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---

### 2.1 Households — preferences, budget, and bonds
- Preferences and optimization:
  - Representative household maximizes lifetime utility: max E_0 ∑_{t=0}^∞ β^t U(c_t,n_t). (2.1)
  - Choice variables: consumption c_t, labor n_t, nominal government bonds B_t.
- Budget constraint (period):
  - c_t + Q_t B_t − (1−κ) B_{t−1} / P_t = B_{t−1} / P_t + (1−τ_t) w_t n_t + Υ_t + z_t + ξ_t. (2.2)
  - c_t ≡ [∫_0^1 c_t(i)^{(θ−1)/θ} di]^{θ/(θ−1)}; P_t composite price; w_t real wage; τ_t labor income tax.
- Government bonds and maturity:
  - Bonds sell at price Q_t and pay (1−κ) dollars t+1 periods later.
  - Average bond maturity: (1−β(1−κ))^{−1} quarters.
  - Short-term bond case: κ = 1.
  - Transversality condition: lim_{T→∞} E_t { q_{t,T} D_T } = 0, (2.3), with q_{t,T} = R_{T−1} (P_T / P_t), D_T ≡ B_{T−1} [1 + (1−κ) Q_T ].

### 2.4 Functional forms, parameterization, and solution method
- Household utility and shocks:
  - U(c_t, n_t) = (c_t − ν_t)^{1−σ} / (1−σ) − χ n_t^{1+φ} / (1+φ). (Equation 2.13)
  - Taste shock: ν_t = ρ_ν ν_{t−1} + ε^ν_t. (Equation 2.14), ν_t ∼ i.i.d. N(0, σ^2_ν).
  - Negative taste shocks injected to generate recessions for policy experiments.
- Baseline quarterly calibration (selected parameters):
  - β = 0.992
  - σ = 2
  - φ = 2
  - θ = 7.66
  - ψ = 78
  - κ = 1 (baseline; alternative κ = 0.05)
  - π^* = 1.005
  - b_{4y} = 0.6
  - g_y = 0.15
  - τ = 0.28
  - α_M = 1.5; γ_M = 0.15
  - α_F = 0.5; γ_F = 0
  - ρ_ν = 0.8; σ_ν = 0.0025
  - ρ_g = 0.9; σ_g = 0.01
- Notable mappings and notes:
  - ψ = 78 implies nominal price rigidity ≈ one year; θ = 7.66 implies 15% price markup.
  - Alternative κ = 0.05 → average debt maturity ≈ 20 quarters (about five years).
  - γ_M chosen as smallest value that meets transversality condition (Equation 2.3).
- Solution method:
  - Model solved fully nonlinearly with Euler equation iteration (Coleman (1991); Davig (2004)).
  - Appendices: equilibrium system, deterministic steady state, solution method.

### 3 Government spending effects: Regime F vs. Regime M — baseline experiment and channels
- Baseline spending shock and conventions:
  - Initial government spending increase: 1% of steady-state output.
  - Initial debt level: deterministic steady state = 60% of annualized steady-state output (b_{4y} = 0.6) unless specified.
  - Impulse responses: percent deviations except G/Y plotted as level differences in percent; inflation and interest variables plotted as annualized level differences.
  - One-year-ahead inflation expectations computed by simulating 10,000 draws and averaging differences at t+4.
- Main comparative results (impact and cumulative multipliers):
  - Impact inflation: regime F = 4.2% on impact; regime M = 0.5% on impact.
  - Impact output multiplier (present-value cumulative):
    - regime M: 0.58 (impact), 0.56 (4Q), 0.45 (20Q)
    - regime F: 1.27 (impact), 1.15 (4Q), 0.90 (20Q)
  - Consumption impact multiplier:
    - regime F: 0.27 (impact)
    - regime M: −0.42 (impact)
- Channels explaining differences:
  - Wealth-effect channel:
    - Regime M (γ_M > 0): households expect higher future tax burden → discourage consumption → output multiplier < 1.
    - Regime F (γ_F = 0): no expected future tax increases → consumption supported by higher real wages.
  - Intertemporal substitution channel:
    - Regime M (α_M > 1): nominal rate rises more than inflation → real interest rate increases → crowding-out.
    - Regime F (α_F = 0.5): nominal rate does not rise sufficiently → real interest rate falls → crowding-in; combined with higher inflation amplifies expansion.

### Sensitivity and robustness (selected experiments)
- Cumulative output multipliers under different calibrations (impact, 4Q, 20Q):
  - Baseline:
    - regime M: 0.58, 0.56, 0.45
    - regime F: 1.27, 1.15, 0.90
  - High government debt (b^0_{4y} = 1):
    - regime M: 0.58, 0.56, 0.44
    - regime F: 1.19, 1.08, 0.85
  - High income tax rate (τ = 0.5):
    - regime M: 0.58, 0.56, 0.43
    - regime F: 1.07, 0.98, 0.80
  - Long-term debt (κ = 0.05):
    - regime M: 0.59, 0.57, 0.49
    - regime F: 1.11, 1.01, 0.81
  - Long-term debt & more responsive MP (κ = 0.05, α_F = 0.8):
    - regime F: 0.89, 0.86, 0.78 (impact falls below one)
- Interpretations:
  - Regime F is substantially more expansionary than regime M under baseline calibration.
  - Factors that reduce expansionary effects in regime F: higher initial debt, higher τ, longer debt maturity, and a more responsive α_F.

