PC(Y_1, π_1) and 0 = BSC(K_H2, B_H2, K_G, Y_1, N_0, N_G)
- When BSC binds, a wedge σ opens between efficient and actual allocations; σ tightly related to β_K P_2 K_H2 / K and exacerbated by increases in R_2.
- Policymakers should deploy credit policy or equity injection to close the wedge and moderate policy rate hikes when these tools are costly.

Findings / formulae:
- Lemma 4 (Constrained - optimal interest rate). When β_G, β_N > 0:
  R_2 = Ω R̄_2  (32)
  with Ω = 1 / (1 + Ω_0 σ (dK_H2/dR_2)) < 1  (33)
  Ω_0 = 1 + r_K2 P_2 [ 1−θ_2 π_2^1 ] [ 1−χ/α ] / Y_1  (34)
- Interpretation:
  - The term Ω_0 σ (dK_H2/dR_2) induces lowering of the optimal interest rate relative to R̄_2 (optimal without financial frictions).
  - Deviation larger when spread σ = β_K P_2 K_H2 / K (1 + r_K2) is larger, when household holdings K_H2/K are larger, and when β_K is larger (costlier household holdings).
  - Sensitivity dK_H2/dR_2 captures how banking intermediation capacity responds to R_2; mathematically linked to BSC partials and the indirect effect through inflation.

Policy tools and optimal use:
- Lemma 5 (Constrained - Credit Policies). If β_G > 0, optimal credit policy:
  K_G = (β_K / β_G) K_H2 (BSC_{K_G} / BSC_{K_H2})
  - Credit policy used in proportion to relative cost 1/β_G, households’ holdings K_H2, households’ inefficiency β_K, and relative efficacy BSC_{K_G}/BSC_{K_H2}.
- If β_G = 0: interest rate equals baseline (equation 31) and K_G > K_{G2} with K_{G2} given by the long expression in the source (do not omit; exact expression preserved in source).
- Extreme cases:
  - If other tools prohibitively costly (β_K → +∞), implement K_H2 = 0 and largest deviations of R_2 to preserve intermediation—separation of objectives impossible.
  - If β_G = 0, perfect separation achieved: public intermediation at no cost restores first best; interest rate set per baseline while K_G addresses BSC.
- Equity injection:
  - If β_N > 0, optimal R_2 same as baseline (equation 31) and optimal equity injection:
    N_G = (β_N / β_G) (K_H2 / K) (BSC_{N_G} / BSC_{K_H2})
    (formula as presented in source)
  - If β_N = 0, optimal R_2 same as baseline and minimum N_G given by the exact expression in the source (preserves full formula).
- Macroprudential and other tools:
  - Relaxing capital requirements ex post by increasing φ_G can raise intermediaries’ capacity if φ_G < φ_P (i.e., regulation binding); ineffective if incentive-based constraint binds.
  - Deposit insurance and lender-of-last-resort less effective when distortion is binding collateral constraint.
- Historical example: U.S. Savings and Loan crisis — Federal Reserve increased lending support to solvent banks, helping maintain operations amid liquidity shortages while keeping tight monetary stance.

### Inside the run zone (ξ > 0)
- Regime description: positive probability of systemic run occurs when inflation shock requires big rate hikes or banks poorly capitalized (section 2.5).
- Policymakers choose policy rate before run outcome; maximize expected welfare over good and run equilibria:
  W = max_{Y_1, Y_1^*, π_1, π_1^*, K_H2, K_H2^*, N_G, K_G} [
    (1−ξ)[ (1−χ/α) log Y_1 + log(1−θ_2 π_2^1) + β log C_2 ] + ξ[ (1−χ/α) log Y_1^* + log(1−θ_2 (π_1^*)^2) + β log C_2^* ]
  ] s.t. 0 = PC(Y_1, π_1) = PC(Y_1^*, π_1^*)
- Run introduces additional wedge ξ between efficient and actual allocations from coordination failures.
- When other tools available, they should be used proportionally to costs to reduce distortions and run probability; if costly, central bank must internalize effect of R_2 on run probability ξ.

