## wp1778

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### 2.1 Households — preferences, production and optimisation
- Preferences and consumption aggregator
  - Households j ∈ [0,1] derive utility from (X_t)_j = X((X^c_t)_j, G_t), where (X^c_t)_j = C^j_t − θ C^j_{t−1}, and θ ∈ (0,1) is the degree of internal habit formation.
  - Composition allows substitutability or complementarity between private and public goods.

- Labour supply, wages and wage adjustment costs
  - Labour supply: H^j_t = (w^j_t / w_t)^{−e} W_t η H_t, with η the intratemporal elasticity of substitution, e^W_t a wage mark-up shock, and H_t average labour demand.
  - Rotemberg quadratic wage adjustment cost (real value of labour services): ξ_W / 2 (W^j_t / W^j_{t−1} − Π)^2 w_t H_t = ξ_W / 2 (w^j_t / w^j_{t−1} Π_t − Π)^2 w_t H_t, where ξ_W is the wage adjustment cost parameter and Π is steady-state inflation.

- Capital accumulation, investment and utilization
  - Capital law of motion: K^j_{t+1} = (1−δ) K^j_t + e^I_t I^j_t [1 − S(I^j_t / I^j_{t−1})], with depreciation rate δ, investment-specific shock e^I_t, and adjustment cost S(·) satisfying S(1) = S′(1) = 0 and S′′(1) > 0.
  - Capital utilization choice u^j_t with utilization cost a(u^j_t) K^j_t and a(u) = 0 at steady state utilization.

- Assets, taxes, transfers and budget constraint
  - Households purchase C^j_t, I^j_t, hold private bonds B^j_t (zero net supply), and nominal short-term government bonds (B^S_t)^j.
  - Taxes: lump-sum τ^L_t and distortionary τ^C_t, τ^W_t, τ^K_t; receive wages, rental income on utilized capital, returns on bonds (government bonds discounted by ex-ante expected haircut rate ∆^g_t), firms’ profits, transfers ˜Ξ_t, and tax depreciation allowance δQ_t τ^K_t K^j_t.
  - Budget constraint (exact expression preserved):
    - (1 + τ^C_t) C^j_t + I^j_t + τ^L_t + ξ_W / 2 (w^j_t / w^j_{t−1} Π_t − Π)^2 w_t H_t + a(u^j_t) K^j_t + B^j_t / P_t + (B^S_t)^j / P_t
      ≤ (1 − τ^W_t) W^j_t / P_t H^j_t + (1 − τ^K_t) R^K_t u^j_t K^j_t + R_{t−1} B^j_{t−1} / P_t + (1 − ∆^g_t) R^S_{t−1} (B^S_{t−1})^j / P_t + ∫_0^1 J_{it} di + ˜Ξ_t + δQ_t τ^K_t u_t K^j_t.

- Household optimisation problem
  - Choice set: {C^j_t, B^j_t, (B^S_t)^j, K^j_{t+1}, u_t, I^j_t, w^j_t}.
  - Objective: max E_t Σ_{s=0}^∞ { e^B_{t+s} β_{t+s} U(X^j_{t+s}, H^j_{t+s}) }, subject to habits, labour supply, capital accumulation and the budget constraint, with β ∈ (0,1) and e^B_t a preference shock.

- Related firm-side components (introduced in following subsection)
  - Firms rent capital services ˜K_it ≡ u_t K_it, hire labour H_it, and produce Y_it = F(e^A_t H_it, ˜K_it).
  - Consumption aggregator (Dixit-Stiglitz): Y_t = [∫_0^1 (Y_it)^{1−1/e^P_t ζ} di]^{1/(1−1/e^P_t ζ)} and Y_it = (P_it / P_t)^{−e^P_t ζ} Y_t.
  - Firms face Rotemberg price adjustment cost ξ_P / 2 (P_it / P_{it−1} − Π)^2 Y_t and maximise discounted profits J_it = E_t Σ_{s=0}^∞ D_{t,t+s} [ P_{it+s} / P_{t+s} Y_{it+s} − w_{it} H_{it+s} − R^K_{t+s} ˜K_{it+s} − ξ_P / 2 (P_{it+s} / P_{it+s−1} − Π)^2 Y_{t+s} ], subject to Y_it = F(e^A_t H_it, ˜K_it) − FC.

### 2.3 Government — financing, default, and policy rules
- Sovereign default setup and probability
  - Government finances G_t via taxes T_t and issuing one-period bonds B^S_t repaid at gross nominal rate R^S_t.
  - Debt-to-GDP Γ_t ≡ B^S_t / (P_t Y_t).
  - Fiscal limit Γ^*_t drawn each period; default occurs if Γ_{t−1} ≥ Γ^*_t.
  - Default probability p^*_t (logistic CDF): p^*_t = P(Γ^*_t ≤ Γ_{t−1}) = exp(η1 + η2 Γ_{t−1}) / (1 + exp(η1 + η2 Γ_{t−1})). (Equation (11))
  - Default form: proportional haircut ̄Δ ∈ [0,1].

- Expected haircut and tractability
  - Ex-ante expected haircut Δ^g_t specified:
    - Δ^g_t = 0 with probability 1 − p^*_t;
    - Δ^g_t = ̄Δ^g with probability p^*_t (Equation (12)),
    - implying Δ^g_t = p^*_t ̄Δ^g. (Equation (13))
  - ̄Δ^g ∈ (0,1] is the haircut applied in default.

- Real government budget constraint and revenues
  - Define b^S_t ≡ B^S_t / P_t and Π_t ≡ P_t / P_{t−1}.
  - Real budget constraint: b^S_t = (1 − Δ^g_t) R^S_{t−1} / Π_t b^S_{t−1} + G_t − T_t + ̃Ξ_t. (Equation (14))
  - Total government revenue: T_t = τ^C_t C_t + τ^W_t w_t h_t + τ^K_t [ (R^K_t − δ Q_t) u_t K_t ] + τ^L_t. (Equation (15))

- Transfers at default
  - Transfers set to ̃Ξ_t ≡ Δ^g_t R^S_{t−1} / Π_t b^S_{t−1} so sovereign default does not alter the actual debt level; rationale given to avoid lower ex-ante premia from implicit post-default lower debt.

- Fiscal instruments and reduction to one tax instrument
  - Uniform proportional tax instrument τ_t with τ^C_t = τ_t τ^C, τ^W_t = τ_t τ^W, τ^K_t = τ_t τ^K, τ^L_t = τ_t τ^L.
  - Two fiscal instruments: τ_t and G_t.
  - Adjustment/Taylor-type rules:
    - log(τ_t / τ) = ρ_τ log(τ_{t−1} / τ) + ρ_{τb} log(b^S_{t−1} / b^S) + ρ_{τy} log(Y_t / Y). (Equation (16))
    - log(G_t / G) = ρ_g log(G_{t−1} / G) − ρ_{gb} log(b^S_{t−1} / b^S) − ρ_{gy} log(Y_t / Y). (Equation (17))

- Calibration (selected parameter values)
  - Haircut rate ̄Δ = 0.0246.
  - Default probability scaling η1 = −9.7480; slope η2 = 1.9921.
  - Steady-state government spending share G/Y = 0.19.
  - Steady-state government debt-to-GDP Γ = 4 × 0.70.
  - Steady-state tax rates: τ^C = 0.05; τ^W = 0.24; τ^K = 0.32.
  - Fiscal-rule parameters: ρ_τ = 0.6252; ρ_{τb} = 0.0777; ρ_{τy} = 0.1478; ρ_g = 0.7662; ρ_{gb} = 0.1197; ρ_{gy} = 0.1624.

