## 4. Optimal Policy Determinacy Condition

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### Introduction: motivation and scope
- Many central banks combine price stability and some degree of exchange rate stability using the policy interest rate and sterilized FX intervention.
- Canonical New Keynesian model features the “divine coincidence” (stable inflation and zero output gap coincide) and hence no role for FX intervention; this paper examines whether international capital market shocks create a micro-founded rationale for sterilized FX intervention and how intervention affects the credibility of inflation targeting.
- Framework incorporates three key features:
  - shocks in international capital markets that vary the cost of capital to the economy;
  - financial frictions in international capital markets that break uncovered interest parity (UIP), rendering sterilized intervention effective via a portfolio balance channel;
  - a working-capital requirement for firms that amplifies capital market shocks on the real economy.

### Main contributions and findings
- Microfoundation:
  - Sterilized intervention is effective because domestic and foreign investors internalize risk premia differently (domestic investors expect a bail-out; foreign investors expect a bail-in), breaking UIP.
- Closed-form discretionary optimal policy:
  - FX intervention can “lean against the wind” and offset changes in foreign investors’ risk appetite, but intervention carries welfare costs (affecting international asset position) so the central bank only partially offsets shocks.
  - Welfare benefits of using FX intervention in addition to interest rate policy are sizable—around 0.4 percent of permanent consumption for a standard calibration.
  - Benefits are null for fully closed economies (α→0) or for economies where all goods are imported (α→1 with domestic goods share zero), and maximal when two thirds of the consumption basket is imported (α such that two thirds imported).
- Determinacy:
  - The financial accelerator matters for equilibrium indeterminacy; availability and willingness to use both interest rate policy and FX intervention reduce the zone of equilibrium indeterminacy and the risk of self-fulfilling exchange rate bubbles.

### Model framework and mechanics
- Small open economy New Keynesian model extended with:
  - endogenous risk premium Ψt as a function of reserves (∫1 0 Ei,t Ri,t di / R̄)^φ where φ is the elasticity of the risk premium to reserves and R̄ is equilibrium reserves with no risk premium;
  - working-capital requirement: firms must borrow a fixed fraction μ ∈ [0,1] of the nominal value of intended output at the beginning of each period, charged interest it;
  - household preferences:
    - U(Ct, Nt) = Σ_{t=0}^∞ β^t [Ct^{1−σ}/(1−σ) − Nt^{1+φ}/(1+φ)];
    - Ct = [ (1−α)^{1/η} CH,t^{(η−1)/η} + α^{1/η} CF,t^{(η−1)/η} ]^{η/(η−1)} with openness coefficient α.
- Financial structure:
  - domestic households can only hold domestic bonds (Dh_{t−1}) and sterilization bonds (Sb_{t−1}) paying home interest rate it−1;
  - foreign investors face a perceived gross return 1 + it Ξt that differs from 1 + it due to VIX-type shock Ξt;
  - exogenous shock Ξt captures changes in foreign investors’ interest for local bonds (VIX shocks), generating risk premium effects that specifically affect domestic consumption.
- Central bank:
  - chooses interest rate and performs sterilized FX intervention; sterilization keeps money supply unchanged via issuance of sterilization bonds Sb_t, with balance condition:
    - ∫1 0 Ei,t Ri,t di − ∫1 0 Ei,t (1 + i^{i}_{t−1}) Ξ_{t−1} R_{i,t−1} di = Sb_t − (1 + i_{t−1}) Sb_{t−1}.
- Government:
  - partial default on domestic-held bonds produces lump-sum bailout tax Tfin_t = (Ψ_{t−1}/Ψ_t) (Dh_t + Sb_t) that finances the domestic investor bail-out; Tgov_t = τ Wt Nt rebates labor tax revenues.

### Asymmetric investor perceptions and UIP
- Foreign investors rationally expect to be bailed in; UIP for a foreign investor in country i:
  - 1 + it Ψt Ξt = (1 + i^i_t) E_{i,t+1} / E_{i,t}
- Domestic households expect to be bailed out; they perceive the risk premium as increasing their effective return in the domestic Euler equation.
- Consequence: FX intervention affects returns faced by foreign investors but not domestic households’ perceived return, creating a wedge between domestic and foreign consumption plans (Backus-Smith distortion).

### Policy intuition: why FX intervention can be useful
- A positive shock to Ξt (lower foreign appetite) requires foreign investors to be compensated (higher Ψt). Home central bank options:
  - accept a large instantaneous currency depreciation (and subsequent appreciation);
  - increase the interest rate;
  - sell foreign exchange reserves to lower the risk premium Ψt and appreciate the currency.
- FX intervention targets domestic consumption directly (it affects the wedge between domestic and foreign consumption) and can therefore better offset VIX-type shocks that monetary policy alone cannot fully address.

### Equilibrium conditions and key relations (log-linearized)
- Backus-Smith-type relation:
  - Ct = Θt C∗_t Q_t^{1/σ}
- Law of motion for Θt:
  - (Θ_{t+1}/Θ_t)^σ = (1 + it)/(1 + i∗_t) (E_t / E_{t+1}) = Ψt Ξt
- Goods market (Cole-Obstfeld parametrization σ = η = γ = 1):
  - y_t = (1−α) [ −(q_t − s_t) + c_t ] + α s_t = (1−α) θ_t + s_t
  - IS equation:
    - y_t = y^e_{t+1} − (i_t − π^e_{H,t+1} − ρ) + α(ψ_t + ξ_t)
  - UIP (log-linearized):
    - e_t = e^e_{t+1} − (i_t − i^∗_t) + ψ_t + ξ_t
- New Keynesian Phillips Curve:
  - π_{H,t} = β π^e_{H,t+1} + κ_y y_t + κ_θ (ĥθ_t + ̄θ_t) + κ_f (i_t − ρ)
    - κ_y = λ(1 + φ), κ_θ = λ α, κ_f = λ μ.

