## Section V.C.

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---

### Data series, data transformations, and assumptions on measurement error
- Table 1 series, symbols, units, measurement error, transform:
  - GDP/capita — Y — Per capita — No — Linear
  - GDP deflator — Π — Ann. % quarterly chg — Yes — ∆
  - HICP — Π — Ann. % quarterly chg — Yes — ∆
  - Consumption — C — Per capita — No — Linear
  - Investment — I — Per capita — No — Linear
  - Employment — E — Prop. of population — No — Linear
  - Real wages — W — Deflated by GDP price — No — Linear
  - Nominal short-term interest rate — R × Π — Ann. % points — No — Linear
  - Average loan rate to businesses — R_s × Π — Ann. % points — Yes — Demean
  - Investment grade corp. bond spread — R_s / R — Ann. % points — Yes — Demean
  - High yield corp. bond spread — R_s / R — Ann. % points — Yes — Demean
- Notes:
  - Sample runs 1980Q1 to 2016Q4, except for the high yield bond spread, which is quarterly from 1998Q1.
  - A ∆ in the transform column indicates a first difference.
- Modeling choices:
  - Both HICP and GDP deflator inflation map into model inflation (Π).
  - Bank lending rates and yields on two corporate bond indexes map into banks’ return on assets (R_s).
  - Structure implies two common factors and three idiosyncratic disturbances; loading factors are estimated, with the investment-grade corporate bond yield loading calibrated to one.

### Calibrated parameters and steady state targets (Table 2)
- Calibrated parameter values and remarks:
  - α Effective capital share = .330 — Gertler and Karadi (2011)
  - β Discount factor = .9942 — 2.48% avg. annualized real interest rate
  - δ Depreciation rate = .178 — 21.8% investment-GDP ratio
  - ε Goods elasticity of substitution = 4.33 — 30% steady state markup
  - ε_w Labor elasticity of substitution = 4.33 — 30% steady state markup
  - φ Inverse of Frisch elasticity of lab. sup. = 1.0 — Chetty et al. (2011)
  - σ Quarterly survival rate of bankers = .940 — Average lifetime of 16 quarters
- Steady state targets:
  - Steady state spreads = 1.93% — Annualized rate. Avg. of investment-grade spreads.
  - Steady state hours worked = 1/3 — Proportion of time endowment
  - Steady state G/Y ratio = 22.2% — Average of govt. spending + net exports as a proportion of GDP
- Calibration notes:
  - α, steady state labor supply, and inverse Frisch elasticity set to literature standard values 0.33, 1/3 and 1 respectively.
  - δ set to .178 to match steady state investment-GDP ratio equal to sample average.
  - Elasticities set to 4.33 to match a 30% steady state markup.
  - σ calibrated so average time to disburse bank’s net worth is 16 quarters.
  - β calibrated to sample average real interest rate of 2.48%.
  - Sample average of investment-grade corporate bonds determines steady state spreads.

### Priors and posterior distributions of estimated parameters (Table 3) — selected entries
- Structural parameters (posterior Mode [90% HPD Interval]):
  - γ Calvo prices: Mode = .7975 [0.75430 0.8870]
  - γ_w Calvo wages: Mode = .6478 [0.61500 0.7601]
  - γ_e Calvo employment: Mode = .8350 [0.80770 0.8619]
  - ι_p Price indexation: Mode = .0315 [0.01380 0.0908]
  - ι_w Wage indexation: Mode = .2274 [0.11180 0.4030]
  - h Internal habit parameter: Mode = .7976 [0.72290 0.8432]
  - S/N Steady-state leverage: Mode = 2.2181 [2.05582 2.5034]
  - κ Investment adjustment costs: Mode = 1.4562 [1.15401 1.9215]
  - α_2 Convexity of variable capital util.: Mode = .7990 [0.64340 0.9005]
- Taylor rule parameters (posterior Mode):
  - φ_π Inflation response: Mode = 1.6815 [1.58961 1.8696]
  - φ_y Output gap response: Mode = .0189 [0.00370 0.0468]
  - φ_∆y Output gap growth response: Mode = .1512 [0.07280 0.2048]
  - ρ_r Smoothing: Mode = .8384 [0.82160 0.8741]
- Forcing processes (posterior Modes):
  - ρ_a TFP: .9358 [0.90880 0.9551]
  - ρ_ivt Inv. Spec. Technology: .9583 [0.95030 0.9730]
  - ρ_gov Government spending: .9906 [0.97960 0.9982]
  - ρ_θ Bank diversion: .9936 [0.98770 0.9988]
- Standard deviations of exogenous shocks (posterior Modes, Inv. Gamma priors):
  - σ_a TFP Inv.: .0087 [0.00760 0.0119]
  - σ_ivt Inv. Spec. Technology Inv.: .0433 [0.03960 0.0537]
  - σ_θ Bank diversion Inv.: .0200 [0.01710 0.0240]
  - σ_r Monetary policy Inv.: .0013 [0.00120 0.0015]
- Data-rich parameters and measurement error (posterior Mode [90% HPD Interval]):
  - Λ_Loan Loading on loan rate: 1.5949 [1.45441 1.7417]
  - Λ_HY Loading on high-yield: 1.0262 [0.73181 1.1872]
  - ε_Loan Meas. err. on loan rate: .0015 [0.0014 0.0018]
  - ε_HY Meas. err. on high-yield: .0076 [0.0066 0.0086]
  - ε_CPI Meas. err on CPI: .0032 [0.0029 0.0035]
- Estimation details:
  - Priors mostly follow Smets and Wouters (2003).
  - Posterior distributions obtained via MCMC with two chains of 300,000 iterations each and burning the first 150,000.
  - Posterior steady state leverage S/N = 2.2; prior set to 4 as in Gertler and Karadi (2011) and Meeks, Nelson, and Alessandri (2017).

### Monetary policy design: loss function, mandates, and welfare comparison
- Central bank per-period quadratic loss, parameterized by mandate M = (λ, ω):
  - L_t(M) = π̂_t^2 + λ x̂_t^2 + ω f̂_t^2
    - π̂_t denotes (annualized) price inflation (log-deviation)
    - x̂_t denotes resource utilization (log-deviation)
    - f̂_t denotes a financial variable (log-deviation)
  - λ ∈ [0, ∞) captures relative preference for resource utilization vs inflation stabilization.
  - ω ∈ [0, ∞) captures weight on financial volatility.
- Social welfare evaluation:
  - Use quadratic approximation of representative household’s utility (Benigno and Woodford (2012) method).
  - Unconditional expected loss: Loss_P ≡ 1/2 trace[W_H Σ(P)] + t.i.p., where Σ(P) is variance-covariance matrix under policy P.
  - Central bank chooses policy optimally under commitment (optimal control policy P* for mandate M).
  - Compare M-optimal policy P* to Ramsey (H-optimal) policy R via Loss_R − Loss_M = 1/2 trace[W_H Σ(R)] − 1/2 trace[W_H Σ(P*)].
  - Performance expressed in consumption equivalent variation (CEV) units — percentage reduction in lifetime consumption that leaves households indifferent between allocations.

