## _wp1230

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---

### Purpose and main contributions
- Objective: shed light on the optimal value of fiscal rules’ parameters and on the effects of imposing fiscal rules when a government can commit to enforce them.
- Primary contributions:
  - Determine optimal fiscal-rule parameter values within a sovereign-default framework à la Eaton and Gersovitz (1981) with long-duration bonds.
  - Analyze the trade-off between larger bondholders’ capital gains and the fiscal cost of stronger adjustment when the government conducts a voluntary debt restructuring.
  - Measure how fiscal rules affect the procyclicality of fiscal policy and, through that channel, the volatility of private consumption.

### Key quantitative findings and policy-relevant results
- Pre-rule calibration and typical values:
  - Pre-rule debt level: 38 percent of mean income (typical in the data for the pre-2001-default Argentina used to calibrate the model).
- Optimal rule in benchmark (no-rule) economy:
  - Government-preferred rule: a debt ceiling of 30 percent of mean income, with a delay of four years between the rule announcement period and the first period in which the ceiling is enforced.
  - Mean borrowing level: falls only from 3.6 percent to 3.1 percent of average income under the rule.
- Voluntary debt restructuring scenario:
  - Government chooses a 25 percent ceiling to be enforced less than two years after its announcement in the voluntary restructuring case.
  - Creditor debt forgiveness increases with the pre-rule spread: with a pre-rule spread of 5 (15) percent, bondholders are willing to forgive up to 21 (44) percent of the government’s debt.
- Effects on borrowing costs and access:
  - Committing to a fiscal rule reduces expected future debt and lowers the interest rate (sovereign spread) at which the government borrows, creating capital gains for bondholders and more favorable borrowing opportunities for the government.
- Effects on cyclicality and consumption volatility:
  - Without a fiscal rule, a high and countercyclical interest rate induces the government (a sequential decision-maker) to borrow less when income is low, producing procyclical fiscal policy.
  - Lower debt ceilings make the interest rate less responsive to income shocks, reducing fiscal procyclicality and lowering consumption volatility.
  - A countercyclical debt ceiling (higher when income is lower) increases both the level and countercyclicality of the interest rate; thus, a countercyclical ceiling does not necessarily reduce consumption volatility.
  - Among linear functions of current income studied, the government prefers procyclical ceilings (e.g., a ceiling on the debt-to-income ratio) over constant debt ceilings.
  - Countercyclical ceilings are preferred only among ceilings that impose excessively low average debt levels; the government prefers the no-rule benchmark over such low ceilings.

### Mechanisms and interpretation
- Why rules help:
  - Commitment to a rule changes lenders’ expectations about future indebtedness, reducing the sovereign risk premium and borrowing costs for a given debt level.
- Trade-off under voluntary restructuring:
  - Stronger and quicker fiscal consolidation raises bondholders’ capital gains (via higher market values of existing bonds) but increases the fiscal cost to the sovereign; the government can use voluntary restructuring to capture capital gains by obtaining creditor debt relief in exchange for announcing and enforcing a stricter rule.
- Limits to cyclicality analysis:
  - Analysis excludes production and therefore is limited in studying the full mechanics of cyclicality of fiscal policy.

### Model framework and assumptions
- Environment:
  - Single tradable good; stochastic endowment y_t with log(y_t) = (1−ρ)μ + ρ log(y_{t−1}) + ε_t, |ρ|<1, ε_t ∼ N(0, σ^2_ε).
  - Government maximizes E[∑_{t=0}^∞ β^t u(c_t)], with u(c) = c^(1−γ) −1 / (1−γ) (constant relative risk aversion γ).
- Debt instrument:
  - Long-term bonds: bond issued in period t promises one unit in period t+1 and (1−δ)^{s−1} units in period t+s for s ≥ 2; bonds follow Hatchondo and Martinez (2009) style long-duration specification.
- Default and costs:
  - Default is all-or-nothing across all current and future obligations; recovery rate in default is zero.
  - Two costs of default:
    - Exclusion from capital markets; after default, the country regains access with probability ψ ∈ [0,1] each period.
    - Per-period income loss while excluded: φ(y) = d_0 y + d_1 y^2.
- Pricing and risk premium:
  - Lenders’ stochastic discount factor: M(y′, y) = exp(−r − α ε′ − 0.5 α^2 σ^2_ε), where r denotes the risk-free rate; formulation yields a risk premium driven by the borrowing economy’s income shock.
  - Discount factor specification is a discrete-time version of the Vasicek one-factor model of the term structure.
- Solution concept and computational approach:
  - Markov Perfect Equilibrium.
  - Avoid multiplicity by solving a finite-horizon version and extending horizon until convergence of first- and second-period value functions and bond prices; first-period functions used as infinite-horizon equilibrium.

