## 1. Final good producers

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---

### Final good production technology and price
- Final goods YF t combine tradable intermediate input YDT t and non-tradable intermediate input YDN t via a CES function:
  - YF t = { βY 1/ηY (YDT t)^(ηY−1)/ηY + (1−βY) 1/ηY (YDN t)^(ηY−1)/ηY }^(ηY/(ηY−1)).
  - Parameters: βY is the share of tradable inputs; ηY is the elasticity of substitution between tradable and non-tradable inputs.
- Final good price:
  - PF t = { βY (P T t)^(1−ηY) + (1−βY) (P N t)^(1−ηY) }^(1/(1−ηY)).
  - P T t and P N t are prices of tradable and non-tradable inputs, respectively.
- Key calibrated values:
  - βY = 0:50 (Share of Tradable Inputs - Final Good Sector)
  - ηY = 0:50 (Elasticity of Substitution - Final Good Sector)

### Intermediate good producers and retailers
- Sectoral production (J = T, N):
  - YJ t = AJ t [ KJ t^αJ (LJ t)^(1−αJ) ].
  - Capital shares:
    - αN = 0:30 (Capital Share - Non-tradable Sector)
    - αT = 0:40 (Capital Share - Tradable Goods Sector)
- Retailing in the non-tradable sector (two-stage):
  - Assembler CES aggregation with elasticity εN and demand YDN t(j) = ( P N t(j) / P N t )^(−εN) YDN t.
  - Price index: P N t = { ∫_0^1 P N t(j)^(1−εN) dj }^(1/(1−εN)).
  - Calvo price setting: fraction (1−θN) reset optimally each period; θN is Calvo survival probability.
  - Optimal price P N* t maximizes E_t [ ∑_{i=0}^∞ θN^i Λ_{t;t+i} (P N* t − P WN t+i) YN t+i(j) ], where Λ_{t;t+i} = β^i (C_t / C_{t+i}) (P F t / P F t+i).
  - Aggregate price evolution:
    - P N t = { θN P N t−1^(1−ςp) + (1−θN) P N* t^(1−ςp) }^(1/(1−ςp)).
  - Calibrated values:
    - εN = 6 (Elasticity of Substitution - Non-tradable Sector; implies average markup 20 percent)
    - θN = 0:75 (Calvo Parameter - Non-tradable Sector)
    - ςp is the Calvo pricing curvature parameter as used in price aggregation.

### Capital producers (sector-specific)
- Representative firm maximization and capital law of motion:
  - VJ t = max_{KJ_{t+i}, IJ_{t+i}} E_t ∑_{i=0}^∞ Λ_{t;t+i} ( R^K J,t+i KJ_{t+i} − P F t+i I J_{t+i} )
  - KJ_{t+1} = (1−δ) KJ t + S( I J t / I J_{t−1} ) I J t
- Calibration and assumptions:
  - δ = 0:02 (Depreciation Rate)
  - S(1) = 1; S′(1) = 0; S″(1) = −κS < 0 with κS = 2:5 (Investment adjustment cost elasticity, consistent with Christiano et al. (2005)).

### Monetary and FX reserves policies
- Central bank budget constraint:
  - E_t F* t − B t = E_t F*_{t−1} (1 + i*_{t−1}) − B_{t−1} (1 + i_{t−1}) − T_t.
  - Sterilized FX intervention: E_t F* t = −B t.
  - Interest income and debt servicing flows remitted to households via lump-sum transfers T_t.
- Policy rules:
  1. Taylor-type interest rate rule (central bank type 1):
     - (1 + i_t)/(1 + i) = ( (1 + i_{t−1})/(1 + i) )^ρi × ( Y_t / Y )^{(1−ρi) y} × ( π_t / π )^{(1−ρi) φ} × exp( εmp;t ).
     - Calibrated values: ρi = 0:7; φ = 1:5; y = 0:5.
  2. FX reserves accumulation rule:
     - F* t / F* = ( (1 + i*_{t})/(1 + i*) )^{ζi*} × exp( εfx;t ).
     - Calibrated value: ζi* = 9:7 (Foreign Interest Rate Coefficient - FXI Rule)
- Foreign interest rate shock:
  - i* follows AR(1) with persistence ρ_{i*} = 0:95.

