## _wp03220

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---

### Introduction: scope and motivation
- Fiscal rules—legal restrictions on government borrowing, spending, or debt accumulation—have recently been adopted in several countries and are being discussed in several others (both industrialized and developing).
- Central claims in the literature:
  - Opponents: fiscal rules prevent the government from smoothing tax rates and expenditures over the business cycle and may prohibit discretionary countercyclical policy.
  - Proponents: fiscal rules supplement weak institutions to promote fiscal responsibility and credibility.
- Relevance for Latin America: several Latin American countries that have suffered from chronic fiscal indiscipline have enacted or proposed fiscal rules; examples include Argentina, Brazil, Colombia, Peru and Chile.

### Theoretical framework and policy regimes
- Ramsey/Barro-based insight: governments should equate marginal deadweight losses across tax instruments; tax smoothing minimizes deadweight loss.
- Alternatives compared:
  - Benchmark (Ramsey) policy: tax rates are completely smooth.
  - General fiscal reaction function: τ_t = γ − κ + β b_{t-1}, linking tax rates to debt with possible constant deficit component κ.
  - Restrictive (GRH-like) rule: permits just enough borrowing to keep debt-GDP constant ex-ante; ex-post variations arise from forecasting errors.
- Definitions and budget identities:
  - Government per-period constraint: b_{t-1} θ − p s_t = b_t, where θ = (1+r)/(1+λ).
  - Primary surplus and borrowing requirement expressions differ by regime (R0, R1, R2) and whether borrowing constraints bind.

### Model extensions: favoring present taxpayers and uncertainty
- Authority with present bias:
  - Loss function separates present φP(τ) (t = 0..J) and future φF(τ) (t = J+1..∞); φi′>0, φi′′>0.
  - Interperiod tax ratio: τP/τF = μPF.
  - Sustainable one-time tax cut/surcharge linkage: κP = -κF/{θ^{J+1} – 1}.
  - Optimal κP*: κP* = {γ(1-μPF) + β(bP - bF)}/{1 + μPF(θ^{J+1} – 1)} (depends on μPF, γ, r, λ, β, bP, bF).
- Uncertainty and borrowing access:
  - Output: Yt = Yt^P + υt; Yt^P = Yt^P (1+λ) is permanent trend; υt mean-zero with known variance constant relative to Yt^P.
  - Random cutoff: country faces cutoff from borrowing with probability πC uniformly distributed on [0,1].
  - Unconstrained tax: τ(U) = γ + (r-λ)/(1+λ) bP.
  - Constrained tax: τ(C)_t = γ + (r-λ_t)/(1+λ_t) b_{t-1}; taxes are min{τ(U), τ(C)_t}.
  - Precautionary policy: with πC > 0, optimal κ may be nonzero to self-insure; in two-period example κ1* satisfies φ′(τ(U) − κ1) = (1 − πC) E{φ′(τ(U))} + πC E{φ′(τ(C)_2)} and ∂κ1*/∂πC < 0.

### Fiscal regimes (definitions and mechanics)
- Regimes evaluated:
  - R0 (Smoothing / benchmark): τ(0) = γ + (r-λ)/(1+λ) bP; ps(0)_t = γ[1 − w_t] + (r-λ)/(1+λ) bP; br(0)_t = γ[w_t − 1] + (r-λ)/(1+λ)[b_{t-1} − bP].
  - R1 (Balanced-budget-like, legally stipulated): τ(1)_t = γ + (1+r)/(1+λ^*_t) b_{t-1}, λ^*_t = [Yt^P / Y_{t-1} − 1]; ps(1)_t = γ[1 − w_t] + [(1+r)/(1+λ^*)] b_{t-1}; br(1)_t = γ[w_t − 1] + [(1+r)/(1+λ^*) ε_t] b_{t-1}, ε_t = (λ_t − λ^*_t)/(1+λ_t).
  - R2 (General fiscal reaction): τ(2)_t = γ − κ + β b_{t-1}; ps(2)_t = −κ + β b_{t-1}; br(2)_t = κ + [(r-λ)/(1+λ) − β] b_{t-1}.
- Interaction with borrowing constraints:
  - Constrained borrowing and taxes computed as max/min with R1: e.g., br(0C)_t = max{br(1)_t, br(0)_t}, τ(0C)_t = min{τ(1)_t, τ(0)_t}.

