## _wp1215 - 1.    Final-good producers

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---

### Introduction: motivation and main results
- Central banks distinguish short-term CPI movements from underlying or core inflation; multiple ad hoc core inflation definitions are in use (e.g., excluding energy, food, administered prices).
- Literature baseline: Aoki (2001) — in two-sector models with flexible versus sticky prices, optimal policy fully stabilizes distorted (sticky) sector inflation and disregards flexible-price sector inflation; this maps to excluding volatile items from CPI as core inflation, but the mapping flexible = volatile is not guaranteed.
- Paper’s extension: introduce a subsidized share of private consumption expenditures and model government intervention that stabilizes subsidized goods’ prices around historical average headline inflation.
- Key qualitative results:
  - Strict headline inflation targeting can be dominated by sectoral inflation targeting, output gap stabilization, or hybrids.
  - Optimal monetary policy with subsidized prices and costly price adjustments is sensitive to the relative importance of two distortions: nominal price inertia (stickiness) and price subsidies (fiscal distortion).
  - Financing of subsidies matters: subsidies financed via distortionary taxes amplify welfare costs; positive comovement of labor supply elasticity and average tax rates magnifies employment and consumption volatility.
  - Asymmetric government reactions (bias toward smoothing upward price changes, i.e., upward stickiness) dampen negative tax-rate changes and amplify taxation distortion following shocks.
  - Empirical estimation across countries shows heterogeneity in sectoral distortions; country-specific thresholds identified numerically for the degree of government intervention under which core inflation targeting is optimal.
- Empirical sample: fourteen countries — Argentina, Brazil, Czech Republic, Egypt, Hungary, Indonesia, Mexico, Morocco, Philippines, Poland, South Africa, Thailand, Tunisia, and Turkey.

### Model setup: agents and key functional forms
- Economy structure:
  - Representative household, representative final-good firm, monopolistically competitive intermediate-good firms, monetary authority setting short-term nominal interest rate via a Taylor rule.
  - Government intervention: some consumed goods are subsidized/administered; government follows a preannounced rule to minimize subsidized-goods inflation deviations from historical average headline inflation. Government reactions can be asymmetric (upward smoothing bias).
- Household preferences and budget:
  - Objective: max E_0 sum_{t=0}^∞ β^t [ u(c_t,c_{t−1}, M_t/P_t) − η V(h_t) ], with β∈(0,1) and η measures leisure weight.
  - Period utility: u(c_t,c_{t−1}, M_t/P_t) = log(c_t − a c_{t−1}) + b log(M_t/P_t).
  - Leisure: V(h_t) = log(1 − h_t).
  - Budget constraint:
    P_t c_t + P_t i_t + M_t + P_t CAC_t + B_t / R_t ≤ (1 − τ_t) W_t h_t + Q_t k_t + M_{t−1} + D_t + B_{t−1} + T_t,
    where R_t is gross nominal interest rate on one-period debt, τ_t is distortionary tax on nominal labor revenues (endogenously determined), T_t is lump-sum transfer, CAC_t is capital adjustment cost.
  - Capital accumulation: k_{t+1} = (1 − δ) k_t + i_t.
  - Capital adjustment cost: CAC_t = Γ(k_{t+1} / k_t) k_t with Γ(1) = 0, Γ′(1) = 0, Γ′′(1) = ψ_k.
- Final-good production and demand:
  - Final-good Cobb–Douglas: y_t = (y^S_t)^φ (y^{nS}_t)^{(1−φ)}, with φ>0 weight of S composite good.
  - Composite indexes:
    y^S_t ≡ ( ∫_0^1 y^S_t(i)^{(θ_S−1)/θ_S} di )^{θ_S/(θ_S−1)}, y^{nS}_t ≡ ( ∫_0^1 y^{nS}_t(i)^{(θ_{nS}−1)/θ_{nS}} di )^{θ_{nS}/(θ_{nS}−1)}.
  - Individual demands:
    y^S_t(i) = (P^S_t(i) / P^S_t)^{−θ_S} y^S_t; y^{nS}_t(i) = (P^{nS}_t(i) / P^{nS}_t)^{−θ_{nS}} y^{nS}_t.
- Government subsidy rule for S-good final price (notation: ̃P^S_t denotes final price for S goods; π is steady-state headline inflation):
  - ̃P^S_t = κ / (1 + exp[ −ν (P^S_t / ̃P^S_{t−1} − π)]) ̃P^S_{t−1} + (1 − κ / (1 + exp[ −ν (P^S_t / ̃P^S_{t−1} − π)])) P^S_t + ε_{P^S_t},
    where κ represents the degree of government intervention, ν controls asymmetric response, and ε_{P^S_t} is an iid shock.
  - Implications: larger κ makes S-good prices stickier; ν>0 generates upward stickiness — increases in market price face exponentially larger smoothing than decreases.
  - Long-run assumption: π_S = π_{nS} = π to solve for relative prices.
- Final-good firm optimization:
  - y^S_t = φ (P_t / ̃P^S_t) y_t.
  - y^{nS}_t = (1 − φ) (P_t / P^{nS}_t) y_t.
  - Zero-profit condition (normalized): P_t = (̃P^S_t^φ) (P^{nS}_t^{1−φ}).