### Market clearing and calibration (baseline)
- Market clearing:
  - Labor: L_t = L N t + L T t.
  - Non-tradable goods: YDN t / ςN t = Y N t.
  - Aggregate domestic demand: YF t = C_t + I T t + I N t.
  - Law of one price for tradables: P T t = E_t P* t.
  - Balance of payments (aggregate): E_t (B* t + F* t) = (1 + i*_{t−1}) [ −B*_{t−1} − E_t B*_{t−1} + E_t F*_{t−1} ] + P T t Y T t − P T t Y DT t.
- Calibration (baseline, quarterly):
  - β = 0:99 (discount factor)
  - Household preferences: u(C_t, L_t) = log(C_t) − ϕ (L_t)^{1+ξ}/(1+ξ), with ξ = 5/3.
  - Wage rigidity: ζw = 0:875 (half-life ≈ 5 quarters).
  - Investment adjustment cost elasticity: κS = 2:5.
  - Risk premium elasticity: % = 0:2.
  - Interest rate smoothing: ρi = 0:7.
  - Foreign interest rate coefficient in Taylor rule: i* = 1.
  - Persistence of foreign interest rate: ρ_{i*} = 0:95.

---

### III. Capital inflows and FX intervention

### Experiment setup and regimes
- Shock: capital inflow shock modeled as a 1 percent drop in the foreign interest rate i* (i* falls by 1 percent).
- Two policy regimes:
  - One instrument: monetary policy only (ζi* = 0).
  - Two instruments: monetary policy + FX intervention (ζi* > 0), with ζi* chosen to minimize L = var(y_t) + var(π_t).

### A. Full information (central bank type 1)
- One instrument (ζi* = 0) — effects:
  - Foreign interest rate drop → nominal exchange rate appreciation → lower headline inflation.
  - Resources reallocated from tradable to non-tradable sector.
  - Short-lived boost in GDP followed by contraction as tradable production declines with appreciation.
  - Monetary policy rate falls but cannot stabilize output and inflation simultaneously; output and inflation move in opposite directions.
  - Transitory decline in current account due to drop in tradable production and higher imports.
- Two instruments (optimal ζi* > 0) — effects relative to one instrument:
  - Accumulation of FX reserves engineers a real exchange rate depreciation relative to the one-instrument case.
  - Boosts tradable output.
  - Improves current account.
  - Raises inflation.
  - Outcome: two instruments achieve better stabilization of both output and inflation; macroeconomic volatility is unambiguously lower with two instruments under full information (Tinbergen principle).

### B. Policy uncertainty (agents uncertain about central bank type)
- Central bank type 2 (fear of floating) rule includes response to i* with coefficient ι* = 1; type 2 moves domestic rate with foreign rate to limit nominal appreciation.
- Learning and Bayesian inference:
  - Agents do not know central bank type; observe dev_mp;t (log deviation of interest rate rule).
  - dev_mp;t = ln( (1 + i_t)/(1 + i) ) − ρi ln( (1 + i_{t−1})/(1 + i) ) − (1−ρi) y ln( Y_t / Y ) − (1−ρi) φ ln( π_t / π ).
  - Agents assign probabilities pr1;t and pr2;t and use Bayesian updating with a Kalman Filter; prior pr1;0|0 = 0:5 (baseline).
- Effects under policy uncertainty:
  - With monetary policy only:
    - Inflation response is more muted under imperfect credibility (weaker nominal exchange rate movement); output response is more pronounced.
    - Expectations that central bank may be type 2 damp nominal appreciation; spike in inflation can induce nominal rate hike, but real rates fall more under imperfect credibility.
  - With monetary policy + FX intervention:
    - FX intervention mitigates output effects but raises inflation.
    - If agents expect fear-of-floating, households anticipate larger depreciation; nominal depreciation from FX intervention stabilizes output but amplifies inflationary effects because monetary policy lacks credibility to anchor inflation expectations.
    - Net effect under policy uncertainty: ambiguous benefits from FX intervention — trade-off between output stabilization and higher inflation volatility.
- Volatility outcomes (described):
  - Full information: moving from one instrument (point A) to two instruments (point B) unambiguously reduces output and inflation volatility.
  - Policy uncertainty: moving from one instrument (point C) to two instruments (point D) reduces output volatility but increases inflation volatility.
  - Gains from second instrument increase with prior probability pr1;0|0; for pr1;0|0 close to 1, adding FX intervention reduces loss L unambiguously; as pr1;0|0 falls, inflation and output volatility increase continuously.

