## _wp1658 — Section 2: Analytical framework and main results

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### Context and motivations
- Since the 2008 global financial crisis, central banks have expanded policy toolkits to include macro-prudential tools, balance sheet operations, credit policy, quantitative easing, and foreign exchange intervention.
- The crisis revived questions about monetary policy objectives (targeting asset prices or financial stability measures) and the appropriate set of instruments.
- Pre-crisis results (e.g., “divine coincidence”) can break down when models include additional frictions or constraints.

### Framework and modelling approach
- Model class: extended linear New Keynesian Model with a Phillips curve, an expected IS curve, and a quadratic loss function.
- Key variables: π_{Ht} (domestic inflation), y_{t} (output gap), i_{t} (policy interest rate), θ_{t} (unconventional instrument), u_{t} and v_{t} (exogenous shocks).
- General reduced-form dynamics include forward-looking terms 1 e_{t+1} y and 1 e_{t+1} θ in the Phillips Curve.
- Unconventional instrument θ candidates include public spending, fiscal policy, capital controls, foreign exchange intervention, quantitative easing, macroprudential policy.
- Social welfare loss function (output gap weight normalized to unity):
  - Σ_{t=0}^{∞} β^{t} [ (π_{Ht+t})^{2} α_{π} + (y_{t})^{2} + (θ_{t})^{2} α_{θ} ]
- Discretionary framework: central banker minimizes current-period term:
  - (π_{Ht})^{2} α_{π} + (y_{t})^{2} + (θ_{t})^{2} α_{θ}
- Intertemporal budget constraint for θ (if applicable):
  - Σ_{t=0}^{∞} β^{t} θ_{t} = 0
  - Γ denotes Lagrange multiplier on this constraint (Γ can be set to 0 if constraint not relevant).

### Frictions and circumstances that complicate standard monetary policy
- Mechanisms that break the “divine coincidence” or limit interest-rate policy:
  - Reduced-form cost-push shocks (New Keynesian Phillips Curves).
  - Real wage rigidities.
  - Cases where interest rates affect marginal costs (indeterminacy).
  - Limits like the zero-lower bound, risk premia, disruptions in financial intermediation.
- Secondary instruments discussed and their targeted frictions:
  - Capital controls for volatile capital flows.
  - Fiscal policy to support constrained monetary policy.
  - Quantitative easing to reduce credit spreads.
  - Macroprudential policy for financial instability or aggregate demand externalities.

### How secondary instruments enter dynamics (examples)
- Capital controls: introduce wedge between domestic and foreign consumption → enter IS and Phillips curves.
- Sterilized FX interventions: generate endogenous risk premium → affect consumption and enter IS and Phillips curves.
- VAT and labor tax: VAT affects consumption (inflation and output); labor tax affects firms’ marginal cost → enters Phillips Curve as an “endogenous cost-push shock.”
- Financial frictions: heterogeneous marginal utilities/leverage → endogenous state variable affecting IS and Phillips curves.

### Equilibrium determinacy and role of θ
- First-order conditions (symbolic forms preserved in source) deliver a law of motion for π_{Ht} with forward expectations.
- Proposition 1 (Equilibrium Determinacy under Discretionary Policy):
  - Determinacy ensured when Blanchard-Kahn type condition (expressed symbolically in source) holds.
  - When θ unavailable (θ_{t}=0 or α_{θ} → +∞) determinacy condition reduces to a symbolic inequality involving forward coefficient on y and α_{π}.
  - Reintroducing θ defines marginal effect X_{θ} (symbolic) and yields determinacy condition (9) (symbolic).
- Graphical insight (Figure 1): high α_{θ} or large X_{θ} can eliminate indeterminacy risk; trade-off between engineered recessions and activism with θ.

