## Appendix A: Proofs

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---

### Overview and purpose
- Contains the proofs of the propositions referenced in the main text.
- Supports results on monetary policy, bank risk taking, and the "leaning against the wind" policy prescription.
- Proofs relate to the general-form model in which the monetary authority may place weight on financial stability alongside standard objectives.

### Key model elements used in proofs
- Aggregate demand function:
  - y_t = B_t(" _t; r^f_t; r^f_{t-1}; ...; r^f_0)
  - Notation: y_t (·) is the output gap; r^f_t; r^f_{t-1}; ...; r^f_0 are current and all past interest rates.
- IS-type sensitivity assumption:
  - ∂y_t(·)/∂r^f_{t-s} < 0 for 0 ≤ s ≤ t.
- Persistent demand shock:
  - " _t = ρ " _{t-1} + ζ_t with ρ ∈ (0,1) and ζ_t iid.
- Shock impact on output:
  - ∂y_t(·)/∂" _t > 0.
- Bank risk profile:
  - β_t denotes bank risk profile; social optimum β^w.
  - ∂y_t(·)/∂β_t > 0 for β_t ∈ [0; β^w).
  - ∂y_t(·)/∂β_t < 0 for β_t ∈ (β^w; 1].

### Monetary authority objective (used in proofs)
- Intertemporal loss minimized:
  - min_{r^f_t, t ≥ 0} E[L] = min_{r^f_t, t ≥ 0} E[ Σ_{t=0}^∞ δ^t [(1−α) f(y_t(·)) + α g(β_t − β^w) ] ]
  - Discount rate: δ < 1.
  - Weight on financial-stability-related losses: α.
  - Weight on standard output-gap losses: (1−α).
- Functional assumptions:
  - f(0) = 0; f''(y_t(·)) = c_1 > 0.
  - g(0) = 0; g''(·) = c_2 > 0.
  - f(·) and g(·) continuous and twice differentiable.
  - Constant second derivatives imply local symmetry around minima (used to simplify analysis with regulation).

### Main theoretical findings (as supported by the proofs)
- Two distinct effects of a negative shock on optimal policy rates:
  - Impact effect: the authority that "leans against the wind" cuts interest rates deeper upon impact than an authority with standard objectives.
  - Dynamic effect: the authority that places weight on financial stability returns interest rates to equilibrium quicker after the initial cut.
- Intuition formalized:
  - Deeper initial cut compensates for persistence of bank risk (β_t) due to long asset maturities.
  - Faster subsequent tightening limits banks' renewed appetite for risk once the trough is passed.
- Interpretation related to "low for too long":
  - The standard-objectives authority path is "low for too long" relative to an authority internalizing financial stability costs; persistence of β_t and intertemporal trade-offs generate this outcome.

### Extensions and robustness (as established in proofs)
- Anticipation channel:
  - Banks adjust risk taking only if they foresee cuts lasting long because unwinding risk takes time; proofs formalize this forward-looking behavior under persistent β_t.
- Role of regulation:
  - With a perfect regulatory tool (perfect targeting of bank risk and full dynamic adjustment), the difference between "leaning against the wind" monetary policy and standard-objectives policy disappears: regulation targets bank risk while interest rate policy targets standard objectives.
  - If the regulatory tool is imperfect in precision and dynamic adjustability (as typical capital requirements may be), the distinction re-emerges.
- Model scope:
  - General-form model establishes qualitative mechanism; Appendix D provides a parameterized example to highlight mechanisms, supported by Appendix A proofs.

### Banking sector: axioms and modeling setup (from related appendices used in proofs)
- Axiom 1: Bank optimal risk taking is larger than the social optimum (b_t > b^w).
- Axiom 2: Risk taking is procyclical: ∂ b^b_t / ∂ y_t (·) > 0.
- Axiom 3: Risk is persistent; modeled by constraint β_t ≥ β̲ β_{t-1} with β̲ ∈ (0,1) and β̲ assumed "small enough" so constraint binds in at least one period (used in Proposition 1).
- One representative/continuum of banks, risk-neutral managers, choose β_t to maximize profit P_t(β_t; y_t(·)) taking monetary policy as given.

