## Macroeconomic and Fiscal Consequences of Quantitative Easing — Working Paper No. WP/2025/158

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### 2.1 Households — preferences, assets, and QE transmission
- Household classes and expectations:
  - Two types: “restricted” and “unrestricted”, j ∈ {r,u}; ω_r ∈ (0,1) is share of restricted households.
  - Deviations from rational expectations modeled via Gabaix (2020) cognitive discounting: perceived law of motion X_{t+1} − X = m^j_G_X( X^s_t − X^s, ε_{t+1} ), with 0 ≤ m^j ≤ 1 and m^j = 1 = rational expectations.
- Lifetime utility (type j):
  - U^j_t = E^j_t ∑_{s=0}^∞ β^s_j exp{ε^d_{t+s}} [ exp{ε^c_{t+s}} log(c^j_{t+s} − κ ̄c^j_{t−1+s}) − (n^j_{t+s})^{1+φ}/(1 + φ) ],
  - Parameters: β_j ∈ [0,1), κ ∈ [0,1) (external habit), φ > 0 (inverse Frisch elasticity).
- Assets and budget constraints:
  - Two nominal assets: short-term bonds and long-term bonds (perpetuities with coupons 1, κ, κ^2, ...; κ ∈ (0,1]).
  - Long-term bond price relation: P_{L−s,t} = κ^s P_{L,t}; yield R_{L,t} = κ + 1/P_{L,t}.
  - Unrestricted households trade both bond types but pay transaction costs ζ_t on long-term positions; flow constraint:
    - P_t(1 + τ_{c,t}) c^u_t + B^u_t + (1 + ζ_t) P_{L,t} B^u_{L,t} + T^u_t = R_{t−1} B^u_{t−1} + (1 + κ P_{L,t}) B^u_{L,t−1} + W_t(1−τ_{n,t}) ̄n^u_t + D^u_t + Ξ^u_t.
  - Adjustment-cost rebate: (1 + ζ_t)/(1 + ζ) = ( b^u_{L,t} / b^u_L )^ξ, ξ > 0; larger private long-term holdings boost term premium; QE reduces term premium by reducing private holdings.
  - Restricted households trade only long-term bonds (negligible transaction costs); flow constraint:
    - P_t(1 + τ_{c,t}) c^r_t + P_{L,t} B^r_{L,t} + T^r_t = (1 + κ P_{L,t}) B^r_{L,t−1} + W_t(1−τ_{n,t}) ̄n^r_t + D^r_t.
- First-order conditions and QE intuition:
  - Unrestricted (abstracting from cognitive discounting, habit, shocks):
    - 1 = β_u E_t [ (c^u_t / c^u_{t+1}) { (P_{L,t+1} / P_{L,t}) R_{L,t+1} / Π_{t+1} } ]^{1/ζ_t }.
  - Restricted:
    - 1 = β_r E_t [ c^r_t / c^r_{t+1} { (P_{L,t+1} / P_{L,t}) R_{L,t+1} / Π_{t+1} } ].
  - QE mechanism:
    - Reducing ζ_t (central bank purchases) lowers real holding return on long-term bonds → equivalent to a fall in the term premium when short-term real rates unchanged.
    - Restricted agents smooth consumption only via long-term bonds; lower real return on long-term bonds stimulates their consumption → aggregate demand shifts out. Magnitude depends on ω_r.

### 2.2 Wage setting, firms, fiscal and monetary frameworks
- Wage setting and firms:
  - Labor differentiated and aggregated by Kimball aggregator parameterized by φ_w > 0; aggregate labor: n_t = (∫_0^1 n_t(h)^{1/(1+φ_w)} dh)^{1+φ_w}.
  - Wages set by Calvo unions with probability 1−θ_w to reoptimize; non-reset wages indexed to steady state inflation π.
  - Firms: Kimball final goods aggregator G(x) with ψ ≤ 0; intermediate firms y_t(i) = exp{ε^z_t} n_t(i) − f; Calvo price reoptimization probability θ_p.
- Fiscal authority and debt:
  - Nominal flow budget: B^f_t + P_{L,t} B^f_{L,t} = R_{t−1} B^f_{t−1} + (1+κ P_{L,t}) B^f_{L,t−1} + P_t g_t − T_t − Φ^c_t.
  - Consolidated government debt (face-value): GD^{con}_t = B^f_t + 1/(1−κ) B^f_{L,t} / (4 P_t Y).
  - Government consumption: g_t = g exp{ε^g_t} with ∆ε^g_t = ρ_{g,1} ∆ε^g_{t−1} − ρ_{g,2} (ε^g_{t−1} − 1) + u_{g,t}.
  - Tax rule: T_t = τ_{c,t} P_t c_t + τ_{n,t} W_t n_t + T_t; τ_{n,t} − τ_n = ψ_τ (τ_{n,t−1} − τ_n) + (1−ψ_τ) ψ_b (GD^{con}_t − GD^{con}).
  - Fiscal authority keeps composition constant: B^f_t / B^f_{L,t} = b / b^f_L.
  - Fiscal deficit D^f_t and real net debt issuance relations defined (equations (17)–(18)).
- Monetary authority and QE implementation:
  - Policy rule with ELB = 1: R_t = max{1, ˜R_t}, ˜R_t / R^*_t = (˜R_{t−1}/R^*_{t−1})^{γ_r} [ (π^{yoy}_t/π)^{γ_π} (y_t/y_{t−1})^{γ_y} ]^{1−γ_r} exp{ε^r_t}.
  - Time-varying neutral gross policy rate: R^*_t = R E_t exp{ε^d_{t+1}/ε^d_t}.
  - QE defined: QE_t ≡ P_{L,t} b^c_{L,t} = − b^c_t; QE rule: QE_t = (1 + (1−%) (κ P_{L,t}/P_{L,t−1} − 1)) QE_{t−1} + ε^{QE}_t, with 0 ≤ % < 1 and ε^{QE}_t AR(1).
  - Central bank profits transferred to government: Φ^c_t ≡ R_{t−1} B^c_{t−1} + (1 + κ P_{L,t}) B^c_{L,t−1}; accumulated CB profits CBPROF^{acc}_{t+h} = Σ_{s=0}^h Φ^c_{t+s}.

