## wpiea2019166-print-pdf - Sections 2–6 (extract)

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---

### The model — framework and key structural ingredients
- Framework: bounded rationality / sparse representations following Gabaix (2014, 2019) with departures:
  - decreasing returns to scale (parameter  > 0) in production (contrast to Gabaix (2019) which assumes  = 0);
  - differentiated forms of myopia: general myopia m, real interest rate myopia m_r, real income myopia m_y, inflation/myopia by firms m_f′ and marginal-cost myopia m_fx.
- Behavioral information and perceptions:
  - behavioral perceived deviations: ^r_BR_t = m_r ^r_t(S_t), ^y_BR_t = m_y ^y_t(S_t) + w_t (N_t − N(S_t)).
- State evolution laws:
  - rational-economy law: S_{t+1} = f(S_t; ε_{t+1}) (Eq. 6).
  - behavioral-economy law: S_{t+1} = m f(S_t; ε_{t+1}) where m ∈ [0;1] (Eq. 7).

### Households — preferences, budget, and behavioral IS
- Preferences and wealth dynamics:
  - U(c_t; N_t) = c_t^{1−ρ} −1 / (1−ρ) − N_t^{1+φ}/(1+φ)  (Eq. 1).
  - k_{t+1} = (1 + r_t) (k_t − c_t + y_t) (Eq. 3).
  - y_t = w_t N_t + y^f_t (Eq. 4).
- Behavioral perceptions:
  - m_r, m_y ∈ [0;1]; m_r = m_y = 1 recovers rational budget constraint (discussion near Eq. 5).
- Behavioral IS (output-gap terms) (Eq. 8):
  - ~y_t = M E_t[~y_{t+1}] − β (i_t − E_t[π_{t+1}] − r^n_t),
    - M = m / (R − m_Y ϖ_r) (notation preserved),
    - β = m_r / (ρ R (R − m_Y ϖ_r)) (notation preserved).
- Rational IS special case (Eq. 10):
  - ~y_t = E_t[~y_{t+1}] − β_re (i_t − E_t[π_{t+1}] − r^n_t) with β_re = 1/(ρ R).
- Comparative insights:
  - M < 1 ⇒ future output affects current output less for behavioral agents.
  - Transmission of monetary policy weaker under behavioral agents (β_re ≥ β).

### Firms, price-setting, and behavioral Phillips curve
- Production and shocks:
  - Y_t(i) = A_t N_t(i)^{1−}, with  > 0; a_t = ln A_t evolves a_t = ζ_a a_{t−1} + ε^a_t (Eq. 11).
- Price-setting and behavioral expectations:
  - Calvo probability of non-adjustment ; firms maximize expected discounted profits using subjective operator E^{BR}_t (Eq. 12) subject to demand (Eq. 13).
  - Behavioral Phillips curve (welfare-relevant form) (Eq. 15 / Eq. 20):
    - π_t = Φ M_f E_t[π_{t+1}] + Λ ~y_t  (behavioral form, Eq. 15)
    - welfare-relevant: π_t = M_f E_t[π_{t+1}] + Λ x_t + u_t (Eq. 20), with u_t = ζ_u u_{t−1} + ε^u_t.
- Definitions preserved:
  - M_f = ϕ  m_f′ / (1 − (1 − ) m_f′).
  - Λ = (1 − )(1 − ϕ ) m_fx / (1 − (1 − ) m_f′) − (χ + φ + /(1 − )).
- Rational Phillips curve (Eq. 16) recovered when m_f′ = m_fx = m = 1.
- Comparative observations:
  - M_f < 1 ⇒ behavioral agents put less weight on future inflation.
  - Λ_re > Λ ⇒ sensitivity of inflation to output gap larger in rational case.
  - Λ is decreasing in : ∂Λ/∂ = m_fx β̄ < 0 (expression preserved).
  - Decreasing returns to scale ( > 0) introduces dependence of Λ on inflation myopia m_f′ (absent when  = 0).

### Welfare-relevant representation and loss function
- Welfare gaps and relations:
  - x_t = y_t − y^e_t; ~y_t = x_t + (y^e_t − y^n_t).
  - Behavioral IS in welfare terms (Eq. 19):
    - x_t = M E_t[x_{t+1}] − β (i_t − E_t[π_{t+1}] − r^e_t).
  - Behavioral Phillips (Eq. 20) reiterated: π_t = M_f E_t[π_{t+1}] + Λ x_t + u_t.
- Central bank objective (Eq. 21):
  - W = 1/2 E_0 ∑_{t=0}^∞ ϕ^t [ π_t^2 + w_x / w_π x_t^2 ],
    - w_π = ς (1 − ϕ )  / (1 − ϕ )(1 − ) (symbol combinations preserved),
    - w_x = χ + φ + /(1 − ).

### Methodology, myopia parameters, and calibration (Section 3)
- Treatment of myopia:
  - Myopia parameters exogenous in [0;1]; endogenous determination left for future research (Section 3.1).
- Main calibration:
  - “Our main experiment uses calibrated values at 15% of myopia, corresponding to a calibration of the myopia parameters at 0.85.” (Section 3.2)
- Table 1 myopia calibrations (No myopia | Myopia):
  - m_r = 1 | 0.85
  - m_fx = 1 | 0.85
  - m_f′ = 1 | 0.85
  - m_y = 1 | 0.85
  - m = 1 | 0.85
- Table 2 parameter calibrations (values preserved):
  - ϕ = 0.996
  - ρ = 2
  - " = 9 (elasticity of substitution between goods)
  -  = 1/3
  - φ = 5
  -  = 0.75
  - ζ_a = 0.75
  - ζ_u = 0.75
- Robustness checks with alternative calibrations in Appendix B.

### Optimal policy problem and commitment (Section 4)
- Central bank maximizes W (Eq. 21) subject to behavioral IS and Phillips; commitment solution analyzed.
- First-order conditions under commitment (Eqs. 22–23):
  - π_t + λ_t − M_f ϕ λ_{t−1} = 0 (Eq. 22).
  - w_x / w_π x_t − Λ λ_t = 0 (Eq. 23).
- Derived targeting rule (Eq. 26):
  - π_t = − (w_x / Λ w_π) [ x_t − M_f ϕ x_{t−1} ].
- Price level under commitment (Eq. 27):
  - p_t = − (w_x / Λ w_π) [ x_t +  (1 − M_f ϕ)^{-1} ∑_{j=0}^{t−1} x_j ].
- Proposition 1 (Section 4.1) — main policy result:
  - PLT optimal when m = 1 and m_f′ = 1 (M_f ϕ = 1) ⇒ stationary price level.
  - IT optimal when inflation myopia present (m_f′ ≠ 1) because PLT requires excessive future disinflationary engineering.
- Central takeaway:
  - Optimal monetary targeting depends on which myopia parameters are active; PLT is optimal only when agents fully attend to inflation expectations.

