## Annex 3. Calibration (wpiea2019159)

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---

### I. Introduction — fiscal stance and debt limit
- Main insight:
  - The fiscal stance should be less counter-cyclical at high debt levels than at low debt levels because the distance to a public debt limit acts as a buffer-stock with insurance value.
- Buffer-stock mechanism:
  - The difference between current debt and the debt limit raises the marginal value of an extra unit of buffer when the buffer is small, inducing governments to preserve buffer rather than fully offset shocks.
- Model features (five key ones):
  - Automatic stabilizers: output affects the primary balance (higher output → higher tax revenue).
  - Hysteresis: downturns can have long-term negative effects on potential output.
  - Interest-elastic debt demand: level and change of debt affect interest rates (risk premium / bond price effect).
  - Fiscal multiplier sensitive to the output gap and market-access risk with an uncertain debt limit.
  - Realistic fiscal-stance setting: government decides stance one-year ahead, faces forecast uncertainty, and faces quadratic adjustment costs.
- Stochastic debt limit taken as exogenous; monetary policy is implicit and embedded in calibration.

### II. Simple models and qualitative implications
- Two contrasting cases:
  - Problem A: forward-looking government with Intertemporal Budget Constraint (IBC), no market-access risk — reaction to shocks is separable from debt level under a quadratic instantaneous utility.
  - Problem B: buffer-stock model with a debt limit d — introduces interaction between debt level and optimal reaction to shocks.
- Key numerical and qualitative points:
  - Example shock band used for illustration: positive and negative output shocks of 5%.
  - Without a debt limit:
    - Primary balance reacts to debt: ∂pb_t/∂d_{t-1} = 1 + r (equation 12).
    - No dependence of primary balance on shock in the fully constrained market-access loss case: ∂pb_t/∂e_t = 0 (equation 13).
  - With a debt limit:
    - Primary balance becomes convex in debt as debt approaches limit; the width of the shock-response band narrows at high debt.
    - Forward-looking government reduces sensitivity to shocks at higher debt because of precautionary preservation of buffer.
- Formal propositions (qualitative):
  - Proposition 1: With quadratic instantaneous utility (1/2) ω (y_t - y*)^2, the optimal fiscal reaction to Problem A is separable; explicit reaction function provided.
  - Proposition 2: For any government facing uncertainty on market access in the future, the optimal primary balance function is not separable: the sensitivity to shocks is lower at higher debt levels.
  - Proposition 3: A government facing sufficiently large negative shocks (condition using P(e_t < d_eq expression) in text) is always at risk to lose market access at some time in the future.
- Numerical illustration notes:
  - Figures illustrate bands: blue = without debt limit; orange/yellow = with debt limit; thin solid lines indicate ± one-standard deviation shock responses.
  - Debt limit thought experiment example: debt limit = 200% of GDP.
  - Example prudent government parameters mentioned: β = 0.99 and growth-adjusted interest rate = 2% used in illustration.
  - Equilibrium debt example: d_eq = 0% in illustrative parameterization.

### III. Augmented model — structure and calibration anchors
- Objective and state variables:
  - Government chooses change in structural primary balance pb^st_t to maximize discounted welfare with per-period utility u(c,L) = (1/(1-ρ)) c^{1-ρ} - ξ* L^{1+σ}/(1+σ).
  - State variables: debt d_t, output gap gap_t, structural primary balance pb^st_t.
- Market access / risk specification:
  - Probability of losing market access:
    - p_t = 1 - exp( - [1 + e^{ - ( (d_t - d)/(d_1) + (1/ d_2) * Δd_t ) } ] )
    - Parameters described: d_1 (debt limit uncertainty), d_2 (effect of debt change), d (debt level with 50% probability), d_3 (adverse shock size in sd units).
  - If market access lost, government must keep debt constant under adverse shock of d_3 * σ.
- Budget constraint and risk premium:
  - Standard debt accumulation with deterministic stock-flow adjustment sf_t implicit in equation: 1/(1+r*) * (1 + r_t) d_t = d_{t-1} + pb_t - g_t + sf_t.
  - Risk premium modeled as linear in past debt and in debt change in baseline; non-linear variants discussed.
- Output, hysteresis, and multiplier:
  - Production: Y_t = A_t * L_t; potential Y_t uses A_t at equilibrium L.
  - Hysteresis: permanent potential output loss when gap below threshold gap^{th}, governed by parameter h ≥ 0 and threshold h^{th}.
  - Shock process v_t and cyclical component with persistence; shocks normalized so gap = v_t if pb = 0.
  - Fiscal multiplier state-dependence:
    - Marginal effect of pb on gap: ∂gap/∂pb = - m_1 - m_2 * gap (multiplier larger in downturns).
    - Closed-form solution for gap as function of shock and structural balance uses Lambert W function.
  - Primary balance decomposition: pb_t = pb^st_t + a * gap_t, where a is automatic stabilizer semi-elasticity.
- Adjustment costs:
  - Resource cost of fiscal adjustment captured by χ * (Δpb^st_t)^2 /2.

