## _wp1233

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### 2.1 Households — model structure and key relations
- Household population: unit measure indexed i∈[0,1]; types τt(i)∈{b,s} with τt(i)=b as “borrowers” and τt(i)=s as “savers”.
- Type redrawing: households redraw type with probability 1−δ, δ∈(0,1); conditional redraw probabilities: borrower πb, saver πs = 1−πb.
- Time preference shock: et is a unit-mean shock to discount factor β∈(0,1).
- Preferences and consumption aggregator:
  - Expected utility: E0 ∑t=0∞ (e t β t) [ (ξ τ) σ−1 τ [c t(i)] 1−σ−1 τ /(1−σ−1 τ) − ψ τ (1 +ν)−1 h t(i) 1+ν ].
  - c t(i) = [∫1 0 c t(j,i) θ−1/θ dj] θ/(θ−1); θ>1.
- Beginning-of-period wealth (Equation 2):
  - A t(i) = [B t−1(i)]+ (1+i d t−1) + [B t−1(i)]− (1+i b t−1) + (1−θ t) B g t−1(i)(1+i g t−1) + D int t + T t(i) + T c t.
  - Definitions: [B t−1(i)]+ deposits earn i d t−1; [B t−1(i)]− debt pays i b t−1; θ t ∈{0,θ def} with θ def∈[0,1); D int t profits of intermediaries; T t(i) state-contingent transfers; T c t lump-sum compensation to bond holders on sovereign default.
- End-of-period wealth (Equation 3):
  - B t(i) = A t(i) − P t c t(i) + (1−τ w t) P t w t h t(i) + D t − T g t.
- Asymptotic risk sharing — post-transfer identical wealth for redrawing households (Equation 4):
  - A t = [d t−1 (1 +i d t−1) + (1−θ t) b g t−1 (1 +i g t−1) − b t−1 (1 +i b t−1)] P t−1 + D int t + T c t.
- Law of motion for aggregate private borrowing b t (Equation 5):
  - b t = δ b t−1 (1 +ω t−1)(1 +i d t−1)/Π t − π b ω t b t + π b [ δ b g t−1 (1 +i g t−1)/Π t − b g t ] + π b π s [ (c b t − c s t) − (1−τ w t)(w t h b t − w t h s t) ].
- Consumption and marginal utilities:
  - c s t = ξ s (λ s t) −σ s.  (Equation 6)
  - c b t = ξ b (λ b t) −σ b.  (Equation 7)
- Euler equations (Equations 8–10) — preserve exact forms for savers and borrowers:
  - e t λ s t = β E t [ e t+1 (1 + i d t)/Π t+1 { (1−δ) π b λ b t+1 + [δ + (1−δ) π s] λ s t+1 } ].
  - e t λ s t = β E t [ e t+1 (1−θ t+1)(1 + i g t)/Π t+1 { (1−δ) π b λ b t+1 + [δ + (1−δ) π s] λ s t+1 } ].
  - e t λ b t = β E t [ e t+1 (1 + i b t)/Π t+1 { (1−δ) π s λ s t+1 + [δ + (1−δ) π b] λ b t+1 } ].
- Labor supply (Equations 11–13):
  - h s t = ( λ s t ψ s (1−τ w t) w t ) 1/ν.
  - h b t = ( λ b t ψ b (1−τ w t) w t ) 1/ν.
  - h t = π b h b t + (1−π b) h s t = ( Λ t ψ (1−τ w t) w t ) 1/ν.
  - Λ t := ψ [ π b (λ b t ψ b) 1/ν + π s (λ s t ψ s) 1/ν ] ν; ψ −1/ν = π b ψ −1/ν b + π s ψ −1/ν s.
- Average marginal utility (Equation 15):
  - λ t = π b λ b t + (1−π b) λ s t.

### 2.2 Financial intermediaries — spreads and sovereign spillovers
- Intermediation setup:
  - Intermediation requires resources Ξ t b t and endogenous fraud fraction χ t of unrecoverable loans.
  - Deposits d t riskless; intermediaries choose deposits repaid from loan proceeds: (1 + i d t) d t = (1 + i b t) b t.
  - Intermediary cash flow: d t − b t − χ t b t − Ξ t b t.
- Spread definition (Equation 16):
  - 1 + ω t = (1 + i b t)/(1 + i d t).
- First-order condition for loan origination (Equation 17):
  - ω t = χ t + Ξ t.
- Sovereign risk spillovers into private spreads — two modeling choices (Equations 18–19):
  - Choice A: χ t = χ ψ [ (1 + i g t)/(1 + i d t) ] α ψ −1 and Ξ t = 0.
  - Choice B: χ t = 0 and Ξ t = χ ψ [ (1 + i g t)/(1 + i d t) ] α ψ −1.
- Parameter roles:
  - χ ψ > 0 scales private spread in steady state; α ψ measures spillover strength.
  - Linearized model: set Ξ t = 0 and χ t > 0. Dynamic simulations: set χ t = 0 and Ξ t > 0.
- Transfer from intermediaries to households:
  - D int t = P t ( ω t b t − Ξ t b t ).

### 2.3 Firms — production, price setting, and inflation
- Production: continuum j∈[0,1], y t(j) = z t h(j) 1/φ (Equation 20); z t aggregate productivity shock.
- Price setting (Calvo-type): fraction (1−α) can reoptimize; non-reoptimizers adjust price by steady-state inflation Π.
- Optimal reset price (Equation 21):
  - (P ∗ t /P t) 1+θ(φ−1) = K t F t.
  - K t = λ t e t μ p φ w t ( y t / z t ) φ + α β E t [ (Π t+1 /Π) θ φ K t+1 ].  (Equation 22)
  - F t = λ t e t y t + α β E t [ (Π t+1 /Π) (θ−1) F t+1 ].  (Equation 23)
  - μ p = θ/(θ−1).
- Price-setting inflation law (Equation 24):
  - (1−α) (Π t /Π) θ−1 = (1−α) (P ∗ t /P t) 1−θ.
- Price dispersion Δ t and evolution (Equation 25).
- Profits distributed to households:
  - D t = ∫1 0 P t(j) y t(j) − P t w t h t(j) dj; equilibrium expression: D t = P t ( y t − w t (y t / z t) φ Δ t ).

### 2.4 Government — debt, default, and policy rules
- Real government debt evolution:
  - b g t = (1−θ t) b g t−1 (1 + i g t−1)/Π t + g t + T c t /P t − T g t /P t − τ w t w t h t.
- Transfers to ensure ex post neutrality of default:
  - T c t = P t θ t b g t−1 (1 + i g t−1)/Π t.
- Consolidated government flow constraint (Equation 26):
  - b g t = b g t−1 (1 + i g t−1)/Π t + g t − T g t /P t − τ w t w t h t.
- Cyclical/stabilization tax component (Equation 27): tr t = τ w t w t h t + T g t /P t.
- Fiscal rule for tr t (Equation 28):
  - (tr t − t) = [ φ T,y (y t − y) + φ T,b g (b g t−1 − b g) ], φ T,y ≥ 0, φ T,b g > 0; φ T,b g assumed large enough to stabilize debt.
- Sovereign default probability / haircut (Equations 29–30):
  - Fiscal limit drawn from generalized beta with α b g, β b g, b g,max.
  - p t = F beta ( b g t / (4 y 1 b g,max) ; α b g, β b g ). (Equation 29)
  - θ t = θ def with probability p t; 0 with probability 1−p t. (Equation 30)
  - b g,max denotes upper end of support for debt-to-GDP ratio.
- Monetary policy (Equation 31):
  - log(1 + i d,∗ t) = log(1 + i d) + φ Π log(Π t /Π) − φ ω log((1 + ω t)/(1 + ω)).
  - φ Π > 1, φ ω > 0.
  - Lower bound: if i d,∗ t < 0 then i d t = 0; otherwise i d t = i d,∗ t.

### 2.5 Market clearing and equilibrium
- Goods market clearing (Equation 32):
  - y t = ∫1 0 c t(i) di + g t + Ξ t b t = π b c b t + π s c s t + g t + Ξ t b t.
- Aggregate supply relation (Equation 33):
  - y t Δ 1/φ t = z t h 1/φ t.
- Equilibrium characterized by system of equations:
  - Households (5)–(15), intermediation (16)–(19), price setting (21)–(25), fiscal behavior (26)–(30), tax/stabilization rule (27)–(28), Taylor rule (31) with lower bound, market clearing (32)–(33).
- Endogenous sequences pinned down for exogenous {e t, g t, z t}: { b t, b g t, c b t, c s t, χ t, Δ t, F t, h t, h b t, h s t, i b t, i d t, i d,∗ t, i g t, K t, λ t, λ b t, λ s t, Λ t, ω t, p t, P ∗ t /P t, Π t, τ w t, T g t /P t, tr t, θ t, w t, Ξ t, y t }.

