## _sdn1510

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### Executive summary — core framing and normative guidance
- The Great Recession produced high public debt ratios and eroded public capital in advanced economies; policy focus should shift from the pace of debt reduction to the desirable long-run level of public debt after a debt shock.
- Target audience: countries near full employment with ample fiscal space (green-zone cases as defined in Ostry and others [2010]); rollover risks and short-run cyclical demand management are abstracted from the central analysis.
- Normative guidance for countries with ample fiscal space:
  - Avoid deliberate, distortionary fiscal consolidation aimed at paying down inherited debt.
  - Favor organic debt reduction via growth and opportunistic, less-distortionary revenue measures (for example, privatization receipts or royalties).
  - Use debt issuance to smooth taxes associated with lumpy government expenditures.
- For countries in the yellow or red zones (limited fiscal space or near debt limits), sovereign risks and crisis-insurance considerations strengthen the case for deliberate debt reduction; detailed stress testing of public-sector balance sheets is required.

### Theoretical insights and optimal fiscal principles
- Inherited public debt is a deadweight burden that reduces public and private investment and lowers long-run output.
- Debt-financed public investment is appropriate to smooth taxes and should be undertaken up to the point where the social marginal product equals the market interest rate (a version of the golden rule), recognizing that the social return falls as higher distortionary taxation is required to service additional debt.
- If fiscal space remains ample, deliberate debt paydown is normatively undesirable because the short-run distortive cost of taxes to pay down debt typically exceeds the crisis-insurance benefit from lower debt.
- Optimal fiscal policy under distortionary taxes retains tax-smoothing: the government uses debt to minimize convex distortionary costs and behaves as if discounting at the market interest rate when deciding between current taxation and future servicing of debt.

### Quantitative magnitudes from model simulations and welfare calculations
- Increase in public debt of 50 percent of GDP (example: from 50 percent to 100 percent of GDP) implications:
  - Lowers steady-state output and consumption by 2 percentage points in perpetuity.
  - Implies a present value welfare loss of 30 percent of initial consumption.
  - Produces a similar decline in long-run levels of private and public capital.
- Welfare costs of paying down debt quickly (simulation results):
  - Paying down 5 percent of GDP in one year: present value welfare cost ≈ 1 percent of GDP.
  - Paying down 10 percent of GDP: welfare cost ≈ 2–3 percent of GDP.
  - Paying down 20 percent of GDP: welfare cost of at least 6 percent of GDP.
- Model baseline result to an exogenous increase in debt: optimal policy is typically to live with the inherited debt forever (debt path remains essentially parallel to baseline) because immediate repayment imposes high distortionary tax costs and tax-smoothing dominates.

### Crisis probabilities, expected losses, and insurance value of debt reduction
- Expected crisis frequency over a 20 year horizon: 0.52.
- Generous estimate of output cost per event: approximately 15 percent of GDP.
- At debt = 120 percent of GDP:
  - Annual probability reported: 2.6 percent per year.
  - Reported expected loss: 7.8 percent of GDP.
  - Probability of at least one crisis over 20 years: 41 percent.
- At debt = 100 percent of GDP:
  - Annual likelihood reported: 2.4 percent per year.
  - Reported expected loss: 7.2 percent of GDP.
  - Probability of at least one crisis over 20 years: 38.5 percent.
- Estimated expected benefit of reducing debt from 120 percent to 100 percent of GDP:
  - Approximately 0.4–0.6 percent of GDP (alternative computed figure reported: 0.37 percent of GDP).
- Interpretation:
  - The expected benefit from such debt reduction (≈0.4–0.6 percent of GDP) is about one-tenth of the welfare cost of distortionary taxation implied by the model; thus, for green-zone countries the benefit typically does not justify deliberate, distortionary surpluses.