### Market clearing, calibration targets, and key steady states
- Market clearing:
  - Goods: y_t = ω_r c^r_t + (1−ω_r) c^u_t + g_t.
  - Production: y_t Δ_t = exp{ε^z_t} n_t − f.
  - Bond market: (1−ω_r) B^u_t = B^f_t − B^c_t; ω_r B^r_{L,t} + (1−ω_r) B^u_{L,t} = B^f_{L,t} − B^c_{L,t}.
- Selected calibration values (exact):
  - ω_r = 0.2
  - φ = 2
  - κ = 0.8
  - β_u = 0.99875
  - β_r = 0.99625
  - m_u = 0.95
  - m_r = 1
  - φ_w = 0.5
  - θ_w = 0.82
  - ξ = 0.02
  - μ = 1.15
  - ψ = -12
  - θ_p = 0.75
  - τ_c = 0.15
  - τ_n = 0.35
  - ψ_τ = 0.98
  - ψ_b = 0.01
  - ρ_{g,1} = 0.7, ρ_{g,2} = 0.02
  - D = 40
  - γ = 0.9
  - γ_π = 2
  - γ_y = 0.5
  - % = 0
  - ρ_{QE} = 0.4
- Targeted steady-state ratios (exact):
  - g/y = 0.2
  - b^f / 4y = 0.75
  - Net inflation (Annualized): 2.04 (00(π−1))
  - Nominal Policy Rate (Annualized): 2.54 (00(R−1))
  - Term-premium: 1.04 (00(R_L − R))
  - Share of Long-term Bonds in Total Bonds: 0.65 (P_L b^f_L / b^f)
  - Central Bank Assets: 0 (P_L b^c_L = b^c)
- Model-implied transmission (normalized to 100 basis cut in short-term policy rate):
  - 1 percent policy rate cut → output rises about 0.7 percent after about 1.5 years.
  - Fiscal AR(2) government spending → output multiplier ≈ 0.8 over first two years (peak after ~6 quarters); AR(1) frontloaded spending → multiplier ≈ 1.
  - QE: requires about 13 percent of baseline GDP in LSAP to match output expansion of a 100 basis cut; peak output impulse ≈ 0.05 percent per 1 percent of baseline GDP QE.
  - QE reduces term premium by about 70 basis points for purchases equal to 13 percent of baseline GDP.
  - QE and rate cuts produce very similar inflation effects in this calibration; QE slightly larger due to more persistent output effects.
  - For a given boost to output, monetary tools are more effective in raising inflation than fiscal tools.

### 2.8 Solution method, behavioral discounting, and QE transmission summaries
- Solution approach:
  - Nonlinear model solved using extended path / Fair and Taylor (1983) two-point boundary value (time-stacking) algorithm as implemented in Dynare (Juillard, 1996); preserves nonlinearities from Kimball aggregator and ELB.
  - Certainty equivalence imposed on nonlinear model; future shock uncertainty not explicitly modeled in nonlinear solution.
- Behavioral discounting treatment:
  - Derive linearized FOCs for behavioral agents producing Euler conditions with forward-looking terms multiplied by cognitive discount m plus small additive asset-holding term.
  - Approximate nonlinear model first-order conditions so linearization matches these linear derivations up to the small additive term.
- QE transmission in deep liquidity traps (overview and scenarios):
  - Deep trap specification: short-term policy rate pinned at ELB for prolonged period.
  - Baseline (slow recovery, deep trap):
    - ELB for 20 quarters; discount factor shock → output decline nearly 8 percent.
    - QE announcement scaled so central bank balance sheet peaks at 10 percent of baseline GDP after four quarters.
    - Key quantitative effects:
      - QE boosts output ≈ 0.8 percent relative to baseline after six quarters.
      - QE raises core inflation by 0.2 percentage points.
      - Term premium declines initially by about 50 basis points and persists lower.
      - Central bank capital rises close to 0.2 percent of annual GDP after five years.
      - Consolidated government debt falls by about 6 percent of baseline GDP after five years.
  - Faster recovery (deep trap with shocks in period 7):
    - Policy rate remains at ELB for > two years after shock; liftoff earlier and faster than slow recovery.
    - QE output stimulus slightly smaller than slow recovery.
    - Inflation may rise to nearly 3 percent even without QE; QE induces small additional overshoot.
    - Central bank profits reduced relative to slow recovery but consolidated fiscal position improves almost as much.
  - Fiscal vs QE (deep trap):
    - For same 30-quarter output path, fiscal expansion stimulates inflation less than QE because fiscal stimulus raises potential output.
    - Fiscal stimulus raises term premium slightly; QE sharply reduces term premium.
    - Consolidated debt after 30 quarters: fiscal stimulus worsens debt by ~3 percent of steady state GDP; QE improves consolidated government debt by ~7 percent.
    - Under QE the debt decline driven >50% by higher labor income tax revenues, with contributions from consumption taxes, lower debt service, debt deflation, and some CB profits; fiscal expansion deteriorates debt largely via government purchases and rising debt service.
- QE in shallow liquidity traps:
  - Shallow trap baseline: output gap ≈ −2 percent; inflation ≈ 0.8 percentage points below target; shadow rate slightly below ELB for ~3 years; QE = 10 percent of baseline GDP.
  - Quantitative contrasts with deep trap:
    - QE peak output ≈ 0.7 percent (vs 0.8 percent deep trap); stimulus less persistent.
    - Consolidated fiscal improvement ≈ 4 percent of GDP after 20 quarters (≈ two-thirds of deep trap improvement).
  - Risks: greater chance QE becomes counterproductive if upside inflation shocks induce rapid recovery; illustrated faster recovery produced sizable CB losses (~0.4 percent of annual GDP) and substantially reduced fiscal improvement.