### Role of initial government debt in regime F
- Impact output multiplier falls with higher initial debt:
  - b^0_{4y} = 0.6 → impact = 1.27
  - b^0_{4y} = 1.0 → impact = 1.19
- Mechanism:
  - Higher initial debt raises nominal base for inflation taxation → given shock, inflation rises less → smaller decline in real interest rate → smaller intertemporal substitution boost → smaller output multiplier.
- Additional notes:
  - Initial debt has almost no influence on multipliers in regime M in these simulations.
  - If government debt were held abroad, the role of indebtedness in regime F would be more pronounced.

### Role of steady-state labor income tax rate in regime F
- Simulated τ values: τ = 0, τ = 0.28 (baseline), τ = 0.5:
  - Higher τ → smaller inflation increase for given spending shock → smaller decline in real interest rate → smaller consumption and output multipliers.
  - Consumption and real wage rise most when τ = 0.
- Regime M: steady-state τ has little effect on output multipliers.

### Debt maturity, monetary responsiveness, and multipliers (Section 3.4)
- Debt maturity effects:
  - Short-term debt baseline (κ = 1): impact output multiplier = 1.27.
  - Longer-term debt (κ = 0.05, ≈ five years): impact output multiplier = 1.11.
  - Mechanism: longer maturity spreads required inflation to future periods → lower current inflation → smaller multipliers.
- Monetary policy responsiveness in regime F:
  - Increasing α_F (while remaining in regime F) limits short-run inflation → reduces decline in real interest rate → reduces multipliers.
  - Combination κ = 0.05 and α_F = 0.8 → impact output multiplier = 0.89 (multipliers fall below one; consumption multipliers can turn negative).
- Empirical reference:
  - Average maturity of total U.S. outstanding Treasury marketable debt (2000–2018) ≈ five years.

### Policy regime uncertainty: setup and calibrated switching probabilities
- Policymakers follow two-state Markov chain with transition probabilities ρ_MM and ρ_FF.
- ρ_FF is increasing in last-period inflation π_{t−1} via logistic Φ(π_{t−1}); households do not expect switching to M until inflation rises above π^*.
- Calibration for simulations:
  - α_1 = 0.05 in Φ(π_{t−1}).
  - ρ_MM fixed at 0.98 → average duration of 50 quarters in regime M.

### Policy regime uncertainty effects on multipliers
- Regime F with expectations of switching to M (initial debt-to-annual output varied; baseline initial debt = 70%):
  - Fixed regime F: impact = 1.27, 4Q = 1.15, 20Q = 0.90.
  - Initial debt = 0.7: impact = 0.90, 4Q = 0.87, 20Q = 0.86.
  - Initial debt = 0.8: impact = 0.76, 4Q = 0.74, 20Q = 0.78.
- Interpretation:
  - Expectations of switching to regime M lower inflation expectations and current inflation, raise current real interest rate, reduce consumption, and lower multipliers.
  - Policy uncertainty amplifies the negative effect of high debt burden on multipliers in regime F.
  - Results robust to longer debt maturity (κ = 0.05).
- Regime M with expectations of switching to F (initial debt = 0.6):
  - Fixed regime M (ρ_MM = 1): impact = 0.58, 4Q = 0.56, 20Q = 0.45.
  - ρ_MM = 0.98: impact = 0.53, 4Q = 0.50, 20Q = 0.29.
  - ρ_MM = 0.95: impact = 0.46, 4Q = 0.39, 20Q = −0.08.
- Mechanisms in regime M:
  - Two offsetting forces: expectation of switching to F raises expected inflation (reducing multipliers) but reduces expected future tax rates (increasing multipliers); net effect generally smaller than in regime F.

### Government spending effects in recessions and the ZLB
- Recession experiment:
  - Negative taste shocks from t = 1 to t = 5 induce recession; nonlinear simulations start in regime M at t = 0; at t = 3 one economy remains in M and the other switches to F.
- Dynamics:
  - Staying in regime M: ZLB binds from t = 2 to t = 8; higher debt-to-GDP; nominal rate at ZLB.
  - Switching to regime F at t = 3: economy exits ZLB immediately as inflation adjusts; nominal rate rises because monetary authority in F still responds to inflation.
- Fiscal multipliers (cumulative output multipliers from Table 5):
  - Regime M:
    - Normal times: impact = 0.56, 4Q = 0.53, 8Q = 0.50, 20Q = 0.39.
    - Recession: impact = 1.27, 4Q = 1.34, 8Q = 1.18, 20Q = 1.04.
  - Regime F (α_F = 0.5):
    - Normal times: impact = 1.27, 4Q = 1.14, 20Q = 0.90.
    - Recession: impact = 1.28, 4Q = 1.15, 20Q = 0.90.
  - Regime F — pegged interest rate (α_F = 0):
    - Recession: impact = 1.45, 4Q = 1.18, 20Q = 0.84.
- Interpretation:
  - In regime M, multipliers are much larger in recessions because ZLB binds: higher inflation expectations lower real interest rate and crowd in consumption.
  - In regime F, switching to F can allow exit from ZLB and multipliers in recessions are similar to normal times when α_F = 0.5; pegging nominal rate in F (α_F = 0) raises multipliers further.

