## Introduction5

## Source details

**Canonical URL:** [Introduction5](https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020293-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2020/english/wpiea2020293-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2020/english/wpiea2020293-print-pdf.pdf.json)

---

### Motivation and empirical facts
- Sovereign debt crises coincide with pronounced output contractions: during the Eurozone crisis, Spanish output fell by about 10% of its 2008 peak.
- Spain experienced a steeper fall in private consumption, by up to 15% of the pre-crisis maximum; unemployment doubled.
- In the 2000s, Spanish households held about 94% of GDP in net worth.
- Empirical finding: increases in sovereign spreads are associated with declines in output accompanied by larger declines in consumption for EU countries during the crisis.

### Mechanism and model overview
- Key interaction: sovereign default risk × precautionary savings motive of heterogeneous households.
  - When sovereign default becomes more likely, households anticipate negative consequences and optimally cut consumption in favor of higher savings.
  - Downward nominal wage rigidities combined with a currency peg transmit insufficient aggregate demand into higher unemployment and output contractions.
  - This generates an aggregate-demand doom loop: high spreads → demand shortages → further increases in debt and spreads, amplifying underlying shocks.
- Even without realized default, the possibility of default endogenously increases volatility of consumption and output and generates large welfare costs of sovereign risk.

### Stylized models and intuition
- Two minimal models illustrate channels:
  - TFP costs of default: default-induced TFP losses increase income volatility from an individual household’s perspective, boosting precautionary saving; households cut consumption more than a planner would because they do not internalize the demand externality.
  - Redistribution at default: defaults transfer resources between creditors (including foreigners) and domestic households via lower taxes. Marginal propensities to consume differ across agents and time horizons; anticipation of transfers can have the opposite effect on aggregate demand than the actual transfer.

### Quantitative model and calibration (overview)
- Small open economy with heterogeneous households (Bewley framework), uninsurable idiosyncratic income risk.
- Households choose a portfolio including government debt; distributions of wealth and exposures interact with government default decisions.
- Government default costs: lost TFP Δ and exclusion from capital markets; some debt is domestically held so defaults produce redistribution.
- Nominal wage rigidities + currency peg represent the Euro and produce an aggregate-demand externality.
- Calibrated to Spain in the 2000s, targeting standard sovereign-debt moments plus wealth-to-GDP ratio, wealth Gini, and share of sovereign debt held domestically.

### Key quantitative findings and statistics
- Counterfactual without default:
  - Simulating the model without the possibility of sovereign default yields much milder recessions for episodes the benchmark model labels as crises.
  - About 30% of the output contraction during a typical simulated crisis can be directly attributed to the presence of sovereign risk.
  - Consumption behavior differs markedly: benchmark matches Spanish experience while no-default model shows much more smoothing.
- Welfare costs:
  - On average, households would give up about 4.2% of permanent consumption to make defaults impossible.
  - At the height of a crisis, moving to a world without the possibility of default is worth about 12% of permanent consumption to the average household.
  - Heterogeneous welfare impacts: welfare costs range between 5% for the bottom 10% of the wealth distribution and 3.7% for the top 10%.

---

### Motivating evidence (Spain and cross-country)

### Spain: 2008–2013 episode
- Peak-to-trough declines:
  - Output: about 10%
  - Consumption: about 15%
- Comparing trough to early 2011 to isolate sovereign-risk effects:
  - Output contraction: of the order of 5%
  - Consumption contraction: of the order of 9%
- Spanish 10-year government bond spread over German Bund increased substantially during the episode.
- Survey evidence: proportion of Spanish firms reporting “demand” as main limiting factor is elevated during crisis.
- Slack increases in two phases: private deleveraging followed by sudden stop interpretation.

### Cross-country panel (11 European countries, 2010–2013)
- Estimated panel: Q_jt = β Spread_jt + γ X_jt + μ_j + δ_t + ε_jt for Q = logY, logC.
- Coefficients on Spread (standard errors in parentheses; significance noted):
  - logY (col 1): -0.007 (0.001) ⋆⋆⋆
  - logY (col 2): -0.006 (0.001) ⋆⋆⋆
  - logC (col 3): -0.014 (0.002) ⋆⋆⋆
  - logC (col 4): -0.009 (0.001) ⋆⋆⋆
  - logC (col 5): -0.007 (0.001) ⋆⋆⋆
  - logC (col 6): -0.004 (0.001) ⋆⋆⋆
- Additional regressor B_t/Y_t coefficients: -0.001 (0.000) ⋆⋆ ; -0.002 (0.000) ⋆⋆⋆ ; -0.002 (0.000) ⋆⋆⋆
- Autoregressive control logY_t in consumption regressions: 0.995 (0.091) ⋆⋆⋆ ; 0.807 (0.067) ⋆⋆⋆
- Sample: N = 143 in each regression; Within-R^2 values: 0.274, 0.325, 0.420, 0.677, 0.715, 0.857.
- Interpretation: a typical country experiencing average increase in spreads (about 200bps) saw:
  - Fall in output between 1.2% and 1.4%
  - Fall in consumption between 1.8% and 2.7%
- Controlling for output shows consumption responses larger than output responses for countries with larger increases in spreads.

---

### Monetary policy, external sector, and timing

### Exchange rate peg and default
- Economy defends a pegged exchange rate; price of nontraded goods normalized p_T ≡ 1.
- Model assumes peg is not abandoned upon default. Allowing devaluation would:
  - Reduce aggregate income losses from default by allowing real wages to fall.
  - Introduce wealth effects from currency denomination of assets, requiring richer currency-choice modeling.

### Foreign borrowing and debt pricing
- Foreigners access funds at fixed international risk-free rate r⋆ implying q_h(S) = 1/(1+r⋆).
- q_g(S) reflects coupon claims, haircut ℏ, unmatured fraction (1−ρ), and foreign pricing of coupon as (1−ξ′)κ with ξ ∈ (0,1) to match home bias.
- Promised interest rate r_b defined by q_g(S) = Σ_{s=0}^∞ κ(1−ρ)^s (1 + r_b(S))^{−s−1} ⇒ r_b(S) = κ/q_g(S) − ρ.
- Spread spr(S) = κ [1/q_g(S) − 1] given κ = r⋆ + ρ.
- Consolidated intertemporal constraint and net foreign inflows NFI_t given (Equation (12)); market clearing conditions in traded and nontraded goods given (Equation (13)).

### Timing and solution approach
- Period timeline: past state → nature chooses TFP and risk premia → government repayment decision → foreign lenders set prices → distribution of wealth realized → government issues/transfers → firms and households act → period ends.
- Distribution λ treated via bounded rationality: agents believe joint distribution of ω and ε is lognormal [ω_t ; ε_t] ∼ logN( [μ_t ; 0], [σ_t ρ_t; ρ_t σ_ε] ), with ρ_t = 0 as assumed.
- Approximation: restrict equilibrium to lognormal family and project policy-implied λ_{t+1} back to (μ,σ,ρ) by matching mean and variance.