### 4.2 Commitment aspects of QE — risks, simulations, and policy implications
- Commitment risks and simulation setup:
  - Forward guidance and pledge to delay hikes can amplify overheating if central bank becomes “locked into” low rates.
  - Simulation: shock in period 7 → interest rate smoothing γ rises from 0.9 to 0.94 for two years; central bank keeps policy rate unchanged for four quarters → short-term rate at ELB for total of 10 quarters.
  - Wage indexation can further amplify overheating if triggered by rapid inflation increase.
- Effects of commitment (delayed lift-off):
  - Overheating in output and amplified inflation via steeper Phillips curve region.
  - Delaying lift-off reduces central bank losses, boosts CB revenue, lowers debt-servicing costs → can improve consolidated fiscal position more than immediate rate rise.
  - If central bank later shifts to stronger tightening, this may amplify CB losses and deteriorate consolidated fiscal position; nonlinear Phillips curve increases this risk.
- Sensitivities to initial term premium and QE scale:
  - Baseline steady state term premium: 100 basis points.
  - If initial term premium = 0 basis points:
    - QE can push term premium negative; CB can make losses even under modal outlook.
    - Improvement in consolidated fiscal position after 5 years ≈ 1 percent of GDP less than when initial term premium = 100 bps.
    - In faster recovery with low initial term premium, CB losses ≈ twice as large and consolidated improvement after 5 years ≈ 2 percent.
  - Doubling QE from 10 percent to 20 percent of pre-shock GDP when initial term premium low:
    - Output and inflation responses roughly linear in QE size.
    - Cumulated CB profits become significantly negative (≈ -1.5 percent of GDP under modal outlook).
    - Consolidated fiscal improvement only ≈ 50 percent bigger despite doubling QE.
    - Larger balance sheet greatly increases exposure to early recovery shocks; CB losses may exceed 2 percent of GDP after 5 years in adverse paths.
- Stochastic uncertainty and risk assessment:
  - For shallow trap with initial term premium 0 bps and QE = 20 percent of annual GDP, 500 simulated trajectories for 30 periods show sizeable probability of early lift-off from ELB (percentiles 68th, 80th, 95th).
  - Drivers: asymmetric upside inflation risk from Kimball curvature, large shock calibration to historical volatility (1960–2019).
  - Consequence: swift repricing can produce sizable CB portfolio losses; however, consolidated fiscal losses often small because QE raises tax revenue and lowers financing costs; risks smaller under Great Moderation calibration or deep trap.
- Policy recommendations:
  - Forward guidance or commitment can increase QE effectiveness but should include escape clauses to allow policy-rate adjustments if recovery outpaces expectations.
  - Allowing policy rate to adjust more quickly when warranted helps contain overheating and improves consolidated fiscal position.
  - Consider alternative arrangements for sharing CB profits and losses with fiscal authority to mitigate central bank reluctance to deploy QE due to loss/independence concerns.

### Appendix highlights (A.1–A.4)
- A.1 Term premium:
  - TP_t = R_{L,t} − R^{EH}_{L,t}; TP approximately proportional to discounted sum of portfolio adjustment costs.
- A.2 Central bank balance sheet and QE mechanics:
  - Passive change in value of long-term assets Ψ_t and decomposition into runoff and revaluation provided.
  - Reinvesment governed by % and active term ^c_t; evolution: P_{L,t} B^c_{L,t} = (1 + (1−%) (−1/P_{L,t−1} + Π_{L,t} R_{L,t} − 1)) P_{L,t−1} B^c_{L,t−1} + ^c_t.
  - Real-equivalent QE_t evolution (Equation (A.1)) preserved.
- A.3 Commitment under wage indexation:
  - Wage indexation to previous-period inflation can materially amplify overheating and CB losses under faster recovery + commitment scenarios.
- A.4 QE under uncertainty in deep trap:
  - Stochastic simulations display distributions for output, inflation, policy rate, cumulative CB profits, and consolidated government debt highlighting risk quantiles (No Uncert., Mean, 68%, 80%, 95%).

*Source: wpiea2025158-source-pdf — selected sections 2.1, 2.2, 2.8, 3–4, 4.2, Appendix A of the IMF PDF chapter.*

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

### 2.1    Households

### Household types, preferences, and expectation formation
- Two household types: “restricted” and “unrestricted”, indexed j∈{r,u}, with ω_r∈(0,1) denoting the share of restricted households.
- Lifetime utility for household of type j:
  - U^j_t = E^j_t ∑_{s=0}^∞ β^s_j exp{ε^d_{t+s}} [ exp{ε^c_{t+s}} log(c^j_{t+s} − κ ̄c^j_{t−1+s}) − (n^j_{t+s})^{1+φ}/(1 + φ) ],
  - where β_j∈[0,1), κ∈[0,1) (external habit parameter), and φ > 0 (inverse Frisch elasticity).
- Preference shocks and expectation formation:
  - Consumption preference shock ε^c_t follows a stationary AR(1) process.
  - Discount factor shock ε^d_t perturbs preferences.
  - Deviations from rational expectations are modeled via Gabaix (2020) cognitive discounting: for any variable X_t perceived law of motion X_{t+1} − X = m^j_G_X( X^s_t − X^s, ε_{t+1} ), with 0 ≤ m^j ≤ 1 and m^j = 1 corresponding to rational expectations.

