## wpiea2020122-print-pdf — Aggregate Demand, Aggregate Supply, Endogenous Risk-Premiums, FXI and CFM Policy Rules

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### Aggregate Demand — model structure and key mechanisms
- Resource constraint (share-weighted):
  - y_t = c_y c_t + g_y g_t + m_y (m^*_c,t − m_c,t). (1)
- Import and export demand:
  - m_c,t = c_t − ε_c γ_{m,c,t}. (2)
  - m^*_c,t = c^*_t − ε^*_c γ_{x,*,t}. (3)
  - γ_{m,c,t} = p_{m,t} − p_{c,t}; γ_{x,*,t} = p_{x,t} − p^*_t.
- Consumption Euler and habit persistence:
  - λ_{c,t} = δ_c λ_{c,t+1} + r_{b,t}. (4)
  - λ_{c,t} = −1/̂σ (c_t − κ_c c_{t−1} − ε_{c,t}). (5)
  - Discounted long-term real rate: c_t − κ_c c_{t−1} − ε_{c,t} = ̂σ E_t ∑_{j=0}^∞ δ_c^j r_{b,t+j} = ̂σ r^L_{b,t}. (6)
  - Discounting mutes forward guidance potency.
- Long-term borrowing rate decomposition:
  - r^L_{b,t} = r^L_t + Ψ_t, where Ψ_t = E_t ∑_{j=0}^∞ δ_c^j (i_{b,t+j} − i_{t+j}). (7)
- Balance-sheet channel and UIP risk premium:
  - Ψ_t = f(q_{c,t}), f(·) upward sloping and locally convex; large depreciations can trigger nonlinear tightening. (8)
  - q_{c,t} = δ_c q_{c,t+1} + (i^*_t − π^*_{c,t+1}) − (i_t − π_{c,t+1}) + Φ(d_t, d_t, b_t) − τ_t. (9)
  - Φ(·) rises when net foreign liabilities d_t exceed a threshold or during stress; government can reduce Φ by FXI; τ_t is capital outflow tax.
- Net foreign liabilities evolution and real-interest determination:
  - d_t = (1 + r_{d,t}) d_{t−1} − m_y (m^*_c,t + γ_{x,*,t} − m_c,t). (10)
  - (1 + r_{d,t}) = (1 + i^*_{t−1}) / (1 + π_{d,t}) * (S_t / S_{t−1}) [ 1 + Φ(d_{t−1}, d_{t−1}, b_{t−1}) ]. (11)

### Aggregate Demand — nonlinear features and policy tradeoffs
- Nonlinearities emphasized:
  - Φ(·) time-varying and nonlinear in net foreign liabilities and stress; can jump once liabilities exceed thresholds.
  - Ψ_t nonlinear in q_{c,t}, capturing sharp tightening of domestic financial conditions with large depreciations.
  - Discounting (δ_c < 1) weakens forward guidance.
- Policy tradeoffs:
  - In AEs with well-anchored expectations, UIP risk-premium driven depreciation acts like expansionary AD shock: output rises and inflation effects are short-lived.
  - In EMEs with less-anchored expectations, same shock produces persistent inflation; tighter monetary policy to contain inflation can depress output and amplify contractions via higher Ψ_t.
  - FXI and CFMs can potentially improve tradeoffs by reducing Φ(·) (FXI) or raising τ_t (CFM), but carry risks:
    - FX sales may cause large reserve losses if shocks are persistent and equilibrium real exchange rate needs to depreciate.
    - Stabilizing exchange rate may impede hedging market development and encourage FX debt accumulation.
    - Frequent CFMs may deter foreign investment and slow financial market development.
- Model purpose:
  - Provides quantitative input on when FXI and CFMs could help; not an unconditional endorsement.

### Aggregate Supply — price/wage setting and calibration
- Modified New Keynesian Phillips curves with indexation:
  - Structural persistence via indexation parameter ιd and discounting δπ.
  - Slope κmc inversely related to mean duration of price contracts.
- Marginal cost and production:
  - mc_t = ξ_t − mpl_t = ξ_t − (y_t − l_t).
  - y_t = (1−α)(l_t + z_t).
- Wage formation and wage Phillips curve:
  - mrs_t = χ l_t^{−λ_{c,t}}.
  - ξ_t = ω/(1−ω) q_{c,t} + mrs_t under full wage flexibility and full pass-through.
  - Wage inflation: ω_t − ι_w π^L_{c,t} = β δ_π (ω_{t+1} − ι_w π^L_{c,t+1}) + κ_w (mrs_t − ξ_{c,t}).
- Sectoral Phillips curves and imperfect law of one price:
  - Separate Calvo-derived Phillips curves for consumer-goods imports (m) and domestic exports (x).
  - Consumer price aggregation: π_{c,t} = (1 − m_c) π_t + m_c π_{m,t}; special case full pass-through π_{c,t} = π_t + m_c Δq_{c,t}.
- Calibration (quarterly) — key parameter values:
  - AE vs EME differences: price/wage formation and two nonlinear balance-sheet channels (Φ and Ψ) in EMEs.
  - AE calibration:
    - κ_p = .005; κ_w = .02; ι_p = .5; ι_w = 0.
  - EME calibration:
    - ι_p = ι_w = 0.75; κ_p = .02; κ_w = .03.
  - Common parameters:
    - Φ = .05; ξ = 1.
    - β = 0.995.
    - Annualized steady-state inflation rate 2 percent (π = 0.005), steady-state annualized nominal interest rate 4 percent (i = 0.01).
    - σ = 1; κ_c = 0.8; χ = 0.5; α = 0.3; g_y = 0.2.
    - δ_c = 0.95; δ_π = 0.92.
  - Monetary policy baseline coefficients:
    - γ_π = 1.5; γ_y = .0625; γ_i = 0.
  - Trade and price-setting:
    - Local currency pricing for import/export prices.
    - ε_c = 0.8; ε^*_c = 0.8; m_y = 0.2.
    - Price adjustment frequencies: EMEs ξ_x = ξ_m = .667; AEs ξ_x = ξ_m = 0.8.
    - Indexation ι_x and ι_m: 0.75 in EMEs; 0.5 in AEs.
- Solution method: model solved using Dynare (Adjemian et al., 2011).

### Shock transmission — UIP risk premium shock (linear model comparisons)
- Shock setup:
  - Investor risk tolerance shock d_t follows AR(1) with ρ̄_d = 0.9 and scaled so AE real exchange rate depreciates by 10 percent.
- AE transmission:
  - Depreciation stimulates net exports → output rises; transient inflation rise is “looked through.”
- EME transmission:
  - Depreciation raises inflation persistently; central bank tightens → output contracts slightly; balance-sheet channel (if active) amplifies output contraction via higher Ψ_t.

