## _wp03225

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---

### I. Introduction and objective
- Examines implications of debt composition (nominal/fixed-rate versus indexed—inflation-indexed) for optimal fiscal and monetary policy over the business cycle and in the long run.
- Builds on Bohn (1988) and Chari, Christiano, and Kehoe (1991) to assess value of different debt structures in a stochastic monetary economy with distortionary fiscal and monetary policy.
- Combines general equilibrium framework with Ramsey (1927) public finance approach; model is a cash-in-advance/stochastic growth hybrid.

### Model structure and solution method
- Closed-economy specification: all indexed debt modeled as inflation-indexed.
- Agents and technologies:
  - Representative household: utility from leisure and two consumption goods (cash good C1, credit good C2); cash-in-advance constraint for C1.
  - Representative firm: production Yt = exp(θt) Ht^α Kt^(1−α), 0<α<1.
  - Technology: θt = ρθ θt−1 + εθ,t, 0<ρθ<1; εθ,t ~ Normal(0, σθ,t).
  - Capital stock fixed: Kt = K.
- Government instruments and constraints:
  - Revenue via labor tax τt (levied at rate αYt), money creation, and debt issuance.
  - Two debt instruments: nominal (BN) and indexed (BL).
  - Government budget constraint: Mt/Pt + BNt/Pt RNt−1 + BLt RLt−1 = τt α Yt − Gt + BNt+1/Pt + BLt+1 + Mt+1/Pt.
  - Government consumption: Gt = exp(gt) Gt−1 with gt = ρg gt−1 + εg,t; money evolves Mt+1 = exp(μt+1) Mt.
- Preferences and closed-form private-sector allocations:
  - Utility: Et Σβ^t [ a log C1t + (1−a) log C2t − γ Ht ], with 0<β,a<1 and γ>0.
  - Closed-form consumption solutions (money supply equals money demand):
    - C1t = (Yt − Gt) β (a/(1−a)) exp(−μt+1) / [1 + β (a/(1−a)) exp(−μt+1)]
    - C2t = (Yt − Gt) / [1 + β (a/(1−a)) exp(−μt+1)]
  - Cash-in-advance constraint: Pt C1t ≤ Mt.
  - Implicit labor condition: h(Ht, gt, θt, μt+1, τt) = 0.
  - Euler-derived interest-rate relations:
    - RLt = (1/(β C2t)) [1 / Et(1/C2t+1)]
    - RNt = (1/(β C2t)) [1 / Et( (1/C2t+1) Pt/Pt+1 )]
- Solution method:
  - Ramsey equilibrium solved using projection methods (Chebyshev polynomials, collocation, Gauss-Hermite quadrature) to capture nonlinear labor distortions.

### Ramsey problem, calibration, and numerical procedure
- Ramsey objective: choose a competitive equilibrium that maximizes household utility by choosing ∆_t = (τ_t, μ_{t+1}, B^N_{t+1}, B^L_{t+1}) given private responses and price system.
- State vector: s_t = (B^N_t/P_{t-1}, B^L_t, M^d_t/P_{t-1}, θ_{t-1}, g_{t-1}, τ_{t-1}, R^N_{t-1}, R^L_{t-1}).
- First-order conditions link tax, money growth, and debt Euler conditions; λ^g_t is the government budget multiplier (shadow value).
- Calibration targets:
  - U.S. post-Korean War averages.
  - Debt-to-income baseline (low debt) = 0.349; "high" debt = twice this ratio.
  - U.S. composition: ratio of inflation-indexed debt to total U.S. debt ≈ 5 percent (indexed) and 95 percent nominal (fixed-rate).
- Numerical solution specifications:
  - Chebyshev polynomial order n = 2 → each policy function has n_θ x n_g = 4 coefficients.
  - Collocation: m_θ = 2, m_g = 2 → 4 nodes → 16 projection equations for four policy functions (H_t, μ_{t+1}, τ_t, λ^g_t) → 16 coefficients.
  - Expectation evaluation: Gauss-Hermite Quadrature with r_θ = r_g = 11 → 121 combinations.
  - Convergence criterion: norm of residual vector < 1.0x10^{-9}.
- Steady-state monetary policy: implements the Friedman rule; expected gross nominal interest rate = 1.0; expected real return on nominal debt and money equals inverse of time preference.