### Assets, bonds, and household budget constraints
- Two nominal assets: short-term bonds and long-term bonds (perpetuities).
- Long-term bond structure:
  - Perpetuities pay coupons 1, κ, κ^2, ... with κ∈(0,1].
  - Price of a long-term bond issued s periods ago: P_{L−s,t} = κ^s P_{L,t}.
  - Yield to maturity / long-term rate: R_{L,t} = κ + 1/P_{L,t}.
- Unrestricted households:
  - Can trade both short- and long-term bonds but pay transaction costs ζ_t on long-term bond positions.
  - Flow budget constraint:
    - P_t(1 + τ_{c,t}) c^u_t + B^u_t + (1 + ζ_t) P_{L,t} B^u_{L,t} + T^u_t = R_{t−1} B^u_{t−1} + (1 + κ P_{L,t}) B^u_{L,t−1} + W_t(1−τ_{n,t}) ̄n^u_t + D^u_t + Ξ^u_t.
    - Definitions: τ_{c,t} = sales (VAT) tax rate; τ_{n,t} = labor income tax rate; T^u_t = lump-sum taxes; D^u_t = dividends; R_t = short-term policy rate; W_t ̄n^u_t = pre-tax labor income inclusive of insurance; Ξ^u_t = rebate of bond holding adjustment costs.
  - Bond-holding adjustment cost rebate relationship:
    - (1 + ζ_t)/(1 + ζ) = ( b^u_{L,t} / b^u_L )^ξ, with ξ > 0.
    - Implication: larger holdings of long-term bonds by unrestricted agents boost the term premium; a decrease (as with QE) reduces the term premium.
- Restricted households:
  - Trade only in long-term bonds and face negligible transaction costs.
  - Flow budget constraint:
    - P_t(1 + τ_{c,t}) c^r_t + P_{L,t} B^r_{L,t} + T^r_t = (1 + κ P_{L,t}) B^r_{L,t−1} + W_t(1−τ_{n,t}) ̄n^r_t + D^r_t.

### First-order conditions, term premium, and QE transmission mechanism
- First-order condition for unrestricted agents (abstracting from cognitive discounting, habit, and preference shocks):
  - 1 = β_u E_t [ (c^u_t / c^u_{t+1}) { (P_{L,t+1} / P_{L,t}) R_{L,t+1} / Π_{t+1} } ]^{1/ζ_t }.
  - Π_t = P_t / P_{t−1} is gross inflation.
- First-order condition for restricted agents (no transactions costs):
  - 1 = β_r E_t [ c^r_t / c^r_{t+1} { (P_{L,t+1} / P_{L,t}) R_{L,t+1} / Π_{t+1} } ].
- Intuition for QE effects:
  - If only unrestricted agents exist, reducing ζ_t via central bank asset purchases lowers the real holding return on long-term bonds; with short-term real rates unchanged, this is equivalent to a fall in the term premium.
  - With restricted agents (who smooth consumption only via long-term bonds), a fall in the real holding return on long-term bonds stimulates their consumption through the first-order condition for restricted agents, shifting out aggregate demand. The magnitude of this aggregate demand shift depends on ω_r, the share of restricted agents.
- Appendix note:
  - Appendix A.1 provides a definition of the term premium in this setup and its relation to transaction costs.

*Source: 2.1 Households section of the provided IMF PDF chapter.*

### 2.2    Wage Setting

### 2.2    Wage Setting

### Wage setting framework
- Labor supplied by individual households is differentiated and aggregated by perfectly competitive labor unions via a constant elasticity of substitution aggregation controlled by parameter φ_w > 0.
- Aggregated homogeneous labor services:
  - n_t = (∫_0^1 n_t(h)^{1/(1+φ_w)} dh)^{1+φ_w} (equation (6))
- Wages are set by labor unions in Calvo-style staggered fashion:
  - Each period a randomly selected fraction 1−θ_w of households reset nominal wages; remaining households mechanically index wages to steady state inflation π.
  - Labor unions reset wages on behalf of all households using aggregate preferences and population-weighted marginal utility of consumption Λ_t and discount factor β.
- Wage reoptimization problem (rational expectations, unions not myopic):
  - max_{˜W_t} E_t Σ_{s=0}^∞ (β θ_w)^s [ Λ_{t+s} exp{ε^w_{t+s}} Π_s ˜W_t / P_{t+s} − exp{ε^d_{t+s}} 1/(1+φ) (˜W_t / W_{t+s})^{φ_w/(1−φ_w)} φ_n φ_{t+s} ] (˜W_t / W_{t+s})^{φ_w/(1−φ_w)} n_{t+s} (equation (7))
  - ˜W_t is the newly set wage and ε^w_t is a wage cost-push shock following a stationary AR(1).
- Perfect insurance against idiosyncratic income risk assumed so all households of a given type make the same consumption and asset choices; labor income net of insurance payments written as W_t ̄n^j_t.

### Firms (complementary to wage setting)
- Final goods aggregator:
  - ∫_0^1 G(y_t(i)/y_t) di = 1 (equation (8))
  - Kimball aggregator parameterization G(x) ≡ φ/(1+ψ) [(1+ψ)x^{−ψ}]^{1/φ} − φ/(1+ψ) + 1 with ψ ≤ 0 (equation (9)), implying steady state gross markup μ = φ/[(1−φ)(1+ψ)+φ]; nests Dixit-Stiglitz for ψ = 0.
- Intermediate good firms (monopolistic competition) production:
  - y_t(i) = exp{ε^z_t} n_t(i) − f (equation (10)), where ε^z_t is productivity shock (AR(1)), f > 0 fixed cost.
- Price setting:
  - Firms face Calvo price reoptimization with probability θ_p each period; non-resetting firms index prices to steady state inflation.
  - Firms’ reoptimization problem (no myopia; ownership mix of restricted/unrestricted agents):
    - max_{˜P_t} E_t Σ_{s=0}^∞ (θ_p)^s Λ_{t+s}/P_{t+s} (˜P_t π^s − exp{ε^p_{t+s} − ε^z_{t+s}} W_{t+s}) y_{t+s}(i) (equation (11))
  - ε^p_t is a marginal-cost cost-push shock (AR(1)).

