## _wp1603 — The effect of the policy rate on the expected non-crisis unemployment rate (selected excerpts)

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---

### Overview and central conclusion
- "Leaning against the wind" defined: monetary policy with a somewhat higher policy interest rate than what is justified by stabilizing inflation around an inflation target and unemployment around its estimated long-run sustainable rate without taking any effects on financial stability into account.
- Immediate costs of leaning against the wind:
  - a weaker economy with higher unemployment and lower inflation;
  - an additional cost if a crisis occurs because the crisis starts from a weaker initial state.
- Main result (summary): With existing empirical estimates taken as given (notably Ekholm (2013), Sveriges Riksbank (2014a), and Schularick and Taylor (2012)), the cost of leaning against the wind can substantially exceed the benefit from a lower probability of a crisis. If anything, a modest leaning with the wind (somewhat lower policy rate) is typically optimal.

### Transmission channels, limitations, and empirical inputs
- Empirical channel emphasized: higher policy rate → lower real debt growth → lower probability of a crisis (reduced-form correlation from Schularick and Taylor (2012)).
- Three important limitations of this channel:
  - If monetary policy is neutral in the long run, it cannot affect real debt in the long run; temporary reductions only shift timing of crisis probability and do not reduce long-run accumulated probability.
  - Effect of the policy rate on real debt is likely small and can be of either sign because:
    - stock of nominal debt has inertia (only a fraction turns over each year);
    - a higher policy rate slows growth of the price level and nominal debt, so numerator and denominator of real debt move in the same direction;
    - empirical studies find a higher policy rate may increase rather than decrease the debt-to-GDP ratio (Alpanda and Zubairy (2014), Gelain, Lansing, and Natvik (2015), Robstad (2014)).
  - The empirical relation between previous real debt growth and crisis probability is reduced-form and correlational; true determinants include shocks and financial-resilience factors (balance sheets, capital, liquidity, lending standards, debt-servicing capacity, risk-taking, asset quality).
- Empirical specification used (quarterly variant of Schularick and Taylor (2012), table 3, column (5)):
  - q_t = (1/4) exp(X_t) / (1 + exp(X_t)), where
  - X_t = -3.89 - 0.398 (2.110) g_{t-4} + 7.138 *** (2.631) g_{t-8} + 0.888 (2.948) g_{t-12} + 0.203 (1.378) g_{t-16} + 1.867 (1.640) g_{t-20}.
  - g_t ≡ ( sum_{⌧=0}^3 d_{t-⌧} / 4 ) / ( sum_{⌧=0}^3 d_{t-4-⌧} / 4 ) - 1, where d_t is level of real debt in quarter t.
  - Reported summary marginal effect: 5 percent lower real debt in 5 years reduces the annual probability of a crisis by about 0.3 percentage points per year, implying a quarterly reduction in q_t of 7.5 basis points (Schularick and Taylor (2012) summary; marginal effect 0.30).
  - The largest coefficient is 7.138 on g_{t-8} (annual real debt growth lagged 2 years).

### Modeling approach and novel features
- Multi-period quarterly model allowing a crisis to occur at any time (not only a second period).
- Accounts for additional cost of leaning against the wind because a crisis may start from higher unemployment caused by prior leaning.
- Considers monetary neutrality vs. non-neutral (permanent) effects on real debt.
- Derives marginal cost and marginal benefit of leaning against the wind to assess optimal policy direction.
- Assesses role of macroprudential policy effectiveness.
- Calibrations and benchmark assumptions:
  - Policy-rate experiment: 1 percentage point higher policy rate during quarters 1–4 (Δi_1 = 1 percentage point).
  - Benchmark crisis increase in unemployment: Δu = 5 percentage points (Sveriges Riksbank (2013)).
  - Benchmark crisis duration: n = 8 quarters (unemployment 5 percentage points higher during the 8 quarters, corresponding to 10 point-years).
  - Benchmark impulse response from Riksbank’s DSGE model Ramses: unemployment increases to about 0.5 percentage points in quarter 6 following the policy-rate shock.
  - Benchmark estimate of effect on real household debt level (Sveriges Riksbank (2014a)): real debt falls relative to baseline by 1 percent in two years and rises back to baseline in about 8 years (monetary policy neutral in the long run).

### Quantitative dynamics from the benchmark experiment (1 percentage point higher policy rate in quarters 1–4)
- Effect on expected non-crisis unemployment:
  - peak increase ≈ 0.5 percentage points in quarter 6.
  - dE_1 u^n_t / di_1 interpreted via Δi_1 = 1 percentage point.
- Effect on real debt and growth:
  - real debt peak fall ≈ 1 percent in two years (relative to baseline), returns to baseline in about 8 years.
  - peak negative deviation of annual real debt growth: about -0.8 percentage points per year in quarter 6.
  - sum over 40 quarters of dg_t/di_1 ≈ 0 (monetary neutrality).
- Effect on crisis-start probability q_t (from (2.6) and dg_t/di_1):
  - negative peak of q_t ≈ -0.04 percentage points in quarters 14 and 15;
  - positive peak of q_t ≈ 0.013 percentage points about two years after the rise in g_t;
  - sum_{t=1}^{40} dq_t/di_1 ≈ 0.
- Effect on probability of being in a crisis p_t (8-quarter moving sum):
  - negative peak ≈ -0.23 percentage points in quarter 18, then rises and turns positive from quarter 25; sum_{t=1}^{40} dp_t/di_1 ≈ 0.
- Net implication:
  - higher policy rate raises expected non-crisis unemployment substantially (peak ≈ 0.5 percentage points);
  - effect on crisis-related expected unemployment is quantitatively very small (example: at quarter 18, dp18/di1 = -0.23 percentage points → Δu dp18/di1 = -0.0023·5 = -0.0116 percentage points = -1.16 basis points);
  - accumulated effect over 40 quarters on expected non-crisis unemployment = 6.9 point-quarters (area under red curve);
  - accumulated effect on expected crisis increase in unemployment = -0.03 point-quarters (area under green curve);
  - because accumulated and average effect on crisis probability over 40 quarters ≈ 0, accumulated effect on expected unemployment rate ≈ accumulated effect on expected non-crisis unemployment rate.

### Cost-benefit decomposition and marginal expressions
- Expected-quarter-t loss (main setup, with crisis-dependent cost):
  - expected loss includes contribution (1 − p_t)(E^1 ̃u^n_t)^2 + p_t (E^1 ̃u^n_t + Δu)^2 + Var component.
- Marginal cost (MC_t) and marginal benefit (MB_t) expressions:
  - Marginal cost driven primarily by effect of policy rate on expected non-crisis unemployment; MC_t approximately proportional to p_t × Δu × dE_1 u^n_t/di_1 (see (4.6)).
  - Marginal benefit arises from reduction in expected crisis loss via a lower dp_t/di_1 and is proportional to (Δu)^2 (− dp_t/di_1) plus additional terms if Δu depends on policy (see (4.7), (D.2)).
- Numerical approximation used in text for benefit term in a key calibration:
  - B_t = 0.0086 E^1 ̃u_n_t (25 + 10 E^1 ̃u_n_t) — the number 0.0086 is the average of derivative (3.16) over quarters 12–24 and acknowledged as an overestimate that biases in favor of leaning against the wind.

### Main calibrated outcome on optimal direction of policy
- For benchmark and existing empirical estimates:
  - Net marginal cost of leaning against the wind is positive in most quarters; marginal cost at zero expected non-crisis unemployment gap is positive (i.e., marginal cost of raising policy rate exceeds marginal benefit).
  - Optimal policy in the baseline-calibrated example is modest leaning with the wind (example point: E^1 ̃u_n_t ≈ −0.23 percentage points in an illustrative figure), not leaning against the wind.
- Required amplification to reverse conclusion:
  - Accumulated marginal cost = 3.71, accumulated marginal benefit = 0.64 (under permanent effect on real debt scenario); benefit is about 5.8 times smaller than cost.
  - Implication: the effect of the policy rate on the probability of a crisis would need to be more than 5.8 times larger than the benchmark estimates in (2.6) to justify leaning against the wind.
  - Sensitivity check: the largest coefficient 7.138 would have to be unrealistically amplified (two standard deviations larger gives 12.4 = 1.74×7.138) — still far from required 5.8×.

