## wpiea2020196-print-pdf — 2.1  A Ricardian Agent,r

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---

### Model structure: agents, preferences, and constraints
- Ricardian agent
  - Preferences: max E0 ∑_{t=0}^∞ β^t [ ln C_{rt} − χ N_{t}^{1+φ}/(1+φ) ].
  - Budget constraint: C_{rt} + b_{rt} = b_{rt−1} R_{t−1}/Π_{t} + Y_{rt}.
  - Euler equation: C_{rt}^{−1} = β R_{t} E_{t}( C_{rt+1}^{−1} / Π_{t+1} ).
  - Implicit borrowing constraint never binds for Ricardians.
- Ricardian income composition
  - Y_{rt} = (1−λ) (A_{t}/A)^{−γ/(1−λ)} w_{t} N_{t} + (1−δ)/(1−λ) d_{t} + t_{rt} − T_{p} Y_{t}.
  - Labor income proportional to aggregate labor income w_{t} N_{t}; TFP shocks shift resources across agent types depending on γ.
  - Dividends distributed equally among Ricardians and taxed at rate δ.
  - Transfers t_{rt} chosen by fiscal authority and financed by taxes on dividends.
- Keynesian agent (contrast)
  - Constrained: b_{kt} = 0 (hand-to-mouth).
  - Same preference functional form as Ricardians.
  - Keynesian income: Y_{kt} = (A_{t}/A)^{−γ} w_{t} N_{t} − T_{p} Y_{t} + t_{kt}.
  - No dividend income for Keynesians; if γ > 0 a positive TFP shock reduces Keynesians’ share of labor income.

### Labor, production, and price setting
- Labor supply rule: w_{t} = χ N_{t}^{φ} Y_{t}.
  - All workers supply same amount of labor by assumption.
- Goods markets
  - Final goods: CES aggregator Y_{t} = [ ∫_{0}^{1} y_{jt}^{θ_{p}−1/θ_{p}} dj ]^{θ_{p}/(θ_{p}−1)}.
  - Intermediate producers: y_{jt} = A_{t} L_{jt}^{1−α}; demand y_{jt} = (p_{jt}/P_{t})^{−θ_{p}} Y_{t}; sales subsidies at rate T_{p}; quadratic price adjustment cost ψ_{p}/2 Y_{t} [p_{jt}/p_{jt−1} − 1]^2.
  - Symmetry yields New Keynesian Phillips curve (equation (7)) linking inflation, expected inflation, consumption growth, and marginal costs.
- Aggregate production and dividends
  - Aggregate production: Y_{t} = A_{t} N_{t}^{1−α}.
  - Real dividends: d_{t} = (1+T_{p}) Y_{t} − w_{t} N_{t} − ψ_{p}/2 Y_{t} (Π_{t}−1)^2.

### Market clearing and fiscal policy
- Market clearing
  - Goods market: Y_{t} = λ C_{kt} + (1−λ) C_{rt} + ψ_{p}/2 Y_{t} (Π_{t}−1)^2.
  - Bond market clearing: ∫ b_{it} di = 0 (with b_{kt} = b = 0).
- Fiscal policy
  - Sales subsidy T_{p} financed by lump-sum taxes on all households.
  - Profits taxed at rate δ and redistributed:
    - t_{kt} = (1−τ) δ d_{t}.
    - t_{rt} = δ d_{t} − λ t_{kt}/(1−λ).
  - τ controls extent to which taxed profits δ d_{t} are distributed to Keynesians; higher τ means less redistribution to Keynesians (textual statement preserved).

### Natural output and equilibrium definitions
- Natural (flex-price) output: Y_{t}^{n} = A_{t} [ (1+T_{p}) (1−α) (θ_{p}−1)/θ_{p} 1/χ ]^{(1−α)/(1+φ)}.
- Competitive equilibrium: sequence {Y_{t}, Y_{t}^{n}, Y_{rt}, Y_{kt}, C_{rt}, C_{kt}, N_{t}, R_{t}, Π_{t}, w_{t}, d_{t}, t_{rt}, t_{kt}, b_{rt}, b_{kt}}_{t=0}^∞ and exogenous {A_{t}}_{t=0}^∞ satisfying individual optimality, market clearing, production, Phillips curve, transfers, natural output, and bond conditions.

### Linearized equilibrium (selected conditions)
- Log-linearization: ˆx_{t} ≡ ln x_{t} − ln x.
- Selected linearized equations (preserved exactly):
  - Ricardian Euler (15): ˆC_{rt} = E_{t} ˆC_{rt+1} − ˆR_{t} + E_{t} ˆΠ_{t+1}.
  - Keynesian consumption/income relation (16): C_{k} wN ˆC_{kt} + T_{p} Y wN ˆY_{t} − t_{k} wN ˆt_{kt} = −γ ˆA_{t} + ˆw_{t} + ˆN_{t}.
  - Labor market (17): ˆw_{t} = φ ˆN_{t} + ˆY_{t}.
  - Production (18): ˆY_{t} = ˆA_{t} + (1−α) ˆN_{t}.
  - Phillips curve linearized (20): ˆΠ_{t} = β E_{t} ˆΠ_{t+1} + θ_{p} ψ_{p} (ˆw_{t} + α/(1−α) ˆY_{t} − 1/(1−α) ˆA_{t}).
  - Taylor rule (25): ˆR_{t} = φ_{π} ˆΠ_{t} + φ_{y} ˆY_{t}.
- Simplifications: Ricardian budget constraint ignored via Walras Law; constant bond positions yield Y_{rt} = C_{rt}, Y_{kt} = C_{kt}.

