## _wp1552

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---

### Introduction — objectives and context
- Purpose: use a simple model and numeric simulations to examine conditions under which delaying fiscal consolidation increases the present value (PV) of GDP.
- Builds on DeLong and Summers (2012) but:
  - Considers an initial stimulus followed by fiscal consolidation that returns the debt-to-GDP ratio to its baseline path.
  - Is explicit about how multipliers vary over time and the persistence of the output gap.
- Key illustrative assumptions:
  - Interest rate equals the time-preference discount rate in baseline analysis.
  - Focus on case where initial output gap is negative.
- High-level findings:
  - If the multiplier is constant over time and there is no hysteresis, delaying consolidation has no effect on the PV of GDP.
  - Under a constant multiplier with hysteresis, delaying consolidation increases the PV of GDP, but the magnitude is quite small.
  - If the multiplier is larger today than during the future consolidation phase, gains from delaying consolidation can be considerable, especially with hysteresis and slow natural closing of the output gap.
  - Exploitable time-variation in multipliers depends on how multipliers vary (e.g., with slowly adjusting variables like the output gap versus faster variables like the growth rate).
- Caveats explicitly noted:
  - Model omits effects on external balance; potential high-return public investments; risks from higher sovereign yields; political-economy risk that stimulus may not be reversed; stochastic risks that shocks could push the economy back into recession while fiscal space is reduced.
  - Framework clarifies criteria rather than advocates a specific fiscal path.

### Model structure and assumptions
- Timing and policy actions:
  - Stimulus at time 1, delayed consolidation at time 2; consolidation size chosen to return debt-to-GDP to level it would have had absent stimulus.
  - Discrete periods: time 1, time 2, time 3, time 4, and from time 4 onward.
- Core mechanisms and notation:
  - Fiscal multipliers: μ_t (μ_1, μ_2).
  - Hysteresis parameter: h (example: h = 0.1 implies an output gap of -5 percent for one period permanently reduces potential GDP by 0.5 percent).
  - Natural closing assumption initially: output gap fully closes from one period to the next (later relaxed).
- Baseline (no discretionary fiscal policy):
  - Time-1 actual GDP = y_1; potential GDP = p_1; output gap γ_1 = p_1 − y_1.
  - p_2 = p_1(1 + g) + h γ_1, with 1 + g underlying potential growth.
  - Under full natural closing: γ_2 = 0 and y_2 = p_2; from time 2 onwards actual GDP = potential GDP growing at rate g.
  - Debt dynamics baseline: d_t+1 = d_t + (1 − τ) y_t, with τ the tax rate and R = 1 + r.
- Alternative scenario (stimulus S at time 1, consolidation C at time 2):
  - Y_1 = y_1 + μ_1 S; Γ_1 = γ_1 + μ_1 S.
  - Stimulus raises potential GDP in time 2 by μ_1 S h.
  - Time-2 GDP after consolidation depends on μ_2 and (C + S); consolidation reduces debt stock but reduces tax revenues via lower GDP, widening the output gap and lowering potential output at time 3 via hysteresis.
  - Dynamics continue through time 3 and time 4, then return to potential growth at g.
- Present value of GDP:
  - Discount factor β with β R = 1; PV = Y_1 + β Y_2 + β^2 Y_3 + β^3 Y_4.

### Analytical condition for PV gains from delayed consolidation
- The net PV impact of stimulus and delayed consolidation is derived; key steps:
  - Impact of stimulus on PV summarized in equation (20); sign determined by inequality in equation (21).
  - Consolidation C chosen so debt-to-GDP at start of period 4 equals baseline (equation (22)).
  - Required consolidation size given explicitly (equation (24)); substituting ∂C/∂S into PV condition yields final inequality (equation (25)) determining when delayed consolidation raises PV of GDP.
- Comparative-static implications:
  - Constant multipliers, no hysteresis: PV-neutrality of timing.
  - Constant multipliers with hysteresis: delays can raise PV but gains typically small.
  - Time-varying multipliers (μ_1 > μ_2), with hysteresis and slow natural closing: delaying consolidation can generate sizable PV gains.
  - Exploitable advantage depends on whether multipliers link to slow-moving state variables (easier to exploit) or fast-moving variables (harder).

