## 2.1    Outline

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---

### Introduction and motivation
- Volatile international capital flows in small open economies with shallow FX markets:
  - exacerbate exchange rate fluctuations, distort external financing conditions, and impede international risk-sharing.
- Foreign exchange interventions (FXI) mitigate distortive effects but involve intertemporal trade-offs because of:
  - risk of depleting FX reserves and a lower bound on FX reserves (LBR).
- Key policy questions:
  - To what extent should portfolio flows be offset?
  - What average level of FX reserves is associated with optimal interventions?

### Model setup and key mechanisms
- Framework:
  - Small open economy model extended with endogenous FX market depth and a lower bound on FX reserves.
  - FX market shallowness proportional to conditional exchange rate volatility; financiers are risk-averse intermediaries.
- Portfolio capital outflow mechanism:
  - Exogenous fall in demand for domestic currency → financiers sell foreign currency assets and buy domestic bonds → financiers’ net long FX exposure rises → UIP premium increases and domestic currency depreciates.
  - Sterilized FX intervention (purchase of domestic bonds financed by sale of foreign bonds) lowers financiers’ exposure and mitigates UIP movements.
- First-best (FB) benchmark:
  - If central bank could act unconstrained as financier, optimal FXI fully offsets portfolio flow shocks and eliminates UIP deviations, allowing exchange rate responses to fundamentals.

### Binding reserve constraint and time-consistency
- Lower bound on FX reserves (LBR) limits central bank’s ability to sell foreign currency bonds and creates intertemporal trade-offs: future reserve depletion risk matters for current FXI.
- Occasional binding of the reserve constraint makes optimal commitment plans time-inconsistent; analysis focuses on optimal time-consistent (discretionary) policy.
- Endogenous FX market depth:
  - Conditional exchange rate volatility increases when the exchange rate depreciates and falls when it appreciates, implying shallower markets during outflows.

### Main theoretical contributions and normative insights
- Characterization of optimal time-consistent FXI policy in a nonlinear global framework (avoiding linear-quadratic approximation).
- Time-consistent optimal policy departs from perfect stabilization of UIP risk premium to preserve reserves for potential constrained future states — a precautionary “keep the powder dry” motive.
- Implications:
  - Optimal policy may hold less FX reserves than first-best when outflows are mild and expected to abate.
  - Precautionary reserve hoarding dominates when outflows intensify; central bank may not fully offset outflows even when contemporaneous reserves permit full stabilization.
  - Asymmetry: FX purchases (during inflows) affect the real exchange rate less than FX sales (during outflows) of the same magnitude because purchases occur in relatively deep FX market states and sales in relatively shallow ones.

### Quantitative calibration and key numerical results (Malaysia calibration)
- Calibration target: Malaysia; exogenous portfolio flow process backed out using model UIP condition applied to data.
- Estimated standard deviation of the portfolio flow process: 4% of annual GDP.
- Optimal time-consistent policy numerical implications:
  - Average level of FX reserves under optimal policy: around 5% of GDP above the effective lower bound.
  - Unconditional probability of hitting the lower bound under optimal policy: 2%.
  - Optimal policy reduces exchange rate volatility relative to no-intervention but does not fully stabilize it (allows expenditure switching).
  - FX markets become significantly deeper under optimal intervention.
- Welfare implications under different LBR assumptions (consumption-equivalence gains):
  - With LBR = 0: welfare gains from optimal time-consistent policy relative to no-intervention: 0.25% of permanent steady-state consumption; first-best upper bound: 0.29% of permanent steady-state consumption.
  - With higher (more realistic) LBRs, scope for decumulating reserves shrinks and welfare differences between optimal time-consistent policy and simple contemporaneous-response rules narrow.
- Contextual calibration note:
  - Malaysia’s gross international reserves assessed as adequate (109 percent of the ARA metric) in the March 2025 Malaysia Staff Report (IMF Country Report No. 25/57).
  - Model abstracts from sudden stops, domestic financial sector disruptions, and firm-specific funding pressures; LBR variations used to assess their potential roles.

### Decentralized vs. constrained-efficient equilibrium (central bank problem)
- Decentralized equilibrium: central bank holds no FX reserves (B*_M,t = 0); households and financiers optimize; goods and bond markets clear.
- Constrained-efficient equilibrium: central bank solves discretionary time-consistent maximization problem subject to:
  - household Euler equation, expenditure switching condition, resource constraint, international risk sharing with financiers’ mean-variance preferences, conditional exchange rate volatility, and B*_M,t ≥ 0.
- First-best condition (if frictionless access to foreign bonds): R* E_t[Θ_{t+1}] = 1 and B*_M,t = B*_t − B*_P,t (fully eliminates international risk-sharing wedge).

### Financial-constraint mechanisms and implicit borrowing limit
- Financiers’ mean-variance preferences produce the risk-augmented UIP:
  - R_t E_t[Θ_{t+1} E_t E_{t+1}] − R* E_t[Θ_{t+1}] = −ω σ^2_t B^*_{F,t},
    where σ^2_t = R_t^2 var_t(E_t E_{t+1}) and ω>0.
- LBR implies an implicit borrowing limit:
  - B^*_t ≥ B^*_{P,t} − 1/(ω σ^2_t) (1 − R* E_t[Θ_{t+1}]) ≡ Ψ_t.
- Interpretation of Ψ_t:
  - Exogenous portfolio outflows B^*_{P,t} increase Ψ_t and tighten the borrowing limit.
  - When LBR binds (R* E_t[Θ_{t+1}] < 1), the term involving financiers’ lending position decreases Ψ_t.

### Value of commitment and time inconsistency
- Implicit borrowing limit is forward-looking (depends on conditional volatility and stochastic discount factor expectations).
- Credible commitment to future FX interventions could relax current borrowing limits by steering expectations, but commitment is not time-consistent; central bank may optimally reoptimize.
- Focus on fully time-consistent discretionary FXI policies.

### Intertemporal trade-offs in second-best (first-order condition summary)
- Second-best first-order condition (schematic):
  - u1,t = β R* E_t[u1,t+1] (FB)
    + λt [positive terms via volatility channel and negative via SDF channel]
    − β E_t[ λ_{t+1} (positive volatility channel) ]
    − β E_t[ λ_{t+1} (negative SDF channel) ].
- Complementary slackness: λt ≥ 0, B^*_t − Ψ_t ≥ 0, (B^*_t − Ψ_t) λt = 0.
- Two opposing effects of current savings B^*_t on future constraints:
  - Volatility channel: ∂σ^2_{t+1}/∂B^*_t < 0 ⇒ saving today lowers future σ^2 and benefits future constraints.
  - SDF channel: ∂Θ_{t+2}/∂B^*_t > 0 ⇒ saving today supports future tradable consumption, raises SDF, tightens implicit borrowing limit next period.

### Optimal time-consistent FXI under second-best (Theorem 1 and interpretation)
- Theorem 1 (when λt = 0 but E_t[λ_{t+1}] > 0) yields closed-form policy formula:
  - B^*_{M,t} = B^*_t − B^*_{P,t}
    + β R* ω σ^2_t u_{1,t} E_t[ λ_{t+1} ( −u_{11,t+1} / u_{1,t+1} ) ( (B^*_{P,t+1} − B^*_{t+1}) − 1/(ω σ^2_{t+1}) ) ].
- Implications:
  - Optimal intervention at t can be liquidity-injecting or liquidity-absorbing depending on sign of expectation term.
  - If LBR not expected to bind in t+1, optimal intervention equals first-best.
- Intuition:
  - Central bank trades off:
    - increasing expected return and financiers’ capacity to lend in t+1 (via liquidity-injecting interventions), versus
    - reducing σ^2_{t+1} and making financiers more willing to lend (via liquidity-absorbing interventions).

### Precautionary accumulation of FX reserves (liquidity-absorbing motive)
- Liquidity-absorbing motive arises when higher reserves today reduce future σ^2 and deepen FX market liquidity.
- Accumulating reserves beyond first-best:
  - reduces probability that future portfolio outflow depletes reserves,
  - compresses non-fundamental loading on exchange rate risk,
  - smooths UIP premium over time at cost of increasing it contemporaneously.
- “Keeping powder dry” — smoothing over time by compressing UIP premium in potentially constrained period.

### Conditions for liquidity-absorbing motive dominance (Proposition 2)
- Condition for reserves higher than first-best when λt = 0 and E_t[λ_{t+1}] > 0:
  - R* E_t[Θ_{t+2}] < 2/3 − Cov_t( λ_{t+1} ( −u_{11,t+1} / u_{1,t+1} ), R* E_{t+1}[Θ_{t+2}] ) / E_t[ λ_{t+1} ( −u_{11,t+1} / u_{1,t+1} ) ].
- Interpretation:
  - Liquidity-absorbing motive prevails if SDF between t+1 and t+2 is expected to drop sufficiently in possibly constrained period t+1.
- Amplifying factors:
  - Higher portfolio outflows B^*_{P,t+1}.
  - Larger existing debt burden (weaker NFA position).
  - Lower tradable endowment Y_{T,t+1}.

