## WP/20/33 — A Model-based Fiscal Taylor Rule and a Toolkit to Assess the Fiscal Stance

## Source details

**Canonical URL:** [WP/20/33 — A Model-based Fiscal Taylor Rule and a Toolkit to Assess the Fiscal Stance](https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020033-print-pdf.pdf)

## Other formats

- [Markdown version](/-/media/files/publications/wp/2020/english/wpiea2020033-print-pdf.pdf.md)
- [Structured JSON version](/-/media/files/publications/wp/2020/english/wpiea2020033-print-pdf.pdf.json)

---

### Abstract and purpose
- Presents a model-based fiscal Taylor rule and a toolkit to assess the fiscal stance, defined as the change in the structural primary balance.
- Built on the normative buffer-stock model of the government (Fournier, 2019) including hysteresis, cycle-dependent multipliers, implementation lags, adjustment costs, buffer-stock concern, and a debt-dependent risk premium.
- Proposes a simple fiscal Taylor rule prescribing the fiscal stance as a function of past government debt, past output gap and the past structural primary balance.
- Applications indicate several advanced economies could have better managed their fiscal stance over the last 20 years; simulations provide fiscal stance recommendations over the medium-term.
- JEL Classification Numbers: E32, E62, H62, H63. Paper identification: WP/20/33; Prepared by Jean‑Marc Fournier and Philipp Lieberknecht; Authorized for distribution by Catherine Pattillo; February 2019.

### Conceptual framework and model features
- Definition:
  - Fiscal stance = change in the structural primary balance (encompasses all discretionary decisions).
- Government objective:
  - Maximize intertemporal welfare by choosing the change in structural primary balance to stabilize output fluctuations subject to economic structure and a government budget constraint with a debt-dependent market-access risk.
- Key model features incorporated:
  - Hysteresis: long-run negative effects of recessions on potential output.
  - Cycle-dependent multipliers: fiscal multipliers larger in recessions.
  - Implementation lags: fiscal plans decided before the year starts.
  - Adjustment costs: reflect difficulty implementing large stimulus and reversing it.
  - Buffer-stock concern: governments may have insufficient buffers in bad times.
  - Risk premium: rising in debt, raising marginal cost of increasing debt.
  - Utility: representative-agent CRRA utility in consumption and labor with parameters such as ρ and ξ described in the model.

### Main theoretical insight
- Fiscal stance should be less counter-cyclical at high debt levels than at low debt levels:
  - At low debt: government can increase debt at moderate cost and should smooth the cycle.
  - At high debt: marginal cost of debt is higher (higher interest rates and market-access risk); moderate counter-cyclical responses are optimal.
  - Interaction with hysteresis: at low debt, stronger hysteresis → larger counter-cyclical reactions; at high debt, hysteresis magnifies crisis costs → reinforces moderation.

### Fiscal Taylor rule and toolkit design
- Fiscal Taylor rule: linear approximation in past primary balance, debt, output gap, with an interaction term debt*gap.
- Toolkit components:
  - Nonlinear fiscal stance function taking output gap, public debt, and last primary balance as inputs; returns recommended fiscal stance for next year.
  - Calibration includes common and country-specific parameters; users can adjust parameters (e.g., fiscal multiplier).
  - Simplified fiscal Taylor rule for practical use.
  - Historical-data comparison of past actual fiscal stance with model prescriptions.
  - Country forecast (output gap, debt, initial primary balance) feeds the stance function for forward advice.
  - Decomposition of recommendations into output-gap versus debt/primary-balance contributions.
  - Scenario analysis capability (recessions, reforms boosting potential growth).
- Practicalities:
  - Toolkit implemented in Matlab with documentation available upon request.
  - Intended as a starting point for discussion and to be used alongside unmodeled considerations (output gap uncertainty, instrument composition, monetary policy interaction, fiscal rules, inequality, aging, political constraints).

### Model structure and fiscal dynamics (key equations and mechanisms)
- Government budget constraint (schematic): 1/(1+r) Δdt = dt+1* − dt = pb_t + g_t + sf_t (equation structure described).
- Risk premium and market-access:
  - Risk premium increases in public debt and in change in debt.
  - Probability of losing market access specified as a logistic-type function of debt level and debt change.
  - If market access lost, government must keep debt constant under an adverse shock of d3 σ (σ = standard deviation of economic shocks; d3 calibrated once for all countries).
- Solution approach:
  - Baseline: risk premium linear in past debt and debt change so implicit new debt level can be solved directly; nonlinear alternatives can be linearized.

### Production, potential output, and hysteresis
- Production: Yt = At Lt; potential output Yt* = At L̄.
- Hysteresis:
  - Permanent loss of potential output if output gap is below threshold h_th.
  - Parameters: h governs size of hysteresis; h_th is output gap threshold.
  - Calibrated to reflect long-run effects in line with literature (long-term effects around 0 – 20%).

### Output gap, fiscal multipliers, and primary balance dynamics
- Output gap driver: shocks vt and structural primary balance pb^s_t.
- Fiscal multiplier specification:
  - Marginal effect of pb on gap = m1 at equilibrium, amplified in downturns by m2.
  - Local representation: ∂gap/∂pb = −m1 (1 + m2 (−gap,·)).
- Primary balance decomposition:
  - pb_t = pb_t^s + a * gap_t (a = automatic stabilizer coefficient).
- Approximate equilibrium for small shocks:
  - gap_t ≈ (v_t + m pb_t^s) / (1 + m a).
  - Structural balance offsetting shocks: pb^s = v_t / m1.
- Interpretation:
  - m1 captures the causal effect of the primary balance on the output gap.