### Steady state, exogenous process, and calibration
- Steady state depends on parameters: T_{p}, θ_{p}, δ, τ, α, A, β, φ, λ; does not depend on γ.
- Exogenous productivity shock: A_{t} = A e^{ε_{t}}; linearized ˆA_{t} = ε_{t}.
  - Shock process: ε_{t} = ρ ε_{t−1}, with ε_{0} = 0.01 (deterministic evolution); ρ = 0.9.
- Key calibrated parameters (Table 1 preserved exactly):
  - λ = 0.4 (Share of Keynesian agents)
  - τ = 0.93 (Redistribution of Profits)
  - δ = 1 (Redistribution of Profits)
  - γ = 1.67 (Degree of Skill Bias)
  - α = 0.25 (Profits Share)
  - β = 0.9925 (Discount Factor)
  - χ = 1 (Labor Disutility)
  - φ = 1 (Inverse Frisch Elasticity)
  - θ_{p} = 9 (CES Elasticity)
  - ψ_{p} = 372.8 (Price Adjustment)
  - ρ = 0.9 (Persistence of Shock)
- Calibration notes:
  - λ = 0.4 chosen as midpoint from literature.
  - τ = 0.93 chosen to match non-labor income ratio statistic (U.S., 2016) equal to 24.5, with τ chosen so t_{r}/t_{k} = 24.5 and δ = 1.
  - γ calibrated to match estimated effect of productivity shock on consumption across income distribution, assuming Taylor rule φ_{π} = 1.5, φ_{y} = 0.125.

### Optimal monetary policy: objective and weights
- Social welfare (26): W = E_{0} ∑_{t=0}^∞ β^t [ λ ln C_{kt} + (1−λ) ln C_{rt} − N_{t}^{1+φ}/(1+φ) ].
- Quadratic approximation (27): W ≈ E_{0} ∑_{t=0}^∞ β^t { −1/2 W_{Π} ˆΠ_{t}^2 − 1/2 W_{Y} (ˆY_{t} − ˆY_{t}^{*})^2 − 1/2 W_{∆c} ˆ∆_{ct}^2 } + T_{0} + t.i.p.
  - Definitions:
    - ˆΠ_{t} = inflation gap.
    - ˆY_{t} − ˆY_{t}^{*} = output gap relative to welfare-optimizing level.
    - ˆ∆_{ct} = ˆC_{rt} − ˆC_{kt} = consumption inequality gap.
- Baseline welfare weights (section 3.2, preserved exactly):
  - W_{∆c} = 0.2, W_{Y} = 5, W_{Π} = 310.
- Observations on weights:
  - Weight on consumption inequality is non-zero but very small relative to output gap and especially inflation gap.
  - Weight on inflation gap declines with larger τ; weight on output gap rises with larger τ.
  - Economic intuition: with larger τ, profits accrue more to Ricardians and inflation's cost is borne by firms; central bank tolerates some inflation as redistributive; higher τ makes stabilizing labor supply and output more important.

### RANK-optimal policy and comparison with fully optimal policy
- RANK-optimal policy: central bank behaves as if representative agent (all Ricardian), ignoring inequality.
  - RANK weights coincide with τ = 0 case.
- Responses to positive TFP shock (section 3.4):
  - Fully optimal policy cushions rise in consumption inequality more strongly than RANK-optimal policy by allowing slightly larger inflation and output gaps on average.
  - Impulse responses under two policies nearly identical; fully optimal policy only marginally better.
- Quantitative welfare result (Welfare gains and decomposition):
  - Welfare gain from moving to fully optimal policy is about 2.5 x 10−6 percent in yearly consumption-equivalent terms.
  - In relative terms, this means that moving to fully optimal policy only increases welfare by 1 percent.
  - Optimal policy increases welfare overwhelmingly through reduced consumption inequality (largest positive component) at the expense of larger losses due to inflation and, to some extent, output gaps.
  - Social planner’s emphasis on output gap under optimal policy depends crucially on τ but not γ; in baseline, optimal policy delivers smaller output gap when τ or γ are high.

### Implementable rules: Taylor rules and inequality targeting
- Standard Taylor rule specification studied:
  - ˆR_t = φ_π ˆΠ_t + φ_y ˆY_t + φ_c (ˆC_rt − ˆC_kt).
  - Baseline parameters: φ_π = 1.5, φ_y = 0.125; initial analysis sets φ_c = 0.
- Table 2 excerpts (selected numeric entries preserved exactly where presented):
  - τ00000.930.930.930.93
  - γ00001.671.671.671.67
  - φ_π 1.51.551.51.51.551.5
  - φ_y 0.1250010.125001
  - Cons.  Equiv.  Loss0.40.101.42.71.90.34.6
  - Cons.  Equiv.W−T0 0.50.201.910.60.12.1
  - Inflation98.298.298.198.287.183.62490.1
  - Output1.81.81.91.83.43.21.53.7
  - Inequality00009.613.274.56.3
- Key qualitative conclusions when τ = γ = 0:
  - Taylor rules with zero weight on output fare better.
  - A large weight on inflation fares even better.
  - Welfare losses are disproportionately due to non-zero inflation gaps (“divine coincidence” result).
- With inequality introduced, the aboveThree observations continue to hold; higher inflation parameter reduces total welfare losses from all three gaps.
- The welfare loss due to consumption inequality can be sizeable, always larger than that of the output gap and, under a strong inflation mandate, even larger than that from inflation gaps.