### Key quantitative steady-state and volatility statistics (selected reported series)
- Parameter Values (as reported in source):  
  αβaγρ θ σ θ ρ g σ g 0.40.9910.841.8770.950.0070.960.021

- Selected steady-state values (reported across Percent Nominal Debt columns):
  - Output: 1.740 1.740 1.741 1.742 1.744 1.745 1.740 1.742 1.745 1.748 1.750
  - Cash Good: 1.198 1.199 1.200 1.201 1.202 1.203 1.199 1.201 1.202 1.205 1.207
  - Credit Good: 0.228 0.228 0.228 0.229 0.229 0.229 0.228 0.229 0.229 0.230 0.230
  - Labor: 0.310 0.310 0.310 0.311 0.311 0.311 0.310 0.311 0.312 0.312 0.313
  - Multiplier: 0.096 0.098 0.098 0.098 0.098 0.098 0.100 0.100 0.100 0.099 0.099
  - Gov. Spending: 0.313 0.313 0.313 0.313 0.313 0.313 0.313 0.313 0.313 0.313 0.313
  - Inflation: 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991
  - Nominal Interest Rate: -1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
  - Real Interest Rate: -1.009 1.009 1.009 1.009 1.009 1.009 1.009 1.009 1.009 1.009
  - Money Growth Rate: -0.009 -0.009 -0.009 -0.009 -0.009 -0.009 -0.009 -0.009 -0.009 -0.009 -0.009
  - Tax Rate 1/ (In percent of total income): 0.186 0.190 0.189 0.189 0.189 0.189 0.193 0.193 0.192 0.192
  - Tax Rate 2/ (In percent of labor income): 0.311 0.316 0.316 0.315 0.315 0.315 0.321 0.321 0.321 0.320 0.320

- Standard deviations (examples across Percent Nominal Debt for U.S. Debt-to-Income):
  - Output: 0.92 0.90 0.85 0.79 0.75 0.71 0.90 0.79 0.71 0.64 0.61
  - Cash Good: 1.28 1.26 1.19 1.10 1.03 0.98 1.27 1.11 0.98 0.86 0.79
  - Credit Good: 1.28 1.17 1.13 1.07 1.02 0.98 1.11 1.04 0.99 0.92 0.84
  - Labor: 0.00 0.02 0.12 0.24 0.33 0.40 0.05 0.23 0.41 0.55 0.65
  - Multiplier: 4.54 4.30 3.72 3.18 2.72 2.51 4.22 3.15 2.40 2.12 2.25
  - Price Level: 1.28 1.30 1.22 1.11 1.04 0.98 1.48 1.27 1.15 1.04 0.88
  - Inflation: 0.98 0.93 0.88 0.82 0.78 0.74 0.92 0.83 0.75 0.68 0.62
  - Nominal Interest Rate: - 0.13 0.10 0.07 0.04 0.01 0.24 0.19 0.16 0.15 0.09
  - Real Interest Rate: - 0.06 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.04
  - Debt (std): - 0.12 0.34 0.58 0.79 0.94 0.12 0.32 0.52 0.66 0.76
  - Money Growth Rate: 0.00 0.13 0.10 0.07 0.04 0.01 0.25 0.19 0.17 0.16 0.10
  - Tax Rate 1/ (std, percent of total income): 2.83 2.13 1.92 1.78 1.74 1.82 1.62 1.29 1.17 1.35 1.84

### Main quantitative and qualitative findings
- Nonlinear simulation methods yield substantial welfare effects of debt composition that linear-quadratic methods understate.
- Moving from real (indexed) to nominal debt:
  - Welfare gain equivalent to increasing consumption growth by 0.6 percent.
  - Gain nearly as large as welfare gain from the ability to issue debt identified in Barro (1979, 1987).
- Hedging role of nominal debt:
  - Nominal debt functions as state-contingent debt: inflation in adverse states reduces real value of nominal debt and partially finances budget without raising distortionary taxes.
  - Economies with higher ratios of nominal debt are less volatile than those with higher ratios of indexed debt.
  - Higher ratios of nominal debt are associated with higher steady-state output and consumption.
  - Hedging contribution rises with both the level of debt and the percentage of nominal debt, up to diminishing marginal returns.
- Quantitative scenario comparisons (moving from mostly indexed debt to mostly nominal debt):
  - At U.S. debt-to-income ratio:
    - Output: 0.29 percent higher.
    - Consumption of the cash good: 0.33 percent higher.
    - Consumption of the credit good: 0.44 percent higher.
    - Leisure: reduced by 0.14 percent.
    - Monetary translation: increase in output ≈ US$29 billion (contextual comparison in source).
  - At high debt-to-income ratio (twice U.S. ratio):
    - Output: 0.57 percent higher.
    - Consumption of the cash good: 0.67 percent higher.
    - Consumption of the credit good: 0.88 percent higher.
    - Monetary translation: increase in output ≈ US$57 billion (contextual comparison in source).
- Volatility and cyclicality:
  - Economies with debt exhibit lower standard deviations of macro variables than no-debt baseline.
  - Money growth volatility is low; most volatility in distortionary revenue is absorbed by labor taxes.
  - Monetary policy cyclicality differs by debt composition:
    - Countercyclical to technology shocks and procyclical to government consumption in economies with mostly indexed debt.
    - Relation reverses in economies with mostly nominal debt; monetary policy is procyclical for economies with mostly nominal debt.