### Simulation design and common parameter values
- Simulation setup:
  - Horizons J = 5, 10, 20 years; 500 random draws.
  - Key parameters: b(initial) = bP = 0.5; permanent growth λ = 4%; variance of temporary income = 0.5 * permanent output; constant interest rate r = 7% (r = .07); permanent spending ratio γ = 0.2.
  - πC considered at 0, 0.3, 0.5.
  - R2 parameterizations: κ = .03, β = .8 and κ = .05, β = .8.
- Reported outputs: mean, standard deviation, minimum, maximum for τ_t, ps_t, and b_J.

### Key simulation findings (selected numeric outcomes and comparisons)
- R0 (Benchmark smoothing), πC = 0.0:
  - Constant tax: τ = 0.2144 (21.44%) across horizons.
  - J = 5: τ Average = 0.2144; Standard Deviation = 0.0000; Minimum = 0.2144; Maximum = 0.2144.
    - ps Average = 0.0137; Standard Deviation = 0.0093; Minimum = 0.0029; Maximum = 0.0238.
    - b_J Average = 0.5033; Standard Deviation = 0.0417; Minimum = 0.3931; Maximum = 0.6476.
  - J = 20: τ Average = 0.2144; b_J Average = 0.5065 (other τ statistics unchanged).
- R1 (Balanced-budget-like), πC = 0.0, J = 5:
  - τ Average = 0.2127; Standard Deviation = 0.0203; Minimum = 0.1889; Maximum = 0.2347.
  - ps Average = 0.0119; Standard Deviation = 0.0197; Minimum = -0.0104; Maximum = 0.0334.
  - b_J Average = 0.5105; Standard Deviation = 0.0752; Minimum = 0.3305; Maximum = 0.7395.
  - Moving from R0 to R1 slightly reduces average τ but introduces tax-rate variability.
- R2 (κ = .03, β = .8), πC = 0.0, J = 5:
  - τ Average = 0.1812; Standard Deviation = 0.0183; Minimum = 0.1598; Maximum = 0.2012.
  - ps Average = -0.0196; Standard Deviation = 0.0179; Minimum = -0.0395; Maximum = 0.0000.
  - b_J Average = 0.6425; Standard Deviation = 0.0720; Minimum = 0.4614; Maximum = 0.8703.
- R2 (κ = .05, β = .8), πC = 0.0, J = 5:
  - τ Average = 0.1618; Standard Deviation = 0.0197; Minimum = 0.1390; Maximum = 0.1834.
  - ps Average = -0.0389; Standard Deviation = 0.0190; Minimum = -0.0601; Maximum = -0.0180.
  - b_J Average = 0.7237; Standard Deviation = 0.0748; Minimum = 0.5325; Maximum = 0.9624.

### Comparative quantitative insights and tradeoffs
- Infinite-horizon optimum: κ = 0 and β = (r-λ)/(1+λ) → constant τ_t and constant debt ratio.
- Regime tradeoffs (selected highlights):
  - R0 vs R1: similar average tax levels over infinite horizon; R1 produces greater tax variability.
  - R2 (nonzero κ) lowers average current taxes but increases debt accumulation and tax-rate variability over finite horizons.
  - Debt accumulation under R2 (illustrative averages):
    - J = 5: about 60% of GDP for κ = 0.03 and 65% for κ = 0.05.
    - J = 10: about 72% (κ = 0.03) and 85% (κ = 0.05).
    - J = 20: just under 100% (κ = 0.03) and 127% (κ = 0.05).
- Tradeoff Ratios (Δ Average / Δ Standard Deviation) — selected values:
  - For κ = .03, β = .8:
    - πc=0, J=5: 15.950
    - πc=0, J=10: -34.149
    - πc=0, J=20: -3.159
    - πc=0.3, J=5: -5.857
    - πc=0.5, J=20: -1.420
  - For κ = .05, β = .8:
    - πc=0, J=5: 78.562
    - πc=0, J=10: -10.619
    - πc=0, J=20: -2.476
    - πc=0.3, J=10: -2.533
    - πc=0.5, J=20: -1.084
- Interpretation: negative Tradeoff Ratio indicates a tax hike required to obtain a one-percent decrease in standard deviation. Required tax increases to reduce volatility fall with higher πc and with longer horizons in many cases.

