## Box 1. Notation Glossary

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### Notation and variable definitions
- Variables without a time sub-index may be treated as constant over time.
- A tilde denotes cyclically adjusted magnitudes (e.g., ̃b). A circumflex denotes inflation adjusted magnitudes (e.g., ˆb).
- Key variable definitions (as ratios to GDP where indicated):
  - p_t = Primary balance in t, as a ratio to GDP at t.
  - b_t = Overall balance at t, as a ratio to GDP at t.
  - d_t = Debt at the end of period t, as a ratio to GDP at t.
  - π_t = Change in the GDP deflator between t – 1 and t.
  - γ_t = Nominal GDP growth rate between t – 1 and t.
  - g_t = Real GDP growth rate between t – 1 and t. Notice that 1+γ_t = (1+g_t)(1+π_t).
  - i_t = Nominal interest rate in period t; paid in period t on the debt stock outstanding at the end of t – 1.
  - r_t = Real interest rate in period t. Defined as r_t ≡ [(1+ i_t)/(1+ π_t)]–1. Thus, 1+i_t = (1+ r_t)(1+ π_t).
  - Y, y = Nominal and real GDP respectively.
  - R, E = Nominal revenue and expenditure.
  - X = Net nominal budgetary aggregates (revenue less expenditure) that do not depend on GDP.
  - α = Output gap as a ratio to potential output.
  - v = Revenue-to-GDP ratio.
  - e = Expenditure-to-GDP ratio.
  - η = Elasticity of revenue to GDP (approximate percent increase in nominal revenue per percentage point of gap).
  - κ = Elasticity of expenditure to GDP (approximate percent increase in nominal expenditure per percentage point of gap).
- Auxiliary definition:
  - λ = (i_t – γ_t)/(1+γ_t) = (r_t – g_t)/(1+g_t). In particular, 1+λ = (1+i_t)/(1+γ_t) = (1+r_t)/(1+g_t).

### Debt dynamics (primary-balance formulation)
- Recursive debt equation:
  - d_t = (1+λ_t) d_{t–1} – p_t (equation (5))
- General solution (N periods):
  - d_N = d_0 Π_{t=1}^N (1+λ_t) – Σ_{t=1}^N [p_t Π_{i=t+1}^N (1+λ_i)] (equation (6))
- Time-invariant λ (λ_t = λ) simplifications:
  - d_t = (1+λ) d_{t–1} – p_t (equation (7))
  - d_N = d_0 (1+λ)^N – Σ_{t=1}^N (1+λ)^{N–t} p_t (equation (8))
  - Identity used later:
    - d_0 = (1+λ)^{-N} d_N + Σ_{t=1}^N (1+λ)^{-t} p_t (equation (9))
  - Difference form:
    - d_t – d_{t–1} = λ d_{t–1} – p_t (equation (10))
  - Multi-period:
    - d_N – d_0 = λ Σ_{t=0}^{N–1} d_t – Σ_{t=1}^N p_t (equation (11))
    - or d_N – d_0 = λ N d̄ – N p̄ (equation (12)) where averages d̄ and p̄ are taken over specified periods.

### Overall-balance formulation and debt dynamics
- Relationship between overall and primary balances:
  - i_t/(1+γ_t) d_{t–1} = p_t – [i_t/(1+γ_t)] d_{t–1} (equation (13) rearranged)
- Recursive equation in overall balance:
  - d_t = (1/(1+γ_t)) d_{t–1} – b_t (equation (14))
- General solution (N periods):
  - d_N = d_0 Π_{t=1}^N (1+γ_t)^{–1} – Σ_{t=1}^N b_t Π_{i=t+1}^N (1+γ_i)^{–1} (equation (15))
- Time-invariant γ:
  - d_N = d_0 (1+γ)^{–N} – Σ_{t=1}^N (1+γ)^{–N+t} b_t (equation (16))
- One-period change (constant γ):
  - (d_t – d_{t–1})/(1+γ) = –γ/(1+γ) d_{t–1} – b_t (equation (17) rearranged)

### Balances compatible with a constant debt ratio
- Primary and overall balances that keep debt constant at d*:
  - p* = λ d* (equation (20))
  - b* = –γ/(1+γ) d* (equation (21))
- Key implications:
  - Setting overall balance at b* leads actual debt ratio to asymptotically converge to d* if nominal growth γ is positive.
  - Setting primary balance at p* does NOT in general stabilize debt unless initial debt equals d* or λ<0. If λ>0 and initial debt ≠ d*, a constant p* can produce explosive (λ>0) or diverging (toward –∞) debt paths.