### Sensitivity analysis highlights
- Nominal price rigidity (ψ) effects:
  - Higher ψ increases impact multiplier in both regimes.
  - Regime F: effect nonlinear — multiplier rises rapidly as ψ increases from zero, then stabilizes beyond baseline ψ = 78.
  - If ψ = 10 (very low price rigidity) → impact output multiplier in regime F falls below one.
  - Mechanism: with low ψ, inflation rises more for given spending → income from bond holdings falls → negative income effects can dominate crowding-in.
- Persistence of government spending (ρ_g) — Section 6.2:
  - Regime F: higher ρ_g increases impact output multipliers (more persistent spending → larger expected future inflation → larger current inflation → larger crowding-in).
  - Regime M: higher ρ_g decreases impact output multipliers (more persistent spending → larger expected future tax burden → larger negative wealth effect).
- Steady-state debt-to-output (Section 6.3):
  - Regime M: impact multipliers almost unchanged as steady-state debt varies.
  - Regime F: impact output multiplier decreases as steady-state debt-to-output ratio increases (higher initial debt reduces required inflation response → smaller intertemporal substitution effect).
- Steady-state labor income tax rate τ (Section 6.4):
  - τ = 0 → impact output multiplier ≈ 1.8 (largest), because spending must be fully financed by inflation.
  - As τ increases, multipliers decline substantially.
  - Regime M: impact multipliers essentially flat in τ.

### 7 Conclusion — policy-relevant findings and methodological notes
- Summary of substantive findings:
  - Government spending multipliers under passive monetary policy (regime F) can be larger than under active monetary policy (regime M) but are sensitive to:
    - higher debt levels,
    - longer debt maturity,
    - higher distorting tax rates,
    - more responsive monetary policy to inflation (higher α_F),
    - policy regime uncertainty.
  - Policy regime uncertainty notably reduces expansionary effects in regime F: expectations of switching to regime M lower inflation expectations and multipliers; with high initial debt, multipliers in regime F can fall well below one.
  - While the framework is cashless (cannot simulate money-financed spending directly), the mechanisms distinguishing regimes are similar to those distinguishing money- and debt-financed spending, implying analogous mitigating factors for money-financed stimulus.
- Computational and solution details:
  - Euler equation iteration used to find fixed point of Euler equations; implementation in Fortran 90 parallelized with OpenMP.
  - Policy functions solved: {λ(bt−1; St), Q(bt−1; St), π(bt−1; St)} with bt−1 endogenous state and St = {νt, gt, st}.
  - State space discretized via Tauchen (1986); linear interpolation used for policy functions; quadrature used as robustness check.
  - Root-finding based on Fortran translation of csolve.m (Sims); convergence criterion: max{‖λ(1) − λ(2)‖, ‖Q(1) − Q(2)‖, ‖π(1) − π(2)‖} < 10−6.
  - Practical solving tips: regime M converges faster than regime F; use steady-state or log-linearized solutions as initial guesses; solve first with small nonzero γ_F when targeting γ_F = 0.

*Source: wpiea2020091-print-pdf*

### 2.1  Households. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   7

### 2.1  Households

### Preferences and optimization problem
- Representative household maximizes lifetime utility:
  - max E_0 ∑_{t=0}^∞ β^t U(c_t,n_t). (2.1)
- Choice variables: consumption c_t, labor n_t, nominal government bonds B_t for each period t.

### Budget constraint and income components
- Period budget constraint:
  - c_t + Q_t B_t − (1−κ) B_{t−1} / P_t = B_{t-1} / P_t + (1−τ_t) w_t n_t + Υ_t + z_t + ξ_t. (2.2)
- Definitions and interpretations:
  - c_t ≡ [∫_0^1 c_t(i)^{(θ−1)/θ} di]^{θ/(θ−1)} is the Dixit-Stiglitz consumption basket.
  - P_t is the price of the composite good.
  - w_t is the real wage rate.
  - τ_t is the labor income tax rate.
  - Left-hand side: total expenditure (consumption plus purchase of net newly issued government bond).
  - Right-hand side: household income sources:
    - bond payments from the government (B_{t-1}/P_t),
    - after-tax labor income (1−τ_t) w_t n_t,
    - dividends from firms Υ_t,
    - government transfers z_t,
    - rebate of nominal price adjustment costs ξ_t (from firms’ problem).