### Effects of borrowing-constraint probability πC and expenditure flexibility
- As πc rises from 0 to 0.3 and 0.5:
  - R1 results are invariant to πc.
  - Under R0 and R2 both the level and variability of tax rates rise; R2 becomes less attractive relative to R0 and R1.
  - Debt buildup under R2 falls when πc rises but remains substantially higher than under R0 or R1.
  - As πc → 1, constrained regimes converge toward R1.
- Variable expenditure extension (ω parameter):
  - τt = γP − κ + ω β b_{t-1}; γt = τt + κ − (1−ω) β b_{t-1}; ω ∈ [0,1] allocates adjustment between taxes and expenditures.
  - ω = 1: adjustment entirely on taxes; ω = 0: adjustment entirely on expenditures (τ constant).
  - With πc > 0 and ω = 0, denied borrowing leads to expenditure cuts that period.
  - Higher γP tends to raise variability of both expenditures and revenues; empirical illustrations for Latin America suggest a positive relationship between average public consumption/GDP (1980–2001) and its coefficient of variation, and between average public consumption/GDP and variance of real GDP growth.

### Policy implications and conclusions
- Over an infinite horizon, Ramsey-style tax smoothing yields lower and less variable tax rates.
- Removing persistent current primary deficits (via fiscal rule or once-and-for-all reform) can permit smoother tax rates even over finite horizons.
- Balanced-budget laws or fiscal restrictions can complement one-off tax/expenditure adjustments; GRH-style rules and Ramsey smoothing can be complementary.
- Political incentives favoring current taxpayers and precautionary motives under borrowing-cutoff risk can justify deviations from infinite-horizon optimum (nonzero κ).
- Tradeoffs policymakers face:
  - Lower current taxes (larger κ) reduce current burden but increase future debt and tax volatility.
  - Credibility and initial debt discipline influence how costly borrowing constraints are when they bind.
- Suggested extensions: allow both taxes and expenditures to share adjustment, specify consumer preferences and production technology fully, introduce endogenous borrowing constraints or default motives, include price-level or interest-rate changes.

*Source: _wp03220 — https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2003/_wp03220.pdf*

### 1. Latin America: Public Consumption / GDP ................................................................25

### 1. Latin America: Public Consumption / GDP ................................................................25

### Introduction: scope and motivation
- Fiscal rules—legal restrictions on government borrowing, spending, or debt accumulation—have recently been adopted in several countries and are being discussed in several others (both industrialized and developing).
- Debates over fiscal rules revolve around two central claims from the literature:
  - Opponents: fiscal rules prevent the government from smoothing tax rates and expenditures over the business cycle and may prohibit discretionary countercyclical policy.2
  - Proponents: fiscal rules supplement weak institutions to promote fiscal responsibility and credibility.3
- Relevance for Latin America: several Latin American countries that have suffered from chronic fiscal indiscipline have enacted or proposed fiscal rules; examples include Argentina, Brazil, Colombia, Peru and Chile.4

### Theoretical framework and literature context
- The analysis builds on Ramsey’s (1927) insight that governments should equate the marginal deadweight losses from different tax sources.
- Barro (1979) applied this to conclude that deadweight losses are minimized under tax smoothing; under such a policy the government borrows during downturns and saves during upturns.
- Subsequent work (Lucas and Stokey (1983); Chari and Kehoe (1999); Aiyagari, Marcet, Sargent and Seppälä (2002)) generalized the Ramsey approach; the paper treats a “Ramsey approach” as minimizing a deadweight loss function akin to Barro (1979).
- Empirical evidence is mixed: jurisdictions with more restrictive fiscal rules run smaller deficits but tend to have more procyclical fiscal policy (Bayoumi and Eichengreen (1995)).

### Policies compared in this study
- A restrictive fiscal law akin to Gramm-Rudman-Hollings (GRH) is compared against several alternatives:
  - Benchmark (Ramsey) policy: tax rates are completely smooth.
  - A general fiscal reaction function: links tax rates to debt (ensuring long-run solvency) but allows for a constant (potentially deficit) component; this may resemble actual government policies.
  - A restrictive rule that permits just enough borrowing to keep the debt-GDP ratio constant ex-ante, with ex-post variations due to forecasting errors.9

### Model assumptions and definitions
- Sources of uncertainty: cyclical output and access to credit.
- Permanent output is known and the ratio of government expenditures to permanent output is constant.
- Under these assumptions, countercyclical fiscal policy is defined as smoothing tax rates.
- Welfare is assumed to fall when either the mean or the variance of tax rates rises.