### Fiscal authority: budget, debt definition, and stabilization
- Nominal flow budget constraint:
  - B^f_t + P_{L,t} B^f_{L,t} = R_{t−1} B^f_{t−1} + (1+κ P_{L,t}) B^f_{L,t−1} + P_t g_t − T_t − Φ^c_t (equation (12))
  - Government finances expenditures net of taxation T_t and central bank asset-portfolio profits Φ^c_t by issuing nominal short-term bonds B^f_t and long-term bonds B^f_{L,t}.
- Consolidated government debt (face-value based, not mark-to-market):
  - GD^{con}_t = B^f_t + 1/(1−κ) B^f_{L,t} / (4 P_t Y) (equation (13))
  - This excludes direct revaluation effects from central bank purchases on secondary market prices; alternative definitions checked with no notable longer-term qualitative quantitative effects.
- Government consumption process:
  - g_t = g exp{ε^g_t} with ∆ε^g_t = ρ_{g,1} ∆ε^g_{t−1} − ρ_{g,2} (ε^g_{t−1} − 1) + u_{g,t} (equation (14)); AR(2) used to capture transmission timing and implementation lags.
- Tax revenues:
  - T_t = τ_{c,t} P_t c_t + τ_{n,t} W_t n_t + T_t (equation (15)); τ_{c,t} = τ_c constant, τ_{n,t} varies gradually to stabilize GD^{con}_t via:
    - τ_{n,t} − τ_n = ψ_τ (τ_{n,t−1} − τ_n) + (1−ψ_τ) ψ_b (GD^{con}_t − GD^{con}) (equation (16))
  - ψ_τ set near unity and ψ_b small to ensure long-run debt sustainability with smooth labor income tax adjustment.
- Debt management:
  - Fiscal authority keeps composition of outstanding short- and long-term bonds (valued at steady state prices) constant: B^f_t / B^f_{L,t} = b / b^f_L.
  - Central bank QE reduces duration of outstanding government debt held by private agents when it purchases long-term bonds with short-term assets.
- Fiscal deficit definition:
  - D^f_t = (R_{t−1} − 1) B^f_{t−1} + B^f_{L,t−1} + P_t g_t − T_t − Φ^c_t (equation (17))
  - Real net debt issuance relation (real terms):
    - d^f_t = b^f_t − b^f_{t−1}/π_t + P_{L,t} (b^f_{L,t} − κ b^f_{L,t−1}/π_t) (equation (18))
  - Real face-value debt measure increases with deficits and decreases with higher inflation (which erodes real debt value); higher bond prices typically improve fiscal position by requiring fewer bonds of given face value.

### Monetary authority: policy rule and QE implementation
- Conventional Taylor-type rule with ELB = 1 (gross) / zero net lower bound:
  - R_t = max{1, ˜R_t}, where ˜R_t / R^*_t = (˜R_{t−1}/R^*_{t−1})^{γ_r} [ (π^{yoy}_t/π)^{γ_π} (y_t/y_{t−1})^{γ_y} ]^{1−γ_r} exp{ε^r_t} (equation (19))
  - π^{yoy}_t = (P_t / P_{t−4})^{1/4}; γ_r ∈ (0,1) interest rate smoothing; γ_π and γ_y determine long-run responses; ε^r_t i.i.d. normal.
- Time-varying neutral gross policy rate:
  - R^*_t = R E_t exp{ε^d_{t+1}/ε^d_t} (equation (20)); discount factor process chosen to generate either deep hump-shaped decline in R^*_t below ELB or shallow L-shaped one.
- Central bank large-scale asset purchases (LSAP / QE):
  - QE_t ≡ P_{L,t} b^c_{L,t} = − b^c_t, purchases financed by issuing one-period reserves b^c_t paying R_t.
  - QE rule:
    - QE_t = (1 + (1−%) (κ P_{L,t}/P_{L,t−1} − 1)) QE_{t−1} + ε^{QE}_t (equation (21)), with 0 ≤ % < 1 controlling reinvestment strategy; ε^{QE}_t AR(1) with ρ_{QE} specified.
  - Central bank discretionary purchases ε^c_t follow stationary AR(1).
- Central bank holding profits (transferred to government):
  - Φ^c_t ≡ R_{t−1} B^c_{t−1} + (1 + κ P_{L,t}) B^c_{L,t−1} (equation (22))
  - Accumulated central bank profits for QE portfolio purchased in period t over horizon h:
    - CBPROF^{acc}_{t+h} = Σ_{s=0}^h Φ^c_{t+s} (equation (23))

### Market clearing conditions
- Goods market equilibrium:
  - y_t = ω_r c^r_t + (1−ω_r) c^u_t + g_t (equation (24))
  - y_t Δ_t = exp{ε^z_t} n_t − f (equation (25))
- Price dispersion due to staggered price setting:
  - Δ_t ≡ 1/(1+ψ) [ (∫_0^1 (P_t(i)/P_t)^{1/(1−φ)} di)^{−φ} ∫_0^1 (P_t(i)/P_t)^{φ/(1−φ)} di + ψ/(1+ψ) ] (equation (26))
- Bond market clearing (home government bonds including reserves):
  - (1−ω_r) B^u_t = B^f_t − B^c_t (equation (27))
  - ω_r B^r_{L,t} + (1−ω_r) B^u_{L,t} = B^f_{L,t} − B^c_{L,t} (equation (28))