### Quantitative calibration (baseline) and interpretation
- Frequency: quarterly.
- Calibrated parameter values:
  - β = 0.98 (quarterly subjective discount factor) ⇒ annual nominal interest rate = 8 percent in deterministic steady state.
  - η chosen so fraction of hours worked = 0.25 in steady state.
  - α_{S,nS} = 0.36; δ = 0.025.
  - τ (average tax rate) = 0.35.
  - θ_S = ∞ (S-intermediate goods: perfect competition); θ_nS = 8 ⇒ markup 14 percent.
  - ψ_S = 0; ψ_nS = 40.
  - φ ∈ [0,0.40] considered.
  - κ = 0.75; ν = 500.
  - ρ_{A_S}, ρ_{A_nS}, ρ_ζ = 0.80.
  - σ_{A_S}, σ_{A_nS} = 0.01; σ_ζ = 0.005; σ_{εP_S} = 0.005.
- Interpretation of ψ_nS: ψ_nS = 40 ⇒ probability of not changing prices = 0.67 ⇒ changing prices every three-quarters on average.

### Which inflation to target? — key analytical results
- Terminology: nS-good price inflation = core inflation.
- Main findings from second-order approximation and welfare evaluation:
  - Solely and aggressively reacting to CPI inflation is always dominated by sectoral inflation targeting.
  - Government intervention through subsidies does not advocate aggressive reaction to CPI inflation.
  - Even when reacting to both sectoral inflation rates, optimal weights do not correspond to implicit structural-parameter weights.
- Special cases:
  - If only imperfection is price adjustment costs in nS goods (κ = 0), optimized rule fully stabilizes nS-good inflation (ρ_{πnS} = ∞).
  - CPI inflation is optimal to target when π_t = π_{nS t} (φ = 0); otherwise targeting CPI is suboptimal.
- Government subsidies substantially alter optimality of nS-good targeting even at φ = 0.20.
- Policy implication in context of large distortions: eliminate price subsidies and fully stabilize core inflation.

### Sources of the tradeoff between sectoral inflation stabilization
- Linearized relations:
  - ˆ˜π_S t = (1−κ)/κ (ˆp_S t − ˆ˜p_S t).
  - ˆπ_{nS t} = β E_t ˆπ_{nS t+1|t} + (θ−1)/ψ_nS ˆξ_{nS t}.
- Tradeoff mechanics:
  - Stabilizing S-good inflation requires government to stop intervening (ˆp_S t = ˆ˜p_S t).
  - Stabilizing nS-good inflation requires firms equalizing prices to marginal costs (ˆξ_{nS t} = 0).
  - Presence of both frictions prevents Pareto-optimal social welfare with only the interest rate instrument.
- Optimal targeting depends on relative importance of frictions; targeting S-good inflation can provide higher welfare when S-good share is high and/or nS price rigidity is low; non-monotonicities can arise.

### Role of asymmetric price adjustments and subsidy financing
- Parameter ν (asymmetry):
  - ν = 0 ⇒ ˜P_S t = κ ˜P_S t−1 + (1−κ) P_S t (symmetric smoothing) ⇒ expected first-level tax effects cancel under mean-zero shocks.
  - ν > 0 ⇒ negative market price changes passed to subsidized price, but positive shocks are smoothed ⇒ net positive tax rate changes ⇒ larger distortionary effect.
- Financing scenarios:
  - Subsidies fully financed through lump-sum taxes: −T_t + (M_t − M_{t−1}) = (P_S t − ˜P_S t) y_s t.
  - Subsidies financed via distortionary taxes on labor revenue: τ_t W_t h_t = (P_S t − ˜P_S t) y_s t.
- Interaction effects:
  - Financing via distortionary taxes amplifies welfare impact of S-good price distortions.
  - Higher average tax rates increase net distorting effect of subsidies and make S-good inflation targeting more attractive.
- Empirical sensitivity reported:
  - Welfare cost of strict S-good targeting can be extremely high when ν = 0 (symmetric adjustment).
  - Higher average tax rates increase welfare gains from allowing S-good inflation targeting.

### Estimation strategy and key estimated parameters
- Estimation method:
  - Bayesian Metropolis-Hastings; state-space representation; Kalman filter; harmonic mean estimator for marginal data density.
- Structural shocks estimated: ε_{A_S}, ε_{A_nS}, ε_R, ε_{P_s}; measurement errors added.
- Observables: production growth, consumption growth, CPI inflation. Morocco and Tunisia include output growth and two sectoral inflation rates.
- Sample: seasonally adjusted quarterly series for fourteen countries from earliest availability to 2008Q4.
- Priors and estimated set:
  - Estimated parameters: {a, ψ_k, ψ_nS, φ, κ, ρ_{A_S}, ρ_{A_nS}, ρ_ζ, σ_{A_S}, σ_{A_nS}, σ_ζ, σ_{P_S}} with priors summarized (examples: a ~ Beta(mean 0.50, std 0.15); ψ_k ~ Gamma(mean 5, df 2); φ ~ Beta(mean 0.25, std 0.1); κ ~ Beta(mean 0.75, std 0.1); ρ_{A_S}, ρ_{A_nS}, ρ_ζ ~ Beta(mean 0.8, std 0.1)).
- Estimation outcomes (high-level):
  - Bayesian draws: 200,000.
  - Posterior precision: shocks and price-distortion parameters identified well.
  - Shock volatilities: nS-good supply shocks most volatile; S-good price shocks often second (exceptions: Egypt, Poland).
  - Habit formation a posterior means range: 0.15 (Mexico) to 0.88 (Philippines).
  - ψ_nS^{Calvo} posterior medians vary: example Argentina ≈ 0.284 (about 5 months); Morocco about 20 months in other countries.
  - Estimated φ range: 0.13 (Poland) to 0.47 (Philippines); Tunisia 90% interval [0.25, 0.44] with declared share 0.32; Morocco 90% interval [0.24, 0.45] with declared share 0.31.
  - κ estimates generally > 0.50, indicating active government smoothing.
  - Model comparison: strong evidence for S-good sector (φ>0) in Argentina, Brazil, Egypt, Mexico, Morocco, Tunisia, Turkey; evidence favors φ = 0 in Czech Rep., Hungary, Indonesia, Poland, South Africa, Thailand.