---

### 2. Optimal FX Intervention

### Optimal intervention and credibility
- Optimal degree of FX intervention is increasing in central bank credibility (prior probability of being type 1).
- Greater credibility implies less inflationary effects from FX intervention and allows greater use of the instrument to stabilize output and inflation.
- Non-monotonicity near high pr1,0|0 reflects that some response to the foreign interest rate (implemented excessively by type 2) can be optimal when accounting for behavior and priors.

### Sensitivity analysis (parameters and representative values)
- Parameters explored: price stickiness N; real wage rigidity w; persistence of capital inflow shock i; asset substitutability %.
- Representative values:
  - N: 0:66, 0:75, 0:875
  - w: 0:66, 0:875, 0:95
  - i: 0:7, 0:95, 0:98
  - %: 0:1, 0:2, 0:3
- Selected robustness patterns (reported as in source tables):
  - Table A2 (price rigidities N) examples:
    - Full Credibility, 1 Instrument (MP): 0:26 (Low), 0:27 (Base), 0:29 (High)
    - Full Credibility, 2 Instruments (MP and FXI): 0:11, 0:11, 0:08
    - Imperfect Credibility, 2 Instruments, Imp. Cred. on MP: 0:17, 0:17, 0:22
    - Imperfect Credibility, 2 Instruments, Imp. Cred. on FXI: 0:15, 0:14, 0:12
  - Table A3 (wage rigidities w) examples:
    - Full Credibility, 1 Instrument (MP): 0:24, 0:27, 0:36
    - Full Credibility, 2 Instruments (MP and FXI): 0:11, 0:11, 0:13
    - Imperfect Credibility, 2 Instruments, Imp. Cred. on FXI: 0:11, 0:14, 0:19
  - Table A4 (persistence i) examples:
    - Full Credibility, 1 Instrument (MP): 0:21, 0:27, 0:33
    - Full Credibility, 2 Instruments (MP and FXI): 0:15, 0:11, 0:22
    - Imperfect Credibility, 2 Instruments, Imp. Cred. on FXI: 0:06, 0:14, 0:21
  - Table A5 (portfolio balance % ) examples:
    - Full Credibility, 1 Instrument (MP): 0:36, 0:27, 0:23
    - Full Credibility, 2 Instruments (MP and FXI): 0:19, 0:11, 0:16
    - Imperfect Credibility, 2 Instruments, Imp. Cred. on FXI: 0:16, 0:14, 0:22

### Exchange rate policy uncertainty (FXI rule uncertainty)
- Agents may be uncertain about whether FX intervention supports monetary policy or pursues a separate depreciation objective.
- Alternative perceived intervention rule (type 2):
  - Ft / F = [ (1 + it) / (1 + i) ]^{} exp(fx;t), with  > 1 indicating desire to depreciate beyond stabilization motives.
  - Benchmark:  = 2 (twice the optimal value under certainty); this induces accumulation of FX reserves of 5 percentage points of GDP.
- Learning about intervention rule:
  - devfx;t = ln( Ft / F )  i ln( (1 + it) / (1 + i ) ).
  - Agents update beliefs about FX rule over time via devfx;t.
- Results under FXI uncertainty:
  - Little difference between full and low credibility in benchmark because monetary policy (observed with certainty) offsets costs from misperception of FX intervention.
  - Sensitivity analysis: only in very low credibility cases may FX intervention be counterproductive.
  - Conclusion: uncertainty about conduct of FX intervention is of second order importance provided monetary policy credibly focuses on stabilizing inflation and output.