### Recursive formulation and equilibrium conditions
- State variables and functions:
  - b: number of outstanding coupon claims at beginning of current period (b < 0 implies net issuer in the past).
  - d: default indicator (d = 1 if defaulted in current period, else 0).
  - V(b, y): government’s value function before the default decision.
  - V_0(b, y): value if not in default; V_1(y): value if in default.
- Bellman-like equations:
  - V(b,y) = max_{d ∈ {0,1}} { d V_1(y) + (1−d) V_0(b,y) }.
  - V_1(y) = u(y − φ(y)) + β ∫ [ ψ V(0, y′) + (1−ψ) V_1(y′) ] F(dy′ | y).
  - V_0(b,y) = max_{b′ ≤ 0} { u(y + b − q(b′, y) [ b′ − (1−δ) b ]) + β ∫ V(b′, y′) F(dy′ | y) }.
- Bond price equation:
  - q(b′, y) = ∫ M(y′, y) [1 − h(b′, y′)] F(dy′ | y)
    + (1−δ) ∫ M(y′, y) [1 − h(b′, y′)] q(g(h(b′, y′), b′, y′), y′) F(dy′ | y),
    where h denotes the future default rule and g the future borrowing rule expected by lenders.
- Equilibrium requirement:
  - Optimal default and borrowing rules that solve the government’s problems must equal the rules h and g that lenders expect for all state variable realizations.

### Model structure and equilibrium objects
- Equilibrium consists of:
  - a set of value functions V, V1, and V0;
  - a default rule h and a borrowing rule g;
  - a bond price function q.
- Fiscal policy (stylized):
  - Government choice each period: tax revenues τ (possibly negative).
  - In default: τ = 0; private agents consume c = y − φ(y).
  - When not in default: private agents consume c = y − τ; government budget: τ − q(b′, y) [b′ − (1 − δ)b] = −b.

### Fiscal rules studied
- Fiscal rules parameterized by (b̄(y), n):
  - b̄(y) = a0 + a1 y denotes the debt ceiling (linear in y).
  - n denotes the number of periods between the announcement of the rule and the period after which the ceiling is enforced.
- After first enforcement period, government solves optimization with constraint b′ ≥ b̄(y).

### Calibration
- Case study: Argentina before the 2001 default.
- Numerical solution method: value function iteration and interpolation (linear interpolation for endowment levels and spline interpolation for asset positions).
- Period: quarter.
- Parameter values (Table 1):
  - Borrower’s risk aversion σ: 2
  - Interest rate r: 0.01
  - Income autocorrelation coefficient ρ: 0.9
  - Standard deviation of innovations σϵ: 0.027
  - Mean log income μ(-1/2)σ2ϵ (as presented)
  - Exclusion ψ: 0.282
  - Duration δ: 0.0341
  - Discount factor β: 0.961
  - Default cost d0: -0.69
  - Default cost d1: 1.017
  - Risk premium α: 3
- Additional calibration details:
  - ψ = 0.282 (probability of regaining access to capital markets).
  - With δ = 0.0341, bonds have average duration of 4.19 years in baseline simulations (Macaulay definition of duration D = (1 + r*)/(δ + r*) where r* denotes the constant per-period yield).
- Targets used to calibrate β, default costs, and α:
  - mean spread = 7.4 percent
  - standard deviation of spread = 2.5 percent
  - mean debt to (annual) GDP ratio = 40 percent (pre-default sample)
  - default frequency = three defaults per 100 years

### Results — overview
- Sequence of results discussed:
  1. Benchmark economy simulations versus data.
  2. Optimal fiscal rules.
  3. Capital gains on previously issued debt upon rule announcement and distributional effects.
  4. How fiscal rules create expectations of lower future indebtedness and benefit the government.
  5. Extent to which fiscal rules reduce procyclicality of fiscal policy.
  6. Cyclically adjusted fiscal rules.

### A. The no-rule benchmark — fit to data
- Simulation design:
  - Extract 500 samples of 32 consecutive periods before a default to compare with Argentine data (fourth quarter 1993 to third quarter 2001).
  - Include only samples where last default was at least two periods before sample start.
  - Default frequencies computed using all simulation data.
  - Trends computed using Hodrick-Prescott filter with smoothing parameter 1,600.
  - Debt face value computed dividing b by δ + r.
- Table 2: Business cycle statistics (Data vs Benchmark)
  - E(Rs): 7.44 vs 7.42
  - σ(Rs): 2.51 vs 2.52
  - Mean debt-to-income ratio: 0.40 vs 0.39
  - Defaults per 100 years: 3.00 vs 2.99
  - σ(c)/σ(y): 0.94 vs 1.23
  - ρ(c,y): 0.97 vs 0.99
  - ρ(Rs,y): -0.65 vs -0.79
- Key benchmark findings:
  - Baseline model matches data reasonably well on targeted moments.
  - Consumption and income are highly correlated; spread is countercyclical.
  - Consumption volatility is higher than income volatility in simulations.
  - Fiscal policy in benchmark is procyclical: taxes tend to be higher (or transfers lower) when income is lower.
  - Intuition: In bad times higher borrowing costs lead government to finance more debt service with taxes instead of new issuances.
  - Bond price is lower in bad times; taxes tend to be higher when sovereign interest rate spread is higher.