### Policy tradeoffs in EMEs — alternative monetary responses
- Two monetary-rule scenarios for same-sized UIP shock in EME:
  - More accommodative (benchmark Taylor rule):
    - Keeps output relatively stable but allows large persistent CPI inflation rise; risks undermining credibility.
  - Tighter (aggressive inflation stabilization):
    - Limits inflation increase but causes sharper output contraction of about 1.5 percent.
- Implication:
  - Conventional monetary policy faces poor simultaneous inflation-output performance when expectations are not well-anchored; motivates considering FXI and CFMs to reduce de-anchoring risk.

### Modeling of Endogenous Risk-Premiums — functional forms and calibration
- Private borrowing spread Ψ_t specification (logistic autoregressive):
  - Ψ_t = ρΨ Ψ_{t−1} + μ_t + ̃μ, with μ_t = μ / (1 + e^{−kq (qc,t − ̄qt)}).
  - Threshold ̄q = 22.5; μ is maximum one-period increase; kq governs sensitivity.
- UIP risk premium Φ_t specification (nonlinear logistic AR(1)):
  - Φ_t = ρΦ Φ_{t−1} + ̃φ_t, where ̃φ_t logistic function of d_t relative to threshold ̄d and influenced by central bank foreign asset purchases b_t.
  - ̄d is threshold net foreign liability level.
- Central bank foreign asset purchases: b_t = ρb b_{t−1} + εb,t.
- Debt appetite shock: ε̄d,t = ρ̄d ε̄d,t−1 + ūd,t.
- Calibration for selected EMEs:
  - UIP parameters: ρΦ = .85; k_d = .4; ̄d = 12.5; φ = .6; ρ̄d = .85; σ̄d = 11.
  - Private spread parameters: ρψ = .85; μ = .4; ̄q = 22.5; kq = .4.
  - Private debt-limit decomposition: ε̄q,t = εp,t + 1/2 ε̄d,t; εp,t AR(1) with ρp = .85 and σ̄d = 4.
  - Empirical moment: median correlation between corporate and sovereign spreads ≈ 0.4 (unweighted mean slightly lower: .31).
- Linearization yields near-zero effective coefficients:
  - Effective linearized ρq ≈ 2e−5.
  - UIP linearization: φd = .003; φb ≈ 1e-10; φε̄d ≈ 4e-11.
  - Interpretation: linearized DSGE frameworks miss empirically relevant nonlinearities; FXIs have little linearized effect in normal times but can matter strongly in low risk-tolerance states.

### Effects of FXI in a crisis situation — nonlinear simulations
- Adverse scenario:
  - Reduction in ̄d_t (AR(1) persistence .85) plus fall in foreign import demand c^*_t.
  - Comparison: “vulnerable EME” (high net foreign liabilities) vs “less vulnerable EME.”
- Simulated outcomes:
  - Vulnerable EME: real exchange rate depreciates several times more; despite stronger net-export boost, output contracts considerably more.
  - Drivers: depreciation triggers larger rise in Ψ_t; larger inflation rise forces much tighter monetary policy.
  - Policy dilemma: sharper monetary tightening reduces inflation but causes much larger output decline in vulnerable EME.
- Policy interventions (FXI sales and CFMs raising τ_t):
  - Both reduce exchange rate depreciation, permit more accommodative monetary policy, and mitigate rise in Ψ_t.
  - Near-term outcome: lower inflation and higher output relative to baseline; domestic demand expands more while net-export improvement is smaller.
  - Medium-run tradeoffs: stronger exchange rate and domestic demand under FXI/CFM imply larger deterioration of net foreign liabilities, which can slightly raise UIP premium in longer run.
  - FXIs slightly better than CFMs at depressing UIP premium in medium run because FXIs directly reduce Φ.
  - Effectiveness:
    - FXI and CFMs helpful for fairly transient dt shocks.
    - For protracted shocks, near-term benefits persist but long-run costs (higher risk spreads, lower output, higher inflation) can materialize.
  - Insurance role:
    - Using FXI/CFMs during inflow episodes can prevent endogenous buildup of foreign indebtedness; cost is constraining potentially beneficial borrowing.

### Unconditional distributions and nonlinear dynamics — simulation design and results
- Simulation design:
  - No initial FXI/CFM; central bank uses nominal policy rate.
  - Six exogenous shocks: ec,t, g_t, c^*_t, e_{i,t}, ε̄d,t, ε̄q,t.
  - AR(1) persistence: ec,t, g_t, c^*_t have root .95; e_{i,t} ρi = .8.
  - Shock standard deviations: ec,t and g_t = 1.5; c^*_t = 2; monetary policy shock = 0.25/4.
  - Add median spreads to simulated levels: corporate median spread = 117 basis points; government median spread = 205 basis points.
  - Simulation length T = 20,000 observations.
- Distributional findings (nonlinear model):
  - UIP risk premium: strong positive skew driven by negative risk-off episodes with high external indebtedness.
  - Net foreign liabilities: left-skewed distribution.
  - Real exchange rate: right skew (occasional large depreciations).
  - Output gap and domestic absorption: sizeable negative left skew (absorption stretches to −15 percent; output gap roughly to −10 percent).
  - CPI inflation: longer right tail from depreciations and somewhat elongated left tail due to output depressions.
  - Correlation between corporate and sovereign spreads in simulated data: 0.35 (close to median 0.39 in data).

### FXI and CFM Policy Rules — rule forms, calibration, and effects
- FX intervention rule:
  - b_t = b_μ μ_t − b_d (d_t − d_t¯). (eq. 35)
  - Parameters: b_μ = 5; b_d = k_d.
  - Interpretation: central bank sells reserves (reduces b_t) when μ_t or UIP premium rises; rule reacts only to non-fundamental debt-limit component and slightly to d_t for stationarity.
- Outflow CFM rule:
  - τ_t = τ_μ μ_t + τ_φ (˜φ_t − φ_t − ρ_Φ φ_{t−1}). (eq. 36)
  - Parameters: τ_μ = 5; τ_φ = 2.5.
  - Interpretation: raises capital-flow tax when μ_t rises; steering UIP premium toward linearized counterpart.
- Benchmark dynamic assumptions:
  - b_μ = 5; b_d = k_d; τ_μ = 5; τ_φ = 2.5.
  - AR(1) persistence for shocks: consumption and export demand 0.95; debt appetite and b_t in liquidity-trap exercise 0.85.
  - Implicit assumption: central bank has sufficient reserves and determination to implement rules.
- Effects in vulnerable EME crisis (nonlinear model):
  - Full commitment to FXI and CFM rules:
    - Efficiently stabilize private borrowing spreads and produce a much lower initial UIP premium path.
    - Delay adjustment toward lower net foreign liabilities, implying UIP premium eventually slightly more elevated.
    - When private spreads and UIP premium are high simultaneously, both rules yield nearly same macro outcome despite different instruments.
    - Small adjustments to CFMs suffice if adoption is credibly communicated.
- Effects on unconditional distributions with combined rule:
  - Marked reduction in tail risk of large real depreciations and large output contractions.
  - Mitigates need for sharp policy tightening and reduces incidence of large trade-balance swings into surplus.
- Liquidity-trap exercise (AE calibration; linearized model):
  - With private spreads and UIP shocks set to zero and large adverse shocks pushing i to zero:
    - Central bank purchases FX (b_t > 0, AR(1) root 0.85) to induce persistent depreciation.
    - FXI-induced depreciation raises net exports and stimulates output; in no-FXI baseline output falls over 10 percent below baseline.
    - At ELB, nominal rates pinned at zero; real rates decline, crowding in demand and keeping core CPI closer to 2 percent target.
    - EMEs near ELB with low indebtedness benefit more from same-sized intervention due to larger second-round inflation effects.
- Policy implications and cautions:
  - FXIs and CFMs can improve tradeoffs, especially with less-anchored expectations and FX mismatches.
  - Risks: frequent use may slow FX derivatives market development and encourage FX debt buildup.
  - Suggested extensions: add housing/banking sector, fiscal interactions, multilateral spillovers, empirical estimation of IPF transmission and heterogeneity.

