## _wp12101

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---

### Introduction and key finding
- Context: Literature on macro-prudential regulation and pecuniary externalities motivates study of institutional design for macro-prudential policy (e.g. Bianchi (2010); Bianchi and Mendoza (2010); Jeanne and Korinek (2010, 2011)).
- Central question: Whether assigning the central bank formal responsibility for financial stability is socially optimal, given monetary–macroprudential interactions and time-inconsistency.
- Central result:
  - A dual-mandate central bank (price and financial stability) can be socially suboptimal because of time inconsistency: ex-ante optimal inflation is not optimal ex-post when the central bank can “fine-tune” monetary policy after credit shocks, creating incentives to use inflation to reduce the real burden of private debt.
  - If institutions are politically independent, separation or combination of objectives can achieve the social optimum.
  - If institutions are not politically independent, neither separate nor combined arrangements necessarily deliver the social optimum; welfare comparisons depend on parameter values.

### Model setup (loss function, policy tools, timing)
- Loss function:
  - L = a/2 (y − y^*)^2 + b/2 (π − π^*)^2 + c/2 (φ − φ^*)^2, where a > 0, b > 0, c > 0.
- Output (augmented Lucas supply):
  - y = y + α(π − π^e) + βδ.
- Credit growth decomposition:
  - δ = δ_0 + ε, with E[ε] = 0 and variance σ^2_ε.
- Inflation decomposition:
  - π = π_0 + γδ.
- Expectations:
  - π^e = π^e_0 + γδ^e = π^e_0 + γδ^e_0.
- Policy timing:
  - Two-stage decision: first stage simultaneous choices of π_0 and δ_0 before realization of ε; second stage monetary policy π_0 can be revised after realization of ε, while δ_0 cannot.
- Leverage and log-linear approximation:
  - Φ = D/PY, δ is rate of change in nominal debt; log leverage: ln(Φ) = φ ≈ φ − (π − π^e) − (y − y^e) + δ.

### Social planner benchmark (commitment)
- Commitment expectations:
  - δ^e_0 = δ_0 and π^e_0 = π_0.
- First-order-condition solutions:
  - δ^*_0 = (aβ(y^* − y) + c(φ^* − φ)) / (c + aβ^2).
  - π^*_0 = π^* − γδ^*_0.
- Social planner equilibrium (with π^* = 0, φ^* > 0):
  - y_SP = y + β(δ^*_0 + ε).
  - π_SP = π^*.
  - δ^*_0 = φ^* − φ.
  - φ_SP = φ + δ^*_0 + (1 − β)ε.
- Expected welfare loss under social planner:
  - L_SP = σ^2/2 (aβ^2 + c(1 − β)^2).

### Time inconsistency in a dual-mandate central bank (DUCB)
- DUCB objective (discretion, takes private expectations as given):
  - Minimize E[b/2 π^2 + c/2 (φ − φ^*)^2].
- First-stage (with rational expectations) yields same first-stage solutions as social planner:
  - π_0 = −γδ_0.
  - δ_0 = φ^* − φ.
- Second-stage (after ε realized) DUCB first-order condition:
  - bπ = c(1 + α)(φ − φ^*).
- Private inflation equilibrium response to shock:
  - π = [c(1 + α)(1 − β) / (b + c(1 + α)^2)] ε = ω ε,
    where ω = c(1 + α)(1 − β) / (b + c(1 + α)^2).
- DUCB monetary response relative to social planner:
  - π^DUCB_0 = −γδ_0 − γε + ωε = π^SP_0 + ωε.
- Interpretation:
  - DUCB allows positive inflation ex-post (ω ε > 0 for β < 1) to reduce real private debt burden; it under-responds to inflationary pressures compared with the social planner.
- Comparative statics on ω:
  - dω / d(b/c) = −(1 + α)(1 − β) / [ (b/c + (1 + α)^2)^2 ] < 0 (for β < 1): greater relative weight on price stability lowers ω.
- DUCB equilibrium outcomes:
  - y_DUCB = y^* + βε + αωε = y_SP + αωε.
  - π_DUCB = ωε.
  - φ_DUCB = φ^* + (1 − β)ε − ε ω (1 + α).
- Expected welfare loss under DUCB:
  - L_DUCB = a σ^2 ω^2 (2 a α β + ω(a α^2 + b) + c(1 + α)(2(1 − β) + (1 + α))) + L_SP.
- Implication: welfare loss is positively related to σ^2 (variance of credit shocks).