### Quantitative results — Dual mandate (M_IT = (λ, 0))
- Dual mandate: stabilize inflation and real economy (typical flexible inflation targeting).
- Welfare losses (relative to Ramsey) as function of λ:
  - Strict inflation targeting (λ = 0) produces welfare losses amounting to 1% of consumption (CEV = 1%).
  - Extreme dovishness (λ ≫ 1) is also suboptimal, but to a lesser extent.
  - Losses minimized at λ = 1.005 (denoted λ†), very close to DKLN (2018) value 1.042 for Smets and Wouters (2007) model.
  - A balanced mandate (equal weight on inflation and output gap stabilization) yields welfare outcomes close to Ramsey in this model with nominal frictions and financial frictions.
- Policy frontier observations:
  - Monetary policy frontier between output gap volatility (x-axis) and inflation volatility (y-axis) is convex.
  - For λ < λ†, reducing inflation volatility requires larger increases in output gap volatility.
  - Ramsey outcome lies well inside the efficient frontier under the dual mandate, reflecting Ramsey accounting for many welfare-relevant variables beyond inflation and output gap.
- Volatility relationships:
  - Output gap stabilization correlates with reduced volatility of other welfare-relevant variables.
  - Volatility of nominal wage inflation decreases (almost) monotonically with output gap volatility as λ increases toward λ†.
  - Loan-to-deposit spread volatility and output gap volatility move together for almost all λ (except near strict inflation targeting) — no trade-off between them for most λ values.
- Summary conclusions on dual mandate:
  - Strict inflation targeting performs poorly in this empirically rich model despite being near-optimal in simpler textbook models.
  - An almost perfectly balanced dual mandate (λ ≈ 1.005) performs best among dual mandates.
  - Even the optimal dual mandate is materially inferior, in welfare terms, to Ramsey policy because it focuses only on inflation and output gap while important variables (wage inflation, credit spreads) are only imperfectly proxied by the output gap.

### Financial extension to the mandate — preview of findings
- Extended mandate M_FF = (λ, ω) includes a non-zero weight ω on the loan-to-deposit spread (financial frictions).
- Welfare comparison (text summary referring to Table 4):
  - Optimal extended mandate M_FF yields welfare loss (CEV) = 0.05%.
  - Optimal dual mandate M_IT yields welfare loss (CEV) = 0.13%.
  - Conclusion: Explicit objective of spread stabilization under the extended mandate brings welfare benefits relative to the dual mandate; in this economy financial frictions render lending spreads welfare-relevant and internalizing them improves welfare.

### Main result: extended mandate reduces reliance on output-gap stabilization
- Intuition:
  - Under an extended mandate that directly stabilizes a financial variable, the need to lower output gap volatility is smaller than under the dual mandate because at least one welfare-relevant variable previously proxied by the output gap is now stabilized directly.
- Practical benefit:
  - The extended mandate down-weights the importance of the output gap, a variable that can be challenging to measure in real time.

### Welfare outcomes and optimal mandates (Table 4)
- Strict inflation targeting: λ = 0, ω = 0, CEV(%) = −1.007
- Optimal dual mandate: λ = 1.005, ω = 0, CEV(%) = −0.130
- Optimal extended mandate: λ = 0.475, ω = 1.397, CEV(%) = −0.050
- Reference: estimated Taylor rule (without monetary policy shocks) relative loss = −0.179% CEV
- Interpretation:
  - An interest rate spread is welfare-relevant not only as a predictor of inflation/output but as an objective in itself.
  - The optimal extended mandate M?FF = (λ?, ω?) corresponds to the black circle in Figure 4 (left-centre).

### Robustness and qualitative features from the welfare surface (Figure 4)
- Main findings:
  - For a wide range of λ, moving from a dual mandate (ω = 0) to an extended mandate (ω > 0) improves welfare.
  - Extreme hawkishness and (to a lesser extent) extreme dovishness cause large welfare losses relative to Ramsey for both dual and extended mandates.
  - A large set of extended mandates yields welfare outcomes superior to the optimal dual mandate.
- Insensitivity region:
  - For ω in excess of 0.2, welfare outcomes are not significantly affected by further increases in ω.
  - Similar welfare outcomes are achieved for values of λ roughly between 0.2 and 1.
- Threshold for shading:
  - Differences in CEV outcomes smaller than 0.05% are depicted with like shading.

### Quantifying the macro-financial trade-off (Figures 5 and 6)
- Principal observation:
  - The macro-financial trade-off is modest—monetary policy can reduce credit spread volatility considerably with limited impact on macroeconomic volatility.
- Examples:
  - λ fixed at λ? = 0.475 (pink line). Increasing ω reduces spread volatility from near 2% to 0.5% while inflation volatility rises by barely 0.1%.
  - When ω > 0.065 (versus ω? = 1.397), output gap volatility rises very little as spread volatility falls.
- Insensitivity to λ:
  - The slope of the policy frontiers is near-identical when λ is held at its dual mandate level (λ? = 1.005), indicating little dependence on λ for the macro-financial trade-off.
- Limits to financial stabilization:
  - For ω > ω? subsequent gains are nearly nil; sufficiently high ω values produce similar welfare outcomes.
- Inflation-output frontier:
  - The inflation-output frontier shifts out under the optimal extended mandate, but the shift is modest; the extended mandate frontier lies within the set of output-inflation outcomes feasible under the dual mandate.
- Financial volatility relative to Ramsey:
  - Financial volatility under the optimal extended mandate is close to that under the Ramsey policy.
  - Spreads are less volatile under the optimal extended mandate than under the dual mandate.
  - Because the macro-financial trade-off is small in the model, reducing spread volatility toward Ramsey is optimal.

### Summary of findings on the extended mandate
- Mandating monetary policymakers to stabilize a financial variable alongside inflation and output gap stabilization is desirable from a welfare standpoint.
- Even a small weight on a financial objective results in an improved welfare outcome relative to the dual mandate.
- Because the structure of the economy generates only a modest macro-financial trade-off, gains from reducing financial volatility under an extended mandate more than offset losses from increased macroeconomic volatility.
- Under the extended mandate, policymakers optimally down-weight output gap fluctuations, since the output gap is an imperfect proxy for welfare-relevant fluctuations in financial frictions.

