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---

### Empirical motivation and historical context
- FOMC projection trajectory and realized inflation:
  - Dec. 2020 projection for end-2021 inflation: less than 2 percent.
  - June 2021 projection for end-2021 inflation: 3.4.
  - Actual year-over-year inflation for 2021: 6.2 percent.
  - FOMC projected return to roughly 2 percent by end-2022; actual end-2022 inflation: 5.4 percent.
  - Projections through June 2023 showed consistent upward revisions at each meeting.
- Historical parallel:
  - 1960–1985 U.S. data: two large oil price shocks in the 1970s and a long upward drift of inflation from around 2 percent to almost 10 percent.
  - Disinflation in the U.S. associated with recessions in 1980 and 1982; major policy shifts were required to restore low inflation.

### Research question and approach
- Key questions:
  - How costly is delay when fighting a surge in inflation caused by a transitory shock?
  - If policy is delayed, should the subsequent response be more aggressive?
  - Can a credible promise to fight inflation later substitute for immediate policy action?
- Approach:
  - New Keynesian model with endogenous persistence sources.
  - Monetary policy: inertial Taylor rule; agents hold rational expectations and, in extensions, behavioral expectations (cognitive discounting).
  - Model the 2021 surge as a positive, persistent but transitory shock to flexible-price output pushing up firms’ real marginal costs.
  - Policies satisfy the Taylor principle and ensure eventual reversion of inflation to steady state.
  - Excludes COVID-related fiscal contributions and ignores the zero lower bound on nominal policy rate.

### Main theoretical findings (rational expectations)
- General conclusions:
  - Waiting to respond can amplify the rise in inflation while simultaneously boosting output and dampening any subsequent recession.
  - Delay is always costly on both loss measures used; the marginal cost of additional delay is increasing.
  - Conditional on delay, loss is reduced by:
    - Implementing a rule with a larger coefficient on inflation (higher φπ).
    - Adopting a less inertial policy rule (lower ρi).
  - Loss is lowest when policy reacts both more strongly to inflation and with less inertia, conditional on delay.
- Role of expectations:
  - A credible promise to respond in the future can substitute for immediate action because of forward guidance effects under rational expectations.

### Behavioral expectations (cognitive discounting) — modeling and core effects
- Modeling approach:
  - Use Gabaix (2014; 2020) cognitive discounting: E_CD_t q_{t+d} = m̄_d E_t q_{t+d}, 0 < m̄ ≤ 1.
  - Baseline m̄ = 0.85 (implying M_f = 0.79 with ω = 0.75, β = 0.995).
- Impacts of cognitive discounting:
  - Reduces the power of forward guidance.
  - Both inflation and the output gap are less volatile when agents display cognitive discounting.
  - Main policy conclusions under rational expectations still hold:
    - Both loss measures are lowest when there is no delay.
    - Conditional on delay, a more aggressive policy reaction still lowers loss.

### Simple three-period illustrative example (parameters and qualitative outcomes)
- Setup:
  - Periods i = 1,2,3; positive inflation shock e1>0 in period 1, e2 = ρ e1 with 0 ≤ ρ < 1; π3 = x3 = 0.
  - Policy parameters: β = 0.995, κ = 0.17, σ = 1, φ = 2, e1 = 1, ρ = 0.9.
- Comparative outcomes (qualitative):
  - Immediate response (γ1 = γ2 = 1): inflation stabilized; output gap falls sharply and remains negative in period 2.
  - One-period delay (γ1 = 0, γ2 = 1): anticipated rate hike limits rise in π1; output gap similar to immediate response with somewhat smaller period-2 drop.
  - No response (γ1 = γ2 = 0): inflation much more volatile; output gap rises in period 1 and returns to steady state in period 2.
- Takeaways:
  - Greater delay ⇒ larger rise in inflation but smaller decline (or a rise) in output gap.
  - Under quadratic loss in inflation and output-gap volatility, whether delay is optimal depends critically on the weight placed on output-gap volatility.

### New Keynesian model specification — core calibration and structural persistence
- Preferences, firms, and policy rule components summarized by core equations (6), (7), (10), (14), (15) and definitions.
- Key structural sources of persistence:
  - Consumption habits (h).
  - Policy inertia (ρi).
  - Serial correlation in shock (ρ).

### Basic calibration (section 3.1) — exact parameter values
- Discount factor: β = 0.995 (steady-state real interest rate ρ equal to 2% at an annual rate).
- Goods elasticity: σ = 1.
- Frisch elasticity of labor: η = 5.
- Habit parameter: h = 0.761.
- Capital share: α = 0.25.
- Markup parameter: ε = 9 (implies steady-state price markup of 12.5%).
- Calvo parameter: ω = 0.75 (average duration of price spells of one year, four periods in the model).
- Resulting κ in equation (14): κ = 0.169.
- Policy rule coefficients:
  - φπ = 1.5 (baseline).
  - φx = 0.5/4 = 0.125.
- Policy inertia: ρi = 0.85 (baseline).
- Policy delay considered: k ∈ {0 2 4 6}.
- Alternative policy aggressiveness:
  - Doubling φπ from 1.5 to 3.0 evaluated.
  - Reducing ρi from 0.85 to 0.5 evaluated.
- Contemporaneous impact effect of inflation on the policy rate: (1 − ρi)φπ.
  - Example: with φπ = 1.5 and ρi reduced from 0.85 to 0.5, the impact effect increases from 0.225 to 0.75.
- Inflation shock calibration:
  - AR(1) coefficient: 0.85.
  - Variance set to generate a peak rise of inflation of 9 percent (deviation from target and at an annual rate) when policy delays four periods.
  - Inflation target: 2 percent.
  - This calibration roughly captures a jump in inflation to over 11 percent in early 2022.