### Certainty-equivalent welfare measures (selected)
- Lump-sum present value and per-period equivalents measured in cash good units:
  - Moving from no debt to U.S. debt-to-income ratio with mostly indexed debt (5 percent nominal, 95 percent indexed):
    - Lump-sum present value gain = 1.38 units of the cash good in the current period (equal to 115 percent of one-period steady-state consumption).
  - Moving from mostly indexed to mostly nominal debt under U.S. debt-to-income ratio:
    - Gain = 2.11 − 1.38 = 0.73 units of the cash good (61 percent of one-period consumption).
- Table 5: Certainty Equivalence (Cash Good Equivalent / Percent of Steady State)
  - U.S. Debt-to-Income (Cash Good Equivalent): 1.38 1.52 1.70 1.92 2.11
  - U.S. Debt-to-Income (Percent of Steady State): 115 127 142 160 176
  - High Debt-to-Income (Cash Good Equivalent): 1.90 2.30 2.91 3.69 4.46
  - High Debt-to-Income (Percent of Steady State): 159 192 243 308 372
- Per-period certainty-equivalent consumption over no-debt economy:
  - U.S. Debt-to-Income: 0.0130 0.0140 0.0150 0.0170 0.0190
  - High Debt-to-Income: 0.0170 0.0210 0.0260 0.0330 0.040
- Interpretation: certainty-equivalent gains rise with a higher share of nominal debt and are larger in the high debt-to-income economy.

### Dynamic correlations and impulse patterns (selected)
- Example output impulse response shape (MoneyGrowth scenario): 0.272 0.470 0.712 1.000 0.712 0.470 0.272
- Cross-correlation examples (MoneyGrowth):
  - Cash Good: 0.232 0.406 0.617 0.870 0.622 0.413 0.240
  - Labor: -0.230 -0.401 -0.611 -0.862 -0.616 -0.409 -0.238
  - Price Level: -0.159 -0.339 -0.562 -0.822 -0.610 -0.428 -0.276
  - Inflation: -0.195 -0.245 -0.305 -0.372 0.296 0.255 0.214
  - Debt (MoneyGrowth): -0.086 -0.222 -0.388 -0.587 -0.823 -0.589 -0.392
- Cross-correlation patterns differ materially between Majority Nominal Debt (95% nominal) and Majority Indexed Debt (5% nominal); the shadow value of reducing debt (λ^g) is highest under indexed compositions and declines as nominal debt share increases.

### Policy implications and recommendations
- Debt composition matters: nominal debt provides a hedge against government budget shocks and can reduce the need for distortionary tax adjustments.
- Sovereign debt management should:
  - Consider second-order costs (variance of stock adjustments) in addition to first-order financing costs.
  - Aim to create a debt structure that includes sufficient nominal debt to reduce macroeconomic volatility and minimize business-cycle costs.
- Commitment and time consistency:
  - Absence of commitment can lead governments to index debt heavily to avoid inflationary erosion of fixed-rate liabilities.
  - Shifting from an indexed to a nominal debt structure may require commitment mechanisms and reputational strategies.
- Broader implications:
  - Results support a negative relation between economic growth and macroeconomic volatility (Ramey and Ramey 1995).
  - Welfare gains from reduced volatility are large due to model nonlinearities and convexities; the hedging value of nominal debt is roughly equal to the value from the mere ability to issue debt.

*Source: IMF working paper section "3. Certainty Equivalent Gains" and related sections from _wp03225 (content as provided).*

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

### _wp03225 - References...............................................................................................................................35

### Tables
- 1.  Parameter Values Corresponding to U.S. Economy..........................................................28
- 2.  Selected Simulations: Steady-State Values and Standard Deviations...............................28
- 3.  Simulated Economy with U.S. Debt-to-Income Ratio.......................................................30
- 4.  Simulated Economy with High Debt-to-Income Ratio......................................................31
- 5.  Certainty Equivalence........................................................................................................33

### Figures
- 1.  Model Simulations.............................................................................................................29
- 2.  Selected Cross Correlations...............................................................................................32

*Source: _wp03225 - References..............................................................................................................*

### 3.  Certainty Equivalent Gains..........................................................................................

### 3.  Certainty Equivalent Gains

### I. Introduction and objective
- Examines implications of debt composition (nominal/fixed-rate versus indexed—here inflation-indexed) for optimal fiscal and monetary policy over the business cycle and in the long run.
- Builds on Bohn (1988) and Chari, Christiano, and Kehoe (1991) to assess value of different debt structures in a stochastic monetary economy with distortionary fiscal and monetary policy.
- Combines general equilibrium framework with Ramsey (1927) public finance approach; model is a cash-in-advance/stochastic growth hybrid.