### Section IV — Constant fiscal multipliers: analytical cases and insights
- Case 1: No hysteresis and full natural closing
  - Delaying consolidation increases PV of GDP iff the discount rate is higher than the interest rate (condition in equation (26): ߚ/൏1ܴ).
  - With constant multipliers and no hysteresis, multiplier size is inconsequential for timing.
- Case 2: Hysteresis and full natural closing
  - Threshold condition modified (equation (27)); additional term typically positive so hysteresis relaxes the interest-rate threshold under which delay is beneficial.
  - Quantitative examples (parameter values in main text):
    - With fiscal multiplier = 0.8, tax rate = 1/3, growth rate = 2 percent, baseline debt-to-GDP = 0.8, and hysteresis parameter = 10 percent, the interest rate threshold exceeds the discount rate ߚ/1 by only 5 basis points.
    - Increasing the fiscal multiplier to 1.2 or hysteresis to 20 percent raises the threshold over the discount rate by only 10 basis points.
  - Conclusion: hysteresis has limited bearing on decision to delay under constant multipliers and full natural closing.
- Case 3: Slow natural closing of the output gap (no closing until time 4)
  - Assumption formalized in equation (28); output gap carries over for t ≤ 3, closing at time 4.
  - Initial stimulus has persistent impacts on potential output via hysteresis and on actual GDP; time-2 GDP includes an extra persistent term (equation (29)) that can offset contraction from withdrawal.
  - Modified condition is equation (30):
    - If h = 0, inequality (30) reduces to (26): no change when multipliers constant and no hysteresis.
    - With hysteresis and plausible parameters, no-closing assumption can imply a significantly higher interest-rate threshold: delaying consolidation may increase PV even if the interest rate exceeds the discount rate by several percentage points.
  - Comparison with DeLong and Summers:
    - DeLong and Summers allow permanently higher debt-to-GDP (pay growth-adjusted interest on new debt), yielding much higher interest-rate thresholds.
    - Requiring common ultimate debt-to-GDP as here reduces the threshold relative to DeLong and Summers.
  - Risk note: even if current interest rates are below model thresholds, delaying consolidation may not be optimal unless expected gains compensate unmodeled risks.

### Quantitative gains under slow output-gap closing (constant multipliers)
- Definitions:
  - Percentage increase in PV of GDP over first three periods: ߙَଵ,ଶ,ଷ (equation (31)).
  - Percentage increase at time 4: ߙَସ (equation (32)); time-4 differences are permanent once output gap closes.
- Simulation example (no natural closing until time 4):
  - Stimulus: 1 percent of baseline GDP.
  - Assumed initial negative output gap: 4 percent.
  - Parameter values (as in Figure 4):
    - μ = 0.8, τ = 1/3, g = 2 percent, d_svy/s = 0.8, and r̄ = 1 percent, where r̄ = 1/β − 1.
  - Results:
    - Delaying consolidation produces only small increases in GDP under constant multipliers.
    - With multipliers up to 1.2 or hysteresis up to 20 percent, a 1 percent of GDP stimulus raises PV of GDP in first three periods by only about 0.08 percent.
    - Impact on time-4 GDP is less than 0.02 percent.
  - Summary: with constant multipliers, PV gains from delay are modest even under substantial multipliers, substantial hysteresis, and slow closing.

### Time-varying multipliers: motivation and G7 simulation approach
- Motivation: stronger case for delay if μ_1 > μ_2 (e.g., severe recession, monetary policy constraint).
- Model extensions:
  - Multipliers depend on state of the economy.
  - Calibrate using multi-year delayed consolidation paths.
  - Allow partial natural closing of output gap each period.
- Approach: numeric simulations because closed-form solutions no longer feasible.