### Role of alternative measures in the mandate (Table 5 and Figures 7)
- Table 5 (optimal dual and extended mandates for alternative measures)
  - Columns: xt: Output gap | xt: Output growth (ann.)
  - Strict inflation targeting:
    - Output gap: λ = 0, ω = 0, CEV(%) = −1.007
    - Output growth: λ = 0, ω = 0, CEV(%) = −1.007
  - Optimal dual:
    - Output gap: λ = 1.005, ω = 0, CEV(%) = −0.130
    - Output growth: λ = 0.4000, ω = 0, CEV(%) = −0.083
  - f_t: Loan-deposit spread, Optimal extended:
    - Output gap: λ = 0.475, ω = 1.397, CEV(%) = −0.050
    - Output growth: λ = 0.3505, ω = —, CEV(%) = −0.071
  - f_t: Leverage, Optimal extended:
    - Output gap: λ = 0.5000, ω = 0.010, CEV(%) = −0.059
    - Output growth: λ = 0.3600, ω = 0.024, CEV(%) = −0.078
- Findings:
  - An optimal dual mandate with output growth instead of the output gap still outperforms strict inflation targeting in welfare terms.
  - The welfare performance of the optimal dual mandate depends on which measure of resource utilization is included.
  - Output growth volatility is a better proxy for credit spread volatility than output gap volatility—low values of λ can achieve substantial reductions in credit spread volatility, which in turn lower inflation volatility.
  - Consequently, the optimal dual mandate that includes output growth performs better (smaller welfare loss relative to Ramsey) despite higher output gap volatility.
  - The welfare difference between optimal dual and optimal extended mandates exceeds the 0.05% threshold only when output gap volatility is the measure of resource utilization; output growth being a better proxy for financial volatility reduces the marginal benefit of explicitly including a financial variable in the mandate.
  - Conditional on the chosen measure of resource utilization, similar welfare outcomes are achieved for the two financial measures considered (loan-deposit spread and leverage).

### The role of shocks
- Shocks and the superiority of the extended mandate:
  - The presence of financial shocks is not the reason why the optimal extended mandate leads to significantly better welfare outcomes than the optimal dual mandate. The difference in welfare outcomes between the optimal dual and extended mandate remains roughly unchanged in their absence.
  - The superiority of the extended mandate reflects the endogenous propagation of the standard shocks, rather than financial shocks being the main driving force.
  - Even in the absence of investment specific shocks (which could be interpreted as financial shocks), welfare outcomes under an extended mandate are notably better than those under a dual mandate.
  - The superiority of the extended mandate appears insensitive to shocks resembling financial disturbances, even if a particular financial disturbance is too small to be informative.
- Table 6. The role of shocks (as reported)
  - Optimal dual mandateOptimal extended mandate
  - λCEV(%)λωCEV(%)
  - All shocks1.005-0.1300.4751.397-0.050
  - No financial shocks1.010-0.1300.4661.680-0.046
  - No inv. specific tech. shocks1.010-0.1130.4901.750-0.045
  - No markup shocks1.450-0.0490.8751.750-0.030
- Note (verbatim): This table shows the optimal dual and optimal extended mandate for different assumptions about the shocks, along with their respective welfare losses relative to the Ramsey policy, expressed in CEV units(%). In the "All shocks"-case the variance of each shock is set to its estimated value; in the "No financial shocks"-case, the variance of the financial shock is set to zero while the variances of the remaining shocks are kept at their estimated value; in the "No inv. specific tech. shocks"-case, the variance of the investment specific technology shock is set to zero while the variances of the remaining shocks are kept at their estimated value; and in the "No markup shock"- case, the variances of both the price and wage markup shock are set to zero, while the variances of the remaining shocks are kept at their estimated value. The measure of resource utilization in the mandate is the output gap, and the financial variable in the mandate is spreads.

### The role of inefficient mark-up shocks
- Welfare gains of an extended mandate compared to a dual mandate are particularly pronounced in the presence of inefficient mark-up shocks.
- In the absence of mark-up shocks:
  - Welfare outcomes under the optimal extended mandate are still better than under the optimal dual mandate, but differences are small because the dual mandate’s welfare performance improves notably.
  - The output gap serves as a proxy for stabilizing other welfare-relevant variables under a dual mandate, making it welfare optimal to increase the weight on that variable in the mandate (λ) even if it increases inflation volatility.
  - There is a substantial reduction in the trade-off between inflation and output gap stabilization.
  - The relation between output gap and credit spread volatility is altered in the absence of mark-up shocks: combinations achievable without mark-up shocks are not achievable with them.
  - The inflation–output gap trade-off is still sufficiently pronounced in the absence of mark-up shocks for extreme hawkishness to lead to significantly larger welfare losses.
- Note (verbatim): Descriptions of panels in Fig. 8 preserved verbatim in source.

### Interest rate smoothing
- Motivation and loss function:
  - Central banks have proposed financial stability concerns as a motivation for interest rate smoothing.
  - Loss function considered: M = ˆπ_t^2 + λ ˆx_t^2 + ρ (∆mro_t)^2, where mro is the ECB’s short-term policy interest rate.
- Main findings:
  - Interest rate smoothing is optimal and produces similar welfare outcomes to an optimal extended mandate with the loan-to-deposit spread.
  - Welfare outcomes with interest rate smoothing are notably better than under an optimal dual mandate, despite restricting the policy instrument.
  - Inflation and output gap volatility are lower under the optimal dual mandate (black diamond) than under the optimal extended mandate with interest rate smoothing (empty black dot), but spread volatility is considerably lower when interest rates are smoothed optimally.
  - Smoothing the policy instrument is a reasonable proxy for reducing spread volatility in and of itself.
  - Both inflation and output gap volatility are comparable under the optimal extended mandates considered, explaining why similar welfare outcomes can be achieved.
- Table 7. The role of interest rate smoothing (as reported)
  - x_t: Output gap
  - λωρCEV(%)std(mro)
  - Optimal dual mandate1.00500-0.13013.17
  - f_t: Change in policy rate
  - Optimal extended mandate0.400017-0.0560.54
  - f_t: Loan-deposit spread
  - Optimal extended mandate0.4751.3970-0.0504.04
- Note (verbatim): This table shows the optimal (welfare-maximising) dual mandate and the optimal extended mandate along with their respective welfare losses relative to the Ramsey policy, expressed in CEV units (%). Two ternary stabilization objectives are considered: the (ex-ante) loan-deposit spread and the change in the (quarterly) policy rate (∆mro_t). The volatility of the (quarterly) policy rate under each respective mandate is also reported.
- Figure 9 note (verbatim): Description of the green line, diamonds, and dots preserved as in source.