### Simulation and welfare (Section 4.2)
- Impulse responses under commitment to a 1 percent cost-push shock:
  - Trade-off between output gap and inflation; intensity varies by myopia type.
  - Full myopia: substantial inflation increase, significant output drop; price level never returns to steady state.
  - Inflation myopia: PLT suboptimal, IT optimal.
  - Output gap, revenue, and interest-rate myopia: inflation rises on impact then decreases to deflation; price level can reach steady-state implying PLT optimal in these cases.
  - Central bank reaction amplitudes:
    - Output-gap, inflation, revenue myopia: impulse response amplitudes very close to rational case.
    - Interest-rate, general, full myopia: require strong central bank reactions; price level typically does not recover original steady state.
  - Robustness: impulse response results robust across calibrations (Appendix B).
- Welfare losses under commitment (Table 3) by myopia type (No myopia / Myopia):
  - Rational: 0.174
  - Interest rate: 0.174
  - Output gap: 0.227
  - Inflation: 0.190
  - Revenue: 0.174
  - General: 0.176
  - Full: 0.248
- Interpretation:
  - Rational case lowest welfare loss.
  - Interest-rate and revenue myopia equal rational benchmark because central bank loss does not penalize interest rate or revenue deviations.
  - General myopia yields welfare losses close to interest-rate and revenue myopia cases.
  - Bounded rationality not necessarily welfare-decreasing.

### Discretionary regime — analytical characterization and simulations (Sections 5.1–5.2)
- Discretionary targeting criterion (Eq. 28):
  - π_t = -(w_x / w_π) x_t.
- Solved forward (Eqs. 29–30):
  - π_t = (w_x / w_π) / (w_x / w_π + λ^2 - w_x / w_π M_f ) u_t.
  - x_t = -λ / (w_x / w_π + λ^2 - w_x / w_π M_f ) u_t.
- Key implications:
  - Central bank lets output gap and inflation deviate proportionally to u_t; inflation determinate, price level indeterminate ⇒ IT preferred under discretion.
  - Bounded rationality affects magnitudes via λ and M_f but not choice of regime under discretion.
- Impulse responses under optimal discretionary policy to a 1 percent cost-push shock:
  - Policy rate generally increased aggressively relative to rational benchmark (except revenue myopia).
  - Price level not stationary in any myopia case under discretion.
- Welfare losses under discretion (Table 4):
  - Rational: 0.270
  - Interest rate: 0.270
  - Output gap: 0.386
  - Inflation: 0.287
  - Revenue: 0.270
  - General: 0.236
  - Full: 0.341
- Interpretation:
  - General myopia can yield lower welfare loss than rational benchmark under discretion.

### Simple rules: regimes and optimal coefficients (Section 6)
- Rule taxonomy: flexible and strict forms of IT, PLT, NGDP growth, NGDP level (F1–F4; S1–S4); instrument rules include monetary policy shock "ε_mp,t".
- Selected optimal coefficients (Table 6) — preserved values:
  - No (rational):
    - F1: φ_π = 1.96, φ_y = 0.25
    - F2: φ_p = 0.33, φ_y = 0.0
    - F3: φ_g = 2.62, φ_y = 0.5
    - F4: φ_n = 0.17, φ_y = 0.0
    - S1: φ_π = 2.37
    - S2: φ_p = 0.34
    - S3: φ_g = 3.90
    - S4: φ_n = 0.17
  - Interest rate myopia:
    - F1: φ_π = 2.44, φ_y = 0.20
    - S1: φ_π = 3.11
    - S3: φ_g = 4.00
  - Output gap myopia:
    - F1: φ_π = 1.39, φ_y = 0.32
    - S1: φ_π = 2.02
    - S3: φ_g = 3.43
  - Inflation myopia:
    - F1: φ_π = 1.43, φ_y = 0.27
    - S1: φ_π = 1.99
    - S3: φ_g = 3.26
  - Revenue myopia:
    - F1: φ_π = 2.03, φ_y = 0.21
    - S1: φ_π = 2.37
    - S3: φ_g = 3.91
  - General myopia:
    - F1: φ_π = 2.05, φ_y = 0.14
    - F2: φ_p = 0.56
    - S1: φ_π = 2.38
    - S3: φ_g = 3.34
  - Full myopia:
    - F1: φ_π = 1.54, φ_y = 0.18
    - F2: φ_p = 0.49
    - S1: φ_π = 2.10
    - S3: φ_g = 2.82
- General findings on coefficients:
  - Inflation coefficients under F1 and S1 > 1 in all myopia cases.
  - Interest-rate myopia increases instrument sensitivity to targets (stronger signaling).
  - Output-gap myopia decreases sensitivity to target, increases reaction to output gap in F1.
  - General and full myopia induce aggressive central bank reactions to curb expectations.
  - Strict targeting yields higher sensitivities than flexible targeting.
  - S3 coefficients exceed one across myopia types (Taylor principle); F2, S2, F4, S4 coefficients positive but < 1.
  - Flexible PLT (F2) and NGDP level targeting (F4) show null optimal output gap coefficients.

