## 1. A common fear factor provides a unified explanation for the classic asset pricing puzzles, including

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### Main findings and unified explanation
- A single time-varying fear factor (subjective tail risk perceived by agents) provides a unified explanation for:
  - the equity premium puzzle,
  - the riskfree rate puzzle, and
  - the equity volatility puzzle.
- Observed variation in fear explains the variation in the risk premium over the business cycle.
- Once the bias from persistent output gaps is accounted for, the long-run neutral rate may have declined by only 0.5 percentage points since the 1980s.
- At the zero lower bound (ZLB), the long-run neutral rate can decline persistently after recessions because of a sharp rise in long-term fear, which can take several years to unwind.
- Model implications and regularities:
  - model generates consistent variation in bond yields over time;
  - recessions follow an inverted yield curve because the inverted curve is associated with higher fear in the future, which is contractionary;
  - dividend-price ratio predicts equity returns, with predictive power greater for longer horizons due to high persistence in output gap, fear, and the neutral rate.

### Quantitative mappings and calibration highlights
- Benchmark calibration choices:
  - η = 0.5
  - α = 0.33
  - γ = 4 (CRRA)
  - χ = 1.67 (implied by η, α, γ)
  - ρ = 0.015
  - β = 0.67
  - ν = 0.05
- Asset-pricing approximations (model-implied magnitudes):
  - risk-free rate approximation: r∗t ∼ 1%
  - market Sharpe Ratio: ≈ 40%
  - equity volatility: ≈ 15%
  - a common fear factor wt ≈ 0.25 can match these empirically-observed values.
- Empirical decomposition (market-implied long-run neutral rate r∗,LRt):
  - Between 1983 and 2022, r∗,LRt has fallen by about 0.5 percentage points once accounting for bias per the paper’s methodology.
  - The decline was sharpest between 2008–13 (about 1.25 percentage points), with nearly all the 2008–10 decline attributable to a rise in long-term fear.
  - By 2019 the long-term fear effects had zeroed out; nearly all the remaining decline (0.5 percentage point) explained by decline in long-run consumption growth.
  - COVID-19 caused a temporary decline in r∗,LRt due to a rise in long-term fear that lasted until early 2022.
  - Regression in the decomposition achieved R2 ≈ 0.9 under the paper’s co-variation assumption.

### SWIM model primitives and structure (key elements)
- Production and resource constraints:
  - Yt = At Kt^α Lt^(1−α)
  - ct + it + gt = 1
  - Notional output gap: Ot ≡ Yt / Yp_t = (Lt / Lp_t)^(1−α)
- Households:
  - CRRA preferences with risk aversion γ and discount factor e^{−ρ}; labor externality χ and Frisch elasticity parameter ω; individual first-order conditions preserved as in source.
  - Complementarity: consumption and labor are complements with elasticity γ/χ > 0.
- Two departures from textbook RBC:
  1. labor externality lowering disutility of work when aggregate employment is higher (Benhabib and Farmer (2000) style);
  2. sticky safe real rate: central bank can fix real interest rate on safe assets r^S_t exogenously so r_t = r^S_t.
- Fear modeled as subjective tail risk (Weitzman framework); safety s_t and fear w_t defined relative to a fail-safe scenario.
- Four-curve representation (SWIM):
  - S: demand for safety (S curve)
  - W: fear (W curve)
  - IS: investment-savings (IS curve)
  - MP: monetary policy (MP curve)
  - Intersection logic: S∩W → r∗t; IS∩MP → rS t and output gap; all three markets clear when these coincide.

### Fear, safety, and the neutral rate (closed-form relations emphasized)
- Neutral rate (S-curve form): r∗t = ρ + γ μt − γ^2/12 · wt^2
- Fear (W-curve form): wt = % · (1 − ζ) · (1 − st)
  - % ≡ 1 + η θ / [ η(1 − β) − ν ]
  - ζ measures social insurance (government countercyclicality)
  - st measures supply of safe tranche (safety)
- IS curve (closed form under MP rule): rS t = r∗t − γ [ η(1 − β) − ν ] gapt
- MP rule (Assumption 3): rS t = Et−1[r∗t] + β [ rS t−1 − r∗t−1 ]
- Output gap dynamics (closed form, equation (29)):
  - gapt = β · gapt−1 + ( (% − 1 ) / ( η % ) ) wt−1 · εt − %^2 / wt−1^(2θ) · ψt
- Key comparative statics (Proposition 2):
  - output gap declines when fear increases, when central bank sets rS t higher than r∗t, and after negative productivity shocks;
  - neutral rate declines when productivity growth falls, patience rises, or fear rises;
  - fear rises when fewer safe assets exist in worst case (s_t low) or social insurance ζ is smaller.

### Six policy and mechanism insights (Section 5) — preserved wording and implications
- First: actively regulating the safe interest rate (in both directions) can mitigate the fluctuations generated by fear cycles.
- Second: recessions will be deeper and longer with an amplified impact of fear if central banks accept the zero lower bound and are unwilling to use negative rates.
- Third: a commitment to use negative interest rate policy in recessions raises interest rates over the entire yield curve and moderates the business cycle (even without having to implement negative rates).
- Fourth: policies to increase the counter-cyclicality of fiscal policy are expansionary; the effects are amplified at the lower bound or when fear is high.
- Fifth: quantitative easing is only narrowly effective when fear is high at the lower bound—by satisfying fear-driven demand for short safe debt and by possibly constraining fiscal borrowing in stressed episodes.
- Sixth: when fear is high, especially at the lower bound, policies that boost productivity also have positive multipliers for fighting recessions.

### Effective lower bound, central bank sluggishness, and business-cycle persistence (Section 5.2 and 5.3)
- Higher β (more sluggish central bank adjustment / more binding effective lower bound) implies:
  - larger contraction from an interest rate gap rS t − r∗t: d(gapt)/d(rS t − r∗t) = −1/[γ(η(1−β) − ν)];
  - amplification of contraction when β rises: d2(gapt)/d(rS t − r∗t)d(β) = −η/[γ(η(1−β) − ν)2] < 0.
- Half-life of recessions Nhalf solves Nhalf = ln 2/(−ln β); dNhalf/dβ = ln 2/[β·(ln β)2] > 0; relationship is convex for relevant β.
- Numerical calibration examples:
  - increasing β from 0.7 to 0.8 → half-life increases by ≈ 1 quarter (if period = quarter);
  - increasing β from 0.8 to 0.9 → half-life increases by ≈ 3.5 quarters.
- Commitment to negative rates reduces perceived β and can raise long-term rates (dEt[r∗t+∞]/dβ < 0 in model), thereby moderating business cycles even if negative rates are not immediately used.

### Fiscal policy, QE, and structural reform multipliers
- Fiscal countercyclicality (raising ζ) reduces fear wt and raises neutral rate r∗t; output effects:
  - d(gapt)/dζ = (1/(1−ζ)) · [γ·w2t/6 · 1/(η(1−β) − ν)] > 0
  - benefits amplified when β is high and when fear wt is high (second derivatives positive as per source).
- Quantitative easing (QE) as maturity transformation:
  - when fear is high, demand for short-term safe assets increases; QE that reduces public long-term debt and boosts short-term safe supply can satisfy fear-driven demand and—if credible—raise ζ by easing fiscal constraints in crises.
  - limits: diminishing returns; QE most effective narrowly when fear is high and the lower bound binds.
- Structural reform / productivity policies:
  - business-cycle multiplier for productivity shocks:
    - dgap_{t+N} / dε_t = β^N ( %−1 / η% ) w_{t−1}
  - multiplier rises with current fear w_{t−1}, with β, and with higher investment-capital ratio a (detailed derivatives in source).
  - Implication: when fear is high, especially at the lower bound, policies that boost productivity have positive multipliers for fighting recessions.

### Asset-pricing and term-structure implications (selected equations and empirical coefficients preserved)
- Neutral rate (long-run measure): r∗,LR t ≡ Et[rS t+10 ] ≈ ρ + γ μ − γ^2/12 · w^2
- Yield-curve slope expression (approximate, equation (36) structure) links slope to gap, μ deviations, and w^2 deviations with Λ factors.
- Inverted yield curve predicts recessions because it reflects higher expected future fear and lower future real rates when central bank sluggishness β is high.
- Price-dividend ratio relations (equations (40)–(41)):
  - ln[Pe t/Dt] ≈ − ln k0 + (γ−λ)[η(1−β)−ν]/(1−β) gapt − (γ−λ)/(1−θ)(μt − μ) + 1/12 (γ−λ)2/(1−θ) (w2t − w2)
  - alternative: ln[Pe t/Dt] ≈ − ln k0 − k1(r∗t − r∗,LRt) + k2·gapt − k3(w2t − w2)
- Empirical regression results (exact coefficients from Table 3 preserved):
  - Output Gap coefficients: 2-0Y Spread 0.131 ∗∗∗ (0.018); 10-2Y Spread 0.152 ∗∗∗ (0.013); 30-10Y Spread 0.044 ∗∗∗ (0.007); 6M Eq. Return −0.024 ∗∗∗ (0.004); 1Y Eq. Return −0.026 ∗∗∗ (0.006); Log P/D 0.082 ∗∗∗ (0.009) and 0.087 ∗∗∗ (0.009).
  - Consumption growth coefficients: Column (1) −1.564 ∗∗∗ (0.101); Column (2) −1.534 ∗∗∗ (0.070); Column (3) −0.521 ∗∗∗ (0.039); Column (4) −0.021 (0.020); Column (5) −0.136 ∗∗∗ (0.029); Column (6) −0.128 ∗∗∗ (0.047).
  - Fear coefficients: Column (1) 36.972 ∗∗∗ (2.743); Column (2) 64.169 ∗∗∗ (1.908); Column (3) 21.360 ∗∗∗ (1.049); Column (4) −2.501 ∗∗∗ (0.562); Column (5) −1.695 ∗∗ (0.777); Column (6) 3.391 ∗∗∗ (1.291); Column (7) −0.754 (1.009).
  - D/P Ratio coefficients (Columns 4–5): 1.708 ∗∗ (0.822); 5.094 ∗∗∗ (1.136).
  - Short-Long r* coefficient (Column 6): −0.044 ∗∗∗ (0.014).
  - Observations: Columns (1)–(3): 458; Column (4): 455; Column (5): 449; Columns (6)–(7): 458.
  - R2 values: 0.433, 0.842, 0.769, 0.195, 0.299, 0.579, 0.581 (Columns 1–7 respectively).
  - Note: standard errors in parentheses; significance: ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01.