---

### Calibration and model fit

### Fiscal rules (data and estimated responses)
- Data: quarterly Spain, Eurostat, 1999Q1 to 2017Q4.
- Regress government consumption and net issuances (% of GDP) on state vector; fitted rules track observed series.
- Fit: Adjusted R2 government consumption: 90% (0.916); debt issuances: 70% (0.776).
- Selected coefficients (examples):
  - Constant examples: 13.194*** (1.350); 14.352*** (0.982).
  - Unemployment t: 1.078*** (0.086); 0.330*** (0.028); 0.410* (0.197); 0.286*** (0.042).
  - Unemployment squared t: -0.020*** (0.002).
  - B_t/Y_t: -0.187*** (0.028); -0.021* (0.010); -0.099 (0.063); -0.020 (0.015).
  - (B_t/Y_t)^2: 0.001*** (0.000).
  - Net Exports t: -0.309*** (0.070) (others reported).

### Key calibration parameters (selected)
- Risk-free rate r^⋆: 4% ann.
- Haircut ℏ: 45%.
- TFP loss Δ in default: 10%.
- Reentry probability θ: 0.04167 (expected duration of default = 25 quarters).
- Share of nontraded π: 0.7397.
- Labor share α_N, α_T: 0.67.
- Elasticity of nontraded consumption η: 0.74.
- Idiosyncratic income persistence ρ_ε = 0.978; σ_ε = 0.022.
- Elasticity of intertemporal substitution ψ: 1.
- Discount rate of HHs 1/β − 15.615% ann.
- Risk aversion γ: 3.
- Tax progressivity τ: 28%.
- Wage minimum w̄: 1.175.
- TFP process ρ_z, σ_z: (0.63, 0.011).
- Mean risk premium ξ̄: 0.05%.
- Risk premium AR(1) ρ_ξ, σ_ξ: (0.95, 0.00025).
- Mean utility cost of default μ_g: 0.012.

### Model fit vs data (averages over simulations)
- AR(1) autocorr. log(Y_t): Model 0.937, Data 0.966.
- AR(1) std log(Y_t): Model 0.658%, Data 0.617%.
- AR(1) autocorr. log(C_t): Model 0.964, Data 0.954.
- AR(1) std log(C_t): Model 0.789%, Data 1.22%.
- Avg spread (bps): Model 122, Data 106.
- Avg Debt-to-GDP: Model 46.3%, Data 64.6%.
- Avg unemployment: Model 10.3%, Data 15.9%.
- Median domestic holdings: Model 63.8%, Data 56.5%.
- Avg wealth-to-GDP: Model 94%, Data 94.5%.
- Avg wealth Gini: Model 48.5%, Data 57.5%.

### Accuracy of simulation
- Average absolute relative discrepancies:
  - Price of nontraded goods: 0.14%.
  - Consumption: 0.59%.

---

### Crises, amplification, and distributional effects

### Crisis definition and model dynamics
- High-spread episode defined as 11 quarters (matching Spanish 2010–mid 2012) ending with spread > 400 bps, without default, conditioned on lower initial spreads.
- Model dynamics around high-spread episodes:
  - TFP, output, and consumption significantly below normal.
  - Government finances deteriorate; lump-sum taxes increase; tax collections worsen as unemployment rises.
  - Typical household consumes about 1 pp. of disposable income less during crisis.
  - Output trough: about 9% below long-run mean.
  - TFP trough: between 0% and -2%.
  - Consumption trough: about 11% below long-run mean.
  - Government debt accumulates rapidly due to high spreads.

### Benchmark vs no-default experiments
- No-default economy (government always repays) simulated with same shocks:
  - Contractions in output and consumption are muted relative to benchmark.
  - Benchmark crisis peak losses: output and consumption contractions between 50% and 60% larger than no-default.
  - Average spread (bps): Benchmark 122 vs No-default 1.2.
  - Avg Debt-to-GDP: Benchmark 46.3% vs No-default 36.6%.
  - Avg unemployment: Benchmark 10.3% vs No-default 8.12%.
  - Welfare in repayment: Benchmark 0.906 vs No-default 0.944.
- Welfare implications:
  - Time-averaged willingness to pay to abolish default: about 4.19% of permanent consumption (average across distribution).
  - Crisis-period willingness to pay at height of crisis: about 12% of permanent consumption (average/median measures).

### Distributional impacts
- Welfare gains from abolishing default (time-averaged, Table 7):
  - p10: Gains 5.04%
  - p25: Gains 4.55%
  - p50: Gains 4.18%
  - p75: Gains 3.92%
  - p90: Gains 3.68%
  - Average: Gains 4.19%
- Crisis-period gains (examples cited):
  - Bottom 10%: 13.2% (bottom 25%: 12.5%)
  - Median household: would pay 12.2% of permanent consumption to abolish default; average ≈ 12.3%.
- Summary: welfare gains from eliminating sovereign risk decline with wealth; crisis-period gains are much larger, especially at the bottom of the distribution.

---

### Channels of household response and wage rigidity

### Household channels to sovereign risk
- Three main channels:
  1. Aggregate income losses in default: default reduces TFP by Δ, lowering market-clearing wage and increasing unemployment when wage floor binds → lower labor income → lower consumption.
  2. Capital losses via debt price q_g(S): higher default probability lowers q_g(S), shifting wealth leftward and affecting households proportionally to domestic bond holdings.
  3. Insurance properties of government bonds worsen in crises: bond returns become more volatile and comove with aggregate income → bonds become poorer hedges for households when spreads are high.

### Wage rigidities and amplification
- Nontraded goods supply: L_N^d = [ α_N p_N / max{w, w̄} ]^{1/(1−α_N)} ; when wage constraint binds, supply curve is flatter so demand shocks produce larger quantity falls and smaller price falls.
- Wage lower bound specification: constant lower bound w̄ (instead of w_t ≥ γ_w w_{t−1}) to reduce state dimension and avoid certain paradoxes in presence of default-driven TFP drops.
- Amplification: precautionary saving in response to sovereign risk interacts with wage rigidity and currency peg to produce large output and employment contractions even without realized default.

---

### Policy implications and conclusions

### Policy implications
- Rapid crisis resolution is welfare-improving:
  - Prolonged periods of heightened sovereign risk impose costs in terms of output and employment via the spreads–demand feedback loop.
  - Swift resolution mechanisms that avoid delays and associated uncertainty could enable significant welfare gains.
- Fiscal tradeoff: when tax and debt-issuance resources are low (spreads high), government faces choice between default or low lump-sum transfers (regressive austerity), with implications for demand via redistribution and precautionary responses.

### Main conclusions
- Sovereign risk amplifies underlying shocks through precautionary household behavior and aggregate-demand externalities, producing substantial welfare costs even if default does not occur.
- Time-averaged welfare cost ≈ 4.2% of permanent consumption; crisis-peak cost ≈ 12% for average household.
- Mechanism especially relevant when private sector is a net saver (as in Spain in the 2000s) or for emerging markets with positive private international investment positions despite government indebtedness.