Findings / formulae:
- Definitions: R̄_2 = θ π_1 C_2 (1−χ/α) β (1−θ_2 (π_1)^2)^2 − PC_Y / PC_π (shadow optimal in no-run state); R_2 = analogous in run state with starred variables.
- Lemma 7 (Run - optimal interest rate policy). When β_G, β_N > 0:
  R_2 = (1 − ξ Ω_1) R̄_2 + ξ Ω_1 R_2 − ξ′ Ω_2 log(C_2^* / C_2)  (35)
  with
    Ω_1 = χ/α [ 1 − (1 − χ/α) R_2 / R_2 ]^{-1}
    Ω_2 = α / χ (1 + β) (1 − χ/α)^{-1} Y_1^* Ω_1
    ξ′ = ξ_Y + ξ_π − PC_Y / PC_π
  (ξ′ is total derivative of run probability w.r.t. Y_1 through R_2, including inflation effect.)
- Interpretation:
  - Optimal R_2 should be below R̄_2 to account for run risk via two mechanisms:
    1) ξ Ω_1 (R̄_2 − R_2) : weight on gap between shadow optimal rates; central bank tolerates higher inflation in good state to avoid deeper run-state recession. Acts like weighted average of shadow rates with weight (1−ξ Ω_1) on good state and ξ Ω_1 on run state (Ω_1 < 1).
    2) − ξ′ Ω_2 log(C_2^* / C_2) : because ξ′ < 0 and log(C_2^* / C_2) < 0, this term lowers R_2 further; captures how R_2 affects run probability.
  - First mechanism implies monetary policy should adopt a risk-management stance even if ξ′ = 0.
  - Second mechanism scales with welfare losses given a run (1 + β) log(C_2^* / C_2).

Policy tools and optimal use in run zone:
- Lemma 8 (Run - other tools). If β_G, β_N > 0:
  - Optimal equity injection:
    N_G = (ξ_{N_G} / ξ_{K_H}) (β_H / β_N) (K_H / K)
  - Optimal asset purchases:
    K_G = (ξ_{K_G} / ξ_{K_H}) (β_K / β_G) K_H2
  - Intuition: equity injections and asset purchases should be used in proportion to their costs 1/β_N, 1/β_G and efficacy at reducing run probability ξ′_{N_G}, ξ′_{K_G}. Both tools lower run risk by strengthening balance sheets and preventing fire-sale driven price declines.
- Extreme cases:
  - If β_N → +∞ (equity injections prohibitively costly), N_G = 0 and separation impossible: central bank must moderate tightening to account for run risk.
  - If β_N = β_G = 0 (no-cost tools), policymakers can fully achieve financial stability objective and separate it from price/output stabilization: interest rate equals baseline (equation 31) and government injects equity or purchases assets until ξ = 0.
- Lemma 9 (Run - complete separation):
  - If β_N = 0, optimal R_2 = baseline (equation 31) and equity injection N_G > N̄ − N_0.
  - If β_G = 0, optimal R_2 = baseline and asset purchases K_G satisfy the exact inequality given in the source (preserves formula).
- Deposit insurance and lender-of-last-resort:
  - Viewed as state-contingent asset purchases (contingent credit lines); because they are activated only when run occurs, they correspond to a lower effective β_N and hence greater separation and more aggressive use of tools.
- Historical example: 2023 U.S. regional banking turmoil — FDIC invoked systemic risk exception to guarantee insured and uninsured deposits at Silicon Valley Bank and Signature Bank; U.S. Treasury allocated $25 billion to support the Federal Reserve’s newly established lending facility.

*Italic source: wpiea2025035-print-pdf - Section 3 described the financial-frictions-implied wedges(σ,ξ)and the trade-offs faced by central banks*

### 5.5    Robustness

### 5.5    Robustness

### Strict Inflation Targeting
- All previous qualitative results hold when the central bank follows a strict inflation targeting mandate.
- If other tools are costly to use, the central bank should adopt a less aggressive policy stance. (Formal proof in Appendix C.)
- When tools are costly:
  - The optimal interest rate is lower in the "constrained" zone than outside and is strictly decreasing in K_H2 and B_H2.
  - Mechanism: a binding balance sheet constraint depresses future consumption (C2 < Y2 + K), which depresses current consumption (C1 = C2/(βR2)). Hence the interest rate does not need to be as high to control inflation.
  - Credit policy and equity injection can help offset the loss of consumption in period 2.
- If other tools are not costly:
  - Governments should use them to address financial distortions from intermediaries’ financial constraints.
  - In equilibrium: households shouldn’t hold any assets, there shouldn’t be any spread between the policy rate and asset returns, and C2 = Y2 + K.
  - This implies the policy rate should follow the same path as outside the "constrained" zone.

