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---

### Major themes and model features
- Embeds a nonlinear Phillips Curve into a DSGE framework that arises from a quasi-kinked demand schedule; the Phillips curve is flat when inflationary pressures are subdued and steepens as inflationary pressures rise.
- Three key extensions to the HLT framework:
  - Learning about cost-push shocks: agents observe a_t but not its transitory and persistent components and solve a signal-extraction problem; initial misperception of a persistent shock as transitory helps explain large early forecast errors during the post-COVID inflation surge.
  - Endogenous indexation: the fraction of prices and wages set in a backward-looking manner is higher when inflation runs persistently and significantly above target; the stock of past inflation forecast errors affects intrinsic persistence in the Phillips Curve.
  - Policy reaction functions: comparison of a forecast-based targeting rule (responding to inflation forecasts one or two years ahead) with a standard contemporaneous-inflation instrument rule; average inflation targeting (AIT) can share many features with forecast targeting if not paired with escape clauses.

### Empirical motivations and data observations
- Post-COVID facts motivating the model:
  - Central banks, forecasters, and academics were surprised by the post-COVID inflation surge despite large supply shocks that were widely viewed as transient.
  - Evidence suggested second-round effects would be small during the Great Moderation; the recent surge saw larger second-round effects.
  - Central banks largely “looked through” the initial supply-driven inflation surge and maintained accommodative policy for about 1 to 1.5 years after inflation rose above 2 percent in the U.S., Euro Area, and U.K.; BoE began raising rates in late 2021 and into 2022, followed by the Fed (Fed began tightening in March 2022).
- Professional forecasters (SPF) underestimated:
  - both the size and persistence of the U.S. inflation increase beginning in early 2021;
  - the initial resilience of economic activity;
  - how much the Fed would need to raise policy rates (proxied by forecasts for 3-month T-bills).

### Mechanisms producing an "inflation cycle" and activity dynamics
- Interaction of learning, endogenous indexation, and forecast-based policy produces an inflation cycle:
  - If a shock is initially perceived as transitory and intrinsic persistence is viewed as low, inflation is expected to revert and a forecast-based rule leads the central bank to keep policy rates nearly unchanged.
  - If shocks prove more persistent, the central bank hikes rates materially, but endogenous indexation and the nonlinear Phillips Curve amplify inflation (“the inflation ghost is out of the bottle”).
  - Because nominal policy rates are set based on medium-term forecasts and rise gradually, short-term real interest rates can fall initially, producing an initial expansion in activity followed by a later contraction as real rates rise.
- This channel can account for observed resilience in activity during the initial inflation surge even without material demand shocks.

### Quantitative model structure and calibrations
- Core modeling choices:
  - Nonlinear variant of Erceg, Henderson and Levin (2000) with sticky wages and prices; excludes endogenous capital accumulation.
  - Kimball quasi-kinked demand in goods and labor aggregators (parameters κ_p ≥ 0, κ_w ≥ 0).
  - State-dependent indexation: Ṽ_t = Ṽ_{w,t} = Π̄^{1 - ι_t} (Π_{t-1})^{ι_t} with ι_t = e^{-α max(Π̃_t - Π̄; 0.0001)} - e^{-α·0.0001} and Π̃_t = (Π̃_{t-1})^θ (Π_{t-1})^{1-θ}.
  - Learning: unobserved-components representation a_t = a_{P,t} + a_{T,t} with Kalman filtering; agents assumed to have infinite past data so recursions converge to fixed point P.
- Selected calibration values (as presented):
  - Π̄ 1:005 Steady state gross inflation rate
  - κ_p 0:1 Net price markup in steady state
  - ξ_p 2=3 Calvo price stickiness parameter
  - κ_p -12 Parameter Kimball aggregator prices
  - % 0:002 Curvature parameter endogenous indexation (α = 0.002 in text)
  - θ 0:8 Parameter in endogenous. indexation
  - { 0 Inflation indexation parameter in linear model
  - κ_w 0:1 Net wage markup in steady state
  - ξ_w 0:75 Calvo wage stickiness parameter
  - κ_w -6 Parameter Kimball aggregator wages
  - ρ_R 0:85 Taylor rule: interest rate smoothing
  - ϕ_E 1:5 Taylor rule: coef. on expected inflation
  - x 0:125 Taylor rule: coef. on output gap
  - β 0:995 Household discount factor
  - h 0:7 Household consumption habit
  - φ 0 Inverse Frisch elasticity of labor supply
  - ζ_P 0:9 AR(1) persistent markup shock
  - ζ_T 0 AR(1) transitory markup shock
  - σ_εP 1 Standard deviation persistent markup shock
  - σ_εT 10 Standard deviation transitory markup shock
  - ρ_ϵ 0:9 AR(1) discount factor shock
- Calibration notes:
  - Unobserved-components calibration set so a_t matches Smets and Wouters (2007) ARMA(1,1).
  - Endogenous indexation implies no dynamic indexation in steady state ({ = 0), but nonlinear model generates indexation once inflation surges.

### Key model results and comparisons with linearized variants
- Nonlinear vs linearized model behavior:
  - Linear model approximates well for small shocks but fails quantitatively and qualitatively for large shocks or when inflation is already high.
  - Nonlinear model reproduces U.S. 2021–2023 contours and SPF forecast revisions when fed with large adverse supply shocks and moderate demand shocks; linear model does not.
- Baseline impulse-response experiment (cost-push shock driven by persistent component):
  - Uses "P0 = 0:0025 and "T0 = 0 so realized cost-push shock is driven by the persistent component.
  - Cost-push shock produces a hump-shaped inflation boom and a gradual rise in the policy rate; output increases in the short run.
- Mechanisms for short-run expansion despite adverse cost-push shock:
  1. Agent misperception that shocks are transitory.
  2. Forecast-based policy inertia (central bank responds to 4-quarter ahead inflation with gradualism).
  3. Nonlinear rise in intrinsic Phillips Curve persistence via endogenous indexation.
- Shock-size dependence:
  - For the baseline (larger) cost-push shock, nonlinear vs linear differences are very large.
  - For smaller shocks (3/4, 1/2, 1/4 sizes), differences diminish; linear variants capture transmission well for small shocks.

### Monetary policy experiments and welfare-type comparisons
- Policy-rule variations:
  - Contemporaneous inflation rule (respond to π_t).
  - One-year ahead expected inflation (E_t π_{t+4}) — baseline.
  - Two-year ahead expected inflation (E_t π_{t+8}).
- Representative quantitative comparison (ad hoc loss example):
  - Expected loss measured as sum of squared deviations of annualized inflation from target plus sum of squared output gaps for first 5 years falls by about 30 percent from a loss of about 255 under the forecast-based rule to a loss of about 174 under a rule responding to actual inflation.
- Policy-response trade-offs:
  - Reacting to realized inflation raises real rates more rapidly; reduces size and persistence of inflation runup but with sizeable output costs.
  - Reacting to two-year ahead expected inflation can allow real rates to decline initially, inducing larger inflation and output runups.
  - Interest rate smoothing (ρ_R = 0:85) moderates differences between rules; lower smoothing would amplify differences.
- Misperception sensitivity:
  - Faster filtering (higher σ_P / σ_T) leads central bank to recognize persistence earlier, tighten faster, and reduce inflation and output expansion.
  - Misperception problems are much less acute when the underlying shock is actually transitory.