### Which rule best describes the first-best solution? (Section 6.2)
- Welfare comparisons:
  - Flexible targeting rules often generate similar welfare losses to strict counterparts (e.g., F1 and S1 identical in some cases).
  - Strict PLT delivers the lowest welfare among considered rules across myopia types when used as an instrument-rule proxy.
  - Rational case delivers similar welfare losses to interest rate and revenue myopia cases.
  - Output gap and full myopia imply largest welfare losses across bounded rationality cases.
  - General myopia combined with appropriate policy can yield smaller welfare losses than rational case in some regimes.
- Reported welfare loss values (Figure 3) — preserved list:
  - 0.2093, 0.2093, 0.2264, 0.2093, 0.1997, 0.1766, 0.1766, 0.1923, 0.1766, 0.1773, 0.2161, 0.2162, 0.2361, 0.2161, 0.2110, 0.1855, 0.1857, 0.2016, 0.1855, 0.1840, 0.2093, 0.2094, 0.2264, 0.2093, 0.1997, 0.1762, 0.1763, 0.1919, 0.1762, 0.1772, 0.2167, 0.2168, 0.2378, 0.2167, 0.2134, 0.1852, 0.1854, 0.2013, 0.1853, 0.1838, 0.2848, 0.2849, 0.2317, 0.2518, 0.2976, 0.3091, 0.2456, 0.2612, 0.2848, 0.2849, 0.2310, 0.2517, 0.2993, 0.3205, 0.2450, 0.2609.
- Role of bounded rationality:
  - First-best (commitment) depends on myopia type: PLT optimal in some cases, IT in others.
  - No single regime universally optimal.
  - High general myopia can improve welfare under commitment, discretion, and optimal simple rules; extreme general myopia demonstrates robustness.
  - Output gap myopia notably increases welfare losses relative to rational case.
- Limits of simple instrument rules:
  - Optimal simple rules favor strict PLT as instrument-rule proxy in all bounded rationality cases, despite commitment first-best calling for either IT or PLT depending on myopia.
  - Mechanical instrument rules inadequate for forward-looking policymaking in a behavioral world.
- Policy recommendations and institutional implications:
  - Central banks should assess, monitor, and analyze agents' myopia as core input for monetary policy design.
  - Communication and public education can attenuate myopia; some myopia may raise welfare and need not always be eliminated.
  - Central banks should condition choice of targeting regime on empirical assessments of agents' myopia.
- Conclusion:
  - First-best in behavioral New Keynesian framework depends on myopia type; neither IT nor PLT universally optimal.
  - Optimal simple rules tend to favor strict PLT, but simple rules fail to replicate commitment first-best, underscoring need to incorporate targeting rules and myopia assessment into policymaking.

### Appendix A — derivations (selected technical results)
- IS derivation highlights:
  - Behavioral Taylor expansion of consumption deviations: ^c_{t} = E_{t} sum_{j=0}^{∞} β^{j} m^{j-t} / R^{j+1} [ b_{r} m_{r} ^r_{t+j} + b_{y} m_{y} ^y_{t+j} ].
  - Market clearing and resulting IS-like expression: ~y_{t} = M E_{t}[~y_{t+1}] - ϕ ^r_{t}, with M and ϕ functional forms preserved.
- Phillips derivation highlights:
  - Term structure of behavioral expectations: E^{BR}_{t}[ζ_{t+k}] = m_{f}^{ζ} m^{k} E_{t}[ζ_{t+k}].
  - Behavioral Phillips: ζ_{t} = β M_{f} E_{t}[ζ_{t+1}] + Λ ~y_{t} (Eq. 68), with M_{f} and Λ expressions preserved.
- Natural output and efficient interest rate:
  - y^{n}_{t} expression preserved (Eqs. 73–74).
  - Efficient interest rate deviations: ^r^{e}_{t} = (1/ϕ) [ M E_{t}[ ^y^{e}_{t+1} ] - ^y^{e}_{t} ] (Eq. 85).
- Welfare loss derivation:
  - Period loss and aggregate W expressions preserved (Eqs. 90–91), with identification of variance term Var_{i}[ p_{t}(i) ].

### Robustness checks (Appendix B)
- Overview:
  - Robustness checks across alternative model and myopia calibrations yield similar qualitative conclusions.
- Model calibrations (Table 7) — preserved calibration rows:
  - GalÌ (2008)  0.99   1  1  6  1/3  0.66
  - Relative risk aversion  0.99   2  1  6  1/3  0.66
  - Frisch elasticity  0.99   1  5  6  1/3  0.66
  - Constant return to scale  0.99   1  1  6   0   0.66
  - Sticky prices  0.99   1  1  6  1/3  3/4
  - Time preferences  0.996  1  1  6  1/3  0.66
  - Demand elasticity  0.99   1  1  9  1/3  0.66
  - GalÌ (2015)  0.996  2  5  9  1/3  3/4
- Key robustness findings:
  - When  ≠ 0, trade-off between inflation and output worsens; central bank acts more aggressively compared to  = 0 calibration.
  - Increasing Frisch elasticity or decreasing returns to scale lowers welfare losses across myopia types.
  - Welfare-improving effects of general myopia robust under discretion and optimal simple rules; under commitment clearer for extreme myopia values.
- Table 8 welfare losses under alternative calibrations — preserved numeric blocks (commitment and discretion values listed in source).
- Myopia calibrations (Table 9) and extreme cases:
  - More pronounced myopic agents with approximately 80% myopia and an extreme case with almost fully myopic agent (99%) considered.
  - Figure 12 and 13 impulse responses reported for these calibrations.
- Table 10 (GalÌ (2015) calibration) welfare losses — preserved numeric entries:
  - Commitment: 0.174 1.446 0.257 0.174 0.143 0.372 0.302
  - Discretion: 0.270 3.357 0.348 0.270 0.145 0.372 0.302
- Robustness conclusion:
  - There exists general myopia levels that improve welfare losses across regimes; qualitative conclusions robust across calibrations.

*Italic source attribution: wpiea2019166-print-pdf — Extracted content from Sections 2–6, Appendix A and Appendix B as provided.*

### Section 9 presents the concluding remarks, and Section 10 presents our derivations

### wpiea2019166-print-pdf - Section 9 presents the concluding remarks, and Section 10 presents our derivations

### The model (Section 2) — framework and key structural ingredients
- Framework: bounded rationality / sparse representations following Gabaix (2014, 2019) with departures:
  - decreasing returns to scale (parameter  > 0) in production (contrast to Gabaix (2019) which assumes  = 0);
  - differentiated forms of myopia (general myopia m, real interest rate myopia m_r, real income myopia m_y, inflation/myopia by firms m_f′ and marginal-cost myopia m_fx).
- Agents observe only a subset of variables and form perceived deviations from steady state:
  - behavioral perceived deviations: ^r_BR_t = m_r ^r_t(S_t), ^y_BR_t = m_y ^y_t(S_t) + w_t (N_t − N(S_t)).
- State evolution:
  - rational-economy law: S_{t+1} = f(S_t; ε_{t+1}) (Eq. 6).
  - behavioral-economy law: S_{t+1} = m f(S_t; ε_{t+1}) where m ∈ [0;1] (Eq. 7).