### Calibration details and numerical solution (key parameter values)
- Quarterly calibration to Malaysia, data 2010–2023.
- Key parameters:
  - R* = 1.01 (World interest rate, quarterly)
  - σ = 2
  - ξ = 0.83
  - α = 0.39
  - β = 0.9871 (targets Malaysian NFA-to-GDP)
  - ω = 28 (financiers’ risk aversion)
  - Targeted average FX market depth: ω σ^2 = 0.05
- VAR and state discretization:
  - Exogenous states s_t = [log Y_T,t, log Y_N,t, sinh^{-1}(B_{P,t} − B̄_P)]′ modeled as first-order VAR with estimated V and ρ (VAR estimates reported in text).
  - Numerical grids: 4 grid points for Y_T and Y_N each; 8 grid points for B^*_P; 1000 grid points for endogenous B^*.

### Policy functions and ergodic implications (numerical findings)
- Policy function for NFA B^*:
  - Constrained planner’s policy function lies above decentralized-equilibrium policy function; both intersect 45-degree line indicating stationary equilibria.
- Optimal FX reserves response to portfolio outflows:
  - For high NFA: optimal policy offsets outflows approximately one-for-one; deviations from first-best minor.
  - For low NFA: lower bound becomes relevant; central bank optimally exhausts reserves in response to sufficiently large outflows.
  - Average policy exhibits precautionary hoarding: average policy function lies above first-best across portfolio outflow levels.
- Ergodic (steady-state) simulation (2×10^6 quarters):
  - Optimal FXI increases average savings and raises net foreign asset position.
  - Positive average FX reserves deepen FX markets; financiers expand balance sheets allowing partial offset between NFA increases and reserves.
  - Optimal FXI stabilizes exchange rate versus no-FXI but still allows efficient responses to endowment shocks.

### Dynamics: portfolio flow episodes and asymmetries
- Simulation episodes: non-overlapping 17-quarter samples with peak outflow/inflow at period 0.
- Portfolio outflow episodes (typical peak ≈ 8% of steady-state GDP):
  - No FXI:
    - Fall in demand for domestic currency → tighter external financial conditions, weaker exchange rate, sharp consumption contraction, FX markets become shallower (σ^2_t rises).
  - Optimal time-consistent FXI:
    - Reserves fall by around 5% of GDP at peak (t = 0) — significant but not full offset.
    - Intervention reduces depreciation and consumption fall; largely eliminates spike in FX market depth by strengthening currency and preserving reserves.
- Asymmetry across outflow sizes:
  - Strong outflows ≈ 4 percentage points of GDP larger than small outflows.
  - Optimal intervention is similar across small and large episodes, implying relatively bigger offset for large outflows.
  - For strong outflows: central bank partially absorbs outflows; financiers increase long exposure B^*_F,t.
  - For small outflows: central bank FX sales nearly fully offset portfolio outflows.
  - Asymmetry persists excluding states with exhausted reserves — evidence of “keeping powder dry”.
- Portfolio inflow episodes:
  - Central bank offsets inflows almost one-for-one (mirrors first-best) because LBR less relevant during inflows.
  - Inflows deepen FX markets.
  - Quantification:
    - FX sales: central bank conducts FX sales worth 4.8% of GDP between t = −8 and t = 0 (outflow episodes).
    - FX purchases: central bank conducts FX purchases worth 6.6% of GDP between t = −8 and t = 0 (inflow episodes).
    - Real exchange rate at peak of outflows is 3.5% more appreciated under optimal FXI compared to decentralized equilibrium.
    - Real exchange rate at peak of inflows is around 3.4% weaker under optimal FXI.
    - FX sales have a 44% bigger impact on the real exchange rate than FX purchases in simulations.
  - Remark: FX sales are more effective than FX purchases because sales occur when FX markets are relatively shallower.

### Response to fundamental shocks
- For tradable endowment decreases:
  - No FXI: sharp consumption contraction; real exchange rate appreciates; portfolio capital flees.
  - Optimal FXI: focuses on stabilizing international risk-sharing wedge; not effective in limiting consumption fall driven by fundamentals; small effect on exchange-rate adjustment since exchange rate allows efficient expenditure switching.

### Welfare implications and policy comparisons (consumption-equivalence κ_p)
- Welfare metric κ_p defined via consumption-equivalence (equation (16)).
- Policies compared: No FXI; RER Rule; Portfolio Rule; UIP Rule; Optimal FXI; First Best.
- Two LBR levels considered:
  - LBR = 0% of GDP.
  - LBR = 21% of annual GDP (minimum level observed in Malaysia after trend-seasonality adjustment).
- Welfare gains (percentages) — LBR = 0%:
  - No FXI: -0.0089
  - RER Rule: 0.0064
  - Portfolio Rule: 0.0452
  - UIP Rule: 0.1500
  - Optimal FXI: 0.2484
  - First Best: 0.2880
- Welfare gains (percentages) — LBR = 21% of GDP:
  - No FXI: -0.0089
  - RER Rule: 0.0063
  - Portfolio Rule: 0.0449
  - UIP Rule: 0.0955
  - Optimal FXI: 0.0960
  - First Best: 0.2880
- Interpretation:
  - With LBR = 0, optimal time-consistent FXI improves welfare by 0.25% of consumption, close to first-best 0.29%.
  - With LBR = 21%, gains from optimal time-consistent FXI shrink; UIP rule nearly matches optimal policy.
  - Initial NFA position ≈ 2% of annual GDP in computations; high initial reserves partly crowd out domestic borrowing in the model.
  - Gradual decumulation of reserves during transition contributes to welfare gains for policies that allow average reserves to change.

### Ergodic moments under alternative FXI regimes (LBR = 21% of annual GDP; 2×10^6-quarter simulations)
- Consumption (Mean, Std):
  - No FXI: 1.001, 0.025
  - RER Rule: 1.001, 0.025
  - Portfolio Rule: 1.000, 0.022
  - UIP Rule: 0.998, 0.020
  - Optimal FXI: 0.998, 0.020
  - First Best: 0.987, 0.023
- Net Foreign Assets (Mean, Std) (fraction of annual GDP):
  - No FXI: 0.020, 0.008
  - RER Rule: 0.020, 0.012
  - Portfolio Rule: -0.006, 0.008
  - UIP Rule: -0.061, 0.014
  - Optimal FXI: -0.059, 0.011
  - First Best: -0.337, 0.007
- FX Reserves (Mean, Std) (fraction of annual GDP):
  - No FXI: 0.328, 0.000
  - RER Rule: 0.327, 0.045
  - Portfolio Rule: 0.301, 0.043
  - UIP Rule: 0.247, 0.037
  - Optimal FXI: 0.248, 0.036
  - First Best: -0.029, 0.044
- Real Exchange Rate (Mean, Std):
  - No FXI: 0.390, 0.014
  - RER Rule: 0.389, 0.009
  - Portfolio Rule: 0.390, 0.011
  - UIP Rule: 0.392, 0.012
  - Optimal FXI: 0.392, 0.012
  - First Best: 0.400, 0.011
- FX Market Depth (Mean, Std):
  - No FXI: 0.078, 0.012
  - RER Rule: 0.034, 0.005
  - Portfolio Rule: 0.047, 0.008
  - UIP Rule: 0.057, 0.009
  - Optimal FXI: 0.054, 0.008
  - First Best: 0.042, 0.010
- UIP Premium (Mean, Std) (annualized):
  - No FXI: 0.007, 0.120
  - RER Rule: 0.011, 0.059
  - Portfolio Rule: 0.012, 0.012
  - UIP Rule: 0.012, 0.029
  - Optimal FXI: 0.012, 0.027
  - First Best: 0.000, 0.000
- Binding LBR Frequency:
  - No FXI: 0.000
  - RER Rule: 0.000
  - Portfolio Rule: 0.000
  - UIP Rule: 0.181, 0.049
  - Optimal FXI: – (not reported)
  - First Best: – (not applicable)

### Interpretation of ergodic results
- Optimal time-consistent FXI:
  - Yields precautionary accumulation of FX reserves (average reserves slightly higher than UIP rule), reducing frequency of hitting LBR.
  - Produces deeper FX markets and lower volatility of the risk-sharing wedge compared to UIP rule.
- Portfolio rule:
  - Implicates very high FX reserves and virtually eliminates reserve depletion risk; deepens markets and lowers exchange rate volatility but reduces international borrowing (nearly balanced NFA).
- First-best:
  - No precautionary reserve accumulation; lower average NFA; eliminates risk-sharing wedge.
- RER rule:
  - Delivers most stable real exchange rate and highest market depth but generates high UIP premium volatility and excessive FX reserves; ranks worst among active FXI regimes.
- Policy takeaway: FXI should target inefficient volatility in the international risk-sharing wedge rather than respond directly to exchange rate movements driven by fundamentals.