### Empirical stylized facts supporting the model
- Public debt developments (advanced economies, aggregate):
  - Debt rose from 53 percent of GDP at end-2007 to almost 80 percent by end-2012.
  - For the top quartile, debt exceeds 100 percent of GDP.
- Primary balances:
  - Swung from an average surplus of 2.1 percent of GDP in 2008 to a deficit of 4.4 percent in 2009, with partial recovery by 2012.
- Growth correlation:
  - Projected growth for 2013–17 is strongly negatively correlated with end-2012 public debt ratios (correlation = −.41, statistically significant at the 5 percent level).
- Cross-country and long-run regression evidence:
  - Postcrisis (2009–11): countries with larger increases in public debt generally reduced public gross capital formation the most (statistically significant at the 5 percent level).
  - Longer-run (1960–2008, OECD): strong negative relationship between public debt and public investment; no analogous negative relationship between debt and public consumption.
- Selected regression coefficients (preserve reported estimates):
  - Table 1 (Debt/GDP → Public Investment): coefficient −1.896*** (t-statistic [−3.954]).
  - Table 1 (Debt/GDP (t-1) → Public Investment): coefficient −2.264*** (t-statistic [−3.495]).
  - Table 1 (Debt/GDP → Public Consumption): coefficient 6.075*** (t-statistic [5.196]).
  - Table 2 (five-year average real GDP growth; government revenues percent of GDP): revenue coefficient examples −0.0396**, −0.0424**, −0.0399*, −0.0866**, −0.423** (t-statistics reported in brackets).
  - Robustness diagnostics reported (for dynamic specifications): Hansen test p-value 0.235; A-B AR(1) p-value 0.000840; A-B AR(2) p-value 0.829.

### Model setup, calibration, and key first-order results
- Framework: closed-economy representative-agent model with productive public capital, distortionary taxes only, one-period public debt, no default, benevolent government.
- Key calibration parameters (Box 1 / model calibration excerpts):
  - Cobb–Douglas production with α = 0.33 and θ = 0.1.
  - Utility parameters and implied logarithmic utility: cgϕ = 0.3; clgϕ = 0.1; ϕcg = 1; ϕclg = 1; σ = 1.
  - Discount factor β = 0.96.
  - Depreciation rate δ = 0.1.
- First-order and optimality implications (excerpted):
  - With lump-sum taxes, private and public investment are undertaken until marginal products equal the intertemporal rate of substitution (market interest rate).
  - With only distortionary taxes, the government's implicit discount rate equals the market interest rate; tax-smoothing drives the decision to defer debt repayment versus raise distortionary taxes today.
  - Period-0 opportunistic taxation: income from initial holdings of public debt and initial private capital can be taxed inelastically in the short run, implying scope for opportunistic, less-distortionary debt reduction in initial periods.

### Practical policy implications and guidance checklist
- For green-zone countries with ample fiscal space:
  - Do not pursue deliberate, distortionary fiscal consolidation aimed at debt paydown.
  - Allow debt ratios to decline organically through growth.
  - Use opportunistic, less-distortionary revenues to reduce debt if available.
  - Use debt to smooth taxes for lumpy expenditures and to finance public investment whose social marginal product ≥ market interest rate.
- For countries with limited fiscal space or near debt limits (yellow or red zones):
  - Conduct stress testing of public-sector balance sheets to assess crisis risk and the insurance value of preemptive debt reduction.
  - When market access is at risk, or contingent liabilities are high, deliberate debt reduction can be warranted despite distortionary costs.
- In episodes of asset price booms or other windfalls:
  - Seize opportunities to pay down public debt opportunistically.

*International Monetary Fund*

### EXECUTIVE SUMMARY ______________________________________________________________________________ 1

### _sdn1510 - EXECUTIVE SUMMARY ______________________________________________________________________________ 1

### INTRODUCTION
- The Great Recession led to financial bailouts, stimulus spending, and lower revenues, resulting in some of the highest public debt ratios seen in advanced economies in the past 40 years.
- Recent debates have focused on the pace of debt reduction, with less attention to the desirable level of public debt to which the economy should converge following a debt shock.
- The paper abstracts from rollover risks faced by countries near their debt limits and from shorter-run cyclical considerations, focusing instead on countries near full employment that enjoy considerable fiscal space.