### Decomposition of the consumption wedge and channels
- θ_t = ĥθ_t + ̄θ_t:
  - ̄θ_t: distortion from exogenous VIX shock ξ_t;
  - ĥθ_t: distortion from FX intervention’s risk premium ψ_t.
- Demand channel: θ_t affects aggregate demand and Home prices; effect on y_t proportional to (1−α).
- Supply channel: ̄θ_t > 0 raises real wages and marginal costs, increasing inflation; FX intervention can counteract by making ĥθ_t < 0 and depressing consumption.
- Financial accelerator: κ_f (i_t − ρ) links interest rate and exchange rate expectations to marginal costs; contributes to indeterminacy when present.

### Intertemporal budget constraint and implications
- Intertemporal budget constraint (equation (17)):
  - ∞∑t=0 β^t ˆθ_t = 0.
- Selling reserves decreases the endogenous risk premium and increases domestic consumption, worsening the trade balance (nx_t = −α( ̄θ_t + ˆθ_t)); a larger trade deficit must be compensated by larger surpluses in the future, so FX intervention cannot permanently raise consumption.
- Central bank is assumed to value the international position at a constant (shadow) price even under discretion.

### Welfare loss function and decomposition of FX intervention costs
- Welfare loss (second-order approximation):
  - (1/2) ∞∑t=0 β^t [ α_π π_{H,t}^2 + y_t^2 + α_θ [ˆθ_t + α_ψ ̄θ_t]^2 ].
- Definitions preserved:
  - α_π = ε_P λ_P (1 + φ) / P (as given in text),
  - α_θ = α/(1+φ) (2−α/(1−α) + (1−α)),
  - α_ψ = (1−α)/(2−α) (1−α + 1−α) — (preserve exact phrasing and symbols as in source).
- Decomposition of α_θ [ˆθ_t + α_ψ ̄θ_t]^2:
  - α_θ ˆθ_t^2 — cost of external imbalance.
  - 2 α_θ α_ψ ˆθ_t ̄θ_t — costs of distorting consumption.
  - plus exogenous terms.
- Sources of welfare costs:
  - Cost of external imbalances: selling reserves worsens trade balance; welfare cost proportional to α_θ, increasing in α; welfare cost → ∞ as α → 1 given output gap cost normalized to 1.
  - Costs of distorting consumption: FX intervention stabilizes consumption and terms of trade but distorts consumption; welfare loss proportional to α_θ α_ψ = α(1−α)/(1+φ).

### Planning problem and optimal stabilization policy (risk premium shocks, μ = 0)
- Planner minimizes α_π π_{H,t}^2 + y_t^2 + α_θ [ˆθ_t + α_ψ ̄θ_t]^2 subject to model constraints and ∞∑t=0 β^t ˆθ_t = 0.
- Proposition 1 (for ξ_t = ξ_0 ρ^t, μ = 0) gives closed-form optimal paths:
  - π_{H,t} = κ_θ (1−α_ψ) ξ_0/(1−ρ) [ (1−β/(1−ρβ)) 1/D(α_θ,1) − (1/D(α_θ,ρ)) ρ^t ] + κ_θ α_θ D(α_θ,1) / Γ
  - y_t = −α_π/κ_y π_{H,t}
  - ˆθ_t = ξ_0/(1−ρ) [ 1 − (1−α_ψ) D(+∞,ρ)/D(α_θ,ρ) ] ( ρ^t − (1−β)/(1−ρ) ξ_β )
  - ψ_t = ξ_0 [ (1−α_ψ) D(+∞,ρ)/D(α_θ,ρ) − 1 ] ρ^t
  - e_{t+1} − e_t = i_t − i^*_t − (ψ_t + ξ_t), with e_0 = y_0 − (1−α)(ˆθ_0 + ̄θ_0)
- D(α_θ, ρ_ξ) = 1 − ρ_ξ β + α_π ( κ_y^2 + κ_θ^2/α_θ )
- Interpretation:
  - Positive risk premium shock decreases domestic consumption in short run by increasing domestic interest rate and lowers inflation even if output stabilized due to Home bias.
  - Without FX intervention: increase interest rate and let currency depreciate instantly.
  - With FX intervention (selling reserves): reduces endogenous risk premium, lowering the interest rate faced by domestic households; limits required interest rate hike and exchange rate devaluation; stabilizes output, inflation, and supports domestic consumption.
- Decomposition of ψ_t:
  - ψ_t = −[ α_ψ (terms of trade effect) + (1−α_ψ) (1 − D(+∞,ρ)/D(α_θ,ρ)) (trade-off between stabilizing consumption and generating external imbalance) ] ξ_t.
  - Ratio D(+∞,ρ)/D(α_θ,ρ) (always < 1) measures FX intervention efficiency.

