## wpiea2024208-print-pdf

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---

### 2.1 Calibration — baseline parameterization and preferences
- Objective: demonstrate large output effects from fiscal transfers to households in the presence of myopia; use standard parameters except for the discounting parameter u.
- Preferences and rates:
  - β Discount rate: 0.995 (set to closely match an annualized real steady state interest rate of 2%)
  - σ Intertemporal elasticity: 1
  - χ Inverse Frisch elasticity of labour: 1
- Production and capital:
  - α Capital share in production: 0.3
  - δ Depreciation rate: 0.025
  - φk Capital adjustment cost: 5
- Price and wage stickiness:
  - θ Calvo parameter, consumer goods: 0.75
  - θw Calvo parameter, wages: 0.75
  - Implied price/wage reset frequency: prices and wages are reset after 3 quarters on average
  - Phillips curve slope κ evaluates at 0.08 (κ = (1−θ)(1−βθ)/(θ) )
- Other elasticities:
  - ε Elasticity of demand for consumption varieties: 10
  - εw Elasticity of demand for labor varieties: 10
- Policy rule parameters:
  - φπ Taylor rule: inflation parameter: 1.5
  - φb Fiscal rule: elasticity of tax to debt: 0.05 (tax adjustment of just 5% of additional debt as a proportion of GDP each quarter; used to approximate a relatively ‘weak’ fiscal rule)

### 2.1 Calibration — fiscal-rule rationale and debt financing implications
- φb = 0.05:
  - Close to the rule estimated in Gali et al. (2007).
  - Produces slow adjustment of taxes: only 5% of additional debt (as a proportion of GDP) is adjusted each quarter.
  - Weak fiscal rule delays tax adjustment and reduces immediate dissaving from transfers because future tax liabilities are pushed into the future.

### 2.1 Calibration — calibrating myopia (u) and HTM share (ω)
- Calibration targets:
  - Primary parameter: degree of myopia of wealthy households (u).
  - Joint calibration of HTM share ω to match empirical spending time-profiles.
- Empirical targets guiding calibration:
  - Short-run: about 1/4 of transfers and transitory income shocks are spent in the quarter received.
  - Longer-term: evidence that about half of transitory shocks are spent within one year and ~90% over 5 years in some lottery studies.
  - Specific empirical MPC profiles: initial MPCs around 0.15 in first quarter rising to ~0.40 after one year (Colarieti et al., 2024); MPCs ~40% after one year (Christelis et al., 2019).
- Model translation (wealthy partial-equilibrium response to transfer T):
  - ∆C^W = (1−β(1−u))(T − Σ_{τ=0}^{∞} ((1−u)β)^τ t(T)_{t+τ})
  - Debt-financed transfer long-run balance: T = Σ_{τ=0}^{∞} Q_{t+τ}^T_{t+τ}
  - With fiscal rule T_t − T_ss = φb (B_t − B_ss): Σ_{τ=0}^{∞} t(T)_{t+τ} = φb T Σ_{τ=0}^{∞} (1−φb)^τ 1/Q_{t+τ}
  - Impact coefficient for consumption (μ0^T): μ0^T = (1−β(1−u))(1− φ / [1−(1−φ)(1−u)])

### 2.1 Calibration — baseline choices and implications
- Baseline HTM share:
  - ω = 0.2 (HTM households spend 100% of transfer immediately)
- Myopia parameter range used in IRFs and simulations:
  - u = 0.06 − 0.175
  - Implications by value:
    - Wealthy households spend about 3-14% of a windfall in the initial quarter (impact coefficient μ0^T).
    - Combined with ω = 0.2, initial impact multiplier ≈ 1/4–1/3 from a transitory income shock.
    - u = 0.175: about 90% of the transfer is spent over 5 years.
    - u = 0.06: about half of transfers are spent over 5 years.
  - Literature anchors within range: Gabaix cognitive discounting u = 0.15; Angeletos et al. u = 0.13.

### 2.4 Baseline response to a 1% of GDP transfer shock — shock specification and output
- Shock:
  - Fiscal transfer shock initially equivalent to 1% of GDP and decaying at rate 0.7.
- Output response:
  - Output rises persistently and remains above zero at the 5 year (20-quarter) horizon.
  - At u = 0.175 the short-term response of output is close to the size of the transfer shock as a share of GDP despite a partial equilibrium MPC of just 1/3.
  - At u = 0.06 the immediate impact multiplier is 0.4.

### 2.4 Wealthy household linearized consumption (reported exact form)
- c_{w,t} = β 1−ω B_{ss} C_{ss} (1−β(1−u))(b_{t−1}+r_{t−1}) + (1−β(1−u)) Σ_{τ=0}^{∞} ((1−u)β)^{τ} (− 1/σ r_{t+τ} + y_{t+τ} − t_{t+τ} )

### 2.4 Multipliers — NPV benchmarking and comparison to SVARs
- Multiplier calculation method:
  - Mountford and Uhlig (2009) NPV method:
    - Multiplier = [ Σ_{t=0}^{T} β^{t} (Y_{t}/Y_{ss} − 1) ] / [ Σ_{t=0}^{T} β^{t} (T_{t}/Y_{ss} − T_{ss}/Y_{ss}) ]
- Distortionary tax-cut model results:
  - NPV multiplier range: 2.5 (u = 0.06) to 68 (u = 0.175)
  - Mid-range u = 0.11–0.12: NPV multiplier ≈ 8–11
- Non-distortionary transfer (lump-sum) results:
  - 5-year NPV multiplier ranges 1.5–6.7 for the u range used.
- Investment response:
  - Model produces positive investment responses to transfer stimulus, matching several SVAR findings.