### Balances that hit a target debt ratio in finite time
- Constant primary balance p* to reach target d*_N in N periods:
  - p* = [λ / ((1+λ)^{-N} – 1)] ((1+λ)^{-N} d*_N – d_0) (equation (22))
- Constant overall balance b* to reach target d*_N in N periods:
  - b* = [–γ / ((1+γ)((1+γ)^N – 1))] ((1+γ)^N d*_N – d_0) (equation (23))
- Note: p* and b* that solve (22) and (23) are not generally consistent with each other via equation (13) unless d_0 = d*_N.

### Decomposition of changes in the debt ratio
- From (1) and (5):
  - (d_t – d_{t–1}) = [i_t/(1+γ_t)] d_{t–1} – [γ_t/(1+γ_t)] d_{t–1} – p_t (equation (24))
- Real-term decomposition:
  - d_t – d_{t–1} = [r_t/(1+g_t)] d_{t–1} – [g_t/(1+g_t)] d_{t–1} – p_t (equation (28))
- Interpretation:
  - Evolution of the debt ratio depends only on real interest rate r_t, real growth g_t, and fiscal adjustment p_t. Inflation affects debt only insofar as it alters the real interest rate paid by the government.

### Stability, budget constraint, and no-Ponzi game condition
- Conditions:
  - Condition 1 (Boundedness): ∃ H such that for all t, d_t ≤ H.
  - Condition 2 (No-Ponzi game/transversality): lim_{N→∞} (1+λ)^{–N} d_N = 0 (equation (29)).
  - Condition 3 (Government inter-temporal budget constraint): d_0 = Σ_{t=1}^∞ (1+λ)^{–t} p_t (equation (30)).
  - Condition 4 (Modified golden rule): λ>0 (equation (31)). Implies, asymptotically, real interest rate r exceeds real growth g.
  - Condition 5 (Primary balance bounded above): ∃ M such that for all t, p_t ≤ M.
- Propositions:
  - Proposition 1: Condition 2 ⇔ Condition 3.
  - Proposition 2: If λ>0 and primary balance bounded above, then Condition 2 implies Condition 1.
  - Proposition 3: If λ>0 and debt ratio bounded above and below, then Condition 1 implies Condition 2.
- Interpretations:
  - Under λ>0, equivalence of: (i) debt and interest not rolled over systematically; (ii) existing debt eventually repaid via future primary surpluses; (iii) debt ratio kept below a ceiling.
  - If λ<0 (growth exceeds interest), these equivalences break down; a stable debt ratio need not imply the no-Ponzi game condition.
  - Practical assumption often used: λ = 0.01 (1%).

### Sustainability indicator (s)
- Definition: fixed annual addition (ratio to contemporaneous GDP) to primary balances that renders the primary-balance sequence sustainable under the inter-temporal budget constraint.
- Formulation:
  - d_0 = Σ_{t=1}^∞ (1+λ)^{–t} (p_t + s) (equation (32))
  - Since Σ_{t=1}^∞ (1+λ)^{–t} = λ^{–1}, then:
    - s = λ d_0 – λ Σ_{t=1}^∞ (1+λ)^{–t} p_t (equation (33))
  - Alternative with dp_t ≡ p_t – p_0:
    - s = λ d_0 – p_0 – λ Σ_{t=1}^∞ (1+λ)^{–t} dp_t (equation (34))
- Practical finite-horizon variant (long-term costs forecast t = 1,...,N and constant thereafter):
  - s = λ d_0 – p_0 – λ Σ_{t=1}^N (1+λ)^{–t} dp_t – (1+λ)^{–N} dp_N (equation (35))
- Usage note: s is a benchmark indicator (used as “s2” by the European Commission), not necessarily a year-by-year policy prescription.