### Simulation with learning (mechanics)
- Log-linear solution conditional on type j:
  - bX t = P bX t1 + Q i;j b i t + Q mp " mp;t  (equation (28))
- Under limited credibility private sector inference:
  - bX t = P bX t1 + (  pr1;tjt Q i;1 + pr2;tjt Q i;2 ) b i t + Q mp " mp;tjt  (equation (29))
- Bayesian updating via Kalman Filter:
  - State vector [ pr1;tjt, b i t, pr2;tjt, b i t, " mp;tjt ]' updated with state-transition F, covariance Q, observation matrix H0.
  - Kalman gain Kg obtained as limiting value of iterative algorithm described in appendix.
- Learning implication:
  - Credibility problems prevent immediate full inference (pr1;tjt = 1 and pr2;tjt = " mp;tjt = 0); learning about central bank type is gradual.

---

### Conclusions and policy implications

- Under full information, FX intervention as an additional stabilization instrument unambiguously improves macroeconomic outcomes.
- Under uncertainty about monetary policy goals, FX intervention creates a trade-off: it stabilizes output but can destabilize inflation.
- Benefits of FX intervention decline with greater policy uncertainty; the optimal degree of intervention is increasing in central bank credibility (prior pr1;0|0).
- If monetary policy is credibly focused on stabilizing inflation and output, uncertainty about FX intervention conduct is of second order importance.
- Policy implication: enhancing the credibility of the monetary policy regime maximizes the stabilization benefits of using FX intervention as an additional policy instrument.

*Source: _wp1667 (PDF chapter/section).*

### 1.    Final good producers

### 1.    Final good producers

### Final good production technology and price
- Final goods YF t combine tradable intermediate input YDT t and non-tradable intermediate input YDN t via a CES function:
  - YF t = { βY 1/ηY (YDT t)^(ηY−1)/ηY + (1−βY) 1/ηY (YDN t)^(ηY−1)/ηY }^(ηY/(ηY−1)).
  - Parameters: βY is the share of tradable inputs; ηY is the elasticity of substitution between tradable and non-tradable inputs.
- Final good price:
  - PF t = { βY (P T t)^(1−ηY) + (1−βY) (P N t)^(1−ηY) }^(1/(1−ηY)).
  - P T t and P N t are prices of tradable and non-tradable inputs, respectively.

### Key calibrated values used in the model
- βY = 0:50 (Share of Tradable Inputs - Final Good Sector)
- ηY = 0:50 (Elasticity of Substitution - Final Good Sector)

---

### Intermediate good producers
- Sectoral production (J = T, N):
  - YJ t = AJ t [ KJ t^αJ (LJ t)^(1−αJ) ].
  - AJ t: aggregate productivity; KJ t: capital; LJ t: labor in sector J.
- Capital shares:
  - αN = 0:30 (Capital Share - Non-tradable Sector)
  - αT = 0:40 (Capital Share - Tradable Goods Sector)

---

### Retailers in the non-tradable sector
- Two-stage retailing:
  1. Assembler combines differentiated intermediate non-tradable goods YDN t(j), j ∈ [0,1], via CES:
     - YDN t = { ∫_0^1 [YDN t(j)]^((εN−1)/εN) dj }^(εN/(εN−1))
     - εN is the elasticity of substitution between varieties.
     - Demand for variety j: YDN t(j) = ( P N t(j) / P N t )^(−εN) YDN t.
     - Aggregate price: P N t = { ∫_0^1 P N t(j)^(1−εN) dj }^(1/(1−εN)).
  2. Retailers differentiate a homogeneous intermediate good into a continuum of goods and set prices with Calvo pricing:
     - Each period a fraction (1−θN) of retailers reset optimally; θN is the Calvo survival probability.
     - Optimal price P N* t maximizes E_t [ ∑_{i=0}^∞ θN^i Λ_{t;t+i} (P N* t − P WN t+i) YN t+i(j) ], where Λ_{t;t+i} = β^i (C_t / C_{t+i}) (P F t / P F t+i).
     - Aggregate price evolution:
       - P N t = { θN P N t−1^(1−ςp) + (1−θN) P N* t^(1−ςp) }^(1/(1−ςp)).
- Calibrated parameters:
  - εN = 6 (Elasticity of Substitution - Non-tradable Sector; implies average markup 20 percent)
  - θN = 0:75 (Calvo Parameter - Non-tradable Sector)
  - ςp is the Calvo pricing curvature parameter as used in price aggregation.