### Impulse-response patterns under rational expectations — key quantitative outcomes
- Reporting convention: interest rates and inflation at annual rates.
- Reference case k = 0:
  - Policy rate rises immediately and peaks two periods after the shock.
  - Inflation peaks immediately, reaching just over 3 percent at an annual rate.
  - Inflation is less than 10 basis points above its steady state after seven periods.
  - Output gap initially declines, trough four periods after the shock, then slowly returns to steady state from below.
- Effects of policy delay (increasing k):
  - When delay > 1 period, the output gap in the period of the shock increases rather than decreases.
  - Delaying four periods increases peak inflation from 3 percent (k = 0) to 9.1 percent.
  - Delaying one additional period, to k = 5, raises peak inflation to 13.5 percent.
  - Delay produces a larger and more persistent fall in the real interest rate, stimulating output in the shock period and raising inflation.
  - A long delay leaves both the output gap and inflation higher at t + k − 1 than with immediate reaction.
- Notable pattern: k = 3 can appear as an “immaculate disinflation” with inflation higher than when delay is shorter and a brief positive output gap before a milder recession than under immediate reaction.

### Reacting more strongly to inflation — quantitative and qualitative effects
- Interpretation 1 — larger φπ (doubling from 1.5 to 3.0):
  - When k = 0:
    - Stronger response stabilizes inflation more; output declines rather than increases.
    - Policy rate peaks at a higher level; real interest rate higher; larger fall in the output gap.
    - After t + 5, nominal policy rate can end up slightly lower than under less aggressive response.
  - When k = 4:
    - Larger φπ reduces output gap and inflation even if nominal rate path is little affected.
    - Conditional on delay, higher φπ moves π and x closer to steady state for several periods compared to baseline φπ.
- Interpretation 2 — lower ρi (from 0.85 to 0.5) holding φπ = 1.5:
  - When k = 0:
    - Less inertia ⇒ much larger initial rise in nominal rate and higher real interest rate path; small effect lowering output gap until t + 5; small effect on inflation.
  - When k = 4:
    - Less inertia front-loads interest-rate increase; combined with higher φπ, dampens rise in output gap and inflation but causes the output gap to turn negative sooner and for longer.

### Costs and risks of extended delay — example statistics
- Extending delay beyond 4 periods under baseline:
  - Increasing delay from k = 4 to k = 6:
    - Peak inflation rises from 9.1 percent to 24.6 percent.
    - Peak output gap rises from 0.9 percent to 3.4 percent.
- Implication: more than one year of delay can substantially increase fluctuations of the output gap and inflation.

### Loss-function framework and quantitative measuring of delay costs (section 4.2)
- Loss function forms:
  - L = (1/2)(ε/κ)[var(π_t) + λ var(x_t)] (equation (16)).
  - Adopt λ = 0.25 and loss simplified to L = (1/2)(ε/κ)[var(π_t) + 0.25 var(x_t)] (equation (17)).
- Two single-shock impulse-response metrics (T-horizon averages):
  - L1_T(k, φ_π, ρ_i) = (1/2)(ε/κ) E_t[ (1/T) Σ_{j=0}^T β^j π_{t+j}^2 + λ (1/T) Σ_{j=0}^T β^j x_{t+j}^2 ]. (equation (18))
  - L2_T(k, φ_π, ρ_i) = (1/2)(ε/κ) E_t[ (1/T) Σ_{j=0}^T β^j π_{t+j} − λ (1/T) Σ_{j=0}^T β^j x_{t+j} ]. (equation (19))
- Key loss-ranking findings under rational expectations:
  - Loss is increasing in delay k for fixed policy-rule parameters; marginal increase in loss rises with k.
  - A one-period delay (k = 1) leads to only a small increase in loss relative to k = 0 in many specifications.
  - Doubling φπ from 1.5 to 3.0 reduces overall loss for all k.
  - Reducing inertia (ρ_i from 0.85 to 0.5) lowers volatility for all k and T.
  - Combining higher φπ and lower ρ_i produces the lowest losses for all k and T and preserves the ranking across k.
- Quantitative example (L1_0 when k = 4):
  - Baseline policy → L1_0 = 0.82.
  - Stronger response (φπ = 3.0) with baseline inertia → L1_0 = 0.43 (a 48 percent reduction).
  - Stronger response combined with less inertia (φπ = 3.0 and ρ_i = 0.5) → L1_0 = 0.29.
  - Total reduction moving from baseline to most aggressive rule when k = 4 is 65 percent (from 0.82 to 0.29).
- Forward guidance:
  - Under rational expectations a credible future promise can substitute for current action; this explains why a year-long delay (k = 4) might not be very costly if combined with an aggressive policy rule.

