## _wp10296

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---

### I. Introduction — motivation and research question
- Micro and macro motivation:
  - Micro evidence: some goods change prices every quarter or once a month (food, energy, airfares); services change prices much less often than once a year. Bils and Klenow (2004) find about half of U.S. products display prices that last 4.3 months or less; for energy related goods and raw products, prices change about 54 percent of the months considered.
  - Macro evidence: services are the most price‑rigid sector; non‑services often display flexible prices.
- Central policy question:
  - Which inflation measure should an interest‑rate rule respond to — headline (includes flexible‑price goods) or core/sticky‑price inflation?
- Paper objective:
  - Select the inflation measure for an interest‑rate rule that best avoids macroeconomic instability by:
    - (i) shielding the economy against fluctuations driven by self‑fulfilling expectations (sunspots), and
    - (ii) ensuring boundedly rational agents (adaptive learning) can learn the monetary‑policy‑targeted equilibrium (E‑stability).

### III. Model setup — two‑sector New‑Keynesian DSGE (high‑level)
- Economy and preferences:
  - Closed economy with two goods (Φ and Σ); aggregate consumption CES with intratemporal elasticity ε and weight χ (share of good Φ).
  - Intertemporal parameters: β∈(0,1), σ>0; sector‑specific inverse Frisch elasticities ηΦ and ηΣ.
  - Goods can be Edgeworth complements (ε < 1/σ), neutral (ε = 1/σ), or substitutes (ε > 1/σ).
- Prices and rigidities:
  - Sectoral gross inflations πΦt and πΣt; CPI Πt; relative price θt.
  - Rotemberg adjustment costs with coefficients ξΦ and ξΣ; allows ξΦ < ξΣ (Φ relatively flexible); fully flexible case ξΦ = 0 and ϱΦ → +∞.
- Technology and shocks:
  - Linear technology yιj,t = zt hιj,t; ln(zt) follows AR(1) with persistence ρ and iid innovation εt ~ Normal(0,σ²).
- Monetary policy rule:
  - Interest‑rate feedback (13): Rt = R̄[ E_t(πo,t+κ / π̄o ) ]^φπ with φπ > 1 (Taylor principle).
  - Timing parameter κ = −1, 0, 1 (backward‑, contemporaneous, forward‑looking).
  - Inflation measure πot = α πΦt + (1−α) πΣt with α ∈ [0,1]; special cases α = 1 (flexible only), α = 0 (sticky only), α = χ nests CPI.

### Main analytical focus — determinacy and learnability
- Two selection criteria for the REE:
  - Determinacy: unique equilibrium driven by fundamentals (no sunspot equilibria).
  - E‑stability (learnability): equilibrium is stable under recursive least squares (Evans and Honkapohja (2001)).
- Policy rule timing considered:
  - Forward‑looking (respond to E_t πo,t+1), contemporaneous (πo,t), backward‑looking (πo,t−1).

### Key findings (determinacy, sunspots, and learning)
- General result:
  - Active rules (φπ > 1) that place positive weight on flexible‑price inflation (α > 0) are more prone to macro instability than rules responding exclusively to sticky‑price inflation (α = 0).
- Timing‑specific results:
  - Forward‑looking rules:
    - Positive weight on expected flexible‑price inflation increases likelihood of inducing fundamental equilibria that are not learnable (non‑E‑stable).
    - Such rules can produce learnable sunspot (CF) equilibria.
  - Backward‑looking rules:
    - Weight on past flexible‑price inflation can generate explosive equilibria with potentially unbounded fluctuations.
  - Contemporaneous rules:
    - Contemporaneous rules guarantee a unique and learnable equilibrium (MSV) when φπ > 1, and are insensitive to α.
- Numerical thresholds (Bils‑Klenow calibration, complements case examples):
  - Fix φπ = 2:
    - Unique equilibrium for 0 ≤ α ≤ 0.44.
    - Multiple equilibria & E‑stable MSV rep. for 0.44 < α ≤ 0.48.
    - Multiple equilibria & E‑unstable MSV rep. for 0.48 < α ≤ 1.
  - Fix α = χ = 0.41:
    - Unique equilibrium for 0 < φπ < 2.18.
    - Multiple equilibria & E‑stable CF sunspot for 2.18 < φπ < 2.3.
    - E‑unstable MSV for φπ ≥ 2.3.
  - Threshold α* = 0.92: for complements (ε < 1/σ) α ∈ (α*,1) implies multiple local REE for any φπ > 1 and MSV‑REE not learnable.
- Policy implication across timings:
  - Safe policy choice to avoid sources of instability is to respond exclusively to sticky‑price inflation (α = 0).

### Mechanism and economic intuition
- Role of intratemporal elasticity ε:
  - ε determines comovement of sectoral inflation and affects how the real interest rate responds to inflation shocks under different πo measures.
  - Determinacy and E‑stability conditions tighten under complementarity (ε < 1/σ).
- Dichotomous economy intuition (fully flexible Φ, sticky Σ):
  - Expected flexible‑price inflation is negatively correlated with current sticky‑price output.
  - Weight on expected flexible inflation makes the rule respond negatively to current sticky output (policy lowers real rate when expected flexible inflation rises).
  - That negative response can reinforce deviations of sticky inflation from the REE → non‑learnability or explosive dynamics.
- Compact expression (complements case):
  - bρτ − Eτ bπSτ+1 = −[ (1 − φπ)/(1 − τ α φπ) ] Eτ bπSτ+1 with τ ≡ α σ — illustrates how φπ and α interact to produce destabilizing real rate responses.

### Model variants and robustness checks
- Strong dichotomy: ξΦ = 0 (fully flexible) vs sticky Σ — yields analytical insight; maps to Aoki (2001), Galí and Monacelli (2005).
- Less extreme heterogeneity: two sticky sectors calibrated to Bils and Klenow (2004) micro evidence for quantitative relevance.
- Extensions:
  - Adding explicit response to output in rule reduces admissible α and φπ ranges for determinacy and E‑stability.
  - Different timing of real money balances (CIA vs CWID): CWID forward‑looking rule is isomorphic to CIA contemporaneous rule for determinacy/learnability.
  - Learning frameworks: main results derived under “Euler Equation” learning (Evans and Honkapohja (2001)); plausible parallels to infinite horizon (IH) approaches are discussed but not fully explored.