### G7 simulation setup (25-year horizon)
- Two fiscal scenarios compared through 2027:
  - Baseline:
    - Uses aggregate G7 projections through 2017 from the October 2012 WEO.
    - CAPB improves from -4¼ percent of potential GDP in 2011 to -½ percent of potential GDP by 2015.
    - After 2017, CAPB converges to zero through end of simulation (2027).
  - Alternative (delayed consolidation):
    - Stimulus in 2012 lowers the CAPB-to-potential-GDP ratio by 1 percent of GDP relative to 2011 and keeps it at this lower level in 2013 — cumulative stimulus of 4¼ percent of potential GDP over 2 years relative to baseline tightening.
    - Gradual consolidation begins in 2014 to bring debt-to-GDP and primary balance-to-GDP back to baseline by 2027.
- Mechanics:
  - Output gap deviations respond to fiscal impulse differences per multiplier assumptions.
  - Potential GDP deviates via hysteresis from output-gap differences.
  - Fiscal balances deviate with fiscal impulses and GDP growth (automatic stabilizers).
- Parameter values summarized in calibration.

### Calibration and baseline assumptions (numerical values)
- Potential GDP growth before hysteresis:
  - 2012-2017: WEO assumptions
  - 2018-2027: 2 percent
- Hysteresis effect (h): 0.10
- Derivative of the PB to potential output (߬1): 1/3
- Derivative of the PB to the output gap (߬2): 0.7
- Persistence of the output gap (ρ): 0.6
- Real interest rate (r):
  - 2012-2017: 1.1 percent on average
  - 2018-2027: 1 percent
- Intertemporal discount factor (β): 1/1.02
- Additional assumptions:
  - Hysteresis effects equal 10 percent of the output gap each period.
  - 40 percent of the output gap closed naturally each year (persistence 0.6).
  - Real interest rates follow 2012 WEO implicit rates until 2017 and 1 percent from 2018.
  - Intertemporal discount rate set to 2 percent.

### Alternative fiscal-multiplier specifications (three alternatives)
- Alternative 1: multiplier constant at 0.8.
- Alternative 2: multiplier varies inversely and linearly with the output gap.
- Alternative 3: multiplier varies inversely and linearly with real GDP growth (per Auerbach and Gorodnichenko (2012)).
- Calibration note: under Alternatives 2 and 3 the average multiplier during delayed consolidation equals 0.8 to isolate cycle-dependence effects.

### Simulation dynamics under delayed consolidation
- Three GDP phases with delayed consolidation:
  - Phase 1 (initial stimulus): loosening raises GDP growth and reduces output gap faster than baseline; especially pronounced under Alternatives 2 and 3.
  - Phase 2 (deeper consolidation): deeper and more prolonged consolidation relative to baseline, causing weak growth and renewed widening of output gap.
  - Phase 3 (reduction of surplus): fiscal surplus reduced once debt-to-GDP approaches baseline, generating stimulative impulse that raises growth and rapidly closes output gap; thereafter CAPB at baseline and output gap converges to zero at rate ρ^1.

### Key quantitative simulation findings (G7, through 2027)
- Alternative 1 (constant multiplier = 0.8):
  - Delaying consolidation has virtually no effect on the PV of GDP through 2027; output losses from consolidation offset stimulus gains.
  - Virtually no effect on potential GDP at end of simulations.
- Alternative 2 (multiplier varies with output gap):
  - PV of GDP through 2027 is 0.3 percent higher than baseline.
  - Potential output in 2027 is permanently 0.2 percent higher than baseline.
- Alternative 3 (multiplier varies with GDP growth):
  - Both PV of GDP and potential GDP in 2027 are lower under delayed consolidation than baseline.
  - Mechanism: stimulus increases subsequent consolidation magnitude and duration; lower growth during consolidation raises the multiplier (since multiplier varies negatively with growth), creating a downward spiral that more than offsets initial benefits.

### Intuition on multiplier variability and "second-derivative effects"
- If multipliers vary with growth (Alternative 3):
  - Stimulus has diminishing marginal benefits (multiplier shrinks as stimulus boosts growth).
  - Consolidation has increasing marginal costs (multiplier increases as consolidation reduces growth).
  - Rapid reaction of growth to policy amplifies adverse dynamics.
- If multipliers vary with output gap (Alternative 2):
  - Output gap is more sluggish than growth, so "second-derivative effects" are weaker; benefits of stimulus when output gap is negative can dominate.
- Empirical uncertainties: whether multipliers depend on output gap or growth, and their functional forms, remain open questions.