### Simulations: calibration and quantitative results
- Calibration:
  - σ = γ = η = 1.
  - φ = 3 → Frisch elasticity = 1/3.
  - Price mark-up = 1.2 → ε = 6.
  - β = 0.96.
  - δ = 0.75.
  - α = 0.2.
  - μ = 0.2.
  - Exogenous shock: 5% risk premium shock with half-life of 2 years.
- Comparative dynamics (monetary policy only vs. monetary + FX intervention):
  - Monetary policy only:
    - Risk premium shock → large exchange rate depreciation and central bank increases interest rate; large trade surplus and large drop in domestic consumption; small net effects on domestic output and PPI inflation.
  - With FX intervention:
    - Central bank sells foreign currency reserves reducing endogenous risk premium by about 1.4% initially.
    - Impact on domestic consumption: fall mitigated (3.5% vs. 4.8% without FX intervention).
    - Trade surplus on impact smaller (2.2% vs. 3%).
    - Exchange rate depreciation on impact smaller (9.2% vs. 12.7%).
- Welfare gains:
  - Steady-state consumption gains from FX intervention around 0.4 percent of permanent consumption for α = 0.2.
  - FX intervention cancels about 40 percent of the welfare losses for α = 0.2.
  - As α → 1, FX intervention becomes useless and welfare losses identical with or without intervention.

### FX intervention and speculative attacks — determinacy under financial accelerator
- Reduced problem for speculative shocks:
  - min α_π π_{H,t}^2 + y_t^2 + α_θ ˆθ_t^2 subject to model dynamics and ∞∑t=0 β^t ˆθ_t = 0.
- First-order conditions (y_t and θ_t) lead to a law of motion for inflation that preserves coefficients.
- Key mechanism:
  - With financial friction (κ_f > 0), increasing the interest rate raises firms’ marginal costs; the recession required to lower inflation must be deeper or inflation weight larger to ensure determinacy.
- Proposition 2 — Equilibrium Determinacy under Discretionary Policy:
  - Blanchard-Kahn determinacy condition (equation (25)):
    - α_π > [ β + κ_f − 1 ] / [ κ_y (κ_y − κ_f) + κ_θ (κ_θ − α κ_f) / α_θ ]
  - Without FX intervention (equation (26)):
    - α_π > [ β + κ_f − 1 ] / [ κ_y (κ_y − κ_f) ]
  - Interpretation:
    - Define X_y = α_π (κ_y − κ_f): size of output loss engineered by central bank when inflation is 1 percent.
    - Determinacy requires κ_y X_y > β + κ_f − 1.
    - If κ_f = 0, β − 1 < 0 ensures uniqueness; with financial accelerator, indeterminacy can arise if κ_f is large relative to 1−β.
  - With FX intervention reintroduced:
    - Define X_θ = α_π/α_θ (κ_θ − α κ_f) > 0.
    - Determinacy condition (equation (27)):
      - κ_y X_y + κ_θ X_θ > β + κ_f − 1.
    - Rationale: central bank can use reserves to increase the risk premium (accumulate reserves) to depreciate the currency for a given interest rate increase; this mitigates current inflation.
    - If reserves are costless (α_θ = 0) or reserves have a strong effect on inflation (X_θ high), indeterminacy can be eliminated even without aggressive conventional monetary policy.
- Edge cases:
  - If (κ_y − κ_f)(2 κ_f − κ_y) + (κ_θ − α κ_f)(2 α κ_f − κ_θ)/α_θ > 0 (implying μ > 1/2), the condition has an upper bound (source presents an upper bound inequality).
  - As κ_f increases, the risk of multiple equilibria increases; FX intervention reduces the parameter range for which indeterminacy occurs.

### Policy-relevant takeaways (conclusions)
- FX intervention can be an effective stabilization tool in the presence of exogenous international risk premium shocks:
  - It stabilizes domestic consumption, exchange rate, and trade balance.
  - It can yield meaningful steady-state consumption gains (example: ≈ 0.4 percent of permanent consumption for α = 0.2) and offset a substantial share of welfare losses (≈ 40 percent for α = 0.2).
- FX intervention creates external imbalances that generate future welfare costs; these costs grow with openness (α), and in the limit α → 1 the intervention becomes ineffective.
- The presence of a financial accelerator (working capital/frictions) can generate equilibrium indeterminacy under discretionary policy — a possibility absent in the standard open-economy New Keynesian model.
- FX intervention reduces the range of parameters for which indeterminacy occurs:
  - If FX intervention is very effective (large X_θ) or the central bank has strong preferences for using FX intervention (small α_θ), indeterminacy can be ruled out even without an aggressive conventional monetary policy stance.

*Source: IMF Working Paper “4. Optimal Policy Determinacy Condition” (excerpt, wp17207).*

### References  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

### References

### Appendices
- I.First-order Conditions for Calvo Price Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
- II.Derivation of the Intertemporal Budget Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
- III.Derivation of the Loss Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
- IV.Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

### Figures
- 1. The Effects of a Risk-Premium Shock under Optimal Discretionary Policy. . . . . . . . . . . . . . . . . . 37
- 2. Welfare Losses following an Exogenous Risk Premium Shock (5 percent shock, two years’ half-life). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
- 3. Welfare Gains from using FX Intervention following an Exogenous Risk Premium Shock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

*Source: wp17207 - References (PDF).*

### 4. Optimal Policy Determinacy Condition .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 

### 4. Optimal Policy Determinacy Condition

### Introduction: motivation and scope
- Many central banks combine price stability and some degree of exchange rate stability using the policy interest rate and sterilized FX intervention.
- Canonical New Keynesian model features the “divine coincidence” (stable inflation and zero output gap coincide) and hence no role for FX intervention; this paper examines whether international capital market shocks (e.g., “risk-on/risk-off” episodes) create a micro-founded rationale for sterilized FX intervention and how intervention affects the credibility of inflation targeting.
- Framework incorporates three key features:
  - shocks in international capital markets that vary the cost of capital to the economy;
  - financial frictions in international capital markets that break uncovered interest parity (UIP), rendering sterilized intervention effective via a portfolio balance channel;
  - a working-capital requirement for firms that amplifies capital market shocks on the real economy.
- Paper focuses on the role of sterilized FX intervention under discretionary policy and on equilibrium determinacy (reducing the zone where speculative attacks/self-fulfilling exchange rate moves are possible).