### State-dependent amplification and numerical examples
- Table 2 example (amplification across initial states):
  - A cost-push shock that raises inflation by 4.7 percent when inflation is initially at target would raise inflation by 9 percent if the shock occurs against a backdrop where inflation was initially 3.2 percent and the output gap is slightly over 5 percent.
- Monetary tightening experiment (reduce inflation by one percentage point APR):
  - Monetary policy shock follows AR(1) with persistence coefficient 0:75.
  - To attain a one percentage point (APR) lower inflation trajectory, the nonlinear model requires notably larger nominal policy rate tightening than the linearized model, producing a considerably lower output path.
  - With a nonlinear (kinked) Phillips curve, bringing inflation back faster requires much stronger tightening than a linear Phillips curve implies, producing substantially larger output costs.
- Timing of intervention (Appendix B.5):
  - "The earlier the central bank intervenes, the larger the reduction in inflation for a given hike in the policy rate. Put differently, monetary policy becomes less effective the higher inflation is to begin with."

### Stochastic simulations and distributional outcomes
- Simulation design:
  - Starting at baseline at t= 8, economy hit by random unexpected cost-push and discount factor shocks each period t ≥ 8.
  - Variances chosen to match unconditional SD of core PCE inflation, SD of real consumption per capita growth, and correlation between consumption growth and inflation in post-war/pre-Covid U.S. data.
  - Density plots use 500 random sequences and show the {2.5;10;20;...;90;97.5} percentiles and the median.
- Nonlinear model — standard monetary policy:
  - Inflation density is asymmetric with more realizations of high inflation than low inflation; amplification from Kimball aggregation and endogenous indexation creates upward skew.
- Nonlinear model — more aggressive monetary policy (increase weight on inflation by factor of three and reduce output-gap weight by factor of three from t=8):
  - Inflation is lower on average and upward asymmetry is nearly eliminated.
  - Trade-off: improved inflation stabilization comes with substantially lower mean output and evident downside output risks.

### Identification and empirical implications
- Forecast-based rules with agent misperceptions can produce persistent positive comovement of inflation and output in response to a supply (cost-push) shock, a pattern commonly attributed to demand shocks.
- This challenges identification assumptions (e.g., sign restrictions in SVARs) that cost-push shocks necessarily push inflation and output in opposite directions.

### Policy implications and practical takeaways
- "Looking through" supply shocks:
  - Appropriate for small shocks when inflation is near target.
  - Risky when activity is strong and large adverse shocks push inflation well above target due to amplification and increased persistence.
- Weight on point inflation forecasts:
  - Placing high weight on point inflation forecasts in forecast-based rules can be dangerous when shocks are large and intrinsic persistence is high.
  - Rules that place some weight on realized inflation can yield better inflation outcomes and may be welfare-preferred despite larger output declines.
- Costs of aggressive disinflation:
  - The nonlinear framework implies economic costs of “going the last mile” (rapid return to target) can be sizeable; the steeper Phillips Curve in high-inflation states does not make rapid disinflation easier.
- Timing and state dependence:
  - Earlier intervention is more effective; delaying tightening when inflation and expectations are elevated increases the sacrifice ratio and reduces policy efficacy.

### Robustness, extensions, and suggested future research
- Model robustness and suggested extensions:
  - Consider firms choosing desired rate of indexation (intensive margin) or a state-dependent fraction of non-reoptimizers fully indexing to past inflation (extensive margin).
  - Explore separate indexation rules for prices and wages and discipline wage-rule parameterization with prevalence of cost-of-living-adjustment clauses in high-inflation episodes.
  - Pseudo-linearized variants where indexation is nonlinear while other equations are linearized show the central role of endogenous indexation in amplifying dynamics.

- Representative reported numeric/statistical lines in source:
  - α = 0.002; Π̄ = 1.005; ρ_R = 0:85; ϕ_E = 1:5; x = 0:125; β = 0:995; h = 0:7; φ = 0; ξ_w = 0:75; ζ_P = 0:9; σ_εP = 1; σ_εT = 10; ρ_ϵ = 0:9.
  - Example peak-inflation numbers: 4.7 percent (from target baseline) vs 9 percent (when initial inflation = 3.2 percent and output gap ≈ 5 percent).
  - Ad hoc loss example: loss ≈ 255 under forecast-based rule vs ≈ 174 under realized-inflation rule (first 5 years, sum of squared deviations).

*Source: wpiea2024260-print-pdf (Working Paper No. WP/2024/260), Sections 3.1, 3, 4, Appendix A and B.*

### 3.1  Households  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   8

### 3.1  Households

### Major themes and model features
- The paper embeds a nonlinear Phillips Curve into a DSGE framework that arises from a quasi-kinked demand schedule; the Phillips curve is flat when inflationary pressures are subdued and steepens as inflationary pressures rise.
- Three key extensions to the HLT framework:
  - Learning about cost-push shocks: agents do not observe separately transitory and persistent components and must solve a signal-extraction problem; initial misperception of a persistent shock as transitory helps explain large early forecast errors during the post-COVID inflation surge.
  - Endogenous indexation: the fraction of prices and wages set in a backward-looking manner is higher when inflation runs persistently and significantly above target; the stock of past inflation forecast errors affects intrinsic persistence in the Phillips Curve.
  - Policy reaction functions: comparison of a forecast-based targeting rule (responding to inflation forecasts one or two years ahead) with a standard contemporaneous-inflation instrument rule; average inflation targeting (AIT) can share many features with forecast targeting if not paired with escape clauses.

### Key empirical observations and model motivations
- Post-COVID inflation surge surprised central banks, forecasters, and academics despite large supply shocks; these shocks were widely viewed as transient.
- Evidence and conventional wisdom suggested second-round effects would be small due to well-anchored expectations and structural features (e.g., relatively flexible labor markets).
- Central banks largely “looked through” the initial supply-driven inflation surge and maintained accommodative policy until perceptions of broader inflationary pressures changed, prompting rapid tightening.
- The model aims to explain:
  - Why forecasters predicted rapid dissipation of high inflation.
  - Why second-round effects were small during the Great Moderation but larger in the recent surge.
  - How large shocks and state-dependent transmission interact with policy rules.

### Mechanisms producing an inflation cycle and activity dynamics
- Interaction of learning, endogenous indexation, and forecast-based policy can produce an “inflation cycle”:
  - If the shock is initially perceived as transitory and intrinsic persistence is viewed as low, inflation is expected to revert and the central bank keeps policy rates nearly unchanged under a forecast-based rule.
  - If shocks prove more persistent, the central bank hikes rates materially, but by then endogenous indexation and the nonlinear Phillips Curve have amplified inflation (“the inflation ghost is out of the bottle”).
  - Because nominal policy rates are set based on medium-term forecasts and rise gradually, short-term real interest rates can fall initially (near-term inflation rises more), producing an initial expansion in activity followed by a later contraction as real rates rise.
- This channel can account for observed resilience in economic activity during the initial inflation surge even without material demand shocks.