- Calibration of fiscal limit CDF and implied premia
  - Anchor points: set p^*_2 = 0.99 at Γ_2 = 4 × 1.8; set Γ_1 = 4 × 0.7; sovereign premia assumptions ABP1 = 15 annual basis points at Γ_1 and ABP2 = 1,000 annual basis points at Γ_2.
  - From equation (13), p^*_1 = 1 − 1 / (ABP1/40000 + 1) ̄Δ^g yields p^*_1 = 0.0152.
  - Parameters η1 = −9.7480 and η2 = 1.9921 recover the chosen (Γ_1, p^*_1) and (Γ_2, p^*_2).
  - Implied sovereign risk premium rises from 15 annual basis points at government debt 70% of annual GDP to:
    - 143 annual basis points at government debt ratio 100%.
    - 452 annual basis points at government debt ratio 120%.

- Role in optimal policy analysis
  - Fiscal rules τ_t and G_t are key instruments in the LQ approximation for normal-times stabilisation and crisis management.
  - Transfers ̃Ξ_t and expected haircut Δ^g_t shape sovereign risk premia dynamics and feed into optimal monetary and fiscal stabilisation policy.

### 4.2 Optimal monetary-fiscal rules for normal times — methodology and quantitative results
- Framework and interpretation
  - Instruments considered jointly: taxation τ_t (equation (16)), government spending G_t (equation (17)), and monetary policy rule (equation (18)).
  - “Optimality” may refer to Ramsey (welfare-optimal), time-consistent, or optimised Taylor-type rules; assignment considered where monetary authority stabilises inflation and fiscal authority stabilises debt.
  - Active vs. passive following Leeper (1991): active monetary rule satisfies Taylor principle; passive fiscal rule stabilises government debt.

- Feasibility constraints and ZLB treatment
  - Feasibility bounds: ρ_{rπ} ≤ 5; ρ_{τb}, ρ_{gb} ≤ 1.
  - Baseline: constrain monetary inertia ρ_r ≤ 1; superinertial (ρ_r > 1) explored separately.
  - ZLB handling: add penalty −w_r (R_t − R)^2 to welfare objective to reduce interest-rate variance and set w_r to make ZLB hit probability 0.0025 (once every 400 quarters / 100 years).
  - After optimisation, utility corrected by adding back w_r var(R_t); adjusted welfare loss reported as original welfare less w_r var(R_t).

- Main welfare results (consumption-equivalent ce)
  - Metric: consumption-equivalent permanent percentage fall in private consumption below Ramsey-optimal policy, denoted c^e.
  - Selected verbatim table entries (for alternative B/4Y values):
    - B/4Y=0.7
      - Optimal n/an/an/a371-0.0038
      - TCT n/an/an/a1350.00830.45
      - Simple (PF) 1, 0.232, 00.051, 0.25, 00.789, 0.018, 03850.00000.14
      - Simple (AF) 0,0,00,0,00,0,01350.00690.40
    - B/4Y=0.9
      - Optimal n/an/an/a296-0.0046
      - TCT n/an/an/a1020.00880.49
      - Simple (PF) 1, 0.236, 00.504, 0.25, 00.733, 0.046, 03810.00000.17
      - Simple (AF) 0,0,00,0,00,0,0790.00660.41
    - B/4Y=1.2
      - Optimal n/an/an/a221-0.0021
      - TCT n/an/an/a630.00320.20
      - Simple (PF) 1,0.261,00.579,1,00.161, 0.749,02910.00150.13
      - Simple (AF) 0,0,00,0,00,0,0600.00540.28
    - B/4Y=1.2 Bailout
      - Optimal n/an/an/a310-0.0047
      - TCT n/an/an/a2440.0110.58
      - Simple (PF) 1,0.232,00,0.25,00,0,03820.00020.18
      - Simple (AF) 0,0,00,0,00,0,0600.00540.38

  - Key textual interpretations:
    - The optimal simple passive fiscal (PF), active monetary rules generate an average welfare loss of 0.14% in consumption-equivalent terms.
    - Time-consistent or optimal simple active-fiscal, passive-monetary rules produce considerably larger consumption-equivalent welfare losses.
    - In passive fiscal cases, the optimised simple monetary policy rule is a price-level rule; monetary and fiscal instruments do not react to output fluctuations.
    - Low steady-state debt/GDP (70%) implies taxes are more responsive relative to government spending; high steady-state debt (120%) makes government spending more reactive to debt unless sovereign risk premia are suppressed.

- Policy and economic dynamics findings
  - Price-level (inertial) monetary rules:
    - Price-level stabilization reduces inflation variability, lowers required interest-rate variance, reduces ZLB probability, and reduces indeterminacy.
    - For the calibrated model, benefits of stabilizing inflation and avoiding ZLB outweigh costs from reduced debt-deflating inflation volatility—resulting in optimality of the price-level rule.
  - Effect of public debt and risk premia:
    - Higher public debt and risk premia increase optimal fiscal reactivity to shocks; composition shifts from tax adjustments (low premia) to government spending adjustments (high premia).
  - Time-consistent vs. commitment:
    - Discretion induces faster debt consolidation and larger short-run output costs relative to commitment/optimal policies.
    - Optimal simple passive fiscal rules produce debt trajectories similar to optimal timeless rules; simple active fiscal rules generate large price jumps consistent with FTPL.
  - Super-inertial monetary policy (allowing ρ_r > 1):
    - Example (B/4Y=0.7): allowing super-inertial rule yields a consumption-equivalent welfare gap of 0.05% versus 0.14% under baseline simple (PF).
    - Mechanism: temporary interest-rate rises and long-run inflation undershoots enhance price-setting incentives and stabilization.

- Summary conclusion
  - Optimal monetary policy remains largely inertial/price-level oriented; ZLB concerns reinforce inertia.
  - Optimal fiscal policy reacts more when sovereign risk premia are larger; shift from tax-based to spending-based consolidation with higher premia.
  - Simple passive fiscal rules with inertial monetary rules can perform close to Ramsey-optimal policies, especially with super-inertial monetary persistence.

### 4.4 Nonlinearities, larger shocks and an interest-rate peg — robustness and extensions
- Nonlinear vs linear responses to larger debt jumps
  - Simulation: debt/annual GDP ratio jumps from 90% to 105%; compared nonlinear perfect-foresight (Dynare) with linear LQ approximation.
  - Key finding: linear LQ approximation "provides a sufficiently good guide to policy design for larger debt shocks," though nonlinear model shows longer output recovery and slower debt consolidation.