---

### Capital producers (sector-specific)
- Representative firm for sector J = {T, N} maximizes:
  - VJ t = max_{KJ_{t+i}, IJ_{t+i}} E_t ∑_{i=0}^∞ Λ_{t;t+i} ( R^K J,t+i KJ_{t+i} − P F t+i I J_{t+i} )
  - Subject to capital law of motion:
    - KJ_{t+1} = (1−δ) KJ t + S( I J t / I J_{t−1} ) I J t
  - δ is the depreciation rate; S(.) is investment adjustment cost.
- Calibration and assumptions:
  - δ = 0:02 (Depreciation Rate)
  - S(1) = 1; S′(1) = 0; S″(1) = −κS < 0 with κS = 2:5 (Investment adjustment cost elasticity, consistent with Christiano et al. (2005)).

---

### Monetary and Foreign Exchange Reserves Policies
- Central bank budget constraint:
  - E_t F* t − B t = E_t F*_{t−1} (1 + i*_{t−1}) − B_{t−1} (1 + i_{t−1}) − T_t.
  - Sterilized FX intervention: issue −B t units of domestic bonds and purchase E_t F* t units of foreign assets (E_t F* t = −B t).
  - Interest income and debt servicing flows are remitted to households via lump-sum transfers T_t.
- Policy rules:
  1. Taylor-type interest rate rule (central bank type 1):
     - (1 + i_t)/(1 + i) = ( (1 + i_{t−1})/(1 + i) )^ρi × ( Y_t / Y )^{(1−ρi) y} × ( π_t / π )^{(1−ρi) φ} × exp( εmp;t ).
     - Parameters: ρi is interest rate smoothing; y and φ are output and inflation weights; εmp;t is an i.i.d. shock, mean 0, variance σ^2_mp.
     - Calibrated values: ρi = 0:7; φ = 1:5; y = 0:5.
  2. FX reserves accumulation rule:
     - F* t / F* = ( (1 + i*_{t})/(1 + i*) )^{ζi*} × exp( εfx;t ).
     - ζi* governs intensity of FX intervention; εfx;t is i.i.d., mean 0, variance σ^2_fx.
     - Calibrated value: ζi* = 9:7 (Foreign Interest Rate Coefficient - FXI Rule)
- Foreign interest rate shock:
  - The only source of fluctuations is the foreign interest rate i* which follows AR(1) with persistence ρ_{i*} = 0:95.

---

### Market Clearing Conditions
- Labor market:
  - L_t = L N t + L T t.
- Non-tradable goods:
  - YDN t / ςN t = Y N t, where ςN t captures retailers' price dispersion.
- Aggregate domestic demand for final goods:
  - YF t = C_t + I T t + I N t.
- Law of one price for tradables:
  - P T t = E_t P* t (P* t is foreign-currency price of tradables).
- Balance of payments (combining households and government constraints; equilibrium B* t = B* t):
  - E_t (B* t + F* t) = (1 + i*_{t−1}) [ −B*_{t−1} − E_t B*_{t−1} + E_t F*_{t−1} ] + P T t Y T t − P T t Y DT t.

---

### Calibration (baseline)
- Frequency: quarterly; discount factor β = 0:99 (consistent with annual real interest rate 4 percent).
- Household preferences: u(C_t, L_t) = log(C_t) − ϕ (L_t)^{1+ξ}/(1+ξ), with ξ = 5/3 (inverse Frisch elasticity).
- Wage rigidity parameter: ζw = 0:875 (consistent with half-life of wage adjustment ≈ 5 quarters).
- Investment adjustment cost elasticity: κS = 2:5.
- Risk premium elasticity: % = 0:2 (foreign risk premium elasticity).
- Interest rate smoothing: ρi = 0:7.
- Foreign interest rate coefficient in Taylor rule: i* = 1.
- Persistence of foreign interest rate: ρ_{i*} = 0:95.