### Model structure and assumptions
- Closed-economy specification: all indexed debt modeled as inflation-indexed.
- Agents and technologies:
  - Representative household derives utility from leisure and two consumption goods (cash good C1, credit good C2); cash-in-advance constraint for C1.
  - Representative firm chooses labor; production Yt = exp(θt) Ht^α Kt^(1−α), 0<α<1.
  - Technology follows θt = ρθ θt−1 + εθ,t, 0<ρθ<1; εθ,t ~ Normal(0, σθ,t).
  - Capital stock fixed: Kt = K.
- Government instruments and constraints:
  - Revenue raised via labor tax τt (levied at rate αYt), money creation, and debt issuance.
  - Two debt instruments: nominal (BN) and indexed (BL).
  - Government budget constraint: Mt/Pt + BNt/Pt RNt−1 + BLt RLt−1 = τt α Yt − Gt + BNt+1/Pt + BLt+1 + Mt+1/Pt.
  - Government consumption evolves as Gt = exp(gt) Gt−1 with gt = ρg gt−1 + εg,t; money evolves Mt+1 = exp(μt+1) Mt.
- Preference and functional-form assumptions:
  - Utility: Et Σβ^t [ a log C1t + (1−a) log C2t − γ Ht ], with 0<β,a<1 and γ>0.
  - Log preferences and fixed capital allow closed-form solutions for private-sector allocations given policy.
- Solution method:
  - Ramsey equilibrium (optimal sequences of money growth, taxes, debt and shadow price of debt) solved using projection methods (Judd 1998, 1992) to capture nonlinear labor distortions.

### Private-sector equilibrium (key equations and implications)
- Cash-in-advance constraint: Pt C1t ≤ Mt.
- Household budget (period t): C1t + C2t + Md t+1/Pt + BN t+1/Pt + BL t+1 ≤ (1−α τt) Yt + Mt/Pt + BNt/Pt RN t−1 + BLt RL t−1.
- Closed-form consumption solutions (money supply equals money demand):
  - C1t = (Yt − Gt) β (a/(1−a)) exp(−μt+1) / [1 + β (a/(1−a)) exp(−μt+1)]
  - C2t = (Yt − Gt) / [1 + β (a/(1−a)) exp(−μt+1)]
- Implicit equilibrium labor condition: h(Ht, gt, θt, μt+1, τt) = 0; contemporaneous τt and money growth determine optimal labor supply, embedding a loss function over distortionary taxes and inflation in the nonlinear labor supply equation.
- Euler-derived interest-rate relations:
  - RLt = (1/(β C2t)) [1 / Et(1/C2t+1)]
  - RNt = (1/(β C2t)) [1 / Et( (1/C2t+1) Pt/Pt+1 )]

### Ramsey equilibrium and policy interactions
- Government maximizes household welfare subject to the budget constraint and private-sector equilibrium responses.
- Key Ramsey policy features documented in simulations:
  - Tax rates on labor are relatively constant over the business cycle, more so in economies with debt.
  - Government policy attempts to follow the Friedman rule, yielding an expected gross nominal interest rate equal to 1.0.
  - In expectation the government equates the real gross rate of return across money, nominal debt, and indexed debt, satisfying Euler conditions.
  - Monetary policy cyclicality:
    - Countercyclical to technology shocks and procyclical to government consumption in economies with mostly indexed debt.
    - The relation reverses for economies with mostly nominal debt; monetary policy is procyclical for economies with mostly nominal debt.
- Role of debt:
  - Existence of debt allows smoothing of distortionary taxes and money growth over time (tax smoothing), consistent with Barro (1979, 1987).
  - Debt provides households with smoother consumption/leisure by enabling response to transitory income changes.

### Main quantitative and qualitative findings
- Nonlinear simulation methods reveal substantial welfare effects of debt composition that linear-quadratic methods understate.
- Welfare comparison:
  - Gain in welfare from moving from real (indexed) to nominal debt is equivalent to increasing consumption growth by 0.6%.
  - This gain is nearly as large as the welfare gain from the ability to issue debt identified by Barro (1979, 1987).
- Hedging role of nominal debt:
  - Nominal debt functions as state-contingent debt: inflation in adverse states reduces real value of nominal debt and partially finances the budget without increasing distortionary taxes.
  - Economies with higher ratios of nominal debt are less volatile than those with higher ratios of indexed debt.
  - Higher ratios of nominal debt are associated with higher steady-state output and consumption.
  - The greater the level of debt and the larger the percentage of nominal debt, the larger the hedging contribution.
- Implications for sovereign debt management:
  - Debt management should consider second-order effects (variance of stock adjustments), not only first-order financing costs.
  - Depending on shock types and magnitudes, optimal debt structure should include nominal debt in sufficient quantities to reduce macroeconomic volatility and minimize business-cycle costs.
- Broader welfare and growth implications:
  - Results support a negative relation between economic growth and macroeconomic volatility (Ramey and Ramey 1995).
  - Reductions in macroeconomic volatility can produce significant welfare increases; estimated gains here are larger than those in Lucas (1987) due to model nonlinearities and convexities.
- Commitment and policy consistency:
  - Time consistency matters: absence of commitment can lead governments to index debt heavily to avoid inflationary erosion of fixed-rate liabilities.
  - Moving from an indexed debt structure to a nominal debt structure may require commitment mechanisms and reputational strategies.

