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---

### Overview and research question
- Studies optimal monetary policy responses to supply shocks in an environment with wage rigidity and boundedly rational expectations.
- Two modelling ingredients emphasized:
  - Wage rigidities (wages set before observing current productivity shock; prices set after).
  - Level-k thinking (LKT) for inflation expectations, nesting adaptive expectations (AE, k = 0) and rational expectations (RE, k → ∞).
- Supply shocks modelled as productivity shocks; appendix shows formal equivalence to oil price shocks for a small, oil-importing economy.

### Model setup — firms, households, and equilibrium
- Production:
  - Final goods: Y_t = [∫_0^1 Y_ft^{(γ−1)/γ} df]^{γ/(γ−1)}, γ > 1.
  - Intermediate goods: Y_ft = θ_t [∫_0^1 N_sft^{(ρ−1)/ρ} ds]^{ρ/(ρ−1)}, ρ > 1.
  - Productivity: ln θ_t = ln θ_{t−1} + ε_t, ε_t ~ i.i.d. N(0, σ^2_θ).
- Pricing and labour:
  - Labour demand: N_sft = (Y_ft / θ_t) (W_st / W_t)^{−ρ}.
  - Mark-up pricing: P_ft = (γ/(γ−1)) (W_t / θ_t).
  - Producer profits: Profits = (P_ft − W_t / θ_t) Y_ft.
- Households and unions:
  - Utility: E_HH(h)_t [∑_{τ=0}^∞ β^τ (ln C_{h,t+τ} − η ∫_0^1 Ñ_{s,h,t+τ} ds ) ], β ∈ (0,1), η > 0.
  - Consumption Euler: β E_HH(h)_t [ (C_ht / C_{h,t+1}) (1 + i_t)^{-1} (P_{t+1}/P_t) ] = 1.
  - Wages set by service-specific unions before observing θ_t; common expectations across unions imply W_st = W_t ∀ s.
- Key equilibrium relations:
  - Linearized Phillips curve: ˆπ_t = E_WS_{t−1}(ˆN_t) + E_WS_{t−1}(ˆπ_t) − ˆθ_t (equation 2.12).
  - IS curve under RE: ˆN_t = E_RE_t(ˆN_{t+1}) − [ˆi_t − E_RE_t(ˆπ_{t+1})] (equation 2.13).
  - Central bank feedback rule (control of employment): ˆN_t = − φ_t ˆπ_t (equation 2.14).

### Expectations frameworks (RE, AE, LKT)
- RE (k → ∞):
  - E_WS_{t−1}(ˆN_t) = 0 ⇒ E_WS_{t−1}(ˆπ_t) = 0.
  - Equilibrium: (ˆπ_t, ˆN_t) = (− ˆθ_t, φ_t ˆθ_t).
  - Optimal policy implication: full look-through of supply shocks is optimal under RE.
- AE (k = 0):
  - E_WS_{t−1}(ˆπ_t) = ˆπ_{t−1}, E_WS_{t−1}(ˆN_t) = ˆN_{t−1} (equation 3.16).
  - Dynamics:
    - ˆπ_t = ˆN_{t−1} + ˆπ_{t−1} − ˆθ_t.
    - ˆN_t = − φ_t (ˆN_{t−1} + ˆπ_{t−1} − ˆθ_t).
  - Optimal policy: constant φ_t (no pivoting); forward-looking β_CB > 0 implies φ_t > 0; β_CB = 0 ⇒ φ_t = 0 (full look-through).
- Level-k thinking (LKT), k ∈ {1,2,...}:
  - Iterative expectations mapping yields (equations 3.17–3.18):
    - E^k_{t−1}(ˆπ_t) = (1 − φ_t)^{k−1} [ E^0_{t−1}(ˆN_t) + E^0_{t−1}(ˆπ_t) ].
    - With initialization E^0_{t−1}(ˆπ_t) = ˆπ_{t−1}, E^0_{t−1}(ˆN_t) = ˆN_{t−1}, equilibrium:
      - ˆπ_t = (1 − φ_t)^k (ˆN_{t−1} + ˆπ_{t−1}) − ˆθ_t.
      - ˆN_t = − φ_t [ (1 − φ_t)^k (ˆN_{t−1} + ˆπ_{t−1}) − ˆθ_t ].
  - LKT nests AE (k = 0) and RE (k → ∞) as limiting cases.

### Policy problem and reduced form (Section 3.2)
- Central-bank objective (choose φ_t announced before observing θ_t):
  - Minimize E_RE_{t−1} [ Σ_{τ=0}^∞ β_CB^τ (ˆπ_{t+τ}^2 + μ ˆN_{t+τ}^2) ], β_CB ∈ [0,1), μ > 0.
- Define overheating x_t ≡ ˆπ_t + ˆN_t.
- Reduced policy problem (choose φ_t) becomes:
  - Minimize E_RE_{t−1} [ Σ_{τ=0}^∞ β_CB^τ (1 + μ φ_{t+τ}^2) [ (1−φ_{t+τ})^{2k} x_{t+τ−1}^2 + σ_θ^2 ] ],
  - Subject to x_t = (1−φ_t)[(1−φ_t) k x_{t−1} − ˆθ_t].

### Optimal policy characterizations
- RE (k → ∞):
  - Losses reduce to E_RE_{t−1} [ Σ β_CB^τ (1 + μ φ_{t+τ}^2) σ_θ^2 ] and are increasing in φ_t^2 ⇒ optimal φ_t is "vanishingly low" (interpretation: arbitrarily small positive φ_t).
- AE (k = 0):
  - Bellman solution yields constant φ_t = β_CB a_1 / (μ + β_CB a_1), with a_1 solving a_1 = 1 + μ β_CB a_1 / (μ + β_CB a_1).
  - Optimal policy time-invariant; φ_t = 0 if β_CB = 0.
- LKT (k ≥ 1):
  - Myopic central bank (β_CB = 0): first-order condition (equation 3.23), inverse mapping f(φ_t) (equation 3.24).
  - Key analytical results (Lemmas and Propositions):
    - Lemma 3.1: Optimal φ_t ∈ [0,1] and f′(φ_t) > 0.
    - Lemma 3.2: Behavior of quadratic d(φ_t) = μ(1 + k)φ_t^2 − μφ_t + k; if μ < 4k(1 + k) then d(φ_t) > 0 ∀ φ_t ∈ [0,1]; otherwise d has two roots φ_{d1}, φ_{d2}.
    - Lemma 3.3: Existence of μ̄ ∈ (0,4k(1 + k)) governing sign of cubic n(φ_t); non-monotonicity of f arises when μ ≥ μ̄.
    - Proposition 3.1 (pivot existence): φ^*(x_{t−1}^2) strictly increasing in x_{t−1}^2; if μ ≤ μ̄ continuous; if μ > μ̄ there exists cutoff x_{t−1}^{2*} ∈ [σ_θ^2 f(φ_{n2}), ∞) at which φ^* jumps discontinuously from lower branch φ_1 to upper branch φ_2.
    - Proposition 3.2: μ̄ strictly increasing in k and lim_{k→∞} μ̄ = ∞; higher k reduces the parameter range where pivots emerge.
    - Proposition 3.3: If μ > μ̄, pivots produce discontinuous increases in employment gap variance var_RE_{t−1}(ˆN_t) = φ_t^2 σ_θ^2; for k = 1 pivots have no effect on expected employment E_RE_{t−1}(ˆN_t) = − φ_t (1−φ_t)^k x_{t−1}.

### Mechanism behind pivots
- Two channels when tightening:
  - Direct channel: tighter policy reduces activity at given private expectations (employment cost).
  - Indirect channel: tighter policy re-anchors inflation expectations, reducing slack needed to stabilize realized inflation (benefit).
- Cross-term μ φ_t^2 (1−φ_t)^{2k} x_{t−1}^2 can be hump-shaped in φ_t, producing W-shaped total losses and multiple local optima when x_{t−1}^2 and μ large.
- Elasticity interpretation:
  - ζ(φ_t) ≡ k φ_t / (1−φ_t) measures sensitivity of private-sector expectations to φ_t (equation 3.26).
  - First-order condition re-expressed (equation 3.27) links central bank’s marginal cost/benefit to ζ(φ_t) and relative importance of expectations-driven losses.

### Generalized Phillips curve and calibration (Section 4.1)
- Generalization introduces λ ≡ χ/(1 + ρχ) so Phillips curve: ˆπ_t = λ E^WS_{t−1}(ˆN_t) + E^WS_{t−1}(ˆπ_t) − ˆθ_t.
- Redefine x_t ≡ ˆπ_t + λ ˆN_t; effective parameters ˜μ = μ/λ^2, ˜φ_t = λ φ_t.
- Policy problem equivalent to unit-slope case with replacements (μ, φ_t) → (˜μ, ˜φ_t).
- Baseline calibration:
  - Time period = 1 year.
  - β_CB = 0.995.
  - μ = 1.
  - λ = 0.092.
  - k = 2.
  - σ^2_θ chosen to match CPI inflation variance in Canada since mid-1990s.
- Solution method: value function iteration (detailed algorithm in Appendix D).