### Comparisons with linearized models and empirical performance
- The nonlinear model contrasts sharply with an otherwise-identical linearized model:
  - The linear model approximates well for small shocks but fails quantitatively and qualitatively for large shocks or when inflation is already high.
  - The nonlinear model, when fed with large adverse supply shocks and moderate demand shocks, reproduces key contours of U.S. data in 2021-2023 and forecast revisions from professional forecasters; the linear model does not.

### Policy implications and recommendations
- Caution about “looking through” supply shocks:
  - For small adverse cost-push shocks that do not push inflation far from target, looking through is often appropriate.
  - For large shocks, especially when inflation is already above target, looking through can be costly due to amplification and increased persistence.
- Caution about placing high weight on point inflation forecasts in forecast-based rules when shocks are large and intrinsic persistence is high:
  - Misperceiving a persistent shock as transitory can produce very large and costly inflation runups.
  - Rules that place some weight on realized inflation (i.e., not solely on point forecasts) can yield better inflation outcomes and may be preferable on welfare grounds despite larger output declines.
- Trade-off for aggressive disinflation:
  - When inflation has peaked and is receding gradually, the central bank faces a more severe inflation-output stabilization trade-off if it tries to push inflation back to target quickly.
  - The steeper Phillips Curve in high-inflation states does not imply it is easy to “ride down” inflation because higher inflation can amplify persistence; the economic costs of “going the last mile” to bring inflation quickly back to target can be sizeable.

### Data evidence summarized
- Cross-country monthly facts (U.S., Euro Area, U.K.):
  - Inflation surged in all three major economies, with U.S. inflation leading the Euro Area and U.K. by roughly half a year.
  - Central banks in these economies kept policy rates unchanged for about 1 to 1.5 years after inflation rose above their 2 percent targets; the BoE began raising rates in late 2021 and into 2022, followed by the Fed and ECB (Fed began tightening in March 2022).
- U.S. quarterly facts (PCE inflation, PCE growth, 3-month T-bill rate, SPF projections):
  - Professional forecasters underestimated both the size and persistence of the U.S. inflation increase beginning in early 2021.
  - Professional forecasters underestimated the initial resilience of economic activity.
  - Professional forecasters underestimated how much the Fed would need to raise policy rates (proxied by the forecast for 3-month T-bills) to cool the economy and bring inflation back to target.

### Model provenance and calibration notes
- The model is a nonlinear variant of the Erceg, Henderson and Levin (2000, EHL) model with sticky wages and prices; it excludes endogenous capital accumulation but shares most features with Christiano, Eichenbaum and Evans (2005) and Smets and Wouters (2007).
- The parameterization follows HLT (2023), allowing a more prominent role for Kimball (1995) quasi-kinked demand in goods markets; this feature increases marginal data density when the average markup aligns with micro- and macroeconomic evidence.

*Source: wpiea2024260-print-pdf, Section headings and text excerpts (Introduction and Data).*

### 3. Quantitative Model

### 3. Quantitative Model

### Households
- Continuum of households j ∈ [0,1]; each supplies specialized labor type n_{j,t}.
- Objective: maximize
  - E_0 Σ_{t=0}^∞ β^t χ_t [ ln(c_t - h C_{t-1}) - 1/(1+φ) n_{j,t}^{1+φ} ]
  - Parameters: 0 ≤ β < 1; 0 ≤ h < 1; habit persistence h; inverse Frisch elasticity parameter φ.
- Budget constraint: P_t c_t + B_t = W_{j,t} n_{j,t} + R_{t-1} B_{t-1} - T_t + π_t + a_{j,t}
  - B_t: risk-free bonds (zero net supply)
  - P_t: aggregate price level
  - W_{j,t}: wage of household j
  - R_t: gross nominal interest rate on bonds purchased in t-1
  - T_t: lump-sum taxes net of transfers
  - π_t: share of profits received by household
  - a_{j,t}: insurance payments/receipts ensuring equal consumption across households
- Exogenous shock to discount factor: χ_t with ϵ_t = χ_{t+1}/χ_t following AR(1): ϵ_t - ϵ̄ = ρ_ϵ (ϵ_{t-1} - ϵ̄) + ε_{ϵ,t}

### Labor Contractors and Wage Setting
- Labor contractors aggregate specialized labor n_{t,j} into homogeneous labor n_t using a Kimball aggregator:
  - G_w(n_{t,j}/n_t) specification with Kimball parameter κ_w ≥ 0; markups ζ_w and markup parameters ϵ_w = 1 + κ_w.
  - Special cases: κ_w = 0 → Dixit-Stiglitz.
- Demand for labor (equation (2)):
  - n_{t,j}/n_t = [ 1/(1+κ_w) ] × [ (W_{t,j}/W_t)^{...} ... ] (see model for full functional form)
- Aggregate wage index (equation (3)) and zero-profit condition for contractors (equation (4)) derive from the aggregator.
- Wage-setting frictions: Calvo-style with probability 1 - ξ_w to re-optimize; otherwise indexation rule:
  - W_{j,t} = Ṽ_{w,t} W_{j,t-1}
  - Ṽ_{w,t} is an indexation factor (determined in Section 3.5)
- For tractability the model sets φ = 0 in wage optimality derivations.

### Final Goods Producers and Pricing
- Final goods producers use a Kimball aggregator G_π(y_{t,i}/y_t) for varieties; Kimball parameter κ_p ≥ 0; markups ζ_p and ϵ_p = 1 + κ_p.
- Optimality yields:
  - Relative demand y_{t,i}/y_t as function of P_{t,i}/P_t and aggregate multiplier #
  - Price index and zero-profit conditions analogous to the labor side
- Calvo pricing: firms re-optimize with probability ξ_p and otherwise follow indexation P_{i,t} = Ṽ_t P_{i,t-1} with Ṽ_t specified in Section 3.5.

### Intermediate Goods Producers and Cost-Push Shock
- Production: y_{t,i} = n_{t,i}
- Total costs: TC_{t,i} = ϕ_{1/λ_t} W_t n_{t,i} (equation (6))
  - ϕ_{1/λ_t} is an exogenous shifter of total costs (cost-push shock); denoted a shock to ϕ_t (also referred to as a markup shock).
  - The shock is scaled so that, after log-linearization, it enters Phillips curves additively with unit coefficient through λ_p = 1/(1+β{...}) (full expression in text).
- Firms minimize costs / set prices taking marginal cost MC_{t,i} into account; MC_{t,i} = ϕ_{1/λ_t} W_t.