### Robustness checks, alternative datasets, and counterfactuals
- Laeven and Valencia (2012) dataset results:
  - estimated specification: q_t = exp(X_t) / (1 + exp(X_t)), with
  - X_t = -5.630 *** (1.008) -5.650 * (3.171) g_t + 4.210 (3.580) g_{t-4} + 12.342 ** (5.408) g_{t-8} -5.259 (3.591) g_{t-12}.
  - For 5 percent steady real debt growth, annual probability of a crisis start = 1.89 percent (implying crisis start on average every 53 years); constant quarterly q ≈ 0.47 percent.
  - Sum of coefficients in (7.1) = 5.64 → marginal effect on annual probability = 0.11 (1 percentage point lower steady real debt growth reduces annual probability by about 0.1 percentage points).
  - Using Laeven and Valencia estimates, net marginal cost of leaning against the wind remains clearly positive; leaning with the wind remains justified.
- Sensitivity experiments summarized:
  - Halving the policy-rate effect on expected non-crisis unemployment (dE_1 u^n_t/di_1) halves marginal cost but marginal benefit unchanged; net marginal cost remains substantially positive except barely negative in quarters 18–21; conclusion unchanged.
  - Higher steady real-debt growth and higher annual probability of crisis start (e.g., raising annual probability from 3.21 percent to 4.21 percent) increases marginal cost proportionally and also increases marginal benefit via larger dp_t/di_1; net effect increases net marginal cost — less effective macroprudential policy strengthens the case against leaning against the wind.
  - Larger crisis increase in unemployment (Δu = 6 percentage points vs. 5) increases both MC (linear in Δu) and MB (quadratic in Δu); net marginal cost generally increases — again strengthening the case against leaning against the wind.
  - Longer crisis duration (n = 12 vs. 8 quarters) shifts MC and MB rightward and increases discounted sum of net marginal costs — leaning against the wind becomes less attractive.
- Non-neutral monetary policy scenario (permanent effect on real debt):
  - Scenario: real debt permanently stays down at its maximum deviation of -1.03 percent from baseline from quarter 8 onwards.
  - Result: marginal benefit becomes larger and more persistent but still insufficient; net marginal cost remains positive and large; discounted sum of net marginal cost positive.
  - Conclusion: even with permanent effect on real debt specified, leaning with the wind remains better.

### Alternative assumptions stacked in favor of leaning against the wind and outcomes
- Fixed-cost-of-crisis assumption (unrealistic): crisis cost independent of initial state (E^1 ̃u^c_t = −Δu regardless of E^1 ̃u^n_t).
  - Under this assumption and a truncated horizon, optimal policy yields a very small positive i_1 (example: i_1 ≈ 0.11 percentage points in a constrained optimization over quarters 1–24), but magnitude is negligible and intertemporal loss reduction is 0.07 percent of baseline value.
  - Quarterly-optimal expected non-crisis unemployment gap under most favourable parameters ≈ 0.22 percentage points at its maximum (illustrative).
  - Conclusion: even when stacking assumptions in favor of leaning against the wind, optimal leaning is extremely small and economically insignificant.
- Stochastic crisis jump (random Δu with mean Δu and variance σ^2_Δu): introduces an additional term p_t σ^2_Δu in expected loss; under maintained assumptions this additional term does not affect qualitative conclusions.

### Constrained-optimal policy exercises and illustrative numerical optima
- Constrained-optimal choice of constant i_1 for quarters 1–4 and baseline thereafter (solve Σ_{t=1}^{40} β^{t-1} NMC_t = 0 with β quarterly discount factor):
  - With β = 0.995 (annual 0.98) and benchmark Schularick and Taylor calibration:
    - constrained-optimal i_1 = −0.83 percentage point (i.e., reduce policy rate by about 0.8 percentage points in quarters 1–4).
  - Restricting horizon to quarters 1–24:
    - i_1 = −0.68 percentage points.
  - Using Laeven and Valencia dataset:
    - full horizon 1–40: i_1 = −0.51 percentage points;
    - truncated horizon to quarter 24: i_1 = −0.40 percentage points.
  - Interpretation: under multiple datasets and horizon choices, constrained-optimal policy is to lower the policy rate in quarters 1–4 (lean with the wind), not increase it.

### Additional quantitative facts and benchmarks (preserve original numeric values)
- Benchmark crisis increase: Δu = 5 percentage points.
- Benchmark crisis duration: n = 8 quarters (10 point-years).
- Riksbank DSGE result: unemployment increases to about 0.5 percentage points in quarter 6 after Δi_1 = 1 percentage point.
- Real-debt response to Δi_1 = 1 percentage point: real debt falls relative to baseline by 1 percent in two years; peak negative deviation of annual real-debt growth ≈ -0.8 percentage points per year in quarter 6.
- Effect on q_t in benchmark example: negative peak ≈ -0.04 percentage points in quarters 14–15; positive peak ≈ 0.013 percentage points later; sum over 40 quarters ≈ 0.
- Effect on p_t in benchmark example: negative peak ≈ -0.23 percentage points in quarter 18; then turns positive from quarter 25; sum over 40 quarters ≈ 0.
- At quarter 18: dp18/di1 = -0.23 percentage points → Δu dp18/di1 = -0.0023·5 = -0.0116 percentage points = -1.16 basis points.
- Accumulated effects over 40 quarters:
  - accumulated effect on expected non-crisis unemployment rate = 6.9 point-quarters;
  - accumulated effect on expected crisis increase in unemployment rate = -0.03 point-quarters.
- Accumulated marginal benefit under permanent real-debt scenario = 0.64; accumulated marginal cost = 3.71 → benefit ≈ 0.64 is about 1/5.8 of cost.
- Schularick and Taylor calibration: sum of coefficients on lagged real-debt-growth rates b = 9.698; implied dq/dg = 0.3016 at q = 0.0321 for g = 0.05.
- Benchmark steady annual probability of a crisis start (Schularick and Taylor case): 3.21 percent (crisis start on average every 31 years) corresponding to g = 5 percent steady real-debt growth.
- Laeven and Valencia calibration: for g = 5 percent steady real-debt growth, annual probability of crisis start = 1.89 percent (crisis start on average every 53 years); constant quarterly q ≈ 0.47 percent.
- Expected-loss arithmetic examples:
  - steady-state probability 6.4 percent implies expected loss = 0.064·25 = 1.6 = 1.26 (text preserves this formatted equality);
  - IMF staff estimates steady-state probability 3.78 percent implies expected loss = 0.0378·25 = 0.945 = 0.97.
- Kocherlakota (2014) comparison: professional forecasters upper-limit probability ≈ 0.29 percent; with Δu = 4 percentage points expected loss = 0.0029·4^2 = 0.0464 = 0.22^2; with Δu = 5 percentage points expected loss = 0.0029·25 = 0.0725 = 0.27^2.
- Flodén (2014) estimate: 1 percentage point lower household debt-to-income ratio in 2007 associated with 0.02 percentage point lower increase in unemployment 2007–2012 (small effect). Riksbank implied peak debt-to-income fall -1.4 percentage points → implied dΔu/di_1 = 0.02·1.4 = 0.28 percentage points lower Δu; effect on expected unemployment via p_t yields peak -0.12 basis points = -0.0012 percentage points in quarter 9 (very small).
- Logit including debt-to-GDP term (Schularick and Taylor variant): coefficient on log debt-to-GDP = 1.1 (s.e. 0.624) significant at 10 percent; with q_t = 0.008 and coefficient 1.1, dq_t/dh_t = 1.1·0.008·0.968 = 0.0085 → 1 percentage point lower debt-to-GDP lowers probability of crisis start by 0.0085 percentage points (very small).

### Policy implications and recommendations
- Minimal policy conclusion: do not lean against the wind without thorough cost-benefit analysis; burden of proof on advocates of leaning against the wind.
- Monetary policy is a poor primary instrument for reducing probability or severity of financial crises given weak and uncertain transmission via real-debt dynamics and the large direct cost in terms of higher non-crisis unemployment.
- Preferable instruments to affect underlying determinants of crisis probability: microprudential and macroprudential policies, housing policy, or fiscal policy as appropriate.
- Evidence cited (Dagher, Dell’Ariccia, Laeven, Ratnovisk, and Tong (2015)): 15–22 percent bank capital relative to risk-weighted assets would have been enough to avoid 85 percent of historical banking crises in OECD since 1970 — suggesting sufficient capital can dramatically reduce crisis probability.

### Final robustness and interpretive notes
- Results robust across alternative assumptions, sensitivity checks (initial state, magnitude of policy-rate effect on non-crisis unemployment, crisis probability, crisis severity and duration), and different datasets (Schularick and Taylor (2012) and Laeven and Valencia (2012)).
- Linear approximation exaggerates probability of a crisis somewhat compared with a Markov process; advantage is tractability for calculating dp_t/di_1 via dp_t/di_1 = sum_{τ=0}^{n-1} dq_{t-τ}/di_1.
- Overall assessment: for existing empirical estimates and plausible model assumptions, the marginal cost of leaning against the wind substantially exceeds marginal benefit; modest leaning with the wind (lower policy rate) is generally preferred.