### Sensitivity to persistence and hysteresis
- Varying persistence (ρ):
  - If multipliers constant (Alternative 1): delaying consolidation negligible regardless of ρ.
  - If multipliers vary cyclically (Alternatives 2 and 3): higher persistence amplifies both gains (Alternative 2) and losses (Alternative 3).
- Varying hysteresis strength (h):
  - Stronger hysteresis magnifies gains and losses from asymmetric multipliers; little effect when multipliers constant.
- Note: linear hysteresis modeled; time-varying or asymmetric hysteresis could magnify gains from delaying consolidation into periods with positive output gaps but make achieving debt sustainability harder under large negative initial gaps and structural deficits.

### Liquidity traps and discontinuous multipliers (stylized experiment)
- Setup: multiplier normally 0.2, rising to 1.5 when output gap < -2 percent; other parameters as baseline.
  - "Temporary stimulus" scenario: a 1 percent of GDP temporary fiscal stimulus can push economy out of slump, shrinking multiplier to 0.2; resulting PV of GDP boost ≈ 2 percent of GDP.
  - "Slower consolidation" scenario: smaller stimulus insufficient to exit slump; multiplier remains high during consolidation, prolonging wide output gap/high multiplier state and lowering PV of GDP relative to baseline.
- Caveats:
  - Debate exists whether zero lower bound is binding; unconventional monetary policies could keep monetary policy effective, invalidating discontinuous-multiplier assumption.
  - A richer model capturing interest-rate dynamics could yield different outcomes; stimulus that raises full-employment rates might later require reversal that re-enters liquidity trap.

### Summary of simulation results
- Delaying consolidation yields no substantive gains if multipliers are constant, even with hysteresis.
- If multipliers vary over the business cycle and are higher during stimulus than consolidation, delaying consolidation can produce meaningful gains.
- Sign and magnitude of gains depend crucially on:
  - Whether multipliers vary with the output gap or with growth.
  - Persistence of output gaps (ρ).
  - Strength and functional form of hysteresis (h).
  - Other macroeconomic and fiscal conditions.

### Risks and considerations omitted or hard to quantify
- Crowding-out: temporary higher debt-to-GDP may absorb financial resources, reducing private investment; effect likely limited with negative output gap but may rise with recovery.
- Higher sovereign risk premia and external borrowing costs: temporary debt increases (roughly 5 percent of GDP in the scenario) could be misread as weak discipline, raising costs and harming debt dynamics; conversely, large asymmetric multipliers and hysteresis could reduce risk premia by raising PV of GDP.
- Political economy: delays risk never implementing required consolidation, leading to higher long-run deficits and debt.
- Loss of option value: postponing consolidation reduces fiscal space to respond to future negative shocks.
- Risk of negative shocks during delay: if shocks hit, countries may be unable to delay further; higher multipliers during shocks can amplify contractionary effects of consolidation.
- Additional policy factors beyond PV of GDP: effects on external balance; direct effects of public investment on potential GDP.

*Source: _wp1552 (Appendix A and Section IV) — provided PDF excerpt.*

### Appendix A .............................................................................................................