### B. Fiscal rules — government preferences and announcement effects
- Experiment focus:
  - Imposing simple fiscal rules with constant debt ceiling and specified delay between announcement and enforcement.
  - Pre-rule starting debt level: 38 percent of mean income.
  - Consider three pre-rule spread cases: 5.1 percent (low-risk), 7.4 percent (normal-risk), 15 percent (high-risk).
- Welfare measure:
  - Welfare gain from moving from benchmark to rule economy: ((VR(b,y)/VB(b,y))^(1/(1−σ)) − 1)
- Main numerical findings (Table 3):
  - For initial spreads 5.1%, 7.4%, 15.0%:
    - Optimal debt ceiling: 30% (for all three cases)
    - Optimal transition length (quarters): 18, 17, 17 (respectively)
    - Welfare gain: 0.23%, 0.23%, 0.22% (respectively)
    - Spread decline at announcement period: 0.9%, 1.4%, 0.3% (respectively)
- Interpretation:
  - Pre-rule default risk has little effect on the government-preferred rule: all prefer a 30% ceiling with about a four-year delay.
  - Government prefers delaying enforcement to reduce cost of adjusting debt; typically delays reduction until last two years of transition.
  - Announcement of the rule bids up sovereign bond prices immediately, reducing spreads even before debt reduction occurs.
  - Spread decline at announcement is smaller for higher initial risk.
- Dynamic paths (when income shocks equal zero):
  - Post-announcement debt-to-income ratio path shows gradual adjustment; debt reduction concentrated in last two years of transition.
  - Post-announcement spread declines continuously; for the high-risk economy spread expected to decline to almost half pre-rule value after one year.
- Tradeoff between transition length and spread decline:
  - For 7.4 percent pre-rule spread, choosing a two-year transition instead of four years could double the spread decline implied by announcement.
  - Spread decline implies capital gains for holders of previously issued debt; government does not capture these gains in this baseline scenario, creating a conflict of interest between government and creditors who prefer shorter transitions and lower ceilings.

### C. Fiscal rules after a debt restructuring (government captures capital gains)
- Exercise:
  - Government makes a take-it-or-leave-it debt buyback offer before rule announcement; rule announced only if offer accepted. Government buys back bonds at pre-rule price (extreme case where government captures all capital gains).
- Main numerical findings (Table 4):
  - For initial spreads 5.1%, 7.4%, 15.0%:
    - Optimal debt ceiling: 25% (for all three cases)
    - Optimal transition length (quarters): 7, 6, 3 (respectively)
    - Debt forgiveness (percent of government’s debt that lenders forgive): 21%, 25%, 44% (respectively)
    - Welfare gain: 1.02%, 1.29%, 2.28% (respectively)
- Interpretation:
  - When government captures capital gains, it chooses a lower debt ceiling and enforces it sooner.
  - Shorter transition periods for higher pre-rule risk are consistent with larger debt forgiveness required.
  - Welfare gains from combining rule announcement with a voluntary debt restructuring increase with pre-rule risk and can be substantial.

### D. Fiscal rules and borrowing opportunities (debt dilution and commitment)
- Mechanism:
  - Commitment to lower future debt levels reduces lenders’ expectations of future indebtedness and lowers current borrowing rates.
  - Imposing a debt ceiling can create new borrowing opportunities: the rule allows government to pay a lower interest rate for the same debt level.
- Illustration and quantitative implication:
  - Spread as a function of face value of next-period debt shifts down when a debt ceiling is imposed.
  - Economies with debt ceilings that imply significantly lower debt levels do not necessarily display much lower borrowing levels because the ceiling allows access to significantly lower interest rates.
  - Example (partial): with a ceiling of 25 percent of mean income, mean interest rate spread decreases (text truncates in source), while mean borrowing only decreases from 3.6 to 3.1 percent of mean income.

### Fiscal rules and the cyclicality of fiscal policy
- Lower debt levels implied by the imposition of a fiscal rule weaken the government’s incentives to implement a procyclical fiscal policy.
- Debt ceilings reduce default risk and thus reduce both:
  - the mean spread and
  - the responsiveness of the spread to income shocks (as reflected in a lower standard deviation of the spread).
- As the cost of borrowing becomes less responsive to income shocks, the incentive for procyclical fiscal policy weakens, leading to lower consumption volatility relative to income.
- These findings support the view that targeting low sovereign debt levels creates room for the implementation of countercyclical fiscal policy.