### Conclusion (from the chapter)
- A simple mandate that includes a stabilization objective for spreads—an observable proxy for the intensity of the underlying financial friction—produces outcomes that are comparable to the Ramsey policy.
- This finding is robust and provides a rationale for central banks that identify trade-offs between macroeconomic and financial variables as relevant for setting monetary policy, even in the absence of discrete financial ‘crisis’ events in the model.
- Compared to existing literature, the role identified for financial stabilization is quantitatively much more important.
- In the absence of a ternary financial stabilization objective, when financial frictions are present a dual mandate ‘makes sense’ because a high weight on stabilizing the output gap implies greater stability in spreads.
- Smoothing policy interest rates, which double as bank funding costs, raises welfare when financial frictions are present.
- The analysis omits other concerns central banks may have about pursuing a financial stabilization objective, including capturing the many sources and precise forms of frictions present in complex financial systems.
- Continued research is needed to establish robustness of these results to alternative characterizations of financial frictions and to realistic treatments of macro-prudential measures used with monetary and fiscal policies.

*Source: wpiea2020244-print-pdf - Section V.C.*

### Section V.C.

### Section V.C.

### Data series, data transformations, and assumptions on measurement error
- Table 1 series, symbols, units, measurement error, transform:
  - GDP/capita — Y — Per capita — No — Linear
  - GDP deflator — Π — Ann. % quarterly chg — Yes — ∆
  - HICP — Π — Ann. % quarterly chg — Yes — ∆
  - Consumption — C — Per capita — No — Linear
  - Investment — I — Per capita — No — Linear
  - Employment — E — Prop. of population — No — Linear
  - Real wages — W — Deflated by GDP price — No — Linear
  - Nominal short-term interest rate — R × Π — Ann. % points — No — Linear
  - Average loan rate to businesses — R_s × Π — Ann. % points — Yes — Demean
  - Investment grade corp. bond spread — R_s / R — Ann. % points — Yes — Demean
  - High yield corp. bond spread — R_s / R — Ann. % points — Yes — Demean
- Notes:
  - Sample runs 1980Q1 to 2016Q4, except for the high yield bond spread, which is quarterly from 1998Q1.
  - A ∆ in the transform column indicates a first difference.
- Modeling choices:
  - Both HICP and GDP deflator inflation map into model inflation (Π).
  - Bank lending rates and yields on two corporate bond indexes map into banks’ return on assets (R_s).
  - Structure implies two common factors and three idiosyncratic disturbances; loading factors are estimated, with the investment-grade corporate bond yield loading calibrated to one.

### Calibrated parameters and steady state targets (Table 2)
- Calibrated parameter values and remarks:
  - α Effective capital share = .330 — Gertler and Karadi (2011)
  - β Discount factor = .9942 — 2.48% avg. annualized real interest rate
  - δ Depreciation rate = .178 — 21.8% investment-GDP ratio
  - ε Goods elasticity of substitution = 4.33 — 30% steady state markup
  - ε_w Labor elasticity of substitution = 4.33 — 30% steady state markup
  - φ Inverse of Frisch elasticity of lab. sup. = 1.0 — Chetty et al. (2011)
  - σ Quarterly survival rate of bankers = .940 — Average lifetime of 16 quarters
- Steady state targets:
  - Steady state spreads = 1.93% — Annualized rate. Avg. of investment-grade spreads.
  - Steady state hours worked = 1/3 — Proportion of time endowment
  - Steady state G/Y ratio = 22.2% — Average of govt. spending + net exports as a proportion of GDP
- Calibration notes:
  - α, steady state labor supply, and inverse Frisch elasticity set to literature standard values 0.33, 1/3 and 1 respectively.
  - δ set to .178 to match steady state investment-GDP ratio equal to sample average.
  - Elasticities set to 4.33 to match a 30% steady state markup.
  - σ calibrated so average time to disburse bank’s net worth is 16 quarters.
  - β calibrated to sample average real interest rate of 2.48%.
  - Sample average of investment-grade corporate bonds determines steady state spreads.

### Priors and posterior distributions of estimated parameters (Table 3) — selected entries
- Structural parameters (posterior Mode [90% HPD Interval]):
  - γ Calvo prices: Mode = .7975 [0.75430 0.8870]
  - γ_w Calvo wages: Mode = .6478 [0.61500 0.7601]
  - γ_e Calvo employment: Mode = .8350 [0.80770 0.8619]
  - ι_p Price indexation: Mode = .0315 [0.01380 0.0908]
  - ι_w Wage indexation: Mode = .2274 [0.11180 0.4030]
  - h Internal habit parameter: Mode = .7976 [0.72290 0.8432]
  - S/N Steady-state leverage: Mode = 2.2181 [2.05582 2.5034]
  - κ Investment adjustment costs: Mode = 1.4562 [1.15401 1.9215]
  - α_2 Convexity of variable capital util.: Mode = .7990 [0.64340 0.9005]
- Taylor rule parameters (posterior Mode):
  - φ_π Inflation response: Mode = 1.6815 [1.58961 1.8696]
  - φ_y Output gap response: Mode = .0189 [0.00370 0.0468]
  - φ_∆y Output gap growth response: Mode = .1512 [0.07280 0.2048]
  - ρ_r Smoothing: Mode = .8384 [0.82160 0.8741]
- Forcing processes (posterior Modes):
  - ρ_a TFP: .9358 [0.90880 0.9551]
  - ρ_ivt Inv. Spec. Technology: .9583 [0.95030 0.9730]
  - ρ_gov Government spending: .9906 [0.97960 0.9982]
  - ρ_θ Bank diversion: .9936 [0.98770 0.9988]
- Standard deviations of exogenous shocks (posterior Modes, Inv. Gamma priors):
  - σ_a TFP Inv.: .0087 [0.00760 0.0119]
  - σ_ivt Inv. Spec. Technology Inv.: .0433 [0.03960 0.0537]
  - σ_θ Bank diversion Inv.: .0200 [0.01710 0.0240]
  - σ_r Monetary policy Inv.: .0013 [0.00120 0.0015]
- Data-rich parameters and measurement error (posterior Mode [90% HPD Interval]):
  - Λ_Loan Loading on loan rate: 1.5949 [1.45441 1.7417]
  - Λ_HY Loading on high-yield: 1.0262 [0.73181 1.1872]
  - ε_Loan Meas. err. on loan rate: .0015 [0.0014 0.0018]
  - ε_HY Meas. err. on high-yield: .0076 [0.0066 0.0086]
  - ε_CPI Meas. err on CPI: .0032 [0.0029 0.0035]
- Estimation details:
  - Priors mostly follow Smets and Wouters (2003).
  - Posterior distributions obtained via MCMC with two chains of 300,000 iterations each and burning the first 150,000.
  - Posterior steady state leverage S/N = 2.2; prior set to 4 as in Gertler and Karadi (2011) and Meeks, Nelson, and Alessandri (2017).