### Illustrative episodes
- Czech National Bank (CNB): signaled consideration of FX intervention in September 2012; in November 2013 calibrated exchange rate target to its effect on aggregate demand and inflation and began intervening—intervention anchored exchange rate and inflation expectations and boosted aggregate demand (Alichi et al. (2015)).
- Swiss National Bank (SNB): reserves swelled above US$ 500 billion in 2013; in January 2015 SNB cut its policy rate from -0.25 to -0.75 percent after reserve accumulation became costly and revaluation expectations rose.

### Main contributions and findings
- Threefold contribution:
  - Microfoundation: a simple model where sterilized intervention is effective because domestic and foreign investors internalize risk premia differently (domestic investors expect a bail-out; foreign investors expect a bail-in), breaking UIP.
  - Closed-form discretionary optimal policy: FX intervention can “lean against the wind” and offset changes in foreign investors’ risk appetite, but intervention carries welfare costs (affecting international asset position) so the central bank only partially offsets shocks. Welfare benefits of using FX intervention in addition to interest rate policy are sizable—around 0.4 percent of permanent consumption for a standard calibration. Benefits are:
    - null for fully closed economies (α→0) or for economies where all goods are imported (α→1 with domestic goods share zero),
    - maximal when two thirds of the consumption basket is imported (α such that two thirds imported).
  - Determinacy: the financial accelerator matters for equilibrium indeterminacy; availability and willingness to use both interest rate policy and FX intervention reduce the zone of equilibrium indeterminacy and the risk of self-fulfilling exchange rate bubbles.

### Model framework and mechanics
- Small open economy New Keynesian model extended with:
  - endogenous risk premium Ψt as a function of reserves (∫1 0 Ei,t Ri,t di / R̄)φ where φ is the elasticity of the risk premium to reserves and R̄ is equilibrium reserves with no risk premium;
  - working-capital requirement: firms must borrow a fixed fraction μ ∈ [0,1] of the nominal value of intended output at the beginning of each period, charged interest it;
  - household preferences:
    - utility U(Ct, Nt) = Σ_{t=0}^∞ β^t [Ct^{1−σ}/(1−σ) − Nt^{1+φ}/(1+φ)];
    - aggregate consumption Ct = [ (1−α)^{1/η} CH,t^{(η−1)/η} + α^{1/η} CF,t^{(η−1)/η} ]^{η/(η−1)} with openness coefficient α.
- Financial structure:
  - domestic households can only hold domestic bonds (Dh_{t−1}) and sterilization bonds (Sb_{t−1}) paying home interest rate it−1;
  - foreign investors can buy domestic or foreign bonds and face a perceived gross return 1 + it Ξt that differs from 1 + it due to the VIX-type shock Ξt;
  - exogenous shock Ξt captures changes in foreign investors’ interest for local bonds (VIX shocks), generating risk premium effects that specifically affect domestic consumption.
- Central bank: chooses interest rate and performs sterilized FX intervention; sterilization keeps money supply unchanged via issuance of sterilization bonds Sb_t, with the central bank balance condition:
  - ∫1 0 Ei,t Ri,t di − ∫1 0 Ei,t (1 + i^{i}_{t−1}) Ξ_{t−1} R_{i,t−1} di = Sb_t − (1 + i_{t−1}) Sb_{t−1}.
- Government: issues bonds to domestic and foreign investors; partial default on domestic-held bonds produces lump-sum bailout tax Tfin_t = (Ψ_{t−1}/Ψ_t) (Dh_t + Sb_t) that finances the domestic investor bail-out; Tgov_t = τ Wt Nt rebates labor tax revenues.

### Asymmetric investor perceptions and UIP
- Foreign investors: rationally expect to be bailed in; UIP for a foreign investor in country i:
  - 1 + it Ψt Ξt = (1 + i^i_t) E_{i,t+1} / E_{i,t}
- Domestic households: expect to be bailed out; they perceive the risk premium as increasing their effective return in the domestic Euler equation; lump-sum tax Tfin_t finances the bailout.
- Consequence: FX intervention affects returns faced by foreign investors but not the domestic households’ perceived return, creating a wedge between domestic and foreign consumption plans (Backus-Smith distortion).

### Why FX intervention can be useful (policy intuition)
- A positive shock to Ξt (lower foreign appetite) requires foreign investors to be compensated (higher Ψt). The Home central bank has three strategies:
  - accept a large instantaneous currency depreciation (and subsequent appreciation);
  - increase the interest rate;
  - sell foreign exchange reserves to lower the risk premium Ψt and appreciate the currency.
- FX intervention targets domestic consumption directly (it affects the wedge between domestic and foreign consumption) and can therefore better offset VIX-type shocks that monetary policy alone cannot fully address.
- Empirical support: interventions that reduce reserves can protect consumption by limiting required depreciation and interest rate increases (Jeanne and Rancière (2011)).