### Numerical results and policy implications (Section 4.3)
- Baseline (forward-looking central bank, blue lines in Figure 5):
  - Optimal φ_t as function of x_{t−1} shows pivot threshold around x_{t−1} ≈ 4% where policy stance jumps sharply.
  - Pivot associated with sharp renormalizations in both rational and LKT-based expectations.
- Myopic central bank (β_CB = 0, magenta lines):
  - Later pivot threshold than baseline; forward-looking CB "pulls forward" pivot (pivots earlier with smaller re-normalizations).
- Role of k (levels of thinking):
  - k ∈ {1,2,3} produce pivots around 3–5% thresholds.
  - Large k (e.g., k = 40, rational-expectations limit) show no pivots even at x_{t−1} up to 10%.
  - Higher k → smaller jumps in policy stance and expectations; reduce employment variance and hard-landing risk.
- Role of effective weight ˜μ = μ/λ^2:
  - Pivoting requires ˜μ exceed threshold.
  - Higher ˜μ → delayed pivots at higher x_{t−1} and larger swings in φ_t once triggered.
- Costs of deviating from optimal pivot (Section 5 simulations):
  - "Too late" pivot (delay 3 periods) → significantly worse inflation and employment than optimal.
  - "Too gradual" tightening (smooth 3-period increase to same peak) → worse employment outcomes even relative to "too late".
  - Conclusion: gradual approaches generally suboptimal; mistiming and partial tightening costly.
- Demand shocks (Section 6):
  - IS augmented by stochastic intercept d_t; appropriate stimulus measured by cumulative expected future differences between real rate and time-varying natural rate (equation 6.32).
  - Policy setting is two-stage: (1) insulate employment from demand shocks; (2) adjust to address supply shocks/expectations anchoring.
  - Results carry over if policymakers can distinguish demand vs supply shocks; mis-identification introduces signal-extraction risks.

### Empirical evidence and interpretation of recent inflation surge (Section 7)
- Model implication: policymakers initially look through supply shocks; pivot sharply if shocks cumulate beyond threshold to re-anchor expectations; expected inflation and wage growth should turn before realized inflation peaks.
- U.S. evidence (2020Q1–2023Q4):
  - Two one-year-ahead expected-inflation series and wage-growth proxy rose through 2022Q2 then turned down while inflation continued to rise for quarters after 2022Q2 — consistent with model.
- Empirical regression decomposition (equations 7.33–7.36) constructs residual u_t capturing departure from long-run φ̄; residuals rose from 2020Q2 to 2022Q2 and declined thereafter in multiple specifications (Figures 10–11).
- Robustness: controlling for Global Supply Chain Pressure Index (GSCPI) does not change qualitative patterns.
- Conclusion: evidence supports the model’s mechanism that timely pivots can re-anchor expectations without immediate collapse in realized inflation.

### Robustness and extensions (Appendices summary)
- Appendix A: analytical proofs for lemmas and propositions; key expressions for d(φ_t), n(φ_t), discriminants, and limiting behaviour lim_{k→∞} μ̄ = ∞.
- Appendix B–C: generalizations to non-rational household expectations and GHH preferences; equivalence mapping to modified λ and ˜μ, ˜φ_t.
- Appendix D: numerical solution algorithm (value function iteration steps 1–5).
- Appendix E: hybrid Phillips curves with fraction α ∈ (0,1) of adaptive wage-setters:
  - If non-adaptive are RE, myopic CB produces no pivots (unique continuous φ^*(·)).
  - If non-adaptive are level-k thinkers, pivots can re-emerge when μ large and k positive but small.
- Appendix F: econometric learning (constant and decreasing gains):
  - Pure learning-only yields smooth φ_t (no pivots); higher gain → tighter policy.
  - Mixing learning (level-0 beliefs follow gain rule) with LKT re-introduces pivots for low k and high ˜μ; decreasing gains can produce pivots early and persistence.
- Appendix G: alternative level-0 specifications (signal-informed, fully anchored):
  - Pivots sensitive to level-0 properties (A0, B0, γ) and share α; pivots can disappear or re-emerge depending on signal informativeness and anchoring.
- Appendix H: alternative bounded-rationality models:
  - Reflective expectations (continuous analogue): E_WS_{t−1}(x_t) = exp(−k φ_t) x_{t−1}; pivots occur when μ large and k small.
  - Cognitive hierarchy (heterogeneous levels): pivots occur when μ large and population sophistication low; elasticity of aggregate expectations with respect to φ_t rises with φ_t.
- Appendix I: oil price shocks map to productivity shocks in small oil-importing economy via θ_t = Z_t h(P_O t).

### Key policy takeaways and scenarios
- Optimal policy under LKT is threshold-based:
  - Small inflation overshoots → central bank optimally mostly looks through supply shocks.
  - When cumulative overheating x_{t−1}^2 exceeds threshold x_{t−1}^{2*}, optimal φ_t may jump discontinuously (pivot) to a more hawkish stance to re-anchor expectations.
- Pivots:
  - Can enable "soft landings" in expected employment (special case k = 1 yields exact offset); generally increase employment variance (riskier soft landings).
  - Timing and magnitude critical: mistiming or gradualism can be materially costlier than a timely sharp pivot.
- Role of private-sector sophistication:
  - Higher k reduces need for pivots and makes pivots smaller and less risky.
  - Low single-digit k (experimentally supported) implies pivots in empirically relevant overheating ranges (3–5%).
- Practical implication:
  - Central banks facing supply-driven inflation episodes should weigh benefits of a rapid, sizable re-anchoring move against higher short-run employment uncertainty; the model formalizes the trade-off and identifies conditions under which abrupt pivots are optimal.

*Source: wpiea2026097-source-pdf - 2.1    Firms (excerpt).*

### 2.1    Firms   .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  .  

### 2.1    Firms

### Overview
- Section 2.1 is part of a broader paper that studies optimal monetary policy responses to supply shocks in an environment with wage rigidity and boundedly rational expectations.
- The paper emphasizes two modelling ingredients: (i) wage rigidities; and (ii) a flexible approach to modelling inflation expectations via level-k thinking (LKT), which nests adaptive expectations (AE, k = 0) and rational expectations (RE, k → ∞).

### Model ingredients and setup (as described across Sections 2 and 3)
- Wage rigidity specification:
  - Wages are set before observing the current productivity shock; prices are set after observing the shock.
  - Wages are rigid for one period in advance; prices and wages reset each period.
  - This timing creates intratemporal coordination frictions and potential wage-price spirals: wage-setters form expectations about prices, which firms then use (together with average wages) to set mark-ups.
- Expectations specification:
  - Level-k thinking (LKT) is used to model bounded rationality.
  - k = 0 corresponds to adaptive expectations (AE).
  - k → ∞ corresponds to rational expectations (RE).
  - Intermediate k implies expectations combine backward- and forward-looking elements and reflect an imperfect understanding of monetary policy’s effects.
- Shocks and interpretation:
  - The paper models supply shocks as productivity shocks but notes these can be interpreted as subsuming a range of supply shocks (e.g., oil price shocks); Appendix shows a formal equivalence for a small, oil-importing economy.

### Key mechanisms and theoretical implications
- Under RE (k → ∞):
  - It is optimal for central banks to fully look through supply shocks.
- Under AE (k = 0):
  - The optimal policy is generally tighter than under RE but involves a constant degree of "look-through" — the trade-off rate between the inflation and employment gaps never changes over time.
- Under LKT (k ∈ {1,2,...}):
  - The optimal policy can display "pivoting" behaviour: policymakers initially mostly look through supply shocks, but may switch suddenly to aggressive tightening if shocks cumulate beyond a threshold.
  - Non-convexity in the central bank’s loss function arises because tightening affects employment through two offsetting channels:
    - Direct channel: tighter policy reduces economic activity at a given level of private-sector expectations.
    - Indirect channel: tighter policy helps re-anchor inflation expectations, reducing the actual slack needed to stabilize realized inflation.
  - The indirect channel tends to be stronger when there have been large inflation overshoots (expectations already elevated), and can also strengthen as policy becomes tighter, because private-sector expectations under LKT are more sensitive to policy changes when policy is already tight.
  - These dynamics generate benefits to tightening by a sufficiently wide margin, but only after sufficiently large overshoots have occurred, producing abrupt pivots.

### Implications for landing dynamics and risks
- Pivots need not produce hard landings:
  - If the indirect channel is sufficiently strong, pivots can re-anchor expectations without requiring large amounts of slack; a special case exists where the channels offset in expectation, implying pivots have no impact on expected employment.
  - Outside that special case, expected impacts on employment are consistently small in numerical exercises.
- Costs and risks:
  - Pivots increase the variance of employment by making monetary policy more responsive to supply shocks; pivots are risky and accept some probability that poor draws of the supply shock could lead to hard landings.
  - Simulations show costs of mistiming pivots can be substantial, as can costs of attempting to achieve similar tightening more gradually.

### Scope and focus choices
- The paper downplays intertemporal coordination frictions from staggered price- and wage-setting to focus on intratemporal coordination frictions driven by wage-setting timing.
- The analysis mostly disregards demand shocks to isolate the policy dilemmas from supply shocks; results should be viewed as describing optimal adjustments dictated by supply shocks over and above a central bank’s response to demand shocks (formalized in Section 6).
- The issue studied is especially relevant for emerging economies, where food and fuel represent a larger consumption share and supply shocks are frequent and volatile.

### Relation to literature
- Connects to:
  - New Keynesian macroeconomic models and recent studies on the global inflation episode.
  - Literature on bounded rationality and level-k thinking, with LKT supported by experimental evidence and previous macro applications.
  - Literature on successful disinflations: expectation management can enable disinflations with smaller slack; LKT provides a complementary mechanism where disinflations occur without large shifts in both slack and long-run expectations.
- Appendices:
  - Confirm robustness of main results under reflective expectations and cognitive hierarchy frameworks.
  - Provide proofs and additional formal equivalences (e.g., productivity shocks and oil price shocks).

### Organization (as stated)
- Section 2 presents the model.
- Section 3 formalizes the optimal policy problem and presents analytically solvable special cases.
- Section 4 presents the full dynamic policy problem and numerical simulations.
- Section 5 explores consequences of policy errors.
- Section 6 discusses demand shocks.
- Section 7 interprets the recent inflation surge.
- Section 8 concludes.
- Appendices contain proofs and additional material.