### Numerical calibration and quantitative results (benchmarks and key statistics)
- Benchmark calibration (Table 1):
  - Annual steady‑state inflation = 3 percent.
  - β = 0.99, σ = 1, χ = 0.4.
  - η_F = η_S = 8 (steady‑state mark‑up = 15%).
  - ζ_F = ζ_S = 2 (Frisch elasticity = 0.5).
  - ρ = 0.78.
  - Price adjustment costs (from Bils and Klenow (2004) mapping):
    - ψ_F = 0.67, ψ_S = 74 (sector F almost flexible).
  - π̄ = 1.0074 (steady state inflation factor reported).
- Aggregated micro evidence mapped to quarterly frequencies:
  - Aggregate per‑quarter probability of a price change φ = 0.52.
  - Expected duration δ = 2.7 quarters.
  - Mapping from monthly per‑month probabilities α_i to quarterly probabilities φ_i via φ_i ≡ α_i [ (1−α_i)^2 + 3(1−α_i) ].
  - Calibration steps yield ψ_F = 0.67 and ψ_S = 74 to match φ and δ with share of flexible firms set to χ = 0.4.
- Sensitivity and heterogeneity measures:
  - Aggregate frequency φ is considered over [0.2, 0.8]; benchmark φ = 0.52.
  - Degree of heterogeneity Δφ defined by equation (35).
- Quantitative findings (Appendix C illustrations):
  - With equally‑sized sectors (χ = 0.5) and complementarity (ε = 0.5), forward‑looking rule with φπ = 2 and α = χ = 0.5:
    - Aggregate instability arises endogenously when sticky‑price heterogeneity is sufficiently high for average φ just above 0.45.
    - The upper threshold on heterogeneity that induces instability is strictly decreasing in aggregate rigidity: more flexible average prices (higher φ) require a lower Δφ to trigger instability.
  - Responding to CPI inflation (α = χ) does not guarantee determinacy or E‑stability.

### Aggregation bias, frequency mapping, and adjustment cost calibration
- Aggregation bias:
  - Bils and Klenow use 350 categories; two‑sector model is a parsimonious representation calibrated to match key moments to limit aggregation bias.
- Frequency mapping (monthly → quarterly):
  - Per‑month probability α_i converted to per‑quarter φ_i as above; expected duration δ_i = (φ_i)^−1.
  - Weighted averages across 350 categories produce φ = 0.52 and δ = 2.7.
- Mapping frequencies into Rotemberg adjustment costs:
  - Use equivalence between Rotemberg and Calvo formulations: choose ψ_i so (ζ_i^−1) ψ_i (π̄_i)^2 = (1−p_i)(1−β p_i)/p_i with p_i = 1 − φ_i.
  - Calibrated ψ_F = 0.67 and ψ_S = 74 to match computed φ_F and φ_S given χ = 0.4.

### Policy recommendations (instrumental‑rule perspective)
- Prefer targeting the inflation measure of the sticky‑price sectors (α = 0) when designing Taylor‑type interest‑rate rules to avoid:
  - Fundamental‑driven non‑learnable equilibria,
  - Learnable sunspot equilibria, and
  - Explosive paths caused by weight on flexible‑price inflation.
- Particular cautions:
  - Forward‑looking and backward‑looking rules that place positive weight on flexible‑price inflation are especially prone to instability.
  - Contemporaneous rules are more robust, but the safe choice remains core (sticky‑price) inflation targeting.
- Additional caveats and extensions:
  - Allowing the rule to respond to expected output (positive φψ) reduces admissible α and φπ ranges; for sufficiently large φψ no active rule yields a determinate and E‑stable equilibrium.
  - Sectoral heterogeneity beyond price rigidity (e.g., cost channel, durability) may alter determinacy and E‑stability boundaries and warrant further study.

*Source: Excerpted content from the provided PDF chapter/section (_wp10296).*

### References .............................................................................................................

### _wp10296 - References .............................................................................................................

### I. Introduction — motivation and research question
- Microeconomic evidence (Klenow and Malin (2010); Bils and Klenow (2004); Nakamura and Steinsson (2008); Klenow and Kryvtsov (2008)) shows large heterogeneity in price stickiness across goods:
  - Some goods change prices every quarter or once a month (food, energy, airfares).
  - Services change prices much less often than once a year.
  - Bils and Klenow (2004) find that about half of U.S. products display prices that last 4.3 months or less; for energy related goods and raw products, prices change about 54 percent of the months considered.
- Macro estimates (e.g., Bouakez et al. (2009)) confirm heterogeneity: some sectors (typically non-services) display flexible prices; services are the most price-rigid sector.
- Central policy question: which inflation measure should monetary policy (interest rate rules) respond to — an inflation measure that includes flexible-price goods (headline) or one that focuses on sticky-price goods (core)?
- Paper objective: select the inflation measure of an interest rate rule that is more prone to avert macroeconomic instability, defined by two criteria:
  - (i) shield the economy against fluctuations driven by self-fulfilling expectations (sunspots), and
  - (ii) ensure that boundedly rational agents (adaptive learning as in Evans and Honkapohja (2001)) can learn the monetary-policy-targeted equilibrium (E-stability).

### III. Model setup — two-sector New‑Keynesian DSGE (high-level)
- Economy: closed, two goods (denoted Φ and Σ), each produced by a continuum of intermediate varieties; representative household-firm unit consumes and produces both types.
- Preferences and aggregation:
  - Intertemporal preferences given by equation (1) with parameters: β∈(0,1), σ>0, and sector-specific inverse Frisch elasticities ηΦ and ηΣ.
  - Aggregate consumption CES aggregator (2) with intratemporal elasticity of substitution ε and weight χ (share of good Φ).
  - Goods can be Edgeworth complements, neutral, or substitutes depending on ε vs 1/σ: complements if ε < 1/σ, neutral if ε = 1/σ, substitutes if ε > 1/σ.
- Prices and rigidities:
  - Sectoral gross inflations πΦt and πΣt, CPI Πt as in (3), relative price θt in (5).
  - Nominal price rigidities introduced as Rotemberg adjustment costs: quadratic resource cost with coefficients ξΦ and ξΣ; model allows ξΦ < ξΣ (sector Φ relatively more flexible); fully flexible case obtained by ξΦ = 0 and ϱΦ → +∞.
- Production and markets:
  - Linear technology yιj,t = zt hιj,t with aggregate technology shock zt evolving as ln(zt) with persistence ρ (equation (6)) and iid innovation εt ~ Normal(0,σ²).
  - Monopolistic competition; demand-determined output; NK Phillips curves for each sector derived (12).
- Monetary policy rule:
  - Interest-rate feedback rule (13): Rt = R̄[ E_t(πot+κ / π̄o ) ]^ϕπ with ϕπ > 1 (Taylor principle, active rule), and timing parameter κ = −1, 0, 1 to allow backward-looking, contemporaneous, or forward-looking rules.
  - Inflation measure in the rule defined as πot = α πΦt + (1−α) πΣt (14); central bank can pick weight α ∈ [0,1]. Special cases: α = 1 (flexible-price inflation only), α = 0 (stickiest-price/core inflation only), α = χ nests CPI when α = χ.