### Separation of objectives achieves social optimum
- Institutional arrangement: central bank handles price stability only; macro-prudential regulator handles financial stability only (independent agencies).
- Objectives:
  - Central bank: minimize E[b/2 π^2].
  - Macro-prudential regulator: minimize E[c/2 (φ − φ^*)^2].
- First-order conditions yield social optimum:
  - E[π] = 0.
  - π_0 = −γδ_0.
  - δ_0 = φ^* − φ (assuming 1 − β ≠ γ(1 + α)).
- Second-stage monetary revision:
  - π = 0 and π_0 = −γ(δ_0 + ε).
- Equilibrium matches social planner:
  - y = y_SP.
  - π = 0.
  - φ = φ_SP.
- Robustness: result holds under a Stackelberg (leader–follower) specification.

### Role of political independence (Barro and Gordon style extension)
- Government target: stabilize output at y^* + k with k > 0 representing political pressure to generate expansions.
- Main message: political independence of institutions is sufficient for achieving the social optimum; if institutions are not politically independent, suboptimal outcomes arise regardless of separation vs. combination of objectives.

A. Non-independent central bank (NICB) with independent macro-prudential regulator
- NICB objective (first stage, set ε = 0 for focus on political distortion):
  - Minimize a/2 (y − y^* − k)^2 + b/2 π^2.
- Macro-prudential regulator objective:
  - Minimize c/2 (φ − φ^*)^2.
- Equilibrium:
  - y_NICB = y^*.
  - π_NICB = a α k / b.
  - φ_NICB = φ^*.
- Welfare loss due to political bias:
  - L_NICB = a^2 k^2 α^2 / 2 b.

B. Non-independent macro-prudential regulator (NIMR) with independent central bank
- Central bank objective: minimize b/2 π^2.
- NIMR objective: minimize a/2 (y − y^* − k)^2 + c/2 (φ − φ^*)^2.
- First-order conditions yield:
  - π_NIMR = 0.
  - Equilibrium output, leverage:
    - y_NIMR = y^* + β k / [ β + (c/a) (1 − γ / (β + αγ) − 1 ) ].
    - φ_NIMR = φ^* + k / [ β + (c/a) (1 − γ / (β + αγ) − 1 ) ].
- Sufficient condition for higher-than-optimal debt and output:
  - dφ / dδ_0 = 1 − β − γ(1 + α) > 0 (leverage increases with looser macro-prudential regulation).
- Conclusion: lack of political independence in either institution generates welfare losses; ensuring political independence is central if political interference is the primary distortion.

### Analytical results on political pressures, regulation, and monetary response
- Non-independent macro-prudential regulator (NIMR) first-order-condition solutions (as stated):
  - δ0 = δ*0 + k β + c a (1 − γ β + αγ −1)
  - π0 = π*0 − γ k β + c a (1 − γ β + αγ −1)
- Interpretation:
  - An un-committed macro-prudential regulator loosens macro-prudential regulation because she expects higher inflation and output to partially undo increased debt.
  - The monetary authority must respond more strongly to contain inflation given regulatory laxity.
  - The anticipated increase in inflation does not materialize because the monetary authority responds to achieve the social optimal inflation level.
  - Consequence: debt and output are excessive; the central bank attains the socially optimal inflation rate but by maintaining a tighter stance than the social planner would.
- Overall loss for NIMR:
  - L_NIMR = (aβ^2 + c) k^2 / 2 ( β + c a (1 − γ β + αγ −1) )^2 . (Equation (28))

### Welfare comparisons across institutional arrangements (qualitative)
- Under Debt Monetization Incentives:
  - Independent CB, Independent MR: 0 welfare loss.
  - Dual Mandate Central Bank (DUCB): Loss.
- Under Expansionary Political Pressures:
  - Independent CB, Independent MR: 0 welfare loss.
  - Non-Independent CB, Independent MR (NICB): Loss.
  - Non-Independent MR, Independent CB (NIMR): Loss.
  - Independent and Dual Mandate CB (Financial and Price Stability): 0 welfare loss.
- Key implication:
  - If both political pressures to generate expansions and ex-post debt monetization incentives are present, only a regime with politically independent and separate agencies delivers the social optimum.