- Role of shock size and sovereign risk premia
  - Exercise: start at debt/GDP = 90%; simulate a 15 percentage points of GDP shock and halve successively down to about 2 percentage points; rescale IRFs.
  - Nonlinearities intensify with shock size, chiefly via sovereign risk premium elasticity to debt stock.
  - Larger shocks imply:
    - more than proportional increase in interest payments;
    - larger debt roll-over and more protracted consolidation;
    - government spending more persistently below steady state and tax rate more persistently above steady state;
    - lengthier output contraction and slower debt consolidation.

- Interest-rate peg (ZLB-like constraint) and expectations
  - Setup: demand-driven recession preference shock that increases debt by ~1% when central bank is free (steady-state debt/GDP = 90%); impose nominal interest-rate peg for 4, 6, and 8 periods.
  - Findings:
    - Peg durations of 4 and 6 periods magnify inflation fall, deepen recession, and increase level and persistence of government debt.
    - An 8-period peg can reverse effects: perfect-foresight expectation of persistent peg raises expected inflation, increases current inflation, stimulates private demand, and reduces government debt.
  - Policy messages:
    - (i) Constrained monetary policy exacerbates fiscal costs of recessions.
    - (ii) If central banks credibly keep rates at the effective lower bound longer, inflation expectations may aid debt consolidation.
  - Numerical/algorithmic note: bringing nominal rate to zero requires very large shocks; Dynare nonlinear solver may not converge with very large shocks and non-negativity constraints.

- Long-term government debt and portfolio adjustment costs
  - Model extension: government issues short-term bonds B^S_t and consols B^C_t with value V_t and long bonds B^L_t = V_t B^C_t.
  - Portfolio adjustment cost in household utility: −ν_B / 2 (δ_B (B^S_{t+s})^j (B^L_{t+s})^j − 1)^2 with δ_B = 3 and ν_B = 0.1 (following Harrison (2012)).
  - Policy rule for consols: log(b^C_t / b^C) = ρ_{bc} log(b^C_{t−1}/b^C) + ρ_{bcb} log(b^g_t / b^g) − ρ_{bcy} log(Y_t / Y).
  - Results:
    - Welfare rankings and optimised simple rules qualitatively similar to short-bonds-only case.
    - Optimal response to positive debt shock: increase long-to-short bond ratio.
    - Under commitment, issuing long-term bonds slows optimal debt consolidation relative to short-term-only case; effect larger with higher sovereign risk premia.
    - Under time-consistent policy without commitment, consolidation remains fast as in the short-term-only model.

- Quantitative illustration and policy implication
  - Example: initial debt/GDP 90% jumps to 105% under optimised simple fiscal-monetary (passive fiscal, active monetary) commitment rule with short-term debt.
  - Optimal policy: sovereign risk implies debt ratio may rise for about 10 quarters, then slowly fall back to 105% after about a further 20 quarters — implying a 7-8 year adjustment period, much slower than many post-crisis consolidation programs.
  - General conclusion: optimal consolidation should be slow in realistic specifications; rapid consolidation mainly arises without commitment or when initial public debt overhang is very high and bailouts are unavailable.

_Italic source attribution: wp1778 — IMF Working Paper (excerpts from sections 2.1–2.3, 4.2, 4.4 and Appendices A–D)_

### 2.1  Households  . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .    9

### 2.1  Households

### Preferences and consumption aggregator
- Households j ∈ [0,1] have preferences over differentiated consumption varieties i ∈ [0,1] and derive utility from (X_t)_j = X((X^c_t)_j, G_t), a composite of habit-adjusted private consumption (X^c_t)_j and public consumption G_t, allowing for substitutability or complementarity between private and public goods.
- The private component of the habit-adjusted consumption composite is given by:
  - (X^c_t)_j = C^j_t − θ C^j_{t−1}, where θ ∈ (0,1) is the degree of internal habit formation.
- Habits are internal (rather than external) to avoid counterfactual welfare externalities.

### Labour supply, wages and wage adjustment costs
- Each household j is a monopolistic provider of a differentiated labour service and supplies labour H^j_t:
  - H^j_t = (w^j_t / w_t)^{−e} W_t η H_t,
    where w^j_t is the real wage charged by household j, w_t is the average real wage, η is the intratemporal elasticity of substitution across labour services, e^W_t is a wage mark-up shock, and H_t is average labour demand.
- Rotemberg quadratic wage adjustment cost (proportional to average real value of labour services):
  - ξ_W / 2 (W^j_t / W^j_{t−1} − Π)^2 w_t H_t = ξ_W / 2 (w^j_t / w^j_{t−1} Π_t − Π)^2 w_t H_t,
    where ξ_W is the wage adjustment cost parameter and Π is the steady state value of inflation.

### Capital accumulation, investment and utilization
- Capital evolves according to:
  - K^j_{t+1} = (1−δ) K^j_t + e^I_t I^j_t [1 − S(I^j_t / I^j_{t−1})],
    where δ is the capital depreciation rate, S(·) is an investment adjustment cost satisfying S(1) = S′(1) = 0 and S′′(1) > 0, and e^I_t is an investment-specific shock.
- Households control capital utilization u^j_t with a utilization cost a(u^j_t) K^j_t, satisfying a(u) = 0 at steady state utilization.

### Assets, taxes, transfers and household budget constraint
- Households buy consumption C^j_t, invest I^j_t, hold private bonds B^j_t (zero net supply), hold nominal short-term government bonds (B^S_t)^j, and face:
  - net lump-sum taxes τ^L_t and distortionary taxes τ^C_t, τ^W_t, τ^K_t;
  - receive wage income, rental income on utilized capital, returns on private and government bonds (the latter discounted at the ex-ante expected haircut rate ∆^g_t), firms’ profits, transfers ˜Ξ_t, and a tax depreciation allowance δQ_t τ^K_t K^j_t.
- Household j’s budget constraint:
  - (1 + τ^C_t) C^j_t + I^j_t + τ^L_t + ξ_W / 2 (w^j_t / w^j_{t−1} Π_t − Π)^2 w_t H_t + a(u^j_t) K^j_t + B^j_t / P_t + (B^S_t)^j / P_t
    ≤ (1 − τ^W_t) W^j_t / P_t H^j_t + (1 − τ^K_t) R^K_t u^j_t K^j_t + R_{t−1} B^j_{t−1} / P_t + (1 − ∆^g_t) R^S_{t−1} (B^S_{t−1})^j / P_t + ∫_0^1 J_{it} di + ˜Ξ_t + δQ_t τ^K_t u_t K^j_t.

### Household optimisation problem
- Household j chooses {C^j_t, B^j_t, (B^S_t)^j, K^j_{t+1}, u_t, I^j_t, w^j_t} to:
  - max E_t Σ_{s=0}^∞ { e^B_{t+s} β_{t+s} U(X^j_{t+s}, H^j_{t+s}) },
    subject to equations for habits, labour supply, capital accumulation and the budget constraint, where β ∈ (0,1) is the discount factor, U(·) is instantaneous utility, and e^B_t is a preference shock.

