## Tax on Inflation Policy at the Zero Lower Bound — sections 2.1–3.5, robustness and counterfactual analyses (content unit)

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### 2.1 Household: preferences and optimality conditions
- Representative infinitely-lived household maximizes discounted CRRA utility:
  - U(B_{t−1}) = max_{C_t,N_t,B_t} ( C_t^{1−σ}/(1−σ) − N_t^{1+ψ}/(1+ψ) + β_t E_t U(B_t) )
- Key constraints and conditions:
  - Flow budget: P_t C_t + Q_t B_t = B_{t−1} + W_t N_t + T_t
  - No-Ponzi: lim_{T→+∞} Π_{j=0}^T Q_j B_T ≥ 0
  - First-order conditions:
    - W_t / P_t = C_t^σ N_t^ψ
    - Q_t = E_t[ β_t (C_{t+1}/C_t)^{−σ} P_t / P_{t+1} ]
- Economic interpretation:
  - Labor supply determined by W_t/P_t given C_t.
  - Euler equation links consumption to nominal bond returns 1/Q_t, inflation P_{t+1}/P_t, and β_t.
- Role of stochastic discount factor:
  - Time discount factor β_t is stochastic and, in equilibrium, equals the neutral rate of interest; increases in β_t (declines in the neutral rate) are one potential cause of the liquidity trap.

### 2.2–2.3 Final and intermediate goods firms: technology, pricing, and TIP
- Final goods:
  - CES aggregation: Y_t = [ ∫_0^1 Y_{ti}^{1−1/ε_t} di ]^{ε_t/(ε_t−1)} with ε_t elasticity of substitution.
- Intermediate goods:
  - Production: Y_{ti} = A_t N_{ti}^{1−α}
  - Firms face Rotemberg quadratic price adjustment costs:
    - C_t(P_{t−1i},P_{ti}) = (θ/2) [ (P_{ti}/P_{t−1i}) − 1 ]^2 P_t Y_t
  - Novel firm-level tax/subsidy linked to price changes:
    - Tax = τ_t (P_{ti} − P_{t−1i}), τ_t ∈ R; total tax payment scales with Y_{ti}
    - Equivalently interpretable as price-increase permits (Capelle and Liu (forthcoming)).
- Profit net of taxes:
  - Π(P_{t−1i},P_{ti}) = P_{ti} Y_{ti} − W_t N_{ti} − τ_t (P_{ti} − P_{t−1i}) Y_{ti} − C_t(P_{t−1i},P_{ti})
- Symmetric equilibrium price condition (π_t = P_t/P_{t−1} − 1):
  - (ε_t − 1)(M_t/MC_t − 1) + E_t[ Q_t (Y_{t+1}/Y_t) (τ_{t+1} + θ(π_{t+1}+1)π_{t+1}) ] = τ_t [ (1−ε_t) π_t / (1+π_t) ] + θ π_t (π_t+1)
  - MC_t = (W_t/P_t)(1−α) Y_t^{α/(1−α)} A_t^{1/(1−α)}
  - M_t = ε_t/(ε_t − 1)
- Interpretation:
  - Negative cost-push shocks can act as one of three causes of a liquidity trap (conflicting wage/price aspirations or improved anchoring).

### 2.4 Central bank policy and the ZLB
- Policy rule with ZLB:
  - i_t = − log(Q_t) = max(0, φ_π π_t − log β_t)
  - Monetary policy active outside ZLB: φ_π > 1
- Nonlinearity and multiplicity:
  - ZLB nonlinearity combined with steep Phillips curve can yield multiple equilibria including deflationary traps.
- Fiscal bookkeeping:
  - Government runs balanced budget funding net transfers due to TIP and wage-bill subsidy via lump-sum household taxes; markets clear.

### 2.5 Three-equation linearized representation (Lemma 1)
- Definitions and parameter:
  - ȳ_t = log(y_t / y^e_t); u_t deviation of markup; β ∈ (0,1); κ = (ε−1)/θ [ σ + (ψ+α)/(1−α) ]
- Linearized system around zero inflation and zero TIP:
  - ȳ_t = E_t ȳ_{t+1} − (1/σ)( i_t − E_t π_{t+1} + log β_t )
  - π_t = β E_t π_{t+1} + κ ȳ_t + (1/θ)[ β E_t τ_{t+1} − τ_t + u_t ]
  - i_t = max(0, φ_π π_t − log β_t)
- Calvo-type frictions yield identical first-order representation.

### 3 Optimal TIP in an r^*-driven liquidity trap: setup
- Liquidity trap driven by decline in neutral rate: discount factor shock Δβ > 0.
  - β_t ∈ { β, β + Δβ } with β < 1 < β + Δβ; ZLB may arise when β_t = β + Δβ.
- Markov persistence:
  - Economy starts in liquidity trap (β_0 = β + Δβ); probability p that high β persists each period; with probability (1−p) economy exits and stays in normal state β (absorbing).
- Minimal state-variable equilibrium: allocations and prices are functions of β_t only.

### 3.1 Liquidity-trap laissez-faire equilibrium (Lemma 2) — characterization and closed form
- In ZLB (β_t = β + Δβ), i = 0 and (π_L, ȳ_L) solve:
  - ȳ_L = p ȳ_L + (1/σ)( p π_L − log(β + Δβ) )
  - π_L = p β π_L + κ ȳ_L + (1/θ)[ β p τ − τ ]
- Existence of unique equilibrium with π_L < 0 and ȳ_L < 0 iff:
  1. log(β + Δβ) > 0
  2. (1 − pβ)/κ > p / [ σ(1 − p) ]
- Decentralized closed-form:
  - π_L = − [ κ log(β + Δβ) + (1 − pβ)σ(1 − p) (τ/θ) ] / [ (1 − pβ)σ(1 − p) − κ p ]
  - ȳ_L = (1 / [ σ(1 − p) ]) ( p π_L − log(β + Δβ) )
- Economic note: persistence p < 1 makes Euler equation upward-sloping in (ȳ_L, π_L), necessary for stationary liquidity trap.

### 3.2 Constrained-efficient allocation and pecuniary externality
- Planner minimizes quadratic loss (LQ approximation):
  - Loss L = Σ_{t=0}^∞ E_0 β^t [ π_t^2 + η_y (ȳ_t)^2 ], where η_y = (1/θ)[ σ + (α+ψ)/(1−α) ]
- Constrained-efficient allocation (Proposition 1):
  - ȳ_L^{LF} < ȳ_L^{CS} = − [ (1/σ^2) (1−p)^2 + p^2 η_y ]^{-1} log(β + Δβ) < 0
  - π_L^{LF} < 0 < π_L^{CS} = − [ η_y p / (σ(1−p)) ] (1/σ^2 (1−p)^2 + p^2 η_y)^{-1} log(β + Δβ)
- Mechanism:
  - Firms’ price-cutting imposes pecuniary externality (raises real rate via deflation), depressing output; planner raises inflation (π_L^{CS} > 0) to relax ZLB and lower real rate.

### 3.3 Ramsey allocation: welfare-maximizing TIP (Proposition 2)
- Ramsey problem chooses τ_{SB} (Markov, decentralized-consistent) to maximize household utility subject to IS and Phillips curve.
- Implementation result:
  - π_{SB}^L = π_{CS}^L and ȳ_{SB}^L = ȳ_{CS}^L
  - Optimal TIP:
    - τ_{SB} = − θ log(β + Δβ) / [ (1 − pβ)σ(1 − p) ] × [ κ + η_y p ( (1 − pβ)σ(1 − p) − κ p ) / ( (1 − p)^2 σ^2 + η_y p^2 ) ]
- Intuition:
  - TIP shifts firms’ price-setting to internalize the pecuniary externality and break the deflationary spiral.

### 3.4 Remarks, alternative objectives, uniqueness
- First-best unattainable with TIP alone: TIP cannot lower nominal rate sufficiently; first-best requires intertemporal relative-price policies (consumption taxes/labor subsidies).
- Alternative objective—price stability (η_y = 0):
  - Optimal TIP to set π_L = 0:
    - τ_{π=0} = − [ κ θ / (1 − pβ) ] × [ log(β + Δβ) / (σ(1 − p)) ] < 0
  - Output gap ordering: ȳ_L^{LF} < ȳ_L^{π=0} = − log(β + Δβ) / [ σ(1 − p) ] < ȳ_L^{SB} < 0
- Determinacy and uniqueness:
  - Outside ZLB: Taylor principle φ_π > 1 ensures determinacy.
  - At ZLB: multiplicity of forward-stable solutions; TIP responding to output gap can ensure global uniqueness iff:
    - φ_y / θ p > κ / (1 − pβ) + [ 1 + (φ_π / θ) ] / [ σ(1 − p) ] (Proposition 4).