### Cyclical adjustment (methodology and approximations)
- Purpose:
  - (i) Estimate underlying fiscal position (what would prevail at potential output).
  - (ii) Measure discretionary fiscal policy contribution to demand (fiscal stance).
- Convention: output gap α positive if actual output above potential. Y = (1+α) ̃Y ; y = (1+α) ̃y (equations (36),(37)).
- Cyclically adjusted ratios to GDP defined relative to potential GDP; correction to express as ratio to actual GDP uses division by 1+α (equation (38)).
- Revenue elasticity and constant-elasticity solution:
  - η = d ln R(Y) / d ln Y (equation (40))
  - ln R(Y) = η ln Y + constant (equation (41))
  - R( ̃Y ) = R(Y) (1+α)^{–η} (equation (43))
- Main adjustment equations (exact):
  - ṽ = v (1+α)^{1–η} (equation (44))
  - ẽ = e (1+α)^{1–κ} (equation (45))
  - b̃ = ṽ – ẽ (equation (46))
- First-order approximations (for small α):
  - ṽ ≈ v (1 + (1–η) α) (equation (47))
  - ẽ ≈ e (1 + (1–κ) α) (equation (48))
  - b̃ ≈ b + v α (1–η) – e α (1–κ) (equation (49))
- “Poor man’s” cyclical adjustment (when η ≈ 1 and κ ≈ 0):
  - ṽ ≈ v (equation (50))
  - ẽ ≈ e (1+α) (equation (51))
  - b̃ ≈ b – α e (equation (52))
  - Rule of thumb: cyclically adjusted balance (percent of potential output) ≈ actual balance (percent of actual output) minus one expenditure ratio per percentage point of gap.
- OECD methodology:
  - b̃ = [Σ_{i=1}^4 R_i (1+α)^{–η_i} – E (1+α)^{–κ} + X] / ̃Y (equation (53))
  - Equivalent ratios-to-GDP form: b̃ = Σ v_i (1+α)^{1–η_i} – e (1+α)^{1–κ} + x (1+α) (equation (54))
  - OECD elasticities and semi-elasticity measures summarized in Table 2 (Girouard et al. (2005)).

### Commodity exporters
- Procedure:
  - Adjust expenditure and non-commodity revenue using equations (47),(48).
  - Compute structural non-commodity balance as in (49).
  - Compute structural commodity revenue based on expected long-term average commodity prices or structural rate of return on commodity funds.
  - Add structural commodity revenue to structural non-commodity balance to obtain overall structural balance.

### Inflation adjustment
- Inflation distorts revenue and expenditure via multiple channels (e.g., Tanzi effect, bracket creep, indexation). Adjustments depend on country-specific institutional arrangements and data.
- Inflation adjustment of interest payments (operational balance) is standardized:
  - Operational balance ˆb_t equals actual overall balance increased by the inflation-induced erosion of the real value of debt:
    - ˆb_t = b_t + [π_t/(1+γ_t)] d_{t–1} (equation (55) first line)
  - Expressed in primary-balance terms and real components as in the text.
- Numerical illustration:
  - If real growth is zero, debt is 50 percent of GDP, real interest rate is 5 percent, and inflation is 10 percent, the increase in the interest bill due to inflation could be of close to 200 percent or almost 5 points of GDP—assuming inflation is fully anticipated and debt is rolled over annually.

### Annex — Summary of main formulas (selected)
- Growth-adjusted interest rate:
  - λ_t = (i_t – γ_t) / (1+γ_t) = (r_t – g_t) / (1+g_t)
  - r_t ≡ [(1+i_t)/(1+π_t)]–1; g_t ≡ [(1+γ_t)/(1+π_t)]–1
- Debt dynamics (time-varying and time-invariant forms) and aggregated forms (equations (5)–(12)).
- Overall/primary balance relation and no-Ponzi game condition (equations (13), (29), (30)).
- Sustainability indicator formulas (equations (33)–(35)).
- Cyclical adjustment exact and approximated equations (equations (36)–(54)).
- Inflation-adjusted operational balance (equation (55)).

