## wpiea2022016-print-pdf — Introduction and selected chapters

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### 1. Introduction — definitions, accounting backbone, and operational sustainability criteria
- Definitions and conceptual framing
  - Sovereign debt is unsustainable if it cannot be repaid without altering contractual terms or rendering them irrelevant via default, restructuring, or hyperinflation.
  - Fiscal policy is unsustainable if preventing such an event requires a change in (fiscal) policy. If the required policy change is feasible, debt is sustainable although status-quo policies are not.
  - Feasibility of policy change depends on social, political, and economic costs (e.g., efficiency or growth costs of cutting essential government spending or raising already high taxes).
  - Recent literature links debt sustainability to the liquidity services provided by sovereign debt (beyond future primary balances) and to central bank policy credibility.
- Historical triggers of interest in debt sustainability
  - 1980s Latin American debt crisis introduced debt overhang and self-fulfilling crises; motivated techniques to distinguish insolvency from illiquidity.
  - 1980s U.S. widening fiscal deficits raised U.S. fiscal sustainability concerns.
  - Recent deficits and sharply higher debts (partly preceding and partly triggered by the Covid crisis) revived debates in an environment where real interest rates in many countries are lower than real growth rates.
- Core analytical tension: the 푟<푔 debate
  - If real interest rates are below real growth rates (푟<푔), debt as a share of GDP can fall even if governments never run primary surpluses.
  - Broad agreement: fiscal policy remains subject to sustainability constraints in a 푟<푔 environment, but disagreement exists on how constraints should be defined and when they bind.
  - Sovereign debt’s role as a “safe asset” (store of value and source of liquidity) matters: liquidity services can allow governments to run larger deficits without expected future primary surpluses, but this is limited by the value of the safe-debt service flow.
- Accounting backbone and key equations (nominal or real, abstracting from foreign-currency debt)
  - Debt evolution: 퐵_{t+1} = (1+푟_{t}) 퐵_{t} − 푆_{t+1}, where 퐵_{t} = debt at time t (end of period); 푟_{t} = government borrowing rate; 푆_{t} = primary balance (revenues including seignorage minus non-interest expenditures).
  - Debt-to-GDP dynamics (define (1+푔_{t}) ≡ 푌_{t+1}/푌_{t}, lower-case letters denote shares of GDP):
    - 푏_{t+1} = (1+푟_{t})/(1+푔_{t}) * 푏_{t} − 푠_{t+1}.
  - Iterated form with growth-adjusted discount factors 푞_{t,t+j}:
    - 푏_{t+n} = (1/푞_{t,t+n}) 푏_{t} − (1/푞_{t,t+n}) ∑_{j=1}^{n} 푞_{t,t+j} 푠_{t+j}.
    - Solving for current debt: 푏_{t} = 푞_{t,t+n} 푏_{t+n} + ∑_{j=1}^{n} 푞_{t,t+j} 푠_{t+j}.
  - Asymptotic analysis requires lim_{j→∞} 푞_{t,t+j} = 0 (growth-adjusted discount factors vanish asymptotically), a weaker condition than requiring 푟_{t}>푔_{t} every period.
- Two operational definitions of sustainability
  - IGBC / NPGC (Hamilton and Flavin, 1986)
    - Fiscal policy sustainable if intertemporal government budget constraint holds:
      - 푏_{t} = lim_{n→∞} ∑_{j=1}^{n} 푞_{t,t+j} 푠_{t+j}.
    - Transversality / no-Ponzi-game condition (NPGC): lim_{n→∞} 푞_{t,t+n} 푏_{t+n} = 0.
    - Economic interpretation: current debt equals expected present discounted value of future primary surpluses; violation implies adjustment or default/hyperinflation.
  - Non-explosive debt-ratio criterion (Kremers 1988, 1989; based on Blanchard 1984)
    - Fiscal policy sustainable if lim_{n→∞} 푏_{t+n} = 푏 (a constant).
    - Stronger condition than NPGC/IGBC; non-explosive condition implies NPGC/IGBC if lim_{n→∞} 푞_{t,t+n} = 0, but not vice versa.
- Empirical testing tradition and mixed evidence
  - Early tests (1980s–early 1990s) examined unit roots and cointegration implied by NPGC/IGBC and non-explosive conditions with mixed conclusions:
    - Examples: Hamilton and Flavin (1986), Wilcox (1989), Trehan and Walsh (1991), Hakkio and Rush (1991), Kremers (1989), Ahmed and Rogers (1995).
  - Mixed results: Hamilton and Flavin (1986) and Trehan and Walsh (1991) find NPGC holds in the U.S.; Wilcox (1989) and Hakkio and Rush (1991) conclude it is violated.
  - Long-sample studies suggest episodes of unsustainability can be followed by institutional restoration of IGBC validity (e.g., U.S. surplus in the late 1990s).