*Source: Introduction and selected sections (content unit "wpiea2020293-print-pdf - Introduction5" and related sections) of the provided IMF working paper PDF.*

### Introduction5

### Introduction

### Motivation and empirical facts
- Sovereign debt crises coincide with pronounced output contractions: during the Eurozone crisis, Spanish output fell by about 10% of its 2008 peak.
- Spain experienced a steeper fall in private consumption, by up to 15% of the pre-crisis maximum; unemployment doubled.
- In the 2000s, Spanish households held about 94% of GDP in net worth.
- I document that increases in sovereign spreads are associated with declines in output accompanied by larger declines in consumption for EU countries during the crisis.

### Mechanism and model overview
- Key interaction: sovereign default risk × precautionary savings motive of heterogeneous households.
  - When sovereign default becomes more likely, households anticipate negative consequences and optimally cut consumption in favor of higher savings.
  - Downward nominal wage rigidities combined with a currency peg transmit insufficient aggregate demand into higher unemployment and output contractions.
  - This creates an aggregate-demand doom loop: high spreads → demand shortages → further increases in debt and spreads, amplifying underlying shocks.
- Even without realized default, the possibility of default endogenously increases volatility of consumption and output and generates large welfare costs of sovereign risk.

### Stylized models and intuition (Section 3 summary)
- Two minimal models illustrate channels:
  - TFP costs of default: default-induced TFP losses increase income volatility from an individual household’s perspective, boosting precautionary saving; households cut consumption more than a planner would because they do not internalize the demand externality.
  - Redistribution of wealth at default: defaults transfer resources between creditors (including foreigners) and domestic households (via lower taxes). The marginal propensities to consume (MPC) differ across agents and across time horizons; agents with larger MPCs out of current income tend to have smaller MPCs out of future income, so anticipation of transfers can have the opposite effect on aggregate demand than the actual transfer.

### Quantitative model and calibration (Sections 4–6 summary)
- Model setup:
  - Small open economy with heterogeneous households subject to uninsurable idiosyncratic income risk (Bewley framework).
  - Households can save and choose exposure to government debt; distributions of wealth and exposures interact with government default decisions.
  - Government faces default costs as lost TFP and exclusion from capital markets; some debt is domestically held so defaults produce redistribution.
  - Nominal wage rigidities + currency peg represent the Euro and produce an aggregate-demand externality.
  - Government cannot make taxes agent-specific, providing a constraint preventing it from fully using households as off-balance-sheet borrowers.
- Calibration:
  - Calibrated to Spain in the 2000s, targeting standard sovereign-debt literature moments plus wealth-to-GDP ratio, Gini index for wealth, and share of sovereign debt held domestically.
  - Simulate long time-series and identify crisis episodes resembling the Spanish experience of 2010–2013: no default accompanied by elevated sovereign spreads.

### Key quantitative findings and statistics (Section 7–8 summary)
- Counterfactual without default:
  - Model solved without the possibility of sovereign default but simulated with the same shocks shows much milder recessions at times when the benchmark model is in crisis.
  - About 30% of the output contraction during a typical simulated crisis can be directly attributed to the presence of sovereign risk.
  - Consumption behavior differs markedly: the benchmark model matches the Spanish experience while the model without default exhibits much more smoothing in aggregate consumption.
- Welfare costs:
  - On average, households would give up about 4.2% of permanent consumption to make defaults impossible.
  - At the height of a crisis, moving to a world without the possibility of default is worth about 12% of permanent consumption to the average household.
  - Heterogeneous welfare impacts across the wealth distribution: welfare costs of sovereign risk range between 5% of permanent consumption for the bottom 10% of the wealth distribution and 3.7% for the top 10%.

### Distributional considerations and government incentives
- The government’s repayment decision accounts for redistribution that defaults bring about.
- Default incentives increase with higher debt and lower productivity, and are also heightened when the distribution of wealth is more unequal or when the private sector is poorer as a whole.
- The potential redistribution at default varies over time as payments shift between bondholders (including foreigners) and domestic households facing lower taxes; effective transfers depend on distributions of wealth and exposures and the exogenous distribution of the tax burden.

### Policy implications
- Rapid crisis resolution is welfare-improving:
  - Prolonged periods of heightened sovereign risk impose costs in terms of output and employment via the spreads–demand feedback loop.
  - Swift resolution mechanisms that avoid delays and the associated uncertainty could enable significant welfare gains.

### Connection to existing literature
- Builds on canonical sovereign debt models (Eaton and Gersovitz,1981; Aguiar and Gopinath,2006; Arellano,2008) and papers emphasizing internal costs of default (e.g., Mendoza and Yue,2012).
- Contrasts with one-sector small open economy models that abstract from domestic demand and household saving choices.
- Relates to literature on nominal rigidities and aggregate-demand externalities (Schmitt-Grohé and Uribe,2016); complements work by Anzoategui(2020), Bianchi, Ottonello, and Presno(2016), and Arellano, Bai, and Mihalache(2020).
- Connects to redistribution and MPC channels in demand contractions (Eggertsson and Krugman,2012; Auclert,2017; Korinek and Simsek,2016) and to studies of distributional considerations in sovereign policy (Ferriere,2016; Deng,2020).

### Paper layout
- Section 2: motivating evidence.
- Section 3: two stylized settings for intuition.
- Section 4: quantitative model.
- Section 5: equilibrium and model mechanics.
- Section 6: calibration.
- Section 7: results from the model solution.
- Section 8: focus on crises and main results.
- Section 9: conclusions.

*Source: Introduction (content unit "wpiea2020293-print-pdf - Introduction5")*

### 2.  Motivating Evidence

### 2. Motivating Evidence

### Spain: output and consumption during the 2000s crisis
- Output and households’ consumption are plotted relative to their values at the start of 2008; both strongly contract during the crisis years.
- Consumption contracts more than output as the crisis unfolds.
- Peak-to-trough declines:
  - Output: about 10%
  - Consumption: about 15%
- Comparing the trough of the crisis to early 2011 to isolate sovereign-risk effects:
  - Output contraction: of the order of 5%
  - Consumption contraction: of the order of 9%
- The increase in the Spanish 10-year government bond spread over a comparable German Bund is substantial during the episode.
- Survey evidence (Eurostat) shows the proportion of Spanish firms reporting “demand” as the main limiting factor is elevated during the crisis.
- Slack measures in the Spanish economy increase significantly during the crisis in two phases, consistent with a private deleveraging followed by a sudden stop interpretation.