### Welfare evaluation and policy rankings
- Welfare metric: compensating variation ω (percentage of lifetime consumption change to equate expected utility), computed from second-order approximation.
- Policies compared (relative to headline CPI targeting):
  - (i) S-good inflation stabilization.
  - (ii) core (nS-good) inflation stabilization.
  - (iii) optimized Taylor rule (ρ_{πS}, ρ_{πnS}, ρ_y, ρ_R chosen to maximize unconditional expected utility), subject to uniqueness/stability constraint.
- Welfare gains from optimized rules (permanent consumption equivalents) relative to headline targeting range:
  - From 0.0013 percent (Tunisia) to 0.1455 percent (Argentina).
- Country-specific optimal policy examples:
  - Tunisia and Indonesia: full core-inflation stabilization optimal (optimized rule with log(R_t/R) = ∞ log(π_{nS t}/π_{nS})).
  - Brazil optimized rule: log(R_t/R) = 2.5947 log(π_S t/π_S) + 1.9917 log(π_{nS t}/π_{nS}) + 0.4296 log(Y_t/Y).
  - Argentina optimized coefficients: ρ_{πS} = 1.3079, ρ_{πnS} = 0.0000, ρ_y = 0.0000, ρ_R = 0.0016; headline targeting loses −0.1455 percent permanent consumption.
  - Turkey example: ρ_{πS} = 3.6436, ρ_{πnS} = ∞ in optimal rule for some calibrations.
- Welfare losses of aggressive core-only targeting can be multiple times the loss under the optimized rule when φ and τ are high.

### Fiscal thresholds and policy implications
- Required reductions in φ for core inflation targeting to be optimal (selected entries from Table 6):
  - Argentina: estimated φ = 0.356 → threshold φ = 0.186.
  - Brazil: 0.307 → 0.204.
  - Czech Republic: 0.246 → 0.068.
  - Egypt: 0.336 → 0.182.
  - Hungary: 0.226 → 0.088.
  - Mexico: 0.373 → 0.262.
  - Morocco: 0.347 → 0.200.
  - Philippines: 0.472 → 0.360.
  - Poland: 0.134 → 0.094.
  - South Africa: 0.248 → 0.196.
  - Turkey: 0.355 → 0.132.
  - Indonesia, Thailand, Tunisia: thresholds not binding (“—”).
- Practical obstacles to achieving required fiscal reforms:
  - Weak capacity to target the poor.
  - Lack of transparency in subsidy reporting.
  - Opposition from vested interests.
  - Spillover effects.
  - Ad hoc price setting mechanisms.
- Policy recommendation: reduce the extent of price subsidies to enable an optimized, comprehensive, credible, and transparent monetary policy rule that targets core inflation; however, fiscal reforms face significant implementation challenges.

### Conclusion: policy-relevant summary
- In a New Keynesian framework with subsidized (S) and non-subsidized (nS) sectors (nS sticky), optimized monetary reaction functions that account for sectoral inflation and output can yield welfare gains relative to headline CPI targeting.
- Optimal policy choice depends critically on:
  - Share of subsidized goods φ.
  - Degree of nominal rigidity in nS sector ψ_nS.
  - Degree of government price smoothing κ and asymmetry parameter ν.
  - Financing of subsidies (distortionary vs lump-sum) and average tax rate τ.
- Practical implication: reducing government price interventions (subsidies) can expand feasibility and optimality of core inflation targeting, but fiscal reforms face significant implementation challenges.

*Italic source: IMF Working Paper — _wp1215 - 1.    Final-good producers (chapter extract).*

### 1.    Final-good producers   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .8

### _wp1215 - 1.    Final-good producers

### Introduction: motivation and main results
- Central banks distinguish short-term CPI movements from underlying or core inflation; multiple ad hoc core inflation definitions are in use (e.g., excluding energy, food, administered prices).
- Literature view: Aoki (2001) — in two-sector models with flexible versus sticky prices, optimal policy fully stabilizes distorted (sticky) sector inflation and disregards flexible-price sector inflation; this maps to excluding volatile items from CPI as core inflation, but the mapping flexible = volatile is not guaranteed.
- Paper’s extension: introduce a subsidized share of private consumption expenditures and model government intervention that stabilizes subsidized goods’ prices around historical average headline inflation.
- Key qualitative results (from model and second-order welfare computations):
  - Strict headline inflation targeting can be dominated by sectoral inflation targeting, output gap stabilization, or hybrids.
  - Optimal monetary policy in presence of subsidized prices and costly price adjustments is sensitive to the relative importance of two distortions: nominal price inertia (stickiness) and price subsidies (fiscal distortion).
  - Financing of subsidies matters: subsidies financed via distortionary taxes amplify welfare costs; positive comovement of labor supply elasticity and average tax rates magnifies employment and consumption volatility.
  - Asymmetric government reactions (bias toward smoothing upward price changes, i.e., upward stickiness) dampen negative tax-rate changes and amplify taxation distortion following shocks.
  - Empirical estimation across countries shows heterogeneity in sectoral distortions; country-specific thresholds identified numerically for the degree of government intervention under which core inflation targeting is optimal.
- Empirical sample: fourteen countries — Argentina, Brazil, Czech Republic, Egypt, Hungary, Indonesia, Mexico, Morocco, Philippines, Poland, South Africa, Thailand, Tunisia, and Turkey.