### 3. The Bohn revolution and its aftermath — stochastic discounting, stabilization feedback, and fiscal fatigue
- Bohn (1995): stochastic discounting and the IGBC
  - Key point: with uncertainty and risk-averse agents, the standard NPGC is incorrect as derived from individual optimization; discounting uses a stochastic discount factor tied to marginal rates of substitution, not government borrowing rates.
  - Stochastic discount factor (equation (8)):
    - q̃_{t,t+j} ≡ E_t[ μ_{t,t+j} ∏_{h=t}^{t+j-1} (1+g_h)^{-1} ] with μ_{t,t+j} ≡ β^{j} U′(S_{t+j}) / U′(S_t).
  - Correct transversality condition in complete-markets model (equation (9)):
    - lim_{n→∞} E_t[ q̃_{t,t+n} b_{t+n} ] = 0.
  - Model-based IGBC (equation (10)):
    - b_t = E_t[ ∑_{j=1}^{n} ( q̃_{t,t+j} s_{t+j} ) ].
  - Relation to accounting-based IGBC (equations (11) and (12)):
    - b_t = ∑_{j=1}^{n} { E_t[ q̃_{t,t+j} ] E_t[ s_{t+j} ] + cov( q̃_{t,t+j}, s_{t+j} ) }.
    - b_t = E_t[ ∑_{j=1}^{n} { q_{t,t+j} s_{t+j} + cov( q̃_{t,t+j}, s_{t+j} ) } ].
  - Because cov( q̃_{t,t+j}, s_{t+j} ) is typically negative (marginal utility higher in recessions when governments run deficits), the model-based IGBC is typically tighter than the accounting-based IGBC.
- Bohn (1998): empirical test using fiscal feedback
  - Empirical specification (equation (13)): s_t = ρ b_t + ε_t (ε_t includes business cycle controls and temporary expenditures).
  - Estimated magnitude: ρ is on the order of 0.05 (a 10 percent of GDP rise in the debt ratio leads to a rise in the primary balance by about half a percent of GDP).
  - Substituting (13) into debt dynamics yields (equation (14)):
    - b_{t+1} = [ (1 + r_t) / (1 + g_t) * (1 + ρ)^{-1} ] b_t + υ_{t+1}, with υ_{t+1} ≡ −ε_{t+1} (1+ρ)^{-1}.
  - Implication: debt is mean-reverting if (1 + r_t)/(1 + g_t) < 1 + ρ on average; with ρ ≈ 0.05 and U.S. rates near or below growth rates, this condition holds.
  - Additional implication: ρ>0 implies the stochastic IGBC holds as well as the deterministic version, because debt growth is reduced by factor 1−ρ relative to a Ponzi scheme so E_t[ q̃_{t,t+n} b_{t+n} ] ≈ (1−ρ)^{n} b_t → 0 for any small ρ>0.
  - Caveat: result depends on linearity (f′(b_t) ≥ ρ > 0 for all b_t ≥ b*); if feedback becomes concave at high b, sustainability can fail.
- Fiscal fatigue, nonlinearities, and implied debt limits
  - Empirical extensions (Mendoza and Ostry 2008; Mauro et al. 2015) confirm Bohn’s basic result but find the fiscal response weakens at high debt levels.
  - Ghosh et al. (2013): identify “fiscal fatigue” — the positive relationship between primary balance and debt ratio weakens and can turn zero or negative at high debt.
  - Formal debt-limit condition in an r > g world (equation (15)):
    - α + f(b̅) = (r − g)/(1 + g) * b̅, and for any b > b̅, α + f(b) < (r − g)/(1 + g) * b.
  - Ghosh et al. (2013) quantitative findings (cubic approximation, 1985-2007, 18 advanced countries):
    - Median steady-state debt-to-GDP ratio b* is 49 percent.
    - Median debt limit b̅ is 186 percent.
    - Five cases—Greece, Iceland, Italy, Japan, and Portugal—have no steady state debt level given country-specific α and r−g projections and the common f(b_t) estimate; during 2008-2012 all except Japan suffered debt crises.
    - Projected median r−g was 1.3 percent, but actual median r−g in most advanced countries during 2013-2021 has been negative.
  - Implication: fiscal fatigue can generate threshold behavior where debt converges to b* if b < b̅ but explodes if shocks push b > b̅ unless exceptional adjustments or temporary cheaper financing occur.

### Rollover risk, self-fulfilling crises, and policy/institutional responses
- Nature of rollover risk and self-fulfilling crises
  - Governments may lose market access at acceptable borrowing rates for non-fundamental reasons (debt market panics, pure liquidity crises, rollover crises, self-fulfilling debt crises).
  - Sachs (1984): if refinancing needs exceed cash reserves and feasible short-term adjustment, failure to roll over leads to default; creditor coordination failures can produce multiple equilibria.
  - Calvo (1988): taxation to repay may be too distortionary, yielding coexistence of a “good” equilibrium (θ=0, default-free) and a “bad” equilibrium (partial repudiation); which equilibrium prevails depends on investor expectations.
  - Lorenzoni and Werning (2019): with long-term debt, marginal borrowing costs can rise sharply while defaults materialize slowly, producing “slow moving” debt crises.
- Policy and institutional implications from literature
  - Lower debt reduces rollover risk; calibrations by Blanchard, Huertas, and Kister (2021) suggest safe levels around 20 percent of GDP or less when rollover risk dominates.
  - Longer debt maturities reduce likelihood of self-fulfilling crises by lowering frequency of large rollover needs and slowing the path from sentiment shocks to default, allowing time for policy adjustment.
  - Limiting exposure to private, non-resident investors lowers run probabilities.
  - A credible central bank can potentially rule out the “bad” equilibrium by acting as lender of last resort or capping sovereign borrowing costs; asset purchase programs can swap privately-held finite-maturity bonds for central bank reserves and reduce rollover needs.
- Conditions for effective central bank rescues
  - Central bank must be credible in controlling inflation and exert some control over real interest rates; absent credibility, bailouts risk high inflation or fail to keep borrowing costs below default thresholds.
  - Debt issued in domestic currency is necessary for central bank to supply liquidity and act as lender of last resort; foreign-currency borrowing constrains such interventions.
- Practice in debt sustainability analysis
  - Applied DSA (e.g., IMF practice) incorporates rollover risk in addition to solvency risk, focusing on flow concepts such as gross financing needs as a share of GDP and quantifying interest rate risk, not only fundamental risk.
  - IMF (2013) definition: public debt is sustainable when the primary balance needed to at least stabilize debt under both baseline and realistic shock scenarios is economically and politically feasible, consistent with an acceptably low rollover risk and preserving potential growth.