### Endogenous Price and Wage Indexation (state-dependent)
- Define gross inflation rate Π_t = P_t / P_{t-1}.
- Indexation for non-optimizing firms and unions:
  - Ṽ_t = Ṽ_{w,t} = Π̄^{1 - ι_t} (Π_{t-1})^{ι_t}  (equation (7))
  - ι_t = e^{-α max(Π̃_t - Π̄; 0.0001)} - e^{-α·0.0001}  (equation (8))
  - Π̃_t = (Π̃_{t-1})^θ (Π_{t-1})^{1-θ}  (equation (9))
  - Parameters: 0 ≤ θ < 1; α ≥ 0; Π̄ denotes steady state inflation.
- Interpretation:
  - Indexation depends on a geometric lag of past inflation; called endogenous indexation because it depends on endogenous aggregate inflation.
  - In the nonlinear model indexation is state-dependent and can activate when inflation runs well above target; in the log-linearized model the state-dependence disappears around steady state (equation (10): ĉ_Ṽ_t = ι_t · ĥΠ_{t-1} and ι = 0 in linear approximation).
- Numerical illustration parameters given in text: set α = 0.002 and Π̄ = 1.005; the max operator uses 0.0001 for numerical stability.

### Aggregate Resources and Market Clearing
- Aggregate resource constraint: c_t = y_t = (p̃_t)^{-1} (w̃_t)^{-1} l_t
  - p̃_t and w̃_t are measures of price and wage dispersion (expressions in appendix).
- Insurance and bond markets: ∫ a_{j,t} dj = 0; B_t = 0.

### Monetary Policy
- Notional interest rate rule (equation (11)):
  - R_{not,t} / R̄ = ρ_R (R_{not,t-1} / R̄) + (1-ρ_R) [ ϕ_E E_t Π_{t+4} / Π̄ + (1-ϕ_E) (Y_t / Ȳ^{pot}_t)^{χ} ] + x e_{R,t}
  - Policy shock e_{R,t} is i.i.d. zero mean with positive variance.
  - Actual nominal rate: R_t = max(0, R_{not,t}) (zero lower bound).
- Fiscal: net lump-sum taxes adjust to balance budget; Ricardian equivalence assumed (government budget not explicitly modeled).

### Learning about the Cost-Push Shock (signal-extraction)
- Cost-push deviation: a_t ≡ ϕ_t - 1. Decomposed into transitory a_{T,t} and persistent a_{P,t} components; agents observe a_t but not components.
- State-space form (H, F, Q matrices) with
  - a_t = H [ a_{P,t}; a_{T,t} ]
  - [a_{P,t+1}; a_{T,t+1}] = F [a_{P,t}; a_{T,t}] + Q [ε_{P,t+1}; ε_{T,t+1}]
- Kalman filter recursions provided: state update and forecast with gain matrices L_t and K_t; agents assumed to have infinite past data so recursions converge to fixed point P.
- Forecast propagation: ^a_{t+j|t} = H' F^j [^a_{P,t|t}; ^a_{T,t|t}]'.

### Equilibrium, Solution, and Parameters
- Appendix contains full nonlinear and linear equilibrium equations, steady state, and solution details.
- Calibration choices:
  - Parameters largely from HLT (2022, 2023).
  - Cost-push unobserved components calibrated to match Smets and Wouters (2007) ARMA(1,1) for a_t, implying:
    - ζ_P = 0.9
    - ζ_T = 0
    - σ_εP = σ_εT = 1/10
  - Indexation parameters chosen so nonlinear model captures hump-shaped inflation dynamics.
- Table 1: Model parameter values (as used in analysis)
  - Π̄ 1:005 Steady state gross inflation rate
  - κ_p 0:1 Net price markup in steady state
  - ξ_p 2=3 Calvo price stickiness parameter
  - κ_p -12 Parameter Kimball aggregator prices
  - % 0:002 Curvature parameter endogenous indexation
  - θ 0:8 Parameter in endogenous. indexation
  - { 0 Inflation indexation parameter in linear model
  - κ_w 0:1 Net wage markup in steady state
  - ξ_w 0:75 Calvo wage stickiness parameter
  - κ_w -6 Parameter Kimball aggregator wages
  - ρ_R 0:85 Taylor rule: interest rate smoothing
  - ϕ_E 1:5 Taylor rule: coef. on expected inflation
  - x 0:125 Taylor rule: coef. on output gap
  - β 0:995 Household discount factor
  - h 0:7 Household consumption habit
  - φ 0 Inverse Frisch elasticity of labor supply
  - ζ_P 0:9 AR(1) persistent markup shock
  - ζ_T 0 AR(1) transitory markup shock
  - σ_εP 1 Standard deviation persistent markup shock
  - σ_εT 10 Standard deviation transitory markup shock
  - ρ_ϵ 0:9 AR(1) discount factor shock
- Notes on calibration:
  - The unobserved-components calibration (ζ_P, ζ_T, σ_εP, σ_εT) is chosen so a_t matches Smets and Wouters (2007) ARMA(1,1).
  - Endogenous indexation implies no dynamic indexation in steady state ({ = 0) in both nonlinear and linear models, but nonlinear model generates endogenous indexation once inflation surges.

*Source: 3. Quantitative Model — wpiea2024260-print-pdf*

### 4. Results

### 4. Results

### U.S. Data-Model Comparison
- The nonlinear model is calibrated and simulated so that model agents’ expected liftoff of the policy rate matches SPF expectations proxied by the 3 month T-bill rate; model variables are expressed relative to their steady states except for inflation and the policy rate.
- The model captures key features of the post-COVID inflation surge, including:
  - progressive ratcheting up of inflation through mid-2022; and
  - “optimistic” SPF forecasts projecting fairly quick receding of inflation.
- Two important implications highlighted:
  - Endogenous propagation of underlying demand shocks is very strong because the shocks required to match the observed runup in inflation over 2021-23 are large enough that the model nonlinearities “kick in.” The same structural shocks in a linearized model would generate only a small and relatively transient rise in inflation.
  - Large but temporary adverse cost-push shocks can by themselves account for many salient data features and can produce expansionary short-run output effects while triggering an inflation cycle.

### How Adverse Cost-push Shocks can be Expansionary
- Baseline impulse-response experiment:
  - Assume "P0 = 0:0025 and "T0 = 0 so realized cost-push shock is driven by the persistent component.
  - Central bank follows the inflation-forecast-based Taylor rule (eq. (11)) and must sequentially filter persistent vs transitory components in periods t = 0,1,2,...
- Key empirical model responses:
  - Cost-push shock produces a hump-shaped inflation boom and a gradual rise in the policy rate.
  - Output increases in the short run (contrary to the conventional expectation that a markup shock causes output contraction).
- Three mechanisms accounting for short-run output expansion:
  1. Agents misperceive shocks: calibrated belief that cost-push shocks are usually transitory leads private agents and the central bank to forecast quick receding of inflation and output returning to steady state.
  2. Forecast-based policy inertia: the central bank’s forecast rule responds to 4-quarter ahead inflation with gradualism, so nominal interest rates rise very little initially; higher short-term expected inflation can dominate higher real rates at longer horizons, supporting output.
  3. Nonlinear rise in intrinsic Phillips Curve persistence: large and ongoing misses of the inflation target raise intrinsic persistence (indexation), amplifying subsequent markup-shock effects on inflation and inflation expectations; this drives a fall in real rates and sizeable short-term output expansion.
- Comparison with linear model:
  - Linearized model implies a far smaller, front-loaded inflation response and modest output effects; modest gradual policy tightening in the linear model keeps output near potential and brings inflation monotonically back to target.
  - Nonlinear features (Kimball aggregator and endogenous indexation) substantially amplify and hump-shape the inflation response.