### Calibration: key parameter choices and targeted steady states
- Calibration goals:
  - Match empirical transmission of short-term interest rates, QE, and government spending in large economies (U.S., Euro area approximated as closed).
  - Time period = quarter.
- Key parameter values (selected, exact values preserved):
  - ω_r = 0.2 (Share of restricted households)
  - φ = 2 (Inv. Frisch elasticity of labor supply)
  - κ = 0.8 (Habit Persistence)
  - β_u = 0.99875 (Discount factor, unrestricted households)
  - β_r = 0.99625 (Discount factor, restricted households)
  - m_u = 0.95 (Cognitive discounting, unrestricted households)
  - m_r = 1 (Cognitive discounting, restricted households)
  - φ_w = 0.5 (Wage markup)
  - θ_w = 0.82 (Calvo wage probability)
  - ξ = 0.02 (Transaction cost on long-term bonds)
  - μ = 1.15 (Gross price markup)
  - ψ = -12 (Kimball parameter)
  - θ_p = 0.75 (Calvo price probability)
  - τ_c = 0.15 (Steady state Consumption Sales Tax)
  - τ_n = 0.35 (Steady state Labor income Tax)
  - ψ_τ = 0.98 (Tax smoothing coeff. in eq.(16))
  - ψ_b = 0.01 (Gov’t debt response coeff. in eq.(16))
  - ρ_{g,1} = 0.7, ρ_{g,2} = 0.02 (AR(2) coefficients for government consumption)
  - D = 40 (Long-term bond duration)
  - γ = 0.9 (Interest rate smoothing)
  - γ_π = 2 (Interest rate response to inflation)
  - γ_y = 0.5 (Interest rate response to output gap)
  - % = 0 (Reinvestment strategy)
  - ρ_{QE} = 0.4 (AR(1) coefficient on QE shock)
  - Note: D and μ are composite parameters defined as D = π β_r^{−1} / (π β_r^{−1} − κ) and μ = φ/[(1−φ)(1+ψ) + φ], so calibrating them pins down κ and φ, respectively.
- Targeted steady state ratios (exact values preserved):
  - Government Consumption to GDP: 0.2 (g/y)
  - Government Debt to Annual GDP: 0.75 (b^f / 4y)
  - Net inflation (Annualized): 2.04 (00(π−1))
  - Nominal Policy Rate (Annualized): 2.54 (00(R−1))
  - Term-premium: 1.04 (00(R_L − R))
  - Share of Long-term Bonds in Total Bonds: 0.65 (P_L b^f_L / b^f)
  - Central Bank Assets: 0 (P_L b^c_L = b^c)
- Additional calibration choices and implications:
  - Steady state sovereign bonds in annual GDP set to 0.75, central bank initially holds zero government bonds.
  - Long-term bond duration set to 10 years, share in total issuance calibrated at 0.65 → effective duration of outstanding public debt ≈ 7 years.
  - Bond market segmentation governed by ω_r = 0.2 and ξ = 0.02 to match term premium and output responses to QE.
  - Inflation target π = 1.005 (2% annualized), β_u = 0.99875, β_r = 0.99625 pinned to U.S. averages.
  - Cognitive myopia parameter m_u = 0.95 induces small deviation from rational expectations, reducing potency of forward guidance in prolonged liquidity traps.
  - Kimball curvature ψ = −12 to capture state-dependence in Phillips curve slope.
  - Calvo probabilities: θ_p = 0.75 (prices), θ_w = 0.82 (wages).
- Model-implied transmission around steady state (normalized so each policy instrument moves output at peak equally; output effect normalized to a 100 basis cut in short-term policy rate):
  - A one percent policy rate cut raises output by about 0.7 percent after about 1.5 years (consistent with empirical VAR evidence).
  - Fiscal AR(2) government spending: output multiplier ≈ 0.8 on average over first two years when spending peaks after ~6 quarters; AR(1) frontloaded spending → multiplier ≈ 1.
  - QE: requires about 13 percent of baseline GDP in LSAP to match the output expansion of a 100 basis point cut when short-term rate adjusts; implies peak output impulse ≈ 0.05 percent per 1 percent of baseline GDP QE (conservative estimate).
  - QE reduces term premium by about 70 basis points for asset purchases equal to 13 percent of baseline GDP.
  - Inflation transmission: QE and conventional rate cuts produce very similar inflation effects in this calibration; QE has slightly larger impact due to more persistent output effects.
  - Fiscal vs monetary stimulus: for a given boost to output, monetary tools are more effective in raising inflation because higher government spending increases labor supply and reduces real wages, driving up potential output and producing smaller output gap increases than monetary policy for the same output stimulus.

*Source: wpiea2025158-source-pdf - 2.2    Wage Setting*

### 2.8    Solution

### 2.8    Solution

### Solution method and treatment of behavioral discounting
- Nonlinearities associated with the Kimball aggregator and the effective lower bound (ELB) constraint are preserved by solving the nonlinear model using the extended path approach of Fair and Taylor (1983) as implemented in Dynare (Juillard, 1996).  This method is also known as a two-point boundary value or time-stacking algorithm.
- The Fair-Taylor solution method imposes certainty equivalence on the nonlinear model and therefore does not account for future shock uncertainty. All relevant information is captured by the current state of the economy, including contemporaneous and future realizations of exogenous shocks that are known by agents.
- Behavioral discounting is not tractable in a fully nonlinear setting. The approach taken:
  - Derive linearized first-order conditions describing decisions of behavioral agents, yielding aggregate Euler conditions in which forward-looking terms are multiplied by the cognitive discounting parameter m, plus an additive term that depends on agents’ asset holdings (quantitatively very small).
  - Approximate the relevant first-order conditions in the nonlinear model with formulas that, after linearization, match the linear derivations up to the small additive term.