### Key simulation results (selected exact figures)
- From Table 5 (selected columns):
  - E(Rs): 7.42, 6.64, 3.6, 1.66, 0.54
  - σ(Rs): 2.52, 2.85, 1.93, 1.29, 0.7
  - Mean debt-to-income ratio: 0.39, 0.38, 0.34, 0.35, 0.24
  - Defaults per 100 years: 2.99, 2.86, 1.33, 0.42, 0.09
  - σ(c)/σ(y): 1.23, 1.12, 1.08, 1.05, 1.02
  - ρ(c,y): 0.99, 0.99, 0.99, 0.99, 0.99
  - ρ(Rs,y): -0.79, -0.72, -0.68, -0.62, -0.53
  - Mean borrowing /average income (%): 3.65, 3.63, 3.48, 3.14, 2.63
- From Table 6 (average ceiling = 25 percent of mean annual income, varying slope a1):
  - E(Rs): 0.09, 0.59, 1.00, 1.66, 2.10, 2.41, 2.63
  - σ(Rs): 0.03, 0.46, 0.81, 1.29, 1.40, 1.41, 1.47
  - Mean debt-to-income ratio: 0.23, 0.25, 0.27, 0.30, 0.30, 0.31, 0.31
  - Defaults per 100 years: 0.03, 0.13, 0.25, 0.42, 0.65, 0.75, 0.79
  - σ(c)/σ(y): 1.73, 1.49, 1.27, 1.05, 0.97, 1.04, 1.05
  - ρ(Rs,y): -0.86, -0.65, -0.62, -0.62, -0.62, -0.67, -0.67
  - Mean borrowing /average income (%): 2.67, 2.76, 2.97, 3.14, 3.26, 3.30, 3.34

### Cyclically adjusted fiscal rules
- Ex-ante optimal ceiling enforced immediately (n = 0) for a government that currently does not have debt:
  - Optimal rule: limit on debt-to-income ratio of 25 percent of annualized income (i.e., a1 = 1 and a0 = 0), therefore procyclical.
- Among rules with average ceiling = 25 percent:
  - More countercyclical ceilings (lower a1) do not necessarily imply lower consumption volatility.
  - Allowing more debt in low-income periods increases default probability sensitivity and can increase borrowing-cost countercyclicality, making countercyclical fiscal policy more difficult.
- For very low average ceilings (below 10 percent of mean annual income):
  - Rules that allow extra room for countercyclical fiscal policy may be preferred, but such ceilings leave the government worse off compared with the no-rule benchmark; no defaults observed in simulations with such low ceilings.

### Conclusions and policy implications
- A sovereign-default framework indicates a government may benefit from implementing a debt ceiling because the expectation of lower future debt levels can make small borrowing declines lead to substantial interest rate declines, which in turn lead to non-trivial debt declines.
- If the government could capture the capital gains from the increase in the market value of its debt implied by the rule announcement (instead of these gains accruing to foreign investors), it would choose lower debt ceilings to be enforced sooner.
- Fiscal rules can promote a decline of consumption volatility.
- Rules that allow for more borrowing in bad times may not further decrease consumption volatility and can increase borrowing-cost countercyclicality.
- Results caution against “escape clauses” in fiscal rules that allow for extra discretionary borrowing during crisis periods.

### Extensions and avenues for future research
- Study less-than-perfectly-credible fiscal rules, including political shocks that could threaten enforcement of a rule.
- Allow the government to choose the duration of its debt; this could change insights but would significantly increase computation cost.
- Extend the framework to include production so that interest rate effects on investment, productivity, and amplification of shocks can be captured.
- Study setups where the sovereign simultaneously holds assets and liabilities.
- Explore models where difficulties in promising contingent debt payments are consistent with difficulties to establish contingent fiscal-rule limits; current analysis assumes the government can commit to fiscal rules contingent on realized income but does not allow state-contingent debt issuance.

*Source: _wp1230 - References (PDF), https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1230.pdf*

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

### _wp1230 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

### Purpose and main contributions
- Objective: shed light on the optimal value of fiscal rules’ parameters and on the effects of imposing fiscal rules when a government can commit to enforce them.
- Primary contributions:
  - Determine optimal fiscal-rule parameter values within a sovereign-default framework à la Eaton and Gersovitz (1981) with long-duration bonds.
  - Analyze the trade-off between larger bondholders’ capital gains and the fiscal cost of stronger adjustment when the government conducts a voluntary debt restructuring.
  - Measure how fiscal rules affect the procyclicality of fiscal policy and, through that channel, the volatility of private consumption.