### (Related model components introduced in following subsection)
- Firms (section 2.2) operate under monopolistic competition, rent capital services ˜K_it ≡ u_t K_it, hire labour H_it, and produce differentiated goods Y_it with production function F(e^A_t H_it, ˜K_it), where e^A_t is a labour-augmenting technology shock.
- Consumption aggregator (Dixit-Stiglitz):
  - Y_t = [∫_0^1 (Y_it)^{1−1/e^P_t ζ} di]^{1/(1−1/e^P_t ζ)},
    implying demand for each variety:
  - Y_it = (P_it / P_t)^{−e^P_t ζ} Y_t,
    where e^P_t is a price mark-up shock and ζ is the intratemporal elasticity of substitution across varieties.
- Firms face Rotemberg quadratic price adjustment costs:
  - ξ_P / 2 (P_it / P_{it−1} − Π)^2 Y_t,
    and solve a dynamic profit maximisation problem:
  - max_{˜K_{it+s}, H_{it+s}, P_{it+s}} J_it = E_t Σ_{s=0}^∞ D_{t,t+s} [ P_{it+s} / P_{t+s} Y_{it+s} − w_{it} H_{it+s} − R^K_{t+s} ˜K_{it+s} − ξ_P / 2 (P_{it+s} / P_{it+s−1} − Π)^2 Y_{t+s} ],
    subject to demand and firm resource constraint Y_it = F(e^A_t H_it, ˜K_it) − FC, where FC are fixed production costs.

*Source: wp1778 - 2.1  Households (sections 2.1–2.2) — IMF Working Paper*

### 2.3  Government

### 2.3  Government

### Government financing and sovereign default setup
- Government finances expenditures, Gt, by levying taxes, Tt, and by issuing one-period bonds, BS t.
- One-period bonds promise repayment next period at gross nominal interest rate RS t.
- Government bond contracts are not enforceable; sovereign risk premium introduced via a stochastic fiscal limit Γ* t drawn each period.
- Fiscal limit distribution CDF p* t is logistical with parameters η1 and η2:
  - p* t = P(Γ* t ≤ Γt−1) = exp(η1 + η2 Γt−1) / (1 + exp(η1 + η2 Γt−1)). (Equation (11))
- Debt-to-GDP ratio defined as Γt ≡ BS t / (Pt Yt).
- If Γt−1 ≥ Γ* t, government defaults. p* t represents the probability of default.
- Default takes the form of a haircut ̄Δ ∈ [0,1] applied proportionally to outstanding government debt.

### Expected haircut and tractability
- Agents consider the ex-ante expected haircut rate Δg t specified as:
  - Δg t = 0 with probability 1 − p* t
  - Δg t = ̄Δg with probability p* t (Equation (12)),
  - implying Δg t = p* t ̄Δg. (Equation (13))
- ̄Δg ∈ (0,1] is the haircut rate applied in case of default.
- This expectation-based specification follows Corsetti et al. (2013) to enable solution via perturbation methods.

### Real government budget constraint and revenues
- Let bS t ≡ BS t / Pt and Πt ≡ Pt / Pt−1 be gross inflation.
- Real government budget constraint:
  - bS t = (1 − Δg t) RS t−1 / Πt bS t−1 + Gt − Tt + ̃Ξt. (Equation (14))
- Total government revenue Tt is:
  - Tt = τC t Ct + τW t wt ht + τK t [ (RK t − δ Qt) ut Kt ] + τL t. (Equation (15))
  - τC t, τW t, τK t are tax rates on private consumption Ct, labour income wt ht, and net capital rental rate (RK t − δ Qt) ut Kt, respectively.
  - τL t are lump-sum taxes.

### Government transfers at default
- Transfers ̃Ξt are set to ̃Ξt ≡ Δg t RS t−1 / Πt bS t−1 so that sovereign default does not alter the actual debt level.
- Rationale: absence of such transfers would imply lower ex-ante risk premia prior to default because lower post-default debt would already be priced in.

### Fiscal instruments and Taylor-type rules
- To reduce instruments to one, τC t, τW t, τK t and τL t deviate from steady state by the same proportion: τC t = τt τC, τW t = τt τW, τK t = τt τK, τL t = τt τL. The uniform proportional tax change τt is a fiscal instrument.
- The other fiscal instrument is government spending Gt.
- Adjustment rules (Taylor-type):
  - log(τt / τ) = ρτ log(τt−1 / τ) + ρτb log(bS t−1 / bS) + ρτy log(Yt / Y). (Equation (16))
  - log(Gt / G) = ρg log(Gt−1 / G) − ρgb log(bS t−1 / bS) − ρgy log(Yt / Y). (Equation (17))
- Interpretation of rule parameters:
  - ρτ: persistence in tax instrument.
  - ρτb: responsiveness of tax instrument to deviation of government debt from steady state.
  - ρτy: responsiveness of tax instrument to output deviations.
  - ρg, ρgb, ρgy: analogous parameters for the expenditure rule.
- Rules respond to deviations from deterministic steady-state values (not flexi-price outcomes), facilitating implementation using readily available macro-data.

### Parameter calibration related to government features (selected values)
- Haircut rate ̄Δ = 0.0246.
- Scaling factor in default probability η1 = −9.7480.
- Slope parameter in default probability η2 = 1.9921.
- Steady-state government spending share of output G/Y = 0.19.
- Steady-state government debt-to-GDP ratio Γ = 4 × 0.70.
- Steady-state consumption tax rate τC = 0.05.
- Steady-state labor income tax rate τW = 0.24.
- Steady-state capital tax rate τK = 0.32.
- Fiscal-rule parameters:
  - Persistence of tax rates ρτ = 0.6252.
  - Tax response to government debt ρτb = 0.0777.
  - Tax response to output ρτy = 0.1478.
  - Persistence of government spending ρg = 0.7662.
  - Government spending response to government debt ρgb = 0.1197.
  - Government spending response to output ρgy = 0.1624.

### Calibration of the fiscal limit CDF and implied sovereign premia
- Calibration anchors:
  - Set p*2 = 0.99 at Γ2 = 4 × 1.8.
  - Set Γ1 = 4 × 0.7 (average US post-WWII public-debt-to-GDP).
  - Assume sovereign risk premium at Γ1 corresponds to ABP1 = 15 annual basis points.
  - Assume sovereign risk premium at Γ2 corresponds to ABP2 = 1,000 annual basis points.
- From equation (13), probability of default at Γ = Γ1 is:
  - p*1 = 1 − 1 / (ABP1/40000 + 1) ̄Δg, yielding p*1 = 0.0152.
- Using equations (24) and (25) with the chosen (Γ1, p*1) and (Γ2, p*2) recovers parameters η1 = −9.7480 and η2 = 1.9921.
- This parameterisation implies sovereign risk premium rises from 15 annual basis points when government debt is 70% of annual GDP to:
  - 143 annual basis points when government debt ratio increases to 100%.
  - 452 annual basis points when government debt ratio increases to 120%.