### Augmented Taylor rule (targeting consumption inequality) — results
- Allowing φ_c ≠ 0 changes welfare outcomes.
  - When τ and γ are not zero, standard Taylor rules achieve higher welfare when φ_c becomes negative.
  - Maximum welfare typically achieved around φ_c ∈ [−0.2, −0.05], except for the rule with output weight of 1 (which prefers a much more negative φ_c).
- Quantitative gains from augmentation:
  - Consumption-equivalent gain of moving to augmented Taylor rule (with optimized φ_c) is 7 x 10−3 percent.
  - This gain is about 96 percent of the loss under the standard Taylor rule (as reported).
- Mechanism:
  - Negative φ_c leads the central bank to set lower interest rates following a positive TFP shock.
  - Lower rates increase wages and stimulate demand, benefiting poor agents who rely on labor income; reduces increase in inequality while closing inflation and output gaps.
  - Optimized augmented rule typically produces small positive inflation and output gaps—trading off complete macro stabilization against reducing consumption inequality.
- Robustness:
  - Qualitative results hold under wage rigidities (appendix results preserved).

### Policy implications and recommendations
- If central bank already implements optimal policy ignoring inequality, explicitly targeting inequality yields only a small additional welfare improvement (about 1 percent of the loss under optimal monetary policy targeted to the average agent).
- If central bank uses a standard Taylor rule, augmenting it with an inequality target (negative φ_c) can deliver large welfare gains:
  - Augmented Taylor rule can recoup about 96 percent of the welfare loss relative to the standard Taylor rule.
  - Practically: following a positive TFP shock that increases consumption inequality, central bank should on the margin loosen policy rate (use a negative φ_c).
- Suggested extensions (model-specific caveats):
  - Consider broader set of shocks (including demand-side shocks).
  - Model richer sources of inequality (differentiated labor income, richer wealth distributions, savings behavior, illiquid assets like housing).
  - Study interactions of monetary policy with other policy tools.

### Appendix A — analytical steps, weights, and wage rigidity extensions (selected preserved expressions)
- Deriving the loss function (overview)
  - Utility approximation (Equation (31)): U_t ≈ λ ˆC_kt + (1−λ) ˆC_rt − ˆY_t − 1/2 (1 + φ)/(1−α) (ˆY_t − ˆA_t)^2.
  - Dynamic inequality term: ˆΔ_ct ≡ ˆC_rt − ˆC_kt.
  - Final welfare function after collecting terms: W = Σ_{t=0}^∞ β^t { −1/2 W_Π ˆΠ_t^2 − 1/2 W_Δc ˆΔ_ct^2 − 1/2 W_Y (ˆY_t − ˆY^*_t)^2 } + T_0, where ˆY^*_t ≡ W_AY / W_Y ˆA_t.
- Final welfare weights (preserved exactly)
  - W_Π = ψ_p [ 1 − (λ − ˜λ)/(1 − ˜λ) [ 1 − (1−τ)δ / ˜C_k ] ].
  - W_Δc = ˜λ (1 − ˜λ).
  - W_Y = 1 + φ 1−α + (λ − ˜λ) (1 − ˜λ)^−2 ( (1−α) (1 − (1−τ)δ) ˜C_k )^2 (1 + φ 1−α )^2.
    - where ˜λ ≡ λ C_k / Y = λ ˜C_k.
- Central bank maximization problem (preserved exactly)
  - max W = Σ_{t=0}^∞ β^t { −1/2 W_Π ˆΠ_t^2 − 1/2 W_Δc ˆΔ_ct^2 − 1/2 W_Y (ˆY_t − ˆY^*_t)^2 } + T_0 subject to Phillips curve and dynamic inequality equation.
  - Single-equation FOC (Equation (35)) preserved exactly in source.
- RANK case (λ = ˜λ = 0) weights simplify (preserved exactly)
  - W_Π = ψ_p.
  - ˜W_YY = 1 + φ 1−α.
  - RANK welfare and constraint as in appendix (preserved exactly).
- Wage rigidity extension (selected preserved expressions)
  - Wage rule: W_t = W_t^{ψ_w} W_t^{* 1−ψ_w}; real wage linearized: ˆw_t = ψ_w ˆw_{t−1} − ψ_w ˆΠ_t + (1 − ψ_w) ˆw^*_t.
  - Welfare function under wage rigidity final form (Equation (41)) preserved exactly: W = Σ_{t=0}^∞ β^t { −1/2 W_Π ˆΠ_t^2 − 1/2 W_Δc ˆΔ_t^2 − 1/2 1 + φ 1−α (ˆY_t − ˆA_t)^2 − 1/2 W_LS^2 (ˆLS_t − ˆLS^*_t)^2 } + T_0.
  - Wage rigidity calibration: ψ_w = 0.75; calibrated γ obtained as 2.27 under wage rigidity experiment.
- Appendix reports figures and tables (weights as functions of τ, impulse responses under wage rigidity, Table A1 welfare in Taylor rules) with preserved numeric results as in source.

*Source: wpiea2020196-print-pdf — 2.1  A Ricardian Agent,r — https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020196-print-pdf.pdf*

### 2.1  A Ricardian Agent,r

### 2.1  A Ricardian Agent,r

### Ricardian agent: preferences, constraints, and Euler equation
- Preferences: max E0 ∑_{t=0}^∞ β^t [ ln C_{rt} − χ N_{t}^{1+φ}/(1+φ) ].
- Budget constraint (equation (1)): C_{rt} + b_{rt} = b_{rt−1} R_{t−1}/Π_{t} + Y_{rt}, with real bonds b_{rt} = B_{rt}/P_{t}.
- Consumption Euler equation (equation (2)): C_{rt}^{−1} = β R_{t} E_{t}( C_{rt+1}^{−1} / Π_{t+1} ).
- Implicit borrowing constraint in background that never binds for Ricardians.