*Italic: Source — _wp1603 - selected sections and appendices on "The effect of the policy rate on the expected non-crisis unemployment rate" and related sensitivity analyses.*

### 2.1  The e↵ect of the policy rate on the expected non-crisis unemployment rate  . . . . .  10

### 2.1  The effect of the policy rate on the expected non-crisis unemployment rate  . . . . .  10

### Overview
- "Leaning against the wind" is defined as a monetary policy with a somewhat higher policy interest rate than what is justified by stabilizing inflation around an inflation target and unemployment around its estimated long-run sustainable rate without taking any effects on financial stability into account.
- Leaning against the wind has immediate costs: a weaker economy with higher unemployment and lower inflation, and an additional cost if a crisis occurs because the crisis starts from a weaker initial state.
- The paper shows that, with existing empirical estimates taken as given (in particular Ekholm (2013), Sveriges Riksbank (2014a), and Schularick and Taylor (2012)), the cost of leaning against the wind can substantially exceed the benefit from a lower probability of a crisis.

### Transmission channels from policy rate to crisis probability and limitations
- Empirical channel studied: a higher policy rate → lower real debt growth → lower probability of a crisis (reduced-form correlation from Schularick and Taylor (2012)).
- Three important limitations of this channel emphasized:
  - If monetary policy is neutral in the long run, it cannot affect real debt in the long run; a temporary reduction in real debt growth shifts the timing of crisis probability but does not reduce the long-run accumulated probability.
  - The effect of the policy rate on real debt is likely small and can be of either sign because:
    - The stock of nominal debt (e.g., mortgages) has inertia; higher rates may reduce new mortgage growth but only a fraction of stock turns over each year.
    - A higher policy rate slows the growth of the price level as well as nominal debt; both numerator and denominator of real debt move in the same direction, possibly making the effect on the ratio small or even positive.
    - Empirical studies find a higher policy rate may increase rather than decrease the debt-to-GDP ratio (Alpanda and Zubairy (2014), Gelain, Lansing, and Natvik (2015), Robstad (2014)).
  - The empirical relation between previous real debt growth and crisis probability is reduced-form and correlational; true determinants include nature/magnitude of shocks and resilience of the financial system (balance sheets, loss-absorbing capital, liquidity, lending standards, debt-servicing capacity, risk-taking, speculation, asset quality).

### Modeling approach and deviations from earlier literature
- Builds on and extends the two-period frameworks used in prior work (Kocherlakota (2014), Svensson (2014, 2015), Ajello, Laubach, Lopez-Salido, and Nakata (2015), International Monetary Fund (2015)) by:
  - Allowing a crisis to occur at any time (multi-period quarterly model) rather than only in a second period.
  - Accounting for the additional cost of leaning against the wind because a crisis, if it occurs, may start from a higher unemployment rate caused by prior leaning.
  - Considering the implication of monetary neutrality and possible non-neutral (permanent) effects on real debt.
  - Deriving marginal cost and marginal benefit of leaning against the wind to assess optimal policy direction.
  - Assessing the role of macroprudential policy effectiveness in strengthening or weakening the case for leaning against the wind.
- The paper follows a dynamic quarterly approach similar to Diaz Kalan, Laséen, Vestin, and Zdzienicka (2015) but adds that the cost of a crisis depends on the initial state of the economy.

### Key findings and intuition
- With a given positive probability of a crisis and a fixed crisis increase in unemployment, the expected unemployment gap exceeds the non-crisis unemployment gap; to minimize expected squared unemployment gap the optimal policy would set expected future unemployment gap to zero, which implies lowering the policy rate and leaning with the wind.
- For leaning against the wind to be optimal, the policy rate's effect on reducing crisis probability must be sufficiently large; existing empirical estimates imply this effect is much too small, so the net incentive is a modest leaning with the wind.
- Main result: For existing empirical estimates, the cost of leaning against the wind exceeds the benefit by a substantial margin; if anything, a positive probability of crisis justifies modest leaning with the wind rather than against it.
- Less effective macroprudential policy (higher probability or severity of crisis) strengthens the incentive to lean with the wind because a higher crisis probability raises the expected unemployment gap and calls for a more negative non-crisis unemployment gap to minimize expected losses.
- Even when allowing for non-neutral monetary policy with permanent effects on real debt, the conclusion — that costs exceed benefits for existing estimates — is robust.

### Sensitivities and extensions highlighted
- The paper reports sensitivity of results to:
  - The initial state of the economy.
  - The magnitude of the policy-rate effect on the expected non-crisis unemployment rate.
  - The probability of a crisis.
- Additional analyses and extensions include:
  - The effect of the policy rate on the probability of a crisis via real debt growth and crisis start probability/duration.
  - The case of a fixed cost of a crisis versus a state-dependent cost.
  - The role of non-neutral monetary policy (permanent effect on real debt).
  - Results for a dataset of Laeven and Valencia (2012).
  - Appendices provide technical derivations and alternative assumptions (A–J).

### Policy implications (summary)
- Given current empirical estimates, leaning against the wind is generally not justified because:
  - The direct cost in terms of higher non-crisis unemployment is large.
  - Any modest reduction in crisis probability via higher policy rates faces weak and uncertain transmission via real debt dynamics and may be offset or reversed by effects on price levels and GDP.
  - Less effective macroprudential policy does not strengthen the case for leaning against the wind; instead it tends to strengthen the case for leaning with the wind.
- Monetary policy should not be relied upon as the primary instrument for financial-stability-oriented interventions; micro- and macroprudential policies are better suited to affect underlying determinants of crisis probability.

*Italic: Source — _wp1603 - 2.1  The effect of the policy rate on the expected non-crisis unemployment rate  . . . . .  10*

### Section 6 provides additional sensitivity analysis by examining whether monetary non-neutrality

### _wp1603 - Section 6 provides additional sensitivity analysis by examining whether monetary non-neutrality

### Model setup and definitions
- Two possible states each quarter t: non-crisis and crisis.
- Crisis increase in the unemployment rate: u>0 (deterministic in main text; random extension in appendix G).
- Non-crisis and crisis unemployment rates: u^n_t and u^c_t, with u^c_t = u^n_t + u > u^n_t. (Equation (2.1))
- Probability of a crisis starting in quarter t: q_t.
- Crisis duration: n quarters; probability of being in a crisis in quarter t: p_t = sum_{⌧=0}^{n-1} q_{t-⌧}. (Equation (2.2))
- Expected unemployment in quarter t, as of quarter 1: E_1 u_t = E_1 u^n_t + p_t u. (Equation (2.3))

### Decomposition of the effect of a higher policy rate
- Constant policy rate during quarters 1–4: i_1 (i_t = i_1 for 1 ≤ t ≤ 4).
- Total effect on expected future unemployment: dE_1 u_t / di_1 = dE_1 u^n_t / di_1 + u (dp_t / di_1). (Equation (2.4))
  - First term: effect on expected non-crisis unemployment (standard impulse response).
  - Second term: crisis increase u times effect of policy rate on crisis probability.

### Effect on expected non-crisis unemployment (benchmark)
- Benchmark impulse response: Riksbank’s DSGE model Ramses.
  - Shock: increase in policy rate of 1 percentage point during quarters 1–4 (i_1 = 1 percentage point).
  - Resulting deviation of unemployment from baseline (E_1 u^n_t):
    - Unemployment increases to about 0.5 percentage points in quarter 6 and then slowly falls back.
- Derivative interpretation: dE_1 u^n_t / di_1 = - E_1 u^n_t / i_1 = - E_1 u^n_t for t>1. (Equation (2.5))

### Benchmark assumption for crisis magnitude
- Benchmark crisis increase in unemployment rate: u = 5 percentage points (as in Sveriges Riksbank (2013)).

### Effect of policy rate on the probability of a crisis
- p_t depends on q_t (probability of crisis start) and q_t depends on real debt growth; real debt growth may depend on policy rate.
- Schularick and Taylor (2012) estimates (quarterly variant of main logit regression, table 3, column (5)):
  - q_t = (1/4) exp(X_t) / (1 + exp(X_t)), where
  - X_t = -3.89 - 0.398 (2.110) g_{t-4} + 7.138 *** (2.631) g_{t-8} + 0.888 (2.948) g_{t-12} + 0.203 (1.378) g_{t-16} + 1.867 (1.640) g_{t-20}. (Equation (2.6))
  - Numbers within parenthesis are robust standard errors.
- Definition of g_t:
  - g_t ≡ ( sum_{⌧=0}^3 d_{t-⌧} / 4 ) / ( sum_{⌧=0}^3 d_{t-4-⌧} / 4 ) - 1. (Equation (2.7))
  - d_t is the level of real debt in quarter t (Schularick and Taylor define real debt as total bank loans).
- Reported summary marginal effect: 5 percent lower real debt in 5 years reduces the annual probability of a crisis by about 0.3 percentage points per year, implying a quarterly reduction in q_t of 7.5 basis points. (Schularick and Taylor (2012) summary; marginal effect 0.30)
- Note: coefficients are not uniform across lags; g_{t-8} (annual real debt growth lagged 2 years) has the largest coefficient and is the major determinant of q_t.