### Aggregate resource constraint and adjustment costs
- Resource constraint: c_t + pb_t + y_t χ = −Δb_t.
- Adjustment costs χ represent implementation costs relative to output; χ = 3 calibrated as moderate value.
- Adjustment costs moderate speed of changes in structural primary balance.

### Calibration — common parameters (reported exactly)
- Welfare function parameters:
  - Discount factor β = 0.99
  - Risk aversion σ = 2
  - Labor elasticity η = 1/0.3
  - Weight of labor ξ = 1
- Fiscal parameters:
  - Fiscal multiplier sensitivity to shocks m2 = 3
  - Adjustment cost χ = 3
- Interest rate and debt parameters:
  - Effect of debt level on the risk premium α1 = 1.5%
  - Effect of debt change on the risk premium α2 = 0.5%
  - Debt level at which the risk to lose market access is 50% d = 150%
  - Debt limit accuracy d1 = 3
  - Effect of debt change on the risk to lose market access d2 = 1
  - Adverse scenario coefficient in case of loss of market access d3 = 0
- Economy parameters:
  - Hysteresis = 10%
  - Hysteresis threshold = -1%
- Additional calibration notes:
  - Discount factor and long-term per capita growth imply overall discount rate 0.975 with growth rate at 1.5%.
  - α1 = 1.5% chosen as midpoint of empirical estimates.

### Country-specific calibration (exact country parameter values)
- Table of country-specific parameters (columns: Country; m1; Automatic stabilizers; potential growth per capita; r-g; Shock persistence ρ; Shock size εσ)
  - Canada: m1 = 0.70; Automatic stabilizers = 0.56; potential growth per capita = 0.36%; r-g = 1.77%; Shock persistence ρ = 0.72; Shock size εσ = 0.021
  - France: m1 = 0.98; Automatic stabilizers = 0.61; potential growth per capita = 1.08%; r-g = 1.05%; Shock persistence ρ = 0.58; Shock size εσ = 0.024
  - Germany: m1 = 0.84; Automatic stabilizers = 0.48; potential growth per capita = 1.21%; r-g = 1.28%; Shock persistence ρ = 0.32; Shock size εσ = 0.025
  - Italy: m1 = 1.00; Automatic stabilizers = 0.5; potential growth per capita = 0.42%; r-g = 2.47%; Shock persistence ρ = 0.58; Shock size εσ = 0.022
  - Japan: m1 = 0.87; Automatic stabilizers = 0.41; potential growth per capita = 0.90%; r-g = 0.63%; Shock persistence ρ = 0.47; Shock size εσ = 0.040
  - Sweden: m1 = 0.62; Automatic stabilizers = 0.66; potential growth per capita = 0.95%; r-g = 0.16%; Shock persistence ρ = 0.53; Shock size εσ = 0.027
  - United Kingdom: m1 = 0.72; Automatic stabilizers = 0.59; potential growth per capita = 0.91%; r-g = 1.00%; Shock persistence ρ = 0.70; Shock size εσ = 0.028
  - United States: m1 = 0.86; Automatic stabilizers = 0.5; potential growth per capita = 1.31%; r-g = -0.29%; Shock persistence ρ = 0.67; Shock size εσ = 0.044
- Note: m1 is the fiscal multiplier when the output gap is zero; r-g reported at the 2018 debt level.

### Comparative dynamics, sensitivities, and policy implications
- Global solution:
  - Government chooses change in structural primary balance as Θ−1(d_{t−1}, gap_{t−1}, pb_{t−1}); global solution method used so results valid across debt levels.
- Qualitative simulation findings:
  - As debt increases, appropriate fiscal tightening is sharper (convex relationship); slope steeper at high debt.
  - At very high debt levels, substantial tightening recommended reflecting sizeable interest rate risk premia and calibrated low adjustment cost.
  - At low debt, fiscal policy should be used widely to offset shocks (wider band of allowable structural balance changes).
  - At high debt, reaction to shocks is weaker (narrower band); capacity to offset shocks reduced.
- Drivers of recommendations:
  - Interest-rate growth-rate differential, hysteresis, fiscal multipliers, preference parameters, and shock persistence materially affect recommendations.
  - Hysteresis: stronger countercyclical response at low debt; higher marginal cost of debt at high debt moderates stimulus.
  - Higher fiscal multipliers increase reaction to shocks; lower discounting reduces reaction to debt; higher shock persistence favors less stimulus for negative shocks at high debt.
- Uncertainty:
  - Output gap measurement uncertainty and multiplier uncertainty imply policymakers should react less aggressively (Brainard principle).

### Fiscal stance decomposition and the linear fiscal Taylor rule
- Decomposition of recommended Δpb into:
  - Stabilization component (response to output gap),
  - Sustainability component (debt and lagged primary balance relative to long-run d* and pb*),
  - Short-run considerations (e.g., temporarily lower interest environment).
- Computation method:
  - Shut off inputs sequentially to isolate effects (output gap shut off first, then replace debt and lagged pb with d* and pb*).
- Fiscal Taylor rule specification:
  - Linear form with coefficients β0–β4 where β1 reaction to past pb (expected negative), β2 reaction to debt (positive), β3 reaction to output gap (positive), β4 interaction debt*gap (expected negative).
  - Coefficients obtained by OLS regression of global solution on state variables over debt range 30%–150% of GDP; linear rule approximates global solution very well (R2 close to 1).
- Reported country-specific fiscal Taylor rule coefficients (Table 3 — reported exactly)
  - Canada: β0 -0.030, β1 -0.446, β2 0.083, β3 1.401, β4 -0.628, 2R 0.970
  - France: β0 -0.028, β1 -0.493, β2 0.055, β3 0.833, β4 -0.190, 2R 0.975
  - Germany: β0 -0.033, β1 -0.651, β2 0.078, β3 0.578, β4 -0.242, 2R 0.970
  - Italy: β0 -0.044, β1 -0.533, β2 0.111, β3 0.897, β4 -0.426, 2R 0.961
  - Japan: β0 -0.037, β1 -0.544, β2 0.016, β3 0.417, β4 0.062, 2R 0.998
  - Sweden: β0 -0.026, β1 -0.509, β2 0.050, β3 0.815, β4 -0.031, 2R 0.993
  - United Kingdom: β0 -0.026, β1 -0.413, β2 0.054, β3 1.056, β4 -0.185, 2R 0.987
  - United States: β0 -0.033, β1 -0.425, β2 0.030, β3 0.662, β4 0.040, 2R 0.997
- Country-specific notes:
  - Germany: lower shock persistence reduces output-gap coefficient.
  - Japan and United States: negative r-g at these debt levels → lower reaction to debt and near-zero interaction term (more fiscal space).