### Equilibrium conditions and key log-linearized relations
- Backus-Smith-type relation (definition of Pareto weight Θt linking domestic consumption to foreign consumption and real exchange rate):
  - Ct = Θt C∗_t Q_t^{1/σ}
- Law of motion for Θt (no-arbitrage for foreign households):
  - (Θ_{t+1}/Θ_t)^σ = (1 + it)/(1 + i∗_t) (E_t / E_{t+1}) = Ψt Ξt
- Goods market (log-linearized IS representation under Cole-Obstfeld parametrization σ = η = γ = 1):
  - y_t = (1−α) [ −(q_t − s_t) + c_t ] + α s_t = (1−α) θ_t + s_t
  - IS equation (demand side):
    - y_t = y^e_{t+1} − (i_t − π^e_{H,t+1} − ρ) + α(ψ_t + ξ_t)
    - alternative form: y_t = y^e_{t+1} − (i^∗_t − π^e_{H,t+1} − ρ) + e_t − e^e_{t+1} − (1−α)(ψ_t + ξ_t)
  - UIP (log-linearized):
    - e_t = e^e_{t+1} − (i_t − i^∗_t) + ψ_t + ξ_t
- New Keynesian Phillips Curve (supply side):
  - π_{H,t} = β π^e_{H,t+1} + κ_y y_t + κ_θ (ĥθ_t + ̄θ_t) + κ_f (i_t − ρ)
    - where κ_y = λ(1 + φ), κ_θ = λ α, κ_f = λ μ; κ_θ captures openness effects on the supply channel of risk premia, κ_f captures financial accelerator (cost channel).

### Decomposition of consumption wedge and effects
- Pareto weight distortion θ_t decomposed as:
  - θ_t = ĥθ_t + ̄θ_t
    - ̄θ_t: distortion from exogenous VIX shock ξ_t;
    - ĥθ_t: distortion from FX intervention’s risk premium ψ_t.
- Demand channel: θ_t affects aggregate demand and thus Home prices and terms of trade; effect on y_t is proportional to (1−α).
- Supply channel: ̄θ_t > 0 (positive shock to domestic consumption) raises real wages and firms’ marginal costs, increasing inflation; FX intervention can counteract by increasing endogenous risk premium (ĥθ_t < 0) and depressing consumption.
- Financial accelerator: term κ_f (i_t − ρ) links interest rate and exchange rate expectations to marginal costs; exchange rate expectations can have real effects and contribute to indeterminacy in presence of financial frictions.

### Policy implications and determinacy
- Sterilized FX intervention is a potentially valuable second instrument when:
  - exogenous shocks to foreigners’ risk appetite create a wedge between domestic and foreign consumption that monetary policy alone cannot offset (breaks divine coincidence);
  - financial frictions (financial accelerator) increase the risk of equilibrium indeterminacy and self-fulfilling exchange rate movements.
- Under discretionary policy:
  - availability of FX intervention helps reduce the “stabilization bias” associated with lack of commitment (Clarida, Gali and Gertler (1999)); the benefit depends on degree of openness α;
  - central banks more willing to use both interest rate policy and FX intervention are less likely to face risks of equilibrium indeterminacy.
- Welfare gains and limits:
  - welfare benefits from using FX intervention (in addition to interest rate policy) are around 0.4 percent of permanent consumption for a standard calibration;
  - benefits are null for fully closed economies or for economies where all goods are imported, and maximal when two thirds of the consumption basket is imported;
  - FX intervention carries welfare costs because it affects the international asset position and is limited by budget/Reserve constraints.

*Source: IMF Working Paper “4. Optimal Policy Determinacy Condition” (excerpt).*

### Section V, we explore the impact of the financial accelerator on equilibrium determinacy

### wp17207 - Section V, we explore the impact of the financial accelerator on equilibrium determinacy

### Intertemporal budget constraint and implications for FX intervention
- Intertemporal budget constraint (equation (17)): ∞∑t=0 β^t ˆθ_t = 0.
- The intertemporal budget constraint requires that the discounted value of the distortions to domestic consumption be zero.
- When the central bank sells reserves:
  - The endogenous risk-premium decreases and domestic consumption increases, worsening the trade balance (nx_t = −α( ̄θ_t + ˆθ_t)).
  - A larger trade deficit must be compensated by larger surpluses in the future; hence FX intervention cannot permanently raise the path of consumption.
- Assumption on policy institutions: even under discretionary policy, the central bank values the international position of the country at a constant (shadow) price (the intertemporal budget constraint is taken into account by the central bank).
- Parameter relation noted: λ = (1−δ)(1−βδ) / δ. As a result, λ → ∞ when prices are fully flexible.

### Welfare loss function and decomposition of FX intervention costs
- Welfare loss (second-order approximation):
  - (1/2) ∞∑t=0 β^t [ α_π π_{H,t}^2 + y_t^2 + α_θ [ˆθ_t + α_ψ ̄θ_t]^2 ].
- Definitions preserved as in text:
  - α_π = ε_P λ_P (1 + φ) / P (as given in text presentation),
  - α_θ = α/(1+φ) (2−α/(1−α) + (1−α)),
  - α_ψ = (1−α)/(2−α) (1−α + 1−α) — (preserve exact phrasing and symbols as in source).
- The first two terms match Gali and Monacelli (2005). The third term captures direct distortions induced by FX intervention on domestic consumption.
- Decomposition of α_θ [ˆθ_t + α_ψ ̄θ_t]^2:
  - α_θ ˆθ_t^2 — cost of external imbalance.
  - 2 α_θ α_ψ ˆθ_t ̄θ_t — costs of distorting consumption.
  - plus exogenous terms.
- Sources of welfare costs from FX intervention:
  - Cost of external imbalances:
    - Selling reserves decreases the endogenous risk premium, increasing domestic consumption without changing foreign demand or Home exports; this worsens the trade balance.
    - The second-order approximation of the budget constraint is captured by α_θ ˆθ_t^2.
    - Welfare cost is proportional to α_θ = α/(1+φ) (2−α/(1−α) + 1−α), an increasing function of α.
    - The welfare cost of external imbalances exists even when the country is fully open; because the welfare cost of non-zero output gap was normalized to 1, the welfare cost of external imbalances goes to infinity when α → 1.
  - Costs of distorting consumption:
    - A positive exogenous risk premium shock increases the interest rate faced by domestic consumers and decreases domestic consumption and domestic prices (because of Home bias), decreasing the terms of trade.
    - FX intervention can stabilize consumption and terms of trade when used to offset exogenous risk premium shocks.
    - The welfare loss of distorting consumption is proportional to α_θ α_ψ = α(1−α)/(1+φ) (preserve source expression α(1−α)).
    - Limits:
      - As α → 0 (closed economy): no extra cost from distorting domestic consumption beyond the output gap cost.
      - As α → 1 (fully open): Home consumption entirely imported; distorting it has no terms-of-trade cost, making FX intervention useless.