### IV. Results — baseline and sensitivity analysis
- Baseline calibration represents an average advanced economy (calibration follows Fournier and Lieberknecht, 2019).
- Baseline findings:
  - Reaction to debt:
    - Structural primary balance change increases in past debt; relationship is convex (steeper response as debt approaches high levels).
    - Bands for ± one-standard deviation shocks narrow as debt increases.
  - Reaction to output gap:
    - Reaction to output gap roughly linear; width of shock-response band captures extent of counter-cyclicality.
- Hysteresis:
  - Higher hysteresis magnifies the difference between low- and high-debt responses:
    - At low debt: wider shock-response band (greater use of stimulus to avoid costly long-run output loss).
    - At high debt: ambiguous net effect; in calibration, bandwidths at high debt are similar for low and high hysteresis.
  - Higher hysteresis implies a lower optimal debt target (buffer more valuable).
- Interest-rate sensitivity to debt:
  - Greater sensitivity of risk premium to debt raises marginal cost of debt at high debt → governments should refrain from offsetting shocks at high debt.
  - Non-linear risk-premium specifications amplify convexity: little reaction at low debt, larger reaction at high debt.
- Growth sensitivity:
  - Higher potential growth gives leeway for larger debt and less tightening at high debt.
  - With higher growth, reactions to shocks at high debt can be more pronounced since debt concerns are milder.
- Fiscal multiplier:
  - Lower multipliers require larger primary-balance changes to stabilize output (both at low and high debt).
  - If multipliers are very low, governments should not react strongly to cycles; composition of fiscal instruments matters (spending vs. taxes).
- Automatic stabilizers:
  - Size of automatic stabilizers has limited quantitative effect:
    - Lower stabilizers → slightly more discretionary action at low debt and less steep reaction of structural primary balance to debt.
- Shock persistence:
  - Less persistence → lower reaction to previous-year gap.
  - Very high persistence and high debt → a highly indebted government may tighten in response to a negative shock to avoid future market-access loss.
- Shock size:
  - Greater shock standard deviation → larger short-term changes in primary balance (scale effect).
  - Baseline systematic primary balance when shock is null is similar across shock-size scenarios in this calibration.
- Numerical examples and parameter remarks preserved from text:
  - Example illustrated shocks: ±5% output shock bands.
  - Proposition 3 numerical illustration: negative shocks would need not exceed 2% of GDP for the peculiar case where government can ensure it never loses market access (given the example parameters).
  - Adverse shock upon market-access loss calibrated as d_3 * σ.

### V. Policy implications and applications
- Core policy recommendation:
  - Fiscal stance advice should be tailored to the country-specific debt level — higher debt implies a lower optimal degree of counter-cyclicality because preserving the buffer against market-access risk is valuable.
- Policy-relevant channels to consider in calibration:
  - Growth rate, interest-rate sensitivity to debt, hysteresis parameter, fiscal multiplier and its gap sensitivity, automatic stabilizers, shock persistence and size, adjustment costs.
- Use cases and extensions:
  - Tool can produce medium-run appropriate paths conditional on short-term forecasts and can be adjusted for institutional features (e.g., credibility-enhancing institutions → lower elasticity of interest rate to debt and higher effective debt limit).
  - Model can be embedded in DSGE frameworks to endogenize instrument choice and market learning processes; credibility-building over time could reduce market reaction to debt.
- Empirical guidance:
  - Expect non-linear (convex) reaction to debt and interaction between debt level and cyclicality of fiscal policy — standard separable fiscal-reaction specifications may miss these effects.