### Appendix A

### Introduction — objectives and context
- Purpose: use a simple model and numeric simulations to examine conditions under which delaying fiscal consolidation increases the present value (PV) of GDP.
- Builds on DeLong and Summers (2012) which posits substantial fiscal multipliers and hysteresis in advanced economies; differs by:
  - Considering an initial stimulus followed by fiscal consolidation that returns the debt-to-GDP ratio to its baseline path (rather than permanently higher debt).
  - Being explicit about how multipliers vary over time and the persistence of the output gap.
- Key illustrative assumptions discussed:
  - Interest rate equals the time-preference discount rate in baseline analysis.
  - Focus on case where initial output gap is negative (consistent with many advanced economies).
- Summary of principal findings (high-level):
  - If the multiplier is constant over time and there is no hysteresis, delaying consolidation has no effect on the PV of GDP.
  - Under a constant multiplier with hysteresis, delaying consolidation increases the PV of GDP, but the magnitude is quite small.
  - If the multiplier is larger today than during the future consolidation phase, gains from delaying consolidation can be considerable, especially with hysteresis and slow natural closing of the output gap.
  - Exploitable time-variation in multipliers depends on how multipliers vary (e.g., with slowly adjusting variables like the output gap versus faster variables like the growth rate).
- Caveats and omitted considerations (explicitly stated):
  - The model does not incorporate all policy-relevant factors: effects on external balance; that certain public investments may have high returns; risks from higher sovereign yields from higher debt; political-economy risks that stimulus may not be reversed; stochastic risks that shocks could push the economy back into recession while fiscal space is reduced.
  - The paper is not an advocacy for any specific fiscal path; it aims to clarify criteria for assessing how fiscal paths affect PV of GDP.

### Model structure and assumptions
- Time framework and policy timing:
  - Government implements fiscal stimulus at time 1 and delays consolidation to time 2; consolidation size brings debt-to-GDP back to the level it would have had absent the stimulus.
  - Model tracks actual GDP, potential GDP, output gap, and public debt across discrete time periods (time 1, time 2, time 3, time 4, and from time 4 onward).
- Core mechanisms:
  - Fiscal multipliers determine the impact of fiscal policy on GDP. The multiplier at time t is denoted μ_t (e.g., μ_1, μ_2).
  - Hysteresis: output gap has permanent effects on potential GDP. Parameter h captures hysteresis strength; example: h = 0.1 implies an output gap of -5 percent for one period permanently reduces potential GDP by 0.5 percent.
  - Natural closing of the output gap: initial presentation assumes the output gap fully closes “naturally” from one period to the next (closing not driven by discretionary fiscal policy); this assumption is relaxed in later sections.
- Baseline (no discretionary fiscal policy):
  - Time-1 actual GDP = y_1 and potential GDP = p_1; time-1 output gap γ_1 = p_1 − y_1 (equation (1)).
  - Time-2 potential GDP: p_2 = p_1(1 + g) + h γ_1, where 1 + g is underlying potential growth (equation (2)).
  - Under full natural closing, γ_2 = 0 and y_2 = p_2 (equation (3)); from time 2 onwards actual GDP = potential GDP growing at rate g (equation (4)).
  - Debt dynamics baseline: d_t+1 = d_t + (1 − τ) y_t, where τ is the tax rate and R = 1 + r with r the real sovereign yield (equation (5)).
- Alternative scenario (stimulus S at time 1, consolidation C at time 2):
  - Time-1 GDP after stimulus: Y_1 = y_1 + μ_1 S (equation (6)); output gap after stimulus: Γ_1 = γ_1 + μ_1 S (equation (7)).
  - Debt after stimulus: D_1 = R^{-1}[D_0 + τ Y_1] = R^{-1}[D_0 + S + μ_1 τ(1 − S?)]. (See full expression in equation (8).)
  - Stimulus raises potential GDP in time 2 through reduced hysteresis: p_2 after stimulus equals p_2 + μ_1 S h (equation (9)).
  - Time-2 policy: stimulus withdrawn and additional consolidation C applied so debt-to-GDP returns to baseline level; GDP at time 2 after consolidation depends on μ_2 and (C + S) (equation (10)).
  - The consolidation reduces the debt stock at end of time 2 (equation (11)) but some fiscal gains dissipate due to lower tax revenues from the contraction in GDP.
  - The contraction at time 2 widens the output gap (equation (12)), which via hysteresis lowers potential output at time 3 (equation (13)).
  - Subsequent time-3 and time-4 dynamics show the ramp-down of consolidation and final return to no further fiscal impulses; after time 4 GDP equals potential GDP growing at g (equations (14)–(18)).
- Present value of GDP:
  - Denote β the intertemporal discount factor and impose β R = 1; PV of GDP is PV = Y_1 + β Y_2 + β^2 Y_3 + β^3 Y_4 (equation (19)).