### Model setup: agents and key functional forms
- Economy: representative household, representative final-good firm, monopolistically competitive intermediate-good firms, monetary authority setting short-term nominal interest rate via a Taylor rule.
- Government intervention: some consumed goods are subsidized/administered; government follows a preannounced rule to minimize subsidized-goods inflation deviations from historical average headline inflation. Government reactions can be asymmetric (upward smoothing bias).
- Household:
  - Objective: max E_0 sum_{t=0}^∞ β^t [ u(c_t,c_{t−1}, M_t/P_t) − η V(h_t) ], where β∈(0,1) and η measures leisure weight.
  - Period utility: u(c_t,c_{t−1}, M_t/P_t) = log(c_t − a c_{t−1}) + b log(M_t/P_t).
  - Leisure utility: V(h_t) = log(1 − h_t).
  - Parameters: a is habit-formation degree; b measures weight of real balances.
  - Budget constraint:
    P_t c_t + P_t i_t + M_t + P_t CAC_t + B_t / R_t ≤ (1 − τ_t) W_t h_t + Q_t k_t + M_{t−1} + D_t + B_{t−1} + T_t,
    where R_t is gross nominal interest rate on one-period debt, τ_t is distortionary tax on nominal labor revenues (endogenously determined by fiscal rule), T_t is lump-sum transfer, CAC_t is capital adjustment cost.
  - Capital accumulation: k_{t+1} = (1 − δ) k_t + i_t, where δ∈(0,1).
  - Capital adjustment cost: CAC_t = Γ(k_{t+1} / k_t) k_t with Γ satisfying Γ(1) = 0, Γ′(1) = 0, and Γ′′(1) = ψ_k.
- Firms:
  - Final-good producers: perfectly competitive, combine subsidized (S) and non-subsidized (nS) intermediate goods via Cobb–Douglas:
    y_t = (y^S_t)^φ (y^{nS}_t)^{(1−φ)}, where φ>0 is weight of S composite good.
  - Composite indexes:
    y^S_t ≡ ( ∫_0^1 y^S_t(i)^{(θ_S−1)/θ_S} di )^{θ_S/(θ_S−1)} and y^{nS}_t ≡ ( ∫_0^1 y^{nS}_t(i)^{(θ_{nS}−1)/θ_{nS}} di )^{θ_{nS}/(θ_{nS}−1)},
    with θ_S (θ_{nS}) > 1 elasticities of substitution.
  - Individual demands:
    y^S_t(i) = (P^S_t(i) / P^S_t)^{−θ_S} y^S_t and y^{nS}_t(i) = (P^{nS}_t(i) / P^{nS}_t)^{−θ_{nS}} y^{nS}_t for i∈(0,1).
- Government subsidy rule for S-good final price (notation: ̃P^S_t denotes final price for S goods; π is steady-state headline inflation):
  ̃P^S_t = κ / (1 + exp[ −ν (P^S_t / ̃P^S_{t−1} − π)]) ̃P^S_{t−1} + (1 − κ / (1 + exp[ −ν (P^S_t / ̃P^S_{t−1} − π)])) P^S_t + ε_{P^S_t},
  where κ represents the degree of government intervention to smooth sector price changes over time, ν controls asymmetric response to sign/magnitude of market price P^S_t, and ε_{P^S_t} is an iid shock.
  - Implications: larger κ makes S-good prices stickier; ν>0 generates upward stickiness — increases in market price face exponentially larger smoothing/frictional cost than decreases.
  - Long-run assumption: sectoral inflation rates equal (π_S = π_{nS} = π) to allow solving for relative prices.
- Final-good firm maximization:
  - Problem: max_{y^S_t,y^{nS}_t} P_t y_t − ̃P^S_t y^S_t − P^{nS}_t y^{nS}_t, with y_t from production function.
  - First-order conditions imply:
    y^S_t = φ (P_t / ̃P^S_t) y_t,
    y^{nS}_t = (1 − φ) (P_t / P^{nS}_t) y_t.
  - Zero-profit condition for final good price:
    P_t = (̃P^S_t^φ) (P^{nS}_t^{1−φ}) normalized as P_t = (̃P^S_t φ)^φ (P^{nS}_t 1−φ)^{(1−φ)}.

### Quantitative and policy implications (qualitative summary from provided text)
- Heterogeneity across countries in sectoral distortions implies multiple optimal monetary-policy scenarios; a one-size-fits-all inflation-targeting approach is likely inappropriate where price subsidies and sticky prices coexist.
- Policy recommendation drawn from model estimation: reduce the extent of price subsidies to enable an optimized, comprehensive, credible, and transparent monetary policy rule that targets core inflation.
- Numerical identification of country-specific subsidy thresholds under which core inflation targeting is optimal (threshold results are computed numerically in paper; tables and figures provided in the full chapter).