### Ricardian income composition (equation (3))
- Y_{rt} = (1−λ) (A_{t}/A)^{−γ/(1−λ)} w_{t} N_{t} + (1−δ)/(1−λ) d_{t} + t_{rt} − T_{p} Y_{t}.
- Labor income proportional to aggregate labor income w_{t} N_{t}, allowing TFP shocks to shift resources across agent types depending on γ.
- Dividends distributed equally among Ricardians and taxed at rate δ.
- Taxes T_{p} Y_{t} are raised from households and redistributed as sales subsidies to intermediate firms.
- Transfers t_{rt} chosen by fiscal authority and financed by taxes on dividends.

### Keynesian agent (section 2.2) — contrast
- Keynesian agent constrained: b_{kt} = 0 (binding borrowing constraint); effectively hand-to-mouth.
- Preferences: same functional form as Ricardian.
- Budget constraint (equation (4)) simplifies with b_{kt} = 0.
- Keynesian income (equation (5)): Y_{kt} = (A_{t}/A)^{−γ} w_{t} N_{t} − T_{p} Y_{t} + t_{kt}.
- No dividend income for Keynesians; if γ > 0 a positive TFP shock reduces Keynesians’ share of labor income.

### Labor supply (section 2.3)
- Aggregate labor rule (equation (6)): w_{t} = χ N_{t}^{φ} Y_{t}, where w_{t} is the real wage, φ is inverse Frisch elasticity, χ is labor disutility parameter.
- All workers supply same amount of labor by assumption to abstract from type-dependent labor supply distortions.

### Markets for goods (section 2.4)
- Final goods:
  - CES aggregator: Y_{t} = [ ∫_{0}^{1} y_{jt}^{θ_{p}−1/θ_{p}} dj ]^{θ_{p}/(θ_{p}−1)}; price index and demand functions as standard CES.
- Intermediate goods producers:
  - Production y_{jt} = A_{t} L_{jt}^{1−α}.
  - Face demand y_{jt} = (p_{jt}/P_{t})^{−θ_{p}} Y_{t}, nominal wage W_{t}, sales subsidies at rate T_{p}, and quadratic price adjustment cost ψ_{p}/2 Y_{t} [p_{jt}/p_{jt−1} − 1]^2.
  - Symmetry (p_{jt} = P_{t}) yields New Keynesian Phillips curve (equation (7)):
    - Π_{t}[Π_{t−1}] = β (C_{rt+1}/C_{rt})^{−1} (Y_{t+1}/Y_{t}) Π_{t+1}[Π_{t+1}^{−1}] + (θ_{p} ψ_{p})[ 1/(1−α) w_{t} A_{t}^{1/(1−α)} Y_{t}^{α/(1−α)} − (1+T_{p}) (θ_{p}−1)/θ_{p} ].
- Aggregate production (equation (8)): Y_{t} = A_{t} N_{t}^{1−α}.
- Real dividends (equation (9)): d_{t} = (1+T_{p}) Y_{t} − w_{t} N_{t} − ψ_{p}/2 Y_{t} (Π_{t}−1)^2.

### Market clearing (section 2.5)
- Goods market (equation (10)): Y_{t} = λ C_{kt} + (1−λ) C_{rt} + ψ_{p}/2 Y_{t} (Π_{t}−1)^2.
- Bond market clearing (equation (11)): ∫ b_{it} di = 0.

### Fiscal policy (section 2.6)
- Two parts:
  - Sales subsidy T_{p} to intermediate producers, financed by lump-sum taxes on all households.
  - Profits taxed at rate δ and redistributed:
    - t_{kt} = (1−τ) δ d_{t} (equation (12)).
    - t_{rt} = δ d_{t} − λ t_{kt}/(1−λ) (equation (13)).
- τ controls extent to which taxed profits δ d_{t} are distributed to Keynesians; higher τ means less (more) inequality (textual statement preserved).

### Natural output (section 2.7)
- Natural (flexible-price) output (equation (14)): Y_{t}^{n} = A_{t} [ (1+T_{p}) (1−α) (θ_{p}−1)/θ_{p} 1/χ ]^{(1−α)/(1+φ)}.

### Equilibrium (section 2.8)
- Competitive equilibrium defined by sequence {Y_{t}, Y_{t}^{n}, Y_{rt}, Y_{kt}, C_{rt}, C_{kt}, N_{t}, R_{t}, Π_{t}, w_{t}, d_{t}, t_{rt}, t_{kt}, b_{rt}, b_{kt}}_{t=0}^∞ and exogenous {A_{t}}_{t=0}^∞ such that:
  - Ricardians and Keynesians solve respective problems and constraints (equations (1), (2), (3), (4), (5)).
  - Labor supply (6), aggregate production (8), dividends (9), Phillips curve (7), transfers (12)–(13), goods market (10), natural output (14), bond conditions b_{kt} = b = 0 and bond market clearing (11) hold.
- Monetary policy closes the model through Ricardian Euler and general equilibrium effects.