### Monetary policy response to a persistent negative shock (short, deep vs longer, smoother cuts)
- Shock setup:
  - At t = 1 a random shock ε_1 occurs; persistence parameter η (or ρ in general form) determines path.
- Definition of policy profile ς:
  - ς = 0: optimal policy with ζ = 0 (no leaning against the wind).
  - Higher ς denotes a deeper but shorter-lived policy: deeper initial cuts and faster return to steady state.
- Proposition 1:
  - Following a negative shock (ε_1 < 0) and assuming constraint binds in at least one period, a monetary authority that "leans against the wind" (ζ > 0) chooses a profile ς > 0 for its interest rates — i.e., opts for a deeper but shorter response compared to an authority with standard objectives (ζ = 0):
    - dς/dζ > 0.
- Intuition:
  - Banks build up risk when the economy recovers while rates are still low. A financial-stability-weighted authority raises rates quickly after an initial cut to mitigate incentives to build up risk later.
  - Trade-off: shorter response reduces later risk at the cost of less optimal output-gap stabilization; larger ζ increases willingness to accept output-gap costs and thus increases dς/dζ.
- Asymmetry for positive shocks:
  - Corollary 1: Proposition 1 does not extend to an upturn (ε_1 > 0); no unambiguous effect of higher ζ following a positive shock because reducing risk is time consuming while increasing risk can be quick.
- Two-period parametric example (Appendix D) confirms mechanism: with constraint binding first period and released second, a financial-objective authority cuts more in period 1 then returns to steady state.

### Regulation: interaction with monetary policy (as proven)
- A. Benchmark (perfect, dynamically adjustable tool)
  - Central bank jointly sets r^f_t and β_t to minimize loss subject to y_t(·), β_t ≤ β_t and model assumptions.
  - Proposition 2: r^f_t is the same for any ζ. The optimal policy path equals that of the standard-objectives (ζ = 0) authority.
  - Interpretation: perfect regulation eliminates need for leaning against the wind.
- B. Imperfect regulatory tool (precision and dynamic inflexibility)
  - Tool set once at t = 0: chosen cap b̄, realized β = b̄ + ξ with ξ symmetric zero-mean ⇒ E[β] = b̄.
  - Central bank chooses r^f_t and b̄ to minimize expected loss subject to β_t ≤ b̄ + ξ.
  - Proposition 3:
    - Optimal b̄ > β^w; central bank allows bank risk to exceed social optimum on average.
    - Interest rate policy qualitatively as in Proposition 1: ζ > 0 authority chooses deeper and shorter-lived response (ς > 0).
  - Intuition: setting b̄ > β^w reduces expected regulatory loss given imprecision; monetary policy still must lean against the wind.
- C. Dynamic socially optimal risk (time-varying β^w_t and only dynamic inflexibility)
  - Relax constant β^w assumption; require ∂(β^b_t − β^w_t)/∂y_t(·) > 0 (excess risk procyclical).
  - Central bank sets single cap β̄ at t = 0 and cannot adjust over cycle.
  - Proposition 4: Central bank sets β̄ > min_t β^w_t; there are periods of excessive bank risk where Proposition 1 arguments apply and monetary policy can still lean against the wind.
  - Intuition: regulation less frequently adjustable than monetary policy; monetary authorities can still affect financial stability via rate dynamics, though quantitative effects are smaller.

### Appendix C and D: bank optimization and parametric illustration (proof-supporting material)
- Appendix C: bank optimization setup with continuum of banks, constraint β_{it} ≥ β̲ β_{it-1}, and Proposition 59 establishing existence of interest-rate paths (shorter and deeper cuts) that produce unambiguous period-by-period reductions in risk for appropriately chosen parameters.
- Appendix D: two-period parametric example
  - IS curve:
    - y_1 = −φ r^f_1 + ε_1
    - y_2 = −(ψ r^f_2 + ρ r^f_1) + ε_2 with ε_2 = η ε_1 and ρ, φ, η ∈ (0,1).
  - Bank risk:
    - β_1 = max{ A + κ y_1 ; β̲ β_0 }
    - β_2 = max{ A + κ y_2 ; β̲ β_1 } with κ ∈ (0,1).
  - Central bank two-period objective:
    - min_{r^f_1, r^f_2} L = min_{r^f_1, r^f_2} ( (1−ζ)( y_1^2 + y_2^2 ) + ζ( β_1 + β_2 ) ).
  - ζ = 0 authority:
    - r^f_1^* = η ε_1 (equation (21))
    - r^f_2^* = η (η − ρ) ε_1 (equation (22))
    - y_1^* = 0, y_2^* = 0, steady-state r^f = 0.
  - ζ > 0 authority with constraint binding in period 1:
    - r^f_1^* = η ε_1 (equation (26))
    - r^f_2^* = ζ κ (2 φ) (1−ζ)^{-1} + η (η − ρ) ε_1 (equation (27))
    - equilibrium steady-state r^f = ζ κ (2 φ) (1−ζ)^{-1}.
  - Comparison for ε_1 < 0:
    - ( r^f_1^* − r^f )|_{ζ>0} − ( r^f_1^* − r^f )|_{ζ=0} = ζ κ (2 φ) (1−ζ)^{-1} (equation (28)): initial cut is deeper for ζ > 0 authority.
  - Numerical illustration parameters in source: ε_3 = 0, ε_1 = −1, η = 0.5, φ = 0.5, ζ = 0.5, κ = 0.5, ρ = 0.25; confirms deeper initial cut and faster return for financial-stability-concerned authority.