### Households — preferences, budget, and behavioral IS
- Household utility (representative, infinitely lived) (Eq. 1):
  - U(c_t; N_t) = c_t^{1−ρ} −1 / (1−ρ) − N_t^{1+φ}/(1+φ)  (notation preserved as in source).
  - ρ is the coefficient of relative risk aversion; φ is the inverse of the Frisch elasticity.
- Wealth dynamics and income (Eqs. 3–4):
  - k_{t+1} = (1 + r_t) (k_t − c_t + y_t) (Eq. 3).
  - y_t = w_t N_t + y^f_t (Eq. 4).
- Behavioral household:
  - perceives only fractions m_r and m_y of deviations; m_r, m_y ∈ [0;1]; m_r = m_y = 1 recovers rational budget constraint (discussion near Eq. 5).
  - behavioral IS (in output-gap terms) (Eq. 8):
    - ~y_t = M E_t[~y_{t+1}] − β (i_t − E_t[π_{t+1}] − r^n_t), where
      - M = m / (R − m_Y ϖ_r) (notation as in source; exact functional forms provided in text),
      - β = m_r / (ρ R (R − m_Y ϖ_r)) (source expressions preserved).
- Rational IS as special case (Eq. 10):
  - ~y_t = E_t[~y_{t+1}] − β_re (i_t − E_t[π_{t+1}] − r^n_t) with β_re = 1/(ρ R).
- Comparative insight:
  - M < 1 ⇒ future output affects current output less for behavioral agents.
  - Transmission of monetary policy weaker under behavioral agents (β_re ≥ β).

### Firms, price-setting, and behavioral Phillips curve
- Production technology (Eq. 11):
  - Y_t(i) = A_t N_t(i)^{1−}, with decreasing returns parameter  > 0; a_t = ln A_t evolves a_t = ζ_a a_{t−1} + ε^a_t.
- Calvo pricing with probability of non-adjustment :
  - Firms maximize expected discounted profits using subjective operator E^{BR}_t (Eq. 12) subject to demand (Eq. 13).
- Firm FOC expanded around zero-inflation steady state yields a pricing relation (Eq. 14).
- Behavioral Phillips curve (welfare-relevant form) (Eq. 15 / Eq. 20):
  - π_t = Φ M_f E_t[π_{t+1}] + Λ ~y_t  (behavioral form, Eq. 15)
  - welfare-relevant: π_t = M_f E_t[π_{t+1}] + Λ x_t + u_t (Eq. 20), where u_t is a cost-push AR(1): u_t = ζ_u u_{t−1} + ε^u_t.
  - Definitions:
    - M_f = ϕ  m_f′ / (1 − (1 − ) m_f′) (notation as in source);
    - Λ = (1 − )(1 − ϕ ) m_fx / (1 − (1 − ) m_f′) − (χ + φ + /(1 − ))  (exact composite expression preserved in text).
- Rational Phillips curve (special case m_f′ = m_fx = m = 1) (Eq. 16):
  - π_t = ϕ E_t[π_{t+1}] + Λ_re ~y_t with Λ_re given in text.
- Key comparative observations:
  - M_f < 1 ⇒ behavioral agents put less weight on future inflation in current inflation.
  - Λ_re > Λ ⇒ sensitivity of inflation to output gap is larger in rational case.
  - Λ is decreasing in : ∂Λ/∂ = m_fx β̄ < 0 (source expression for β̄ given).
  - Decreasing returns to scale ( > 0) introduces dependence of Λ on inflation myopia m_f′ (absent in Gabaix (2019) where  = 0).

### Welfare-relevant representation and loss function
- Welfare-relevant output gap defined: x_t = y_t − y^e_t; relation to natural output gap ~y_t = x_t + (y^e_t − y^n_t).
- Behavioral IS in welfare-relevant terms (Eq. 19):
  - x_t = M E_t[x_{t+1}] − β (i_t − E_t[π_{t+1}] − r^e_t) with r^e_t defined in the text.
- Behavioral Phillips in welfare terms (Eq. 20) repeated above; u_t defined as cost-push shock.
- Central bank objective: microfounded welfare loss (Eq. 21):
  - W = 1/2 E_0 ∑_{t=0}^∞ ϕ^t [ π_t^2 + w_x / w_π x_t^2 ] (Eq. 21), with
    - w_π = ς (1 − ϕ )  / (1 − ϕ )(1 − ) (exact symbol combinations preserved as in source),
    - w_x = χ + φ + /(1 − ) (text preserves algebraic definitions).

### Methodology, myopia parameters, and calibration (Section 3)
- Myopia parameters treated as exogenous in [0;1]; endogenous determination left for future research (Section 3.1).
- Main calibration:
  - “Our main experiment uses calibrated values at 15% of myopia, corresponding to a calibration of the myopia parameters at 0.85.” (Section 3.2)
  - Table 1 (myopia calibrations):
    - Models: No myopia vs Myopia (columns described).
    - m_r = 1 | 0.85 (Rational | Myopia)
    - m_fx = 1 | 0.85
    - m_f′ = 1 | 0.85
    - m_y = 1 | 0.85
    - m = 1 | 0.85
  - Table 2 (other parameter calibrations):
    - ϕ = 0.996 (static discount factor)
    - ρ = 2 (household’s relative risk aversion)
    - " = 9 (elasticity of substitution between goods)
    -  = 1/3 (return to scale)
    - φ = 5 (Frisch elasticity inverse)
    -  = 0.75 (probability firms do not adjust prices)
    - ζ_a = 0.75 (technology shock persistence)
    - ζ_u = 0.75 (cost-push shock persistence)
  - Robustness checks with alternative calibrations available in Appendix B (referenced in source).

### Optimal policy problem and commitment (Section 4)
- Central bank maximizes microfounded welfare (Eq. 21) subject to behavioral IS and Phillips constraints; commitment solution analyzed analytically.
- First-order conditions under commitment (Eqs. 22–23):
  - π_t + λ_t − M_f ϕ λ_{t−1} = 0 (Eq. 22)
  - w_x / w_π x_t − Λ λ_t = 0 (Eq. 23), where λ_t is the Lagrange multiplier.
- Derived central-bank targeting rule (Eq. 26):
  - π_t = − (w_x / Λ w_π) [ x_t − M_f ϕ x_{t−1} ] (Eq. 26)
- Price level expression under commitment (Eq. 27):
  - p_t = − (w_x / Λ w_π) [ x_t +  (1 − M_f ϕ)^{-1} ∑_{j=0}^{t−1} x_j ] (Eq. 27) (equation form preserved).
- Proposition 1 (main policy result, Section 4.1):
  - A form of Price-Level Targeting (PLT) is optimal when m = 1 and m_f′ = 1.
  - A form of Inflation Targeting (IT) is optimal when this condition is not satisfied.
- Proof sketch and interpretation (as given in source):
  - When M_f ϕ = 1 (which obtains when m = 1 and m_f′ = 1), the first-order condition implies a stationary price level — PLT is optimal.
  - If agents are myopic to inflation (m_f′ ≠ 1), PLT requires excessive future disinflationary engineering by the central bank because the transmission from output to the price level is incomplete; thus IT becomes optimal.
- Policy implications and nuance:
  - Contrary to a blanket conclusion that bounded rationality makes PLT suboptimal, the result depends on which forms of myopia prevail.
  - If households and firms are fully attentive to inflation (no inflation myopia) but are myopic in other dimensions (interest rate, output, revenue), PLT can still deliver first-best outcomes.
  - If inflation myopia exists (m_f′ < 1), commitment-based PLT is socially costly because monetary-induced recessions transmit imperfectly to the price level, necessitating successive rounds of de‑inflationary policy at the expense of output.
- Central takeaway (Section 4.1 final paragraph):
  - No universal prescription: optimal monetary targeting depends on the specific configuration of behavioral attentiveness (which myopia parameters are active). PLT is optimal in behavioral environments only when agents fully attend to inflation expectations; otherwise IT is optimal.