*Source: wpiea2026097-source-pdf - 2.1    Firms (excerpt).*

### 2.1    Firms

### 2.1    Firms

### Production structure
- Two-stage production with distinct final and intermediate goods.
- Final goods:
  - Technology: Y_t = [∫_0^1 Y_ft^{(γ−1)/γ} df]^{γ/(γ−1)}, where γ >1 denotes the elasticity of substitution among intermediate goods.
  - Demand for intermediate good f (final goods producers perfectly competitive, zero profits):
    - Y_ft = Y_t (P_ft / P_t)^{−γ}  (equation 2.1)
    - Aggregate price level: P_t ≡ [∫_0^1 P_ft^{1−γ} df]^{1/(1−γ)}.
- Intermediate goods:
  - Technology: Y_ft = θ_t [∫_0^1 N_sft^{(ρ−1)/ρ} ds]^{ρ/(ρ−1)}, where ρ >1 denotes the elasticity of substitution among services and θ_t is aggregate productivity (stochastic).
  - Productivity process (convenient specification): ln θ_t = ln θ_{t−1} + ε_t, where ε_t is i.i.d. normal with mean zero and variance σ^2_θ.

### Labour demand, profits, and pricing
- Labour demand (wage-taking assumption):
  - N_sft = (Y_ft / θ_t) (W_st / W_t)^{−ρ}, with W_t ≡ [∫_0^1 W_st^{1−ρ} ds]^{1/(1−ρ)} (equation 2.2).
- Producer profits:
  - Profits = P_ft Y_ft − ∫_0^1 W_st N_sft ds = (P_ft − W_t / θ_t) Y_ft (equation 2.3).
- Price-setting (monopolistic competition, flexible prices observed after θ_t):
  - Mark-up rule: P_ft = (γ/(γ−1)) (W_t / θ_t) (equation 2.4).
  - Implications:
    - Inverse relationship between prices and productivity.
    - All intermediate good producers behave identically: (P_ft, Y_ft) = (P_t, Y_t) ∀ f ∈ [0,1].
    - Aggregate labour demand simplifies to ∫_0^1 (Y_ft / θ_t) (W_st / W_t)^{−ρ} df = (Y_t / θ_t) (W_st / W_t)^{−ρ} (equation 2.5).

### Natural levels (flexible-wage benchmark)
- Flexible-wage (wages set after observing θ_t) and equal consumption across households (C_ht = Y_t ∀ h) imply flexible wage-setting:
  - W_t = (ρ η / (ρ−1)) P_t Y_t.
- From mark-up rule multiplied by total output and defining N_t ≡ Y_t / θ_t:
  - P_t Y_t = (γ/(γ−1)) W_t N_t.
- Natural level of employment:
  - N_t = ρ^{−1} (ρ η) (γ−1)/γ ≡ N̄. (text gives N_t = ρ−1 ρη γ−1 γ ≡ ̄N; preserve notation as presented)
  - Natural employment constant, decreasing in disutility of labour and monopoly distortions.
- Potential output: Ȳ_t ≡ θ_t N̄.

---

### 2.2    Households

### Preferences and budget constraint
- Unit measure of households indexed by h ∈ [0,1]; each supplies range of labour services.
- Household utility:
  - E_HH(h)_t [∑_{τ=0}^∞ β^τ (ln C_{h,t+τ} − η ∫_0^1  Ñ_{s,h,t+τ} ds ) ] (equation 2.6)
  - Parameters: β ∈ (0,1); η > 0.
  - C_ht denotes household consumption; Ñ_sht denotes supply-side hours (tilde distinguishes supply-side labour).
- Budget constraint:
  - ∫_0^1 W_st Ñ_sht ds + D_t + T_t + B_{h,t−1} = P_t C_ht + B_ht / (1 + i_t) (equation 2.7).
  - D_t = dividends (distributed equally); T_t = government transfers (distributed similarly); B_ht = household purchases of risk-free one-period bonds; i_t = nominal interest rate.

### Consumption-saving decision
- Consumption decision observed after current θ_t (affects expectations set):
  - Euler equation: β E_HH(h)_t [ (C_ht / C_{h,t+1}) (1 + i_t)^{-1} (P_{t+1}/P_t) ] = 1 (equation 2.8).
  - Subscript on expectation indicates θ_t is in information set.

### Labour-supply decision (unions)
- Supply decisions made by service-specific unions; household head handles consumption-saving.
- Wages set by unions on monopolistically competitive basis before observing θ_t (wage stickiness: can adjust every period but before θ_t).
- For union for service s ∈ [0,1], hours per member:
  - Ñ_hst = Ñ_st = (Y_t / θ_t) (W_st / W_t)^{−ρ} ∀ h ∈ [0,1].
- Union objective (values wages and disutility using marginal preferences of members):
  - E_WS(s)_{t−1} [ W_st Ñ_st ∫_0^1 1/(P_t C_ht) dh − η Ñ_st ] (equation 2.9), where E_WS(s)_{t−1}(·) excludes θ_t.
- Wage-setting rule:
  - W_st = (ρ η / (ρ−1)) E_WS(s)_{t−1} [ Y_t / θ_t ] / E_WS(s)_{t−1} [ (Y_t / θ_t) ∫_0^1 1/(P_t C_ht) dh ] (equation 2.10).
- Common expectations across unions assumed: E_WS(s)_{t−1}(X_t) = E_WS(s̃)_{t−1}(X_t) for aggregate X_t and any (s, s̃) ∈ [0,1].
  - Implies W_st = W_t ∀ s, allowing aggregate wage-setting condition:
    - W_t = (ρ η / (ρ−1)) E_WS_{t−1} [ Y_t / θ_t ] / E_WS_{t−1} [ (Y_t / θ_t) ∫_0^1 1/(P_t C_ht) dh ] (equation 2.11).

---

### 2.3    Phillips and IS curves

### Phillips curve (derivation and form)
- Linearizing the wage-setting condition (2.11) and the mark-up rule (2.4) around the symmetric flexible-wage benchmark yields:
  - ˆπ_t = E_WS_{t−1}(ˆN_t) + E_WS_{t−1}(ˆπ_t) − ˆθ_t  (equation 2.12),
    - where ˆπ_t ≡ ln P_t − ln P_{t−1} − π^* (inflation deviation from target π^*),
    - ˆN_t ≡ ln N_t − ln N̄ (employment gap),
    - ˆθ_t ≡ ln θ_t − E_WS_{t−1}(ln θ_t) (supply shock).
- Key features:
  - Residual is a “true” supply shock rather than a mark-up shock — a consequence of wage rigidities.
  - Form derived without specifying private-sector expectation formation due to Lucas-style nominal rigidities (wages set one period in advance).

### IS curve
- Under rational expectations for households (main text focus), linearizing (2.8) and market clearing gives:
  - ˆN_t = E_RE_t(ˆN_{t+1}) − [ˆi_t − E_RE_t(ˆπ_{t+1})]  (equation 2.13),
    - where E_RE_t(·) denotes rational expectation, and ˆi_t ≡ i_t − (π^* − ln β) (deviation of nominal rate from steady state).
- Authors note IS curve depends on how household expectations are specified; main results are independent of IS details because model solved recursively for (ˆπ_t, ˆN_t, φ_t) first, then ˆi_t from IS.

---

### 2.4    Monetary and fiscal policy

### Central bank (policy stance and feedback)
- Central bank modeled as directly controlling employment level, setting it as function of inflation:
  - ˆN_t = − φ_t ˆπ_t  (equation 2.14), where φ_t is time-varying policy stance reflecting tightening in response to off-target inflation.
- Policy-makers update φ_t each period, announcing before observing θ_t, aiming to minimize an ad-hoc loss function (policy problem elaborated in next section).
- Interpretation via forward-solving IS:
  - ˆN_t = − E_RE_t [ ∑_{τ=0}^∞ (ˆi_{t+τ} − ˆπ_{t+τ+1}) ].
  - Thus φ_t ˆπ_t = E_RE_t [ ∑_{τ=0}^∞ (ˆi_{t+τ} − ˆπ_{t+τ+1}) ]  (equation 2.15).
- Time variation in φ_t allows analysis of “pivoting” from low to high φ_t.

### Fiscal authority
- Government operates a balanced budget each period and keeps bonds in zero net supply.
- Authors note these fiscal assumptions are not essential; results hold for wide range of Ricardian fiscal policies.

---

### 2.5    Summary of equilibrium conditions
- Key variables: ˆπ_t (inflation gap), ˆN_t (employment gap), ˆi_t (nominal rate gap), φ_t (policy stance).
- Equilibrium pinned down by:
  - Phillips curve (2.12),
  - IS curve (2.13),
  - Policy feedback rule (2.14),
  - Solution to the policy problem (next section).
- Recursive solution property:
  - Solve for (ˆπ_t, ˆN_t, φ_t) independent of IS curve, then back out ˆi_t using IS.
  - Main results independent of IS curve details and household expectation formation.
- Focus of analysis: private-sector expectations of wage-setters and how monetary policy shapes them.