### Conclusions and policy implications (as proven)
- Core finding:
  - Accounting for financial imbalances (procyclical, persistent bank risk) alters optimal monetary policy: following negative shocks, authorities with a financial stability objective prefer brief but deeper interest-rate cuts (frontloading stimulation at the trough).
- Mechanism:
  - Short, deep cuts minimize later incentives for banks to build up risk when the cycle recovers and rates remain low.
  - Trade-off: this increases rate variability and requires accepting larger short-run output-gap deviations to reduce financial-stability losses.
- Role of regulation:
  - A perfect, dynamically adjustable regulatory tool obviates the need for leaning against the wind (monetary policy unchanged by ζ).
  - Imperfect or inflexible regulation leaves scope for monetary policy to partially compensate; optimal regulation may relax constraints (b̄ > β^w) to minimize expected regulatory costs when tool precision is limited.
- Practical considerations:
  - Policymakers' preference for smooth rate paths conflicts with the implications of internalizing financial stability: more fluid, potentially more aggressive rate dynamics may be warranted.
  - Definition and measurement of risk remain considerable practical obstacles for implementation.

*Source: Appendix A: Proofs (from the supplied content of the PDF).*

### Appendix A: Proofs  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   15

### Appendix A: Proofs

### Overview and purpose
- Contains the proofs of the propositions referenced in the main text.
- Supports results summarized in the paper on monetary policy, bank risk taking, and the "leaning against the wind" policy prescription.
- Proofs relate to the general-form model in which the monetary authority may place weight on Önancial stability alongside standard objectives.

### Key model elements used in proofs
- Aggregate demand function:
  - y_t = B_t(" _t; r^f_t; r^f_{t-1}; ...; r^f_0)
  - Notation: y_t (·) is the output gap; r^f_t; r^f_{t-1}; ...; r^f_0 are current and all past interest rates.
- IS-type assumption on sensitivities:
  - ∂y_t(·)/∂r^f_{t-s} < 0 for 0 ≤ s ≤ t.
- Persistent demand shock:
  - " _t = ρ " _{t-1} + ζ_t.
  - Persistence parameter: ρ ∈ (0,1).
  - ζ_t is iid.
- Impact of shocks on output:
  - ∂y_t(·)/∂" _t > 0.
- Bank risk profile variable:
  - β_t denotes bank risk profile; social optimum denoted β^w.
  - ∂y_t(·)/∂β_t > 0 for β_t ∈ [0; β^w).
  - ∂y_t(·)/∂β_t < 0 for β_t ∈ (β^w; 1].

### Monetary authority objective (used in proofs)
- Intertemporal loss minimized:
  - min_{r^f_t, t ≥ 0} E[L] = min_{r^f_t, t ≥ 0} E[ Σ_{t=0}^∞ δ^t [(1−α) f(y_t(·)) + α g(β_t − β^w) ] ]
  - Discount rate: δ < 1.
  - Weight on financial-stability-related losses: α.
  - Weight on standard output-gap losses: (1−α).
- Functional assumptions:
  - f(0) = 0; f''(y_t(·)) = c_1 > 0.
  - g(0) = 0; g''(·) = c_2 > 0.
  - Both f(·) and g(·) are continuous and twice differentiable.
  - Symmetry around minima implied by constant second derivatives (used to simplify analysis with regulation).