*Source: _wp1215 - 1.    Final-good producers (chapter extract).*

### 2.    Intermediate-good producers

### 2.    Intermediate-good producers

### Production technology and price setting
- S and nS intermediate-good producers have identical Cobb-Douglas production functions:
  - y_jt(i) = A_jt k_jt(i)^{α_j} h_jt(i)^{1−α_j}, where j = S,nS and α_j ∈ (0,1).
- Aggregate technology shock process:
  - log(A_jt) = (1−ρ_j^A) log(A) + ρ_j^A log(A_j,t−1) + ε_jt, with ρ_j^A ∈ (−1,1), ε_jt ~ N(0, σ_j^A).
- Firms are monopolistically competitive and price-setters, segmenting markets by destination.
- Quadratic price adjustment (Rotemberg 1982):
  - Adjustment cost = ψ_j/2 (P_jt(i)/(π_j P_j,t−1(i)) − 1)^2, ψ_j ≥ 0, π_j steady-state of π_jt ≡ P_jt/P_j,t−1.
- Firm dynamic problem maximizes discounted profits with demand functions given (see text).
- First-order conditions (representative):
  - w_t/p_jt = (1−α_j) ξ_jt(i) y_jt(i) / h_jt(i).
  - q_t/p_jt = α_j ξ_jt(i) y_jt(i) / k_jt(i).
  - Pricing FOC (generalized Rotemberg expression) detailed in equation (18) relating markups, adjustment costs, and expected future variables.

### Monetary and fiscal authorities (model rules)
- Monetary authority (historical specification):
  - ζ_t = M_t / M_{t−1}, log(ζ_t) = ρ_ζ log(ζ_{t−1}) + ε_{ζ,t}, with ρ_ζ ∈ (0,1), ε_{ζ,t} ~ iid N(0, σ_ζ).
- Government budget constraint:
  - (P_S t − ˜P_S t) y_s t + T_t = τ_t W_t h_t + (M_t − M_{t−1}).
- Two financing scenarios highlighted:
  - Subsidies fully financed through lump-sum taxes: −T_t + (M_t − M_{t−1}) = (P_S t − ˜P_S t) y_s t.
  - Subsidies financed via distortionary taxes on labor revenue: τ_t W_t h_t = (P_S t − ˜P_S t) y_s t.
- Optimized monetary rule (hybrid Taylor rule) used in counterfactuals substitutes money-growth rule:
  - log(R_t/R) = ρ_R log(R_{t−1}/R) + (1−ρ_R)[ρ_π log(π_t/π) + ρ_{πS} log(˜π_t^S/˜π_S) + ρ_{πnS} log(π_{nS t}/π_{nS}) + ρ_y log(y_t/y)].

### Calibration (baseline)
- Frequency: quarterly.
- Key calibrated parameter values:
  - β = 0.98 (quarterly subjective discount factor) ⇒ annual nominal interest rate = 8 percent in deterministic steady state.
  - η chosen so fraction of hours worked = 0.25 in steady state.
  - α_{S,nS} = 0.36 (capital shares), δ = 0.025 (quarterly depreciation rate).
  - τ (average tax rate) = 0.35.
  - θ_S = ∞ (S-intermediate goods: perfect competition); θ_nS = 8 ⇒ markup 14 percent.
  - ψ_S = 0 (S-goods prices fully flexible); ψ_nS = 40 (Rotemberg price-adjustment cost for nS).
  - φ (share of S goods) ∈ [0,0.40] considered in sensitivity analysis.
  - κ = 0.75 (first order government price control).
  - ν = 500 (asymmetric price adjustment parameter for S-good sector).
  - ρ_{A_S}, ρ_{A_nS}, ρ_ζ = 0.80 (first-order serial correlations).
  - σ_{A_S}, σ_{A_nS} = 0.01; σ_ζ = 0.005; σ_{εP_S} = 0.005.
- Interpretation of ψ_nS:
  - Imposing equivalence with Calvo, ψ_nS = 40 ⇒ probability of not changing prices = 0.67 ⇒ changing prices every three-quarters on average.

### Which inflation to target? — key results
- Terminology: nS-good price inflation = core inflation.
- Main findings from second-order approximation and welfare evaluation:
  - Solely and aggressively reacting to CPI inflation is always dominated by sectoral inflation targeting.
  - Government intervention through subsidies does not advocate aggressive reaction to CPI inflation.
  - Even when reacting to both sectoral inflation rates, optimal weights do not correspond to implicit structural-parameter weights (equation (12)).
- Special cases:
  - If only imperfection is price adjustment costs in nS goods (κ = 0), optimized rule fully stabilizes nS-good inflation (ρ_{πnS} = ∞).
  - CPI inflation is the right variable to target when π_t = π_{nS t} (φ = 0). Otherwise, targeting CPI becomes suboptimal.
- Government subsidies substantially alter optimality of nS-good targeting even at φ = 0.20.
- The optimal policy in the context of large distortions: eliminate price subsidies and fully stabilize core inflation.