---

### 3    Expectations and optimal policy

### Expectations frameworks considered
- Rational Expectations (RE)
  - Under RE, E_WS_{t−1}(ˆN_t) = 0. Combining with ˆN_t = −φ_t ˆπ_t and φ_t > 0 implies E_WS_{t−1}(ˆπ_t) = 0.
  - Equilibrium:
    - (ˆπ_t, ˆN_t) = (− ˆθ_t, φ_t ˆθ_t).
  - Implication: inflation follows an i.i.d. process; any φ_t > 0 anchors inflation expectations at target; employment stabilized as φ_t → 0.
- Adaptive Expectations (AE)
  - Simple AE specification: [E_WS_{t−1}(ˆπ_t), E_WS_{t−1}(ˆN_t)] = (ˆπ_{t−1}, ˆN_{t−1}) (equation 3.16).
  - Equilibrium:
    - ˆπ_t = ˆN_{t−1} + ˆπ_{t−1} − ˆθ_t
    - ˆN_t = − φ_t (ˆN_{t−1} + ˆπ_{t−1} − ˆθ_t)
  - Special case φ_t = 0: inflation collapses to a random walk.
  - Authors note extensions (appendix) with hybrid mixes and more sophisticated backward-looking rules.
- Level-k thinking (LKT)
  - LKT formalizes iterative “mutual forecasting” with finite iterations k̃ = 1,2,…,k (bounded computing power).
  - Iteration from level-zero guesses E^0_{t−1}(ˆπ_t), E^0_{t−1}(ˆN_t) updated via Phillips curve and feedback rule:
    - E^1_{t−1}(ˆπ_t) = E^0_{t−1}(ˆN_t) + E^0_{t−1}(ˆπ_t)
    - E^1_{t−1}(ˆN_t) = − φ_t [ E^0_{t−1}(ˆN_t) + E^0_{t−1}(ˆπ_t) ]
  - Iterating to level k yields:
    - E^k_{t−1}(ˆπ_t) = (1 − φ_t)^{k−1} [ E^0_{t−1}(ˆN_t) + E^0_{t−1}(ˆπ_t) ]  (equation 3.17)
    - E^k_{t−1}(ˆN_t) = − φ_t (1 − φ_t)^{k−1} [ E^0_{t−1}(ˆN_t) + E^0_{t−1}(ˆπ_t) ]  (equation 3.18)
  - Main-text assumption for initialization: E^0_{t−1}(ˆπ_t) = ˆπ_{t−1} and E^0_{t−1}(ˆN_t) = ˆN_{t−1} (agents seed with simple adaptive benchmark).
  - Feeding level-k expectations into true Phillips curve and feedback rule gives equilibrium:
    - ˆπ_t = (1 − φ_t)^k (ˆN_{t−1} + ˆπ_{t−1}) − ˆθ_t
    - ˆN_t = − φ_t [ (1 − φ_t)^k (ˆN_{t−1} + ˆπ_{t−1}) − ˆθ_t ]
  - Links across frameworks:
    - k = 0 → coincides with AE.
    - k → ∞ with φ_t ∈ (0,1] → converges to RE solution (ˆπ_t, ˆN_t) = (− ˆθ_t, φ_t ˆθ_t).
  - LKT nests AE and RE as ends of an iterative spectrum.

*Source: wpiea2026097-source-pdf - 2.1    Firms*

### 3.2    Optimal policy

### 3.2    Optimal policy

### Policy problem and loss function
- Policy-makers choose the policy stance φ_t announced before observing current productivity θ_t to minimize the quadratic loss:
  - E_RE_{t−1} [ Σ_{τ=0}^∞ β_CB^τ (ˆπ_{t+τ}^2 + μ ˆN_{t+τ}^2) ],
  - where β_CB ∈ [0,1) may differ from private sector discounting; μ > 0 is the weight on employment stabilization.
- Optimization is subject to the Phillips curve (line 2.12) and the feedback rule (line 2.14), with commitment to φ_t even if suboptimal ex post.

### Simplification: overheating measure and law of motion
- Define x_t ≡ ˆπ_t + ˆN_t (total overheating).
- Reduced policy problem: choose φ_t to minimize
  - E_RE_{t−1} [ Σ_{τ=0}^∞ β_CB^τ (1 + μ φ_{t+τ}^2) [(1−φ_{t+τ})^2k x_{t+τ−1}^2 + σ_θ^2] ],
  - subject to law of motion:
    - x_t = (1−φ_t)[(1−φ_t) k x_{t−1} − ˆθ_t].

### Optimal policy under RE (k → ∞)
- Under RE:
  - Losses: E_RE_{t−1} [ Σ β_CB^τ (1 + μ φ_{t+τ}^2) σ_θ^2 ].
  - Law: x_t = −(1−φ_t) ˆθ_t.
- Since losses independent of x_t and strictly increasing in φ_t^2, optimal φ_t is "vanishingly low" (interpret as positive but arbitrarily small) → full anchoring of inflation expectations and vanishingly low employment variance.
- Interpretation: even very weak commitment keeps expected inflation at target under fully rational agents.

### Optimal policy under AE (k = 0)
- Bellman representation:
  - V(x_{t−1}) = min_{φ_t} { (1 + μ φ_t^2)(x_{t−1}^2 + σ_θ^2) + β_CB E_RE_{t−1}[V(x_t)] },
  - with x_t = (1−φ_t)(x_{t−1} − ˆθ_t).
- Guess V(x) = a_1 x^2 + a_2 σ_θ^2 leads to:
  - φ_t = β_CB a_1 / (μ + β_CB a_1),
  - a_2 = [1 + μ φ_t^2 + β_CB a_1 (1−φ_t)^2] / (1−β_CB),
  - a_1 satisfies a_1 = 1 + μ β_CB a_1 / (μ + β_CB a_1).
- Features:
  - Optimal policy is constant (no pivoting).
  - If β_CB = 0 (myopic central bank), φ_t = 0 (fully look through shocks).
  - Forward-looking β_CB > 0 ⇒ φ_t > 0 (more hawkish).

### Optimal policy under LKT (k ∈ {1,2,...})
- No closed-form solution; proceed in two steps:
  1. Myopic central bank β_CB = 0 — derive qualitative properties and dependence on parameters and x_{t−1}.
  2. Forward-looking β_CB > 0 — solve dynamic problem numerically to test robustness.

- Decomposition of period expected losses (myopic case):
  - (1 + μ φ_t^2)[ (1−φ_t)^2 k x_{t−1}^2 + σ_θ^2 ] = E_RE_{t−1}(ˆπ_t^2) + μ E_RE_{t−1}(ˆN_t^2).
  - Interpret components:
    - (1−φ_t)^2 k x_{t−1}^2: inflation gap cost from poor anchoring.
    - σ_θ^2: inflation gap cost from supply shocks.
    - μ φ_t^2 (1−φ_t)^2 k x_{t−1}^2: employment cost of tightening in response to poor anchoring (cross-term).
    - μ φ_t^2 σ_θ^2: employment cost of tightening in response to supply shocks.

- Cross-term μ φ_t^2 (1−φ_t)^2k x_{t−1}^2 is hump-shaped in φ_t and can create W-shaped total losses as a function of φ_t when x_{t−1}^2 is large and μ is large.

### First-order condition and inverse mapping
- First-order condition (myopic):
  - 0.5 L_1(φ_t, x_{t−1}^2) = μ φ_t[(1−φ_t)^{2k} x_{t−1}^2 + σ_θ^2] − k(1−φ_t)^{2k−1} x_{t−1}^2 (1 + μ φ_t^2) = 0.  (3.23)
- Inverse mapping useful: define f(φ_t) by
  - μ φ_t (1−φ_t)^{2k−1} [ k(1 + μ φ_t^2) − μ φ_t (1−φ_t) ] = x_{t−1}^2 / σ_θ^2.  (3.24)
- Lemma 3.1 constraints:
  - Optimal φ_t ∈ [0,1] and f′(φ_t) > 0.

### Characterization lemmas and scenarios
- Define d(φ_t) as the square-bracket term in denominator of f(·); d(·) is quadratic.
  - Lemma 3.2: If μ < 4k(1 + k), then d(φ_t) > 0 ∀ φ_t ∈ [0,1]. Otherwise d(·) has two roots φ_{d1} and φ_{d2} with specified expressions and signs; d(φ_t) < 0 iff φ_t ∈ (φ_{d1}, φ_{d2}).
- Derivative f′(φ_t) shares sign with cubic numerator n(φ_t) ≡ 2μ(k+1) φ_t^3 − 3μ φ_t^2 + 2(k−1) φ_t + 1.
  - Lemma 3.3: There exists μ̄ ∈ (0,4k(1 + k)) such that:
    - If μ < μ̄, n(φ_t) > 0 ∀ φ_t ∈ [0,1].
    - If μ ≥ μ̄, n(·) has two roots φ_{n1} > 0 and φ_{n2} ∈ [φ_{n1},1) with n(φ_t) < 0 for φ_t ∈ (φ_{n1}, φ_{n2}), and n(φ_t) > 0 otherwise; roots distinct when μ > μ̄.
    - If μ ≥ 4k(1 + k) as well, roots can be ordered relative to φ_{d1}, φ_{d2}: φ_{d1} ≤ φ_{n1} ≤ φ_{d2} < φ_{n2}.

- Three possible shapes for f(·) on [0,1]:
  1. μ ≤ μ̄: f(·) continuous and strictly increasing → unique φ_t = f^{-1}(x_{t−1}^2 / σ_θ^2).
  2. μ ∈ (μ̄, 4k(1 + k)): f(·) continuous but non-monotonic → two increasing branches:
     - Lower branch: domain [0, φ_{n1}] → range [0, f(φ_{n1})].
     - Upper branch: domain [φ_{n2}, 1) → range [f(φ_{n2}), ∞) (higher φ_t).
  3. μ ≥ 4k(1 + k): f(·) discontinuous; lower branch domain [0, φ_{d1}) → range [0, ∞); similar upper branch as above.