### 2.6 Calibration — key parameter choices and targets
- Time unit: one quarter.
- Average inflation: 2 percent per year.
- Taylor rule inflation coefficient: φΠ = 1.5.
- Central bank steady-state target interest rate: id = 4.5 percent annualized; β follows from id.
- ZLB treatment:
  - Nominal rate stuck at zero in initial period(s); returns to Taylor rule next period with probability 1 − μ; expected length of ZLB episode: 1/(1 − μ).
  - Temporary increase in effective discount factor at ZLB: 0 < et = eL < 1, Γt = μ log(eL) − log(eL) = −(1 − μ) log(eL) ≥ 0.
- Fiscal calibration:
  - g/y = 0.2.
  - Steady-state gross public debt: 60 percent of annual GDP.
  - Baseline distortionary tax rates set to zero; taxes adjust via lump-sum.
  - φT,y = 0.34; φT,bg large enough to bound debt.
  - Fiscal limit parameters: αbg = 3.70, βbg = 0.54, bg,max = 2.56.
  - Sovereign default haircut θdef = 0.5.
  - Target steady-state normal spread: 2.1 percent annualized → pins down χψ.
- Real-side calibration:
  - ν = 1/1.9.
  - θ = 7.6 → μp = 1.15.
  - Steady-state aggregate hours h = 1/3.
  - σb/σs = 0.53; ψb/ψs = 0.82.
  - σ := πb·(cb/y)·σb + πs·(cs/y)·σs.
  - φ = 1; target steady-state output = 1 → pins down z.
  - α = 0.9 → implied Phillips curve slope κ = 0.012 (empirical estimates range 0.007 to 0.047; recent estimate 0.014).
- Private debt and household types:
  - Target private debt ratio: b/4y = 80 percent.
  - δ = 0.975 → households redraw on average every 40 quarters (≈10 years without asymptotic risk sharing).
  - Baseline πb construction:
    - Survey metrics: debt-holders 76% → πb candidate 0.76; savers 57% → πb candidate 0.43.
    - Corporations account for 50% of private borrowing; baseline πb set as (1 − 0.17)·0.43 + 0.17·1 = 0.53 (0.17 = share of nonresidential private investment).
  - ξb and ξs determined to satisfy market clearing and private debt target.
- Sovereign-to-private spillovers:
  - Baseline αψ = 0.55 (Harjes (2011): 100-basis-point sovereign → 50–60 basis-point private spread).
  - αψ may understate spillovers for smaller/domestic firms; higher αψ considered in simulations.
- Analytical special case (linearized):
  - NK Phillips curve: ̂Πt = β Et ̂Πt+1 + κy ̃yt − κg ̃gt, with κy = κ(ν + ̄σ−1), κg = κ ̄σ−1, κ = (1 − α)(1 − αβ)/α.
  - Dynamic IS: ̃yt − ̃gt = Et ̃yt+1 − Et ̃gt+1 − ̄σ [ ̂idt + (πb + sΩ) ̂ωt − Et ̂Πt+1 + Γt ].
  - sΩ := πb πs (σb cb/y − σs cs/y) / ̄σ; cb > cs ⇒ sΩ > 0.
  - Under three-equation mapping, only current interest rate spread enters; output and inflation dynamics independent of private debt level in this special case.
- Linking fiscal deficits to private spreads — ξ and ξ′:
  - ξ′ expression (text): ξ′ = αψ (πb + sΩ) θdef 1 − θdef Fbeta(bg/4y 1 / bg,max ; αbg, βbg) 1 / 4y 1 ̄bg,max fbeta(b/4y 1 / bg,max ; αb, βb).
  - Mapping ξ = (1 + μ(1 − μ)) / (μ(1 − μ)) ξ′ for μ > 0.5.
  - Representative Table 1 values (preserve reported numbers):
    - Debt/GDP = 60 percent → ξ′ = 0.0005 → ξ = 0.0040, 0.0050, 0.005 (for 6, 7, 8 qtrs).
    - Debt/GDP = 90 percent → ξ′ = 0.0016 → ξ = 0.0140, 0.0150, 0.0170.
    - Debt/GDP = 110 percent → ξ′ = 0.0030 → ξ = 0.0250, 0.0280, 0.0310.
    - Debt/GDP = 130 percent → ξ′ = 0.0051 → ξ = 0.0420, 0.0470, 0.0520.
    - Debt/GDP = 140 percent → ξ′ = 0.0065 → ξ = 0.0540, 0.0600, 0.0660.
    - Debt/GDP = 150 percent → ξ′ = 0.0083 → ξ = 0.0680, 0.0760, 0.0840.
  - Interpretation: ξ′ small but ξ (mapping deficits over expected ZLB horizon) can be substantially larger; ξ up to 0.1 possible with large initial debt and persistent shock; ξ scales linearly with αψ.

### 3.3 Sovereign risk channel and equilibrium determinacy — propositions and implications
- Baseline insight: sovereign risk channel materially shrinks determinacy region while ZLB binds.
- Proposition 1 (exogenous g):
  - ZLB in initial period; remains at zero with probability μ each subsequent period; monetary policy returns with probability 1 − μ.
  - Locally unique bounded equilibrium iff:
    - a) a < 1/(βμ), and
    - b) (1−βμ)(1−a) > μ ̄σκ_y,
    - where a := μ + μ ξ φ_{T,y} ̄σ and κ_y := κ[ν + 1/ ̄σ].
  - Remarks:
    - If ξ = 0 then condition a) always satisfied and determinacy hinges on b): (1−βμ)(1−μ) > μ ̄σκ_y.
    - Determinacy easier with smaller μ, smaller ̄σ, smaller κ_y.
    - With ξ > 0, large ξ or large φ_{T,y} may violate condition a).
- Proposition 2 (endogenous spending at ZLB: ̃g_t = φ ̃y_t when at ZLB, φ < 1):
  - Define a^∗ := μ + μ ξ φ^∗_{T,y} ̄σ, κ^∗_y = κ_y − φ κ_g, φ^∗_{T,y} := φ_{T,y} − φ, ̄σ^∗ = ̄σ/(1 − φ).
  - Locally unique bounded equilibrium conditions split by sign of a^∗:
    - If a^∗ > 0: (a) a^∗ < 1/(βμ), (b) (1−βμ)(1−a^∗) > μ ̄σ^∗ κ^∗_y.
    - If a^∗ < 0: (a) (1 + βμ)(1 + a^∗) > −μ ̄σ^∗ κ^∗_y, (b) (1−βμ)(1−a^∗) > μ ̄σ^∗ κ^∗_y.
  - Key unconventional result: with ξ > 0, procyclical spending response φ∈(0,1) can expand determinacy region under certain parameter restrictions (especially smaller φ_{T,y} and larger ξ), i.e., procyclical fiscal cuts during ZLB can reduce risk of indeterminacy.
- Mechanism:
  - Belief-driven equilibria arise because expected output declines → expected deficits rise → spreads widen → higher real rates at ZLB → confirm pessimism.
  - Procyclical spending cuts can offset expected tax revenue shortfalls, prevent deficits from rising, and break the self-fulfilling spiral.
- Qualitative depiction (Figure 3):
  - Determinacy regions depend on μ (expected ZLB durations: 6, 7, 8 quarters correspond to μ = 5/6, 6/7, 7/8).
  - For large ξ and prolonged ZLB, no simple countercyclical spending rule ensures determinacy; procyclical cuts can help but not universally guarantee determinacy.

### 4.1 Deep recessions and sovereign risk — nonlinear dynamics and ZLB interactions
- Perfect-foresight simulations calibrated to U.S. 2007–2009 output loss:
  - CBO estimates cited: output gap 6.7 percent in 2009; 1.7 percent in 2014; 0.5 percent in 2015.
  - AR shock: log(e_t) = ρ_e log(e_{t−1}) + u_t with u_0 = −0.1525, u_t = 0 thereafter; ρ_e = 0.93.
  - φ_{T,b_g} = 0.014 for first 30 quarters; beyond quarter 30 it doubles.
- Nonlinear sovereign risk channel results (Figure 5 summary):
  - Simulated for initial b g/y = 60 percent (black solid), 90 percent (blue dashed), 115 percent (red dots).
  - Higher initial debt → much larger output decline and interest rate spread rise.
  - Mechanism: shock raises deficits → public debt builds → sovereign risk premium rises → private interest rates rise; at ZLB monetary policy cannot offset → larger fall in private expenditure and output.
  - Nonlinear debt–risk relation: higher initial debt amplifies spread response.
  - Persistently high risk premia lengthen ZLB episode by up to 11 quarters for high debt.