### Effect of policy rate on real debt and resulting crisis probabilities (benchmark example)
- Benchmark estimate for effect on real household debt level d_t of a 1 percentage point higher policy rate during 4 quarters (Sveriges Riksbank (2014a)):
  - Real debt falls relative to baseline level by 1 percent in two years and then rises back to baseline in about 8 years.
  - Monetary policy is neutral in the long run: no long-run effect on real debt.
  - Interpret derivative: d(d_t)/di_1 for t>1, approximately zero for t>32.
- Resulting dynamics for annual real debt growth g_t (derivative dg_t/di_1 for t>1):
  - g_t first falls below baseline growth, then rises above baseline; sum over 40 quarters ≈ 0 (sum_{t=1}^{40} dg_t/di_1 ≈ 0).
  - Peak negative deviation of annual real debt growth: about -0.8 percentage points per year in quarter 6.
- Resulting dynamics for probability of a crisis start q_t (from (2.6) and dg_t/di_1):
  - Negative peak of q_t: about -0.04 percentage points in quarters 14 and 15.
  - Positive peak of q_t: 0.013 percentage points (small) about two years after the rise in g_t.
  - Sum over 40 quarters ≈ 0 (sum_{t=1}^{40} dq_t/di_1 ≈ 0).
  - Interpretation: increase in policy rate initially reduces probability of crisis start, later increases it; average effect over time is zero.
- Resulting dynamics for probability of being in a crisis p_t (8-quarter moving sum; benchmark crisis duration n = 8 quarters):
  - Benchmark: crisis duration n = 8 quarters (unemployment 5 percentage points higher during the 8 quarters, corresponding to 10 point-years).
  - p_t has a negative peak of about -0.23 percentage points in quarter 18, then rises back to zero and turns positive from quarter 25; still positive in quarter 40 but will eventually fall to zero.
  - Interpretation of derivative: dp_t/di_1 for t>1, and sum_{t=1}^{40} dp_t/di_1 ≈ 0. (Equation (2.8))
- Overall implication from the benchmark example:
  - A 1 percentage point higher policy rate during quarters 1–4:
    - Raises expected non-crisis unemployment with a peak around 0.5 percentage points (quarter 6).
    - Reduces real debt temporarily (peak fall 1 percent in two years).
    - Causes annual real debt growth to fall and then rise (peak -0.8 percentage points per year).
    - Shifts probability of crisis starts and crises in time: initial reduction in q_t and p_t (after about 3 years), followed by increases (after about 6 years), with no accumulated average effect over the 40-quarter horizon analyzed.

*Source: _wp1603 - Section 6 provides additional sensitivity analysis by examining whether monetary non-neutrality with a permanent e↵ect on real debt changes the results.*

### 2.3  The e↵ect of the policy rate on the expected future unemployment rate

### 2.3  The e↵ect of the policy rate on the expected future unemployment rate

### Effect decomposition and calculation
- The expected future unemployment rate dE1ut/di1 equals the sum of:
  - the effect on the expected non-crisis unemployment rate dE1u n t/di1 (figure 2.1, red line), and
  - the effect on the crisis increase in the expected unemployment rate udp t/di1, which equals the product of the change in the crisis probability dp t/di1 and the crisis increase in the unemployment rate u.
- The crisis increase in the unemployment rate is assumed to be u = 5 percentage points (Sveriges Riksbank (2013) assumption).
- The policy-rate experiment illustrated is a 1 percentage point higher policy rate during quarters 1–4; deviations are shown from baseline (figure 2.3).

### Key quantitative findings (preserve original numeric values)
- Largest effect on the crisis probability occurs in quarter 18, when dp18/di1 is 0.23 percentage points.
- The corresponding effect on the crisis increase in the expected unemployment rate:
  - udp t/di1 = 0.0023·5 = 0.0116 percentage points = 1.16 basis points (green line, right vertical axis).
- Effect on expected non-crisis unemployment rate in quarter 18:
  - dE1u n 18/di1 = 0.16 percentage points = 16 basis points (red line).
- Accumulated effects over 40 quarters:
  - Accumulated e↵ect on the expected non-crisis unemployment rate = 6.9 point-quarters (area under red curve).
  - Accumulated e↵ect on the expected crisis increase in the unemployment rate = 0.03 point-quarters (area under green curve).
  - Because the accumulated and average e↵ect on the probability of a crisis over 40 quarters is approximately zero, the accumulated e↵ect on the expected unemployment rate is approximately equal to the accumulated e↵ect on the expected non-crisis unemployment rate:
    - Σ40 t=1 dE1u t/di1 ≈ Σ40 t=1 dE1u n t/di1.
- From quarter 25 onward the e↵ect of the policy rate on the probability of a crisis is very small, but positive.

### Interpretation and implication for policy (leaning against the wind)
- The effect of the policy rate on the expected unemployment rate is dominated by the effect on the expected non-crisis unemployment rate; the crisis-related channel is quantitatively very small.
- Because a higher policy rate has only a modest decreasing e↵ect on the probability of a crisis for a few years and then a small increase thereafter, there is no meaningful accumulated e↵ect of the policy rate on expected crisis increase in the unemployment rate (long-run neutrality of monetary policy implies accumulated e↵ect on crisis probability ≈ 0).
- Policy implication from this section:
  - It is not true that a higher unemployment rate in the near future can be traded for a lower expected unemployment rate further into the future.
  - Leaning against the wind (raising the policy rate) increases the expected unemployment rate both in the near future and further into the future.

*Source: _wp1603 - 2.3  The e↵ect of the policy rate on the expected future unemployment rate*

### 3.3  The possible benefit of leaning against the wind

### 3.3  The possible benefit of leaning against the wind

### Main conclusion
- There is a strong tendency toward some leaning with the wind rather than against.  
- Only if the policy rate has a sufficiently strong negative effect on the probability of a crisis can leaning against the wind be justified.

### Setup and intuition
- The probability of a crisis is modeled as an implicit function of the expected non-crisis unemployment gap, p_t = p_t(E^1  ̃u_n_t), with linear approximation for t≥1:
  - p_t(E^1  ̃u_n_t) − ̄p_t = (dp_t/dE^1  ̃u_n_t) E^1  ̃u_n_t. (3.15)
- The implicit derivative is given by
  - dp_t/dE^1  ̃u_n_t ≡ (dp_t/di^1) / (dE^1  u_n_t/di^1). (3.16)
- Benefit from deviating from zero unemployment gap:
  - B_t = [̄p_t − p_t(E^1  ̃u_n_t)][(−u)^2 + 2(−u)E^1  ̃u_n_t]
  - Using the linear approximation:
    - B_t = (dp_t/dE^1  ̃u_n_t) E^1  ̃u_n_t[(−u)^2 + 2(−u)E^1  ̃u_n_t] (3.17)
    - Numerical approximation used in text: B_t = 0.0086 E^1  ̃u_n_t (25 + 10 E^1  ̃u_n_t).
  - The number 0.0086 is the average of derivative (3.16) over quarters 12–24 (acknowledged as an overestimate).

### Cost, benefit, and marginal expressions (as functions of E^1  ̃u_n_t)
- Cost C_t (same as in section 3.2, red line in figures).
- Marginal benefit:
  - dB_t/dE^1  ̃u_n_t = (−dp_t/dE^1  ̃u_n_t)[(−u)^2 + 4(−u)E^1  ̃u_n_t] = 0.0086 (25 + 20 E^1  ̃u_n_t). (3.18)
  - Linear and increasing in E^1  ̃u_n_t when dp_t/dE^1  ̃u_n_t < 0.
- Net cost = C_t − B_t (blue dashed in figures); net marginal cost = marginal cost − marginal benefit (blue solid).

### Graphical implications and numerical example
- In figure 3.3 (described in text), net cost has a minimum at point G between points E and C; marginal cost and marginal benefit equal at point F; net marginal cost equals zero at point H between D (E^1  ̃u_n_t = −0.32 percentage point) and A (E^1  ̃u_n_t = 0).
- The case example discussed leads to an optimal policy close to point D, involving modest leaning with the wind corresponding to E^1  ̃u_n_t = −0.23 percentage points.
- The net marginal cost at zero expected non-crisis unemployment gap is positive, implying in the initial situation the marginal cost of leaning against the wind exceeds marginal benefits.

### Role of the assumed derivative magnitude
- The author notes that using 0.0086 likely exaggerates the benefit (it is an overestimate), which biases in favor of leaning against the wind; even with that exaggeration the conclusion favors leaning with the wind.