### 2.4 Mechanisms required for large persistent multipliers
- Necessary model features beyond labor-only NK:
  - Sticky wages (staggered wage resetting) to amplify labor quantity adjustment and second-round spending.
  - Capital and investment with adjustment costs to add persistence through investment responses.
  - Interest payments/wealth channel: wealthy households’ government-debt holdings earn higher real returns when interest rates rise; myopia reduces the offsetting weight of future tax liabilities.
- Calibration point:
  - Steady state government debt assumed equal to steady state output (100% debt-to-GDP).
- Counterfactual:
  - Taxing wealthy households on interest receipts to recover interest each period materially reduces IRF persistence.
- Variant experiment (all use u = 0.12): removing sticky wages, capital, or interest reduces persistence and scale of GDP response to a 1% of GDP transfer shock.

### 2.4 Pandemic EIP simulation — design and key outcomes
- Pandemic context and inputs:
  - Excess savings peaked at about 10% of household income (cumulative difference relative to 2016-19 average savings rate).
  - Pandemic EIP fiscal transfers total simulated: $850bn, or about 4% of US GDP (2019 levels).
  - 2008-era cash transfers ≈ $100bn (for comparison).
- Simulation setup:
  - EIP payments between 2020Q1–2021Q4 using BEA quarterly profile as share of potential GDP (potential GDP grown at 2% counterfactual).
  - Three cases:
    1. Partial equilibrium: first-round consumption spending by HTM and myopic wealthy households only (u = 0.06).
    2. Full general equilibrium, Ricardian wealthy households (standard Ricardian TANK).
    3. Full general equilibrium, myopic wealthy households (u ∈ [0.06−0.175], midpoint u = 0.12 shown).
- Key simulation findings:
  - Ricardian: effects short-lived; spending increases mainly among HTM households then quickly slows.
  - Myopic wealthy households: materially larger and prolonged general equilibrium effects:
    - Investment rises sharply in response to persistent consumption demand.
    - HTM and wealthy households receive higher incomes, amplifying spending.
    - Endogenous monetary response (higher interest rates) increases wealthy households’ assets and further stimulates consumption via the wealth channel; initial inter-temporal substitution may reduce wealthy consumption in general equilibrium relative to partial-equilibrium first-round effects.
  - Midpoint u = 0.12 results:
    - Output rises persistently above baseline by three percentage points for several years, fading only by 2030.
    - Inflation rises persistently, about 1 1/4 percentage points above baseline early in the simulation and only begins to decline in 2024.
- Sensitivity to Phillips slope κ:
  - Baseline κ ≈ 0.08.
  - Choosing θ = 0.9 (thus κ ≈ 0.01) lowers average annualized inflation impact during 2021-2023 from 1.2 percentage points to 0.75 percentage points; inflation remains about 0.5 percentage point higher in 2024 with the shallower curve.
- Dynamics across u:
  - High u (top of range): dissaving progresses rapidly then slows as assets deplete.
  - Mid u (u = 0.12): consumption slows less rapidly due to slower drawdown of excess assets.

### 2.4 Monetary policy transmission with myopia and high debt
- Conceptual mechanism:
  - Endogenous interest rate reactions to fiscal transfers increase wealthy households’ interest income; with myopia this wealth channel makes monetary policy shocks less contractionary.
  - Effect depends on initial debt-to-output and fiscal rule responsiveness.
- Model modifications:
  - HTM households introduced as credit-constrained net debtors; wealthy households own this debt.
  - HTM borrowing constraint: R_{t} b^{H}_{t} / ω ≤ Ψ π_{t+1}, with Ψ set at 20% of GDP.
- Monetary shock experiment:
  - Shock: 1 percentage point to policy rate, autoregressive decay rate 0.5.
  - Three cases (all use u = 0.12):
    - High debt, lax rule: Debt:Output = 100%, φ_{b} = 0.05.
    - Medium debt, medium rule: Debt:Output = 75%, φ_{b} = 0.1.
    - Low debt, strict rule: Debt:Output = 50%, φ_{b} = 0.3.
- Key results:
  - High debt & weak fiscal rule: higher interest rates are less contractionary and can become mildly stimulative as higher interest receipts are spent by myopic wealthy households.
  - Low debt & strict fiscal rule: monetary policy is more traditionally contractionary.

### Conclusion — main takeaways
- A New Keynesian TANK model augmented with:
  - myopic households with realistic dissaving behavior,
  - sticky wages enabling large quantity responses with subdued marginal cost increases, and
  - capital and investment with adjustment costs,
  can reproduce highly persistent output responses to wealth shocks and fiscal transfers similar to VAR-based empirical tax-shock estimates.
- Calibrated implications:
  - Large and persistent output and inflationary responses to COVID-era transfers (2020–21) consistent with accumulated ‘excess savings’ supporting post-COVID output and consumption resilience.
  - The wealth channel via higher interest receipts is a key amplification mechanism when households are myopic.
  - That same wealth channel can offset contractionary effects of monetary tightening when steady state debt is high and fiscal rules are weak, implying important fiscal–monetary interaction effects.