### 5. Debt sustainability and fiscal risks when 푟 < 푔 — accounting vs. model-based views, liquidity services, and central bank credibility
- Overview of the debate
  - Blanchard (2019): very low real interest rates, particularly 푟 < 푔, lower costs of issuing public debt; with 푟 < 푔 on average, “higher debt may not imply a higher fiscal cost.”
  - Pushback arguments:
    - Higher debt raises rollover risks even if 푟 < 푔 because more debt needs refinancing and does not reduce probability of rollover shocks conditional on debt ratio (Mauro and Zhou, 2020; Moreno Badia et al., 2020).
    - Model-based analyses indicate current debt levels can still raise sustainability concerns even if 푟 < 푔 on a sustained basis (Jiang et al. (2019), Olijslagers et al. (2020)).
- Accounting exercise vs. constraints
  - Under constant s_t, r_t, g_t, equation (2) becomes (2'):
    - b_{t+1} = (1+r)/(1+g) b_t − s.
  - With r < g, (1+r)/(1+g) < 1, implying a stable difference equation and steady-state:
    - b = (1+g)/(r − g) s.
  - Accounting implications:
    - If s = 0 and the debt ratio is positive, the debt ratio will fall over time because interest expenditures grow slower than output.
    - A larger primary deficit leads to stabilization at a higher debt ratio; as long as primary deficit does not keep rising and r remains below g, the debt ratio stabilizes eventually.
  - Limitations not captured by the accounting view:
    - Steady-state primary deficit might exceed the size of the economy (−s > 1).
    - Steady-state debt might exceed available private savings.
    - Such contradictions show the accounting exercise omits economic feedbacks and feasibility constraints (Cochrane (2021a) highlighted examples).
- Model-based IGBC and empirical findings
  - Model-based IGBC uses stochastic discounting that can be substantially higher than government borrowing rates.
  - Jiang et al. (2019) and Olijslagers et al. (2020):
    - Derive/estimate model-based IGBC using arbitrage pricing theory or calibrated general equilibrium models plus VARs to estimate cyclical properties.
    - Procyclical nature of primary balances implies average discount rates substantially higher than growth rates, making the effective environment like 푟 > 푔 for sustainability concerns.
    - Conclusion: perpetual primary deficits—or even perpetual primary balance of zero—would violate the model-based IGBC; positive market values of debt require strictly positive future primary balances on average if primary balances remain procyclical.
- Liquidity services, convenience yield, and limited bubbles
  - In incomplete-markets frameworks (Berentsen and Waller (2018), Brunnermeier et al. (2021), Reis (2021)), government debt provides liquidity and safety, generating a convenience yield or liquidity premium.
  - Modified valuation equation:
    - Standard stochastic IGBC (16): value of debt stock = E{PV(future primary surpluses)}.
    - Expanded IGBC with service flow (17): value of debt stock = E{PV(future primary surpluses)} + E{PV(future service flow)}.
  - Consequences:
    - Transversality condition can fail under (17), permitting a finite “bubble” where permanent primary deficits may be consistent with sustainability if the service flow component is sufficiently large.
    - Sustainability of policies such as permanent zero primary balances depends on the magnitude of the service flow value.
    - Jiang et al. (2019) estimate the service flow component to be worth about 65 percent of GDP on average during their sample period — close to the market value of U.S. debt at the time — making the thought experiment that U.S. debt could be sustainable under permanent zero primary balances not implausible.
    - Nevertheless, maintaining sustainability from recent pre-pandemic positions would require large fiscal adjustments: pre-pandemic (2019) primary deficit of about 3.5 percent; 2020 and 2021 primary deficits are over 10 percent of GDP.
- Policy channels and implications: central bank credibility as a fiscal asset
  - Central bank credibility expands sovereign debt-carrying capacity via multiple channels:
    - Prevents rollover crises, raising the safe level of sovereign debt.
    - Countercyclical monetary policy (cutting rates in recessions) allows capital gains on fixed-rate, longer-term bonds in downturns, giving government debt a cyclical insurance property and potentially making it a “negative beta” asset (Brunnermeier et al., 2021; Cochrane, 2021a).
      - Credibility enables central banks to cut rates in recessions without de-anchoring inflation expectations.
    - Enhances liquidity services (the second term in (17)) by:
      - Ex ante market development: credible low and stable inflation supports local-currency debt market development.
      - Ex post market-maker role: credible central banks can inject liquidity as “market maker of last resort” to prevent secondary market freezes without de-anchoring inflation expectations.
  - Heterogeneity across country groups
    - Advanced economies: generally more central bank credibility and deeper local-currency markets, enabling higher sustainable debt levels than decades ago.
    - EMDEs: lower central bank credibility limits their ability to prevent rollover crises, lower rates in downturns, develop deep local-currency secondary markets, and act as market makers of last resort — tightening fiscal constraints and requiring higher expected future primary surpluses for a given investor valuation.
    - Notwithstanding, several EMDE central banks (Guatemala, Indonesia, The Philippines) bought government bonds after COVID onset without runaway inflation or exchange rate collapse, implying some had reputational capital entering 2020.
  - Credibility is reputational capital: can be accumulated by low and stable inflation records and is finite and depletable; exceeding the envelope defined by equation (17) could destroy the premises of credibility.

### Select quantitative findings and exact numerical values cited in the text
- Bohn (1998) empirical feedback parameter: ρ is on the order of 0.05.
- Ghosh et al. (2013) estimates (1985–2007, 18 advanced countries):
  - Median steady-state debt-to-GDP ratio b* is 49 percent.
  - Median debt limit b̅ is 186 percent.
  - Projected median r−g was 1.3 percent; actual median r−g in most advanced countries during 2013–2021 has been negative.
- Blanchard thought experiment context:
  - Pre-pandemic (2019) primary deficit of about 3.5 percent.
  - 2020 and 2021 primary deficits are over 10 percent of GDP.
- Jiang et al. (2019) estimated service flow component: about 65 percent of GDP on average during their sample period.
- Policy calibration note:
  - Blanchard, Huertas, and Kister (2021) suggest safe debt levels around 20 percent of GDP or less in the presence of rollover risk.

*Source: wpiea2022016-print-pdf — 1. Introduction; 3. The Bohn revolution and its aftermath; 5. Debt sustainability and fiscal risks when r < g*

### 1. Introduction

### 1. Introduction

### Definitions and conceptual framing
- Sovereign debt is unsustainable if it cannot be repaid without altering contractual terms or rendering them irrelevant via default, restructuring, or hyperinflation.
- Fiscal policy is unsustainable if preventing such an event requires a change in (fiscal) policy. If the required policy change is feasible, debt is sustainable although status-quo policies are not.
- Feasibility of policy change depends on social, political, and economic costs (e.g., efficiency or growth costs of cutting essential government spending or raising already high taxes).
- Recent literature links debt sustainability to the liquidity services provided by sovereign debt (beyond future primary balances) and to central bank policy credibility.

### Historical triggers of interest in debt sustainability
- 1980s Latin American debt crisis: introduced concepts such as debt overhang and self-fulfilling debt crises; prompted development of techniques to distinguish insolvency from illiquidity.
- Widening U.S. fiscal deficits in the 1980s: raised questions about U.S. fiscal sustainability.
- Recent phase of deficits and sharply higher debts, partly preceding and partly triggered by the Covid crisis, revived debates similar to the 1980s/early 1990s debates but in an environment where real interest rates in many countries are lower than real growth rates.