*Italic source attribution: wpiea2019166-print-pdf — Extracted content from Sections 2–4 as provided.*

### 4.2  Simulation and welfare

### 4.2  Simulation and welfare

### Impulse responses under commitment
- A 1 percent cost-push shock generates a trade-off between the output gap and inflation; the intensity of the trade-off differs slightly by form of myopia.
- Full myopia:
  - Entails a substantial increase in inflation with a significant drop in output.
  - Requires a strong reaction from the central bank.
  - The price level never returns to its steady state after a cost-push shock, corroborating the analytical result about the suboptimality of PLT in this case.
- Inflation myopia:
  - PLT is suboptimal while IT is optimal because stabilizing the price level in a behavioral world imposes welfare costs via central bank decisions.
- Output gap myopia, revenue myopia, and interest-rate myopia:
  - Following a cost-push shock, inflation rises on impact but decreases to deflation after some periods.
  - The price level reaches its steady-state value, implying a form of PLT is an optimal monetary policy.
- Central bank reaction amplitudes:
  - Output-gap myopia, inflation myopia, and revenue myopia: impulse response amplitudes are very close to the rational case.
  - Interest-rate myopia, general myopia, and full myopia: require strong central bank reactions; optimal policy sharply offsets the shock and typically does not allow the price level to recover its original steady-state value (finds a new steady-state instead).
  - In cases where reactions are smoother, the central bank allows deflation to operate and the price level to be stationary.
- Robustness: impulse response results confirm analytical results from Section 4.1 and are robust to different model and myopia calibrations reported in Appendix B.

### Welfare losses under commitment (Table 3)
- Welfare losses under commitment by myopia type (No myopia / Myopia):
  - Rational: 0.174
  - Interest rate: 0.174
  - Output gap: 0.227
  - Inflation: 0.190
  - Revenue: 0.174
  - General: 0.176
  - Full: 0.248
- Interpretation:
  - The rational case generates the lowest welfare loss (consistent with perfect foresight).
  - Interest-rate myopia and revenue myopia produce the same welfare losses as the rational benchmark because the central bank loss does not penalize deviations of interest rate or revenue, and agents remain well-informed about output and inflation.
  - General myopia yields welfare losses very close to the interest-rate and revenue myopia cases.
  - Bounded rationality is not necessarily welfare-decreasing.

### Discretionary regime — analytical characterization (Section 5.1)
- Under discretion the central bank minimizes the period welfare loss taking expectations as given; the targeting criterion is:
  - π_t = -(w_x / w_π) x_t  (Eq. 28 as presented)
- Solving forward with the Phillips curve yields:
  - π_t = (w_x / w_π) / (w_x / w_π + λ^2 - w_x / w_π M_f ) u_t  (expression form as shown in Eq. 29)
  - x_t = -λ / (w_x / w_π + λ^2 - w_x / w_π M_f ) u_t  (expression form as shown in Eq. 30)
- Key implications:
  - The central bank lets output gap and inflation deviate proportionally to the cost-push shock u_t.
  - Bounded rationality influences magnitudes through λ (which depends on output-gap and inflation myopias, m_f^x and m_f^π) and through M_f (which depends on general and inflation myopia, m and m_f^π).
  - The optimal policy under discretion yields determinate inflation but an indeterminate price level, implying IT as the preferred regime under discretion.
  - Bounded rationality affects magnitudes of reactions but not the choice of policy regime under discretion because policy takes expectations as exogenous and accommodates the current-period shock.

### Discretionary regime — simulation and welfare (Section 5.2)
- Impulse responses to a 1 percent cost-push shock under optimal discretionary policy:
  - Optimal policy generally increases the policy rate to accommodate higher inflation, and does so more aggressively than the rational benchmark except for revenue myopia.
  - The price level is not stationary in any myopia case under discretion, supporting IT as desirable.
- Welfare losses under discretion (Table 4):
  - Rational: 0.270
  - Interest rate: 0.270
  - Output gap: 0.386
  - Inflation: 0.287
  - Revenue: 0.270
  - General: 0.236
  - Full: 0.341
- Interpretation:
  - General myopia yields lower welfare loss than the rational benchmark under discretion, reflecting that distorted expectations across all macroeconomic variables can be consistent with a discretionary policymaker.