### Transmission of QE in Deep Liquidity Traps — overview
- Focus on a deep liquidity trap: output gap substantially negative, inflation projected well below the central bank’s target for some time, short-term policy rate constrained by the ELB for a prolonged period.
- Deep liquidity trap calibrated to generate a severe recession via a sequence of discount factor shocks ε^d_t that produce a persistent increase in desired household savings.
- Modal baseline uses a mix of AR(2) and AR(1) processes to generate the shadow rate path (AR(2) for U-shaped path; AR(1) to ensure variable falls below zero in first period).

### 3.1 Baseline scenario with slow recovery (deep trap)
- Scenario mechanics and calibration:
  - Short-term policy rate becomes pinned at the ELB for 20 quarters.
  - The discount factor shock triggers a sharp decline in output of nearly 8 percent.
  - Central bank purchases are implemented in an anticipated yet gradual fashion per process in eq. (21); purchase announcement innovation in period 1 scaled so the stock of assets held by the central bank peaks at 10 percent of baseline GDP after four quarters.
- Key quantitative effects of QE (deep trap, baseline):
  - QE boosts output by about 0.8 percent relative to baseline after six quarters.
  - QE raises core inflation by 0.2 percentage points.
  - Term premium declines persistently and initially by about 50 basis points.
  - Stimulus from QE induces only a slightly steeper liftoff of the policy rate.
  - Central bank capital rises close to 0.2 percent of annual GDP after five years.
  - Consolidated government debt falls by about 6 percent of baseline GDP after five years.
- Transmission channels emphasized:
  - Markets expect progressive expansion of the central bank balance sheet → lower term premium → pass-through to long-term yields.
  - Fiscal consolidation via QE reflects higher tax revenues from stimulus, reduced real value and servicing cost of past debt, and lower cost of issuing new debt.
  - Central bank profits arise because long-term yields, while falling substantially, remain well above the policy rate pinned at zero for several years.

### 3.2 Scenario with faster recovery (deep trap with unexpected shocks)
- Shock timing and composition:
  - Unexpected mix of cost-push impulses for firms ε^p_t and stronger consumption demand ε^c_t hit the economy in period 7.
- System response and quantitative findings:
  - Policy rate remains tethered to the ELB for more than two years after the shock; liftoff is earlier and faster than under slow recovery but still constrained for a time.
  - Stimulus to output from QE in the faster recovery scenario is only slightly smaller than in the slow recovery scenario.
  - Inflation in the faster recovery scenario rises to nearly 3 percent even without QE; deploying QE induces a bigger inflation overshoot (quantitatively small amplification).
  - Central bank profits are reduced relative to the slow recovery scenario, but the consolidated fiscal position still improves almost as much as under baseline.
- Implication:
  - QE benefits in a deep liquidity trap are robust even to sizeable subsequent shocks that call for faster policy rate adjustment; consolidated fiscal outcomes remain favorable despite potential reductions in central bank profits.

### 3.3 Comparison with conventional fiscal stimulus (deep trap)
- Policy comparison setup:
  - Fiscal instrument: increase in government consumption.
  - Policies normalized to imply the same boost to aggregate output over the subsequent 30 quarters.
  - Government spending shocks ε^g_t (fully anticipated after announcement) calibrated to match the output path produced by QE over the first 30 quarters.
- Main comparative findings (slow recovery):
  - Fiscal policy stimulates inflation considerably less than QE for the same output boost, because higher government spending increases potential output while QE leaves potential output nearly unchanged. Thus fiscal stimulus yields a smaller positive output gap for a given increase in output and therefore a smaller inflation response.
  - Fiscal stimulus raises the term premium slightly (increased issuance of long-term government debt), whereas QE sharply declines the term premium.
  - Consolidated government debt outcomes after 30 quarters:
    - Fiscal stimulus worsens the consolidated government debt position by about 3 percent of steady state GDP.
    - QE improves the consolidated government debt position by about 7 percent of steady state GDP.
- Decomposition of debt dynamics (Figure 5 summary):
  - Under QE (slow recovery), more than half of the overall decline in public debt reflects higher labor income tax revenues, with additional contributions from consumption tax revenue, lower debt service and issuance costs, debt deflation from rising price level, and some gains from central bank profits.
  - Under fiscal expansion, accumulation of government purchases is the key driver of fiscal deterioration, with later amplification from rising debt service costs; offset from higher tax revenue is quantitatively much smaller than under QE.
- Faster recovery scenario:
  - Drivers of debt dynamics are qualitatively similar; QE’s improvement in the debt-to-GDP ratio is smaller, mainly due to noticeably smaller labor income tax revenues.

### 4 QE in shallow liquidity traps — overview
- Shallow liquidity trap defined: economic activity much closer to potential than in deep trap, but central bank still constrained by the ELB from persistent below-target inflation.
- Baseline for shallow trap uses the same discount factor shock ε^d_t but modeled as AR(1) with high persistence.