### Optimal stabilization after cost-push shocks
- Consider u_{t} following u_{t} = ρ_{u} u_{t-1} with u_{0} ≠ 0.
- Proposition 2 (Optimal Policy Following Cost-Push Shocks):
  - Optimal paths for π_{Ht+t}, y_{t}, θ_{t} presented in closed symbolic form in the source (equation (10) and auxiliary D term).
- Key implications:
  - Availability and use of θ can reduce the impact of cost-push shocks when stabilization term in D_{uρ} is positive.
  - If expected future use of θ offsets current use (symbolic condition), committing not to use θ can be welfare-improving — availability may increase volatility.
  - Tightness of intertemporal budget constraint (larger |Γ|) reduces stabilization capacity.

### Central banker’s preferences: stabilization and inflationary biases
- Discretionary policy and lack of commitment generate stabilization bias and inflationary bias.
- Central banker’s design problem: choose {α_{π}~, α_{θ}~} to minimize welfare loss subject to determinacy.
- Proposition 3 (A Conservative and Interventionist Central Banker):
  - If social preferences ensure determinacy:
    (i) Central banker’s optimal preferences cannot induce indeterminacy.
    (ii) If 1 u k_{π} ρ < , optimal preferences minimizing welfare losses given in closed symbolic forms (source).
    (iii) If 1 u k_{π} ρ > , optimal preferences become: α_{π}~ = ∞ and α_{θ}~ specified symbolically in source.
    (iv) When intertemporal budget constraint not applicable, optimal weight for θ: α_{θ}~ = 1 (symbolic relation preserved in source).
- Corollary 1 (properties of optimal preferences):
  - (i) α_{π}~ ≥ α_{π} (central banker more inflation-averse than social planner).
  - (ii) α_{π}~ increases with shock persistence and with effect of future output on current inflation (e_{y} k).
  - (iii) α_{θ}~ < α_{θ} if (//) e e_{y} y k k kk θ θ > (central banker uses θ more actively when future θ has larger effect on current inflation).
  - (iv) α_{θ}~ decreases with shock persistence if e e_{y} y k k kk θ θ >.

- Policy intuition:
  - Optimal central banker increases inflation aversion to improve credibility and reduce discretionary inflation.
  - When future use of θ strongly influences current inflation, the optimal banker is both more conservative (higher α_{π}~) and more interventionist with θ.
  - Intertemporal constraints on θ can alter optimal α_{θ}~ relative to social α_{θ} even for one-off shocks.

### Stabilization bias when optimal preferences would trigger multiple equilibria
- Constrained design problem: minimize welfare subject to determinacy constraint when unconstrained optimum is indeterminate.
- Proposition 4 (Optimal Preferences under Equilibrium Indeterminacy):
  - If unconstrained optimal preferences are indeterminate, the constrained optimum {α_{π}^c, α_{θ}^c}:
    (i) lies on the determinacy frontier;
    (ii) has α_{π}^c > α_{π}^{opt};
    (iii) has α_{θ}^c < α_{θ}^{opt} if and only if (//) e e_{y} y k k kk θ θ >.
- Policy implication: appoint a central banker whose preferences lie on the determinacy frontier — more inflation-averse and, depending on forward-looking effects, possibly less inclined to use θ.

### The inflationary bias (structure and cases)
- Social welfare loss setup (symbolic expression preserved).
- Proposition 5 — The inflationary bias:
  - Distinguishes models where intertemporal budget constraint on θ applies vs. not.
  - (i) If instrument not constrained intertemporally:
    - a. If current inflation depends weakly on expected inflation (1 k_{π} < ), central banker’s minimizing preferences are given symbolically in source.
    - b. If current inflation depends strongly on expected inflation (1 k_{π} > ), central banker’s preferences may drive α_{π}~ → +∞ and α_{θ}~ specified symbolically.
  - (ii) If instrument is constrained intertemporally:
    - a. It is optimal not to use it, i.e., θ_{t} = 0.
- Intuition: with a binding intertemporal budget constraint and permanent shocks (1 u ρ = ), any use today must be repaid later so optimal use is zero.