*Source: wpiea2020122-print-pdf — selected sections 2.1, 2.2, 4.1–4.4*

### 2.1    Aggregate Demand  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  . 

### 2.1 Aggregate Demand

### Overview
- The home economy resource constraint is expressed as a share-weighted average of home consumption c_t, government spending g_t, and net exports (m^*_c,t − m_c,t) weighted by the steady-state trade share m_y:
  - y_t = c_y c_t + g_y g_t + m_y (m^*_c,t − m_c,t). (1)
- The model is a New Keynesian small open economy framework extended to capture EME features: habit persistence, sticky wages and prices, imperfect pass-through, local/producer/dominant currency pricing, adaptive inflation expectations, incomplete markets, discounting (Gabaix, 2016), nonlinear UIP risk-premium, nonlinear balance-sheet driven borrowing spreads, and the effective lower bound on policy rates.

### Key equations and mechanisms (as presented)
- Import demand:
  - m_c,t = c_t − ε_c γ_{m,c,t}. (2)
  - γ_{m,c,t} = p_{m,t} − p_{c,t}; under full pass-through p_{m,t} = p^*_t + s_t, γ_{m,c,t} reduces to q_{c,t} = p^*_t + s_t − p_{c,t}.
- Export demand:
  - m^*_c,t = c^*_t − ε^*_c γ_{x,*,t}. (3)
  - γ_{x,*,t} = p_{x,t} − p^*_t; with foreign currency prices moving one-to-one with the exchange rate γ_{x,*,t} = p_t − s_t − p^*_t and q_{p,t} = s_t + p^*_t − p_t.
- Consumption Euler (linking marginal utilities and short-term real rates):
  - λ_{c,t} = δ_c λ_{c,t+1} + r_{b,t}. (4)
- Marginal utility with habit persistence:
  - λ_{c,t} = −1/̂σ (c_t − κ_c c_{t−1} − ε_{c,t}). (5)
- Discounted long-term real rate relation (behavioral discounting):
  - c_t − κ_c c_{t−1} − ε_{c,t} = ̂σ E_t ∑_{j=0}^∞ δ_c^j r_{b,t+j} = ̂σ r^L_{b,t}. (6)
  - Discounting mutes the effect of future short rates (reduces forward guidance potency).
- Long-term borrowing rate decomposition with a private borrowing spread Ψ_t:
  - r^L_{b,t} = E_t ∑_{j=0}^∞ { δ_c^j (i_{t+j} − π_{c,t+j+1}) + δ_c^j (i_{b,t+j} − i_{t+j}) } = r^L_t + Ψ_t. (7)
  - Ψ_t = E_t ∑_{j=0}^∞ δ_c^j (i_{b,t+j} − i_{t+j}).
- Balance-sheet channel: private borrowing spread rises with depreciation:
  - Ψ_t = f(q_{c,t}). (8)
  - f(·) is upward sloping and locally convex; depreciations above a threshold can trigger large nonlinear tightening of financial conditions.
- Risk-augmented UIP:
  - q_{c,t} = δ_c q_{c,t+1} + (i^*_t − π^*_{c,t+1}) − (i_t − π_{c,t+1}) + Φ(d_t, d_t, b_t) − τ_t. (9)
  - UIP risk premium Φ(·) rises when net foreign liabilities d_t exceed a threshold or during market stress; government can reduce Φ by foreign exchange intervention (FXI). τ_t is a tax on capital outflows.
- Net foreign liabilities evolution (foreign-currency borrowing):
  - d_t = (1 + r_{d,t}) d_{t−1} − m_y (m^*_c,t + γ_{x,*,t} − m_c,t). (10)
- Nonlinear determination of the period real interest affecting debt service:
  - (1 + r_{d,t}) = (1 + i^*_{t−1}) / (1 + π_{d,t}) * (S_t / S_{t−1}) [ 1 + Φ(d_{t−1}, d_{t−1}, b_{t−1}) ]. (11)
  - Domestic real interest influenced by foreign nominal rate, domestic inflation, nominal exchange rate change, and previous-period risk premium.

### Nonlinear features emphasized
- UIP risk premium Φ(·) is time-varying and nonlinear in net foreign liabilities and stress conditions, allowing sharp increases in required returns once liabilities exceed thresholds.
- Borrowing spreads Ψ_t are nonlinear in the real exchange rate q_{c,t}, capturing sharp tightening of domestic financial conditions when depreciations are large due to unhedged foreign-currency debt.
- Discounting of future rates (Gabaix-style) reduces the impact of forward guidance and weakens the channel from future short rates to current consumption.

### Policy implications and tradeoffs (from Aggregate Demand discussion and Introduction)
- In AEs with well-anchored inflation expectations a UIP risk premium shock that depreciates the currency acts like an expansionary aggregate demand shock: boosts output via expenditure switching, short-lived inflation effects that monetary policy can “look through.”
- In EMEs with less-anchored inflation expectations the same shock causes persistent inflation increases; tighter monetary policy to contain inflation may depress output, and output contraction can be amplified by balance-sheet tightening via higher Ψ_t.
- FXI and CFMs can potentially improve policy tradeoffs:
  - FXI (foreign exchange sales or purchases) can directly reduce Φ(·) by strengthening demand for home currency.
  - FXI and CFMs may mitigate capital inflow-driven domestic risk taking and reduce the need for monetary tightening during outflow episodes.
- Caveats and risks when using FXI/CFMs:
  - FX sales to support the exchange rate can entail large potentially destabilizing reserve losses if shocks are more persistent and the equilibrium real exchange rate needs to depreciate.
  - Stabilizing the exchange rate may impede hedging market development and encourage excessive foreign-currency debt buildup.
  - Frequent CFMs may deter foreign investment and slow financial market development.
- The model is intended to provide quantitative input on when FXI and CFMs could help; results are not an unconditional endorsement and should be considered alongside uncertainty about shocks and policy efficacy.