### Sensitivity to initial reserves and value of commitment
- When initial FX reserves are far from LBR, optimal time-consistent FXI delivers robust welfare gains close to first-best; committing to a UIP-targeting rule also improves welfare but less so.
- When reserves are initially small, optimal time-consistent policy achieves welfare very similar to a simple UIP rule; value of commitment increases when reserves are relatively low.

### Final takeaways (Conclusions)
- Optimal time-consistent FXI can reduce exchange rate volatility caused by portfolio flow shocks, shrink UIP deviations, and improve FX market depth.
- Optimal response is state-dependent:
  - Strong NFA position and/or moderate expected outflows: holding FX reserves below first-best can be optimal.
  - Large and persistent outflows with weak NFA: precautionary accumulation of reserves is optimal.
- Asymmetry in intervention effectiveness:
  - FX purchases during inflows have lower impact on exchange rate (markets deeper).
  - FX sales during outflows are more effective (markets shallower).
- Welfare gains from optimal time-consistent FXI substantial when starting far from LBR; a UIP-targeting rule narrows the gap when LBR is binding or reserves are low.
- Emphasis on precautionary motive and state-dependency in FXI; suggested future research: include additional frictions or endogenize effective LBR.

*Source: wpiea2025261-source-pdf (sections 2.1–4.4 and appendices).*

### 2.1    Outline .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  

### 2.1    Outline

### Introduction and motivation
- Volatile international capital flows in small open economies with shallow FX markets exacerbate exchange rate fluctuations, distort external financing conditions, and impede international risk-sharing.
- Foreign exchange interventions (FXI) are a key policy tool to mitigate these distortive effects, but their implementation involves intertemporal trade-offs because of the risk of depleting FX reserves and a lower bound on reserves.
- Key policy questions addressed: the extent to which portfolio flows should be offset, and the average level of FX reserves associated with optimal interventions.

### Model setup and key mechanisms
- Framework: a standard small open economy model extended with endogenous FX market depth and a lower bound on FX reserves, following the spirit of Itskhoki & Mukhin (2023).
- FX markets are imperfectly elastic due to risk-averse intermediaries (financiers) exposed to FX risk; FX market shallowness is proportional to conditional exchange rate volatility.
- Portfolio capital outflow shock example:
  - Exogenous fall in demand for domestic currency → financiers accommodate by selling foreign currency assets and buying domestic-currency assets → increases financiers’ net long FX exposure.
  - Increased exposure raises UIP premium and depreciates the domestic currency.
  - Sterilized FX intervention (purchase of domestic currency bonds financed by sale of foreign currency bonds) lowers financiers’ exposure and mitigates UIP movements.
- First-best (FB) benchmark: if the central bank could unconstrainedly act as financiers, optimal FXI would fully offset portfolio flow shocks and eliminate UIP deviations, while still allowing exchange rate responses to fundamentals.

### Binding reserve constraint and time-consistency
- A lower bound on FX reserves (LBR) limits the central bank’s ability to sell foreign currency bonds and implies intertemporal considerations for FXI policy: even if unconstrained now, future reserve depletion risk matters.
- Occasional binding of the reserve constraint makes optimal plans under commitment time-inconsistent; the analysis focuses on optimal time-consistent (discretionary) policy.
- Endogenous variation in FX market depth: conditional exchange rate volatility increases when the exchange rate depreciates and falls when it appreciates, implying shallower markets during outflows.

### Main theoretical contributions and normative insights
- Analytical and quantitative characterization of optimal time-consistent FXI policy in a nonlinear global framework (avoiding the linear-quadratic approximation).
- Time-consistent optimal policy departs from perfect stabilization of the UIP risk premium primarily to preserve reserves for potential constrained future states—i.e., a precautionary “keep the powder dry” motive.
- Consequence: optimal policy may hold less FX reserves than first-best when outflows are mild and expected to abate, but it emphasizes precautionary reserve hoarding when outflows intensify, sometimes choosing not to fully offset outflows even when contemporaneous reserves would permit full stabilization.
- Asymmetric effects: optimal FX purchases (during inflows) affect the real exchange rate less than FX sales (during outflows) of the same magnitude because purchases occur in relatively deep FX market states and sales in relatively shallow ones.

### Quantitative calibration and key numerical results (Malaysia calibration)
- Calibration target: Malaysia; the exogenous portfolio flow process is backed out using the model UIP condition applied to data.
- Estimated standard deviation of the portfolio flow process: 4% of annual GDP.
- Optimal time-consistent policy implications:
  - Average level of FX reserves under optimal policy: around 5% of GDP above the effective lower bound.
  - Unconditional probability of hitting the lower bound under optimal policy: 2%.
  - Optimal policy reduces exchange rate volatility relative to no-intervention, but does not fully stabilize it (allows expenditure switching).
  - FX markets become significantly deeper under optimal intervention.
- Welfare implications under different LBR assumptions:
  - With LBR set to zero (ignoring non-modeled considerations), welfare gains from optimal time-consistent policy relative to no-intervention: 0.25% of permanent steady-state consumption.
  - First-best welfare upper bound: 0.29% of permanent steady-state consumption.
  - With higher (more realistic) LBRs, scope for decumulating reserves shrinks and welfare differences between optimal time-consistent policy and simple contemporaneous-response rules narrow.
- Additional contextual note in the calibration discussion: Malaysia’s gross international reserves were assessed as adequate (109 percent of the ARA metric) in the March 2025 Malaysia Staff Report (IMF Country Report No. 25/57). (Model abstraction caveat: sudden stops, domestic financial sector disruptions, and firm-specific funding pressures are abstracted from; LBR variations are used to assess their potential roles.)

### Comparative literature and methodological links
- The FXI mechanism builds on portfolio balance channels and segmentation in international financial markets (Gabaix & Maggiori, 2015; Itskhoki & Mukhin, 2021).
- Related work on FXI and reserve constraints: Basu et al. (2018); Itskhoki & Mukhin (2023) (linear-quadratic approach finds more aggressive interventions; this paper’s nonlinear approach highlights a dominant precautionary motive implying less-than-one-for-one response to outflows).
- Empirical challenges in estimating FX market depth noted; the paper leverages recent FXI proxies constructed on monthly and quarterly basis (Adler et al., 2025) in calibration.
- Methodological connection to literature computing decentralized and constrained-efficient equilibria with occasionally binding constraints using global methods (Mendoza, 2010; Bianchi, 2011; Bianchi & Mendoza, 2018; Schmitt-Grohé & Uribe, 2016, 2021; Davis et al., 2023).

### Paper structure (outline)
- Section 2: model description (decentralized equilibrium and constrained-efficient equilibrium with discretionary time-consistent FXI facing a lower bound on reserves).
- Section 3: optimal FXI problem analysis.
- Section 4: quantitative analysis (calibration, policy functions, ergodic implications, dynamics).
- Section 5: welfare implications.
- Section 6: conclusions.
- Appendices: Proofs; Calibration Details; Simulation Details.

*Source: wpiea2025261-source-pdf (sections 2.1–2 and outline material).*

### 2.1    Outline

### 2.1    Outline

### Utility Function
- Households: identical, infinitely-lived; preferences
  - ∞
    X
    t=0
    E
    0
    
    β
    t
    u(C
    T,t
    ,C
    N,t
    )
    
  - u(C
    T,t
    ,C
    N,t
    ) =
    1
    1−σ
    Ü
    h
    α(C
    T,t
    )
    ξ−1
    ξ
    + (1−α) (C
    N,t
    )
    ξ−1
    ξ
    i
    ξ
    ξ−1
    |{z}
    C
    t
    ê
    1−σ
- Interpretation and parameters:
  - C
    t
    is total consumption composed of tradable goods C
    T,t
    and nontradable goods C
    N,t
  - β∈(0,1) is the subjective discount factor
  - 1/σ is the intertemporal elasticity of substitution
  - ξ is the elasticity of substitution between tradable and nontradable goods
  - α∈(0,1) controls the share of tradables in the total consumption basket

### Households’ Budget Constraint
- Endowments: stochastic Y
  T,t
  and Y
  N,t
  follow a first-order Markov process
- One-period local currency bond B
  t
  pays gross nominal interest rate R
  t
- Sequential budget constraint:
  - P
    T,t
    C
    T,t
    +P
    N,t
    C
    N,t
    ≤P
    T,t
    Y
    T,t
    +P
    N,t
    Y
    N,t
    −B
    t
    +B
    t−1
    R
    t−1
    + Π
    M,t
    + Π
    F,t
    + Π
    P,t
- Financial sector is fully domestically owned (implied by formulation)
- Law of one price for tradables with foreign price normalized to unity: P
  T,t
  =E
  t
- Price of nontradables normalized to unity: P
  N,t
  = 1