### Determinacy of optimal preferences and constrained-optimum characterization
- If social preferences {α_{π}, α_{θ}} are on or above the determinacy frontier, optimal central banker preferences are determinate.
- Appendix characterizes frontier parametrization and algebra proving uniqueness of constrained optimum.
- Constrained-optimum properties:
  - Frontier parametrization fr_{θπ}: θ α = 1 a + b (symbolic a, b defined in appendix).
  - If unconstrained optima yield indeterminacy, constrained optimum lies on frontier and α_{π}^c > α_{π}^{opt}.
  - Direction of change in α_{θ}^c relative to α_{θ}^{opt} depends on whether the unconventional instrument is more forward-looking than output: if instrument is more forward-looking (ratio < 1), constrained optimum uses it more aggressively; otherwise, less aggressively.

### Broader implications and policy relevance
- Tinbergen principle: need as many independent instruments as objectives; divine coincidence can fail with cost-push shocks.
- When policy rate is sole instrument and divine coincidence fails, optimal response may involve maintaining a positive output gap while inflation is below target.
- Main contributions:
  - Additional instruments can ensure determinacy and reduce volatility in presence of cost-push shocks.
  - Under some parameterizations, committing not to use an unconventional instrument can be optimal due to intertemporal constraints.
  - If future use of θ matters more for inflation than for future output gap, the optimal central banker will be more interventionist with θ than social preferences imply.
- Suggested extensions:
  - Incorporate stochastic setups and non-linear dynamics relevant for financial stability and regime-switching.
  - Empirical task to better identify Phillips Curve shape and effects of unconventional instruments on activity and inflation.

*Source: IMF Working Paper section provided in content unit*

### References. ............................................................................................................

### References.

### Context and motivations
- Since the 2008 global financial crisis, central banks have rethought monetary and financial stability frameworks, using a variety of policy instruments including macro-prudential tools, balance sheet operations, credit policy, quantitative easing, and foreign exchange intervention.
- The crisis revived questions about monetary policy objectives, targeting asset prices or financial stability measures, and the set of instruments appropriate for central banks.
- Prevailing pre-crisis results (e.g., “divine coincidence” where inflation targeting alone suffices) can break down when models include additional frictions or constraints.

### Frictions and circumstances that complicate standard monetary policy
- Additional model elements that break the “divine coincidence” include:
  - Reduced-form, exogenous, cost-push shocks (New Keynesian Phillips Curves) leading to trade-offs between output and inflation volatility (Taylor (1979), Clarida et al. (1999)).
  - Real wage rigidities (Blanchard and Gali (2007)).
  - Cases where interest rates affect marginal costs and standard rules may lead to indeterminacy (Surico (2008)); monetary policy inefficiency where output gap and inflation fluctuate after productivity or demand shocks (Ravenna and Walsh (2006)).
  - Limits to interest-rate policy efficacy: zero-lower bound (Eggertsson and Woodford (2003)), risk premia in international capital markets (Farhi and Werning (2014)), disruptions in financial intermediation (Curdia and Woodford (2010)).
- Secondary instruments discussed in the literature depend on the source of friction:
  - Capital controls for volatile capital flows (Farhi and Werning (2014)).
  - Fiscal policy to support constrained monetary policy (Correia et al. (2013)).
  - Quantitative easing to reduce credit spreads impeding intermediation (Curdia and Woodford (2011)).
  - Macroprudential policy for financial instability or aggregate demand externalities (De Paoli and Paustian (2013), Farhi and Werning (2013)).
  - Debates about monetary policy capacity alone and the need for coordination (Woodford (2012), Svensson (2014)).

### Framework and modelling approach
- Many studies reduce to an extended New Keynesian Model where:
  - The linearized expected IS curve and Phillips curve are affected by a “friction” and by a new policy instrument.
  - The quadratic approximation of welfare explicitly includes the unconventional policy instrument (typically penalizing its use).
- The paper presents a unifying framework casting New Keynesian Models with multiple instruments to derive general results applicable across a wide range of models.

### Key findings and results stated
- Additional policy instruments can:
  - Help rule out equilibrium indeterminacy.
  - Reduce welfare losses after exogenous shocks or in the presence of a distorted steady state.
- Under some circumstances, committing not to use the unconventional instrument may be welfare improving.
- Aggressive use of a secondary instrument mitigates both the inflationary bias and the stabilization bias.
- The paper characterizes optimal preferences for the central bank governor in cases where societal preferences would result in indeterminacy.