*Source: wpiea2020122-print-pdf — 2.1 Aggregate Demand*

### 2.2    Aggregate Supply

### 2.2    Aggregate Supply

### Supply-side specification and Phillips curves
- Price-setting is given by a modified New Keynesian Phillips Curve with Calvo-style price setting and indexation:
  - Inflation equation embeds structural persistence via the indexation parameter ιd and discounting δπ that damps the influence of future marginal cost.
  - Sensitivity of inflation to marginal cost is governed by slope κmc, which varies inversely with mean duration of price contracts.
  - Full indexation in the non-stochastic formulation implies the steady-state embeds no price distortions.
- Marginal cost and production technology:
  - Marginal cost: mc_t = ξ_t − mpl_t = ξ_t − (y_t − l_t).
  - Production function (Cobb-Douglas): y_t = (1−α)(l_t + z_t).
- Labor supply and wage formation:
  - Marginal rate of substitution: mrs_t = χ l_t^{−λ_{c,t}} (rises with hours worked or decline in marginal utility of consumption).
  - Product real wage under full wage flexibility and full pass-through (producer currency pricing):
    - ξ_t = ω(p_{m,t} − p_t) + ξ_{c,t} = ω/(1−ω) q_{c,t} + mrs_t.
    - Exchange rate depreciation raises relative price of imported goods and hence required real wage ξ_t; effect larger with greater openness and larger pass-through.
- Wage Phillips curve with Calvo-style wage stickiness:
  - Nominal wage inflation: ω_t − ι_w π^L_{c,t} = β δ_π (ω_{t+1} − ι_w π^L_{c,t+1}) + κ_w (mrs_t − ξ_{c,t}).
  - Wage indexation π^L_{c,t} is a long moving average: π^L_{c,t} = (1−Φ) π^L_{c,t} + Φ(ξ π_{c,t−1} + (1−ξ) Δq_{c,t−1}).
  - Product real wage identity: ξ_t = ξ_{t−1} + ω_t − π_t.
- Imperfect law of one price and sectoral Phillips curves:
  - Separate Calvo-derived Phillips curves for consumer-goods imports (m) and domestic exports (x), with sectoral indexation parameters ι_j and Calvo parameters ξ_j.
  - Sectoral inflation deviation definition: π_{j,t} ≡ dΠ_{j,t} / Π_j with Π_{j,t} ≡ P_{j,t} / P_{j,t−1}.
  - Time-varying sectoral markup %_{j,t} assumed i.i.d. N(0, σ^2_{%_j}).
- Marginal costs definitions and relative prices:
  - mc_{x,t} = p_t − s_t − p_{x,t} ≡ mc_{x,t} + π_t − π_{x,t} − Δs_t.
  - mc_{m,t} = p^*_t + s_t − p_{m,t} ≡ −mc_{x,t} − γ_{x,∗t} − γ_{m,d t}.
  - Relative prices evolve: γ_{x,∗t} ≡ p_{x,t} − p^*_t ≡ γ_{x,∗t−1} + π_{x,t} − π^*_t; γ_{m,d t} ≡ p_{m,t} − p_t ≡ γ_{m,d t−1} + π_{m,t} − π_t.
- Consumer price inflation aggregation:
  - π_{c,t} = (1 − m_c) π_t + m_c π_{m,t}, where m_c = m_y (1 − g_y).
  - Special case of full pass-through: π_{c,t} = π_t + m_c Δq_{c,t}.
  - Modeling nests local, producer, and dominant currency pricing as special cases.

### Monetary and fiscal policy in the model
- Taylor-style monetary policy rule:
  - i_t = max{−i, (1−γ_i)[γ_π( ̄π_{c,t+4|t} + ̄π_{c,t}) + γ_y y_t] + γ_i i_{t−1} + e_{i,t}}.
  - Central bank responds to expected and current four-quarter change in core CPI and to domestic output gap; γ_i allows interest rate smoothing; e_{i,t} is an interest rate shock.
  - ̄π_{c,t} is the four-quarter average: ̄π_{c,t} = (̄π_{c,t} + ̄π_{c,t−1} + ̄π_{c,t−2} + ̄π_{c,t−3}) / 4.
- Fiscal policy:
  - Government expenditure g_t follows an exogenous AR(1) process.
  - Steady-state g financed by labor income taxes; deviations financed by lump-sum taxes (abstracts from fiscal dynamics).

### Calibration (quarterly frequency) — key parameter values and distinctions AE vs EME
- Two exceptions across AE and EME calibrations:
  - Parameters governing price and wage formation differ to capture less-well anchored inflation expectations in EMEs.
  - EME formulation allows two key nonlinear balance-sheet channels: endogenous UIP risk premium Φ(d_t, d_t, b_t) and endogenous private borrowing spread Ψ_t.
- AE calibration (literature-based low slopes and low structural persistence):
  - κ_p = .005
  - κ_w = .02
  - ι_p = .5
  - ι_w = 0
  - Interpretation note: equivalent to firms (unions) probability of not re-optimizing prices (wages) equals 0.92 (0.87) in standard Calvo setup.
- EME calibration (less-anchored expectations, higher slopes/persistence):
  - ι_p and ι_w set to 0.75.
  - κ_p = .02
  - κ_w = .03
  - Firms and unions re-optimize prices and wages on average once per year in this parametrization.
- Common parameters across AE and EME:
  - Φ = .05 and ξ = 1 for trend inflation measure in wage indexation (long distributed lag of past inflation).
  - β = 0.995 (steady-state real interest rate of 2 percent annualized).
  - Annualized steady-state inflation rate 2 percent (π = 0.005), steady-state annualized nominal interest rate equals 4 percent (i = 0.01).
  - Intertemporal substitution elasticity σ = 1.
  - Habit parameter κ_c = 0.8.
  - Frisch elasticity χ = 0.5.
  - Capital share α = 0.3.
  - Government share of steady state output g_y = 0.2.
  - Discounting parameters: δ_c = 0.95, δ_π = 0.92.
- Monetary policy baseline coefficients (policy rule (26)):
  - γ_π = 1.5
  - γ_y = .0625 (coefficient on output gap equals 0.25 at annual rate)
  - γ_i = 0
- Trade and price-setting for trade sectors:
  - Local currency pricing assumed for import and export prices.
  - Import/export price responsiveness and structural persistence higher in EMEs than AEs.
  - Price adjustment frequencies: EMEs adjust on average every 3 quarters (ξ_x = ξ_m = .667); AEs every 5 quarters (ξ_x = ξ_m = 0.8).
  - Indexation coefficients ι_x and ι_m: 0.75 in EMEs and 0.5 in AEs.
  - Import/export real exchange rate sensitivity parameters: ε_c = 0.8 and ε^*_c = 0.8.
  - Import share m_y = 0.2.
- Exogenous shock processes:
  - s_t = {g_t, ε_{c,t}, e_{i,t}, c^*_t, r^*_t, d_t, b_t, τ_{i,t}, τ_{o,t}} follow AR(1): s_t = ρ_s s_{t−1} + e_{s,t}, e_{s,t} ∼ i.i.d. (0, σ^2_s).
- Solution method: model solved using Dynare (Adjemian et al., 2011).