### Appendix A — model setup (households, wages, investment, government, firms, retailers)
- Households:
  - Two types: HTM (subscript “H”) consumes all labor income net of lump-sum taxes each period; Wealthy (subscript “W”) optimizes inter-temporally and die with probability u each period.
  - Fractions: HTM share ω; Wealthy share 1−ω.
  - Aggregate consumption: C_t = ω C_{H,t} + (1−ω) C_{W,t}
  - Wealthy survivor lifetime utility: ∞∑_{τ=0} ((1−u)β)^τ U(C^W_{s,t+τ}, N_{t+τ}), with U(C^W_s, N) = C_{W_s,t}^{1−σ}/(1−σ) − N_t^{1+χ}/(1+χ)
  - Wealthy budget constraint (scaled) and Euler condition preserved as in source.
  - Linearized wealthy consumption (preserved exact equations) shows wealthy consume fraction (1−β(1−u)) of additional wealth and face inter-temporal substitution (−1/σ r_t) and wealth channels.
- Wage determination:
  - Labor supplied via a union, differentiated with elasticity ε_w; sticky wage resetting with θ_w and survival-weighted discounting ((1−u)βθ_w).
  - Optimal wage-resetting condition, recursions for x_{1,t} and x_{2,t}, and aggregate wage index preserved as in source.
- Investment:
  - Wealthy own capital; capital accumulation and first-order conditions (Tobin’s q and investment Euler) preserved exactly.
  - Aggregate capital accumulation: K_{t+1} = I_t − φ/2 (I_t/I_{t−1} − 1)^2 I_t + (1−δ) K_t
- Government sector:
  - Budget constraint: B_t + T_t = R_{t−1}/π_t B_{t−1} + G_t + T_t
  - Fiscal rule: T_t/Y_t − T_{ss}/Y_{ss} = φ_b ( B_t/Y_t − B_{ss}/Y_{ss} )
- Firms and retailers:
  - Production technology, first-order conditions, real marginal cost MC_t, final goods aggregator, Calvo pricing with survival discounting ((1−u)β) and stickiness θ preserved with recursions and price-reset conditions as in source.
- Full list of model equations:
  - Equations (2)–(24) provided in source are preserved, including exact functional forms and stationarized recursions for price-setting and wage-setting.

### Appendix B — distortionary and non-distortionary taxes
- Alternative specification: distortionary income taxes at rate τ on wage income instead of lump-sum taxes.
- Budget constraints for wealthy and HTM with distortionary tax preserved as in source.
- Union optimization and wage-resetting first-order condition incorporate (1−τ_t) factors; flexible-wage special case with taxes preserved exactly.

*Source: wpiea2024208-print-pdf — IMF working paper content (sections 2.1, 2.4, Appendix A and Appendix B) as supplied.*

### 2.1    Calibration

### 2.1    Calibration

### Baseline parameterization and preferences
- Objective: demonstrate large output effects from fiscal transfers to households in the presence of myopia; use standard parameters except for the discounting parameter u.
- Preferences and rates:
  - β Discount rate: 0.995 (set to closely match an annualized real steady state interest rate of 2%)
  - σ Intertemporal elasticity: 1
  - χ Inverse Frisch elasticity of labour: 1
- Production and capital:
  - α Capital share in production: 0.3
  - δ Depreciation rate: 0.025
  - φk Capital adjustment cost: 5
- Price and wage stickiness:
  - θ Calvo parameter, consumer goods: 0.75
  - θw Calvo parameter, wages: 0.75
  - Implied price/wage reset frequency: prices and wages are reset after 3 quarters on average
  - Phillips curve slope κ evaluates at 0.08 (κ = (1−θ)(1−βθ)/(θ) )
- Other elasticities:
  - ε Elasticity of demand for consumption varieties: 10
  - εw Elasticity of demand for labor varieties: 10
- Policy rule parameters:
  - φπ Taylor rule: inflation parameter: 1.5
  - φb Fiscal rule: elasticity of tax to debt: 0.05 (tax adjustment of just 5% of additional debt as a proportion of GDP each quarter; used to approximate a relatively ‘weak’ fiscal rule)

### Rationale for fiscal-rule choice and implications
- φb = 0.05:
  - Close to the rule estimated in Gali et al. (2007).
  - Produces slow adjustment of taxes: only 5% of additional debt (as a proportion of GDP) is adjusted each quarter.
  - This weak fiscal rule delays a large proportion of the adjustment of taxes to fund transfers years into the future, affecting how much of transfers are spent once future tax liabilities are accounted for.

### Calibrating the rate of dissaving from wealth and fiscal transfer shocks
- Calibration targets:
  - The degree of myopia of wealthy households (u) is the primary parameter of interest, governing their disposition to spend in response to income or asset shocks.
  - The share of HTM (hand-to-mouth) households (ω) must be calibrated jointly with u to match empirical spending time-profiles.
- Empirical facts guiding calibration:
  - Short-run evidence: about 1/4 of transfers and transitory income shocks are spent in the quarter received (Kaplan and Violante, 2014; Whalen and Reichling, 2015; Commault, 2022). Parker et al. (2022) find short-term spending rates close to 25% for COVID-related stimulus.
  - Heterogeneity: Misra and Surico (2014) find around 20% of consumers spent the majority of checks; higher-wealth/income consumers show smaller immediate spending rates.
  - Longer-term evidence:
    - Lottery wins in Norway: about half of transitory income shocks are spent within one year; approximately 90% are spent over 5 years (Fagereng et al., 2021).
    - Colarieti et al. (2024): initial MPC of 0.15 in the first quarter, rising to about 0.40 after one year.
    - Christelis et al. (2019): MPCs around 40% after one year (Netherlands surveys).
  - Implication: realistic modeling should include wealthy households spending a material proportion of transfer income over time in addition to HTM households spending all at receipt.