### Monetary policy design: loss function, mandates, and welfare comparison
- Central bank per-period quadratic loss, parameterized by mandate M = (λ, ω):
  - L_t(M) = π̂_t^2 + λ x̂_t^2 + ω f̂_t^2
    - π̂_t denotes (annualized) price inflation (log-deviation)
    - x̂_t denotes resource utilization (log-deviation)
    - f̂_t denotes a financial variable (log-deviation)
  - λ ∈ [0, ∞) captures relative preference for resource utilization vs inflation stabilization.
  - ω ∈ [0, ∞) captures weight on financial volatility.
- Social welfare evaluation:
  - Use quadratic approximation of representative household’s utility (Benigno and Woodford (2012) method).
  - Unconditional expected loss: Loss_P ≡ 1/2 trace[W_H Σ(P)] + t.i.p., where Σ(P) is variance-covariance matrix under policy P.
  - Central bank chooses policy optimally under commitment (optimal control policy P* for mandate M).
  - Compare M-optimal policy P* to Ramsey (H-optimal) policy R via Loss_R − Loss_M = 1/2 trace[W_H Σ(R)] − 1/2 trace[W_H Σ(P*)].
  - Performance expressed in consumption equivalent variation (CEV) units — percentage reduction in lifetime consumption that leaves households indifferent between allocations.

### Quantitative results — Dual mandate (M_IT = (λ, 0))
- Dual mandate: stabilize inflation and real economy (typical flexible inflation targeting).
- Welfare losses (relative to Ramsey) as function of λ:
  - Strict inflation targeting (λ = 0) produces welfare losses amounting to 1% of consumption (CEV = 1%).
  - Extreme dovishness (λ ≫ 1) is also suboptimal, but to a lesser extent.
  - Losses minimized at λ = 1.005 (denoted λ†), very close to DKLN (2018) value 1.042 for Smets and Wouters (2007) model.
  - A balanced mandate (equal weight on inflation and output gap stabilization) yields welfare outcomes close to Ramsey in this model with nominal frictions and financial frictions.
- Policy frontier observations:
  - Monetary policy frontier between output gap volatility (x-axis) and inflation volatility (y-axis) is convex.
  - For λ < λ†, reducing inflation volatility requires larger increases in output gap volatility.
  - Ramsey outcome lies well inside the efficient frontier under the dual mandate, reflecting Ramsey accounting for many welfare-relevant variables beyond inflation and output gap.
- Volatility relationships:
  - Output gap stabilization correlates with reduced volatility of other welfare-relevant variables.
  - Volatility of nominal wage inflation decreases (almost) monotonically with output gap volatility as λ increases toward λ†.
  - Loan-to-deposit spread volatility and output gap volatility move together for almost all λ (except near strict inflation targeting) — no trade-off between them for most λ values.
- Summary conclusions on dual mandate:
  - Strict inflation targeting performs poorly in this empirically rich model despite being near-optimal in simpler textbook models.
  - An almost perfectly balanced dual mandate (λ ≈ 1.005) performs best among dual mandates.
  - Even the optimal dual mandate is materially inferior, in welfare terms, to Ramsey policy because it focuses only on inflation and output gap while important variables (wage inflation, credit spreads) are only imperfectly proxied by the output gap.

### Financial extension to the mandate — preview of findings
- Extended mandate M_FF = (λ, ω) includes a non-zero weight ω on the loan-to-deposit spread (financial frictions).
- Welfare comparison (text summary referring to Table 4):
  - Optimal extended mandate M_FF yields welfare loss (CEV) = 0.05%.
  - Optimal dual mandate M_IT yields welfare loss (CEV) = 0.13%.
  - Conclusion: Explicit objective of spread stabilization under the extended mandate brings welfare benefits relative to the dual mandate; in this economy financial frictions render lending spreads welfare-relevant and internalizing them improves welfare.

*Source: wpiea2020244-print-pdf - Section V.C.*

### 1.005 toλ

### wpiea2020244-print-pdf - 1.005 toλ

### Main result: extended mandate reduces reliance on output-gap stabilization
- Intuition: under an extended mandate that directly stabilizes a financial variable, the need to lower output gap volatility is smaller than under the dual mandate because at least one welfare-relevant variable previously proxied by the output gap is now stabilized directly.
- Practical benefit: the extended mandate down-weights the importance of the output gap, a variable that can be challenging to measure in real time.

### Welfare outcomes and optimal mandates (Table 4)
- Strict inflation targeting: λ = 0, ω = 0, CEV(%) = −1.007
- Optimal dual mandate: λ = 1.005, ω = 0, CEV(%) = −0.130
- Optimal extended mandate: λ = 0.475, ω = 1.397, CEV(%) = −0.050
- Reference: estimated Taylor rule (without monetary policy shocks) relative loss = −0.179% CEV
- Interpretation:
  - An interest rate spread is welfare-relevant not only as a predictor of inflation/output but as an objective in itself.
  - The optimal extended mandate M?FF = (λ?, ω?) corresponds to the black circle in Figure 4 (left-centre).

### Robustness and qualitative features from the welfare surface (Figure 4)
- Main findings:
  - For a wide range of λ, moving from a dual mandate (ω = 0) to an extended mandate (ω > 0) improves welfare.
  - Extreme hawkishness and (to a lesser extent) extreme dovishness cause large welfare losses relative to Ramsey for both dual and extended mandates.
  - A large set of extended mandates yields welfare outcomes superior to the optimal dual mandate.
- Insensitivity region:
  - For ω in excess of 0.2, welfare outcomes are not significantly affected by further increases in ω.
  - Similar welfare outcomes are achieved for values of λ roughly between 0.2 and 1.
- Threshold for shading: differences in CEV outcomes smaller than 0.05% are depicted with like shading.

### Quantifying the macro-financial trade-off (Figures 5 and 6)
- Principal observation: the macro-financial trade-off is modest—monetary policy can reduce credit spread volatility considerably with limited impact on macroeconomic volatility.
- Example (Figure 5, panel a):
  - λ fixed at λ? = 0.475 (pink line). Increasing ω reduces spread volatility from near 2% to 0.5% while inflation volatility rises by barely 0.1%.
- Example (Figure 5, panel b):
  - When ω > 0.065 (versus ω? = 1.397), output gap volatility rises very little as spread volatility falls.
- Insensitivity to λ:
  - The slope of the policy frontiers is near-identical when λ is held at its dual mandate level (λ? = 1.005), indicating little dependence on λ for the macro-financial trade-off.
- Limits to financial stabilization:
  - Although small increases above zero in ω yield material reductions in volatility, for ω > ω? subsequent gains are nearly nil. This explains why sufficiently high ω values produce similar welfare outcomes (Figure 4).
- Inflation-output frontier:
  - The inflation-output frontier shifts out under the optimal extended mandate, but the shift is modest; the extended mandate frontier lies within the set of output-inflation outcomes feasible under the dual mandate (Figure 6).
- Financial volatility relative to Ramsey:
  - Financial volatility under the optimal extended mandate is close to that under the Ramsey policy (black dot near black square in Figure 6).
  - Spreads are less volatile under the optimal extended mandate than under the dual mandate (black diamond).
  - Because the macro-financial trade-off is small in the model, reducing spread volatility toward Ramsey is optimal.