### Government bonds, maturity, and transversality condition
- Households have access to a portfolio of government bonds B_t selling at price Q_t at t and paying (1−κ)_t dollars t+1 periods later for each t ≥ 0 (Woodford (2001) style).
- Average bond maturity:
  - (1−β(1−κ))^{−1} quarters.
- Short-term bond case is nested with κ = 1.
- Transversality condition for bonds:
  - lim_{T→∞} E_t { q_{t,T} D_T } = 0, (2.3)
  - where q_{t,T} = R_{T−1} (P_T / P_t) and D_T ≡ B_{T−1} [1 + (1−κ) Q_T ].

*Source: wpiea2020091-print-pdf — 2.1 Households.*

### 2.4 Functional Forms, Parameterization, and the Solution Method

### 2.4 Functional Forms, Parameterization, and the Solution Method

### Household preferences and shocks
- Representative household utility:
  - U(c_t, n_t) = (c_t − ν_t)^{1−σ} / (1−σ) − χ n_t^{1+φ} / (1+φ). (Equation 2.13)
  - ν_t affects consumption taste as in Erceg and Lindé (2014) and Battistini et al. (2019).
- Taste shock process:
  - ν_t = ρ_ν ν_{t−1} + ε^ν_t. (Equation 2.14)
  - ν_t ∼ i.i.d. N(0, σ^2_ν).
- Usage:
  - Negative taste shocks are injected to generate a recession when analyzing government spending effects in such a state.

### Baseline calibration (quarterly frequency)
- Calibration choices (Table 1 / Table 2.4):
  - discounting factor (β): 0.992 — annualized real interest rate of 3%
  - risk aversion (σ): 2
  - inverse of Frisch elasticity (φ): 2
  - elasticity of substitution (θ): 7.66 — 15% price markup at the steady state
  - price adjustment cost (ψ): 78 — implied price rigidity: one year
  - government debt maturity (κ): 1 — short-term debt
  - targeted inflation (π^*): 1.005 — annualized inflation target 2%
  - steady state debt to GDP ratio (b_{4y}): 0.6 — Drautzburg and Uhlig (2015)
  - steady state government spending to GDP ratio (g_y): 0.15 — Drautzburg and Uhlig (2015)
  - steady state labor tax rate (τ): 0.28 — Drautzburg and Uhlig (2015)
  - interest rate response to inflation in regime M (α_M): 1.5 — Bianchi and Melosi (2017)
  - tax rate response to debt in regime M (γ_M): 0.15 — Bianchi and Melosi (2017)
  - interest rate response to inflation in regime F (α_F): 0.5 — Bianchi and Melosi (2017)
  - tax rate response to debt in regime F (γ_F): 0 — by definition
- Additional calibration choices and targets:
  - steady-state technology A = 1 (normalized steady-state yearly output of 1)
  - labor disutility parameter χ is endogenously set to produce steady-state labor n = 0.25
  - κ = 0.05 used in an alternative case to represent longer debt maturity (average debt maturity ≈ 20 quarters; matches average maturity of total U.S. outstanding Treasury marketable debt from 2000 to 2018 of about five years — Office of Debt Management (2018))
  - Rotemberg–Calvo mapping note: ψ = (θ−1) * ω / [(1−ω)(1−ωβ)] under first-order approximation and zero net inflation target; here model is fully nonlinear and net inflation target is not zero so ψ cannot be backed out precisely.
- Exogenous process parameters:
  - persistence of taste (ρ_ν): 0.8 — conditional probability of hitting the ZLB about 5% in regime M
  - standard deviation of the taste shock (σ_ν): 0.0025
  - persistence of government purchase (ρ_g): 0.9 — Shen and Yang (2018)
  - standard deviation of government purchase (σ_g): 0.01 — Shen and Yang (2018)
- Notes on regime-dependent policy rule parameterization:
  - α(s_t) and γ(s_t) set per Leeper (1991) active/passive definitions:
    - s_t = F: α(s_t) = α_F, γ(s_t) = γ_F
    - s_t = M: α(s_t) = α_M, γ(s_t) = γ_M. (Equation 2.15)
  - γ_M is set to the smallest value that meets the transversality condition for government debt (Equation 2.3).

### Solution method
- The model is solved fully nonlinearly with Euler equation iteration, following Coleman (1991) and Davig (2004).
- Appendices:
  - Appendix A: equilibrium system
  - Appendix B: deterministic steady state calculation
  - Appendix C: solution method description

### Key model features and interpretations
- Price rigidity and markup:
  - θ = 7.66 implies a 15% price markup at steady state.
  - ψ = 78 implies nominal price rigidity of about one year (in line with Smets and Wouters (2007) Calvo estimates for probability firms can choose prices optimally).
- Debt maturity:
  - Baseline κ = 1 (short-term debt); alternative κ = 0.05 implies average debt maturity ≈ 20 quarters.
- Fiscal steady-state targets follow Drautzburg and Uhlig (2015) for U.S. data: b_{4y} = 0.6, g_y = 0.15, τ = 0.28.