### Optimal pivots and Proposition 3.1
- When μ > μ̄, multiple local extrema imply global optimum can jump between lower branch φ_1(·) and upper branch φ_2(·).
- Limits:
  - For very large x_{t−1}^2, φ^*(x_{t−1}^2) = φ_2(x_{t−1}^2) (upper branch preferred).
  - For x_{t−1}^2 ∈ [0, σ_θ^2 f(φ_{n2})), φ^*(x_{t−1}^2) = φ_1(x_{t−1}^2) (upper branch inaccessible).
- Existence of unique pivot:
  - Proposition 3.1: φ^*(x_{t−1}^2) is strictly increasing in x_{t−1}^2. If μ ≤ μ̄ it is continuous. Otherwise there exists cutoff x_{t−1}^{2*} ∈ [σ_θ^2 f(φ_{n2}), ∞) at which φ^* jumps discontinuously:
    - φ^*(x_{t−1}^2) = φ_1(x_{t−1}^2) for x_{t−1}^2 ∈ [0, x_{t−1}^{2*}),
    - φ^*(x_{t−1}^2) = φ_2(x_{t−1}^2) for x_{t−1}^2 ∈ (x_{t−1}^{2*}, ∞).
- Interpretation: with high μ, policy-makers prefer low φ_t until overheating x_{t−1}^2 exceeds threshold, then pivot to a much more aggressive stance to re-anchor expectations without inducing excessive employment gaps.

### Role of elasticity ζ(φ_t) and parameter k
- Define elasticity of private-sector expectations to policy stance:
  - dE_WS_{t−1}(x_t)/dφ_t * (φ_t / E_WS_{t−1}(x_t)) = k φ_t / (1−φ_t) ≡ ζ(φ_t).  (3.26)
- Re-expressing first-order condition:
  - μ φ_t^2 / (1 + μ φ_t^2) = ζ(φ_t) * (1−φ_t)^{2k} x_{t−1}^2 / [ (1−φ_t)^{2k} x_{t−1}^2 + σ_θ^2 ].  (3.27)
  - Left ratio = elasticity of central bank expected losses w.r.t. policy stance (holding expectations constant).
  - Right ratio r(φ_t, x_{t−1}^2) = (1−φ_t)^{2k} x_{t−1}^2 / [ (1−φ_t)^{2k} x_{t−1}^2 + σ_θ^2 ] = elasticity of expected losses w.r.t. expectations (holding direct effects constant).
- Key points:
  - ζ(φ_t) is increasing in φ_t → marginal benefits of tightening are non-monotonic.
  - r(φ_t, x_{t−1}^2) increases with x_{t−1}^2 → incentives to delay tightening until overheating is sufficiently large.
  - Higher k (more levels of thinking) raises ζ(φ_t) and makes policy more powerful; in the RE limit pivots are less likely.
  - Proposition 3.2: μ̄ is strictly increasing in k and lim_{k→∞} μ̄ = ∞; increasing k narrows the μ-range where pivots emerge.

### Pivots and soft landings
- Expected employment gap:
  - E_RE_{t−1}(ˆN_t) = − φ_t (1−φ_t)^k x_{t−1}.  (3.28)
- Employment variance:
  - var_RE_{t−1}(ˆN_t) = φ_t^2 σ_θ^2.  (3.29)
- Effects around pivot:
  - For k = 1, direct and indirect channels offset exactly in expectation → pivots have no effect on expected employment level (φ_t (1−φ_t)^k unchanged) → pivots compatible with "soft landings" in expectation.
  - However, pivots increase φ_t discontinuously → variance of employment (φ_t^2 σ_θ^2) increases discontinuously → pivots produce "risky" or "narrow" soft landings: inflation re-anchored without expected employment loss but with higher employment uncertainty.
- Proposition 3.3: If μ > μ̄, pivots produce discontinuous increases in employment gap variance as given by (3.29); if k = 1, they have no effect on expected employment level as given by (3.28).

### Next steps: dynamic policy (numerical solution)
- The paper proceeds to solve the full dynamic problem numerically for β_CB > 0 (forward-looking central bank) and generalizes the Phillips curve to allow an arbitrary slope on expected employment gap to aid calibration and interpretation.

*Source: wpiea2026097-source-pdf - 3.2    Optimal policy*

### 4.1    Generalizing the Phillips curve

### 4.1    Generalizing the Phillips curve

### Modified Phillips curve and overheating measure
- Household utility assumed:  
  E_HH(h)_t [ Σ_{τ=0}^∞ β^τ ln( C_{h,t+τ} − η θ_{t+τ} ∫_0^1 ˜N_{h,s,t+τ}^{1+χ} ds ) ], with χ > 0 the inverse Frisch elasticity of labour supply.
- Resulting modified Phillips curve (from the Appendix):  
  ˆπ_t = λ E^WS_{t−1}(ˆN_t) + E^WS_{t−1}(ˆπ_t) − ˆθ_t, where λ ≡ χ/(1 + ρχ).
- Overheating measure re-defined as:  
  x_t ≡ ˆπ_t + λ ˆN_t.
- Policy-maker effective weight and stance defined as:  
  ˜μ = μ/λ^2, and ˜φ_t ≡ λ φ_t.

### Central bank optimization problem (generalized)
- Central bank chooses ˜φ_t to minimize:  
  E^{RE}_{t−1} [ Σ_{τ=0}^∞ β^τ CB (1 + ˜μ ˜φ_{t+τ}^2) { (1−˜φ_{t+τ})^2 k x_{t+τ−1}^2 + σ^2_θ } ], equation (4.30).
- Law of motion for overheating x_t:  
  x_t = (1−˜φ_t)[(1−˜φ_t) k x_{t−1} − ˆθ_t], equation (4.31).
- Equivalence statement: solving the policy problem with a non-unit slope λ is equivalent to solving the original program with (μ, φ_t) replaced by (˜μ, ˜φ_t), and inferring the true policy stance via ˜φ_t = λ φ_t.

### Key implications
- Qualitative results from the unit-slope case carry over to an arbitrarily sloped Phillips curve.
- Since the literature favors values for λ much less than one:
  - There is a significant widening in the range of values for the “true weight” μ within which the pivoting behaviour described in Proposition 3.1 emerges.
  - For λ < 1, pivots should be associated with wider swings in the true policy stance φ_t = ˜φ_t / λ.

### Baseline parameterization and solution method
- Time period interpretation: model periods identified with years due to wage-setting frequency.
- Discount factor for policy-makers: β_CB = 0.995, aligning with the midpoint of Bank of Canada staff’s current assessed range of 0 to 1% for the real neutral rate in Canada.
- Baseline parameters:
  - μ = 1 (relative weight on employment in central bank’s loss function).
  - λ = 0.092 (Phillips curve slope, broadly in line with Canadian estimates in Djeutem et al. (2022) and Wagner et al. (2022)).
  - k = 2 (levels of thinking in the private sector; mid-range of k = 1 to 4 considered in Farhi and Werning (2019)).
- Supply-shock variance σ^2_θ calibrated to match the variance of CPI inflation in Canada, using a data sample beginning when the country’s current 2% inflation target first formally started applying in the mid-1990s.
- Solution method: value function iteration. Details on the solution algorithm are given in the Appendix.

*Source: wpiea2026097-source-pdf - 4.1    Generalizing the Phillips curve*

### 4.3    Results

### 4.3    Results

### Baseline versus myopic central bank
- Baseline parameterization (blue lines in Figure 5):
  - Optimal policy stance is a function of the past overheating measure x_{t−1}.
  - Pivots remain a key feature, with a threshold value for x_{t−1} around 4% triggering a sudden increase in the policy stance.
  - Pivots are associated with sharp re-normalizations in both rational and LKT-based inflation expectations (middle and right-hand panels of Figure 5).
- Myopic parameterization (β_{CB} = 0; magenta lines in Figure 5):
  - Compared with baseline, myopic central bank leads to a later pivot threshold.
  - Allowing for a realistic degree of forward-looking behaviour by the central bank produces a “pull forward” of the pivot threshold: forward-looking central banks pivot earlier (and thus with smaller re-normalizations in expectations).
  - Interpretation: forward-looking central bank internalizes future benefits from stabilizing inflation today via better-anchored expectations; non-zero β_{CB} strengthens pivoting behaviour.

### Role of the levels-of-thinking parameter k
- Parameter sweep reported in Figure 6:
  - Varying k between one and five while holding other parameters at baseline.
  - Dashed red lines show k = 40 (extreme parameterization); dashed blue reports rational expectations limit k → ∞.
- Key patterns:
  - k ∈ {1,2,3} lead to pivots around thresholds in the 3-5% range.
  - Higher-k cases (large k) exhibit no pivots even when x_{t−1} is as high as 10%.
  - Consistent with Proposition 3.2: policy-makers can make do without pivots when facing a sufficiently sophisticated private sector.
  - Within low single-digit k values typically supported by experimental literature:
    - Higher k associated with smaller jumps in the policy stance around the pivot point (left-hand panel).
    - Higher k associated with less sharp adjustments in expectations of wage-setters (middle panel) and policy-makers (right-hand panel).
    - Reason: policy-makers with higher k have greater leverage over private-sector expectations and thus need not tighten as aggressively to re-stabilize inflation.
- Implications for employment risk:
  - Smaller jumps in policy stance around pivots reduce sensitivity of employment outcomes to supply shocks, mitigating the “risky soft landing” problem when k is high.
  - Figure 7 zooms on k = 1 through 3:
    - Solid lines depict rationally expected inflation and employment before observing supply shock θ̂_t; surrounding bands depict realized outcomes for a range of shock values.
    - Pivots have negligible impacts on expected employment across parameterizations; main impact is increased uncertainty (variance) of employment outcomes, especially when k is small.
    - Higher k improves the trade-off between stabilizing expectations and reducing employment variance and hard-landing risk.