### Analytical condition for PV gains from delayed consolidation
- The net impact on PV from the initial stimulus and subsequent consolidation is derived analytically:
  - The impact of fiscal stimulus on PV is given by an expression summarized in equation (20); positive if and only if inequality in equation (21) holds, which balances the positive initial stimulus effect against contractionary effects from delayed consolidation.
  - The consolidation C is chosen so the debt-to-GDP ratio at the start of period 4 equals the baseline (equality expressed in equation (22)); this ensures fair comparison between scenarios.
  - The required consolidation size is given explicitly (equation (24)).
  - Substituting the partial derivative ∂C/∂S into the PV condition yields a final inequality (equation (25)) that determines when the stimulus offset by delayed consolidation increases PV of GDP.
- Interpretation of key comparative-static implications (as summarized in the main text):
  - Constant multipliers, no hysteresis: PV-neutrality of timing (gains from stimulus offset exactly by losses from consolidation).
  - Constant multipliers with hysteresis: delays can raise PV but gains are typically small.
  - Time-varying multipliers (μ_1 > μ_2), hysteresis present, and slow natural closing: delaying consolidation can generate sizable PV gains.
  - The exploitable advantage from time-variation in multipliers depends on whether multipliers are linked to slow-moving state variables (easier to exploit) or fast-moving variables (harder to exploit).

### Modeling choices, empirical limitations, and policy implications
- Modeling choices highlighted:
  - Explicit modeling of consolidation that returns debt-to-GDP to baseline to account for costs of permanently higher debt.
  - Explicit modeling of time-variation in multipliers and persistence of output gap.
- Empirical limitations noted:
  - Existing empirical literature largely estimates average multipliers (VARs with constrained multipliers independent of state); evidence on time-variation is limited but improving (Auerbach and Gorodnichenko; Christiano, Eichenbaum, and Rebelo; others cited).
- Policy-relevant takeaways (as framed in the source):
  - Decisions about delaying fiscal consolidation should focus less on the average size of the multiplier and more on (i) how multipliers vary over the cycle and (ii) the degree to which policymakers can exploit that time-variation.
  - Potential gains from delayed consolidation via multipliers and hysteresis may be modest under many plausible parameterizations and must be weighed against unmodeled risks (higher sovereign yields, political-economy risks, stochastic shocks).
  - The framework is parsimonious and intended to clarify criteria rather than to recommend specific fiscal paths for particular countries.

*Source: _wp1552 - Appendix A (https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp1552.pdf)*

### Section IV.

### _wp1552 - Section IV.

### Constant fiscal multipliers — Overview and key analytical cases
- Case 1 (No hysteresis and full natural closing of the output gap)
  - Delaying consolidation increases the PV of GDP if and only if the discount rate is higher than the interest rate (condition expressed in equation (26): ߚ/൏1ܴ).
  - Intuition: stimulus raises GDP today but requires future consolidation (and interest payments) to return debt ratio to baseline; higher interest rates increase future consolidation costs, while higher discount rates reduce the weight on future costs.
  - Under constant multipliers and no hysteresis, the size of the fiscal multiplier is inconsequential for the decision to delay consolidation because stimulative and subsequent contractionary effects exactly offset.

- Case 2 (Hysteresis and full natural closing of the output gap)
  - Allowing for hysteresis modifies the threshold condition (see equation (27)); the additional term on the right-hand side is typically positive, meaning hysteresis relaxes the interest rate threshold under which delaying consolidation can be beneficial.
  - Intuition: initial stimulus raises potential output via hysteresis and thus increases future tax revenue, reducing required consolidation.
  - Quantitative examples (parameter values as in main text):
    - With fiscal multiplier = 0.8, tax rate = 1/3, growth rate = 2 percent, baseline debt-to-GDP = 0.8, and hysteresis parameter = 10 percent, the interest rate threshold exceeds the discount rate ߚ/1 by only 5 basis points.
    - Increasing the fiscal multiplier to 1.2 or the hysteresis parameter to 20 percent raises the interest rate threshold over the discount rate by only 10 basis points.
  - Conclusion: under constant multipliers and full natural closing of the output gap, hysteresis has limited bearing on the decision to delay consolidation.