### Regime definitions and ZLB incidence
- Regimes:
  - Regime M: monetary policy active (α_M > 1) and fiscal policy responds to debt (γ_M > 0).
  - Regime F: monetary policy less responsive (α_F < 1) and fiscal policy passive on debt (γ_F = 0).
- ZLB incidence:
  - Calibration with ρ_ν = 0.8 and σ_ν = 0.0025 implies a conditional probability of hitting the ZLB of about 5% in regime M — consistent with typical NK models subject to the ZLB.

---

### 3 Government Spending Effects: Regime F vs. Regime M

### Baseline simulation setup
- Government spending shock used in simulations:
  - Size: an initial government spending increase of 1% of steady-state output.
  - Initial debt level: deterministic steady state = 60% of annualized steady-state output unless specified otherwise.
- Simulation conventions:
  - Impulse responses report percent deviations from a path without the government spending shock, except:
    - G/Y (government spending-to-steady-state output ratio) plotted as level differences in percent.
    - Inflation, nominal rate, real interest rate, and expected inflation plotted as annualized level differences.
  - One-year ahead inflation expectations:
    - Calculated by simulating the impulse responses 10,000 times for both paths (with and without the spending shock), drawing taste shock each period; the one-year ahead inflation expectation at t is the average of the inflation differences at t+4 between the two paths across simulations.

### Main comparative findings (regime F vs. regime M)
- Aggregate responses to a 1% of steady-state output government spending increase (baseline calibration):
  - Impact inflation: 4.2% in regime F vs. 0.5% in regime M (on impact).
  - Impact output multiplier (present-value cumulative, see formula 3.1):
    - regime M: 0.58 (impact), 0.56 (4Q), 0.45 (20Q)
    - regime F: 1.27 (impact), 1.15 (4Q), 0.90 (20Q)
  - Consumption impact multiplier:
    - regime F: 0.27 (impact)
    - regime M: −0.42 (impact)
- Channels driving differences:
  - Wealth-effect channel:
    - Regime M: households expect higher future tax burden (γ_M > 0), discouraging consumption and encouraging saving → output multiplier < 1.
    - Regime F: no expectation of future tax increases (γ_F = 0), so negative wealth effect does not operate → consumption supported by higher real wage income.
  - Intertemporal substitution channel:
    - Regime M: monetary authority raises nominal interest rate more than inflation increase (α_M > 1) → real interest rate increases → crowding-out of consumption.
    - Regime F: monetary authority does not raise nominal rate sufficiently (α_F = 0.5) → real interest rate falls → crowding-in of consumption; this, combined with higher inflation, amplifies expansionary effect.

### Sensitivity and robustness (selected experiments summarized in Table 2)
- Cumulative output multipliers (impact, 4Q, 20Q) under different calibrations:
  - Baseline:
    - regime M: 0.58, 0.56, 0.45
    - regime F: 1.27, 1.15, 0.90
  - High government debt (b^0_{4y} = 1):
    - regime M: 0.58, 0.56, 0.44
    - regime F: 1.19, 1.08, 0.85
  - High income tax rate (τ = 0.5):
    - regime M: 0.58, 0.56, 0.43
    - regime F: 1.07, 0.98, 0.80
  - Long-term debt (κ = 0.05):
    - regime M: 0.59, 0.57, 0.49
    - regime F: 1.11, 1.01, 0.81
  - Long-term debt & more responsive MP (κ = 0.05, α_F = 0.8):
    - regime F: 0.89, 0.86, 0.78
- Interpretation:
  - Government spending is substantially more expansionary in regime F than in regime M under baseline calibration.
  - Factors that reduce expansionary effects in regime F include higher initial debt, higher steady-state labor income tax rate, and longer debt maturity combined with a more responsive monetary policy in regime F.

### Role of higher initial government debt in regime F
- Comparing initial debt-to-annual-output b^0_{4y} = 0.6 vs. 1.0:
  - Impact output multiplier in regime F falls from 1.27 (60%) to 1.19 (100%).
- Mechanism in regime F:
  - Higher initial debt provides a bigger base for inflation taxes; therefore, inflation needs to increase less for a given spending shock.
  - Lower inflation increase → smaller decline in real interest rate → smaller intertemporal substitution boost to consumption → smaller output multiplier.
  - Real bond income b_{t−1} / π_t falls with higher inflation; with 100% initial debt inflation increases less, so the drop in real bond income is smaller than with 60% initial debt.
  - Net effect: intertemporal substitution effect dominates bond-income effect; overall consumption increase is smaller with higher initial debt.
- Additional notes:
  - Initial debt has almost no influence on multipliers in regime M in these simulations.
  - If government debt is held abroad, the role of government indebtedness in regime F would be more pronounced.