### Cognitive discounting — equations, calibration, and effects on volatility (section 4.2 and 5.3)
- Modified equations under cognitive discounting (select):
  - x_t = [(1/(1+h))] m̄ E_t x_{t+1} + [h/(1+h)] x_{t−1} − (1/σ)(i_t − m̄ E_t π_{t+1} − r^*_t). (equation (20))
  - π_t = β M_f E_t π_{t+1} + κ[ ... ] + ê_t. (equation (21))
  - ĩ_t = ρ_i ĩ_{t−1} + (1−ρ_i)[φ_π π_{t−k} + φ_x x_{t−k}]. (equation (22))
  - M_f = m̄[ ω + (1−ω) ( (1−βω)/(1−βω m̄) ) ] ≤ 1. (equation (23))
- Calibration evidence and choices:
  - Gabaix (2020) suggests m̄ = 0.85 ⇒ M_f = 0.79 with ω = 0.75, β = 0.995.
  - Ilabaca, Meggiorini, and Milani (2020): m̄ = 0.71; M_f = 0.41.
  - Andrade, Coredeiro, and Lambais (2019): m̄ in [0.013, 0.645].
  - Paper adopts m̄ = 0.85 for baseline cognitive-discounting results.
- Effects of cognitive discounting on impulse responses and loss:
  - Nominal and real interest rates rise more and remain higher for longer compared with rational expectations in the baseline no-delay policy.
  - Output gap falls less (muted recession) and can rise slightly initially.
  - Inflation is somewhat higher and more persistent than with rational expectations.
  - Volatility measures (18) and (19) are always lower under cognitive discounting than under rational expectations.
  - Cognitive discounting reduces marginal impact of increasing k; e.g., increasing k from 4 to 5 has about same impact as from 3 to 4 under cognitive discounting.
  - Forward guidance is less powerful; promised future policy actions are less effective.
- Quantitative illustration of reduced amplitude (output gap x):
  - Under cognitive discounting: x varies between a high of just under 0.006 when k = 5 and a low of −0.003 when k = 0.
  - Under rational expectations: x varies between a high of 0.021 (k = 5) and a low of −0.008 (k = 0).

### Can aggression compensate for delay? (comparison across expectations)
- Under rational expectations (loss measured by (18)):
  - A delay of two periods combined with a more aggressive response (φπ = 3.0, ρi = 0.5) matches the baseline, no-delay policy; k = 1 with aggressive policy can outperform baseline k = 0.
- Under cognitive discounting:
  - An aggressive policy rule can replicate the baseline, no-delay policy even if delaying three periods.
- General conclusion:
  - A more aggressive rule (higher φπ and/or lower ρi) substantially compensates for delay under both expectation frameworks, with cognitive discounting allowing longer delays to be compensated relative to rational expectations.

### Policy implications and recommendations (synthesis)
- Immediate action against a transitory inflation surge minimizes losses on both considered metrics (quadratic loss and soft-landing preference).
- If delay occurs, policymakers should:
  - React more aggressively to inflation (raise φπ).
  - Reduce policy inertia (lower ρi).
  - Combine stronger response with less inertia to minimize loss conditional on delay.
- Credible promises to act later can partly substitute for immediate action, but effectiveness is muted when agents exhibit cognitive discounting.
- Evaluations of policy delay must account for the tradeoff:
  - Faster disinflation reduces inflation volatility but increases short-run output costs.
  - Delay raises inflation but cushions output.
- Practical takeaway on timing:
  - Costs of delay are small for one or two periods but grow rapidly if delay extends to four periods or longer; excessive delay is particularly costly under rational expectations.

*Italic source: Content unit: wpiea2024042-print-pdf — The Consequences of Falling Behind the Curve: Inflation Shocks and Policy Delays under Rational and Behavioral Expectations, Working Paper No. WP/2024/042*

### 2023.  In Dec.  2020, the FOMC was projecting inflation at the end of 2021 would be less than its 2

### wpiea2024042-print-pdf - 2023.  In Dec.  2020, the FOMC was projecting inflation at the end of 2021 would be less than its 2 percent target.  By June 2021, this estimate had risen to 3.4.  The actual year-over-year inflation for 2021 was 6.2 percent.  The FOMC was still projecting inflation to return to roughly 2 percent by the end of 2022; it ended up equaling 5.4 percent by the end of 2022.  The sequence of projections until June of 2023 showed a consistent upward shift at each new meeting.  These projections reveal underestimates of inflation and overestimates of the speed with which inflation would decline.  The initial 2021 surge in inflation persisted to a degree that policymakers failed to foresee.

### Empirical motivation and historical context
- FOMC projection trajectory and realized inflation:
  - Dec. 2020 projection for end-2021 inflation: less than 2 percent.
  - June 2021 projection for end-2021 inflation: 3.4.
  - Actual year-over-year inflation for 2021: 6.2 percent.
  - FOMC projected return to roughly 2 percent by end-2022; actual end-2022 inflation: 5.4 percent.
  - Projections through June 2023 showed consistent upward revisions at each meeting.
- Historical parallel:
  - 1960–1985 U.S. data show two large oil price shocks in the 1970s and a long upward drift of inflation from around 2 percent to almost 10 percent.
  - Disinflation in the U.S. associated with recessions in 1980 and 1982; major policy shifts were required to restore low inflation.