### Extensions, robustness, and conceptual variants
- Fear-Avoidance specification (online appendix) where subjective perceived volatility depends inversely on r∗t:
  - modified W-curve: wt = %·(1−ζ)·(1−st)·(1 − φ·r∗t)
  - leads to potential nonlinearities and two-branch solutions where “fear breeds fear” via convex dependence on an underlying resilience term; can generate multiple equilibria under low resilience.
- Appendix outlines:
  - formal proofs of Propositions and Lemmas, balanced growth conditions, derivations of S/IS/MP/W curves, term structure, price-dividend formula, and risk-premium expressions using cumulant expansions and approximations preserved in source.

*Source: wpiea2022175-print-pdf — excerpted sections and equations as provided.*

### 1. A common fear factor provides a unified explanation for the classic asset pricing puzzles, including

### 1. A common fear factor provides a unified explanation for the classic asset pricing puzzles, including

### Main findings and unified explanation
- A single time-varying fear factor (subjective tail risk perceived by agents) provides a unified explanation for:
  - the equity premium puzzle,
  - the riskfree rate puzzle, and
  - the equity volatility puzzle.
- Observed variation in fear explains the variation in the risk premium over the business cycle.
- Once the bias from persistent output gaps is accounted for, the long-run neutral rate may have declined by only 0.5 percentage points since the 1980s.
- At the zero lower bound (ZLB), the long-run neutral rate can decline persistently after recessions because of a sharp rise in long-term fear, which can take several years to unwind.
- The model:
  - generates consistent variation in bond yields over time;
  - predicts that recessions follow an inverted yield curve because the inverted curve is associated with higher fear in the future, which is contractionary.
- The model provides a novel representation of the price-dividend ratio and expected returns linked to macro variables and long-run risk:
  - The dividend-price ratio predicts equity returns;
  - Predictive power is greater for longer horizons due to high persistence in output gap, fear, and the neutral rate of interest.

### Six novel policy and mechanism insights (Section 5)
- First, actively regulating the safe interest rate (in both directions) can mitigate the fluctuations generated by fear cycles.
- Second, recessions will be deeper and longer with an amplified impact of fear if central banks accept the zero lower bound and are unwilling to use negative rates.
- Third, a commitment to use negative interest rate policy in recessions raises interest rates over the entire yield curve and moderates the business cycle (even without having to implement negative rates).
- Fourth, policies to increase the counter-cyclicality of fiscal policy are expansionary; the effects are amplified at the lower bound or when fear is high.
- Fifth, quantitative easing is only narrowly effective when fear is high at the lower bound—by satisfying fear-driven demand for short safe debt and by possibly constraining fiscal borrowing in stressed episodes.
- Sixth, when fear is high, especially at the lower bound, policies that boost productivity also have positive multipliers for fighting recessions.

### SWIM model: basic intuition and structure
- Two departures from textbook RBC:
  1. Utility feature: the disutility of work is lower when aggregate employment is higher (labor externality à la Benhabib and Farmer (2000)), implying aggregate consumption and employment are complements.
     - Aggregate labor supply elasticity conceptually differs from individual-level Frisch elasticity because of nonconvexity from the labor externality.
  2. Sticky safe real rate: the central bank can exogenously fix the real interest rate on safe assets, distorting consumption-savings decisions; the safe rate set by the central bank (given fear) becomes a central driver of the economy, generating positive co-movements in consumption, labor, investment, and output.
- Fear is modeled as subjective tail risk (Weitzman (2007; 2009) framework) rather than objective rare-disaster probabilities.
- The model is labeled the "SWIM model" and represented by four curves: demand for safety (S), fear (W), investment-savings (IS), and monetary policy (MP). (W stands for worry.)
  - Goods and labor market equilibrium depends on IS and MP curves.
  - Market for safe assets (government bonds) determines the neutral rate at the intersection of S and W curves.

### Quantitative approach and empirical testing
- The quantitative assessment introduces:
  - an interest rate rule for the central bank, and
  - a stochastic process for technology with time-varying volatility.
- The time-varying fear implied by the model is quantified and tested using macro-financial data.

### Relation to existing literature and mechanisms
- Builds on RBC tradition (Kydland and Prescott (1982); Long and Plosser (1983)) while emphasizing Keynesian mechanisms through a sticky safe rate.
- Consistent with a variety of nominal-rigidity theories (New Keynesian sticky prices, money illusion, imperfect information).
- Distinctive elements relative to other approaches:
  - Unlike rare-disasters literature, fear is subjective and time-varying (Weitzman).
  - Addresses consumption-investment co-movement via labor externality rather than relying on sticky nominal prices or intermediate inputs.
  - Links to literature on safe assets and safe-asset shortages, but focuses on fear cycles driven by time-varying safety rather than only safe-asset supply shortages.
- Draws on psychology literature for the definition of fear (Ellis (1962); Beck (1979)) and on historical/philosophical perspectives (Hobbes (1651); Locke (1689); Keynes).

### Benchmark model primitives and definitions
- Aggregate production: Cobb-Douglas Yt = At Kt^α Lt^(1−α).
- Aggregate resource constraint (scaled): ct + it + gt = 1. (Equation (1) where ct, it, gt are Ct, It, Gt scaled by Yt.)
- Government spending share is exogenous: Gt = gt Yt, with gt given exogenously.
- Potential/frictionless equilibrium variables: {Lp_t, Cp_t, Ip_t} denote potential labor, consumption, and investment; potential output Yp_t ≡ At Kt^α (Lp_t)^(1−α).
- Notional output gap: Ot ≡ Yt / Yp_t = (Lt / Lp_t)^(1−α). (Equation (2))
- Assumptions on evolution: ∆Lp_t = ξ_L,t; ∆g_t = ξ_t; cp_t ip_t = φ_t with Lp_0 and g_0 known and cp_t + ip_t + gt = 1. (Equation (3))
- Resource constraint implication for investment ratios: It / Ip_t = (1 + φ_t) (Yt / Yp_t)^(−φ_t) (Ct / Cp_t). (Equation (4))
- Government finances: spending financed by lump-sum tax Tt or issuing riskless debt Dt paying gross return Rt with rt ≡ ln Rt.
- Household problem (individual): maximize E_t ∑_{t=0}^∞ e^{−ρ t} [ C_{i,t}^{1−γ}/(1−γ) − (υ_t L^{ω+χ}_t)· L_{i,t}^{1+ω}/(1+ω) ] subject to budget constraint (Equation (5) and (6)).
  - Utility parameters: γ > 0; discount factor e^{−ρ}; time-varying relative cost of labor υ_t; Frisch elasticity 1/ω ≥ 0; aggregate externality χ > 0.
  - First-order conditions (individual): C_{i,t}^{−γ} = exp(r_t − ρ)·E_t[C_{i,t+1}^{−γ}]; C_{i,t}^{−γ} = υ_t L_{i,t}^ω W_t L_{i,t}^{ω+χ} (Equation (7) notation preserved).
  - Aggregate relations: C_t^{−γ} = exp(r_t − ρ)·E_t[C_{t+1}^{−γ}]; C_t^{−γ} = υ_t L_t^{−χ} W_t (Equation (8)).
  - Complementarity: consumption and labor are complements with elasticity γ/χ > 0.
- Firm: law of motion K_{t+1} = I_t + (1−δ) K_t; production Y_t = A_t K_t^α L_t^{1−α}. (Equation (9))
  - Wage condition: W_t = (1−α) Y_t / L_t. (Equation (11))
  - Capital return first-order condition: 1 = E_t[M_{t+1} R_{K,t+1}].
- Central bank: has power to fix the safe real rate r_t at exogenous level r^S_t ≡ ln R^S_t, so r_t = r^S_t. (Equation (12))
- Shadow neutral rate r^*_t is the interest that would prevail when agents expect no central bank intervention today or tomorrow; when r^S_t = r^*_t, Y_t = Y^p_t (Ot = 1).
  - Expression for r^*_t: r^*_t = ρ − ln E_t[(C_{t+1}/C_t)^{−γ} | Ot = E_t Ot+1 = 1]. (Equation (13))

*Source: wpiea2022175-print-pdf - 1. A common fear factor provides a unified explanation for the classic asset pricing puzzles, including*

### 2.6  Conjectured Solution

### 2.6–3.2.3 Conjectured Solution; Investment, Euler Equation, Equilibrium, Calibration, Balanced Growth, Fear & Safety, Shocks, Central Bank

### Conjectured Solution (2.6)
- Conjecture: aggregate consumption gap is proportional to the output gap for exogenous and known C^p_t, Y^p_t:
  - C_t / C^p_t = (Y_t / Y^p_t)^η  (equation (14))
- η is pinned down so optimality conditions in (8) and (11) hold.
- Substituting (14) into labor supply and wage conditions yields (after algebra):
  - η = (χ − α) / [ γ(1 − α) ]  (equation (16))
- Interpretation: when r^S_t = r^*_t for all t, equilibrium reduces to basic RBC with adjusted labor supply elasticity (−χ instead of ω). For deviations, the conjecture-and-verify method around potential values {Y^p_t, C^p_t, I^p_t} yields (14) with η as in (16).