### Cross-country panel evidence: spreads and macro outcomes (11 European countries, 2010–2013)
- Estimated specification: Q_jt = β Spread_jt + γ X_jt + μ_j + δ_t + ε_jt for Q = logY, logC with country and time fixed effects.
- Key regression coefficients and statistics (Table 1):
  - Coefficients on Spread:
    - logY (col 1): -0.007 (0.001) ⋆⋆⋆
    - logY (col 2): -0.006 (0.001) ⋆⋆⋆
    - logC (col 3): -0.014 (0.002) ⋆⋆⋆
    - logC (col 4): -0.009 (0.001) ⋆⋆⋆
    - logC (col 5): -0.007 (0.001) ⋆⋆⋆
    - logC (col 6): -0.004 (0.001) ⋆⋆⋆
  - Additional regressor B_t/Y_t in columns with controls:
    - Coefficients: -0.001 (0.000) ⋆⋆ ; -0.002 (0.000) ⋆⋆⋆ ; -0.002 (0.000) ⋆⋆⋆
  - Autoregressive control logY_t in consumption regressions:
    - Coefficients: 0.995 (0.091) ⋆⋆⋆ ; 0.807 (0.067) ⋆⋆⋆
  - Sample and fit:
    - N = 143 in each regression
    - Within-R^2 values: 0.274, 0.325, 0.420, 0.677, 0.715, 0.857
  - Significance notation:
    - ⋆⋆⋆ p<0.01, ⋆⋆ p<0.05, ⋆ p<0.1
- Interpretation of coefficients:
  - A typical country experiencing the average increase in spreads (about 200bps) saw:
    - A fall in output between 1.2% and 1.4%
    - A fall in consumption between 1.8% and 2.7%
  - Controlling for output in consumption regressions shows consumption responses are larger than output responses for countries with larger increases in spreads.

### Stylized mechanisms motivating the quantitative model
- Two minimal models illustrate core mechanisms:
  1. Expectations of income losses in case of default depress aggregate demand in the present.
  2. Redistribution induced by default when domestic agents hold sovereign debt contributes to the contraction in demand.

- Model 1: Costs of default (small open economy, two periods)
  - Setup highlights:
    - Default reduces output by factor Δ in period 2.
    - Representative agent can save s_1 ≥ 0 in period 1; legacy debt d is due in period 2.
    - Consumption in period 2:
      - If government repays: c_2 = y_2 − d + s_1
      - If government defaults: c_2 = y_2(1−Δ) + s_1
    - Default occurs iff y_2 < d/Δ.
    - Nontraded production with wage rigidities: y_N1 = F(h_1) = h_1^α where α < 1.
    - CES preferences across traded and nontraded goods with elasticity of substitution η and nontraded weight π imply labor demand and relation (2):
      - h = H(c_T, w) = [ (π/(1−π)) (α/w) ]^{1/(1+αη)} c_T^{(1+η)/(1+αη)}
    - Wage floor w̄ induces a constrained regime where equilibrium employment is increasing in traded consumption when the wage floor binds (H'_c(c_T, w̄) > 0).
  - Planner’s Euler equation (planner problem (3)):
    - u'_T(c_T, F(h)) + H'_c(c_T, w̄) μ = β(1+r) E[ u'(y_2 + s − min{d, y_2 Δ}) ]
    - Both the default cost Δ and debt level d enter the Euler equation by increasing the marginal value of consumption in period 2, thereby boosting precautionary savings in period 1.
    - Increased savings can reduce employment when H'_c(c_T, w̄) > 0, amplifying the demand contraction.
  - If planner can repurchase debt at price q(d−s_1, Δ), the Euler equation adjusts and increases in Δ raise the debt price, partially offsetting precautionary incentives; increases in d lower the debt price, magnifying effects.

- Model 2: Propensities to consume and anticipation of redistribution (closed economy)
  - Population: measure χ of hand-to-mouth agents and 1−χ of savers; each endowed with one unit of labor each period; output y = h.
  - In period 2, with probability π there is a transfer k from savers to hand-to-mouth; consumptions in period 2:
    - (c_s^2, c_h^2) = (1−k, 1+k) with probability π; (1, 1) with probability 1−π.
  - Savers’ Euler equation with CARA utility and absolute risk aversion γ yields first-period consumption:
    - c_s^1 = 1 − (1/γ) log(β(1+r)) − (1/γ) log(1−π + π e^{γ k})
  - Comparative statics:
    - Larger potential transfer k unambiguously reduces savers’ consumption.
    - The reduction increases with probability π.
    - The effect of γ is non-monotonic: γ amplifies k inside the exponential but also dampens substitution incentives because γ is the inverse of the intertemporal elasticity of substitution.
  - Hand-to-mouth consumption is c_h^1 = w h (dictated by first-period income).
  - Market-clearing and implications:
    - Output y = χ c_h^1 + (1−χ) c_s^1; with y = h, output is a multiple of savers’ consumption when unemployment exists.
    - Expected redistribution depresses first-period output; demand and output are decreasing in π and k (Figure 3 illustrates percent deviations).
  - Policy implication:
    - Anticipated transfers (future redistribution) reduce savers’ demand today (precautionary saving) while hand-to-mouth agents’ MPC = 1 makes actual transfers expansionary if implemented today. Expected future transfers therefore can be contractionary today.

### Links to quantitative model and fiscal policy tradeoffs
- The quantitative model needs to endogenize:
  - Amount of debt outstanding when spreads increase.
  - Distribution of that debt across foreign and domestic agents and across domestic agents.
  - Distribution of MPCs for various current and future transfers.
- Key modeling elements summarized:
  - Heterogeneous households with idiosyncratic labor shocks ε and asset holdings (a for risk-free asset, b for government debt); a′ ≥ ā and b′ ≥ 0; no shorting government debt.
  - CES aggregator for traded and nontraded consumption with elasticity η and weight π.
  - Epstein-Zin preferences with parameters β, γ, ψ.
  - Aggregate state S_t = (B_t, λ_t, ξ_t, ζ_t, z_t): total government debt B_t; distribution of households λ_t; sovereign spread shock ξ_t; international credit market state ζ_t; productivity shock z_t.
  - Government debt is long-term with geometrically-decaying coupon κ(1−ρ)^{t−1}; default applies haircut ℏ and suspends coupons until market reaccess with probability θ.
  - Households’ idiosyncratic asset portfolio ω_t = a′_{t−1} + R^b_{t−1,t} b′_{t−1}; productivity follows log AR(1): log ε_{t+1} = ρ_ε log ε_t + σ_ε ν^ε_{t+1}.
  - Firms produce traded and nontraded goods with technologies Y_Nt = f_N(z_t, ζ_t) L_Nt^α and Y_Tt = f_T(z_t, ζ_t) L_Tt^α; benchmark where z_t affects traded production and default reduces TFP by Δ when ζ ≠ 1:
    - f_N(z, ζ) = 1 − Δ 1(ζ̸=1)
    - f_T(z, ζ) = z [ 1 − Δ 1(ζ̸=1) ]
  - Nominal rigidities: wage w_t cannot fall below w̄; when constraint binds labor is rationed proportionally.
  - Government budget constraint (8):
    - q_g(S_t) [ B′(S_t) − (1−ρ) B(S_t) ] + τ w(S_t) L(S_t) = κ 1(ζ=1) B(S_t) + g(S_t) − T(S_t)
  - Government follows exogenous, estimated fiscal rules for g and B′ but chooses default vs repayment with discretion; transfers T(S_t) can be obtained residually.
  - In default (ζ = 0): coupon payments suspended; defaulted debt trades; no new debt issuance (B′ = B). Haircut ℏ applies so B(S_{t+1}) = (1−ℏ) B′(S_t) if default occurs.
- Fiscal tradeoff emphasized:
  - When tax and debt-issuance resources are low (spreads high), government chooses between default or low lump-sum transfers (a regressive austerity option), with implications for aggregate demand via redistribution and precautionary responses.

