## _wp13227

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---

### 2.1 Evidence on the sovereign risk channel
- CDS and corporate spillovers:
  - From 2010 through mid-2012, sovereign 5-year CDS spreads rose markedly in the stressed economies but fairly little elsewhere; corporate credit spreads in stressed economies exhibited the same divergence.
  - As financial stress abated in the second half of 2012, sovereign spreads in the stressed economies fell and corporate credit spreads fell as well.
  - Daily correlation between sovereign and corporate CDS spreads:
    - stressed economies: 0.83
    - rest of the euro area: 0.34
  - Caveat: corporate sample likely understates spillovers because it includes only the largest firms with good capital market access; smaller firms reliant on domestic banks tend to face much tighter funding during domestic stress.
- Empirical pass-through estimates:
  - Neri (2013): April 2010–end-2011 sovereign spreads led to increases in borrowing costs of:
    - nonfinancial firms: 130 basis points
    - households: 60 basis points
  - Zoli (2013) for Italy: some 50-60 percent of the increase in sovereign spreads transmitted to firms’ borrowing rates within six months.
- Fiscal and monetary backdrop (mid-2012 vantage):
  - Fiscal: since 2010 both stressed and other euro area economies adopted a procyclical fiscal stance with substantial real spending cuts; fiscal policy since 2010 much tighter than usual and tight stance expected to persist, particularly in stressed economies.
  - Monetary: ECB policy rates cut to record-low levels; Eonia basically zero by mid-2012; market-based Eonia forward rates as of July 31, 2012 implied overnight rates expected at, or very close to, zero for at least another eight to ten quarters.
  - Corsetti et al. (2013): episodes of ultra-low interest rates may persist for a significant time in presence of elevated sovereign risk.
- Nonstandard measures and partial pooling (ESM and OMT):
  - Timeline:
    - July 26, 2012: Draghi “to do whatever it takes to preserve the euro”
    - September 2012: formal announcement of OMT
    - October 8, 2012: ESM inauguration
  - ESM design/scale:
    - upfront capital contributions: EUR 80 billion
    - maximum potential lending capacity (EFSF + ESM): EUR 700 billion
    - EUR 700 billion ≈ 7 percent of euro area GDP, or 22 percent of the stressed economies’ GDP
  - OMT design/scale:
    - potentially open-ended ECB purchases of government paper with remaining maturity 1 to 3 years (limited only by amount outstanding in that bracket)
    - potential bond purchases for stressed economies alone exceed EUR 640 billion or 20 percent of their GDP
    - ECB clarified OMT-acquired claims will rank pari passu with other creditor claims
  - Illustrative pooling calculations (mid-2012):
    - IMF WEO April 2012: general government debt (end-2012) at 109.0 percent of GDP (stressed) and 80.5 percent of GDP (rest)
    - Contingent liabilities (Arslanalp and Liao, 2012): add 17 percentage points of GDP (stressed) and 12 percentage points of GDP (other)
    - Assumed ECB capital shares: stressed 37 percent, other 63 percent
    - ESM potential exposures for other countries: EUR 440 billion (63 percent of EUR 700 billion) ≈ 7 percent of the group’s GDP
    - Debt eligible for OMT in stressed economies: about EUR 640 billion
    - Full OMT purchases could raise exposure of rest of euro area by about EUR 400 billion (63 percent of total exposure) ≈ 6.3 percent of its GDP
    - Extreme full write-down scenario: stressed economies’ liabilities would fall by:
      - 13.6 percent of GDP for the ESM
      - 12.5 percent of GDP under OMT
- Table 1 highlights (debt maturing 2014–2015; Bloomberg as of Feb. 4, 2013):
  - Rest of the euro area totals:
    - Billions of euro: 699.4
    - Percent of total debt: 20.2
    - Percent of 2012 GDP: 11.0
  - Stressed economies totals:
    - Billions of euro: 643.6
    - Percent of total debt: 21.3
    - Percent of 2012 GDP: 20.0
  - Select entries:
    - France: 253.1; 18.6; 12.2
    - Germany: 243.0; 21.8; 9.2
    - Italy: 369.2; 22.6; 23.5
    - Spain: 199.9; 27.0; 18.8
- Table 2 (government debt end-2012, percent of GDP) under pooling assumptions:
  - no pooling (before pooling): Rest of the euro area 92.5; Stressed economies 126.0
  - ESM only: Rest of the euro area 99.4; Stressed economies 112.4
  - OMT only: Rest of the euro area 98.8; Stressed economies 113.4
  - ESM and OMT: Rest of the euro area 105.8; Stressed economies 99.9
- Interpretation: ESM/OMT could materially alter liability distribution; illustrative “debt pooling” equalizing indebtedness is within program scope but extremely unlikely under current institutions.

### Modeling implications (preview)
- Model embeds sovereign risk channel: intermediaries’ lending costs to residents rise with sovereign funding costs so that sovereign risk increases private-sector borrowing costs and depresses activity.
- Setup: two-country monetary union (Home and Foreign) with sticky prices, CW-style borrowers and savers, competitive intermediaries funding through deposits, country risk in lending markets, and limited cross-country risk sharing via household mobility assumptions.