### Key analytical points and insights
- Prior literature has emphasized that GRH-like fiscal rules hinder countercyclical public borrowing and tax smoothing.
- This paper highlights a complementary point: removing persistent current primary deficits—either via a fiscal rule or a once-and-for-all fiscal reform—permits smoother tax rates than otherwise.10
- Over an infinite horizon, moving closer to a Ramsey regime implies welfare gains from lower and less variable tax rates.
- Over finite horizons the tradeoff is ambiguous: financing current expenditures by accumulating debt can lower taxes today but increase tax variability.
- Simulations in the paper quantify the magnitude of the tradeoff between lower mean tax levels and higher tax volatility.11
- Consequently, under certain conditions a fiscal rule may reduce tax-rate variability even over a finite horizon; thus, Ramsey-style tax smoothing and GRH-like fiscal rules are not necessarily in conflict.12

### Organization of the paper (from the supplied content)
- Section II presents basic identities and discusses optimal fiscal policy over an infinite horizon under certainty.
- A general fiscal rule is introduced thereafter.

*Source: _wp03220 - 1. Latin America: Public Consumption / GDP ................................................................25*

### Section III presents an alternative model in which the authority may favor current taxpayers

### _wp03220 - Section III presents an alternative model in which the authority may favor current taxpayers

### III. Favoring the present over the future — model setup and implications
- Governments may systematically favor present taxpayers over future ones by using a loss function with two components: φP(τ) for the present (t = 0 to J) and φF(τ) for the future (t = J+1 to ∞), with φi′>0 and φi′′>0 for i = P, F.
- The authority minimizes
  - Σ_{t=0}^{J} φP(τt) θ^{−t} + Σ_{t=J+1}^{∞} φF(τt) θ^{−t}
  subject to the transversality / no-Ponzi condition lim b_t/(1+r)^{t-1} = 0.
- Across the two periods, the authority equates the ratio of average tax rates between periods τP/τF to a constant interperiod marginal substitution μPF. Thus τP/τF = μPF.
- A fiscal rule compatible with this behavior can deliver a one-time tax cut for current taxpayers κP and a surcharge for future taxpayers κF linked by the sustainability condition:
  - κP = -κF/{θ^{J+1} – 1}
- For constant β, the optimal tax break for current taxpayers is:
  - κP* = {γ(1-μPF) + β(bP - bF)}/{1 + μPF(θ^{J+1} – 1)}
  - κP* depends on μPF, γ, r, λ, β, bP and bF.

### IV. Optimization under uncertainty — output and access to credit
- Output decomposition:
  - Yt = Yt^P + υt
    - Yt^P = Yt^P (1+λ) is permanent trend output
    - υt is mean-zero temporary income; its known variance is constant relative to Yt^P
  - Government spending ratio γ is constant relative to permanent income.
- Random cutoff from access to borrowing:
  - Country faces a cutoff from borrowing with probability πC uniformly distributed between 0 and 1.
  - If πC = 0, access is certain; if πC = 0.5, there is a 50 percent chance of no borrowing in a period.
- If denied access to credit when it would otherwise borrow, the government must raise taxes in that period; borrowing cutoffs limit deficits but not surpluses.
- Under unfettered access the tax rate is τ(U) = γ + (r-λ)/(1+λ) bP.
- If constrained, taxes link to previous-period debt: τ(C)_t = γ + (r-λ_t)/(1+λ_t) b_{t-1}; taxes in any period are min{τ(U), τ(C)_t}.
- Precautionary motive: with πC > 0, optimal policy may choose κ ≠ 0 to self-insure. In a two-period example, κ1* satisfies:
  - φ′(τ(U) − κ1) = (1 − πC) E{φ′(τ(U))} + πC E{φ′(τ(C)_2)}
  - If πC = 0, κ1* = 0. More generally, ∂κ1*/∂πC < 0: as borrowing becomes more restricted probabilistically, the optimal primary surplus rises.