### Role in optimal policy analysis
- Fiscal rules (τt and Gt) become key policy instruments in the LQ approximation for both normal-times stabilisation and crisis management.
- The specification of transfers ̃Ξt and expected haircut Δg t affects sovereign risk premia dynamics and thus feeds into optimal monetary and fiscal stabilisation policy analysis.

*Source: wp1778 - 2.3  Government (IMF Working Paper chapter).*

### 4.2  Optimal monetary-fiscal rules for normal times

### 4.2  Optimal monetary-fiscal rules for normal times

### Framework and policy-rule setup
- Consideration of monetary and fiscal instruments together: taxation τt (equation (16)), government spending Gt (equation (17)), and monetary policy rule (equation (18)).
- Policy interpretations:
  - “Optimality” can mean welfare-optimal (Ramsey), time-consistent, or optimised Taylor-type interest-rate and fiscal rules.
  - Assignment: monetary authority stabilises inflation and fiscal authorities stabilise debt; both can respond to output fluctuations.
  - Active vs. passive (Leeper, 1991): active monetary rule satisfies the Taylor principle; passive fiscal rule stabilises government debt; both possibilities are allowed by the simple rules used.
- Steady-state and LQ solution remarks:
  - Small quadratic cost on fiscal deviations pins fiscal steady states to historically observed values; lump-sum taxes adjust government debt-to-GDP ratio.
  - LQ solutions for the three policy regimes are standard; iterative algorithm rules out “point-in-time” multiple discretionary equilibria for this model.

### Feasibility constraints and ZLB treatment
- Feasibility constraints imposed (Schmitt-Grohe and Uribe, 2007):
  - ρrπ ≤ 5
  - ρτb, ρgb ≤ 1
- Baseline bounds:
  - Monetary-rule inertia constrained: ρr ≤ 1 in baseline; superinertial monetary policy (ρr > 1) explored separately (Table 3).
- ZLB (zero lower bound) concern and penalty-function approach:
  - Add penalty −wr(Rt − R)2 to welfare objective to lower interest-rate variance and reduce ZLB hitting probability.
  - Choose wr in each regime so probability of hitting the ZLB is 0.0025 (once every 400 quarters / 100 years).
  - After optimisation, representative agent utility is corrected by adding back wr var(Rt); adjusted welfare loss reported in tables is original welfare less wr var(Rt).
- Role of persistence:
  - Higher persistence ρr reduces interest-rate variability needed for stabilization, lowering ZLB probability.
  - Price-level rules (ρr = 1, ρry = 0) emerge as limiting/inertial cases with desirable properties for inflation stabilization and ZLB avoidance.

### Main quantitative results (welfare losses; consumption-equivalent c e)
- Welfare loss metric: consumption-equivalent permanent percentage fall in private consumption below Ramsey-optimal policy, denoted ce.
- Table 2 (Optimal policy results for alternative government debt/GDP ratios) — reported entries verbatim from source:

  - B/4Y=0.7
    - Rule[ρr, ρrπ, ρry][ρτ, ρτb, ρτy][ρg, ρgb, ρgy]wr Adjusted Los scons eq %
    - Optimal n/an/an/a371-0.0038
    - TCT n/an/an/a1350.00830.45
    - Simple (PF) 1, 0.232, 00.051, 0.25, 00.789, 0.018, 03850.00000.14
    - Simple (AF) 0,0,00,0,00,0,01350.00690.40

  - B/4Y=0.9
    - Rule[ρr, ρrπ, ρry][ρτ, ρτb, ρτy][ρg, ρgb, ρgy]wr Adjusted Losscons eq %
    - Optimal n/an/an/a296-0.0046
    - TCT n/an/an/a1020.00880.49
    - Simple (PF) 1, 0.236, 00.504, 0.25, 00.733, 0.046, 03810.00000.17
    - Simple (AF) 0,0,00,0,00,0,0790.00660.41

  - B/4Y=1.2
    - Rule[ρr, ρrπ, ρry][ρτ, ρτb, ρτy][ρg, ρgb, ρgy]wr Adjusted Losscons eq %
    - Optimal n/an/an/a221-0.0021
    - TCT n/an/an/a630.00320.20
    - Simple (PF) 1,0.261,00.579,1,00.161, 0.749,02910.00150.13
    - Simple (AF) 0,0,00,0,00,0,0600.00540.28

  - B/4Y=1.2 Bailout
    - Rule[ρr, ρrπ, ρry][ρτ, ρτb, ρτy][ρg, ρgb, ρgy]wr Adjusted Losscons eq %
    - Optimal n/an/an/a310-0.0047
    - TCT n/an/an/a2440.0110.58
    - Simple (PF) 1,0.232,00,0.25,00,0,03820.00020.18
    - Simple (AF) 0,0,00,0,00,0,0600.00540.38

- Interpretation of key numerical findings (from text and table entries):
  - The optimal simple passive fiscal (PF), active monetary rules generate an average welfare loss of 0.14% in consumption-equivalent terms (as noted in the text).
  - Time-consistent or optimal simple active fiscal, passive monetary rules produce considerably larger consumption-equivalent welfare losses.
  - In all cases with passive fiscal policy, the optimised simple monetary policy rule is a price-level rule and both monetary and fiscal instruments do not react to output fluctuations.
  - Low steady-state debt/GDP (70%) implies stronger responsiveness of taxes relative to government spending (government spending displays high inertia and low elasticity to government debt).
  - High steady-state debt (120% of GDP) makes government spending much more reactive to debt unless the sovereign risk premium is suppressed.

### Policy and economic dynamics findings
- Price-level (inertial) monetary rules:
  - Price-level stabilization reduces inflation variability because forward-looking agents expect compensating below-target inflation after above-target inflation, dampening firms’ price adjustments.
  - Price-level rules lower required interest-rate variance, reducing ZLB probability, and reduce indeterminacy, enlarging policy space.
  - For the calibrated model, welfare benefits of stabilizing inflation and avoiding the ZLB outweigh the welfare cost from reduced debt-deflating inflation volatility — resulting in optimality of the price-level rule.
- Effect of public debt and risk premia:
  - Higher public debt and associated risk premia do not significantly alter optimal monetary policy features found in the literature.
  - Optimal fiscal policy becomes much more reactive to shocks when risk premia are higher.
  - Under low risk premia most debt stabilization is achieved via tax changes; under high risk premia it is optimal to adjust government expenditures more.
- Time-consistent vs. commitment outcomes for debt shocks:
  - Figure 3 (described in text) and associated discussion:
    - For debt/GDP ratios of 70% and 120% with no sovereign risk, optimal policy yields debt that barely shifts over time (debt follows a “random walk” under optimal policy when inverse discount rate matches gross real interest rate).
    - Optimal simple passive fiscal rules produce trajectories similar to optimal timeless rules.
    - Optimal simple active fiscal rules produce a large jump in prices consistent with FTPL, with price jumps two to three times larger than under optimal time-consistent rules.
    - Under time-consistent policy, debt consolidation is always very fast — discretion implies quick consolidation because inability to commit raises the immediate consolidation cost.
  - Commitment vs. high debt:
    - If government can commit, optimal and optimal passive simple rules display slow consolidation unless steady-state debt is very high (e.g., 120% of GDP), where risk premia force debt to drop quickly and fiscal adjustment is more painful (output drops more and more persistently).
    - A successful bailout that suppresses risk premia reverses fast consolidation pressure.
- Super-inertial monetary policy (ρr > 1):
  - Allowing ρr > 1 (super-inertial) can make simple rules better mimic Ramsey-optimal debt paths.
  - Table 3 (B/4Y=0.7 allowing for superinertial monetary policy) — reported verbatim:
    - B/4Y=0.7
      - Rule[ρr, ρrπ, ρry][ρτ, ρτB, ρτy][ρG, ρGB, ρGy]wr Adjusted Losscons eq %
      - Optimal n/an/an/a371-0.0038
      - Simple (PF) 1, 0.232, 00.051, 0.25, 00.789, 0.018, 03850.00000.14
      - Superinertial 1.59,0.215,0.0340,0.15,00.39,0.18,03700.00250.05
  - For B/4Y=0.7 allowing super-inertial monetary policy, the consumption-equivalent welfare compared with optimal policy is reduced to 0.05%.
  - Mechanism: a super-inertial rule commits to only temporarily raise the interest rate in response to above-target inflation; in equilibrium inflation undershoots its target in the long run, enhancing firms’ price-setting incentives and further stabilizing outcomes.