### Shock transmission: linear model — UIP risk premium shock (AEs vs EMEs)
- Shock description:
  - Investor risk tolerance shock d_t that reduces willingness to hold home bonds → UIP risk premium shock.
  - Shock follows AR(1) with ρ̄_d = 0.9 and is scaled so that the real exchange rate depreciates by 10 percent in the AE economy.
- Advanced Economy (AE) transmission summary:
  - Exchange rate depreciation stimulates net exports → output rises.
  - Higher output and higher import prices raise inflation, but well-anchored inflation expectations allow monetary policy to “look through” transient inflation and focus on output.
  - Shock resembles a standard aggregate demand shock despite transient inflation boost.
- Emerging Market Economy (EME) transmission summary:
  - Depreciation boosts net exports but, with less well-anchored inflation expectations, exchange rate depreciation has large and persistent effects on inflation.
  - Central bank raises nominal and real interest rates, crowding out domestic demand; output contracts slightly on balance while inflation rises substantially.
  - If balance sheet channel allowed (exchange rate depreciation increases private borrowing spreads), output contraction would be larger.

### Policy tradeoffs in EMEs (alternative monetary rules)
- Comparative scenarios for same-sized UIP risk premium shock in EME:
  - More accommodative policy (benchmark Taylor rule — red-dotted line):
    - Central bank keeps output relatively stable.
    - Allows exchange rate depreciation to have large and persistent effects on CPI inflation and domestic inflation → potential undermining of monetary credibility.
  - Tighter policy (aggressive inflation stabilization — blue line):
    - Central bank responds more vigorously to both domestic and imported inflation.
    - Limits size and persistence of inflation increase but causes sharper output contraction of about 1.5 percent.
- Policy implication:
  - Conventional monetary policy faces poor performance in simultaneously achieving inflation and output objectives when inflation expectations are not well-anchored.
  - Rationale for using additional instruments (FX intervention or capital controls) to reduce risk of de-anchoring inflation expectations, permitting monetary policy to focus more on output stabilization and improving output-inflation trade-offs.

### Nonlinear model and balance-sheet channels
- Nonlinear features included: UIP risk premium Φ(d_t, d_t, b_t) and private borrowing spread Ψ_t.
- Intuition from Figure 3:
  - UIP risk premium as function of net foreign liabilities:
    - High risk tolerance environment (blue dotted line): risk premium rises modestly as net foreign liabilities increase.
    - Lower risk tolerance environment (red dotted line): risk premium jumps much more for highly indebted economies during market stress.
  - Private borrowing spread as function of real exchange rate:
    - Lower risk tolerance regime exhibits larger jumps in private borrowing spread for given exchange rate movements.
- Empirical stylized fact captured:
  - In run-up to crises spreads may be low even with high vulnerabilities; during crisis when risk tolerance declines, spreads rise sharply for vulnerable economies (pattern observed in euro area sovereign debt crisis and recent COVID crisis).

*Source: IMF working paper chapter "2.2 Aggregate Supply" (canonical document contents).*

### 4.1    Modeling of Endogenous Risk-Premiums

### 4.1 Modeling of Endogenous Risk-Premiums

### Functional forms and interpretation
- Private borrowing spread Ψt specified as a logistic autoregressive process:
  - Ψt = ρΨ Ψt−1 + μt + ̃μ, with μt = μ / (1 + e^{−kq (qc,t − ̄qt)})
  - ̄qt ≡ ̄q + ε̄q,t is a threshold real exchange rate level (E(̄qt) = E(̄q + ε̄q,t) = ̄q).
  - μ is the maximum one-period increase in Ψt; kq governs contemporaneous sensitivity as qc,t approaches ̄qt.
  - Calibration feature: ̄q = 22.5 implies substantial real exchange rate depreciation relative to steady state will not induce much rise in private borrowing spread unless ε̄q,t is sufficiently negative so that ̄qt ≤ qc,t.
- UIP risk premium Φt specified as a nonlinear logistic AR(1):
  - Φt = ρΦ Φt−1 + ̃φt, with ̃φt given by a logistic function of dt relative to threshold dt ≡ ̄d + ε̄d,t and influenced by central bank foreign asset purchases bt.
  - ̄d is a threshold net foreign liability level (E(̄dt) = E(̄d + ε̄d,t) = ̄d) at which UIP premium elevates.
  - Constant ̃φ is set so Φt = 0 in the deterministic steady state.
- Central bank foreign asset purchases follow AR(1): bt = ρb bt−1 + εb,t.
- Debt appetite/limit shock follows AR(1): ε̄d,t = ρ̄d ε̄d,t−1 + ūd,t.
- Key qualitative differences between Ψt and Φt:
  - Φt can have non-zero slope when dt = 0 and can be negative when dt < 0 (net foreign assets), ensuring stationarity of net foreign liabilities.

### Calibration and parameter values (as chosen to match moments for selected EMEs)
- UIP risk premium parameters:
  - ρΦ = .85
  - kd = .4
  - ̄d = 12.5
  - φ = .6
  - ρ̄d = .85
  - σ̄d = 11
- Private borrowing spread parameters:
  - ρψ = .85
  - μ = .4
  - ̄q = 22.5
  - kq = .4
- Private debt-limit shock decomposition and idiosyncratic component:
  - ε̄q,t = εp,t + 1/2 ε̄d,t
  - εp,t = ρp εp,t−1 + up,t, with ρp = .85 and σ̄d = 4
  - Median correlation between corporate and sovereign spreads in sample ≈ 0.4 (unweighted mean of individual country correlations slightly lower than the median: .31 instead of .39).
- Small effective linear coefficients implied by nonlinear parametrization:
  - Effective linearized coefficient on real exchange rate for private spread ρq ≈ 2e−5.
  - For UIP risk premium linearization: φd = .003, φb ≈ 1e-10, φε̄d ≈ 4e-11.
- Interpretation of small linearized coefficients:
  - Implies exogenous risk premium shocks ε̄d,t and central bank foreign asset purchases bt would have very limited effects on risk premia if relationships were log-linearized.
  - Nonlinear reduced-form consistent with Chang (2019): FX interventions (FXIs) have essentially no effect on UIP risk premium in normal times but can have substantial effects when investor risk tolerance is low and economy has weak initial conditions.