### Research question and objectives
- Key questions:
  - How costly is delay when fighting a surge in inflation caused by a transitory shock?
  - If policy is delayed, should the subsequent response be more aggressive?
  - Can a credible promise to fight inflation later substitute for immediate policy action?
- Approach:
  - Use a new Keynesian model with endogenous persistence sources.
  - Monetary policy modeled by an inertial Taylor rule; agents hold rational expectations (and later behavioral expectations).
  - Model the 2021 inflation surge as a positive, persistent but transitory shock to flexible-price output pushing up firms’ real marginal costs.
  - Policies considered satisfy the Taylor principle and ensure eventual reversion of inflation to steady state.
  - Experiment excludes COVID-related fiscal contributions and ignores the zero lower bound on nominal policy rate.

### Main theoretical findings (rational expectations)
- General conclusions:
  - Waiting to respond can amplify the rise in inflation while simultaneously boosting output and dampening any subsequent recession.
  - Delay is always costly on both loss measures used; the marginal cost of additional delay is increasing.
  - Conditional on delay, loss is reduced by:
    - Implementing a rule with a larger coefficient on inflation (stronger response to inflation).
    - Adopting a less inertial policy rule (less persistence in policy).
  - Loss is lowest when policy reacts both more strongly to inflation and with less inertia, conditional on delay.
- Role of expectations:
  - A credible promise to respond in the future can substitute for immediate action because of forward guidance effects under rational expectations.

### Behavioral expectations (cognitive discounting) results
- Modeling approach:
  - Follow Gabaix (2014; 2020) cognitive discounting framework: agents systematically discount the future relative to rational expectations.
  - Evidence and related literature cited showing underreaction and bounded rationality in expectations.
- Impacts of cognitive discounting:
  - Reduces the power of forward guidance.
  - Both inflation and the output gap are less volatile when agents display cognitive discounting.
  - Despite reduced forward guidance effectiveness, main policy conclusions under rational expectations still hold:
    - Both loss measures are lowest when there is no delay.
    - Conditional on delay, a more aggressive policy reaction still lowers loss.

### Simple three-period illustrative example
- Setup:
  - Periods i = 1,2,3 with inflation πi and output gap xi.
  - Positive inflation shock e1>0 in period 1; persists and decays to e2 = ρ e1, 0 ≤ ρ < 1; no shocks in period 3, π3 = x3 = 0.
  - Policy rule parameterized by φ (response to inflation) and γt indices indicating whether policy responds in given period.
- Key algebraic and comparative results (no approximations):
  - When policy acts in period 2, inflation in period 2 is attenuated by a factor involving κ, φ, and σ as in
    - π2 = − (γ2 κ φ / (2 σ)) π2 + e2  (expression presented in text).
  - Comparative outcomes:
    - π1(0,0) > π1(0,1) > π1(1,1): rise in inflation in period 1 increases with delay.
    - π2(0,0) > π2(0,1) = π2(1,1): delay increases inflation in period 2.
    - Output gap behavior differs: absence of policy reaction (γ1 = γ2 = 0) leads to output expansion in period 1 and return to zero in period 2; delayed or immediate reactions produce output declines at different magnitudes.
- Numerical illustration parameters and shock:
  - β = 0.995, κ = 0.17, σ = 1, φ = 2, e1 = 1, ρ = 0.9.
- Numerical scenario outcomes (described qualitatively in text):
  - Immediate response (γ1 = γ2 = 1): policy interest rate increased immediately; inflation stabilized but output gap falls sharply and remains negative in period 2.
  - One-period delay (γ1 = 0, γ2 = 1): anticipated rate hike limits rise in π1; output gap behaves similarly to immediate response with somewhat smaller period-2 output drop.
  - No response (γ1 = γ2 = 0): inflation much more volatile; output gap rises in period 1 and returns to steady state in period 2.
- Takeaways:
  - Delay affects inflation and the output gap differently: greater delay => larger rise in inflation but smaller decline (or a rise) in output gap.
  - Under a standard quadratic loss in inflation and output-gap volatility, whether delay is optimal depends critically on the weight placed on output-gap volatility.

### New Keynesian model specification (core components)
- Households and preferences:
  - External habits in consumption with parameter h.
  - Utility: Ui,t = νt[(Ci,t − h Ct−1)^(1−σ)/(1−σ) − χt (Ni,t)^(1+η)/(1+η)].
  - Intertemporal condition and intratemporal condition derived; log deviations used.
- Demand side and output gap:
  - Log-linearized Euler equation with habits yields (in deviation form)
    - xt = (1/(1 + h)) Ẽt xt+1 + (h/(1 + h)) xt−1 − (1/σ)((1 − h)/(1 + h))(it − Ẽt πt+1 − r̃∗t).
  - Flex-price output ŷf t follows AR(1) (explicit expression (10) in text).
- Firms and inflation dynamics:
  - Calvo pricing with fraction 1 − ω adjusting; κ given by
    - κ = (1 − ω)(1 − ω β)/ω * (1 − α)/(1 − α + ε α).
  - New Keynesian Phillips curve with indexation and marginal cost:
    - πt = β Ẽt πt+1 + κ(cmc t + êt).
  - Average real marginal cost expression and its linearization provided.
- Monetary policy rule (inertia and delay):
  - Policy can delay by k periods; when t ≥ t0 + k, inertial Taylor rule applies:
    - it = ρi it−1 + (1 − ρi)(φπ πt + φx xt) for t ≥ t0 + k.
  - For t = t0, t0 + 1, ..., t0 + k − 1: it = 0 (no policy response).
- Model summary:
  - Core equations: (6), (7), (10), (14), and (15) plus definitions of output gap and exogenous shocks.
  - Key structural sources of persistence: consumption habits (h), policy inertia (ρi), and serial correlation in shock (ρ).