### Sources of the tradeoff between sectoral inflation stabilization
- Linearized relations (mechanisms):
  - ˆ˜π_S t = (1−κ)/κ (ˆp_S t − ˆ˜p_S t).
  - ˆπ_{nS t} = β E_t ˆπ_{nS t+1|t} + (θ−1)/ψ_nS ˆξ_{nS t}.
- Tradeoff:
  - Stabilizing S-good inflation requires government to stop intervening in price setting (ˆp_S t = ˆ˜p_S t).
  - Stabilizing nS-good inflation requires firms equalizing prices to marginal costs (ˆξ_{nS t} = 0).
  - Presence of both frictions prevents Pareto-optimal social welfare with only interest rate instrument.
- Optimal targeting depends on relative importance of frictions; targeting S-good inflation can provide higher welfare when S-good share is high and/or nS price rigidity is low, but non-monotonicities can arise (e.g., ψ_nS = 0.67, φ = 0.3).

### Role of asymmetric price adjustments and financing subsidies
- Asymmetric price adjustment parameter ν matters:
  - ν = 0 ⇒ ˜P_S t = κ ˜P_S t−1 + (1−κ) P_S t (symmetric smoothing) ⇒ expected first-level tax effects cancel under mean-zero shocks.
  - ν > 0 ⇒ negative market price changes passed to subsidized price, but positive shocks are smoothed by government ⇒ net positive tax rate changes ⇒ larger distortionary effect.
- Interaction with financing:
  - Financing subsidies via distortionary taxes amplifies impact of S-good price distortions on welfare.
- Tax rate level matters:
  - Higher average tax rates increase net distorting effect of subsidies and make S-good inflation targeting more attractive.
- Empirical sensitivity:
  - Top panel of Figure 5: welfare cost of strict S-good targeting can be extremely high when ν = 0 (symmetric adjustment).
  - Lower panel of Figure 5: higher average tax rates increase welfare gains from allowing S-good inflation targeting.

### Caveat on sectoral average inflation rates
- Assumption: equal average inflation rates across sectors (as in Erceg et al. 2000; Aoki 2001; etc.), corresponding to central bank’s target π.
- Empirical note:
  - Tunisia: average inflation S vs nS = 3.70 and 3.96 percent.
  - Morocco: average inflation S vs nS = 4.08 and 3.80 percent.
- If some subsidized prices are kept constant (zero average inflation), in a frictionless context this policy is suboptimal and causes non-stationary relative prices, severe resource misallocations, and long-term welfare deterioration. Monetary policy response: fully stabilize relative price of subsidized and non-subsidized goods.

### Estimation strategy and identification
- Estimation approach:
  - Bayesian techniques with Metropolis-Hastings; state-space representation and Kalman filter to evaluate likelihood L(θ|Y_T).
  - Posterior p(θ|Y_T) ∝ L(θ|Y_T) p(θ).
  - Marginal data density p(Y_T) approximated via harmonic mean estimator.
- Structural shocks in estimation:
  - ε_{A_S}, ε_{A_nS} (sector-specific technology shocks); ε_R (monetary policy shock); ε_{P_s} (shock to degree of price subsidy).
  - Measurement errors added to observables to capture misspecification and data quality issues.
- Observables:
  - Production growth, consumption growth, CPI inflation. Exceptions: Morocco and Tunisia where sectoral price data available; there the observables include output growth and two sectoral inflation rates.
- Sample:
  - Seasonally adjusted quarterly series for fourteen countries (Argentina, Brazil, Czech Republic, Hungary, Mexico, Philippines, Poland, South Africa, Thailand, Turkey, Egypt, Morocco, Tunisia, Indonesia) from earliest availability to 2008Q4.

### Choice of priors (summary)
- Fixed/calibrated parameters: β, π, η, δ, α_S, α_nS, θ_nS. Average tax rates τ calibrated per country (tax revenues / total revenues).
- Estimated parameter set: {a, ψ_k, ψ_nS, φ, κ, ρ_{A_S}, ρ_{A_nS}, ρ_ζ, σ_{A_S}, σ_{A_nS}, σ_ζ, σ_{P_S}}.
- Priors examples (Table 2):
  - a ~ Beta( mean 0.50, std 0.15 ).
  - ψ_k ~ Gamma( mean 5, df 2 ).
  - ψ_nS^{Calvo} ~ Beta( mean 0.67, std 0.1 ).
  - φ ~ Beta( mean 0.25, std 0.1 ).
  - κ ~ Beta( mean 0.75, std 0.1 ).
  - ρ_{A_S}, ρ_{A_nS}, ρ_ζ ~ Beta( mean 0.8, std 0.1 ).
  - σ_{A_S}, σ_{A_nS} ~ Inverse Gamma (mode 0.02).
  - σ_ζ, σ_{P_S} ~ Inverse Gamma (mode 0.005).