### Key quantitative findings and policy-relevant results
- Pre-rule calibration and typical values:
  - Study economies with a pre-rule debt level of 38 percent of mean income (typical in the data for the pre-2001-default Argentina used to calibrate the model).
- Optimal rule in benchmark (no-rule) economy:
  - Government-preferred rule: a debt ceiling of 30 percent of mean income, with a delay of four years between the rule announcement period and the first period in which the ceiling is enforced.
  - Mean borrowing level: falls only from 3.6 percent to 3.1 percent of average income under the rule.
- Voluntary debt restructuring scenario:
  - Government chooses a 25 percent ceiling to be enforced less than two years after its announcement in the voluntary restructuring case.
  - Creditor debt forgiveness increases with the pre-rule spread: with a pre-rule spread of 5 (15) percent, bondholders are willing to forgive up to 21 (44) percent of the government’s debt.
- Effects on borrowing costs and access:
  - Committing to a fiscal rule reduces expected future debt and lowers the interest rate (sovereign spread) at which the government borrows, creating capital gains for bondholders and more favorable borrowing opportunities for the government.
- Effects on cyclicality and consumption volatility:
  - Without a fiscal rule, a high and countercyclical interest rate induces the government (a sequential decision-maker) to borrow less when income is low, producing procyclical fiscal policy.
  - Lower debt ceilings make the interest rate less responsive to income shocks, reducing fiscal procyclicality and lowering consumption volatility.
  - A countercyclical debt ceiling (higher when income is lower) increases both the level and countercyclicality of the interest rate; thus, a countercyclical ceiling does not necessarily reduce consumption volatility.
  - Among linear functions of current income studied, the government prefers procyclical ceilings (e.g., a ceiling on the debt-to-income ratio) over constant debt ceilings.
  - Countercyclical ceilings are preferred only among ceilings that impose excessively low average debt levels; the government prefers the no-rule benchmark over such low ceilings.

### Mechanisms and interpretation
- Why rules help: Commitment to a rule changes lenders’ expectations about future indebtedness, reducing the sovereign risk premium and borrowing costs for a given debt level.
- Trade-off under voluntary restructuring: Stronger and quicker fiscal consolidation raises bondholders’ capital gains (via higher market values of existing bonds) but increases the fiscal cost to the sovereign; the government can use voluntary restructuring to capture capital gains by obtaining creditor debt relief in exchange for announcing and enforcing a stricter rule.
- Limits to cyclicality analysis: analysis excludes production and therefore is limited in studying the full mechanics of cyclicality of fiscal policy.

### Model framework and assumptions
- Environment:
  - Single tradable good; stochastic endowment y_t with log(y_t) = (1−ρ)μ + ρ log(y_{t−1}) + ε_t, |ρ|<1, ε_t ∼ N(0, σ^2_ε).
  - Government maximizes E[∑_{t=0}^∞ β^t u(c_t)], with u(c) = c^(1−γ) −1 / (1−γ) (constant relative risk aversion γ).
- Debt instrument:
  - Long-term bonds: bond issued in period t promises one unit in period t+1 and (1−δ)^{s−1} units in period t+s for s ≥ 2; bonds follow Hatchondo and Martinez (2009) style long-duration specification.
- Default and costs:
  - Default is all-or-nothing across all current and future obligations (consistent with acceleration and cross-default clauses); recovery rate in default is zero.
  - Two costs of default:
    - Exclusion from capital markets; after default, the country regains access with probability ψ ∈ [0,1] each period.
    - Per-period income loss while excluded: φ(y) = d_0 y + d_1 y^2 (quadratic loss function).
- Pricing and risk premium:
  - Lenders’ stochastic discount factor: M(y′, y) = exp(−r − α ε′ − 0.5 α^2 σ^2_ε), where r denotes the risk-free rate; the formulation yields a risk premium driven by the borrowing economy’s income shock.
  - The discount factor specification is a discrete-time version of the Vasicek one-factor model of the term structure.
- Solution concept and computational approach:
  - Markov Perfect Equilibrium (government strategies depend only on payoff-relevant state variables).
  - Potential multiplicity of equilibria avoided by solving a finite-horizon version and extending horizon until convergence of first- and second-period value functions and bond prices; first-period functions used as infinite-horizon equilibrium.

### Recursive formulation and equilibrium conditions
- State variables:
  - b: number of outstanding coupon claims at the beginning of the current period (b < 0 implies net issuer in the past).
  - d: default indicator (d = 1 if defaulted in current period, else 0).
  - V(b, y): government’s value function before the default decision.
  - V_0(b, y): value if not in default; V_1(y): value if in default.
- Bellman-like equations:
  - V(b,y) = max_{d ∈ {0,1}} { d V_1(y) + (1−d) V_0(b,y) }.
  - V_1(y) = u(y − φ(y)) + β ∫ [ ψ V(0, y′) + (1−ψ) V_1(y′) ] F(dy′ | y).
  - V_0(b,y) = max_{b′ ≤ 0} { u(y + b − q(b′, y) [ b′ − (1−δ) b ]) + β ∫ V(b′, y′) F(dy′ | y) }.
- Bond price equation (no arbitrage with default risk and long-term structure):
  - q(b′, y) = ∫ M(y′, y) [1 − h(b′, y′)] F(dy′ | y)
    + (1−δ) ∫ M(y′, y) [1 − h(b′, y′)] q(g(h(b′, y′), b′, y′), y′) F(dy′ | y),
    where h denotes the future default rule and g the future borrowing rule expected by lenders.
- Equilibrium requirement:
  - Optimal default and borrowing rules that solve the government’s problems must equal the rules h and g that lenders expect for all state variable realizations.