### Role of Endogenous Indexation and Shock Size
- Endogenous indexation (indexation parameter depends on geometric average of inflation deviations from target) is critical:
  - When inflation is low, indexation is near zero; once inflation rises well above target and stays high, indexation starts to increase.
  - Combination of Kimball aggregator nonlinearity and endogenous indexation produces large hump-shaped inflation dynamics.
- Shock-size dependence:
  - For the baseline (larger) cost-push shock, nonlinear vs linear differences are very large.
  - For smaller shocks (3/4, 1/2, 1/4 shock sizes), differences between nonlinear and linearized models diminish; when cost-push shocks are small, linearized variants capture transmission well.

### Monetary Policy Rule
- Policy-rule experiments compare central bank reacting to:
  - contemporaneous inflation (πt = π),
  - one-year ahead expected inflation (E_t π_{t+4} = π) — baseline,
  - two-year ahead expected inflation (E_t π_{t+8} = π).
- Responses:
  - Reacting to realized inflation: real interest rates rise substantially, output contracts, and both the size and persistence of the inflation runup are reduced — but with sizeable output costs.
  - Reacting to two-year ahead expected inflation: allows real rates to decline more sharply than baseline, inducing a bigger runup in inflation and output; inertia in the rule slows policy response, causing the central bank to fall further “behind the curve.”
- Trade-offs and context:
  - Forecast-based rules can inadvertently amplify inflation and output when shocks are large and persistence is high because misperceptions delay recognition of persistence.
  - When shocks are small and transient (as in the Great Moderation), forecast-based rules that “look through” shocks can avoid excessive output volatility and are appealing.
- Quantitative illustrative comparison (ad hoc loss example):
  - Expected loss measured as sum of squared deviations of annualized inflation from target plus sum of squared output gaps for first 5 years falls by about 30 percent from a loss of about 255 under the forecast-based rule to a loss of about 174 under a rule responding to actual inflation.
- Additional note: assumption of sizeable interest rate smoothing (ρ_R = 0:85) moderates differences between rules; without such smoothing differences would be further amplified.

### Role of Misperceptions
- If agents and the central bank could immediately identify a persistent markup shock as persistent, interest rates would rise much faster even under a forecast-based rule, causing output contraction and limiting inflation rise.
- Learning speed depends on the “signal-to-noise” ratio given by standard deviations of persistent and transitory components (σ_P / σ_T).
  - Faster filtering (higher σ_P / σ_T) leads the central bank to recognize persistent upward pressure earlier, tightening policy faster and more forcefully, reducing inflation and output expansion.
- Misperception problems are much less acute when the underlying shock is actually transitory.

### Implications for Identification of Shocks
- Forecast-based rules with agent misperceptions can produce persistent positive comovement of inflation and output in response to a supply (cost-push) shock.
  - This pattern resembles demand shocks despite the driving force being adverse cost-push shocks.
- This challenges empirical identification assumptions (e.g., sign restrictions used in structural VARs) that cost-push shocks necessarily drive inflation and output in opposite directions, and the assumption that only demand shocks drive positive short-term co-movement of output and inflation.
- For the recent inflation surge, the model suggests that initial strength in activity may partly reflect muted monetary policy response to adverse supply shocks.

### State-dependent Amplification of Cost-push Shocks
- Cost-push shocks have amplified effects when demand is initially strong and inflation already elevated.
- Experiment design:
  - Alternative baselines constructed with progressively larger demand shocks (via discount factor shocks) that raise the potential real interest rate but not potential output.
  - First row baseline: zero positive demand shock — inflation at steady state and output at potential.
  - Last row baseline: largest demand shock considered — peak inflation rises to 3.2 percent and the output gap rises (table referenced for details).
- Conclusion:
  - Initial macroeconomic conditions (strong demand, elevated inflation) raise vulnerability to amplification from adverse cost-push shocks through the model’s nonlinear mechanisms (misperceptions, forecast-based policy inertia, endogenous indexation).

*Italic: Source — chapter "4. Results" from wpiea2024260-print-pdf.*

### 5.4 percent.

### wpiea2024260-print-pdf - 5.4 percent.

### Amplification of cost-push shocks (Table 2)
- The same-sized underlying discount factor shock is applied across alternative baselines; peak inflation responses increase with the shock size due to the nonlinear Phillips Curve.
- Example from Table 2: a cost-push shock that raises inflation by 4.7 percent when inflation is initially at target would raise inflation by 9 percent if the shock occurs against a backdrop where inflation was initially 3.2 percent and the output gap is slightly over 5 percent.
- Interpretation: stronger initial demand conditions amplify the inflationary impact of a given-sized cost shock — i.e., state-dependence matters for the amplification of cost-push shocks.

### Effects of monetary tightening (nonlinear vs linearized models)
- Policy experiment design:
  - The monetary policy shock R;t in the Taylor rule follows an AR(1) process with a persistence coe¢ cient of0:75.
  - The shock is sized in both nonlinear and linearized models so that inflation is reduced by one percentage point (APR) below its baseline path. The baseline uses the cost-push shock in Figure 5 plus a one percent discount factor shock to generate a "high inflation" baseline.
  - The monetary policy intervention is assumed to start when inflation attains its peak in the nonlinear model.
- Key quantitative result:
  - To attain a one percentage point (APR) lower inflation trajectory, the nonlinear model requires notably larger nominal policy rate tightening than the linearized model.
  - That larger tightening in the nonlinear model produces a considerably lower output path than implied by the linearized model.
- Intuition (Figure 12):
  - With a nonlinear (kinked) Phillips curve, bringing inflation back faster requires much stronger tightening than a linear Phillips curve implies, producing substantially larger output costs.
- Additional sensitivity notes:
  - Appendix B.5: earlier central bank intervention yields a larger reduction in inflation for a given policy rate hike — monetary policy becomes more effective the higher inflation is to begin with.
  - A pseudo-linearized model is considered where indexation is nonlinear/endogenous while other equations are linearized.