### Main analytical focus — equilibrium determinacy and learnability
- Two selection criteria for the policy-targeted REE:
  - Determinacy: unique equilibrium driven solely by fundamentals (rules out sunspot-driven equilibria).
  - E-stability (learnability): under recursive least-squares learning (Evans and Honkapohja (2001)) agents converge to the fundamental REE.
- Analysis considers different policy rule timings:
  - Forward-looking: responds to expected future inflation E_t(πo,t+1).
  - Contemporaneous: reacts to current inflation πo,t.
  - Backward-looking: reacts to past inflation πo,t−1.

### Key findings (determinacy and learning)
- General result: Active rules (ϕπ > 1) that put some weight on flexible-price inflation (headline measures; α > 0) are more prone to generate macroeconomic instability than rules that respond exclusively to sticky-price inflation (core measures; α = 0).
- Timing-specific results:
  - Forward-looking rules:
    - Assigning positive weight to expected flexible-price inflation increases the likelihood of inducing fundamental-driven equilibria that are not learnable (non-E-stable).
    - Such rules can also produce learnable sunspot (extrinsic) equilibria of the "sunspot" type.
  - Backward-looking rules:
    - Weight on past flexible-price inflation can be destabilizing by creating explosive equilibria with potentially unbounded fluctuations.
  - Contemporaneous rules:
    - Contemporaneous rules appear to guarantee a unique and learnable equilibrium regardless of the inflation measure (even if the rule puts weight on current flexible-price inflation).
- Overall policy implication across timings:
  - The safe policy choice to rule out potential sources of instability is to respond exclusively to sticky-price inflation (α = 0).

### Mechanism and economic intuition
- Intratemporal elasticity of substitution ε (i.e., whether goods are complements, neutral, or substitutes) is central:
  - ε determines comovement of sectoral inflation rates and affects the equilibrium response of the real interest rate to inflation shocks under different inflation measures.
  - Determinacy and learnability conditions are generally more stringent under complementarity (ε < 1/σ).
  - There exists a threshold weight on expected flexible-price inflation beyond which any active rule responding to headline inflation will always induce non-learnability of the fundamental REE (threshold depends on model parameters and elasticity assumptions).
- Intuition in dichotomous economy (fully flexible Φ sector and sticky Σ sector):
  - Expected flexible-price inflation is negatively correlated with current sticky-price output.
  - Putting weight on expected flexible-price inflation makes the rule respond more negatively to current sticky-price output, i.e., the policy reduces the real interest rate in response to an increase in expected flexible inflation.
  - That negative response can reinforce deviations of sticky-price inflation from the REE, adding to sticky-price inflationary pressures and pushing the economy away from the targeted equilibrium — leading to non-learnability.

### Model variants and robustness
- Two price-setting cases analyzed:
  - Strong dichotomy: fully flexible-price sector (ξΦ = 0) and sticky-price sector — yields analytical intuition and maps to frameworks such as Aoki (2001) and some small open-economy models (Galí and Monacelli (2005)).
  - Less extreme heterogeneity: two sticky-price sectors calibrated to micro evidence (Bils and Klenow (2004)) to assess quantitative relevance.
- Robustness checks:
  - Extensions include explicit response to output in the policy rule and different timing of real money balances in utility; main conclusions persist.

### Relationship to literature and novelty
- Distinguishes itself by treating the inflation measure (α) as a central bank policy choice affecting determinacy and learnability — a dimension largely absent from prior New‑Keynesian interest-rate rule literature.
- Contrasts with Carlstrom et al. (2006a): while they claim the inflation measure is irrelevant for contemporaneous rules if the Taylor principle holds, this paper:
  - Tests robustness across forward- and backward-looking rules, and
  - Adds learnability as a selection criterion, revealing that the inflation measure can matter for determinacy and learnability for degrees of heterogeneity consistent with micro data.
- Complements optimal-policy results (Aoki (2001); Benigno (2004); Mankiw and Reis (2003)) that recommend targeting the stickiest inflation from a welfare perspective, by providing an instrumental-rule perspective focusing on determinacy and learnability.

### Policy recommendation (instrumental-rule perspective)
- Prefer targeting the inflation measure of the truly sticky-price sectors (α = 0) when designing Taylor-type interest-rate rules aimed at avoiding:
  - Fundamental-driven non-learnable equilibria, and
  - Sunspot-driven indeterminacy or explosive paths caused by weight on flexible-price inflation.
- Particularly cautious about forward-looking and backward-looking rules that place positive weight on flexible-price inflation; contemporaneous rules are more robust but the safe choice remains core (sticky-price) inflation targeting.

*Source: Excerpted content from the provided PDF chapter/section.*

### Introduction, reflects the popular two-sector price setting assumption in both the optimal monetary

### _wp10296 - Introduction, reflects the popular two-sector price setting assumption in both the optimal monetary

### A. The Methodology
- Model representation (log-linearized):
  - Eτ bξτ+1 = J bξτ + Λ bζτ (equation (15))
  - bξτ is a σ×1 vector of endogenous variables; bζτ is the technology shock.
- REE characterization (Blanchard and Kahn (1980)):
  - Number of eigenroots of J outside the unit circle compared to number of non‑predetermined variables determines:
    - determinacy (equal),
    - indeterminacy (less),
    - no non‑explosive solution (greater).
- Learning framework (Evans and Honkapohja (2001)):
  - Agents use recursive least squares (RLS) updating and Perceived Law of Motion (PLM).
  - E-stability as learning criterion: a representation of an equilibrium is learnable if it is E-stable.
  - MSV PLM: bξτ = A + F bξτ−1 + N bζτ; implied ALM and T-mapping T(A,F,N) = (Ā,F̄,N̄) with
    - Ā = B(Υ + ΩA), F̄ = B Γ, N̄ = B(Ω N ρ + Π), B = (I − Ω F)−1.
  - E-stability conditions for MSV representation: all eigenvalues of derivatives
    - ΔT_A = (I − Ω F)−1 Ω,
    - ΔT_F = [(I − Ω F)−1 Γ] 0 ⊗ [(I − Ω F)−1 Ω],
    - ΔT_N = ρ ⊗ [(I − Ω F)−1 Ω]
    evaluated at the REE fixed point must have real parts less than one (equation (17)).
  - CF (Common Factor) PLM for sunspot equilibria: bξτ = A + N bζτ + G ϵτ with ϵτ = ω ϵτ−1 + ξτ; T-mapping yields E-stability matrices
    - ΔT_A = Ω, ΔT_N = ρ ⊗ Ω, ΔT_G = ω ⊗ Ω (equation (18)); eigenvalues must have real parts less than one.