### 3.5 Numerical illustration — baseline calibration and results
- Period: a quarter.
- Baseline parameters (Table 1):
  - α: 0.25
  - β: 0.99
  - σ: 1
  - ψ: 5
  - ε: 6
  - θ: 372.8
  - β + ∆β: 1.001
  - p: .7
- Calibration notes:
  - θ = 372.8 targeting Galí (2015) Phillips curve slope.
  - β + ∆β = 1.001 (neutral rate −.4% per annum following Bilbiie (2019)).
  - p = .7.
- Numerical outcomes (laissez-faire):
  - Output gap in liquidity trap: -1.8%
  - Inflation gap in liquidity trap: -.6%
- Second-best allocation:
  - Output gap: -.3%
  - Inflation: above target by .01pp
  - τ_SB = -44%
  - Fiscal cost of τ_SB: .004% of firms’ sales
- Comparative statics:
  - Optimal TIP (τ_SB, τ_{π=0}) more negative as Δβ increases.
  - Degree of price stickiness has negligible impact on optimal TIP (offsetting effects).
  - Higher p and smaller σ worsen recession/deflation, requiring more negative TIP.

### TIP versus Forward Guidance and Government Spending (numerical comparisons)
- Forward guidance modeling:
  - Three states: L (β + ∆β, i = 0), FG (β, i = 0), N (β, i = − log β > 0).
  - Persistence q for FG state; forward guidance approximated by keeping i = 0 with probability q.
- Main numerical findings:
  - Welfare losses are smaller under TIP than under forward guidance in most parameter space.
  - Forward guidance generates future boom that distorts future allocation; TIP avoids intertemporal distortion by operating intratemporally.
  - TIP remains relatively more effective as p increases.
- Government spending comparison:
  - Utility includes G_t with σ_G = 1.5 and w_c = .2.
  - Welfare losses under TIP are higher but of same order of magnitude as under government spending in most parameter space.
  - Caveat: comparisons assume lump-sum financing and abstract from distortionary taxation/fiscal sustainability.

### TIP in alternative trap mechanisms: self-fulfilling and Phillips-curve-driven traps
- Self-fulfilling trap setup:
  - β fixed at β < 1; liquidity trap is one stationary equilibrium indexed by sunspot x_t ∈ {L, FB}; economy initially in L with persistence p.
- Existence conditions (Lemma 3):
  - Requires Taylor coefficient condition and Phillips curve steeper than Euler equation: (1 − pβ)/κ < p / [σ(1 − p)].
- Optimal TIP (Proposition 5):
  - There exists τ_SF such that if τ_L > τ_SF and τ_FB = 0, first-best allocation is unique.
  - τ_SF = − θ log β / [(1 − pβ)(1 − p)σ] [ κ + ((1 − pβ)σ(1 − p) − κ p)/φ_π ] .
  - Counterintuitive: sufficiently high (positive) TIP can eliminate the self-fulfilling liquidity trap.
- Phillips-curve-driven trap:
  - Cost-push shock u < 0 triggers trap; existence requires u < threshold and (1 − pβ)/κ > p / [σ(1 − p)] (Lemma 4).
  - Optimal TIP (Proposition 6) restoring first-best:
    - τ_FB = 0 and τ_L = u / (1 − pβ) < 0.

### Robust inflation-targeting TIP rule and policymaker uncertainty
- Robust TIP specification:
  - τ = τ0 + φπ π
  - Core recommendation: aggressive φπ and negative τ0 are robust across trap types.
  - In r^*- and PC-driven traps: optimal τ0 < 0.
  - In SF trap: TIP should be positive and sufficiently high.
- Policymaker ambiguity setup:
  - Policymaker chooses τ0 ∈ [τ0, τ0] and φπ ∈ [0, φπ] minimizing worst-case expected losses over distributions g ∈ G consistent with observed laissez-faire (equations (35)–(38)).
  - Dimensionality reduction yields uncertainty over (θ, σ).
- Comparative statics (Lemma 5):
  1. r^* trap: output gap, inflation, welfare increase monotonically with φπ; as φπ → +∞, inflation → 0 and output gap → ˆyL = − 1 / [σ (1 − p)] log(β + ∆β).
  2. SF trap: first-best unique iff φπ ≥ φSFπ with φSFπ = θ [ κ p / ( (1 − pβ) σ (1 − p) ) − 1 ] > 0.
  3. PC trap: inflation increases monotonically to 0 with φπ; output gap non-monotonic in φπ (increases for φπ < φPCπ then decreases; converges to 0 as φπ → +∞).
- Proposition 7:
  - If φπ ∈ [0, φ̄π) with φ̄π > max(φSFπ, φPCπ), setting φπ = φ̄π is weakly welfare-maximizing across all three trap types.
  - There exist τ0 < 0 such that welfare strictly higher than laissez-faire in r^* and PC traps.

### Numerical simulations under alternative policymaker objective formulations
- Three approaches:
  1. Expected-welfare maximization with prior g∗: θ and σ log-normal with mean (stdev) 372 (80) and 2 (1); bounds τ0 ∈ [−1, 1]; φπ = 1500; exclude inflation > 1pp above target.
  2. Multiplier preferences (entropy penalty) with ν and reference g∗.
  3. Max-min (Gilboa–Schmeidler) with θ ∈ [50, 700], σ ∈ [.5, 10]; Agent A selects worst-case (θ, σ).
- Common numerical result:
  - Optimal rule features φπ = ̄φπ (strong positive reaction to inflation) combined with τ0 = τ0 (negative intercept).
  - Under multiplier preferences, welfare surface smoother than in other approaches.

### Paradox of flexibility and policy calibration
- Inflation-targeting TIP increases aggregate price stickiness (via φπ) and can improve welfare at the ZLB (generalizes paradox of flexibility).
- Condition: increase in aggregate stickiness must be large enough; small increases can worsen self-fulfilling equilibria.
- Rebuttal to concern on relative-price distortion:
  - TIP is a linear tax on price changes and does not distort relative prices in the way convex taxes or price controls do.

### Quantitative assessment for Japan (medium-scale DSGE counterfactuals)
- Model extensions: bonds-in-utility, consumption habits h, investment with adjustment cost ψI, capital utilization, sticky wages, indexation ιp, ιw, fixed costs, sunspot for indeterminacy.
- TIP rule used in calibration:
  - τ_t = ρ τ_{t−1} + τ0 + φπ (π_t − π∗)
  - Calibration for counterfactual TIP: π∗ = 1%, ρ = 0.9, τ0 = -0.51%, φπ = 50
- Key internal calibration choices (two scenario values: self-fulfilling / r^* decline):
  - δ: 0.0398 / 0.0764
  - ψp: 314.2 / 312.2
  - ψw: 583.2 / 595.0
  - p: 0.998 / 0.943
- Counterfactuals and main quantitative findings (Section 7.2):
  - Self-fulfilling trap: φπ must exceed 28.0 to eliminate the trap with ρ = 0.9; when φπ = 28.0, a 1 percentage-point annual inflation increase raises TIP by 7.00 percentage points on impact; left-hand limit as φπ → 28.0: Inflation −3.7%, Output loss −13.2%.
  - r^*-driven trap: with ρ = 0.9 and φπ = 50, required τ0 to restore steady-state inflation to 1% is −0.51% (implying steady-state TIP rate of −5.1%).
  - Simulation under calibrated TIP:
    - TIP raises inflation by about 2.1 percentage points and raises output by as much as 4.5 percentage points.
    - TIP rate typically fluctuates between -30% and 20%, except spikes to 40% in Q4 2008 and Q1 2009.
    - Fiscal impact fluctuates between −0.1% and 0.1% of GDP (except Q4 2008); effective transfers are small because tax base (price changes) is small and contracts as TIP rate increases.
    - TIP as % of sales computed as 100 τ_t π_t.

### Determinacy, uniqueness, and policy implications in the medium-scale model
- Determinacy:
  - Outside ZLB and without state switching: Taylor principle required (φ_Nπ > 1).
  - With state switching and a liquidity-trap state (φ_Lπ = 0), Taylor principle may not suffice; TIP can restore uniqueness.
  - Proposition 8: with τ_Nt = 0, equilibrium unique if θκ/(1 − pβ) < φ_Ly and φ_Nπ > 1.
- Policy implications:
  - Robust recommendation: inflation-targeting TIP rule with large φπ and negative τ0 is effective and robust across trap types.
  - Negative TIP in r^*-driven traps and PC-driven traps encourages higher firm-level prices, mitigates deflationary spiral, raises inflation and output; in SF traps a large positive TIP can eliminate the bad equilibrium.
- Implementation caveats and suggested research:
  - Risks of tax avoidance (product relabeling, quality changes) merit quantitative study.
  - Examine TIP designs in settings with endogenous product creation, information asymmetries, and costly monitoring.
  - Study political economy and implications for monetary independence.

- Exact parameter and numerical excerpts preserved from the source where reported (examples):
  - Baseline θ = 372.8; β + ∆β = 1.001; p = .7; laissez-faire outcomes: output gap −1.8%, inflation gap −.6%; τ_SB = −44%; fiscal cost .004% of firms’ sales.
  - Japan counterfactual TIP calibration: π∗ = 1%, ρ = 0.9, τ0 = −0.51%, φπ = 50; required φπ > 28.0 to eliminate self-fulfilling trap with ρ = 0.9; when φπ = 28.0, 1 pp annual inflation raises TIP by 7.00 pp; left-hand limit inflation −3.7%, output loss −13.2%.