### Approximate (linearized) equilibrium (section 2.9)
- Log-linearization around deterministic steady state; define ˆx_{t} ≡ ln x_{t} − ln x.
- Approximate equilibrium conditions include (selected equations as in text):
  - Ricardian Euler (15): ˆC_{rt} = E_{t} ˆC_{rt+1} − ˆR_{t} + E_{t} ˆΠ_{t+1}.
  - Keynesian consumption/income relation (16): C_{k} wN ˆC_{kt} + T_{p} Y wN ˆY_{t} − t_{k} wN ˆt_{kt} = −γ ˆA_{t} + ˆw_{t} + ˆN_{t}.
  - Labor market (17): ˆw_{t} = φ ˆN_{t} + ˆY_{t}.
  - Production (18): ˆY_{t} = ˆA_{t} + (1−α) ˆN_{t}.
  - Phillips curve linearized (20): ˆΠ_{t} = β E_{t} ˆΠ_{t+1} + θ_{p} ψ_{p} (ˆw_{t} + α/(1−α) ˆY_{t} − 1/(1−α) ˆA_{t}).
  - Dividends and transfers relations (21)–(22) and goods aggregation (23), natural output (24), and Taylor rule (25): ˆR_{t} = φ_{π} ˆΠ_{t} + φ_{y} ˆY_{t}.
- Budget constraint of Ricardians ignored due to Walras Law; constant bond positions permit simplifications Y_{rt} = C_{rt}, Y_{kt} = C_{kt}.

### Steady state (section 2.10)
- Steady state depends on parameters governing monopolistic distortions (T_{p}, θ_{p}), fiscal redistribution (δ, τ), production (α, A), preferences (β, φ), and population shares (λ).
- Steady state does not depend on γ (skill-bias parameter) as γ governs only cyclical skill-bias.

### Exogenous process (section 2.11)
- Productivity shock: A_{t} = A e^{ε_{t}}, linearized to ˆA_{t} = ε_{t}.
- Shock process: ε_{t} = ρ ε_{t−1}, with ε_{0} = 0.01 (deterministic evolution); ρ = 0.9 as calibrated.

### Calibration (section 2.12 and Table 1)
- Key calibrated parameters (as in Table 1):
  - λ = 0.4 (Share of Keynesian agents)
  - τ = 0.93 (Redistribution of Profits)
  - δ = 1 (Redistribution of Profits)
  - γ = 1.67 (Degree of Skill Bias)
  - α = 0.25 (Profits Share)
  - β = 0.9925 (Discount Factor)
  - χ = 1 (Labor Disutility)
  - φ = 1 (Inverse Frisch Elasticity)
  - θ_{p} = 9 (CES Elasticity)
  - ψ_{p} = 372.8 (Price Adjustment)
  - ρ = 0.9 (Persistence of Shock)
- Calibration notes:
  - λ = 0.4 chosen as midpoint from literature.
  - τ = 0.93 chosen to match non-labor income ratio statistic from the Survey of Consumer Finances (U.S., 2016); this ratio equals 24.5 and τ chosen so t_{r}/t_{k} = 24.5 with δ = 1.
  - γ calibrated to match estimated effect of productivity shock on consumption across income distribution (De Giorgi and Gambetti, 2017), assuming standard Taylor rule φ_{π} = 1.5, φ_{y} = 0.125 in equation (25).

### Optimal monetary policy (section 3 and subsections)
- Social welfare (equation (26)) for central bank valuing all agents equally:
  - W = E_{0} ∑_{t=0}^∞ β^t [ λ ln C_{kt} + (1−λ) ln C_{rt} − N_{t}^{1+φ}/(1+φ) ].
- Quadratic approximation (equation (27)): welfare approximated as expected discounted sum of three quadratic gaps plus predetermined term T_{0}:
  - W ≈ E_{0} ∑_{t=0}^∞ β^t { −1/2 W_{Π} ˆΠ_{t}^2 − 1/2 W_{Y} (ˆY_{t} − ˆY_{t}^{*})^2 − 1/2 W_{∆c} ˆ∆_{ct}^2 } + T_{0} + t.i.p.
  - Definitions:
    - ˆΠ_{t} = inflation gap.
    - ˆY_{t} − ˆY_{t}^{*} = output gap (deviation from welfare-optimizing level).
    - ˆ∆_{ct} = ˆC_{rt} − ˆC_{kt} = consumption inequality gap.
  - T_{0} predetermined; t.i.p. = terms independent of monetary policy.
- Welfare weights and dependence on τ (section 3.2):
  - Baseline calibration weights: W_{∆c} = 0.2, W_{Y} = 5, W_{Π} = 310 (stated as "the weights in our baseline calibration are 0.2, 5 and 310, respectively").
  - Two main observations:
    - Weight on consumption inequality is non-zero but very small relative to output gap and especially inflation gap.
    - Weight on inflation gap declines with larger τ; weight on output gap rises with larger τ.
  - Economic intuition:
    - Inflation gap cost borne by firms reduces profits; when τ is large (less redistribution to Keynesians), profits accrue to Ricardians, so central bank tolerates some inflation gap as redistributive.
    - Higher τ implies Keynesians rely on wages, so stabilizing labor supply and output becomes relatively more important.
- RANK-optimal policy (section 3.3):
  - Central bank ignores inequality and maximizes welfare as if representative agent (RANK), objective simplifies to quadratic in inflation and output gap relative to ˆA_{t}.
  - Weights W_{RANK,Π}, W_{RANK,Y} coincide with τ = 0 case in Figure 1.
- Responses to TFP shocks (section 3.4 and Figures)
  - Impulse responses comparing fully optimal vs RANK-optimal policy to positive TFP shock:
    - Optimal policy cushions rise in consumption inequality more strongly than RANK-optimal policy, allowing slightly larger inflation and output gaps on average.
    - Impulse responses under the two policies are nearly identical; fully optimal policy only marginally better than RANK-optimal.
  - Welfare implications:
    - Welfare loss measured in consumption-equivalent terms computed using equation (27) across different τ and γ (details in text and appendices).