### Aggregate Investment (2.7)
- Using C_t / C^p_t = (Y_t / Y^p_t)^η and the resource constraint (4):
  - I_t / I^p_t = (1 + φ_t)(Y_t / Y^p_t) − (φ_t)(Y_t / Y^p_t)^η  (equation (17))
- With O_t ≡ Y_t / Y^p_t and I^p_t ≡ i^p_t Y^p_t:
  - I_t = [ (1 + φ_t) O_t − (φ_t) O_t^η ] · i^p_t · Y^p_t  (equation (18))
- Conclusion: investment is procyclical with output gap (as long as η < 1). Termed the "investment accelerator".

### Euler Equation and the Output Gap (2.8)
- With C_t = C^p_t O_t^η and r^S_t given by (12), the Euler equation implies:
  - O_t^(−γη) = exp(r^S_t − ρ) · E_t[ (C^p_{t+1} / C^p_t · O_{t+1}^{−η})^(−γ) ]  (equation (19))

### Equilibrium and Proposition 1 (2.9)
- If there exists a unique O_t = Ô_t solving (19), then a Walrasian competitive equilibrium exists with η = (χ − α)/[γ(1 − α)].
- Equilibrium allocations (Proposition 1):
  - Consumption: C_t = Ô_t^η · c^p_t · Y^p_t
  - Investment: I_t = [ (1 + φ_t) Ô_t − (φ_t) Ô_t^η ] · i^p_t · Y^p_t
  - Employment: L_t = (Ô_t)^{1/(1−α)} · L^p_t
  - Output: Y_t = Ô_t · Y^p_t
- Three markets clear: government debt D_t (price 1/R^S_t), labor L_t (price W_t), goods Y_t (price unity).
- Intuition: higher exogenous desire to save lowers consumption, aggregate demand, investment, and employment — consumption, investment, and labor co-move positively.

### Calibration (2.10)
- Focus: US economy co-movements using observed output gap O_t and exogenous trends c^p_t, i^p_t with φ_t = c^p_t / i^p_t.
- Calibration parameters chosen:
  - η = 0.5
  - α = 0.33
  - These impose γ and χ via η = (χ − α)/[γ(1 − α)].
  - For asset pricing applications: γ = 4 implies χ = 1.67.
- Six-step calibration uses CBO measure of O_t; derives model-implied C_t/Y_t = c^p_t O_t^{η−1}, C_t/Y^p_t = c^p_t O_t^η, I_t/Y_t = [1 + φ_t − φ_t O_t^{−(1−η)}] i^p_t, labor gap L_t/L^p_t = O_t^{1/(1−α)}, and plots relationships.
- Empirical findings:
  - I_t/Y_t and C_t/Y_t are negatively correlated (after scaling by potential).
  - I_t/Y_t and O_t are positively correlated.
  - Model fits directions and magnitudes of key macro variable dynamics for 1955–2021.
- Table 1 parameter summary:
  - Capital Share of Income α = 0.33
  - Consumption Elasticity η = 0.5
  - Note: underlying parameters chosen as CRRA γ = 4 and aggregate labor elasticity χ = 1.67.

### Balanced Growth and Consumption Growth (2.11)
- Specification for disutility growth ensures potential labor L^p_t unaffected by output gap:
  - υ_{t+1} / υ_t = (c^p_{t+1} / c^p_t)^{−γ} · [ (A_{t+1} / A_t)(K_{t+1} / K_t)^α ]^{1−γ}
  - Ensures d ln L^p_t / dt = 0 and balanced growth path for capital, output, productivity.
- Notation for potential-next-period:
  - Y^{pp}_{t+1} ≡ [ Y^p_{t+1} | O_t = 1 ], C^{pp}_{t+1} ≡ c^p_{t+1} · Y^{pp}_{t+1}
  - Potential consumption growth: X_{t+1} ≡ ln[ C^{pp}_{t+1} / C^p_t ]
- Lemma 1 (Consumption Growth & Euler Equation):
  - ln[C_{t+1} / C_t] ≈ X_{t+1} + η · ∆gap_{t+1} + ν · gap_t
  - r^S_t = ρ − ln E_t[ (C_{t+1} / C_t)^{−γ} ]
  - gap_t ≡ ln O_t
  - ν ≡ α a [1 + (1 − η) φ] / (1 + a − δ)
  - φ, a are average values of φ_t and the investment-capital ratio
- Interpretation: consumption growth is weighted sum of potential consumption growth, output gap growth, and the dynamic effect of output gap on next period’s capital.

### The SWIM Model: Fear & Safety (3.1)
- Two states: normal and extreme. Extreme ≈ 1% of the time. Boundary = "fail-safe scenario".
- Definitions:
  - Fail-Safe Scenario: A^{safe}_t is the notional worst 1st percentile outcome for productivity.
  - Safety s_t: 1 − s_t ≡ E_t ln[ A_{t+1} / A^{safe}_{t+1} ]
  - Fear w_t: w_t ≡ E_t ln[ C_{t+1} / C^{safe}_{t+1} ]
  - Safe Assets: securities free of default in normal states, guarantee one unit consumption when A_t ≥ A^{safe}_t
- Interpretation:
  - w_t measures expected decline in consumption in the fail-case; e.g., w_t = 0.25 implies fail-safe consumption ≈ 25% lower than average.
  - s_t captures perceived TFP risk; more safe economies have higher s_t (lower tail risk) and lower fear.
  - Innovation and institutions can reduce worst-case severity and thus fear by increasing supply of safe assets.

### Setup: Government and Social Insurance (3.2.1)
- Assumption 1 (Social Insurance): government spending path countercyclical and depends on realization of A_{t+1}:
  - ∆g_{t+1} = (1 − g_t) [ 1 − ( A_t / A_{t+1} )^ζ ] with 0 ≤ ζ < 1
- When A_{t+1} = A_t, ∆g_{t+1} = 0. For falls in A_{t+1}, government consumption falls (social transfers in bad times), which raises household potential consumption share c^p_t.
- ζ = 0 implies ∆g_{t+1} = 0.
- No default risk for government debt; short safe real rate represented by one-period government debt r^S_t.

### Shocks: Productivity Process (3.2.2)
- Assumption 2 (Productivity Process):
  - ln(A_{t+1} / A_t) = μ_A + e_{t+1}
  - e_{t+1} = θ e_t + σ_t · ε_{t+1}
  - σ_t = (σ)^{1−θ} (σ_{t−1})^θ · √(1 + ψ_t)
- Joint shocks {ε_t, ψ_t} stationary with no correlation.
  - ε_{t+1} drawn from truncated student-t with n degrees of freedom, E_t[ε_{t+1}] = 0, V_t[ε_{t+1}] = n/(n−2), implying σ^2_{A,t} = (n/(n−2)) σ^2_t.
  - ψ_t has E_t[ψ_{t+1}] = 0, V_t[ψ_{t+1}] = σ^2_ψ and bounded below by zero.
  - 0 ≤ θ ≤ 1 measures persistence of e_t and σ_t, with long-run mean zero and σ respectively.
- Fail-safe normalization:
  - Agents act as if 1st percentile is k standard deviations below mean; normalize k = √6
  - Safety representation:
    - 1 − s_t √6 ≡ σ_{A,t} = √(n/(n−2)) · σ_t  (equation (21))
- Potential consumption growth (combining Assumptions 1 & 2 with Lemma 1):
  - X_{t+1} = b + (1 − ζ) · ln[A_{t+1} / A_t]  (equation (22))
  - where b ≡ α ln(1 + a − δ); E_t[X_{t+1}] ≡ μ_t = b + (1 − ζ)(μ_A + θ e_t); V_t[X_{t+1}] = (1 − ζ)^2 σ^2_{A,t}
  - Greater social insurance (higher ζ) shields consumption volatility.
- Lemma 2 (Consumption Process):
  - E_t ln[C_{t+1} / C_t] = μ_t + η · E_t[gap_{t+1}] − (η − ν) · gap_t
  - V_t ln[C_{t+1} / C_t] = V_t[ X_{t+1} + η · gap_{t+1} ]
  - In equilibrium V_t[gap_{t+1}] ≈ ( η / [ η(1 − β) − ν ] )^2 σ^2_{X,t}, implying V_t ln[C_{t+1} / C_t] ≈ w_t^2 / 6 with
    - w_t ≡ % · (1 − ζ) (1 − s_t)
    - % ≡ 1 + η θ / [ η(1 − β) − ν ]
- Conclusion: fear w_t ≡ %·(1 − ζ) (1 − s_t) governs consumption growth volatility; more safe output in fail-safe lowers w_t.

### Central Bank (3.2.3)
- Prices fully flexible, normalized to unity; inflation not modeled here.
- Central bank controls the real safe rate r^S_t and sets it before observing ε_t.
- Assumption 3 (Monetary Policy Rule):
  - r_t = r^S_t with r^S_t = E_{t−1}[ r^*_t ] + β [ r^S_{t−1} − r^*_{t−1} ] with 0 ≤ β < 1

*Source: wpiea2022175-print-pdf — sections 2.6–3.2.3 as provided.*

### 3.3  Market Clearing Conditions

### 3.3 Market Clearing Conditions

### Overview
- Three markets must clear in equilibrium: the goods market, the labor market, and the safe government debt market (the “safe assets market”).
- Along the MP curve, the safe assets market is in equilibrium. Along the IS curve, both the goods and labor markets are in equilibrium. The intersection of IS and MP implies all three markets clear.
- The S curve and the W curve serve as shadow market-clearing conditions for the safe assets market under the hypothetical that the output gap is closed. The intersection of S and W determines the neutral rate of interest r∗t, which enters the IS curve.