### Summary conclusions
- Optimal monetary policy remains largely inertial/price-level oriented even with public debt considerations; ZLB concerns reinforce inertia.
- Optimal fiscal policy reacts more to shocks when sovereign risk premia are larger; composition of adjustment shifts from taxes (low risk premia) to greater reliance on government spending adjustments (high risk premia).
- Simple passive fiscal rules coupled with inertial monetary rules can perform close to Ramsey-optimal policies, particularly when augmented with super-inertial monetary persistence to better mimic optimal debt dynamics.
- Time-consistency (discretion) induces faster debt consolidation and larger short-run output costs relative to commitment/optimal policies; bailouts that remove sovereign risk premia can materially change optimal consolidation dynamics.

*Source: wp1778 - 4.2  Optimal monetary-fiscal rules for normal times (IMF working paper content unit).*

### 4.4  The role of nonlinearities, larger shocks and an interest-rate peg

### 4.4  The role of nonlinearities, larger shocks and an interest-rate peg

### Nonlinear versus linear model responses to larger debt jumps
- Comparative simulation: shock that brings the debt/annual GDP ratio from 90% to 105%.
- Method: nonlinear perfect foresight solver (Dynare) versus linear LQ approximation.
- Key finding:
  - Despite model nonlinearities, the linear approximation of the LQ methodology "provides a sufficiently good guide to policy design for larger debt shocks."
  - Quantitative differences exist: in the nonlinear simulations output takes longer to return to its base level, and government debt as a fraction of GDP is consolidated at an even slower pace in the nonlinear model—reinforcing the slow fiscal consolidation conclusion from the linear model.
- Example summary statistics from the simulation:
  - Initial steady-state government debt-to-GDP ratio: 90%.
  - Jump to: 105%.

### The role of shock size and sovereign risk premia
- Experiment: nonlinear model responses to shocks of different magnitudes.
  - Procedure: start from steady state debt/GDP = 90%; simulate a 15 percentage points of GDP shock; progressively halve shock size down to about 2 percentage points of GDP; rescale impulse responses by the same factors for comparability.
- Mechanism and outcomes:
  - Nonlinearities become more evident as shock size increases, especially via the sovereign risk premium.
  - Because the interest rate paid on debt is elastic to the size of the stock of debt, greater shocks imply a more than proportional increase in interest payments.
  - Higher interest payments translate into:
    - larger debt roll-over;
    - more protracted fiscal consolidation (government spending more persistently below—and the tax rate more persistently above—their respective steady states);
    - a lengthier output contraction;
    - a slower consolidation of debt as a fraction of GDP.

### Interest-rate peg, ZLB-like constraints, and expectations
- Setup: simulate a demand-driven recession (preference shock) that reduces output and inflation and increases government debt; impose a nominal interest-rate peg for a fixed number of periods so the nominal rate cannot adjust following the shock.
- Numerical elements:
  - Preference shock inducing an increase in the debt-to-GDP ratio of approximately 1% when the central bank is free to follow the optimized rule for steady-state debt/GDP = 90%.
  - Interest-rate peg durations simulated: constrained constant nominal rate for 4 periods, 6 periods, and 8 periods.
- Key findings:
  - Increasing the peg duration from zero to 4 and 6 periods magnifies the fall in inflation, deepens the recession, and increases both the level and persistence of government debt as a fraction of GDP.
  - An 8-period peg can have the opposite effect: because agents in a perfect foresight context expect the central bank to keep the rate constant even after the recession ends, expected inflation increases, which raises current inflation, stimulates private demand, and reduces government debt.
- Policy messages:
  - (i) A constrained monetary policy exacerbates the negative fiscal implications of a recession.
  - (ii) If central banks can credibly keep interest rates at the effective lower bound for a longer time, inflation expectations can help the government debt consolidation process.
- Practical note on nonlinear solver limitations:
  - Bringing the nominal interest rate from its steady state value to zero would require a very large shock and Dynare’s nonlinear algorithm may not converge in presence of very large shocks and a non-negativity constraint; alternative procedures using linearised models and exogenous interest-rate shocks are suggested.

### Introducing long-term government debt and portfolio adjustment costs
- Modeling approach:
  - Government issues short-term bonds B^S_t and consols B^C_t; value of a consol is V_t; total value of long bonds B^L_t = V_t B^C_t.
  - Real government budget constraint with long bonds (equation labels in source: (28), (29), (30)).
  - Policy instrument: b^C_t (real stock of consols) follows a simple rule:
    - log(b^C_t / b^C) = ρ_bc log(b^C_{t-1}/b^C) + ρ_bcb log(b^g_t / b^g) − ρ_bcy log(Y_t / Y).
- Impediments to arbitrage:
  - Households perceive long-term bonds as less liquid and demand additional holdings of short-term bonds when long-term holdings increase.
  - Implemented via a convex portfolio adjustment cost added to households’ utility: −ν_B/2 (δ_B (B^S_{t+s})^j (B^L_{t+s})^j − 1)^2.
  - Parameter choices (following Harrison (2012)):
    - δ_B = 3
    - ν_B = 0.1
- Effects on optimal policy and consolidation:
  - Qualitatively similar welfare rankings and optimized simple rules as the short-bonds-only case (see Table 4 in source).
  - When the government can issue long-term bonds, it is always optimal to increase the ratio of long-to-short bonds in response to a positive debt shock.
  - Under commitment (Ramsey), issuing long-term bonds leads to a slower optimal debt consolidation pace than in the short-term-bonds-only case; the difference is more visible at higher debt levels (greater sovereign risk premia).
  - Under time-consistent policy without commitment, the pace of consolidation remains fast, similar to the short-term-bonds-only model.