### 4.1 Baseline scenario with slow and fast recovery (shallow trap)
- Baseline (slow recovery) specifics:
  - Baseline without QE has a slightly negative output gap of about 2 percent.
  - Inflation is about 0.8 percentage points below target.
  - Shadow rate falls just slightly below the ELB for about three years.
  - Central bank purchases set equal to 10 percent of baseline GDP (same as deep trap case).
- Quantitative effects and contrasts with deep trap:
  - QE remains beneficial under the modal outlook (boosts inflation and closes the output gap).
  - QE peak effect on output in shallow trap: about 0.7 percent (compare with 0.8 percent peak in deep trap).
  - QE stimulus is less persistent in shallow trap than in deep trap.
  - Improvement in consolidated fiscal position under QE: about 4 percent of GDP after 20 quarters (about two-thirds of the improvement achieved in the deep trap).
- Risk considerations in shallow trap:
  - Greater risk that QE can be counterproductive if upside inflation risks materialize and induce a rapid recovery and inflation surge.
  - If unexpected shocks push inflation well above target and switch the output gap from negative to positive, policy rates can lift off immediately and rise steeply, producing:
    - Sizable central bank losses (about 0.4 percent of annual GDP in the illustrated faster recovery scenario).
    - A substantially reduced improvement in the consolidated fiscal position (though the improvement need not be overturned in the scenario shown).
  - Conclusion: QE in a shallow liquidity trap can still yield macroeconomic and fiscal benefits but carries a significantly larger ex post risk of being counterproductive than in a deep liquidity trap.

*Source: wpiea2025158-source-pdf (chapter section 2.8 and sections 3–4).*

### 4.2    Commitment Aspects of QE

### 4.2    Commitment Aspects of QE

### Commitment risks and simulation setup
- QE typically involves forward guidance and some form of pledge to delay hiking interest rates until well after QE ends; if the central bank feels “locked into” keeping policy rates low, overheating may be amplified (Orphanides, 2023; Eggertsson and Kohn, 2022).
- Simulation scenario for commitment:
  - When shocks triggering a faster recovery hit in period 7, the interest rate smoothing coefficient γ increases from 0.9 to 0.94 for two years.
  - The central bank keeps the policy rate unchanged for four quarters.
  - These assumptions imply the short-term interest rate is at the ELB for a total of 10 quarters, coinciding with the ELB duration under the slow recovery outlook when QE is first implemented.
- Appendix A.3: overheating could be further exacerbated (and a surge in inflation much stronger) if increased inflation pressure triggers wage indexation mechanisms.

### Effects on output, inflation, and policy rates
- Commitment path (delayed lift-off) leads to:
  - Overheating in output.
  - An amplified reaction of inflation as the economy enters a steeper part of the Phillips curve.
  - Both factors make QE more counterproductive ex post relative to cases where policy rates adjust immediately.
- Allowing the policy rate to adjust more quickly can help contain overheating risks and tends to improve the consolidated fiscal position.

### Fiscal and central bank implications
- By delaying lift-off, commitment:
  - Reduces central bank losses.
  - Boosts central bank revenue and lowers debt-servicing costs.
  - Together, these improve the consolidated fiscal position even more than when interest rates rise immediately.
- However, if the central bank later shifts its reaction function and acts more forcefully, this could:
  - Amplify central bank losses.
  - Cause the consolidated fiscal position to deteriorate.
  - Be more likely if nonlinearities in the Phillips Curve are more pronounced.
- Baseline calibration and sensitivity (from section 4.3 relevant to commitment context):
  - Baseline steady state term premium: 100 basis points.
  - If initial term premium is 0 basis points, QE drives it into negative territory and the central bank can make losses even under the modal outlook.
  - When initial term premium is 0 basis points, the improvement in the consolidated fiscal position after 5 years is about 1 percent of GDP less than if the initial term premium was 100 basis points.
  - In the faster recovery scenario with low initial term premium, central bank losses are about twice as large as with a higher initial term premium, and the improvement in the consolidated fiscal position after 5 years is merely 2 percent.
  - Doubling QE from 10 percent to 20 percent of pre-shock GDP when the initial term premium is low:
    - Transmission to output and inflation is approximately linear in QE size.
    - Cumulated central bank profits become significantly negative, cumulating to about -1.5 percent of GDP under the modal outlook.
    - The improvement in the consolidated fiscal position is only about 50 percent bigger despite QE being twice as large.
    - Bigger balance sheets expose the central bank to larger risks with an earlier recovery; unexpected shocks can generate very large central bank losses that cumulate to more than 2 percent of GDP after 5 years.

### Stochastic uncertainty and risk assessment (contextual implications)
- To capture upside and downside risks, stochastic simulations are used (section 5 context):
  - For the shallow liquidity trap with initial term premium 0 bps and QE = 20 percent of annual GDP, 500 trajectories are simulated for 30 periods, starting from period 2 of the deterministic simulation.
  - Uncertainty is sizeable; percentiles (68th, 80th, 95th) indicate a material probability of early lift-off from the ELB.
  - Drivers of high probability of early lift-off:
    - Asymmetric upside inflation risk from Phillips Curve nonlinearities (Kimball (1995) aggregator implies more upward than downside inflation risk).
    - Large shock sizes calibrated to match historical volatility (1960-2019), allowing substantial near-term movements in the shadow rate and long-term yields.
  - Consequence: swift repricing of long-term assets and rising short-term rates can cause sizable central bank losses on its portfolio.
  - However, even when central bank portfolio losses can be large, the risk of sizeable consolidated fiscal losses is small in many calibrations because QE raises tax revenue and reduces financing costs; risks are smaller if calibrated over the Great Moderation subsample or in a deep liquidity trap.

### Policy implications and recommendations
- Guidance that includes some degree of commitment can boost QE effectiveness but should be complemented with escape clauses to allow the central bank to mitigate overheating risk if recovery is faster than envisaged at QE implementation.
- Allowing the policy rate to adjust more quickly when warranted helps contain overheating and improves the consolidated fiscal position.
- Consideration of alternative arrangements for sharing central bank profits and losses with the fiscal authority may mitigate central bank reluctance to use QE due to concerns about losses, credibility, and independence (see concluding remarks for further context).

*Source: wpiea2025158-source-pdf - 4.2    Commitment Aspects of QE (IMF).*

### References

### wpiea2025158-source-pdf - References

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_Source: https://www.imf.org/-/media/files/publications/wp/2025/english/wpiea2025158-source-pdf.pdf_