### Translating empirical spending profiles into model behavior
- Wealthy-household partial-equilibrium consumption response to a one-off government transfer T:
  - ∆C^W = (1−β(1−u))(T − Σ_{τ=0}^{∞} ((1−u)β)^τ t(T)_{t+τ})
  - Future expected tax liabilities t(T)_{t+τ} enter the expression and reduce immediate dissaving when discounted (even with elevated discounting rates).
- Debt-financed transfer condition (long-run stability of debt):
  - T = Σ_{τ=0}^{∞} Q_{t+τ}^T_{t+τ}
- Under the fiscal rule T_t − T_ss = φb (B_t − B_ss) (assuming constant output), the sequence of tax payments for additional debt financing of transfer T is:
  - Σ_{τ=0}^{∞} t(T)_{t+τ} = φb T Σ_{τ=0}^{∞} (1−φb)^τ 1/Q_{t+τ}
- For this tax-rule specification, the impact multiplier (impact coefficient μ0^T) for consumption spending as a proportion of the transfer is:
  - μ0^T = (1−β(1−u))(1− φ / [1−(1−φ)(1−u)])  (denoted |{z} Impact coefficient, μ0^T in text)
- Dynamics:
  - In the next period, households retain (1−μ0^T) of the initial transfer, earn interest, and pay tax while outstanding debt decreases; the fiscal rule and u jointly determine the evolution of spending out of transfers.

### Baseline choices for u and ω and implied spending profiles
- Baseline share of HTM households:
  - ω = 0.2 (HTM households spend 100% of transfer immediately)
  - Rationale: micro evidence that about 1/4 of transfers are spent immediately, heterogeneity across consumers, and HTM share of around 20% matches substantial immediate spending by a minority.
- Range chosen for the myopia parameter u:
  - u = 0.06 − 0.175 used in IRFs and simulations
  - Implications of the range:
    - Wealthy households will spend about 3-14% of a windfall in the initial quarter (impact coefficient μ described above).
    - Combined with ω = 0.2, the model produces an initial impact multiplier of about 1/4–1/3 from a transitory income shock, consistent with empirical evidence.
    - Upper bound u = 0.175: about 90% of the transfer is spent over 5 years (aligned with Fagereng et al., 2021).
    - Lower bound u = 0.06: implies about half of transfers are spent over 5 years, reflecting evidence that about 50% of households show little detectable change in spending after transfers (Misra and Surico, 2014; Kaplan and Violante, 2014).
    - The u range encompasses other literature values: Gabaix (2020) cognitive discounting u = 0.15 and Angeletos et al. (2024) u = 0.13.
- Empirical consistency:
  - SVAR evidence on general equilibrium effects of fiscal stimulus (discussed later in paper) will show this u range produces plausible IRFs.

*Source: wpiea2024208-print-pdf - 2.1    Calibration*

### 2.4    Baseline response to a transfer shock

### 2.4    Baseline response to a transfer shock

### Baseline impulse responses to a 1% of GDP transfer shock
- Shock specification: fiscal transfer shock initially equivalent to 1% of GDP and decaying at rate 0.7.
- Output response:
  - Output rises persistently and remains above zero at the 5 year (20-quarter) horizon.
  - At u = 0.175 (top of myopic discounting range), the short-term response of output is close to the size of the transfer shock as a share of GDP despite a partial equilibrium MPC of just 1/3.
  - At u = 0.06 the immediate impact multiplier is 0.4.
- Comparison to micro and DSGE estimates:
  - The u = 0.06 multiplier is closer to empirically estimated multipliers from micro econometric literature and standard DSGE models, but the model produces much more persistent output effects.

### Exact linearized wealthy household consumption expression (as reported)
- c_{w,t} = β 1−ω B_{ss} C_{ss} (1−β(1−u))(b_{t−1}+r_{t−1}) + (1−β(1−u)) Σ_{τ=0}^{∞} ((1−u)β)^{τ} (− 1/σ r_{t+τ} + y_{t+τ} − t_{t+τ} )

### Note on reported units in figures
- Percent deviations from steady state following a fiscal transfer shock to HTM and wealthy households initially equivalent to 1% of output.
- Interest and inflation deviations are provided at annualized rates and in percentage point units rather than percent deviations.

---

### Estimated multipliers and relation to the SVAR literature
- Benchmarking approach:
  - Multipliers calculated using Mountford and Uhlig (2009) NPV method:
    - Multiplier = [ Σ_{t=0}^{T} β^{t} (Y_{t}/Y_{ss} − 1) ] / [ Σ_{t=0}^{T} β^{t} (T_{t}/Y_{ss} − T_{ss}/Y_{ss}) ], where T reflects the fiscal cost of the stimulus.
- Comparison to recent SVAR tax-cut literature:
  - Literature reports substantial 5-year NPV tax multipliers (examples in the source include values such as 8, 5, 3, 2 for various papers as summarized in Table 2).
  - Tax-cut multipliers generally larger than government-spending multipliers (government spending closer to 1).
- Model variants and NPV multipliers:
  - Distortionary tax-cut model:
    - NPV multiplier ranges from 2.5 (u = 0.06) to 68 (u = 0.175).
    - In the middle of discount factor range (u = 0.11–0.12), NPV multiplier ≈ 8–11, very close to Mertens and Ravn (2013).
  - Non-distortionary transfer (applicable to lump-sum transfers):
    - 5-year NPV multiplier ranges 1.5–6.7 for the range of discount factors used.
- Interpretation:
  - The model’s range of myopic discounting and spending time profiles from microdata literature is broadly consistent with SVAR general equilibrium effects.
  - Very high MPCs (e.g., Fagereng et al. (2021)) are toward the top of plausible general equilibrium outturns as judged by implied NPV multipliers and IRF confidence intervals.
- Investment response:
  - Model produces a positive response of investment to transfer stimulus, consistent with several SVAR studies finding large positive residential and non-residential investment responses to personal income tax cuts.