### V. Alternative fiscal regimes — definitions and comparative mechanics
- Government budget constraint, per-period:
  - b_{t-1} θ − p s_t = b_t
  - where b = debt/GDP, θ = (1+r)/(1+λ), r constant, λ < r, ps_t primary surplus (ratio to GDP)
- General fiscal reaction rule (used to summarize fiscal regimes):
  - τ_t = γ − κ + β b_{t-1}
  - κ is a “tax gap” (Blanchard et al. (1990)); β determines responsiveness to past debt.
- Three regimes evaluated:
  - R0 (Smoothing / benchmark): τ(0) = γ + (r-λ)/(1+λ) bP. Access to borrowing unfettered; constant tax rate in absence of constraints.
    - Primary surplus: ps(0)_t = γ[1 − w_t] + (r-λ)/(1+λ) bP
    - Borrowing requirement beyond minimum: br(0)_t = γ[w_t − 1] + (r-λ)/(1+λ)[b_{t-1} − bP]
  - R1 (Balanced-budget-like, legally stipulated): τ(1)_t = γ + (1+r)/(1+λ^*_t) b_{t-1}, where λ^*_t = [Yt^P / Y_{t-1} − 1]
    - Aims ex-ante to maintain constant debt/GDP. Primary surplus: ps(1)_t = γ[1 − w_t] + [(1+r)/(1+λ^*)] b_{t-1}
    - Ex post borrowing requirement: br(1)_t = γ[w_t − 1] + [(1+r)/(1+λ^*) ε_t] b_{t-1}, ε_t = (λ_t − λ^*_t)/(1+λ_t)
  - R2 (General fiscal reaction, unconstrained by law): τ(2)_t = γ − κ + β b_{t-1}, κ > 0, β > 0
    - Primary surplus: ps(2)_t = −κ + β b_{t-1}
    - Incremental new borrowing: br(2)_t = κ + [(r-λ)/(1+λ) − β] b_{t-1}
- Interaction with borrowing constraints:
  - Under constraints, for R0 and R2: borrowing is max{br(1)_t, br(regime)_t}; tax is min{τ(1)_t, τ(regime)_t}.
  - Formally: br(0C)_t = max{br(1)_t, br(0)_t}, τ(0C)_t = min{τ(1)_t, τ(0)_t}; br(2C)_t = max{br(1)_t, br(2)_t}, τ(2C)_t = min{τ(1)_t, τ(2)_t}.

### V.C. Simulation design and parameter assumptions
- Simulations compare regimes (R0), (R1), (R2) over horizons J = 5, 10, 20 years using 500 random draws.
- Key parameters (for all simulations):
  - initial debt ratio b(initial) = bP = 0.5
  - permanent growth λ = 4%
  - variance of temporary income = 0.5 * permanent output
  - constant interest rate r = 7% (r = .07)
  - permanent spending ratio γ = 0.2
- Probability of borrowing constraint πC considered at 0, 0.3, 0.5 (Tables 2, 3, 4 respectively).
- For R2 simulations, two parameterizations: κ = .03, β = .8 and κ = .05, β = .8.
- Outputs reported: mean, standard deviation, minimum, maximum for τ_t (tax rate), ps_t (primary surplus ratio), and b_J (end-of-period debt).

### V.C. Key simulation findings (selected outcomes for πC = 0.0 from Table 2)
- Benchmark smoothing (R0):
  - Constant tax rate τ = 0.2144 (21.44%) across horizons.
  - 5-year results (J=5):
    - τ Average = 0.2144; Standard Deviation = 0.0000; Minimum = 0.2144; Maximum = 0.2144
    - ps Average = 0.0137; Standard Deviation = 0.0093; Minimum = 0.0029; Maximum = 0.0238
    - b_J Average = 0.5033; Standard Deviation = 0.0417; Minimum = 0.3931; Maximum = 0.6476
  - 10-year and 20-year rows show similar τ Average = 0.2144 with small changes in ps and b_J statistics (e.g., b_J Average = 0.5065 at J=20).
- "Balanced Budget" rule (R1):
  - 5-year results (J=5):
    - τ Average = 0.2127; Standard Deviation = 0.0203; Minimum = 0.1889; Maximum = 0.2347
    - ps Average = 0.0119; Standard Deviation = 0.0197; Minimum = -0.0104; Maximum = 0.0334
    - b_J Average = 0.5105; Standard Deviation = 0.0752; Minimum = 0.3305; Maximum = 0.7395
  - Moving from R0 to R1 reduces the average τ slightly but introduces positive tax-rate variability (e.g., τ Standard Deviation = 0.0203 at J=5).
- General fiscal reaction (R2), κ = .03, β = .8:
  - 5-year results (J=5):
    - τ Average = 0.1812; Standard Deviation = 0.0183; Minimum = 0.1598; Maximum = 0.2012
    - ps Average = -0.0196; Standard Deviation = 0.0179; Minimum = -0.0395; Maximum = 0.0000
    - b_J Average = 0.6425; Standard Deviation = 0.0720; Minimum = 0.4614; Maximum = 0.8703
- General fiscal reaction (R2), κ = .05, β = .8:
  - 5-year results (J=5):
    - τ Average = 0.1618; Standard Deviation = 0.0197; Minimum = 0.1390; Maximum = 0.1834
    - ps Average = -0.0389; Standard Deviation = 0.0190; Minimum = -0.0601; Maximum = -0.0180
    - b_J Average = 0.7237; Standard Deviation = 0.0748; Minimum = 0.5325; Maximum = 0.9624
- Comparative insights highlighted by the simulations:
  - Over an infinite horizon, R0 and R1 yield similar average tax levels but R1 produces greater tax variability.
  - Regime R2 with κ and β differing from infinite-horizon optima (0, (r-λ)/(1+λ)) yields lower average taxes but substantially higher debt accumulation and greater tax-rate variability over finite horizons.
  - For finite horizons, choosing R2 can increase both tax variability and debt accumulation; the welfare tradeoff depends on the implicit marginal rate of substitution between tax level and tax variability in φ(τ).
  - Credibility and initial debt discipline matter: lower initial debt implies smaller tax increases when constrained.