### The S Curve (Demand for Safe Assets)
- Definition: the S curve represents the demand for safe assets; the neutral rate r∗t is the short-term interest rate for a riskless asset that pays 1 unit of consumption in the future when gapt = Et gapt+1 = 0.
- Key representation (equation (23)):
  - r∗t = ρ + γ μt − γ^2/12 · wt^2
- Economic interpretation of terms:
  - ρ: pure rate of time preference (higher ρ → higher risk-free rate).
  - γ μt: product of coefficient of relative risk aversion γ and expected growth in disposable income μt; γ measures the price of intertemporal substitution, μt measures the quantity.
  - −γ^2/12 · wt^2: captures risk in the next period; variance of consumption growth Vt[ln Ct+1/Ct] ≈ wt^2/6, and wt measures fear/uncertainty.

### The W Curve (Fear or Uncertainty)
- Definition: fear wt is the log distance between expected consumption and fail-safe consumption: wt ≡ Et ln[Ct+1/Csafe t+1].
- Key representation (equation (24)):
  - wt = % · (1 − ζ) · (1 − st)
  - with % ≡ 1 + η θ / [η(1 − β) − ν]
- Interpretation: fear depends on:
  - safe tranche of productivity st (net supply of safe assets),
  - degree of social insurance ζ provided by the government,
  - persistence in technology shocks (θ) and in central bank interest rates (β).

### The MP Curve (Interest Rate Rule)
- Assumption 3: the central bank pins the safe real interest rate so that rt = rS t each period.
- Key representation (equation (25)):
  - rS t = Et−1[r∗t] + β [ rS t−1 − r∗t−1 ]
- Expected neutral rate used in MP:
  - Et−1[r∗t] = ρ + γ [ at + (1 − ζ) ( μA + θ^2 et−1 ) ] − γ^2/12 · ( wt−1^2 ) θ ( w^2 )^(1−θ)
- Implication: “monetary non-neutrality” arises from the central bank’s role in the safe assets market via sticky safe rates.

### The IS Curve (Euler Equation)
- Starting from the Euler equation and substituting r∗t from (23), the IS relation is (equation (26)):
  - rS t = r∗t − γ(η − ν) gapt + γ η Et[gapt+1]
- Eliminating expectations under the assumption that for some future t+J, Et(gapt+J) = 0, and iterating backward as J → ∞, yields the IS curve in closed form (equation (27)):
  - rS t = r∗t − γ [ η(1 − β) − ν ] gapt
- S, W, and IS are a decomposition of the household Euler equation; intersection of S and W pins r∗t which shifts the IS intercept.

### Four Key Equations (as presented in Figure 5 / equation (28))
- S Curve: r∗t = ρ + γ μt − γ^2/12 · wt^2
- W Curve: wt = % · (1 − ζ) · (1 − st)
- IS Curve: rS t = r∗t − γ[η(1 − β) − ν] gapt
- MP Curve: rS t = Et−1[r∗t] + β [ rS t−1 − r∗t−1 ]

### Equilibrium (five-step characterization)
1. Realizations of {εt, ψt} determine expected productivity growth μt and safety st. Intersection of S and W determines neutral rate r∗t (shadow safe rate).
2. The safe assets market clearing condition is given by the MP curve, which pins the price of government debt 1/rS t. Government debt supplied must be held by households at that price.
3. The goods market equilibrium (IS curve) determines the combination of output Yt and safe rate rS t such that investment equals total savings. Deviations of rS t from r∗t force households to adjust consumption-savings to hold the fixed quantity of safe assets.
4. The IS curve pins down the output gap gapt ≡ ln Ot. Given potential labor Lp t, gapt implies labor supply Lt; the wage Wt adjusts so firms hire Lt and the labor market clears.
5. All markets clear when S and W intersection (determining r∗t) and IS and MP intersection (determining rS t, gapt, Lt) coincide. The four equations represent a dynamic general equilibrium.

- If rS t = r∗t (no sticky safe rates), the IS curve is vertical with gapt = 0. Sticky safe rates allow gapt ≠ 0.
- Closed-form for the output gap (equation (29)):
  - gapt = β · gapt−1 + ( (% − 1 ) / ( η % ) ) wt−1 · εt − %^2 / wt−1^(2θ) · ψt
  - with %^2 ≡ γ · w^2(1 − θ) / [ 12 [ η(1 − β) − ν ] ]

### Proposition 2 — Role of Fear in Output, Interest & Safe Assets
- Output gap is given by (29).
- (i) The output gap declines when:
  - fear increases;
  - the central bank sets the interest rate higher than the neutral rate;
  - there is a negative productivity shock.
  - The contractionary effect of fear is larger when the central bank reacts slowly to changes in the neutral rate (higher β).
- (ii) The neutral rate declines when:
  - productivity growth falls;
  - patience rises;
  - fear rises.
- (iii) Fear rises when:
  - there are fewer safe assets in the reasonable worst-case scenario (st is low);
  - social insurance ζ provided by governments is smaller.

### Numeric Parameter Values and Calibration Inputs (from Table 2 and text)
- Coefficient of Relative Risk Aversion γ = 4
- Discount rate ρ = 0.015
- Elasticity of Capital w.r.t. the Output Gap ν = 0.05
- Persistence of the Interest Rate Rule β = 0.67
- In addition, η = 0.5 is chosen the same as in Section 2.

### Key Quantitative Insights and Asset-Pricing Mapping (selected results)
- Risk-free rate approximation (from (30)):
  - r∗t ≈ ρ + γ μt − γ^2/2 · ( wt^2 / 6 ) ∼ 1%
- Market Sharpe Ratio approximation:
  - [ Et(Re t+1) − RS t ] / σt(Re t+1) ≈ γ σt(Δ ln Ct+1) = γ wt / √6 ∼ 40%
- Equity volatility approximation:
  - σt(ln Re t+1) ≈ wt / √6 · [ 2 − (Ct/Pe t) + (Ct/Pe t)^2 ]^1/2 ∼ 15%
- Empirical calibration points cited:
  - ρ ≈ 1.5 percent, γ ≈ 4, μt ≈ 2 percent, unlevered dividend-price ratio Ct/Pe t ≈ 0.03.
- A common fear factor wt ≈ 0.25 can match the empirically-observed:
  - risk-free rate ≈ 1%,
  - market Sharpe Ratio ≈ 40%,
  - equity return volatility ≈ 15%.
- Risk-free rate puzzle resolution: subjective uncertainty (fear wt) calibrated at wt ≈ 0.25 gives r∗t ≈ 1%.
- Long-run neutral rate (definition and mapping):
  - N-period average real short rate rS t,N = (1/N) Et[ rS t + ... + rS t+N−1 ] (equation (31))
  - N-period yield decomposition: y(N) t = rS t,N + risk premium (N) t (equation (32))
  - Expected short-rate N years ahead via equilibrium conditions (equation (33)):
    - Et[ rS t+N ] = Et[ r∗t+N ] − γ[ η(1 − β) − ν ] β^N Et[ gapt ]
  - For sufficiently large N (e.g., N ≈ 10 when t is in years), β^10 ≈ 0 and Et[ rS t+10 ] ≈ Et[ r∗t+10 ].
  - Preferred long-run neutral rate measure:
    - r∗,LR t ≡ Et[ rS t+10 ] ≈ Et[ r∗t+∞ ] = ρ + γ μ − γ^2/12 · w^2  (equation (35))
- Drivers of decline in long-run neutral rate:
  - decline in long-run mean consumption growth μ (dr∗,LR/dμ = −γ < 0),
  - increase in long-run fear w (dr∗,LR/dw = − γ^2/6 · w < 0).
- Long-run fear w expression:
  - w ≡ % · (1 − ζ) · (1 − s)
  - Long-run fear rises if government disaster insurance ζ declines, expected long-run safety s declines, or central bank reaction becomes more sluggish (higher β and ∂%/∂β > 0).

*Italic source: Excerpt from "3.3 Market Clearing Conditions" (and related passages) of the SWIM model chapter in the provided PDF.*

### 1. Use the Fed Board’s staff estimates of the yield curve (based on Gürkaynak et al. (2007)), to obtain

### wpiea2022175-print-pdf - 1. Use the Fed Board’s staff estimates of the yield curve (based on Gürkaynak et al. (2007)), to obtain

### Methodology (steps used to obtain market-implied long-run neutral rate)
- Step 1: Use the Fed Board’s staff estimates of the yield curve (based on Gürkaynak et al. (2007)), to obtain the 10-year nominal yields.
- Step 2: Use the Cleveland Fed’s inflation model estimates of average expected inflation (based on Haubrich et al. (2012)) and subtract it from #1 to obtain the implied real yields, y(N)t, and forward rates, f(N)t.
- Step 3: Use the New York Fed’s staff model (based on Adrian et al. (2013)) estimates of the term premiums, and the estimates of f(N)t from #2, obtain an estimate of the market-implied long-run neutral rate, r∗,LRt ≈ Et[r∗t+10] as per (34).
- Step 4: Estimate a linear regression of the form: r∗,LRt = α0 + α1(μt − μ0) + α2(w2t − w20) + εt, under the assumption that long-run values of {μ, w2} co-vary with their short-term values (R2 ≈0.9). Here {μ0, w20} correspond to values in 1983.
- Step 5: Use the estimated regression to decompose the decline in r∗,LRt between changes in long-run consumption growth vs. long-run fear (plus residual).

### Main empirical decomposition results and magnitudes
- The decomposition suggests that the neutral rate tends to decline persistently after recessions largely due to a sharp and persistent rise in long-term fear.
- Result 2 (summary bullets):
  - Between 1983 and 2022, the market-implied long-run neutral rate, r∗,LRt, has fallen by about 0.5 percentage points once accounting for the bias as per (34).
  - The decline was sharpest immediately after the Great Recession between 2008–13 when the size of the decline was about 1.25 percentage points, which partly reversed over the subsequent 5 years.
  - Between 2008–10 nearly all the decline can be attributed to a rise in long-term fear.
  - By 2019 the long-term fear effects had zeroed out, with nearly all the remaining decline (of 0.5 percentage point.) explained by a decline in long-run consumption growth.
  - The COVID-19 shock led to a temporary decline in r∗,LRt due to a rise in long-term fear, which lasted only until early 2022.
- Ancillary quantitative detail: regression in Step 4 achieved R2 ≈0.9 in the assumed co-variation.