### Historical fit and empirical findings
- Empirical test (sample 1996–2016 using July 2019 WEO data):
  - Regressed realized stance (change in structural primary balance observed ex-post) on recommended stance (toolkit prescription using lagged state variables); two regressions per country (without and with lagged structural primary balance) using Prais-Winsten with AR(1) residuals.
- Key results:
  - About half of advanced economies have fiscal stance decisions broadly consistent with model prescriptions.
  - Significant positive association between recommended and actual decisions: Canada, Germany, Sweden, United Kingdom.
  - In other countries association weak or nonexistent.
  - Stance adjustments are smaller than model recommendation (coefficients below one).
  - In all G7 countries, observed primary balance is below model recommendation on average (consistent with political economy debt bias).
- Robustness:
  - Using real-time output gap estimates yields similar results, indicating decision-process weaknesses beyond real-time measurement issues.

### Forward-looking simulation approach and Monte Carlo analysis
- Setup:
  - Countries assumed to follow infinite-horizon appropriate fiscal stance in long run.
  - WEO forecast period of five years followed by a 10-year linear transition to long-run parameters; short-run finite-horizon problem solved by backward iteration.
- Stochastic simulations:
  - Simulations run from 2020 to 2024 (WEO forecast horizon), taking 2019 as last observed year.
  - 5,000 Monte-Carlo simulations drawing shocks from estimated distribution.
  - Output gap driven by shocks and can be partially countered by appropriate stance.
  - Distributions of recommended stance and associated debt/output gap trajectories provide medium-run uncertainty assessment.
- Benchmark comparison:
  - Fiscal Taylor rule outcomes compared to Carnot (2014) rule-of-thumb where change in primary balance equals average of a primary gap P_t and macro score S_t.

### Case studies and policy implications
- Germany (summary):
  - Fiscal Taylor rule recommends an expansionary fiscal stance for 2020–2022.
  - Decomposition for 2020: lagged primary balance well above sustainability requirement and very low interest rates → supports easing; cyclical assessment alone would warrant small tightening but is outweighed by easing drivers.
  - Lower shock persistence increases implementation-lag risk → discount positive output gap signal from 2019.
  - Policy implication: near-term easing recommended; stabilize fiscal stance over medium run as interest rates normalize.
  - Quantitative note: tax and social contribution to GDP ratio about 1.5 points above level observed five years earlier.
  - Comparison: recommended easing larger than government plan (WEO) and close to Carnot (2014) recommendation; optimal debt path still downward given very low interest rates.
- France (summary):
  - Fiscal Taylor rule recommends frontloaded consolidation over the medium run for 2020.
  - Drivers: higher debt-to-GDP and lower structural primary balance than Germany → sustainability considerations dominate.
  - Recommended tightening broadly in line with IMF and Carnot (2014); tightening moderate due to adjustment costs.
  - Differences from Germany: higher debt implies less reaction to shocks; greater shock persistence implies higher sensitivity to lagged output gap information.

### France — sensitivity, severe recession, and reform scenarios (Section 4 highlights)
- Robustness of 2020 tightening recommendation:
  - Tested parameter variations including interest rate ±0.5%, automatic stabilizers 3/4 of baseline, shock persistence set to 0.4 (from 0.6), faster interest rate normalization (5 years after 2024), potential growth lower by 0.5%, hysteresis 50% larger, debt limit at 170% of GDP, debt limit uncertainty d1 = 4.
  - Key robustness findings:
    - Low sensitivity to elasticity of interest rates to debt due to offsetting effects.
    - Results not very sensitive to market-access risk parameters, shock persistence, automatic stabilizers, or extent of hysteresis.
    - Results sensitive to average interest rate, potential growth, and fiscal multipliers (higher rates → tighter consolidation; permanently higher growth → easier stance; higher multipliers → larger recommended consolidation).
- Severe recession simulation for France:
  - One-year shock triggers annual GDP decline by 1% in France.
  - Process vt subject to one-off exogenous shock of about 4% of GDP in 2019; dampened by automatic stabilizers so growth is 2.3% below baseline; about 40% of shock dissipates each year.
  - Government reacts in 2020 due to implementation delays.
  - Policy implication: discretionary stimulus around ¼–1 percent of GDP could be considered in this severe recession scenario to reduce short-term and long-term (hysteresis) costs.
  - Stimulus size depends on fiscal multiplier, hysteresis, and financing cost; shock and stimulus have permanent effects on debt requiring future consolidation.
- Growth-enhancing reforms scenario:
  - Illustrative permanent increase in potential growth of 0.25% each year considered.
  - Findings: permanent higher potential growth allows less tightening but gain is moderate; model still recommends sizeable tightening even under permanent higher growth.
  - Emphasis: fiscal gain hinges on permanence of growth effect; temporary growth boosts deliver much less benefit for fiscal stance.
- Toolkit functionality reiterated:
  - Provide guidance on appropriate fiscal stance, compare to historical stances, recommend short- and medium-run stances, decompose recommendations, simulate shocks, and gauge effect of growth-enhancing policies.
  - Customizable parameters: interest-rate environment, fiscal multiplier, automatic stabilizers, hysteresis, shock size and persistence.