### THREE THEORETICAL INSIGHTS
- Inherited public debt represents a deadweight burden on the economy, dimming both investment and growth prospects; a corollary is that an economy that has inherited a lot of public debt will rationally choose to invest less in public capital than one with a lower level of debt.
- If fiscal space remains ample, policies to deliberately pay down debt are normatively undesirable because the distortive cost of deliberate debt reduction is likely to exceed the crisis-insurance benefit from lower debt; in such cases, debt-to-GDP ratios should be reduced organically through growth or opportunistically when less distortionary sources of revenue are available.
- Public debt should be issued to smooth taxes necessary to finance lumpy expenditures, yielding a version of the golden rule: public investment is debt-financed and undertaken to the point that social returns equal the market interest rate, with the twist that the social return will itself be reduced by the need to raise distortive taxation on labor and capital to service the higher debt.

### POLICY FRAMEWORK AND NORMATIVE GUIDANCE
- The analysis is targeted at green-zone cases—countries with ample fiscal space as defined in Ostry and others [2010]—where reducing debt deliberately is likely to be normatively undesirable because the costs will be larger than the benefits.
- For countries with ample fiscal space:
  - Avoid deliberate, distortionary fiscal consolidation aimed at paying down inherited debt.
  - Favor organic debt reduction via growth or opportunistic revenue measures that are less distortionary.
  - Use debt issuance to smooth tax distortions associated with lumpy public expenditures.
- For countries outside the green zone (yellow or red zones), the analysis recognizes sovereign risks and fiscal constraints but does not provide specific prescriptions in this summary.

### ANALYTICAL APPROACH AND CONCEPTS
- The paper develops a pure theory of public debt and investment, exploring:
  - Optimal Fiscal Policy
  - A Debt Shock
  - Paying Down the Debt
- Emphasis is placed on the interaction between public debt, public investment, social returns to public capital, and the distortionary effects of taxation required to service debt.

### PRACTICAL IMPLICATIONS
- Determining a safe level of debt is difficult and cannot be established by a mechanical rule or threshold.
- Stress testing public-sector balance sheets is essential for country-level judgments about safe public debt levels.
- It may be helpful to conceptualize debt levels as falling into three zones:
  - Green zone: fiscal space is ample.
  - Yellow zone: space is positive but sovereign risks are salient.
  - Red zone: fiscal space has run out.
- The paper is concerned primarily with green-zone cases where deliberate debt reduction is likely undesirable.

*INTERNATIONAL MONETARY FUND*

### INTRODUCTION

### INTRODUCTION

### Key framing and tradeoffs
- High public debt in advanced economies is an important legacy of the global financial crisis, as is an erosion of the public capital stock.
- The policy tradeoff:
  - Reducing public debt lowers sovereign risk and provides margins to cope with contingent risks.
  - Increasing public investment addresses public capital shortfalls and can be debt-financed given low real interest rates and demand shortfalls.
- The paper focuses on cases near full employment and with ample fiscal space, abstracting from Keynesian demand management and crisis risk to analyze interactions among public debt, public investment, and growth.

### Rationale for reducing public debt
- Risk-management: the option value of lower debt is high if catastrophic events (for example, a financial crisis requiring a public backstop) could force massive borrowing.
- Growth channel: high public debt implies higher distortionary taxation (or cuts in productive spending) to service it, which dampens investment and economic growth.
- For countries with demand shortfalls, infrastructure gaps, and high/risky public debt, there is an evident tradeoff between building public capital and containing sovereign risks.

### Exception and normative caution
- An exception arises if debt-financed public investment actually lowers the debt ratio (requires large Keynesian multipliers, minuscule interest burden, and super-efficient investment). These conditions are not generally neutral assumptions for policy.

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### When should public debt be reduced?
- Policy priorities depend on country circumstances:
  - Low-debt countries with big infrastructure needs (and idle resources and low interest rates): building infrastructure should be the greater priority.
  - Countries with significant risk of fiscal distress: unlikely to afford major borrowing for investment.
  - Less clear-cut cases (high debt but no plausible risk of fiscal distress; some infrastructure needs; near full employment and normal real interest rates): the paper develops an analytical framework to address this.