### Stochastic simulations and distributional outcomes (Figure 13)
- Simulation setup:
  - Starting at the baseline path at t= 8, the economy is hit by random unexpected cost-push and discount factor shocks in each period t8.
  - Cost-push shocks follow the model’s unobserved components specification with transitory and persistent realizations.
  - Variances of cost-push and discount factor shocks are chosen to match roughly: the unconditional standard deviation of core PCE inflation, the unconditional standard deviation of real consumption per capita growth, and the correlation between consumption growth and inflation in post-war/pre-Covid U.S. data.
  - Density plots use 500 random sequences of shocks and show the f2:5;10;20;::::;90;97:5g percentiles and the median.
- Nonlinear model — standard monetary policy:
  - Inflation density is asymmetric with more realizations of high inflation than low inflation.
  - Mechanism: amplification from Kimball aggregation and endogenous indexation causes bursts of inflation to occur more often when inflation is already above the central bank’s target.
- Nonlinear model — more aggressive monetary policy:
  - The central bank increases the weight of inflation in the Taylor rule by a factor of three and reduces the weight of the output gap by a factor of three starting in period t= 8.
  - Outcomes: inflation is lower on average and almost no upward asymmetry in the inflation distribution; improved inflation stabilization comes with substantial economic costs — the distribution of output shows a significantly lower mean and evident downside risks.
- Trade-off highlighted: improved stabilization of inflation (lower mean, less upward skew) under aggressive policy is achieved at the expense of larger downside output risks.

### Policy implications and practical takeaways
- "Looking through" supply shocks:
  - May be appropriate for small shocks when inflation is near the central bank’s target.
  - Is risky when economic activity is strong and large adverse shocks push inflation well above target.
- “Going the last mile” (tighter-than-normal policy to return inflation quickly to target):
  - The nonlinear model implies that the economic costs of such additional tightening can be considerable relative to linearized assessments.
- Timing matters:
  - Earlier policy intervention is more effective; late disinflation after inflation and inflation expectations have become entrenched is costlier.

### Relation to model features and broader contribution
- The paper’s model combines:
  - i) nonlinear price and wage Phillips curves,
  - ii) an unobserved components representation for cost-push shocks,
  - iii) endogenous intrinsic price and wage indexation,
  - iv) an inflation forecast-based Taylor rule.
- With these elements the model generates jointly:
  - steep surges in inflation, resilient economic activity, a slow central bank response, and a fall of the real wage in the post-COVID episode.
- The nonlinear framework also addresses other episodes (e.g., deep recessions, "missing deflation" puzzle) due to the boomerang-shaped nonlinear price and wage Phillips curves.

### Conclusion and directions for future research
- Main conclusions:
  - A nonlinear model with endogenous indexation and a persistent cost-push component can explain the 2021-23 post-COVID inflation surge where linearized models do not.
  - Adverse cost-push shocks can be expansionary in the short run when the central bank misjudges persistence and "looks through" shocks.
  - The costs of stronger-than-baseline monetary tightening are larger in a nonlinear framework.
- Suggested future work:
  - Allow firms to choose desired rate of indexation (intensive margin) and compare nonlinear vs linear outcomes.
  - Consider an extensive-margin interpretation where a state-dependent fraction of non-reoptimizers fully index to past inflation, calibrated to price adjustment frequency across inflation regimes.
  - Explore separate indexation rules for prices and wages, and discipline wage rule parameterization with observed prevalence of cost-of-living-adjustment clauses in high-inflation episodes.

*Italic: Source — wpiea2024260-print-pdf - 5.4 percent.*

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### Appendix A.Derivations, Equilibrium Equations, and Additional Re-

### Appendix A.Derivations, Equilibrium Equations, and Additional Results

### A.1. Households
- First-order conditions (scaled form):
  - c_t : 1 / c_t - h C_{t-1} = λ_t
  - B_t : λ_t = β_t E_t[R_t π_{t+1} λ_{t+1}]  
    where π_t = P_t / P_{t-1} and λ_t = Λ_t / P_t.
- Steady state: R = π = β and in equilibrium c_t = C_t.

### A.2. Labor Contractors and Wage Setting
- Labor contractor optimization and aggregator yield relations for wage index and labor demand:
  - Definitions and intermediate identities involve parameters w, ζ_w, ϕ_w and ratios W_{t;j}/W_t.
- Free-entry (zero profits) condition for labor contractors:
  - 1 = 1/(1 + w) χ_{w,t} + w/(1 + w) ∫ W_{t;j}/W_t dj
  - χ_{w,t} = 1 + w - w ∫ W_{t;j}/W_t dj
- Optimal wage-setting problem (stylized):
  - max_{~W_{j;t}} E_t Σ_{i=0}^{∞} (β_w)^i Λ_{t+i} n_{t+i} π_{t+i}^{-1} [complex expression in ~W_{j;t} and wages]
- Key definitions (scaled variables):
  - ~w_t = ~W_t / W_t; w_t = W_t / P_t
- Compact first-order condition (nonlinear wage-setting):
  - S_{w,t} = F_{w,t} ~w_t - A_{w,t} ~w_t^{1+((1+ζ_w)(1+w))/ζ_w}
  - Scaled: s_{w,t} = f_{w,t} ~w_t - a_{w,t} ~w_t^{1+((1+ζ_w)(1+w))/ζ_w}
- Recursive expressions:
  - F_{w,t} = Λ_t n_t π_t w_t * [(1+ζ_w)(1+w)/ζ_w] w_{;t} + β_w E_t[... F_{w,t+1}]
  - a_{w,t}, s_{w,t} have analogous recursive forms with same discounting structure.
- Optimality condition in scaled terms:
  - s_{w,t} = f_{w,t} ~w_t - b_{w,t} ~w_t^{1+((1+ζ_w)(1+w))/ζ_w}

### A.3. Final Goods Producers
- Price aggregator and first-order condition (analogous to wage section) with parameters p, ζ_p:
  - P_{t;i}/P_t = Γ_t * [-(1 + p) (y_{t;i}/y_t - p) ]^{1-θ_p} ... (derivation steps)
- Zero-profit condition after aggregation:
  - 1 = 1/(1 + p) χ_{t} + p/(1 + p) ∫ P_{t;i}/P_t di
  - χ_t = 1 + p - p ∫ P_{t;i}/P_t di

### A.4. Intermediate Goods Producers
- Optimization (pricing) problem (stylized):
  - max_{~P_{t;i}} E_t Σ_{j=0}^{∞} β_p^j Λ_{t+j} y_{t+j} [demand term in ~P_{t;i}] - [~P_{t;i} p - MC_{t+j}] * [...]
- First-order condition (rearranged):
  - 0 = E_t Σ_{j=0}^{∞} β_p^j Λ_{t+j} y_{t+j} { F_t ~p_{t;i} - S_t - A_t ~p_{t;i}^{1+((1+ζ_p)(1+p))/ζ_p} }
- All resetting firms set same price: ~p_{t;i} = ~p_t.
- Scaled optimality condition:
  - s_t = f_t ~p_t - b_t ~p_t^{1+((1+ζ_p)(1+p))/ζ_p}
- Definitions (S_t, F_t, A_t) involve expectations over future Λ_{t+j}, y_{t+j}, Π-terms and mc_{t+j}.
- Recursive relations (examples):
  - s_t = ((1 + p) (1 + ζ_p)/(1 + p + ζ_p) ) w.r.t current variables + β_p E_t[ ... s_{t+1} ] (explicit form in text).
- Note: mc_{t} = MC_t / P_t; definitions for f_t, b_t, s_t follow similarly.