---

### III.    Capital inflows and FX intervention

### Experiment setup
- Shock analyzed: a capital inflow shock modeled as a 1 percent drop in the foreign interest rate i* (i* falls by 1 percent).
- Two policy regimes compared:
  - One instrument: monetary policy only (ζi* = 0).
  - Two instruments: monetary policy + FX intervention (ζi* > 0), with ζi* chosen to minimize L = var(y_t) + var(π_t).

### A. Full information (central bank type 1)
- With one instrument (ζi* = 0):
  - Foreign interest rate drop → nominal exchange rate appreciation → lower headline inflation; resources reallocated from tradable to non-tradable sector.
  - Short-lived boost in GDP followed by contraction as tradable production declines with appreciation.
  - Policy trade-off: monetary policy rate falls but is insufficient to stabilize output and inflation simultaneously; output and inflation move in opposite directions.
  - Transitory decline in current account due to drop in tradable production and higher imports.
- With two instruments (optimal ζi* > 0):
  - Accumulation of FX reserves (purchasing foreign assets) engineers a real exchange rate depreciation relative to the one-instrument case.
  - Effects relative to one instrument:
    - Boosts tradable output.
    - Improves current account.
    - Raises inflation.
  - Result: two instruments achieve better stabilization of both output and inflation (Tinbergen principle). Macroeconomic volatility is unambiguously lower with two instruments under full information.

### B. Policy uncertainty (agents uncertain about central bank type)
- Alternative central bank type 2 (fear of floating):
  - Interest rate rule includes response to foreign interest rate:
    - (1 + i_t)/(1 + i) = ( (1 + i_{t−1})/(1 + i) )^ρi × ( Y_t / Y )^{(1−ρi) y} × ( π_t / π )^{(1−ρi) φ} × ( (1 + i*_{t})/(1 + i*) )^{(1−ρi) ι* } × exp( εmp;t ).
    - Coefficient on foreign interest rate ι* set to 1.
  - Type 2 tends to move domestic policy rate with foreign rate to limit nominal appreciation.
- Learning and Bayesian inference:
  - Private agents do not know central bank type; they observe dev_mp;t (log deviation of interest rate rule) which is an imperfect signal.
  - dev_mp;t = ln( (1 + i_t)/(1 + i) ) − ρi ln( (1 + i_{t−1})/(1 + i) ) − (1−ρi) y ln( Y_t / Y ) − (1−ρi) φ ln( π_t / π ).
  - Observed deviations may arise from εmp;t or from the central bank being type 2.
  - Agents assign probabilities pr1;t and pr2;t and use Bayesian updating with a Kalman Filter to infer prj;t and εmp;t.
  - Prior probability of type 1: pr1;0|0 = 0:5 (baseline); sensitivity analysis conducted on this parameter.

- Effects under policy uncertainty:
  - With monetary policy only:
    - Inflation response is more muted under imperfect credibility (weaker nominal exchange rate movement) while output response is more pronounced.
    - Private expectations that central bank may be type 2 dampens nominal appreciation; spike in inflation can induce a nominal rate hike, but real rates fall more under imperfect credibility.
  - With monetary policy + FX intervention:
    - FX intervention helps mitigate output effects but at the expense of higher inflation.
    - Households anticipate larger depreciation if central bank might pursue fear-of-floating: nominal depreciation from FX intervention stabilizes output but amplifies inflationary effects because monetary policy lacks credibility to anchor inflation expectations.
    - Under policy uncertainty, using FX intervention yields ambiguous benefits: trade-off between output stabilization and higher inflation volatility.