- Two central questions in the framework:
  1. Normative implications of high public debt itself: is it optimal to pay down debt in a reasonable timeframe given the tradeoff between perpetual distortionary taxation and the short-run distortion of raising taxes to pay down debt?
  2. Impact of a debt shock on the warranted level of public investment: higher debt implies higher distortionary taxation that reduces the productivity of factors complementary to public capital, lowering warranted public investment and long-run output.

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### Core policy-relevant findings (summary bullets drawn from model and simulations)
- Inherited public debt represents a deadweight burden, reducing both investment potential and growth prospects; the larger the inherited debt, the lower will be both public and private investment and the lower will be output growth.
- Where countries retain ample fiscal space, governments should generally not pursue policies aimed at deliberately paying down the debt (through distortionary tax increases or cuts in productive spending), and should instead allow the debt ratio to decline through growth and “opportunistic” revenues, while pursuing cuts in unproductive spending where possible.
- Debt should be used to smooth taxes for lumpy government expenditures. For public investment, debt-finance is appropriate for projects whose social marginal product earns at least the market interest rate, but the required social rate of return rises as public debt increases because servicing additional debt requires distortionary taxation of factors complementary to public capital.
- Quantitative magnitudes from simulations:
  - An increase in public debt of 50 percent of GDP (e.g., from 50 percent to 100 percent of GDP) would lower steady-state output and consumption by 2 percentage points in perpetuity, implying a present value welfare loss of 30 percent of initial consumption, with a similar decline in long-run levels of private and public capital.
  - Paying down 5 percent of GDP (one-tenth of the additional inherited debt in that example) in one year incurs a present value welfare cost equal to about 1 percent of GDP.
  - Paying down 10 percent of GDP would imply a welfare cost equal to 2–3 percent of GDP.
  - Paying down 20 percent of GDP would incur a cost of at least 6 percent of GDP.

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### Public finances and growth in advanced economies (empirical snapshot)
- Public debt developments:
  - On average, debt rose from 53 percent of GDP at end-2007 to almost 80 percent by end-2012.
  - For the top quartile, debt now exceeds 100 percent of GDP.
- Primary balances:
  - Swung from an average surplus of 2.1 percent of GDP in 2008 to a deficit of 4.4 percent in 2009, before partial recovery by 2012.
  - The deterioration mainly reflected loss of revenues and automatic stabilizers; very little represented discretionary stimulus, and of that, only a small fraction was investment in public infrastructure.
- Growth:
  - Real GDP growth in advanced economies turned sharply negative in 2009, rebounded in 2010, and remained moderate thereafter.
  - Projections for 2017 suggest advanced economies will have barely half their precrisis growth rates.
  - Projected growth for 2013–17 is strongly negatively correlated with end-2012 public debt ratios (correlation is –.41, statistically significant at the 5 percent level).

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### A pure theory of public debt and investment — model setup and implications
- Framework features:
  - Closed-economy representative-agent model of domestic debt (abstracts from inter- and intragenerational distribution and international transfer problems).
  - Government spending includes productive public capital.
  - Only distortionary taxes are available to finance spending.
  - Public debt consists of one-period bonds; no possibility of default; government is benevolent (maximizes representative agent’s lifetime utility).
- Key analytical results:
  - Government’s implied discount rate equals the market interest rate despite wedges between private and social valuations.
  - Optimal policy exhibits tax smoothing: because distortionary costs are convex in tax rates, minimizing total distortion involves smoothing tax rates; public investment is largely debt financed to the point that its marginal product equals the market interest rate (golden rule), recognizing the tax-distortion feedback on returns.
  - Simulated dynamics under low, medium, and high initial debt show higher initial debt leads to lower public capital, private consumption, and output along transitional and steady-state paths.

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### A debt shock: optimal response and welfare implications
- Conceptual options to an exogenous increase in public debt:
  - Pay down the debt immediately.
  - Pay down the debt gradually.
  - Live with the higher debt forever.
- Model result:
  - Optimal policy in the baseline closed-economy model is to live with the inherited debt forever; the new debt path is essentially parallel to the baseline path.
  - Intuition: paying down $1 of public debt today incurs distortionary tax costs; deferring repayment leads to servicing $(1+r) tomorrow but the government discounts at rate (1+r), so there is no gain from early repayment (tax-smoothing argument).
- Consequence:
  - Although living with debt is optimal in this setup, higher inherited debt permanently lowers steady-state output, consumption, and capital through higher taxation needed to service the debt.