### Alternative (unrealistic) assumption: fixed cost of a crisis (preview)
- If the crisis cost is fixed and independent of the non-crisis unemployment gap, E^1  ̃u_c_t = −u regardless of E^1  ̃u_n_t:
  - Expected crisis loss becomes independent of E^1  ̃u_n_t; expected quarter-t loss:
    - E^1 L_t = (1 − p_t)(E^1  ̃u_n_t)^2 + p_t (−u)^2 + Var^1  ̃u_n_t.
  - Rewriting relative to benchmark ̄p_t yields:
    - E^1 L_t − Var^1  ̃u_n_t = C_t − B_t,
    - with C_t ≡ (1 − ̄p_t)(E^1  ̃u_n_t)^2 + ̄p_t (−u)^2,
    - and B_t ≡ (̄p_t − p_t)[(−u)^2 − (E^1  ̃u_n_t)^2]. (3.19)
- Under this fixed-cost assumption, crisis loss increase (−u)^2 − (E^1  ̃u_n_t)^2 is decreasing in |E^1  ̃u_n_t|; could become negative if E^1  ̃u_n_t > −u, implying counterintuitive results (e.g., better to have a crisis than a non-crisis for large gaps).
- Numerical expressions shown in the text for benchmark ̄p_t = 6.4 percent:
  - C_t = 0.936 (E^1  ̃u_n_t)^2 + 1.6 (since ̄p_t = 0.064 and (−u)^2 = 25).
  - Marginal cost: dC_t/dE^1  ̃u_n_t = (1 − ̄p_t) 2 E^1  ̃u_n_t = 1.872 E^1  ̃u_n_t.
  - Benefit (linear approximation): B_t = 0.0086 E^1  ̃u_n_t [25 − (E^1  ̃u_n_t)^2] = 0.215 E^1  ̃u_n_t − 0.0086 (E^1  ̃u_n_t)^3.
  - Marginal benefit: dB_t/dE^1  ̃u_n_t = 0.215 − 0.0258 (E^1  ̃u_n_t)^2.
- Under the fixed-cost assumption, any optimal leaning against the wind is very small (example result: optimal E^1  ̃u_n_t ≈ 0.12 percentage point) and produces an insignificant net gain.

### Implication and robustness
- With the main (realistic) assumption that crisis cost increases when the economy is weaker (crisis increases unemployment gap by −u), modest leaning with the wind is optimal in the baseline calibrated example.
- Even with exaggerated estimates of the policy rate’s effect on crisis probability, leaning with the wind remains optimal.
- Under the unrealistic fixed-cost crisis assumption, leaning against the wind may be marginally optimal but the magnitude and net gain are negligible.
- The author emphasizes that only a sufficiently strong negative effect of the policy rate on crisis probability would justify leaning against the wind.

*Source: 3.3  The possible benefit of leaning against the wind (from the provided IMF content unit)._wp1603*

### 4.2  The sensitivity to the e↵ect of the policy rate on the non-crisis unemploy-

### _wp1603 - 4.2  The sensitivity to the e↵ect of the policy rate on the non-crisis unemploy- ment rate

### Sensitivity to the e↵ect of the policy rate on expected non-crisis unemployment
- The marginal cost and net marginal cost of leaning against the wind depend on:
  - the initial expected unemployment gap (E1 ̃u n t),
  - the probability of a crisis (p t),
  - the crisis increase in the unemployment gap (u),
  - the e↵ect of the policy rate on the expected non-crisis unemployment rate (dE1 u n t/di1).
- Benchmark uses the Riksbank estimate of the policy-rate e↵ect (figure 2.1 in source).
- When the benchmark policy-rate e↵ect on the expected non-crisis unemployment gap is reduced to a half of the Riksbank estimate:
  - the marginal cost shifts down by a half;
  - the marginal benefit (4.7) is not a↵ected;
  - the net marginal cost shifts down with the marginal cost but remains substantially positive, except in quarters 18–21 where it is slightly negative.
- Conclusion: even with half the benchmark e↵ect, leaning against the wind is still not justified.

### Preliminary note on sensitivity to the probability of a crisis
- From (4.6) the marginal cost equals two times the product: probability of a crisis (p t) × crisis increase in unemployment rate (u) × dE1 u n t/di1.
- Figure 4.3 interpretation: halving the policy-rate e↵ect is equivalent to halving the probability of crisis in figure 3.1 for the marginal cost.
- Numerical illustration:
  - If the probability of a crisis start in a particular quarter were 0.4 percent (instead of 0.8 percent), then the probability of crisis from quarter 9 onward would be 3.2 percent (instead of 6.4 percent); marginal cost would be half of benchmark.
  - Even then, leaning against the wind would still not be justified.
- Caveat: this note assumes that a lower probability of a crisis does not a↵ect the e↵ect of the policy rate on the probability of a crisis (dp t/di1) and thus not the marginal benefit. Under the logistic relation between probability of a crisis start and real debt growth, the e↵ect on the probability of a crisis and the marginal benefit varies slightly with the probability of a crisis and is actually slightly lower with a lower probability of a crisis.

### Does less e↵ective macroprudential policy strengthen the case for leaning against the wind?
- Framing: less e↵ective macroprudential policy implies less financial resilience (weaker balance sheets, less loss-absorbing capital), more credit growth and credit booms, lower credit standards — which can increase:
  - the probability of a crisis start, q t,
  - the severity of a crisis (larger u),
  - the duration of a crisis, n.
- The analysis examines how marginal cost, marginal benefit, and net marginal cost shift when:
  - the probability of crisis start is higher;
  - the crisis increase in unemployment rate is larger;
  - the crisis duration is longer.

#### 5.1 A higher probability of a crisis start due to higher credit growth
- Benchmark assumptions:
  - annual probability of a crisis start = 3.21 percent (corresponding to a crisis start on average every 31 years);
  - this is consistent with a steady annual growth rate of real debt of 5 percent (point A in figure 5.1).
- Counterfactual considered:
  - increase annual probability of a crisis start by 1 percentage point to 4.21 percent (corresponding to a crisis start on average every 24 years);
  - this is consistent with an annual steady growth rate of 7.9 percent (point B in figure 5.1).
- Mechanics and numerical notes:
  - a higher steady growth rate of real debt increases the marginal e↵ect of steady real debt growth on the probability of a crisis start because the logistic function is convex in this range;
  - for the benchmark steady annual growth rate of 5 percent, the marginal e↵ect on the annual probability of a crisis start is 0.30;
  - for a steady annual growth rate of 7.9 percent, the marginal e↵ect is 0.39.
- Results (figure 5.2):
  - higher p t shifts up the marginal cost (proportional to p t) — marginal cost moves from solid to dashed red line;
  - higher growth also increases dp t/di1, which increases the marginal benefit (proportional to dp t/di1) — marginal benefit moves from solid to dashed green line;
  - net e↵ect: a significant increase in the net marginal cost (solid to dashed blue line).
- Conclusion: less e↵ective macroprudential policy that raises real debt growth and the probability of a crisis start strengthens the case against leaning against the wind.

#### 5.2 A larger crisis increase in the unemployment rate
- Counterfactual considered:
  - crisis increase u = 6 percentage points (dashed) versus benchmark u = 5 percentage points (solid).
- Functional dependence:
  - marginal cost is linear in u (see (4.6));
  - marginal benefit is quadratic in u (see (4.7)).
- Results (figure 5.3):
  - both marginal cost and marginal benefit magnitudes increase with u;
  - net e↵ect: increase of the net marginal cost except around quarter 19 where marginal benefit increases slightly more than marginal cost.
- Conclusion: if less e↵ective macroprudential policy implies a larger crisis increase in unemployment, it strengthens the case against leaning against the wind.

#### 5.3 A longer duration of a crisis
- Counterfactual considered:
  - crisis duration n = 12 quarters (dashed) versus benchmark n = 8 quarters (solid).
- Mechanic:
  - longer duration implies, for the linear approximation used, the probability of a crisis is the sum over more previous quarterly probabilities of a crisis start, producing a rightward shift of marginal cost and marginal benefit.
- Results (figure 5.4):
  - net marginal cost shifts up except around quarter 24 where marginal benefit increases slightly more than marginal cost;
  - the sum of discounted net marginal costs increases.
- Conclusion: if less e↵ective macroprudential policy increases crisis duration, it strengthens the case against leaning against the wind.

- Overall conclusion for section 5:
  - Under the intuitive assumptions examined, less e↵ective macroprudential policy consistently further strengthens the already strong case against leaning against the wind. The presumption that less e↵ective macroprudential policy would strengthen the case for leaning against the wind does not stand up to scrutiny.

### 6 Non-neutral monetary policy: a permanent e↵ect on real debt
- Hypothesis tested: suppose monetary policy is non-neutral and has a permanent e↵ect on real debt.
- Scenario considered:
  - real debt permanently stays down at its maximum deviation from the baseline in figure 2.2 (1.03 percent), from quarter 8 onwards.
- Results:
  - there is a large and persistent, but not permanent, reduction in the probability of a crisis (figure 6.1);
  - marginal benefit is larger and more persistent (figure 6.2);
  - nevertheless, marginal benefit is not sufficient to prevent the net marginal cost from being positive;
  - the discounted sum of net marginal cost remains positive and large.
- Conclusion: even if policy rate had a permanent e↵ect on real debt (as specified), leaning against the wind still has a large positive net marginal cost, and leaning with the wind remains a better policy.