### Key calibration values (Table 1. Baseline Calibration)
- Welfare function
  - Discount factor β: 0.99
  - Risk aversion σ: 2
  - Labor elasticity η: 1/0.3
  - Weight of labor ξ: 1
- Fiscal parameters
  - Fiscal multiplier when the gap is null m1: 0.8
  - Fiscal multiplier sensitivity to shocks m2: 3
  - Automatic stabilizers (primary balance semi-elasticity to the gap) a: 0.52
  - Adjustment cost χ: 3
- Interest rate and debt parameters
  - Growth-adjusted interest rate at current debt level: 0.49%
  - Effect of debt level on the risk premium α: 1.5%
  - Effect of debt change on the risk premium α2: 0.5%
  - Debt level at which the risk to lose market access is 50% d: 150%
  - Debt limit accuracy d1: 3
  - Effect of debt change on the risk to lose market access d2: 1
  - Adverse scenario coefficient in case of loss of market access d3: 0
- Economy parameters
  - Potential GDP per capita growth: 1.6%
  - Shock persistence ρ: 0.60
  - Shock size σ: 3.2%
  - Hysteresis: 10%
  - Hysteresis threshold: -1%
- Calibration notes (exact interpretations preserved):
  - Overall discount rate with per capita growth at 1.6% implies the overall discount rate is about 0.975.
  - Fiscal multiplier m1 = 0.8 is the cross-country average from a continuous fiscal multiplier bucket approach; with m2 = 3, a negative output gap of five percent lowers the standard multiplier by 0.15.
  - Automatic stabilizers a = 0.52 is the cross-country average of primary balance semi-elasticities to GDP.
  - Adjustment cost χ = 3 is a moderate adjustment cost enabling sizable recommended adjustments if previous primary balance far from appropriate level.
  - Risk premium slope α = 1.5% is calibrated in the middle of empirical estimates; risk to lose market access uses d1 = 3, d2 = 1, d3 = 0, and d = 150%.
  - Potential growth assumption equals average WEO potential growth over 2017-21; growth-interest rate differential calibrated with 20-year averages across advanced economies from IMF WEO April 2018 data.
  - Hysteresis calibration: h = 0.1 and h_th = -1%, chosen so long-run effect aligns with literature postulating long-term effects of around 0–20%.

*Source: wpiea2019159 - References / Annex 3. Calibration*

### References .............................................................................................................

### wpiea2019159 - References .............................................................................................................

### References
- References ......................................................................................................................................................................... 29

### Annexes
- Annex 1. Analytic Results with the Simple Model .............................................................................................. 33
- Annex 2. The Gap as a Function of the Underlying Shock and the Structural Primary Balance ...... 36

*Source: wpiea2019159 - References (pages and annex listings as provided).*

### Annex 3. Calibration ...................................................................................................

### Annex 3. Calibration

### I. Introduction — fiscal stance and debt limit
- Main insight: the fiscal stance should be less counter-cyclical at high debt levels than at low debt levels because the distance to a public debt limit acts as a buffer-stock with insurance value.
- Buffer-stock mechanism: the difference between current debt and the debt limit raises the marginal value of an extra unit of buffer when the buffer is small, inducing governments to preserve buffer rather than fully offset shocks.
- Model features (five key ones):
  - Automatic stabilizers: output affects the primary balance (higher output → higher tax revenue).
  - Hysteresis: downturns can have long-term negative effects on potential output.
  - Interest-elastic debt demand: level and change of debt affect interest rates (risk premium / bond price effect).
  - Fiscal multiplier sensitive to the output gap and market-access risk with an uncertain debt limit.
  - Realistic fiscal-stance setting: government decides stance one-year ahead, faces forecast uncertainty, and faces quadratic adjustment costs.
- Stochastic debt limit taken as exogenous; monetary policy is implicit and embedded in calibration.