### Calibration, scenarios, and simulation setup (overview)
- Baseline: economy without debt.
- Low-debt economy: calibrated to prevailing U.S. debt-to-income ratio.
- High-debt economy: calibrated to twice the prevailing U.S. debt-to-income ratio.
- Debt composition scenarios: from 95 percent nominal / 5 percent indexed (matching current U.S. composition) to 5 percent nominal / 95 percent indexed.
- For each scenario, Ramsey plan solved and economy simulated under technology and government spending shocks to examine consequences for optimal policy and macroeconomic activity.

*Source: IMF working paper section "3.  Certainty Equivalent Gains" (content as provided).*

### 1. A feasible allocation is a sequence of{C

### 1. A feasible allocation is a sequence of{C
1t
}
∞
t=1
,{C
2t
}
∞
t=1
,{H
t
}
∞
t=1
,{G
t
}
∞
t=1
that satisfy
theresourceconstraintin13.

### Definitions and primitives
- A feasible allocation is a sequence of{C
1t
}
∞
t=1
,{C
2t
}
∞
t=1
,{H
t
}
∞
t=1
,{G
t
}
∞
t=1 that satisfy the resource constraint in 13.

- A price system is a set of nonnegative bounded sequences{P
t
}
∞
t=1
,{w
t
}
∞
t=1
,
©
R
N
t
ª
∞
t=1
,
©
R
L
t
ª
∞
t=1
.

- A government policy is a set of sequences{τ
t
}
∞
t=1
,{M
t+1
}
∞
t=1
,
©
B
N
t+1
ª
∞
t=1
,
©
B
L
t+1
ª
∞
t=1
.

### Competitive equilibrium
- Given the exogenous sequences{g
t
}
∞
t=1
and{θ
t
}
∞
t=1
; initial stocks of money, nominal
bonds, and indexed bonds; andM
0
=M
d
0
; a competitive equilibrium is a feasible
allocation, a price system, and a government policy such that:
  - (a) given the price system and government policy, the allocation solves both the firm’s problem and the household’s problem; and
  - (b) given the allocation and price system, the government policy satisfies the sequence of government budget constraints.

*Source: _wp03225 - 1. A feasible allocation is a sequence of{C (PDF chapter/section)*

### 5. The Ramsey Problem is to choose a competitive equilibrium that maximizes

### 5. The Ramsey Problem is to choose a competitive equilibrium that maximizes household utility in 4. The competitive allocation that solves the Ramsey Problem is called the Ramsey plan or Ramsey equilibrium.

### Ramsey problem setup and optimality conditions
- Government maximizes V(s_t) (equation 22) by choosing ∆_t = (τ_t, μ_{t+1}, B^N_{t+1}, B^L_{t+1}) taking as given household and firm optimal responses (equations 1, 14, 15, 17) and price system (equations 11, 12, 16).
- State vector s_t = (B^N_t/P_{t-1}, B^L_t, M^d_t/P_{t-1}, θ_{t-1}, g_{t-1}, τ_{t-1}, R^N_{t-1}, R^L_{t-1}).
- Government budget multiplier λ^g_t is the shadow value of relaxing the government budget constraint (value to households of a lump-sum revenue source).
- First-order conditions (tax, money growth, and debt Euler conditions) are given in equations (23)–(26), linking:
  - The distortionary effects of labor income taxation (∂H_t/∂τ_t, ∂C_{1t}/∂τ_t, ∂C_{2t}/∂τ_t all negative).
  - The effects of money growth on labor, consumption, prices, and seigniorage (∂H_t/∂μ_{t+1} negative; ∂C_{1t}/∂μ_{t+1}, ∂C_{2t}/∂μ_{t+1} negative).
  - The trade-offs between taxation, seigniorage, issuing nominal debt, and issuing indexed debt, scaled by λ^g_t.
- Interpretation highlights:
  - Tax Euler (23): trade-off between current distortionary taxation and issuing indexed debt; tax increases affect current price level and the real value of prior nominal assets.
  - Money Euler (24): trade-off between money growth and issuing debt that matures next period; money growth directly enters via exp(μ_{t+1}) and indirectly via labor and consumption responses.
  - Debt pricing conditions (25)–(26): standard intertemporal pricing with λ^g_t scaling.

### Calibration and numerical solution procedure (key specifications and numerical tolerances)
- Calibration targets U.S. post-Korean War averages; debt-to-income baseline (low debt) = 0.349. This ratio is doubled for the "high" debt economy.
- U.S. composition: ratio of inflation-indexed debt to total U.S. debt ≈ 5 percent (indexed) and 95 percent nominal (fixed-rate) debt.
- Numerical solution uses projection approach with Chebyshev polynomials:
  - Polynomial order n = 2 (second-order polynomials), so each policy function has n_θ x n_g = 4 coefficients (two random variables: θ and g).
  - Collocation: m_θ = 2 technology levels, m_g = 2 spending levels → m_θ x m_g = 4 nodes → 16 projection equations for four policy functions (H_t, μ_{t+1}, τ_t, λ^g_t) totaling 16 coefficients.
  - Expectation evaluation via Gauss-Hermite Quadrature with r_θ = r_g = 11 → 121 possible future combinations for expectations.
  - Convergence criterion: norm of residual vector < 1.0x10^{-9}. Under this stopping criterion, a mistake in the optimal policy function would cost the government less than $1 per billion in nominal GDP.
  - Higher polynomial orders (e.g., n = 3) would increase coefficients to n_θ x n_g = 9 per policy function (36 coefficients total) and raise computational difficulty.