### B. The Strong Dichotomy: A Flexible-Price and Sticky-Price Economy (analytical results)
- Setup:
  - Sector F: perfect competition (η_F → +∞) and fully flexible prices (ψ_F = 0).
  - Sector S: standard New‑Keynesian sticky-price sector.
  - Focus on forward-looking rules (κ = 1).
  - Steady-state relative price θ = 1, hence π_F = π_S = π̄.
- Key log-linearized relationships and definitions:
  - Interest-rate rule (forward-looking): bρτ = φπ Eτ bπoτ+1 (19), with bπoτ+1 = α bπFτ+1 + (1−α) bπSτ+1 (20).
  - Demand relation between flexible- and sticky‑price consumption: b cFτ = ϖ b cSτ + κ1 bζτ (21) with ϖ ≡ (1−χ)(1−εσ)/( (ζ_F + σ χ) ε + 1 − χ ) and κ1 ≡ (1+ζ_F) ε/( (ζ_F + σ χ) ε + 1 − χ ). Sign(ϖ) = sign(1−εσ).
  - Sticky-price consumption growth: b cSτ − b cSτ−1 = (1/α)(bπFτ − bπSτ) + κ1/(α ε) (bζτ − bζτ−1) (22) with α ≡ σ + ζ_F((ζ_F + σ χ) ε + 1 − χ)−1 > 0.
  - Euler (aggregate): b cτ = Eτ b cτ+1 − (1/σ)(bρτ − Eτ bπτ+1) (23).
  - CPI inflation: bπτ+1 = χ bπFτ+1 + (1−χ) bπSτ+1 (24).
  - Sticky‑price NKPC: bπSτ = β Eτ bπSτ+1 + μ c m cSτ (26) with μ ≡ (η_S − 1) ψ_S (π̄_S)2 > 0.
  - Real marginal cost expressed in sticky‑price consumption: c m cSτ = ζ̄ b cSτ + h bζτ (27) with ζ̄ ≡ ζ_S + 1/α > 0 and h ≡ −(1+ζ_S) − κ1/χ (1 − σ ε)/ε .
  - Combined sticky‑price NKPC: bπSτ = β Eτ bπSτ+1 + ϖ̃ b cSτ + s bζτ (28) with ϖ̃ ≡ μ ζ̄ > 0 and s ≡ μ h < 0.
- Definition 1: Fundamental-driven equilibrium consists of stochastic processes {bρτ, bπoτ, bπFτ, bπSτ, b cSτ}∞τ=0 satisfying (19)-(20), (22), (25), (28).
- Determinacy and E-stability results (Propositions 1–3):
  - Proposition 1 (determinacy): Let φ1(α) ≡ (ϖ α + 2(1+β))/(ϖ α + 2(1+β) τ α) > 0 and α* = 1/τ, τ ≡ α σ.
    - a) For α ∈ (0,1) when ε ≥ 1/σ (substitutes or neutral), or α ∈ (0, α*) when ε < 1/σ (complements):
      - unique local REE if 1 < φπ < φ1(α),
      - multiple local REE if φπ > φ1(α).
    - b) For α ∈ (α*,1) when ε < 1/σ (complements): multiple local REE for any φπ > 1.
  - Proposition 2 (learnable CF sunspot equilibria): For α as above, CF sunspot equilibria are learnable (E-stable) if φ1(α) < φπ < φ2(α) where φ2(α) ≡ 1/(τ α) > 0.
  - Proposition 3 (learnability of MSV REE): For α as above:
    - a) For α ∈ (0,1) when ε ≥ 1/σ, or α ∈ (0, α*) when ε < 1/σ:
      - MSV-REE learnable if 1 < φπ < φ2(α),
      - MSV-REE not learnable if φπ > φ2(α).
    - b) For α ∈ (α*,1) when ε < 1/σ: MSV-REE not learnable for any φπ > 1.
- Key comparative statics and intuition:
  - Increasing weight on flexible-price inflation α reduces the range 1 < φπ < φ1(α) of unique & learnable equilibria.
  - If φ1(α) < φπ < φ2(α): learnable sunspot equilibria (CF).
  - If φπ > φ2(α): fundamental equilibria with non‑learnable MSV representation.
  - Lower elasticity of substitution (ε) tightens determinacy and E-stability conditions (frontiers φ1(α), φ2(α) increase with ε).
  - For complements, there exists α* beyond which any active rule leads to indeterminacy and non‑learnable MSV.
  - Intuition (complements case): real interest rate expression
    - bρτ − Eτ bπSτ+1 = −[ (1 − φπ)/(1 − τ α φπ) ] Eτ bπSτ+1 with τ ≡ α σ (29).
    - If agents initially overestimate sticky inflation and α ∈ (1/(τ), 1] then for φπ > 1 real rate falls, stimulating sticky consumption and further inflation → instability. If 1 < φπ < 1/(τ α) initial deviations are self‑correcting.

- Policy interpretation:
  - A rule placing positive weight on expected flexible‑price inflation (α > 0) is equivalent to responding negatively to current sticky-price output for given φπ and σ, since bρτ = φπ Eτ bπSτ+1 + φψ (Eτ bψSτ+1 − bψSτ) with φψ ≡ φπ α σ. This negative response can be destabilizing (contrary to “leaning against the wind”).

### C. The General Set-up: The Bils‑Klenow Calibration (numerical setup)
- Full system (log-linearized):
  - Aggregate Euler: b cτ = Eτ b cτ+1 − (1/σ)[ bρτ − χ Eτ bπFτ+1 − (1−χ) Eτ bπSτ+1 ] (30).
  - CPI inflation: bπτ = χ bπFτ + (1−χ) bπSτ (31).
  - Sectoral NKPCs:
    - bπFτ = β Eτ bπFτ+1 + μ_F (ζ_F + σ) b cτ − μ_F (1 + ε ζ_F) (1−χ) bθτ − μ_F (1 + ζ_F) bζτ (32).
    - bπSτ = β Eτ bπSτ+1 + μ_S (ζ_S + σ) b cτ + μ_S (1 + ε ζ_S) χ bθτ − μ_S (1 + ζ_S) bζτ (33).
  - Relative price law: bθτ = bθτ−1 + bπFτ − bπSτ (34).
  - Interest‑rate rule: bρτ = φπ α Eτ bπFτ+κ + φπ (1−α) Eτ bπSτ+κ.
- Calibration choices (benchmark):
  - Annual steady-state inflation = 3 percent; β = 0.99; σ = 1; χ (share flexible) = 0.4; η_F = η_S = 8 (steady-state mark‑up = 15%); ζ_F = ζ_S = 2 (Frisch elasticity = 0.5); ρ = 0.78.
  - Price adjustment cost calibration (from Bils and Klenow (2004) micro evidence):
    - χ = 0.4, ψ_F = 0.67, ψ_S = 74 (result: significant sectoral heterogeneity; sector F almost flexible).
  - Table 1 summary:
    - β σ χ A ζ_F ζ_S η_F η_S ψ_F ψ_S π̄ ρ = 0.99 1 0.4 2 2 8 8 0.67 74 1.0074 0.78