### 4.2 Fiscal retrenchment during deep recessions — scenarios and effects
- Retrenchment scenario (immediate):
  - Fiscal restraint starts at recession onset, lasts 2 years (8 quarters); government spending falls by 2 percent of steady-state GDP during episode.
- Comparative ZLB timing cases:
  - Left: endogenous ZLB exit.
  - Middle: ZLB fixed exogenously 7 quarters.
  - Right: ZLB fixed exogenously 18 quarters (matching durations for b g/y = 60% and 115% in Figure 5).
- Effects of 8-quarter, −2 percent ss-GDP spending cuts:
  - Deficit and debt decline for all initial debt levels; risk premium falls; monetary policy escapes ZLB earlier.
  - Output/private expenditure responses differ by initial debt:
    - Initial debt low (60%): spending cuts cause initial output decline > 2 percent due to fall in private expenditure.
    - Initial debt high (115%): initial output response is positive (private expenditure rises) because sovereign-risk channel dominates.
  - Two channels:
    - Channel 1: lower public spending → deflationary → raises real rates (if policy constrained at ZLB) → contractionary.
    - Channel 2: debt reduction → lower sovereign risk → lower private spreads → expansionary.
  - With higher initial debt and long ZLB (18 quarters), Channel 2 can dominate: spending cuts can be expansionary and reduce ZLB duration.
  - When ZLB expected short (7 quarters), spending cuts effects nearly independent of debt level.

### 4.3 Further considerations — timing, distortionary taxation, monetary offset
- Timing of spending cuts:
  - Immediate retrenchment: 2 years at −2 percent ss-GDP.
  - Medium-term retrenchment: starts two years later, lasts 10 years (years 3–12).
  - Persistent retrenchment: immediate + medium-term, lasting 12 years total.
  - Findings:
    - Medium-term retrenchment yields immediate output gains via fiscal forward guidance: anticipated future lower real rates crowd in current consumption.
    - Medium-term retrenchment reduces sovereign risk premium immediately via higher expected future tax revenue.
    - Persistent retrenchment achieves largest risk premium reduction but smaller output gains than medium-term due to upfront contraction.
    - For b g/y = 115%, persistent retrenchment crowds in activity during most of ZLB but with smaller initial boost than medium-term retrenchment.
    - Credibility of announced future consolidation crucial: if not credible, medium-term strategy fails.
- Distortionary taxation:
  - Baseline: lump-sum taxes.
  - Alternative: labor taxes throughout, steady-state labor tax = 35 percent; set φ_T,b_g = 0.0193 for all periods.
  - Hybrid: lump-sum for first 18 quarters, labor taxes thereafter.
  - Findings:
    - Under persistent distortionary taxes, net effects of spending cuts on output resemble baseline; future reduction in distortionary tax burden partly offsets contemporaneous effects.
    - Hybrid scenario: for high initial debt (115%), early retrenchment can crowd in output much more strongly because deferred distortionary tax increases worsen baseline without early retrenchment.
- Degree of monetary policy offset for risk premia:
  - Baseline: post-ZLB central bank can fully neutralize sovereign risk premia (full offset).
  - Alternative: φ_ω reduced to 0.75(π_b + s_Ω) (three-quarters of benchmark).
  - Findings:
    - Limited neutralization → deeper baseline recession due to stronger headwinds from persistent premia.
    - Under limited neutralization, early retrenchment less detrimental at high initial debt: reduced central bank offset raises relative benefit of retrenchment via sovereign risk channel.
  - Policy implication: effectiveness of retrenchment depends on central bank capacity/willingness to offset sovereign risk post-ZLB.
- Overarching conclusions:
  - Rising sovereign risk raises private borrowing costs unless offset by looser monetary policy; if central bank constrained, sovereign risk crucially shapes macro outcomes.
  - Sovereign risk increases vulnerability to indeterminacy; procyclical fiscal cuts can help ensure determinacy in some parameter regimes.
  - Fiscal multipliers depend on responsiveness of private-sector risk premia to fiscal indicators (ξ, ξ′) and expected ZLB duration (μ).
  - Upfront retrenchment less harmful (and possibly expansionary) when sovereign risk significant and ZLB prolonged; however, retrenchment is not a panacea and deep recessions can persist.
  - External financial support or credible announcements of future adjustment can improve outcomes and allow delayed consolidation to be preferable.

- Analytical appendices and supporting results (selected highlights)
  - Three-equation representation (IS, NK Phillips, Taylor) derived under conditions: Ξ = 0; [ψ_Ω + s_Ω(1 − ̄δ)] = 0; ̂τ^w_t = 0; [s_Ω + π_b − γ_b] = 0.
  - Determinacy proofs: Blanchard-Kahn conditions applied to matrix A; det(A) = 1/(a μ β); tr(A) expression given in proofs.
  - Corollaries on spending cyclicality and multipliers:
    - Corollary 4: ξ = 0 → larger determinacy region with countercyclical spending; ξ > 0 → procyclical spending can enlarge determinacy region under conditions.
    - Corollary 5: conditions for government spending multiplier θ_g > 0 and self-financing thresholds; θ_g can be negative if ξ large; θ_g > 1 conditions depend on φ_{T,y} and ξ.
  - Delayed retrenchment results (Appendix E):
    - Mapping between ξ and ξ′: ξ = [1 + μ(1 − μ)] / [μ(1 − μ)] ξ′.
    - Proposition 6: closed-form expression for output at ZLB when austerity scheduled after exit; Corollary 7: future austerity can boost ZLB-period output, effect amplified by sovereign risk (ξ).

*Source: _wp1233 - 2.1  Households; 2.2 Financial intermediaries; 2.3 Firms; 2.4 Government; 2.5 Market clearing and equilibrium; 2.6 Calibration; 3.3 Sovereign risk channel and equilibrium determinacy; 4.1–4.3 Deep recessions, retrenchment, and further considerations.*

### 2.1  Households

### 2.1 Households

### Model setup and household types
- Economy populated by a unit measure of households indexed by i∈[0,1].
- Household i is of one of two types, τt(i)∈{b,s}; in equilibrium τt(i)=b are “borrowers” and τt(i)=s are “savers”.
- Households redraw type with probability 1−δ, where δ∈(0,1). Conditional on redrawing, borrower probability is πb and saver probability is πs = 1−πb.
- Time preference shock e t is a unit-mean shock to discount factor β∈(0,1).
- Preferences (per household i):
  - E0 ∑t=0∞ (e t β t) [ (ξ τ) σ−1 τ [c t(i)] 1−σ−1 τ /(1−σ−1 τ) − ψ τ (1 +ν)−1 h t(i) 1+ν ],
  - consumption aggregator: c t(i) = [∫1 0 c t(j,i) θ−1/θ dj] θ/(θ−1); θ>1.
- Parameters ξτ, στ, ψτ, ν are positive.

### Wealth, transfers, and government debt haircut
- Beginning-of-period household wealth:
  - A t(i) = [B t−1(i)]+ (1+i d t−1) + [B t−1(i)]− (1+i b t−1) + (1−θ t) B g t−1(i)(1+i g t−1) + D int t + T t(i) + T c t.  (Equation 2)
  - [B t−1(i)]+ = deposits earning deposit rate i d t−1; [B t−1(i)]− = debt charging borrowing rate i b t−1.
  - Government debt may be partially defaulted: θ t = 0 (honor) or θ t = θ def with θ def∈[0,1) (haircut).
  - D int t are profits from competitive financial intermediaries distributed lump-sum.
  - T t(i) are state-contingent contract transfers (nonzero only on redraw).
  - T c t is a lump-sum transfer that compensates bond holders in case of sovereign default (not proportional to individual holdings).
- End-of-period household wealth:
  - B t(i) = A t(i) − P t c t(i) + (1−τ w t) P t w t h t(i) + D t − T g t.  (Equation 3)
  - P t is consumption price index; τ w t labor tax rate; w t real wage; D t profits of intermediary goods producers; −T g t lump-sum government transfers.