### Estimation results (high-level)
- Bayesian draws: 200,000.
- Posterior precision: parameters related to shocks and price distortions identified well (posterior deviates from priors).
- Shock volatilities: nS-good supply shocks generally most volatile, followed by S-good price shocks for many countries (exceptions: Egypt, Poland).
- Habit formation a: posterior means range from 0.15 (Mexico) to 0.88 (Philippines).
- ψ_nS^{Calvo} (average duration of price contracts) posterior medians vary by country:
  - Example: Argentina posterior median ≈ 0.284 (about 5 months); Morocco about 20 months in other countries.
- Estimated share φ of S goods in consumption:
  - Range across countries from 0.13 (Poland) to 0.47 (Philippines).
  - For Tunisia: 90% interval [0.25, 0.44]; declared share 0.32 lies within interval.
  - For Morocco: 90% interval [0.24, 0.45]; declared share 0.31 lies within interval.
- κ estimates generally > 0.50, indicating active government smoothing of subsidized prices.
- Model comparison: posterior odds comparing benchmark (φ>0) vs φ = 0 show strong evidence for existence of S-good sector in Argentina, Brazil, Egypt, Mexico, Morocco, Tunisia, Turkey. For Czech Rep., Hungary, Indonesia, Poland, South Africa, Thailand evidence favors φ = 0.

### Quantitative implications — welfare evaluation (overview)
- Welfare metric: compensating variation ω (percentage of lifetime consumption change to equate expected utility between policies), using second-order approximation of unconditional expected lifetime utility.
- Policies compared (relative to headline CPI inflation targeting):
  - (i) S-good inflation stabilization.
  - (ii) core (nS-good) inflation stabilization.
  - (iii) optimized Taylor rule (parameters ρ_{πS}, ρ_{πnS}, ρ_y, ρ_R chosen to maximize unconditional expected utility).
- Existence/stability constraint: optimized rules restricted to guarantee unique stable equilibrium near deterministic steady state.

### Welfare results (selected quantitative outcomes from Table 5)
- Headline inflation targeting is always suboptimal relative to at least one optimized rule; compensating variation welfare gains (permanent consumption equivalent) from optimized rules range:
  - From 0.0013 percent (Tunisia) to 0.1455 percent (Argentina) permanent increase in consumption.
- Country groupings by optimal policy:
  - Full core-inflation stabilization optimal:
    - Tunisia and Indonesia with optimized rule log(R_t/R) = ∞ log(π_{nS t}/π_{nS}) (i.e., fully stabilizing nS inflation).
    - Consistent with low average tax rates (< 20 percent for Tunisia and Indonesia) and dominant nS distortion.
  - Mild reactions to both sectoral inflation rates and output gap:
    - Brazil’s optimized rule: log(R_t/R) = 2.5947 log(π_S t/π_S) + 1.9917 log(π_{nS t}/π_{nS}) + 0.4296 log(Y_t/Y).
    - Brazil: ψ_nS^{Calvo} = 0.696, φ = 0.307, τ = 0.35 (high average tax rate exacerbates subsidy distortion).
  - Other heterogenous cases:
    - Argentina optimized coefficients: ρ_{πS} = 1.3079, ρ_{πnS} = 0.0000, ρ_y = 0.0000, ρ_R = 0.0016; headline targeting loses −0.1455 percent (permanent consumption).
    - Egypt, South Africa, Philippines, Turkey: optimal rules stabilize core inflation and react strongly to S-good inflation (e.g., Turkey ρ_{πS} = 3.6436, ρ_{πnS} = ∞).
- Welfare losses of aggressive core-targeting can be multiple times the loss under optimized rule in some countries; higher φ and high τ increase welfare loss under core-only targeting.

### Preconditions for optimal core inflation targeting (fiscal thresholds)
- Table 6 reports reductions in φ required for core inflation targeting to be optimal:
  - Argentina: estimated φ = 0.356 → threshold φ = 0.186 (reduce by almost half).
  - Brazil: 0.307 → 0.204.
  - Czech Republic: 0.246 → 0.068.
  - Egypt: 0.336 → 0.182.
  - Hungary: 0.226 → 0.088.
  - Mexico: 0.373 → 0.262.
  - Morocco: 0.347 → 0.200.
  - Philippines: 0.472 → 0.360.
  - Poland: 0.134 → 0.094.
  - South Africa: 0.248 → 0.196.
  - Turkey: 0.355 → 0.132.
  - For Indonesia, Thailand, Tunisia thresholds not binding (“—”).
- Practical obstacles to achieving required fiscal reforms:
  - Weak capacity to target the poor.
  - Lack of transparency in subsidy reporting.
  - Opposition from vested interests.
  - Spillover effects.
  - Ad hoc price setting mechanisms.

### Conclusion (policy-relevant summary)
- In a New Keynesian framework with subsidized (S) and non-subsidized (nS) sectors (nS sticky), optimized monetary reaction functions that account for sectoral inflation and output can yield welfare gains relative to headline CPI targeting.
- Optimal policy choice depends critically on:
  - Share of subsidized goods φ.
  - Degree of nominal rigidity in nS sector (ψ_nS).
  - Degree of government price smoothing κ and asymmetry parameter ν.
  - Financing of subsidies (distortionary vs lump-sum) and average tax rate τ.
- Practical implication: reducing government price interventions (subsidies) can expand feasibility and optimality of core inflation targeting, but fiscal reforms face significant implementation challenges.