*Source: _wp1230 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (PDF), https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1230.pdf*

### 1. a set of value functionsV,V

### 1. a set of value functionsV,V

### Model structure and equilibrium objects
- Equilibrium consists of:
  - a set of value functions V, V1, and V0;
  - a default rule h and a borrowing rule g;
  - a bond price function q;
- Conditions for equilibrium:
  - (a) given h and g, V, V1, and V0 satisfy functional equations (1), (2), and (3), when the government can trade bonds at q;
  - (b) given h and g, the bond price function q is given by equation (4);
  - (c) the default rule h and borrowing rule g solve the dynamic programming problem defined by equations (1) and (3) when the government can trade bonds at q.

### Fiscal policy (stylized)
- Government choice each period: tax revenues τ (possibly negative).
- In default:
  - τ = 0
  - private agents consume c = y − φ(y)
- When not in default:
  - private agents consume c = y − τ
  - government uses tax revenues and issuance revenues to service debt:
    - τ − q(b′, y) [b′ − (1 − δ)b] = −b

### Fiscal rules studied
- Fiscal rules parameterized by (b̄(y), n) where:
  - b̄(y) = a0 + a1 y denotes the debt ceiling (linear in y);
  - n denotes the number of periods between the announcement of the rule and the period after which the ceiling is enforced.
- After first enforcement period, government solves optimization in equation (3) with constraint b′ ≥ b̄(y).

### Calibration
- Case study: Argentina before the 2001 default.
- Numerical solution method: value function iteration and interpolation (linear interpolation for endowment levels and spline interpolation for asset positions).
- Period: quarter.
- Parameter values (Table 1):
  - Borrower’s risk aversion σ: 2
  - Interest rate r: 0.01
  - Income autocorrelation coefficient ρ: 0.9
  - Standard deviation of innovations σϵ: 0.027
  - Mean log income μ(-1/2)σ2ϵ (as presented)
  - Exclusion ψ: 0.282
  - Duration δ: 0.0341
  - Discount factor β: 0.961
  - Default cost d0: -0.69
  - Default cost d1: 1.017
  - Risk premium α: 3
- Additional calibration details:
  - ψ = 0.282 (probability of regaining access to capital markets)
  - With δ = 0.0341, bonds have average duration of 4.19 years in baseline simulations (Macaulay definition of duration D = (1 + r*)/(δ + r*) where r* denotes the constant per-period yield).
- Targets used to calibrate β, default costs, and α:
  - mean spread = 7.4 percent
  - standard deviation of spread = 2.5 percent
  - mean debt to (annual) GDP ratio = 40 percent (pre-default sample)
  - default frequency = three defaults per 100 years

### Results — overview
- Sequence of results discussed:
  1. Benchmark economy simulations versus data.
  2. Optimal fiscal rules.
  3. Capital gains on previously issued debt upon rule announcement and distributional effects.
  4. How fiscal rules create expectations of lower future indebtedness and benefit the government.
  5. Extent to which fiscal rules reduce procyclicality of fiscal policy.
  6. Cyclically adjusted fiscal rules.

### A. The no-rule benchmark — fit to data
- Simulation design:
  - Extract 500 samples of 32 consecutive periods before a default to compare with Argentine data (fourth quarter 1993 to third quarter 2001).
  - For comparison, include only samples where last default was at least two periods before sample start.
  - Default frequencies computed using all simulation data.
  - Trends computed using Hodrick-Prescott filter with smoothing parameter 1,600.
  - Debt face value computed dividing b by δ + r.
- Table 2: Business cycle statistics (Data vs Benchmark)
  - E(Rs): 7.44 vs 7.42
  - σ(Rs): 2.51 vs 2.52
  - Mean debt-to-income ratio: 0.40 vs 0.39
  - Defaults per 100 years: 3.00 vs 2.99
  - σ(c)/σ(y): 0.94 vs 1.23
  - ρ(c,y): 0.97 vs 0.99
  - ρ(Rs,y): -0.65 vs -0.79
- Key benchmark findings:
  - Baseline model matches data reasonably well on targeted moments.
  - Consumption and income are highly correlated; spread is countercyclical.
  - Consumption volatility is higher than income volatility in simulations.
  - Fiscal policy in benchmark is procyclical: taxes tend to be higher (or transfers lower) when income is lower.
  - Intuition: In bad times higher borrowing costs lead government to finance more debt service with taxes instead of new issuances.
  - Bond price is lower in bad times; taxes tend to be higher when sovereign interest rate spread is higher.