### Asymptotic risk sharing and aggregate variables
- Post-transfer identical wealth for households selected to redraw:
  - A t = [d t−1 (1 +i d t−1) + (1−θ t) b g t−1 (1 +i g t−1) − b t−1 (1 +i b t−1)] P t−1 + D int t + T c t.  (Equation 4)
  - b g t denotes government debt in real terms; d t aggregate savings deposited; b t aggregate private borrowing (real terms).
- Law of motion for aggregate private borrowing b t:
  - b t = δ b t−1 (1 +ω t−1)(1 +i d t−1)/Π t − π b ω t b t + π b [ δ b g t−1 (1 +i g t−1)/Π t − b g t ] + π b π s [ (c b t − c s t) − (1−τ w t)(w t h b t − w t h s t) ].  (Equation 5)
  - Intuition: b t evolves from (i) prior private debt plus interest for non-redrawers, (ii) gain from fraudulent loans −πb ω t b t, (iii) sovereign indebtedness adjustment, (iv) consumption vs wage-income differences across types.

### Consumption, Euler equations, and labor supply
- Asymptotic risk sharing ⇒ common marginal utility λ τ t within type and common expenditure:
  - c s t = ξ s (λ s t) −σ s.  (Equation 6)
  - c b t = ξ b (λ b t) −σ b.  (Equation 7)
- Euler equations (borrowing/lending):
  - e t λ s t = β E t [ e t+1 (1 + i d t)/Π t+1 { (1−δ) π b λ b t+1 + [δ + (1−δ) π s] λ s t+1 } ].  (Equation 8)
  - e t λ s t = β E t [ e t+1 (1−θ t+1)(1 + i g t)/Π t+1 { (1−δ) π b λ b t+1 + [δ + (1−δ) π s] λ s t+1 } ].  (Equation 9)
  - e t λ b t = β E t [ e t+1 (1 + i b t)/Π t+1 { (1−δ) π s λ s t+1 + [δ + (1−δ) π b] λ b t+1 } ].  (Equation 10)
- Optimal labor supply:
  - h s t = ( λ s t ψ s (1−τ w t) w t ) 1/ν.  (Equation 11)
  - h b t = ( λ b t ψ b (1−τ w t) w t ) 1/ν.  (Equation 12)
- Average labor supply:
  - h t = π b h b t + (1−π b) h s t = ( Λ t ψ (1−τ w t) w t ) 1/ν.  (Equation 13)
  - Λ t := ψ [ π b (λ b t ψ b) 1/ν + π s (λ s t ψ s) 1/ν ] ν.  (Equation 14)
  - ψ −1/ν = π b ψ −1/ν b + π s ψ −1/ν s.
- Average marginal utility:
  - λ t = π b λ b t + (1−π b) λ s t.  (Equation 15)

---

### 2.2 Financial intermediaries

### Intermediation, spreads, and loan defaults
- Competitive financial intermediaries intermediate saving and borrowing across types.
- Intermediation requires resources Ξ t b t and a fraction χ t of loans cannot be recovered (fraud), independent of borrower characteristics.
- Deposits d t are riskless; intermediaries choose largest deposits repayable from loan proceeds: (1 + i d t) d t = (1 + i b t) b t.
- Cash flow of intermediary in period t: d t − b t − χ t b t − Ξ t b t.
- Spread ω t between lending and deposit rates:
  - 1 + ω t = (1 + i b t)/(1 + i d t).  (Equation 16)
- First-order condition for loan origination (maximizing intermediary profits):
  - ω t = χ t + Ξ t.  (Equation 17)

### Sovereign risk spillovers into private spreads
- Either χ t or Ξ t depends on sovereign risk to capture monitoring/enforcement deterioration under fiscal strain.
- Two modeling choices:
  - χ t = χ ψ [ (1 + i g t)/(1 + i d t) ] α ψ −1 and Ξ t = 0.  (Equation 18)
  - χ t = 0 and Ξ t = χ ψ [ (1 + i g t)/(1 + i d t) ] α ψ −1.  (Equation 19)
- χ ψ > 0 scales private spread in steady state; α ψ measures spillover strength from sovereign to private risk premium.
- In linearized model (Section 3): set Ξ t = 0 and let χ t > 0. In dynamic simulations (Section 4): set χ t = 0 and Ξ t > 0.
- Transfers from intermediaries to households include unrecovered loans:
  - D int t = P t ( ω t b t − Ξ t b t ).

---

### 2.3 Firms

### Production, price setting, and inflation dynamics
- Continuum of firms j∈[0,1]; production technology:
  - y t(j) = z t h(j) 1/φ.  (Equation 20)
  - z t aggregate productivity shock.
- Only fraction (1−α) of firms can reoptimize prices each period; non-reoptimizers adjust price by steady-state inflation Π.
- Optimal reset price P ∗ t satisfies:
  - (P ∗ t /P t) 1+θ(φ−1) = K t F t.  (Equation 21)
  - K t = λ t e t μ p φ w t ( y t / z t ) φ + α β E t [ (Π t+1 /Π) θ φ K t+1 ].  (Equation 22)
  - F t = λ t e t y t + α β E t [ (Π t+1 /Π) (θ−1) F t+1 ].  (Equation 23)
  - μ p = θ/(θ−1).
- Price-setting inflation law:
  - (1−α) (Π t /Π) θ−1 = (1−α) (P ∗ t /P t) 1−θ.  (Equation 24)
- Price dispersion Δ t:
  - Δ t := ∫1 0 (P j,t /P t) −θ φ dj; evolves as
  - Δ t = α Δ t−1 (Π t /Π) θ φ + (1−α) [ (1−α(Π t /Π) θ−1)/(1−α) ] θ φ /(θ−1).  (Equation 25)
- Profits distributed to households:
  - D t = ∫1 0 P t(j) y t(j) − P t w t h t(j) dj; in equilibrium D t = P t ( y t − w t (y t / z t) φ Δ t ).

---

### 2.4 Government

### Debt dynamics, transfers, and fiscal policy rule
- Real government debt evolution:
  - b g t = (1−θ t) b g t−1 (1 + i g t−1)/Π t + g t + T c t /P t − T g t /P t − τ w t w t h t.
- Transfers T c t ensure default is ex post neutral regarding distribution and debt level:
  - T c t = P t θ t b g t−1 (1 + i g t−1)/Π t.
- Consolidated government flow budget constraint:
  - b g t = b g t−1 (1 + i g t−1)/Π t + g t − T g t /P t − τ w t w t h t.  (Equation 26)
- Define cyclical/stabilization tax component:
  - tr t = τ w t w t h t + T g t /P t.  (Equation 27)
  - Policy rule for tr t deviations:
  - (tr t − t) = [ φ T,y (y t − y) + φ T,b g (b g t−1 − b g) ], with φ T,y ≥ 0, φ T,b g > 0.  (Equation 28)
  - φ T,b g assumed large enough to stabilize public debt.

### Sovereign default probability and haircut
- Fiscal limit approach: default occurs when debt exceeds stochastic fiscal limit.
- Fiscal limit drawn each period from a generalized beta distribution with parameters α b g, β b g, and b g,max.
- Ex ante default probability for debt level b g t:
  - p t = F beta ( b g t / (4 y 1 b g,max) ; α b g, β b g ).  (Equation 29)
- θ t is stochastic:
  - θ t = θ def with probability p t; 0 with probability 1−p t.  (Equation 30)
- Note b g,max denotes upper end of support for debt-to-GDP ratio.

### Monetary policy
- Central bank follows Taylor-type rule cushioning activity from risk spread fluctuations:
  - log(1 + i d,∗ t) = log(1 + i d) + φ Π log(Π t /Π) − φ ω log((1 + ω t)/(1 + ω)).  (Equation 31)
  - φ Π > 1, φ ω > 0.
- Deposit rate constrained by lower bound: if implementation would require i d t < 0, then i d t = 0; otherwise i d t = i d,∗ t. This limits offsetting of spread increases when policy rate hits zero.

---

### 2.5 Market clearing and equilibrium

### Market clearing conditions
- Goods market clearing:
  - y t = ∫1 0 c t(i) di + g t + Ξ t b t = π b c b t + π s c s t + g t + Ξ t b t.  (Equation 32)
- Aggregate supply relation:
  - y t Δ 1/φ t = z t h 1/φ t.  (Equation 33)

### Equilibrium characterization
- Equilibrium determined by system comprising:
  - Households: equations (5)-(15).
  - Financial intermediation: equations (16)-(19).
  - Price setting: equations (21)-(25).
  - Fiscal behavior: equations (26)-(30).
  - Tax evolution assumption and stabilization rule (27)-(28).
  - Interest rate target rule (31) with lower-bound constraint.
  - Market clearing: equations (32)-(33).
- For exogenous realizations {e t, g t, z t}, the model pins down sequences for endogenous variables:
  - { b t, b g t, c b t, c s t, χ t, Δ t, F t, h t, h b t, h s t, i b t, i d t, i d,∗ t, i g t, K t, λ t, λ b t, λ s t, Λ t, ω t, p t, P ∗ t /P t, Π t, τ w t, T g t /P t, tr t, θ t, w t, Ξ t, y t }.