### A.5. Aggregate Resources
- Aggregate output and labor:
  - y_{sum,t} = ∫ y_{t;i} di = ∫ n_{t;i} di = n_t.
  - y_t = (p̃_t)^{-1} (w̃_t)^{-1} l_t, so y_t = (p̃_t)^{-1} (w̃_t)^{-1} l_t.
  - Denote p̃_t and w̃_t as aggregate dispersion indices (definitions provided).
- Aggregate resource constraint:
  - c_t = y_t = (p̃_t)^{-1} (w̃_t)^{-1} l_t
- Zero-profit condition for final goods producers:
  - χ_t = 1 + p - p ∫ P_{t;i}/P_t di = 1 + p - p τ_{t;2}
- Aggregate price and wage index equations expressed with dispersion terms τ_{t;k}, ω_{t;k} and indexation terms.
- Relation between real wage, price inflation and wage inflation:
  - π_{w,t} = Π_t^w = (W_t / W_{t-1}) / (P_t / P_{t-1}) etc., compactly: π_{w,t} = π_t w_t / w_{t-1}.

### A.6. Solution and Implementation
- Solver approach:
  - Use the nonlinear (’simul’) two-point boundary value solver in Dynare.
  - Model written as time T system: f(y_{t-1}; y_t; y_{t+1}; a_t) = 0 for t = 0,...,T.
  - y_{-1} and y_{T+1} given (usually steady state).
- Expectation formation:
  - For each realization of shocks, agents form expectations using the Kalman filter.
  - Simulation procedure: observe a shock at t=0, solve t=0..T; then move to t=1 with updated initial state and repeat until no new shocks.
- Certainty-equivalence assumption:
  - Solve the deterministic solution in each simulation period; interactions between non-linearities and uncertainty (Jensen's inequality) are not accounted for.

### A.7. Nonlinear Equilibrium Equations
- Core nonlinear equilibrium system (selected equations labeled n1–n33):
  - Marginal utility (n1): 1 / c_t - h c_{t-1} = λ_t
  - MRS (n2): mrs_t = 1 / λ_t
  - Euler equation (n3): λ_t = β E_t[λ_{t+1} R_t π_{t+1}]
  - Resource Constraint (n4): c_t = y_t
  - Production (n5): y_t = (p̃_t)^{-1} (w̃_t)^{-1} l_t
  - Non.lin. pricing/wage-setting equations (n6–n9; n17–n20): s_t = f_t ~p_t - b_t ~p_t^{1+...}; s_{w,t} = f_{w,t} ~w_t - b_{w,t} ~w_t^{1+...}
  - Zero profit conditions and indexation constraints (n10–n15; n21–n26).
  - Indexation definitions (n27–n30): ~π_t = Π_{1-ζ}(...), ~π_{w,t} = ~π_t, {t} = e^{- % max(π̃_t - π̄; 0:0001)} - e^{- % 0:0001}, π̃_t = π̃_{t-1}^{1-α}(π_{t-1})^{α}
  - Marginal cost (n31): mc_t = χ^{-1} w_t
  - Taylor rule (n32): R^{not}_t = R̄^{ζ_R} R^{not}_{t-1}^{1-ζ_R} exp{ φ E_t[π_{t+4} = π̄ ] } ... (full expression in text)
  - ZLB (n33): R_t = max(1, R^{not}_t)
- Flex-price / flex-wage (potential) economy (ζ_p = ζ_w = 0; cost-push shock set to zero):
  - Real potential rate (n34): 1 / y^{pot}_t - h y^{pot}_{t-1} = β E_t[ r^{pot}_t (1 / y^{pot}_{t+1} - h)]
  - Potential output (n35): y^{pot}_t - h y^{pot}_{t-1} = 1 / (1 + ζ_p) * 1 / (1 + ζ_w)
  - Potential output is constant; only the real potential rate moves with discount factor shock.
- Model dimension:
  - 35 equations in 35 endogenous variables (list provided in text).
  - Exogenous variables: Λ_t, χ_t-1 = a_t and "R_t".

### A.8. Steady State
- Solve steady state given central bank steady-state inflation π̄:
  - (n3): R = 1 / β π̄
  - (n27): ~π = π̄ and π̃ = π̄
- Key steady-state identities (selected):
  - (n10): χ = 1 + p - p τ_{2}
  - (n11): τ_{3} = χ
  - (n12): p̃ = χ^{(1+ζ_p) ζ_p (1+ p) ... } (full algebra in text)
  - From n6–n9: explicit mc expressed as a function of ζ_p, p, ~p and parameters.
  - (n31): w = mc
  - Steady-state real variables:
    - y = 1 / (1 - h) * w / (1 + ζ_w)
    - c = y
    - λ = 1 / (1 - h) c
  - Prices/wages steady state: ~p = 1; ~w = 1; various dispersion indices = 1; χ = 1; π_{w} = π̄.
  - Flex-price flex-wage potential economy:
    - (n34) rr^{pot} = 1 / β
    - (n35) y^{pot} = 1 / (1 - h) * 1 / (1 + ζ_p) * 1 / (1 + ζ_w)

### A.9. Log-Linearized Equilibrium Equations
- Log-linearized identities for price-dispersion variables:
  - ^χ_t = 0; ^p̃_t = 0; ^τ_{t;1} = 0; ^τ_{t;2} = 0; ^τ_{t;3} = 0
  - b ~π_t = ζ_p/(1-ζ_p) ( ^π_t - b ~π_t )
- Log-linearized nonlinear pricing (n6–n9) condensed to New Keynesian Phillips Curve:
  - ^π_t = ω_π ^π_{t-1} + β/(1 + β ω_π) E_t[^π_{t+1}] + ς cmc_t
  - where ς = 1/(1 + β ω_π) - (1 - ζ_p)/(1 - β ζ_p) * ζ_p/(1 - ζ_p) * ζ_p/(1 + ζ_p) * p (exact expression in text)
  - The coefficient 1/(1 - (1+ζ_p) p/(1+ p) ) matches Levin, Lopez-Salido and Yun (2007) structure.
- Log-linearized wage-setting leads to Wage Phillips Curve:
  - ^π_{w;t} - b ~π_{w;t} = β E_t[^π_{w;t+1} - b ~π_{w;t+1}] + ς_w (dmrs_t - ^w_t)
  - with ς_w = (1 - ζ_w)(1 - β ζ_w) ζ_w / (1 - (1 + ζ_w) ζ_w) ... (full algebra in text)
- Log-linearized system (l1–l11) — 13 equations in 13 endogenous variables:
  - Euler equation (l1): ^y_t = (1/(1 + h)) E_t[^y_{t+1}] + (h/(1 + h)) ^y_{t-1} - ((1 - h)/(1 + h)) E_t[^R_t - ^π_{t+1} + ^ε_{t+1}]
  - Marginal cost (l2): cmc_t = 1/χ ^ξ_t + ^w_t
  - Marg. rate of substitution (l3): dmrs_t = 1/(1 - h) (^y_t - h ^y_{t-1})
  - Taylor rule and ZLB (l4–l5): ^R^{not}_t = ζ_R ^R_{t-1} + (1 - ζ_R)[φ E_t ^π_{t+4} + x(^y_t - ^y^{pot}_t)] + ε_{R;t}; ^R_t = max(-ln R̄, ^R^{not}_t)
  - Price Phillips Curve (l6): ^π_t - b ~π_t = β E_t[^π_{t+1} - b ~π_{t+1}] + (1 + β ω_π) ς cmc_t
  - Wage Phillips Curve (l7) and wage inflation (l8): analogous to price case (full forms in text)
  - Real potential rate and potential output (l9–l10): b rr^{pot}_t = - E_t ^ε_{t+1}; ^y^{pot}_t = 0
  - Indexation (l11): b ~π_t = 0
- Phillips curve slopes:
  - ς = - [1 - β ζ_p]^{-1} [1 - ζ_p]^{-1} ζ_p * p * [1 / (1 - (1 + ζ_p) p/(1 + p) )] (text provides precise expression)
  - ς_w = (1 - ζ_w) (1 - β ζ_w) ζ_w / [1 - (1 + ζ_w) ζ_w] * w (precise algebra in text)
- Model dimension and exogenous processes:
  - 13 log-linear equations; endogenous variables: ^R_t, ^R^{not}_t, ^π_t, b ~π_t, ^y_t, ^w_t, cmc_t, dmrs_t, ^π_{w;t}, b rr^{pot}_t, ^y^{pot}_t.
  - Exogenous: ^ε_t, ^ξ_t = χ_t - 1 = a_t and ε_{R;t}.