### B. Fiscal rules — government preferences and announcement effects
- Experiment focus:
  - Imposing simple fiscal rules with constant debt ceiling and specified delay between announcement and enforcement.
  - Pre-rule starting debt level: 38 percent of mean income.
  - Consider three pre-rule spread cases: 5.1 percent (low-risk), 7.4 percent (normal-risk), 15 percent (high-risk).
- Welfare measure:
  - Welfare gain from moving from benchmark to rule economy: ((VR(b,y)/VB(b,y))^(1/(1−σ)) − 1)
- Main numerical findings (Table 3):
  - For initial spreads 5.1%, 7.4%, 15.0%:
    - Optimal debt ceiling: 30% (for all three cases)
    - Optimal transition length (quarters): 18, 17, 17 (respectively)
    - Welfare gain: 0.23%, 0.23%, 0.22% (respectively)
    - Spread decline at announcement period: 0.9%, 1.4%, 0.3% (respectively)
- Interpretation:
  - Pre-rule default risk has little effect on the government-preferred rule: all prefer a 30% ceiling with about a four-year delay.
  - Government prefers delaying enforcement to reduce cost of adjusting debt; typically delays reduction until last two years of transition.
  - Announcement of the rule bids up sovereign bond prices immediately, reducing spreads even before debt reduction occurs.
  - Spread decline at announcement is smaller for higher initial risk.
- Dynamic paths (when income shocks equal zero):
  - Post-announcement debt-to-income ratio path shows gradual adjustment; debt reduction concentrated in last two years of transition.
  - Post-announcement spread declines continuously; for the high-risk economy spread expected to decline to almost half pre-rule value after one year.
- Tradeoff between transition length and spread decline:
  - For 7.4 percent pre-rule spread, choosing a two-year transition (time to adjust debt) instead of four years could double the spread decline implied by announcement.
  - Spread decline implies capital gains for holders of previously issued debt; government does not capture these gains in this baseline scenario, creating a conflict of interest between government and creditors who prefer shorter transitions and lower ceilings.

### C. Fiscal rules after a debt restructuring (government captures capital gains)
- Exercise: Government makes a take-it-or-leave-it debt buyback offer before rule announcement; rule announced only if offer accepted. Government buys back bonds at pre-rule price (extreme case where government captures all capital gains).
- Main numerical findings (Table 4):
  - For initial spreads 5.1%, 7.4%, 15.0%:
    - Optimal debt ceiling: 25% (for all three cases)
    - Optimal transition length (quarters): 7, 6, 3 (respectively)
    - Debt forgiveness (percent of government’s debt that lenders forgive): 21%, 25%, 44% (respectively)
    - Welfare gain: 1.02%, 1.29%, 2.28% (respectively)
- Interpretation:
  - When government captures capital gains, it chooses a lower debt ceiling and enforces it sooner.
  - Shorter transition periods for higher pre-rule risk are consistent with larger debt forgiveness required.
  - Welfare gains from combining rule announcement with a voluntary debt restructuring increase with pre-rule risk and can be substantial.

### D. Fiscal rules and borrowing opportunities (debt dilution and commitment)
- Mechanism:
  - Commitment to lower future debt levels reduces lenders’ expectations of future indebtedness and lowers current borrowing rates.
  - Imposing a debt ceiling can create new borrowing opportunities: the rule allows government to pay a lower interest rate for the same debt level.
- Illustration:
  - Spread as a function of face value of next-period debt (present value of future payment obligations discounted at risk-free rate, as percentage of mean annual income) shifts down when a debt ceiling is imposed.
- Quantitative implication:
  - Economies with debt ceilings that imply significantly lower debt levels do not necessarily display much lower borrowing levels because the ceiling allows access to significantly lower interest rates.
  - Example (partial): with a ceiling of 25 percent of mean income, mean interest rate spread decreases from 7.4 ... (text truncates at supplied content)

*Source: Excerpt from PDF chapter _wp1230 - 1. a set of value functionsV,V*

### 1.7 percent, while mean borrowing only decreases from 3.6 to 3.1 percent of mean income.

### _wp1230 - 1.7 percent, while mean borrowing only decreases from 3.6 to 3.1 percent of mean income.

### Fiscal rules and the cyclicality of fiscal policy
- Lower debt levels implied by the imposition of a fiscal rule weaken the government’s incentives to implement a procyclical fiscal policy.
- In the benchmark economy fiscal policy is procyclical because in low-income periods the higher cost of issuing debt induces the government to finance more of its debt service obligations with taxes.
- Debt ceilings reduce default risk and thus reduce both:
  - the mean spread and
  - the responsiveness of the spread to income shocks (as reflected in a lower standard deviation of the spread).
- As the cost of borrowing becomes less responsive to income shocks, the incentive for procyclical fiscal policy weakens, leading to lower consumption volatility relative to income.
- These findings support the view that targeting low sovereign debt levels creates room for the implementation of countercyclical fiscal policy.