### C.1 Forward-Looking Rules — numerical findings (Bils‑Klenow calibration)
- Qualitative replication of analytical strong‑dichotomy results:
  - Increasing α reduces set of forward‑looking rules that induce a unique and learnable fundamental REE.
  - There exist parameter combinations (φπ, α) with learnable sunspot equilibria.
  - For complements, any active rule induces non‑learnable fundamental equilibria if α > α*.
- Quantitative example (Edgeworth complements):
  - Row a) Fix φπ = 2:
    - Multiple equilibria arise for α > 0.44.
    - Fundamental MSV-REE ceases to be learnable for α > 0.48.
    - Interpretation: aggregate instability occurs for α just above χ = 0.4.
  - Row b) Fix α = χ = 0.41:
    - Multiple equilibria for φπ > 2.18.
    - MSV-REE non‑learnable for φπ > 2.3.
  - Threshold α* = 0.92 (note).
- Table 2 (determinacy and E-stability bounds for complement goods):
  - a) φπ = 2: 0 ≤ α ≤ 0.44 → Unique Equilibrium; 0.44 < α ≤ 0.48 → Multiple Equilibria & E-Stable MSV Rep.; 0.48 < α ≤ 1 → Multiple Equilibria & E-Unstable MSV Rep.
  - b) α = χ = 0.41: 0 < φπ < 2.18 → Unique Equilibrium; 2.18 < φπ < 2.3 → Multiple Equilibria & E-Stable CF Sunspot; φπ ≥ 2.3 → E-Unstable MSV.
  - Note: α* = 0.92. Legend: UE = Unique equilibrium; ME = Multiple Equilibria; ES-MSV = E-stable MSV; ES-CF Sunspot = E-stable CF sunspot; EU-MSV = E-unstable MSV.

### C.2 Other Timings for the Policy Rule
- Contemporaneous rule (bρτ = φπ bπoτ):
  - Determinacy and E-stability are insensitive to α across goods types.
  - If φπ > 1 (Taylor principle), there is a unique and learnable equilibrium (MSV).
- Backward‑looking rule (bρτ = φπ bπoτ−1):
  - Responding to past flexible‑price inflation can induce explosive equilibria (unbounded fluctuations).
  - Raising α increases propensity for explosive equilibria, particularly for Edgeworth complements.
  - When equilibrium is determinate, it is also learnable.
- Policy lesson: to avoid potential instability, target inflation of sectors with higher degree of price stickiness.

### C.3 Quantitative Sensitivity (aggregate price stickiness and heterogeneity)
- Measures:
  - Aggregate frequency of price change φ = χ φ_F + (1−χ) φ_S.
  - Degree of heterogeneity Δφ = sqrt[ χ(φ_F − φ)2 + (1−χ)(φ_S − φ)2 ] / φ (35).
  - Higher φ_i implies lower ψ_i; one‑to‑one mapping between φ_i and ψ_i (calibration via micro‑evidence).
- Calibration range considered: φ ∈ [0.2, 0.8] including benchmark φ = 0.52.
- Procedure:
  - For each (φ, Δφ) pair compute φ_F and φ_S satisfying (35), then compute corresponding ψ_F and ψ_S via the calibration procedure described in Appendix C (mapping frequency → adjustment cost).

*Source: _wp10296 - Introduction, reflects the popular two-sector price setting assumption in both the optimal monetary; canonical PDF: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2010/_wp10296.pdf*

### Appendix C. We consider a forward-looking rule and an economy of equally-sized sectors, i.e.,=05

### Appendix C. We consider a forward-looking rule and an economy of equally-sized sectors, i.e.,=05 and set=.

### Setup and parametrization
- Economy: two-sector model with equally-sized sectors (=0.5) and =0.5 (Edgeworth complementarity).
- Policy rule: forward-looking interest-rate rule with 

=2 and weight on expected headline inflation given by  (here set to ).
- All remaining parameters are as in Table 1 (not reproduced here).
- Figures referenced: Figure 5 (varying  and ∆

), Figure 6 (varying  and ), Figure 7 (varying 

and  when responding to CPI inflation).

### Main findings on determinacy and expectational stability
- Aggregate instability arises endogenously when sticky-price heterogeneity is sufficiently high for a given average frequency of price change just above 0.45.
- The upper threshold on heterogeneity (above which aggregate instability arises) is strictly decreasing in the aggregate level of rigidity: the more flexible prices are on average (higher ), the lower is the minimum degree of heterogeneity (lower ∆

) that induces aggregate instability.
- If the economy is sufficiently sticky in aggregate terms (low ), the equilibrium is always unique and E-stable, independently from the degree of heterogeneity.
- Parametrization notes: goods are complements (=0.5); rule is forward-looking with 

=2 and ==0.5 (Figure 5 caption).
- Terminology: CF = Common Factor representation of a “sunspot” equilibrium; MSV = Minimum State Variable representation of a fundamental equilibrium.

### The share of flexible-price goods and response to CPI inflation
- The share of flexible-price goods  affects the steady state (including the relative price ̄), so steady state with ̄=1 is no longer the default when varying .
- CPI inflation (log) used in the Euler equation is b

=

b


+

b


with
  

=
∙
+(1−)
³
1
̄
 ́
1−
 ̧
−1
and
  

=(1−)
£
̄
1−
+(1−)
¤
−1.
- Figure 6 (complements, forward-looking rule): as  decreases, the threshold value for  (above which aggregate instability arises) decreases. Hence, a rule responding to expected headline inflation b

+1
=b

+1
+(1−)b

+1
is more prone to induce stability problems the smaller the share of flexible-price goods.
- Responding to CPI inflation (=, the 45-degree line) does not guarantee determinacy and/or E-stability.
- Figure 7 (forward-looking rule responding to CPI inflation b

=

E

b
+1
, complements): qualitatively similar to results for a rule responding to expected headline inflation. Forward-looking rules responding to CPI inflation are more susceptible to non-learnable fundamental-driven equilibria the higher the share . However, there is no threshold share beyond which any active rule necessarily induces instability (contrast with headline-response case).