### Yield curve slope and recession prediction (Yield Curve Inversion Puzzle)
- The model expression for the slope y(T+N)t − y(N)t is given approximately by equation (36):
  - y(T+N)t − y(N)t ≈ Λβ·γ[η(1−β)−ν]·(gapt) − Λθ·γ·(μt − μ) + Λθ·γ2/12·(w2t − w2) + (risk premium)
  - With Λx ≡ 1/(1−x)[(1−xN)/N − (1−xT+N)/(T+N)] ≥ 0.
- Condition for inverted yield curve (equation (37)):
  - Λθ·γ/12·(w2 − w2t) + Λθ·(μt − μ) > Λβ·[η(1−β)−ν]·(gapt)
- Result 3 (interpretation):
  - An inverted yield curve predicts recessions because it is associated with higher fear in the future, especially when central banks are in tightening cycles (higher β).
  - Under inversion, future real rates r∗t+N are expected to be lower (with lower μt and higher wt), and sluggish central bank adjustment (due to β) can raise future interest rate gaps rSt+N − r∗t+N, hence leading to contractions.

### Asset-pricing implications: dividend-price ratio and return predictability
- Key identity and approximations:
  - N-period equity return identity approximated in (38): lnRe t+N ≈ ln(Dt+N/Dt) + 1/f ln[(Pe t+N/Dt+N)/(Pe t/Dt)] + 1/f (N − N(N−1)/2 g t,N)·Dt/Pe t
  - In the SWIM model, expected log returns (equation (39)):
    - Et[lnRe t+N] ≈ Θ0 − Θβ·[η(1−β)−ν] gapt + Θθ·(μt − μ) − Θθ·(w2t − w2) + Θg·Dt/Pe t
  - Log price-dividend ratio (equation (40)):
    - ln[Pe t/Dt] ≈ − ln k0 + (γ−λ)[η(1−β)−ν]/(1−β) gapt − (γ−λ)/(1−θ)(μt − μ) + 1/12 (γ−λ)2/(1−θ) (w2t − w2)
  - Alternative representation (equation (41)):
    - ln[Pe t/Dt] ≈ − ln k0 − k1(r∗t − r∗,LRt) + k2·gapt − k3(w2t − w2)
- Result 4 (summary bullets):
  - The dividend-price (D/P) ratio predicts expected equity returns.
  - Predictive power of D/P is small around N = 1, and rises with horizon N (for N not too large) due to persistence in output gap, fear, and the neutral rate.
  - The price-dividend ratio co-moves positively with the output gap and the slope of the real neutral rate curve, and negatively with fear.

### Empirical evidence (Table 3 highlights, preserved coefficients and statistics)
- Sample: monthly US time series data between 1984 and 2022; Observations vary by regression (see individual columns).
- Selected coefficient estimates and statistics (exact values as in Table 3):
  - Output Gap coefficients:
    - Column (1) 2-0Y Spread: 0.131 ∗∗∗ (0.018)
    - Column (2) 10-2Y Spread: 0.152 ∗∗∗ (0.013)
    - Column (3) 30-10Y Spread: 0.044 ∗∗∗ (0.007)
    - Column (4) 6M Eq. Return: −0.024 ∗∗∗ (0.004)
    - Column (5) 1Y Eq. Return: −0.026 ∗∗∗ (0.006)
    - Column (6) Log P/D: 0.082 ∗∗∗ (0.009)
    - Column (7) Log P/D: 0.087 ∗∗∗ (0.009)
  - Cons. Gr. (consumption growth) coefficients:
    - Column (1): −1.564 ∗∗∗ (0.101)
    - Column (2): −1.534 ∗∗∗ (0.070)
    - Column (3): −0.521 ∗∗∗ (0.039)
    - Column (4): −0.021 (0.020)
    - Column (5): −0.136 ∗∗∗ (0.029)
    - Column (6): −0.128 ∗∗∗ (0.047)
  - Fear coefficients:
    - Column (1): 36.972 ∗∗∗ (2.743)
    - Column (2): 64.169 ∗∗∗ (1.908)
    - Column (3): 21.360 ∗∗∗ (1.049)
    - Column (4): −2.501 ∗∗∗ (0.562)
    - Column (5): −1.695 ∗∗ (0.777)
    - Column (6): 3.391 ∗∗∗ (1.291)
    - Column (7): −0.754 (1.009)
  - D/P Ratio coefficients:
    - Column (4) and (5) include D/P Ratio:
      - Column (4): 1.708 ∗∗ (0.822)
      - Column (5): 5.094 ∗∗∗ (1.136)
  - Short - Long r* coefficient:
    - Column (6): −0.044 ∗∗∗ (0.014)
  - Observations:
    - Columns (1)–(3): 458
    - Column (4): 455
    - Column (5): 449
    - Columns (6)–(7): 458
  - R2 values:
    - Column (1): 0.433
    - Column (2): 0.842
    - Column (3): 0.769
    - Column (4): 0.195
    - Column (5): 0.299
    - Column (6): 0.579
    - Column (7): 0.581
- Note in table: Standard errors in parentheses; significance symbols: ∗ p < 0.10, ∗∗ p < 0.05, ∗∗∗ p < 0.01.

### Policy implication on fear cycles, money supply, and macro stabilization
- Mechanics and quantitative relations:
  - Fear shocks are recessionary: d(gapt)/d(wt) = − γ·wt / [6 · (η(1−β)−ν)] < 0 (equation (43)).
  - Fear shocks are amplified when central bank adjustment is sluggish (higher β) and the negative relationship between output gap and fear is concave:
    - d2(gapt)/d(wt)d(β) = − γη·wt / [6 (η(1−β)−ν)2] − γ·(1−ζ)(1−st)η2 θ / [6 (η(1−β)−ν)3] < 0
    - d2(gapt)/dw2 t = − γ / [6 (η(1−β)−ν)] < 0 (equation (44)).
  - Implication 1 (policy): Actively regulating the interest on safe assets (in either direction) can mitigate the fluctuations generated by fear cycles. Fear-driven contractions are amplified at higher levels of initial fear, and when interest rates are more sluggish (high β).
- Broad money supply evidence and magnitudes:
  - The M3 measure of money declined by about 10 percent during the Great Recession of 2007–09, similar in magnitudes to declines in money supply observed during the Great Depression (as presented in Figure 10).
  - During the same period M2 balances continued to grow, albeit at a slightly slower pace; the divergence implies the contraction in broad money was felt more by institutional investors (non-M2 components) than by retail investors.
  - Both during the Great Depression and the Great Recession the growth in broad money supply declined sharply within a few months and went into negative territory in about two years.

*Italic: Source: wpiea2022175-print-pdf (section content provided).*

### 5.2  Longer & Deeper Business Cycles with Stagnation Episodes

### 5.2  Longer & Deeper Business Cycles with Stagnation Episodes

### Mechanism: effective lower bound and central bank sluggishness
- The zero lower bound (effective lower bound) arises from paper currency guaranteeing a zero nominal interest rate available in unlimited quantities, creating an interest rate floor and making lenders unwilling to lend at significantly lower rates.
- In the SWIM model, a central bank that refuses to cut rates into negative territory corresponds to an increase in the persistence term β in the MP curve; a higher β is called a “more binding lower bound.”
- Key relationships (preserve notation as in source):
  - Interest rate gap: rSt − r∗t.
  - Contractionary effect from the IS curve:
    - d(gapt)/d(rSt − r∗t) = −1/[γ(η(1−β) − ν)].
  - Amplification of contraction when β rises:
    - d2(gapt)/d(rSt − r∗t)d(β) = −η/[γ(η(1−β) − ν)2] < 0 (equation (45)).
  - Expected evolution of the gap:
    - Et[gapt+N] = −βN/[γ(η(1−β) − ν)] · [rSt − r∗t] = βN · gapt for N ≥ 0.
  - Half-life of recessions/booms Nhalf solves 1/2 = βNhalf, so Nhalf = ln 2/(−ln β).
  - Derivatives of Nhalf with respect to β:
    - dNhalf/dβ = ln 2/[β·(ln β)2] > 0 (equation (46)).
    - d2Nhalf/dβ2 = ln 2·[2 + ln β]/[−β2 · (ln β)3] > 0 (convex for β > 1/e2 ≈ 0.14).
- Numerical examples and empirical concordance:
  - If each period is a quarter, typical values for β are around 0.8, but higher after the Great Recession.
  - Increase β from 0.7 to 0.8 → half-life of recessions increases by about 1 quarter.
  - Increase β from 0.8 to 0.9 (roughly observed after 2009) → half-life can increase by 3.5 quarters.
  - Fed model simulations suggest pre-Great Recession policy rules would have constrained short-term rates by zero as much as one-third of the time, producing costlier and longer recessions.

### Asymmetry and broader implications
- The effective lower bound binds when rSt > r∗t, but a high β also implies sluggishness in raising rates when r∗t rises (as seen during high inflation episodes after 2021).
- A more sluggish central bank amplifies expansion when the gap narrows and contraction when it widens; formally d2(gapt)/d(r∗t − rSt)d(β) > 0.
- Empirical policy asymmetry: central banks tolerated inflation below 2 percent but acted strongly against inflation above 2 percent, making high β more consequential when rSt > r∗t and increasing likelihood of long-lasting stagnations.
- Distributional considerations: a negatively skewed distribution of output gap (long left tails) could make secular stagnation outcomes more frequent.