### Planning problem and optimal stabilization policy following risk premium shocks
- Planning problem under discretion (objective and constraints given at date t):
  - min over π_{H,t}, y_t, i_t, ψ_t, ˆθ_t, e_t of α_π π_{H,t}^2 + y_t^2 + α_θ [ˆθ_t + α_ψ ̄θ_t]^2
  - Subject to:
    - π_{H,t} = β π^e_{H,t+1} + κ_y y_t + κ_θ ˆθ_t + κ_θ ̄θ_t + κ_f (i_t − ρ)
    - y_t = y^e_{t+1} − (i_t − π^e_{H,t+1} − ρ) + α(ψ_t + ξ_t)
    - e_t = e^e_{t+1} − (i_t − i^*_t) + ψ_t + ξ_t
    - ˆθ^e_{t+1} = ψ_t + ˆθ_t
    - ∞∑t=0 β^t ˆθ_t = 0
    - y_0 = (1−α)(ˆθ_0 + ̄θ_0) + e_0
- Focus in Section IV on exogenous risk premium shocks (ξ_t), capturing changes in foreign investors’ sentiment (push factors).
- Proposition 1 (optimal policy following risk premium shocks, financial accelerator turned off μ = 0):
  - For ξ_t = ξ_0 ρ^t, optimal paths:
    - π_{H,t} = κ_θ (1−α_ψ) ξ_0/(1−ρ) [ (1−β/(1−ρβ)) 1/D(α_θ,1) − (1/D(α_θ,ρ)) ρ^t ] + κ_θ α_θ D(α_θ,1) / Γ
    - y_t = −α_π/κ_y π_{H,t}
    - ˆθ_t = ξ_0/(1−ρ) [ 1 − (1−α_ψ) D(+∞,ρ)/D(α_θ,ρ) ] ( ρ^t − (1−β)/(1−ρ) ξ_β )
    - ψ_t = ξ_0 [ (1−α_ψ) D(+∞,ρ)/D(α_θ,ρ) − 1 ] ρ^t
    - e_{t+1} − e_t = i_t − i^*_t − (ψ_t + ξ_t), with e_0 = y_0 − (1−α)(ˆθ_0 + ̄θ_0)
  - D(α_θ, ρ_ξ) = 1 − ρ_ξ β + α_π ( κ_y^2 + κ_θ^2/α_θ )
  - Interpretation:
    - Positive risk premium shock decreases domestic consumption in short run by increasing domestic interest rate; lowers inflation even if output stabilized due to Home bias in consumption.
    - Optimal monetary policy without FX intervention: increase interest rate and let currency depreciate instantly → positive output gap.
    - FX intervention (selling reserves) reduces endogenous risk premium, lowering interest rate faced by domestic households; limits required interest rate hike and exchange rate devaluation; stabilizes output, inflation, and supports domestic consumption.
  - Decomposition of optimal endogenous risk-premium ψ_t:
    - ψ_t = −[ α_ψ (terms of trade effect) + (1−α_ψ) (1 − D(+∞,ρ)/D(α_θ,ρ)) (trade-off between stabilizing consumption and generating external imbalance) ] ξ_t.
    - Terms:
      - Terms of trade effect: first component offsets the exogenous risk premium to stabilize terms of trade.
      - Trade-off: perfect offset would be ψ_t = −ξ_t if D(α_θ,·) = +∞ (FX intervention perfectly efficient); but external imbalances prevent full offset so ψ_t cannot fully offset ξ_t.
    - The ratio D(+∞,ρ)/D(α_θ,ρ) (always < 1) measures FX intervention efficiency.

### Simulations: calibration and quantitative results
- Calibration (following Farhi and Werning (2014) and Gali and Monacelli (2005)):
  - σ = γ = η = 1 (Cole-Obstfeld parametrization).
  - φ = 3 → Frisch elasticity = 1/3.
  - Price mark-up = 1.2 → elasticity of substitution ε = 6.
  - β = 0.96 (riskless annual return ≈ 4% in steady state).
  - δ (price stickiness) = 0.75 (average period of one year and a half between price adjustments).
  - Degree of openness α = 0.2 (matches imports/GDP ~15% Brazil, ~20% India as stated).
  - Financial accelerator parameter μ = 0.2 (working capital = 20% of output value).
  - Exogenous shock: 5% risk premium shock with half-life of 2 years.
- Comparative dynamics (monetary policy only vs. monetary + FX intervention):
  - When only monetary policy allowed:
    - Risk premium shock → large exchange rate depreciation and central bank increases interest rate to compensate foreign investors.
    - Large trade surplus (currency depreciation) and large drop in domestic consumption; small net effects on domestic output and PPI inflation.
    - Exchange rate, trade balance, and domestic consumption act as shock absorbers.
  - With FX intervention:
    - Central bank sells foreign currency reserves reducing endogenous risk premium by about 1.4% initially.
    - Impact on domestic consumption: fall is mitigated (3.5% vs. 4.8% without FX intervention).
    - Trade surplus on impact smaller (2.2% vs. 3%).
    - Exchange rate depreciation on impact smaller (9.2% vs. 12.7%).
    - FX intervention proves useful for stabilization of domestic consumption, trade balance, and nominal exchange rate.
- Welfare gains of FX intervention:
  - Welfare losses expressed as compensatory per-period percentage decline in permanent consumption.
  - For reasonable openness values, steady-state consumption gains from FX intervention are sizable: around 0.4 percent of permanent consumption for α = 0.2 (LHS panel Figure 3).
  - Relative impact: FX intervention cancels about 40 percent of the welfare losses for α = 0.2 (RHS panel Figure 3).
  - However, relative cost of FX intervention due to external imbalances increases with openness; as α → 1, FX intervention becomes useless and welfare losses identical with or without intervention.