### Appendix — output gap and multiplier formulae (model specification)
- Multipliers cycle-dependent following Auerbach and Gorodnichenko (2013): ∂gap/∂pb = −m1 (1 + m2 (−gap,·)).
- Integrated and combined expressions provided in source (including expressions involving the Lambert W function) for the output gap and stabilizers.

*Source: WP/20/33 — A Model-based Fiscal Taylor Rule and a Toolkit to Assess the Fiscal Stance (wpiea2020033-print-pdf).*

### Section 1

### WP/20/33 — A Model-based Fiscal Taylor Rule and a Toolkit to Assess the Fiscal Stance (Section 1)

### Abstract and Purpose
- Presents a model-based fiscal Taylor rule and a toolkit to assess the fiscal stance, defined as the change in the structural primary balance.
- Built on the normative buffer-stock model of the government (Fournier, 2019) which includes key channels: hysteresis, cycle-dependent multipliers, and a risk premium.
- Proposes a simple fiscal Taylor rule prescribing the fiscal stance as a function of past government debt, past output gap and the past structural primary balance.
- Applications suggest several advanced economies could have better managed their fiscal stance over the last 20 years.
- Simulations provide fiscal stance recommendations over the medium-term.
- JEL Classification Numbers: E32, E62, H62, H63.
- Paper identification: WP/20/33; Prepared by Jean-Marc Fournier and Philipp Lieberknecht; Authorized for distribution by Catherine Pattillo; February 2019.

### Definition and Conceptual Framework
- Fiscal stance is defined as the change in the structural primary balance, encompassing all discretionary decisions.
- Toolkit objective: inform policy discussions with model-based analysis built on a consistent set of considerations to bring clarity and discipline to fiscal-stance advice.
- Government objective in model: maximize intertemporal welfare by choosing the change in structural primary balance to stabilize output fluctuations, subject to economic structure and a government budget constraint with a debt-dependent risk of losing market access.
- The model is a fiscal variant of the buffer-stock model of the consumer (Carroll, 1992/1997; Deaton 1991).

### Key Model Features Incorporated
- Hysteresis: the model includes long-run negative effects of recessions on output, consistent with arguments by Delong and Summers (2012) and Krugman (2015).
- Cycle-dependent multipliers: fiscal multipliers are magnified in recessions, consistent with Auerbach and Gorodnichenko (2013).
- Implementation lags: government decides fiscal plans before the year starts (Blanchard and Perotti, 2002).
- Adjustment costs: reflect difficulty implementing large fiscal stimulus and reversing it (IMF, 2017).
- Buffer-stock concern: governments may have insufficient buffers in bad times.
- Risk premium: rising in debt, raising the marginal cost of increasing debt (Gruber and Kamin, 2012; Poghosyan, 2012; D’Agostino and Ehrmann, 2014; Fall and Fournier, 2015; Henao-Arbelaez and Sobrinho, 2017).
- Utility specification: representative-agent CRRA utility in consumption and labor; utility peaks at equilibrium output equal to potential output (parameters such as ρ and ξ described in the model).

### Main Theoretical Insight
- Fiscal stance should be less counter-cyclical at high debt levels than at low debt levels:
  - At low debt, government can increase debt at moderate cost (acting like a buffer) and should use fiscal stance to smooth the cycle.
  - At high debt, the marginal cost of debt is higher (via higher interest rates and market-access risk); government should moderate counter-cyclical responses to preserve the small buffer.
  - Hysteresis interacts with debt: at low debt, stronger hysteresis implies larger counter-cyclical reactions; at high debt, hysteresis magnifies the cost of debt crises, reinforcing the case for more moderate reactions.

### Fiscal Taylor Rule and Toolkit Design
- The fiscal Taylor rule: a linear approximation in past primary balance, debt, and the output gap augmented with an interaction term between debt and the output gap.
- Toolkit components:
  - A nonlinear fiscal stance function taking as arguments: output gap, public debt, and last primary balance; returns the recommended fiscal stance for the following year.
  - Calibration includes common and country-specific parameters; users can adjust parameters (e.g., fiscal multiplier) to reflect additional country knowledge.
  - The nonlinear function is simplified into a fiscal Taylor rule for practical use.
  - Historical data can be supplied to compare past actual fiscal stance with model prescriptions.
  - Country forecast (output gap, debt, initial primary balance) feeds the fiscal stance function for forward-looking advice.
  - Decomposition of recommendations shows contributions from output-gap considerations versus debt/primary-balance considerations.
  - Scenario analysis capability (e.g., recession scenarios, reforms that boost potential growth).
- Practicalities:
  - The toolkit consists of a Matlab code with documentation available upon request.
  - The toolkit is intended as a starting point for discussion and can be used alongside considerations not explicitly modeled (uncertainty in output gap, fiscal instrument composition, interaction with monetary policy, fiscal rules, inequality, aging, political constraints).

### Applications and Uses Highlighted
- Historical analysis: comparison of past decisions with model recommendations indicates only a few G7 economies had an appropriate fiscal stance on average over the last 20 years.
- Case studies (summarized in later sections): Germany and France illustrate different prescriptions driven by different initial debt levels and primary balances.
- Policy experiments: variants with higher potential growth (public investment, innovation) and temporary growth boosts illustrate how reform packages affect appropriate fiscal stance.
- Users can test sensitivity to parameters and check consistency with national fiscal rules; the model can seek optimal adjustment paths that respect implementation constraints.