### II. Simple models and qualitative implications
- Two contrasting cases:
  - Problem A: forward-looking government with Intertemporal Budget Constraint (IBC), no market-access risk — reaction to shocks is separable from debt level under a quadratic instantaneous utility.
  - Problem B: buffer-stock model with a debt limit d — introduces interaction between debt level and optimal reaction to shocks.
- Key numerical and qualitative points:
  - Example shock band: a positive and negative output shock of 5% used to illustrate reactions.
  - Without a debt limit:
    - Primary balance reacts to debt: ∂pb_t/∂d_{t-1} = 1 + r (equation 12).
    - No dependence of primary balance on shock: ∂pb_t/∂e_t = 0 (equation 13) in fully constrained market-access loss case.
  - With debt limit:
    - Primary balance becomes convex in debt as debt approaches limit; the width of the shock-response band narrows at high debt (Figure 1 qualitative).
    - Forward-looking government reduces sensitivity to shocks at higher debt because of precautionary preservation of buffer.
- Formal propositions (qualitative statements preserved):
  - Proposition 1: With quadratic instantaneous utility 2() (1/2) ω (y_t - y*)^2, the optimal fiscal reaction to Problem A is separable; explicit reaction function provided (equation (5)).
  - Proposition 2: For any government facing uncertainty on market access in the future, the optimal primary balance function is not separable: the sensitivity to shocks is lower at higher debt levels (statement preserved).
  - Proposition 3: A government facing sufficiently large negative shocks (condition using P(e_t < d_eq expression) as in text) is always at risk to lose market access at some time in the future.

- Numerical illustration notes:
  - Bands in Figure 1: blue = without debt limit; orange/yellow = with debt limit; thin solid lines indicate ± one-standard deviation shock responses.

### III. Augmented (more realistic) model — structure and calibration anchors
- Value function and objective:
  - Government chooses change in structural primary balance pb^st_t (fiscal stance) to maximize discounted welfare with per-period utility u(c,L) = (1/(1-ρ)) c^{1-ρ} - ξ* L^{1+σ}/(1+σ) (as described).
  - State variables: debt d_t, output gap gap_t, structural primary balance pb^st_t.
- Market access / risk specification:
  - Probability of losing market access:
    - p_t = 1 - exp( - [1 + e^{ - ( (d_t - d)/(d_1) + (1/ d_2) * Δd_t ) } ] ) — qualitative functional form described; parameters: d_1 (debt limit uncertainty), d_2 (effect of debt change), d (debt level with 50% probability), d_3 (adverse shock size in sd units).
  - If market access lost, government must keep debt constant under adverse shock of d_3 * σ.
- Budget constraint:
  - Standard debt accumulation with deterministic stock-flow adjustment sf_t: 1/(1+r*) * (1 + r_t) d_t = d_{t-1} + pb_t - g_t + sf_t (implicit form described).
  - Risk premium modeled as linear in past debt and in debt change in baseline; non-linear variants discussed (equation (17) noted).
- Output, hysteresis, and multiplier:
  - Production: Y_t = A_t * L_t; potential Y_t uses A_t at equilibrium L.
  - Hysteresis: permanent potential output loss when gap below threshold gap^{th}, governed by parameter h ≥ 0 and threshold h^{th}.
  - Shock process v_t and cyclical component with persistence; shocks normalized so gap = v_t if pb = 0.
  - Fiscal multiplier state-dependence:
    - Marginal effect of pb on gap: ∂gap/∂pb = - m_1 - m_2 * gap (multiplier larger in downturns).
    - Closed-form solution uses Lambert W (equation (16) and text).
  - Primary balance decomposition: pb_t = pb^st_t + a * gap_t, where a is automatic stabilizer semi-elasticity.
- Adjustment costs:
  - Resource cost of fiscal adjustment captured by χ * (Δpb^st_t)^2 /2 (modeled as part of resource constraint).