### Core analytical tension: the 푟<푔 debate
- When real interest rates are below real growth rates (“푟<푔”), debt as a share of GDP can fall even if governments never run primary (non-interest) fiscal surpluses.
- There is broad agreement that fiscal policy remains subject to sustainability constraints in a 푟<푔 environment, but much less agreement on how constraints should be defined and when they bind.
- The role of sovereign debt as a “safe asset” (store of value and source of liquidity) matters: if agents hold debt despite no expected future primary surpluses because it provides liquidity services, governments may effectively obtain a “free lunch,” but this is limited by the value of the safe-debt service flow.

### Structure of the paper (overview of sections)
- Section 2: introduces fiscal accounting relationships, the no-Ponzi-game condition (NPGC), and the intertemporal government budget constraint (IGBC). Contrasts IGBC-based sustainability (Hamilton and Flavin) with non-explosive debt-ratio-based sustainability (Kremers/Blanchard).
- Section 3: discusses Henning Bohn’s contributions — stochastic discounting in the NPGC under uncertainty and risk aversion; an alternative test based on feedback from debt to fiscal policy; evidence that feedback weakens at high debt levels, implying potential critical debt thresholds.
- Section 4: examines rollover risk and its implications for debt sustainability; contrasts definitions that exclude liquidity problems from those that include them; policy levers to reduce rollover risk include lengthening debt maturity, borrowing in domestic currency, and building central bank credibility.
- Section 5: analyzes implications of 푟<푔 for advanced economies and the safe-asset role of sovereign debt; warns against “overexploitation” that would destroy the safe-asset service flow.
- Section 6: concludes on links between debt sustainability and central bank credibility: credible central banks can expand sustainable debt/deficit boundaries by reducing rollover risk, supporting local-currency sovereign secondary markets, ensuring market liquidity during crises, and improving cyclical properties of debt from consumers’ perspective. Building central bank credibility requires strong monetary frameworks and a track record of fiscal prudence; it is inconsistent with fiscal dominance.

### Accounting backbone and operational definitions (preview of Section 2 material included in this unit)
- Basic debt evolution equation (nominal or real, abstracting from foreign-currency debt):
  - 퐵_{t+1} = (1+푟_{t}) 퐵_{t} − 푆_{t+1}
  - 퐵_{t} = debt at time t (end of period); 푟_{t} = government borrowing rate; 푆_{t} = primary balance (revenues including seignorage minus non-interest expenditures).
- Two interpretations:
  - One-period maturity interpretation: 퐵_{t} is the level of debt that matures each period; 푟_{t} is the contractual interest rate on one-period debt.
  - Market-value interpretation: 퐵_{t} is the market value of outstanding debt; 푟_{t} is the holding return (coupon plus capital gains/losses). The market-value perspective is more flexible and dominates the literature.
- Debt-to-GDP dynamics (define (1+푔_{t}) ≡ 푌_{t+1}/푌_{t}, lower-case letters denote shares of GDP):
  - 푏_{t+1} = (1+푟_{t})/(1+푔_{t}) * 푏_{t} − 푠_{t+1}
- Iteration and compact notation using growth-adjusted discount factors 푞_{t,t+j}:
  - 푏_{t+n} = (1/푞_{t,t+n}) 푏_{t} − (1/푞_{t,t+n}) ∑_{j=1}^{n} 푞_{t,t+j} 푠_{t+j}
  - Solving for current debt:
    - 푏_{t} = 푞_{t,t+n} 푏_{t+n} + ∑_{j=1}^{n} 푞_{t,t+j} 푠_{t+j}
- Key asymptotic assumption for the analysis in this section: while 푟_{t}>푔_{t} need not hold every period, the weaker condition required is lim_{j→∞} 푞_{t,t+j} = 0 (i.e., growth-adjusted discount factors vanish asymptotically). This corresponds to the traditional case where, on average, 푟>푔.

### Two operational definitions of sustainability
- IGBC / NPGC (Hamilton and Flavin, 1986):
  - Fiscal policy sustainable if the intertemporal government budget constraint holds:
    - 푏_{t} = lim_{n→∞} ∑_{j=1}^{n} 푞_{t,t+j} 푠_{t+j}
  - Requires the transversality condition / no-Ponzi-game condition (NPGC):
    - lim_{n→∞} 푞_{t,t+n} 푏_{t+n} = 0
  - Economic interpretation: current debt must equal the expected present discounted value of future primary surpluses; violation implies policy adjustment or default/hyperinflation.
  - Note: in environments with uncertainty and risk-averse agents, the NPGC takes a different (stochastic discounting) form (see Section 3).
- Non-explosive debt-ratio criterion (Kremers 1988, 1989; based on Blanchard 1984):
  - Fiscal policy sustainable if current and projected policy do not lead to an exploding debt ratio:
    - lim_{n→∞} 푏_{t+n} = 푏, where 푏 is a constant
  - This is a stronger condition than NPGC/IGBC; with lim_{n→∞} 푞_{t,t+n} = 0, the non-explosive condition implies the NPGC/IGBC, but not vice versa.

### Empirical testing tradition and mixed evidence
- Early empirical work (1980s–early 1990s) tested stationarity and cointegration properties implied by NPGC/IGBC and non-explosive conditions:
  - Examples and sample windows cited in this unit:
    - Hamilton and Flavin (1986): unit root test on U.S. real debt (sample beginning 1960).
    - Wilcox (1989): constructs 1960-1987 series of present value of U.S. debt using ex-post realized holding returns.
    - Trehan and Walsh (1991): stationarity of first difference of real value of debt.
    - Hakkio and Rush (1991): cointegration of total government spending and revenue.
    - Kremers (1989): U.S. data 1920-1982 finds cointegration of face-value U.S. debt and GDP.
    - Ahmed and Rogers (1995): very long samples — U.S. 1795-1990 and U.K. 1698-1987 — find strong cointegration between revenues, non-interest spending, and interest spending and evidence of stable cointegrating vectors around structural breaks.
- Mixed empirical findings:
  - Hamilton and Flavin (1986) and Trehan and Walsh (1991) conclude that the NPGC holds in the U.S.
  - Wilcox (1989) and Hakkio and Rush (1991) conclude that it is violated.
  - Possible reasons for mixed results: short samples (1960s–late 1980s), regime shifts (e.g., post-1980 fiscal changes).
  - Long-sample studies (Ahmed and Rogers 1995) suggest that while fiscal policy can exhibit unsustainable phases, policy institutions in the U.S. and U.K. eventually restore IGBC validity, evidenced by eventual return to surpluses (e.g., U.S. fiscal balance briefly returned to surplus in the second half of the 1990s).