### FX intervention and speculative attacks — equilibrium determinacy under financial accelerator
- Setup for speculative shocks (deviations in expected values): central bank problem reduces to
  - min α_π π_{H,t}^2 + y_t^2 + α_θ ˆθ_t^2
  - s.t. π_{H,t} = (β + κ_f) π^e_{H,t+1} + (κ_y − κ_f) y_t + κ_f y^e_{t+1} + (κ_θ − α κ_f) ˆθ_t + α κ_f ˆθ^e_{t+1}
  - ∞∑t=0 β^t ˆθ_t = 0
- First-order conditions (y_t and θ_t):
  - α_π (κ_y − κ_f) π_{H,t} + y_t = 0  (equation (22))
  - (κ_θ − α κ_f) π_{H,t} + α_θ θ_t + Γ = 0  (equation (23)), where Γ is Lagrange multiplier for intertemporal budget constraint.
- Law of motion for domestic inflation (equation (24)) preserves coefficients explicitly as in source.
- Key mechanism:
  - With financial friction (κ_f > 0), increasing the interest rate raises firms’ marginal costs; the recession required to lower inflation must be deeper or inflation weight larger to ensure determinacy.
- Proposition 2 — Equilibrium Determinacy under Discretionary Policy:
  - Blanchard-Kahn determinacy condition (equation (25)):
    - α_π > [ β + κ_f − 1 ] / [ κ_y (κ_y − κ_f) + κ_θ (κ_θ − α κ_f) / α_θ ]  (preserve expression ordering as in source).
  - Without FX intervention, determinacy condition simplifies to (equation (26)):
    - α_π > [ β + κ_f − 1 ] / [ κ_y (κ_y − κ_f) ].
  - Interpretation without FX intervention:
    - Define X_y = α_π (κ_y − κ_f): size of output loss engineered by central bank when inflation is 1 percent.
    - Determinacy requires κ_y X_y > β + κ_f − 1.
    - If κ_f = 0 (no financial accelerator), β − 1 < 0 ensures uniqueness; financial accelerator can create indeterminacy if its coefficient larger than 1−β.
  - With FX intervention reintroduced:
    - Define X_θ = α_π/α_θ (κ_θ − α κ_f) > 0 : marginal increase in consumption wedge for a decrease in inflation.
    - Determinacy condition (equation (27)):
      - κ_y X_y + κ_θ X_θ > β + κ_f − 1.
    - Rationale: central bank can use reserves to increase risk premium (accumulate reserves) to depreciate the currency for a given interest rate increase; this mitigates current inflation.
    - If reserves are costless (α_θ = 0) or reserves have a strong effect on inflation (X_θ high), indeterminacy can be eliminated even if central bank is not aggressive with conventional monetary policy.
    - Trade-off depicted: for given impact of interest rate, central bank must either engineer larger recessions (large X_y) or be activist with FX reserves (large X_θ) to avoid indeterminacy.
- Edge cases and notes:
  - If (κ_y − κ_f)(2 κ_f − κ_y) + (κ_θ − α κ_f)(2 α κ_f − κ_θ)/α_θ > 0 (implying μ > 1/2), the condition has an upper bound (source text presents upper bound inequality).
  - As κ_f increases (financial accelerator stronger), the risk of multiple equilibria (self-fulfilling currency movements) increases; FX intervention reduces the parameter range for which indeterminacy occurs.

### Conclusions (policy-relevant takeaways)
- FX intervention can be an effective stabilization tool in presence of exogenous international risk premium shocks:
  - It stabilizes domestic consumption, exchange rate, and trade balance.
  - It can yield meaningful steady-state consumption gains (example: ≈ 0.4 percent of permanent consumption for α = 0.2) and offset a substantial share of welfare losses (≈ 40 percent for α = 0.2).
- However, FX intervention creates external imbalances that generate future welfare costs; these costs grow with openness (α), and in the limit α → 1 the intervention becomes ineffective.
- The presence of a financial accelerator (working capital/frictions) can generate equilibrium indeterminacy under discretionary policy — a possibility absent in the standard open-economy New Keynesian model.
- FX intervention reduces the range of parameters for which indeterminacy occurs:
  - If FX intervention is very effective (large X_θ) or the central bank has strong preferences for using FX intervention (small α_θ), indeterminacy can be ruled out even without an aggressive conventional monetary policy stance.