---

### 3.1 Households — structure, assets, and first-order conditions
- Household types, mobility, and pooling:
  - Types: borrowers (superscript b) and savers (superscript s)
  - Fraction θ∈(0,1) reside in Home; 1−θ in Foreign
  - Each period fraction (1−δ) change type; transfers ensure type changers have same wealth and same ex-ante marginal utility
  - After transfers, redraw location and type: probability θ of Home; conditional on location, probability πb of borrower; πs = 1−πb of saver
  - Assumption: complete pooling of assets within households of a particular location and type
- Preferences, labor supply, and consumption bundle:
  - Lifetime utility: E0 Σt=0^∞ β^t [uτ(ctτ, hτt)], β∈(0,1)
  - Period utility: uτ(ctτ, hτt) = ξτ ctτ^(1−1/στ)/(1−1/στ) − ψτ hτt^(1+ν)/(1+ν)
  - Composite consumption: ct = cH,t^θ cF,t^(1−θ) θ^θ (1−θ)^(1−θ)
  - CES intermediate bundles with elasticity parameter μ>1
  - Price indices and terms of trade: Pt = PH,t^θ PF,t^(1−θ); τt = PH,t / PF,t
- Asset holdings, sovereign haircut transfers, and wealth dynamics:
  - Savers can hold domestic government bonds or one-period risk-free deposits with union-wide intermediaries; borrowers obtain funds subject to country-specific spreads
  - Beginning-of-period combined nominal per-capita wealth of Home savers (before type changes):
    - As−_t = Sp_{t−1}(1 + i_d_{t−1}) + (1−θ_t) B_g_{t−1} (1 + i_g_{t−1}) + Tc_t. (equation (4))
  - Government haircut indicator θ_t = 0 if government honors debt; θ_t = θ_def ∈(0,1) if partial default
  - Transfers compensating savers in case of sovereign default:
    - Tc_t = θ_t B_g_{t−1} (1 + i_g_{t−1}). (equation (5))
  - Beginning-of-period combined nominal per-capita debt of Home borrowers:
    - Ab−_t = Bp_{t−1} (1 + i_b_{t−1}). (equation (6))
  - Beginning-of-period wealth of redraw pool:
    - A†_t = θ Bg_{t−1}(1 + i_g_{t−1}) + (1−θ) Bg*_{t−1}(1 + i_g*_{t−1})
  - End-of-period combined wealth of Home savers (after type changes):
    - Sp_t + Bg_t = δ_h Sp_{t−1}(1 + i_d_{t−1}) + Bg_{t−1}(1 + i_g_{t−1}) − π_s Xs_t + π_s (1−δ) A†_t. (equation (7))
    - Xs_t = Pt c_s_t − wt Pt h_s_t − Df_H,t − Dint_t + Tg_t
  - End-of-period combined debt of Home borrowers:
    - Bp_t = δ Bp_{t−1}(1 + i_b_{t−1}) + π_b Xb_t − π_b (1−δ) A†_t. (equation (8))
- Marginal utilities, Euler equations, and labor supply:
  - Marginal utilities:
    - λs_t = ∂u_s(c_s_t, h_s_t)/∂c_s_t (equation (9))
    - λb_t = ∂u_b(c_b_t, h_b_t)/∂c_b_t (equation (10))
  - Home savers’ Euler (two equivalent forms):
    - λs_t = β E_t [ (1 + i_d_t)/Π_{t+1} { δ λs_{t+1} + (1−δ)[ θ(π_b λb_{t+1} + π_s λs_{t+1}) + (1−θ)(π_b λb*_{t+1} + π_s λs*_{t+1}) ] } ]. (equation (11), first line)
    - λs_t = β E_t [ (1−θ_{t+1})(1 + i_g_t)/Π_{t+1} { δ λs_{t+1} + (1−δ) ̄λ_{t+1} } ]. (equation (11), second line)
  - Home borrowers’ Euler:
    - λb_t = β E_t [ (1 + i_b_t)/Π_{t+1} { δ λb_{t+1} + (1−δ) ̄λ_{t+1} } ]. (equation (12))
  - Labor supply:
    - h_s_t = [ λs_t / (ψ_s w_t) ]^(1/ν). (equation (13))
    - h_b_t = [ λb_t / (ψ_b w_t) ]^(1/ν). (equation (14))
    - Average labor: h_t = π_b h_b_t + (1−π_b) h_s_t = [ Λ_t / ψ w_t ]^(1/ν). (equation (15))
  - Average marginal utility: λ_t = π_b λb_t + (1−π_b) λs_t. (equation (17))

---

### Equilibrium determinacy, fiscal response, and fiscal multipliers (Propositions and implications)
- Proposition 1 (exogenous government spending in both countries; monetary policy per assumption 6):
  - Locally unique bounded equilibrium iff:
    - a) a < 1/(βμ), and
    - b) (1−βμ)(1−a) > μ ̄σκ_y,
  - where a := μ + μ̄ ξ φ_{T,y} ̄σ.
  - Implications:
    - Sovereign risk channel can undermine determinacy when average interest rate spread is sufficiently responsive to government deficit (large ̄ξ and hence large a).
    - Determinacy depends on average slope of the risk premium in the union; pooling can reduce ̄ξ and move from indeterminate to determinate.
- Proposition 2 (symmetric area-wide fiscal response ̃g_t = φ ̃y_t; each country same policy; φ < 1):
  - Define a^* := μ + μ̄ ξ φ^*_{T,y} ̄σ^*; φ^*_{T,y} := φ_{T,y} − φ; ̄σ^* = ̄σ/(1−φ); κ^*_y = κ_y − φ κ_g.
  - Locally unique bounded equilibrium conditions differ depending on sign of a^* (see full statement).
  - Fiscal-response insights:
    - Without sovereign risk (̄ξ = 0), countercyclical area-wide spending (φ < 0) enlarges determinacy region.
    - With endogenous risk premium (̄ξ > 0), procyclical spending (φ ∈ (0,1)) can under certain conditions enlarge determinacy region relative to no fiscal response—intuition: when monetary policy cannot offset widening spread (ZLB), procyclical fiscal stance that offsets anticipated falls in tax revenue can prevent self-fulfilling adverse expectations.
- Proposition 3 (asymmetric fiscal responses; area-wide government spending fixed ̃g_t = 0):
  - Define a^** := μ + ̄σ μ(̄ξ φ_{T,y} − θ ξ_D Δφ).
  - Locally bounded equilibrium exists under conditions dependent on sign of a^** (see full statement).
  - Corollary 6 (central implication):
    - If ξ_D := ξ − ξ^* > 0 (Home spread more responsive to deficit than Foreign), holding area-wide spending constant, determinacy is more likely if fiscal austerity is implemented in Home (higher-spread region) combined with fiscal expansion in Foreign (lower-spread region).
- Fiscal multiplier channels and key results:
  - Channels:
    1. Direct addition to aggregate demand (nominal rigidities amplify).
    2. Higher output raises marginal costs and inflation; at the ZLB real rates decline and private expenditure rises (large multiplier).
    3. Sovereign risk channel: higher government spending may raise deficits and spreads, depressing private expenditure and offsetting multiplier.
  - Union-wide symmetric impulse: multiplier may approach zero or turn negative if sovereign risk very high at union level (high ̄ξ) and expected ZLB duration not too short.
  - Asymmetric fiscal impulse (Proposition 4):
    - With Home expansion g_L and Foreign contraction g^*_L = − θ/(1−θ) g_L (area-wide spending unchanged) and interest rate fixed:
      - ̃y_L = − (̄σ μ θ ξ_D)/(1 − a − ̄σ μ κ_y/(1−βμ)) g_L.
    - If ξ_D > 0, Home expansion with offsetting Foreign contraction reduces area activity via higher local risk premia even though area-wide spending unchanged.
    - Reverse policy (cuts in higher-spread country; expansion in lower-spread country) can raise aggregate activity without changing area-wide spending.
  - Country-specific fiscal impulse (Proposition 5):
    - With Home g_L and Foreign 0, during constant-interest period:
      - ̃y_L = [1 − μ − ̄σ μ κ_g/(1−βμ) − μ ̄σ ξ/(1 − μ − μ̄ ξ φ_{T,y} ̄σ − ̄σ μ κ_y/(1−βμ))] θ g_L.
    - Effect of unilateral fiscal impulse depends on country-specific (ξ) and area-wide (̄ξ) sovereign risk; effect on area-wide activity declines with extent of sovereign risk in country undertaking expansion (ξ).