*Source: wpiea2026059-source-pdf — sections 2.1–3.5, robustness and counterfactual analyses.*

### 2.1    Household

### 2.1 Household

### Preferences and optimality conditions
- Representative infinitely-lived household maximizes discounted CRRA utility over consumption and labor:
  - Utility: U(B_{t−1}) = max_{C_t,N_t,B_t} (C_t^{1−σ}/(1−σ) − N_t^{1+ψ}/(1+ψ) + β_t E_t U(B_t))
  - Time discount factor β_t is stochastic and, in equilibrium, equals the neutral rate of interest; increases in β_t (declines in the neutral rate) are one potential cause of the liquidity trap.
- Flow budget constraint and no-Ponzi condition:
  - P_t C_t + Q_t B_t = B_{t−1} + W_t N_t + T_t
  - lim_{T→+∞} Π_{j=0}^T Q_j B_T ≥ 0
- First-order conditions (optimal labor supply and Euler equation):
  - W_t / P_t = C_t^σ N_t^ψ
  - Q_t = E_t[ β_t (C_{t+1}/C_t)^{−σ} P_t / P_{t+1} ]
- Economic interpretation:
  - First equation determines N_t given C_t and real wage W_t.
  - Second is the Euler equation determining consumption path given nominal bond returns 1/Q_t, inflation P_{t+1}/P_t, and discount factor β_t.

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### 2.2–2.3 Final and Intermediate Goods Firms

### Final good producers
- Final good produced competitively by continuum of firms combining intermediate varieties via CES technology:
  - Y_t = [ ∫_0^1 Y_{ti}^{1−1/ε_t} di ]^{ε_t/(ε_t−1)}
  - ε_t is elasticity of substitution across varieties.
- Final good firms take P_t and {P_{ti}} as given and maximize profits subject to the technology.

### Intermediate goods: technology, pricing, and taxes
- Continuum of mass one firms produce unique varieties with decreasing marginal returns:
  - Y_{ti} = A_t N_{ti}^{1−α}
  - A_t is common total factor productivity; 1−α is elasticity of output to labor.
- Firms are in monopolistic competition, face Rotemberg quadratic price adjustment costs:
  - C_t(P_{t−1i},P_{ti}) = (θ/2) [ (P_{ti}/P_{t−1i}) − 1 ]^2 P_t Y_t
- Novelty: firms pay a tax (or subsidy if negative) proportional to the increase in their price:
  - Tax = τ_t (P_{ti} − P_{t−1i}), τ_t ∈ R, and total tax payment scales with output Y_{ti}.
  - (Capelle and Liu (forthcoming): a market for inflation permits can implement same allocations; τ is equilibrium price of a permit to increase prices.)
- Profit net of taxes:
  - Π(P_{t−1i},P_{ti}) = P_{ti} Y_{ti} − W_t N_{ti} − τ_t (P_{ti} − P_{t−1i}) Y_{ti} − C_t(P_{t−1i},P_{ti})
- Firm’s recursive problem and equilibrium price condition (symmetric P_{ti}=P_t; inflation π_t = P_t/P_{t−1} − 1):
  - (ε_t − 1)(M_t/MC_t − 1) + E_t[ Q_t (Y_{t+1}/Y_t) (τ_{t+1} + θ(π_{t+1}+1)π_{t+1}) ] = τ_t [ (1−ε_t) π_t / (1+π_t) ] + θ π_t (π_t+1)
  - Real marginal cost: MC_t = (W_t/P_t)(1−α) Y_t^{α/(1−α)} A_t^{1/(1−α)}
  - Flexible-price ideal markup: M_t = ε_t/(ε_t − 1)

### Interpretation and role of cost-push shock
- A negative cost-push shock is identified as one of the three potential causes of a liquidity trap analyzed in the paper; interpreted as conflicting worker–firm relative wage/price aspirations or improved anchoring of inflation expectations.

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### 2.4 Central Bank Policy and Zero-Lower-Bound (ZLB)

### Policy rule and ZLB constraint
- Policy rate: i_t = − log(Q_t) = max(0, φ_π π_t − log β_t)
  - − log β_t is equilibrium neutral rate of interest.
- Monetary policy is "active" outside the ZLB:
  - φ_π > 1
- Nonlinearity from ZLB combined with steep Phillips curve can yield multiple rational expectations equilibria, including a deflationary trap (Benhabib, Schmitt-Grohé, and Uribe (2001)).
- Government: balanced budget funds net transfers due to TIP and wage-bill subsidy via lump-sum household taxes. Markets for intermediate goods, final good, labor and bonds clear. Full equilibrium definition in Appendix A.1.

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### 2.5 Three-equation Representation

### Linearized IS, Phillips curve, and Taylor rule with ZLB (Lemma 1)
- Variables and notation:
  - ȳ_t = log(y_t / y^e_t) (log-deviation of output from efficient value)
  - u_t: deviation of markup from steady state
  - β ∈ (0,1): steady-state discount factor
  - κ = (ε−1)/θ [ σ + (ψ+α)/(1−α) ]
- Linearized system around zero inflation and zero TIP:
  - ȳ_t = E_t ȳ_{t+1} − (1/σ)( i_t − E_t π_{t+1} + log β_t )
  - π_t = β E_t π_{t+1} + κ ȳ_t + (1/θ)[ β E_t τ_{t+1} − τ_t + u_t ]
  - i_t = max(0, φ_π π_t − log β_t)
- Calvo-type frictions produce same first-order representation (Capelle and Liu (forthcoming)).

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### 3 Optimal TIP in an r^*-Driven Liquidity Trap

### Setup and motivation
- Focus: liquidity trap driven by decline in neutral rate r^* modeled as a discount factor shock Δβ > 0.
- Discount factor β_t ∈ { β, β + Δβ }, with β < 1 < β + Δβ; ZLB may arise when β_t = β + Δβ.
- Markov process: economy starts in liquidity trap (β_0 = β + Δβ); probability p that high discount factor persists each period; with probability (1−p) economy exits ZLB and then stays in normal state (β is absorbing).
- Minimal state variable equilibria: allocations and prices are functions of β_t only.

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### 3.1 Liquidity Trap laissez-faire equilibrium (Lemma 2)
- In ZLB (β_t = β + Δβ), i = 0 and π_L, ȳ_L solve:
  - ȳ_L = p ȳ_L + (1/σ)( p π_L − log(β + Δβ) )
  - π_L = p β π_L + κ ȳ_L + (1/θ)[ β p τ − τ ]
- Existence of unique equilibrium with π_L < 0 and ȳ_L < 0 when β_t = β + Δβ iff:
  1. log(β + Δβ) > 0
  2. (1 − pβ)/κ > p / [ σ(1 − p) ]
- Decentralized equilibrium closed-form:
  - π_L = − [ κ log(β + Δβ) + (1 − pβ)σ(1 − p) (τ/θ) ] / [ (1 − pβ)σ(1 − p) − κ p ]
  - ȳ_L = (1 / [ σ(1 − p) ]) ( p π_L − log(β + Δβ) )
- Economic insight: persistence p < 1 makes Euler equation upward sloping in (ȳ_L, π_L) space; necessary for stationary liquidity trap equilibrium.

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### 3.2 Constrained-efficient allocation and pecuniary externality
- Constrained-efficient allocation defined as path maximizing household utility subject to technologies, resource constraints, and implementation constraint:
  - E_t[ β (C_{t+1}/C_t)^{−σ} P_t / P_{t+1} ] ≤ 1
- Planner minimizes second-order approximation of welfare losses (LQ setup):
  - Loss L = Σ_{t=0}^∞ E_0 β^t [ π_t^2 + η_y (ȳ_t)^2 ], where η_y = (1/θ)[ σ + (α+ψ)/(1−α) ]
- Planner restricted to Markov, time-consistent allocations that revert to first-best π=ȳ=0 once β_t returns to β.
- Constrained-efficient allocation highlights a pecuniary externality: decentralized firms’ price-cutting that increases aggregate deflation raises real interest rate and depresses output; firms do not internalize this effect.
- Proposition 1 (Constrained-efficient Allocation):
  - ȳ_L^{LF} < ȳ_L^{CS} = − [ (1/σ^2) (1−p)^2 + p^2 η_y ]^{-1} log(β + Δβ) < 0
  - π_L^{LF} < 0 < π_L^{CS} = − [ η_y p / (σ(1−p)) ] (1/σ^2 (1−p)^2 + p^2 η_y)^{-1} log(β + Δβ)
- Planner chooses positive inflation and negative output gap in constrained-efficient allocation to relax ZLB and reduce real rate, trading off inflation cost and output gains.