*Source: Box 1. Notation Glossary, _tnm1002*

### Box 1. Notation Glossary

### Box 1. Notation Glossary

### Notation and variable definitions
- Variables without a time sub-index are sometimes treated as constant over time or when the formula is not dynamic.
- A tilde denotes cyclically adjusted magnitudes (e.g., ̃b). A circumflex denotes inflation adjusted magnitudes (e.g., ˆb).
- When necessary, additional notation specific to a section or formula is defined where it is used.
- Key variable definitions (as ratios to GDP where indicated):
  - p_t = Primary balance in t, as a ratio to GDP at t.
  - b_t = Overall balance at t, as a ratio to GDP at t.
  - d_t = Debt at the end of period t, as a ratio to GDP at t.
  - π_t = Change in the GDP deflator between t – 1 and t.
  - γ_t = Nominal GDP growth rate between t – 1 and t.
  - g_t = Real GDP growth rate between t – 1 and t. Notice that 1+γ_t = (1+g_t)(1+π_t).
  - i_t = Nominal interest rate in period t; paid in period t on the debt stock outstanding at the end of t – 1.
  - r_t = Real interest rate in period t. Defined as r_t ≡ [(1+ i_t)/(1+ π_t)]–1. Thus, 1+i_t = (1+ r_t)(1+ π_t).
  - Y, y = Nominal and real GDP respectively.
  - R, E = Nominal revenue and expenditure.
  - X = Net nominal budgetary aggregates (revenue less expenditure) that do not depend on GDP.
  - α = Output gap as a ratio to potential output.
  - v = Revenue-to-GDP ratio.
  - e = Expenditure-to-GDP ratio.
  - η = Elasticity of revenue to GDP (approximate percent increase in nominal revenue per percentage point of gap).
  - κ = Elasticity of expenditure to GDP (approximate percent increase in nominal expenditure per percentage point of gap).
- Auxiliary definitions and identities:
  - λ = (i_t – γ_t)/(1+γ_t) = (r_t – g_t)/(1+g_t). In particular, 1+λ = (1+i_t)/(1+γ_t) = (1+r_t)/(1+g_t).

### Debt dynamics (primary-balance formulation)
- Recursive debt equation:
  - d_t = (1+λ_t) d_{t–1} – p_t (equation (5))
- General solution (N periods):
  - d_N = d_0 Π_{t=1}^N (1+λ_t) – Σ_{t=1}^N [p_t Π_{i=t+1}^N (1+λ_i)] (equation (6))
- Time-invariant λ (λ_t = λ) simplifications:
  - d_t = (1+λ) d_{t–1} – p_t (equation (7))
  - d_N = d_0 (1+λ)^N – Σ_{t=1}^N (1+λ)^{N–t} p_t (equation (8))
  - Identity used later:
    - d_0 = (1+λ)^{-N} d_N + Σ_{t=1}^N (1+λ)^{-t} p_t (equation (9))
  - Difference form:
    - d_t – d_{t–1} = λ d_{t–1} – p_t (equation (10))
  - Multi-period:
    - d_N – d_0 = λ Σ_{t=0}^{N–1} d_t – Σ_{t=1}^N p_t (equation (11))
    - or d_N – d_0 = λ N d̄ – N p̄ (equation (12)) where averages d̄ and p̄ are taken over specified periods.

### Overall-balance formulation and debt dynamics
- Relationship between overall and primary balances:
  - i_t/(1+γ_t) d_{t–1} = p_t – [i_t/(1+γ_t)] d_{t–1} (equation (13) rearranged)
- Recursive equation in overall balance:
  - d_t = (1/(1+γ_t)) d_{t–1} – b_t (equation (14))
- General solution (N periods):
  - d_N = d_0 Π_{t=1}^N (1+γ_t)^{–1} – Σ_{t=1}^N b_t Π_{i=t+1}^N (1+γ_i)^{–1} (equation (15))
- Time-invariant γ:
  - d_N = d_0 (1+γ)^{–N} – Σ_{t=1}^N (1+γ)^{–N+t} b_t (equation (16))
- One-period change (constant γ):
  - (d_t – d_{t–1})/(1+γ) = –γ/(1+γ) d_{t–1} – b_t (equation (17) rearranged)
- Multi-period generalizations and averages given (equations (18),(19)).