### Implication 2
- Recessions will be deeper & longer when central banks accept the lower bound and are unwilling to use negative rates:
  - A more sluggish central bank function (higher β) amplifies recession size (equation (45)).
  - The half-life of recessions/booms is strictly increasing in β and the β–half-life relationship is highly non-linear (equation (46)).

---

### 5.3  Power of Negative Interest Rate Policy to Overcome Low-for-Long Rates

### Long-run non-neutrality of monetary policy and role of β
- Long-horizon expected safe rate:
  - Et[r∗t+J] = ρ + γμt+J − (γ/2·1/12)·w2t+J. As J → ∞, Et[r∗t+∞] = ρ + γμ − (γ/2·1/12)·w2 where μ is the long-run μt+J.
- Effect of β on long-term expected rates:
  - d%/dβ > 0 (intermediate result noted in source).
  - dEt[r∗t+∞]/dβ = −(γ2 (1−ζ) (1−s) η2 θ 1/12)/[η(1−β) − ν]2 < 0 (equation (47)).
  - d2Et[r∗t+∞]/dβ2 < 0; the effect is non-linear and can be large at high β.
- Paradoxical result: committing to deeper negative rates (reducing perceived β) can raise longer-term rates; conversely, signaling reluctance to use negative rates (higher perceived β) can depress long-term rates by increasing perceived future probability of prolonged lower rates due to binding lower bound.
- Economic intuition: lowering perceived effective lower bound reduces fear that central banks will be stuck at the bound, restoring “hope for the return of a normal yield curve,” and raising long-term rates.

### Policy mechanics to break the zero lower bound
- Breaking the arbitrage between cash (zero nominal return) and bank money requires modifying paper currency policy and making electronic money central.
- One operational approach: adopt/strengthen an electronic-money unit of account and implement a time-varying interest rate (rate of return) on paper currency so cash moves with official policy rates.
- Legal, communication, and political challenges exist for such transitions.

### Commitment effects without immediate use
- A central bank commitment to use negative rates (even without immediate use) lowers perceived β today and in the future, steepening the IS curve and reducing output-gap volatility (shallower recessions, smaller booms).
- Implication 3:
  - Committing to use negative interest rate policy in recessions raises the real neutral rate over the entire yield curve and moderates the business cycle.
  - This long-run monetary non-neutrality holds even away from the zero lower bound if the central bank can credibly commit to negative rates when needed.

---

### 5.4  The Fear-Mitigation Role of Fiscal Policy

### Mechanism: social insurance, fear, and neutral rate
- Greater social insurance (higher ζ) lowers fear and raises the neutral rate r∗t.
- From the S curve: wt = %·(1−ζ)·(1−st), so dwtdζ > 0.
- Output effects of raising ζ:
  - d(gapt)/dζ = (1/(1−ζ)) · [γ·w2t/6 · 1/(η(1−β) − ν)] > 0 (equation (48)).
- The S-curve rightward shift raises r∗t and causes a rightward shift in the IS curve (intersection with zero output gap at higher r∗t).

### Amplification at the lower bound and when fear is high
- The fear-mitigation role of fiscal policy is more potent when β is high or fear wt is high:
  - d2(gapt)/d(ζ)d(β) = (1/(1−ζ)) · d/dβ [γ·w2t/6 · 1/(η(1−β) − ν)] > 0.
  - d2(gapt)/d(ζ)d(wt) = (1/(1−ζ)) · γ·wt/3 · 1/(η(1−β) − ν) > 0 (equation (49)).

### Implication 4
- Policies that increase the counter-cyclicality of fiscal policy (raise ζ) are expansionary.
- The benefits of raising ζ are amplified at the lower bound or when fear is high.

---

### 5.5  The Fear Theory of Quantitative Easing

### Demand shift toward short-maturity safe debt when fear is high
- Yield curve relation for long vs short maturity (from equation (36) specialized):
  - y(∞)t − y(1)t ≈ γ[η(1−β) − ν]·(gapt) − γ·(μt − μ) + (γ2/12)·(w2t − w2) + (risk premium) (equation (50)).
- When fear wt is high, the long-term slope steepens, increasing demand for short-term safe assets relative to long-maturity safe assets:
  - d(y(∞)t − y(1)t)/dwt = γ2/6 · wt > 0.
  - The steepening effect is greatest for the short-term vs very long-term debt:
    - d2(y(T)t − y(1)t)/dwt dT > 0.
- Empirical evidence (as described): during 2007–09 Great Recession and the 2020 COVID-19 pandemic, simultaneous rises occurred in model-implied fear wt, yield spreads, and the share of T-Bills held by the public relative to total Treasury debt held by the public.

### Maturing debt, fiscal constraints, and QE as maturity transformation
- Government gross financing needs include maturing debt plus new borrowing; sovereign auctions face upward-sloping demand curves in stressed episodes.
- Large quantities of maturing debt in a period can limit financing raised, potentially constraining new borrowing for fiscal spending—this constraining effect may be non-binding in normal times but binding during crises (Beetsma et al., 2016).
- QE and maturity-transformation policies (central bank purchases of long-term Treasuries vs. increasing short-term debt issuance) can:
  - Satisfy fear-driven demand for short-term safe assets.
  - Potentially constrain fiscal authorities’ ability to borrow/spend in stressed episodes (raising ζ, the counter-cyclicality of fiscal policy).
- For maturing debt to constrain crisis outcomes, the model requires dζ/d(Maturing Debt t+1) ≥ 0 and dg t+1/d(Maturing Debt t+1) ≤ 0 in crises (with dg t+1/d(Maturing Debt t+1) ≈ 0 in normal times).

### Limits and diminishing returns
- Diminishing returns: if government borrowing constraint is less likely to bind once a given amount of maturing debt has been removed from public hands, then d2ζ/d(Maturing Debt t+1)2 < 0.
- The expansionary effects of raising ζ are smaller when fear wt is low or when β is low; thus QE that shortens public debt maturity is most effective narrowly when fear is high and the lower bound is binding.

### Real-finance counterpart and policy implication
- Market participants face market/duration risk from holding longer maturity bonds in stressed episodes; short-maturity bonds provide an option to convert assets to liquidity without selling into high yields.
- In periods of high fear (Great Recession, COVID-19), demand for short-term debt rose, steepening the real yield curve; substituting long-term debt with short-term debt can be effective in these cases.
- Implication 5:
  - Quantitative easing has limits but may be narrowly effective when fear is high at the lower bound by satisfying demand for safe short-term debt and by credibly constraining fiscal borrowing in stressed episodes.
  - Effectiveness depends on presence of fear-driven demand for short debt and on whether QE credibly raises ζ; diminishing returns and smaller effects in normal times apply.

*Source: wpiea2022175-print-pdf - 5.2  Longer & Deeper Business Cycles with Stagnation Episodes*

### 5.6  The Structural Reforms Multiplier at the Effective Lower Bound

### 5.6  The Structural Reforms Multiplier at the Effective Lower Bound

### Business-cycle multiplier mechanism
- Structural reform policies that boost productivity (εt) or raise the average investment-capital ratio (a) can have positive cyclical effects when fear is high.
- The effect of an increase in productivity today on future output gap is given by:
  - dgap_{t+N} / dε_t = β^N ( %−1 / η% ) w_{t−1}  (Equation (51) as in source)
- There is a “business cycle multiplier”: a boost to productivity increases the output gap, not just trend output.
- The derivative is zero when β = 0, implying the multiplier is zero when central bank rates are not sluggish.
- The size of the multiplier rises with current fear levels w_{t−1} and with β, as shown by:
  - dgap_{t+N} / d(ε_t) d(w_{t−1}) = β^N ( %−1 / η% ) > 0
  - dgap_{t+N} / d(ε_t) d(β) = N β^{N−1} ( %−1 / η% ) w_{t−1} + β^N η / θ ( %−1 / η% )^2 w_{t−1} > 0  (Equations (52) as in source)
- Therefore, the beneficial effect of improving productivity is greater when the effective lower bound is more binding (that is when β is high) or when fear today is high.

### Interaction with investment-capital ratio
- The benefit of the productivity boost is higher when the investment-capital ratio is higher:
  - dgap_{t+N} / d(ε_t) da = β^N w_{t−1} η%^2 ( %−1 / η% )^2 dν/da > 0  (Equation (53) as in source)
- This result relies on dν/da > 0.
- Implication: there may be a cyclical benefit of a higher investment ratio when combined with a productivity boost (relevant for public investment and green investments in low interest rate environments).

### Key implication (Implication 6)
- When fear is high, especially at the lower bound, policies that boost productivity also have positive multipliers for fighting recessions:
  - When fear is high there is a non-negligible “business cycle multiplier” such that boosting productivity increases the output gap, not just trend output. Moreover, the beneficial is higher at the effective lower bound or when the investment-capital ratio is higher.

### Context within the paper’s contributions and policy insights
- The paper presents a fear-based theory of output, interest rates, and safe assets and highlights three main contributions:
  - (i) developing a closed-form dynamic general equilibrium model generating consistent cross-correlations in key macro variables in the postwar US economy;
  - (ii) presenting a unified framework for analyzing macroeconomy and financial markets with a common fear factor driving equity prices, bond prices, and the risk premium; and
  - (iii) deriving six insights on expanding the macro policy toolkit to manage the ‘fear economy’.
- The six key policy implications summarized in the source are:
  - (1) fear is key driver of business cycles and asset prices, which central banks can manage by regulating the interest rate on safe assets;
  - (2) recessions will be deeper and longer when central banks accept the self-imposed zero lower bound and are unwilling to use negative rates;
  - (3) a commitment to use negative rates in recessions—even if never implemented—raises both the short- and long-run real neutral rates, and moderates the business cycle (as seen during the Great Moderation);
  - (4) counter-cyclical fiscal policy provides disaster insurance and is expansionary by reducing fear;
  - (5) quantitative easing can be narrowly effective when fear is high at the lower bound; and
  - (6) when fear is high, especially at the lower bound, policies that boost productivity also have positive multipliers for fighting recessions.