### IV. Results — baseline and sensitivity analysis
- Baseline calibration: represents an average advanced economy (calibration follows Fournier and Lieberknecht, 2019; details in Annex 3).
- Baseline findings (Figures 3 A and B):
  - Reaction to debt:
    - Structural primary balance change increases in past debt; relationship is convex (steeper response as debt approaches high levels).
    - Bands for ± one-standard deviation shocks narrow as debt increases.
  - Reaction to output gap:
    - Reaction to output gap roughly linear; width of shock-response band captures extent of counter-cyclicality.
- Hysteresis (Figure 4):
  - Higher hysteresis magnifies the difference between low- and high-debt responses:
    - At low debt: wider shock-response band (greater use of stimulus to avoid costly long-run output loss).
    - At high debt: ambiguous net effect; in calibration, bandwidths at high debt are similar for low and high hysteresis.
  - Higher hysteresis implies a lower optimal debt target (buffer more valuable).
- Interest-rate sensitivity to debt (Figures 5 and 6):
  - Greater sensitivity of risk premium to debt raises marginal cost of debt at high debt → governments should refrain from offsetting shocks at high debt.
  - Non-linear risk-premium specifications amplify convexity: little reaction at low debt, larger reaction at high debt.
- Growth sensitivity (Figure 7):
  - Higher potential growth gives leeway for larger debt and less tightening at high debt.
  - With higher growth, reactions to shocks at high debt can be more pronounced since debt concerns are milder.
- Fiscal multiplier (Figures 8 and 9):
  - Lower multipliers require larger primary-balance changes to stabilize output (both at low and high debt).
  - If multipliers are very low, governments should not react strongly to cycles; composition of fiscal instruments matters (spending vs. taxes).
- Automatic stabilizers (Figure 10):
  - Size of automatic stabilizers has limited quantitative effect:
    - Lower stabilizers → slightly more discretionary action at low debt and less steep reaction of structural primary balance to debt.
- Shock persistence (Figures 11 and 12):
  - Less persistence → lower reaction to previous-year gap (because government anticipates less persistence when decision materializes).
  - Very high persistence and high debt → counterintuitive possibility: a highly indebted government may tighten in response to a negative shock to avoid future market-access loss.
- Shock size (Figure 13):
  - Greater shock standard deviation → larger short-term changes in primary balance (scale effect).
  - Baseline systematic primary balance when shock is null is similar across shock-size scenarios in this calibration (offsetting forces).
- Numerical examples and parameter values preserved from text:
  - Example illustrated shocks: ±5% output shock bands.
  - Debt limit example: debt limit = 200% of GDP considered in a thought experiment.
  - Example prudent government: β = 0.99 and growth-adjusted interest rate = 2% used in illustration.
  - Equilibrium debt example: d_eq = 0% in illustrative parameterization mentioned.
  - Proposition 3 numerical illustration: negative shocks would need not exceed 2% of GDP for the peculiar case where government can ensure it never loses market access (given the example parameters).
  - Adverse shock upon market-access loss calibrated as d_3 * σ (d_3 parameter described conceptually).

### V. Policy implications and applications
- Core policy recommendation: fiscal stance advice should be tailored to the country-specific debt level — higher debt implies a lower optimal degree of counter-cyclicality because preserving the buffer against market-access risk is valuable.
- Policy-relevant channels to consider in calibration:
  - Growth rate, interest-rate sensitivity to debt, hysteresis parameter, fiscal multiplier and its gap sensitivity, automatic stabilizers, shock persistence and size, adjustment costs.
- Use cases and extensions:
  - Tool can produce medium-run appropriate paths conditional on short-term forecasts and can be adjusted for institutional features (e.g., credibility-enhancing institutions → lower elasticity of interest rate to debt and higher effective debt limit).
  - Model can be embedded in DSGE frameworks to endogenize instrument choice and market learning processes; credibility-building over time could reduce market reaction to debt.
- Empirical guidance:
  - Expect non-linear (convex) reaction to debt and interaction between debt level and cyclicality of fiscal policy — standard separable fiscal-reaction specifications may miss these effects.