*Source: Appendix A. Derivations, Equilibrium Equations, and Additional Results (wpiea2024260-print-pdf).*

### Appendix B.Additional Results

### Appendix B.Additional Results

### B.1. Data-Model Comparison in Linearized Model
- Figure B.1 provides a comparison between the data and the linearized model for:
  - PCE Inflation and SPF Forecasts (APR, Dev. from 2020Q4)
  - 3-Month T-Bill and SPF Forecasts (APR, Dev. from 2020Q4)
  - Real Consumption Growth and SPF Forecasts (APR, Dev. from 2020Q4)
  - Linear Model: Inflation and Forecasts (APR, Dev. From St.St.)
  - Linear Model: Policy Rate and Forecasts (APR, Dev. From St.St.)
  - Linear Model: Real Consumption Growth and Forecasts (APR, Dev. From St.St.)
- Figure caption: "Figure B.1: Comparison of data vs. linear model."

### B.2. Transmission of Discount Factor Shock
- SpeciÖcation and shock size:
  - SpeciÖcation: discount factor shock of" ;0 =0:01, i.e. fall in discount factor of1percent (quarterly) or4percent (annualized).
  - Fall in discount factor implies rise in demand (but no e§ects on potential output).
- Figure B.2 shows impulse responses of the nonlinear model to a discount factor shock for the following variables (plotted over 0–20 on the x-axis as quarters):
  - Real GDP (%)
  - Policy Rate (APR)
  - Real Wage (%)
  - Discount Factor Shock (%)
  - Wage inflation (APR)
  - Cost-push Shock, a=aP+aT (%) — observed vs. true vs. estimated state
  - Unobs. persistent comp. aP (%) — true vs. estimated state
  - Unobs. transitory comp. aT (%) — true vs. estimated state
  - Inflation (APR) — Realization vs. Real-time predictions
- Figure caption: "Figure B.2: Impulse responses to a discount factor shock in the nonlinear model."

### B.3. Transmission of Transitory Cost-push Shock
- Shock specification and interpretation:
  - Here, we consider the case when the cost push shock is transitory, i.e." T 0 = 0:0025 shock to the transitory component of the cost push shock.
  - I.e.1=4percent (quarterly) or1percent (annualized) cost-push shock.
  - The analysis notes: "This analysis could be extended to show (or argue based on the the results below) that it is more optimal to ílook throughí transitory cost push shocks."
- Figure B.3 provides impulse responses of the nonlinear model to a cost-push shock when driven by the transitory (iid) component for:
  - Real GDP (%)
  - Policy Rate (APR)
  - Real Wage (%)
  - Discount Factor Shock (%)
  - Wage inflation (APR)
  - Cost-push Shock, a=aP+aT (%) — Realization
  - Unobs. persistent comp. aP (%) — True vs. Estimated state
  - Unobs. transitory comp. aT (%) — True vs. Estimated state
  - Inflation (APR) — Realization vs. Real-time Prediction
- Figure caption: "Figure B.3: Impulse responses to a transitory (iid) cost-push shock in the nonlinear model."

### B.4. Further Details on the Unobserved Components Representation
- Investigation:
  - Figure B.4 shows the effects of assuming alternative values for the ratio of standard deviations ( P = T ) of the persistent and transitory components of the unobserved components representation of the cost push shock in the nonlinear model.
- Plotted responses and scenarios include:
  - Real GDP (%)
  - Policy Rate (APR)
  - Real Wage (%)
  - Discount Factor Shock (%)
  - Wage inflation (APR)
  - Cost-push Shock, a=aP+aT (%) — Ex-Post Observed vs. Forecast in t=0 with P/T = 1/100, P/T = 1/10, P/T = 1/5, P/T = 1/3
  - Inflation (APR) — Ratio of cost-push standard deviations P/T = 1/100 ... P/T = 1/10 ... P/T = 1/5 ... P/T = 1/3
- Figure caption: "Figure B.4: E§ects of alternative values for the ratio of standard deviations of the persistent and transitory components of the unobserved components representation of the cost push shock in the nonlinear model."

### B.5. Timing of Monetary Policy Intervention
- Experiment:
  - Simulations consider the implications when the central bank becomes more aggressive at different points in time.
  - Figure B.5 shows simulation results for more aggressive monetary policy in the nonlinear model for different start dates of the monetary policy intervention.
  - All impulse responses are displayed in deviations from baseline.
- Key takeaway:
  - "The earlier the central bank intervenes, the larger the reduction in ináation for a given hike in the policy rate. Put di§erently, monetary policy becomes less e§ective the higher ináation is to begin with. In this sense, the e¢ cacy of monetary tightening and the sacriÖce ratio are state-dependent in our model."
- Figure B.5 displays responses (quarters since policy intervention) for:
  - Real GDP (%)
  - Policy Rate (APR)
  - Real Interest Rate (APR)
  - Inflation (APR)
  - Intervention start dates shown: "Monetary policy intervention in quarter 1... in quarter 3... in quarter 7... in quarter 11"
- Figure caption: "Figure B.5: Simulation results for more aggressive monetary policy in nonlinear model with di§erent start dates for the monetary policy intervention. All responses in deviations from baseline."

*Monetary Policy and Inflation Scares — Working Paper No. WP/2024/260*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024260-print-pdf.pdf_