### Numerical calibration and ranking insights
- Calibration parameters used for quantitative welfare comparisons:
  - Welfare-function weights: a = 0.63, b = 173, c = 0.17 (from De Paoli and Paustian (2011)).
  - Alternate equal-weights case: a = b = c = 1/3.
  - Additional parameters: k = 0.01, σ = 0.01, α = 35 (Dixit and Lambertini (2001)); robustness also with α = 5.
- Figure 1 setup (welfare loss as function of β and γ):
  - Solid line = NICOM (Non-Independent CB with Dual Mandate).
  - Dashed line = NICB (Non-Independent CB, Independent MR).
  - Dotted line = NIMR (Non-Independent MR, Independent CB).
- Numerical ranking insights:
  - Calibrated weights (a = 0.63, b = 173, c = 0.17):
    - For γ closer to zero and β < 0.5, NIMR dominates NICB and NICOM.
    - For high values of β, NIMR still dominates NICOM.
    - Only when γ = 1 or γ = −1 is NIMR dominated by NICOM and NICB; for high β, NIMR again dominates NICOM.
  - Equal weights (a = b = c = 1/3):
    - NIMR dominates NICOM and NICB for the values of γ considered and for all but the highest values of β in the analyzed range.
  - Overall takeaway:
    - Which suboptimal institutional regime dominates depends on parameters β, γ, and weight calibration.
    - In almost all cases, a non-independent macro-prudential regulator (with independent monetary authority) dominates a non-independent single authority (NICOM). Thus, when institutions are not politically independent, having separate institutions may be preferable to a single institution in charge of both price and financial stability.

### Main conclusions
- Model features:
  - Monetary policy can re-optimize after credit shocks; macro-prudential regulation cannot be adjusted immediately after the credit shock (macro-prudential regulation is an ex-ante tool; monetary policy can be used ex-ante and ex-post).
- Central bank with price and financial stability mandate (dual mandate) does not deliver the social optimum because of time-inconsistency:
  - Ex-ante the central bank finds it optimal to deliver the social optimal level of inflation; ex-post it lets inflation rise to repair private balance sheets because only monetary policy is available ex-post.
  - Achieving the social optimum in this case requires separating price and financial stability objectives.
- Political independence matters:
  - Separation of price and financial stability objectives delivers the social optimum only if both institutions (central bank and macro-prudential regulator) are politically independent.
  - If either institution is not politically independent, the social optimum is not achieved.
  - Numerical analysis suggests that in most cases a non-independent macro-prudential regulator (with independent monetary authority) delivers a better outcome than a non-independent central bank in charge of both price and financial stability.