### Steady-state characterization and optimal monetary policy
- Optimal monetary policy implements the Friedman rule: money growth set equal to the rate of time preference; expected gross nominal interest rate equals 1.0; expected real return on nominal debt and money balances equals the inverse of time preference in steady state.
- Under the Friedman rule, the government runs a gross-of-interest surplus via labor income taxes sufficient to cover government spending, interest on debt, and withdrawal of money balances.
- As debt-to-income ratio rises, equilibrium labor tax rate increases to produce the gross-of-interest surplus, increasing distortionary welfare costs and raising the shadow value λ^g_t.

### Role of debt and debt composition — quantitative findings
- Debt permits smoothing of distortionary taxes and money growth, reducing macroeconomic volatility relative to the no-debt baseline.
- Simulated economies: baseline (no debt), low-debt (U.S. debt-to-income = 0.349), high-debt (twice U.S. ratio); debt composition varied from 95 percent nominal / 5 percent indexed to 5 percent nominal / 95 percent indexed.
- Key quantitative comparisons (moving from mostly indexed debt to mostly nominal debt at U.S. debt-to-income ratio):
  - Output: 0.29 percent higher.
  - Consumption of the cash good: 0.33 percent higher.
  - Consumption of the credit good: 0.44 percent higher.
  - Leisure: reduced by 0.14 percent (loss of leisure utility more than offset by higher consumption).
  - Monetary translation: increase in output ≈ US$29 billion based on current U.S. economic output (contextual comparison provided in text).
- Under the high debt-to-income ratio (twice U.S. ratio), moving from mostly indexed to mostly nominal debt:
  - Output: 0.57 percent higher.
  - Consumption of the cash good: 0.67 percent higher.
  - Consumption of the credit good: 0.88 percent higher.
  - Monetary translation: increase in output ≈ US$57 billion based on current U.S. economy (contextual comparison provided in text).
- Volatility results:
  - Economies with debt exhibit lower standard deviations (percent) of macro variables than the no-debt baseline.
  - Money growth volatility is low across models; most volatility in distortionary revenue is absorbed by labor taxes.
  - Economies with mostly indexed debt generate negative correlations between money growth and output; those with more nominal debt show positive correlation.
  - Price and inflation volatility more closely match U.S. data due to the cash-in-advance constraint transmitting cash-good volatility into prices.

### Hedging role of nominal debt
- Nominal debt acts as a hedge against government budget shocks because unexpected inflation reduces the real value of existing nominal debt, offsetting the need for distortionary revenue increases.
- Empirical mechanism (simulated):
  - Positive government spending shocks → higher taxes and money growth; government spending positively correlated with prices → higher-than-expected inflation reduces real value of prior nominal debt, partially offsetting revenue needs.
  - Negative technology shocks in economies with mostly indexed debt → output falls, household increases labor → need for higher taxes and money growth increases real/nominal interest costs on indexed and nominal debt; unexpected inflation still provides hedging by reducing real value of nominal debt.
- Hedging effectiveness:
  - The hedging value of nominal debt rises with the level of nominal debt up to a point; marginal hedging returns fall as absolute volatility declines and correlations change.
  - Correlation between the multiplier λ^g and inflation is highest under mostly indexed debt, declining as the percentage of nominal debt increases, reaching zero under the high debt-to-income ratio with mostly nominal debt; beyond this, debt stock value adjustments outweigh direct shock effects.
- Conclusion on hedging: nominal debt is particularly valuable in states where shocks would otherwise necessitate larger distortionary adjustments.

### Welfare gains and certainty-equivalent measures
- Certainty-equivalence measures computed in terms of the cash good: lump-sum present discounted value and constant per-period equivalent.
- Lump-sum present value example:
  - Moving from no debt to U.S. debt-to-income ratio with mostly indexed debt (5 percent nominal, 95 percent indexed) → lump-sum present value gain = 1.38 units of the cash good in the current period (equal to 115 percent of one-period steady-state consumption).
  - Moving from mostly indexed to mostly nominal debt under U.S. debt-to-income ratio: gain = 2.11 − 1.38 = 0.73 units of the cash good (61 percent of one-period consumption).
- Per-period certainty-equivalent gains:
  - Moving from no debt to U.S. debt-to-income ratio with primarily indexed debt → gain = 1 percent of the cash good per period.
  - Changing composition from indexed to nominal under U.S. ratio → additional gain = 0.6 percent of the cash good per period.
  - Under the high debt-to-income ratio, changing debt structure yields an additional gain = 2.0 percent of the cash good per period.
- Overall implications:
  - The value of nominal debt as a hedge is roughly equal to the value from the mere ability to issue debt.
  - Policy implications: optimal debt policy should consider debt composition and second-order volatility costs in addition to debt levels and first-order financing costs; a predominantly indexed debt structure is a second-best solution — including sufficient nominal debt improves welfare and reduces volatility.