### Simple rules: regime definitions and optimal coefficients (Section 6)
- Rule taxonomy (as described):
  - Flexible and strict forms of targeting for Inflation (IT), Price Level (PLT), NGDP growth, and NGDP level (F1–F4; S1–S4).
  - Instrument rules include a monetary policy shock term "ε_mp,t" when the central bank deviates from the rule.
- Optimal coefficients for simple rules (Table 6) — selected highlights:
  - Coefficients reported for rules F1, F2, F3, F4, S1, S2, S3, S4 across myopia cases.
  - No (rational):
    - F1: φ_π = 1.96, φ_y = 0.25
    - F2: φ_p = 0.33, φ_y = 0.0
    - F3: φ_g = 2.62, φ_y = 0.5
    - F4: φ_n = 0.17, φ_y = 0.0
    - S1: φ_π = 2.37
    - S2: φ_p = 0.34
    - S3: φ_g = 3.90
    - S4: φ_n = 0.17
  - Interest rate myopia:
    - F1: φ_π = 2.44, φ_y = 0.20
    - S1: φ_π = 3.11
    - S3: φ_g = 4.00
  - Output gap myopia:
    - F1: φ_π = 1.39, φ_y = 0.32
    - S1: φ_π = 2.02
    - S3: φ_g = 3.43
  - Inflation myopia:
    - F1: φ_π = 1.43, φ_y = 0.27
    - S1: φ_π = 1.99
    - S3: φ_g = 3.26
  - Revenue myopia:
    - F1: φ_π = 2.03, φ_y = 0.21
    - S1: φ_π = 2.37
    - S3: φ_g = 3.91
  - General myopia:
    - F1: φ_π = 2.05, φ_y = 0.14
    - F2: φ_p = 0.56
    - S1: φ_π = 2.38
    - S3: φ_g = 3.34
  - Full myopia:
    - F1: φ_π = 1.54, φ_y = 0.18
    - F2: φ_p = 0.49
    - S1: φ_π = 2.10
    - S3: φ_g = 2.82
- General findings on coefficients:
  - Inflation coefficients under flexible and strict IT (F1 and S1) are strictly greater than one in all myopia cases, consistent with stability conditions.
  - Interest-rate myopia increases the sensitivity of the policy instrument to the central bank target (central bank reacts more aggressively to send an appropriate signal).
  - Output-gap myopia decreases sensitivity of the interest rate to the central bank target but increases reaction to the output gap in flexible IT (F1).
  - Inflation myopia and revenue myopia produce sensitivities similar to or smaller than the rational case, due to altered transmission channels.
  - General and full myopia induce the central bank to react aggressively to curb expectations.
  - Strict targeting generally yields higher sensitivity to targets than flexible targeting.
  - Coefficients for strict NGDP growth targeting (S3) are higher than for flexible NGDP growth targeting (F3) across all myopia types and exceed one (Taylor principle).
  - Coefficients for price level targeting (F2, S2) and NGDP level targeting (F4, S4) are positive but lower than one.
  - Flexible PLT (F2) and NGDP level targeting (F4) show null optimal values for the output gap coefficient, implying the central bank need not respond to real fluctuations under these regimes.
  - All optimal coefficients depend on agent myopia; interest-rate myopia delivers the largest amplitude under IT and NGDP growth targeting, while general myopia delivers the highest coefficients under price level and NGDP level targeting.
  - For F2 and S2, sensitivity of the instrument to the price level does not vary significantly between flexible and strict PLT; the same holds for F4 and S4.

*Source: wpiea2019166-print-pdf*

### 6.2  Which rule best describes the Örst best solution?

### 6.2  Which rule best describes the Örst best solution?

### Welfare comparisons under alternative rules
- Flexible targeting rules do not necessarily induce welfare losses compared to strict rules; many flexible targeting rules generate similar welfare losses to their corresponding strict targeting rules (for instance, welfare losses are identical between F1 and S1).
- Strict PLT delivers the lowest welfare among the considered rules across all types of myopia.
- The welfare losses associated with strict PLT are similar to the flexible PLT rule through different myopia cases because the optimal value of the output gap feedback to the interest rate in rule F2 is null (a case of divine coincidence when the central bank pursues a price level objective).
- The rational case delivers similar welfare losses to interest rate and revenue myopia cases (consistent with Tables 3 and 4 referenced in the text).
- Across bounded rationality cases, output gap and full myopia imply the largest welfare losses compared to other cases; however, general myopia combined with appropriate central bank action sometimes yields smaller welfare losses compared to the rational case (as in the discretion case, Table 4).

### Reported welfare loss values (Figure 3)
- 0.2093
- 0.2093
- 0.2264
- 0.2093
- 0.1997
- 0.1766
- 0.1766
- 0.1923
- 0.1766
- 0.1773
- 0.2161
- 0.2162
- 0.2361
- 0.2161
- 0.2110
- 0.1855
- 0.1857
- 0.2016
- 0.1855
- 0.1840
- 0.2093
- 0.2094
- 0.2264
- 0.2093
- 0.1997
- 0.1762
- 0.1763
- 0.1919
- 0.1762
- 0.1772
- 0.2167
- 0.2168
- 0.2378
- 0.2167
- 0.2134
- 0.1852
- 0.1854
- 0.2013
- 0.1853
- 0.1838
- 0.2848
- 0.2849
- 0.2317
- 0.2518
- 0.2976
- 0.3091
- 0.2456
- 0.2612
- 0.2848
- 0.2849
- 0.2310
- 0.2517
- 0.2993
- 0.3205
- 0.2450
- 0.2609

### Role of bounded rationality and implications for the optimal policy
- The first-best (commitment) optimal policy depends on the type of myopia characterizing agents; a form of PLT is optimal in some myopia cases, while IT is optimal in the remaining cases.
- No single targeting regime (IT or PLT) is consistently optimal across all states of the world.
- Deviations from rationality can corroborate some rational-expectations results (by setting myopia parameters to 1) but also produce results that contrast with the behavioral macro literature when myopia parameters differ from 1.
- Bounded rationality can sometimes improve welfare: high general myopia always improves welfare losses under commitment, discretion, and optimal simple rule regimes; extreme general myopia can increase welfare in all monetary policy regimes considered.
- Output gap myopia notably implies significant welfare losses relative to the rational case.

### Limits of simple instrument rules versus targeting rules
- Optimal simple (instrument) rules favor strict PLT in all bounded rationality cases considered, making strict PLT the desirable instrument-rule proxy for the optimal monetary policy despite the commitment (first-best) solution calling for either IT or PLT depending on myopia.
- This inability of simple rules to replicate the first-best solution highlights the shortcomings of mechanical instrument rules for forward-looking policymaking and supports including targeting rules (as in Proposition 1) in the central bank toolbox.
- Mechanical instrument rules are likely inadequate for optimizing policy in a behavioral world where agents' inattention/myopia changes across circumstances.

### Policy recommendations and institutional implications
- Central banks should assess, monitor, and analyze agents' myopia as a core input for monetary policy design; assessing the degree of myopia is an area for increased investment by central banks.
- Communication and public education campaigns by central banks can be seen as efforts to attenuate myopia and guide agents toward more rational expectations, but bounded rationality is not always a problem to be eliminated—some myopia can raise welfare.
- Given that both PLT and IT can be optimal under different bounded-rationality configurations, central banks should condition their choice of targeting regime on empirical assessments of agents' myopia.

### Conclusion
- The first-best solution in a behavioral New Keynesian framework depends on the type of myopia; neither IT nor PLT is universally optimal.
- Optimal simple rules tend to favor strict PLT across bounded rationality cases, but simple rules fail to replicate the commitment first-best solution, underscoring the need to reconsider the use of simple instrument rules and to incorporate targeting rules and myopia assessment into monetary policymaking.