### Summary of findings on the extended mandate
- Mandating monetary policymakers to stabilize a financial variable alongside inflation and output gap stabilization is desirable from a welfare standpoint.
- Even a small weight on a financial objective results in an improved welfare outcome relative to the dual mandate.
- Because the structure of the economy generates only a modest macro-financial trade-off, gains from reducing financial volatility under an extended mandate more than offset losses from increased macroeconomic volatility.
- Under the extended mandate, policymakers optimally down-weight output gap fluctuations, since the output gap is an imperfect proxy for welfare-relevant fluctuations in financial frictions.

### Role of alternative measures in the mandate (Table 5 and Figures 7)
- Table 5 (optimal dual and extended mandates for alternative measures)
  - Columns: xt: Output gap | xt: Output growth (ann.)
  - Strict inflation targeting:
    - Output gap: λ = 0, ω = 0, CEV(%) = −1.007
    - Output growth: λ = 0, ω = 0, CEV(%) = −1.007
  - Optimal dual:
    - Output gap: λ = 1.005, ω = 0, CEV(%) = −0.130
    - Output growth: λ = 0.4000, ω = 0, CEV(%) = −0.083
  - f_t: Loan-deposit spread, Optimal extended:
    - Output gap: λ = 0.475, ω = 1.397, CEV(%) = −0.050
    - Output growth: λ = 0.3505, ω = —, CEV(%) = −0.071
  - f_t: Leverage, Optimal extended:
    - Output gap: λ = 0.5000, ω = 0.010, CEV(%) = −0.059
    - Output growth: λ = 0.3600, ω = 0.024, CEV(%) = −0.078
- Findings:
  - An optimal dual mandate with output growth instead of the output gap still outperforms strict inflation targeting in welfare terms.
  - The welfare performance of the optimal dual mandate depends on which measure of resource utilization is included.
  - Output growth volatility is a better proxy for credit spread volatility than output gap volatility—low values of λ can achieve substantial reductions in credit spread volatility, which in turn lower inflation volatility.
  - Consequently, the optimal dual mandate that includes output growth performs better (smaller welfare loss relative to Ramsey) despite higher output gap volatility.
  - The welfare difference between optimal dual and optimal extended mandates exceeds the 0.05% threshold only when output gap volatility is the measure of resource utilization; output growth being a better proxy for financial volatility reduces the marginal benefit of explicitly including a financial variable in the mandate.
  - Conditional on the chosen measure of resource utilization, similar welfare outcomes are achieved for the two financial measures considered (loan-deposit spread and leverage).

*Source: wpiea2020244-print-pdf - 1.005 toλ*

### 2.   The role of shocks

### 2.   The role of shocks

### Shocks and the superiority of the extended mandate
- The presence of financial shocks is not the reason why the optimal extended mandate leads to significantly better welfare outcomes than the optimal dual mandate. The difference in welfare outcomes between the optimal dual and extended mandate remains roughly unchanged in their absence.
- The superiority of the extended mandate reflects the endogenous propagation of the standard shocks, rather than financial shocks being the main driving force.
- Even in the absence of investment specific shocks (which could be interpreted as financial shocks), welfare outcomes under an extended mandate are notably better than those under a dual mandate.
- The superiority of the extended mandate appears insensitive to shocks resembling financial disturbances, even if a particular financial disturbance is too small to be informative.

- Table 6. The role of shocks (as reported)
  - Optimal dual mandateOptimal extended mandate
  - λCEV(%)λωCEV(%)
  - All shocks1.005-0.1300.4751.397-0.050
  - No financial shocks1.010-0.1300.4661.680-0.046
  - No inv. specific tech. shocks1.010-0.1130.4901.750-0.045
  - No markup shocks1.450-0.0490.8751.750-0.030

- Note summary (verbatim from source): This table shows the optimal dual and optimal extended mandate for different assumptions about the shocks, along with their respective welfare losses relative to the Ramsey policy, expressed in CEV units(%). In the "All shocks"-case the variance of each shock is set to its estimated value; in the "No financial shocks"-case, the variance of the financial shock is set to zero while the variances of the remaining shocks are kept at their estimated value; in the "No inv. specific tech. shocks"-case, the variance of the investment specific technology shock is set to zero while the variances of the remaining shocks are kept at their estimated value; and in the "No markup shock"- case, the variances of both the price and wage markup shock are set to zero, while the variances of the remaining shocks are kept at their estimated value. The measure of resource utilization in the mandate is the output gap, and the financial variable in the mandate is spreads.

### The role of inefficient mark-up shocks
- Welfare gains of an extended mandate compared to a dual mandate are particularly pronounced in the presence of inefficient mark-up shocks.
- In the absence of mark-up shocks:
  - Welfare outcomes under the optimal extended mandate are still better than under the optimal dual mandate, but differences are small because the dual mandate’s welfare performance improves notably.
  - The output gap serves as a proxy for stabilizing other welfare-relevant variables under a dual mandate, making it welfare optimal to increase the weight on that variable in the mandate (λ) even if it increases inflation volatility.
  - There is a substantial reduction in the trade-off between inflation and output gap stabilization (top left panel of Fig. 8). Consequently, an increase in λ leads to a considerably smaller increase in inflation volatility than in the presence of mark-up shocks.
  - The relation between output gap and credit spread volatility is altered in the absence of mark-up shocks (top right panel of Fig. 8): combinations of credit spread and output-gap volatility achievable without mark-up shocks are not achievable with them.
  - The inflation–output gap trade-off is still sufficiently pronounced in the absence of mark-up shocks for extreme hawkishness to lead to significantly larger welfare losses than those obtained for higher values of λ (bottom left panel of Fig. 8).

- Note (verbatim): The top left panel shows the best achievable combinations of the standard deviation of the output gap (x-axis) and the standard deviation of inflation (y-axis) for alternative weights on output gap stabilization (λ). The top right panel shows the best achievable combinations of volatilities in the credit spread (y-axis) and the output gap (x-axis) that can be achieved for alternative weights on output gap stabilization (λ). The bottom left panel shows the welfare losses of a dual mandate relative to Ramsey, expressed in CEV units (%), as a function of policymakers’ preferences over output gap stabilization (λ). The values ofλrange from0(strict inflation targeting) to2.4(extreme dovishness). In all panels, the dashed blue line represents the case in which all shocks are present, while the yellow line represents the case in which all shocks are present except for the mark-up shocks. The black diamond shows the result under the optimal dual mandate in the presence of all shocks. The unfilled black diamond shows the result under the optimal dual mandate when all shocks are present except for the mark-up shocks.

