## Numerical Fiscal Rules for Economic Unions: the Role of Sovereign Spreads (WP/21/196, Sections 1–2)

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### Introduction
- Study question: Gains from introducing a common numerical fiscal rule in a “Union” of model economies facing sovereign default risk.
- High-level findings:
  - Significant disagreement across economies about a common debt limit; a debt limit preferred by some economies can generate welfare losses in others.
  - A common sovereign spread limit (limit on the difference between a sovereign bond yield and a risk-free interest rate) can result in higher welfare across economies in the Union.
- Context and design trade-offs:
  - Fiscal rules aim to mitigate sovereign risk; the mapping from sovereign debt levels to sovereign risk (debt intolerance) varies across countries and time.
  - Alternative anchors (e.g., interest payments, spreads) have been proposed as complements or substitutes for debt limits.
  - Design trade-offs include simplicity, flexibility, sustainability, operational guidance, and enforceability.

### Model (three-period economy; default risk)
- Time and endowments:
  - Periods t = 1, 2, 3.
  - Endowments: y1 = y2 = 0, y3 ≥ 0. Uncertainty over y3 with c.d.f. F and density f, f(y3) > 0 for all y3 ≥ 0. Standard deviation σy.
- Preferences and technology:
  - Utility u(c) satisfies Inada conditions, consumption ct ≥ 0, government discount factor β < 1, expectation operator E.
- Debt instruments and timing:
  - Bond issued in period 1 promises δ units in period 2 and (1 − δ) units in period 3. If δ = 1, bonds are one-period; if δ < 1, long-term bonds.
  - Bond issued in period 2 promises one unit in period 3.
- Default:
  - Government may default in period 3; default implies losing fraction φ > 0 of y3.
  - Bonds priced by competitive risk-neutral investors discounting at rate 1.
- Default decision (period-3):
  - ˆd(b1,b2,y3) = 1 if y3 < b1(1−δ) + b2 φ; = 0 otherwise.
- Bond prices:
  - q1(b1,b2) = δ + (1−δ)[1 − F(b1(1−δ) + b2 φ)]
  - q2(b1,b2) = 1 − F(b1(1−δ) + b2 φ)
- Consumption allocations:
  - c1 = b1 q1(b1,b2)
  - c2 = b2 q2(b1,b2) − δ b1
  - c3 = y3[1 − ˆd(b1,b2,y3) φ] − [1 − ˆd(b1,b2,y3)][b1(1−δ) + b2]

### Single-economy results: Ramsey vs. Markov and role of debt structure
- Definitions:
  - {bR1, bR2} = Ramsey (commitment) borrowing maximizing expected utility in period 1 subject to feasibility (ct ≥ 0).
  - {bM1, bM2(bM1)} = Markov (sequential) borrowing chosen by governments in periods 1 and 2.
- Key propositions:
  - Proposition 1:
    - When δ = 1 (one-period debt), Markov policies coincide with Ramsey policies; no role for fiscal rules.
  - Proposition 2:
    - When δ < 1 (long-term debt), Markov policies do not coincide with Ramsey policies due to default risk and a time-inconsistency problem (debt dilution).
- Fiscal-rule instruments considered for limiting period-2 choices:
  - Debt-limit rule: ceiling on debt level (1 − δ) b1 + b2 ≤ B̄.
  - Spread-limit rule: ceiling on spread paid in period 2, equivalently floor on bond price q2(b1,b2) ≥ Q̄.
- Proposition 3:
  - For a single economy, imposing either a debt limit with threshold B̄* = (1−δ) bR1 + bR2 or a spread limit with threshold Q* = q2(bR1,bR2) makes Markov policies coincide with Ramsey policies. Thus, either instrument can commit the sequential government to the Ramsey allocation.
- Ramsey first-order conditions (representative):
  - u′(c1(bR1,bR2)) = β u′(c2(bR1,bR2)) = β^2 E[u′(c3(bR1,bR2,y3))[1−ˆd(bR1,bR2,y3)]] q2(bR1,bR2) + bR2 ∂q2(bR1,bR2)/∂b2.
- Divergence source between Ramsey and Markov:
  - The period-2 Markov decision maker ignores the marginal cost that period-2 borrowing imposes on period-1 consumption, leading to higher borrowing and divergence from the Ramsey solution.