---

### Model simulations and calibration (eurosystem baseline)
- Geographic mapping: Home = stressed economies; Foreign = rest of the euro area; Home share θ = 1/3 of union (GDP weight).
- Time unit: one quarter.
- Baseline steady-state and fiscal targets:
  - Ratio of government debt to (annual) GDP across both parts: 60 percent
  - Government spending set at 20 percent of each country’s GDP
- Key calibration parameter values:
  - θ = 0.333
  - θ_def = 0.550
  - α = 0.925
  - α_b_g = 3.702
  - β = 0.991
  - β_b_g = 0.539
  - b_g,max = 2.559
  - δ = 0.950
  - π_b = 0.500
  - π = 1.005
  - μ_p = 1.150
  - ψ = 1.006
  - ψ_b = 1.829
  - ψ_s = 0.760
  - ν = 0.526
  - σ_b = 0.693
  - σ_s = 1.307
  - ̄σ = 0.8
  - φ_{T,y} = 0.500
  - φ_Π = 1.500
  - φ_ω = 0.500
  - ω steady-state target (deposit-lending spread) = 2.5 percent
  - i_d steady-state central bank target interest rate = 4.5 percent
  - z = 3.000
  - ξ_b = 0.296
  - ξ_s = 0.104
  - α_ψ = 0.550
- Sovereign risk and fiscal-limit parameterization:
  - Haircut θ_def = 0.55
  - Fiscal limit parameters: α_b_g = 3.70, β_b_g = 0.54, b_g,max = 2.56
  - Spillovers from sovereign spreads to private credit spreads: α_ψ = 0.55
- Monetary policy and ZLB:
  - Baseline: interest rate fixed in initial period
  - Policy rate expected constant for 9 quarters in baseline (consistent with Eonia OIS as of July 2012)
- Other calibration targets:
  - Share of borrowers π_b = 0.5
  - Target ratio private debt to annual GDP b/4y = 130 percent (1999–2007 average)
  - Households redraw type on average every 40 quarters (δ = 0.95)
  - Disutility of work curvature ν = 1/1.9
  - Gross price markup μ_p = 1.15
  - Steady-state aggregate hours h = 1/3

---

### 5.2 Macroeconomic stability in the euro area — quantitative findings and policy implications
- Motivating question: could a sovereign crisis in the stressed economies (mid-2012) expose the entire union to belief-driven downturn?
- Baseline calibrations and counterfactuals:
  - Sovereign liabilities prior to pooling: 93 and 126 percentof GDP in the two regions
  - Illustrative equalized (pooled) debt stock: some 104 percentof GDP
  - Maastricht-level counterfactual: 60 percentof GDP
  - Constant-interest-rate (ZLB) anticipated durations: 9 quarters (baseline) and 10 quarters
  - Debt in rest of euro area assumed in many exercises: 92.5 percentof GDP
  - Vertical reference: stressed-economies debt = 126.0 percentof GDP
- Fiscal and monetary interaction — major findings:
  - Determinacy depends on expected duration of constant-interest-rate episode and cyclical stance φ ( ̃g_t = φ ̃y_t):
    - With φ = 0: equilibrium determinate if expected constant-interest-rate duration = 9 quarters; indeterminate if duration = 10 quarters
    - Counterfactual with Maastricht debt (60 percentof GDP) yields determinacy for constant-interest-rate horizons up to 10 quarters
    - Sovereign risk under calibration moves frontier of indeterminacy forward by one quarter relative to Maastricht-debt case
  - Role of procyclical spending:
    - Strongly procyclical spending generates indeterminacy largely independently of liabilities in stressed economies
    - Baseline (9 quarters, stressed debt = 126 percentof GDP): equilibrium indeterminate for φ in excess of 0.4
  - Region-specific spending asymmetries:
    - If spending acyclical in rest-of-euro-area and endogenous in stressed economies:
      - Unique determinacy for stressed-economy debt below 130 percentof GDP, irrespective of stressed economies’ fiscal stance
      - For debt above 130 percentof GDP, countercyclical spending expansions in stressed economies can produce macro instability because high debt amplifies sovereign risk channel
    - Longer expected ZLB (10 quarters) worsens determinacy: at stressed-economy debt = 126 percentof GDP, only very procyclical austerity (spending falling at least one-for-one with output) appears able to rule out self-fulfilling crisis dynamics
  - Asymmetric fiscal stances (constant aggregate spending):
    - Determinacy region generally smaller; at high stressed-economy debt levels the combination most likely to restore determinacy is:
      - procyclical stance in stressed economies coupled with countercyclical stance in rest of euro area
    - Intuition: cuts in stressed economies reduce deficits and sovereign risk while expansion elsewhere offsets union-wide demand losses, supporting activity and tax revenues in stressed economies
- Risk pooling — major findings:
  - Nonlinearity: premia increase nonlinearly in debt so pooling can reduce premia in stressed economies by more than it raises premia elsewhere
  - Illustrative pooling to common debt = 104 percentof GDP implies reduction of sovereign risk premium in stressed economies by some 300 basis points relative to pre-pooling
  - Determinacy under pooling:
    - When pooled debt relatively low, determinacy ensured if union-wide spending countercyclical or at most mildly procyclical
    - At high pooled debt, determinacy requires strongly procyclical stance
    - Reversal points:
      - with 9 quarters ZLB: reversal occurs just below 120 percentof GDP
      - with 10 quarters ZLB: reversal occurs at 100 percentof GDP
    - Pooling to 104 percentof GDP (baseline) with 9 quarters ZLB does not materially reduce relevance of cyclical stance for macro stability relative to pre-pooling scenario
  - Implication: debt pooling alone would do little to eliminate threat of self-fulfilling downturns if union-wide fiscal stance remains largely procyclical
- Mechanism emphasized:
  - Rising government risk premia raise private-sector credit spreads; at ZLB public-to-private spillovers shift trade-off between consolidation and demand support and can generate self-fulfilling downturns
- Key numeric thresholds and preserved values:
  - Pre-pooling sovereign liabilities: 93 and 126 percentof GDP
  - Illustrative pooled debt: some 104 percentof GDP
  - Maastricht debt level: 60 percentof GDP
  - Debt threshold in asymmetric middle-row exercise: 130 percentof GDP
  - φ threshold at baseline (9 quarters, stressed debt = 126 percentof GDP) above which equilibrium indeterminate: 0.4
  - Constant-interest-rate durations considered: 9 quarters and 10 quarters
  - Reduction of sovereign risk premium under pooling to 104 percentof GDP: some 300 basis points
  - Reversal points for procyclical requirement under pooling:
    - just below 120 percentof GDP for 9 quarters
    - 100 percentof GDP for 10 quarters
  - Debt in rest of euro area assumed in many figures: 92.5 percentof GDP
  - Vertical dashed reference: stressed-economies debt = 126.0 percentof GDP
- Policy-relevant implications:
  - Avoid strong procyclicality across union when central bank faces extended ZLB
  - Countercyclical or mildly procyclical union-wide spending conducive to determinacy when pooled debt moderate
  - Coordinated asymmetric stance (procyclical consolidation in high-debt/stressed economies combined with countercyclical expansion elsewhere) can restore macro stability under high debt
  - Debt pooling reduces premia in stressed economies substantially but is not sufficient by itself to eliminate risk of self-fulfilling downturns if union-wide fiscal stance remains procyclical
  - Longer anticipated ZLB durations materially worsen determinacy prospects and raise fiscal-behavior requirements to avoid indeterminacy