### Role of higher steady-state taxation in regime F
- Comparing τ = 0, τ = 0.28 (baseline), τ = 0.5 in regime F:
  - Higher steady-state labor income tax rate leads to smaller increase in inflation and hence a smaller decrease in the real interest rate for a given government spending increase.
  - Mechanism:
    - With higher τ, a larger share of the spending increase is financed by higher tax revenue from output expansion (even though γ_F = 0 so tax rate does not change), reducing the required inflation rise.
    - Smaller fall in real interest rate weakens intertemporal substitution channel → smaller consumption and output multipliers.
  - Outcomes in the impulse responses:
    - Consumption and real wage rise most when τ = 0.
- Effect in regime M:
  - Steady-state τ has little effect on output multipliers in regime M because:
    - Lower τ implies the government issues more debt for a given spending increase, increasing negative wealth effects, but lower current tax burden implies bigger increase in after-tax income; net effect on current consumption is roughly offsetting.
    - This is consistent with Leeper et al. (2017) in a linearized model.

*Source: Excerpt from "2.4 Functional Forms, Parameterization, and the Solution Method" and associated simulation results in the provided content.*

### 3.4 A Longer Debt Maturity with a More Responsive Monetary Policy

### 3.4 A Longer Debt Maturity with a More Responsive Monetary Policy

### Debt maturity, inflation responses, and multipliers
- Inflation and inflation expectations respond much more to a government spending increase in regime F than in regime M.
- Short-term debt specification overstates inflation responses and the crowding-in effect of government spending.
- Longer average debt maturity lowers consumption and output multipliers in regime F:
  - Impact output multiplier with an average maturity of approximately five years is 1.11, compared to the baseline short-term debt multiplier of 1.27.
- Mechanism:
  - With only short-term debt, the government must repay a relatively large amount of liabilities next period, requiring an immediate inflation adjustment to stabilize debt.
  - Longer-term debt permits inflation to be spread to future periods, lowering current and near-future inflation and leading to smaller multipliers.
- Role of monetary policy responsiveness in regime F:
  - Even though monetary policy in regime F does not actively respond to inflation, the responsiveness (α_F) shapes current inflation and inflation expectations.
  - With an average maturity of five years (κ = 0.05), a bigger response of the nominal interest rate (a bigger α_F in (2.12) but still in Regime F) limits the increase in short-run inflation and pushes inflation to the future, reducing the decline in the real interest rate and therefore reducing output and consumption multipliers.
- Combined effect of longer maturity and more responsive interest-rate policy:
  - Table (last row of Table 2 in source): with κ = 0.05 and α_F = 0.8 (still in regime F), output multipliers throughout the horizon drop below one and consumption multipliers turn negative as in regime M.
  - On impact, the output multiplier falls to 0.89 with five-year average debt maturity.
- Empirical context:
  - The average maturity of total U.S. outstanding Treasury marketable debt from 2000 to 2018 is five years (Office of Debt Management(2018)).

### Limits of the cashless framework and relevance to money-financed spending
- The analysis focuses on factors that diminish expansionary effects of government spending under passive monetary policy.
- The cashless model cannot simulate money-financed government increases directly.
- Results are likely relevant for expansionary effects of money-financed government spending in related work because the main mechanisms underlying large multipliers are similar to those in regime F.

### Key numerical references (debt maturity and policy responsiveness)
- Short-term debt multiplier (baseline, κ = 1): impact = 1.27.
- Longer-term debt (κ = 0.05, ~five years): impact = 1.11.
- Combination κ = 0.05 and α_F = 0.8: impact output multiplier = 0.89.

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### Policy regime uncertainty: setup and calibrated switching probabilities
- Policymakers’ behavior modeled by a two-state Markov chain with transition probabilities ρ_MM and ρ_FF (equation (4.1)).
- Transition probability from regime F to M, ρ_FF, is an increasing function of last-period inflation π_{t−1} with a logistic form (equation (4.2)); households do not expect switching to regime M until inflation rises above the targeted level π^*.
- Calibration for simulations:
  - α_1 = 0.05 in Φ(π_{t−1}).
  - ρ_MM fixed at 0.98, implying an average duration of 50 quarters in regime M.
- Figure 6 in source: plots ρ_FF as a function of π_{t−1} under different α_1 values.

### Policy regime uncertainty in regime F: effects on multipliers
- Simulation baseline: initial debt-to-annual output ratio at 70%.
- Table 3 (cumulative output multipliers in regime F with expectations of switching to regime M):
  - Baseline (fixed regime F): impact = 1.27, 4Q = 1.15, 20Q = 0.90.
  - Initial debt-to-annual output: 0.7 → impact = 0.90, 4Q = 0.87, 20Q = 0.86.
  - Initial debt-to-annual output: 0.8 → impact = 0.76, 4Q = 0.74, 20Q = 0.78.
- Interpretation:
  - Expectations that monetary authority can switch to active inflation control (regime M) lower inflation expectations and current inflation, raise the current real interest rate, reduce consumption, and lower current goods demand.
  - With some probability of switching to regime M, households may save for a potential future tax hike, further lowering multipliers.
- Interaction with debt burden:
  - Policy uncertainty substantially amplifies the negative effect of high debt burden in lowering government spending multipliers in regime F.
- Debt maturity robustness:
  - Simulations with longer average debt maturity (κ = 0.05, ~five years) produce similar magnitudes of multiplier reductions relative to a fixed regime F with longer maturity.