### Union of heterogeneous economies (heterogeneity in debt intolerance φ)
- Heterogeneity:
  - Economies differ only in cost of defaulting φ (lower φ = less cost of defaulting = more debt intolerance).
  - Constrained Ramsey planner must apply same policy to every economy in the Union (political constraints or ignorance of true φ).
- Assumption 1 (sufficient condition for unique Ramsey policy per φ):
  - ζq(b) = (b/φ) f(b/φ) / [1 − F(b/φ)] is increasing in b, lim b→0 ζq(b) = 0, and lim b→∞ ζq(b) ≥ 1.
  - ζq is the absolute value of elasticity of period-2 bond price with respect to debt level b.
- Proposition 4 (special case u(c) = c, δ = 0):
  - Under Assumption 1, Ramsey policies are {bR1 = η φ, bR2 = 0}, with η ∈ R++ satisfying 1 − η f(η) / [1 − F(η)] = β^2.
  - Interpretation: an impatient risk-neutral borrower front-loads borrowing into period 1; Ramsey borrowing increases with cost of defaulting φ.
- Implication for common debt limit:
  - Since bR1 = η φ is increasing in φ, no single common debt ceiling B̄ can replicate Ramsey allocations for all φ in the Union.
- Advantage of common spread limit:
  - Under Proposition 4 assumptions, the bond price implied by Ramsey debt level is identical across economies: 1 − F(η).
  - Optimal common spread-limit threshold Q* = 1 − F(η) delivers Ramsey allocation in every economy of the Union.
- Proposition 5 (special case u(c) = c, δ = 0, Assumption 1):
  - For any Union differing only in φ, the optimal common spread-limit threshold is Q* = 1 − F(η) and achieves the Ramsey allocation in every economy, with η given by 1 − η f(η) / [1 − F(η)] = β^2.
  - Q* generates larger welfare gains than any common debt limit B̄.
  - Robustness: Q* is the same regardless of the distribution of welfare weights across economies in the Union.

### Numerical examples (relaxing some assumptions)
- Parameterization and assumptions:
  - u(c) = −c^{−1}
  - log(y3) ∼ N(0, 0.1)
  - δ = 0
  - Consumer discount factor βC = 1; government discount factor β = 0.8
  - Union with φ ∼ U[0.05, 0.2]
- Outcomes without a fiscal rule:
  - At end of period 2, debt levels between 3 and 12 percent of average period-3 income.
  - Spreads between 3 and 5 percent.
- Constrained Ramsey policy experiments:
  - Consider three common fiscal rules that maximize Union welfare under different weighting schemes: equal weights, all weight to least debt-intolerant economy, all weight to most debt-intolerant economy.
- Numerical findings:
  - Common spread limits outperform common debt limits.
  - Common debt limits prioritizing economies with less debt intolerance (higher φ) fail to generate welfare gains in the majority of Union economies because the higher debt limit preferred by low-debt-intolerance economies is not binding for many economies.
  - Common debt limits set to satisfy high-debt-intolerance economies impose unnecessarily low debt for low-debt-intolerance economies and cause welfare losses.
  - Common spread limits produce welfare gains for all economies in the Union.
  - The rule that maximizes average welfare achieves more than 80 percent of the welfare gains obtained with the best rule for each economy.

### Policy implications and design considerations
- Main policy implication:
  - A common spread-limit fiscal rule is a more robust and welfare-improving instrument for a Union of heterogeneous economies than a common debt-limit rule.
- Practical considerations for spread rules:
  - Spread rules require a sovereign interest rate freely determined in a liquid market that does not reflect bailout expectations; not every country has such a rate.
  - Concerns: bond prices can be delinked from fundamentals; spreads can react abruptly.
  - Counterpoints from companion quantitative analysis: spread rules can perform well even when spreads are influenced by non-domestic fundamentals, and having a spread rule in place when a large shock hits moderates spread increases and yields substantial gains for both government and bondholders.
  - If sovereign spreads are zero and carry no information, spread anchors could be problematic, but anchored expectations remain useful when shocks increase spreads.
- Institutional complements:
  - Independent fiscal councils and escape clauses enhance enforceability and flexibility; these developments could be applied to spread-based fiscal rules.
- Robustness advantage:
  - The optimal common spread limit Q* does not depend on the distribution of welfare weights across economies and thus is a robust policy choice when welfare weights are unknown or politically constrained.

### Conclusions (Section 2)
- Since levels of debt intolerance are difficult to identify, and seem to vary greatly both across countries and over time, a spread limit is likely to be a more robust fiscal anchor than a debt limit.
- Economies should be allowed to issue more debt when they suffer less of a debt intolerance problem. A spread limit allows for this and a debt limit does not.