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### 3.3 Ramsey allocation: welfare-maximizing TIP (Proposition 2)
- Ramsey problem: choose TIP rate τ_{SB} (Markov, decentralized equilibrium-consistent) to maximize household utility subject to IS and Phillips curve constraints.
- TIP influences firms’ price-setting, shifting Phillips curve; social planner can select τ to make firms internalize externality.
- Proposition 2 (Implementation with TIP):
  - π_{SB}^L = π_{CS}^L and ȳ_{SB}^L = ȳ_{CS}^L
  - Optimal TIP:
    - τ_{SB} = − θ log(β + Δβ) / [ (1 − pβ)σ(1 − p) ] × [ κ + η_y p ( (1 − pβ)σ(1 − p) − κ p ) / ( (1 − p)^2 σ^2 + η_y p^2 ) ]
- Intuition: TIP gives correct social signal about social cost of price changes, internalizing pecuniary externality and breaking deflationary spiral to raise output by lowering real rate.

---

### 3.4 Remarks, alternative objectives, and uniqueness

- First-best allocation:
  - TIP cannot restore first-best (π_L = 0, ȳ_L = 0) because it cannot lower nominal rate to negative value of natural rate; first-best requires policies that change intertemporal relative prices (e.g., consumption taxes/labor subsidies).

- Alternative objective: keeping inflation at target (η_y = 0)
  - Lexicographic preferences (price stability prioritized) lead to optimal TIP that sets π_L = 0.
  - Proposition 3: TIP that sets inflation to 0 at ZLB satisfies:
    - τ_{π=0} = − [ κ θ / (1 − pβ) ] × [ log(β + Δβ) / (σ(1 − p)) ] < 0
  - Resulting output gap ordering:
    - ȳ_L^{LF} < ȳ_L^{π=0} = − log(β + Δβ) / [ σ(1 − p) ] < ȳ_L^{SB} < 0
  - Interpretation: TIP less negative than second-best TIP; inflation returned to target but at cost of larger output gap than second-best.

- Stability and uniqueness
  - Outside ZLB, Taylor principle (φ_π > 1) ensures determinacy given absorbing normal state.
  - At ZLB, multiplicity of forward-stable solutions exists; minimal state variable equilibrium selected.
  - TIP can help pin down a unique equilibrium at the ZLB if it responds to output gap:
    - Assume τ_L = τ_0 + φ_π π_L + φ_y ȳ_L (TIP responds linearly at ZLB), τ=0 outside ZLB, Taylor principle holds outside ZLB.
    - Proposition 4: global uniqueness iff φ_y / θ p > κ / (1 − pβ) + [ 1 + (φ_π / θ) ] / [ σ(1 − p) ]
  - TIP thus provides an alternative instrument to ensure uniqueness at the ZLB (analogous to Taylor principle for monetary policy).

---

*Source: wpiea2026059-source-pdf — sections 2.1–3.4*

### 3.5    Numerical Illustration

### 3.5    Numerical Illustration

### Baseline calibration
- Period length: a quarter.
- Model parameters (Table 1):
  - α One minus the elasticity of output to labor: 0.25
  - β Time discount factor: 0.99
  - σ Elasticity of intertemporal substitution: 1
  - ψ Inverse Frish elasticity of labor: 5
  - ε Elasticity of substitution across varieties: 6
  - θ Adjustment cost: 372.8
  - β + ∆β Discount factor shock: 1.001
  - p Persistence of liquidity trap: .7
- Calibration notes:
  - Rotemberg parameter set to θ = 372.8 following Capelle and Liu (forthcoming), targeting the slope of the Phillips curve in Galí (2015).
  - β + ∆β = 1.001 is set following Bilbiie (2019), which amounts to a neutral rate of interest of -.4% per annum.
  - The persistence probability of the liquidity trap is p = .7.

### Results and sensitivity
- Laissez-faire outcomes (this parametrization):
  - Output gap in the liquidity trap: -1.8%
  - Inflation gap in the liquidity trap: -.6%
- Second-best allocation:
  - Output gap: -.3%
  - Inflation: above target by .01pp
  - Level of TIP that implements second-best, τ_SB = -44%
  - Fiscal cost of τ_SB: .004% of firms’ sales
- Policy comparison remark:
  - The allocation implemented by a policymaker seeking to bring inflation back to target (η_y = 0) is almost identical to the second-best allocation.
- Comparative statics and sensitivity:
  - The optimal TIP (τ_SB, τ_π=0) must be more negative as the discount factor shock increases (i.e., as the neutral rate of interest declines).
  - The degree of price stickiness has a negligible impact on the optimal TIP due to two offsetting effects: stickier prices attenuate negative effects of the decline in the neutral rate, but make a given decrease in TIP less effective at boosting inflation.
  - A higher persistence parameter p and a smaller IES (σ) worsen the recession and deflation, requiring a more aggressive decrease in TIP.

### TIP and Forward Guidance (comparison)
- Modeling of forward guidance:
  - Three states: liquidity trap “L” (β_t = β + ∆β and i_t = 0), forward guidance “FG” (β_t = β and i_t = 0), normal “N” (β_t = β and i_t = − log β > 0).
  - Economy remains in “L” with probability p; once in “FG” it remains there with probability q; “N” is absorbing.
  - Forward guidance approximated by keeping interest rate at 0 with some probability q every period.
- Key trade-off for forward guidance:
  - Between boosting the economy in the liquidity trap and overheating the economy after exiting it; optimal q balances these forces.
- Numerical comparison setup:
  - Welfare losses(20) of second-best forward guidance vs second-best TIP are compared by varying one parameter at a time: p, κ, σ, and β + ∆β. Dashed black vertical line in Figure 3 corresponds to baseline parameter values.
- Main findings:
  - Welfare losses are smaller under TIP than under forward guidance in most of the parameter space.
  - Forward guidance requires generating a future boom, which distorts the future allocation and entails welfare costs.
  - TIP operates through an intratemporal trade-off—boosting inflation today above its first-best level to decrease the real rate and increase output—avoiding the intertemporal distortion of forward guidance.
  - TIP remains relatively more effective than forward guidance as the persistence of the liquidity trap p increases.
- Additional caveats on forward guidance (not modeled here but noted in text):
  - Credibility and commitment issues, and the forward guidance puzzle (strong private responses to future rates absent liquidity constraints or bounded rationality), may limit effectiveness in practice.

### TIP and Government Spending (comparison)
- Government spending modeling:
  - Utility separable in government spending with intertemporal elasticity of substitution σ_G:
    - C_t^(1−σ)/(1−σ) − N_t^(1+ψ)/(1+ψ) + G_t^(1−σ_G)/(1−σ_G) (Equation (27))
  - Outside liquidity trap, government spending chosen to equalize marginal utilities of private and government consumption.
  - In liquidity trap, optimal government spending trades off smaller output gap (boosting output and employment) versus intratemporal distortion (crowding out private consumption), which shows up as a wedge shifting the Phillips curve.
- Calibration for comparison:
  - σ_G = 1.5 and w_c = .2 (consistent with relative size of government consumption).
- Numerical results (Figure 4):
  - Welfare losses are higher under TIP but of the same order of magnitude as under government spending in most of the parameter space.
- Caveats:
  - Comparisons abstract from additional costs of government spending—assume lump-sum financing and omit distortionary taxation and fiscal sustainability concerns that could limit the appeal of government spending.

### TIP in a Self-fulfilling Liquidity Trap
- Setup:
  - Discount factor fixed at β < 1; liquidity trap is one of two possible stationary equilibria.
  - Sunspot x_t ∈ {L, FB}; economy initially in L; Markov chain with probability p to stay at L; FB state absorbing.
  - Equilibrium outside ZLB: π = ŷ = 0 and i = − log β.
  - In liquidity trap equilibrium: i = 0 and ŷ_L, π_L solve Phillips curve (15) and Euler equation with β < 1:
    - ŷ_L = p ŷ_L + 1/σ (p π_L − log β)
- Existence conditions (Lemma 3) for ZLB equilibrium with deflation and negative output gap (τ = 0):
  1. Taylor coefficient condition:
     - φ_π > κ [p − (1 − pβ)σ(1 − p)] / κ
  2. Phillips curve steeper than Euler equation:
     - (1 − pβ) / κ < p / [σ(1 − p)] . (Equation (28))
- Coordination failure:
  - Constrained-efficient and first-best allocations have π_L = ŷ_L = 0; social planner can select first-best among equilibria.
- Optimal TIP and elimination of liquidity trap:
  - By setting sufficiently high TIP whenever agents coordinate on liquidity trap, policymakers can eliminate it.
  - Proposition 5 (Optimal TIP in a Self-Fulfilling Liquidity Trap):
    - There exists a minimum level of TIP τ_SF such that if τ_L > τ_SF and τ_FB = 0, the first-best allocation is the unique equilibrium.
    - τ_SF = − θ log β / [(1 − pβ)(1 − p)σ] [ κ + ((1 − pβ)σ(1 − p) − κ p)/φ_π ] . (Equation (29))
  - Counterintuitive note: a policy that gives incentives to further decrease prices at the firm-level can eliminate the liquidity trap in general equilibrium by making inflation sufficiently high for the central bank to set a positive interest rate.