*Source: WP/20/33 — A Model-based Fiscal Taylor Rule and a Toolkit to Assess the Fiscal Stance, Section 1 (Jean‑Marc Fournier and Philipp Lieberknecht).*

### Section 2

### wpiea2020033-print-pdf - Section 2

### Model structure and fiscal dynamics
- Government budget constraint: debt accumulation with a deterministic stock-flow adjustment sft and interest rate r:
  - 1/(1+r) Δdt = dt+1* − dt = pb t + g t + sf t (equation structure described).
- Risk premium and market-access channels:
  - Risk premium increases in public debt and in the change in debt (empirical references cited).
  - Probability of losing market access is a logistic-type function of debt level and debt change:
    - Plmadd = 1 − exp[−d1 (1 − e(1/d2)(d−d)/d? ) ] (functional form described in text; parameters defined below).
  - If market access is lost, government must keep debt constant under an adverse shock of d3 σ, where σ is the standard deviation of economic shocks and d3 is calibrated once for all countries.
- Solution approach:
  - Baseline uses a risk premium linear in past debt and debt change so the implicit new debt level can be solved directly.
  - Non-linear risk premium alternatives can be linearized for solution.

### Production, potential output, and hysteresis
- Output production: Yt = At Lt (standard linear production in labor).
- Potential output Yt* corresponds to equilibrium labor L̄: Yt* = At L̄.
- Productivity and hysteresis:
  - Productivity is affected by a permanent hysteresis effect when production is substantially below potential (references: Blanchard and Summers, 1987; DeLong and Summers, 2012).
  - Hysteresis modeled as a permanent loss of potential output if the output gap is below a threshold h_th:
    - At = A_{t−1} * (1 + g* ) ^(1−h) ... (hysteresis functional representation described).
  - Parameters: h governs size of hysteresis; h_th is the output gap threshold below which hysteresis affects potential output.
  - Calibrated to reflect long-run effects in line with literature (long-term effects around 0 – 20%).

### Output gap, fiscal multipliers, and primary balance
- Output gap dynamics driven by shocks vt and the structural primary balance pb_st:
  - The fiscal multiplier depends on the output gap: marginal effect of pb on gap is m1 at equilibrium, amplified in downturns by m2.
  - Local representation: ∂gap/∂pb = −m1 (1 + m2 (−gap,·)) (functional description).
- Primary balance decomposition:
  - pb_t = pb_t^s + a * gap_t, where a is automatic stabilizer coefficient and pb^s is structural primary balance.
- Approximate equilibrium relation for small shocks:
  - gap_t ≈ (v_t + m pb_t^s) / (1 + m a)  (approximation showing automatic stabilizers reduce effects of shocks).
  - Structural balance that offsets shocks in approximation: pb^s = v_t / m1.
- Interpretation of m1:
  - m1 captures the causal effect of the primary balance on the output gap. Other formulations (e.g., Batini et al. 2014) map to m1/(1+m1·a).

### Aggregate resource constraint and adjustment costs
- Aggregate resource constraint:
  - c_t + pb_t + y_t χ = −Δb_t (resource accounting with fiscal adjustment costs).
  - Adjustment costs χ represent direct resource costs relative to output (implementation costs, tax uncertainty, difficulty reversing fiscal decisions).
- Role in recommendations:
  - Adjustment costs moderate how quickly governments adjust structural primary balances; χ = 3 is calibrated as a moderate value.

### Calibration: common parameters and key numeric values (Table 1)
- Welfare function parameters:
  - Discount factor β = 0.99
  - Risk aversion σ = 2
  - Labor elasticity η = 1/0.3
  - Weight of labor ξ = 1
- Fiscal parameters:
  - Fiscal multiplier sensitivity to shocks m2 = 3
  - Adjustment cost χ = 3
- Interest rate and debt parameters:
  - Effect of debt level on the risk premium α1 = 1.5%
  - Effect of debt change on the risk premium α2 = 0.5%
  - Debt level at which the risk to lose market access is 50% d = 150%
  - Debt limit accuracy d1 = 3
  - Effect of debt change on the risk to lose market access d2 = 1
  - Adverse scenario coefficient in case of loss of market access d3 = 0
- Economy parameters:
  - Hysteresis = 10%
  - Hysteresis threshold = -1%
- Additional calibration notes:
  - Discount factor and long-term per capita growth imply overall discount rate 0.975 with growth rate at 1.5%.
  - α1 = 1.5% chosen as midpoint of empirical estimates in cited literature.

### Country-specific calibration (Table 2): fiscal multipliers, stabilizers, growth, r−g, and shocks
- Country-specific parameter set (columns: Country; m1; Automatic stabilizers; potential growth per capita; r-g; Shock persistence ρ; Shock size εσ)
  - Canada: m1 = 0.70; Automatic stabilizers = 0.56; potential growth per capita = 0.36%; r-g = 1.77%; Shock persistence ρ = 0.72; Shock size εσ = 0.021
  - France: m1 = 0.98; Automatic stabilizers = 0.61; potential growth per capita = 1.08%; r-g = 1.05%; Shock persistence ρ = 0.58; Shock size εσ = 0.024
  - Germany: m1 = 0.84; Automatic stabilizers = 0.48; potential growth per capita = 1.21%; r-g = 1.28%; Shock persistence ρ = 0.32; Shock size εσ = 0.025
  - Italy: m1 = 1.00; Automatic stabilizers = 0.5; potential growth per capita = 0.42%; r-g = 2.47%; Shock persistence ρ = 0.58; Shock size εσ = 0.022
  - Japan: m1 = 0.87; Automatic stabilizers = 0.41; potential growth per capita = 0.90%; r-g = 0.63%; Shock persistence ρ = 0.47; Shock size εσ = 0.040
  - Sweden: m1 = 0.62; Automatic stabilizers = 0.66; potential growth per capita = 0.95%; r-g = 0.16%; Shock persistence ρ = 0.53; Shock size εσ = 0.027
  - United Kingdom: m1 = 0.72; Automatic stabilizers = 0.59; potential growth per capita = 0.91%; r-g = 1.00%; Shock persistence ρ = 0.70; Shock size εσ = 0.028
  - United States: m1 = 0.86; Automatic stabilizers = 0.5; potential growth per capita = 1.31%; r-g = -0.29%; Shock persistence ρ = 0.67; Shock size εσ = 0.044
- Note: m1 is the fiscal multiplier when the output gap is zero. The interest-growth-rate differential r-g is reported at the 2018 debt level.