*Source: WP/21/196 — Numerical Fiscal Rules for Economic Unions: the Role of Sovereign Spreads (Sections 1–2).*

### Section 1

### Numerical Fiscal Rules for Economic Unions: the Role of Sovereign Spreads

### Introduction
- Study question: Gains from introducing a common numerical fiscal rule in a “Union” of model economies facing sovereign default risk.
- Key high-level findings:
  - There is significant disagreement across economies about a common debt limit; a debt limit preferred by some economies can generate welfare losses in others.
  - A common sovereign spread limit (limit on the difference between a sovereign bond yield and a risk-free interest rate) can result in higher welfare across economies in the Union.
- Contextual notes:
  - Fiscal rules aim to mitigate sovereign risk; the mapping from sovereign debt levels to sovereign risk (debt intolerance) varies across countries and time.
  - Prior critiques of debt limits exist; alternative anchors like interest payments have been proposed.
  - Design trade-offs for fiscal rules include simplicity, flexibility, sustainability, operational guidance, and enforceability.

### Model (three-period economy; default risk)
- Time structure:
  - Periods t = 1, 2, 3.
  - Endowments: y1 = y2 = 0, y3 ≥ 0. Uncertainty only about y3 with c.d.f. F and density f, f(y3) > 0 for all y3 ≥ 0. Standard deviation σy.
- Preferences and technology:
  - Utility u(c) satisfies Inada conditions, consumption ct ≥ 0, government discount factor β < 1, expectation operator E.
- Debt instruments:
  - Bond issued in period 1 promises δ units in period 2 and (1 − δ) units in period 3. If δ = 1, bonds are one-period; if δ < 1, long-term bonds.
  - Bond issued in period 2 promises one unit in period 3.
- Default:
  - Government may default in period 3; default implies losing fraction φ > 0 of y3.
  - Bonds priced by competitive risk-neutral investors discounting at rate 1.
- Default decision (equation (1)):
  - ˆd(b1,b2,y3) = 1 if y3 < b1(1−δ) + b2 φ; = 0 otherwise.
- Bond prices:
  - q1(b1,b2) = δ + (1−δ)[1 − F(b1(1−δ) + b2 φ)]  (equation (2))
  - q2(b1,b2) = 1 − F(b1(1−δ) + b2 φ)  (equation (3))
- Consumption allocations (equation (4)):
  - c1 = b1 q1(b1,b2)
  - c2 = b2 q2(b1,b2) − δ b1
  - c3 = y3[1 − ˆd(b1,b2,y3) φ] − [1 − ˆd(b1,b2,y3)][b1(1−δ) + b2]

### Results — single economy: Ramsey vs. Markov and role of debt structure
- Definitions:
  - {bR1, bR2} = Ramsey (commitment) borrowing maximizing expected utility in period 1 subject to feasibility (ct ≥ 0).
  - {bM1, bM2(bM1)} = Markov (sequential) borrowing chosen by governments in periods 1 and 2.
- Proposition 1:
  - When δ = 1 (one-period debt), Markov policies coincide with Ramsey policies; no role for fiscal rules.
- Proposition 2:
  - When δ < 1 (long-term debt), Markov policies do not coincide with Ramsey policies due to default risk and a time-inconsistency problem (debt dilution).
- Two fiscal rule instruments considered for limiting period-2 choices:
  - Debt-limit rule: ceiling on debt level (1 − δ) b1 + b2 ≤ B̄.
  - Spread-limit rule: ceiling on spread paid in period 2, equivalently floor on bond price q2(b1,b2) ≥ Q̄.
- Proposition 3:
  - For a single economy, imposing either a debt limit with threshold B̄* = (1−δ) bR1 + bR2 or a spread limit with threshold Q* = q2(bR1,bR2) makes Markov policies coincide with Ramsey policies. Thus, either instrument can commit the sequential government to the Ramsey allocation.