### TIP in a Phillips-curve Driven Trap
- Setup:
  - Introduce cost-push shock u_t ∈ {u, 0} with u < 0; negative cost-push shock drives the liquidity trap.
  - Shock starts at u_0 = u, follows Markov chain with probability p of remaining u; once reverts to 0 it stays forever.
  - Equilibrium outside ZLB: π = ŷ = τ = u = 0 and i = − log β.
  - In ZLB equilibrium, Phillips curve:
    - π_L = pβ π_L + κ ŷ_L + 1/θ [ β p τ_L − τ_L + u ] . (Equation (30))
- Existence conditions (Lemma 4) for unique ZLB equilibrium with deflation and negative output gap (u < 0):
  1. Negative cost-push shock large enough:
     - u < θ log β / [σ(1 − p)] [ κ(1 + p) − (1 − pβ)σ(1 − p) ]
  2. Phillips curve less steep than Euler equation:
     - (1 − pβ)/κ > p / [σ(1 − p)] . (Equation (31))
- Constrained-efficient and first-best allocations:
  - Both feature π_L = ŷ_L = 0; source of inefficiency is the cost-push shock that opens a wedge between private and social returns to decreasing prices.
- Optimal TIP:
  - A simple negative TIP can restore the first-best allocation in the liquidity trap.
  - Proposition 6 (Optimal TIP in a Phillips-curve induced Liquidity Trap):
    - There exists a Markov policy for TIP such that the first-best allocation is the unique equilibrium π_L = ŷ_L = 0.
    - τ_FB = 0 and τ_L = u / (1 − pβ) < 0. (Equation (33))
  - Interpretation: given u < 0, the optimal τ_L is negative; a negative TIP incentivizes higher prices, closing the wedge between private and social returns and correcting excessive aggregate deflation.

*Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026059-source-pdf.pdf*

### Section 3 where TIP could only address the pecuniary externality and the amplifying

### Section 3 where TIP could only address the pecuniary externality and the amplifying

### A Robust Inflation-Targeting Rule
- Core message: An inflation-targeting TIP rule that responds aggressively to inflation and has a negative intercept is a robust policy to increase welfare in a liquidity trap.
- Key characterization of the TIP rule:
  - τ = τ0 + φπ π(34)
  - τ0 is the intercept; φπ parametrizes the strength of TIP’s reaction to inflation.
- Comparative statement on optimal TIP by trap type:
  - In r∗- and Phillips-curve-driven (PC) liquidity traps, the optimal policy is to set a negative TIP.
  - In the self-fulfilling (SF) trap, TIP should be positive and sufficiently high.
- Robustness advantage: an aggressive inflation-targeting TIP can address the root cause when the trap stems from a shift in the Phillips curve; it outperforms policies like forward guidance and government spending which lack robustness across trap types.

### Environment: an Uncertain Policymaker with a TIP rule
- Policymaker’s rule set and choice:
  - Policymaker chooses τ0 ∈ [τ0, τ0] and φπ ∈ [0, φπ] to minimize expected losses subject to ambiguity aversion.
  - Objective with ambiguity aversion (general form):
    - W(τ∗0, φ∗π) = maxτ0,φπ ming∈G [ ∫ −L(θ, σ, τ0, φπ) g(dθ, dσ) + c(g) ] s.t. (35)–(38). (39)
    - c(g) is an ambiguity index and a convex function on the simplex.
- Policymaker’s uncertainty and observed information:
  - Unknown parameters: elasticity of intertemporal substitution σ, degree of price stickiness θ, cost-push shock u, discount factor shock ∆β.
  - A state s is (σs, θs, ∆βs, us).
  - Policymaker observes the laissez-faire allocation (πLF, ˆyLF), which imposes restrictions:
    - log(β + ∆βs) − p πLF + ˆyLF / [σs (1 − p)] = 0 (35)
    - us − θs [ πLF (1 − β p) − κs ˆyLF ] = 0 (36)
  - Allocation under rule:
    - log(β + ∆βs) − p πL + ˆyL / [σs (1 − p)] = 0 (37)
    - us − θs [ πL (1 − β p) − κs ˆyL ] = (1 − β p)(τ0 + φπ π). (38)
- Dimensionality reduction:
  - Combining (35)/(37) and (36)/(38) eliminates log(β + ∆β) and u, reducing free parameters to (θ, σ). Expectation is over prior distributions of θ and σ. G denotes the set of probability distributions consistent with these restrictions.
- Game interpretation:
  - Maxmin representation: Planner (P) chooses (τ0, φπ); Agent (A) chooses a density g over (∆β, u, σ, θ) to minimize welfare, subject to constraints.

### Preliminary Results: Comparative Static
- Setup for intuition:
  - Consider three states s ∈ {r∗, SF, PC} with characteristics:
    - r∗-driven trap: ∆βr∗ > 0, ur∗ = 0; (σr∗, θr∗) such that Phillips curve is less steep than IS curve.
    - SF trap: ∆βSF = uSF = 0; slopes consistent with Lemma (3).
    - PC trap: uPC < 0, ∆βPC = 0; (σPC, θPC) such that Phillips curve is less steep than IS curve.
- Lemma 5 (Comparative statics - φπ):
  1. In an r∗ liquidity trap, the output gap, inflation and welfare increase monotonically with φπ. In the limit φπ → +∞, inflation converges to 0 and the output gap to ˆyL = − 1 / [σ (1 − p)] log(β + ∆β).
  2. In an SF liquidity trap, the first-best equilibrium π = ˆy = 0 is unique if and only if φπ ≥ φSFπ with φSFπ = θ [ κ p / ( (1 − pβ) σ (1 − p) ) − 1 ] > 0.
  3. In a PC liquidity trap, inflation increases monotonically to 0 with φπ. The output gap increases with φπ for φπ < φPCπ (defined in Appendix) then decreases for φπ > φPCπ and converges monotonically to 0 as φπ → +∞.
- Mechanism: Increasing φπ flattens the Phillips curve (decreases sensitivity of inflation to output and cost-push shocks). Phillips curve under inflation-targeting rule:
  - πL = κ ˆyL + (u − (1 − pβ) τ0)/θ (1 − pβ) [ 1 + φπ / θ ]. (40)
- Graphical implications (Figure 7 summary):
  - r∗-driven: rotation of Phillips curve lowers deflation and output gap.
  - SF: sufficiently large rotation can eliminate the liquidity-trap intersection, leaving only first-best.
  - PC: rotation reduces deflation; output gap effect is non-monotonic due to real rate changes as inflation rises.
- Proposition 7 (optimality of strong φπ and negative τ0):
  - Assume φπ ∈ [0, φ̄π) with φ̄π > max(φSFπ, φPCπ).
  - In all three liquidity trap types, setting φπ = φ̄π is (weakly) welfare-maximizing.
  - There exist τ0 < 0 such that welfare is strictly higher with τ0 than in laissez-faire in both r∗ and PC liquidity traps.
  - Interpretation: negative intercept τ0 shifts Phillips curve upward, reducing deflation and output gap in r∗ and PC traps.

### Numerical Simulations
- Three complementary approaches (robust control literature):
  1. Policymaker maximizing expected welfare with a prior:
     - Objective: W(τ∗0, φ∗π) = maxτ0,φπ ∫ −L(θ, σ, τ0, φπ) g∗(dθ, dσ). (41)
     - Calibration of priors and choice sets:
       - θ and σ log-normally distributed with mean (standard deviation) 372 (80) and 2 (1), respectively.
       - Laissez-faire allocation: πLFL = − .6, ˆyLFL = −1.8.
       - Policy parameter bounds: [τ0, τ0] = [−1, 1]; φπ = 1500.
       - Exclude allocations where inflation is 1 percentage point above target.
  2. Policymaker with multiplier preferences (Hansen and Sargent style):
     - Objective: W(τ∗0, φ∗π) = maxτ0,φπ ming∈G ∫ −L(θ, σ, τ0, φπ) g(dθ, dσ) + ν ∫ ln( dg / dg∗ ) dg. (42)
     - c(g) = ν ∫ ln( dg / dg∗ ) dg (relative entropy).
     - Same calibration as approach 1; g∗ is joint log-normal reference prior.
  3. Max-min policymaker (Gilboa and Schmeidler style):
     - Objective: W(τ∗0, φ∗π) = maxτ0,φπ ming∈G ∫ −L(θ, σ, τ0, φπ) g(dθ, dσ). (43)
     - Restrict support of G: σ ∈ [σ, ̄σ], θ ∈ [θ, ̄θ] to bounded sets.
     - Calibrated bounds: [σ, ̄σ] = [.5, 10]; [θ, ̄θ] = [50, 700].
- Numerical result common across all three approaches:
  - Optimal rule features φπ = ̄φπ (strong positive reaction to inflation) combined with τ0 = τ0 (negative intercept).
  - Under multiplier preferences, the welfare surface over parameter space is smoother than in the other approaches.
  - In the maxmin case, the paper reports optimal policy rules for σ and θ chosen by Agent A (details in Appendix).