### Structure and supplementary materials (as listed)
- Figures:
  - 1. Optimal Policy Determinacy Condition (page 12)
  - 2. Optimal Preferences Determinacy (page 27)
  - 3. Welfare Loss Variations in the Determinacy Area (page 28)
- Appendix:
  - A. Proof of Proposition 3 (page 23)
  - B. Proof of Proposition 4 (page 27)
- References section begins on page 32.

*Source: _wp1658 - References. ........................................................................................................................*

### Section 2 presents the analytical framework, which is a general linear New Keynesian Model, and

### _wp1658 - Section 2 presents the analytical framework, which is a general linear New Keynesian Model, and

### Analytical Framework — overview
- Presents a general linear New Keynesian Model comprising: a Phillips curve, an expected IS curve, and a quadratic loss function.
- Focus: discretionary framework (expected future values taken as given) and mechanisms by which a central bank can reinforce credibility.
- Unconventional policy instrument denoted θ; candidates include public spending, fiscal policy, capital controls, foreign exchange intervention, quantitative easing, macroprudential policy.

### The Extended New Keynesian Framework
- Core variables: Hπ (domestic inflation), y (output gap), i (policy interest rate), θ (unconventional instrument), u and v (exogenous shocks).
- General form: Φ(πH t, πH t+1 e, y t, y t+1 e, i t, θ t, θ t+1 e, u t, v t) and Ψ(...) are linear functions; model allows interest rate to enter the NKPC.
- Reduced-form dynamics (after substituting for the interest rate where possible) summarized by equation (1):
  - ,,111 eee HtHty ttttt y kky k ykku πθ θ π πθθ + ++ = ++ ++ +
- Key modeling point: presence of expected terms 1 e t y + and 1 e t θ + in the Phillips Curve is the crucial forward-looking feature.

### How secondary instruments enter dynamics (examples)
- Capital controls (Farhi and Werning (2014)): introduce wedge between domestic and foreign consumption → enter IS curve; affect firms’ marginal costs → enter Phillips Curve.
- Sterilized foreign exchange interventions (Alla et al. (2016)): generate endogenous risk premium → portfolio balance channel; affect domestic consumption and enter IS and Phillips curves.
- VAT and labor tax (Alla (2015)): VAT affects consumption (inflation and output); labor tax affects firms’ marginal cost → enters Phillips Curve as an “endogenous cost-push shock.”
- Financial friction (Woodford (2012), Ravenna and Walsh (2006)): heterogeneous marginal utilities / leverage → endogenous state variable affecting IS and Phillips curves.

### Objective Function and intertemporal constraint
- Social welfare loss function (with output gap weight normalized to unity):
  - Σ_{t=0}^{∞} β^{t} [ (π_{Ht+t})^{2} α_{π} + (y_{t})^{2} + (θ_{t})^{2} α_{θ} ]
- In discretionary framework, central banker minimizes current-period term (equation (2)):
  - (π_{Ht})^{2} α_{π} + (y_{t})^{2} + (θ_{t})^{2} α_{θ}
- Intertemporal budget constraint for θ (if applicable) (equation (3)):
  - Σ_{t=0}^{∞} β^{t} θ_{t} = 0
- Γ denotes Lagrange multiplier on (3); Γ can be set to 0 if constraint not relevant. Constraint implies commitment not to default; otherwise secondary instrument could not be used.

### Rationale for unconventional policy instruments
- Two cases where monetary policy alone cannot perfectly stabilize:
  - Exogenous shocks: additive factors in Phillips or IS curve breaking the divine coincidence; examples include risks to financial intermediation, risk premia.
  - Financial frictions defined broadly: any element that implies the interest rate enters the Phillips curve (cost channel), which can lead to monetary policy indeterminacy and reduce efficacy of interest rate policy.
- Forward-looking determinants of inflation: presence of 1 e t y + and 1 e t θ + matters; literature often restricts to future inflation or output gap, but other forward-looking instruments (θ expectations) can affect current inflation.