### Results — Union of heterogeneous economies (heterogeneity in debt intolerance φ)
- Setup:
  - Union of economies differ only in cost of defaulting φ (lower φ = less cost of defaulting = more debt intolerance).
  - Constrained Ramsey planner must apply same policy to every economy in the Union (political constraints or ignorance of true φ).
- Assumption 1 (sufficient condition for unique Ramsey policy per φ):
  - ζq(b) = (b/φ) f(b/φ) / [1 − F(b/φ)] is increasing in b, lim b→0 ζq(b) = 0, and lim b→∞ ζq(b) ≥ 1.
  - ζq is the absolute value of elasticity of period-2 bond price with respect to debt level b.
- Proposition 4 (special case u(c) = c, δ = 0):
  - Under Assumption 1, Ramsey policies are {bR1 = η φ, bR2 = 0}, with η ∈ R++ satisfying 1 − η f(η) / [1 − F(η)] = β^2 (equation (5)).
  - Interpretation: an impatient risk-neutral borrower front-loads borrowing into period 1; Ramsey borrowing increases with cost of defaulting φ.
- Implication for common debt limit:
  - Since bR1 = η φ is increasing in φ, no single common debt ceiling B̄ can replicate Ramsey allocations for all φ in the Union; economies with lower debt intolerance (lower φ) should be allowed to borrow more, which a common debt limit cannot accommodate.
- Advantage of common spread limit:
  - Under Proposition 4 assumptions, the bond price implied by Ramsey debt level is identical across economies: 1 − F(η).
  - Optimal common spread-limit threshold Q* = 1 − F(η) delivers Ramsey allocation in every economy of the Union.
- Proposition 5 (special case u(c) = c, δ = 0, Assumption 1):
  - For any Union differing only in φ, the optimal common spread-limit threshold is Q* = 1 − F(η) and achieves the Ramsey allocation in every economy, with η given by equation (5).
  - Q* generates larger welfare gains than any common debt limit B̄.
  - Robustness: Q* is the same regardless of the distribution of welfare weights across economies in the Union.

### Numerical examples (relaxing some assumptions)
- Parameterization and assumptions:
  - u(c) = −c^{−1}
  - log(y3) ∼ N(0, 0.1)
  - δ = 0
  - Consumer discount factor βC = 1; government discount factor β = 0.8 (captures political myopia)
  - Union with φ ∼ U[0.05, 0.2]
- Outcomes without a fiscal rule:
  - At end of period 2, debt levels between 3 and 12 percent of average period-3 income.
  - Spreads between 3 and 5 percent.
- Constrained Ramsey policy experiments:
  - Consider three common fiscal rules that maximize Union welfare under different weighting schemes: equal weights, all weight to least debt-intolerant economy, all weight to most debt-intolerant economy.
- Numerical findings:
  - Common spread limits outperform common debt limits.
  - Common debt limits prioritizing economies with less debt intolerance (higher φ) fail to generate welfare gains in the majority of Union economies because the higher debt limit preferred by low-debt-intolerance economies is not binding for many economies.
  - Common debt limits set to satisfy high-debt-intolerance economies impose unnecessarily low debt for low-debt-intolerance economies and cause welfare losses.
  - Common spread limits produce welfare gains for all economies in the Union.
  - The rule that maximizes average welfare achieves more than 80 percent of the welfare gains obtained with the best rule for each economy.

### Policy implications and design considerations
- Main policy implication: A common spread-limit fiscal rule is a more robust and welfare-improving instrument for a Union of heterogeneous economies than a common debt-limit rule.
- Practical considerations for spread rules:
  - Spread rules require a sovereign interest rate freely determined in a liquid market that does not reflect bailout expectations; not every country has such a rate.
  - Concerns raised: bond prices can be delinked from fundamentals; spreads can react abruptly.
  - Counterpoints from companion quantitative analysis: spread rules can perform well even when spreads are influenced by non-domestic fundamentals, and having a spread rule in place when a large shock hits moderates spread increases and yields substantial gains for both government and bondholders.
  - If sovereign spreads are zero and carry no information, spread anchors could be problematic, but anchored expectations remain useful when shocks increase spreads.
- Institutional complements:
  - Independent fiscal councils and escape clauses enhance enforceability and flexibility, and these developments could be applied to spread-based fiscal rules.
- Robustness advantage: The optimal common spread limit Q* does not depend on the distribution of welfare weights across economies and thus is a robust policy choice when welfare weights are unknown or politically constrained.

*Source: WP/21/196 — Numerical Fiscal Rules for Economic Unions: the Role of Sovereign Spreads (Section 1).*

### Section 2

### Section 2

### Conclusions
- Since levels of debt intolerance are difficult to identify, and seem to vary greatly both across countries and over time, a spread limit is likely to be a more robust fiscal anchor than a debt limit.
- Economies should be allowed to issue more debt when they suffer less of a debt intolerance problem. A spread limit allows for this and a debt limit does not.

### Empirical / graphical finding (Figure 1)
- Figure 1 reports the ratio of welfare gain obtained with a common rule for the Union relative to the welfare gain delivered by the optimal idiosyncratic rules.
- Axis and plotted markers from the figure (as labeled in the source):
  - Horizontal axis markers: 0.05 0.1 0.15 0.2
  - Vertical axis markers (left panel): -1100 -800 -500 -200 100
  - Vertical axis markers (right panels): 0 20 40 60 80 100
  - Series labels shown in panels:
    - With common debt rule lowest
    - With common debt rule average (rhs)
    - With common debt rule highest (rhs)
    - With common spread rule lowest
    - With common spread rule average
    - With common spread rule highest
- Caption: "Figure 1: Ratio of welfare gain obtained with a common rule for the Union relative to the welfare gain delivered by the optimal idiosyncratic rules."