### Main theoretical findings (as supported by the proofs)
- Two distinct effects of a negative shock on optimal policy rates:
  - Impact effect: The authority that "leans against the wind" cuts interest rates deeper upon impact than an authority with standard objectives.
  - Dynamic effect: The authority that places weight on financial stability returns interest rates to equilibrium quicker after the initial cut.
- Intuition formalized in proofs:
  - Deeper initial cut compensates for persistence of bank risk (β_t) due to long asset maturities.
  - Faster subsequent tightening limits banks' renewed appetite for risk once the trough is passed.
- Interpretation related to "low for too long":
  - The path of a standard-objectives authority is characterized as "low for too long" relative to the path of an authority that internalizes financial stability costs.
  - Proofs show this outcome arises from persistence of β_t and the intertemporal trade-offs captured by the objective function.

### Extensions and robustness (as established in proofs)
- Anticipation channel:
  - Banks adjust risk taking only if they foresee cuts lasting long because unwinding risk takes time; proofs formalize this forward-looking behavior under persistent β_t.
- Role of regulation:
  - With a perfect regulatory tool (perfect targeting of bank risk and full dynamic adjustment), the difference between a "leaning against the wind" monetary policy and standard-objectives monetary policy disappears:
    - The regulatory tool targets bank risk; interest rate policy targets standard objectives.
  - If the regulatory tool is imperfect in precision and dynamic adjustability (as typical capital requirements may be), the distinction between policy types re-emerges.
- Model scope in proofs:
  - General-form model establishes qualitative mechanism.
  - Appendix D provides a parameterized example to highlight mechanisms; proofs in Appendix A underpin both general results and the parameterized illustration.

### Relationship to existing literature (proofs support)
- Proofs formalize an analytical interpretation of empirical findings that low interest rates strengthen banks' incentives to take on risky assets, especially when cuts are prolonged (citations in main text support empirical claims).
- The persistence of bank risk (β_t) is the key modeling ingredient that generates the two-part optimal policy response proven in Appendix A.

*Source: Appendix A: Proofs (from the supplied content of the PDF).*

### 5.  Note that for the output gap these assumptions nest the standard speciÖcation of a

### _wp1386 - 5.  Note that for the output gap these assumptions nest the standard speciÖcation of a

### Banking sector: axioms and modeling setup
- Axiom 1: Bank optimal risk taking is larger than the social optimum.
- Axiom 2: Risk taking is procyclical.
- Axiom 3: Risk is persistent.
- Rationale and implications:
  - Axiom 1: bank moral hazard — the bank does not fully internalize social costs of risky loans (limited liability, bailouts, deposit insurance, lost bank-specific relations). Operationalized as: b t > w :(6)
  - Axiom 2: returns on risky projects rise with the business cycle; operationalized as: @b t @y t (
) >0:(7)
  - Axiom 3: long maturities and maturity mismatch imply slow reduction of risk on balance sheets; operationalized with the constraint t  t1 :(8) where 2(0;1) and  is assumed "small enough" so constraint (8) binds in at least one period (see Proposition 1).
- Model structure:
  - One representative bank, risk neutral management, chooses risk profile  t to maximize profit P t ( t ; y t (
)); profit-maximizing risk profile denoted b t.
  - Inflation ignored without loss of generality for demand shocks (policy to close output gap closes inflation gap).

### Monetary policy response to a persistent negative shock (short, deep vs longer, smoother cuts)
- Shock and setup:
  - At time t = 1 a random shock 1 occurs; path of " is determined by persistence parameter .
  - Consider only a one-period shock; dynamic choice concerns how policy is "spread" over time.
- Definition of policy profile :
  - Baseline:  = 0 corresponds to optimal policy with  = 0 (no leaning against the wind).
  - Higher  denotes a deeper but shorter-lived policy:
    - Definition 1: Policy profile i has a higher  than policy profile j if, for some t < b t and some t > b t, the deviations of r f t from steady state r f satisfy the inequalities given in the source (comparisons of r f t;i  r f and r f t;j  r f with respect to b t).
- Main analytical result:
  - Proposition 1: Following a negative shock (1 <0), and assuming  is sufficiently small such that constraint (8) binds in at least one period: a monetary authority that "leans against the wind" ( >0) chooses a profile  >0 for its interest rates — i.e., opts for a deeper but shorter response compared to an authority with standard objectives ( = 0):
    - d d >0.
  - Intuition:
    - Banks build up risk when the economy recovers while rates are still low. A monetary authority with a financial stability objective raises rates quickly after an initial cut to mitigate incentives to build up risk later.
    - Trade-off: shorter response reduces later risk at the cost of less optimal output-gap stabilization in aggregate; higher weight on financial stability (larger ) increases willingness to accept output-gap costs and thus increases d/d.
- Asymmetry for positive shocks:
  - Corollary 1: Proposition 1 does not extend to an upturn (1 >0). No unambiguous statement about the effect of higher  on dynamics following a positive shock.
  - Intuition: moving portfolio from shorter to longer maturities is quick; building down risk is time consuming (driven by constraint (8)), creating asymmetry between negative and positive shocks.
- Two-period parametric example:
  - Appendix D: with constraint binding in first period and released in second, a financial-objective authority indeed cuts interest rates more in the first period before returning to steady state, consistent with Proposition 1.