### Balances compatible with a constant debt ratio
- Primary and overall balances that keep debt constant at d*:
  - p* = λ d* (equation (20))
  - b* = –γ/(1+γ) d* (equation (21))
- Key implications:
  - Setting overall balance at b* leads actual debt ratio to asymptotically converge to d* if nominal growth γ is positive.
  - Setting primary balance at p* does NOT in general stabilize debt unless initial debt equals d* or λ<0. If λ>0 and initial debt ≠ d*, a constant p* can produce explosive (λ>0) or diverging (toward –∞) debt paths.

### Balances that hit a target debt ratio in finite time
- Constant primary balance p* to reach target d*_N in N periods:
  - p* = [λ / ((1+λ)^{-N} – 1)] ((1+λ)^{-N} d*_N – d_0) (equation (22))
- Constant overall balance b* to reach target d*_N in N periods:
  - b* = [–γ / ((1+γ)((1+γ)^N – 1))] ((1+γ)^N d*_N – d_0) (equation (23))
- Note: p* and b* that solve (22) and (23) are not generally consistent with each other via equation (13) unless d_0 = d*_N.

### Decomposition of changes in the debt ratio
- From (1) and (5):
  - (d_t – d_{t–1}) = [i_t/(1+γ_t)] d_{t–1} – [γ_t/(1+γ_t)] d_{t–1} – p_t (equation (24))
  - Expressed to separate interest, inflation, real growth, fiscal adjustment:
    - (d_t – d_{t–1}) = [i_t/(1+γ_t)] d_{t–1} – [π_t/(1+γ_t)] d_{t–1} – [g_t/(1+g_t)(1+γ_t)^{-1}] d_{t–1} – p_t (equation (26) structure)
  - More enlightening equivalent decomposition in real terms:
    - d_t – d_{t–1} = [r_t/(1+g_t)] d_{t–1} – [g_t/(1+g_t)] d_{t–1} – p_t (equation (28))
  - Interpretation: evolution of debt ratio depends only on real interest rate r_t, real growth g_t, and fiscal adjustment p_t. Inflation affects debt only insofar as it alters the real interest rate paid by the government.

### Stability, budget constraint, and no-Ponzi game condition
- Conditions defined:
  - Condition 1 (Boundedness): ∃ H such that for all t, d_t ≤ H.
  - Condition 2 (No-Ponzi game/transversality): lim_{N→∞} (1+λ)^{–N} d_N = 0 (equation (29)).
  - Condition 3 (Government inter-temporal budget constraint): d_0 = Σ_{t=1}^∞ (1+λ)^{–t} p_t (equation (30)).
  - Condition 4 (Modified golden rule): λ>0 (equation (31)). Implies, asymptotically, real interest rate r exceeds real growth g.
  - Condition 5 (Primary balance bounded above): ∃ M such that for all t, p_t ≤ M.
- Propositions connecting conditions:
  - Proposition 1: Condition 2 ⇔ Condition 3 (equivalence via identity (9) and limits).
  - Proposition 2: If λ>0 and primary balance bounded above, then Condition 2 implies Condition 1.
  - Proposition 3: If λ>0 and debt ratio bounded above and below, then Condition 1 implies Condition 2.
- Interpretations and implications:
  - Under λ>0, equivalence of: (i) debt and interest not rolled over systematically; (ii) existing debt eventually repaid via future primary surpluses; (iii) debt ratio kept below a ceiling.
  - If λ<0 (growth exceeds interest), debt can be rolled over without snowballing; many standard equivalences break down—e.g., a stable debt ratio need not imply satisfaction of the no-Ponzi game condition.
  - Practical assumption often used: λ = 0.01 (1%).