*Source: 2.  Motivating Evidence (wpiea2020293-print-pdf).*

### 4.5  Monetary policy

### 4.5  Monetary policy

### Exchange rate peg and default
- The small open economy defends a pegged exchange rate. This amounts to a normalization of the (constant) price of nontraded goods p_T ≡ 1.
- The model assumes the economy does not abandon the peg upon default. Relaxing this assumption (allowing devaluations with defaults) would:
  - Reduce aggregate income losses from default by allowing real wages to fall.
  - Create wealth effects from the currency denomination of contracts and assets that households own, which would require a richer model of currency choice and is beyond the scope of this paper.
- The wage bound could be made dependent on the default state ζ to capture the first consequence.

### Foreign borrowing and the external sector (section 4.6)
- Foreigners access funds at a fixed international risk-free rate r⋆, which implies q_h(S) = 1/(1+r⋆). (Equation (9))
- When foreigners hold government debt in state S, no-arbitrage implies an expression for q_g(S) that reflects:
  - Debt claims to coupon payments while there is no default.
  - A default haircut ℏ.
  - The unmatured fraction (1−ρ) of the bond can be resold in secondary markets. (Equation (10))
- Foreigners price the coupon as if it were (1−ξ′)κ, with ξ constrained to remain in (0,1); this artificially depresses the price of government debt to match home bias in holdings.
- If the government was already in default at state S_t, equation (10) specializes to:
  q_g(S|ζ̸=1) = 1/(1+r⋆) [ θ E[(1−ξ′|S) κ + (1−ρθ) E[q_g(S′)|S] ].
- The implicit (promised) interest rate r_b on a government bond is measured by:
  q_g(S) = Σ_{s=0}^∞ κ(1−ρ)^s (1 + r_b(S))^{−s−1}  (Equation (11))
  - which yields r_b(S) = κ/q_g(S) − ρ.
  - The spread on government debt is spr(S) = κ [1/q_g(S) − 1] given the normalization κ = r⋆ + ρ.
- Consolidated intertemporal budget constraint: value of debt obligations equals expected discounted value of trade surpluses. Net foreign inflows are:
  NFI_t = q_h,t A_f,t+1 + q_g,t [B′_t^f − (1−ρ) B_f,t] − [κ B_f,t + A_f,t]  (Equation (12))
  - Resources flow in when domestic agents borrow from foreigners and when foreigners purchase debt; flow out when government makes coupon payments to foreigners and when domestic agents repay debts.
- Recasting flows yields:
  NFI_t = q_g,t B′_t^f − [A_f,t + (κ + (1−ρ) q_g,t) B_f,t] + q_h,t A_f,t+1
        = ∫[ω − q_h,t φ_a − q_g,t φ_b] dλ_t − κ B_t + q_g,t (B′_t − (1−ρ) B_t)
- Market clearing requires:
  Y_Nt = C_Nt + θ_N p_Nt G_t
  and
  Y_Tt + NFI_t = C_Tt + (1−θ_N) G_t  (Equation (13))

### Timing (section 4.7)
- Within a period:
  - Past state carried from previous period.
  - Nature chooses current TFP and risk premia.
  - Government decides repayment if not already in default; if in default, nature may choose reentry to markets.
  - Foreign lenders set asset prices.
  - Distribution of wealth across households is determined.
  - Government implements issuance and transfer policies.
  - Firms choose employment and prices; households choose consumption and savings; period ends.
- Timeline summarized in Figure 4 of the source.

### Evolution of the distribution and solution approach (section 4.8)
- The full state S includes the entire distribution λ (infinite-dimensional). The author uses a bounded rationality equilibrium where agents have limited knowledge of λ and believe the joint distribution of wealth ω and individual labor productivity ε is lognormal:
  [ω_t ; ε_t] ∼ logN( [μ_t ; 0], [σ_t ρ_t; ρ_t σ_ε] )  (Equation (14))
  - As of this writing, ρ_t = 0 is assumed.
  - λ_t summarized by (μ_t, σ_t, ρ_t); μ_t and σ_t vary over time and are state variables.
- Two approximations:
  1. Equilibrium solved over subset of distributions (lognormal).
  2. Bounded rationality: when policy-implied λ_{t+1} is not lognormal, project back onto (μ,σ,ρ) space by matching mean and variance.
- Given functions and current distribution, substituting λ_t yields system for joint evolution of distribution parameters and debt price (which depends on future distribution through default incentives). Key relations in (15) include expressions for R_b(S_{t+1}), first and second moments of ω under λ_{t+1}, and cross-moment ω ε under λ_{t+1}.
- This approach avoids the usual simulation step; simulations are used to check forecasting-rule accuracy (Section 6.3).

### Competitive equilibrium (section 5.1)
- Definition: Given government policies h′(S,ξ′,z′), B′(S), and g(S), a competitive equilibrium consists of value and policy functions {v, φ_a, φ_b, φ_c}(s,S), aggregates L_T(S), L_N(S), Π(S), Y_N(S), Y_T(S), prices p_C(S), p_N(S), w(S), q_g(S), q_h(S), taxes T(S), and laws of motion for distribution parameters {μ′,σ′}(S,ξ′,z′,ζ′) such that:
  - Policy functions solve household problem (4).
  - Relative price p_N(S) satisfies intratemporal FOC (5).
  - Aggregates L_T(S), L_N(S) maximize firms' profits given w(S), p_N(S); quantities satisfy production functions (6,7).
  - Lump-sum taxes T(S) satisfy government budget constraint (8).
  - Asset prices q_h(S) and q_g(S) satisfy no-arbitrage conditions (9,10).
  - Market clearing in traded and nontraded goods (13); in labor either w(S) = w̄ or L_T(S) + L_N(S) = ∫ ε dλ_S.
  - Laws of motion for distribution parameters satisfy consistency requirement (15).

### Government strategy (section 5.2)
- Government maximizes current welfare with equal weights on every agent. Without commitment, in each state it maximizes:
  W(S,h′) = ∫ v(s,S) dλ_S(s) + 1(ζ=1) [ μ_g + σ_g ξ_def ]  (Equation (16))
  - ξ_def iid ∼ N(0,1) is a preference shock to smooth default policy.
  - μ_g disciplines average default frequency; discount factor and risk aversion are disciplined by moments of private wealth distribution.
- A policy h′ for repayment is part of an equilibrium if, at each (S,z′), the probability of repayment satisfies:
  h′(S,z′) = P[ μ_g + σ_g ξ_def ≤ W(Ψ(S,ξ′,z′,ζ′=1), h′) − W(Ψ(S,ξ′,z′,ζ′̸=1, h′)) ]  (Equation (17))
  - This is a Nash/recursive-equilibrium restriction: the policy h′ expected by agents must coincide with the policy the government would choose given those expectations.