---

*Source: _wp13227*

### 2.1  Evidence on the sovereign risk channel

### 2.1  Evidence on the sovereign risk channel

### CDS evidence and spillovers to private borrowing
- From 2010 through mid-2012, sovereign 5-year CDS spreads rose markedly in the stressed economies but fairly little elsewhere; credit spreads for nonfinancial corporates in stressed economies exhibited the same divergence.
- As financial stress abated in the second half of 2012, sovereign spreads in the stressed economies fell and corporate credit spreads fell as well, consistent with strong spillovers from sovereign distress to private-sector financial conditions.
- Daily correlation between sovereign and corporate CDS spreads:
  - stressed economies: 0.83
  - rest of the euro area: 0.34
- The paper frames these spillovers as a “sovereign risk channel”: fears about potential sovereign default (including a disorderly exit) can cause a general retrenchment of private credit as lenders anticipate broad-based economic disruption.
- Caveat: Figure 1 likely understates spillovers because the corporate sample includes only the largest firms with good capital market access; smaller firms—more reliant on domestic banks—tend to face much tighter funding during domestic financial stress.

### Empirical estimates from the literature
- Neri (2013): between April 2010 and end-2011, sovereign spreads in crisis countries led to an increase in borrowing costs of:
  - nonfinancial firms: 130 basis points
  - households: 60 basis points
- Zoli (2013) for Italy: some 50-60 percent of the increase in sovereign spreads is transmitted to firms’ borrowing rates within six months.
- Additional supporting studies: Albertazzi and Signoretti (2012), Neri and Ropele (2013); literature reviews in Cavallo and Valenzuela (2007) and Harjes (2011).

### Fiscal and monetary backdrop relevant to the sovereign risk channel
- Fiscal policy shift since 2010: both stressed and other euro area economies adopted a procyclical fiscal stance with substantial real spending cuts during the downturn; Figure 2 shows fiscal policy since 2010 has been much tighter than usual and the tight stance is expected to persist, particularly in the stressed economies.
- Monetary policy: ECB policy rates were cut to record-low levels, driving overnight money market rates (Eonia) to basically zero by mid-2012; market-based Eonia forward rates as of July 31, 2012 implied overnight rates expected to be at, or very close to, zero for at least another eight to ten quarters.
- Corsetti et al. (2013) finding: episodes of ultra-low interest rates may persist for a significant time in the presence of elevated sovereign risk.

### Nonstandard measures and partial pooling of sovereign risk (ESM and OMT)
- Two policy innovations interpreted as partial pooling of debt: the ESM and the ECB’s OMT (both announced or foreshadowed by end-July 2012; formally established later in 2012).
- Timeline markers:
  - July 26, 2012: ECB President Draghi’s announcement “to do whatever it takes to preserve the euro” (foreshadowing a bond purchase program).
  - September 2012: formal announcement of OMT.
  - October 8, 2012: ESM inauguration.
- ESM design and scale:
  - upfront capital contributions: EUR 80 billion
  - maximum potential lending capacity (EFSF + ESM): EUR 700 billion
  - EUR 700 billion amounts to about 7 percent of euro area GDP, or 22 percent of the stressed economies’ GDP.
- OMT design and scale:
  - potentially open-ended ECB purchases of government paper with remaining maturity of 1 to 3 years (limited only by amount outstanding in that bracket).
  - Based on debt figures in Table 1, potential bond purchases for stressed economies alone exceed EUR 640 billion or 20 percent of their GDP.
  - ECB clarified OMT-acquired claims will rank pari passu with other creditor claims, implying a transfer of fiscal risks.
- Methodology for quantifying pooling (as of mid-2012 vantage point):
  - IMF WEO April 2012 projections: general government debt (end-2012) at 109.0 percent of GDP (stressed economies) and 80.5 percent of GDP (rest of the euro area).
  - Contingent liabilities related to potential financial sector support (Arslanalp and Liao, 2012): add 17 percentage points of GDP (stressed) and 12 percentage points of GDP (other economies).
  - Assumed ECB capital shares: stressed economies 37 percent, other economies 63 percent (used to allocate potential liabilities arising from ESM/OMT).
  - To determine hypothetical maximum pooling scope, assume ESM and ECB acquire exposures only to stressed economies.
- Illustrative calculations and magnitudes:
  - ESM potential exposures for other countries: EUR 440 billion (63 percent of EUR 700 billion) ≈ 7 percent of the group’s GDP.
  - Debt eligible for OMT in stressed economies: about EUR 640 billion (Table 1).
  - Under assumptions, full OMT purchases could raise the exposure of the rest of the euro area by about EUR 400 billion (63 percent of total exposure) ≈ 6.3 percent of its GDP.
  - In the extreme scenario of a full write-down on these exposures, stressed economies’ liabilities would fall by:
    - 13.6 percent of GDP for the ESM
    - 12.5 percent of GDP under OMT
- Table 1 (debt maturing in 2014 and 2015; Bloomberg as of Feb. 4, 2013; GDP from IMF):
  - Rest of the euro area totals:
    - Billions of euro: 699.4
    - Percent of total debt: 20.2
    - Percent of 2012 GDP: 11.0
  - Stressed economies totals:
    - Billions of euro: 643.6
    - Percent of total debt: 21.3
    - Percent of 2012 GDP: 20.0
  - Select country entries from Table 1 (examples):
    - France: 253.1 (Billions of euro), 18.6 (Percent of total debt), 12.2 (Percent of 2012 GDP)
    - Germany: 243.0, 21.8, 9.2
    - Italy: 369.2, 22.6, 23.5
    - Spain: 199.9, 27.0, 18.8
- Table 2 (government debt under different pooling assumptions; figures refer to end-2012, percent of GDP):
  - no pooling (before pooling): Rest of the euro area 92.5; Stressed economies 126.0
  - ESM only: Rest of the euro area 99.4; Stressed economies 112.4
  - OMT only: Rest of the euro area 98.8; Stressed economies 113.4
  - ESM and OMT: Rest of the euro area 105.8; Stressed economies 99.9
- Interpretation: under the illustrative assumptions, the potential scope of ESM and OMT could materially alter the distribution of government liabilities across the union; an illustrative “debt pooling” scenario assumes risk-sharing effectively equalizes indebtedness (relative to GDP) between the two parts of the union—a scenario within the potential scope of the programs analyzed, though noted as extremely unlikely under current institutions.