*wpiea2020196-print-pdf — 2.1  A Ricardian Agent,r*

### 2011.  Specifically, we computexas the consumption equivalent representing the permanent increase in consumption that

### wpiea2020196-print-pdf - 2011.  Specifically, we computexas the consumption equivalent representing the permanent increase in consumption that

### Welfare gains and decomposition
- Under the paper’s calibration, the welfare gain from moving to fully optimal policy is only about 2.5 x 10−6 percent in yearly consumption-equivalent terms.
- In relative terms, this means that moving to fully optimal policy only increases welfare by 1 percent.
- Optimal policy increases welfare overwhelmingly through reduced consumption inequality (the yellow area in the decomposition being the largest and positive) at the expense of larger losses due to inflation and, to some extent, output gaps.
- The social planner’s emphasis on the output gap under optimal policy depends crucially on τ but not γ; in the baseline calibration, optimal policy delivers a smaller output gap when profits are not much distributed or when the skill-bias is more pronounced (τ or γ are high).

### Augmented Taylor rules (motivation and mechanism)
- Optimal monetary policy is difficult to implement in practice because it requires extensive information and credible pre-commitment to full paths of all three gaps.
- Practical rules (Taylor rules) are widely used as implementable approximations.
- Mechanism: a policy of lower interest rates following a positive TFP shock leads to higher wages on the margin, benefiting disproportionately the poor (who rely more on labor income). Lower rates thus reduce inequality and can improve inflation and growth outcomes by avoiding excessive tightening.

### Standard Taylor rules and inequality (specification and calibration)
- The studied Taylor rule specification:
  - ˆR_t = φ_π ˆΠ_t + φ_y ˆY_t + φ_c (ˆC_rt − ˆC_kt)
  - ˆY_t = deviation of output from steady state; ˆΠ_t = deviation of inflation from target; ˆC_rt − ˆC_kt = consumption inequality (workers minus capitalists).
- Baseline standard parameterization examined: φ_π = 1.5, φ_y = 0.125 (as in Galí, 2015, Chapter 3); initial analysis sets φ_c = 0.
- Table 2 excerpts (as presented in source):
  - τ00000.930.930.930.93
  - γ00001.671.671.671.67
  - φ_π 1.51.551.51.51.551.5
  - φ_y 0.1250010.125001
  - Cons.  Equiv.  Loss0.40.101.42.71.90.34.6
  - Cons.  Equiv.W−T0 0.50.201.910.60.12.1
  - Inflation98.298.298.198.287.183.62490.1
  - Output1.81.81.91.83.43.21.53.7
  - Inequality00009.613.274.56.3
- Key qualitative conclusions when τ = γ = 0 (no inequality):
  - Taylor rules with zero weight on output fare better.
  - A large weight on inflation fares even better.
  - Any welfare losses are disproportionately due to non-zero inflation gaps (“divine coincidence” result).
- When inequality is introduced (columns shown in Table 2), these three observations continue to hold; importantly, a higher parameter on inflation reduces total welfare losses from all three gaps.
- The welfare loss due to the consumption inequality gap can be sizeable, always larger than that of the output gap and, under a strong inflation mandate, even larger than that stemming from inflation gaps.

### Consumption inequality targeting (augmented Taylor rule and quantitative effects)
- Allowing φ_c ≠ 0 (targeting consumption inequality) changes welfare outcomes.
- When τ and γ are not zero, standard Taylor rules achieve higher welfare when φ_c becomes negative; the maximum welfare is typically achieved around φ_c ∈ [−0.2, −0.05], except for the rule with an output weight of 1 (which prefers a much more negative φ_c).
- The paper reports a consumption-equivalent gain of moving to an augmented Taylor rule (with optimized φ_c) of 7 x 10−3 percent.
  - This gain is about 96 percent of the loss under the standard Taylor rule (as reported in Table 2).
- Mechanism under augmentation:
  - A negative coefficient on consumption inequality leads the central bank to set lower interest rates following a positive TFP shock.
  - Lower rates increase wages and stimulate demand, closing inflation and output gaps while disproportionately helping the poor and reducing the increase in inequality.
  - The optimized augmented Taylor rule not only closes negative inflation and output gaps induced by a positive TFP shock but goes beyond to produce small positive gaps—trading off complete stabilization of macro gaps against stabilizing the consumption inequality gap.
- The qualitative results hold under wage rigidities (see referenced appendices).

### Policy implications and recommendations
- If a central bank already implements optimal monetary policy ignoring inequality, explicitly targeting inequality yields only a small additional welfare improvement (about 1 percent of the loss under optimal monetary policy targeted to the average agent).
- If the central bank uses a standard Taylor rule, augmenting it with an inequality target (negative φ_c) can deliver large welfare gains:
  - The augmented Taylor rule can recoup about 96 percent of the welfare loss relative to the standard Taylor rule.
  - Practically, following a positive TFP shock that increases consumption inequality, the central bank should on the margin loosen its policy rate (i.e., use a negative φ_c).
- These conclusions are model-specific; extensions suggested include:
  - Considering a broader set of shocks (including demand-side shocks).
  - Modeling richer sources of inequality (differentiated labor income, richer wealth distributions, savings behavior, illiquid assets like housing).
  - Studying interactions of monetary policy with other policy tools.