### Illustrative example (open economy with capital controls)
- Policy minimizes (equation (4)):
  - min_{i,t,θ,t} (π_{Ht}^{2} α_{π} + y_{t}^{2} + θ_{t}^{2} α_{θ})
- Phillips Curve and IS curve specified (see section for full expressions); substitution yields dynamic equation with expected terms 1 e t y + and 1 e t θ + and financial friction increasing forward coefficient on inflation.
- Intertemporal budget constraint represents no-Ponzi condition on net foreign assets; capital controls distort consumption and trade balance; present value of distortion must be zero.

### Need for unconventional policy instruments — Equilibrium determinacy
- First-order conditions for y_{t} and θ_{t}:
  - (5) ,t y: y_{k} π_{Ht} = -α_{π}
  - (6) ,t θ: k_{θ} π_{Ht} = -α_{θ} θ_{t} - Γ
- Law of motion for domestic inflation (equation (7)) (preserved symbolically in source).
- Proposition 1 (Equilibrium Determinacy under Discretionary Policy):
  - Determinacy ensured when Blanchard-Kahn condition satisfied; condition expressed as inequality (8) and alternative form (1() ... ) in the source.
  - Intuition: when expected impact of output and second instrument on inflation exceeds current impact, indeterminacy possible; ensure α_{π} not too high or use second instrument to offset.
- Determinacy when θ unavailable (θ_{t}=0 or α_{θ} → +∞):
  - Condition: 1() e y y y k kk k π π α − > + (preserved symbolic form).
  - Denote X_{y} as recession engineered when inflation is 1 percent: 0 y y X k π α =>.
  - Determinacy requires total impact 1() e y y k k X ++ > k_{π}.
- Reintroducing θ and marginal effect X_{θ} defined as 0 X k π θ α θ θ α => leads to determinacy condition (9):
  - (   )(  )1 e e yy y k  k X   k  kX   k θ θθ π +   + +   >−
- Graphical insight (Figure 1 in source): when α_{θ}=0 or X_{θ} high, indeterminacy risk eliminated; trade-off between engineered recessions and activism with θ.

### Optimal stabilization policy following cost-push shocks
- Analyze cost-push shocks u_{t} following u_{t} = ρ_{u} u_{t-1} with u_{0} ≠ 0.
- Proposition 2 (Optimal Policy Following Cost-Push Shocks):
  - Optimal paths (equation (10)):
    - π_{Ht+t} = (1) ... (preserved symbolic forms) and y_{t}, θ_{t} expressions in closed form (source presents exact symbolic expressions).
  - Auxiliary D term defined in source (preserved symbolic form).
- Key results:
  - Availability and use of unconventional instrument reduce impact of cost-push shock when stabilization term ()_{e u kkk θ απ θθθ ρ +} in D_{u ρ} is positive.
  - If expected use of θ offsets current use ((()0 e u kkk θ θ θ ρ + < )), better to commit not to use θ — availability may increase volatility.
  - Tightness of intertemporal budget constraint (larger |Γ|) reduces policymakers’ stabilization capacity.

### Central banker’s preferences — stabilization and inflationary biases
- Discretionary policy and lack of commitment create stabilization bias and inflationary bias.
- Investigate which central banker’s preference weights (α_{π}~, α_{θ}~) minimize welfare losses arising from:
  - Stabilization bias (due to inability to commit and forward-looking Phillips Curve).
  - Inflationary bias (special case with permanent shock).