### Households’ Optimality
- Euler equation for B
  t
  :
  - R
    t
    E
    t
    ï
    Θ
    t+1
    E
    t
    E
    t+1
    ò
    = 1,(1)
- Stochastic discount factor (SDF) Θ
  t+1
  :
  - Θ
    t+1
    =β
    u
    1
    (C
    T,t+1
    ,C
    N,t+1
    )
    u
    1
    (C
    T,t
    ,C
    N,t
    )
    =β
    Å
    C
    t+1
    C
    t
    ã
    1−σξ
    ξ
    Å
    C
    T,t+1
    C
    T,t
    ã
    −
    1
    ξ
- Expenditure switching condition pinning down the exchange rate E
  t
  :
  - E
    t
    =
    u
    1
    (C
    T,t
    ,C
    N,t
    )
    u
    2
    (C
    T,t
    ,C
    N,t
    )
    =
    α
    1−α
    Å
    C
    N,t
    C
    T,t
    ã
    1
    ξ
    .(2)

### Financiers
- Hold zero capital portfolio: B
  F,t
  +E
  t
  B
  ∗
  F,t
  = 0
- Mean-variance preferences:
  - E
    t
    î
    Θ
    t+1
     ̃
    R
    ∗
    t+1
    B
    ∗
    F,t
    ó
    −
    ω
    2
    var
    t
    Ä
     ̃
    R
    ∗
    t+1
    B
    ∗
    F,t
    ä
- Parameters and definitions:
  - ω>0 measures financiers’ additional degree of risk aversion
  - R
    ∗
    is the (constant) world interest rate
  -  ̃
    R
    ∗
    t+1
    =R
    ∗
    −R
    t
    E
    t
    E
    t+1
    denotes the carry trade return in foreign currency
- First-order condition yields risk-augmented UIP:
  - R
    t
    E
    t
    ï
    Θ
    t+1
    E
    t
    E
    t+1
    ò
    −R
    ∗
    E
    t
    [Θ
    t+1
    ] =
    −ωσ
    2
    t
    B
    ∗
    F,t
    ,(3)
    where σ
    2
    t
    =R
    2
    t
    var
    t
    Ä
    E
    t
    E
    t+1
    ä
- Interpretation:
  - Ex-ante UIP deviation (international risk-sharing wedge) driven by B
    ∗
    F,t
    and FX market depth ωσ
    2
    t
  - Higher long exposure to domestic currency (more negative B
    ∗
    F,t
    ) raises required excess return
  - Higher (lower) ωσ
    2
    t
    corresponds to shallower (deeper) FX markets
- Financiers’ domestic currency profits:
  - Π
    F,t
    =
    Ä
    R
    t−1
    −R
    ∗
    t−1
    E
    t
    E
    t−1
    ä
    B
    F,t−1

### Portfolio Investors
- Hold zero capital portfolio: B
  P,t
  +E
  t
  B
  ∗
  P,t
  = 0
- Non-optimizing: B
  ∗
  P,t
  exogenous first-order Markov process; buy (sell) foreign currency bonds randomly
- Profits:
  - Π
    P,t
    =
    Ä
    R
    t−1
    −R
    ∗
    t−1
    E
    t
    E
    t−1
    ä
    B
    P,t−1

### Central Bank
- Sterilized FX interventions via foreign reserves B
  ∗
  M,t
  and sterilization bonds B
  M,t
  with B
  M,t
  +E
  t
  B
  ∗
  M,t
  = 0
- Lower bound constraint on reserves initially set to zero:
  - B
    ∗
    M,t
    ≥0
- Profits:
  - Π
    M,t
    =
    Ä
    R
    t−1
    −R
    ∗
    t−1
    E
    t
    E
    t−1
    ä
    B
    M,t−1
- Central bank assumed to fully stabilize nontradable goods prices, allowing P
  N,t
  = 1

### Bond Market Clearing
- Domestic bond market clearing:
  - B
    F,t
    +B
    t
    +B
    P,t
    +B
    M,t
    = 0
- Net foreign asset (NFA) position in foreign currency:
  - B
    ∗
    t
    =B
    ∗
    F,t
    +B
    ∗
    M,t
    +B
    ∗
    P,t
    , which implies B
    ∗
    t
    =B
    t
    /E
    t

### Resource Constraint
- Nontradable goods consumption equals endowment:
  - C
    N,t
    =Y
    N,t
    .(5)
- Consolidated resource constraint:
  - B
    ∗
    t
    −B
    ∗
    t−1
    R
    ∗
    =Y
    T,t
    −C
    T,t
    .(6)
- Implication: at aggregate level, domestic economy borrows in foreign currency at the world interest rate due to full domestic ownership of the financial sector

### Decentralized and Constrained-Efficient Equilibrium
- Decentralized equilibrium (no FX interventions; central bank holds no FX reserves)
  - Definition 1: Given exogenous process {B
    ∗
    P,t
    ,Y
    T,t
    ,Y
    N,t
    }
    ∞
    t=0
    and initial B
    ∗
    −1
    , a competitive equilibrium is sequences {E
    t
    ,R
    t
    }
    ∞
    t=0
    , {σ
    2
    t
    }
    ∞
    t=0
    , allocations {C
    T,t
    ,C
    N,t
    }
    ∞
    t=0
    , bond positions {B
    ∗
    t
    ,B
    ∗
    F,t
    }
    ∞
    t=0
    , and FXI policy {B
    ∗
    M,t
    }
    ∞
    t=0
    such that:
  - 1. Households and financiers optimize, implying (1), (2), and (3)
  - 2. Central bank holds no FX reserves, ∀t: B
    ∗
    M,t
    = 0
  - 3. Goods and bond markets clear, implying (5), (6), and (4)
  - 4. Transversality condition: lim
    T→∞
    B
    ∗
    T
    (R
    ∗
    )
    T
    = 0
- Constrained-efficient equilibrium (central bank optimally conducts FX interventions, discretionary time-consistent policy)
  - Central bank maximization problem:
    - max
      {C
      T,t
      ,B
      ∗
      t
      ,E
      t
      ,R
      t
      ,B
      ∗
      M,t
      ,σ
      2
      t
      }
      E
      t
      ∞
      X
      s=0
      β
      t+s
      [u(C
      T,t+s
      ,Y
      N,t+s
      )](7)
    - Subject to constraints (Household Euler Equation, Expenditure Switching, Resource Constraint, International Risk Sharing, Conditional Exchange Rate Volatility, Constraint on Reserves)
      - R
        t+s
        E
        t+s
        ï
        Θ
        t+s+1
        E
        t+s
        E
        t+s+1
        ò
        = 1
      - E
        t+s
        =
        u
        1
        (C
        T,t+s
        ,Y
        N,t+s
        )
        u
        2
        (C
        T,t+s
        ,Y
        N,t+s
        )
      - B
        ∗
        t+s
        −B
        ∗
        t+s−1
        R
        ∗
        =Y
        T,t+s
        −C
        T,t+s
      - R
        ∗
        E
        t+s
        [Θ
        t+s+1
        ] = 1 +ωσ
        2
        t+s
        
        B
        ∗
        t+s
        −B
        ∗
        M,t+s
        −B
        ∗
        P,t+s
        

      - σ
        2
        t+s
        =R
        2
        t+s
        var
        t+s
        Å
        E
        t+s
        E
        t+s+1
        ã
      - B
        ∗
        M,t+s
        ≥0
  - Definition 2: Constrained-efficient equilibrium is sequences satisfying (1), (2) (central bank solves (7)), (3) market clearing, and (4) transversality condition lim
    T→∞
    B
    ∗
    T
    (R
    ∗
    )
    T
    = 0

### 3    Optimal FX Interventions

#### First-Best
- If households had direct frictionless access to foreign currency bonds, first-order condition:
  - R
    ∗
    E
    t
    [Θ
    t+1
    ] = 1.(8)
- In constrained-efficient equilibrium without an LBR this condition holds if financial sector is entirely domestically owned
- Associated optimal FXI policy:
  - B
    ∗
    M,t
    =B
    ∗
    t
    −B
    ∗
    P,t
    ,(9)
  - This fully eliminates the international risk-sharing wedge and makes financiers’ intermediation redundant (B
    ∗
    F,t
    = 0)
- Note: strict stabilization of the UIP premium would not be optimal if financial market participants were partially foreign owned

#### Financial Conditions Under Second-Best
- Presence of an LBR generally prevents complete elimination of the risk-sharing wedge
- Proposition 1 (period t with a binding LBR: B
  ∗
  t
  −B
  ∗
  P,t
  <0, and portfolio outflow associated with improvement in the NFA position: ∂B
  ∗
  t
  /∂B
  ∗
  P,t
  >0):
  - (a) ∂σ
    2
    t
    /∂B
    ∗
    P,t
    >0: A portfolio outflow leads to an elevated conditional volatility of the exchange rate.
  - (b) ∂E
    t
    [Θ
    t+1
    ]/∂B
    ∗
    P,t
    <0: A portfolio outflow leads to an expected decrease of the SDF between periods t and t+1.
- Mechanism:
  - When LBR binds, portfolio outflow cannot be fully offset by selling FX reserves; financiers absorb imbalance, increasing their required premium
  - Higher UIP wedge tightens households’ financial conditions, decreasing consumption of tradable goods and lowering the SDF
- State-dependent FX market depth ωσ
 2
  t
  amplifies the impact of portfolio outflows on the risk-sharing wedge: increased conditional exchange rate volatility raises financiers’ compensation needs, tightening borrowing conditions further