*Source: wpiea2020196-print-pdf - 2011. Specifically, we computexas the consumption equivalent representing the permanent increase in consumption that — https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020196-print-pdf.pdf*

### References

### References and Appendix A (Details in Deriving Optimal and RANK-Optimal Policies)

### Key literature cited
- Works on HANK, monetary policy and inequality: Acharya, Sushant, Edouard Challe, and Keshav Dogra (2020); Kaplan, Greg, Benjamin Moll, and Giovanni L. Violante (2018); Kaplan et al. (2018) — “Monetary Policy According to HANK”.
- Papers on optimal monetary policy, fiscal interactions, and debt deleveraging: Benigno, Pierpaolo, Gauti B. Eggertsson, and Federica Romei (Apr. 2020); Eggertsson, Gauti B and Paul Krugman (2012).
- Foundational New Keynesian and policy-rule literature: Clarida, Richard, Jordi Galí, and Mark Gertler (1999); Taylor, John B (1993); Woodford, Michael (2002).
- Literature on distributional effects and heterogeneity: Coibion, Olivier, Yuriy Gorodnichenko, Lorenz Kueng, and John Silvia (2017); Gornemann, Nils, Keith Kuester, and Makoto Nakajima (2016); Broer, Tobias et al. (2020).
- Additional theoretical and methodological references: Blanchard and Galí (2007); Rotemberg and Woodford (1997); Erceg, Henderson, and Levin (2000); Ravenna and Walsh (2011).

### Appendix A — Overview of analytical steps and main results

A.1 Deriving the Loss Function
- Procedure:
  - Step 1: Approximate period utility and obtain expression involving a linear term in output gap ˆY_t − ˆA_t.
  - Step 2: Take a second order approximation to the New Keynesian Phillips curve featuring a linear term in the output gap.
  - Step 3: Substitute the Phillips curve into the discounted sum of the period utility approximation.
- Notation and approximations used:
  - Use of t.i.p. to denote terms independent of monetary policy.
  - Use of ≈ for Taylor approximations and ignoring terms of order higher than 2.
  - Definition: ˆx_t ≡ ln x_t − ln x and x_t − x x ≈ ˆx_t + 1/2 ˆx_t^2.
- Key intermediate expressions (selected, preserved exactly as in source):
  - Utility approximation (Equation (31)):
    - U_t ≈ λ ˆC_kt + (1−λ) ˆC_rt − ˆY_t − 1/2 (1 + φ)/(1−α) (ˆY_t − ˆA_t)^2
  - Dynamic inequality term definition:
    - ˆΔ_ct ≡ ˆC_rt − ˆC_kt
  - Welfare aggregation form (before Phillips substitution):
    - W = Σ_{t=0}^∞ β^t { −1/2 ˜W_Π ˆΠ_t^2 − 1/2 W_Δc ˆΔ_t^2 − 1/2 ˜W_YY ˆY_t^2 + ˜W_AY ˆA_t ˆY_t + ˜W_Y ˆY_t } (intermediate expression)
- Final welfare function after collecting terms (preserving symbols and definitions):
  - W = Σ_{t=0}^∞ β^t { −1/2 W_Π ˆΠ_t^2 − 1/2 W_Δc ˆΔ_ct^2 − 1/2 W_Y (ˆY_t − ˆY^*_t)^2 } + T_0
  - ˆY^*_t ≡ W_AY / W_Y ˆA_t

A.1 Final welfare weights (preserved exactly)
- W_Π = ψ_p [ 1 − (λ − ˜λ)/(1 − ˜λ) [ 1 − (1−τ)δ / ˜C_k ] ]
- W_Δc = ˜λ (1 − ˜λ)
- W_Y = 1 + φ 1−α + (λ − ˜λ) (1 − ˜λ)^−2 ( (1−α) (1 − (1−τ)δ) ˜C_k )^2 (1 + φ 1−α )^2
  - where ˜λ ≡ λ C_k / Y = λ ˜C_k

A.2 The Central Bank’s Maximization Problem
- Problem statement (preserved exactly):
  - max W = Σ_{t=0}^∞ β^t { −1/2 W_Π ˆΠ_t^2 − 1/2 W_Δc ˆΔ_ct^2 − 1/2 W_Y (ˆY_t − ˆY^*_t)^2 } + T_0
  - subject to the New-Keynesian Phillips curve (first order) and the dynamic inequality equation (first order).
- First-order conditions and Lagrangian setup (preserved exactly):
  - L = Σ_{t=0}^∞ β^t { −1/2 W_Π ˆΠ_t^2 − 1/2 W_Δc ˆΔ_t^2 − 1/2 W_Y (ˆY_t − ˆY^*_t)^2 } + Σ_{t=0}^∞ β^t μ_t [ ˆΠ_t − β E_t ˆΠ_{t+1} − θ_p ψ (1 + φ)/(1−α) (ˆY_t − ˆY^*_t) ] + Σ_{t=0}^∞ β^t η_t [ ˆΔ_t + 1/(1 − ˜λ) 1 − (1−τ)δ ˜C_k (1−α) (1 + φ)/(1−α) (ˆY_t − ˆY^*_t) ]
- Resulting single-equation FOC (preserved exactly, Equation (35)):
  - −W_Π ˆΠ_t − W_Y θ_p ψ (1+φ)/(1−α) (ˆY_t − ˆY^*_t) + 1/(1−˜λ) 1−(1−τ)δ ˜C_k (1−α) (1+φ)/(1−α) W_Δc θ_p ψ (1+φ)/(1−α) ˆΔ_t + W_Y θ_p ψ (1+φ)/(1−α) (ˆY_{t−1} − ˆY^*_{t−1}) − 1/(1−˜λ) 1−(1−τ)δ ˜C_k (1−α) (1+φ)/(1−α) W_Δc θ_p ψ (1+φ)/(1−α) ˆΔ_{t−1} = 0