### Comparative dynamics, sensitivities, and policy implications
- Fiscal reaction function and global solution:
  - Government chooses change in structural primary balance as a function Θ−1(d_{t−1}, gap_{t−1}, pb_{t−1}).
  - Global solution method used so results are valid for countries with high or low debt.
- Reaction to debt and shocks (qualitative findings from baseline simulations):
  - As debt increases, the appropriate fiscal tightening is sharper (convex relationship); slope steeper at high debt.
  - At very high debt levels, substantial tightening is recommended reflecting sizeable interest rate risk premia and calibrated low adjustment cost.
  - At low debt levels, governments should use fiscal policy widely to offset shocks (wider band of allowable structural balance changes).
  - At high debt levels, reaction to shocks is weaker (narrower band); capacity to offset shocks is reduced.
- Drivers of recommended fiscal stance:
  - Interest-rate growth-rate differential, hysteresis, fiscal multipliers, preference parameters, and shock persistence materially affect recommendations.
  - Hysteresis implies stronger countercyclical response at low debt but implies higher marginal cost of debt at high debt (moderating countercyclical stimulus when indebtedness is high).
  - Higher fiscal multipliers increase reaction to shocks; higher present-bias (lower discounting) reduces reaction to debt; higher shock persistence favors less stimulus for negative shocks at high debt.
- Uncertainty and use of output gap:
  - Output gap measurement uncertainty discussed; if uncertainty is unbiased, Brainard principle applies and uncertainty on policy effects affects decisions.

*Source: wpiea2020033-print-pdf - Section 2*

### Section 3

### Section 3 — Fiscal Stance Guidance, Decomposition, and Fiscal Taylor Rule

### Output gap uncertainty and implications for fiscal policy
- Output gap uncertainty affects the fiscal multiplier; governments should do less consolidation during observed booms and less stimulus in observed downturns than a face-value reading of the output gap would suggest.
- A government concerned about a systematic optimistic bias would increase the balance relative to a face-value reading of the output gap not only during downturns but also during booms; the tool can use an output gap adjusted for the historical bias or compare recommendations for different estimates of the output gap.
- The output gap remains useful as it captures a mismatch between short-term demand and a long-term supply level.
- Recent research to improve real-time measures of the output gap suggests using structural methods (Coibion et al 2018) or including information on capacity utilization rates (Turner, 2016).
- The Brainard principle should also lead policy makers to react less to shocks if uncertainty surrounding the fiscal multiplier is large.

### A simple decomposition to describe fiscal stance advice
- The recommended change in the primary balance (Δpb) is decomposed into contributions from:
  - Stabilization considerations (response to the output gap),
  - Sustainability considerations (debt and lagged primary balance relative to long-run equilibrium values d* and pb*),
  - Short-run considerations (e.g., temporarily lower interest environment).
- Method to compute contributions:
  - Shut off inputs one by one to isolate effects.
  - First, shut off the output gap input to isolate the stabilization effect (done at last year’s debt level).
  - Second, isolate sustainability by replacing debt and lagged primary balance with their long-run equilibrium stable values d* and pb*.
  - Third, remaining deviation from a neutral stance at equilibrium captures short-run factors (e.g., temporarily lower interest rates).
- The sustainability component can be negative if sustainability gains from high lagged primary balance are dominated by welfare costs (high taxation or low public spending).
- The decomposition embeds a welfare consideration: high lagged primary balance can impose welfare costs despite improving sustainability.

### The Fiscal Taylor Rule (linear approximation)
- The fiscal Taylor rule approximates the global model solution as a linear function of state variables augmented with an interaction between debt and the output gap:
  - The paper represents this as a linear rule with coefficients β0–β4 where:
    - β1 governs reaction to past primary structural balance (expected negative),
    - β2 reaction to debt (positive, satisfying the Bohn (1998) principle),
    - β3 reaction to the output gap (positive),
    - β4 interaction term (debt*gap) expected negative since counter-cyclicality decreases at high debt levels.
- The parameter values β0–β4 are obtained by OLS regression of the global solution on the state variables; the fiscal Taylor rule is estimated over a common debt range: 30% of GDP to 150% of GDP.
- The linear fiscal Taylor rule is a very good approximation of the global solution (the R2 is close to 1 in almost all countries).
- Country-specific explanations in the model:
  - Germany: lower shock persistence reduces the output-gap coefficient.
  - Japan and the United States: negative interest-growth-rate differentials at these debt levels lead to a lower reaction to debt and a near-zero interaction term (more fiscal space).