### Policy regime uncertainty in regime M: smaller effects
- Switching probability from regime M to F is assumed constant for this analysis.
- Table 4 (cumulative output multiplier in regime M with expectations of switching to regime F; initial debt-to-annual output = 0.6):
  - Fixed regime (ρ_MM = 1): impact = 0.58, 4Q = 0.56, 20Q = 0.45.
  - Middle switching probability (ρ_MM = 0.98): impact = 0.53, 4Q = 0.50, 20Q = 0.29.
  - High switching probability (ρ_MM = 0.95): impact = 0.46, 4Q = 0.39, 20Q = −0.08.
- Mechanisms offsetting each other in regime M:
  - Expectation of switching to regime F raises expected inflation (increasing current inflation and nominal-rate responses), which reduces multipliers.
  - Expectation of switching to regime F reduces expected future tax rates, which raises multipliers.
  - Net effect is generally small relative to regime F.

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### Government spending effects in recessions and the ZLB
- Recession scenario setup:
  - Consider a recession where the ZLB can bind for a sustained period in regime M, induced by a series of negative taste shocks from t = 1 to t = 5.
  - Non-linear simulations start both economies in regime M at t = 0; at t = 3 one economy remains in regime M and the other switches to regime F.
- Macroeconomic dynamics (Figure 7 summary):
  - Staying in regime M:
    - Negative taste shocks lower consumption, output, and inflation; government debt-to-GDP rises.
    - Monetary authority lowers nominal interest rate; ZLB binds from t = 2 to t = 8.
  - Switching to regime F at t = 3:
    - Economy exits binding ZLB immediately as inflation adjusts to stabilize rising debt; nominal interest rate rises.
    - Rising inflation in regime F drives up the nominal interest rate because monetary authority still responds to inflation in regime F.
- Fiscal multipliers in recessions (Table 5):
  - Table 5 compares cumulative output multipliers across scenarios (multipliers calculated by (3.1)):
    - Regime M:
      - Normal times: impact = 0.56, 4Q = 0.53, 8Q = 0.50, 20Q = 0.39 (baseline, no negative taste shocks).
      - Recession: impact = 1.27, 4Q = 1.34, 8Q = 1.18, 20Q = 1.04.
    - Regime F (α_F = 0.5):
      - Normal times (baseline): impact = 1.27, 4Q = 1.14, 20Q = 0.90.
      - Recession: impact = 1.28, 4Q = 1.15, 20Q = 0.90.
    - Regime F — pegged interest rate (α_F = 0):
      - Recession: impact = 1.45, 4Q = 1.18, 20Q = 0.84.
- Interpretation:
  - In regime M, multipliers are much bigger in recessions than in normal times mainly because of the binding ZLB: with nominal rate stuck at ZLB, higher inflation and inflation expectations lower the real interest rate and crowd in current consumption.
  - In regime F, switching to F allows the economy to escape the ZLB; hence multipliers in recessions are similar to normal times when α_F = 0.5.
  - If the monetary authority pegs the nominal interest rate in regime F during the recession (α_F = 0), multipliers increase (impact = 1.45) because a more passive monetary policy lowers the real interest rate more and generates larger consumption responses.
- Key point:
  - Whether spending multipliers are larger during recessions in regime F depends on monetary policy behavior (degree of response to inflation). If the authority refrains from responding to inflation (more passive), multipliers can be larger.

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### Sensitivity analysis
- Dimensions examined: 1) degree of nominal rigidity; 2) persistence of government spending; 3) steady-state debt-to-output ratio; 4) steady-state labor income tax rate.
- Nominal price rigidity (Rotemberg adjustment cost coefficient ψ):
  - Impact multipliers plotted as a function of ψ (left panel of Figure 9).
  - Higher price rigidity increases the impact multiplier in both regimes.
  - Effect is nonlinear in regime F:
    - Multiplier in regime F increases rapidly as ψ rises from zero, then stays almost unchanged when ψ exceeds the baseline value of 78 (corresponding to price rigidity of roughly one year).
    - Impact output multiplier in regime F falls below one when ψ = 10 (a very low degree of price rigidity).
  - Mechanism for low ψ:
    - Less price stickiness makes inflation rise more for a given government spending increase, lowering income from bond holdings; when price stickiness is sufficiently low, negative income effects can dominate the crowding-in effect in regime F and drive the output multiplier below one.