*Source: wp17207 - Section V, we explore the impact of the financial accelerator on equilibrium determinacy*

### REFERENCES

### REFERENCES

### APPENDIX I — FIRST-ORDER CONDITIONS FOR CALVO PRICE SETTING
- Non-linearized Calvo price-setting first-order condition:
  - 1−δ(ΠH,t)ε−1
    1−δ
    =
    (FtKt)ε−1
- Definitions:
  - Kt = ε ε−1 1 +τ A H Yt Nφ t +δβΠε H,t+1 Kt+1
  - Ft = Yt C−σ t S−1 t Qt +δβΠε−1 H,t+1 Ft+1
  - PPI inflation: ΠH,t+1 = PH,t+1 PH,t
  - Price dispersion law of motion:
    - ∆t = h(∆t−1, ΠH,t), h(∆,Π) = δ∆Πε + (1−δ) ( 1−δΠε−1 1−δ ) ε/(ε−1)

### APPENDIX II — DERIVATION OF THE INTERTEMPORAL BUDGET CONSTRAINT
- Consolidation approach: add budget constraints of households, central bank, and government.
- Household budget constraint:
  - PtCt + Dh t+1 + Sb t+1 + Tfin t ≤ (1 + it−1)(Dh t + Sb t) + Wt Nt + Πt + Tfirm t + Tgov t
- Central bank constraint:
  - Et R∗ t+1 − Et (1 + i∗ t−1) Ξt−1 R∗ t = Sb t+1 − (1 + it−1) Sb t
- Government constraint:
  - 1 + it−1 Ψt−1 Df t + (1 + it−1)(Dh t + Sb t) + Tgov t ≤ Df t+1 + Dh t+1 + τ Wt Nt + Tfin t
- Consolidated budget constraint:
  - Et R∗ t+1 − Et (1 + i∗ t−1) Ξt−1 R∗ t + 1 + it−1 Ψt−1 Df t ≤ Df t+1 + PH,T Yt − PtCt

### APPENDIX III — NET FOREIGN ASSET POSITION, TRADE BALANCE, AND LOSS FUNCTION DERIVATION
- Definitions:
  - NFA t and NX t:
    - NFA t = (1 + i∗ t−1) Ξt−1 Et PF,t C∗ t σ [R∗ t − Df t Et−1]
    - NX t = PH,t Yt − PtCt
- Consolidated budget constraint using NFA and NX:
  - PF,t+1 C∗ t+1 σ (1 + i∗ t) Ξt NFA t+1 − PF,t C∗ t σ NFA t = NX t
- Euler equation for foreign households:
  - 1 + i∗ t = β −1 ( C∗ t+1 C∗ t ) σ PF,t+1 PF,t
- Resulting expression:
  - NFA t = − 1 PF,t C∗ t NX t + β Ξt NFA t+1
- Goods market clearing second-order approximation:
  - Yt = St C∗ [(1−α) Θt + α]:
  - yt = st + (1−α) θt + 1 2 α(1−α) θ2 t
- Derived consumption relation:
  - ct = (1−α) y t + α(2−α) θ t − 1 2 α(1−α)2 θ2 t
- Labor market clearing up to second-order:
  - nt = yt + log ∆P t + 1 2 y2 t
- By Woodford (2003):
  - ∞ ∑ t=0 β t log ∆P t = εP 2λP ∫ ∞ 0 e−ρt π2 H,t
- Integrated objective function (exact expression given):
  - ∞ ∑ t=0 β t (Ut − ̄U CU c) =
    − (1−α)(1+φ) 2 ∫ ∞ 0 e−ρt { α π π2 H,t + y2 t − 2α(2−α) (1−α)(1+φ) θt + α 1−α 1 +φ θ2 t }
- Second-order approximation of budget constraint eliminates linear θt terms leading to welfare loss (up to additive/multiplicative constants):
  - 1 2 ∞ ∑ t=0 β t [ α π π2 H,t + y2 t + α θ [ ˆ θ t + α ψ ̄ θ t ] 2 ]
- Coefficient definitions:
  - α π = εP λP (1 + φ)
  - α θ = α 1 + φ ( 2−α 1−α + 1−α )
  - α ψ = 1−α 2−α 1−α + 1−α

### APPENDIX IV — PROOF OF PROPOSITION 1 (RISK-PREMIUM SHOCKS AND OPTIMAL POLICY)
- Optimization problem (minimization):
  - min πH,t, yt, it, ψt, θt, et α π π2 H,t + y2 t + α θ [ θt − (1−α ψ ) ̄ θ t ] 2
  - s.t. πH,t = β π e H,t+1 + κ y y t + κ θ θ t
  - ∞ ∑ t=0 β t θ t = 0
- First-order conditions on instruments yield:
  - α π κ y πH,t + yt = 0
  - α π κ θ πH,t + α θ [ θt − (1−α ψ ) ̄ θ t ] − Γ = 0
- For exponentially decreasing risk premium shocks ξt = ξ0 ρ t with
  - ̄ θ t = ξ0 1−ρ ( 1−β 1−ρβ − ρ t )
- Iteration and substitution give:
  - πH,t = κ θ (1−α ψ ) ξ0 1−ρ ( 1−β 1−ρβ 1 D(1) − 1 D(ρ) ρ t ) + κ θ α θ D(1) Γ
  - Γ = (1−α ψ ) ξ0 (1−ρ) D(1) Dy (1) 1−β 1−ρβ α π κ2 θ ( 1 D(1) − 1 D(ρ) )
  - yt = −α π κ y πH,t
  - θt = (1−α ψ ) ξ0 1−ρ Dy (ρ) D(ρ) ( 1−β 1−ρβ − ρ t )
- Impact of foreign exchange intervention on consumption:
  - ˆ θ t = ξ0 1−ρ [ (1−α ψ ) Dy (ρ) D(ρ) − 1 ] ( 1−β 1−ρβ − ρ t )

*Source: wp17207 - REFERENCES (IMF working paper PDFs, appendices I–IV).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp17207.pdf_