### Extensions and further research avenues
- The model can be extended to include a banking system to study macro-financial linkages and potential financial cycles with boom-bust dynamics due to the link between banking shocks and safety.
- It can be extended to open economy settings to generate implications for exchange rates and capital flows.
- The SWIM model can be enriched to consider subjective uncertainty and integrate insights from clinical psychology into macro:
  - Relaxing rational expectations by adding structure on subjective belief formation can yield a downward sloping W curve with multiple equilibria and ‘fear breeds fear’ dynamics.
  - In that extension, the safe rate of return acts as a signal about how bad the reasonable worst-case scenario can be, generating fear cycles with episodes of euphoria followed by panic.

*Source: 5.6  The Structural Reforms Multiplier at the Effective Lower Bound (from the supplied content).*

### References

### wpiea2022175-print-pdf - References

### Key Bibliographic Sources Cited
- Extensive list of foundational and recent works in macroeconomics, asset pricing, monetary and fiscal policy, uncertainty, safe assets, business cycle theory, and behavioral/clinical literature (examples include Abel (1999); Adrian, Crump, and Moench (2013); Agarwal & Kimball (2015, WP/15/224; 2019, WP/19/84); Barro (2006); Basu & Bundick (2017); Bloom et al. (2018); Caballero, Farhi, and Gourinchas (2016); Christiano, Eichenbaum, and Evans (2005); Cochrane (2009); Eggertsson, Mehrotra, and Robbins (2019); Gorton (2017); Jaimovich & Rebelo (2009); Kydland & Prescott (1982); Lucas (1972, 1978); Mehra & Prescott (1985); Weitzman (2007, 2009)).  
- Clinical and psychological references related to fear, anxiety, and intolerance of uncertainty (examples include Abramowitz & Blakey (2020); Beck (1979); Dugas, Buhr, & Ladouceur (2004); Vlaeyen & Linton (2012); Vahratian et al. (2021)).

### Appendix A — Analytical Framework, Proofs, and Derivations (Overview)
- A1. Select overview of business cycle theories: Table A.1 maps model features (Rigidities, Source of Shocks, Non-Standard Preferences or Beliefs, Production or Consumption Externality, Capacity Utilization or Adjustment Costs) to whether models can produce positive co-movements in output, consumption, investment, and labor. The table highlights models including Real Business Cycle, News Shocks, Uncertainty shocks, Increasing Returns with Sunspots, SWIM Model (this paper), Uncertainty shocks with sticky prices/wages, New Keynesian models with sticky prices.
- A2. Equilibrium conditions of the frictionless benchmark: equilibrium represented via a system of conditions including
  - C_{t}^{-γ} = e^{-ρ} E_{t}[C_{t+1}^{-γ}](1 + r_{t}^{S})
  - C_{t}^{-γ} = υ_{t} L_{t}^{-χ}
  - 6W_{t} C_{t} = (Y_{t}/Y_{p t})^{η} c_{p t} Y_{p t}
  - W_{t} = (1−α) Y_{t}/L_{t}
  - 1 + r_{t}^{S} = E_{t}[r_{t+1}]
  - I_{t}/I_{p t} − Cov(M_{t+1}, r_{t+1})
  - K_{t} = I_{t−1} + (1−α)K_{t−1}
  - Y_{p t} = A_{t} K_{t}^{α} (L_{p t})^{1−α}
  - I_{t} Y_{t} = i_{p t}[1 + φ_{t} − φ_{t}(Y_{t}/Y_{p t})^{−(1−η)}]
  - Y_{t} = O_{t} Y_{p t}; Y_{t} = C_{t} + I_{t} + G_{t}
  - r_{t+1} ≡ [α Y_{t+1}/K_{t+1} − δ], M_{t+1} = (C_{t+1}/C_{t})^{−γ}, η ≡ (χ − α)/(γ(1 − α))

### Appendix A3–A5 — Existence of Conjectured Solution, Balanced Growth Path, and Fear Measure
- A3. Proof of Proposition 1: shows existence of a conjectured equilibrium O_{t} = O_{t}^{*} yielding optimal consumption C_{t}^{*} = c_{t} Y_{t}^{*} = (Y_{t}^{*}/Y_{p t})^{η} c_{p t} Y_{p t}, derives optimal labor supply L_{t}^{*} and condition η = (χ − α)/(γ(1 − α)) for consistency.
- A4. Balanced growth path and Lemma 1:
  - Choice of υ_{t} such that L_{p t} growth is zero in steady state and a balanced growth path holds.
  - Derives υ_{t+1}/υ_{t} = (c_{p t+1}/c_{p t})^{−γ} (A_{t+1} K_{t+1}^{α} / A_{t} K_{t}^{α})^{1−γ}; interprets A_{t} ≡ z_{t}^{1−α}.
  - Lemma 1: ln[C_{t+1}/C_{t}] ≈ X_{t+1} + η ln[O_{t+1}/O_{t}] + ν ln[O_{t}] where ν ≡ α a[1 + (1−η)φ]/(1 + a − δ) and X_{t+1} ≡ ln[C_{p t+1}/C_{p t}].
- A5. Proof of Lemma 2 and definition of fear:
  - Model for X_{t+1}: X_{t+1} = b + (1−ζ) ln(A_{t+1}/A_{t}), with E_{t}[X_{t+1}] = μ_{A,t}, V_{t}[X_{t+1}] = (1 − ζ)^{2} σ_{A,t}^{2}.
  - Defines 1 − s_{t} as a tail quantile relation: 1 − s_{t} = k σ_{A,t} with k ≡ −Φ^{−1}(.01) (k = 2.33 for normal).
  - Relates consumption growth variance σ_{C,t}^{2} = (1 + η θ/(η(1 − β) − ν))^{2} σ_{X,t}^{2} and fear w_{t} ≡ E_{t} ln[C_{t+1}/C_{t}^{safe}] with w_{t} = k σ_{C,t} = (1 − ζ) (1 − s_{t}) (1 + η θ/(η(1 − β) − ν)).

### Appendix A6–A7 — Asset Pricing: IS Curve and S Curve (Neutral Rate)
- A6. IS curve derivation from Euler equation using cumulant expansion (ignoring higher cumulants):
  - r_{t}^{S} ≈ ρ + γ E_{t}[X_{t+1} + η ln(O_{t+1}/O_{t}) + ν ln O_{t}] − (γ^{2}/2) V_{t}[X_{t+1} + η ln(O_{t+1}/O_{t}) + ν ln O_{t}]
  - Defines r_{t}^{*} ≡ ln R_{t}^{*} = ρ + γ E_{t}[X_{t+1}] − (γ^{2}/2) V_{t}[X_{t+1} + η·gap_{t+1}]
  - Then r_{t}^{S} = r_{t}^{*} − γ(η − ν) gap_{t} + γ η E_{t}[gap_{t+1}].
- A7. S curve derivation (neutral rate as demand for safe assets):
  - Uses approximation V_{t}[X_{t+1} + η·gap_{t+1}] ≈ (1 + θ η/(η(1 − β) − ν))^{2} σ_{X,t}^{2}.
  - With σ_{X,t}^{2} = (1 − ζ)^{2} (1 − s_{t})^{2} 6^{−1} and w_{t} = k σ_{X,t} with k^{2} = 6, shows V_{t}[...] ≈ w_{t}^{2}/6.
  - Neutral rate expressed as r_{t}^{*} ≈ ρ + γ μ_{t} − (γ^{2}/12) w_{t}^{2}.