A.3 Details in Deriving RANK-Optimal Policies
- RANK central bank assumption: all agents Ricardian, i.e. λ = ˜λ = 0.
- Under λ = ˜λ = 0 the weights simplify (preserved exactly):
  - W_Π = ψ_p
  - ˜W_YY = 1 + φ 1−α
  - ˜W_AY = 1 + φ 1−α
  - ˜W_Y = 1 + φ 1−α
- RANK welfare function and constraint (preserved exactly):
  - W = Σ_{t=0}^∞ β^t { −1/2 ψ_p ˆΠ_t^2 − 1/2 1 + φ 1−α (ˆY_t − ˆA_t)^2 } subject to ˆΠ_t = β E_t ˆΠ_{t+1} + θ_p ψ_p 1 + φ 1−α (ˆY_t − ˆA_t)
- Derived targeting rule for pseudo-optimal policy (preserved exactly):
  - −ψ_p ˆΠ_t − 1/(θ_p ψ_p) (ˆY_t − ˆA_t) + 1/(θ_p ψ_p) (ˆY_{t−1} − ˆA_{t−1}) = 0

A.4 Wage Rigidity — extension and implications
- Wage rigidity specification (preserved exactly):
  - W_t = W_t^{ψ_w} W_t^{* 1−ψ_w}
  - W^*_t = χ N_t^φ Y_t / P_t
  - Real wage condition: w_t = w_{t−1}^{ψ_w} Π_t^{−ψ_w} w_t^{* 1−ψ_w}  (Equation (36))
  - w^*_t = φ N_t + ˆY_t (Equation (37) linearized)
- Linearized real wage and wage*:
  - ˆw_t = ψ_w ˆw_{t−1} − ψ_w ˆΠ_t + (1 − ψ_w) ˆw^*_t  (Equation (38))
  - ˆw^*_t = φ ˆN_t + ˆY_t  (Equation (39))
- Welfare approximation under wage rigidity (selected preserved expressions):
  - U_t starting point as in Equation (33) with extra terms that depend on ˆLS_t (labor-share deviations).
  - Definitions and coefficients (preserved exactly):
    - ˆΠ_t^2 coefficient: −1/2 ψ_p [ 1 − (λ − ˜λ)/(1 − ˜λ) [ 1 − (1−τ)δ / ˜C_k ] ] ≡ −1/2 W_Π
    - ˆΔ^2 coefficient: −1/2 ˜λ (1 − ˜λ) ≡ −1/2 W_Δc
    - ˆY^2 coefficient: −1/2 1 + φ 1−α ≡ −1/2 ˜W_Y2
    - ˆLS^2 coefficient: 1/2 (λ − ˜λ)/(1 − ˜λ) [1 − (1−τ)δ] (1−α) ˜C_k [ 1 − ... ] ≡ 1/2 ˜W_LS
    - Cross-term ˆA_t ˆLS_t coefficient: −(λ − ˜λ)/(1 − ˜λ) γ/(1−α) ˜C_k [ ... ] ≡ W_ALS
    - ˆLS_t linear coefficient: (λ − ˜λ)/(1 − ˜λ) [1 − (1−τ)δ] (1−α) ˜C_k ≡ W_LS
  - New Keynesian Phillips curve second-order approximation linking Σ β^t ˆLS_t to ψ_p θ_p ˜T_0 and quadratic terms in ˆLS_t and cross-terms with ˆA_t.
- Welfare function with wage rigidity final form (preserved exactly, Equation (41)):
  - W = Σ_{t=0}^∞ β^t { −1/2 W_Π ˆΠ_t^2 − 1/2 W_Δc ˆΔ_t^2 − 1/2 1 + φ 1−α (ˆY_t − ˆA_t)^2 − 1/2 W_LS^2 (ˆLS_t − ˆLS^*_t)^2 } + T_0
- Definitions (preserved exactly):
  - W_LS^2 = (λ − ˜λ) (1 − ˜λ)^−2 ( 1 − (1−τ)δ ˜C_k (1−α) )^2  (Equation (42))
  - ˆLS^*_t = −(1 − ˜λ) γ ( 1 − [1 − (1−τ)δ] (1−α) ˜C_k ) / [ (1 − (1−τ)δ)^2 (1−α) ˜C_k ] ˆA_t  (Equation (43))

A.4.2–A.4.3 Optimal and RANK-Optimal Policy under Wage Rigidities
- Lagrangian with wage rigidity constraints (preserved exactly, Equation (44)) includes multipliers μ_t, η_t, ψ_t, ν_t and the wage dynamics constraint.
- First-order conditions and system of equations are provided (preserved exactly; see Equations (45)–(49) and subsequent system).
- RANK-optimal policy under wage rigidity: central bank believes λ = 0, implying W_Π = ψ_p and W_LS^2 = 0 in the system.

A.4.4 Results and calibration notes (preserved)
- Wage rigidity calibration: ψ_w = 0.75 (only one quarter of the gap between flex and rigid wage closed every period).
- Calibrated γ obtained as 2.27 under wage rigidity experiment (preserved numeric result).
- Figures and tables (reported in the source) present:
  - Figure A1: Welfare weights as functions of redistribution parameter τ (panels for Weight on Inflation Gap; Weight on Output Gap; Weight on Consumption Inequality Gap; Weight on Labor Share Gap).
  - Figure A2: Impulse responses to a positive TFP shock: Optimal vs RANK-Optimal Policy, with Wage Rigidity (panels for Inflation; Output Gap; Inequality; Labor Share).
  - Table A1: Welfare in Taylor Rules — Standard Parametrization (reports τ, γ, φ_π, φ_y and outcomes including Cons. Equiv. Loss, Cons. Equiv. W−T_0, Inflation, Output, Inequality, Labor Share; preserved numeric entries as in the source table).
  - Figures A3–A5 and further impulse-response and welfare-comparison plots under different rules and calibrations (details and axes preserved in the source material).

*Content unit: wpiea2020196-print-pdf - References (Appendix A).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020196-print-pdf.pdf_