*Source: wpiea2022016-print-pdf - 1. Introduction*

### 3. The Bohn revolution and its aftermath

### 3. The Bohn revolution and its aftermath

### Bohn (1995): theoretical insights on stochastic discounting and the IGBC
- Main point: In an environment with uncertainty and risk averse consumers/investors, the NPGC (6) is incorrect as it cannot be derived from individual optimization; the IGBC (5) may not be the relevant intertemporal budget constraint.
- Consumers/investors discount future income using a stochastic discount factor reflecting the marginal rate of substitution between consumption today and future consumption derived from investing in government debt, not the government borrowing rate.
- Stochastic discount factor definition (equation (8)):
  - q̃_{t,t+j} ≡ E_t[ μ_{t,t+j} ∏_{h=t}^{t+j-1} (1+g_h)^{-1} ] with μ_{t,t+j} ≡ β^{j} U′(S_{t+j}) / U′(S_t).
- Correct transversality condition in the complete-markets model (equation (9)):
  - lim_{n→∞} E_t[ q̃_{t,t+n} b_{t+n} ] = 0.
- Model-based IGBC (equation (10)):
  - b_t = E_t[ ∑_{j=1}^{n} ( q̃_{t,t+j} s_{t+j} ) ].
- Comparison with accounting-based IGBC: rewriting leads to (equation (11)) and (equation (12)):
  - b_t = ∑_{j=1}^{n} { E_t[ q̃_{t,t+j} ] E_t[ s_{t+j} ] + cov( q̃_{t,t+j}, s_{t+j} ) }
  - b_t = E_t[ ∑_{j=1}^{n} { q_{t,t+j} s_{t+j} + cov( q̃_{t,t+j}, s_{t+j} ) } ].
- Because covariance between future marginal utility and future primary balances is generally negative (marginal utility higher in recessions when governments tend to run deficits), the model-based IGBC (12) typically implies a tighter constraint on sustainable debt levels than the accounting-based IGBC (5).

### Bohn (1998): empirical test of debt sustainability and stabilization feedback
- Empirical specification (equation (13)):
  - s_t = ρ b_t + ε_t, where ε_t includes controls (business cycle measures, temporary government expenditures such as wars).
- Estimated magnitude: ρ is on the order of 0.05 (a 10 percent of GDP rise in the debt ratio leads to a rise in the primary balance by about half a percent of GDP).
- First reason the test is stronger: it encapsulates debt stabilization. Substituting (13) into debt dynamics yields (equation (14)):
  - b_{t+1} = [ (1 + r_t) / (1 + g_t) * (1 + ρ)^{-1} ] b_t + υ_{t+1}, with υ_{t+1} ≡ −ε_{t+1} (1+ρ)^{-1}.
  - (14) is mean-reverting as long as (1 + r_t)/(1 + g_t) ≈ 1 + r_t − g_t < 1 + ρ on average. With ρ ≈ 0.05 and given U.S. government borrowing rates generally near or below the growth rate, the inequality holds.
- Second reason: ρ>0 implies the IGBC holds in its stochastic version (10) as well as the deterministic version (5). Intuition (Bohn 2008): debt growth is reduced by 1−ρ relative to a Ponzi scheme, so E_t[ q̃_{t,t+n} b_{t+n} ] ≈ (1−ρ)^{n} b_t → 0 for any small ρ>0.
- Caveat: the result relies on the surplus-debt relationship being at least linear for high debt levels (f′(b_t) ≥ ρ > 0 for all b_t ≥ b*). If the relationship becomes concave above some b*, sustainability may fail.

### Fiscal fatigue, nonlinearities, and implied debt limits
- Empirical extensions (Mendoza and Ostry 2008; Mauro et al. 2015) broadly confirm Bohn’s result but find the positive reaction of the fiscal balance to the debt ratio diminishes at higher debt ratios.
- Ghosh et al. (2013): estimate non-linear (concave) relationships between primary balance and debt ratio for 23 industrial countries and identify “fiscal fatigue” — the positive relationship weakens at high debt levels and can turn zero or negative.
- In an r > g world, concavity implies a debt limit b̅ beyond which primary balances cannot stabilize debt. Formal condition (equation (15)):
  - α + f(b̅) = (r − g)/(1 + g) * b̅, and for any b > b̅, α + f(b) < (r − g)/(1 + g) * b.
- Ghosh et al. (2013) quantitative findings based on a cubic approximation of f(b_t) using 1985-2007 data for 18 advanced countries:
  - Median steady-state debt-to-GDP ratio b* is 49 percent.
  - Median debt limit b̅ is 186 percent.
  - In five cases—Greece, Iceland, Italy, Japan, and Portugal—no steady state debt level exists given country-specific α and r−g projections and the common f(b_t) estimate. During 2008-2012, all of these countries except Japan suffered debt crises.
  - Projected median r−g was 1.3 percent, but actual median r−g in most advanced countries during 2013-2021 has been negative.
- Implication: fiscal fatigue can create a threshold behavior where debt converges to b* if b < b̅ but explodes if a bad shock or temporary fiscal irresponsibility pushes b > b̅, unless exceptional fiscal adjustments or temporarily cheaper financing are secured.