### Role of the effective employment weight ̃μ = μ/λ^2
- Results reported in Figure 8 for a range of ̃μ, holding other parameters at baseline.
- Key patterns:
  - Optimal policy stance is discontinuous when ̃μ is large; takes on a smooth profile for small ̃μ.
  - ̃μ must exceed some threshold for pivoting to occur.
  - Conditional on being above that threshold, higher ̃μ associated with higher pivot points: policymakers place greater weight on employment and thus delay pivots (more willing to look through high inflation).
  - When ̃μ is high, pivots occur later but are associated with larger swings in the policy stance once triggered.

### Costs of deviating from the optimal policy (Section 5)
- Setup:
  - Simulations initialize baseline parameterization at a modest recent overheating level and assume a supply shock large enough to trigger pivoting under the optimal policy.
  - Figure 9 shows time paths for key variables under the optimal policy and two deviations.
- Two deviation scenarios:
  1. “Too late” pivot:
     - Central bank stays on the lowest local optimum after the supply shock (though no longer globally optimal) and only corrects after three periods.
     - Result: significantly worse inflation and employment outcomes relative to the globally optimal policy (middle and bottom panels, Figure 9).
  2. “Too gradual” tightening:
     - Central bank smoothly increases its stance over the three periods following the supply shock, reaching the same peak as the “too late” scenario, then reverts to the globally optimal policy.
     - Result: worse employment outcomes relative to both the optimal policy and the “too late” scenario.
- Interpretation:
  - Gradual approaches are generally sub-optimal in the model.
  - Intermediate policy stances can raise the risk of hard landings; the mapping from policy stance to aggregate outcomes is rich and non-monotonic.

### Demand shocks and interpretation (Section 6)
- Introducing demand shocks via stochastic intercept d_t in the IS curve (line 2.13) yields:
  - Forward-solving: N̂_t = −E^{RE}_t [∑_{τ=0}^{∞} (î_{t+τ} − π̂_{t+τ+1})] + E^{RE}_t [∑_{τ=0}^{∞} d_{t+τ}] (equation 6.32).
  - The appropriate measure of stimulus is the cumulative sum of expected future differences between the real rate and a time-varying natural rate linked to the demand shock.
  - In special cases where the central bank ensures an average difference of zero, this implies full stabilization of the employment gap.
- Policy interpretation:
  - Feedback rule on line 2.14 can be interpreted so the level of stimulus is set as a function of prevailing inflation, with stimulus measured by equation (6.32): φ_t π̂_t = E^{RE}_t [∑_{τ=0}^{∞} (î_{t+τ} − π̂_{t+τ+1})] + E^{RE}_t [∑_{τ=0}^{∞} d_{t+τ}].
  - Core policy problem from Section 3 is unaltered; demand shocks shift nominal rate settings implied by the solution.
  - Policy-setting is a two-stage process when both demand and supply shocks exist:
    1. Gauge rate path needed to insulate employment from demand shocks.
    2. Judge if adjustments are needed to address supply shocks or poor anchoring of expectations.
  - All model results carry over but inform stage 2.
- Caveat:
  - This interpretation assumes policymakers can distinguish demand from supply shocks. Mis-identification (e.g., interpreting demand shocks as supply shocks) can lead to overly stimulative stances; this signal-extraction problem is left for future work.

### Model mechanism and empirical evidence around the recent inflation surge (Section 7)
- Mechanism:
  - Optimal policy: initially look through supply shocks but be ready to pivot sharply if shocks cumulate beyond a threshold.
  - Expected inflation is primary channel: with unchanging policy stance, inflation expectations co-move with a lagged overheating measure; pivots from loose to tight stance should quickly moderate expected inflation even if realized inflation remains high for some time.
  - Wage growth should drift with expected inflation.
- US evidence (Figure 10):
  - Series 2020Q1–2023Q4: two measures of one-year-ahead inflation expectations (University of Michigan; Federal Reserve Bank of Cleveland), wage growth proxied by employment cost index changes, a trimmed measure of inflation, and the federal funds rate.
  - Pattern: expected inflation measures and wage growth trended up until 2022Q2 (first full quarter of the Fed’s tightening cycle); following 2022Q2 expected inflation and wage growth immediately turned around while inflation continued rising for a couple more quarters — consistent with the model’s mechanism.
- Formal relationships (equations reproduced from text):
  - π̂_t = Ŵ_t − θ̂_t (7.33)
  - Ŵ_t = (1 − λ φ_t) E^{WS}_{t−1} (π̂_t) (7.34)
  - E^{WS}_{t−1} (π̂_t) = (1 − λ φ_t)^{k−1} (π̂_{t−1} + λ N̂_{t−1}) (7.35)
  - Rewrite with long-run average φ̄: E^{WS}_{t−1} (π̂_t) = (1 − λ φ̄)^{k−1} (π̂_{t−1} + λ N̂_{t−1}) + u_t (7.36), where u_t ≡ [(1 − λ φ_t)^{k−1} − (1 − λ φ̄)^{k−1}] (π̂_{t−1} + λ N̂_{t−1}).
  - During overheating with φ_t < φ̄ and π̂_{t−1} + λ N̂_{t−1} > 0, residual u_t should be positive and rising; pivoting should revert u_t toward zero or negative.
- Empirical test using expected-inflation regressions:
  - Baseline: Michigan series for inflation expectations; trimmed inflation for realized inflation; vacancy-to-unemployment ratio (V/U) as labour market tightness proxy.
  - Residuals from regressions (empirical analogue of u_t) plotted in Figure 11 for sample periods:
    - 1982Q1-2023Q4
    - 1990Q1-2023Q4
    - 2000Q1-2023Q4
  - Some specifications include non-linear V/U dummies (V/U > 1) or use contemporaneous V/U instead of lagged.
  - Common finding across specifications:
    - Residuals start rising around 2020Q2 and keep rising until 2022Q2.
    - Residuals begin declining back toward zero from 2022Q2 onward (first full quarter of the US tightening cycle).
    - Authors view this as strong evidence supporting the model’s main mechanism.
  - Robustness: adding controls for the New York Fed’s Global Supply Chain Pressure Index (GSCPI) does not change qualitative patterns (results available from authors upon request).
- Broader literature:
  - Complementary findings cited in the text (e.g., Patzelt and Reis (2024); Hilscher, Raviv, and Reis (2025); Di Pace et al. (2025)) indicate renormalization in expectations and non-linearities amplifying impacts of larger hikes—consistent with model predictions that larger pivots have disproportionate impacts on expectations.

### Conclusion (Section 8)
- Main theoretical result:
  - Under level-k thinking, optimal policy is a threshold rule: slight inflation above target → look through supply shocks; cumulative shocks past a threshold → discontinuous pivot to a more hawkish stance to control inflation expectations.
  - The pivot aims to directly decrease inflation expectations and thereby inflation, enabling possible soft landings by avoiding large engineered slack.
- Policy implications:
  - Pivots can be optimal; timing and magnitude matter.
  - In 2022H1 many central banks enacted policy changes interpretable as pivots; these responses align with the model’s prescriptions but whether they were sufficiently aggressive is an open question.
  - Less aggressive or more gradual pivots may have lengthened tightening cycles and reduced the odds of a soft landing relative to an optimally timed and sized pivot.

*Italic: Source: wpiea2026097-source-pdf - 4.3    Results*

### References

### wpiea2026097-source-pdf - References

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### Appendix A — Proofs (key mathematical claims and expressions)
- d(φ_t) = μ(1+k)φ_t^2 − μφ_t + k, a convex quadratic with discriminant μ^2 − 4μk(1 + k).
- n(·) is a cubic satisfying n(0) > 0, n′(0) ≥ 0, n(1) > 0, and n′(1) > 0; its discriminant takes the form μ[108μ^2 + B(k)μ + C(k)], where:
  - B(k) ≡ 36(k − 1)^2 − 216(k + 1)(k − 1) − 108(k + 1)^2,
  - C(k) ≡ −64(k + 1)(k − 1)^3 ≤ 0.
- B(k) is decreasing and satisfies B(1) < 0; thus B(k) < 0.
- The discriminant on line A.37 is strictly negative when μ ∈ (0, μ) and strictly positive with φ_n2 > φ_n1 when μ > μ.
- μ < 4k(1 + k).
- Ordering result when μ ≥ 4k(1 + k): φ_d1 < φ_n1 < φ_d2 < φ_n2, with boundary equality when μ = 4k(1 + k).
- Existence and uniqueness of pivot x_2^∗ established by showing continuity and monotonicity of L_i(x_2t−1) and by identifying x_2^∗ solving L_1(x_2^∗) = L_2(x_2^∗) (equation A.40), with crossing properties implied by envelope theorem.
- μ is the upper root of convex quadratic 108μ^2 + B(k)μ + C(k) (A.41); dμ/dk > 0 since (dμ/dk)[216μ + B(k)] + B′(k)μ + C′(k) = 0 and B′(k) < 0, C′(k) < 0 ∀ k ∈ [1, ∞).
- In the RE limit, lim_{k→∞} {μ} = lim_{k→∞} ((−B(k) + sqrt[B(k)^2 − 432C(k)])/216) = lim_{k→∞} (1 + k)^2 = ∞.
- For k = 1 and μ > μ, explicit solution pair (φ_1, φ_2) = [(μ − sqrt(μ^2 − 4μ)) / (2μ), (μ + sqrt(μ^2 − 4μ)) / (2μ)] is a solution; when μ = 4 for k = 1 the pair is real and satisfies φ_2 = 1 − φ_1, implying expected employment level E^RE_{t−1}(ĤN_t) = −φ_t(1 − φ_t)x_{t−1} is constant around the pivot.