### Appendix A8–A12 — Equilibrium Dynamics, Risk Premium, Term Structure, Price-Dividend Ratio, Expected Returns
- A8. Proof of Proposition 3:
  - Uses X_{t+1} dynamics: X_{t+1} = μ_{t} + (1/% ) w_{t} ε_{t+1} with % ≡ 1 + η θ/(η(1 − β) − ν), and combines IS and MP curves to obtain gap dynamics and consumption growth:
    - gap_{t+1} ≈ β·gap_{t} + (%^{−1}/η%) w_{t} ε_{t+1} − γ (w_{t}^{2})^{θ} (w^{2})^{1−θ} 12[η(1 − β) − ν] [ψ_{t}]
    - ln[C_{t+1}/C_{t}] ≈ μ_{t} − [η(1 − β) − ν] gap_{t} + w_{t} ε_{t+1} − γ η (w_{t}^{2})^{θ} (w^{2})^{1−θ} 12 [η(1 − β) − ν] [ψ_{t}]
  - Identifies channels by which shocks affect output gap and consumption (see Proposition 3 implications).
- A9. Risk premium and Sharpe ratio:
  - Equity premium ep_{t} ≈ γ V_{t}[X_{t+1} + η ln(O_{t+1}/O_{t}) + ν ln O_{t}] ≈ γ w_{t}^{2}/6.
  - Market Sharpe ratio ≈ γ σ_{t}(Δ ln C_{t+1}) ≈ γ w_{t}/√6.
  - Multi-period equity volatility: V_{t}[ln R_{e t+1}] ≈ λ^{2} w_{t}^{2}/6 + additional terms; for λ = 1 recovers main-text value.
- A10. Term structure derivation:
  - Yield y_{t}^{(N)} ≈ ρ + (γ/N) E_{t} ln[C_{t+N}/C_{t}] − (γ^{2}/2N) V_{t} ln[C_{t+N}/C_{t}].
  - Multi-period consumption growth approximations lead to:
    - E_{t} ln[C_{t+N}/C_{t}] = N·μ + [(1 − θ^{N})/(1 − θ)] [μ_{t} − μ] − [η(1 − β) − ν] [(1 − β^{N})/(1 − β)] gap_{t}
    - V_{t} ln[C_{t+N}/C_{t}] ≈ (1/6)[N w^{2} + ( (1 − θ^{N})/(1 − θ) )(w_{t}^{2} − w^{2}) ] + (risk premium)
  - Yield approximation:
    - y_{t}^{(N)} ≈ r_{t}^{*} +∞ − (γ/N)[η(1 − β) − ν] ((1 − β^{N})/(1 − β))·gap_{t} + (γ/N)((1 − θ^{N})/(1 − θ))[μ_{t} − μ] − (γ^{2}/12N)((1 − θ^{N})/(1 − θ))(w_{t}^{2} − w^{2}) + (risk premium)
  - Yield-curve slope approximation includes terms in gap_{t}, μ_{t} − μ, and (w_{t}^{2} − w^{2}) with Λ_{x} factors.
- A11. Price-dividend ratio (modified Gordon growth representation):
  - For dividends D_{t} = C_{t}^{λ}, derived approximation:
    - ln[P_{e t}/D_{t}] ≈ − ln k_{0} − k_{1} (r_{t}^{*} − r_{t}^{*,LR}) + k_{2} · gap_{t} − k_{3} (w_{t}^{2} − w^{2})
    - Constants: k_{0} ≡ ρ·exp{ (1/ρ)[(γ − λ) μ − (1/12) ρ (γ − λ)^{2} w^{2}] }, k_{1} ≡ (1/γ)((γ − λ)/(1 − θ)), k_{2} ≡ (γ − λ)[η(1 − β) − ν]/(1 − β), k_{3} ≡ (γ − λ)/12 [γ/(1 − θ) − (γ − λ)/(1 − θ)]
  - Exponentiated form: P_{e t} ≈ D_{t} k_{0}·(1 + k_{1}[r_{t}^{*} − r_{t}^{*,LR}])·(1 − k_{2}·gap_{t})·(1 + k_{3}[w_{t}^{2} − w^{2}]).
  - Expected change in log price-dividend ratio across horizons given by terms in gap_{t}, μ_{t} − μ, and w_{t}^{2} − w^{2}.
- A12. Expected equity returns (N-period):
  - Leads to approximation (A.3):
    - E_{t}[ln R_{e t+N}] ≈ Θ_{0} − Θ_{β}·[η(1 − β) − ν] gap_{t} + Θ_{θ}·(μ_{t} − μ) − Θ_{θ}·(w_{t}^{2} − w^{2}) + Θ_{g}·(D_{t}/P_{e t})
  - Constants: Θ_{0} ≡ N λ μ; Θ_{x} ≡ [λ + (1/f)(γ − λ)]((1 − x^{N})/(1 − x)) for x = β, θ; Θ_{θ} = (1/12)[λ + (1/f)(γ − λ)^{2}]((1 − θ^{N})/(1 − θ)); Θ_{g} ≡ (1/f)(N − N(N − 1)/2 E_{t}[g_{t,N}]).
  - Uses dividend-growth averaging g_{t,N} and approximations for f (typical f ≈ 3/2 for calibrations).

*Italic: Source — wpiea2022175-print-pdf - References (canonical PDF content provided).*

### appendix is mainly to lay down the ingredients of a simple setup with inflation (π

### appendix is mainly to lay down the ingredients of a simple setup with inflation (πt)

### SWIM model extension with backward-looking inflation
- Setup:
  - Backward-looking Phillips curve: πt = πet + τ·gapt + επ,t with πet = πt−1 and επ,t white noise.
  - Micro-foundation described: price-setting under adaptive expectations could justify the central bank fixing the real interest rate on safe assets as in the basic SWIM model.
  - Alternative nominal-rigidity micro-foundations noted: sticky information (Mankiw and Reis, 2002); rigidity in numeraire price (Lucas, 1972); money illusion (Reis and Watson, 2007).
  - MP curve modification parameter: κ ≥ 0.
  - Modified MP curve: rSt = Et−1[r∗t] + β [ rSt−1 − r∗t−1 ] + κ Et−1[πt − π∗].

- Equilibrium characterized by five equations:
  - S Curve: r∗t = ρ + γ μt − γ2 12 w2t
  - W Curve: wt = %·(1−ζ)·(1−st)
  - IS Curve: rSt = r∗t − κ φ3 [πt−1 − π∗ + επ,t] − (1/φ1 + κ τ φ3) gapt
  - MP Curve: rSt = Et−1[r∗t] + β(1 − κ τ φ1) [ rSt−1 − r∗t−1 ] + κ(1 − τ φ2) [πt−1 − π∗]
  - P Curve: πt = πt−1 + τ·gapt + επ,t

- Definitions:
  - φ1 ≡ 1 / γ[η(1−β) − ν]
  - φ2 ≡ (η − ν) / η[γ(η − ν) − 1] γ[η(1−β) − ν]
  - φ3 ≡ 1 / γ(η − ν) − 1

- Special case:
  - When κ = 0 the equilibrium conditions reduce to those in the SWIM model without inflation.

- Equilibrium output gap expression:
  - gapt = − φ1 { β(1 − κ τ φ1) [ rSt−1 − r∗t−1 ] + κ(1 − τ φ2 + φ3) [πt−1 − π∗] + κ φ3 [επ,t] + γ2 12 [ w2t − Et−1(w2t) ] }

- Illustration:
  - Figure .1: Example where MP sets interest rates too “low” relative to r∗t, producing a positive output gap and higher inflation. Graphical depiction uses three panels plus an inflation-output tradeoff panel.

### The Fear-Avoidance Model of Uncertainty (Online Appendix C)
- Conceptual motivation:
  - Parameter uncertainty about consumption growth depends on observed real rate r∗t.
  - Draws on Weitzman (2007) and clinical-psychology concepts of catastrophizing and the Fear-Avoidance Model of pain.
  - Agents’ subjective estimate of wt bounded between [w, w].
  - Non-negative constant φ ≥ 0 links perceived volatility to r∗t.

- Subjective volatility specification (equation C.1):
  - m2t = 1 6 ( n − 2 n ) (1 − φ·r∗t )2
  - Limit behavior: as n → ∞, mt → 1 / √6.

- Economic intuition:
  - Low real interest rates signal greater risk of underlying weakness → higher perceived volatility.
  - Finite sample / subjective information leads agents to perceive higher volatility than econometrician measures.
  - mt depending inversely on neutral rate implies lower confidence (higher perceived uncertainty) when real rates are low.

- Modified W curve (Fear-Avoidance specification):
  - wt = %·(1−ζ) (1−st) (1−φ·r∗t)  (equation C.2)

- System of market-clearing conditions under Fear-Avoidance:
  - S Curve: r∗t = ρ + γ μt − γ2 12 w2t
  - W Curve: wt = %·(1−ζ) (1−st) (1−φ·r∗t)
  - IS Curve: rSt = r∗t − γ[η(1−β) − ν] gapt
  - MP Curve: rSt = Et−1[r∗t] + β [ rSt−1 − r∗t−1 ]

- Solving for wt (two-branch solution, equation C.3):
  - wt = 6 γ2 φ% (1−ζ) (1−st) ± √( (6 γ2 φ% (1−ζ) (1−st))2 − 12 [1 − φ(ρ + γ μt)] γ2 φ )
  - Equivalently: wt = St ± √( (St)2 + Resiliencet )
    - Resiliencet ≡ 2 k2 φ γ2 [ φ(ρ + γ μt) − 1 ]
    - St ≡ k2 γ2 φ wt,0
    - wt,0 ≡ [wt | r∗t = 0] = φ% (1−ζ) (1−st)
    - k2 = 6 (by assumption as per Assumption 1 in the main text)

- Interpretation of components:
  - Resiliencet: summary measure of underlying robustness of consumption growth for agents.
    - Moves positively with mean potential growth μt and with k (size of tail risk).
    - Moves negatively with γ (coefficient of relative risk aversion).
  - St: inverse measure of subjective tail risk.
    - Positive association with k.
    - Negative association with γ, baseline fear wt,0, and dependence parameter φ.

- Equilibrium selection and uniqueness:
  - If upper bound on wt eliminates multiple equilibria (St + √(St)2 + Resiliencet > w), unique equilibrium: wt = St − √(St)2 + 2·Resiliencet.
  - Graphical representation: Figure C.1 shows downward-sloping W curve; rightward shift (higher fear) can generate two equilibria (high and low r∗t).

- Key implications (qualitative and comparative-static results):
  - “Fear breeds fear” dynamic (first channel):
    - When wt = St − √(St)2 + Resiliencet, fear falls when Resiliencet rises: dwt / dResiliencet < 0.
    - Fear is convex in Resiliencet: d2wt / dResiliencet2 > 0.
    - At high Resiliencet, declines have small impact on fear; at low Resiliencet, declines generate larger fear increases.
    - Lower neutral rate can raise fear disproportionately through this amplification.
  - Multiple equilibria channel:
    - At low Resiliencet or St, condition for uniqueness fails and two equilibria can exist.
    - Economy may jump from lower wt to higher wt, producing large increases in perceived uncertainty.
    - Rightward shifts of the W curve increase chance of jumping to a “perverse fearful equilibrium.”

- Formal proposition (stated):
  - Proposition (The Fear Avoidance Model of Uncertainty). When the agent’s subjective beliefs about fear (wt) depend inversely on the neutral rate of interest as given in (C.1) with bounds [w, w], then the equilibrium market clearing conditions of the SWIM model are given by (C.2). In this model, there are two types of ‘fear breeds fear’ dynamic. First, uncertainty wt is a convex and negative function of the underlying resilience in the economy, Resiliencet as given in (C.3). Second, this SWIM model extension features the possibility of multiple equilibria, with the chance of the economy jumping to a perverse fearful equilibrium becoming higher when the underlying resilience or safety in the economy deteriorates.

*The Fear Economy: A Theory of Output, Interest, and Safe Assets — Working Paper No. WP/2022/175*

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_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022175-print-pdf.pdf_