### Appendix highlights (selected closed-form expressions)
- Non-independent single authority (NICOM) equilibrium:
  - π_NICOM = a c k (α + β) / b ( c (1 − β) + a β^2 ).
  - δ_COM = δ* + a β k / b ( c (1 − β) + a β^2 ).
  - φ_COM = φ* + β a k / ( c (1 − β) + a β^2 ).
  - y_COM = y* + β^2 a k / ( c (1 − β) + a β^2 ).
  - Welfare loss: L_COM = 1 / 2 b ( a k / ( c (1 − β) + a β^2 ) )^2 ( (α + β)^2 + b β^2 (a + c) ). (Equation (A2))
- Distortionary macro-prudential regulation (social planner):
  - L = a/2 (y − y*)^2 + b/2 (π − π*)^2 + c/2 (φ − φ*)^2 + χ δ0, with χ < 0.
  - δ_SP0 = ( a β ( y* − y ) + c ( φ* − φ ) − χ ) / ( c + a β^2 ). (Equation (A2))
  - π_SP0 = π* − γ δ_SP0 . (Equation (A3))
- Independent agencies with distortion (IND3−Dist):
  - y_IND3−Dist = y* + β χ / c ( 1 / (1 − β − γ (1 + α)) ).
  - π_IND3−Dist = 0.
  - φ_IND3−Dist = φ* − χ / c ( 1 / (1 − β − γ (1 + α)) ).
  - Implication: independent agencies achieve socially optimal inflation but output and debt deviate from social optimum under 1 − β − γ (1 + α) > 0.
- Dual-mandate central bank with distortion (IND2b−Dist):
  - y_IND2b−Dist = y* − β χ / c ( 1 / (1 − β) ).
  - π_IND2b−Dist = − (1 + α) χ / b ( 1 / (1 − β) ).
  - φ_IND2b−Dist = φ* − χ / c ( 1 / (1 − β) ).
  - Conclusion: dual mandate does not deliver the social optimum under distortionary macro-prudential regulation; relative superiority of separation versus dual mandate depends on magnitude of distortions and model parameters.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12101.pdf*

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

### _wp12101 - References .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .

### Introduction and key finding
- Context: Literature on macro-prudential regulation and pecuniary externalities (e.g. Bianchi (2010), Bianchi and Mendoza (2010), Jeanne and Korinek (2010, 2011)) motivates study of institutional design for macro-prudential policy.
- Central question: Whether assigning the central bank formal responsibility for financial stability is socially optimal, given potential monetary–macroprudential interactions and time-inconsistency problems.
- Central result: A dual-mandate central bank (price and financial stability) can be socially suboptimal because of time inconsistency: ex-ante optimal inflation is not optimal ex-post when the central bank can “fine-tune” monetary policy after credit shocks, creating incentives to use inflation to reduce the real burden of private debt.
- Political independence result: If institutions are politically independent, separation or combination of objectives can achieve the social optimum. If institutions are not politically independent, neither separate nor combined arrangements necessarily deliver the social optimum; welfare comparisons depend on parameter values.

### Model setup (loss function, policy tools, timing)
- Loss function (from Carlstrom, Fuerst and Paustian (2010)):
  - L = a/2 (y − y^*)^2 + b/2 (π − π^*)^2 + c/2 (φ − φ^*)^2, where a > 0, b > 0, c > 0.
- Output (augmented Lucas supply):
  - y = y + α(π − π^e) + βδ.
- Credit growth decomposition:
  - δ = δ_0 + ε, with E[ε] = 0 and variance σ^2_ε.
- Inflation decomposition:
  - π = π_0 + γδ.
- Expectations:
  - π^e = π^e_0 + γδ^e = π^e_0 + γδ^e_0.
- Policy tools and timing:
  - Two-stage decision: first stage simultaneous choices of π_0 and δ_0 before realization of ε; second stage monetary policy π_0 can be revised after realization of ε, while δ_0 cannot.
- Leverage and log-linear approximation:
  - Φ = D/PY, δ is rate of change in nominal debt; log leverage: ln(Φ) = φ ≈ φ − (π − π^e) − (y − y^e) + δ.

### Social planner benchmark (commitment)
- Commitment: δ^e_0 = δ_0 and π^e_0 = π_0.
- First-order conditions yield:
  - δ^*_0 = (aβ(y^* − y) + c(φ^* − φ)) / (c + aβ^2).
  - π^*_0 = π^* − γδ^*_0.
- Social planner equilibrium (with π^* = 0, φ^* > 0):
  - y_SP = y + β(δ^*_0 + ε),
  - π_SP = π^*,
  - δ^*_0 = φ^* − φ,
  - φ_SP = φ + δ^*_0 + (1 − β)ε.
- Expected welfare loss under social planner:
  - L_SP = σ^2/2 (aβ^2 + c(1 − β)^2).