### Policy implications and recommendations (from model results)
- Immediate action against a transitory inflation surge minimizes losses on both considered metrics (quadratic loss and soft-landing preference).
- If delay occurs, policymakers should:
  - React more aggressively to inflation (raise φ).
  - Reduce policy inertia (lower ρi).
  - Combine stronger response with less inertia to minimize loss conditional on delay.
- Credible promises to act later can partly substitute for immediate action, but effectiveness is muted when agents exhibit cognitive discounting.
- Evaluations of policy delay must account for the tradeoff: faster disinflation reduces inflation volatility but increases short-run output costs; delay raises inflation but cushions output.

*Italic source: Content unit: wpiea2024042-print-pdf - 2023.  In Dec.  2020, the FOMC was projecting inflation at the end of 2021 would be less than its 2 percent target.  By June 2021, this estimate had risen to 3.4.  The actual year-over-year inflation for 2021 was 6.2 percent.  The FOMC was still projecting inflation to return to roughly 2 percent by the end of 2022; it ended up equaling 5.4 percent by the end of 2022.  The sequence of projections until June of 2023 showed a consistent upward shift at each new meeting.  These projections reveal underestimates of inflation and overestimates of the speed with which inflation would decline.  The initial 2021 surge in inflation persisted to a degree that policymakers failed to foresee.*

### 3.1    Basic Calibration

### 3.1    Basic Calibration

### Calibration of model parameters
- Discount factor: β = 0.995 (implying a steady-state real interest rate ρ equal to 2% at an annual rate).
- Goods elasticity: σ = 1.
- Frisch elasticity of labor: η = 5.
- Habit parameter: h = 0.761.
- Capital share: α = 0.25.
- Markup parameter: ε = 9 (implies steady-state price markup of 12.5%).
- Calvo parameter: ω = 0.75 (consistent with an average duration of price spells of one year, four periods in the model).
- Resulting κ in equation (14): κ = 0.169.
- Policy rule coefficients:
  - φπ = 1.5 (baseline coefficient on inflation).
  - φx = 0.5/4 = 0.125 (coefficient on the output gap).
- Policy inertia: ρi = 0.85 (baseline persistence in the policy rule).
- Policy delay considered: k ∈ {0 2 4 6} (allowing policy delay of zero to 6 periods).
- Alternative policy aggressiveness:
  - Doubling φπ from 1.5 to 3.0 evaluated.
  - Reducing ρi from 0.85 to 0.5 evaluated.
- Contemporaneous impact effect of inflation on the policy rate: (1 − ρi)φπ.
  - When φπ = 1.5 and ρi reduced from 0.85 to 0.5, the impact effect increases from 0.225 to 0.75.

### Calibration of inflation shock and target
- Inflation shock AR(1) coefficient: 0.85.
- Variance of inflation shock innovation set to generate a peak rise of inflation of 9 percent (as a deviation from the target and at an annual rate) when policy delays four periods before reacting.
- Inflation target: 2 percent.
- This calibration roughly captures a jump in inflation to over 11 percent in early 2022.

### Impulse responses under rational expectations (̃E = ̃ERE)
- Definition: An inflation surge is an unexpected positive innovation to εt, the disturbance term in the inflation equation.
- Impulse responses shown for nominal interest rate (policy rate), real interest rate, output gap, and inflation.
- Reporting convention: interest rates and inflation are reported at annual rates.
- Reference case k = 0 (policy reacts contemporaneously):
  - Policy rate rises immediately and peaks two periods after the shock.
  - Real interest rate falls reflecting a jump in inflation that is much larger than the rise in the nominal rate.
  - Inflation peaks immediately, reaching just over 3 percent at an annual rate.
  - Inflation is less than 10 basis points above its steady state after seven periods.
  - Output gap initially declines, reaching a trough four periods after the shock before slowly returning to steady state from below.
  - Trade-off: the rise in inflation is limited but at the cost of a recession.
- Effects of policy delay (increasing k):
  - When delay is more than 1 period, the output gap in the period of the shock increases rather than decreases.
  - As k increases, the increase in the output gap rises at an increasing rate; increment from k = 4 to k = 5 is much larger than from k = 3 to k = 4.
  - Delaying four periods increases peak inflation from 3 percent (annual rate) when k = 0 to 9.1 percent.
  - Delaying one additional period, to k = 5, raises peak inflation to 13.5 percent.
  - Delay produces a larger and more persistent fall in the real interest rate, stimulating output in the period of the shock due to forward-looking demand and thereby raising inflation.
  - A long delay leaves both the output gap and inflation higher at t + k − 1, ensuring persistence above outcomes with immediate policy reaction.
- Notable pattern: the k = 3 policy can appear as an “immaculate disinflation” with inflation higher than when delay is shorter and a brief positive output gap before a milder recession than under immediate reaction.