### Runs and policy under run risk
- In the case of a run, the inflation rate depends on whether a run materializes.
- Since policies are set before the realization of a run, policy-makers cannot achieve perfect ex post price stabilization; they minimize expected squared deviations from target.
- In Appendix C it is shown that the optimal policy rate is strictly lower than the one outside of the run zone.
  - Rationale: the central bank tolerates some inflation in the no-run path to avoid a drop in inflation in a run period, reducing average deviation.
  - This illustrates a risk-management approach: monetary policy should be less aggressive when there is run risk.
- When credit policy, equity injection, or deposit insurance are available:
  - They should be used to decrease run risk by boosting asset prices, strengthening intermediaries’ balance sheets, or directly reassuring depositors.
  - These actions improve intermediation and raise consumption in period 2, which boosts inflation in the run scenario and leads the central bank to raise its policy rate.

### Large financial crisis (run with probability one)
- Modelled as ξ = 1 (run happens with probability one).
- In equilibrium, banks lose all their net worth: N1 = 0.
- Absent government interventions, all capital is intermediated by households, resulting in a drop in consumption in period 2, and hence in period 1.
- Given the run and output disruption occur independently of monetary policy stance, the central bank’s only trade-off is between inflation stabilization and preserving output—the same trade-off outside the constrained and run zones.
- The optimal interest rate policy in this case is:
  - R2 = − θπ ∗1 (1− θ2 (π∗1)2 )(1− χ α ) PC ∗ Y PC ∗ π C ∗2 .
- When credit policies are available:
  - They should intermediate part of private assets and minimize the drop in C∗2.
  - In the extreme case where the government can intermediate assets βG = 0, it should hold all assets in the economy K G = K. (Further details in Appendix B.4.)

### Key methodological and policy implications (from Conclusion)
- The paper presents a two-period NK framework with a financial sector financed by short-term deposits investing in long-term assets, facing a leverage constraint and run risk.
- The model explains how monetary tightening during high inflation and financial vulnerability can increase financial instability, erode bank equity, and trigger banking panics.
- The model characterizes optimal combinations of interest rate policy, credit policy, equity injections, deposit insurance, and macroprudential measures, yielding intuitive expressions for optimal policy design.
- Policymakers can theoretically fully separate price and financial stability goals using alternative tools (credit policy, equity injections, deposit insurance, macroprudential measures) only if these tools can be deployed without costs.
- With implementation challenges (costly tools), policymakers should adopt a mixed approach: a less aggressive interest rate policy combined with sound use of alternative tools.
- The degree to which interest rate policy must accommodate financial stability, and the extent to expand alternative tools, depends on the severity of financial vulnerabilities and the types of tools available.