### Rollover risk, self-fulfilling crises, and policy/institutional responses
- Rollover risk: governments may lose market access at acceptable borrowing rates for non-fundamental reasons (“debt market panics”, “pure liquidity crises”, “rollover crises”, “self-fulfilling debt crises”).
- Sachs (1984) canonical rollover crisis: if refinancing needs exceed cash reserves and maximum feasible short-term adjustment, failure to roll over leads to default; coordination failure among creditors can produce multiple equilibria.
- Calvo (1988): government may choose to default when taxation to repay is too distortionary, producing coexistence of a “good” equilibrium (θ=0, default-free) and a “bad” equilibrium (partial repudiation). Which equilibrium prevails depends on investor expectations.
- Lorenzoni and Werning (2019): extension with long-term debt can generate “slow moving” debt crises where marginal borrowing costs rise sharply but defaults materialize slowly because average borrowing costs adjust more slowly.
- Policy and institutional implications drawn from the literature:
  - Lower debt reduces rollover risk; calibrations by Blanchard, Huertas, and Kister (2021) suggest safe levels around 20 percent of GDP or less in the presence of rollover risk.
  - Longer debt maturities reduce the likelihood of self-fulfilling crises (less frequent large rollover needs) and slow the path from sentiment shocks to default, giving time for policy adjustment.
  - Limiting exposure to private, non-resident investors reduces probability of runs.
  - A credible central bank can potentially rule out the “bad” equilibrium by acting as lender of last resort or capping sovereign borrowing costs; asset purchase programs can swap privately-held finite-maturity bonds for central bank reserves and reduce rollover needs.
- Two important conditions for central bank rescues to be effective:
  - The central bank must be credible in controlling inflation and exerting some control over real interest rates; otherwise bailouts risk high inflation or fail to keep borrowing costs below default thresholds.
  - Debt must be issued in domestic currency; many emerging and developing economies borrow in foreign currency, constraining the central bank’s ability to supply foreign liquidity and act as lender of last resort.
- Policy practice: applied debt sustainability analyses by institutions such as the IMF incorporate rollover risk in addition to solvency risk, focusing on flow concepts such as gross financing needs as a share of GDP and quantifying interest rate risk, not only risk related to economic fundamentals. IMF definition (2013): public debt is sustainable when the primary balance needed to at least stabilize debt under both baseline and realistic shock scenarios is economically and politically feasible, consistent with an acceptably low rollover risk and preserving potential growth at a satisfactory level.

*Source: 3. The Bohn revolution and its aftermath (wpiea2022016-print-pdf)*

### 5. Debt sustainability and fiscal risks when r < g

### 5. Debt sustainability and fiscal risks when r < g

### Overview of the debate
- Blanchard (2019) argued that very low real interest rates, and particularly real interest rates below real growth rates, lowered the costs of issuing public debt and that with r < g on average, “higher debt may not imply a higher fiscal cost.”
- Empirical and theoretical pushback makes two key arguments:
  - Higher debt creates fiscal risks even when r < g because higher debt increases rollover risks (more debt to refinance) and does not reduce the probability of rollover shocks conditional on the debt ratio (Mauro and Zhou, 2020; Moreno Badia et al., 2020).
  - Model-based analyses suggest current debt levels can still raise fiscal sustainability concerns even if 푟 < 푔 on a sustained basis (Jiang et al. (2019), Olijslagers et al. (2020)).

### Accounting exercise vs. constraints
- Under constant s_t, r_t, g_t, equation (2) becomes:
  - b_{t+1} = (1+r)/(1+g) b_t − s   (labelled (2'))
- With r < g, (1+r)/(1+g) < 1, implying a stable difference equation and a steady-state:
  - b = (1+g)/(r − g) s
- Implications of this accounting view:
  - If s = 0 and the debt ratio is positive, the debt ratio will fall over time because interest expenditures grow slower than output.
  - A larger primary deficit leads to stabilization at a higher debt ratio; as long as the primary deficit does not keep rising and r remains below g, the debt ratio will stabilize eventually.
- Limitations and constraints not captured by the simple difference equation:
  - The steady-state primary deficit might exceed the size of the economy (−s > 1).
  - The steady-state debt might exceed the value of all available private savings.
  - Such contradictions show the accounting exercise omits economic feedbacks and feasibility constraints (Cochrane (2021a) cited for examples).

### Model-based IGBC and empirical findings
- Alternative view: debt sustainability determined by a model-based Intertemporal Government Budget Constraint (IGBC) where discounting uses stochastic (“risky”) discount rates that can be substantially higher than government borrowing rates.
- Jiang et al. (2019) and Olijslagers et al. (2020):
  - Derive/estimate a model-based IGBC using arbitrage pricing theory or a calibrated general equilibrium model plus a VAR to estimate cyclical properties of primary balances and stochastic discount factors.
  - The procyclical nature of primary balances implies average discount rates that are substantially higher than growth rates, making the environment effectively like r > g for sustainability considerations.
  - Conclusion: running perpetual primary deficits—or even a perpetual primary balance of zero—would violate the model-based IGBC; positive market values of debt require strictly positive future primary balances on average (if primary balances remain procyclical).

### Liquidity services, convenience yield, and limited bubbles
- Papers in incomplete-markets frameworks (Berentsen and Waller (2018), Brunnermeier et al. (2021), Reis (2021)) argue government debt may enjoy additional demand because it provides liquidity and safety, generating a convenience yield or liquidity premium.
- This yields a modified valuation equation:
  - Standard stochastic IGBC (16): value of debt stock = E{PV(future primary surpluses)}
  - Expanded IGBC with service flow (17): value of debt stock = E{PV(future primary surpluses)} + E{PV(future service flow)}
- Consequences:
  - The transversality condition (TVC) can fail under (17), permitting a finite “bubble” in which permanent primary deficits may be consistent with debt sustainability if the service flow component is sufficiently large.
  - Sustainability of a given fiscal policy (e.g., zero primary balances forever as in Blanchard’s thought experiment) depends on the magnitude of the service flow value.
  - Jiang et al. (2019) estimate the service flow component to be worth about 65 percent of GDP on average during their sample period, close to the market value of U.S. debt at the time — making the notion that U.S. debt could be sustainable under permanent zero primary balances not implausible.
  - Even so, maintaining sustainability from recent pre-pandemic fiscal positions would require large fiscal adjustments: pre-pandemic (2019) primary deficit of about 3.5 percent; 2020 and 2021 primary deficits are over 10 percent of GDP.