### Reacting more strongly to inflation (two interpretations)
- Motivation: empirical events in 2022 when the Fed lifted its policy rate more aggressively than normal (e.g., a 25 basis point increase in March 2022, a 50 basis point hike in May 2022, and four further increases of 75 basis points).
- Interpretation 1 — larger φπ:
  - Doubling φπ from 1.5 to 3.0:
    - When k = 0:
      - Stronger response stabilizes inflation more; output declines rather than increases.
      - Policy rate peaks at a higher level and real interest rate is higher, causing a larger fall in the output gap.
      - Lower inflation and output gap reduce upward pressure on the policy rate, causing it to fall and end up slightly lower than under the less aggressive response after t + 5.
    - When k = 4:
      - Dampening effect on the output gap and inflation is apparent even though the nominal rate path is little affected.
      - Lower inflation produces a smaller fall in the real interest rate and a smaller expansion in the output gap, accounting for the lower inflation path.
      - Conditional on delay, a higher φπ results in lower output and inflation paths, moving π and x closer to steady-state for several periods compared to the baseline φπ.
      - Delayed case shows larger changes toward steady state with larger φπ, implying stronger policy responses under delay help stabilize both inflation and the output gap.
- Interpretation 2 — lower ρi (less inertia):
  - Reducing ρi from 0.85 to 0.5 while holding φπ = 1.5:
    - When k = 0:
      - Less inertia leads to a much larger rise in the nominal rate initially and throughout convergence.
      - Higher real interest rate path and a small effect in lowering the output gap until t + 5.
      - Small effect on inflation.
    - When k = 4:
      - Less inertia front-loads the interest rate increase.
      - Combined with higher φπ (e.g., φπ = 3.0), less inertia dampens the rise in the output gap and inflation, but the output gap then turns negative sooner and for longer than under more inertial policy.
      - Trade-off: lower inflation versus a deeper and more prolonged negative output gap.

### Costs and risks of extended delay
- Extending policy reaction delay beyond 4 periods under the baseline calibration significantly increases peak rises in the output gap and inflation.
- Example statistics:
  - Increasing delay from k = 4 to k = 6:
    - Peak inflation rises from 9.1 percent to 24.6 percent.
    - Peak output gap rises from 0.9 percent to 3.4 percent.
- Implication: significantly delayed policy responses — more than one year of delay — can substantially increase fluctuations of the output gap and inflation.

*Source: 3.1 Basic Calibration (extracted from the provided IMF content unit).*

### 4.2    Measuring the costs of delay under rational expectations

### 4.2    Measuring the costs of delay under rational expectations

### Loss-function framework and calibration
- Adopt an ad hoc loss function emphasizing inflation volatility and output gap volatility:
  - General form used: L = (1/2)(ε/κ)[var(π_t) + λ var(x_t)]. (equation (16) in source)
- Weighting choices and calibration:
  - Under Woodford (2003) calibration: if weight on inflation volatility is normalized to equal 1, the weight on the volatility of the output gap is 0.048.
  - The paper adopts λ = 0.25 in line with Debortoli, Kim, Lindé, and Nunes (2019) and to approximate a dual-mandate perspective (Yellen 2012).
  - With Okun’s Law coefficient of 2: var(π_t) + var(u_t) = var(π_t) + 0.25 var(x_t).
  - Resulting simple loss function used in many comparisons: L = (1/2)(ε/κ)[var(π_t) + 0.25 var(x_t)]. (equation (17) in source)
- Two metrics focused on single-shock impulse responses (paths from shock time t to t+T):
  - L1_T(k, φ_π, ρ_i) = (1/2)(ε/κ) E_t[ (1/T) Σ_{j=0}^T β^j π_{t+j}^2 + λ (1/T) Σ_{j=0}^T β^j x_{t+j}^2 ]. (equation (18))
    - Interprets loss as cumulative average discounted squared deviations (balanced loss); λ = 0.25.
  - L2_T(k, φ_π, ρ_i) = (1/2)(ε/κ) E_t[ (1/T) Σ_{j=0}^T β^j π_{t+j} − λ (1/T) Σ_{j=0}^T β^j x_{t+j} ]. (equation (19))
    - Penalizes positive inflation deviations and negative output gap deviations; allows positive output gap to reduce loss; λ = 0.25.

### Main findings under rational expectations (impulse-response based ranking)
- Policy rule parameters and delays examined:
  - φ_π values considered: 1.5 (baseline) and 3.0 (more aggressive).
  - ρ_i values considered: 0.85 (baseline inertia) and 0.5 (less inertial).
  - Delay lags k evaluated: k = [0 1 3 4 5].
- Loss behavior (L1_T and L2_T):
  - Loss is increasing in delay k for fixed policy-rule parameters.
  - Marginal increase in loss as k rises is increasing; delays beyond 4 quarters cause very significant deterioration in performance.
  - A one-period delay (k=1) leads to only a small increase in the loss measure relative to k=0 in many specifications.
- Benefits of a more aggressive policy response:
  - Doubling φ_π from 1.5 to 3.0 reduces overall loss for all k at each horizon T.
  - Reducing inertia (ρ_i from 0.85 to 0.5) lowers volatility for all k and T.
  - Combining higher φ_π and lower ρ_i produces the lowest losses for all k and T and does not change the ranking across k.
  - Quantitative example (measured by L1_0 when k = 4):
    - Baseline policy → L1_0 = 0.82.
    - Stronger response (φ_π = 3.0) with baseline inertia → L1_0 = 0.43 (a 48 percent reduction from 0.82 to 0.43).
    - Stronger response combined with less inertia (φ_π = 3.0 and ρ_i = 0.5) → L1_0 = 0.29.
    - Total reduction moving from baseline to most aggressive rule when k = 4 is 65 percent (from 0.82 to 0.29).
- Summary implications under rational expectations:
  - Holding the policy rule fixed, any delay increases loss; the marginal cost of delaying an additional period increases with k.
  - Aggressive policy (higher φ_π and lower ρ_i) can compensate substantially for delay:
    - The loss for k = 1 under the most aggressive policy can outperform the baseline policy with k = 0.
    - The aggressive policy with k = 2 yields essentially the same loss as the baseline rule with no delay.
  - The best outcomes (per L1_T and L2_T) are obtained by responding aggressively, with less inertia and with at most a short delay.