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### Paying down the debt: costs versus crisis-insurance benefits
- Cost estimates (from simulations with plausible parameters):
  - Paying down 5 percent of GDP in one year incurs a present value welfare cost ≈ 1 percent of GDP.
  - Paying down 10 percent of GDP implies a welfare cost ≈ 2–3 percent of GDP.
  - Paying down 20 percent of GDP incurs a cost of at least 6 percent of GDP.
- Benefit depends critically on fiscal space (distance to sovereign debt limit):
  - If fiscal space is ample (green zone), the benefit in lowering crisis probability by reducing debt is likely small.
  - As fiscal space narrows (transition from green to yellow to red zones), the crisis-insurance benefit of paying down debt increases; the optimal choice depends on detailed stress testing of public-sector balance sheets and judgments about resilience.
- The model pertains to countries in the green zone (ample fiscal space), where living with debt is generally optimal; practical application requires assessing where countries lie along the fiscal-space continuum.

---

*Source: INTRODUCTION, _sdn1510*

### 2.6 percent per year: over a 20 year horizon, the expected number of crises is 0.52, with a very

### 2.6 percent per year: over a 20 year horizon, the expected number of crises is 0.52, with a very

### Expected costs and probabilities of crises
- Expected crisis frequency over a 20 year horizon: 0.52.
- Generous estimate of output cost associated with these events (most not full sovereign debt crises): approximately 15 percent of GDP.
- At debt of 120 percent of GDP:
  - Annual probability of event implied in text: 2.6 percent per year.
  - Expected loss = 7.8 percent of GDP (computed as 0.026 × 20 × 0.15? — preserve reported result: 7.8 percent of GDP).
  - Probability that at least one crisis occurs over 20 years: 41 percent (reported calculation: 100 × (1 − (1 − 0.026)^20) = 41 percent).
- At debt of 100 percent of GDP:
  - Annual likelihood of an event: 2.4 percent per year.
  - Expected loss: 7.2 percent of GDP.
  - Probability that at least one crisis occurs over 20 years: 38.5 percent.
- Estimated expected benefit of reducing debt from 120 percent to 100 percent of GDP:
  - Approximately 0.4–0.6 percent of GDP (in text: about 0.4–0.6 percent of GDP).
  - Alternative probability-based benefit: 0.37 percent of GDP (reported: 100 × ((0.41 − 0.385) × 0.15) = 0.37 percent of GDP).

### Interpretation: costs versus distortionary taxation
- The expected benefit from reducing debt (≈0.4–0.6 percent of GDP) is around one-tenth of the welfare cost due to distortionary taxation (textual comparison).
- Conclusion for countries with ample fiscal space (clearly in the green zone):
  - The benefit of deliberately running surpluses to reduce debt is unlikely to exceed the cost of the necessary distortionary taxation.
  - Debt-to-GDP ratios should be reduced organically (through output growth) or opportunistically (when less distortionary revenues are available, e.g., privatization receipts or royalties).

### Insurance value near debt limit and fiscal space considerations
- When sovereigns are at or very near the debt limit (yellow zone, close to red zone threshold), there can be enormous gains from reducing debt before a crisis occurs.
- With uncertain precise location of the debt limit, maintaining fiscal space has insurance value against future shocks that increase sovereign debt (applies to yellow zone countries).
- For countries firmly in the green zone, preemptive debt reduction via distortionary taxation is usually not justified by the modest reduction in expected crisis costs.

### Opportunistic short-run revenue sources and taxation implications
- Two short-run inelastic revenue sources identified in the model:
  - Income from initial holdings of public debt.
  - Income from the initial stock of private capital.
- Short-run policy implications:
  - Taxation of initially inelastic capital income can be non-distortionary in initial periods; optimal fiscal program may call for heavy taxation initially to reduce inherited debt, but optimal tax rate on capital falls to zero quickly as private capital becomes elastic.
  - “Taxation” on income from government bonds can take the form of very low—perhaps negative—real interest rates in the initial period (in monetary models with sticky prices by cutting nominal rates; in the real model achieved by cutting taxes on labor).
  - Low real interest rates or cuts in taxes on more elastic tax bases give a fillip to output and consumption in the initial period and reduce the rate of return payable on inherited debt.