### Regulation: interaction with monetary policy
- General approach:
  - Introduce a regulatory tool that implements a risk cap  (interpretable as a risk-weighted capital requirement).
  - Benchmark assumes the tool is perfect and dynamically adjustable ( t ).
  - Subsequent sections relax precision and dynamic adjustability.
- A. Benchmark (perfect, dynamically adjustable tool)
  - Policy problem (central bank jointly sets r f t and  t):
    - min r f t ;  t t0 E[L] = min r f t ;  t t0 ( E 1 X t=0  t [(1)f(y t (
)) +  g( t   w )] ) s.t.: y t (
);  t   t ; assumptions in (1)-(4), (6)-(8).
  - Proposition 2: r f t is the same for any . The optimal policy path equals that of the standard objectives ( = 0) authority.
  - Interpretation: a perfect regulatory tool eliminates the need to "lean against the wind" because regulation can manage bank risk to its optimum, freeing monetary policy to focus on output-gap objectives.
- B. Imperfect regulatory tool (lack of precision and dynamic inflexibility)
  - Assumptions:
    - Regulatory tool can be set only once at the beginning of time.
    - Central bank targets b but realizes  = b +  where  is symmetric, zero-mean random shock ⇒ E[] = b. (Equation (10))
    - Central bank chooses r f t and b to minimize expected loss:
      - min r f t ; b t0 E[L] = min r f t ; b t0 ( E 1 X t=0  t [(1)f(y t (
)) +  g( t   w )] ) s.t.: y t (
);  t  b + .
  - Proposition 3:
    - Optimal b >  w ; central bank allows bank risk to exceed social optimum on average.
    - Interest rate policy qualitatively as in Proposition 1:  >0 authority chooses a profile with deeper and shorter-lived response ( >0).
  - Intuition:
    - Imprecision is less costly on average when b is set above  w because when realized  is large it may hit a natural cap  b t and not increase g( t   w ) further (due to inequality in Axiom 1). Setting b >  w reduces expected regulatory loss.
    - Because the regulator relaxes constraints expecting excess risk, monetary policy still needs to "lean against the wind" with deeper, shorter cuts; regulatory policy relieves but does not eliminate monetary policy's financial-stability burden.
- C. Dynamic socially optimal risk (time-varying  w t and only dynamic inflexibility)
  - Relax constant  w assumption; require only:
    - @ ( b t   w t ) @y t (
) >0 :(12) — gap between bank-desired and socially optimal risk expands during booms (procyclicality of the excess).
  - Central bank can set a single regulatory cap  at t = 0 but cannot adjust it over the cycle:
    - min r f t ;  t0 E[L] = min r f t ;  t0 ( E 1 X t=0  t [(1)f(y t (
)) +  g( t   w t )] ) s.t.: y t (
);  t   .
  - Proposition 4: Central bank sets  > min t  w t . Hence, there are periods when banks take excessive risk; in those periods arguments of Proposition 1 apply and monetary policy can still "lean against the wind".
  - Intuition: because regulation is less frequently adjustable than monetary policy, monetary authorities can still play a role in financial stability by adjusting rates more frequently; quantitative differences are expected (smaller v-shape) relative to a world without any bank regulation.