---

### Importance of sticky wages, capital, and interest payments to generate large and persistent multipliers
- Key model features required (beyond labor-only New Keynesian baseline) to replicate SVAR-like persistence and scale:
  - Sticky wages (staggered wage resetting) increase labor quantity adjustment and support larger amplification via increased wage income and second-round spending.
  - Presence of capital and investment (with adjustment costs) adds persistence as investment responds to sustained consumption demand.
  - Interest income on wealth amplifies longer-horizon multipliers via higher government-debt interest receipts to wealthy households when interest rates rise endogenously.
- Quantitative calibration points and mechanisms:
  - Steady state government debt assumed equal to steady state output (a 100% debt-to-GDP ratio).
  - Wealth effect operates because wealthy households' holdings of government debt earn higher real returns when interest rates rise; with myopia, discounted future tax liabilities are down-weighted so wealth gains are not fully offset.
  - Counterfactual exercise: when government taxes are increased to recover all interest payments each period (wealthy households taxed on interest), IRFs are substantially less persistent.
- Figure variant experiment (all variants use u = 0.12):
  - Baseline vs. No sticky wages vs. No capital vs. No interest: removing sticky wages, capital, or interest reduces persistence and scale of GDP response to a 1% of GDP transfer shock.

---

### Simulating the effects of the pandemic EIP transfers on output
- Pandemic excess savings and transfer overview:
  - Households accumulated ‘excess savings’ peaking at about 10% of household income (calculated as cumulative difference in household savings relative to counterfactual where savings rate remained at 2016-19 average).
  - Pandemic EIP fiscal transfers simulated: total $850bn, or about 4% of US GDP (2019 levels).
  - By comparison, 2008-era cash transfers ≈ $100bn.
- Simulation design:
  - EIP payments between 2020Q1–2021Q4 simulated using BEA quarterly profile as share of potential GDP (with potential GDP grown at 2% counterfactual).
  - Three simulated cases shown in Figure 6:
    1. Partial equilibrium: first-round consumption spending by HTM and myopic wealthy households only (myopia u = 0.06).
    2. Full general equilibrium, Ricardian wealthy households (standard Ricardian TANK).
    3. Full general equilibrium, myopic wealthy households (myopic discounting u ∈ [0.06−0.175], midpoint u = 0.12 shown).
- Key simulation findings:
  - Ricardian framework: effects short-lived; spending increases mainly among HTM households then quickly slows.
  - With myopia: transfers have materially larger and prolonged general equilibrium effects:
    - Investment rises sharply in response to persistent consumer spending.
    - HTM and wealthy households receive higher incomes, amplifying spending.
    - Endogenous monetary response (higher interest rates) increases wealthy households’ assets and further stimulates consumption via wealth channel; initially, however, inter-temporal substitution can reduce wealthy households’ consumption in general equilibrium relative to partial-equilibrium first-round effects.
  - Midpoint calibration (u = 0.12) outcomes:
    - Output rises persistently above baseline by three percentage points for several years, fading only by 2030.
    - Inflation rises persistently, about 1 1/4 percentage points above baseline early in the simulation and only begins to decline in 2024.
- Sensitivity to Phillips curve slope:
  - Baseline uses κ ≈ 0.08.
  - Choosing θ = 0.9 (thus κ ≈ 0.01) lowers average annualized inflation impact during 2021-2023 from 1.2 percentage points to 0.75 percentage points; inflation remains about 0.5 percentage point higher in 2024 with the shallower curve.
- Dynamics across myopia range:
  - At high myopia (u top of range), dissaving progresses rapidly but then slows as assets deplete.
  - At midpoint (u = 0.12), consumption slows less rapidly due to slower drawdown of excess assets.

---

### Monetary policy transmission with myopia and high debt
- Conceptual mechanism:
  - Endogenous interest rate reactions to fiscal transfers increase wealthy households’ interest income; with myopia this wealth channel makes monetary policy shocks less contractionary.
  - Effect depends on initial debt-to-output and fiscal rule responsiveness.
- Model adjustment for monetary transmission exercise:
  - Introduce HTM households as credit-constrained net debtors; wealthy households own this debt.
  - HTM borrowing constraint: R_{t} b^{H}_{t} / ω ≤ Ψ π_{t+1}, with Ψ set at 20% of GDP (roughly average US consumer credit outstanding per Flow of Funds). Mortgage debt ignored.
- Monetary policy shock experiment:
  - Shock: 1 percentage point to policy rate, autoregressive decay rate 0.5.
  - Three cases (all simulations use u = 0.12):
    - High debt, lax rule: Debt:Output = 100%, φ_{b} = 0.05.
    - Medium debt, medium rule: Debt:Output = 75%, φ_{b} = 0.1.
    - Low debt, strict rule: Debt:Output = 50%, φ_{b} = 0.3.
- Key results:
  - High debt with weak fiscal rule: higher interest rates are less contractionary and can become somewhat stimulative as higher interest receipts are spent by wealthy households who heavily discount future tax liabilities.
  - Low debt with strict fiscal rule: monetary policy is more traditionally contractionary, weighing on activity in near and medium term.
- Policy implication:
  - Fiscal-monetary interactions matter: the effectiveness of monetary policy in influencing inflation and activity depends on steady state debt and how fiscal policy responds to interest-driven debt increases.