### Sustainability indicator (s)
- Definition: fixed annual addition (ratio to contemporaneous GDP) to the primary balances that renders the primary-balance sequence sustainable under the inter-temporal budget constraint.
- Formulation:
  - d_0 = Σ_{t=1}^∞ (1+λ)^{–t} (p_t + s) (equation (32))
  - Since Σ_{t=1}^∞ (1+λ)^{–t} = λ^{–1}, then:
    - s = λ d_0 – λ Σ_{t=1}^∞ (1+λ)^{–t} p_t (equation (33))
  - Alternative with dp_t ≡ p_t – p_0:
    - s = λ d_0 – p_0 – λ Σ_{t=1}^∞ (1+λ)^{–t} dp_t (equation (34))
- Practical finite-horizon variant when long-term costs explicitly forecast for t = 1,...,N and constant thereafter:
  - s = λ d_0 – p_0 – λ Σ_{t=1}^N (1+λ)^{–t} dp_t – (1+λ)^{–N} dp_N (equation (35))
- Usage note: s is a benchmark indicator (used as “s2” by the European Commission), not necessarily a year-by-year policy prescription.

### Cyclical adjustment (methodology and approximations)
- Purpose:
  - (i) Estimate underlying fiscal position (what would prevail at potential output).
  - (ii) Measure discretionary fiscal policy contribution to demand (fiscal stance).
- Convention: output gap α positive if actual output above potential. Y = (1+α) ̃Y ; y = (1+α) ̃y (equations (36),(37)).
- Cyclically adjusted ratios to GDP are defined relative to potential GDP; correction to express as ratio to actual GDP uses division by 1+α (equation (38)).
- Revenue elasticity definition and constant-elasticity solution:
  - η = d ln R(Y) / d ln Y (equation (40))
  - ln R(Y) = η ln Y + constant (equation (41))
  - R( ̃Y ) = R(Y) (1+α)^{–η} (equation (43))
- Main adjustment equations (exact):
  - ṽ = v (1+α)^{1–η} (equation (44))
  - ẽ = e (1+α)^{1–κ} (equation (45))
  - b̃ = ṽ – ẽ (equation (46))
- First-order approximations (for small α):
  - ṽ ≈ v (1 + (1–η) α) (equation (47))
  - ẽ ≈ e (1 + (1–κ) α) (equation (48))
  - b̃ ≈ b + v α (1–η) – e α (1–κ) (equation (49))
- “Poor man’s” cyclical adjustment (common shortcut when η ≈ 1 and κ ≈ 0):
  - ṽ ≈ v (equation (50))
  - ẽ ≈ e (1+α) (equation (51))
  - b̃ ≈ b – α e (equation (52))
  - Rule of thumb: cyclically adjusted balance (percent of potential output) ≈ actual balance (percent of actual output) minus one expenditure ratio per percentage point of gap.
- OECD methodology (separate adjustment by revenue categories and unemployment-related expenditure):
  - Cyclically adjusted balance expressed as:
    - b̃ = [Σ_{i=1}^4 R_i (1+α)^{–η_i} – E (1+α)^{–κ} + X] / ̃Y (equation (53))
    - Equivalent ratios-to-GDP form (equation (54))
  - OECD elasticities and semi-elasticity measures summarized in Table 2 (Girouard et al. (2005)).

### Commodity exporters
- Approach: adjust expenditure and non-commodity revenue as in (47),(48); compute structural non-commodity balance as in (49); compute structural commodity revenue based on expected long-term average commodity prices or structural rate of return on commodity funds; add to structural non-commodity balance to obtain overall structural balance.

### Inflation adjustment
- Inflation distorts revenue and expenditure via multiple channels (e.g., Tanzi effect, bracket creep, indexation).
- No single general methodology; adjustments depend on country-specific institutional arrangements and data (collection lags, indexation, tax payment timing).
- Inflation adjustment of interest payments (operational balance) is more standardized:
  - Operational balance ˆb_t equals actual overall balance increased by the inflation-induced erosion of the real value of debt:
    - ˆb_t = b_t + [π_t/(1+γ_t)] d_{t–1} (equation (55) first line)
  - Expressed in primary-balance terms:
    - i_t/(1+γ_t) π_t = p_t – [i_t/(1+γ_t)] d_{t–1} + [π_t/(1+γ_t)] d_{t–1} (equation (55) structure)
    - Alternatively: r_t/(1+g_t) = p_t – [real component of interest payments] (end of section)
- Numerical illustration from text:
  - If real growth is zero, debt is 50 percent of GDP, real interest rate is 5 percent, and inflation is 10 percent, the increase in the interest bill due to inflation could be of close to 200 percent or almost 5 points of GDP—assuming inflation is fully anticipated and debt is rolled over annually.