### Conclusions and policy implications
- Core finding:
  - Accounting for financial imbalances (procyclical, persistent bank risk) alters optimal monetary policy: following negative shocks, authorities with a financial stability objective prefer brief (short-lived) but deeper interest-rate cuts (frontloading stimulation at the trough).
- Mechanism:
  - Short, deep cuts minimize later incentives for banks to build up risk when the cycle recovers and rates remain low.
  - Trade-off: such conduct increases rate variability and requires accepting larger short-run output-gap deviations to reduce financial-stability losses.
- Role of regulation:
  - A perfect, dynamically adjustable regulatory tool obviates the need for leaning against the wind (monetary policy unchanged by ).
  - Imperfect or inflexible regulation leaves scope for monetary policy to partially compensate; optimal regulation may be set to relax constraints (b >  w ) to minimize expected regulatory costs when tool precision is limited.
- Practical considerations:
  - Policymakers often prefer smooth interest rate paths; incorporating financial stability objectives argues against slow, smooth policy dynamics and suggests the need for more fluid, potentially more aggressive monetary policy when financial stability targets are included.
- Implementation challenge:
  - Definition and measurement of risk remain considerable practical obstacles for implementing the paper's prescriptions.

*Source: _wp1386 - 5.  Note that for the output gap these assumptions nest the standard speciÖcation of a (IMF working paper content provided).*

### Appendix B: Central bank objective in general form

### _wp1386 - Appendix B: Central bank objective in general form

### Central bank objective — general form
- The monetary authority's optimization problem can be written in the general form:
  - min_{r^f_t; t≥0} E[L] = min_{r^f_t; t≥0} ( E ∑_{t=0}^∞ δ^t H_ζ( f(y_t(·)); g(β_t; β̄_w) ) )  (equation (14) structure retained)
- Constraint: y_t(·) (as specified in the model).
- H_ζ is a set of functions such that:
  - when f(y_t(·)) > g(β_t; β̄_w) the value of H_ζ( f(y_t(·)); g(β_t; β̄_w) ) decreases in ζ,
  - when f(y_t(·)) < g(β_t; β̄_w) the value of H_ζ( f(y_t(·)); g(β_t; β̄_w) ) increases in ζ.

### Properties imposed on g(β_t; β̄_w)
- g(β_t; β̄_w) satisfies:
  - ∂g(β_t; β̄_w)/∂β_t |_{β_t > β̄_w} > 0
  - ∂g(β_t; β̄_w)/∂β_t |_{β_t < β̄_w} < 0
  - ∂g(β_t; β̄_w)/∂β̄_w |_{β_t > β̄_w} < 0
  - ∂g(β_t; β̄_w)/∂β̄_w |_{β_t < β̄_w} > 0

### Additional regularity for f and g
- f(y_t(·)) and g(β_t; β̄_w) are:
  - continuous and twice differentiable,
  - f(0) = 0,
  - f''(y_t(·)) = c_1 > 0,
  - g(0) = 0,
  - g''(·) = c_2 > 0.

### Appendix C: Bank optimization — setup and main proposition
- Setup:
  - Continuum of atomistic banks; each bank i solves:
    - max_{β_{it}} ∑_{t=0}^∞ δ^t P_{it}( β_{it}; y_t(β_t; r^f_t, r^f_{t-1}, …, r^f_0) )
    - subject to β_{it} ≥ β̲ β_{it-1}  (constraint on risk path)
  - β_{it} is bank i's risk profile; β_t is average banking sector risk.
  - Banks take monetary policy path r^f_t; r^f_{t-1}; … as given.
  - P_{it}(·) satisfies condition (7): banks take more risk when the output gap is larger:
    - ∂ β^b_{it} / ∂ y_t(·) > 0  (with β^b_{it} being bank i's optimal risk in period t).
  - With identical banks, β_t = β_{it} for all i.
- Proposition 59 (paraphrased preserving notation):
  - For b̄ ζ > 0 : β_{it}|_{ζ = b̄ ζ} < β_{it}|_{ζ = 0} ∀ t: interest rate paths can be found which involve shorter and deeper cuts than the ζ = 0 path, and bring about an unambiguous (i.e. each period) reduction in risk taking.
- Proof sketch (as in source):
  - For t > b̄ t we have β^b_{it}|_{ζ = b̄ ζ} < β^b_{it}|_{ζ = 0}.
  - For t < b̄ t there is a trade-off: lower r^f_t under the ζ = b̄ ζ path raises β^b_{it} relative to ζ = 0, but the desire for lower future risk when faced with ζ = b̄ ζ and the constraint β_{it} ≥ β̲ β_{it-1} can make current risk lower.
  - One can choose b̄ ζ > 0 such that the latter argument dominates, producing β_{it}|_{ζ = b̄ ζ} < β_{it}|_{ζ = 0} ∀ t.
  - Construction: pick b̄ ζ so that at t = 1, r^f_1|_{ζ = b̄ ζ} = r^f_1|_{ζ = 0} − ε and r^f_t|_{ζ = b̄ ζ} = r^f_t from t = 2 onward; ε arbitrarily small but sufficient to ensure intertemporal losses offset instantaneous gain.