### Policy implications and concluding quantitative illustration
- General conclusion: optimal consolidation should be slow in most realistic specifications; rapid consolidation arises mainly when the government lacks commitment or when initial public debt overhang is very high and bailout options are unavailable.
- Example illustration from optimized simple fiscal-monetary (passive fiscal, active monetary) commitment rule with short-term debt:
  - Initial steady-state debt/GDP ratio: 90% that jumps to 105%.
  - Optimal policy implication: sovereign risk implies a debt reduction policy that nevertheless allows the debt-GDP ratio to rise for about 10 quarters, and then to slowly fall back to 105% after about a further 20 quarters.
  - Time span: this implies a 7-8 year adjustment period, a markedly slower consolidation than many post-financial-crisis fiscal programs. 

*Source: IMF Working Paper wp1778, section 4.4 (excerpt).*

### References

### wp1778 - References

### References cited
- List of cited works (authors and years) includes: Adam (2011); Andrés, López-Salido, and Nelson (2004); Austin and Miller (2011); Batini, Callegari, and Melina (2012); Benigno and Woodford (2004, 2012); Bi, Leeper, and Leith (2013); Bi and Traum (2014); Blake and Kirsanova (2012); Bouakez and Rebei (2007); Cantore, Levine, and Melina (2014); Cantore et al. (2012); Christiano, Motto, and Rostagno (2014); Cochrane (2011); Cogan et al. (2010); Correia et al. (2013, 2008); Corsetti et al. (2013); Denes, Eggertsson, and Gilbukh (2013); Dennis and Kirsanova (2013, 2015); Erceg and Linde (2013); Furlanetto (2011); Giannoni (2014); Harrison (2012); Jaimovich and Rebelo (2008); Kirsanova, Leith, and Wren-Lewis (2009); Kirsanova and Wren-Lewis (2012); Leeper (1991); Leeper, Plante, and Traum (2010); Leith, Moldovan, and Rossi (2012); Leith and Wren-Lewis (2013); Levine, McAdam, and Pearlman (2008a); Levine and Pearlman (2011); Levine, Pearlman, and Pierse (2008b); Michal (2011); Ostry, Ghosh, and Espinoza (2015); Pappa (2009); Rotemberg (1982); Schmitt-Grohe and Uribe (2004a, 2004b, 2007); Sims (2013); Siu (2004); Smets and Wouters (2007); Taylor (1993); Woodford (2003); Zubairy (2014).

### Appendix A — Equilibrium conditions (key equations and definitions)
- Utility and marginal utilities:
  - U(X_t, H_t) = X_{c,t}^{1−σ_c}/(1−σ_c) (1−ψ H_t^{θ})^{1−σ_c} − ν_B/2 (δ_B b^S_t b^L_{t−1} − 1)^2 (A.1)
  - X_t = { ν^{1/σ_x}_x [(X^c_t)]^{(σ_x−1)/σ_x} + (1−ν_x)^{1/σ_x} G_t^{(σ_x−1)/σ_x} }^{σ_x/(σ_x−1)} (A.2)
  - U_{X^c,t} = e^{B_t} ν^{1/σ_x}_x X_{c,t}^{−σ_c} (1−ψ H_t^{θ})^{1−σ_c} (X_t/X^c_t)^{1/σ_x} (A.3)
  - U_{H,t} = −e^{B_t} θ ψ X_{c,t}^{1−σ_c} (1−ψ H_t^{θ})^{−σ_c} H_t^{θ−1} (A.4)

- Consumption/saving conditions:
  - X^c_t = C_t − θ C_{t−1} (A.5)
  - U_{X^c_t} − (1 + τ^C_t) λ_t = θ β E_t[ U_{X^c_{t+1}} ] (A.6)
  - 1 = E_t[ D_{t,t+1} R_t Π_{t+1} ] (A.7)
  - D_{t,t+1} = β λ_{t+1}/λ_t (A.8)
  - Euler-like conditions with bond spreads and debt frictions (A.9) and (A.10).

- Investment and capital:
  - K_{t+1} = (1−δ) K_t + e^I_t I_t [1 − S(I_t/I_{t−1})] (A.11)
  - Q_t = E_t{ D_{t,t+1}[ (1−τ^K_{t+1}) u_{t+1} R^K_{t+1} + δ Q_{t+1} τ^K_{t+1} u_{t+1} − a(u_{t+1}) + (1−δ) Q_{t+1} ] } (A.12)
  - First-order condition linking Q_t, adjustment costs S(·), and expected future Q_{t+1} (A.13)
  - a′(u_t) = (1−τ^K_t) R^K_t + δ Q_t τ^K_t (A.14)
  - S(.) and a(.) functional forms: S(x) = γ_2/2 (x−1)^2 (A.15); S′(x) = γ (x−1) (A.16); a(u_t) = γ_1 (u_t−1) + γ_2/2 (u_t−1)^2 (A.17); a′(u_t) = γ_1 + γ_2 (u_t−1) (A.18)

- Wage setting and bargaining:
  - (1−τ^W_t) (e^W_t)^{η−1} w_t − e^W_t^{η} w_t ̃μ_t + ξ_W (Π^W_t − Π) w_t/Π^W_t = E_t[ D_{t,t+1} ξ_W (Π^W_{t+1} − Π) w_{t+1}/Π^W_{t+1} H_{t+1}/H_t ] (A.19)
  - ̃μ_t = w_t / MRS_t (A.20); MRS_t = −U_{H,t} / λ_t (A.21)
  - Π^W_t = w_t / w_{t−1} Π_t (A.22)

- Production and pricing:
  - F(e^A_t H_t,  ̃K_t) = (e^A_t H_t)^α (u_t K_t)^{1−α} (A.23)
  - Factor marginal products defined: F_{H,t} and F_{K,t} (A.24, A.25)
  - Y_t = F(e^A_t H_t,  ̃K_t) − FC (A.26)
  - R^K_t = MC_t F_{K,t} and w_t = MC_t F_{H,t} (A.27, A.28)
  - Price-setting condition with price adjustment costs and forward-looking inflation terms (A.29)

- Government and fiscal policy rules:
  - Default probability p^*_t = P(Γ^*_t ≤ Γ_{t−1}) = exp(η_1 + η_2 Γ_{t−1}) / [1 + exp(η_1 + η_2 Γ_{t−1})] (A.30)
  - ∆_t = p^*_t ∆ (A.31)
  - Γ_t = b^g_t / Y_t (A.32); b^g_t = b^S_t + b^L_t (A.33)
  - Debt composition and law of motion for short- and long-term debt (A.36–A.37)
  - Taxes and tax rules: T_t = τ^C_t C_t + τ^W_t w_t h_t + τ^K_t [(R^K_t − δQ_t) u_t K_t] + τ^L_t (A.38)
  - Tax and government spending adjustment rules: τ^C_t = τ_t τ^C (A.39) through (A.42); log(τ_t/τ) dynamic rule (A.43); log(G_t/G) dynamic rule (A.44)