### Role of forward guidance and expectations under rational expectations
- Forward guidance is powerful in basic New Keynesian models: a credible promise of future rate increases can be more powerful than an actual current rate rise.
- This helps explain why even a delay as long as a year (k = 4) might not be very costly if combined with an aggressive policy rule under rational expectations.

### Cognitive discounting: model modification and calibration
- Replace rational expectations by cognitive discounting: Ê = E_CD with
  - E_CD_t q_{t+d} = m̄_d E_t q_{t+d},  0 < m̄ ≤ 1 (Gabaix 2014; 2020).
  - When m̄ = 1, expectations equal rational expectations.
- Modified model equations under cognitive discounting (as stated in source):
  - x_t = [(1/(1+h))] m̄ E_t x_{t+1} + [h/(1+h)] x_{t−1} − (1/σ)(i_t − m̄ E_t π_{t+1} − r^*_t). (equation (20))
  - π_t = β M_f E_t π_{t+1} + κ[ (σ/(1−h) + (α+η)/(1−α) ) x_t − (σ h/(1−h)) x_{t−1} ] + ê_t. (equation (21))
  - ĩ_t = ρ_i ĩ_{t−1} + (1−ρ_i)[φ_π π_{t−k} + φ_x x_{t−k}]. (equation (22))
  - M_f = m̄[ ω + (1−ω) ( (1−βω)/(1−βω m̄) ) ] ≤ 1. (equation (23))
- Calibration and empirical guidance:
  - Gabaix (2020) suggests m̄ = 0.85; with typical NK parameters (ω = 0.75, β = 0.995), m̄ = 0.85 implies M_f = 0.79.
  - Ilabaca, Meggiorini, and Milani (2020) estimate m̄ = 0.71 from Euler equation and M_f = 0.41 from Phillips curve.
  - Andrade, Coredeiro, and Lambais (2019) report a confidence range for m̄ between 0.013 and 0.645.
  - The paper adopts m̄ = 0.85 for the baseline cognitive-discounting results, with robustness checks reported separately.

### Results under cognitive discounting (comparisons and implications)
- Impulse-response comparison for baseline policy with no delay:
  - Under cognitive discounting versus rational expectations:
    - Nominal and real interest rates rise more and remain higher for longer.
    - Output gap falls less (muted recession) and in some cases rises slightly initially.
    - Inflation is somewhat higher and more persistent than with rational expectations.
  - Mechanism:
    - Dampened impact of expected future output on current output (because of cognitive discounting) weakens the immediate contraction in output gap and thus contributes to higher inflation, eliciting higher nominal rates via the policy rule.
- Effects across delays k under cognitive discounting:
  - With cognitive discounting, nominal interest rate reaches highest peak when k = 0; delay reduces the policy-rate peak as k increases.
  - The magnitudes of output-gap and inflation swings are smaller under cognitive discounting than under rational expectations.
    - Example numerical ranges for output gap x:
      - Under cognitive discounting: x varies between a high of just under 0.006 when k = 5 and a low of −0.003 when k = 0.
      - Under rational expectations: x varies between a high of 0.021 (k = 5) and a low of −0.008 (k = 0).
  - Cognitive discounting reduces the marginal impact of increasing k:
    - Increasing k from 4 to 5 has about the same impact as increasing k from 3 to 4 under cognitive discounting, unlike under rational expectations where increases at higher k have larger incremental costs.
  - Forward guidance is less powerful with cognitive discounting, reducing the effectiveness of promised future policy changes and weakening the benefits of delayed promises to act.
- Aggressive response under cognitive discounting:
  - Qualitatively similar to rational-expectations findings: responding more strongly to inflation (higher φ_π) and being less inertial (lower ρ_i) reduces volatility and loss.
  - Overall amplitude of output swings is smaller and inflation rises are dampened under cognitive discounting.
  - Direct comparison under the most aggressive policy (φ_π = 3.0, ρ_i = 0.5) for k = 0 and k = 4:
    - Aggressive policies with no delay result in policy-rate rises sooner and higher, particularly under cognitive discounting, stabilizing inflation but producing persistent declines in the output gap.
    - The decline in the output gap is smaller under cognitive discounting than under rational expectations.
    - A delay of 4 periods leads to higher inflation under either expectations assumption, but yields a brief rise in the output gap followed by a smaller subsequent recession than the policy that responds immediately.
- Illustrative contextual note included in source text:
  - “Recall that the Fed waiting 4-quarters before raising its policy rate in March 2022.” (statement in source text referencing a 4-quarter delay example)