### Summary figures and policy implications (described results)
- Figure 2 (referred): impulse response functions comparing one vs two instruments under full information.
- Figure 3/4 (referred): dynamics under imperfect credibility; trade-offs when adding FX intervention under uncertainty.
- Figure 5 (referred): volatility outcomes:
  - Panel A (full information): moving from one instrument (point A) to two instruments (point B) unambiguously reduces output and inflation volatility.
  - Panel B (policy uncertainty): moving from one instrument (point C) to two instruments (point D) reduces output volatility but increases inflation volatility — trade-off.
- Gains from the second instrument:
  - The gain from using FX reserves as a second instrument depends on the prior probability of central bank being type 1 (pr1;0|0).
  - For pr1;0|0 close to 1, adding the second instrument yields unambiguous gains (reductions in the loss L).
  - As pr1;0|0 falls (lower credibility), inflation and output volatility increase continuously, illustrating the cost of policy uncertainty.
  - Panel results (Figure 6 referred):
    - Orange dots: outcomes with 1 instrument across varying priors.
    - Blue dots: outcomes with 2 instruments across varying priors.
    - The benefit of a second instrument (reduction in L) is increasing in policy credibility.

---

*Source: _wp1667 - 1.    Final good producers (PDF chapter).*

### 2.    Optimal FX Intervention

### 2.    Optimal FX Intervention

### Optimal degree of intervention and credibility
- The optimal degree of FX intervention is increasing in the degree of central bank credibility (the prior probability of being type 1).
- Greater credibility implies less inflationary effects of using FX intervention and thus the instrument can be used to a greater extent to stabilize both output and inflation.
- Figure 7 shows the optimal FX response (percent of GDP) rising with the prior prob. of type 1 (pr1,0|0).

### Key mechanism
- When central bank type 1 optimizes the FX intervention rule it takes into account the behavior, and the prior probability, of central bank type 2.
- Non-monotonicity around high values of pr1,0|0 reflects that some degree of response to the foreign interest rate (which only central bank type 2 implements, although in excess) is optimal.

---

### Sensitivity Analysis of Model Parameters (summary)
- The paper conducts sensitivity analysis on four key parameters: 
  - (ii) the degree of price stickiness (N);
  - (iii) the degree of real wage rigidity (w);
  - (iv) the persistence of the capital inflow shock (i);
  - (v) the degree of asset substitutability (%).
- The main results hold for a wide range of values, as shown in Appendix Tables A2-A5.

- Representative parameter values explored in Appendix Tables A2-A5:
  - Price stickiness N: 0:66, 0:75, 0:875
  - Wage rigidity w: 0:66, 0:875, 0:95
  - Persistence of capital inflows i: 0:7, 0:95, 0:98
  - Portfolio balance channel %: 0:1, 0:2, 0:3

- Selected robustness patterns from Tables A2–A5 (reported exactly as in the source tables):
  - Table A2 (sensitivity to price rigidities N): examples
    - Full Credibility, 1 Instrument (MP): 0:26 (Low), 0:27 (Base), 0:29 (High)
    - Full Credibility, 2 Instruments (MP and FXI): 0:11, 0:11, 0:08
    - Imperfect Credibility, 2 Instruments, Imp. Cred. on MP: 0:17, 0:17, 0:22
    - Imperfect Credibility, 2 Instruments, Imp. Cred. on FXI: 0:15, 0:14, 0:12
  - Table A3 (sensitivity to wage rigidities w): examples
    - Full Credibility, 1 Instrument (MP): 0:24, 0:27, 0:36
    - Full Credibility, 2 Instruments (MP and FXI): 0:11, 0:11, 0:13
    - Imperfect Credibility, 2 Instruments, Imp. Cred. on FXI: 0:11, 0:14, 0:19
  - Table A4 (sensitivity to persistence i): examples
    - Full Credibility, 1 Instrument (MP): 0:21, 0:27, 0:33
    - Full Credibility, 2 Instruments (MP and FXI): 0:15, 0:11, 0:22
    - Imperfect Credibility, 2 Instruments, Imp. Cred. on FXI: 0:06, 0:14, 0:21
  - Table A5 (sensitivity to portfolio balance channel %): examples
    - Full Credibility, 1 Instrument (MP): 0:36, 0:27, 0:23
    - Full Credibility, 2 Instruments (MP and FXI): 0:19, 0:11, 0:16
    - Imperfect Credibility, 2 Instruments, Imp. Cred. on FXI: 0:16, 0:14, 0:22

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### Exchange Rate Policy Uncertainty
- The analysis is extended to imperfect credibility about the FX intervention rule while assuming monetary policy conduct is observed with certainty.
- Interpretation: uncertainty on whether FX intervention supports monetary policy in its inflation/output stabilization goals or whether the instrument is used for a separate objective.