*Source: Box 1. Notation Glossary, _tnm1002*

### References

### _tnm1002 - References

### References cited
- Bartolini and Cottarelli (1994): “Government Ponzi Games and the sustainability of Public Deficits under Uncertainty,” Ricerche Economiche 48, 1-22.
- Blanchard, O. J. (1984), “Current and Anticipated Deficits, Interest Rates and Economic Activity,” European Economic Review 25, 7-27.
- Blanchard, O. J. and S. Fischer (1989), Lectures on Macroeconomics, (MIT Press, Cambridge, Massachusetts).
- Blanchard, O. J. and Weil, P. (1992): “Dynamic Efficiency and Debt Ponzi Games under Uncertainty,” NBER Working Paper No. 3992.
- Fedelino, A., A. Ivanova, and M. Horton (2009): “Computing Cyclically Adjusted Balances and Automatic Stabilizers,” IMF Technical Notes and Manuals 09/05.
- Girouard, N. and C. André (2005), “Measuring Cyclically adjusted Budget Balances for OECD Countries”, OECD Economics Department Working Papers, No. 434, OECD publishing. doi:10.1787/787626008442
- Tanzi, V. (1977): “Inflation, Lags in Collection, and the Real Value of Tax Revenue,” Staff Papers, IMF, Vol. 24 (March), pp. 154-67.
- Tanzi, V., M. I. Blejer, and M. O. Teijeiro (1987): “Inflation and the Measurement of Fiscal Deficits,” Staff Papers, IMF, Vol. 34 (December), pp. 711-38.

### Annex. Summary of Formulas — main formulas for reference

- Growth-adjusted interest rate:
  - λt = (it – γt) / (1+γt) = (rt – gt) / (1+gt) (1,4)
  - rt ≡ [(1+it)/(1+πt)]–1; gt ≡ [(1+γt)/(1+πt)]–1

- Debt dynamics and primary balance: annual difference equation and its solutions
  - Time-varying λt:
    - dt = (1+λt)dt–1 – pt (5)
    - dN = d0π(1+λt)– S[π(1+λi)] p t  (6)  (indices: i=1 t=1 i=t+1)
  - Time-invariant λt = λ:
    - dt = (1+λ)dt–1 – pt (7)
    - dN = d0(1+λ)N – S(1+λ)N–t pt  (8)  (t=1)
  - Aggregated forms:
    - dN – d0 = λ S d t – S pt  (11)  (t=0 t=1; sums implied)
    - dN – d0 = λN – d – N – p (12)  (expressions with averages defined below)
    - d̄ ≡ (1/N) S d t ; p̄ ≡ (1/N) S pt  (definitions for averages)

- Overall and primary balances:
  - ibt = pt – (1/(1+γ)) d t–1 (13)

- Debt dynamics and overall balance: annual difference equation and its solutions
  - Time-varying γt:
    - (1/(1+γt)) dt = – dt–1 – bt (14)
    - dN = d0π(1+λt)– S[π(1+λi)] p t  (15)  (t=1 t=1 i=t+1)
  - Time-invariant γt = γ:
    - (1/(1+γ)) dt = – dt–1 – bt (16)
    - dN = d0(1+γ)–N – S(1+γ)–N+t bt  (t=1)
    - –γ/(1+γ) (dN – d0) = S dt – S bt  (18)  (t=0 t=0)
    - –γ/(1+γ) (dN – d0) = – (N – d – N – b) (19)
    - d̄ ≡ (1/N) S d t ; b̄ ≡ (1/N) S b t  (definitions)