### The Stabilization Bias — central banker design
- Central banker’s objective (using Proposition 2) expressed symbolically in source (preserve structure).
- Optimization problem (11): choose {α_{π}~, α_{θ}~} = argmin W_{π~,θ~} subject to equilibrium determinacy constraint (preserved symbolic inequality).
- Proposition 3 (A Conservative and Interventionist Central Banker):
  - If social preferences ensure determinacy then:
    (i) Central banker’s optimal preferences cannot induce indeterminacy.
    (ii) When shock not highly persistent (1 u k_{π} ρ < ), optimal preferences that minimize welfare losses given in closed symbolic forms in source.
    (iii) If shock sufficiently persistent (1 u k_{π} ρ > ), optimal preferences become:
      - α_{π}~ = ∞ (inflation nutter); α_{θ}~ = finite expression (preserved in source).
    (iv) When intertemporal budget constraint not applicable, optimal weight for θ:
      - α_{θ}~ = 1 ( (/)... ) (preserved symbolic exact expression from source).
- Corollary 1 — characteristics of optimal preferences:
  - (i) α_{π}~ ≥ α_{π}  (central banker more inflation-averse than social planner).
  - (ii) α_{π}~ increases with shock persistence and with effect of future output on current inflation (e_{y} k).
  - (iii) α_{θ}~ < α_{θ} if (//) e e y y k k kk θ θ > (i.e., central banker uses θ more actively than social preference when future θ has larger effect on current inflation).
  - (iv) α_{θ}~ decreases with shock persistence if e e y y k k kk θ θ >.

- Intuition and policy implications:
  - Optimal central banker increases inflation aversion (Clarida et al. (1999) style) to improve credibility and reduce discretionary inflation.
  - When future use of θ influences current inflation strongly, optimal banker is both conservative and more interventionist with θ.
  - Budget constraint on θ implies optimal α_{θ}~ may differ from social α_{θ} even for one-off shocks.

### Stabilization bias when optimal preferences would trigger multiple equilibria
- Constrained design problem (12): minimize W subject to determinacy constraint and knowing unconstrained optimum may be indeterminate.
- Proposition 4 (Optimal Preferences in Situations of Equilibrium Indeterminacy):
  - If unconstrained optimal preferences are indeterminate, the constrained optimal choice {α_{π}^c, α_{θ}^c}:
    (i) lies on the determinacy frontier;
    (ii) features α_{π}^c > α_{π}^{opt} (higher inflation weight);
    (iii) features α_{θ}^c < α_{θ}^{opt} if and only if (//) e e yy kk kk θ θ >.
- Policy implication: when indeterminacy risk is present, appoint a central banker with preferences on determinacy frontier — more inflation-averse and, depending on relative effects of future vs current θ, less inclined to use unconventional instrument.

*Italic: Source — IMF Working Paper section provided in content unit*

### 3. If the central banker has an instrument whose future use matters a lot, he should be more

### 3. If the central banker has an instrument whose future use matters a lot, he should be more interventionist with this instrument, even though the constraint on determinacy forces him to adopt “second-best”' preferences.

### C. The Inflationary Bias — setup and social welfare loss
- Social welfare loss minimized: ,222{, ,},0 1 min()2 Ht t t t yHttt t yy πθπθ β απαθ ∞ = +−+∑
- Constraint (policy transmission / structural): ,,,111 ee eee HtHty tttt y kky k ykk πθ θ π πθθ + ++ = ++ ++
- Intertemporal budget constraint: 0 0 t t t βθ ∞ = = ∑

### Proposition 5 — The inflationary bias (structure and cases)
- Assumption: optimal preferences are determinate.
- Distinction: models without the intertemporal budget constraint vs. those with it.

(i) Instrument not constrained intertemporally (constraint (3) does not apply)
- a. If current inflation depends weakly on expected inflation (1k π <), central banker’s preferences that minimize welfare losses are:
  - 1( / )1( / ); 11( / ) ee yyyy e kkkk kkk ππθθ πθθ αααα ++ == −+ 
- b. If current inflation strongly depends on expected inflation (1k π >), central banker's preferences become:
  - 1( / ); 1( / ) e yy e kk kk πθθ θθ ααα += +∞ = + 

(ii) Instrument constrained intertemporally
- a. It is optimal not to use it, i.e., 0 t θ = (its weight is then irrelevant).

- Intuition: when the intertemporal budget constraint applies and the shock is permanent (1 u ρ =), any use today must be paid back later; therefore optimal use is 0 t θ =.