*Source: Annex 3. Calibration (wpiea2019159) — text supplied.*

### REFERENCES

### wpiea2019159 - REFERENCES

### References (selected)
- A list of cited works covering government debt, fiscal multipliers, fiscal policy, sovereign default, DSGE models, hysteresis, and debt sustainability. Notable authors and works include Adam (2011); Aiyagari et al. (2002); Alesina, Favero, and Giavazzi (2015); Auerbach and Gorodnichenko (2013); Ball (2014); Barro (1979); Batini et al. (2014); Baum, Poplawski-Ribeiro, and Weber (2012); Blanchard (2018); Blanchard and Perotti (2002); Bohn (1998, 2007, 2008); Canzoneri et al. (2016); Carroll (1992, 1997); Cerra and Saxena (2008); Christiano, Eichenbaum, and Evans (2005); Coenen et al. (2012); Cuadra, Sanchez and Sapriza (2010); Deaton (1991); Delong and Summers (2012); Eyraud and Weber (2013); Fournier and Fall (2017); Ghosh et al. (2013); Henao-Arbelaez and Sobrinho (2017); IMF Fiscal Monitor (2012/1, 2017/1); Lucas and Stokey (1983); Mendoza and Oviedo (2009); Mourougane (2017); Perotti (2004); Price, Dang and Botev (2015); Ramey and Zubairy (2018); Smets and Wouters (2007); Woodford (2011); Zubairy (2014). 

### Annex 1 — Analytic results with the simple model (key points)
- Analytic solution with a quadratic utility function and no debt limit (Proposition 1):
  - Utility: 2() (1/2) ttt uyyy ω = − + .
  - Conjectured linear primary balance reaction: 01 21 ttt pb ed β β β − = + + .
  - Two solutions for β2: β2 = 0 or β2 = 2 (1 )1(1) rr β β β − = + + .
  - If (1 ) 1 r β + < , first solution (β2 = 0) respects the weak transversality condition and corresponds to complete shock offsetting (government reaches global maximum of instantaneous utility: 1 ttt y e mpb ω + − ==).
  - If (1 ) 1 r β + ≥ , only the second solution respects the transversality condition.
  - Expressions for β1 and β0 are derived by extracting terms with et and terms without et and d_{t-1} respectively (see full derivation in source).

- Properties with a debt limit:
  - Initialization and backward iteration: at final period T either government reaches the instantaneous utility maximum or hits the debt limit:
    - 1 (1 1 max), T TT e pbr dd m ω − + + − (stylized final-period condition).
  - Backward iterations converge to the fixed-point infinite horizon solution (illustrated in Figure A1.1: Appropriate primary balance as a function of past debt level).
  - Derivative of the primary balance with respect to past debt:
    - For a government at risk of losing market access, the derivative with respect to debt is strictly increasing.
    - Induction with backward iteration shows at any period: 1 01 t t pb r d − − ∂ ≤ ≤ + ∂ and equivalently 11 01 1 tt tt dpb rr dd − − − − ∂ ∂ ≤ = + − ≤ + ∂ ∂ .
    - When government has market access, derivative is bounded strictly below 1 + r:
      - 1 2 1 011 1 1 (1 ) t m t pb r rr d r β − − ∂ + ≤ ≤ = + < + ∂ + + .
    - If government always has market access next period, iterative expressions for second derivatives show monotonicity properties (see source).

- Proposition 2 proof (sketch):
  - Lemma: 2 2 1 0 t t pb d − − ∂ > ∂ for a government with market access facing uncertainty on future market access.
  - By contraposition, define d_a as highest debt level with some shock where market access holds and 2 2 1 0 t t pb d − ∂ = ∂. Then show government never loses market access beyond that bound; hence derivative conditions imply monotone behavior.
  - Proposition 2 follows from equation (15): as derivative with respect to debt increases in debt, derivative with respect to shock decreases in debt.