### Robustness and discussion
- Extended interest-rate rules: adding a positive response to (expected) output lowers the upper bound on  above which multiple equilibria arise and the fundamental MSV-REE ceases to be stable under learning — aggregate instability becomes more pervasive. Intuition: higher  implies a more negative implicit response to current sticky-price output, undermining determinacy and E-stability.
- Cash-in-Advance (CIA) timing vs. Cash-When-I’m-Done (CWID): the determinacy and learnability conditions for a forward-looking rule under CWID are identical to those for a contemporaneous rule under CIA. The CWID Euler equation under a forward-looking rule is isomorphic to the CIA Euler equation with a contemporaneous rule. Therefore, graphical characterizations (Figures 1 and 2) apply to contemporaneous CIA rules: active contemporaneous rules responding to inflation measures that include some flexible-price inflation are more prone to indeterminacy and non-learnable fundamental equilibria than rules that respond exclusively to sticky-price inflation.
- Learning approaches:
  - Adopted approach: “Euler Equation” learning a la Evans and Honkapohja (2001), requiring agents to form one-period-ahead forecasts based on first-order conditions.
  - Infinite horizon (IH) approach (Preston, 2006): decision rules depend on infinite-horizon expectations; long-horizon forecasts are not reducible to one-step-ahead predictions.
  - Inference: results might still hold under the IH approach provided plausible conditions (drawing on Honkapohja et al. (2003)): if all agents form the same forecasts (same econometric model and data), the IH approach can collapse to the Euler equation approach, yielding symmetric equilibrium and similar implications for determinacy and learnability.
  - Caveat: for general IH conditions there are no equivalence convergence results; an extensive IH analysis would require long-horizon expectations of marginal costs, sectoral inflations, and future relative prices — beyond current scope.

### Conclusions and policy implications
- Main conclusion: the inflation measure included in an interest-rate rule matters for equilibrium determinacy and learnability. Active rules that place some weight on the inflation of the sector with low price stickiness (flexible-price sector) are more susceptible to inducing aggregate instability than rules responding exclusively to the inflation of the sector with high price stickiness.
- Sources of instability identified: (i) fundamental-driven equilibria that are not learnable; (ii) learnable equilibria driven by self-validating beliefs (“sunspots”); and (iii) equilibria with unbounded fluctuations.
- Policy relevance: guidance for whether central banks should respond to headline inflation (including flexible energy prices) versus core inflation (dominated by sticky prices). Forward-looking rule results support reacting to core inflation rather than headline inflation because responding to headline (which includes flexible-price inflation) can induce greater volatility associated with learnable sunspot equilibria and non-learnable fundamental equilibria.
- Extensions noted: sectoral heterogeneity beyond price rigidity (e.g., differing impact of policy rates on marginal costs — the cost channel), and the role of durability of goods, which may affect determinacy and E-stability results.

### Selected analytical results and lemmas (high-level)
- Lemma 1: defines ≡ and characterizes  depending on  relative to 1

: a) if ≤1

 (Edgeworth complements or neutral) then =1; b) if >1

 (Edgeworth substitutes) then 01.
- Lemma 2: provides E-stability conditions for the MSV and CF representations of equilibria based on the sign of Det(Ω−I) and Trace(Ω−I), and the order of indeterminacy.
- Proposition results (summarized):
  - Proposition 1: characterizes regimes of uniqueness vs. multiplicity of equilibria depending on parameters including 

 and functions 
1
(), 
2
() with critical thresholds; multiple equilibria arise when 

exceeds certain 
k
() thresholds.
  - Proposition 2: when indeterminacy of order one occurs (
1
()


2
()), the CF “sunspot” representation is E-stable under specified conditions (Trace(Ω−I)<0 and Det(Ω−I)>0).
  - Proposition 3: describes E-stability vs. E-unstability of the MSV solution depending on 

relative to 
2
() and parameter regimes where 1 imply 
2
()<1.
- Steady state expressions: relative price ̄ given by equation (39) and CPI inflation log-linearization b

=

b


+

b


with weights 

and 

defined in Appendix A.

*Source: Appendix C of the supplied content unit.*

### 1. Aggregation bias: Bils and Klenow refer to a sample of 350 categories of goods and services,

### _wp10296 - 1. Aggregation bias: Bils and Klenow refer to a sample of 350 categories of goods and services,

### Aggregation bias
- Bils and Klenow use a sample of 350 categories of goods and services, while the model in the text is a two-sector two-good economy; going from micro-evidence to the macro model may induce aggregation bias.
- Extending the model to incorporate 350 sectors would help address aggregation bias but would make the model intractable without adding further insights on the inflation measure.
- The two-sector set-up is retained and the parameters  and _Σ are calibrated to match key moments of Bils and Klenow’s evidence.

### Frequency (mapping monthly evidence to quarterly calibration)
- Bils and Klenow’s statistics are at monthly frequencies; a quarterly calibration is used to compare with the one-sector New-Keynesian literature.
- Procedure:
  - For each sector i = 1,2,...,350 set the per-month probability of a price change, _i, equal to the observed monthly frequency.
  - The per-quarter probability of a price change for sector i is given by:
    - _i ≡ _i [ (1−_i)^2 + 3(1−_i) ] (as obtained by summing the probability of at least one monthly price change within a quarter).
  - Expected duration between price spells in sector i: _i = (_i)^−1 quarters.
  - Weighted averages across sectors (weights _i from the sample):
    -  ≡ sum_{i=1}^{350} _i _i
    -  ≡ sum_{i=1}^{350} _i _i
- Empirical averages obtained:
  -  = 052
  -  = 27
  - Interpretation: on average, firms change their price at least once with 52% probability per quarter, while expected duration is between 2 and 3 quarters after a price change.

### Mapping frequencies into price adjustment costs
- Goal: assign values to adjustment cost coefficients _F and _S consistent with the computed quarterly frequencies.
- Steps:
  1. Compute quarterly frequencies for the flexible sector F and sticky sector S, denoted _F and _S, to match aggregate moments:
     -  _F + (1 − ) _S = 052
     -  _F + (1 − ) _S = 27
     - where _i ≡ (_i)^−1 and  is the size of the relatively more flexible sector F.
  2. Set  = 04 (corresponds to share of firms with expected price duration ≤ one quarter in the sample).
  3. Use reduced-form augmented Phillips Curve for sector i:
     - b_i_t =  E_t b_i_{t+1} + _i c_i_t
     - where _i ≡ (_i^−1) _i (_i)^2 and c_i_t is log-deviation of marginal cost ratio.
  4. Exploit equivalence between Rotemberg and Calvo forms: set _i so that
     - (_i^−1) _i (_i)^2 = (1−p_i)(1− p_i)/p_i with p_i = 1 − _i.
- Calibrated adjustment cost coefficients obtained:
  - _F = 067
  - _S = 74.:
  - (These values are set to match the computed _i.)