*Source: _wp1233 - 2.1  Households*

### 2.6  Calibration

### 2.6  Calibration

### Monetary policy calibration
- Time period in the model is one quarter.
- Average inflation rate: 2 percent per year.
- Coefficient on inflation in the Taylor rule: φΠ = 1.5.
- Consider different values for φω (response of monetary policy to the interest rate spread); specific parameterization discussed in subsequent sections.
- Central bank steady-state target interest rate: id = 4.5 percent annualized.
- Time discount factor β follows from id.
- Monetary policy rule (linearized, normal times): 
  - ̂idt = φπ ̂Πt − φω ̂ωt.
- Special analytical case considered with φω = (πb + sΩ) where the central bank fully neutralizes the effect of the sovereign risk premium on aggregate outcomes.
- ZLB treatment:
  - Nominal interest rate stuck at zero in initial period(s).
  - Monetary policy returns to Taylor rule in the next period with probability 1 − μ; otherwise ZLB persists.
  - Expected length of ZLB episode: 1/(1 − μ).
  - Shocks to time discount factor Γt follow same Markov structure.
  - Temporary increase in effective discount factor when at ZLB: 0 < et = eL < 1, so Γt = μ log(eL) − log(eL) = −(1 − μ) log(eL) ≥ 0.

### Fiscal and government debt calibration
- Steady-state government spending (consumption and investment) relative to GDP: g/y = 0.2.
- Steady-state gross public debt: 60 percent of annual GDP.
- Baseline distortionary tax rates set to zero; taxes adjust via lump-sum taxes.
- Taxes react to debt sufficiently strongly (φT,bg large enough) to ensure debt remains bounded.
- Sensitivity of taxes to output: φT,y = 0.34.
- Fiscal limit / sovereign risk premium calibration parameters chosen to match empirical relationship in Figure 2:
  - αbg = 3.70
  - βbg = 0.54
  - bg,max = 2.56
- Sovereign default haircut parameter: θdef = 0.5 (average value).
- Target steady-state normal spread between deposit and lending rates: 2.1 percent (annualized) — pins down χψ.

### Preferences, production, and other real-side calibration
- Curvature of disutility of work: ν = 1/1.9.
- Elasticity of demand: θ = 7.6 → gross price markup μp = 1.15.
- Aggregate hours worked in steady state: h = 1/3.
- Relative intertemporal elasticities and scaling parameters chosen so the linearized model maps to canonical three-equation New Keynesian form:
  - σb/σs = 0.53
  - ψb/ψs = 0.82
- Average intertemporal elasticity of substitution σ defined as:
  - σ := πb·(cb/y)·σb + πs·(cs/y)·σs (preserves log-utility case for representative household).
- Production parameters:
  - φ = 1 (linear production function).
  - Target steady-state output = 1 → pins down productivity z.
  - α = 0.9 → implied Phillips curve slope κ = 0.012.
    - Paper notes empirical estimates for slope in range 0.007 to 0.047 and a recent estimate of 0.014.

### Private debt and household types
- Target ratio of private debt to annual GDP: b/4y = 80 percent (Great Moderation U.S. average, nonfinancial nonmortgage nongovernment credit market debt).
- Households redraw type on average every 40 quarters:
  - δ = 0.975 → average time without asymptotic risk sharing ≈ 10 years.
- Share of borrowers in baseline calibration (πb):
  - Survey of Consumer Finances: 76 percent of U.S. families hold some kind of debt → πb = 0.76, πs = 0.24 (but primary residence loans may overstate borrowing importance).
  - Alternative savings-based metric: 57 percent report having saved on average → πb = 1 − 0.57 = 0.43.
  - Corporations account for 50 percent of private borrowing in nonfinancial nonmortgage nongovernment credit measure.
  - Baseline πb set as (1 − 0.17)·0.43 + 0.17·1 = 0.53.
    - Here 0.17 = share of nonresidential private domestic fixed and inventory investment in private domestic economic activity.
- Scaling parameters ξb and ξs determined by market clearing and the private debt target.

### Sovereign-to-private spillovers and αψ
- Figure 1 and cross-country evidence motivate a sovereign risk channel from sovereign spreads to household and corporate spreads.
- Baseline spillover parameter: αψ = 0.55 (based on Harjes (2011): a 100-basis-point increase in sovereign spreads raises private firms’ credit spreads by about 50 to 60 basis points).
- Rationale and caveats:
  - αψ = 0.55 may understate spillovers because:
    - Based on large, internationally active firms with better market access.
    - Smaller, more domestically financed firms likely experience stronger spillovers.
    - Comovement of spreads stronger in countries with intense fiscal strain.
  - Higher values of αψ are considered in simulations.

### Analytical special case — linearized IS and New Keynesian Phillips curve
- New Keynesian Phillips curve (linearized around deterministic steady state):
  - ̂Πt = β Et ̂Πt+1 + κy ̃yt − κg ̃gt,
    - where κy = κ(ν + ̄σ−1) and κg = κ ̄σ−1 with κ = (1 − α)(1 − αβ)/α.
    - Notation: ̃yt = yt − y, ̃gt = gt − g, ̂Πt = log(Πt/Π).
- Dynamic IS relationship:
  - ̃yt − ̃gt = Et ̃yt+1 − Et ̃gt+1 − ̄σ [ ̂idt + (πb + sΩ) ̂ωt − Et ̂Πt+1 + Γt ].
    - ̂ωt := log((1 + ωt)/(1 + ω)), ̂idt := log((1 + idt)/(1 + id)), Γt := Et log(et+1) − log(et).
- Degree to which private-sector spread affects activity for a given policy rate determined by πb + sΩ.
  - sΩ := πb πs (σb cb/y − σs cs/y) / ̄σ; in calibration cb > cs and sΩ > 0.
- Under calibration that maps to three-equation NK form, only current value of interest rate spread enters the dynamic IS and NK Phillips curve; evolution of output and inflation is independent of level of private debt in this special case.
- Special-case assumption for analytical tractability: probability of sovereign default (and sovereign risk premium) depends on expected primary deficit rather than level of debt. Postulate linear relationship:
  - ̂$ t = ξ Et (̃gt+1 − φT,y ̃yt+1),
    - where ̂$ t := (πb + sΩ) ̂ωt; ξ ≥ 0; φT,y ∈ [0,1).

### Quantifying parameter ξ (linking fiscal deficits to private spreads)
- Define ξ′ as slope of risk premium with respect to debt at a specific debt level (steady state).
- Derived expression for ξ′ (from sovereign spread specification):
  - ξ′ = αψ (πb + sΩ) θdef 1 − θdef Fbeta(bg/4y 1 / bg,max ; αbg, βbg) 1 / 4y 1 ̄bg,max fbeta(b/4y 1 / bg,max ; αb, βb).
- Table 1 reports ξ′ for different debt/GDP ratios and maps to ξ using accumulation over expected ZLB horizon.
- Mapping from ξ′ to ξ for empirically reasonable μ > 0.5 (ZLB expected ≥ 2 periods):
  - ξ = (1 + μ(1 − μ)) / (μ(1 − μ)) ξ′.
- Table 1 (described):
  - First column: ξ′ for different debt/GDP ratios using fiscal limit distribution calibration.
  - Subsequent columns: ξ for expected ZLB durations of 6, 7, and 8 quarters (computed via the formula above).
- Representative values from Table 1 (reported structure preserved in source):
  - Debt/GDP = 60 percent → ξ′ = 0.0005 → ξ = 0.0040, 0.0050, 0.005 (for 6, 7, 8 qtrs).
  - Debt/GDP = 90 percent → ξ′ = 0.0016 → ξ = 0.0140, 0.0150, 0.0170.
  - Debt/GDP = 110 percent → ξ′ = 0.0030 → ξ = 0.0250, 0.0280, 0.0310.
  - Debt/GDP = 130 percent → ξ′ = 0.0051 → ξ = 0.0420, 0.0470, 0.0520.
  - Debt/GDP = 140 percent → ξ′ = 0.0065 → ξ = 0.0540, 0.0600, 0.0660.
  - Debt/GDP = 150 percent → ξ′ = 0.0083 → ξ = 0.0680, 0.0760, 0.0840.
- Interpretation:
  - ξ′ values are fairly small, but ξ (which maps deficits accumulated over an expected ZLB horizon into spreads) can be substantially larger.
  - A value of ξ as high as 0.1 cannot be ruled out if initial debt is large and recessionary shock persists.
  - Larger αψ would scale up ξ′ and ξ linearly.