### Key analytical results on optimal coefficients and corner solutions
- If determinacy holds and 1 u k π ρ <, there is an interior optimum with finite optimal π α .
- If 1 u k π ρ >, the partial derivative is negative for any values and the optimal solution is π α = +∞  (welfare loss bounded below by zero).
- For the interior solution when 1 u k π ρ > (case examined), the optimal inflation coefficient satisfies:
  - 1; 1 e y y k u k opt u k ππ π ρ α α ρ + = − 
- When 1 u k π ρ < (other regime), optimal π α  → +∞ and opt θ α  satisfies a limiting relation:
  - 2 2 1(/) (1) lim()1(/) (1) e uy yu opt e uu kk kk π α θ π θ θ ρ βρ ααα ρβρ →+∞ +− = ++−  
- Resulting optimal unconventional-tool coefficient in that limit:
  - 2 2 1 (1) ,; (1) 1 e y y e k u k optopt u k u u k θ θ π θ θ ρ βρ ααα βρ ρ + − = +∞ = − + 

### Determinacy of optimal preferences
- If social preferences { π θ α α } are on or above the determinacy frontier, the optimal preferences chosen by the central banker are also determinate.
- Frontier concavity argument: since opt π π α α ≥ , if opt θ θ α α ≤  then determinacy is preserved.
- Even when the unconventional instrument is less forward-looking than output ((/)(/) ee yy kkkk θθ >), the slope S of the optimal deviation compared to the frontier derivative shows the optimal preferences lie in the determinacy area under the stated inequalities.

### IV–V. Broader implications and conclusion (policy relevance)
- Tinbergen principle reminder: need as many independent instruments as independent objectives. In New Keynesian models with divine coincidence, one instrument suffices, but cost-push shocks break divine coincidence.
- When policy rate is sole instrument, optimal response is to maintain a positive output gap while inflation is below target.
- Crisis-era developments motivated central banks to use additional instruments: balance sheet operations (quantitative easing), sterilized FX intervention, macroprudential policy, fiscal devaluations, etc.
- Main contributions of the paper:
  - Additional instruments can ensure equilibrium determinacy and reduce volatility in presence of cost-push shocks.
  - Under some parameterizations, committing not to use an unconventional instrument can be optimal (intertemporal constraint implication).
  - If future use of the unconventional instrument matters relatively more for inflation than for future output gap, the optimal central banker will be more interventionist with that instrument than social preferences imply (conservative central banker intuition extended).
- Policy-relevant extensions highlighted:
  - Incorporate explicitly stochastic setups and non-linear dynamics (relevant for financial stability problems, abrupt transitions, regime-switching).
  - Empirical task: improve knowledge of the Phillips Curve shape and the impact of unconventional instruments on economic activity and inflation.

### Appendix — methodological highlights and constrained optimization results
- Planning problem and first-order conditions laid out in Appendix A.1: central banker minimizes a welfare functional with parameters α, β, ρ, k, etc., subject to structural constraints.
- Definitions: N, D, W functions used to express derivatives and interior solution conditions (determinacy requires 0 D >).
- Solution characterization:
  - When both partial derivatives change sign once, a unique interior global minimum exists.
  - When the unconstrained optimal preferences yield indeterminacy (inequality (13)), the constrained problem solution lies on the determinacy frontier (boundary) and is unique.
- Constrained-optimum properties:
  - Frontier parametrization: fr θ π θ α α = 1 a b +   with a and b defined in the appendix.
  - Welfare loss along the frontier and along rays from origin analyzed; uniqueness of constrained optimum shown through algebraic conditions.
  - Comparison of constrained vs. unconstrained optima: constrained optimal inflation weight is always above the unconstrained optimal when the unconstrained preferences would be indeterminate.
  - For the unconventional instrument, whether the constrained optimum uses it more or less aggressively depends on the ratio of forward-looking impacts; if the unconventional instrument is more forward-looking than output (ratio < 1) the constrained optimum uses it more aggressively, and vice versa.

*Source: IMF working paper chapter (Appendix and Proposition statements and proofs reproduced from the provided content).*

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