*Source: wpiea2025035-print-pdf - 5.5    Robustness*

### References

### wpiea2025035-print-pdf - References (Appendices A–D)

### Global game microfoundation (Appendix A)
- Setup and information:
  - Depositors enter period 1 with holdings K_H1, B_H1, D_1.
  - True bank equity N_0 drawn from log-normal around ̄N_0 with dispersion σ_N:
    - log N_0 ∼ N(log ̄N_0, σ_N) (equation (36)).
  - Depositors receive idiosyncratic log-normal signals η about N_0:
    - log η ∼ N(log N_0, σ_η) (equation (37)).
  - Posterior on N_0 conditional on signal η is log-normal with mean μ_N0(η) and variance σ^2_NP (equation (38)).
- Run outcome condition:
  - A run is successful if bank liquidation cannot repay running depositors:
    - R^*_k1 Q_K0 (K−K_H1) + R^*_L1 Q_L0 (L−L_H1) < ̄R_1 D_1 δ (equation (39)).
  - Definitions used:
    - Q^*_K1 = 1 + r^*_K2 − β_K + (β_K − β_G) K_G / (K R_2) and Q^*_L1 = 1 + r_L R_2 (equation (40)).
    - R^*_k1 = r_K1 + Q^*_K1 / Q_K0 and R^*_L1 = r_L + Q^*_L1 / Q_L0 (equation (41)).
  - Threshold share for successful run:
    - δ > ̄δ(N_0) with ̄δ(N_0) given by equation (42).
  - Define ̄N as the net worth level such that even if all depositors run, the run is unsuccessful.
- Trigger strategy equilibrium:
  - Depositors run iff their signal is below common threshold ̄η.
  - Equilibrium run mass δ^*(N_0) = F(̄η|N_0).
- Depositor payoffs and indifference condition:
  - Payoffs depend on successful vs unsuccessful run and on running/not running; indirect utility U(I) defined.
  - Indifference condition for threshold signal ̄η given by equation (43).
- Ex ante run probability:
  - ξ = P(δ^*(N_0) ̄R_1 D_1 > R^*_k1 Q_K0 (K−K_H1) + R^*_L1 Q_L0 (L−L_H1) | ̄N_0) (equation (44)).
  - D_1 = Q_K0 (K−K_H1) + Q_L0 (L−L_H1) − N_0 (equation (45)).
- Dependences emphasized:
  - ξ is a function of ̄N_0, Y^*_1, π^*_1 via r^*_k1, and of K_G and R_2 through Q^*_K1.