#### Implicit Borrowing Limit
- The LBR implies an upper bound on net foreign liabilities — an implicit borrowing limit
- Combining international risk sharing (3) with the LBR yields:
  - B
    ∗
    t
    ≥B
    ∗
    P,t
    −
    1
    ωσ
    2
    t
    (1−R
    ∗
    E
    t
    [Θ
    t+1
    ])≡Ψ
    t
    .(10)
- Interpretation:
  - Ψ
    t
    has two components:
    - Exogenous portfolio outflows B
      ∗
      P,t
      increase Ψ
      t
      and tighten the borrowing limit
    - The term involving financiers’ domestic currency lending position decreases Ψ
      t
      whenever the LBR is binding (i.e., when R
      ∗
      E
      t
      [Θ
      t+1
      ] < 1)

*Source: wpiea2025261-source-pdf - 2.1 Outline*

### 1. Due to their mean-variance preferences, the financiers’ position in equilibrium is a function

### wpiea2025261-source-pdf - 1. Due to their mean-variance preferences, the financiers’ position in equilibrium is a function

### Financiers’ equilibrium position and the implicit borrowing limit
- Financiers’ position in equilibrium is a function of the expected excess return on domestic currency 1−R∗Et[Θt+1] and the risk factor ωσ^2_t.
- If the expected return increases and/or the risk factor decreases, financiers are willing to lend more in domestic currency, relaxing the implicit borrowing limit.
- Optimal FX reserves reflect the difference between net foreign assets and the borrowing limit: B∗_M,t = B∗_t − Ψ_t.
- The borrowing limit is binding (B∗_t = Ψ_t) if and only if the central bank runs out of reserves (B∗_M,t = 0).

### Value of commitment and time inconsistency of optimal plans
- The implicit borrowing limit is forward-looking: it depends on the conditional volatility of the exchange rate and on the stochastic discount factor used by households to value future flows, both depending on expectations at time t about t+1 events.
- Credible commitment to future FX interventions could relax the current-period borrowing limit by steering expectations, akin to “forward guidance” about future interest rates.
- Committing to future interventions is not time-consistent: once new shocks arrive, the central bank may optimally reoptimize and renege on past promises.
- The analysis focuses on fully time-consistent but discretionary FX intervention (FXI) policies.

### Intertemporal tradeoffs under second-best (first-order condition summary)
- First-order condition for the second-best policy problem (equation (11)):

  u1,t = βR∗ Et[u1,t+1] (FB)
  + λt [ (1 − 1−R∗ Et[Θt+1]) ω(σ^2_t)/2 ∂σ^2_t/∂B∗_t − R∗ ωσ^2_t Et[∂Θt+1/∂B∗_t] (≥0) ]
  − β Et[ λt+1 (1 − R∗ Et+1[Θt+2]) ω (σ^2_{t+1})/2 ∂σ^2_{t+1}/∂B∗_t (≥0) ]
  − β Et[ λt+1 R∗ ωσ^2_{t+1} Et+1[∂Θt+2/∂B∗_t] ] (≤0)

- Notation: u1,t ≡ u1(CT,t, YN,t); λt is the Lagrange multiplier for the implicit borrowing constraint (10).
- Observations:
  - If the lower bound on reserves is never binding (∀t, λt = 0), condition (11) reduces to the first-best (equation (8)) and the central bank can eliminate the international risk sharing wedge.
  - If the LBR binds at time t (λt > 0), financial conditions tighten (UIP premium positive), borrowing falls, and consumption must be restricted.
  - Future possibility of a binding LBR (λt+1 > 0 in some states) affects current allocations even if current reserves are positive — central bank internalizes how current savings affect future financial conditions.

- Two opposing effects of current savings B∗_t on future constraints:
  - Effect via conditional exchange rate volatility:
    - ∂σ^2_{t+1} / ∂B∗_t < 0: more saving today lowers future σ^2_{t+1}, reducing the probability reserves deplete and lowering non-fundamental shock impact on next period’s exchange rate.
    - Because 1 − R∗ Et[Θt+2] > 0 whenever FXI is constrained in t+1 (λt+1 > 0), the volatility channel yields a positive contribution (saving more today benefits future constraints).
  - Effect via stochastic discount factor:
    - ∂Θt+2 / ∂B∗_t > 0: more saving today supports future tradable consumption, increasing the aggregate stochastic discount factor Θt+2, which decreases financiers’ expected profits (1 − R∗ Et[Θt+2]) and tightens the implicit borrowing limit next period.
    - This channel yields a negative contribution (saving more today can amplify intermediation friction next period).

- Complementary slackness conditions: λt ≥ 0, B∗_t − Ψ_t ≥ 0, (B∗_t − Ψ_t) λt = 0.

### Optimal time-consistent FXI under second-best (Theorem 1 and interpretation)
- Theorem 1 (policy formula when λt = 0 but Et[λt+1] > 0):
  B∗_M,t = B∗_t − B∗_P,t + βR∗ ωσ^2_t u1,t Et[ λt+1 ( −u11,t+1 / u1,t+1 ) ( (B∗_P,t+1 − B∗_t+1) − 1/(ωσ^2_{t+1}) ) ]

- Implication:
  - Optimal intervention in period t can be either liquidity-injecting (B∗_M,t < B∗_t − B∗_P,t) or liquidity-absorbing (B∗_M,t > B∗_t − B∗_P,t) depending on sign of the third term on the RHS.
  - If LBR is not expected to bind in any states in t+1, the optimal intervention equals the first-best.

- Intuition:
  - When t+1 is potentially constrained (binding LBR), intermediated funds B∗_F,t+1 = B∗_t+1 − B∗_P,t+1 < 0 and the risk sharing wedge −ωσ^2_{t+1} B∗_F,t+1 > 0.
  - Central bank can affect expected return 1 − R∗ Et+1[Θt+2] and risk factor ωσ^2_{t+1} by FXI in t:
    - Liquidity-injecting interventions can increase expected return and financiers’ capacity to lend in t+1.
    - Liquidity-absorbing interventions can decrease σ^2_{t+1}, making financiers more willing to lend.

- FXI neutrality vs. market depth:
  - FX reserves influence domestic bond market liquidity because interventions are sterilized: central bank net buyer of domestic bonds when holding less reserves vs. net seller when holding more.

### Precautionary accumulation of FX reserves (liquidity-absorbing motive)
- Key difference from prior first-order approximations: the liquidity-absorbing motive (precautionary accumulation) arises when higher reserves today reduce future σ^2 and deepen FX market liquidity.
- Mechanism:
  - Accumulating reserves beyond first-best forces the economy to save more, reducing the likelihood that a given portfolio outflow in t+1 depletes reserves.
  - This compresses the loading of non-fundamental forces on exchange rate risk, smoothing the international risk-sharing wedge over time, at the cost of increasing it contemporaneously in t.
- Liquidity-absorbing interventions are “keeping powder dry” — smoothing over time by compressing UIP premium in the potentially constrained period t+1.

### Conditions under which liquidity-absorbing motive dominates (Proposition 2)
- Proposition 2 (condition for reserves higher than first-best when λt = 0 and Et[λt+1] > 0):

  R∗ Et[Θt+2] < 2/3 − Covt( λt+1 ( −u11,t+1 / u1,t+1 ), R∗ Et+1[Θt+2] ) / Et[ λt+1 ( −u11,t+1 / u1,t+1 ) ]

- Interpretation:
  - Liquidity-absorbing motive prevails if the stochastic discount factor between t+1 and t+2 is expected to drop sufficiently in the possibly constrained period t+1.
  - If households are expected to suffer a severe drop in tradable consumption when the LBR binds, it is optimal for the central bank to hold more reserves than first-best as precaution.
- Amplifying factors that strengthen the incentive to accumulate reserves at t:
  - Higher portfolio outflows B∗_P,t+1 increase UIP premium and the likelihood of severe consumption cuts in t+1.
  - Larger existing debt burden (weaker NFA position) increases the need for precautionary reserves.
  - Lower tradable endowment YT,t+1 reduces trade balance TBt+1 = YT,t+1 − Ct+1, reinforcing the case for holding more reserves at t.

### Quantitative analysis — calibration overview
- Model calibrated to Malaysia, quarterly frequency, data 2010–2023; central bank FX interventions used to prevent undue exchange rate fluctuations.
- Key calibrated parameter values:
  - R∗ = 1.01 (World interest rate, quarterly)
  - σ = 2 (Relative risk aversion)
  - ξ = 0.83 (Elasticity of substitution of T-NT goods)
  - α = 0.39 (Weight on traded goods in CES aggregator)
  - β = 0.9871 (Subjective discount factor, quarterly)
  - ω = 28 (Financiers’ risk aversion)
  - Targeted average FX market depth: ω σ^2 = 0.05

- Calibration notes:
  - ξ follows Bianchi (2011) and is at the upper bound of empirical estimates.
  - α equals the average tradable share of Malaysian GDP (39%).
  - β targets the Malaysian NFA-to-GDP ratio (IMF IIP statistics).
  - ω set to imply average FX market depth consistent with Davis et al. (2023).
  - Parameter values imply reasonable exchange rate reaction to capital flows: simulated regression of log change in exchange rate on portfolio outflows yields coefficient ≈ 0.6; coefficient is 1.3 in decentralized equilibrium and 0.4 under optimal time-consistent policy.