*Source: _wp1233 - 2.6  Calibration*

### 3.3  Sovereign risk channel and equilibrium determinacy

### 3.3  Sovereign risk channel and equilibrium determinacy

### Baseline assumption and main effect
- Assume government spending level is exogenously given.
- Sovereign risk channel significantly alters determinacy properties of the model while the ZLB is binding: the range of parameters that ensure determinacy shrinks in the presence of sovereign risk.

### Proposition 1 — determinacy conditions with exogenous spending
- Setup: economy summarized by equations(34)–(37). Interest rate = 0 in initial period. In each subsequent period, interest rate remains at zero with probability μ∈(0,1); otherwise monetary policy can permanently return to Taylor rule(36).
- There is a locally unique bounded equilibrium if and only if:
  - a) a < 1/(βμ), and
  - b) (1−βμ)(1−a) > μ ̄σκ_y,
- where a := μ + μ ξ φ_{T,y} ̄σ and κ_y := κ[ν + 1/ ̄σ].
- Remarks:
  - In absence of endogenous risk premium (ξ = 0), condition a) is always satisfied; determinacy hinges on condition b), which becomes (1−βμ)(1−μ) > μ ̄σκ_y.
  - The set of parameters satisfying condition b) is larger when: (i) smaller μ (less persistent ZLB), (ii) smaller ̄σ (lower interest sensitivity of demand), (iii) smaller κ_y (flatter Phillips curve).
  - With ξ > 0, condition a) may be violated if either the interest rate spread is sufficiently responsive to the deficit (ξ large) or if tax revenue is sufficiently responsive to output (φ_{T,y} large). The same parameters affect condition b).

### Endogenous government spending at the ZLB
- Contrast baseline with case where government spending adjusts endogenously to output while economy at ZLB.

### Proposition 2 — determinacy with endogenous spending response at ZLB
- Setup: in Proposition 1 economy, let ̃g_t = φ ̃y_t when at ZLB, and ̃g_t = 0 otherwise. Suppose φ < 1. Define:
  - a^∗ := μ + μ ξ φ^∗_{T,y} ̄σ^∗
  - κ^∗_y = κ_y − φ κ_g
  - φ^∗_{T,y} := φ_{T,y} − φ
  - ̄σ^∗ = ̄σ/(1 − φ)
- There exists a locally unique bounded equilibrium if and only if:
  1. with a^∗ > 0:
     - a) a^∗ < 1/(βμ), and
     - b) (1−βμ)(1−a^∗) > μ ̄σ^∗ κ^∗_y
  2. with a^∗ < 0:
     - a) (1 + βμ)(1 + a^∗) > −μ ̄σ^∗ κ^∗_y, and
     - b) (1−βμ)(1−a^∗) > μ ̄σ^∗ κ^∗_y.

- Insights:
  - If ξ = 0 (no sovereign risk), determinacy range is larger if spending is countercyclical (φ < 0).
  - If ξ > 0 and Item 1 conditions hold, then subject to φ_{T,y} < 1 − κν/((1−β μ) ξ), the determinacy region can be at least as large with a procyclical spending response φ∈(0,1) as without any response; it can even be larger.
  - This outcome is more likely when φ_{T,y} is smaller and ξ is larger.
  - Main unconventional conclusion: a procyclical fiscal stance may reduce the risk of equilibrium indeterminacy in the presence of sovereign risk.

### Mechanism and intuition
- Context: model where debt ultimately will be stabilized through one-off tax measures (so debt stabilizes though sovereign default can still occur).
- With endogenous risk premium and constrained central bank, economy can be prone to belief-driven equilibria.
- Systematic spending cuts during recessions (when ZLB binding) may anchor expectations to a unique equilibrium:
  - If agents expect output to drop, tax revenue falls, deficits rise, interest rate spread widens, real interest rate rises (since monetary policy cannot offset at ZLB), confirming adverse expectations → self-fulfilling downturn.
  - Procyclical spending cuts offset expected decline in tax revenue, preventing deficits from rising and avoiding the self-fulfilling spiral.

### Figure 3 summary (determinacy regions; qualitative description)
- Determinacy regions plotted for baseline parameterization across varying μ (expected ZLB durations of 6, 7, 8 quarters: μ = 5/6, 6/7, 7/8).
- Axes: vertical φ (response of government spending to output, ̃g_t = φ ̃y_t); horizontal ξ (response of interest rate spread to deficit).
- Grey areas = determinacy; white = indeterminacy.
- For protracted ZLB episodes with steep risk-premium slope (large ξ), there may be no countercyclical spending policy of this simple form that ensures determinacy.
- Procyclical spending cuts can help reduce risk of belief-driven equilibria but do not ensure determinacy for all ξ.

### Note on specification of risk premium
- Risk premium was modeled as depending on expected next-period deficit (equation (37)), consistent with forward-looking sovereign spreads.
- Quantitatively, results would be very similar if sovereign spread depended on current period deficit.

*Source: IMF Working Paper section 3.3*

### 4.1  Deep recessions and sovereign risk

### 4.1 Deep recessions and sovereign risk

### Model setup and calibration
- The model economy is solved under perfect foresight.
- The recessionary shock is modeled as a first-order autoregressive process for ê_t calibrated to capture the U.S. 2007–2009 output loss.
- Congressional Budget Office (2011) estimates cited in the text:
  - output gap reached 6.7 percent in 2009,
  - output gap at 1.7 percent in 2014,
  - output gap at 0.5 percent in 2015.
- AR process parameters used for the baseline economy (60 percent debt-to-GDP):
  - log(e_t) = ρ_e log(e_{t−1}) + u_t with u_0 = −0.1525 (u_t = 0 for all other periods) and ρ_e = 0.93.
- Tax response parameter for debt stabilization:
  - φ_{T,b_g} = 0.014 for the first 30 quarters,
  - beyond quarter 30, φ_{T,b_g} rises to twice the previous value.

### Nonlinear sovereign risk channel and ZLB interactions (Figure 5 insights)
- Simulated responses compared across three initial government debt levels: 60 percent of GDP (black solid), 90 percent (blue dashed), 115 percent (red dots).
- Key dynamics:
  - Decline in output and rise in interest rate spread are much stronger when initial debt is high.
  - Mechanism: the shock increases the budget deficit → public debt builds up → sovereign risk premium rises → private interest rates rise; with the economy at the ZLB, monetary policy cannot offset this spillover, leading to larger falls in private expenditure and output.
  - The relationship between public debt and sovereign risk is fundamentally nonlinear.
- ZLB timing:
  - A persistently high risk premium lengthens the time the ZLB remains binding.
  - The ZLB episode is extended by as much as 11 quarters if government debt is high.
- Additional insight: sovereign risk is not only amplified by monetary policy constraints, it can also determine how severely monetary policy is constrained in the face of certain shocks.

### 4.2 Fiscal retrenchment during deep recessions

### Retrenchment scenario definition
- Immediate retrenchment scenario:
  - Fiscal restraint starts at the onset of the recession and lasts for two years (8 quarters).
  - Government spending falls below its steady-state level by 2 percent of steady-state GDP during the episode.
- Comparative cases for ZLB exit timing (Figure 6):
  - Left panels: timing of exit from the ZLB is endogenous.
  - Middle panels: ZLB is fixed exogenously to last 7 quarters.
  - Right panels: ZLB is fixed exogenously to last 18 quarters.
  - The 7- and 18-quarter durations correspond to the durations of the ZLB episode for debt-to-GDP ratios of 60 percent and 115 percent in Figure 5.
- For the middle and right panels, monetary policy is assumed to keep the nominal rate at zero for the specified time period (7 and 18 quarters, respectively).

### Effects of spending cuts (8 quarters, −2 percent of ss GDP)
- Deficit and debt dynamics:
  - For all initial debt levels, the retrenchment package is effective in reducing the deficit (level of government debt declines).
  - The risk premium declines as debt falls.
  - Monetary policy escapes the ZLB constraint somewhat earlier for all debt levels (not shown in the figures).
- Output and private expenditure responses vary by initial debt:
  - If initial debt is low (60 percent), the spending cuts prompt an initial output decline of more than 2 percent, as private expenditure also falls.
  - If initial debt is high (115 percent), the initial output response is positive, reflecting a strong increase in private expenditure.
- Channels operating:
  - Channel 1: Deflationary effect of lower public spending raises real interest rates (cannot be offset by policy rate reduction at the ZLB).
  - Channel 2: Reduction of public debt reduces sovereign risk, lowering private interest rate spreads.
  - The sovereign risk channel strengthens with higher initial debt because of the nonlinear debt–risk relationship; in the simulations this channel dominates at 115 percent initial debt.
- Role of ZLB duration:
  - When the lower bound is expected to be binding only for a short period (7 quarters), the effect of spending cuts hardly varies with the debt level—so sovereign risk importance appears limited.
  - When the lower bound is expected to be binding for an extended period (18 quarters), spending cuts have a positive and lasting effect on economic activity at high initial debt levels (dotted red line, right panel).
  - The “output costs” of public spending cuts are noticeably lower than usual even for the intermediate 90 percent debt-to-GDP (dashed blue line in the rightmost panel).
- Overall conclusion from simulations:
  - Fiscal strain alters the fiscal transmission mechanism through the sovereign risk channel, provided that monetary policy is constrained for an extended period.