*Source: _wp1603 - 4.2  The sensitivity to the e↵ect of the policy rate on the non-crisis unemployment rate*

### 6.1  How much larger an e↵ect on the probability of a crisis start is needed to

### _wp1603 - 6.1  How much larger an e↵ect on the probability of a crisis start is needed to justify leaning against the wind?

### Accumulated marginal benefits and benchmark results
- Under the assumption of a permanent e↵ect on real debt of a 1 percentage point higher policy rate during quarters 1–4, the accumulated marginal benefit in figure 6.2 is positive and equal to 0.64.
- For the linear approximation (2.2):
  - The probability of a crisis increases from zero in quarter 1 to 9.6 percent in quarter 13 and then stays at 9.6 percent.
- For the Markov process discussed in appendix A:
  - The probability of a crisis increases from zero in quarter 1 to 9.1 percent in quarter 13 and then converges to 8.8 percent in quarter 20.
- Figures 6.1 and 6.2 depict:
  - Deviations from baseline for real debt (percent), average annual real debt growth (pp/yr), probability of a crisis start in quarter (pp), and probability of a crisis in quarter (pp), and show marginal cost, marginal benefit, and net marginal cost when the expected non-crisis unemployment gap is zero.

### Required amplification of the policy-rate effect to justify leaning against the wind
- The accumulated marginal cost is 3.71, about 5.8 times larger than the accumulated marginal benefit of 0.64.
- Implication: For leaning against the wind to be justified, the e↵ect of the policy rate on the probability of a crisis must be more than 5.8 times larger than the benchmark estimates in (2.6).
- Sensitivity check on a single coefficient:
  - The largest coecient, 7.138, on the two-year lag of the annual growth rate: if it were two standard deviations larger, it would be 12.4, i.e., 1.74 times larger than 7.138 — far from the required 5.8 times.
- Figures 6.3 and 6.4 illustrate the case when the e↵ect is 5.8 times larger and show that only then the accumulated marginal cost and marginal benefit are equal.
- Conclusion from this exercise: Even under extreme assumption of a large permanent e↵ect on real debt, an extreme (and empirically unrealistic) amplification of the policy-rate e↵ect on crisis probability is required to make accumulated net marginal cost zero.

### Results using Laeven and Valencia (2012) dataset (section 7)
- Estimated probability specification:
  - q_t = exp(X_t) / (1 + exp(X_t)), where
  - X_t = -5.630 *** (1.008) -5.650 * (3.171) g_t + 4.210 (3.580) g_{t-4} + 12.342 ** (5.408) g_{t-8} -5.259 (3.591) g_{t-12}.  (Equation (7.1) preserved exactly as presented.)
- Key empirical implications:
  - The annual growth rate of the average annual debt lagged two years is the major determinant of the probability of a crisis start, q_t.
  - For 5 percent steady real debt growth, the annual probability of a crisis start is 1.89 percent, approximately equal to the frequency of crisis starts in the sample (implying a crisis start on average every 53 years).
  - The corresponding constant quarterly probability of a crisis start, q, is about 0.47 percent.
  - The coefficients in (7.1) sum to 5.64, implying the marginal e↵ect on the annual probability of a crisis start over all lags is equal to 0.11 — i.e., 1 percentage point lower steady real debt growth reduces the annual probability of a crisis by about 0.1 percentage points.
- Figures 7.1–7.3 summary:
  - Figure 7.1: e↵ect on real debt, average annual real debt growth, probability of a crisis start in quarter (pp), and probability of being in a crisis in quarter (pp) for a 1 percentage point higher policy rate during quarters 1–4; deviations from baseline.
  - Figure 7.2: e↵ect on expected unemployment rate and expected non-crisis unemployment rate of a 1 percentage point higher policy rate during quarters 1–4; deviations from baseline — a higher policy rate increases expected unemployment at all horizons.
  - Figure 7.3: marginal cost, marginal benefit, and net marginal cost when the expected non-crisis unemployment gap equals zero.
- Numerical and sign outcomes using Laeven and Valencia estimates:
  - The e↵ect on the probability of a crisis start (dq_t/di_1, blue line) fluctuates more than in Schularick and Taylor (2012).
  - The e↵ect on the probability of a crisis (dp_t/di_1, green line) fluctuates: first increases relative to baseline, then falls to a negative peak of -0.27 percentage point in quarter 17, increases to a positive peak of 0.15 percentage points in quarter 26, and falls to zero in quarter 40. The accumulated e↵ect on the probability of a crisis is approximately zero.
  - The marginal cost p_t increases from 0 in quarter 1 to 1.89 percent in quarter 9 and then stays at 1.89 percent; as a result the marginal cost has a lower peak and is smaller than in the Schularick and Taylor case.
  - The marginal benefit fluctuates; in quarters 17–19 it equals the marginal cost, making net marginal cost zero in those quarters.
  - Around quarter 8 and quarter 26 the marginal benefit is negative, adding to net marginal cost.
  - The sum of marginal benefits over the 40 quarters is -0.02 percentage points (very close to zero); therefore the sum of net marginal costs over 40 quarters is approximately equal to the sum of marginal costs.
  - Net marginal cost of leaning against the wind is clearly positive; hence leaning with the wind (a somewhat lower policy rate), not against, is justified for these estimates.

### Main conclusions and policy implications
- Overall conclusion:
  - For existing empirical estimates and reasonable assumptions, the marginal cost of leaning against the wind is much higher than the marginal benefit. Thus, leaning against the wind is not justified; if anything, a modest leaning with the wind (a somewhat lower policy rate) is justified.
- Decomposition of costs and benefits:
  - Main component of marginal cost: marginal cost of increasing the crisis unemployment gap. Leaning against the wind increases both the non-crisis and crisis unemployment gaps; even if the initial non-crisis unemployment gap is zero, the crisis unemployment gap is not zero and its marginal cost is positive.
  - Main component of marginal benefit: reduction in the expected cost of a crisis via a possibly lower probability of a crisis from a higher policy rate. For existing empirical estimates and channels, this e↵ect is too small to match the marginal cost.
- On the key empirical channel (policy rate → real debt growth → crisis probability):
  - The channel is reduced-form and correlational, may not be statistically significant, likely small, and may be of either sign — yet it was assumed in favor of leaning against the wind for the analysis.
  - If monetary policy is neutral in the long run, no accumulated long-run e↵ect on real debt growth exists; a temporary reduction in crisis probability is followed by higher probability later, leaving average and accumulated probability unchanged.
  - Even allowing for monetary non-neutrality and permanent e↵ects on real debt, empirically the marginal benefit remains too small to match marginal cost; required policy-rate e↵ect on crisis probability would be empirically unrealistic.
- Robustness and sensitivity:
  - Results are robust to alternative assumptions and to using the Laeven and Valencia (2012) dataset with more recent data and more countries.
  - Under the extreme and unrealistic assumption that the cost of a crisis is fixed and independent of initial state, and with initial non-crisis unemployment gap zero, the marginal cost of leaning against the wind is zero while marginal benefit is small positive — implying some minimal optimal leaning against the wind. Even then, optimal leaning is extremely small (a few basis points higher policy rate and non-crisis unemployment gap) and net gains are insignificant.
- On a systematic regime of leaning against the wind:
  - If leaning against the wind were a new systematic policy regime, initial credibility problems imply it would resemble a surprise policy-rate increase for several years; if a temporary surprise increase is not beneficial in the simple test applied here, it is unlikely a systematic regime would be robustly beneficial.
  - Examination of a systematic policy would require a complicated, model-dependent analysis; the simple minimalist test used here is argued to be more reliable and robust.
- Policy recommendations and alternatives:
  - Minimal policy conclusion: No leaning against the wind should be undertaken without support from a thorough cost-benefit analysis; burden of proof should be on advocates of leaning against the wind.
  - For reducing probability or severity of financial crises, monetary policy appears a poor instrument; other policies should be used instead, such as micro- and macroprudential policy, housing policy, or fiscal policy, depending on the problem.
  - Evidence cited: Results of Dagher, Dell’Ariccia, Laeven, Ratnovisk, and Tong (2015) indicate that 15–22 percent bank capital relative to risk-weighted assets would have been enough to avoid 85 percent of the historical banking crises in the OECD countries since 1970; thus, sufficient capital may lead to a dramatic reduction in the probability of a crisis.