### Key simulation results (Table 5 and related discussion)
- Debt ceilings reduce default risk and interest-rate volatility; selected exact figures from Table 5:
  - E(Rs): 7.42, 6.64, 3.6, 1.66, 0.54
  - σ(Rs): 2.52, 2.85, 1.93, 1.29, 0.7
  - Mean debt-to-income ratio: 0.39, 0.38, 0.34, 0.35, 0.24
  - Defaults per 100 years: 2.99, 2.86, 1.33, 0.42, 0.09
  - σ(c)/σ(y): 1.23, 1.12, 1.08, 1.05, 1.02
  - ρ(c,y): 0.99, 0.99, 0.99, 0.99, 0.99
  - ρ(Rs,y): -0.79, -0.72, -0.68, -0.62, -0.53
  - Mean borrowing /average income (%): 3.65, 3.63, 3.48, 3.14, 2.63
- Table 6 (debt ceiling b̄ = a0 + a1 y with average ceiling = 25 percent of mean annual income) shows how varying slope a1 affects outcomes; selected exact figures:
  - E(Rs): 0.09, 0.59, 1.00, 1.66, 2.10, 2.41, 2.63
  - σ(Rs): 0.03, 0.46, 0.81, 1.29, 1.40, 1.41, 1.47
  - Mean debt-to-income ratio: 0.23, 0.25, 0.27, 0.30, 0.30, 0.31, 0.31
  - Defaults per 100 years: 0.03, 0.13, 0.25, 0.42, 0.65, 0.75, 0.79
  - σ(c)/σ(y): 1.73, 1.49, 1.27, 1.05, 0.97, 1.04, 1.05
  - ρ(c,y): 0.99, 0.99, 0.99, 0.99, 0.99, 0.99, 0.99
  - ρ(Rs,y): -0.86, -0.65, -0.62, -0.62, -0.62, -0.67, -0.67
  - Mean borrowing /average income (%): 2.67, 2.76, 2.97, 3.14, 3.26, 3.30, 3.34

### Cyclically adjusted fiscal rules
- The authors search for the ex-ante optimal ceiling to be enforced immediately (n = 0) for a government that currently does not have debt.
- The optimal rule found:
  - imposes a limit on the debt-to-income ratio of 25 percent of annualized income (i.e., has a1 = 1 and a0 = 0), and is therefore procyclical.
- Among rules with an average ceiling equal to 25 percent of mean annual income:
  - rules with a more countercyclical ceiling (a lower a1) do not necessarily imply lower consumption volatility.
  - Because default probability is more sensitive to increases in debt during lower-income periods, allowing more debt in low-income periods reduces commitment to low future default probabilities, especially in low-income periods.
  - This increases both the level and the countercyclicality of the government’s borrowing cost, making countercyclical fiscal policy more difficult.
- Figure 8 reports welfare gains from allowing the debt ceiling to depend on current income for average ceiling = 25 percent of mean annual income when current income is:
  - low (one standard deviation below the mean),
  - average, or
  - high (one standard deviation above the mean).
- Among fiscal rules that impose low average debt ceilings, the government prefers rules that allow extra room for countercyclical fiscal policy (a negative a1). However, for the case studied:
  - such debt ceilings are too low (below 10 percent of mean annual income) and leave the government worse off compared with the no-rule benchmark.
  - No defaults are observed in simulations with such low ceilings.
- Figure 9 presents the welfare gain from allowing the debt ceiling to depend on current income among rules with an average debt ceiling of 10 percent of mean annual income.

### Conclusions and policy implications
- A sovereign-default framework indicates a government may benefit from implementing a debt ceiling because:
  - the expectation of lower future debt levels can make small borrowing declines lead to substantial interest rate declines, which in turn lead to non-trivial debt declines.
- If the government could capture the capital gains from the increase in the market value of its debt implied by the rule announcement (instead of these gains accruing to foreign investors), it would choose lower debt ceilings to be enforced sooner.
- Fiscal rules can promote a decline of consumption volatility.
- Rules that allow for more borrowing in bad times may not further decrease consumption volatility and can increase borrowing cost countercyclicality.
- These results caution against “escape clauses” in fiscal rules that allow for extra discretionary borrowing during crisis periods.

### Extensions and avenues for future research
- Study less-than-perfectly-credible fiscal rules, including political shocks that could threaten enforcement of a rule.
- Allow the government to choose the duration of its debt; this could change insights but would significantly increase computation cost.
- Extend the framework to include production so that interest rate effects on investment, productivity, and amplification of shocks can be captured.
- Study setups where the sovereign simultaneously holds assets and liabilities, since fiscal rules often aim at controlling accumulation of both.
- Explore models where difficulties in promising contingent debt payments are consistent with difficulties to establish contingent fiscal-rule limits; current analysis assumes the government can commit to fiscal rules contingent on realized income but does not allow state-contingent debt issuance.

*Source: _wp1230 - 1.7 percent, while mean borrowing only decreases from 3.6 to 3.1 percent of mean income.*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2012/_wp1230.pdf_