### Remark: the Paradox of Flexibility Revisited
- Revisited paradox: Increasing aggregate price stickiness via inflation-targeting TIP can improve welfare at the ZLB—generalizes the paradox of flexibility (Eggertsson and Krugman, 2012; Billi and Galí, 2020).
- Condition: the increase in aggregate price stickiness must be strong enough; small increases can worsen allocation in a self-fulfilling equilibrium.
- Policy implication: the strength of TIP’s response to inflation must be appropriately calibrated.
- Addressing a common concern:
  - Concern: aggressive φπ increases firms’ cost of changing prices and distorts relative prices.
  - Rebuttal (as stated): an inflation-targeting TIP operates as a linear tax and does not distort relative prices, unlike price controls which act as convex taxes.

### Quantitative Assessment for Japan (introductory material)
- Objective: Use a medium-scale DSGE model calibrated to Japan to assess whether a simple inflation-targeting TIP could have restored inflation with negligible fiscal cost and robustness to trap source.
- Model extensions and features:
  - Adds bonds in utility, habits in consumption, investment in physical capital with adjustment costs, sticky nominal wages and indexation of prices and wages, variable capital utilization, fixed costs in production.
  - Multiple temporary shocks: price markup, wage markup, risk premium, investment-specific technology, monetary policy shocks.
  - Includes a sunspot variable indexing inflation expectation errors to address local indeterminacy around expectation-driven ZLB steady state.
- Key departures from CBS:
  1. Introduce discount factor to expectations, p (cognitive discounting).
  2. Partial indexation of prices and wages to lagged inflation πt−1.
  3. Introduce TIP rule:
     - τt = ρ τt−1 + τ0 + φπ (πt − π∗). (44)
     - ρ introduces policy inertia; π∗ is a strictly positive inflation target.
- Considered scenarios: r∗-driven and expectation-driven ZLB explanations (exclude PC-driven in this quantitative section).
- Calibration highlights and chosen parameter values (excerpted from Table 2):
  - Externally calibrated parameters:
    - β Time discount factor 0.942
    - σ Elasticity of intertemporal substitution 1
    - ψ Inverse Frish elasticity of labor 2.27
    - εp Elasticity of substitution across varieties 6
    - εw Elasticity of substitution across workers 6
    - α Capital share 0.37
    - δk Capital depreciation rate 0.015
    - h Consumption habit 0.358
    - ω Labor disutility 0.588
    - A′′(1) Capital utilization elasticity 2.246
    - ψI Investment adjustment cost 5.16
    - ιp Price indexation 0.225
    - ιw Wage indexation 0.295
    - g Autonomous spending 1.333
    - Gz TFP trend growth 0.26
  - Internally calibrated parameters (two scenario values: self-fulfilling / r∗ decline):
    - δ Marginal utility of bonds 0.0398 / 0.0764
    - ψp Price adjustment cost 314.2 / 312.2
    - ψw Wage adjustment cost 583.2 / 595.0
    - p Persistence probability/cognitive discounting 0.998 / 0.943
  - Counterfactual TIP rule parameters:
    - π∗ Inflation target 1%
    - ρ Policy inertia 0.9
    - τ0 Intercept -0.51%
    - φπ Response to deviation of inflation from target 50
- Calibration notes:
  - δ chosen to target steady-state r∗ of 0% in expectation-driven and -1.1% in r∗-driven scenarios (numerical δ: 0.0398 and 0.0764).
  - ψp and ψw set to match slopes of long-run Phillips curves from Hirose (2020); price adjustment cost ψp = 314.2 (expectation-driven) and 312.2 (r∗-driven); wage adjustment cost ψw = 583.2 and 595.0.
  - Discount factor p pinned by steady-state inflation target of -1.06% per annum → p = 0.998 (expectation-driven) and p = 0.943 (r∗-driven).
  - Shock processes estimated assuming no TIP (τt = 0); results in Tables 3 and 4 (not reproduced here).

*Source: wpiea2026059-source-pdf - Section 3 where TIP could only address the pecuniary externality and the amplifying (IMF working paper PDF).*

### 7.2    Counterfactual Analyses

### 7.2    Counterfactual Analyses

### Self-fulfilling liquidity trap: strength of TIP response required
- Question studied: What strength of the TIP’s response to inflation, φπ, is required to eliminate the self-fulfilling trap in the calibrated medium-scale model?
- Method: numerically compute the stationary liquidity-trap equilibrium in the self-fulfilling calibration for different values of φπ. Inflation is annualized; output is normalized relative to the flexible-price steady state.
- Key quantitative findings:
  - φπ needs to exceed 28.0 to eliminate the self-fulfilling liquidity trap when TIP inertia ρin in Equation(44) equals 0.9.
  - When φπ = 28.0, a one-percentage-point increase in annual inflation (equivalent to 0.25 percentage point per quarter) raises the TIP rate by 7.00 percentage points on impact.
  - As φπ approaches 28.0, the left-hand limit converges to:
    - Inflation: -3.7%
    - Output loss: -13.2%
  - The bounded left-hand limit and finite jumps are attributed to non-linearities in the medium-scale model (contrast with small-scale linear model).

### Liquidity trap caused by a decline of r*
- Question studied: What value of the intercept in the TIP rule, τ0, would bring the steady-state level of inflation back to the 1% target in an r*-driven liquidity trap?
- Policy parameter choices for this experiment:
  - Set ρ = 0.9 and φπ = 50 for the TIP rule while keeping monetary policy at the ZLB.
- Key quantitative findings:
  - Required value of τ0 is -0.51%, which implies a steady-state TIP rate of -5.1%.
  - Given an r* of -1.1% and steady-state inflation of 1%, the steady-state nominal rate is -0.1%, which remains below the zero lower bound, consistent with the assumption.

### Counterfactual path and simulated effects under calibrated TIP
- Procedure:
  - Estimate empirical shock series using the log-linearized version of the model and a Kalman filter.
  - Recompute the equilibrium path using the calibrated TIP rule.
- Main simulation outcomes (Figure 9 described in the source):
  - Inflation:
    - Deflation would have been mostly eliminated under TIP.
    - Inflation is positive and close to the 1% target on average.
    - TIP raises inflation by about 2.1 percentage points.
  - Output:
    - TIP raises the output level by as much as 4.5 percentage points.
    - TIP dampens inflation fluctuations due to its response to variations around the inflation target via φπ.
  - TIP rate dynamics:
    - In most periods, the TIP rate is negative and not very large, fluctuating between -30% and 20%.
    - Exceptions: Q4 2008 and Q1 2009, when the TIP rate rises to 40% (reflecting a one-time increase in inflation in Q4 2008 plus inertia embedded in the TIP rule).
  - Fiscal impact:
    - Fiscal impact (ratio of fiscal expenses to GDP) fluctuates between -0.1% and 0.1% of GDP, except in Q4 2008.
    - Effective transfers are small because the tax base—the change in prices—is small and contracts as the TIP rate increases.
  - TIP as % of sales:
    - TIP as % of sales is calculated as 100 τt πt.

### Determinacy and role of TIP
- Model characterization:
  - Outside the ZLB and without state switching, determinacy requires the Taylor principle.
  - With state switching and one state displaying a ZLB (φLπ = 0), the Taylor principle may no longer be necessary or sufficient.
- System equations summarized (outside and inside liquidity trap):
  - Outside (N):
    - ˆyNt = ENt ˆyt+1 − 1/σ (φNπ πNt − ENt πt+1)  (Equation 49)
    - πNt = βENt πt+1 + κ ˆyNt + 1/θ [βENt τt+1 − τNt]  (Equation 50)
    - φNπ > 1 in normal state.
  - At liquidity trap (L):
    - ˆyLt = ELt ˆyt+1 − 1/σ (φLπ πLt − ELt πt+1 + log(β + ∆β))  (Equation 51)
    - πLt = βELt πt+1 + κ ˆyLt + 1/θ [βELt τt+1 − τLt]  (Equation 52)
    - φLπ = 0 in the liquidity-trap state.
- Proposition on uniqueness (Proposition 8):
  - Assume τNt = 0 for all t. The equilibrium path is unique if
    - θκ/(1− pβ) < φLy
    - and φNπ > 1

### Policy implications and robustness
- Policy-design implication emphasized in the source:
  - An inflation-targeting TIP rule that is sufficiently responsive to deflation (large φπ) and with a negative intercept (τ0 < 0) is a robust optimal policy at the ZLB.
- Broader conclusions:
  - In an r*-driven liquidity trap, a negative TIP can encourage firms to raise prices, mitigate the deflationary spiral, increase inflation, and boost output; in conventional calibrations, TIP performs better than forward guidance.
  - In a liquidity trap driven by coordination failures (self-fulfilling expectations), a large enough TIP can eliminate the ZLB equilibrium.
  - Results apply most directly to instruments that affect the Phillips curve (for example, labor subsidies and to some extent government spending).
- Implementation caveats and suggested avenues for future research (as stated in the source):
  - Tax avoidance risks (firms relabeling products, shrinking quality) deserve further quantitative inquiry.
  - Quantifying effects of alternative TIP designs in frameworks with endogenous product creation, information asymmetries about product quality, and costly monitoring is an important next step.
  - Studying the political economy of TIP, including risks of misuse and impacts on monetary independence, is another important research avenue.