- Data construction and estimation:
  - Tradable and nontradable cyclical components derived from sectoral GDP at constant prices; agriculture, mining and quarrying, manufacturing = tradables.
  - Risk sharing wedge empirical measure RSW_t defined as in equation (14).
  - Stochastic discount factor estimate bΘ_{t+1} computed using realized consumption growth and tradable consumption components.
  - Exogenous portfolio outflows ˆB∗_P,t backed out using international risk-sharing condition (equation (15)), Malaysia’s NFA ˆB∗_t (IMF IFS), FXI proxy (Adler et al., 2025), and implied USD/MYR volatility for ˆσ^2_t.
  - Exogenous states modeled as first-order VAR: s_t = ρ s_{t−1} + ε_t, where s_t = [log YT,t, log YN,t, sinh^{-1}(BP,t − B̄P)]′.

- VAR parameter estimates and statistics (period 2010:Q1–2023:Q4):
  - Means and standard deviations:
    - Mean portfolio outflows B̄P = −3.09 (Malaysia experiences portfolio inflows on average)
    - σ_YT = 0.029
    - σ_YN = 0.037
    - σ_B∗P = 0.521
    - Correlations: σ_YT,YN = 0.795; σ_B∗P,YT = −0.036; σ_B∗P,YN = −0.147
  - VAR contemporaneous variance-covariance matrix V:
    - V = [[0.0005447, 0.0005911, 0.0019075],
           [0.0005911, 0.0008851, 0.0013138],
           [0.0019075, 0.0013138, 0.1727534]]
  - VAR autocorrelation matrix ρ:
    - ρ = [[0.8213977, −0.3171368, −0.0201376],
           [0.2110661, 0.3794069, −0.0260989],
           [−0.650205, −0.1477713, 0.4799129]]

- Numerical solution details:
  - Exogenous state discretization: 4 grid points for YT and YN each; 8 grid points for B∗_P; 1000 grid points for the endogenous state B∗.
  - Solve decentralized equilibrium and constrained planner’s problem using time iteration on the Euler equation.

*Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025261-source-pdf.pdf*

### 4.2    Policy Functions

### 4.2    Policy Functions

### Policy functions for net foreign assets
- Figure 1 compares the policy function for the small open economy’s NFA position B∗ between:
  - the decentralized equilibrium without FXI (blue solid line), and
  - the constrained planner’s solution (optimal FXI; red dashed line).
- For low levels of the current net foreign asset position, the economy increases its NFA next period; for relatively high current net foreign asset holdings, the economy decreases its NFA next period.
- Both policy-function lines intersect the 45 degree line, indicating the existence of stationary equilibria.
- Intuition:
  - Financiers’ risk aversion creates an upward-sloping supply of funds to the domestic economy, which limits external borrowing.
  - The constrained planner’s policy function lies above the decentralized-equilibrium policy function, highlighting the importance of FX interventions for the economy’s steady state NFA position.
- Exogenous state used for Figure 1: {YT, YN, B∗P} = {1.01, 1.02, −2.6}.

### Optimal FX reserves and their response to portfolio outflows
- Figure 2 shows the optimal level of FX reserves B∗M as a function of portfolio outflows B∗P, distinguishing:
  - relatively low current NFA (solid blue),
  - relatively high current NFA (dashed red), and
  - an average across all NFA states (green dotted).
- Corresponding first-best policies are indicated by black solid lines.
- Key patterns:
  - For a relatively high NFA position, optimal policy offsets portfolio outflows approximately one-for-one and deviations from the first-best policy are minor because ample FX reserves reduce the relevance of the lower bound.
  - For a relatively low NFA position, the lower bound on reserves becomes relevant: the central bank optimally exhausts its reserves in response to sufficiently large portfolio outflows.
  - Optimal reserve holdings may exceed or fall short of the first-best level depending on the state, but precautionary considerations dominate on average (the green dotted line lies above the first-best across portfolio outflow levels).
  - The flatter section of the average policy function is driven by the precautionary motive and not simply reserve depletion in some states.
- Definitions and values used in Figure 2:
  - Low NFA is defined as the state {YT, YN, B∗} = {1.01, 1.02, −3.18}.
  - High NFA is defined as the state {YT, YN, B∗} = {1.01, 1.02, 0.16}.
  - In terms of annual GDP, low and high NFA are -32% and 2%, respectively.
  - The average is computed over all NFA positions B∗, conditional on {YT, YN} = {1.01, 1.02}.
- Grid points used in numerical solution:
  - YT ∈ {0.96, 0.99, 1.01, 1.04}
  - YN ∈ {0.95, 0.98, 1.02, 1.06}
  - B∗P ∈ {−3.81, −3.58, −3.38, −3.18, −2.99, −2.79, −2.59, −2.36}.

### Remark 1 (time-consistent optimal policy)
- Under the optimal time-consistent policy:
  - Reserve holdings may lie above or below the first-best benchmark, depending on the state.
  - The average policy is shaped by a precautionary motive: conditional on the NFA position, the extent to which FXI offsets contemporaneous portfolio outflows declines with the size of the outflow shock.
- Numerical confirmation: excluding states with zero reserves from the average has a relatively minor impact on the shape of the policy function.

### 4.3    Ergodic Implications
- Simulation design:
  - A long sequence of exogenous variables {YT,t, YN,t, B∗P,t}t=1^{2×10^6} is generated based on the estimated Markov process in Section 4.1.
  - Using this sequence and the policy functions, equilibrium variables are computed for the economy without FXI and with optimally conducted FXI.
- Figure 3 findings (ergodic distributions):
  - The optimal use of FXI significantly affects the steady state: savings are higher on average, resulting in a higher net foreign asset position.
  - Mechanism: active FXI implies a positive average level of FX reserves, increasing demand for foreign currency; bond market equilibrium then requires either lower household borrowing (higher NFAB∗t) or increased intermediation by financiers (higher −B∗F,t). In the model, both occur in parallel.
  - Net foreign assets increase markedly, but not one-for-one with FX reserve holdings, indicating improved FX market conditions that allow financiers to expand balance sheets.
- Figure 4 findings (FX market conditions):
  - The unconditional distribution of FX market depth ωσ^2_t (left panel) and the real exchange rate Q_t (right panel) are shown from the 2×10^6-quarter simulation.
  - Optimal FXI policy stabilizes the exchange rate by counteracting portfolio flows, reducing the non-fundamental and inefficient component of exchange rate dynamics.
  - The planner still allows for efficient exchange rate fluctuations in response to endowment shocks; sizable exchange rate variability remains under optimal policy.
  - The reduction in exchange rate risk significantly deepens the FX market (higher FX market depth).
- Remark 2 (stochastic steady state under optimal FXI):
  - Compared to a no-FXI equilibrium, the stochastic steady state under the optimal time-consistent FXI policy is characterized by:
    - (a) a precautionary level of FX reserves and thus a higher net foreign asset position,
    - (b) deeper FX markets as the central bank acts as an FX liquidity provider.

*Source: wpiea2025261-source-pdf - 4.2    Policy Functions*

### 4.4    Dynamics

### 4.4    Dynamics

### Overview
- Analysis focuses on how portfolio flows affect the economy and how their impact depends on whether foreign exchange interventions (FXI) are deployed.
- Episodes are characterized using stochastic simulations: non-overlapping, 17-period samples with the peak outflow (inflow) in the middle period (period 0). One period corresponds to one quarter.

### Portfolio outflow episodes (Figure 5)
- Typical peak outflow at period t = 0 equals about 8% of steady-state GDP.
- Under no FX interventions (solid blue lines):
  - Fall in demand for domestic currency tightens external financial conditions and weakens the exchange rate.
  - Higher UIP premium discourages borrowing from abroad, causing a sharp contraction of consumption.
  - FX markets become shallower as conditional exchange rate volatility σ_t^2 rises, raising required compensation for risk.
- Under optimal time-consistent FXI (dashed red lines):
  - Central bank reserves fall by around 5% of GDP at the peak (t = 0) — an offset that is significant but not full.
  - Intervention reduces exchange rate depreciation and mitigates deterioration in external financing conditions, substantially reducing the fall in consumption.
  - Intervention largely eliminates a spike in FX market depth via a stronger currency and “keeping powder dry” (precautionary preservation of reserves to offset possible future outflows).

### Asymmetry across small and large outflows (Figure 6)
- Episodes of strong portfolio outflows are around four percentage points of GDP larger than small outflow episodes.
- Optimal FX intervention is similar across small and large episodes, implying a considerably bigger offset for large outflows.
- For strong outflows:
  - Central bank intervention only partially absorbs outflows; financiers increase long exposure to domestic currency (increase in B*_F,t).
- For small outflows:
  - Central bank FX sales nearly fully offset portfolio outflows; net demand for domestic currency is essentially unchanged.
- The asymmetry remains even excluding episodes where reserves are exhausted, highlighting the “keeping powder dry” motive.