### Optimal policy and model proofs (Appendix B)
- Outside "Constrained" and "Run" zones:
  - Condition (27) rewritten as LHS(N_0) > RHS(R_2) with explicit expressions; existence and monotonicity of ̄R_2(N_0) such that LHS = RHS.
  - Regularity conditions (Assumption 1):
    - Households hold positive deposits D_1 > 0.
    - Phillips curve upward-sloping and not too steep; explicit elasticity condition θ π_2 1  1 2 + ε_{π1}/Y_1 .
    - Households’ holdings not too large: φ^{−1}/φ > max(K_H1/K, B_H1/B).
  - r_K1(.) and r_K2(.) shown decreasing in R_2 via market-clearing and Phillips curve.
- Optimal rate outside constraints:
  - First-order condition yields:
    - R_2 = − (PC_Y / PC_π) θ π_1 (1 − θ^2 π_1^2)^{-1} C_2 / βR_2 (expression derived in text; baseline term and additional components shown).
- Inside "Constrained" zone:
  - Define balance-sheet constraint BSC(K_H2, K_G, π_1, Y_1, R_2, N_0, N_G) = LHS − RHS with detailed RHS expression including r_K1(π_1, Y_1).
  - BSC is increasing in Y_1 and π_1.
  - Policy-maker problem:
    - Maximize welfare W s.t. PC(Y_1, π_1) = 0 and BSC = 0; Lagrange multipliers λ, μ.
    - FOCs yield expression for R_2 with components:
      - Baseline term: −PC_Y/PC_π θ π_1 (1 − θ^2 π_1^2) C_2 / β.
      - Cost term −β_K (K_H2 / K).
      - Sensitivity term involving BSC_Y/BSC_{K_H2} and dK_H2/dR_2.
    - Relationship between wedge σ and household holdings:
      - σ = 1 − R_2 / R_k2 = β_K P_2 K_H2 / (K (1+r_K2)).
    - dK_H2/dY_1 and dK_H2/dR_2 relations (equations (46) and (47)).
  - Credit policy FOCs:
    - Optimal K_G when β_G > 0: K_G = (β_K / β_G) (K_H2) × [BSC_{K_G} / BSC_{K_H}] (expression structure provided).
    - When β_G = 0, μ = 0 ⇒ banks’ balance-sheet constraint has no welfare cost and optimal K_H2 = 0; interest-rate rule equals outside constraint case.
  - Minimum K_G required for households to hold no asset K_H2 = B_H2 = 0 given by an explicit formula (in text).
  - Equity injection (N_G) when β_G > 0:
    - N_G = (β_N / β_G) K_H2 × [BSC_{N_G} / BSC_{K_H}] (structure given).
    - Minimum N_G to achieve K_H2 = B_H2 = 0 given by explicit expression in text.
- Inside the "Run" zone:
  - Additional constraint C_1/C_2 = C^*_1 / C^*_2 (Euler equality across states).
  - Planner maximizes weighted welfare (1−ξ) and ξ terms; FOCs yield system tying π_1, π^*_1, C_2, C^*_2, K_G, N_G, K_H2.
  - Derived expression for R_2 in run zone:
    - R_2 = ̄R_2 − ξ Ω_1 (̄R_2 − R_2) + ξ′ Ω_2 log(C^*_2 / C_2) with definitions:
      - ̄R_2 = θ π_1 C_2 (1−χ/α)β (1 − θ^2 π_1^2)^{−2} − PC_Y / PC_π (expressed in text).
      - R_2 (run) = θ π^*_1 C^*_2 (1−χ/α)β (1 − θ^2 (π^*)^2)^{−2} − PC^*_Y / PC^*_π.
      - Ω_1 = χ/α [1 − (1−χ/α) R_2/R_2]^{−1}; Ω_2 = (1+β)(1−χ/α) Y^*_1 [1 − (1−χ/α) R_2/R_2]^{−1}.
      - ξ′ = ξ_Y + ξ_π − PC_Y/PC_π.
  - Optimal credit policy relation:
    - K_G / K = Ω_3 ξ_{K_G} β_G^{-1} log(C^*_2 / C_2) with Ω_3 defined in text.
  - Ratios for N_G and K_H2 relative to K_G:
    - ξ_{N_G} / ξ_{K_G} = K β_N N_G / (β_G K_G).
    - ξ_{K_H2} / ξ_{K_G} = K β_K K_H2 / (β_G K_G).
- Large financial crisis (ξ = 1) — Lemma 10 and 11:
  - Lemma 10 (Large crisis - interest rate policy):
    - When other tools are costly β_K, β_N > 0, optimal rate:
      - R_2 = − θ π^*_1  1 − θ^2 (π^*_1)^2  (1 − χ/α)^{-1} PC^*_Y / PC^*_π C^*_2 (equation given in text).
    - Intuition: two opposing effects — markup shock pushes R_2 up (PC^*_Y / PC^*_π term), large crisis (drop in C^*_2) pushes R_2 down.
  - Lemma 11 (Large crisis - other tools):
    - Optimal credit policy and implied C^*_2:
      - K_G = β_K / (β_G + β_K) K
      - C^*_2 = Y_2 + K − β_K / (β_G (β_K + β_G)^2) K − β_L^2 L (compact expression in text).
    - Full separation case β_G = 0:
      - K_G = K and C^*_2 = Y_2 + K.
    - Intuition: higher government intermediation efficiency (β_K/β_G) ⇒ government should hold higher share of private assets.
- Strict inflation targeting (Appendix C) — key results:
  - Baseline strict target:
    - Lemma 12: Optimal R_2 satisfies PC(C_2/(β R_2) (1 − θ^2 ̄π^2), ̄π) = 0 with C_2 = Y_2 + K (equation (48)).
    - Higher cost-push shock (ε_1/(ε_1 − 1)), steeper Phillips curve (−PC_Y / PC_π), and larger future consumption C_2 ⇒ higher optimal R_2.
  - Inside constrained zone under strict targeting:
    - Lemma 13: Optimal R_2 lower in "constrained" zone than outside and strictly decreasing in K_H2 and B_H2.
    - Expression: PC(C_2/(β R_2) (1 − θ^2 ̄π^2), ̄π) = 0 with C_2 = Y_2 + K − [β_L^2 P_2 B_H2 / B] B_H2 − [β_K^2 P_2 K_H2 / K] K_H2 − [β_G^2 K_G / K] K_G and K_G = β_K / β_G (interpretation provided).
  - Run zone and large crisis under strict targeting:
    - In run/large crisis, R_2 that stabilizes inflation is lower than outside zones; other tools reducing ξ and increasing C^*_2 allow raising R_2 back toward outside level.
    - Full separation (β_G = 0 and B = 0) implies C_2 = Y_2 + K and policy rate equals outside-zone level.