*Source: _wp1233 - 4.1 Deep recessions and sovereign risk*

### 4.3  Further considerations

### 4.3  Further considerations

### The timing of spending cuts
- Scenario definitions and numerical specifics:
  - Immediate retrenchment: spending cuts by 2 percent of steady-state GDP implemented immediately (left panels in Figure 7); immediate retrenchment for first 2 years.
  - Medium-term retrenchment: spending cuts of 2 percent of steady-state GDP that start two years after the initial recessionary impact and last for 10 years (years 3 through 12).
  - Persistent retrenchment: combination of immediate and medium-term packages; spending cuts start immediately and last for 12 years in total.
  - Initial debt-to-GDP ratios considered: 60%, 90%, 115%.
  - Timing of the exit from the ZLB is endogenously determined according to equation (36) in the model.
- Key findings:
  - Medium-term retrenchment produces immediate output gains via fiscal forward guidance: anticipated lower future real rates crowd in current consumption and output before cuts occur.
  - Medium-term retrenchment reduces sovereign risk premium immediately through higher expected tax revenue, stimulating demand via reduced interest rate spreads.
  - The expansionary upfront effect of medium-term retrenchment is stronger the weaker the initial fiscal situation (dotted red line vs dashed blue line in the central panels).
  - Persistent retrenchment achieves the largest reduction in the risk premium (reflecting larger overall cuts) but typically yields smaller output gains than medium-term retrenchment because immediate upfront cuts contract demand.
  - For 115% debt/GDP, persistent retrenchment still crowds in activity for most of the ZLB period but with a more subdued initial boost compared with medium-term retrenchment.
- Policy implication:
  - Delaying fiscal adjustment until after the economy has recovered from the worst of the initial recession is theoretically beneficial, provided announced future consolidation is credible. If credibility is lacking, expected deficits remain higher and sovereign risk premia will be higher, undermining the medium-term retrenchment strategy.

### Distortionary taxation
- Numerical settings and assumptions:
  - Baseline: lump-sum taxes used.
  - Alternative: labor taxes used throughout; steady-state labor tax rate fixed at 35 percent (in line with the 2006 average U.S. marginal income tax rate reported by Barro and Redlick (2011)); for scenarios with labor taxes set φ_T,b_g = 0.0193 for all periods.
  - A hybrid scenario: lump-sum taxes for the first 18 quarters, distortionary (labor) taxes thereafter.
  - Initial debt-to-GDP ratios considered: 60%, 90%, 115%.
- Key findings:
  - When distortionary labor taxes are used throughout, they have little net bearing on how spending cuts affect output: reduced future distortionary taxation stimulates future activity, but without retrenchment distortionary rates would have risen more steeply, which can raise inflation and help stabilize output at the ZLB (per Eggertsson (2011)); the net effect in simulations is that early retrenchment reduces the labor tax burden and can exert a negative effect on activity, outweighing future benefits.
  - In the hybrid scenario (lump-sum for first 18 quarters, labor taxes afterward):
    - For 60% debt/GDP, effects are similar to the baseline.
    - For 90% debt/GDP, retrenchment effects become more favorable, mostly affecting the medium-term retrenchment package.
    - For 115% debt/GDP, the sovereign risk channel may be stronger when changes in distortionary tax rates materialize only after the ZLB: higher required future labor tax increases to stabilize debt worsen the recession absent early retrenchment, so early retrenchment can crowd in output much more strongly than under lump-sum taxation.

### Degree of monetary policy offset for risk premia
- Baseline monetary policy assumption:
  - Once the economy has exited the ZLB, the central bank can set interest rates to prevent sovereign risk from affecting economic activity (i.e., full neutralization of sovereign risk premia).
- Alternative assumption:
  - Central bank response coefficient φ_ω reduced so the central bank’s response is only three-quarters the size of the benchmark calibration: φ_ω = 0.75(π_b + s_Ω).
  - Top-panel baseline comparisons use 60, 90, and 100 percent initial debt-to-GDP ratios; limited-neutralization bottom panels show effects when φ_ω is 3/4 of the benchmark.
- Key findings:
  - When the central bank cannot fully neutralize sovereign spreads, the baseline recession is deeper because of stronger headwinds from persistently high risk premia.
  - Under limited neutralization, early retrenchment is less detrimental to activity at high initial debt levels: reduced central bank offset raises the relative benefit of fiscal retrenchment in lowering long-term real rates via the sovereign risk channel.
- Policy implication:
  - The effectiveness and desirability of fiscal retrenchment depend crucially on the central bank’s capacity or willingness to offset sovereign risk effects on private borrowing costs post-ZLB.

### Overarching conclusions (from the closing section)
- Sovereign risk channel mechanisms and implications:
  - Rising sovereign risk raises private-sector borrowing costs unless offset by looser monetary policy.
  - If the central bank is constrained in counteracting higher risk premia, sovereign risk becomes a critical determinant of macroeconomic outcomes.
  - Sovereign risk increases vulnerability to indeterminacy: private-sector pessimistic beliefs can be self-fulfilling, raising risk premia and choking off demand; in such environments, a procyclical fiscal stance (tighter fiscal policy during downturns) can help ensure determinacy.
  - Sovereign risk tends to exacerbate negative cyclical shocks; it can delay exit from the ZLB and prolong macroeconomic distress.
  - Fiscal multipliers depend on (i) responsiveness of private-sector risk premia to fiscal strain indicators, and (ii) the expected duration of monetary-policy constraints (ZLB).
  - Upfront fiscal retrenchment is less detrimental to activity when sovereign risk is significant (multipliers are smaller) because lower deficits improve private financing conditions; in extreme cases with severe fiscal strains and prolonged ZLB, fiscal tightening can be expansionary relative to an even bleaker baseline.
- Caveats and policy considerations:
  - Fiscal retrenchment is not a cure-all: simulations still feature deep recessions even when retrenchment stimulates activity relative to the baseline.
  - Availability of external financial support at below-market rates could allow countries to spread out fiscal adjustment and benefit from lower funding costs and credibility effects; where credible announcements of future adjustment are feasible, delaying some spending cuts remains superior for protecting short-term growth.
  - How countries choose among these strategies may be a defining feature of global economic developments in the coming years.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1233.pdf*

### References

### _wp1233 - References

### References (bibliographic sources)
- Lists empirical and theoretical literature on fiscal policy, sovereign risk premia, monetary policy at the zero lower bound (ZLB), and related macroeconomic topics. Representative items include:
  - Empirical studies: Aizcorbe, Kennickell, and Moore (2003); Bucks et al. (2009); Congressional Budget Office (2011).
  - Sovereign spreads and default risk: Arellano (2008); Baldacci, Gupta, and Mati (2008); Baldacci and Kumar (2010); Durbin and Ng (2005); Panizza, Sturzenegger, and Zettelmeyer (2009).
  - Fiscal multipliers and fiscal contractions: Alesina and Ardagna (2010); Giavazzi and Pagano (1990); Christiano, Eichenbaum, and Rebelo (2011); Woodford (2011).
  - Liquidity trap / ZLB analyses: Benhabib, Schmitt-Groh́e, and Uribe (2002); Eggertsson (2011); Eggertsson and Woodford (2003); Ćurdia and Woodford (2009).
  - Sovereign risk and macro links: Juessen, Linnemann, and Schabert (2011); Laubach (2009); Mendoza and Yue (2010); Yeyati and Panizza (2011).
- Additional working papers, IMF reports, ECB Financial Stability Review (June 2010), and OECD working papers are cited.