### Model-based guidance on when to reduce debt
- When fiscal space is ample:
  - Prefer living with debt and letting debt ratios decline via growth.
  - Reduce debt opportunistically when non-distortionary or less-distortionary revenue sources are available.
- When fiscal space is limited or market access is at risk:
  - Reducing debt to restore fiscal space or avoid imminent funding crisis is warranted.
- In episodes of asset price booms:
  - Seize opportunity to pay down public debt.

### Stylized facts and empirical evidence
- Postcrisis (2009–11) cross-country evidence:
  - Countries with larger increases in public debt generally reduced public gross capital formation the most (statistically significant at 5 percent level).
  - Public consumption showed no negative relationship with rising debt; relationship marginally positive but statistically insignificant.
  - Negative relationship between increase in public debt and decrease in economywide gross fixed-capital formation is even stronger.
- Longer-run (1960–2008) evidence for OECD countries:
  - Regressions show a strong negative relationship between public debt and current or subsequent public investment (Table 1, columns 1–2).
  - No analogous negative relationship between debt and public consumption (Table 1, columns 3–4).
- Growth regressions (five-year averages, 1960–2008, OECD):
  - Higher tax rates (revenues as percent of GDP) are associated with lower real GDP growth, controlling for population growth, initial per capita income, terms of trade shocks, inflation, and investment (Table 2, column 1).
  - Part of the negative effect of taxes on growth operates through lower investment; dropping investment increases the magnitude of the negative tax coefficient (Table 2, column 2).
  - Instrumental-variable and Arellano-Bond estimations yield similar results (Table 2, columns 4–6).
  - Including current or lagged debt in the regressions (alongside taxes) yields negative but statistically insignificant coefficients for debt.

### Key regression and calibration details (preserve exact parameters and results)
- Table 1 reported coefficients (selected):
  - Debt/GDP → Public Investment: coefficient −1.896*** (t-statistic [−3.954]).
  - Debt/GDP (t-1) → Public Investment: coefficient −2.264*** (t-statistic [−3.495]).
  - Debt/GDP → Public Consumption: coefficient 6.075*** (t-statistic [5.196]).
  - Debt/GDP (t-1) → Public Consumption: coefficient 4.251*** (t-statistic [4.520]).
  - Observations vary by column; R-squared values reported (e.g., 0.168, 0.211, 0.277, 0.174).
- Table 2 reported coefficients (selected, dependent variable: five-year average real GDP growth, 1960–2008):
  - Gvt revenues (percent of GDP) coefficient examples: −0.0396**, −0.0424**, −0.0399*, −0.0866**, −0.423** (t-statistics shown in brackets).
  - Population growth coefficients: 0.424, 0.532*, 0.536*, 0.557*, 1.424**, 1.083* (with t-statistics).
  - Initial GDP coefficients: −0.0386***, −0.0266***, −0.0219***, 0.00238, −0.0848*, −0.0698** (with t-statistics).
  - Log(inflation) coefficients include −0.00728*** and −0.0249*** in some specifications.
  - Observations and R-squared values reported across columns; robustness checks reported (Hansen test p-value 0.235; A-B AR(1) p-value 0.000840; A-B AR(2) p-value 0.829).
- Model calibration in Box 1:
  - Cobb–Douglas production with α = 0.33 and θ = 0.1.
  - Utility parameters: cgϕ = 0.3; clgϕ = 0.1; ϕcg = 1; ϕclg = 1; σ = 1 (text: "0 . 3;0 . 1;1;1;1 cg clg ϕ ϕ σ σ σ = = = = =", implying logarithmic utility).
  - Discount factor β = 0.96.
  - Rate of depreciation δ = 0.1.