### Modeling implications (preview)
- The model to follow embeds the sovereign risk channel by making financial intermediaries’ lending costs to residents of a member state rise with sovereign funding costs, so that sovereign risk increases private-sector borrowing costs and depresses activity.
- The model adopts a two-country monetary union (Home and Foreign) with sticky prices, CW-style borrowers and savers, competitive intermediaries funding through deposits, country risk in lending markets, and limited cross-country risk sharing via household mobility assumptions. Variables are expressed per capita; foreign variables indexed with an asterisk.

*Source: _wp13227 - 2.1  Evidence on the sovereign risk channel*

### 3.1  Households

### 3.1  Households

### Household types, mobility, and pooling
- Households differ in preferences and are indexed by type and country of residence.  
- Types: borrowers (superscript b) and savers (superscript s).  
- At each point in time a fraction θ∈(0,1) of the population resides in Home, and the remaining 1−θ reside in Foreign.  
- In each period, a share (1−δ), δ∈(0,1), of household members change their type. Type changers receive transfers that depend on past type; transfers ensure all type changers have the same wealth and the same ex-ante marginal utility.  
- After transfers, type changers redraw location and type: probability θ of Home, 1−θ of Foreign; conditional on location, probability πb of borrower, πs = 1−πb of saver.  
- Assumption: complete pooling of assets within households of a particular location and type (“family”). The type-changing mechanism partially insures households across types by making changes in wealth levels temporary.

### Preferences, labor supply, and consumption bundle
- Individual household lifetime utility:
  - E0 Σt=0^∞ β^t [uτ(ctτ, hτt)], with β∈(0,1) and expectations over aggregate shocks and future type.
- Period utility functional form:
  - uτ(ctτ, hτt) = ξτ ctτ^(1−1/στ)/(1−1/στ) − ψτ hτt^(1+ν)/(1+ν), where ξτ, στ, ψτ, and ν are positive parameters.
- Composite consumption good ct for household τ:
  - ct = cH,t^θ cF,t^(1−θ) θ^θ (1−θ)^(1−θ).
- Home and Foreign intermediate good bundles (CES):
  - cH,t = [ (1/θ)^(1/μ) ∫_0^θ cH,t(j)^(μ−1)/μ dj ]^(μ/(μ−1)),
  - cF,t = [ (1/(1−θ))^(1/μ) ∫_θ^1 cF,t(j)^(μ−1)/μ dj ]^(μ/(μ−1)),
  - μ>1 is the price elasticity of demand for differentiated output goods.
- Law of one price: consumers in Home and Foreign pay the same price for the same good.
- Price indices:
  - PH,t = [ (1/θ) ∫_0^θ PH,t(j)^(1−μ) dj ]^(1/(1−μ)),
  - PF,t = [ (1/(1−θ)) ∫_θ^1 PF,t(j)^(1−μ) dj ]^(1/(1−μ)).
- Consumer price index:
  - Pt = PH,t^θ PF,t^(1−θ) (equation (2)).
- Terms of trade: τt = PH,t / PF,t (equation (3)).

### Asset holdings, sovereign haircut transfers, and wealth dynamics
- Savers can hold domestic government bonds or one-period risk-free deposits with union-wide intermediaries; borrowers obtain funds from intermediaries subject to country-specific spreads.  
- Beginning-of-period combined nominal per-capita wealth of Home savers (before type changes):
  - As−_t = Sp_{t−1}(1 + i_d_{t−1}) + (1−θ_t) B_g_{t−1} (1 + i_g_{t−1}) + Tc_t. (equation (4))
  - Sp_{t−1} = Home households’ deposits at intermediaries at end of previous period; i_d_{t−1} = deposit rate.
  - B_g_{t−1} ≥ 0 denotes domestic government debt.
- Government debt default specification for individual household: θ_t = 0 if government honors debt; θ_t = θ_def ∈(0,1) indicates haircut size when partially defaulting.
- Transfers compensating savers in case of sovereign default:
  - Tc_t = θ_t B_g_{t−1} (1 + i_g_{t−1}). (equation (5))
- Beginning-of-period combined nominal per-capita debt of Home borrowers:
  - Ab−_t = Bp_{t−1} (1 + i_b_{t−1}). (equation (6))
  - Bp_{t−1} = nominal private debt; i_b_{t−1} = Home borrowing rate.
- Beginning-of-period wealth of the pool of individuals selected to redraw characteristics (per capita):
  - A†_t = θ[Sp_{t−1}(1 + i_d_{t−1}) + Bg_{t−1}(1 + i_g_{t−1}) − Bp_{t−1}(1 + i_b_{t−1})]  
    + (1−θ)[Sp*_{t−1}(1 + i_d_{t−1}) + Bg*_{t−1}(1 + i_g*_{t−1}) − Bp*_{t−1}(1 + i_b*_{t−1})]  
    = θ Bg_{t−1}(1 + i_g_{t−1}) + (1−θ) Bg*_{t−1}(1 + i_g*_{t−1}).
- End-of-period combined wealth of Home savers (after type changes):
  - Sp_t + Bg_t = δ_h Sp_{t−1}(1 + i_d_{t−1}) + Bg_{t−1}(1 + i_g_{t−1}) − π_s Xs_t + π_s (1−δ) A†_t. (equation (7))
  - Xs_t = Pt c_s_t − wt Pt h_s_t − Df_H,t − Dint_t + Tg_t.
  - wt = Home economy-wide real wage; Df_H,t = per-capita profits of goods-producing firms; Dint_t = profits of financial intermediaries; Tg_t = lump-sum taxes by Home government.
- End-of-period combined debt of Home borrowers (per capita):
  - Bp_t = δ Bp_{t−1}(1 + i_b_{t−1}) + π_b Xb_t − π_b (1−δ) A†_t. (equation (8))
  - Xb_t defined analogously to Xs_t.