### Interest rate smoothing
- Central banks have proposed financial stability concerns as a motivation for interest rate smoothing.
- The loss function considered:
  - M = ˆπ_t^2 + λ ˆx_t^2 + ρ (∆mro_t)^2, where mro is the ECB’s short-term policy interest rate.
- Main findings:
  - Interest rate smoothing is optimal and produces similar welfare outcomes to an optimal extended mandate with the loan-to-deposit spread.
  - Welfare outcomes with interest rate smoothing are notably better than under an optimal dual mandate, despite restricting the policy instrument.
  - Inflation and output gap volatility are lower under the optimal dual mandate (black diamond) than under the optimal extended mandate with interest rate smoothing (empty black dot), but spread volatility is considerably lower when interest rates are smoothed optimally.
  - Smoothing the policy instrument is a reasonable proxy for reducing spread volatility in and of itself.
  - Both inflation and output gap volatility are comparable under the optimal extended mandates considered, explaining why similar welfare outcomes can be achieved.

- Table 7. The role of interest rate smoothing (as reported)
  - x_t: Output gap
  - λωρCEV(%)std(mro)
  - Optimal dual mandate1.00500-0.13013.17
  - f_t: Change in policy rate
  - Optimal extended mandate0.400017-0.0560.54
  - f_t: Loan-deposit spread
  - Optimal extended mandate0.4751.3970-0.0504.04

- Note summary (verbatim from source): This table shows the optimal (welfare-maximising) dual mandate and the optimal extended mandate along with their respective welfare losses relative to the Ramsey policy, expressed in CEV units (%). Two ternary stabilization objectives are considered: the (ex-ante) loan-deposit spread and the change in the (quarterly) policy rate (∆mro_t). The volatility of the (quarterly) policy rate under each respective mandate is also reported.

- Figure 9 note (verbatim): The green line shows the volatility of the respective variable under the extended mandate with interest rate smoothing as a function of the weight placed on interest rate smoothing(ρ), while keepingλat its optimal value under that mandate. The solid black diamond is the optimal dual mandate without interest rate smoothing; the empty black dot is the optimal dual mandate with interest rate smoothing; the solid black dot is the optimal extended mandate with the financial variable being the loan-to-deposit spread. In all cases considered, the measure of resource utilization is the output gap.

### Conclusion (from the chapter)
- A simple mandate that includes a stabilization objective for spreads—an observable proxy for the intensity of the underlying financial friction—produces outcomes that are comparable to the Ramsey policy.
- This finding is robust and provides a rationale for central banks that identify trade-offs between macroeconomic and financial variables as relevant for setting monetary policy, even in the absence of discrete financial ‘crisis’ events in the model.
- Compared to existing literature, the role identified for financial stabilization is quantitatively much more important.
- In the absence of a ternary financial stabilization objective, when financial frictions are present a dual mandate ‘makes sense’ because a high weight on stabilizing the output gap implies greater stability in spreads.
- Smoothing policy interest rates, which double as bank funding costs, raises welfare when financial frictions are present.
- The analysis omits other concerns central banks may have about pursuing a financial stabilization objective, including capturing the many sources and precise forms of frictions present in complex financial systems.
- Continued research is needed to establish robustness of these results to alternative characterizations of financial frictions and to realistic treatments of macro-prudential measures used with monetary and fiscal policies.

*Source: 2.   The role of shocks (content unit: wpiea2020244-print-pdf - 2.   The role of shocks).*

### REFERENCES

### REFERENCES and Supplementary Material (wpiea2020244-print-pdf)

### Key bibliographic references
- Major cited works on financial intermediation, monetary policy, DSGE modeling, and macro-financial interactions include:
  - Adrian, Tobias, and Hyun Song Shin, 2011, “Financial intermediaries and monetary economics.”
  - Gertler, Mark, and Peter Karadi, 2011, “A model of unconventional monetary policy.”
  - Gertler, Mark, and Nobuhiro Kiyotaki, 2011, “Financial intermediation and credit policy in business cycle analysis.”
  - Smets, Frank, and Rafael Wouters, 2003 and 2007, DSGE estimation and Bayesian DSGE approaches.
  - Woodford, Michael, 2003, Interest and prices: foundations of a theory of monetary policy.
- Additional notable contributions cited: Bean (1998), Bernanke (2004), Clarida, Galí, and Gertler (1999), Monacelli (2008), Svensson (2003, 2017), Taylor (1979), and numerous papers on leaning against the wind, macroprudential coordination, and banking-friction DSGE models.

### Model structure (Appendix A: MODEL)
- Agents and sectors:
  - Households: continuum j ∈ [0,1], maximize E_t ∑_{t=0}^∞ β^t ψ^b_t { ln[C_t(j) − h C_{t−1}(j)] − χ L_t(j)^{1+φ}/(1+φ) }.
  - Firms: intermediate good producers, capital producers, retailers.
  - Banks: follow Gertler and Karadi (2011) setup; deposits D_t(i), securities S_t(i) priced at Q_t, net worth N_t(i).
- Key frictional features:
  - Investment adjustment costs: S(I_t/I_{t−1}) = κ/2 (I_t/I_{t−1} − 1)^2.
  - Capital utilization costs: a(U_t) = α_1/α_2 [exp(α_2(U_t − 1)) − 1].
  - Price stickiness (Calvo): prices reset with probability 1−γ and partial indexation parameter ι; price mark-up shock ψ^M_t follows ARMA(1,1).
  - Wage stickiness (EHL): wages reset with probability 1−γ_w and indexation parameter ι_w; wage mark-up shock ψ^μ_w_t follows ARMA(1,1).
  - Banking incentive constraint: going-concern value V_t(i) must exceed liquidation/diversion θ_t Q_t S_t(i); log(θ_t) = ρ_θ log(θ_{t−1}) + ε_{θ,t}.
- Representative first-order and equilibrium conditions (select examples preserved verbatim from source):
  - Euler equation: 1 = E_t R_t Λ_{t,t+1}, with U_{c,t} ≡ ψ^b_t (C_t − h C_{t−1})^{−1} − β h E_t ψ^b_{t+1} (C_{t+1} − h C_t)^{−1}, Λ_{t,t+1} ≡ β U_{c,t+1}/U_{c,t}.
  - Firm production: Y_t = A_t (U_t K_{t−1})^α (L^d_t)^{1−α}, with log(A_t) = ρ_a log(A_{t−1}) + ε_{a,t}.
  - Bank net worth accumulation: N_t(i) = (R_{s,t} − R_{t−1}) Q_{t−1} S_{t−1}(i) + R_{t−1} N_{t−1}(i).
  - Banking incentive constraint and multiplier relations: λ_t = μ_t θ_t − μ_t (equation (21)), Q_t S_t(i) = ν_t θ_t − μ_t N_t(i) (equation (22)), with μ_t = E_t Λ_{t,t+1} Ω_{t+1} (R_{s,t+1} − R_t) (equation (25)) and ν_{s,t} Q_t = E_t Λ_{t,t+1} Ω_{t+1} R_{s,t+1} (equation (28)).