### Summary implications and comparatives
- Optimal infinite-horizon tax smoothing implies κ = 0 and β = (r-λ)/(1+λ), yielding constant τ_t and constant debt ratio b_{-1}.
- Political/institutional incentives to favor current taxpayers (nonzero μPF) or precautionary motives under borrowing-cutoff risk (πC > 0) can justify deviations from the infinite-horizon optimum (κ ≠ 0).
- Explicit fiscal rules (R1) can reduce average tax levels only slightly relative to unconstrained smoothing (R0) but introduce tax variability; general ad hoc reaction functions (R2) can lower current tax levels at the cost of higher future debt and greater tax volatility.
- The decision among regimes involves tradeoffs between average tax levels and tax-rate variability; simulation-based tradeoff ratios (∆ Average / ∆ Standard Deviation) summarize how much tax increase is required to buy reductions in volatility.

*Source: _wp03220 - Section III presents an alternative model in which the authority may favor current taxpayers*

### 23.5 percent under (R1). For longer horizons (J = 10, 20) the variance of τ rises, as does the

### Table 3. Alternative Fiscal Regimes: Simulation Results

### Key simulation findings (regimes R0, R1, R2; parameters and common assumptions)
- For all simulations: λ = 4%, r = 7%, b(initial)=bP = 0.5, γ = 0.2, variance of temporary output = 0.5 * permanent output. Number of draws = 500.
- Unconstrained regimes definitions are provided for (R0) smoothing, (R1) near-balanced budget, and (R2) general reaction (τ(2)t = γ − κ + β b_{t-1}, κ >0, β>0).
- Constrained regimes use max/min formulations; πC = probability of constraint in any period.

### Regime comparisons: tax rates, primary surpluses, end-period debt (selected summary points)
- Under (R0) and (R1) for presented horizons, primary surpluses range from 1.1 to 1.3 percent of GDP; end-period debt bJ remains on average close to initial value of 0.5 under these regimes.
- Taxes are lower under regime (R2) than either (R0) or (R1) for the horizons shown, but tax-rate variability is higher under (R2), and debt accumulation is substantially greater under (R2).
- Example (κ = 0.03, primary deficits average about 2 percent of GDP in one presentation):
  - J = 5: tax rates range from just under 16% to just over 20%.
  - J = 10: tax rates range from about 14% to 22%.
  - J = 20: tax rates range from about 12% to about 25%.
- Increasing κ to 0.05 (primary deficits of 3 to 4 percent of GDP) increases tax-rate variability in all cases.

### Debt accumulation under (R2) (illustrative averages)
- J = 5: debt accumulation averages about 60% of GDP for κ = 0.03 and 65% for κ = 0.05.
- J = 10: end-period debt ratio rises to about 72% (κ = 0.03) and 85% (κ = 0.05).
- J = 20: debt ratio rises to just under 100% (κ = 0.03) and 127% (κ = 0.05).