### Euler equations, coupon payments, and household demand for bonds (section 5.3)
- Household Euler equation for purchases of government bonds:
  q_g(S) ≥ β E[ R_b(S,S′) p_C(S)/p_C(S′) ( φ_c(ω′,ε′,S′)/φ_c(ω,ε,S) )^{−1/ψ} ( v(ω′,ε′,S′) E[v(ω′,ε′,S′)^{1−γ}|S]^{−1} )^{1/ψ − γ} ]  (Equation (18))
  - Equality holds if household purchases a positive amount of bonds.
- If ξ = 0, household would not buy many government bonds. Risk-averse households demand a risk premium to hold government debt; shock ξ creates a risk premium in government bond returns relative to the risk-free asset, helping match domestic holdings of sovereign debt and allowing study of spread increases not driven by domestic fundamentals.

### Household responses to sovereign risk (section 5.4)
- Three main channels:
  1. Aggregate income losses in default: Conditional on default, TFP drops by Δ in both sectors for a random number of periods → downward pressure on market-clearing wage; if wage constraint binds, unemployment increases; labor income w(S)L(S)ε is lower in default than in repayment → households feel poorer and reduce consumption.
     - Figure 5 (source) shows expected labor income as a function of next period’s TFP for default and repayment, conditioning on current debt and ξ. Labor income increases in TFP and is higher in repayment than in default. Current government debt and ξ tend to close the gap.
  2. Price of government bonds q_g(S) falls with higher default probability, causing immediate capital losses to past bondholders and shifting wealth distribution left. Strength of this channel depends on proportion of bonds held domestically and inequality in bondholdings. (Figure 11 in source shows debt price as function of state variables.)
  3. Insurance properties of government bonds worsen in crises: R_b(S,S′) = 1(ζ′=1) [ κ + (1−ρ) (1−ℏ 1(ζ=1 ∩ ζ′̸=1)) q_g(S′) ].
     - In normal times, variance of R_b is low and driven by future resale price q_g(S′).
     - As default probability rises, variance of R_b becomes driven by repayment probability, which correlates with aggregate income → conditional covariance between bond return and household stochastic discount factor increases in crises → bond is a poor hedge, worse when spreads are high.
- Figure 5 (source) shows realized bond returns and labor income: variance of returns and covariance with income increase when default uncertainty rises. When default is very unlikely or very likely, next period shocks have limited influence; when debt is intermediate, returns are very volatile and comove with income.
- Heterogeneity in fear of default (robustness interpretation):
  - With unitary elasticity of intertemporal substitution, households act as if logarithmic preferences combined with model-misspecification concerns. Risk aversion maps into a robustness parameter.
  - Define subjective expectation ˜E[X|s,S] = E[ v(ω′,ε′,S′)^{1−γ} / E[v(ω′,ε′,S′)^{1−γ}|S] · X | s,S ]  (Equation (19))
  - Twisted probabilities overweight states where household value is lower. Figure 6 (source) shows twisted default probability by household; richer and higher-income households fear default more, while poorer and low-income households fear prolongation of crisis more.

### Wage rigidities and aggregate demand (section 5.5)
- Sovereign risk reduces consumption demand, feeding back mainly through the nontraded goods market.
- Traded goods firms supply at international price for given wage; they adjust employment given current TFP.
- Nontraded goods supply curve (from firm FOC) is:
  L_N^d = [ α_N p_N / max{w, w̄} ]^{1/(1−α_N)}  (Equation (20))
  - When nontraded firms cut production, workers are expelled, pushing down wages.
  - In normal times, wage falls induce reallocation to traded sector and some return to nontraded sector; when wage constraint binds, second-round effects cannot occur → unemployment increases and nontraded production falls more.
  - Figure 7 (source) shows supply curve is flatter when wage constraint binds: demand shocks cause larger quantity falls and smaller price falls → wage rigidities create price stickiness.
- Wage rigidity specification:
  - Unlike Schmitt-Grohé and Uribe (2016) who constrain wage w_t ≥ γ_w w_{t−1} with γ_w ≤ 1, this paper follows Bianchi, Ottonello, and Presno (2016) and sets a constant lower bound w̄ on nominal wages.
  - Advantages of constant lower bound:
    - Saves one state variable.
    - Avoids paradoxes where good TFP shocks can be welfare-decreasing via future unemployment (the “overborrowing” externality). With defaults depressing TFP, a benevolent government might otherwise default to suppress overconsumption in good times; constraining government to estimated fiscal rules renders the constant lower bound preferable despite being less realistic.

*Source: wpiea2020293-print-pdf - 4.5  Monetary policy*

### 6.  CalibRation

### 6.  CalibRation

### 6.1 Fiscal rules
- Data: quarterly Spain, Eurostat, 1999Q1 to 2017Q4.
- Approach: regress government consumption and net issuances (fractions of GDP) on the whole state vector; endogenous variables used as regressors.
- Fit:
  - Adjusted R2 for government consumption: 90% (0.916 in Table 2).
  - Adjusted R2 for debt issuances: 70% (0.776 in Table 2).
- Key estimated responses (preferred specifications, Table 2):
  - Constant: 13.194*** (std. error 1.350) for one regression; other constants reported (e.g., 14.352*** (0.982)).
  - Unemployment t: positive and significant, e.g., 1.078*** (0.086) in one specification; other coefficients include 0.330*** (0.028), 0.410* (0.197), 0.286*** (0.042).
  - Unemployment squared t: negative, e.g., -0.020*** (0.002) in one specification.
  - B t /Y t (debt-to-GDP): negative response of new issuances, e.g., -0.187*** (0.028) in one specification; other coefficients include -0.021* (0.010), -0.099 (0.063), -0.020 (0.015).
  - (B t /Y t )^2: small positive, e.g., 0.001*** (0.000).
  - Net Exports t: negative coefficients, e.g., -0.309*** (0.070), -0.167 (0.096), 0.233 (0.162), 0.212 (0.138).
- Observations reported: 72, 72, 71, 71 across regressions in Table 2.
- Figure 8: fitted fiscal rules closely track observed government spending and debt issuances (% of GDP).

### 6.2 Model parameters
- Calibration strategy: rely on external calibration where possible; follow Anzoategui (2020) and Stockman and Tesar (1995) for supply side and preferences; CES demand links via p_Nt = π^{1/η} (1−π)^{1/η} (C_Tt / C_Nt)^{1/η}.
- Selected parameter values and sources/targets (Table 3):
  - Risk-free rate r^⋆: 4% ann. (Anzoategui (2020)).
  - Haircut in case of default ℏ: 45% (Philippon and Roldán (2018)).
  - TFP loss in case of default Δ: 10% (Philippon and Roldán (2018)).
  - Reentry probability θ: 0.04167 (Cruces and Trebesch (2013)) — implies expected duration of default of 25 quarters.
  - Share of nontraded in prod π: 0.7397 (Anzoategui (2020)).
  - Labor share in prod α_N, α_T: 0.67 (Anzoategui (2020)).
  - Share of nontraded in G θ_N: 88% (Anzoategui (2020)).
  - Elasticity of nontraded consumption η: 0.74 (Anzoategui (2020)).
  - Idiosyncratic income shocks: persistence ρ_ε = 0.978; std. deviation σ_ε = 0.022 (D’Erasmo and Mendoza (2016)).
  - Elasticity of intertemporal substitution ψ: 1 (Epstein-Zin = robustness; internally calibrated).
  - Discount rate of HHs 1/β − 15.615% ann. (moments in Table 4).
  - Risk aversion γ: 3 (moments in Table 4).
  - Progressivity of tax schedule τ: 28% (moments in Table 4).
  - Wage minimum w̄: 1.175 (moments in Table 4).
  - TFP process ρ_z, σ_z: (0.63, 0.011) (moments in Table 4).
  - Mean risk premium ξ̄: 0.05% (moments in Table 4).
  - Risk premium AR(1) ρ_ξ, σ_ξ: (0.95, 0.00025) (moments in Table 4).
  - Mean utility cost of default μ_g: 0.012 (moments in Table 4).
- Calibration notes:
  - Risk-free rate set standard in literature.
  - Default haircut and TFP loss chosen to mimic Greek-style default.
  - Reentry probability chosen for expected default duration = 25 quarters (lower end of estimate in Cruces and Trebesch (2013)).
  - Household idiosyncratic income shocks follow D’Erasmo and Mendoza (2016) based on Spanish income distribution.