### Broader implications and links to literature
- Results are consistent with tax-smoothing literature (Barro 1979, 1987) and the Friedman rule optimality (Friedman 1969) in the presence of distortionary taxation.
- Welfare gains from reduced volatility are large relative to some prior estimates (e.g., Lucas 1987), reflecting model nonlinearities and convexities in tax-money-growth–labor supply interaction.
- Findings align with literature linking volatility reductions to higher growth (Ramey and Ramey 1995) and the importance of convexities (Bernanke 1983; Ramey and Ramey 1991; Galí, Gertler, and López-Salido 2002; Woodford 2001).

*Source: _wp03225 - 5. The Ramsey Problem is to choose a competitive equilibrium that maximizes (IMF working paper content provided).*

### conclusion is that the role of nominal debt as state-contingent debt can be a significant

### _wp03225 - conclusion is that the role of nominal debt as state-contingent debt can be a significant

### Main conclusion
- Nominal debt functions as state-contingent debt and can be a significant policy tool for reducing volatility of distortionary government policy.
- Reductions in the volatility of fiscal and monetary policy lead to a reduction in macroeconomic volatility and increases in equilibrium output and consumption.
- The gain in welfare from using nominal debt to hedge against shocks to the government budget is as large as the gain in welfare from the ability to issue debt.
- The ability to issue debt is worthwhile even if additional covenants (i.e. indexation) are needed to access capital markets in sufficient quantities to smooth distortionary government policy.
- Sovereign debt management should treat the true cost of debt as more than first-order financing costs and include a second-order concept of variance of stock adjustments.
- Depending on prevailing shocks, optimal debt management should create a debt structure that includes nominal debt in sufficient quantities to reduce macroeconomic volatility and minimize business cycle costs.

### Model, approach, and mechanisms
- Framework: Combines a traditional general equilibrium framework with the Ramsey approach in public finance to calibrate and simulate a stochastic monetary model under various debt-to-income ratios and differing compositions of nominal and indexed debt.
- Key modelling elements:
  - Incorporates a loss function within the nonlinearity of the labor supply equation because contemporaneous tax on labor income and money growth determine optimal household labor supply.
  - Shocks that cause variations in government policy transmit to labor supply, output, household allocations, and the equilibrium price system, feeding back into the government budget constraint through tax revenue.
  - Equilibrium decisions by households, firms, and the government are transferred across time through the price level and interest rates.
- Hedging mechanism: Unexpected shocks that call for higher distortionary government revenue correspond to states with higher inflation; higher-than-expected inflation reduces the real value of existing nominal debt, thereby offsetting the need to increase distortionary revenue.

### Parameterization
- Parameter Values (as reported in source):  
  αβaγρ θ σ θ ρ g σ g 0.40.9910.841.8770.950.0070.960.021

### Selected steady-state values (reported series across Percent Nominal Debt columns)
- Output: 1.740 1.740 1.741 1.742 1.744 1.745 1.740 1.742 1.745 1.748 1.750
- Cash Good: 1.198 1.199 1.200 1.201 1.202 1.203 1.199 1.201 1.202 1.205 1.207
- Credit Good: 0.228 0.228 0.228 0.229 0.229 0.229 0.228 0.229 0.229 0.230 0.230
- Labor: 0.310 0.310 0.310 0.311 0.311 0.311 0.310 0.311 0.312 0.312 0.313
- Multiplier: 0.096 0.098 0.098 0.098 0.098 0.098 0.100 0.100 0.100 0.099 0.099
- Gov. Spending: 0.313 0.313 0.313 0.313 0.313 0.313 0.313 0.313 0.313 0.313 0.313
- Inflation: 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991 0.991
- Nominal Interest Rate: -1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000 1.000
- Real Interest Rate: -1.009 1.009 1.009 1.009 1.009 1.009 1.009 1.009 1.009 1.009
- Money Growth Rate: -0.009 -0.009 -0.009 -0.009 -0.009 -0.009 -0.009 -0.009 -0.009 -0.009 -0.009
- Tax Rate 1/ (In percent of total income): 0.186 0.190 0.189 0.189 0.189 0.189 0.193 0.193 0.192 0.192
- Tax Rate 2/ (In percent of labor income): 0.311 0.316 0.316 0.315 0.315 0.315 0.321 0.321 0.321 0.320 0.320