- Table 3: Fiscal Taylor Rules in Advanced Economies (coefficients reported exactly)
  - Canada: β0 -0.030, β1 -0.446, β2 0.083, β3 1.401, β4 -0.628, 2R 0.970
  - France: β0 -0.028, β1 -0.493, β2 0.055, β3 0.833, β4 -0.190, 2R 0.975
  - Germany: β0 -0.033, β1 -0.651, β2 0.078, β3 0.578, β4 -0.242, 2R 0.970
  - Italy: β0 -0.044, β1 -0.533, β2 0.111, β3 0.897, β4 -0.426, 2R 0.961
  - Japan: β0 -0.037, β1 -0.544, β2 0.016, β3 0.417, β4 0.062, 2R 0.998
  - Sweden: β0 -0.026, β1 -0.509, β2 0.050, β3 0.815, β4 -0.031, 2R 0.993
  - United Kingdom: β0 -0.026, β1 -0.413, β2 0.054, β3 1.056, β4 -0.185, 2R 0.987
  - United States: β0 -0.033, β1 -0.425, β2 0.030, β3 0.662, β4 0.040, 2R 0.997

### Historical fit of the model-based recommended fiscal stance
- Empirical test:
  - Realized stance = change in structural primary balance observed ex-post in the July 2019 WEO database (sample: 1996–2016).
  - Recommended stance = stance prescribed by the toolkit using lagged state variables reported in July 2019 WEO.
  - Tested via regression of observed stance on recommended stance; two regressions per country: without and with lagged structural primary balance (Prais-Winsten estimation with AR(1) residuals).
- Key findings:
  - Results suggest that half of advanced economies considered have fiscal stance decisions broadly consistent with the model prescription.
  - Countries with a significant positive association between recommended and actual decisions: Canada, Germany, Sweden, and the United Kingdom.
  - In other countries, association between appropriate stance and past decisions is weak or nonexistent.
  - Stance adjustments are smaller than the model recommendation (coefficients below one).
  - In all G7 countries, the observed primary balance is below the model recommendation on average, consistent with political economy considerations leading to a debt bias.

- Table 4: Historical Fit (note format and summary statistics are reported in the source; regressions include t-ratios and sample information)

- Robustness with real-time output gap estimates:
  - Repeating the exercise with real time output gap data (Table 5) provides a similar picture.
  - This confirms that weaknesses in the decision-making process play a large role, beyond difficulties in observing the output gap in real time.

### Forward-looking simulation and scenarios
- Simulation setup:
  - For each country, assume the country will follow the appropriate fiscal stance of the infinite horizon problem in the long run.
  - Within the WEO forecast period of five years followed by a transition period of ten years, the government solves a short-run finite horizon problem given this long-run trajectory and WEO projections of interest rates.
  - The transition period is a 10-year linear transition between the last year of WEO forecast and the long-run parameters.
  - Short-run solution obtained by finite horizon backward iteration.
- Stochastic Monte Carlo analysis:
  - Simulations are run from 2020 to 2024 (WEO forecast horizon), taking 2019 as the last observed year.
  - 5,000 Monte-Carlo simulations are run, drawing shocks from the estimated distribution.
  - The output gap is driven by these shocks and can be countered to some extent by the appropriate stance.
  - The distribution of appropriate fiscal stance and associated debt and output gap trajectories deliver a range of possible scenarios and uncertainties over the medium run.
- Comparative benchmark:
  - The paper compares outcomes from the fiscal Taylor rule to recommendations following Carnot (2014), who advocates a rule of thumb where the change in primary balance equals an average of a primary gap Pt (distance to debt target) and St (a score on macroeconomic conditions).

### Case study: Germany
- Fiscal Taylor rule recommendation:
  - Recommends an expansionary fiscal stance for Germany for 2020 to 2022.
- Decomposition for 2020:
  - Primary balance is well above the level required to ensure sustainability and interest rates are very low; reducing the primary surplus can improve welfare.
  - The primary balance and debt contribution is negative (i.e., supports easing).
  - The cyclical assessment alone would warrant a small tightening since the output gap is positive; however, this cyclical tightening is small and does not offset the easing drivers.
  - Lower shock persistence in Germany increases the implementation-lag risk; the government should discount the signal from a positive output gap in 2019.
- Policy implication:
  - The dominant factors (lagged structural primary balance level, moderate debt level, low interest rate) point toward easing the fiscal stance over the near term.
  - This recommendation is largely in line with IMF recommendations and the government plan for 2020.
  - Over the medium run, the government should stabilize the fiscal stance as interest rate conditions normalize.
- Quantitative note:
  - The paper notes the tax and social contribution to GDP ratio is about 1.5 points above the level observed five years ago.
- Comparison to other frameworks:
  - The recommended easing over three years is more pronounced than the government planned (WEO forecast) and quite close to the Carnot (2014) recommendation.
  - Even with recommended easing, the optimal debt path remains on a downward path given very low interest rates.

### Case study: France
- Fiscal Taylor rule recommendation:
  - Proposes a frontloaded consolidation in France over the medium run.
- Drivers:
  - Debt-to-GDP is higher than in Germany and structural primary balance is lower; sustainability considerations are the main driver of 2020 advice.
  - The fiscal Taylor rule therefore recommends a tighter fiscal stance to build buffers for future crises.
- Comparison:
  - The recommended consolidation is broadly in line with IMF and Carnot (2014) advice, recommending more frontloading though tightening is moderate because of assumed adjustment costs.
  - The 90% confidence interval surrounding the recommended stance is not far from the one observed for Germany.
- Offset mechanisms between France and Germany:
  - Higher debt in France implies France should react less to shocks.
  - Greater shock persistence in France implies higher sensitivity to lagged output gap information, increasing sensitivity to shocks.