- Model fit (Table 4; Model column averages over 40 simulations of 700 years each) vs Data:
  - AR(1) autocorr. coef log(Y_t): Model 0.937, Data 0.966.
  - AR(1) std coef log(Y_t): Model 0.658%, Data 0.617%.
  - AR(1) autocorr. coef log(C_t): Model 0.964, Data 0.954.
  - AR(1) std coef log(C_t): Model 0.789%, Data 1.22%.
  - Avg spread (bps): Model 122, Data 106.
  - AR(1) autocorr. coef spread: Model 0.98, Data 0.967.
  - AR(1) std coef spread: Model 23.5, Data 30.1.
  - Avg Debt-to-GDP: Model 46.3%, Data 64.6%.
  - Std Debt-to-GDP: Model 6.73%, Data 23.5%.
  - Avg unemployment: Model 10.3%, Data 15.9%.
  - Std unemployment: Model 2.23%, Data 6.09%.
  - Median domestic holdings: Model 63.8%, Data 56.5%.
  - Avg wealth-to-GDP: Model 94%, Data 94.5%.
  - Avg wealth Gini: Model 48.5%, Data 57.5%.
- Calibration caveats:
  - Spanish sample has one crisis per 17 years — higher than ergodic frequency; calibration targets informally set to lower unemployment and debt-to-GDP to compensate.
  - Gini coefficient on wealth in model falls short of data, gap smaller during crisis (see Figure 14).

### 6.3 Accuracy of the simulation
- Test: compare theoretical endogenous variables x_t from model solution at aggregate state S_t = (B_t, μ_t, σ_t, ξ_t, ζ_t, z_t) with actual market-clearing values given distribution λ_t.
- Table 5: average absolute relative discrepancies:
  - Price of nontraded goods: 0.14%.
  - Consumption: 0.59%.
- Conclusion: assuming a lognormal distribution for idiosyncratic states does not produce large errors.

### 7.1 Government policy (value functions, transfers, price of debt)
- Government value function W(Ψ(S, ξ′, z′, ζ′), h′) analyzed as function of next period TFP z′ and risk premium ξ′; panels vary initial debt B.
  - Higher initial debt increases the value of default relative to repayment.
  - For intermediate initial debt, higher future TFP raises the relative value of repayment.
  - Higher spreads marginally raise the relative value of default.
- Lump-sum transfers that clear government budget constraint (as function of ξ′, z′, ζ′):
  - High indebtedness forces high lump-sum taxes.
  - In repayment, higher future TFP induces high lump-sum taxes driven by relatively low unemployment.
- Price of debt q_g(S):
  - Left panel (fixed distribution, economy not in default): spreads rise steeply (price falls) when debt reaches a threshold; higher z associated with lower spreads.
  - Right panel (fixed B and z, varying distribution): higher spreads when economy is poorer and more unequal.
    - Mechanism: poorer aggregate wealth → more agents close to borrowing limit → larger value of autarky losses from default.
    - Greater inequality → defaults become a more effective redistribution tool → affects value of autarky and spreads.

### 7.2 Macroeconomic conditions
- Unemployment (Figure 12):
  - Unemployment decreases with productivity (z) and increases with government debt (B).
  - Unemployment related negatively to total wealth and positively to inequality.

### 7.3 Ergodic distributions
- Figure 13: estimated densities for normalized output and consumption (demeaned and divided by standard deviation).
  - Output skew = -0.0981.
  - Consumption skew = -1.02 (left skew).
- Appendix analysis (Figure 18, Figure 19): extra left-tail mass in consumption comes from crisis episodes; conditional on default, output and consumption fall to extremely negative levels, with most mass of consumption distribution between 2 and 4 standard deviations below unconditional mean.

### 8. CRises — definition and dynamics
- Definition of episode of high spreads:
  - Period of 11 quarters (matching Spanish 2010–mid 2012) at end of which spread > 400 bps, without a default.
  - Conditioned on lower spreads at start of episode.
- Model performance:
  - Model matches spread volatility but does not generate as sharp spread accelerations as 2010 crisis.
- Dynamics around high-spread episodes (Figure 14; time measured in years; crisis episode starts ~ -1.5):
  - TFP, output, and consumption significantly below normal.
  - Government finances deteriorate; lump-sum taxes increase.
  - As unemployment increases, tax collections worsen.
  - Average propensity to save elevated during crisis: typical household consumes about 1 pp. of disposable income less than in normal times.
  - Output trough: about 9% below long-run mean.
  - TFP trough: varies between 0% and -2%.
  - Consumption trough: about 11% below long-run mean.
  - These recession magnitudes match peak-to-trough contractions in the data.
  - Fiscal adjustments: sustained increase in taxes; slight fall in government spending; government debt accumulates rapidly due to high spreads.
- Dynamics around defaults (Figure 15):
  - Most variables exhibit jumps at default: output falls (directly and via unemployment), level of debt reflects haircut, private wealth is destroyed, inequality increases, and aggregate propensity to save jumps.

### 8.1 Amplification forces (benchmark vs no-default)
- Comparative experiment: benchmark economy (default possible) vs alternate economy where government always repays (no-default) using same shocks.
- Results (Figure 16 summary and Table 6):
  - Contractions in output and consumption are muted in no-default economy.
  - No-default economy has higher average propensity to consume; aggregate demand closer to normal.
  - In benchmark, spreads dynamics induce acceleration of debt-to-GDP.
  - Crisis peak losses: benchmark suffers output and consumption contractions between 50% and 60% larger than no-default.
  - Fiscal policy: benchmark forced to larger increase in lump-sum transfers; no-default can keep fiscal stance closer to neutral.
  - Welfare differences:
    - At height of crisis, average household would give up up to 12% of consumption to move to no-default economy.
    - Averaging over time, average household would give up 4.2% of permanent consumption to make defaults impossible (Table 6: Welfare in repayment 0.906 benchmark vs 0.944 no-default).
- Table 6 (Moments, Benchmark vs No default):
  - AR(1) autocorr. coef log(Y_t): 0.937 vs 0.815.
  - AR(1) std coef log(Y_t): 0.658% vs 0.523%.
  - AR(1) autocorr. coef log(C_t): 0.964 vs 0.898.
  - AR(1) std coef log(C_t): 0.789% vs 0.471%.
  - Avg spread (bps): 122 vs 1.2.
  - AR(1) autocorr. coef spread: 0.98 vs 0.874.
  - AR(1) std coef spread: 23.5 vs 0.134.
  - Avg Debt-to-GDP: 46.3% vs 36.6%.
  - Std Debt-to-GDP: 6.73% vs 1.83%.
  - Avg unemployment: 10.3% vs 8.12%.
  - Std unemployment: 2.23% vs 0.84%.
  - Median dom holdings: 63.8% vs 246%.
  - Avg wealth-to-GDP: 94% vs 83%.
  - Avg wealth Gini: 48.5% vs 50.9%.
  - Default frequency: 0.243% vs 0%.
  - Welfare in repayment: 0.906 vs 0.944.