### Standard deviations and volatility patterns (selected)
- For U.S. Debt-to-Income standard deviations (examples across Percent Nominal Debt):  
  - Output: 0.92 0.90 0.85 0.79 0.75 0.71 0.90 0.79 0.71 0.64 0.61  
  - Cash Good: 1.28 1.26 1.19 1.10 1.03 0.98 1.27 1.11 0.98 0.86 0.79  
  - Credit Good: 1.28 1.17 1.13 1.07 1.02 0.98 1.11 1.04 0.99 0.92 0.84  
  - Labor: 0.00 0.02 0.12 0.24 0.33 0.40 0.05 0.23 0.41 0.55 0.65  
  - Multiplier: 4.54 4.30 3.72 3.18 2.72 2.51 4.22 3.15 2.40 2.12 2.25  
  - Price Level: 1.28 1.30 1.22 1.11 1.04 0.98 1.48 1.27 1.15 1.04 0.88  
  - Inflation: 0.98 0.93 0.88 0.82 0.78 0.74 0.92 0.83 0.75 0.68 0.62  
  - Nominal Interest Rate: - 0.13 0.10 0.07 0.04 0.01 0.24 0.19 0.16 0.15 0.09  
  - Real Interest Rate: - 0.06 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.05 0.04  
  - Debt (std): - 0.12 0.34 0.58 0.79 0.94 0.12 0.32 0.52 0.66 0.76  
  - Money Growth Rate: 0.00 0.13 0.10 0.07 0.04 0.01 0.25 0.19 0.17 0.16 0.10  
  - Tax Rate 1/ (std, percent of total income): 2.83 2.13 1.92 1.78 1.74 1.82 1.62 1.29 1.17 1.35 1.84

- As the percentage of nominal debt increases, the overall volatility of each economy declines, increasing household welfare by reducing volatility of consumption (Figure note).

### Dynamic cross-correlations and impulse structure (selected)
- Cross-correlations reported across lags x(-3) to x(+3) for shocks (examples from U.S. Debt-to-Income simulations):
  - Output impulse response shape (MoneyGrowth scenario): 0.272 0.470 0.712 1.000 0.712 0.470 0.272
  - Cash Good (MoneyGrowth): 0.232 0.406 0.617 0.870 0.622 0.413 0.240
  - Labor (MoneyGrowth): -0.230 -0.401 -0.611 -0.862 -0.616 -0.409 -0.238
  - Multiplier (MoneyGrowth): -0.120 -0.216 -0.331 -0.472 -0.342 -0.232 -0.138
  - Price Level (MoneyGrowth): -0.159 -0.339 -0.562 -0.822 -0.610 -0.428 -0.276
  - Inflation (MoneyGrowth): -0.195 -0.245 -0.305 -0.372 0.296 0.255 0.214
  - Nom. Int. Rate (MoneyGrowth): -0.066 -0.123 -0.191 -0.276 -0.202 -0.137 -0.083
  - Debt (MoneyGrowth): -0.086 -0.222 -0.388 -0.587 -0.823 -0.589 -0.392
- Cross-correlation patterns differ when debt composition is Majority Nominal Debt (95% nominal) versus Majority Indexed Debt (5% nominal), with the shadow value of reducing debt highest under indexed debt compositions and declining as more nominal debt is utilized.

### Certainty-equivalence and welfare gains
- Table 5: Certainty Equivalence — Increase in consumption of the cash good necessary to make the household indifferent between no debt and the selected debt composition (Discounted Present Value / Per Period Value):
  - U.S. Debt-to-Income (Cash Good Equivalent): 1.38 1.52 1.70 1.92 2.11  
  - U.S. Debt-to-Income (Percent of Steady State): 115 127 142 160 176
  - High Debt-to-Income (Cash Good Equivalent): 1.90 2.30 2.91 3.69 4.46  
  - High Debt-to-Income (Percent of Steady State): 159 192 243 308 372
- Per-period certainty-equivalent consumption over no-debt economy (U.S. Debt-to-Income): 0.0130 0.0140 0.0150 0.0170 0.0190
- Per-period certainty-equivalent consumption over no-debt economy (High Debt-to-Income): 0.0170 0.0210 0.0260 0.0330 0.040
- Interpretation: Certainty-equivalent gains rise with a higher share of nominal debt and are larger in the high debt-to-income economy.

### Policy implications and recommendations
- Debt composition matters: Including nominal debt in the sovereign debt structure provides a hedge against shocks to the government budget and can reduce the need for distortionary tax adjustments.
- Debt management strategy should:
  - Consider second-order costs (variance of stock adjustments) in addition to first-order financing costs.
  - Strive over time to create a debt structure that includes nominal debt in sufficient quantities to reduce macroeconomic volatility.
- Even when indexation or additional covenants are needed to access capital markets, the welfare benefits of issuing debt and using nominal debt as a hedge can justify such arrangements.

*Source: _wp03225 - conclusion is that the role of nominal debt as state-contingent debt can be a significant (IMF working paper PDF content).*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2003/_wp03225.pdf_