### Linearization vs nonlinear behavior
- Linearized private spread dynamics (approximation):
  - dΨt = ρΨ dΨt−1 − μ (1 / (1 + e^{kq ̄q}))^2 e^{kq ̄q} (−kq (dqc,t − dε̄q,t))
  - Under steady-state substitutions leads to ψt = ρΨ ψt−1 + ρq (qc,t − ε̄q,t).
  - Under calibration, ρq is extremely small (ρq ≈ 2e−5), demonstrating that linearized DSGE frameworks cannot capture empirically relevant nonlinearities in private borrowing spreads.
- Linearized UIP risk premium dynamics yield φt = ρΦ φt−1 + φd dt + φb bt − φε̄d ε̄d,t with extremely small φd, φb, φε̄d (values above), again implying limited linearized sensitivity.

### Calibration method
- Parameters for endogenous spread functions calibrated via un-conditional stochastic simulations (simulated method of moments) to match moments (persistence, unconditional and conditional volatilities, mean and max values) for sovereign yield spreads on dollar-denominated bonds and corporate spreads (JP Morgan’s EMBI database) for a selected set of emerging market economies.

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### 4.2 Effects of FXI in a Crisis Situation

- Adverse scenario considered:
  - Adverse external shock in global risk tolerance captured by reduction in ̄dt, coupled with a fall in foreign import demand c*t.
  - Compare “vulnerable EME” (high net foreign liabilities) vs “less vulnerable EME” (low net foreign liabilities).
  - Shock to ̄dt follows AR(1) with persistence .85 (half-life about 5 quarters).
- Key simulated outcomes (qualitative and comparative):
  - Vulnerable EME experiences a depreciation of real exchange rate several times larger than less vulnerable EME.
  - Despite larger boost to net exports from depreciation, vulnerable EME experiences a considerably larger output contraction.
    - Drivers: depreciation triggers larger rise in private borrowing spreads; larger rise in inflation forces much tighter monetary policy in vulnerable EME.
  - Policy dilemma: sharper monetary tightening reduces inflation but causes much larger output decline in vulnerable EME.
  - Structural differences (e.g., less well-anchored inflation expectations) would amplify disparity.
- Policy interventions compared: FXI sales and capital flow management (CFM) raising tax on capital outflows τt.
  - Common effects of FXI and CFM versus baseline Taylor rule:
    - Both reduce exchange rate depreciation, allow policy to be more accommodative by lessening inflationary pressure, and mitigate rise in private borrowing spreads.
    - Outcome: reduce “stagflationary” effects of UIP shock—lower inflation and higher output; larger expansionary effects on domestic demand (real net exports improve less).
  - Medium-run tradeoffs:
    - Stronger exchange rate and domestic demand under FXIs or CFMs imply less trade balance improvement and somewhat bigger deterioration of net foreign liabilities.
    - Higher net foreign liabilities can raise UIP risk premium slightly in the longer run, weakening exchange rate relative to baseline.
    - Upward pressure on UIP risk premium slightly higher for CFMs than FXIs because FXIs directly depress the risk premium (reduce dollar borrowing cost for residents) whereas CFMs do not.
  - Temporal considerations:
    - FXIs and CFMs improve near-term inflation-output tradeoffs for fairly transient dt shocks.
    - Appeal diminishes for more protracted shocks: while near-term benefits remain, they can weaken the longer-run turnaround in domestic borrowing needed to lower risk spreads; persistent low investor risk appetite leads to higher long-run risk spreads, lower output, and higher inflation relative to no intervention.
  - Insurance role:
    - Responding to favorable risk premium shocks (surging capital inflows) by using FXIs/CFMs can prevent endogenous buildup of foreign indebtedness and keep economy away from high-debt states vulnerable to sudden spread spikes.
    - Cost: such policies constrain some borrowing that could have benefitted households if high global risk tolerance persisted.

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### 4.3 Unconditional Distributions (simulation design and results)
- Simulation setup:
  - No initial FXI or CFM; central bank uses nominal policy rate for stabilization.
  - Six exogenous shocks driving macro fluctuations: private consumption (ec,t), government spending (gt), foreign demand (c*t), monetary policy shock (ei,t), sovereign risk premium/debt limit shock (ε̄d,t), private risk premium/debt limit shock (ε̄q,t).
  - AR(1) processes: ec,t, gt, c*t, ei,t have persistence root .95 except ei,t with ρi = .8.
  - Shock standard deviations:
    - ec,t and gt: 1.5
    - c*t: 2
    - monetary policy shock: 0.25/4 (so 0.25 at an annualized rate)
  - Calibration of ε̄d,t and ε̄q,t kept as in Section 4.1.
  - Add median spreads observed in EM economies to simulated spread levels to avoid sharply negative sovereign spreads:
    - corporate median spread: 117 basis points
    - government median spread: 205 basis points
  - Long-sample simulation length: T = 20,000 observations.
  - Linearized and nonlinear models compared using identical sampled shocks so differences reflect nonlinear risk-premium propagation.
- Simulated relationships and distributional features:
  - Simulated time series show both risk premiums (annualized percentage points).
  - Scatter relationships:
    - Private borrowing spreads tend to rise when UIP risk premium rises; relationship not tight.
    - Private borrowing spreads tend to rise when real exchange rate depreciates; relationship not tight.
    - Elevated UIP risk premium unlikely when economy holds favorable net foreign asset position; UIP premium much more elevated as net foreign liabilities expand.
  - Unconditional distributions (nonlinear model):
    - UIP risk premium: strong positive skew driven by negative risk-off episodes combined with excessive external indebtedness.
    - Net foreign liabilities: distribution skewed to the left.
    - Real exchange rate: right skew (occasional large depreciations).
    - Output gap and domestic absorption: sizeable negative left skew (domestic absorption stretches to −15 percent; output gap stretches roughly to −10 percent).
    - CPI inflation: longer right tail driven by exchange rate depreciations, but also somewhat elongated left tail because output is sometimes more depressed in nonlinear economy.
  - Correlation statistics:
    - Unconditional correlation coefficient between corporate and sovereign spreads in simulated data: 0.35 (close to median value of 0.39 in the data).

*Source: IMF Working Paper — section 4.1 (Modeling of Endogenous Risk-Premiums) and subsequent results presented in sections 4.2–4.3.*

### 4.4    FXI and CFM Policy Rules

### 4.4    FXI and CFM Policy Rules

### Policy rule specifications
- FX intervention (foreign asset purchases) rule:
  - b_t = b_μ μ_t − b_d (d_t − d_t¯) (eq. 35).
  - Policy parameters: b_μ = 5 and b_d = k_d.
  - Interpretation: central bank sells currency reserves (reduces b_t) when private borrowing spreads or the UIP risk premium on foreign-currency government bonds rise. The rule responds only to the part of the UIP premium that depends on the non-fundamental debt limit shock, not the fundamental part driven by net foreign liabilities. The FX intervention rule must react “slightly” to net foreign liabilities d_t to ensure stationarity of the model.
- Outflow CFM (capital flow management) rule:
  - τ_t = τ_μ μ_t + τ_φ (˜φ_t − φ_t − ρ_Φ φ_{t−1}) (eq. 36).
  - Policy parameters: τ_μ = 5 and τ_φ = 2.5.
  - Interpretation: increases in μ_t raise the capital-output flow tax to reduce incentives to sell domestic currency; the τ_φ term uses CFMs to steer the nonlinear UIP risk premium toward its linearized counterpart (primarily affected by net foreign asset position rather than non-fundamental debt limit shocks).