### Tradeoffs: tax-rate level versus standard deviation (Table 5 highlights)
- For J = 10 (πc = 0): tax-rate increase required to obtain a one percent decrease in standard deviation: 35% for κ = 0.03 and 10% for κ = 0.05.
- For J = 20 (πc = 0): corresponding tradeoffs drop to 3.1% for κ = 0.03 and 2.5% for κ = 0.05.
- With uncertain borrowing constraints (πc = 0.3 and πc = 0.5), required tax increases to obtain a one-percent decrease in standard deviation are substantially lower than for πc = 0. Example (πc = 0.5, J = 10): 2.4% for κ = 0.03 and 1.6% for κ = 0.05. For J = 20: 3.4% and 1.1%, respectively.
- Table 5 reports Tradeoff Ratio = Δ Average / Δ Standard Deviation (negative indicates tax hike required to obtain a one-percent decrease in standard deviation). Selected Tradeoff Ratio values (κ = .03, β = .8):
  - πc=0, J=5: 15.950
  - πc=0, J=10: -34.149
  - πc=0, J=20: -3.159
  - πc=0.3, J=5: -5.857
  - πc=0.5, J=20: -1.420
- Selected Tradeoff Ratio values (κ = .05, β = .8):
  - πc=0, J=5: 78.562
  - πc=0, J=10: -10.619
  - πc=0, J=20: -2.476
  - πc=0.3, J=10: -2.533
  - πc=0.5, J=20: -1.084

### Effects of borrowing-constraint probability πc
- Results for (R1) are invariant to πc.
- As πc rises from 0 to 0.3 and 0.5:
  - Under (R0) both the level and variability of tax rates rise (making (R0) slightly less attractive relative to (R1)).
  - Under (R2) both the level and variability of tax rates rise; (R2) regimes become less attractive relative to both (R0) and (R1).
  - Debt buildup under (R2) falls when πc rises, but remains substantially higher than under (R0) or (R1).
- For πc > 0, tax rates are always more variable under (R2) than (R1). Example (κ = 0.03, primary deficits average between 0.5 and 1 percent of GDP):
  - J = 5: tax rates range about 16.5% to about 21%.
  - J = 10: tax rates range about 15% to just under 24%.
  - J = 20: tax rates range about 13% to about 27%.
- Note: As πc → 1, constrained regimes converge toward (R1) (πc = 1 is the same as an (R1) regime).

### Extension: Variable government expenditures (ω parameter)
- Generalized setup (without borrowing constraints):
  - τt = γP − κ + ω β b_{t-1}  (16a)
  - γt = τt + κ − (1−ω) β b_{t-1}  (16b)
- ω ∈ [0,1] allocates long-run fiscal adjustment between taxes and expenditures:
  - ω = 1: entirety of adjustment falls on taxes (τ adjusts).
  - ω = 0: all adjustment falls on expenditures; τ is constant and γP ≡ τ + κ.
- With πc > 0, constrained expressions (17b) show τtC = max[...] and γtC = min[...]; if ω = 0 and a borrowing government is denied credit, it cuts expenditures that period.

### Extension: Public sector size and volatility
- Under endogenous expenditure regime (ω = 0), average level γP = τ + κ and var(γ) should be positively related.
- More generally (0 ≤ ω ≤ 1) higher γP should raise variability of both expenditures and revenues.
- Empirical illustrations for Latin America (Figures):
  - Figure 1: plot of average public consumption/GDP (1980–2001) against coefficient of variation suggests a positive relationship among Latin American countries.
  - Figure 2: plot of average public consumption/GDP (1980–2001) against variance of real GDP growth also suggests a positive relationship among Latin American countries.

### Summary and conclusions (selected points)
- Over an infinite horizon, Ramsey-style tax smoothing yields lower and less variable tax rates; short-horizon implications depend on eliminating persistent tax gaps.
- Simulations suggest once-and-for-all measures that eliminate a persistent tax gap can make tax rates or expenditures appreciably smoother.
- Balanced-budget laws or other fiscal restrictions may complement one-off tax/expenditure adjustments; Gramm-Rudman-Hollings–style rules and Ramsey tax smoothing can be "friends" rather than "enemies".
- Extensions and future work suggested: allow both taxes and expenditures to share adjustment, specify consumer preferences and production technology more fully, introduce endogenous borrowing constraints or default motives, include price-level or interest-rate changes.

*Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2003/_wp03220.pdf*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2003/_wp03220.pdf_