---

### Conclusion — main takeaways
- A New Keynesian (TANK) model augmented with:
  - a) myopic households with realistic dissaving behavior,
  - b) sticky wages enabling large quantity responses with subdued marginal cost increases, and
  - c) capital and investment with adjustment costs,
  can reproduce highly persistent output responses to wealth shocks and fiscal transfers similar to VAR-based empirical tax-shock estimates.
- The calibrated model implies:
  - Large and persistent output and inflationary responses to the large COVID-era transfers (2020–21), consistent with the role of accumulated ‘excess savings’ in supporting post-COVID output and consumption resilience.
  - The wealth channel from higher interest receipts is a key mechanism: endogenous interest rate rises compound wealth and spending when households are myopic.
  - This same wealth channel can offset contractionary effects of monetary tightening when steady state debt is high and fiscal rules are weak, highlighting important fiscal-monetary interaction implications.

*Source: IMF working paper content (section 2.4 and surrounding panels) from the supplied PDF content.*

### Appendix A    Model setup

### Appendix A    Model setup

### A.1 Households
- Two household types:
  - Hand-to-mouth (HTM) household (subscript “H”): consumes all labor income net of lump-sum taxes each period.
  - Wealthy saving agent (subscript “W”): supplies labor, receives dividends from firm profits and returns on physical capital, holds net positive financial assets in government debt, can optimize inter-temporally.
- Fractions:
  - HTM share: ω
  - Wealthy share: 1−ω
- Aggregate consumption:
  - C_t = ω C_{H,t} + (1−ω) C_{W,t}
- HTM consumption:
  - C_{H,t} = W_t N_t − T_t + T_t
- Wealthy agents die with probability u each period; surviving agents receive assets of the dead (adds factor 1/(1−u) to returns). Wealthy lifetime utility (surviving cohort):
  - ∞∑_{τ=0} ((1−u)β)^τ U(C^W_{s,t+τ}, N_{t+τ})
  - U(C^W_s, N) = C_{W_s,t}^{1−σ}/(1−σ) − N_t^{1+χ}/(1+χ)
- Wealthy budget constraint (scaled by wealthy share):
  - C_{W_s,t} + B_{W_s,t} 1/(1−ω) + T_t + I_{W_s,t} 1/(1−ω) = W_t N_t + R_{t−1}/π_t (1−u) B_{W_s,t−1} 1/(1−ω) + R^k_t (1−u) K_{W_s,t} 1/(1−ω) + T_t − T_b − T_k
- Euler equation for wealthy survivors:
  - C_{W_s,t}^{−σ} = β R_t / π_{t+1} C_{W_s,t+1}^{−σ}
- Permanent-income/wealth representation for surviving wealthy:
  - C_{W,t} ∞∑_{τ=0} ((1−u)β)^{τ} (Q_{t+τ})^{1−1/σ} = R_{t−1}/((1−u)(1−ω)π_t) B_{t−1} + ∞∑_{τ=0} (1−u)^{τ} Q_{t+τ} (Y_t − T_t)
- Linearized consumption equation for wealthy (definitions preserved):
  - (1/(1−β(1−u))) c_{w,t} = β/(1−ω) B_{ss}/C_{ss} (b_{t−1} − q_{t−1}) + ∞∑_{τ=0} ((1−u)β)^{τ} (1/σ q_{t+τ} + y_{t+τ} − t_{t+τ})
  - Simplified form:
    - c_{w,t} = β/(1−ω) B_{ss}/C_{ss} (1−β(1−u))(b_{t−1} + r_{t−1}) + (1−β(1−u)) ∞∑_{τ=0} ((1−u)β)^{τ} (−1/σ r_{t+τ} + y_{t+τ} − t_{t+τ})
- Key economic intuitions:
  - Wealthy consumers consume fraction (1−β(1−u)) of additional wealth (scaled relative to consumption).
  - Interest rate developments affect consumption via inter-temporal substitution (−1/σ r_t) and via wealth accumulation through real interest income; relative importance depends on σ, size of asset holdings relative to consumption, and (1−u).

### A.2 Wage determination
- Labor supplied collectively via a union; both household types supply same labor quantity N_t at prevailing real wage W_t.
- Labor differentiated with elasticity ε_w; demand for individual’s labor:
  - N_t(i) = (W_t(i)/W_t)^{−ε_w} N_t
- Sticky wage resetting: union can reset fraction (1−θ_w) of wages each period; stickiness probability θ_w. Discounting incorporates survival: ((1−u)βθ_w)^s.
- Union maximization (notation Π_{t+s} = (1+π_{t+1})...(1+π_{t+s})):
  - Max ∑_{s=0}^∞ ((1−u)βθ_w)^s [ − N_{t}^{1+χ}/(1+χ) + λ_{t+s} ( w_t(i) Π_{t+s}^{−1} / w_{t+s} Π_{t+s}^{−1} )^{−ε_w} N_{t+s} ]
- First-order condition (optimal resetting wage):
  - w^{1+ε_w}_χ t(i) = ε_w/(ε_w − 1) ( ∑_{s=0}^∞ ((1−u)βθ_w)^s w^{ε_w(1+χ)}_{t+s} Π_t+s^{ε(1+χ)} N_{t+s}^{1+χ} ) / ( ∑_{s=0}^∞ ((1−u)βθ_w)^s λ_{t+s} w^{ε_w}_{t+s} Π_{t+s}^{ε_w−1} N_{t+s} )
- Lagrange multiplier for aggregate marginal utility:
  - λ_t = ω C_{H,t}^{−σ} + (1−ω) C_{W,t}^{−σ}
- Flexible-wage special case (θ_w = 0):
  - w_t = ε_w/(ε_w − 1) N_t^{χ} / λ_t
- Recursions for numerator and denominator (x_1,t and x_2,t):
  - x_{1,t} = w_t^{ε_w(1+χ)} N_t^{1+χ} + (1−u)βθ_w (1+π_{t+1})^{ε_w(1+χ)} x_{1,t+1}
  - x_{2,t} = λ_t w_t^{ε_w} N_t + (1−u)βθ_w (1+π_{t+1})^{ε_w−1} x_{2,t+1}
  - w^{1+ε_w}_χ t(i) = ε_w/(ε_w − 1) x_{1,t} / x_{2,t}
- Aggregate wage index:
  - w^{1−ε_w}_t = (1−θ_w) w^{1−ε_w}_t(i) + θ_w (1+π_t)^{ε_w−1} w^{1−ε_w}_{t−1}