_Italic source: Extracted from "7.2    Counterfactual Analyses" in the provided IMF chapter PDF._

### 0. As a result, we can rewrite the system of equations in the liquidity trap state as

### 0. As a result, we can rewrite the system of equations in the liquidity trap state as

### Determinacy and stability of the liquidity-trap system
- Liquidity-trap state equations (in original notation):
  - ˆyLt = pˆyLt+1 − 1/σ (φLπ πLt − pπLt+1 + log(β +∆β))  (equation (53))
  - πLt = β pπLt+1 + κ ˆyLt + 1/θ [β pτLt+1 − τLt]  (equation (54))
- System is block-recursive because N is an absorbing state; determinacy analyzed via roots of (53) and (54).
- Matrix representation: A [πLt, xLt]' = p B [πLt+1, xLt+1]' + C with
  - A = [ φLπ/σ + φLy , 1 + φLπ/θ ; φLy/θ , −κ ]
  - B = [ 1/σ , β ; 1 + φLπ/θ , φLy/θ β ]
- Inverse A−1 computed and A−1 pB = A′ (denoted A′) with scaling factor Ω = p (φLπ (φLy/θ − κ) − (σ + φLy) (1 + φLπ/θ)).
- Trace and determinant of A′ derived explicitly:
  - Tr A′ = −Ω [ −φLy/θ + κ + (σ + φLy)β (1 + φLπ/θ) + σ (1 + φLπ/θ) − φLπ φLy/θ β ] = p (σ + φLy) (1 + φLπ/θ) + φLπ (κ − φLy/θ) [ −φLy/θ (1 + βφLπ) + κ + (σ(β + 1) + φLy β) (1 + φLπ/θ) ]
  - det A′ = ( p (σ + φLy) (1 + φLπ/θ) + φLπ (κ − φLy/θ) )^2 × β [ (σ + φLy) (1 + φLπ/θ) + φLπ (κ − φLy/θ) ]^(-1) × [ σ (1 + φLπ/θ) − φLy/θ ]
  - Final compact expression: det A′ = ( p^2 (σ + φLy) (1 + φLπ/θ) + φLπ (κ − φLy/θ) )^2 β [ σ (1 + φLπ/θ) − φLy/θ ]
- Assumptions for analysis:
  - φLy, φLπ, φLπ, φLy ≥ 0
  - Restrict to case where determinant is positive (both eigenvalues have same sign and non-imaginary), i.e. assume ( σ (1 + φLπ/θ) − φLy/θ ) ( (σ + φLy) (1 + φLπ/θ) + φLπ (κ − φLπ φLy/θ) ) > 0.
- Determinacy condition (Blanchard and Kahn, 1980): two non-predetermined variables determinate iff both eigenvalues are within the unit circle.
  - Necessary and sufficient conditions: det A′ < 1 and Tr A′ < 1 + det A′.
- Condition det A′ < 1 gives inequality:
  - φLy/θ ( φLπ − p^2 β ) < φLπ κ + ( σ(1 − p^2 β) + φLy ) (1 + φLπ/θ)
  - Rearranged: φLy < θ φLπ κ + θ ( σ(1 − p^2 β) + φLy ) (1 + φLπ/θ) / ( φLπ − p^2 β )
  - Interpretation: φLy must be small enough.
- Second condition Tr A′ < 1 + det A′ leads to a more complex inequality (derived in text).
- Special case imposing φLπ = 0 = φLy yields:
  - [1 − pβ] ( φLy/θ p − (1 + φLπ/θ) σ(1 − p) ) > κ
  - φLy/θ p > κ / (1 − pβ) + (1 + φLπ/θ) σ(1 − p)

### Forward Guidance (TIP and forward-guidance state “F”)
- Assume when economy exits liquidity trap, interest rate is maintained at 0 with probability q every period (Bilbiie, 2019).
- Forward-guidance state “F” closed-form:
  - ˆyF = 1 / (σ(1 − q)) ( q πF − log β )
  - πF = − log β / ( (1 − qβ) σ(1 − q) − κ q )
- Liquidity-trap state “L” solved by backward induction:
  - πL = β (1 − p) q πF + κ ˆyL / (1 − β p)
  - ˆyL = (1 − p) q ˆyF + 1/σ ( p πL + (1 − p) q πF − log(β + ∆β) ) / (1 − p)
- Solved expression for πL:
  - πL = [ β (1 − p)^2 q + κ (1 − p) q / σ ] πF + κ [ (1 − p) q ˆyF − 1/σ log(β + ∆β) ] / [ (1 − p)(1 − β p) − κ p / σ ]

### Welfare measures with forward guidance
- Conditional welfare losses:
  - XF = (πF)^2 + ηy (ˆyF)^2
  - XL = (πL)^2 + ηy (ˆyL)^2
- Lifetime welfare values:
  - WF = EF ∑_{t=0}∞ β^t [ π_t^2 + ηy (ˆyet)^2 ] = XF + β q YF = XF / (1 − β q)
  - WL = EL ∑_{t=0}∞ β^t [ π_t^2 + ηy (ˆyet)^2 ] = XL + β [ p WL + (1 − p) q WF ] = XL + β (1 − p) q WF / (1 − β p)

### Government spending (TIP absent)
- Linearized model equations (selected):
  - Euler: ˆc_t = E_t ˆc_{t+1} − 1/σ ( i_t − E_t π_{t+1} + log β )  (55)
  - Intratemporal: ˆw_t − ˆp_t = σ ˆc_t + ψ ˆn_t  (56)
  - Production: ˆy_t = (1 − α) ˆn_t  (57)
  - Marginal cost: ˆmc_t = ˆw_t − ˆp_t + α/(1 − α) ˆy_t − log(1 − α)  (58)
  - Goods clearing: ˆy_t = w_c ˆc_t + (1 − w_c) ˆg_t  (59)
  - Phillips: π_t = β E_t π_{t+1} + ε−1/θ ˆmc_t + 1/θ ( β E_t τ_{t+1} − τ_t ) + u_t  (60)
- Combined reduced form with ˜κ = (ε − 1)/θ [ σ w_c + ψ/(1 − α) + α/(1 − α) ]:
  - ˆy_t − (1 − w_c) ˆg_t = E_t ( ˆy_{t+1} − (1 − w_c) ˆg_{t+1} ) − w_c σ ( i_t − E_t π_{t+1} + log β )  (61)
  - π_t = β E_t π_{t+1} + ˜κ ˆy_t − (ε − 1)/θ σ(1 − w_c)/w_c ˆg_t + 1/θ ( β E_t τ_{t+1} − τ_t ) + u_t  (62)
- Liquidity trap (without TIP):
  - ˆyL = (1 − w_c) ˆgL + w_c (1 − p) σ ( p πL − log(β + ∆β) )  (63)
  - πL = 1 / (1 − β p) [ ˜κ ˆyL − (ε − 1)/θ σ(1 − w_c)/w_c ˆgL + u_t ]  (64)
- Solved πL (abstracting from cost-push shocks):
  - πL [ 1 − β p − ˜κ w_c p (1 − p) σ ] = ˜κ [ (1 − w_c) ˆgL − w_c (1 − p) σ log(β + ∆β) ] − (ε − 1)/θ σ(1 − w_c)/w_c ˆgL  (65)
  - Hence:
    - πL = − w_c ˜κ (1 − p) σ (1 − β p) − ˜κ p w_c log(β + ∆β) + Γ_g ˆg_t / Γ_g  (66)
    - Γ_g = ˜κ(1 − w_c) − (ε − 1)/θ σ(1 − w_c)/w_c [ 1 − β p − ˜κ w_c p (1 − p) σ ]  (67)
    - ˆyL = (1 − w_c) ˆgL + w_c (1 − p) σ ( p πL − log(β + ∆β) )  (68)

### Second-order approximation of welfare loss
- Welfare loss function depends on inflation, output gap, and government spending gap:
  - L = ∑_{t=0}∞ β^t [ π_t^2 − (1 − σ)/(θ w_c) ( ˆy_t − (1 − w_c) ˆg_t )^2 + (1 + ψ)/(θ(1 − α)) (ˆy_t)^2 − (1 − w_c)/θ (1 − σ_G) ˆg_t^2 − Φ/θ ˆy_t ]
- With no government spending and no steady-state distortion:
  - L = ∑_{t=0}∞ β^t [ π_t^2 + 1/θ ( σ + α + ψ/(1 − α) ) (ˆy_t)^2 ]
- Representative household second-order expansion gives log-utility deviation:
  - U_t − U = [ ˆy_t − (1 − w_c) ˆg_t − 1/2 θ π_t^2 w_c + (1 − σ)/2 ( ˆy_t − (1 − w_c) ˆg_t / w_c )^2 ] − (1 − Φ)/w_c (ˆy_t − ˆa_t) + (1 + ψ)/2 (1/(1 − α))^2 (ˆy_t − ˆa_t)^2 + (1 − w_c)/w_c (ˆg_t + (1 − σ_G)/2 ˆg_t^2) + t.i.p
- Under small-distortion assumption and abstracting from TFP shocks:
  - w_c (U_t − U) / (U_c C) = Φ ˆy_t − 1/2 θ π_t^2 + (1 − σ)/(2 w_c) (ˆy_t − (1 − w_c) ˆg_t)^2 − (1 + ψ)/(2(1 − α)) (ˆy_t)^2 + (1 − w_c) (1 − σ_G)/2 ˆg_t^2 + t.i.p