*Source: wpiea2020033-print-pdf - Section 3*

### Section 4

### Section 4 — Fiscal Stance Analysis for France

### Sensitivity Analysis of Fiscal Tightening Recommendation
- The recommendation to tighten the structural primary balance in France in 2020 is robust across a broad range of assumptions (Figure 7).
- Model sensitivities tested and parameter adjustments described in the source:
  - Interest rate scenarios: interest rate is 0.5% lower (lower scenario) or 0.5% higher (higher scenario).
  - Automatic stabilizers: in the lower automatic stabilizers’ scenario, automatic stabilizers are three fourth of the baseline value.
  - Shock persistence: in the lower shock persistence, the persistence parameter is set to 0.4 instead of 0.6.
  - Interest rate normalization: in the faster interest rate normalization, it takes five years after 2024 for the effective interest rate to reach its long-run value.
  - Potential growth: in the lower potential growth scenario, potential growth is 0.5% lower.
  - Hysteresis: in the higher hysteresis scenario, hysteresis is 50 percent larger than in the baseline.
  - Debt limit: in the higher debt limit scenario, the debt level at which the risk to lose market access is 50 percent is set at 170 percent of GDP.
  - Debt limit uncertainty: in the lower debt limit uncertainty scenario, d1 = 4.
- Key robustness findings:
  - Low sensitivity to the elasticity of interest rates to the debt level due to offsetting effects: (i) higher elasticity raises the marginal cost of debt (leading to targeting lower debt); (ii) it lessens the surplus needed to reduce debt because interest burden drops faster when debt declines.
  - Results not very sensitive to parameters governing market-access risk, persistence of output growth shocks, magnitude of automatic stabilizers, or extent of hysteresis.
  - Results sensitive to assumptions on:
    - Average interest rate (higher interest rates → tighter consolidation).
    - Potential growth (permanently higher growth → easier stance; permanently lower growth → tighter stance).
    - Fiscal multipliers (higher multipliers → recommended consolidation larger because of larger output costs and greater debt aversion).

### Severe Recession Scenario and Discretionary Stimulus
- Shock specification and dynamics:
  - Simulated one-year shock triggers a decline in annual GDP by 1 percent in France.
  - The process vt is subject to a one-off exogenous shock of about 4 percent of GDP in 2019.
  - The shock is dampened by automatic stabilizers so that growth is 2.3 percent below the baseline.
  - About 40 percent of this shock dissipates each year (autoregressive term in vt).
  - The government reacts in 2020, reflecting implementation delays.
- Policy implication from the simulation:
  - A discretionary stimulus around ¼–1 percent of GDP could be considered in this severe recession scenario.
  - Rationale: such a stimulus can help reduce both short-term and long-term costs (hysteresis) of the recession.
  - Size of stimulus depends on:
    - Fiscal multiplier of the instruments used (higher multiplier → smaller required stimulus).
    - Hysteresis associated with the downturn.
    - Cost of financing.
  - Permanent effects and future costs:
    - The adverse shock and the stimulus have a permanent effect on debt.
    - There is a cost in terms of future consolidation as adjustment effort must be sustained longer after the shock dissipates.
  - Contrast: fiscal tightening is unambiguously desirable when debt is high and the economy is booming—reduces debt and avoids overheating.

### Scenario Analysis with Growth-Enhancing Reforms
- Purpose: gauge effect of growth-enhancing policies on appropriate fiscal path since potential growth is an input for fiscal stance calculations.
- Illustrative simulation employed:
  - Permanent increase in potential growth of 0.25% each year is considered to illustrate large implications on fiscal stance.
  - Note: alternative simulation also referenced where higher growth is temporary (effective over 5 years only) would deliver much less in terms of fiscal stance.
- Findings:
  - With permanently higher potential growth, France can afford less fiscal tightening, but the gain (smaller consolidation) is moderate (Figure 9).
  - Higher growth reduces future debt ratio but the model still recommends a sizeable tightening even under permanent higher growth.
  - The fiscal gain crucially hinges on the assumption that growth effect is permanent—permanent change alters debt dynamics permanently and permits a higher debt target.
  - Practical cautions:
    - Implementing large investment plans or ambitious reform packages is difficult and can take time.
    - Growth effects of reforms are delicate to quantify; policymakers may consider prudent ex ante quantification.
    - Many structural reforms affect output levels rather than the pace of technology gains; permanence of growth gains matters for fiscal stance implications.

### Toolkit Functionality and Policy Uses
- The toolkit is a fiscal Taylor rule based on a buffer-stock model featuring an optimizing government and channels linking the economy and fiscal stance.
- Uses and applications:
  - Provide simple guidance on appropriate fiscal stance.
  - Compare model-implied fiscal stance to historical fiscal stances to assess past behavior.
  - Recommend appropriate fiscal stance in the short- and medium-run using current data and forecasts.
  - Decompose recommendations into debt-related versus cyclical considerations.
  - Simulate alternative scenarios with output shocks to determine fiscal responsiveness.
  - Gauge effect of growth-enhancing policies by specifying potential output growth assumptions.
- Customizability:
  - Users can adjust parameters for low interest rate environment, fiscal multiplier, automatic stabilizers, hysteresis, shock size, and persistence.
  - Scenarios can compare mechanisms driving prescriptions and tailor fiscal advice.
  - Example: if spending consolidation plans are associated with lower fiscal multipliers (Alesina et al, 2015), users can assume lower multipliers when calculating optimal fiscal stance.

### Appendix — Output Gap Formula (model specification)
- Multipliers allowed to be cycle-dependent following Auerbach and Gorodnichenko (2013):
  - 12
    (1) tt t gap mm gap pb ∂ = − − ∂
  - where m2 governs sensitivity of the multiplier to the output gap.
- Integrated form:
  - 11 2212 ()( ) ttt gap m v m exp m m pb − − = + −
- Fiscal stabilizers specification:
  - st tt t pb p b a gap = +
- Combined expression for the output gap (with Lambert function W):
  - 1 121 2 2 12 ((1 )(()) st tt t W a mm v exp m a m pb gap m am m − − + = − +

*Source: wpiea2020033-print-pdf - Section 4*

---


_Source: https://www.imf.org/-/media/files/publications/wp/2020/english/wpiea2020033-print-pdf.pdf_