### A.3 Investment
- Wealthy households own capital, receive firm distributions and rental income; pay lump-sum tax T_k to fund newly born households' steady state capital.
- Wealthy household budget constraint (scaled):
  - C_{W_s,t} + B_{W_s,t} 1/(1−ω) + I_{W_s,t} 1/(1−ω) = w_t N_t + Π_t/(1−ω) + r^k_t K_{W_s,t}/((1−u)(1−ω)) + R_{t−1}/π_t B_{W_s,t−1}/((1−ω)(1−u)) + T_t − T_b − T_k
- Capital law of motion for surviving wealthy households (scaled):
  - K_{W_s,t+1}/(1−ω) = I_{W_s,t}/(1−ω) − φ/2 (I_{W_s,t}/I_{t−1} − 1)^2 I_{W_s,t}/(1−ω) + (1−δ) K_{W_s,t}/((1−u)(1−ω))
- First-order conditions with multipliers λ_t and Φ_t yield Tobin’s q and investment Euler:
  - ∂L/∂K_{t+1}: q_t = β (λ_{t+1}/λ_t) (R^k_{t+1} + (1−δ) q_{t+1})
  - ∂L/∂I_t: 1 = q_t [ 1 − φ/2 (I_t/I_{t−1})^2 − φ (I_t/I_{t−1} − 1) (I_t/I_{t−1}) ] + (1−u)βφ [ q_{t+1} (λ_{t+1}/λ_t) (I_{t+1}/I_t − 1)(I_{t+1}/I_t)^2 ]
- Aggregate capital accumulation:
  - K_{t+1} = I_t − φ/2 (I_t/I_{t−1} − 1)^2 I_t + (1−δ) K_t

### A.4 Government sector
- Government budget constraint:
  - B_t + T_t = R_{t−1}/π_t B_{t−1} + G_t + T_t
  - B_t is real stock of government debt; G_t exogenous; T_t lump-sum taxes uniformly levied.
- Fiscal rule (t = T/Y):
  - T_t/Y_t − T_{ss}/Y_{ss} = φ_b ( B_t/Y_t − B_{ss}/Y_{ss} )
  - Taxes adjust proportionally to deviations of debt-to-GDP from steady state.

### A.5 Producing firms
- Firms maximize profits subject to production technology; capital rented from wealthy households.
- Profit maximization constraint:
  - P_t(i) Y_t(i) − W_t L_t(i) − r^k_t K_t(i) − λ_t [ Y_t(i) − A_t K_t(i)^α L_t(i)^{1−α} ]
- First-order conditions imply capital-labor ratio:
  - K_t / L_t = α/(1−α) W_t / r^k_t
- Real marginal cost:
  - MC_t = (1−α)^{−(1−α)} α^{−α} w_t^{1−α} r_{k,t}^{α} A_t^{−1}

### A.6 Retailers and final goods firms
- Final goods aggregator:
  - Y_t = [ ∫_0^1 Y_t(j)^{(ε−1)/ε} dj ]^{ε/(ε−1)}
- Retailer demand for variety j:
  - Y_t(j) = ( P_t(j) / P_t )^{−ε} Y_t
- Price index:
  - P_t^{1−ε} = ∫_0^1 P_t(j)^{1−ε} dj
- Calvo pricing with survival discounting ((1−u)β) and stickiness θ:
  - P^#_{k,t} = ε/(ε−1) [ ∑_{s=0}^∞ ((1−u)β)^s θ^s (C_{t+s}^{−σ}) P_{t+s}^{ε} MC_{t+s} Y_{t+s} ] / [ ∑_{s=0}^∞ ((1−u)β)^s θ^s (C_{t+s}^{−σ}) P_{t+s}^{ε−1} Y_{t+s} ]
- Definitions and recursions (stationarized via inflation terms):
  - ̄XMC_t = C_t^{−σ} P_t^{ε} MC_t Y_t + (1−u)βθ E[ ̄XMC_{t+1} ]
  - ̄X_t = C_t^{−σ} P_t^{ε−1} Y_t + (1−u)βθ E[ ̄X_{t+1} ]
  - Using P-term normalization: ̄xmc_t = C_t^{−σ} Y_t MC_{k,t} + βθ (1+π_{t+1})^{ε} E[̄xmc_{t+1}]
  - ̄x_t = C_t^{−σ} Y_t + βθ (1+π_{t+1})^{ε−1} E[̄x_{t+1}]
  - Optimal reset price:
    - (1+π^#_t) = ε/(ε−1) ̄xmc_t / ̄x_t
  - Aggregate inflation relation (Calvo aggregation):
    - 1 = (1−θ) (1+π^#_t)^{1−ε} + θ (1+π_t)^{ε−1}