### Calibration and dynamic assumptions
- Benchmark parametrization: b_μ = 5, b_d = k_d for FXI; τ_μ = 5, τ_φ = 2.5 for CFMs.
- AR(1) persistence used in related exercises:
  - Consumption preference shock and export demand shocks follow AR(1) with root 0.95.
  - Debt appetite shock and b_t intervention in liquidity-trap exercise follow AR(1) with root 0.85.
- Implicit operational assumption: the central bank has sufficient reserves and determination to follow-through on the rules.

### Effects in a vulnerable EME crisis scenario (nonlinear model)
- Full commitment to the FXI and CFM rules:
  - Deliver efficient stabilization of private borrowing spreads.
  - Produce a much lower initial path for the UIP risk premium.
  - Delay adjustment toward lower net foreign liabilities, which implies the UIP risk premium eventually becomes slightly more elevated.
  - When both private spreads and UIP premium are high simultaneously, both rules produce nearly the same macro outcome, though they may imply very different concrete policy actions.
  - Model result: very small adjustments are needed for outflow CFMs if a rule clarifying how they will be adopted is credibly communicated.
- Additional note: these rules can be sufficiently aggressive to stabilize the UIP premium and private spreads relative to the baseline with no endogenous rules (baseline equals case with high initial FX exposure discussed earlier).

### Effects on unconditional distributions (nonlinear model with combined FXI/CFM rule)
- Combined rule outcomes:
  - Effective in countering “sudden stop” shocks that raise the UIP premium and tighten private borrowing spreads.
  - Markedly reduce the likelihood of large real exchange rate depreciations.
  - Mitigate the rise in private borrowing spreads and the need for sharp policy tightening to control inflationary pressures.
  - Noticeable reduction in tail risk of large output contractions driven by such episodes.
  - Reduced incidence of large corresponding swings of the trade balance into surplus.

### Related liquidity-trap exercise (AE calibration; linearized model)
- Setup:
  - Private spread and UIP risk premium shocks set to zero.
  - Large adverse global and domestic shocks push nominal interest rate to zero (effective lower bound).
  - Central bank purchases FX (b_t > 0) with b_t following AR(1) persistence 0.85 to generate a persistent yet transient depreciation.
- Quantitative effects reported:
  - FXI-induced exchange depreciation raises net exports and stimulates output, cushioning recessionary shocks (output falls over 10 percent below baseline in the no-FXI baseline).
  - Because the policy rate is pinned at zero, the central bank does not raise nominal rates in response to higher inflation: real interest rates decline and crowd in domestic demand initially.
  - Core CPI inflation remains much closer to its 2 percent target during the crisis under FXI, stabilizing both inflation and output better.
- For EMEs near the ELB with more responsive inflation expectations and low indebtedness relative to the debt limit:
  - The same-sized intervention yields notably larger stimulative effects on domestic absorption and the output gap due to larger second-round inflation effects.
  - FXIs (or CFMs) may help alleviate monetary policy tradeoffs stemming from the ELB for EMEs not vulnerable to a risk-off shock.

### Policy implications and limitations
- Potential benefits:
  - FXIs and CFMs can improve policy tradeoffs—particularly for economies with less well anchored inflation expectations and significant foreign currency mismatch.
  - Nonlinear model highlights stronger scope for improving tradeoffs when initial conditions are weak (e.g., high debt levels).
- Cautions and open questions:
  - Frequent use of FXIs and CFMs may slow development of FX derivatives markets and possibly encourage a buildup of FX debt.
  - Additional considerations needed before practical adoption, including potential longer-term costs.
  - Future extensions suggested:
    - Add reduced-form housing and/or banking sector to analyze macroprudential interactions.
    - Consider how fiscal policy affects appropriate use of IPF tools.
    - Assess multilateral spillovers from IPF tool usage.
    - Empirically estimate transmission of IPF tools and heterogeneity across economies; the model is being estimated for a group of emerging and advanced small open economies to deepen quantitative understanding.

*Source: wpiea2020122-print-pdf - 4.4    FXI and CFM Policy Rules*

### References

### wpiea2020122-print-pdf - References and Technical Appendix

### References: scope and thematic coverage
- Bibliographic coverage emphasizes models, empirical work, and policy analysis on:
  - Foreign exchange intervention, exchange rate management, and capital flows (Adler; Blanchard et al.; Cavallino; Daude et al.; Fratzscher et al.; Ghosh et al.; Tong and Wei).
  - DSGE, New Keynesian, and open-economy macro frameworks and estimation methods (Adjemian et al.; Adrian et al. (2020); Del Negro et al.; Lindé et al.; Schmitt-Grohe and Uribe; Erceg et al.; Christiano et al.; Clarida et al.; Gabaix).
  - Price and wage setting, nominal rigidities, and Phillips-curve dynamics (Calvo; Dixit and Stiglitz; Dotsey and King; Yun et al.; Kimball).
  - Fiscal and monetary policy at the ZLB, forward guidance, and government spending multipliers (Eggertsson; Eggertsson and Woodford; Campbell et al.; Erceg and Linde; Fève et al.).
  - International finance and risk premia, reserves, and macroprudential aspects (Jeanne; Jeanne and Korinek; Kalemli-Ozcan; Hoffmann et al.; Bruno and Shin).
- The References list provides the theoretical and empirical foundations cited throughout the Technical Appendix and main text.

### Technical Appendix — Major model components (A The Model Environment)
- The Appendix documents the full non-linear open-economy model used for analysis and references a companion document Adrian et al. (2020) for full derivations.