- Primary balance (p*) and overall balance (b*) compatible with a constant debt ratio (d*):
  - p* = λ d* (20)
  - b* = –γ/(1+γ) d* (21)

- Constant primary balance (p*) and overall balance (b*) that hit a given debt ratio (d*N) in a finite number of periods (N), given initial debt (d0):
  - λ p* = ((1+λ)–N d*N – d0) / ((1+λ)–N – 1) (22)
  - –γ b* = ((1+γ)N d*N – d0) / ((1+γ)((1+γ)N – 1)) (23)

- Decomposition of changes in the debt ratio:
  - (it/(1+γt)) dt – dt–1 = – pt (24)  (presentation of decomposition)
  - it/(1+γt) dt – dt–1 = – (πt/(1+γt)) dt–1 – pt (26)
  - (it/(1+gt)) dt – dt–1 = – (rt/(1+gt)) dt–1 – pt (28)  (variants with r t and g t)

- No-Ponzi game condition:
  - lim (1+λ)–N dN = 0 as N→∞ (29)

- Government’s inter-temporal budget constraint:
  - d0 = Σ (1+λ)–t pt , t=0 to ∞ (30)

- Sustainability indicator:
  - s = λ d0 – λ Σ (1+λ)–t pt , t=1 to ∞ (33)
  - Let dpt ≡ pt – p0. Then,
    - s = λ d0 – p0 – λ Σ (1+λ)–t dpt , t=1 to ∞ (34)
  - If dpt, t = 1,...N, is known, and assumed constant for t = N, N+1,... then
    - sN = λ d0 – p0 – λ Σ (1+λ)–t dpt – (1+λ)–N dpN , t=1 to N (35)

- Cyclical adjustment: Output (nominal, real) and output gap
  - Y = (1+α) Ỹ (36)
  - y = (1+α) ỹ (37)
  - Cyclically adjusted variables as a ratio to actual GDP: divide by 1+α.
    - b̃ ratio to actual GDP = b̃ / (1+α) ; similarly for other variables (38)
  - Actual (unadjusted) variables as a ratio to potential GDP: multiply by 1+α.
    - b ratio to potential GDP = (1+α) b ; similarly for other variables. (39)
  - Revenue function:
    - ln R(Y) = η ln Y + constant; similarly for expenditure. (41)
    - R(Ỹ) = R(Y) (1+α)–η ; similarly for expenditure. (43)
  - Exact cyclical adjustment:
    - ṽ = v (1+α)1–η (44)
    - ẽ = e (1+α)1–κ (45)
    - b̃ = ṽ – ẽ (46)
  - Approximation for small gap (α):
    - ṽ ≈ v (1+(1–η)α) (47)
    - ẽ ≈ e (1+(1–κ)α) (48)
    - b̃ ≈ b + vα(1–η) – eα(1–κ) (49)
  - If η ≈ 1 and κ ≈ 0:
    - ṽ ≈ v (50)
    - ẽ ≈ e (1+α) (51)
    - b̃ ≈ b – α e (52)
  - OECD methodology:
    - b̃ = [Σ Ri (1+α)–ηi – E(1+α)–κ + X] / Ỹ (53)  (i=1 to 4)
    - b̃ = Σ vi (1+α)1–ηi – e(1+α)1–κ + x(1+α) (54)  (i=1 to 4)
    - Country-by-country estimates of ηi, i = 1,...4, and κ are shown in Table 2.

- Inflation adjustment — Operational balance (b̂):
  - b̂t = bt + (πt/(1+γt)) dt–1 (55)
  - it πt = pt – (1/(1+γt)) dt–1 + (πt/(1+γt)) dt–1  (expression relating components)
  - rt = pt – (1/(1+gt)) dt–1  (expression for rt)

### Contact and series identification (from document footer)
- TNM/10/02
- International Monetary Fund, Fiscal Affairs Department
- A Practical Guide to Public Debt Dynamics, Fiscal Sustainability, and Cyclical Adjustment of Budgetary Aggregates
- Julio Escolano, Fiscal Affairs Department

*Source: _tnm1002 - References*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/tnm/2010/_tnm1002.pdf_