### Robustness: responding to output in the interest-rate rule
- Consider an augmented forward-looking policy rule that responds to expected sticky-price output:
  - b_t = _ E_t[ b_F_{t+1} + (1−) b_S_{t+1}] + _ E_t b_S_{t+1}
  - with _ > 1,  ∈ (0,1), and _ > 0.
- Definitions and threshold expressions:
  - ^* ≡ 1 / ()
  - _2() ≡ 1 / ( )
  - _3() ≡ [  + 2(1+)] / [  + 2(1+)  ] − [(1+)] / [  + 2(1+)  ] _
  - _u ≡ ^* [ 1 − ( _)/2 ]
- Proposition on determinacy and E-stability (Edgeworth complementarity case):
  - a) If 0 < _ < 2/, then a unique local REE exists and its MSV representation is learnable (E-stable) when 1 < _ < _3() for  ∈ (0, _u), where _u < ^*.
  - b) If _ ≥ 2/ then:
    1) multiple local REE exist for any _ > 1;
    2) the CF representation of stationary “sunspot” equilibria is learnable (E-stable) if 1 < _ < _2() for any  ∈ (0,1);
    3) the MSV-REE is non-learnable (E-unstable) when _ > _2() for any  ∈ (0,1).
  - Comparative statics:
    - ∂_u / ∂_ < 0
    - ∂_3() / ∂_ < 0
- Qualitative implications:
  - An active forward-looking rule that responds to expected sticky-price output restricts determinacy and E-stability regions (smaller admissible  and lower upper bounds on _).
  - For sufficiently large _ (≥ 2/), no active rules induce a determinate and E-stable equilibrium; the economy displays pervasive aggregate instability.
  - Allowing the rule to respond to aggregate output in the two-stickiness case yields similar destabilizing effects.
- Numerical illustrations and comparative results:
  - A mild response to output, _ = 05, lowers both the _ and the  upper bounds, enlarging indeterminacy/E-instability regions.
  - Example: a rule responding to the CPI inflation measure,  = 04 with coefficient _ = 2 induces a unique equilibrium under the simple rule, but learnable “sunspot” equilibria under the extended rule (both for _ = 0 and _ = ).

### Robustness: cash-in-advance (CIA) timing
- Alternative timing: money in utility is N_t / P_t = M_{t−1} + R_{t−1} B_{t−1} − B_t + T^g_t / P_t (following Carlstrom and Fuerst (2001)); focus is on the Euler equation difference.
- Under CIA timing, log-linearized Euler equation:
  - b_t = E_t b_{t+1} − ^−1 ( E_t b_{t+1} − E_t b_{t+1} )
  - Note: nominal interest rate is scrolled forward one period under CIA (contrast with contemporaneous rate under CWID timing).
- Key isomorphism:
  - The CIA Euler equation with a contemporaneous rule is identical to the CWID Euler equation under a forward-looking rule for determinacy and learnability analyses.
- Proposition on equivalence:
  - For the CIA model with contemporaneous rule b_t = _ b_o_t (_ > 1,  ∈ (0,1)), the determinacy and learnability conditions are the same as those for a forward-looking rule in the CWID model (as stated in Propositions 1, 3, and 2).
- Implication: determinacy and E-stability results derived for forward-looking rules under CWID timing carry over to contemporaneous-rule CIA timing by isomorphism.

*Source: _wp10296 - 1. Aggregation bias: Bils and Klenow refer to a sample of 350 categories of goods and services,*

### References

### References

### Monetary policy rules, learnability, and model uncertainty
- Adam K., and A. Marcet, 2010, “Internal Rationality, Imperfect Market Knowledge and Asset Prices,” Manuscript, Mannheim University.
- Benhabib, J., S. Schmitt-Grohé, and M. Uribe, 2001, “Monetary Policy Rules and Multiple Equilibria,” American Economic Review, Vol. 91(1), pp. 167-184.
- Bullard, J., and K. Mitra, 2002, “Learning About Monetary Policy Rules,” Journal of Monetary Economics Vol. 49(6), pp. 1105-1129.
- Canzoneri, M.R., R. Cumby, and B. Diba, 2003, “Recent Developments in the Macroeconomic Stabilization Literature: Is Price Stability a Good Stabilization Strategy?,” in Dynamic Macroeconomic Analysis: Theory and Policy in General Equilibrium, ed. by S. Altug, J. Chadha, and C. Nolan. (Cambridge: Cambridge University Press).
- Clarida, J., J. Galí, and M. Gertler, 2000, “Monetary Policy Rules and Macroeconomic Stability: Evidence and Some Theory,” Quarterly Journal of Economics, Vol. 115(1), pp. 147-180.
- Cochrane, J.H., 2009, “Can Learnability Save New-Keynesian Models?,” Journal of Monetary Economics, Vol. 56(8), pp. 1109-1113.
- Eusepi, S., 2007, “Learnability and Monetary Policy: a Global Perspective,” Journal of Monetary Economics, Vol. 54(4), pp. 1115-1131.
- Evans, G. W., and S. Honkapohja, 2003, “Adaptive Learning and Monetary Policy Design,” Journal of Money, Credit and Banking, Vol. 35(6), pp. 1045-1072.
- Evans, G. W. and B. McGough, 2005, “Monetary Policy, Indeterminacy and Learning,” Journal of Economic Dynamics and Control, Vol. 29(11), pp. 1809-1840.
- Honkapohja, S., K. Mitra, and G. W. Evans, 2003, “Notes on Agents’ Behavioral Rules Under Adaptive Learning and Recent Studies of Monetary Policy,” Manuscript, University of St. Andrews.
- Marcet, A and T. Sargent, 1989, “Convergence of Least-Square Learning Mechanisms in Self-Referential Linear Stochastic Models,” Journal of Economic Theory, Vol. 48(2), pp. 337—368.
- Preston, B., 2006, “Adaptive Learning, Forecast-based Instrument Rules and Monetary Policy,” Journal of Monetary Economics, Vol. 53(3), pp. 507-535.
- Schorfheide, F., 2005, “Learning and Monetary Policy Shifts,” Review of Economic Dynamics, Vol. 8(2), pp. 392-419.
- Svensson, L. and N. Williams, 2007, “Bayesian and Adaptive Optimal Policy under Model Uncertainty,” NBER Working Papers 13414.
- Taylor, J. B., 1993, “Discretion Versus Policy Rules in Practice,” Carnegie-Rochester Conference Series on Public Policy, Vol. 39(1), pp. 195-214.
- Taylor, J., 1999, “A Historical Analysis of Monetary Policy Rules,” in Monetary Policy Rules, ed. by J. Taylor. (Chicago: University of Chicago Press).
- McCallum, B.T., 2009, “Inflation Determination with Taylor Rules: Is New-Keynesian Analysis Critically Flawed?,” Journal of Monetary Economics, Vol. 56(8), pp. 1101-1108.