*Source: wpiea2019166-print-pdf - 6.2  Which rule best describes the Örst best solution?*

### References

### References

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### Appendix A — Derivations

### A.1 IS curve
- Lagrangian of the optimization problem:
  - L = sum_{t=0}^{1} β^{t} u(c_{t};N_{t}) + sum_{t=0}^{1} β^{t} λ_{t} (k_{t} - (1 + r_{t}) (k_{t-1} - c_{t-1} + y_{t-1}))
  - where r_{t} = \bar r + m_{r} \hat r_{t}, y_{t} = \bar y + m_{y} \hat y_{t}, and λ_{t} = ∂V(k_{t})/∂k_{t}.
- Value function: V(k_{t}) = max_{c} { u(c) + β V(k_{t+1}) }.
- Envelope theorem yields expressions (32)–(36) relating ∂V/∂r_{t}, ∂V/∂y_{t}, and derivatives of consumption with respect to shocks.
- Taylor expansion of consumption deviations:
  - ^c_{t} = E_{t} sum_{j=0}^{∞} β^{j} [ b_{r} ^r_{t+j} + b_{y} ^y_{t+j} ] / R^{j+1}
  - with behavioral agent form (43): ^c_{t} = E_{t} sum_{j=0}^{∞} β^{j} m^{j-t} / R^{j+1} [ b_{r} m_{r} ^r_{t+j} + b_{y} m_{y} ^y_{t+j} ].
- Market clearing y_{t} = c_{t} and output gap notation:
  - ~y_{t} = ^c_{t}/c = ^y_{t}/c.
- Resulting IS-like expression:
  - ~y_{t} = M E_{t}[~y_{t+1}] - ϕ ^r_{t}
  - where M = m / R - ρ r m_{y}, ϕ = m_{r} / (c R (R - r m_{y})), and R = 1/β.

### A.2 Phillips curve
- Firm problem and demand constraint (48)–(49) with notation for stochastic discount factor and cost function.
- First-order condition summarized in (50)–(54).
- Deviation and log notation: lower-case p_{t} = ln P_{t}, cmc_{t+k|t} = mc_{t+k|t} - mc.
- Marginal cost relation (55) used to derive price-setting condition (56)–(61).
- Term structure of behavioral expectations from Gabaix (2019):
  - E^{BR}_{t}[ζ_{t+k}] = m_{f}^{ζ} m^{k} E_{t}[ζ_{t+k}], with distinct myopias m, m_{f}^{ζ}, m_{f}^{x}.
- Behavioral Phillips curve derived:
  - ζ_{t} = β M_{f} E_{t}[ζ_{t+1}] + Λ ~y_{t}  (Eq. 68)
  - with M_{f} = ξ m / (1 - (1-ξ) m_{f}^{ζ}) and Λ = (1-ξ) (1-β ξ) κ m_{f}^{x} / (1 - (1-ξ) m_{f}^{ζ}) - (χ + ϵ + β)/(1-β), where parameters retain their original definitions in the text.
- Note: Rational case m_{f}^{x} = m_{f}^{ζ} = m = 1 recovers the standard Phillips curve as in GalÌ (2015).

### A.3 Natural output
- Marginal cost definition:
  - μ_{t} = - w_{t} - mpn_{t}  (Eq. 69)
  - real wage: -U_{n,t}/U_{c,t} = W_{t}/P_{t} => w_{t} = ϵ n_{t} + ψ c_{t}.
- Marginal productivity: mpn_{t} = a - β n_{t} + ln(1-β).
- Using production function y_{t} = a_{t} + (1-β) n_{t}, marginal cost in terms of y_{t} and a_{t} (72).
- Flexible-price economy marginal cost μ and natural output y^{n}_{t} defined in (73)–(74):
  - y^{n}_{t} = [1 + ϵ]/[ϵ + β + ψ (1-β)] a_{t} + (1-β)(-μ + ln(1-β)) / [ϵ + β + ψ (1-β)].

### A.4 Efficient interest rate
- IS curve in deviation form (75):
  - ^y_{t} = M E_{t}[^y_{t+1}] - ϕ ( i_{t} - E_{t}[ζ_{t+1}] - r^{n}_{t} ).
- Definitions:
  - ^y_{t} = y_{t} - y^{n}_{t}, x_{t} = y_{t} - y^{e}_{t}.
- Welfare-relevant gap dynamics leading to:
  - x_{t} = M E_{t}[x_{t+1}] - ϕ ( i_{t} - E_{t}[ζ_{t+1}] - r^{e}_{t} )  (Eq. 82)
  - r^{e}_{t} = r^{n}_{t} + (1/ϕ) [ M E_{t}[ y^{e}_{t+1} - y^{n}_{t+1} ] - ( y^{e}_{t} - y^{n}_{t} ) ]  (Eq. 83)
- Deviation form for efficient interest rate:
  - ^r^{e}_{t} = (1/ϕ) [ M E_{t}[ ^y^{e}_{t+1} ] - ^y^{e}_{t} ]  (Eq. 85).

### A.5 Welfare loss
- Second-order Taylor expansion of utility U_{t} around steady state (86)–(89).
- Key identities used:
  - U_{cc} = - (1/c) U_{c}, U_{nn} = - ϵ (1/N) U_{n}, and for any z_{t}: z_{t} - \bar z = ^z_{t} + 1/2 ^z^{2}_{t}.
- Expression for labor deviation in terms of output gap:
  - (1-β) ^n_{t} = ~y_{t} - a_{t} + d_{t}, with d_{t} = (1-β) ln ∫_{0}^{1} (P_{t}(i)/P_{t})^{ - ε / (1-β) } di and Lemma 1 result d_{t} = ε^{2}/2 var_{i}[ p_{t}(i) ].
- Welfare loss per period (as fraction of steady-state consumption) derived (90)–(91):
  - U_{t} - U / U_{c} c = -1/2 [ χ ε Var_{i}[ p_{t}(i) ] + ( ψ + ϵ + β ) / (1-β) ~y^{2}_{t} - 2 ξ (1 + ϵ)/(1-β) ~y_{t} a_{t} ]
  - Aggregate welfare loss:
    - W = - E_{0} sum_{t=0}^{∞} β^{t} [ -1/2 ( χ ε Var_{i}[ p_{t}(i) ] + ( κ + ϵ + β )/(1-β) ( ~y_{t} - ^y^{e}_{t} )^{2} ) ]  (Eq. 91)
- Assumption: x_{t} = y_{t} - y^{e}_{t} = ~y_{t} - ^y^{e}_{t} used to express welfare loss in terms of policy-relevant deviations.

*Content derived from "wpiea2019166-print-pdf - References" and Appendix A as provided.*

### chapter 4), we Önd the welfare loss expression

### chapter 4), we Önd the welfare loss expression

### B  Robustness check — overview
- Presents results under alternative model and myopia calibrations.
- Impulse response functions for inflation, output, interest rate and price level are shown for commitment and discretion across calibrations.
- Impulse responses and welfare heatmaps lead to similar conclusions as in Sections 4.2 and 5.2, irrespective of model calibration chosen.