### Portfolio inflow episodes (Figure 7)
- Central bank offsets contemporaneous portfolio inflows almost one-for-one, closely following the first-best policy.
  - Reason: lower bound on reserves is less relevant during inflow episodes (policy mirrors unconstrained case).
- Optimal policy stabilizes the economy during inflows and mutes real exchange rate volatility.
- Inflows deepen FX markets (contrast with outflows which shallow markets).
- Quantification (comparison of Figures 5 and 7):
  - Central bank conducts FX sales worth 4.8% of GDP between periods t = −8 and t = 0 (during outflow episodes).
  - Central bank conducts FX purchases worth 6.6% of GDP between periods t = −8 and t = 0 (during inflow episodes).
  - Real exchange rate at peak of outflows (t = 0) is 3.5% more appreciated under optimal FXI compared to decentralized equilibrium.
  - Real exchange rate at peak of inflows is around 3.4% weaker under optimal FXI.
  - FX sales have a 44% bigger impact on the real exchange rate than FX purchases in these simulations.
- Remark 3: Under the optimal time-consistent FXI policy, FX sales are more effective than FX purchases as they are implemented when FX markets are relatively shallower.

### Response to fundamental shocks (endowment decreases, Figure 8)
- Under no FXI:
  - Sharp contraction in consumption; only partially cushioned by increased borrowing from abroad.
  - Real exchange rate appreciates because fall in nontradable endowment dominates decrease in tradable consumption.
  - Portfolio capital flees the country due to slight negative correlation between endowments and portfolio outflows.
- Under optimal FXI:
  - Policy focuses on stabilizing the international risk-sharing wedge (which otherwise reflects fluctuations in portfolio capital and NFA position).
  - Policy is not effective in limiting consumption fall driven by fundamentals.
  - Effect on exchange rate adjustment is small because exchange rate facilitates efficient expenditure switching.
- Remark 4: In response to fundamental shocks, the optimal time-consistent FXI policy focuses on stabilizing the international risk-sharing wedge, otherwise allowing the exchange rate to float freely.

### Welfare implications (consumption equivalence, equation (16))
- Welfare gains expressed in consumption equivalence κ_p computed as:
  - κ_p = [ Ṽ_p(B* ) / Ṽ_b(B* ) ]^(1/(1−σ)) − 1
  - Ṽ_i denotes household lifetime utility under policy i ∈ {b, p}, conditional on initial bond holdings equal to steady-state value B* and averaged over the stationary distribution.
- Benchmark calibration: version of model calibrated to Malaysia (accounts for proxy of BNM’s FX interventions).
- Policies compared:
  - Optimal time-consistent FXI.
  - Three simple FXI rules (respect lower bound on reserves LBR): UIP premium rule (equation (17)), portfolio flow rule (equation (18)), RER rule (equation (19)).
  - No FXI (special case of RER rule with φ = 0).
  - First-best policy (upper bound, assumes away LBR).
- Two LBR levels considered:
  - LBR = 0 (relevant if only motive for holding reserves is FX market friction in model).
  - LBR = 21 percent of annual GDP (minimum level observed in Malaysia over analyzed period after adjusting for trend and seasonality). This non-zero LBR establishes the ceiling on gains achievable by optimal time-consistent FXI.

### Welfare gains (Table 2) — all values represent welfare gains in percentages, computed using equation (16)
- LBR = 0% of GDP:
  - No FXI: -0.0089
  - RER Rule: 0.0064
  - Portfolio Rule: 0.0452
  - UIP Rule: 0.1500
  - Optimal FXI: 0.2484
  - First Best: 0.2880
- LBR = 21% of GDP:
  - No FXI: -0.0089
  - RER Rule: 0.0063
  - Portfolio Rule: 0.0449
  - UIP Rule: 0.0955
  - Optimal FXI: 0.0960
  - First Best: 0.2880
- Notes on Table 2:
  - No FXI and RER rule defined by equation (19) with φ = 0 under no FXI and φ = 19.29 under RER rule (φ = 19.29 maximizes welfare). Intercept γ_RER calibrated to match average NFA in benchmark.
  - UIP and portfolio flow rules given by equations (17) and (18), respectively.

### Interpretation of welfare results
- With LBR = 0:
  - Optimal time-consistent FXI improves welfare by 0.25% of consumption, close to maximum achievable gain of 0.29% (first-best) — limited additional gains from commitment.
  - Optimal FXI significantly outperforms the three simple rules; UIP rule fares best among simple rules.
- With LBR = 21% of annual GDP:
  - Gains from optimal time-consistent FXI shrink significantly but remain sizable.
  - Simple UIP rule now delivers almost the same welfare as optimal time-consistent FXI.
- Explanation:
  - Initial NFA position used in computation equals about 2% of annual GDP.
  - Initial high reserves (in benchmark) partially crowd out domestic borrowing, yielding an excessive reserve level in the model (given only motive in model).
  - Gradual decumulation of reserves during transition contributes positively to welfare gains for policies that allow average reserves to change (all except no FXI and RER rule, which fix average reserves).
  - Higher LBR limits reserve decumulation, reducing associated gains.

### Value functions and sensitivity to initial NFA (Figure 9)
- LBR = 0:
  - Starting from a lower level of assets, optimal time-consistent FXI deviates more from first-best and the advantage over UIP rule shrinks; can become a small disadvantage for sufficiently high initial debt.
- LBR = 21%:
  - Higher LBR compresses difference between optimal policy and UIP rule.
- Key insight: When average reserves are close to LBR, value of commitment is larger and a simple UIP rule can almost match optimal time-consistent policy.

### Ergodic (unconditional) moments under alternative FXI regimes (Table 3) — LBR = 21% of annual GDP; simulations over 2×10^6 quarters; NFA and FX reserves expressed as fraction of annual GDP; UIP premium annualized
- Consumption (Mean, Std):
  - No FXI: 1.001, 0.025
  - RER Rule: 1.001, 0.025
  - Portfolio Rule: 1.000, 0.022
  - UIP Rule: 0.998, 0.020
  - Optimal FXI: 0.998, 0.020
  - First Best: 0.987, 0.023
- Net Foreign Assets (Mean, Std):
  - No FXI: 0.020, 0.008
  - RER Rule: 0.020, 0.012
  - Portfolio Rule: -0.006, 0.008
  - UIP Rule: -0.061, 0.014
  - Optimal FXI: -0.059, 0.011
  - First Best: -0.337, 0.007
- FX Reserves (Mean, Std):
  - No FXI: 0.328, 0.000
  - RER Rule: 0.327, 0.045
  - Portfolio Rule: 0.301, 0.043
  - UIP Rule: 0.247, 0.037
  - Optimal FXI: 0.248, 0.036
  - First Best: -0.029, 0.044
- Real Exchange Rate (Mean, Std):
  - No FXI: 0.390, 0.014
  - RER Rule: 0.389, 0.009
  - Portfolio Rule: 0.390, 0.011
  - UIP Rule: 0.392, 0.012
  - Optimal FXI: 0.392, 0.012
  - First Best: 0.400, 0.011
- FX Market Depth (Mean, Std):
  - No FXI: 0.078, 0.012
  - RER Rule: 0.034, 0.005
  - Portfolio Rule: 0.047, 0.008
  - UIP Rule: 0.057, 0.009
  - Optimal FXI: 0.054, 0.008
  - First Best: 0.042, 0.010
- UIP Premium (Mean, Std):
  - No FXI: 0.007, 0.120
  - RER Rule: 0.011, 0.059
  - Portfolio Rule: 0.012, 0.012
  - UIP Rule: 0.012, 0.029
  - Optimal FXI: 0.012, 0.027
  - First Best: 0.000, 0.000
- Binding LBR Frequency:
  - No FXI: 0.000
  - RER Rule: 0.000
  - Portfolio Rule: 0.000
  - UIP Rule: 0.181, 0.049
  - Optimal FXI: – (not reported)
  - First Best: – (not applicable)
- Notes: UIP premium is annualized.

### Interpretation of ergodic results
- Optimal time-consistent FXI yields precautionary accumulation of FX reserves (average reserves slightly higher than UIP rule), reducing frequency of hitting LBR.
- Optimal FXI leads to deeper FX markets and lower volatility of risk-sharing wedge compared to UIP rule.
- Portfolio rule leads to very high FX reserves (due to average positive appetite for domestic currency and negative NFA from impatient households βR* < 1), virtually eliminating reserve depletion risk, deepening markets, lowering exchange rate volatility, but at cost of low international borrowing (nearly balanced NFA), yielding modest welfare gains.
- First-best:
  - No precautionary reserve accumulation; typically borrows from abroad to accommodate domestic borrowing needs, yielding lower NFA than other regimes.
  - Perfectly eliminates risk-sharing wedge but does not stabilize exchange rate beyond portfolio rule due to expenditure-switching motive under fundamental shocks.
  - FX markets deeper than other policies, but not perfectly deep.
- RER rule:
  - Delivers most stable real exchange rate and highest market depth.
  - Generates highest UIP premium volatility and implies excessive FX reserves.
  - Consequently, RER rule ranks worst among regimes actively using FXI.
  - Highlights importance: FXI should target inefficient volatility in international risk-sharing wedge, not respond directly to exchange rate driven by fundamentals.