### Aggregation, market clearing, and monetary policy specification (Appendix A & B)
- Banking aggregation and market clearing:
  - N_t = (σ + ξ) Q_{t−1} S_{t−1} R_{s,t} − σ R_{t−1} D_{t−1} (equation (30)).
  - Market clearing identities: Q_t K_t = Q_t S_t, Y^d_t = C_t + I_t + G_t + a(U_t) K_{t−1}, Y^d_t = Y_t ∆p_t, L^d_t = L_t ∆w_t.
- Monetary policy used in estimation (only for estimation):
  - F_t = F^ρ r_{t−1} [ π^{φ_π}_t (Y_t/Y^e_t)^{φ_y} ( (Y_t/Y^e_t)/(Y_{t−1}/Y^e_{t−1}) )^{φ_{∆y}} ]^{1−ρ_r} (equation (61)).

### Potential allocation and steady state (Appendix B & C)
- Efficient allocation conditions (examples preserved):
  - U^e_{c,t} = ψ^b_t (C^e_t − h C^e_{t−1})^{−1} − β h E_t ψ^b_{t+1} (C^e_{t+1} − h C^e_t)^{−1}; 1 = R^e_t E_t β U^e_{c,t+1}/U^e_{c,t}.
  - Q^e_t = 1 in steady state (from capital goods producers’ profit maximization).
- Steady-state calibration assumptions and illustrative steady-state values (as stated):
  - Requal in steady state to 1/β.
  - R_s − R the steady state loan-deposit spread (e.g. 1.20 percent).
  - R_s the steady state loan rate, equal to [R_s/100 + (1/β)]^{1/4}.
  - S/N steady state leverage of commercial banks (e.g. 4.5x).
  - L steady state share of hours worked (e.g. 1/3).
  - M steady state mark-up, equal to ε/(ε − 1).
  - Π steady state price inflation, assumed to be unity.
  - Π^* steady state reset price inflation, assumed to be unity.
  - U steady state utilization, normalized to be unity.
- Derived steady-state banking expressions provided verbatim (examples):
  - λ = S/N · (R_s − R)/R.
  - θ = (1 − σ)(1 − β R_s) / [ σ λ − (1 − σ) [1 + λ]/λ ].
  - ξ = (1 − σ R) − σ (R_s − R) (S/N) / [ R_s (S/N) ].
  - Ω = θ / [ β (R_s − R) ] · λ/(1 + λ).

### Linearized model summary (Appendix D)
- The linearized (approximate) system preserves the banking-friction interactions, nominal and wage rigidities, and real side dynamics.
- Select linearized equations preserved verbatim (representative samples):
  - (1 − β h)(1 − h) û_{c,t} = −(1 + β h^2) ĉ_t + h ĉ_{t−1} + β h E_t{ ĉ_{t+1} } + (1 − h)( ψ̂^b_t − β h E_t ψ̂^b_{t+1} ) (equation (8)).
  - ˆΛ_t = û_{c,t} − û_{c,t−1} (equation (9)).
  - Banking relations: λ̂_t = θ/(θ − μ) ( μ̂_t − θ̂_t ) (equation (10)); μ̂_t = E_t{ ˆΛ_{t+1} + ˆΩ_{t+1} + (R_s r̂_{s,t+1} − R r̂_t)/(R_s − R) } (equation (12)).
  - Price Phillips and Calvo blocks in linearized form: Γ̂_{1,t} and Γ̂_{2,t} relations (equations (23)–(26)).
  - Wage setting linearized block (equations (27)–(31)).
- Numerical note: The linearized version has 28 (instead of 31) endogenous variables because ∆_w and ∆_p drop out as second-order terms; L^d_t = L_t.

### Estimation results and impulse-response evidence (Appendix E)
- Data and fit:
  - Observables plotted and 1-step ahead forecasts (posterior mode calibration) show the model matches real observables “fairly well” though data-rich observables (inflation, spreads) display wedge/persistence issues not captured by i.i.d. measurement errors.
- Variance decomposition of key variables (Table E.1) — preserved exactly as shown (columns corresponding to shock categories: TFPRisk Pr.Govt.Inv-Spec.Mon Pol.W. Mark-upP. Mark-upTheta):
  - Y: 54.5 2.8 9.0 5.2 0.3 1.9 25.4 0.04
  - C: 58.6 24.6 2.4 22.8 0.1 2.6 11.1 0.01
  - I: 29.2 0.4 3.1 17.0 0.7 1.3 47.9 0.51
  - π: 29.1 1.9 1.2 32.2 1.0 17.6 17.1 0.01
  - r: 26.4 1.2 1.1 37.6 5.5 14.0 14.3 0.01
  - SPR: 10.2 0.8 0.4 60.6 2.4 5.4 15.0 5.29
  - LEV: 5.8 1.2 0.9 0.7 0.2 2.8 1.6 6.9
  - Note: For variable mnemonics, see Table 1 (as in source).
- Impulse-response behavior (summary):
  - A one standard deviation negative TFP shock: reduces real activity, raises inflation sharply, raises credit spreads as banks’ net worth falls; TFP shocks account for a sizeable proportion of fluctuations in output and consumption per variance decompositions.
  - Mark-up shocks act as cost-push: increase inflation and dampen real activity.
  - Negative preference (demand) shocks: reduce consumption and investment, lower inflation, loosen monetary stance.
  - Government spending shocks: increase output but consumption does not co-move with output (classic neoclassical property).
  - Monetary policy shocks: contractionary monetary shock reduces bank net worth, raises credit spreads, amplifying real effects via financial accelerator.

### Additional results (Appendix F)
- Welfare comparisons:
  - Figure F.1 shows welfare performance of alternative monetary policy mandates with levels of equal welfare in steps of 0.0025% CEV (an interval 20 times finer than standard), enabling clearer identification of the minimum welfare loss location.
  - Note: For further information see notes to Fig. 4 in the main text.

*Source: REFERENCES and Supplementary Material from wpiea2020244-print-pdf (IMF Working Paper; content preserved exactly as provided).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020244-print-pdf.pdf_