### Price-setting, price stickiness, and inflation measurement
- Airaudo, M., and L. F. Zanna, 2010, “On the Measure of Inflation for Interest Rate Rules with Credit Market Imperfections,” Manuscript, Drexel University.
- Anand, R., and E. Prasad, 2010, “Optimal Price Indices for Targeting Inflation Under Incomplete Markets,” NBER Working Papers 16290.
- Aoki, K., 2001, “Optimal Monetary Policy Responses to Relative-Price Changes,” Journal of Monetary Economics, Vol. 48(1), pp. 55-80.
- Barsky, R. B., C. L. House, and M. S. Kimball, 2007, “Sticky-Price Models and Durable Goods,” American Economic Review, Vol. 97(3), pp. 984-998.
- Bils, M., and P. Klenow, 2004, “Some Evidence on the Importance of Sticky Prices,” Journal of Political Economy, Vol. 112(5), pp. 947-985.
- Bodenstein, M., C. J. Erceg, and L. Guerrieri, 2008, “Optimal Monetary Policy with Distinct Core and Headline Inflation Rates,” Journal of Monetary Economics, Vol. 55, Supplement 1, pp. S18-S33.
- Carvalho, C., 2006, “Heterogeneity in Price Stickiness and the Real Effects of Monetary Shocks,” Frontiers of Macroeconomics, Vol. 2(1), Article 1.
- Carvalho, C. and N. Dam, 2010, “The Cross-sectional Distribution of Price Stickiness Implied by Aggregate Data,” Manuscript.
- Klenow, P.J., and O. Kryvtsov, 2008, “State-Dependent or Time-Dependent Pricing: Does It Matter for Recent U.S. Inflation?,” Quarterly Journal of Economics, Vol. 123(3), pp. 863-904.
- Klenow, P.J., and B. A. Malin, 2010, “Microeconomic Evidence on Price Setting,” NBER, Working Papers 15826.
- Nakamura, E. and J. Steinsson, 2008, “Five Facts About Prices: A Reevaluation of Menu Costs Models,” Quarterly Journal of Economics, Vol. 123(4), pp. 1415-1464.
- Rotemberg, J., 1982, “Sticky Prices in the United States,” Journal of Political Economy, Vol. 90(6), pp. 1187-1211.
- Mankiw, G. and R. Reis, 2003, “What Measure of Inflation Should a Central Bank Target?,” Journal of European Economic Association, Vol. 5(9), pp. 1058-1086.
- Wynne, M.A., 2008, “How Should Central Banks Define Price Stability?,” Working Paper No.8, Globalization and Monetary Policy Institute, Federal Reserve Bank of Dallas.

### Open-economy and sectoral transmission of monetary policy
- Benigno, P., 2004, “Optimal Monetary Policy in a Currency Area,” Journal of International Economics, Vol. 63(2), pp. 293-320.
- Bouakez, H., E. Cardia, and F. J. Ruge-Murcia, 2009, “The Transmission of Monetary Policy in a Multi-Sector Economy,” International Economic Review, Vol. 50(4), pp. 1243-1266.
- Galí, J. and T. Monacelli, 2005, “Monetary Policy and Exchange Rate Volatility in a Small Open Economy,” Review of Economic Studies, Vol. 72(7), pp. 707-734.
- Llosa, G., Tuesta, V., 2008. Determinacy and Learnability of Monetary Policy Rules in Small Open Economies. Journal of Money, Credit and Banking, Vol. 40(5), 1033-1063.
- Llosa, G., and V. Tuesta, 2009, “Learning about Monetary Policy Rules when the Cost Channel Matters,” Journal of Economic Dynamics and Control, Vol. 33(11), pp. 1880-1896.
- Ravenna, F. and C. Walsh, 2006, “Optimal Monetary Policy with the Cost Channel,” Journal of Monetary Economics, Vol. 53(2), pp. 199-216.
- Carlstrom, C., T. Fuerst, F. Ghironi, and K. Hernández, 2006b, “Relative Price Dynamics and the Aggregate Economy,” Manuscript, Boston College.

### Theoretical foundations, methods, and empirical evidence
- Azariadis, C., 1993, Intertemporal Macroeconomics. (Cambridge, Massachusetts: Blackwell).
- Blanchard, O., and C. Kahn, 1980, “The Solution of Linear Difference Models Under Rational Expectations,” Econometrica, Vol. 48(5), pp. 1305-1312.
- Carlstrom, C. and T. Fuerst, 2001, “Timing and Real Indeterminacy in Monetary Models,” Journal of Monetary Economics, Vol. 47(2), pp. 285-298.
- Carlstrom, C., T. Fuerst, and F. Ghironi, 2006a, “Does It Matter (For Equilibrium Determinacy) What Price Index The Central Bank Targets,” Journal of Economic Theory, Vol. 128(1), pp. 214-231.
- Cogley, T., and T. J. Sargent, 2005, “Drifts and Volatilities: Monetary Policy and Outcomes in the Post WWII US,” Review of Economic Dynamics, Vol. 8(2), pp. 262-302.
- Dhawan, R., and K. Jeske, 2007, “Taylor Rules and Headline Inflation: A Bad Idea,” Working Paper 2007-14, Federak Reserve Bank of Atlanta.
- Evans, G. W. and S. Honkapohja, 2001, Learning and Expectations in Macroeconomics, (Princeton: Princeton University Press).
- McCallum, B., 1983, “On Nonuniqueness in Linear Rational Expectations Models: An Attempt at Perspective,” Journal of Monetary Economics, Vol. 11(2), pp. 134-168.
- Mishkin, F.S., 2008, “Does Stabilizing Inflation Contribute to Stabilizing Economic Activity?,” NBER Working Paper 13970.
- Preston, B., 2006, “Adaptive Learning, Forecast-based Instrument Rules and Monetary Policy,” Journal of Monetary Economics, Vol. 53(3), pp. 507-535.
- Ravenna, F. and C. Walsh, 2006, “Optimal Monetary Policy with the Cost Channel,” Journal of Monetary Economics, Vol. 53(2), pp. 199-216.
- Woodford, M., 2003. Interest and Prices: Foundations of a Theory of Monetary Policy. (Princeton: Princeton University Press).

### Sector-specific and policy-related working papers
- Catao, L.A.V. and R. Chang, 2010, “World Food Price and Monetary Policy,” IMF Working Paper 10/161.
- Klenow, P.J., and B. A. Malin, 2010, “Microeconomic Evidence on Price Setting,” NBER, Working Papers 15826.
- Anand, R., and E. Prasad, 2010, “Optimal Price Indices for Targeting Inflation Under Incomplete Markets,” NBER Working Papers 16290.
- Svensson, L. and N. Williams, 2007, “Bayesian and Adaptive Optimal Policy under Model Uncertainty,” NBER Working Papers 13414.
- Mishkin, F.S., 2008, “Does Stabilizing Inflation Contribute to Stabilizing Economic Activity?,” NBER Working Paper 13970.
- Wynne, M.A., 2008, “How Should Central Banks Define Price Stability?,” Working Paper No.8, Globalization and Monetary Policy Institute, Federal Reserve Bank of Dallas.

*Source: _wp10296 - References*

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