### Marginal utilities, Euler equations, and labor supply
- Common marginal utility of real income for households of type τ (family pooling):
  - λs_t = ∂u_s(c_s_t, h_s_t)/∂c_s_t,  (equation (9))
  - λb_t = ∂u_b(c_b_t, h_b_t)/∂c_b_t.  (equation (10))
- Home savers’ Euler equations (intertemporal optimality) linking marginal utility to returns:
  - λs_t = β E_t [ (1 + i_d_t)/Π_{t+1} { δ λs_{t+1} + (1−δ)[ θ(π_b λb_{t+1} + π_s λs_{t+1}) + (1−θ)(π_b λb*_{t+1} + π_s λs*_{t+1}) ] } ]. (equation (11), first line)
  - Equivalent expression emphasizing government debt return:
    λs_t = β E_t [ (1−θ_{t+1})(1 + i_g_t)/Π_{t+1} { δ λs_{t+1} + (1−δ) ̄λ_{t+1} } ]. (equation (11), second line)
  - Home borrowers’ Euler:
    λb_t = β E_t [ (1 + i_b_t)/Π_{t+1} { δ λb_{t+1} + (1−δ) ̄λ_{t+1} } ]. (equation (12))
  - Here ̄λ_{t+1} := θ[π_b λb_{t+1} + π_s λs_{t+1}] + (1−θ)[π_b λb*_{t+1} + π_s λs*_{t+1}].
- Optimal labor supply for each type in Home:
  - h_s_t = [ λs_t / (ψ_s w_t) ]^(1/ν). (equation (13))
  - h_b_t = [ λb_t / (ψ_b w_t) ]^(1/ν). (equation (14))
- Average labor supply across household types:
  - h_t = π_b h_b_t + (1−π_b) h_s_t = [ Λ_t / ψ w_t ]^(1/ν). (equation (15))
  - where Λ_t := ψ [ π_b (λb_t / ψ_b)^(1/ν) + π_s (λs_t / ψ_s)^(1/ν) ]^ν. (equation (16))
  - ψ^(−1/ν) = π_b (ψ_b)^(−1/ν) + π_s (ψ_s)^(−1/ν).
- Average marginal utility of real income in Home:
  - λ_t = π_b λb_t + (1−π_b) λs_t. (equation (17))

*Source: IMF Working Paper, chapter/section 3.1 "Households" (content unit: _wp13227 - 3.1  Households).*

### Section 3.2 in Corsetti et al. (2013) explicitly shows howξrelates to the debt level.

### _wp13227 - Section 3.2 in Corsetti et al. (2013) explicitly shows howξrelates to the debt level.

### Equilibrium determinacy
- Proposition 1 (exogenous government spending in both countries; monetary policy per assumption 6):
  - 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 + μ̄ ξ φ_{T,y} ̄σ.
- Key mechanisms and implications:
  - The sovereign risk channel can undermine equilibrium determinacy in a monetary union when the average interest rate spread is sufficiently responsive to the government deficit (i.e., when ̄ξ, and thus a, is large).
  - Determinacy depends on the average slope of the risk premium in the union: a high sovereign risk premium affecting a sufficiently large part of the union can have systemically destabilizing effects even if other parts do not face elevated risk premia.
  - The convexity of the risk premium in sovereign debt implies that pooling government liabilities across borders may reduce ̄ξ and move the aggregate economy from indeterminate to determinate equilibrium.
  - The degree to which debt pooling can substitute for a missing monetary response (e.g., when the central bank is at the ZLB) depends on how much pooling reduces ̄ξ.

- Proposition 2 (symmetric area-wide fiscal response ̃g_t = φ ̃y_t; each country same policy; φ < 1):
  - Define a^* := μ + μ̄ ξ φ^*_{T,y} ̄σ^*;
    - φ^*_{T,y} := φ_{T,y} − φ;
    - ̄σ^* = ̄σ/(1−φ);
    - κ^*_y = κ_y − φ κ_g.
  - A locally unique bounded equilibrium exists iff:
    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.
- Fiscal-response insights:
  - Without sovereign risk (̄ξ = 0), countercyclical area-wide spending (φ < 0) enlarges the parameter range for determinacy.
  - With endogenous risk premium (̄ξ > 0), procyclical spending (φ ∈ (0,1)) can, under certain conditions (including taxes not too elastic: φ_{T,y} < 1 − κ ν/(1−βμ) ̄ξ), enlarge the determinacy region relative to no fiscal response.
  - Intuition: when monetary policy cannot offset a widening spread (e.g., at the ZLB), a procyclical fiscal stance that offsets anticipated falls in tax revenue can prevent self-fulfilling adverse expectations that raise real rates and depress demand.

- Proposition 3 (asymmetric fiscal responses; area-wide government spending fixed ̃g_t = 0):
  - Government spending when interest rate held constant:
    - Home: ̃g_t = Δφ ̃y_t
    - Foreign: ̃g^*_t = − θ/(1−θ) Δφ ̃y^*_t
  - Define a^** := μ + ̄σ μ(̄ξ φ_{T,y} − θ ξ_D Δφ).
  - A locally bounded equilibrium exists iff:
    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.
  - Central implication (Corollary 6):
    - If ξ_D := ξ − ξ^* > 0 (spread more responsive to deficit in Home than Foreign), then, holding area-wide government spending constant, determinacy is more likely if fiscal austerity is implemented in Home (the higher-spread region) combined with fiscal expansion in Foreign (the lower-spread region).

### Fiscal multiplier
- Channels through which government spending affects activity:
  1. Direct addition to aggregate demand (nominal rigidities amplify).
  2. Higher output → higher marginal costs and inflation; at the ZLB (or constant nominal rates) real rates decline and private expenditure rises (large multiplier per Woodford 2011; Christiano et al. 2011).
  3. Sovereign risk channel: higher government spending may raise deficits and interest rate spreads, depressing private expenditure and offsetting the multiplier.
- Union-wide symmetric impulse:
  - Isomorphic to closed-economy case: multiplier may approach zero or turn negative if sovereign risk is very high at the union level (high ̄ξ) and the expected duration of the ZLB episode is not too short.
- Asymmetric fiscal impulses:
  - Proposition 4 (Home expands g_L; Foreign contracts g^*_L = − θ/(1−θ) g_L; both zero otherwise; determinacy per Proposition 1):
    - Define a := μ + μ̄ ξ φ_{T,y} ̄σ.
    - While interest rate fixed, output:
      - ̃y_L = − (̄σ μ θ ξ_D)/(1 − a − ̄σ μ κ_y/(1−βμ)) g_L.
    - If ξ_D > 0 (spread more responsive in Home), a Home expansion combined with offsetting Foreign contraction has a negative effect on area activity:
      - No direct aggregate demand effect (area-wide spending unchanged).
      - But deficits in Home raise local risk premia more, reducing consumption and lowering aggregate activity.
    - Reverse policy (cuts in higher-spread country; expansion in lower-spread country) can raise aggregate activity even with unchanged area-wide spending.
    - Policy implication: coordinated fiscal packages that internalize asymmetric elasticity of risk premia can be expansionary without changing area-wide spending.
- Country-specific fiscal impulse:
  - Proposition 5 (Home g_L; Foreign 0):
    - During constant-interest period, output:
      - ̃y_L = [1 − μ − ̄σ μ κ_g/(1−βμ) − μ ̄σ ξ/(1 − μ − μ̄ ξ φ_{T,y} ̄σ − ̄σ μ κ_y/(1−βμ))] θ g_L.
    - Interpretation:
      - Effect of unilateral fiscal impulse depends on country-specific (ξ) and area-wide (̄ξ) sovereign risk.
      - Denominator strictly falls in ̄ξ (for φ_{T,y} > 0 and determinacy ensured): larger average spread increases the effect of spending variation on area-wide activity because higher tax revenues reduce spreads.
      - For a given fiscal expansion, the effect on area-wide activity declines with the extent of sovereign risk in the country undertaking the expansion (ξ).