### Appendix B — Demand curve under non-rational expectations
- Weaken full rationality: assume household expectations satisfy law of iterated expectations E^HH(h)_t(·) = E^HH(h)_t[E^HH(h)_T(·)] for each household h ∈ [0,1] and horizon T ∈ {t+1, t+2, ...}; households may be rational on fully exogenous variables.
- Forward-solved linearized household consumption plan:
  - E^HH(h)_t(ĤC_{hT}) = ĤC_{ht} + E^HH(h)_t[Σ_{τ=t}^{T−1} (Ĥi_τ − Ĥπ_{τ+1})].
  - ĤC_{ht} ≡ ln C_{ht} − ln Y_t.
- Linearized expected lifetime budget constraint:
  - B_{h,t−1} / (P_t Y_t) = E^HH(h)_t[Σ_{T=t}^∞ β^{T−t} (ĤC_{hT} − ĤN_T)].
- Aggregate demand (equation B.42):
  - ĤN_t = (1 − β) E^HH_t[Σ_{T=t}^∞ β^{T−t} ĤN_T] − β E^HH_t[Σ_{T=t}^∞ β^{T−t} (Ĥi_T − Ĥπ_{T+1})],
  - where E^HH_t(·) ≡ ∫_0^1 E^HH(h)_t(·) dh.
- Interpretation:
  - Demand depends on households’ expected real policy rate path E^HH_t[Σ_{T=t}^∞ β^{T−t} (Ĥi_T − Ĥπ_{T+1})] and expected permanent income (1 − β) E^HH_t[Σ_{T=t}^∞ β^{T−t} ĤN_T].
  - Recursive solution structure preserved: solve for triplet (Ĥπ_t, ĤN_t, φ_t) using other blocks, then back out policy rate path via equation (B.42).
  - Policy feedback rule on line 2.14 can be interpreted as central bank adjusting policy rate path to regulate demand as function of inflation, accounting for potential permanent income effects.

### Appendix C — Phillips curve under GHH preferences and policy implications
- Preferences (Greenwood, Hercowitz, and Huffman (1988) form) for household h:
  - E^HH(h)_t[Σ_{τ=0}^∞ β^τ ln(C_{h,t+τ} − η θ_{t+τ} ∫_0^1 ̃N_{h,s,t+τ}^{1+χ} ds)],
  - χ > 0 is inverse Frisch elasticity of labour supply.
- Union wage-setting problem maximizes expected real utility subject to labour demand; first-order condition (C.43) expressed with common expectations E^WS_{t−1}.
- Flexible-wage benchmark gives aggregate relationships:
  - W_t = ηρ(1 + χ)/ρ^{−1} θ_t N_t^χ P_t (expression in prose: W_t = ηρ(1 +χ)/ρ−1 θ_t N_t^χ P_t).
  - Mark-up rule: P_t/(θ_t N_t) = γ/(γ − 1) W_t/N_t.
  - Natural rate of employment:
    - N_t = [ρ^{−1} / (ηρ(1 + χ)) * (γ − 1)/γ]^{1/χ} ≡ N̄.
- Linearized Phillips curve under GHH preferences:
  - Ĥπ_t = λ E^WS_{t−1}(ĤN_t) + E^WS_{t−1}(Ĥπ_t) − Ĥθ_t,
  - where λ ≡ χ/(1 + ρχ).
- Under KLT expectations:
  - E^k_{t−1}(Ĥπ_t) = (1 − λ φ_t)^{k−1} (Ĥπ_{t−1} + λ ĤN_{t−1}),
  - E^k_{t−1}(ĤN_t) = −φ_t (1 − λ φ_t)^{k−1} (Ĥπ_{t−1} + λ ĤN_{t−1}).
- Realized outcomes:
  - Ĥπ_t = (1 − λ φ_t)^k (Ĥπ_{t−1} + λ ĤN_{t−1}) − Ĥθ_t,
  - ĤN_t = −φ_t[(1 − λ φ_t)^k (Ĥπ_{t−1} + λ ĤN_{t−1}) − Ĥθ_t].
- Define x_t ≡ Ĥπ_t + λ ĤN_t. Central bank expected lifetime losses:
  - E^RE_{t−1}[Σ_{τ=0}^∞ β^τ CB (Ĥπ_{t+τ}^2 + μ ĤN_{t+τ}^2)]
    = E^RE_{t−1}[Σ_{τ=0}^∞ β^τ CB (1 + μ φ_t^2) ((1 − λ φ_t)^{2k} x_{t+τ−1}^2 + σ_θ^2)].
- Law of motion for overheating measure:
  - x_t = (1 − λ φ_t)[(1 − λ φ_t)^k x_{t−1} − Ĥθ_t].
- Defining ̃μ ≡ μ/λ^2 and ̃φ_t ≡ λ φ_t, the policy problem becomes:
  - Minimize E^RE_{t−1}[Σ_{τ=0}^∞ β^τ CB (1 + ̃μ ̃φ_{t+τ}^2) ((1 − ̃φ_{t+τ})^{2k} x_{t+τ−1}^2 + σ_θ^2)] subject to
  - x_t = (1 − ̃φ_t)[(1 − ̃φ_t)^k x_{t−1} − Ĥθ_t].
- This collapses to the form claimed in main text (equations C.44 and C.45).

### Appendix D — Numerical solution algorithm (value function iteration steps)
- For a given parameter set, convert policy problem to Bellman equation and solve by value function iteration:
  1. Fix a grid X_G of values for state variable x_{t−1}.
  2. Initialize a guess for central bank’s value function V at each x_{t−1} ∈ X_G.
  3. For each x_{t−1} ∈ X_G, use a fine grid search to find implied optimal ̃φ_t:
     - ̃φ^*(x_{t−1}) ∈ arg min_{̃φ_t ∈ ̃Φ} { (1 + ̃μ ̃φ_t^2)[(1 − ̃φ_t)^{2k} x_{t−1}^2 + σ_θ^2] + β_G Σ_{n=1}^N ω_n V((1 − ̃φ_t)^{k+1} x_{t−1} − (1 − ̃φ_t) Ĥθ_n) },
     - where ̃Φ is a fine grid; {ω_n}_{n=1}^N and {Ĥθ_n}_{n=1}^N are weights and nodes of an N-point quadrature approximating Normal(0, σ_θ^2); V(·) denotes linear interpolant from current value-function guess.
  4. For each x_{t−1} ∈ X_G, update the guessed value function using ̃φ^*(x_{t−1}).
  5. (Implied further iterations) Iterate until convergence of the value function (procedure described up to step 4 in source).

*Source: wpiea2026097-source-pdf - References (Appendices A–D).*

### 5.  Repeat steps three and four until successive guesses fall within some small tolerance of one

### 5.  Repeat steps three and four until successive guesses fall within some small tolerance of one another.

### E. Hybrid Phillips curves — setup and main comparative-static findings
- Revisit wage-setting for unions s∈[0,1] (line 2.10 of the main text). Define P∗t ≡ exp(π∗)P t−1 and W∗t ≡ P∗t θ t−1 (γ−1)/γ.
- Linearized individual-union wage gap:
  - ˆWst ≡ lnWst − lnW∗t and ˆWst = EWS(s) t−1 (ˆπ t + ˆN t). (E.46)
- Aggregate Phillips-curve form:
  - ˆπ t = ∫1 0 EWS(s) t−1 (ˆπ t + ˆN t) ds − ˆθ t. (E.47)
- Introduce share α∈(0,1) of adaptive wage-setters; non-adaptive set can be rational or level-k thinkers.
- Case one (non-adaptive are rational):
  - Hybrid Phillips curve: ˆπ t = α x t−1 + (1−α) ERE t−1 (ˆπ t + ˆN t) − ˆθ t. (E.48) with x t−1 ≡ ˆπ t−1 + ˆN t−1.
  - Rational expectations solved to give:
    - ERE t−1 (ˆπ t + ˆN t) = (1−φ t) α / [1 − (1−φ t)(1−α)] x t−1.
    - Final Phillips curve: ˆπ t = [α / {1 − (1−φ t)(1−α)}] x t−1 − ˆθ t ≡ A(φ t) x t−1 − ˆθ t. (E.49)
  - Main result: when central bank is myopic (βCB → 0), no pivots emerge under this hybrid Phillips curve.
  - Lemma E.4 (myopic central bank minimization):
    - Expected loss to minimize: ERE t−1 (ˆπ 2 t + μ ˆN 2 t) = (1 + μ φ2 t)[{A(φ t)}2 x2 t−1 + σ2 θ] ≡ L(φ t, x2 t−1). (E.50)
    - For any x t−1 ∈ R, program solved at unique local minimum φ∗(x2 t−1) ∈ [0,∞), continuous in x2 t−1.
    - Analytical proof uses sign changes of L1(φ t, x2 t−1) and monotonicity of a function f[φ∗(x2 t−1)].

- Case two (non-adaptive are level-k thinkers):
  - Phillips curve: ˆπ t = α x t−1 + (1−α) E k t−1 (x t) − ˆθ t.
  - Iterated expectations define coefficient B(φ t) such that E k t−1 (x t) = B(φ t) x t−1.
  - Central-bank one-period expected loss:
    - ERE t−1 (ˆπ 2 t + μ ˆN 2 t) = (1 + μ φ2 t)[{α + (1−α) B(φ t)}2 x2 t−1 + σ2 θ]. (E.51)
  - Numerical results (Figure 12): pivots emerge when μ is sufficiently large and k is positive but sufficiently small.