### Time inconsistency in a dual-mandate central bank (DUCB)
- DUCB objectives (discretion, takes private expectations as given):
  - Minimize expected b/2 π^2 + c/2 (φ − φ^*)^2.
- First-stage (with rational expectations) delivers same first-stage solutions as social planner:
  - π_0 = −γδ_0,
  - δ_0 = φ^* − φ.
- Second-stage (after ε realized): DUCB first-order condition implies
  - bπ = c(1 + α)(φ − φ^*).
- Solving yields private inflation equilibrium response to shock:
  - π = [c(1 + α)(1 − β) / (b + c(1 + α)^2)] ε = ω ε, where ω = c(1 + α)(1 − β) / (b + c(1 + α)^2).
- DUCB monetary response relative to social planner:
  - π^DUCB_0 = −γδ_0 − γε + ωε = π^SP_0 + ωε.
- Interpretation: DUCB allows positive inflation ex-post (ω ε > 0 for β < 1) to reduce real private debt burden; it under-responds to inflationary pressures compared with the social planner.
- Comparative statics on ω:
  - dω / d(b/c) = −(1 + α)(1 − β) / [ (b/c + (1 + α)^2)^2 ] < 0 (for β < 1): greater relative weight on price stability lowers ω.
- DUCB equilibrium outcomes:
  - y_DUCB = y^* + βε + αωε = y_SP + αωε,
  - π_DUCB = ωε,
  - φ_DUCB = φ^* + (1 − β)ε − ε ω (1 + α).
- Expected welfare loss under DUCB:
  - L_DUCB = a σ^2 ω^2 (2 a α β + ω(a α^2 + b) + c(1 + α)(2(1 − β) + (1 + α))) + L_SP.
- Implication: welfare loss is positively related to σ^2 (variance of credit shocks).

### Separation of objectives achieves social optimum
- Institutional arrangement: central bank handles price stability only; macro-prudential regulator handles financial stability only (independent agencies).
- Central bank objective: minimize E[b/2 π^2]; macro-prudential regulator objective: minimize E[c/2 (φ − φ^*)^2].
- First-order conditions yield social optimum:
  - E[π] = 0,
  - π_0 = −γδ_0,
  - δ_0 = φ^* − φ (assuming 1 − β ≠ γ(1 + α)).
- Second-stage monetary revision yields π = 0 and π_0 = −γ(δ_0 + ε).
- Equilibrium matches social planner:
  - y = y_SP,
  - π = 0,
  - φ = φ_SP.
- Robustness: result holds under Stackelberg (leader–follower) specification.

### Role of political independence (Barro and Gordon style extension)
- Government target: stabilize output at y^* + k with k > 0 representing political pressure to generate expansions. Social planner has k = 0.
- Focus here is on political interference only (abstracting from time inconsistency).
- Main message: political independence of institutions is sufficient for achieving the social optimum; if institutions are not politically independent, suboptimal outcomes arise regardless of separation vs. combination of objectives.

A. Non-independent central bank (NICB) with independent macro-prudential regulator
- NICB objective (first stage, set ε = 0 for focus on political distortion):
  - Minimize a/2 (y − y^* − k)^2 + b/2 π^2.
- Macro-prudential regulator objective:
  - Minimize c/2 (φ − φ^*)^2.
- Equilibrium under NICB:
  - y_NICB = y^*,
  - π_NICB = a α k / b,
  - φ_NICB = φ^*.
- Welfare loss due to political bias:
  - L_NICB = a^2 k^2 α^2 / 2 b.

B. Non-independent macro-prudential regulator (NIMR) with independent central bank
- Central bank objective: minimize b/2 π^2.
- NIMR objective: minimize a/2 (y − y^* − k)^2 + c/2 (φ − φ^*)^2.
- First-order conditions yield:
  - π_NIMR = 0.
  - Equilibrium output, leverage:
    - y_NIMR = y^* + β k / [ β + (c/a) (1 − γ / (β + αγ) − 1 ) ],
    - φ_NIMR = φ^* + k / [ β + (c/a) (1 − γ / (β + αγ) − 1 ) ].
- Sufficient condition for higher-than-optimal debt and output: dφ / dδ_0 = 1 − β − γ(1 + α) > 0 (interpreted as leverage increasing with looser macro-prudential regulation).
- Conclusion: lack of political independence in either institution generates welfare losses; ensuring political independence is central if political interference is the primary distortion.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12101.pdf*