### Key theoretical results (propositions and interpretations)
- Proposition 1: Ramsey policies maximize the government’s expected utility in period 1 and coincide with Markov policies when b_M1 = b_R1 and b_M2(b_M1) = b_R2; there is no room for improving welfare with fiscal rules in this case.
  - Ramsey first-order conditions (representative): u′(c1(b_R1,b_R2)) = β u′(c2(b_R1,b_R2)) = β^2 E[u′(c3(b_R1,b_R2,y3))[1−ˆd(b_R1,b_R2,y3)]] q2(b_R1,b_R2) + b_R2 ∂q2(b_R1,b_R2)/∂b2.
- Proposition 2: Markov policies differ from Ramsey policies in general because a period-2 government does not internalize the marginal cost that period-2 borrowing imposes on period-1 consumption.
  - Relevant condition showing divergence (equations reproduced as in source):
    - Ramsey condition (period 1 vs period 2):
      u′(c1(b_R1,b_R2))[q1(b_R1,b_R2) + b_R1 ∂q1(b_R1,b_R2)/∂b1] = β u′(c2(b_R1,b_R2))[δ − b_R2 ∂q2(b_R1,b_R2)/∂b1] + β^2 (1−δ) E[u′(c3(b_R1,b_R2,y3))[1−ˆd(b_R1,b_R2,y3)]]  (equation (6))
    - Ramsey second condition:
      β u′(c2(b_R1,b_R2))[q2(b_R1,b_R2) + b_R2 ∂q2(b_R1,b_R2)/∂b2] = β^2 E[u′(c3(b_R1,b_R2,y3))[1−ˆd(b_R1,b_R2,y3)]] − u′(c1(b_R1,b_R2)) b_R1 ∂q1(b_R1,b_R2)/∂b2.  (equation (7))
    - Markov period-2 condition for any b1:
      u′(c2(b1,b_M2(b1)))[q2(b1,b_M2(b1)) + b_M2(b1) ∂q2(b1,b_M2(b1))/∂b2] = β E[u′(c3(b1,b_M2(b1),y3))[1−ˆd(b1,b_M2(b1),y3)]].  (equation (8))
  - The second term on the right-hand side of equation (7) is positive and represents the marginal cost that borrowing in period 2 imposes on consumption in period 1; this cost is ignored by period-2 Markov decision makers, causing divergence from Ramsey.
- Proposition 3: Because f(y3) ≥ 0 for all y3 ≥ 0, q2 is strictly decreasing in b2. If the period-1 government chooses b_R1, imposing a spread brake threshold q* = q2(b_R1,b_R2) is equivalent to imposing a debt brake threshold b̄* = (1−δ)b_R1 + b_R2.
  - Under the condition u′(c2(b_R1,b_R2))[q2(b_R1,b_R2) + b_R2 ∂q2(b_R1,b_R2)/∂b_R2] > β E[u′(c3(b_R1,b_R2,y3))[1−ˆd(b_R1,b_R2,y3)]], and with b_M1 = b_R1 and b_M2(b_M1) = b_R2, Markov policies coincide with Ramsey policies.
- Proposition 4 (special case with linear utility): With u(c) = c and β < 1, the government’s expected utility in period 1 is maximized with period-2 consumption equal to zero. Therefore, Ramsey policies are {b_R1, 0}, where b_R1 satisfies
  - 1 − b_R1 φ f(b_R1 φ) / (1 − F(b_R1 φ)) = β^2.  (equation (9))
  - Under Assumption 1 there is a unique level of b_R1 φ that solves equation (9). Let η denote this level. Then, for any economy with cost of defaulting φ, b_R1 = η φ.

### Policy implication emphasized
- Because debt intolerance varies across countries and over time and is hard to identify, policy designs that use a spread limit (spread brake) can adapt to differences in debt intolerance by allowing economies with lower debt intolerance to borrow more. Debt limits do not allow for this flexibility.
- The mathematical equivalence between a spread brake and a debt brake depends on the period-1 choices and the mapping q2(b_R1,b_R2); in particular, a spread brake threshold q* = q2(b_R1,b_R2) corresponds to a debt brake threshold b̄* = (1−δ)b_R1 + b_R2 when the period-1 government chooses b_R1.

*Source: wpiea2021196-print-pdf - Section 2*

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_Source: https://www.imf.org/-/media/files/publications/wp/2021/english/wpiea2021196-print-pdf.pdf_