### A.7 Full list of model equations
- Household consumption and wage setting (preserved equations):
  - C_{W,t} ∞∑_{τ=0} ((1−u)β^{1/σ})^{τ} (Q_{t+τ})^{1−1/σ} = R_{t−1}/((1−ω)π_t) B_{t−1} + ∞∑_{τ=0} (1−u)^{τ} Q_{t+τ} (Y^w_t − T_t + T_t)  (2)
  - C_{H,t} = W_t N_t + T_t − T_t  (3)
  - C_t = ω C_{H,t} + (1−ω) C_{W,t}  (4)
  - Y^w_t = W_t N_t + + R^k_t (1−u) K_t 1/(1−ω) + Π_t 1/(1−u) − I_t 1/(1−ω)  (5)
  - x_{1,t} = Ψ w^{ε_w(1+χ)}_t N^{1+χ}_{t+s} + β θ_w (1+π_{t+1})^{ε_w(1+χ)} x_{1,t+1}  (6)
  - x_{2,t} = λ_t w^{ε_w}_t N_t + β θ_w (1+π_{t+1})^{ε_w−1} x_{2,t+1}  (7)
  - w^{1+ε_w}_χ t(i) = ε_w/(ε_w − 1) x_{1,t} / x_{2,t}  (8)
  - w^{1−ε_w}_t = (1−θ_w) w^{1−ε_w}_t(i) + θ_w (1+π_t)^{ε_w−1} w^{1−ε_w}_{t−1}  (9)
  - K_{t+1} = I_t − φ/2 (I_t/I_{t−1} − 1)^2 I_t + (1−δ) K_t  (10)
  - q_t = β (λ_{t+1}/λ_t) (R^k_{t+1} + (1−δ) q_{t+1})  (11)
  - 1 = q_t [ 1 − φ/2 (I_t/I_{t−1})^2 − φ (I_t/I_{t−1} − 1) (I_t/I_{t−1}) ] + (1−u)βφ [ q_{t+1} (λ_{t+1}/λ_t) (I_{t+1}/I_t − 1)(I_{t+1}/I_t)^2 ]  (12)
  - K_t / L_t = α/(1−α) W_t / r^k_t  (13)
  - MC_t = (1−α)^{−(1−α)} α^{−α} w_t^{1−α} r_{k,t}^{α} A_t^{−1}  (14)
  - ̄xmc_t = C_t^{−σ} Y_t MC_{k,t} + βθ(1+π_{t+1})^{ε} E[ ̄xmc_{t+1} ]  (15)
  - ̄x_t = C_t^{−σ} Y_t + βθ(1+π_{t+1})^{ε−1} E[ ̄x_{t+1} ]  (16)
  - (1+π^#_t) = ε/(ε−1) ̄xmc_t / ̄x_t  (17)
  - 1 = (1−θ)(1+π^#_t)^{1−ε} + θ(1+π_t)^{ε−1}  (18)
  - Π_t = (1 − MC_t) Y_t  (19)
  - Y_t = C_t + I_t + G_t  (20)
  - Y_t = A_t K_t^{α} N_t^{1−α}  (21)
  - B_t + T_t = R_{t−1}/π_t B_{t−1} + G_t + T_t  (22)
  - T_t/Y_t − T_{ss}/Y_{ss} = φ_b ( B_t/Y_t − B_{ss}/Y_{ss} )  (23)
  - R_t = R_{ss} Π^{φ}_{π_t}  (24)

### Appendix B    Distortionary and non-distortionary taxes
- Alternative specification: government imposes distortionary income taxes at rate τ on wage income (instead of lump-sum).
- Budget constraints with distortionary wages:
  - Wealthy:
    - C_{W,t} + B_t 1/(1−ω) + T_t + I_t 1/(1−ω) = (1−τ_t) W_t N_t + R^k_t (1−u) K_t 1/(1−ω) + R_{t−1}/π_t (1−u) B_t 1/(1−ω)
  - HTM:
    - C_{H,t} = (1−τ_t) W_t N_t − T_t + T_t
- Union optimization incorporates post-tax wage income (1−τ_t) factor in objective:
  - ∑_{s=0}^∞ (β θ_w)^s [ − Ψ N_{t}^{1+χ}/(1+χ) + λ_{t+s} (1−τ_t) ( w_t(i) / w_{t+s} Π_{t+s}^{−1} )^{−ε_w} N_{t+s} ]
- First-order condition for resetting wages under distortionary tax (preserved):
  - w^{1+ε_w}_χ t(i) = ε_w/(ε_w − 1) [ ∑_{s=0}^∞ ((1−u)βθ_w)^s Ψ w^{ε_w(1+χ)}_{t+s} N_{t+s}^{1+χ} ] / [ ∑_{s=0}^∞ ((1−u)βθ_w)^s λ_{t+s} (1−τ_t) w^{ε_w}_{t+s} Π_{t+s}^{ε_w−1} N_{t+s} ]
- Flexible-wage special case with taxes (θ_w = 0):
  - w(i) = ε_w/(ε_w − 1) N_t^{χ} (1−τ_t) λ_t

*Appendix A and Appendix B model equations and descriptions as provided in the source content.*

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_Source: https://www.imf.org/-/media/files/publications/wp/2024/english/wpiea2024208-print-pdf.pdf_