### conclusions follow if we assume that political influences appear in the form of generating higher credit availability so

### _wp12101 - conclusions follow if we assume that political influences appear in the form of generating higher credit availability so

### Key analytical results on political pressures, regulation, and monetary response
- First-order-condition solution for a non-independent macro-prudential regulator (NIMR) alters policy choices:
  - δ0 = δ*0 + k β + c a (1 − γ β + αγ −1)
  - π0 = π*0 − γ k β + c a (1 − γ β + αγ −1)
- Interpretation:
  - An un-committed macro-prudential regulator loosens macro-prudential regulation because she expects higher inflation and output to partially undo increased debt.
  - The monetary authority must respond more strongly to contain inflation given regulatory laxity.
  - The anticipated increase in inflation does not materialize because the monetary authority responds to achieve the social optimal inflation level.
  - Consequence: debt and output are excessive; the central bank attains the socially optimal inflation rate but by maintaining a tighter stance than the social planner would.
- Overall loss for the non-independent macro-prudential regulator case:
  - L_NIMR = (aβ^2 + c) k^2 / 2 ( β + c a (1 − γ β + αγ −1) )^2 . (Equation (28))

### Welfare comparisons across institutional arrangements (qualitative summary from Table 1)
- Under Debt Monetization Incentives:
  - Independent CB, Independent MR: 0 welfare loss
  - Dual Mandate Central Bank (Price and Financial Stability) (DUCB): Loss
- Under Expansionary Political Pressures:
  - Independent CB, Independent MR: 0 welfare loss
  - Non-Independent CB, Independent MR (NICB): Loss
  - Non-Independent MR, Independent CB (NIMR): Loss
  - Independent and Dual Mandate CB (Financial and Price Stability): 0 welfare loss
- Key qualitative implication:
  - If both political pressures to generate expansions and ex-post debt monetization incentives are present, only a regime with politically independent and separate agencies delivers the social optimum.

### Numerical calibration and parameter choices used for quantitative welfare comparisons
- Welfare-function weights (from De Paoli and Paustian (2011)):
  - a = 0.63, b = 173, c = 0.17
- Alternate case examined: equal weights (a = b = c = 1/3)
- Additional parameters:
  - k = 0.01
  - σ = 0.01
  - α = 35 (from Dixit and Lambertini (2001)); robustness note: analysis also conducted with α = 5 — changes levels of losses but not ranking
- Figure 1 setup:
  - Welfare loss plotted as function of β and γ
  - Solid line = NICOM (Non-Independent CB with Dual Mandate)
  - Dashed line = NICB (Non-Independent CB, Independent MR)
  - Dotted line = NIMR (Non-Independent MR, Independent CB)
  - Top three graphs: calibrated weights; bottom three: equal weights

### Numerical ranking insights from the calibrated and equal-weight exercises
- Calibrated weights (a = 0.63, b = 173, c = 0.17):
  - For γ closer to zero and β < 0.5, NIMR dominates NICB and NICOM.
  - For high values of β, NIMR still dominates NICOM.
  - Only when macro-prudential regulation has a strong positive or negative impact on inflation (γ equals 1 or -1) is NIMR dominated by NICOM and NICB; for high β, NIMR again dominates NICOM.
- Equal weights (a = b = c = 1/3):
  - NIMR dominates NICOM and NICB for the values of γ considered and for all but the highest values of β in the analyzed range.
- Overall takeaway:
  - Which suboptimal institutional regime dominates depends on parameter values (β, γ, and weight calibration).
  - In almost all cases, a non-independent macro-prudential regulator (with independent monetary authority) dominates a non-independent single authority (NICOM). Thus, when institutions are not politically independent, having separate institutions may be preferable to a single institution in charge of both price and financial stability.