- Proposition 3 proof (sketch):
  - If government never loses market access, solution without a debt limit is solution with debt limit and this occurs only if (1 ) 1 r β + > .
  - For any shock and any debt level, government not affected by debt limit constraint:
    - () () 1 111 1 1 (1)[(1.)](1) tttttt pbeur E uem pbr dd m β − ++ + − ′′ [=] + + + − ≥ + − − .
  - Worst-case with past debt at the limit yields condition:
    - () 2 (1 ) 1 (1 )(1 ) 1 eqt r erddrd mr d r β β β + − − + ≥ + + + .
  - Equivalent condition:
    - () () (1 ) (1 ) 1 eqt dem rr d β ≥ − + + − −  or () () () (1 ) 10(1 ) teq dPe m rrd β − + − + >< .

### Annex 2 — Output gap as a function of shock and structural primary balance
- Output gap identity (rearranged using Lambert W function):
  - Starting relation: 12 22 11 () t mm pb tt gap ve mm = + − .
  - Reformulation leads to an inversion using Lambert W (principal branch) because m2.gap assumed below 1 so principal Lambert function is the only real solution above -1.
  - Closed-form expression:
    - () 12 () 12 122 (1 ) 1 st t mampb t t Wamm ve gap am mm + − = − + .
  - Thus output gap is expressed as an explicit function of the underlying shock and the structural balance via the Lambert W function.

### Annex 3 — Calibration (baseline, following Fournier and Lieberknecht, 2019)
- Calibration purpose: baseline positive assessment reflecting averages across advanced economies; four parameter sets: (a) welfare function, (b) fiscal parameters, (c) cost of debt parameters, (d) economy parameters.
- Table 1. Baseline Calibration (values preserved exactly)

  - Welfare function
    - Discount factor β: 0.99
    - Risk aversion σ: 2
    - Labor elasticity η: 1/0.3
    - Weight of labor ξ: 1

  - Fiscal parameters
    - Fiscal multiplier when the gap is null m1: 0.8
    - Fiscal multiplier sensitivity to shocks m2: 3
    - Automatic stabilizers (primary balance semi-elasticity to the gap) a: 0.52
    - Adjustment cost χ: 3

  - Interest rate and debt parameters
    - Growth-adjusted interest rate at current debt level: 0.49%
    - Effect of debt level on the risk premium α: 1.5%
    - Effect of debt change on the risk premium α2: 0.5%
    - Debt level at which the risk to lose market access is 50% d: 150%
    - Debt limit accuracy d1: 3
    - Effect of debt change on the risk to lose market access d2: 1
    - Adverse scenario coefficient in case of loss of market access d3: 0

  - Economy parameters
    - Potential GDP per capita growth: 1.6%
    - Shock persistence ρ: 0.60
    - Shock size σ: 3.2%
    - Hysteresis: 10%
    - Hysteresis threshold: -1%

- Calibration notes and rationale (preserve exact numerical interpretations):
  - Overall discount rate with per capita growth at 1.6% implies the overall discount rate is about 0.975.
  - Fiscal multiplier m1 = 0.8 is the cross-country average from a continuous fiscal multiplier bucket approach; with m2 = 3, a negative output gap of five percent lowers the standard multiplier by 0.15.
  - Automatic stabilizers a = 0.52 is the cross-country average of primary balance semi-elasticities to GDP (Price et al. (2015)).
  - Adjustment cost χ = 3 is a moderate adjustment cost enabling sizable recommended adjustments if previous primary balance far from appropriate level.
  - Risk premium slope α = 1.5% is calibrated in the middle of empirical estimates; risk to lose market access uses d1 = 3, d2 = 1, d3 = 0, and d = 150%.
  - Potential growth assumption equals average WEO potential growth over 2017-21; growth-interest rate differential calibrated with 20-year averages across advanced economies from IMF WEO April 2018 data.
  - Shock parameters (size σ and persistence ρ) are estimated with past shocks reflecting output gap and primary balance.
  - Hysteresis calibration: h = 0.1 and h_th = -1%, chosen so long-run effect aligns with Blanchard and Summers (1987), Delong and Summers (2012) and Ball (2014) who postulate/find long-term effects of around 0–20%.

*Italic: Source: wpiea2019159 - REFERENCES*

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_Source: https://www.imf.org/-/media/files/publications/wp/2019/wpiea2019159.pdf_