### Concluding welfare remark
- Remark 5: If initial stock of FX reserves is far from effective lower bound, optimal time-consistent FXI delivers robust welfare gains over simple rules, with performance close to first-best. If reserves are initially small, optimal time-consistent policy achieves welfare very similar to a simple rule targeting ex-ante UIP deviations.

### Final takeaways (Conclusions)
- Optimal time-consistent FXI can effectively reduce exchange rate volatility caused by portfolio flow shocks, shrinking UIP deviations and improving FX market depth.
- Optimal response depends on expected path of portfolio flows and existing net foreign asset position:
  - Strong NFA position and/or moderate expected outflows: holding FX reserves below first-best can be optimal.
  - Large and persistent outflows with weak NFA: precautionary accumulation of reserves is optimal.
- Effectiveness of FXI is state-dependent:
  - FX purchases (buying foreign currency) have lower impact on exchange rate during inflows (markets deeper).
  - FX sales (selling reserves) are more effective during outflows (markets shallower).
- Welfare gains from optimal time-consistent FXI are substantial when starting far from LBR; committing to a UIP-targeting rule also improves welfare but less so. Value of commitment increases when reserves are relatively low.
- Emphasizes precautionary motive for reserves and state dependency in FXI conduct; suggests future research directions including additional frictions or endogenizing effective LBR.

*Italic source: Content extracted from "4.4    Dynamics" of the supplied IMF PDF chapter.*

### References

### References and Technical Appendices (wpiea2025261-source-pdf - References)

### Major bibliographic sources
- Lists journal articles, working papers, IMF publications, and memos relevant to foreign exchange intervention, reserve management, integrated policy frameworks, capital flows, and related macroprudential and open-economy theories.
- Key recurring authors and institutions include Adler, Bianchi, Basu, Gopinath, IMF, CEPR, NBER, SNB, and major journals such as Journal of Money, Credit and Banking; Emerging Markets Review; The Review of Economic Studies; American Economic Journal: Macroeconomics; Journal of Political Economy; Journal of International Economics; and others.

### Appendix A — Proofs and analytical results
- Proposition 1
  - Conditional exchange rate volatility σ^2_t expression and its partial derivatives with respect to consumption C_T,t and portfolio outflows B^*_P,t:
    - ∂σ^2_t/∂C_T,t = 2( u_11,t / u_1,t ) σ^2_t.
    - ∂σ^2_t/∂B^*_P,t = 2( u_11,t / u_1,t ) σ^2_t (−∂B^*_t/∂B^*_P,t ) < 0.
  - Contradiction argument establishes ∂E_t[Θ_{t+1}]/∂B^*_P,t < 0 given ∂B^*_t/∂B^*_P,t > 0.
- Derivation of ∂B^*_t/∂B^*_P,t
  - IRS condition differentiated with respect to B^*_P,t yields algebraic expression:
    - ∂B^*_t/∂B^*_P,t = 1 / { 1 − (−u_11,t / u_1,t) [ 3(B^*_P,t − B^*_t) − 1/(ωσ^2_t) ] }.
  - Sufficient condition for ∂B^*_t/∂B^*_P,t > 0:
    - (−u_11,t / u_1,t) [ 3(B^*_P,t − B^*_t) − 1/(ωσ^2_t) ] < 1.
- Theorem 1 (second-best Lagrangian and first-order condition)
  - Lagrangian L = E_t Σ_{s=0}^∞ β^{t+s} [ u( B^*_{t+s−1}/R^* − B^*_{t+s} + Y_{T,t+s}, Y_{N,t+s} ) − λ_{t+s} ( B^*_{P,t+s} + 1/(ωσ^2_{t+s}) (R^* E_{t+s}[Θ_{t+s+1}] − 1) − B^*_{t+s} ) ].
  - First-order condition with respect to B^*_t leads, after substitutions and use of IRS conditions, to closed-form expression for B^*_{M,t}:
    - B^*_{M,t} = B^*_t − B^*_{P,t} + βR^* ω σ^2_t u_{1,t} E_t[ λ_{t+1} ( −u_{11,t+1}/u_{1,t+1} )^3 ( B^*_{P,t+1} − B^*_{t+1} ) − 1/(ωσ^2_{t+1}) ].
- Proposition 2 (when optimal reserves exceed first-best)
  - Condition B^*_{M,t} > B^*_t − B^*_{P,t} is equivalent to an expectation and covariance inequality:
    - R^* E_t[Θ_{t+2}] < 2/3 − cov( λ_{t+1} (−u_{11,t+1}/u_{1,t+1}), R^* E_{t+1}[Θ_{t+2}] ) / E_t[ λ_{t+1} (−u_{11,t+1}/u_{1,t+1}) ].
- Derivations and analytical expressions provided for:
  - Absolute risk aversion: −u_{11,t} / u_{1,t} = (σ ξ − 1)/ξ α C_t^{1−ξ/ξ} C_{T,t}^{−1/ξ} + (1/ξ) C_{T,t}^{−1}.
  - Price index P_t: P_t = [ α^ξ E_t^{1−ξ} + (1−α)^ξ ]^{1/(1−ξ)}.
  - Real exchange rate Q_t: Q_t = [ α^ξ + (1−α)^ξ (1/E_t)^{1−ξ} ]^{−1/(1−ξ)} (price abroad normalized to unity).

### Appendix B — Calibration details and estimated exogenous process
- Observed FXI regime represented by combined exogenous state B̂_PM,t = B̂_P,t + B̂_M,t; baseline B_PM = 0.24.
- AR(1) process estimated for vector s̃_t = [ log Y_{T,t}, log Y_{N,t}, sinh^{-1}( B̂_PM,t − B_PM ) ]′:
  - s̃_t = ρ̃ s̃_{t−1} + ε̃_t, where ε̃_t follows trivariate normal with zero mean and contemporaneous variance-covariance matrix Ṽ and autocorrelation matrix ρ̃.
- Estimated contemporaneous variance-covariance matrix Ṽ:
  - Ṽ = [ [0.0005258, 0.0005685, −0.000789], [0.0005685, 0.0008582, −0.002274], [−0.000789, −0.002274, 0.1716685] ].
- Estimated autocorrelation matrix ρ̃:
  - ρ̃ = [ [0.829771, −0.414713, −0.024469], [0.220326, 0.2561583, −0.031291], [−1.15058, −0.708053, 0.441873] ].
- Empirical correlation between FX interventions and portfolio outflows: σ_{B^*_P,B^*_M} = −0.3077.
- Calibration targets and parameter values:
  - Subjective discount factor β chosen to match mean of observed net foreign asset position as a percentage of annual GDP = 2% → β = 0.9871.
  - Target average FX market depth ω σ^2 = 0.05 achieved with financiers’ risk aversion ω = 28.
  - Note: changing ω affects the estimated exogenous portfolio outflow process B^*_P,t and the risk sharing wedge; higher ω makes FX markets more shallow and magnifies effects of portfolio outflows.

### Appendix C — Simulation details and counterfactuals
- Consumption equivalence κ (additional fraction of consumption to make benchmark households indifferent to alternative policy):
  - κ = [ V^{ap}(B^*,S) / V^b(B^*,S) ]^{1/(1−σ)} − 1.
  - Value functions weighted by stationary distribution: Ṽ_i(B^*) = Σ_{S∈S} ψ_S V_i(B^*,S) with Σ_{S∈S} ψ_S = 1.
  - Final expression used for κ with steady state net foreign asset position B^* as initial condition:
    - κ = [ Ṽ_{ap}(B^*,S) / Ṽ_b(B^*,S) ]^{1/(1−σ)} − 1.
- First-best planner problem (given exogenous states S = { Y_T, Y_N, B^*_P }):
  - V(B^*,S) = max_{B^*′} { u( R^* B^* + Y_T − B^*′, Y_N ) + β E_S V(B^*′, S′) } subject to B^* ≤ B^*′ ≤ R^* B^* + Y_T and market clearing.
  - Lower bound B^* imposed in all policy regimes though it is nonbinding except potentially under first-best where central bank can take negative foreign bond positions.
- Additional simulation outputs (described figures; numeric timing)
  - Figure C.1: average model response to episodes where tradable endowment decreases relative to nontradable endowment; one period = one quarter; period 0 = tradable endowment trough.
  - Figure C.2: average model response during capital outflow episodes under different policy regimes; one period = one quarter; period 0 = peak outflow.

*Source: wpiea2025261-source-pdf - References (Appendices A–C).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025261-source-pdf.pdf_