- Case 3 (Slow natural closing of the output gap, with and without hysteresis)
  - Assumption: no natural closing of the output gap until time 4 (formalized in equation (28)); output gap fully carries over for ݐ൑3, with closing at time 4 so that PV of GDP grows thereafter at rate ݃.
  - Under these assumptions, initial fiscal stimulus has persistent impacts on both potential output (via hysteresis) and actual GDP; time-2 GDP includes an extra persistent term (see equation (29)) that, with constant multipliers, can offset the contraction from withdrawn stimulus.
  - Modified condition for delaying consolidation given by equation (30). Key observations:
    - If h = 0 (no hysteresis), inequality (30) reduces to inequality (26): no change in decision when multipliers are constant and no hysteresis.
    - With hysteresis and plausible parameter values, the no-closing assumption can imply a significantly higher interest rate threshold: delaying consolidation may increase PV of GDP even if the interest rate exceeds the discount rate by several percentage points.
  - Comparison with a model à la DeLong and Summers:
    - DeLong and Summers’ setup allows the economy to permanently carry a higher debt-to-GDP ratio (paying the growth-adjusted interest rate on new debt), avoiding negative multiplier and hysteresis effects from delayed consolidation and yielding much higher interest rate thresholds (see Figure 3).
    - Requiring a common ultimate debt-to-GDP ratio (as in this analysis) substantially reduces the interest rate threshold relative to DeLong and Summers’ setup.
  - Risk and expected gains:
    - Even if current interest rates lie below model thresholds, delaying consolidation may not be optimal unless expected gains are sufficiently large to compensate risks not captured in the model.

### Quantitative gains under slow output-gap closing (constant multipliers)
- Definitions used to measure gains:
  - Percentage increase in PV of GDP over first three periods (times 1, 2, 3): ߙَଵ,ଶ,ଷ defined in equation (31).
  - Percentage increase at time 4: ߙَସ defined in equation (32); since output gap closes at time 4, differences in time-4 GDP are permanent and dominate PV unless discount rate is very high.
- Simulation example (no natural closing until time 4):
  - Fiscal stimulus: 1 percent of baseline GDP (ݕଵ = 1 percent of baseline GDP).
  - Assumed initial negative output gap under the baseline: 4 percent.
  - Parameter values used (as in Figure 4):
    - ߤ = 0.8, ߬ = 1/3, ݃ = 2 percent, ݀ଷݕ/ସ = 0.8, and ൌ1ݎൌߠ percent, where ߠ = 1/ߚെ1.
  - Results:
    - Even with persistent output-gap effects, delaying consolidation produces only small increases in GDP under constant multipliers.
    - With fiscal multipliers up to 1.2 or hysteresis up to 20 percent, a 1 percent of GDP stimulus raises the PV of GDP in the first three periods by only about 0.08 percent.
    - The impact on time-4 GDP is less than 0.02 percent.
  - Summary conclusion: with constant multipliers, potential PV gains from delaying consolidation are modest even under substantial multipliers, substantial hysteresis, and slow natural closing of the output gap. Positive effects from initial stimulus are largely offset by required subsequent consolidation to return debt-to-GDP to baseline.

### Time-varying multipliers: motivation and simulation approach for the G7
- Motivation: the case for delaying consolidation could be stronger if the initial stimulus multiplier is higher than the multiplier during later consolidation (e.g., when economy in severe recession or monetary policy can't offset fiscal policy).
- Model extensions to analyze time variation:
  - Allow multipliers to depend on the state of the economy.
  - Calibrate the model using more realistic, multi-year paths of delayed consolidation to assess whether reasonable delay paths can exploit time-varying multipliers.
  - Allow a fraction of the output gap to close naturally each period (gradual closing) rather than the two extreme prior assumptions.
- Consequence: these enhancements increase realism but preclude closed-form solutions; analysis proceeds via numeric simulations.