### Model simulations and calibration (eurosystem baseline)
- Purpose: quantitative analysis of euro area debt crisis under alternative spending and risk-pooling assumptions; results based on linear approximation of equilibrium conditions.
- Geographic mapping:
  - Home: stressed economies.
  - Foreign: rest of the euro area.
  - Home share θ = 1/3 of the currency union (GDP weight of stressed economies).
- Time period: one quarter.
- Baseline steady-state and fiscal targets:
  - Ratio of government debt to (annual) GDP across both parts: 60 percent.
  - Government spending set at 20 percent of each country’s GDP.
  - When higher initial debt considered, steady-state tax revenue, φ_{T,b_g}, and government spending adjusted accordingly.
- Key calibration choices and parameter values (Table 3 and text):
  - θ = 0.333
  - θ_def = 0.550
  - α = 0.925
  - α_b_g = 3.702
  - β = 0.991
  - β_b_g = 0.539
  - b_g,max = 2.559
  - δ = 0.950
  - π_b = 0.500
  - π = 1.005
  - μ_p = 1.150
  - ψ = 1.006
  - ψ_b = 1.829
  - ψ_s = 0.760
  - ν = 0.526
  - σ_b = 0.693
  - σ_s = 1.307
  - ̄σ (consumption-weighted average intertemporal elasticity) = 0.8 (text)
  - φ_{T,y} = 0.500
  - φ_Π = 1.500
  - φ_ω = 0.500
  - ω steady-state target (deposit-lending spread) = 2.5 percent (annualized)
  - i_d steady-state central bank target interest rate = 4.5 percent (annualized)
  - θ (Home share) = 1/3 (0.333)
  - z = 3.000
  - ξ_b = 0.296
  - ξ_s = 0.104
  - α_ψ = 0.550
  - μ (price markup related) and other derived symbols used in propositions are as in the model (preserved notation).
- Sovereign risk and fiscal-limit parameterization:
  - Haircut in sovereign default set to θ_def = 0.55 (50–60 percent haircut as reasonable average).
  - Fiscal limit parameters: α_b_g = 3.70, β_b_g = 0.54, b_g,max = 2.56 (following Corsetti et al. (2013)).
  - Spillovers from sovereign spreads to private credit spreads: α_ψ = 0.55 (per Harjes (2011)).
- Monetary policy timing and ZLB dynamics:
  - Baseline: interest rate fixed in initial period.
  - Policy rate expected to be kept constant for 9 quarters in baseline (μ set accordingly; consistent with Eonia OIS rates as of July 2012).
- Other calibration targets:
  - Share of borrowers π_b = 0.5.
  - Target ratio private debt to annual GDP b/4y = 130 percent (1999–2007 average).
  - Households redraw their type on average every 40 quarters (δ = 0.95).
  - Disutility of work curvature ν = 1/1.9.
  - Gross price markup μ_p = 1.15.
  - Steady-state aggregate hours h = 1/3.
  - For the two household types, parameters chosen so steady-state consumption of the two types is the same and to satisfy above targets.
- Calibration intention:
  - Parameters chosen to match euro area features and cross-country evidence on sovereign haircuts and CDS spreads.
  - Assumed taxes react sufficiently strongly to debt (φ_{T,b_g} large enough) so debt remains bounded and to rule out fiscal-theory-of-the-price-level equilibria.

*Source: Section 3.2 and related sections of Corsetti et al. (2013), as presented in the supplied content unit.*

### 5.2  Macroeconomic stability in the euro area

### 5.2  Macroeconomic stability in the euro area

### Motivating question and setup
- Primary question: whether, under conditions prevailing in mid-2012, a sovereign crisis in the stressed economies of the euro area could expose the entire monetary union to the risk of a belief-driven downturn.
- Key calibrations and counterfactuals used in the analysis:
  - Sovereign liabilities prior to any de facto pooling: 93 and 126 percentof GDP in the two regions, respectively.
  - Illustrative equalized (pooled) debt stock considered: some 104 percentof GDP.
  - Alternative counterfactual: Maastricht-level debt of 60 percentof GDP.
  - Constant-interest-rate (ZLB) anticipated durations examined: 9 quarters (baseline) and 10 quarters.
  - Debt in the rest of the euro area assumed in many exercises: 92.5 percentof GDP.
  - A vertical reference line in figures marks a stressed-economies debt level of 126.0 percentof GDP.

### Fiscal and monetary interaction — major findings
- Interaction sensitivity:
  - Determinacy of equilibrium depends on both the expected duration of the constant-interest-rate episode and the cyclical stance of government spending, φ, where  ̃g_t = φ ̃y_t and  ̃g*_t = φ ̃y*_t.
  - With no endogenous, region-specific spending response (φ = 0), the equilibrium is:
    - determinate if the expected duration of the constant-interest-rate episode is 9 quarters;
    - indeterminate if the expected duration is 10 quarters.
  - Counterfactual with Maastricht debt (60 percentof GDP) would yield determinacy for constant-interest-rate horizons up to 10 quarters.
  - Sovereign risk under the model calibration moves the frontier of indeterminacy forward by one quarter relative to the Maastricht-debt case.
- Role of procyclical spending:
  - Strongly procyclical spending stances generate equilibrium indeterminacy largely independently of liabilities in stressed economies.
  - Baseline with 9 quarters ZLB and stressed-economy debt = 126 percentof GDP: equilibrium indeterminate for φ in excess of 0.4.
  - Note: φ = 0.4 implies tax revenue is even more responsive to output than government spending in the simple version of the model.
- Region-specific spending asymmetries:
  - Case: spending acyclical in rest-of-euro-area and endogenous in stressed economies:
    - Equilibrium is uniquely determined for stressed-economy debt below 130 percentof GDP, irrespective of the stressed economies’ fiscal stance.
    - For debt above 130 percentof GDP, countercyclical spending expansions in stressed economies can produce macroeconomic instability because high debt amplifies the sovereign risk channel.
  - Longer expected ZLB (10 quarters) worsens determinacy: at stressed-economy debt = 126 percentof GDP, only very procyclical austerity (spending falling at least one-for-one with output) appears able to rule out self-fulfilling crisis dynamics.
  - Reverse exercise (endogenous spending in rest of euro area, constant in stressed economies) yields a picture similar to the symmetric-case top panels: procyclical cuts in the rest of the union have negative demand effects that can dominate beneficial interest-rate effects unless debt is high.
- Asymmetric fiscal stances (constant aggregate spending across union):
  - Considered differential ∆φ where stressed economies and rest-of-euro-area pursue opposite cyclical stances.
  - Determinacy region is similar in shape to unilateral adjustment cases but generally smaller (scope for policy mistakes increases).
  - At high stressed-economy debt levels, the combination most likely to restore determinacy is:
    - procyclical stance in the stressed economies coupled with a countercyclical stance in the rest of the euro area.
  - Intuition: spending cuts in stressed economies reduce deficits and sovereign risk (and private spreads) while countercyclical expansion elsewhere offsets union-wide demand losses, supporting activity and tax revenues in stressed economies.