### Self-fulfilling and Phillips-curve-driven ZLB conditions
- Self-fulfilling liquidity trap (proof conditions):
  - Need inflation given by −1/( (1 − pβ) σ (1 − p) − κ p ) [ κ log β ] to be negative without TIP, requiring:
    - 1 − pβ κ < p / (σ(1 − p))  (i.e., 1 − pβ over κ < p/(σ(1 − p)))
  - Nominal interest rule must imply negative nominal rate:
    - φπ πL − log β < 0 ⇒ πL < log β / (φπ − 1/( (1 − pβ) σ (1 − p) − κ p ) [ κ log β ] )
    - Implies φπ > ( κ p − (1 − pβ) σ (1 − p) ) / κ p − (1 − pβ) σ(1 − p)  (as given in text)
- Shift in Phillips curve (proof conditions):
  - For ZLB to bind, require πL < log β where
    - πL = − 1/( (1 − pβ) σ (1 − p) − κ p ) [ κ log β − σ(1 − p)/θ u ] < log β
  - Leads to threshold inequalities for u:
    - u < θ log β / ( σ(1 − p) ( κ − ((1 − pβ) σ(1 − p) − κ p) ) )
    - u < θ log β / ( σ(1 − p) ( κ(1 + p) − (1 − pβ) σ(1 − p) ) )
  - Also require 1 − pβ / κ > p / (σ(1 − p)) for ZLB to bind.

### Robust TIP (Thresholds and comparative statics)
- Three cases discussed:
  1. r∗-driven liquidity trap: direct proof (text).
  2. Self-fulfilling liquidity trap: to prevent liquidity trap with inflation-targeting TIP, require
     - −1/( (1 − pβ) σ (1 − p) (1 + φπ/θ) − κ p ) [ κ log β ] > 0 ⇒ implies φπ > θ ( κ p / ( (1 − pβ) σ (1 − p) ) − 1 ).
  3. Phillips-curve case: two regions depending on whether π < log β / φπ (ZLB with i = 0) or π > log β / φπ.
     - If π < log β / φπ:
       - πL = − 1/( (1 − pβ) σ (1 − p) (1 + φπ/θ) − κ p ) [ κ log β − σ(1 − p)/θ u ]  (69)
       - ˆyL = 1/(σ(1 − p)) ( p πL − log β )  (70)
       - π is monotonically increasing in φπ; ˆy is monotonically increasing in φπ.
     - If π > log β / φπ:
       - πL = 1/( (1 − pβ) σ (1 − p) ) ( 1 + φπ/θ ) + κ(φπ − p) u σ(1 − p)/θ  (71)
       - ˆyL = 1/(σ(1 − p)) ( p − φπ ) πL  (72)
     - Threshold φPCπ solves π = log β / φπ = − 1/( (1 − pβ) σ (1 − p) ) ( 1 + φPCπ/θ ) − κ p [ κ log β − σ(1 − p)/θ u ] ⇒ φPCπ expression given.

### Numerical simulations (figures referenced)
- Figures reported in D.1 (titles only in source):
  - Figure 10: Welfare losses - Policymaker maximizing expected welfare with a prior
  - Figure 11: Welfare losses - Max-min policymaker
  - Figure 12: Welfare losses - Policymaker with multiplier preferences
  - Figure 13–16: Inflation and Output Gap under different policymaker priors/preferences
  - Figure 17–18: Optimal θ and Optimal σ - Max-min policymaker

### Medium-scale model (appendix E) — model structure and steady states
- De-trending conventions: Y_t = Y*_t / Z_t, C_t = C*_t / Z_t, K_t = K*_t / Z_t, K^u_t = K^{u*}_t / Z_{t−1}, I_t = I^*_t / Z_t, W_t = W^*_t / (Z_t P_t).
- Consumption block first-order conditions (bond-in-utility specification):
  - λ_t = β p (1 + i_t) E_t [ λ_{t+1} G_{Z,t+1} / Π_{t+1} ] + δ_t  (73)
  - λ_t = 1/C_t − h C_{t−1}/G_{Z,t} − h β p E_t [ (C_{t+1} − h C_t) / G_{Z,t+1} ]  (74)
  - G_{Z,t} = Z_t / Z_{t−1}; Π_t = P_t / P_{t−1}; h degree of external habit; δ_t utility gain from bonds.
- Price and wage setting include Rotemberg quadratic adjustment costs and TIP cost τ_t (Pi difference times Y_i,t):
  - Price Phillips curve expression (76) (full expression in source).
  - Wage Phillips curve expression (79) (full expression in source).
- Investment and capital dynamics:
  - K_t = u_t K^u_t G_{Z,t}  (80)
  - Law of motion: K^u_{t+1} = μ_t [ 1 − S( I_t / I_{t−1} G_{Z,t} / G_Z ) ] I_t + (1 − δ_k) K^u_t G_{Z,t}  (81)
  - Tobin’s q conditions (83)-(84).
  - Investment adjustment S with S(1) = S′(1) = 0 and S′′(1) = 5.16; utilization cost A with A(1) = 0 and A′′ = 2.246.
- Monetary policy at binding ZLB: i_t = 0  (85).
- Production relations and market clearing:
  - r_{k,t} = α MC_t Y_t / K_t  (86)
  - W_t = (1 − α) MC_t Y_t / L_t  (87)
  - P_t MC_t = 1 / Z_t^{1−α} ( r_{k,t} / α )^α ( W_t / (1 − α) )^{1−α }  (88)
  - Market clearing: 1/g_t Y_t = C_t + I_t + A(u_t) K^u_t G_{Z,t} + φ_p/2 ( Π_t ˜Π_{t−1} − 1 )^2 Y_t  (89)
- Flexible-price steady state (FSS) obtained by p = 1, Π_t = Π_wt = G_Z = 1, and 1 + i_t / Π_{t+1} = exp(r∗).

### Self-fulfilling deflationary steady state (DSS) and r∗-driven DSS
- DSS solution approach:
  - Free parameters: p_DSS, φ_p, φ_w.
  - p_DSS calibrated so steady-state inflation satisfies exp(−1.06%/4) = 0.99735 (annualized −1.06%).
- Consumption block in DSS:
  - λ_DSS = β p_DSS (1 + 0) 1/G_Z 1/Π_DSS λ_DSS + δ_FSS  (90) ⇒ pins down λ_DSS and C_DSS.
- Price Phillips curve in DSS (τ_DSS = 0 simplifies to expression for MC_DSS) (91)-(92).
- Investment and capital in DSS:
  - q_DSS = 1 and related conditions (94)-(99) yield K^u_DSS / Y_DSS = α MC_DSS G_Z / ( G_Z β^{−1} − (1 − δ_k) ).
  - I_DSS / Y_DSS = K^u_DSS / Y_DSS ( 1 − 1/G_Z (1 − δ_k) )  (99).
  - Market clearing (100) used to solve for Y_DSS given C_DSS.
- Labor block and wage indexation yield equation (104) that pins down unique p_DSS.
- r∗-driven DSS:
  - r∗ falls from 0% to −1.1% per annum; δ_{rDSS} chosen so r∗ = −1.1% in FSS (equation (105)).
  - Then repeat DSS solution using δ_{rDSS}.

### Log-linearization and estimation notes
- Log-linearization performed around the deflationary steady state (separately per case).
- Parameter estimates tables (Tables 3 and 4) provide prior and posterior means, standard deviations, and HPD intervals for shocks and persistence parameters under scenarios:
  - Table 3: Decline in r∗ Liquidity Trap — includes parameters e_w, e_p, e_g, e_μ, e_z, e_b, ρ_w, ρ_p, ρ_g, ρ_μ, ρ_b, ρ_z with posterior means and HPD bands.
  - Table 4: Self-fulfilling Liquidity Trap — includes similar set plus e_sunspot; posterior means and HPD bands reported.
  - Example posterior entries (preserve exact reported numbers):
    - From Table 3: e_μ posterior mean 0.0125 stdev 0.1253 HPD inf 0.0004 HPD sup 0.1654; ρ_g posterior mean 0.8340 stdev 0.0379 HPD inf 0.7723 HPD sup 0.8978.
    - From Table 4: e_sunspot posterior mean 0.0110 stdev 0.0008 HPD inf 0.0093 HPD sup 0.0119; ρ_g posterior mean 0.9410 stdev 0.0231 HPD inf 0.9051 HPD sup 0.9777.

*Source: Tax on Inflation Policy at the Zero Lower Bound — Working Paper No. WP/2026/059 (sections and appendices excerpted from the supplied PDF).*

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_Source: https://www.imf.org/-/media/files/publications/wp/2026/english/wpiea2026059-source-pdf.pdf_