- Monetary policy and resource constraint:
  - Taylor-type rule: log(R_t / R) = ρ_r log(R_{t−1}/R) + ρ_{rπ} log(Π_t / Π) + ρ_{ry} log(Y_t / Y) (A.45)
  - Resource constraint: Y_t = C_t + I_t + G_t + ξ_P/2 (Π_t − ̄Π)^2 Y_t + ξ_W/2 (Π^W_t − ̄Π)^2 w_t H_t + a(u_t) K_t (A.46)

- Exogenous autoregressive processes (examples):
  - log(e^B_t / e^B) = ρ_κ log(e^B_{t−1}/e^B) + ε^B_t (A.47)
  - Similar AR(1) specifications for e^P_t, e^I_t, e^A_t, e^W_t (A.48–A.51)

### Appendix B — The Ramsey problem and the LQ approximation (method and key optimality conditions)
- Problem statement:
  - Maximise E_0 ∑_{∞ t=0} β^t u(y_t, w_t) subject to E_t f(y_t, y_{t+1}, y_{t−1}, w_t, ε_t) = 0 (B.1)
  - y_t partitioned as y_t ≡ [z_t^T x_t^T], where z_t predetermined, x_t forward-looking, w_t instruments, ε_t shocks.
- Lagrangian:
  - L = ∑_{t=0}^∞ β^t [ u(y_t, w_t) + λ^T_{t+1} f(y_t, y_{t+1}, y_{t−1}, w_t, ε_t) ] (B.2)
- First-order conditions (symbolic):
  - ∂L/∂w_t = u_2 + λ^T_{t+1} f_4(·) (B.3)
  - ∂L/∂y_t = u_1 + λ^T_{t+1} f_1(·) + 1/β λ^T_t f_2(·) + β λ^T_{t+2} f_3(·) (B.4)
- Partitioning co-state vector:
  - λ_t = [λ_{1,t} λ_{2,t}] with λ_{1,t} dimension (n−m)×1 (backward-looking) and λ_{2,t} dimension m×1 (forward-looking).
- Critical boundary conditions for forward-looking multipliers:
  - λ_{2,0} = 0 (ex ante optimal) (B.5)
  - λ_{2,0} =  ̄λ_2 (‘timeless’ solution) (B.6)
  - ̄λ_2 denotes deterministic steady state of λ_{2,t}.
- LQ approximation approach:
  - Solve deterministic steady state of the optimum; then maximise second-order approximation to the Lagrangian about steady state subject to linearised constraints.
  - The second-order expansion of L − ̄L is given in (B.7) with remaining uncancelled first-order terms −1/β ̄λ^T f_2 ̃y_0 + ̄λ^T f_3 ̃y_{−1} (B.8).
  - First-order condition linking u_1 and ̄λ with f matrices at time 0: u_1 + ̄λ^T (f_1 + 1/β f_2 + β f_3) = 0 (B.10).
- Practical note:
  - The LQ approximation requires accounting for forward-looking initial terms when optimising policy; see Levine et al. (2008b) and Benigno and Woodford (2012) referenced for method details.

### Appendix C — Data (sources, variables, transformations)
- Original data sources and series (Table C.1):
  - GDP: Nominal GDP — BEA NIPA Table 1.1.5 — Frequency: Q
  - PCE: Personal Consumption expenditure (total) — BEA NIPA Table 1.1.5 — Q
  - PFI: Private Fixed Investment — BEA NIPA Table 5.3.5 — Q
  - GCE: Government consumption expenditure and gross investment — BEA NIPA Table 1.1.5 — Q
  - GR: Government revenues — BEA NIPA Table 3.1 — Q
  - RGDP: Real GDP (base year 2005) — BEA NIPA Table 1.1.6 — Q
  - CNP16OV: Civilian non-institutional population, over 16 — BLS — Q
  - CE16OV: Civilian Employment sixteen years and over — BLS — Q
  - LBMNU: Non-farm business hours worked — BLS — Q
  - LBCPU: Hourly non-farm business compensation — BLS — Q
  - FFR: Federal Funds Rate — St. Louis FRED — Q
- Constructed series and observables:
  - Sample used for matching: 1984:Q1-2008:Q2 (subsample for moment-matching). All real variables filtered using an HP(1600) filter.
  - Variable transformations (selected exact constructions from Table C.2):
    - GDP_Deflator = GDP / RGDP * 100 (index)
    - Population index = CNP16OV / CNP16OV_{2005:2} * 100
    - CE16OV_index = CE16OV / CE16OV_{2005:2} * 100
  - Observables (9) and their constructions:
    - R_GDP: Real per capita gross domestic product = LN( GDP_index / GDP_Deflator ) 100
    - GOV_SP: Real per capita government spending = LN( GCE_index / GDP_Deflator ) 100
    - GOV_RV: Real per capita government revenues = LN( GR_index / GDP_Deflator ) 100
    - HOURS: Per capita hours worked = LN( LBMNU * CE16OV_index / 100_index ) 100  (note: resulting series of hours is demeaned before estimation)
    - WAGE: Real wage = LN( LBCPU / GDP_Deflator ) 100
    - FFF: Quarterly Federal Funds rate = FFR / 4
    - Π: Inflation = ∆GDP_Deflator / 100
    - CON: Real per capita consumption = LN( PCE_index / GDP_Deflator ) 100
    - INV: Real per capita investment = LN( PFI_index / GDP_Deflator ) 100

### Appendix D — Moments (data vs. model)
- Standard deviations (in%):
  - Real output: 1.09 (Data)    1.89 (Model)
  - Private consumption: 0.88    1.40
  - Private investment: 4.06    6.94
  - Inflation: 0.23    0.91
  - Real wage: 1.10    1.82
  - Hours worked: 2.20    1.96
  - Government spending: 1.07    1.84
  - Government revenue: 3.18    3.93
  - Interest rate: 0.61    0.84

- Autocorrelations (Data    Model):
  - Real output: 0.8516    0.8824
  - Private consumption: 0.8387    0.9742
  - Private investment: 0.9148    0.9551
  - Inflation: 0.5960    0.3791
  - Real wage: 0.8197    0.8975
  - Hours worked: 0.9113    0.6480
  - Government spending: 0.7105    0.9811
  - Government revenue: 0.8649    0.7606
  - Interest rate: 0.9560    0.8731

- Cross-correlations with output (Data    Model):
  - Private consumption: 0.8723    0.6414
  - Private investment: 0.9201    0.8779
  - Inflation: 0.1795    0.1260
  - Real wage: −0.0539    0.7476
  - Hours worked: 0.3714    0.4113
  - Government spending: −0.1550    −0.1008
  - Government revenue: 0.7603    0.6920
  - Interest rate: 0.3460    0.0129

- Additional cross-correlations (Data    Model):
  - Inflation/Interest rate: 0.4111    0.6540
  - Private/public consumption: −0.0334    −0.0938

_Italic source attribution: wp1778 - References_

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_Source: https://www.imf.org/-/media/files/publications/wp/2017/wp1778.pdf_