### Risk pooling — major findings
- Nonlinearity of premia and pooling:
  - Because sovereign risk premia increase nonlinearly in debt, pooling liabilities can reduce premia in stressed economies by more than it raises premia in other parts of the union.
- Illustrative pooling exercise:
  - Assume complete equalization → common debt-to-GDP ratio = 104 percentof GDP.
  - Under calibration, pooling to 104 percentof GDP implies a reduction of the sovereign risk premium in the stressed economies by some 300 basis points relative to the pre-pooling scenario.
- Determinacy under pooling:
  - Determinacy regions plotted for symmetric endogenous spending response (φ same across parts of union) and constant-interest-rate durations of 9 and 10 quarters.
  - Results:
    - When pooled (common) debt is relatively low, determinacy is ensured if union-wide spending is countercyclical or at most mildly procyclical.
    - At high pooled debt levels, determinacy requires a strongly procyclical stance.
    - Reversal points in calibration:
      - With 9 quarters of constant interest rates: reversal occurs somewhere just below 120 percentof GDP.
      - With 10 quarters of constant interest rates: reversal occurs at 100 percentof GDP.
  - Pooling to 104 percentof GDP (baseline pooling value) with 9 quarters ZLB does not make the cyclical stance of spending materially less relevant for macroeconomic stability compared to the pre-pooling scenario.
  - Implication: a transfer of government liabilities across member states, in and by itself, would do little to address equilibrium indeterminacy under the conditions considered.

### Mechanism and model emphasis
- Sovereign risk channel:
  - Rising risk premia on government debt raise private-sector credit spreads.
  - When monetary policy is at the ZLB, these public-to-private spillovers alter the trade-off between fiscal consolidation and demand support, potentially generating self-fulfilling downturns.
- Modeling approach:
  - A New Keynesian two-country model with reduced-form relationships linking fiscal outlook to risk premia on public and private debt.
  - Analysis highlights dependence of determinacy on (i) ZLB duration and (ii) cyclicality of fiscal policy across regions.

### Key numeric thresholds and values (preserved exactly)
- Pre-pooling sovereign liabilities: 93 and 126 percentof GDP.
- Illustrative pooled (equalized) debt stock: some 104 percentof GDP.
- Maastricht debt level used in counterfactual: 60 percentof GDP.
- Debt threshold for determinacy in asymmetric middle-row exercise: 130 percentof GDP.
- φ threshold at baseline (9 quarters, stressed debt = 126 percentof GDP) above which equilibrium is indeterminate: 0.4 (φ in excess of 0.4).
- Expected durations of constant-interest-rate episode considered: 9 quarters and 10 quarters.
- Reduction of sovereign risk premium in stressed economies under pooling to 104 percentof GDP: some 300 basis points.
- Reversal points for requirement of procyclical stance under pooling:
  - just below 120 percentof GDP for 9 quarters;
  - 100 percentof GDP for 10 quarters.
- Debt in rest of euro area assumed in many figures: 92.5 percentof GDP.
- Vertical dashed reference in figures: stressed-economies debt = 126.0 percentof GDP.

### Policy-relevant implications and recommendations (implicit in results)
- Priority for fiscal policy cyclicality:
  - Coordinated fiscal policies that avoid strong procyclicality across the union are critical when the central bank faces an extended ZLB period.
  - Countercyclical or mildly procyclical union-wide spending policies are conducive to determinacy when pooled debt is moderate.
- Role of asymmetric coordination:
  - A coordinated asymmetric stance—procyclical consolidation in high-debt/stressed economies combined with countercyclical expansion in lower-debt parts—can help restore macroeconomic stability under high debt.
- Limited sufficiency of debt pooling alone:
  - Institutional steps that pool sovereign risk (e.g., equalizing liabilities) reduce premia in stressed economies but do not, by themselves, eliminate the threat of self-fulfilling downturns if the union-wide fiscal stance remains largely procyclical.
- Importance of ZLB duration:
  - Longer anticipated durations at the ZLB (e.g., 10 quarters vs. 9 quarters) materially worsen determinacy prospects and raise the bar on fiscal policy behavior needed to avoid indeterminacy.

*Source: IMF Working Paper — section 5.2, "Macroeconomic stability in the euro area."*

### 1. Ifa

### 1. Ifa

### Case analysis (signs of a∗∗ and det(A))
- If a∗∗ > 0, det(A) > 0, so only Case I can be satisfied.
  - Conditions (i) and (ii) of that case correspond to conditions a) and b) in Proposition 3.
  - Since tr(A) > 0, condition (iii) of Case I is also satisfied.
- For a∗∗ < 0, det(A) < 0, so Case I cannot hold.
  - The conditions given in the proposition are those pertaining to Case II with a∗∗ < 0.

### Proof of Proposition 4 (summary and key algebraic relations)
- Markov structure implication:
  - Output and inflation (in deviations from the steady state) take on the same respective values, ̃yL and ˆΠL, in every period in which monetary policy is constrained, and values of zero thereafter.
- Phillips curve relation:
  - ˆΠL = κy 1−βμ ̃yL.
- IS equation terms for the spread yield:
  - [1−a− ̄σμκy 1−βμ] ̃yL = − ̄σμξD θ gL.
- The term in square brackets is strictly positive per condition b) for determinacy in Proposition 1.
- Solving for yL yields the expression given in the proposition.
- Note: Here we assume a∗∗ ≠ 0. Establishing the conditions for determinacy for a∗∗ = 0 is straightforward.

### Corollary 6 (statement and proof summary)
- Statement:
  - Under the conditions of Proposition 3, suppose that ξD := ξ − ξ∗ > 0, so that the Home spread is more responsive to the Home deficit than the Foreign spread is to the Foreign deficit.
  - Then, the range of fundamental parameters (that is, parameters other than ∆φ) for which the equilibrium is determinate is at least as large with a suitable choice of ∆φ > 0 (that is, procyclical spending policy in Home and countercyclical spending policy in Foreign) as in the absence of an endogenous spending response, and can be strictly larger.
- Proof (key steps):
  - Absent a fiscal response, a∗∗ > 0, which makes item 1 of Proposition 3 the relevant point of departure.
  - ∂a∗∗/∂∆φ < 0.
  - A positive value of ∆φ, chosen so that a∗∗ will remain positive, means that condition (i) holds for a bigger set of fundamental parameters than with ∆φ = 0.
  - The same is true for condition (ii).

*Source: _wp13227 - 1. Ifa*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2013/_wp13227.pdf_