*Italic: Source: wpiea2024042-print-pdf - 4.2    Measuring the costs of delay under rational expectations*

### 5.3    Measuring the costs of delay under cognitive discounting

### 5.3    Measuring the costs of delay under cognitive discounting

### Overview
- The measures of loss given by (18) and (19), used under rational expectations, are also applied when agents display cognitive discounting.
- The paper follows the position that policy advice should continue to employ the standard loss appropriate under rational expectations even when private agents form non-rational expectations (citing the approach of Gabaix (2020) and Budianto, Nakata, and Schmidt (2023) as examples).

### Volatility and loss under cognitive discounting
- Cognitive discounting dampens the impact of future deviations from steady state on current inflation (π) and the output gap (x), reducing overall volatility regardless of policy-rule parameter values.
- The measure of volatility defined by (18):
  - Is always lower under cognitive discounting than under rational expectations (compare Figure 16 to Figure 10).
  - In the comparisons made, volatility is increasing in k at each T.
  - For each k, loss is lowest under the aggressive policy rule (lower right quadrant in the referenced figures).
  - The lowest loss as measured by (18) is achieved if k = 0 (policy reacts immediately to the shock).
- The measure of loss defined by (19):
  - Also indicates that the policy that responds immediately (k = 0) achieves the lowest loss.
  - Loss is lowest with the most aggressive of the four policy rules and is increasing with k.
  - Because delay tends, initially, to lead to a positive output gap and a shallower subsequent decline in output, (19) — which depends negatively on the cumulative discounted path of x — could, if λ is large enough, do better under a policy that incorporates a delay.
  - In practice, absolute deviations of x from zero are much smaller than deviations of π expressed at annual rates; consequently, the inflation term dominates the measure of loss.

### Comparison with rational expectations and figures referenced
- Figure 16 (volatility by (18) with cognitive discounting) should be compared to Figure 10 (rational expectations): cognitive discounting yields lower scales of volatility.
- Compare Figure 5 to Figure 13 as examples where, across four quadrants, volatility increases in k at each T and the aggressive policy rule minimizes loss.
- Figure 17 (loss by (19) with cognitive discounting) mirrors the qualitative conclusion that immediate response minimizes loss.

### Can aggression compensate for delay? (summary of Section 6 results that use the cognitive-discounting analysis)
- Baseline policy parameters used for comparison:
  - No delay (k = 0), φπ = 1.5, ρi = 0.85.
- Aggressive policy considered:
  - φπ = 3.0, ρi = 0.5, and values of k = 1, 2, 3 (and note: larger values of k produce higher losses than the baseline and so are not shown).
- When loss is measured by (18):
  - Under rational expectations (top panel of Figure 18): a delay of two periods combined with the more aggressive response matches the baseline, no-delay policy. A delay of k = 1 performs better than baseline, k = 3 does worse.
  - Under cognitive discounting (bottom panels of Figure 18): the baseline, no-delay policy can be replicated by an aggressive policy rule that delays for three periods.
  - Similar results (not shown) apply to the loss function defined in (19): with rational expectations a delay of two periods combined with an aggressive policy rule replicates the baseline rule implemented without delay.
- Footnote: 11 states that larger values of k produce higher losses than the baseline policy rule without delay and so are not shown.

### Main conclusions relating to cognitive discounting (extracted from the paper's Conclusions)
- Delay increases peak inflation and leads the output gap to rise initially under both rational expectations and cognitive discounting, due to the fall in the real interest rate.
- Delay mitigates subsequent declines in the output gap during the adjustment back to steady state, producing a trade-off:
  - Reacting more quickly initially boosts the rise in the output gap but exacerbates the rise and persistence of inflation.
- Costs of delay:
  - Not very costly if policy delays by only one or two periods; can be much more costly if delay extends to four periods or longer.
  - Loss is minimized when policy responds without delay (k = 0).
  - Volatility of inflation increases monotonically with delay.
  - For the output gap, volatility is minimized when delay is two periods.
- Should delayed policy be more aggressive?
  - Yes. A more aggressive policy is defined as placing a larger coefficient on inflation (higher φπ) or displaying less inertia (lower ρi).
  - Conditional on falling behind the curve, deviations are smallest when policy reacts strongly to inflation and with less inertia — a finding that holds under both rational expectations and cognitive discounting.
- Is a credible promise to fight inflation in the future a substitute for current action?
  - Yes, but only if the policymaker does not wait too long and responds aggressively when eventually reacting.
  - Under rational expectations, the no-delay baseline loss is replicated with an aggressive response if the policymaker waits one period before responding.
  - Under cognitive discounting, a three-period delay combined with an aggressive response can replicate the loss achieved under the no-delay baseline rule.
- Overall:
  - Stronger policy responses combined with less inertia moderate losses due to delayed policy under both expectation frameworks.
  - Excessive delay is particularly costly when agents hold rational expectations.

*The Consequences of Falling Behind the Curve: Inflation Shocks and Policy Delays under Rational and Behavioral Expectations, Working Paper No. WP/2024/042*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024042-print-pdf.pdf_