*Italic source: IMF Working Paper — _wp1215 - 2.    Intermediate-good producers*

### REFERENCES

### _wp1215 - REFERENCES

### Monetary policy, inflation dynamics, and price-setting
- Al-Mashat, R. and A. Billmeier, 2000. “The Monetary Transmission Mechanism in Egypt,” International Monetary Fund, Working Paper No. 07/285.
- Aoki, K. 2001, “Optimal Monetary Policy Responses to Relative-Price Changes,” Journal of Monetary Economics, 48 (1): 55–80.
- Batini, N. and D. Laxton. 2007. “Under what Conditions can Inflation Targeting be Adopted? The experience of Emerging Markets.” In: F. Mishkin and K. Schmidt-Hebbel (eds), Monetary Policy under Inflation Targeting. (Santiago: Central Bank of Chile).
- Bodenstein, M., C.J. Erceg, and L. Guerrieri. 2008. “Optimal Monetary Policy with Distinct Core and Headline Inflation Rates,” Federal Reserve Board International Finance Discussion No. 941.
- Calvo, G. A. 1983. “Staggered Prices in a Utility-Maximizing Framework,” Journal of Monetary Economics, 12 (3): 383–98.
- Galí, J. and M. Gertler. 1999. “Inflation Dynamics: A Structural Econometric Analysis,” Journal of Monetary Economics, (44): 195–222.
- Huang, K., and Z. Liu. 2004. “Inflation Targeting: What Inflation Rate to Target?" Journal of Monetary Economics, 52 (8): 1435–62.
- Moreno, R. 2009. “Some Issues in Measuring and Tracking Prices in Emerging Market Economies." Bank of International Settlements, Paper No. 49.
- Rotemberg, J. J. 1982. “Monopolistic Price Adjustment and Aggregate Output,” Review of Economic Studies, (49): 517–531.
- Yun, T. 2005. “Optimal Monetary Policy with Relative Price Distortions.” The American Economic Review, 95 (1): 89–109.

### DSGE, welfare, and business cycle theory
- Basu, S. 1995. “Intermediate Goods and Business Cycles: Implications for Productivity and Welfare,” American Economic Review, 85 (3): 512–31.
- Boldrin, M., L.J. Christiano, and J. Fisher. 2001. “Habit Persistence, Asset Returns and the Business Cycle,” American Economic Review, (91): 149–66.
- Christiano, L.J., M. Eichenbaum, and C.L. Evans. 2005. “Nominal Rigidities and the Dynamic Effects of a Shock to Monetary Policy,” Journal of Political Economy, 113 (1): 1–45.
- Erceg, C.J., D.W. Henderson, and A.T. Levin. 2000. “Optimal Monetary Policy with Staggered Wage and Price Contracts,” Journal of Monetary Economics, 46 (2): 281–313.
- Erceg, C.J., and A.T. Levin. 2006. “Optimal Monetary Policy with Durable and Non-Durable Goods,” Journal of Monetary Economics, 53 (7): 1341–59.
- Galí, J. 1994. “Government Size and the Macroeconomic Stability,” European Economic Review, (38): 117–32.
- Kim, J. and S. H. Kim. 2003. “Spurious welfare reversals in international business cycle models," Journal of International Economics, 60 (2): 471–500.
- Kim, J. and F.J. Ruge-Murcia. 2004. “How much Inflation is Necessary to Grease the Wheels?" Journal of Monetary Economics, 56 (3): 365–77.
- Schmitt-Grohé, S. and M. Uribe, 2004. “Solving Dynamic General Equilibrium Models Using a Second-Order Approximation to the Policy Function,” Journal of Economic Dynamics and Control, 28 (4): 755–75.
- Schmitt-Grohé, S. and M. Uribe, 2007. “Optimal Simple and Implementable Monetary and Fiscal Rules,” Journal of Monetary Economics, (54): 1702–25.
- Smets, F. and R. Wouters. 2003. “An Estimated Dynamic Stochastic General Equilibrium Model of the Euro Area,” Journal of the European Economic Association, 1 (5): 1123–75.

### Terms of trade, oil, and sectoral distortions
- Backus, D. K. and M. J. Crucini. 2000. “Oil prices and the terms of trade,” Journal of International Economics, 50 (1): 185-213.
- Taubman, P. and R. Rasche. 1971. “Subsidies, Economic Lives, and Complete Resource Misallocation,” American Economic Review, 61 (5): 938–45.
- Yun, T. 2005. “Optimal Monetary Policy with Relative Price Distortions.” The American Economic Review, 95 (1): 89–109.

### Econometric methods, Bayesian inference, and model evaluation
- Aucremanne, L. and R. Wouters. 1999. “A Structural VAR Approach to Core Inflation and its Relevance for Monetary Policy,” Bank of International Settlements, Measures of Underlying Inflation and their Role in the Conduct of Monetary Policy, Proceedings of the Workshop of Central Bank Model Builders, February 1999, Amsterdam.
- Geweke, J. F. 1999. “Using Simulation Methods for Bayesian Econometric Models: Inference, Development and Communication,” Econometric Reviews, 18 (1): 1–126.
- Jeffereys, H. 1961. Theory of Probability. Oxford University Press.
- Schorfheide, F. 2000. “Function-Based Evaluation of DSGE Models." Journal of Applied Econometrics, 15(6): 645–70.

### Additional theoretical contributions
- Bils, M. and P.J. Klenow. 2004. “Some Evidence on the Importance of Sticky Prices,” Journal of Political Economy, 112 (5): 947–85.
- Calvo, G. A. 1983. “Staggered Prices in a Utility-Maximizing Framework,” Journal of Monetary Economics, 12 (3): 383–98.
- Rotemberg, J. J. 1982. “Monopolistic Price Adjustment and Aggregate Output,” Review of Economic Studies, (49): 517–531.

*Source: _wp1215 - REFERENCES*

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