### Main conclusions (section VIII)
- Model features:
  - Monetary policy can re-optimize after credit shocks; macro-prudential regulation cannot be adjusted immediately after the credit shock (macro-prudential regulation is an ex-ante tool; monetary policy can be used ex-ante and ex-post).
- Central bank with price and financial stability mandate (dual mandate) does not deliver the social optimum because of time-inconsistency:
  - Ex-ante the central bank finds it optimal to deliver the social optimal level of inflation; ex-post it lets inflation rise to repair private balance sheets because only monetary policy is available ex-post.
  - Achieving the social optimum in this case requires separating price and financial stability objectives.
- Political independence matters:
  - Separation of price and financial stability objectives delivers the social optimum only if both institutions (central bank and macro-prudential regulator) are politically independent.
  - If either institution is not politically independent, the social optimum is not achieved.
  - Numerical analysis suggests that in most cases a non-independent macro-prudential regulator (with independent monetary authority) delivers a better outcome than a non-independent central bank in charge of both price and financial stability.

### Appendix I — Non-independent single authority (NICOM): closed-form equilibrium expressions
- Authority minimizes:
  - L = a/2 (y − y* − k)^2 + b/2 π^2 + c/2 (φ − φ*)^2, subject to model constraints
- Equilibrium inflation (NICOM):
  - π_NICOM = a c k (α + β) / b ( c (1 − β) + a β^2 )
- Equilibrium credit growth:
  - δ_COM = δ* + a β k / b ( c (1 − β) + a β^2 )
- Equilibrium debt:
  - φ_COM = φ* + β a k / ( c (1 − β) + a β^2 )
- Equilibrium output:
  - y_COM = y* + β^2 a k / ( c (1 − β) + a β^2 )
- Welfare loss for NICOM:
  - L_COM = 1 / 2 b ( a k / ( c (1 − β) + a β^2 ) )^2 ( (α + β)^2 + b β^2 (a + c) ). (Equation (A2))

### Appendix II — Distortionary macro-prudential regulation: social planner and decentralized agencies
- Social planner problem (with distortionary term χ δ0, χ < 0 to reflect that macro-prudential tightening exacerbates distortion):
  - L = a/2 (y − y*)^2 + b/2 (π − π*)^2 + c/2 (φ − φ*)^2 + χ δ0
  - Constraints: y = y + β δ0 ; π = π0 + γ δ0 ; φ = φ + δ0
- Social planner solutions:
  - δ_SP0 = ( a β ( y* − y ) + c ( φ* − φ ) − χ ) / ( c + a β^2 ). (Equation (A2))
  - π_SP0 = π* − γ δ_SP0 . (Equation (A3))
  - y_SP = y + β δ_SP0
  - π_SP = π*
  - φ_SP = φ + δ_SP0 = φ* − χ / ( a β^2 + c )
- Three independent agencies (IND3−Dist):
  - Central bank objective: minimize b/2 π^2
  - Macro-prudential regulator objective: minimize c/2 (φ − φ*)^2 + χ δ0
  - Equilibrium:
    - y_IND3−Dist = y* + β χ / c ( 1 / (1 − β − γ (1 + α)) )
    - π_IND3−Dist = 0
    - φ_IND3−Dist = φ* − χ / c ( 1 / (1 − β − γ (1 + α)) )
  - Implication: independent agencies achieve socially optimal inflation but output and debt deviate from social optimum; macro-prudential regulator over-compensates due to anticipating private-sector responses, yielding too much debt and output under assumption 1 − β − γ (1 + α) > 0.
- Dual-mandate central bank with distortion (IND2b−Dist):
  - Central bank minimizes b/2 π^2 + c/2 (φ − φ*)^2 + χ δ0
  - Equilibrium:
    - y_IND2b−Dist = y* − β χ / c ( 1 / (1 − β) )
    - π_IND2b−Dist = − (1 + α) χ / b ( 1 / (1 − β) )
    - φ_IND2b−Dist = φ* − χ / c ( 1 / (1 − β) )
  - Conclusion: dual mandate does not deliver the social optimum under distortionary macro-prudential regulation; relative superiority of separation versus dual mandate depends on magnitude of distortions and model parameters.

*Source: _wp12101 - conclusions follow if we assume that political influences appear in the form of generating higher credit availability so*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp12101.pdf_