### F. Econometric learning — learning-only benchmark and mixing with LKT
- Replace simple adaptive rule (3.16) with gain-based learning:
  - EWS t (ˆz t+1) = EWS t−1 (ˆz t) + g t [ˆz t − EWS t−1 (ˆz t)], ∀z∈{π,N}. (F.52)
  - Two gain specifications: constant gain g t = g ∈(0,1) and decreasing gain g t = g/t.
- Central-bank expected lifetime losses:
  - ERE t−1 ∑τ≥0 βτ CB (ˆπ 2 t+τ + μ ˆN 2 t+τ) = ERE t−1 ∑τ≥0 βτ CB (1 + μ φ2 t+τ)[EWS t+τ−1 (x t+τ)]2 + σ2 θ. (F.53)
- Law of motion for state variable EWS:
  - EWS t (x t+1) = EWS t−1 (x t) − g t [λ φ t EWS t−1 (x t) + (1 − λ φ t) ˆθ t]. (F.54)
- Learning-only numerical results (Figure 13) for gains g = 0.05 and g = 0.15, and decreasing-gain snapshots at t = 1, t = 3, t = 10, t = 50:
  - No pivots occur under any parameterizations considered: optimal φ t increases smoothly with EWS t−1 (x t).
  - Higher gains → tighter optimal policy stance.
  - Under decreasing gain, optimal φ t falls over time approaching zero as t → ∞ and g t → 0.
- Mixing econometric learning with LKT:
  - Level-0 beliefs follow gain-based rule, then used as starting point for k rounds of LKT:
    - EWS t−1 (x t) = (1 − λ φ t) k E0 t−1 (x t).
  - Central-bank expected lifetime losses:
    - ERE t−1 ∑τ≥0 βτ CB (1 + ˜μ ˜φ2 t+τ)[(1 − ˜φ t+τ)2k [E0 t+τ−1 (x t+τ)]2 + σ2 θ], where (˜μ, ˜φ t) ≡ (μ/λ2, λ φ t). (F.55)
  - Law of motion for E0:
    - E0 t (x t+1) = E0 t−1 (x t) − g t [{1 − (1 − ˜φ t) k+1} E0 t−1 (x t) + (1 − ˜φ t) ˆθ t]. (F.56)
  - Numerical results (Figures 14–15):
    - Pivots re-emerge for several parameterizations.
    - Pivots more likely when k is low and ˜μ ≡ μ/λ2 is high.
    - Under decreasing-gain, pivots can emerge over time because early periods are when g t is larger and the central bank’s ability to influence expectations weakens rapidly; pivots can persist even when g t is negligibly close to zero (e.g., t = 1000), reflecting static incentives when k ≥ 1.

### G. Other potential specifications for level-0 beliefs — signal-informed and “fully anchored” variants
- Consider modified hybrid economy with myopic central bank, unit-sloped Phillips curve, adaptive share α∈[0,1), and level-k ≥ 1 thinkers.
- Level-k thinkers receive idiosyncratic signal ζ st = ˆθ t + ν st with ν st ∼ N(0, σ2 ν), independent across s and t.
- Level-0 conditional guess:
  - E0 t−1 (x t | ζ st) = A0 x t−1 − B0 ζ st, with (A0, B0) ∈ R2 and γ ≡ σ2 ν/(σ2 θ + σ2 ν).
- For a level-˜k thinker conjecture E˜k t−1 (x t | ζ st) = A˜k x t−1 − B˜k ζ st. (G.57)
- Wage-setting condition for level-˜k:
  - ˆW˜k st = E˜k t−1 (x t + ˆθ t | ζ st) = A˜k x t−1 + (γ − B˜k) ζ st. (G.58–G.60)
- Implied Phillips curve for economy with all non-adaptive at level-˜k:
  - ˆπ˜k t = [α + (1−α) A˜k] x t−1 − [1 − (1−α)(γ − B˜k)] ˆθ t.
- Recursions for beliefs coefficients (for ˜k ∈ {1,...,k}):
  - A˜k = (1 − φ t)[α + (1 − α) A˜k−1]. (G.61)
  - B˜k = γ (1 − φ t)[1 − (1 − α)(γ − B˜k−1)]. (G.62)
- Central-bank expected-period losses:
  - ERE t−1 (ˆπ 2 t + μ ˆN 2 t) = (1 + μ φ2 t)[{α + (1 − α) A k}2 x2 t−1 + {1 − (1 − α)(γ − B k)}2 σ2 θ]. (G.63)
- Case explorations:
  - Case A0 ∈ [0,1], B0 = γ = 0 (level-0 fully anchored variant):
    - Numerical results (Figure 16): pivots occur at high A0 but disappear as A0 → 0; when A0 = 0 and α ≥ 0, pivots can re-emerge at intermediate α, disappear as α → 1.
  - Case γ ∈ [0,1], A0 ∈ (0,1], B0 = 0 (signals informative but level-0 ignores general equilibrium):
    - Numerical results (Figure 17):
      - As γ increases from 0 to 1 with α = 0, pivots remain but threshold falls; when γ = 1 and (α,γ,B0) = (0,1,0):
        - ERE t−1 (ˆπ 2 t + μ ˆN 2 t) = (1 + μ φ2 t)(1 − φ t)2k (A0 x t−1)2, implying unique minimum at φ t = 1 for x t−1 ≠ 0.
      - Introducing adaptive types (α > 0) with γ = 1 can lead to pivots re-emerging at intermediate α.

### H. Alternative models of bounded rationality — reflective expectations and cognitive hierarchies
- H.1 Reflective expectations (continuous analogue of LKT; Garcia‑Schmidt and Woodford (2019) framework):
  - Continuous updating over ˜k ∈ [0,k]: dx g (˜k)/d ˜k = ˆz o (˜k) − ˆz g (˜k).
  - Solution for combined overheating expectation:
    - EWS t−1 (x t) = exp(−k φ t) x t−1. (H.65)
  - Inflation outcome: ˆπ t = exp(−k φ t) x t−1 − ˆθ t.
  - Central-bank expected loss to minimize (myopic CB):
    - ERE t−1 (ˆπ 2 t + μ ˆN 2 t) = (1 + μ φ2 t)[exp(−2k φ t) x2 t−1 + σ2 θ]. (H.66)
  - Numerical results (Figure 18): pivots emerge when μ is sufficiently large and k is sufficiently small.

- H.2 Cognitive hierarchy (heterogeneous sophistication levels; Camerer et al. (2004)):
  - True shares f˜k ≥ 0 for levels ˜k ∈ {0,...,k}; subjective beliefs g˜k(·) consistent with distribution.
  - Normalized wage variables and counterfactual aggregates defined (H.67).
  - Recursion linking counterfactual wages (Lemma H.5, up to higher-order terms):
    - ln w˜k t = [1 − φ t f˜k / ∑˜k f˜k] ln w˜k−1 t, with w0 t = exp(x t−1).
  - True-economy inflation: ˆπ t = ln{w(φ t, x t−1)} − ln(θ t / θ t−1).
  - Central-bank expected loss (myopic):
    - ERE t−1 (ˆπ 2 t + μ ˆN 2 t) = (1 + μ φ2 t)[{ln w(φ t, x t−1)}2 + σ2 θ]. (H.72)
  - Numerical results (Figures 19–20):
    - Pivots occur when μ is large and population sophistication is sufficiently low.
    - The (unsigned) elasticity of ln[w(φ t, x t−1)] with respect to φ t increases in φ t, implying private expectations are more policy-sensitive when φ t is high.

### I. Oil price shocks as productivity shocks — equivalence in small oil-importing economy
- Gross output specification:
  - Q t = F(Z t N t, O t). (I.73) F(·,·) strictly concave, CRS in (Z t N t, O t).
- Value-added per worker and first-order oil demand:
  - Y t / N t = F(Z t, O t / N t) − P O t O t / N t.
  - First-order condition: F2(Z t, O t / N t) = P O t.
  - Because of CRS, define f(·) ≡ F(1,·) and get downward-sloping oil demand:
    - O t / N t = Z t (f′)−1 (P O t). (I.74)
- Plug back to get value-added per worker:
  - Y t = N t Z t h(P O t), where h(P O t) ≡ f[(f′)−1 (P O t)] − P O t (f′)−1 (P O t), and h′(P O t) = −(f′)−1 (P O t) < 0.
- Define labour productivity θ t ≡ Z t h(P O t), giving production Y t = θ t N t. (I.75)
- Conclusion: oil-price shocks map into productivity shocks in a small, oil-importing economy via θ t = Z t h(P O t).

*Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026097-source-pdf.pdf*

### References cited in the Appendix

### References cited in the Appendix

### Cited works
- Camerer,  C.  F.,  T.  Ho,  and  J.  Chong  (2004,  August).   A  cognitive  hierarchy  model  of  games. Quarterly Journal of Economics  119(3), 861–898.
- Farhi, E. and I. Werning (2019, November). Monetary policy, bounded rationality, and incomplete markets.American Economic Review  109(11), 3887–3928.
- Garc ́ıa-Schmidt,  M.  and  M.  Woodford  (2019,  January).   Are  low  interest  rates  deflationary?   A paradox of perfect-foresight analysis.American Economic Review  109(1), 86–120.
- Greenwood, J., Z. Hercowitz, and G. W. Huffman (1988, June).  Investment, capacity utilization, and the real business cycle.American Economic Review  78(3), 402–417.

### Document details
- 77
- The Central Bank’s Dilemma: Look Through Supply Shocks or Control Inflation Expectations 
- Working Paper No. WP/2026/097

*References cited in the Appendix — wpiea2026097-source-pdf*

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_Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026097-source-pdf.pdf_