- Alternative intervention rule perceived by private agents (central bank type 2):
  - Ft / F = [ (1 + it) / (1 + i) ]^{} exp(fx;t)  (equation (26) in the source), where  > 1 indicates a desire to depreciate beyond standard stabilization motives.
  - In the benchmark simulation,  = 2 (twice as large as the optimal value under certainty).

- Learning about the intervention rule:
  - Deviations of FX reserves from the announced rule are measured by:
    - devfx;t = ln( Ft / F )  i ln( (1 + it) / (1 + i ) )  (equation (27) in the source).
  - Agents learn about the intervention rule over time based on devfx;t.

- Results under exchange rate policy uncertainty:
  - Figure 8 indicates little difference between outcomes under full and low credibility in the benchmark calibration because monetary policy (observed with certainty) is able to offset costs associated with misperception of FX intervention.
  - Sensitivity analysis on prior beliefs (Figure 9) indicates that only in very low credibility cases, FX intervention may be counterproductive.
  - Conclusion: policy uncertainty about the conduct of FX intervention (whether aimed at supporting monetary policy goals or at an exchange rate target) is of second order importance provided monetary policy is credibly focused on stabilizing inflation and output.

- Calibration note:
  - The benchmark value  = 2 induces an accumulation of FX reserves of 5 percentage points of GDP, reflecting the “fear of floating” behavior by the type 2 central bank.

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### Simulation with Learning (Appendix I) — mechanics and key equations
- Conditional on being type j monetary authority, the log-linear solution is:
  - bX t = P bX t1 + Q i;j b i t + Q mp " mp;t  (equation (28))
  - bX t is a vector of all endogenous variables (log deviations); b i t is log deviation of foreign interest rate; " mp;t is monetary policy shock.
- Under limited credibility private sector forms inference about probabilities of the two types and the size of the monetary policy shock:
  - bX t = P bX t1 + (  pr1;tjt Q i;1 + pr2;tjt Q i;2 ) b i t + Q mp " mp;tjt  (equation (29))
- Bayesian updating using the Kalman Filter:
  - The vector [ pr1;tjt, b i t, pr2;tjt, b i t, " mp;tjt ]' is updated by a Kalman filter with state-transition matrix F, covariance Q, and observation matrix H0 (definitions provided in the appendix).
  - Prior probabilities pr j;0j0 enter initial conditions; the Kalman gain Kg is obtained as the limiting value of an iterative algorithm described in the appendix (initialization and iteration steps given exactly).

- Learning implication:
  - Credibility problems prevent immediate full inference (i.e., pr1;tjt = 1 and pr2;tjt = " mp;tjt = 0) by the private sector; learning about the true type of the central bank is gradual.

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### Conclusions (key findings and policy implications)
- Under full information, the use of FX intervention as an additional stabilization instrument unambiguously improves macroeconomic outcomes.
- When there is uncertainty about the goals of monetary policy, FX intervention entails a trade-off between stabilizing output and stabilizing inflation.
- The benefits of using FX intervention are decreasing in the degree of policy uncertainty, and so is the degree of intervention for a central bank focused on inflation and output (but perceived otherwise).
- Uncertainty about the conduct of FX intervention appears to be of second order importance provided monetary policy is credibly focused on stabilizing inflation and output.
- Policy implication: the credibility of the monetary policy regime is important to maximize the stabilization benefits of using FX intervention as an additional policy instrument.

*Source: _wp1667 - 2.    Optimal FX Intervention (PDF chapter/section)*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2016/_wp1667.pdf_