### Empirical relationship between monetary policy and financial instability (Appendix D)
- Methodology (D.1):
  - Baseline linear probability specification (equation (49)):
    - C_{i,t+h} = α_{i,h} + β_h × Δr_{i,t} + Σ_{l=0}^L Γ_{h,l} X_{i,t−l} + ε_{i,t+h}.
    - Δr_{i,t} = change in nominal short-term interest rates (yields on three-month government securities and money market rates).
    - Controls X_{i,t} include four lags of: per capita real GDP growth, per capita real consumption growth, per capita real investment growth, CPI inflation, world GDP growth, changes in short-term and long-term interest rates, growth in real stock prices, real house prices, real bank loans, current account-to-GDP ratio, and crisis dummy (contemporaneous values included except crisis dummy).
  - Instrumental variable approach:
    - Instrument Δr_{i,t} with the Trilemma instrument z_{i,t} from Jordà et al. (2017):
      - z_{i,t} ≡ (Δr_{b(i,t),i,t} − Δ ˆ r_{b(i,t),i,t}) × PEG_{i,t} × PEG_{i,t−1} × KOPEN_{i,t}.
      - PEG_{i,t} indicator for exchange-rate peg to base country; KOPEN index of financial openness.
    - Estimation via Local Projections IV (LP-IV) approach.
- Supply vs demand shock decomposition (D.2):
  - Two-equation framework:
    - Aggregate demand: ỹ_t = −α π_t + d_t.
    - Aggregate supply: π_t = β ỹ_t + η_t.
    - Rearranged:
      - ỹ_t = (1/(1+αβ)) (d_t − α η_t).
      - π_t = (1/(1+αβ)) (β d_t + η_t).
    - Demand shocks move output gap and inflation in same direction; supply shocks move them in opposite directions.
  - Empirical SVAR mapping:
    - Az_t = Σ_{j=1}^p A_j z_{t−j} + ε_t (equation (50)), with A = [[1, α], [−β, 1]] and z_t = [Δy_t, π_t]'.
    - Reduced-form residuals v_t related to structural shocks ε_t via v_t = A^{−1} ε_t; sign restrictions characterize demand vs supply shocks.
  - Data: Jordà-Schularick-Taylor Macrohistory Database (Jordà et al., 2017), annual data for 18 advanced economies, 1870–2016 (countries listed in text).
- Empirical findings on monetary policy and crisis risk (D.3–D.4):
  - Overall effect:
    - Monetary policy tightening can exacerbate financial stability risks, particularly when fluctuations are driven by supply shocks.
    - Rate hikes increase likelihood of both equity crashes and banking panics.
  - Magnitudes and timing (empirical points in text and figures):
    - Unconditional effect: financial-crisis risk peaks at 2.5 percent one year after a one percentage point increase in short-term nominal rates (Figure 7 discussion).
    - Effects concentrated following supply shocks; in periods following demand shocks the rise in crisis risk is largely insignificant.
    - Bank stress measures:
      - Average unconditional annual probability: bank equity crash = 3.5 percent; banking panics = 3.4 percent.
      - Rate hikes increase probability of banking panics by 1 percent point in the same year of the rate increase.
      - Bank equity crashes become significant three years after initial rate hike.
  - Channels (Figure 9 discussion):
    - Tightening leads to significant declines in real stock prices, real house prices, and real bank loans.
    - Effect on stock prices largest one year after initial hike and tends to dissipate afterward.
    - Effects on bank credit and housing prices persist up to five years after initial hike; cumulative impacts sizable at five years.
  - Interpretation:
    - Both intermediary capacity constraints and depositor-run risks are important channels linking rate hikes to financial instability.
    - Declines in asset prices and lending capacity are key transmission channels through which rate hikes exacerbate financial instability.
- Additional empirical visuals and descriptions:
  - Figure 10: Equally-weighted averages of short-term nominal rates (LHS) and probability of financial crisis (RHS) for 18 advanced economies, 1870–2016.
  - Figure 11: Average short-term nominal rate in a 5-year window around crisis vs non-crisis events.

*Italic source: wpiea2025035-print-pdf - References (Appendices A–D) — https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025035-print-pdf.pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025035-print-pdf.pdf_