### 8.2 The distributional impact of sovereign risk
- Welfare gains from abolishing default (during crisis episodes; Figure 17):
  - Gains are heterogeneous, decreasing in wealth.
  - Reported gains during crises (examples cited):
    - Bottom 10%: 13.2% (12.5% for bottom 25% in a reported nearby figure).
    - Top 10%: 10.6% (top 25%: 11.8%).
  - Median household: would pay 12.2% of permanent consumption to abolish default; average ≈ 12.3%.
- Average (time-averaged) welfare gains by wealth quantile (Table 7):
  - p10: Benchmark 0.831, No default 0.873, Gains 5.04%.
  - p25: Benchmark 0.859, No default 0.898, Gains 4.55%.
  - p50: Benchmark 0.902, No default 0.944, Gains 4.18%.
  - p75: Benchmark 0.948, No default 0.985, Gains 3.92%.
  - p90: Benchmark 0.988, No default 1.02, Gains 3.68%.
  - Average: Benchmark 0.906, No default 0.944, Gains 4.19%.
- Summary: welfare gains from eliminating sovereign risk decline with wealth and range between 3.68% to just above 5% of permanent consumption in time-averaged statistics; crisis-period gains are larger at the bottom of the wealth distribution.

*Source: Chapter 6 (CalibRation) of the provided IMF working paper PDF.*

### 9.  Concluding RemaRKs

### 9.  Concluding RemaRKs

### Main findings
- The paper analyzes a model in which households’ consumption demand is negatively affected by the presence of sovereign risk.
- The mechanisms in the model generate substantial amplification of underlying shocks even if the risk of default does not materialize.
- Sovereign risk creates endogenous shifts in demand conditions which exacerbate the equilibrium volatility of aggregate consumption.
- Large welfare costs of sovereign risk are found:
  - about 4.2% of permanent consumption in normal times
  - as much as 12% at the height of a crisis
- Households increase their savings in response to sovereign risk because they anticipate income losses and redistribution in case of default.
- The anticipation of income losses in case of default explains most of the amplification; this effect interacts substantially with the anticipation of redistribution.

### Amplification mechanism
- Amplification relies on precautionary motives of households magnified by sovereign risk.
- Sovereign risk induces endogenous demand shifts that raise equilibrium volatility of aggregate consumption.
- The model highlights the interaction between private saving behavior and the interest rate households face, producing amplification and welfare costs when private agents are net savers.

### Calibration and broader relevance
- The model is calibrated to Spain.
- The mechanism can help explain patterns in emerging-market business cycles, which also exhibit sovereign risk as a feature.
- Both the relative volatility of consumption to output and the volatility of output itself are typical calibration targets in the sovereign debt literature for emerging-market economies; the presented setup offers a more complete explanation by explicitly considering private saving behavior and the interest rate faced by those savers.
- The amplification mechanism and welfare costs are natural consequences in cases where private agents are net savers. The paper notes this describes Spain in the 2000s and salient episodes in emerging markets when the private sector’s international investment position is positive even as the government is in debt.

### Evidence and illustrative statistics from appendices and figures
- Figure notes and captions referenced:
  - Densities conditional on a default probability above15% (Figure 18).
  - Ergodic densities for output and consumption during defaults (Figure 19).
  - Measures of slack in the Spanish economy shown via Capacity Utilization and Unemployment (Figure 20).
  - Spanish firms’ self-reported limits to production, broken down by None, Labor, Financial, Demand, Equipment, Other (Figure 21).
  - Interest rates in Spain shown for Government Bonds and various bank lending and deposit rates (Figure 22).
  - Net worth of Spanish households: Mean 94% (Figure 23).
  - Assets and Liabilities of Spanish households: Mean 169% and Mean 75% (Figure 24).
  - Cross-country table "The Cycle is the TRend" reports σ(C), σ(Y), σ(C)/σ(Y), and σ(C)/σ(Y)(AG) for several countries; for Spain the table reports: σ(C)=1.901, σ(Y)=1.396, σ(C)/σ(Y)=1.362, σ(C)/σ(Y)(AG)=1.110 (series logged and HP-filtered with λ=1600; standard deviations in %).
  - Filtered Spanish detrended output and consumption in the 2000s (Figure 25).
  - Spanish trade balance for the 2000s shown via log(Exports) and log(Imports) (Figure 26).
  - Composition of Spanish government debt with Total and Domestic series and dotted lines showing average levels (Figure 27).

### Model solution and computational approach (appendix summary)
- Household problem reformulated by rewriting controls to eliminate the budget constraint and replace it with tractable inequalities; introduces variables s(a′,b′,S), θ(a′,b′).
- Household problem decomposed into two value functions v and w:
  - v(ω,ε,S) = max_{s,c} [ (1−β)c^{(ψ−1)/ψ} + β w(s,ε,S)^{(ψ−1)/ψ} ]^{ψ/(ψ−1)} subject to C(p_N)c + q_h(S)s = ω + ℓ(S,p_N)ε − T(S,p_N) and s ≥ ̄a.
  - w(s,ε,S) = max_{θ} E[ v(a′ + R_b(S′) b′, ε′, S′)^{1−γ} | ω,ε,S ]^{1/(1−γ)} subject to R_b(S) = 1 (ζ=1) κ + (1−ρ) q_g(S) and θ ∈ [0,1], with a′ and b′ functions of (s,θ,S).
- Solution algorithm (nested, consistent with equilibrium definition):
  1. Guess a law of motion for the distribution.
  2. For each state S:
     - (a) Compute q_g(S) from the foreigners’ SDF (equation (10) referenced).
     - (b) Guess a relative price of nontradables p_N, obtain wage rate w, total labor demand L_d, firm profits Π, compute lump-sum taxes T from government budget constraint (with τ_w L in hand), solve household problem at prices w, p_N, profits Π, and transfers T, check market clearing (equation (13)) for nontraded goods.
     - (c) Iterate on p_N(S) to convergence.
  3. Iterate on the law of motion for the distribution using households’ policy functions.
  4. Update the government’s policy according to equation (17) and iterate until a policy that respects it is found.

*Source: 9.  Concluding RemaRKs (from wpiea2020293-print-pdf)*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020293-print-pdf.pdf_