### Policy implications and guiding principles (summary)
- Inherited public debt is a deadweight burden that reduces investment potential and growth prospects; higher inherited debt implies lower public and private investment and slower output growth.
- With ample fiscal space, living with debt and allowing it to decline via growth is generally preferable to paying it down through distortionary taxation.
- Debt-financing is appropriate to smooth taxes for lumpy government expenditures and for public investment projects whose social marginal product is at least the market interest rate, while recognizing that additional debt implies distortionary taxation that may lower the return to public investment.
- Country-specific circumstances determine the appropriate response: imminent market access risk or contingent liabilities justify active debt reduction; countries with fiscal space should consider opportunistic or organic reduction rather than aggressive tax-financed consolidation.

*International Monetary Fund*

### Box 2. Optimal

### Box 2. Optimal

### Problem setup and constraints
- Government commits to a fiscal plan 0{, , , }gtttttgbk (in which gtk is public capital) to maximize the agent’s utility, subject to:
  - The economywide resource constraint and the feasibility constraint that summarizes private sector behavior (Box 1 (1.6)).
- Formal objective and constraints (as in the source):
  - Maximize: ,,,0(,, )gttttt t tt gbk tMaxu c l g      (2.1)
  - Resource/feasibility condition: 1111 ..(1)(1)(,, )ppgggp tttt ttttt stc g kk kk Fkkl     = (2.2)
  - Feasibility/first-order relation: ,,,  ,00101 0()( ) t ct tlt tc t uc ul u Yb Rk      +  (2.3)

### First-order conditions and optimal program
- Lagrange multipliers:
  - μt, λt denote the Lagrange multipliers on the resource and feasibility constraints respectively.
- Optimal fiscal program (conditions):
  - ,0 pg t tt gt kk FFut (2.4)
  - ,, , , {()}{ ()} 1 tt ltll tltctcc tctlt uulu u ucuFt    +   (2.5)
  - ,, ,1,11,1 ()(())((1))1 g t ctcct tctctccttct k uucu u ucuFt     (2.6)

### Lump-sum taxes (non-distortionary) — first-best analog
- When taxes are lump-sum, feasibility constraint does not bind, so 0, and (2.5)–(2.6) reduce to:
  - ,,   , /1 ltctlt uu F t (2.7)
  - ,,1 (/)( (1))( (1)) 1 pg tt ctct kk uu FFt   (2.8)
- Interpretation:
  - (2.7) equates the marginal rate of substitution between consumption and leisure to the marginal product of labor; wage should equal marginal product of labor (from (1.4)).
  - (2.8) states that private and public investment are undertaken until respective marginal products equal the intertemporal rate of substitution, which (from (1.5)) is the market interest rate.

### Distortionary taxes — binding feasibility constraint
- When only distortionary taxes are available, 0, and government must consider the impact of taxes on revenue-raising ability.
  - Social rather than private marginal products and rates of substitution must be used (as in (2.4)–(2.6)).
  - Investment may yield a higher social return than the private return because it raises the marginal product of labor, easing labor taxation.
- Key result:
  - Even with 0, the implicit discount rate used by the government in its optimal fiscal program equals the market interest rate.
  - Holds exactly for iso-elastic utility, approximately otherwise.
- Substituting (1.8) into (2.6) yields:
  - ,,1 (/)(1)(1) pg tt ctctt kk uu RFF   (2.9)
  - Interpretation: market interest rate equals the representative agent’s discount factor — the discount factor the government uses when deciding whether to repay debt or defer taxation.

### Period 0 (opportunistic taxation of inelastic bases)
- First-order conditions in period 0 differ from t ≥ 1 by a term reflecting income from initial stocks:
  - 0 ,00,0,0,0 0,0,00101,0 {()} {(())} p llll ccccccl uulu uucuuYbRkF      +  + (2.10)
  - ,0,0 0,0,00101,1,1 1,1 (())(())((1)) g t p cccccccccc k uucuuYbRk uucuF     (2.11)
- Interpretation:
  - The extra term ,00101 () p cc uYb Rk   pertains to optimal opportunistic taxation of inelastic tax bases — income from the initial stocks of government bonds and of private capital.

*Source: Box 2. Optimal (excerpted equations and discussion).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/sdn/2015/_sdn1510.pdf_