### Policy channels and implications: central bank credibility as a fiscal asset
- Central bank credibility expands sovereign debt-carrying capacity through multiple channels:
  - Prevents rollover crises, raising the safe level of sovereign debt that can be issued.
  - Countercyclical monetary policy (cutting rates in recessions) allows capital gains on fixed-rate, longer-term bonds in downturns, giving government debt a cyclical insurance property and turning it into a “negative beta” asset (Brunnermeier et al., 2021; Cochrane, 2021a).
    - Credibility helps central banks control real interest rates and cut rates in recessions without de-anchoring inflation expectations.
  - Enhances liquidity services (the second term in (17)) by:
    - Ex ante market development: credible low and stable inflation supports local-currency debt market development.
    - Ex post market-maker role: credible central banks can inject liquidity as “market maker of last resort” to prevent secondary market freezes without de-anchoring inflation expectations.
- Heterogeneity across country groups:
  - Advanced economies have generally accumulated more central bank credibility and deeper local-currency debt markets, enabling higher sustainable debt levels than decades ago.
  - EMDEs tend to have lower central bank credibility, which limits their ability to prevent rollover crises, to lower rates in downturns, to develop deep secondary markets in local currency, and to act as market makers of last resort — tightening fiscal constraints and requiring higher expected future primary surpluses for a given level of investor valuation.
  - Notwithstanding, several EMDE central banks (Guatemala, Indonesia, The Philippines) bought government bonds after the COVID onset without runaway inflation or exchange rate collapse, implying some had enough reputational capital entering 2020.
- Central bank credibility is characterized as reputational capital that can be accumulated by establishing a record of low and stable inflation and relied upon to relax fiscal constraints, but it is finite and depletable; exceeding the envelope defined by equation (17) could destroy credibility’s premises.

*From: wpiea2022016-print-pdf — 5. Debt sustainability and fiscal risks when r < g*

### References

### References

### Optimum debt and public debt theory
- Aiyagari, S. Rao and Ellen R. McGrattan (1998), “The Optimum Quantity of Debt”, Journal of Monetary Economics, 42 (3), pp. 447–469.  
- Barro, Robert J. (1979), “On the Determination of the Public Debt”, Journal of Political Economy, 87 (5), pp. 940–971.  
- Blanchard, Olivier J. (1984), “Current and Anticipated Deficits, Interest Rates and Economic Activity”, European Economic Review, 25 (1), pp. 7–27.  
- Blanchard, Olivier J. (2019), “Public Debt and Low Interest Rates”, American Economic Review, 109 (4), pp. 1197–1229.  
- Blanchard, Olivier J. (forthcoming), Fiscal Policy Under Low Interest Rates, Peterson Institute for International Economics.  
- Blanchard, Olivier J., Gonzalo Huertas and Michael Kister (2021), “Notes on Debt Limits, Uncertainty, and Sudden Stops”, Peterson Institute for International Economics.  
- Woodford, Michael (1990), “Public Debt as Private Liquidity”, American Economic Review, 80 (2), pp. 382–388.  
- Woodford, Michael (1994), “Monetary Policy and Price Level Determinacy in a Cash-in-advance Economy”, Economic Theory, 4, pp. 345–380.  

### Debt sustainability, fiscal solvency, and fiscal policy
- Bohn, Henning (1995), “The Sustainability of Budget Deficits in a Stochastic Economy”, Journal of Money, Credit, and Banking, 27 (1), pp. 257–271.  
- Bohn, Henning (1998), “The Behavior of U.S. Public Debt and Deficits”, Quarterly Journal of Economics, 113 (3), pp. 949–963.  
- Bohn, Henning (2008), “The Sustainability of Fiscal Policy in the United States”, in: Reinhard Neck and Jan-Egbert Sturm (eds.), Sustainability of Public Debt, Cambridge, MA: MIT Press.  
- Ghosh, Atish R., Jun I. Kim, Enrique G. Mendoza, Jonathan D. Ostry and Mahvash S. Qureshi (2013), “Fiscal Fatigue, Fiscal Space and Debt Sustainability in Advanced Economies”, Economic Journal, 123 (566), pp. F4–F30.  
- Lian, Weicheng, Andrea Presbitero, and Ursula Wiriadinata (2020), “Public Debt and r – g at Risk”, IMF Working Paper No. 20/137.  
- Mauro, Paolo and Jing Zhou (2021), “r – g < 0: Can We Sleep More Soundly?”, IMF Economic Review, forthcoming.  
- Olijslagers, Stan, Nander de Vette, and Sweder van Wijnbergen (2020), “Debt Sustainability When r − g < 0: No Free Lunch After All”, CEPR Discussion Paper No. 15478.  
- Wilcox, David W. (1989), “The Sustainability of Government Deficits: Implications of the Present-Value Borrowing Constraint”, Journal of Money, Credit and Banking, 21 (3), pp. 291–306.  

### Monetary policy, liquidity, and the price level
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- Andolfatto, David (2021), “Is It Time for Some Unpleasant Monetary Arithmetic?”, Federal Reserve Bank of St. Louis Review, 103 (30), pp. 315–332.  
- Berentsen, Aleksander and Christopher Waller (2018), “Liquidity Premiums on Government Debt and the Fiscal Theory of the Price Level”, Journal of Economic Dynamics and Control, 89, pp. 173–182.  
- Brunnermeier, Markus K., Sebastian Merkel, and Yuliy Sannikov (2020), “Debt as Safe Asset: Mining the Bubble”, mimeo, Princeton University.  
- Cagan, Phillip D. (1956), “The Monetary Dynamics of Hyperinflation”, in: Milton Friedman (ed.), Studies in the Quantity Theory of Money, Chicago, IL: The University of Chicago Press, pp. 25–117.  
- Cochrane, John H. (2005), “Money as Stock”, Journal of Monetary Economics, 52 (3), pp. 501–528.  
- Cochrane, John H. (2021a), “r < g”, manuscript, Hoover Institution.  
- Cochrane, John H. (2021b), The Fiscal Theory of the Price Level, manuscript, Hoover Institution.  
- Holmstrom, Bengt and Jean Tirole (1998), “Private and Public Supply of Liquidity”, Journal of Political Economy, 106 (1), pp. 1–40.  
- Sargent, Thomas J. and Neil Wallace (1981), “Some Unpleasant Monetarist Arithmetic”, Federal Reserve Bank of Minneapolis Quarterly Review, 5 (3), pp. 1–17.  
- Reis, Ricardo (2021), “The Constraint on Public Debt when r < g but g < m”, mimeo, London School of Economics.  