### Appendix D: A parametric example — two-period illustration
- Purpose: illustrate dynamic response to a single shock over a minimum of 2 periods and match to Proposition 1.
- IS curve (two periods):
  - y_1 = −φ r^f_1 + ε_1
  - y_2 = −(ψ r^f_2 + ρ r^f_1) + ε_2
  - with ρ, φ, ε ∈ (0,1).
- Shock persistence:
  - ε_2 = η ε_1
  - Hence:
    - y_1 = −φ r^f_1 + ε_1  (equation (16))
    - y_2 = −(ψ r^f_2 + ρ r^f_1) + ε_2 ε_1  (equation (17) structure retained)
- Bank risk evolution:
  - β_1 = max{ A + κ y_1 ; β̲ β_0 }
  - β_2 = max{ A + κ y_2 ; β̲ β_1 }
  - with κ ∈ (0,1). A relates to axiom 1 (β_t larger than social optimum β_w). The constraint β̲ β_{t-1} prevents risk from immediately unwinding to zero.
- Central Bank objective (two-period):
  - min_{r^f_1, r^f_2} L = min_{r^f_1, r^f_2} ( (1−ζ)( y_1^2 + y_2^2 ) + ζ( β_1 + β_2 ) )  (equation (20))

### ζ = 0 authority (traditional objective)
- First-order conditions yield r^f_1 and r^f_2 solving:
  - r^f_1 = η ε_1 (1 + ρ^2)^{-1} − ρ (1 + ρ^2)^{-1} r^f_2  (equation solving system)
  - r^f_2 = ε_2 ε_1 − ρ r^f_1
- Closed-form solutions (from source):
  - r^f_1^* = η ε_1  (equation (21))
  - r^f_2^* = ε_2 ε_1 − ρ η ε_1 = η (η − ρ) ε_1  (equation (22))
- Result:
  - y_1^* = 0
  - y_2^* = 0
  - Steady-state interest rate r^f = 0.

### ζ > 0 authority (financial-stability concerned)
- Assume risk constraint binds in the first period:
  - β̲ β_0 > A  (equation (23))
  - β̲ β_2 = β̲ β_1 < A  (equation (24))
- Optimization problem (equation (25)):
  - min_{r^f_1,r^f_2} (1−ζ)( φ (−φ r^f_1 + ε_1)^2 + (−(ψ r^f_2 + ρ r^f_1) + ε_2 ε_1)^2 ) + ζ( (β̲ β_0) + ( A + κ (−(ψ r^f_2 + ρ r^f_1) + ε_2 ε_1) ) )
- First-order conditions lead to closed-form:
  - r^f_1^* = η ε_1  (equation (26))
  - r^f_2^* = ζ κ (2 φ) (1−ζ)^{-1} + η (η − ρ) ε_1  (equation (27))
- In equilibrium (no shocks) steady-state interest rate:
  - r^f = ζ κ (2 φ) (1−ζ)^{-1}  (preserved notation from source)

### Comparison and main mechanism (preserving exact expressions)
- For any ε_1 < 0 (negative shock):
  - r^f_1^*|_{ζ=0} − r^f = η ε_1 − 0 = η ε_1 < 0
  - r^f_1^*|_{ζ>0} − r^f = η ε_1 − ζ κ (2 φ) (1−ζ)^{-1} < 0
  - Difference between deviations:
    - ( r^f_1^* − r^f )|_{ζ>0} − ( r^f_1^* − r^f )|_{ζ=0} = ζ κ (2 φ) (1−ζ)^{-1}  (equation (28))
  - Interpretation in example: compared to the steady state rate, the initial rate cut is deeper for the ζ > 0 authority than for the ζ = 0 authority.
- Numerical illustration provided in source (parameter values used for visualization):
  - ε_3 = 0, ε_1 = −1, η = 0.5, φ = 0.5, ζ = 0.5, κ = 0.5, ρ = 0.25.
- Conclusion stated in source:
  - In the parametric example, the authority that cares about bank risk implements a deeper cut and then raises rates back quickly, matching the general-form model result.

*Source: _wp1386 - Appendix B: Central bank objective in general form (including Appendices C and D) — content as provided.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp1386.pdf_