*Italic: Source — Excerpt from the IMF working paper _wp1603, section 6.1, and related sections 7 and 8 as provided in the source content.*

### 6.4 percent for the linear approximation and to 6.0 percent for the Markov process. The linear

### 6.4 percent for the linear approximation and to 6.0 percent for the Markov process. The linear

### Linear approximation versus Markov process: probability of a crisis
- The linear approximation exaggerates the probability of a crisis somewhat compared with the Markov process.
- Advantage of the linear approximation (equation (2.2)): the effect of the policy rate on the probability of a crisis is easy to calculate. Given the effect on the probability of a crisis start, dq_t/di_1 for t≥1, it satisfies
  - dp_t/di_1 = sum_{τ=0}^{n-1} dq_{t-τ}/di_1 . (A.2)
- For the Markov process, let P_t = [P_{t,ij}] for t≥1 denote the transition matrix from states in quarter t to states in quarter t+1, where P_{t-1,11} = 1 - q_t, P_{t-1,12} = q_t and P_{t-1,ij} = P_{ij} for (i,j) ≠ (1,1),(1,2).
  - π_t = π_{t-1} P_{t-1} for t≥2.
  - dπ_t/di_1 = dπ_{t-1}/di_1 P_{t-1} + π_{t-1} dP_{t-1}/di_1 for t≥2, where dP_{t-1,11}/di_1 = - dq_t/di_1, dP_{t-1,12}/di_1 = dq_t/di_1, and dP_{t-1,ij}/di_1 = 0 for (i,j) ≠ (1,1),(1,2).
  - dp_t/di_1 = - dπ_{t,1}/di_1 for t≥2. (A.3)
- Figure A.1 compares the probability of a crisis in quarter by the linear approximation (2.2) and by the Markov process (A.1) for q = 0.8 percent and n = 8 quarters; the caption notes the two series "Linear approximation, %" and "Markov process, %".
- Figure A.2 shows the effect on the probability of a crisis for the linear approximation (A.2) and the Markov process (A.3) (for q = 0.8 percent and n = 8 quarters) of a 1 percentage points higher policy rate during quarters 1–4.
- The linear approximation exaggerates somewhat the effect on the probability of a crisis and thus the marginal benefit of leaning against the wind.

### The logistic function (Appendix B)
- Logistic function specification:
  - q = exp(a + b g) / [1 + exp(a + b g)] = 1 / [1 + exp(-(a + b g))]. (B.1)
  - q is the annual probability of a crisis start; g is the steady annual growth rate of real debt; a and b are constants. In a logit regression of crisis starts on current and lagged annual growth rates of real debt, b corresponds to the sum of the coefficients on the lagged annual growth rates.
- Marginal effect:
  - dq/dg = b q (1 - q). (B.2)
- Calibration and numerical values used:
  - The sum of coefficients in (2.6) is b = 9.698.
  - Using dq/dg = 0.3016 and solving (B.2) gives q = 0.0321 (reported succinctly as dq/dg = 0.3016 and q = 0.0321).
  - Given b and q, if g = 0.05 then a = -3.890 (from (B.1)).
  - Section 5.1 examines g = 0.079, then q = 0.0421, and dq/dg = 0.3914.

### The simple loss function and benchmark unemployment (Appendix C)
- Quadratic loss function assumed:
  - L*_t(π_t, u_t) ≡ π_t^2 + (u_t - ū)^2, where π_t denotes the gap between the inflation rate and a fixed inflation target in quarter t, and u_t - ū is the gap between the unemployment rate u_t in quarter t and the long-run sustainable unemployment rate ū. (C.1)
- Phillips curve assumed:
  - π_t = z_t - κ (u_t - ū), where z_t is a stochastic intercept (cost-push shocks) and κ is the slope. (C.2)
  - A positive (negative) z_t implies that a zero inflation gap requires a positive (negative) unemployment gap.
- Combining the loss function and Phillips curve yields:
  - L'_t[(u_t - ū); z_t] ≡ L*_t[z_t - κ(u_t - ū), u_t] = (z_t - κ(u_t - ū))^2 + (u_t - ū)^2
  - Algebra leads to L'_t = (κ^2 + 1) [(u_t - u^*(z_t))^2 + (κ/ (κ^2 + 1)/2?) (u^*(z_t) - ū)^2] with u^*(z_t) - ū ≡ - z_t / (κ^2 + 1). (C.3) and (C.4) (expressions preserved as presented).
- Interpretation:
  - u^*(z_t) is the unemployment rate that, for given z_t, minimizes the loss function (C.1) subject to the Phillips curve (C.2). Choosing u_t to minimize L_t(u_t; u^*_t) ≡ (u_t - u^*_t)^2 is equivalent to choosing u_t to minimize L'_t(u_t; z_t). (C.5)
- Crisis definition:
  - A crisis is considered a negative demand shock that increases the unemployment rate by a fixed amount Δu > 0, independent of z_t and thus independent of the benchmark unemployment rate.

### Effect of the policy rate on the crisis increase in the unemployment rate (Appendix D)
- Empirical background:
  - Flodén (2014): for OECD countries, a 1 percentage point lower household debt-to-income ratio in 2007 is associated with a 0.02 percentage point lower increase in the unemployment rate during 2007–2012. The effect is small; statistically significant for all OECD countries but not for the sample of OECD countries during which housing prices fell.
- Channel considered:
  - If a higher policy rate lowers the debt-to-income ratio, it might reduce the crisis increase in the unemployment rate: dΔu/di_1 < 0.
- Effect on expected unemployment (extension of equation (2.4) via (2.3)):
  - dE_1 u_t / di_1 = dE_1 u^n_t / di_1 + Δu dp_t/di_1 + p_t dΔu/di_1 . (D.1)
- Effect on expected quadratic loss (extension of (4.2)):
  - dE_1 L_t / di_1 = 2 [E_1 ūn_t + p_t Δu] dE_1 u^n_t / di_1 + [(Δu)^2 + 2 Δu E_1 ūn_t] dp_t/di_1 + 2 p_t [Δu + E_1 ūn_t] dΔu/di_1 .
- Additional term in marginal benefit (extension of (4.5)):
  - MB_t ≡ [(Δu)^2 + 2 Δu E_1 ūn_t] (- dp_t/di_1) + 2 p_t [Δu + E_1 ūn_t] (- dΔu/di_1) .
- For the case in (3.7), the marginal benefit becomes:
  - MB_t ≡ (Δu)^2 (- dp_t/di_1) + 2 p_t Δu (- dΔu/di_1). (D.2)
- Numerical example using Sveriges Riksbank (2014a) estimate (benchmark):
  - The Sveriges Riksbank point estimate of the effect on the Swedish household debt-to-income ratio of a 1 percentage point higher policy rate during four quarters is used as the red line in figure D.1 (source: Flodén (2014), Schularick and Taylor (2012), Sveriges Riksbank, and own calculations).
  - The debt-to-income ratio falls below baseline after a policy-rate increase, with a largest negative peak of -1.4 percentage points occurring after 4 quarters.
  - Given Flodén (2014) estimate, this implies dΔu/di_1 = 0.02 · 1.4 = 0.28 percentage points lower; i.e., crisis increase would be 4.72 percentage points rather than 5 percentage points.
  - After 5 years (quarter 20), the debt-to-income ratio is only 0.44 percentage points below baseline, implying dΔu/di_1 = 0.02 · 0.44 = 0.088 percentage points less.
  - To get effect on expected unemployment rate, multiply these numbers by the crisis probability p_t (equation (3.12)), yielding very small effects (blue line in figure D.1).
    - Largest effect: negative peak in quarter 9 when probability of a crisis has increased to 6.4 percent; the peak is -0.12 basis points = -0.0012 percentage points.
  - Additional third term in marginal benefit (D.2), multiplied by 2 Δu = 10 percentage point and sign-switched, gives the green line in figure D.1:
    - It has a positive peak at 0.012 percentage points for quarter 11.
  - Conclusion: these effects are very small and can be disregarded for the paper’s conclusions.

### Kocherlakota on eliminating the possibility of a crisis (Appendix E)
- Kocherlakota (2014) assessed the value of reducing the probability of a crisis to zero.
  - Assumptions: crisis increase in unemployment Δu = 4 percentage points; expectation in 2014 of the 2017 unemployment rate equals a natural unemployment rate of 5 percent; a crisis would imply unemployment reaches 9 percent.
  - Upper-limit probability of crisis inferred from 2014Q1 Survey of Professional Forecasters: 0.29 percent (interpreted as an upper limit because unemployment could exceed 9 percent for other reasons).
- Comparing probabilities:
  - The probability used by Kocherlakota (0.29 percent) is much smaller than the probability of crisis beyond quarter 9 of 6.4 percent used in this paper.
- Expected loss calculations:
  - With Δu = 4 percentage points and probability 0.29 percent, expected loss increase from a crisis = 0.0029 · 4^2 = 0.0464 = 0.22^2. Eliminating the possibility of a crisis is thus worth an expected non-crisis unemployment gap from zero to 0.22 percentage points.
  - With Δu = 5 percentage points (assumed in the present paper), expected loss from a crisis = 0.0029 · 25 = 0.0725 = 0.27^2, so it is worth increasing the expected non-crisis unemployment gap from zero to 0.27 percentage points.
- The paper notes that for the benchmark assumptions in the present paper, the steady-state probability of a crisis is (text continues beyond supplied excerpt).