### B.1  Model calibrations
- Table 7: Calibration of the model parameters used for the robustness checks.
- Calibration rows (preserve order and numeric entries exactly):
  - GalÌ (2008)  0.99   1  1  6  1/3  0.66
  - Relative risk aversion  0.99   2  1  6  1/3  0.66
  - Frisch elasticity  0.99   1  5  6  1/3  0.66
  - Constant return to scale  0.99   1  1  6   0   0.66
  - Sticky prices  0.99   1  1  6  1/3  3/4
  - Time preferences  0.996  1  1  6  1/3  0.66
  - Demand elasticity  0.99   1  1  9  1/3  0.66
  - GalÌ (2015)  0.996  2  5  9  1/3  3/4

- Key model-calibration findings:
  - When 6=  0, the trade-o§ between inflation and output worsens, and the central bank acts aggressively to accommodate the cost-push shock; this is noted when comparing the baseline calibration to the constant returns to scale calibration= 0.
  - Increasing the Frisch elasticity or decreasing the constant return to scale leads to lower welfare losses whatever the type of myopia.
  - Welfare-improving abilities of the general myopia are clear and robust under discretion and optimal simple rules; under commitment the result is not clear for such myopia levels (85%), but extreme myopia values demonstrate robustness (Appendix B.2).

- Figures referenced (impulse response functions following a 1% cost-push shock):
  - Figure 4: Inflation under commitment for each model and myopia calibration.
  - Figure 5: Output under commitment for each model and myopia calibration.
  - Figure 6: Interest rate under commitment for each model and myopia calibration.
  - Figure 7: Price level under commitment for each model and myopia calibration.
  - Figure 8–11: Corresponding impulse responses under discretion.

### Table 8 — Welfare losses under commitment and discretion for each calibration
- Table 8: Welfare losses under commitment (top) and discretion (bottom) for each calibration.
- Numeric block (preserve exact numeric entries as presented):
  - Myopia
    - 0.2809
    - 0.2809
    - 0.3171
    - 0.2809
    - 0.2834
    - 0.2235
    - 0.2235
    - 0.2533
    - 0.2235
    - 0.2257
    - 0.1126
    - 0.1126
    - 0.1286
    - 0.1126
    - 0.1136
    - 0.1248
    - 0.1248
    - 0.1424
    - 0.1248
    - 0.1260
    - 0.4667
    - 0.4667
    - 0.5039
    - 0.4667
    - 0.4672
    - 0.2832
    - 0.2832
    - 0.3199
    - 0.2832
    - 0.2861
    - 0.2624
    - 0.2624
    - 0.2966
    - 0.2624
    - 0.2648
    - 0.1741
    - 0.1741
    - 0.1901
    - 0.1741
    - 0.1760
    - 0.3599
    - 0.3962
    - 0.2892
    - 0.3223
    - 0.1492
    - 0.1699
    - 0.1649
    - 0.1873
    - 0.5830
    - 0.6043
    - 0.3627
    - 0.4001
    - 0.3372
    - 0.3727
    - 0.2274
    - 0.2478
  - (continued numeric block corresponding to bottom/discretion values)
    - 0.5102
    - 0.5102
    - 0.5324
    - 0.5102
    - 0.4149
    - 0.5625
    - 0.3740
    - 0.3740
    - 0.4005
    - 0.3740
    - 0.3165
    - 0.1528
    - 0.1528
    - 0.1713
    - 0.1528
    - 0.1403
    - 0.1740
    - 0.1740
    - 0.1942
    - 0.1740
    - 0.1583
    - 1.0109
    - 1.0109
    - 0.9864
    - 1.0109
    - 0.7347
    - 0.8907
    - 0.5212
    - 0.5212
    - 0.5432
    - 0.5212
    - 0.4222
    - 0.5721
    - 0.4649
    - 0.4649
    - 0.4892
    - 0.4649
    - 0.3828
    - 0.2697
    - 0.2697
    - 0.2868
    - 0.2697
    - 0.2362
    - 0.71480.5308
    - 0.4484
    - 0.2189
    - 0.2155
    - 0.2494
    - 0.2412
    - 1.34260.73150.6543
    - 0.5264
    - 0.3862
    - 0.3407

- Interpretation from Table 8:
  - Under different model calibrations, myopia does not necessarily increase welfare losses.
  - The welfare‑improving effect of the general myopia is robust across calibrations.
  - The impulse response functions yield similar qualitative conclusions across calibrations.

### B.2  Myopia calibrations
- Table 9: Calibration of the myopia parameters used for the simulations.
- Models (preserve garbled numeric formatting exactly as in source):
  - No myopiaMyopia
  - RationalInterest rate  Output gap  Ináation  Revenue  General  Full  Extreme
  - m
  - r
  - 10.211110.20.01
  - m
  - f
  - x
  - 110.21110.20.01
  - m
  - f
  - 
  - 1110.2110.20.01
  - m
  - y
  - 11110.210.20.01
  - m111110.20.20.01

- Description:
  - Table 9 presents more pronounced myopic agents with approximately 80% myopia and an extreme case with an almost fully myopic agent (99%).
  - Impulse response functions for commitment (Fig. 12) and discretion (Fig. 13) are presented for the calibrations in Table 9.

- Figures referenced:
  - Figure 12: Impulse response functions of inflation, price level, output and interest rate following a 1% cost-push shock under commitment for each model calibration (Table 9).
  - Figure 13: Impulse response functions of inflation, price level, output and interest rate following a 1% cost-push shock under discretion for each model calibration (Table 9).

### Table 10 — Welfare losses under GalÌ (2015) calibration for myopia cases
- Table 10: Welfare losses under commitment and discretion for each myopia calibration.
- Numeric entries (preserve formatting exactly):
  - Myopia
    - Interest rate  Output gap  Ináation  Revenue  General   Full   Extreme
    - Commitment0.1741.4460.2570.1740.143   0.372   0.302
    - Discretion0.2703.3570.3480.2700.145   0.372   0.302

- Interpretation from Table 10:
  - Welfare losses under discretion are always higher than under commitment, except under full and extreme myopia.
  - The general myopia case leads to the best welfare losses under commitment and discretion, confirming that myopia can improve welfare losses.

### Robustness conclusions
- There exists a general myopia level that improves welfare losses whatever the chosen regime: commitment, discretion or optimal simple rule.
- Results for optimal simple rules and additional calibration cases are available upon request (as indicated in source).

*Source: wpiea2019166-print-pdf - chapter 4) (extracted content).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019166-print-pdf.pdf_