### Appendix A — Three-equation representation of the aggregate economy
- Presents a reduced-form three-equation system (IS curve, New Keynesian Phillips curve, Taylor rule) that describes dynamics of output, inflation, and interest rates when certain simplifying conditions hold.
- IS curve (linearized) includes terms:
  - ̂y_t = ̂y_{t+1} −( ̃g_{t+1} − ̃g_t ) − [ Ξ( ̃b_{t+1} − ̃b_t ) + b( ̃Ξ_{t+1} − ̃Ξ_t ) ] − σ[ ̂i^d_t − ̂Π_{t+1} + ̂Γ_t + (π_b + s_Ω)̂ω_t − [ψ_Ω + s_Ω(1− ̄δ)] ̂Ω_{t+1} ], with ̂Ω_t = ̂ω_t + ̄δ ̂Ω_{t+1}.
- New Keynesian Phillips curve:
  - ̂Π_t = β ̂Π_{t+1} + κ[ (φ(1 + ν) + 1/̄σ − 1)̂y_t − φ(1 + ν)̂z_t − 1/̄σ[ ̃g_t + Ξ ̃b_t + b ̃Ξ_t ] − ̂τ^w_t + [ s_Ω + π_b − γ_b ] ̂Ω_t ].
- Taylor rule:
  - ̂i^{d,*}_t = φ_Π ̂Π_t − φ_ω ̂ω_t.
  - ̂i^d_t = max{ ̂i^{d,*}_t, −(1 + i^{d,*}) }.
- The system is valid independently of private debt if the following four conditions hold exactly:
  1. Ξ = 0 (no aggregate resource costs of loan origination).
  2. [ψ_Ω + s_Ω(1 − ̄δ)] = 0.
  3. ̂τ^w_t = 0 (distortionary taxes do not move or are nonexistent).
  4. [s_Ω + π_b − γ_b] = 0.
- The parameters and laws of motion chosen in the paper satisfy these conditions.

### Appendix B — Linking ξ and ξ′ (derivation supporting equation (39))
- Provides back-of-the-envelope derivation linking two measures of the spread response to debt, ξ and ξ′.
- Key steps and expressions:
  - At ZLB: ̃y_t = E_t ̃y_{t+1} − ̄σ ̂$ _t + ....
  - Iteration gives: ̃y_t = − ̄σ E_t{ Σ_{j=0}^∞ ̂$_{t+j} I(economy at ZLB in period t+j) }.
  - If ̂$_{t+j} = ξ′ E_{t+j} ̃b^g_{t+j+1} then ̃y_t = − ̄σ ξ′ E_t{ Σ_{j=0}^∞ I(economy at ZLB in period t+j) E_{t+j} ̃b^g_{t+j+1} }.
  - With assumptions (primary deficit constant at ZLB and E_{t+j} ̃b^g_{t+j+1} = (j + 1 + μ) deficit) one obtains:
    - ̃y_t = − ̄σ ξ′ deficit Σ_{j=0}^∞ μ^j (j + 1 + μ)
    - Equivalent closed-form: ̃y_t = − ̄σ ξ′ 1 + μ(1−μ) (1−μ)^2 deficit.  (Equation (43))
  - Analytical model in Section 3 yields:
    - ̃y_t = − ̄σ ξ Σ_{j=0}^∞ μ^j deficit μ = − ̄σ ξ μ/(1−μ) deficit.  (Equation (44))
  - Comparing (43) and (44) yields equation (39) in the main text:
    - ξ = [1 + μ(1−μ)] / [μ(1−μ)] ξ′.

### Appendix C — Proofs of propositions (determinacy and dynamics)
- C.1 Proof of Proposition 1:
  - Linear system z_{t+1} = A z_t with z_t = [ ̃y_t; ̂π_t ] and A specified.
  - Blanchard-Kahn conditions applied. Determinacy requires matrix A have two roots outside unit circle.
  - Determinacy corresponds to case conditions a) and b) in the proposition; explicit forms derived from det(A) and tr(A):
    - det(A) = 1 / (a μ β) and tr(A) = 1 / (a μ β) [ μ β + ̄σ μ κ_y + a ].
- C.2 Proof of Proposition 2:
  - Similar analysis with starred parameters a^*, ̄σ^*, κ^*_y.
  - If a^* > 0 then det(A) > 0 → Case I applies; conditions a) and b) correspond to Case I requirements.
  - If a^* < 0 then det(A) < 0 → Case II conditions apply; proposition lists those conditions.
- C.3 Proof of Proposition 3:
  - Markov structure with constant values during constrained periods: y_L, π_L, g_L.
  - IS and Phillips equations solve to:
    - y_L = θ_r [ log(1 + i^d) − Γ ] + θ_g g_L,
    - with θ_r > 0 and θ_g expressions as in the proposition (denominator positive under determinacy per condition b) of Proposition 1).

### Appendix D — Two corollaries (comparative statics and multiplier properties)
- D.1 Corollary 4 (effects of spending cyclicality and endogenous risk premium ξ):
  1. If ξ = 0 (no endogenous risk premium):
     - The parameter range for determinacy is larger if government spending is countercyclical (φ < 0) than acyclical.
     - The more negative φ, the larger the range implying determinacy.
  2. If ξ > 0:
     - The range for determinacy is often larger if government spending is procyclical (spending cut during deep recession).
     - More precisely: Under Proposition 2 conditions, if a^* > 0, φ ∈ (0,1), and φ_{T,y} < 1 − κ ν / [ (1−β μ) ξ ], then determinacy range is at least as large as without endogenous spending response and can be larger.
     - This case is more likely the smaller φ_{T,y} and the larger ξ.
- D.2 Corollary 5 (government spending multiplier θ_g and self-financing):
  1. θ_g > 0 iff (1−μ) − μ ̄σ κ_g / (1−β μ) > μ ξ ̄σ.  (Equation (45))
     - Thus θ_g can be negative if ξ is large enough.
  2. Government spending at the lower bound is self-financing if θ_g > 1 / φ_{T,y}.
     - This holds if φ_{T,y} > (1−β μ)(1−μ) − μ ̄σ κ_y  divided by (1−β μ)(1−μ) − μ ̄σ κ_g.
     - The cutoff is independent of ξ.
  3. If ξ = 0 and determinacy conditions hold, θ_g > 1 (consistent with Christiano et al. (2011) and Woodford (2011)).
  4. If ξ > 0, θ_g > 1 if φ_{T,y} > 1 − κ ν / [ ξ(1−β μ) ].

### Appendix E — Delayed fiscal retrenchment (fiscal consolidation implemented after exit from ZLB)
- Analytical setup: ̃g_t = 0 when interest rate constrained at 0; at first period after constraint ceases, ̃g_t = g_a < 0 (austerity). Thereafter, government spending remains at austerity value with probability % ∈ [0,1) each period, otherwise returns to 0 permanently.
- Proposition 6 (output at ZLB while fiscal austerity scheduled after exit):
  - Under determinacy, while interest rates are constrained to zero, output y_L equals:
    - y_L = 1/d [ ̄σ(1−β μ)[ log(1 + i^d) − Γ ] + (1−μ)(1−β μ)(1 + ̄σ ξ φ_{T,y})(y_a − g_a) + ̄σ(1−μ) π_a − (1−μ)(1−β μ) ̄σ ξ(1−φ_{T,y}) g_a ],
    - where d = (1−β μ)[1 − a] − ̄σ μ κ_y, a := μ + μ ξ φ_{T,y} ̄σ, and y_a, π_a denote output and inflation in the austerity period:
      - y_a = (1−%)(1−β%) + ̄σ(φ_π − %) κ_g  divided by (1−%)(1−β%) + ̄σ(φ_π − %) κ_y times g_a.  (Equation (47))
      - π_a = (1−%)(κ_y − κ_g) divided by (1−%)(1−β%) + ̄σ(φ_π − %) κ_y times g_a.  (Equation (48))
- Corollary 7 (implications of delayed retrenchment):
  1. If ξ = 0:
     - Retrenchment that takes effect after monetary policy exits the ZLB boosts economic activity in the ZLB phase unless too much of retrenchment is expected to occur soon after exit.
     - There exists a value of % ∈ [0,1) such that y_L > 0 if g_a < 0. The condition holds for any % > [1 + φ_π (β μ − 1)] / (β μ).
  2. If φ_{T,y} > 0 and determinacy holds:
     - If future austerity would boost y_L when ξ = 0, then the magnitude of the boost is larger when the sovereign risk channel (ξ > 0) operates; the effect increases with ξ.
  3. There exist parameterizations where future austerity raises y_L if ξ > 0, but lowers y_L if ξ = 0.
- Interpretation:
  - Anticipated fiscal consolidation after the ZLB can lower expected future short-term nominal and real rates, reducing long-term rates immediately and potentially crowding in demand while at the ZLB.
  - Timing and persistence of consolidation (% parameter) are crucial: too rapid scheduled consolidation can perversely raise real rates during the constrained period and hamper recovery.
  - Sovereign risk channel (ξ) amplifies crowding-in effects from future retrenchment and can alter the sign of the impact relative to the ξ = 0 case.

*Source: _wp1233 - References (PDF chapter/section).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1233.pdf_