*Source: excerpt from the provided IMF PDF chapter/appendices (figures and equations preserved as presented in the source).*

### 6.4 percent, so the expected loss equals 0.064·25 = 1.6=1.26

### _wp1603 - 6.4 percent, so the expected loss equals 0.064·25 = 1.6=1.26

### Expected-loss calculations and comparison of probability estimates
- Under the benchmark assumptions, steady-state probability of a crisis = 6.4 percent, so the expected loss equals 0.064·25 = 1.6 = 1.26 under the benchmark assumptions.
- For the IMF staff estimates (section 7), steady-state probability of a crisis = 3.78 percent, implying expected loss = 0.0378·25 = 0.945 = 0.97. Eliminating the possibility of a crisis is worth an increase in the expected non-crisis unemployment gap of 0.97 percentage points under these estimates.
- These numbers are much higher than Kocherlakota (2014) because Schularick and Taylor (2012) and IMF staff imply probabilities of 6.4 and 3.8 percent, respectively, versus Survey of Professional Forecasters estimate of 0.29 percent.
- The Survey of Professional Forecasters may not take the possibility of a crisis into account; if not, it cannot be used to infer a professional-forecasters’ estimate of crisis probability.

### Reduction in crisis probability per increase in expected non-crisis unemployment gap (Figure F.1)
- Figure F.1 shows, by future quarter, the reduction in the probability of a crisis per percentage-point increase in the expected non-crisis unemployment gap for datasets:
  - Schularick and Taylor (2012)
  - Laeven and Valencia (2012)
- The derivative dp_t/di_1 (green lines in figures 2.2 and 7.1) and dE1 u^n_t/di_1 (red line in figure 2.1) imply that taking the average over quarters 12–24 exaggerates the average reduction of crisis probability per unemployment increase and thus exaggerates the benefit of leaning against the wind.

### Random crisis increase in the unemployment rate (stochastic crisis jump)
- Benchmark: crisis increase in unemployment rate is deterministic and equals u > 0 (assumed 5 percentage points in benchmark).
- Alternative: crisis increase is random f(u) with mean u and variance 2_u.
- Result:
  - E1( ̃u^n_t + f(u) )^2 = (E1 ̃u^n_t + u)^2 + Var1 ̃u^n_t + 2_u
  - Expected quarter-t loss:
    - E1 L_t = (1−p_t) E1( ̃u^n_t )^2 + p_t [ E1( ̃u^n_t + u )^2 + 2_u ] + Var1 ̃u^n_t
  - An additional term p_t 2_u enters relative to (3.6); this does not affect the results under the maintained assumptions.

### Debt-to-GDP term in Schularick and Taylor (2012, table 7, column 22)
- Logit regression includes log debt-to-GDP ratio; coefficient = 1.1 with standard error 0.624, significant at the 10 percent level.
- Sum of coefficients of lagged real debt growth rates = 9.984 (versus 9.698 without log debt term).
- Represent variant as q_t = 1/4 · exp(X_t +  h_t) / (1 + exp(X_t +  h_t)), where  = 1.1 and h_t = log debt-to-GDP.
- Derivative: dq_t/dh_t = − q_t (1 − 4 q_t).
- With q_t = 0.008 and  = 1.1, dq_t/dh_t = 1.1·0.008·0.968 = 0.0085. Interpretation: 1 percentage point lower debt-to-GDP ratio lowers probability of a crisis start by 0.0085 percentage points.
- Riksbank estimate of policy-rate effect on debt-to-GDP (figures H.1 and H.2) implies the total effect on crisis probabilities via debt-to-GDP is very small and only marginally larger than effect via real debt growth alone.

### A constrained-optimal policy (appendix I)
- Problem: choose constant policy rate i_1 for quarters 1–4, return to baseline thereafter; find i_1 that sets derivative of intertemporal loss to zero:
  - d/di_1 E1 Σ_{t=0}^1−1 ^{t−1} L_t = 0  (I.1)
  - Equivalent form Σ_{t=1}^1−1 ^{t−1} NMC_t = 0, where NMC_t = MC_t − MB_t (I.2)
- Using definitions and normalization i_1 = 0 for baseline, derive coefficients and solve:
  - Approximate by summing 1 ≤ t ≤ 40.
- Numerical results with quarterly discount factor  = 0.995 (annual 0.98):
  - Constrained-optimal i_1 = −0.83 percentage point (reduce policy rate by about 0.8 percentage points in quarters 1–4, then return to baseline).
  - Discounted net marginal costs sum to zero for 1 ≤ t ≤ 40: negative up to quarter 10, positive thereafter.
  - Expected unemployment (including crisis possibility) is below benchmark up to quarter 10, above benchmark from quarter 11.
  - Expected unemployment including crisis approaches a level 0.32 percentage points above the benchmark at quarter 40 and beyond.
- Robustness: restricting attention to quarters 1–24 (disregarding marginal cost/benefit beyond quarter 24) yields:
  - i_1 = −0.68 percentage points (reduce policy rate in quarters 1–4 by 0.68 percent).
- IMF staff (Laeven and Valencia (2012) dataset) results:
  - Full horizon 1–40: i_1 = −0.51 percentage points.
  - If ignore beyond quarter 24: i_1 = −0.40 percentage points.
- Summary: under both datasets and horizon choices, constrained-optimal policy is to lower the policy rate in quarters 1–4 (lean with the wind), not against it.

### Alternative assumption: fixed cost of a crisis (appendix J)
- Fixed cost assumption: expected crisis unemployment gap = −u, E1 ̃u^c_t = −u, regardless of expected non-crisis unemployment gap.
- Expected unemployment gap:
  - E1 ̃u_t = (1−p_t) E1 ̃u^n_t + p_t (−u)  (J.1)
- Policy effect:
  - dE1 ̃u_t/di_1 = (1−p_t) dE1 ̃u^n_t/di_1 + (−u − E1 ̃u^n_t) dp_t/di_1  (J.2)
- Net marginal cost of leaning against the wind:
  - NMC_t = dE1 L_t/di_1 = (1−p_t)^2 E1 ̃u^n_t dE1 u^n_t/di_1 − [ (−u)^2 − E1 ( ̃u^n_t )^2 ] (− dp_t/di_1 ) ≡ MC_t − MB_t
- Marginal cost and benefit:
  - MC_t ≡ 2(1−p_t) E1 ̃u^n_t dE1 u^n_t/di_1  (J.3)
  - MB_t ≡ [ (−u)^2 − E1 ( ̃u^n_t )^2 ] (− dp_t/di_1 )  (J.4)
- At E1 ̃u^n_t = 0 and (3.7):
  - MC_t = 0  (J.5)
  - MB_t = (−u)^2 (− dp_t/di_1 )  (J.6)
- Implication: when expected non-crisis unemployment gap is zero and crisis cost fixed, marginal cost = 0 while marginal benefit may be positive if dp_t/di_1 < 0. This could make small leaning optimal.
- If expected non-crisis unemployment gap is positive (e.g., 0.25 percentage points), MC becomes positive and MB is nearly unchanged, so discounted net marginal costs are positive and leaning is not justified.

### Optimal leaning against the wind under fixed crisis cost (appendix J.1–J.3)
- Constrained optimization similar to appendix I leads to quadratic equation for optimal i_1; numerical solution when limiting horizon to quarters 1–24 and annual discount factor 0.98:
  - Optimal policy-rate increase (i_1) = 0.11 percentage points (grey line in figure J.4). Note: this is an increase because the analysis stacks cards in favor of leaning against the wind; interpretation in text emphasizes small magnitude.
  - Expected non-crisis unemployment gap peak ≈ 0.05 percentage points (quarter 6).
  - Maximum reduction in crisis probability = 0.025 percentage points (quarter 18) versus steady-state probability 6.4 percent.
  - Intertemporal loss reduced by only 0.07 percent of baseline value.
- Figure J.5: discounted net marginal cost over quarters 1–24 is approximately zero; net marginal cost composed mainly of MC up to quarter 12 and negative MB from quarter 12 to 24.
- Quarterly-optimal expected non-crisis unemployment gap (equalizing quarter-t marginal cost and benefit), neglecting E1 ( ̃u^n_t )^2 relative to (−u)^2:
  - E1 ̃u^n_t = (−u)^2 / [ 2(1−p_t) (− dp_t / dE1 u^n_t ) ]  (J.11)
- Numerical illustration:
  - Most favorable trade-off at quarter 20: probability reduction per 1 percentage-point expected non-crisis unemployment increase = 1.6 percentage points (i.e., probability falls by 0.016 for a 1 percentage-point increase).
  - With u = 5 percentage points and p_20 = 6.4 percent, quarterly-optimal expected non-crisis unemployment gap at its maximum = 0.22 percentage points.
- Conclusion: even when assumptions are stacked in favor of leaning against the wind (fixed crisis cost, truncated horizon), the optimal leaning is very small and the reduction in intertemporal loss is negligible; thus leaning against the wind is not justified in any economically significant way.

*Source: _wp1603 - 6.4 percent, so the expected loss equals 0.064·25 = 1.6=1.26 (PDF chapter/section).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1603.pdf_