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*Italic: Source — wpiea2020091-print-pdf: "3.4 A Longer Debt Maturity with a More Responsive Monetary Policy" (chapter/section content provided).*

### 6.2 Persistence in Government Spending Increases

### 6.2 Persistence in Government Spending Increases

### Key findings on persistence and output multipliers
- Government spending persistence (parameter ρg) has opposite effects across regimes:
  - In regime F, a more persistent government spending process increases the impact output multipliers.
  - In regime M, a more persistent government spending process decreases the impact output multipliers.
- Mechanisms:
  - Higher persistence raises the present-value total of the spending stimulus.
  - Regime M: government spending is largely financed by debt and future taxes; more persistent stimulus implies a larger future tax burden, inducing a bigger negative wealth effect that reduces current output multipliers. This is consistent with Shen and Yang (2018).
  - Regime F: a more persistent spending increase produces a higher jump in current inflation because households expect future inflation to rise more. The larger inflation rise lowers the interest rate more, augmenting the crowding-in effect and producing bigger output multipliers.

### 6.3 Government Debt in the Steady State

### Sensitivity to steady-state debt-to-output ratio
- Setup: only non-distorting steady-state transfers vary to satisfy the government budget constraint; all other parameters set to baseline calibration.
- Findings:
  - Regime M:
    - Almost no change in impact output multipliers as the steady-state debt-to-output ratio varies.
    - Explanation: the crowding-out effect in regime M depends mainly on the amount of additional debt issued following the shock (and hence the taxes eventually required), so the stock of existing steady-state debt matters little for spending effects in regime M.
  - Regime F:
    - The impact output multiplier decreases as the steady-state debt-to-output ratio increases.
    - Explanation: given a fixed spending increase financed by inflation, a higher steady-state debt ratio (which is the initial debt in the simulation) provides a higher nominal base, so inflation increases by less than in the low-debt case. The intertemporal substitution effect is smaller and so is the output multiplier.
- Comparative note:
  - This result in regime F differs from Leeper et al. (2017), who find the steady-state debt ratio affects multipliers only with longer maturity debt. With a fully nonlinear model, the steady-state debt-to-output ratio matters in regime F even with only short-term debt.

### 6.4 The Labor Income Tax Rate in the Steady State

### Sensitivity to steady-state labor income tax rate (τ)
- Findings:
  - When τ = 0 (case of only lump-sum taxes), the impact output multiplier is the highest—around 1.8.
    - Mechanism: with only lump-sum taxes, an increase in government spending must be completely financed by inflation, producing the biggest inflation response and crowding-in effect.
  - As the steady-state labor income tax rate increases, the multipliers decrease at a relatively large rate.
  - Regime M: the impact output multipliers are essentially flat with respect to the steady-state labor income tax rate, confirming that the steady-state labor income tax rate does not play a role in the spending multipliers in regime M (as shown in Table 2 and discussed in Section 3.3).

### 7 Conclusion

### Summary of substantive findings
- Government spending multipliers under passive monetary policy can be lower because of:
  - higher debt levels,
  - longer debt maturity,
  - higher distorting tax rates,
  - more responsive monetary policy to inflation,
  - existence of policy regime uncertainty.
- Policy regime uncertainty is particularly important in reducing expansionary effects of government spending in regime F:
  - Expectations of switching to regime M cause multipliers in regime F to decrease because negative wealth effects from regime M spill over into regime F.
  - With expectations of switching to regime M and an initial high debt ratio, government spending multipliers can fall much below one.
  - Policy uncertainty also matters in regime M: higher inflation expectations in regime F spill over into regime M, raising real interest rates in regime M.
- Implication:
  - Government spending in regime F may not always be an effective stimulus.
  - Although the framework does not model money-financed spending, the mechanisms differentiating the two regimes are similar to those distinguishing money- and debt-financed spending; thus large multipliers for money-financed spending can be subject to the same mitigating factors identified for regime F.

### Computational and solution details (methodology)
- Solution method:
  - Model solved with Euler equation iteration (Coleman (1991)) finding the fixed point of the Euler equations directly.
  - Implementation details: Fortran 90, parallelized with OpenMP.
- Policy functions solved: {λ(bt−1; St), Q(bt−1; St), π(bt−1; St)} where bt−1 is the endogenous state and St = {νt, gt, st}.
- Numerical implementation notes:
  - State space discretized using Tauchen (1986).
  - Linear interpolation for policy function approximations; quadrature used as robustness check with small differences.
  - Root-finding routine: Fortran translation from csolve.m by Sims.
  - Convergence criterion: max{‖λ(1) − λ(2)‖, ‖Q(1) − Q(2)‖, ‖π(1) − π(2)‖} < 10−6.
- Practical solving advice:
  - Regime M typically converges faster than regime F.
  - Use steady-state values or log-linearized solution as initial guesses.
  - When γF = 0 (no tax response to debt), solve first with small non-zero γF and use that solution as initial guess for γF = 0.
  - For regime switching or high spending persistence, use fixed-regime or lower-persistence solutions as initial guesses.

*Source: wpiea2020091-print-pdf - 6.2 Persistence in Government Spending Increases*

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_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020091-print-pdf.pdf_