### Sovereign debt crises, original sin, and international aspects
- Ahmed, Shaghil and John H. Rogers (1995), “Government Budget Deficits and Trade Deficits: Are Present Value Constraints Satisfied in Long-term Data?”, Journal of Monetary Economics, 36 (2), pp. 351–374.  
- Calvo, Guillermo (1988), “Servicing the Public Debt: The Role of Expectations”, American Economic Review, 78 (4), pp. 647–661.  
- Cole, Harold L. and Timothy J. Kehoe (1996), “A Self-Fulfilling Model of Mexico's 1994–1995 Debt Crisis”, Journal of International Economics, 41 (3-4), pp. 309–330.  
- Cole, Harold L. and Timothy J. Kehoe (2000), “Self-Fulfilling Debt Crises”, Review of Economic Studies, 67 (1), pp. 91–116.  
- Corsetti, Giancarlo and Luca Dedola (2016), “The Mystery of the Printing Press: Monetary Policy and Self-Fulfilling Debt Crises”, Journal of the European Economic Association, 14 (6), pp. 1329–1371.  
- Jeanne, Olivier, and Jeromin Zettelmeyer (2002), “‘Original Sin, Balance Sheet Crises, and the Roles of International Lending”, Working Paper 02/234, Washington, D.C.: International Monetary Fund.  
- Jeanne, Olivier (2003), “Why Do Emerging Economies Borrow in Foreign Currency?”, Working Paper 03/177, Washington, D.C: International Monetary Fund.  
- Engel, Charles and JungJae Park (forthcoming), “Debauchery and Original Sin: The Currency Composition of Sovereign Debt”, Journal of the European Economic Association.  
- Farhi, Emmanuel and Matteo Maggiori (2018), “A Model of the International Monetary System”, Quarterly Journal of Economics, 133 (1), pp. 295–355.  

### Debt as liquidity, valuation puzzles, and portfolio risks
- Angeletos, George-Marios, Fabrice Collard, and Harris Dellas (2020), “Public Debt as Private Liquidity: Optimal Policy”, mimeo, MIT.  
- Brunnermeier, Markus K., Sebastian Merkel, and Yuliy Sannikov (2020), “Debt as Safe Asset: Mining the Bubble”, mimeo, Princeton University.  
- Du, Wenxin, Carolin E. Pflueger, and Jesse Schreger (2020), “Sovereign Debt Portfolios, Bond Risks, and the Credibility of Monetary Policy”, Journal of Finance, 75 (6), pp. 3097–3138.  
- Jiang, Zhengyang, Hanno Lustig, Stijn Van Nieuwerburgh, and Mindy Z. Xiaolan (2019), “The U.S. Debt Valuation Puzzle”, NBER Working Paper No. 26583.  
- Lorenzoni, Guido and Ivan Werning (2019), “Slow Moving Debt Crises”, American Economic Review, 109 (9), pp. 3229–3263.  

### Measurement, empirical testing, and methodological contributions
- Buiter, Willem H. (1983), “Measurement of the Public Sector Deficit and Its Implications for Policy Evaluation and Design”, IMF Staff Papers, 30 (2), pp. 306–349.  
- Buiter, Willem H. (1985), “A Guide to Public Sector Debt and Deficits”, Economic Policy, 1 (1), pp. 13–79.  
- Hakkio, Craig S. and Mark Rush (1991), “Is the Budget Deficit ‘Too Large’”, Economic Inquiry, 29 (3), pp. 429–445.  
- Hamilton, James D. and Marjorie A. Flavin (1986), “On the Limitations of Government Borrowing: A Framework for Empirical Testing”, American Economic Review, 76 (4), pp. 808–819.  
- Haug, Alfred A. (1991), “Cointegration and Government Borrowing Constraints: Evidence for the United States”, Journal of Business and Economic Statistics, 9 (1), pp. 97–101.  
- Kremers, Jeroen J.M. (1988), “Long-run Limits on the U.S. Federal Debt”, Economics Letters, 28 (3), pp. 259–262.  
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- Leeper, Eric (1991), “Equilibria under 'Active' and 'Passive' Monetary and Fiscal Policies”, Journal of Monetary Economics, 27 (1), pp. 129–147.  
- Phelps, Edmund S. (1973), “Inflation in the Theory of Public Finance”, Scandinavian Journal of Economics, 75 (1), pp. 67–82.  
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- Rogoff, Kenneth S. (2021), “Fiscal Sustainability in the Aftermath of the Great Pause”, Journal of Policy Modeling, forthcoming.  
- Sachs, Jeffrey (1984), “Theoretical Issues in International Borrowing”, Princeton Studies in International Finance, No 54.  
- Sims, Christopher A. (1999), “Domestic Currency Denominated Government Debt as Equity in the Primary Surplus”, mimeo, Princeton University.  
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- Trehan, Bharat and Carl E. Walsh (1988), “Testing Intertemporal Budget Constraints: Theory and Applications to U.S. Federal Budget and Current Account Deficits”, Journal of Money, Credit and Banking, 23 (2), pp. 206–223.  
- Trehan, Bharat and Carl E. Walsh (1991), “Common Trends, the Government’s Budget Constraint, and Revenue Smoothing”, Journal of Economic Dynamics and Control, 12 (2-3), pp. 425–444.  

### Policy guidance, reports, and books
- IMF (2013), “Staff Guidance Note for Public Debt Sustainability Analysis in Market-Access Countries”, Washington D.C.: International Monetary Fund.  
- IMF (2021), “Review of The Debt Sustainability Framework for Market Access Countries”, Policy Paper No. 2021/003, Washington D.C.: International Monetary Fund.  
- Chamon, Marcos and Jonathan D. Ostry (2021), “A Future with High Public Debt: Low-for-Long Is Not Low Forever”. IMFBlog, Washington D.C.: International Monetary Fund.  
- Kelton, Stephanie (2020), The Deficit Myth: Modern Monetary Theory and How to Build a Better Economy, New York: Public Affairs.  
- EBRD (2010), Transition Report 2010: Recovery and Reform, London U.K.: European Bank of Reconstruction and Development.  

*wpiea2022016-print-pdf - References*

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_Source: https://www.imf.org/-/media/files/publications/wp/2022/english/wpiea2022016-print-pdf.pdf_