H3: Households and wage setting (A.1)
- Utility functional (equation (37)):
  - E_t ∑_{j=0}^∞ β^j ς_{t+j} { (C̃_{t+j}(h) − κ C̃_{t+j−1} − C̃_{t+j}^ν)^{1−1/σ}/(1−1/σ) − χ_0 N_{t+j}(h)^{1+χ}/(1+χ) + μ_0 F(M_{t+j+1}(h)/P_{C,t+j}) }.
- Private and public consumption: C̃_t(h) = C_t(h) + η_0 G_t.
  - κ ∈ (0,1) measures external consumption habits; ν_{t+j} is a demand shock.
- Money included to rationalize the zero lower bound (ZLB); sub-utility F(·) has a satiation point for M/P.
- Key parameter restrictions: 0 < β < 1; σ, χ, χ_0, μ_0 > 0.
- Flow budget constraint (equation (39))—nominal expenditures on consumption, money accumulation, state-contingent domestic claims, domestic government bonds, and international bond position B_int:
  - P_{C,t} C_t(h) + (M_{t+1}(h) − M_t(h)) + ∫_{s∈Ω} ξ_{t,t+1}(s) B_{D,t+1}(h)(s) ds + B_{G,t}(h) + ε_t B_{int,t} = W_t N_t(h) + R_{K,t} K + Ξ_t(h) − T_t + B_{D,t}(h) + I_{t−1} B_{G,t−1}(h) + ε_t I^*_ {t−1} Φ(B_{int,t−1}/P_{D,t} Y, φ_{t−1}) B_{int,t−1}.
- International bond premium specification (equation (40)):
  - Φ(b_{int,t}, φ_t) ≡ e^{−φ_{int}(b_{int,t} − b_{int})} + φ_t, with φ_{int} > 0, Φ(0,0) = 1.
  - Risk-premium shock follows AR(1) (equation (41)): φ_t = ρ_φ φ_{t−1} + ε_{φ,t}, ε_{φ,t} ∼ i.i.d. N(0, σ_ε^2).
- Risk-adjusted UIP (log-linearized):
  - i_t − i^*_t = −φ_{int}(b_{int,t} − b_{int}) + φ_t + E_t ε_{t+1} − ε_t.
- Net foreign assets dynamics (equation (42)) and expression for b_{int,t} (equation (43)):
  - B_{int,t} = T B_t + I^*_{t−1} Φ(b_{int,t−1}, φ_{t−1}) B_{int,t−1}.
  - b_{int,t} = [ ω_C s_{cy} ((1+ρ_C)/ρ_C (2−ω^*_C − ω_C)^{−1}) + ω_G s_{gy} ((1+ρ_G)/ρ_G (2−ω^*_G − ω_G)^{−1}) ] γ_{MD,t} − ω_C s_{cy} (c_t − c^*_t) − ω_G s_{gy} (g_t − g^*_t) + (1 + r^*) b_{int,t−1}, where 1 + r^* ≡ (1 + i^*)/(1 + π).

H3: Firms and price-setting (A.2)
- Final goods aggregator (A.2.1):
  - Final domestic price index P_{D,t} given by (equation (44)):
    - P_{D,t} = 1/(1 + ψ) ( ∫_0^1 P_{D,t}(f)^{−1+ψ+ψ θ_p/θ_p} df )^{−θ_p/(1+ψ+ψ θ_p)} + ψ/(1 + ψ) ∫_0^1 P_{D,t}(f) df.
  - Demand for variety f (equation (45)) is a generalization that nests Dixit-Stiglitz when ψ = 0.
- Intermediate goods production (A.2.2):
  - Cobb-Douglas: Y_{D,t}(i) = K_t(i)^α (Z_t L_t(i))^{1−α}.
  - Marginal cost: MC_t = (W_t/(1−α))^{1−α} (R_{K,t}/α)^α 1/Z_t^{1−α}.
  - Return on capital: R_{K,t} = α/(1−α) W_t L_t/K.
- Optimal price setting and Calvo contracts (A.2.3):
  - Probability of reoptimizing domestically: 1 − ξ_p; reoptimizing abroad: 1 − ξ_{p,x}.
  - Firms that cannot reoptimize index prices: P^{ind}_{D,t}(i) = Π_D P_{D,t−1}(i) (equation (46)), where Π_D is steady-state PPI inflation.
  - Reoptimizing firms set P̃_{D,t}(i) to maximize expected discounted profits with household stochastic discount factor.
  - Resulting log-linearized pricing conditions produce Phillips-curve relationships for domestic, imported, and export prices.

H3: Monetary and fiscal policy (A.3)
- Benchmark Taylor-type rule (equation (47)):
  - I_t = I + ψ_π [Π_{C,t} − Π_C] + ψ_x (Y_{D,t}/Y^{pot}_{D,t} − 1), where I denotes steady-state gross nominal interest rate and Π_C steady state consumer price inflation.
- Government budget and fiscal processes:
  - Nominal government debt evolution: B_{G,t} = I_{t−1} B_{G,t−1} + P_t G_t − T_t.
  - Public consumption process (equation (48)): G_t = (1 − ρ_G) G + ρ_G G_{t−1} + ε_{G,t}.
  - Budget is balanced each period via non-distortionary lump-sum taxes T_t.

H3: Importing and exporting firms (A.4)
- Import sector:
  - Importing firms buy homogeneous foreign good at price P^*_t, differentiate it, and face local-currency Calvo stickiness with reoptimization probability 1 − ξ_{m,c}.
  - Objective for reoptimizers: maximize ∑_{j=0}^∞ (β ξ_{m,c})^j E_t ς_{t+j}/ς_t Λ_{t+j}/Λ_t P_{C,t}/P_{C,t+j} [Π^j_D P̃_{M,t} M_{C,t+j|t}(i) − TC^m_{t+j|t}(i)].
  - Import aggregation: M_{C,t} = [ ∫_0^1 (M_{C,t}(i))^{1/λ_{m,t}} di ]^{λ_{m,t}}, with 1 ≤ λ_{m,t} < ∞, and markup process λ_{m,t} = (1 − ρ_{λ_m}) λ_m + ρ_{λ_m} λ_{m,t−1} + ε_{λ_m,t}.
  - Resulting log-linearized pricing yields a Phillips curve for imported goods.
- Export sector:
  - Export firms buy domestic final good at price P_t, differentiate it, sell to foreign market; demand X_t(i) = (P_{X,t}(i)/P_{X,t})^{−λ_{x,t}/(λ_{x,t} − 1)} X_t.
  - Export markup process: λ_{x,t} = (1 − ρ_{λ_x}) λ_x + ρ_{λ_x} λ_{x,t−1} + ε_{λ_x,t}.
  - Export prices sticky in foreign currency under Calvo setup; firms index when unable to reoptimize: P_{X,t}(i) = Π_D P_{X,t−1}(i).
  - Log-linearized optimality conditions map to a Phillips-curve linking export prices to marginal cost.

H3: Market clearing (A.5)
- Domestic intermediate goods market clearing:
  - Y_{D,t} = (P^#_{D,t})^{−1} K^α (Z_t L_t)^{1−α}, where price dispersion P^#_{D,t} satisfies the integral expression given.
- Aggregate goods resource constraint (per-capita to economy-wide):
  - For each firm f: Y_t(f) = Y_{D,t}(f) + X_t(f).
  - Economy-wide identity: ζ Y_{D,t} = ζ C_{D,t} + ζ G_{D,t} + ζ^* M^*_{C,t} + ζ^* M^*_{G,t} ⇐⇒ Y_{D,t} = C_{D,t} + G_{D,t} + (ζ^*/ζ) [M^*_{C,t} + M^*_{G,t}].

- The Appendix notes that intermediate derivations and linearizations used in simulations are available in Adrian et al. (2020).

*Source: wpiea2020122-print-pdf — References and Technical Appendix*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020122-print-pdf.pdf_