### Simulation assumptions for the G7 scenarios (25-year horizon)
- Two fiscal scenarios compared for the next 25 years:
  - Baseline:
    - Uses projections for the aggregate of G7 economies through 2017 from the October 2012 WEO (IMF, 2012).
    - Baseline entails significant medium-term consolidation: CAPB improves from -4¼ percent of potential GDP in 2011 to -½ percent of potential GDP by 2015.
    - After 2017, CAPB converges to zero and remains there through end of simulation (2027).
  - Alternative (delayed consolidation):
    - Examines a stimulus in 2012 that lowers the CAPB-to-potential-GDP ratio by 1 percent of GDP relative to 2011 and keeps it at this lower level in 2013 — providing a cumulative stimulus of 4¼ percent of potential GDP over 2 years relative to the baseline tightening over those two years.
    - Gradual consolidation begins in 2014 to bring debt-to-GDP and primary balance-to-GDP ratios back to baseline levels by 2027.
- Basic simulation mechanics:
  - Output gap deviations respond to differences in fiscal impulse according to multiplier assumptions.
  - Potential GDP deviates according to differences in the output gap via hysteresis assumptions.
  - Fiscal balances deviate in response to fiscal impulse differences and GDP growth (automatic stabilizers).
- Note: parameter values underlying the simulations are presented in Table 1 (in the source text).

*Italic source: _wp1552 - Section IV.*

### 2017. As WEO projections in the October 2012 edition are not available past 2017,

### _wp1552 - 2017. As WEO projections in the October 2012 edition are not available past 2017,

### Calibration and baseline assumptions
- Growth of potential GDP before hysteresis effects:
  - 2012-2017: WEO assumptions
  - 2018-2027: 2 percent
- Hysteresis effect (h): 0.10
- Derivative of the PB to potential output (߬1): 1/3
- Derivative of the PB to the output gap (߬2): 0.7
- Persistence of the output gap (ρ): 0.6
- Real interest rate (r):
  - 2012-2017: 1.1 percent on average
  - 2018-2027: 1 percent
- Intertemporal discount factor (β): 1/1.02
- Additional modeling assumptions:
  - Hysteresis effects equal 10 percent of the output gap each period.
  - 40 percent of the output gap is assumed to close naturally from one year to the next (persistence 0.6 implies this natural closing example).
  - Real interest rates follow 2012 WEO implicit rates until 2017 and 1 percent from 2018 onwards.
  - Intertemporal discount rate set to 2 percent.

### Alternative fiscal-multiplier specifications
- Three alternatives examined:
  - Alternative 1: the multiplier is constant at 0.8.
  - Alternative 2: the multiplier varies inversely and linearly with the output gap.
  - Alternative 3: the multiplier varies inversely and linearly with real GDP growth (per Auerbach and Gorodnichenko (2012)).
- Calibration note: under Alternatives 2 and 3 the average multiplier during the phase of delayed consolidation equals 0.8, matching Alternative 1, to isolate cycle-dependence effects.

### Simulation dynamics and phases under delayed consolidation
- Three main GDP phases when consolidation is delayed:
  - Phase 1 (initial stimulus): substantial loosening raises GDP growth and reduces the output gap faster than baseline; especially pronounced under Alternatives 2 and 3 where multipliers are larger given wide negative output gaps and modest growth.
  - Phase 2 (deeper consolidation): subsequent deeper and more prolonged consolidation relative to baseline, causing an extended period of weak growth and renewed widening of the output gap.
  - Phase 3 (reduction of surplus): fiscal surplus reduced once debt-to-GDP approaches baseline, generating a stimulative impulse that raises growth and rapidly closes the output gap; once CAPB levels at baseline, fiscal impulses end and output gap converges to zero at the ρ^1 rate.

### Key quantitative simulation findings
- If fiscal multipliers are constant (Alternative 1):
  - Delaying consolidation has virtually no effect on the PV of GDP through 2027; output losses from consolidation offset stimulus gains.
  - Delaying consolidation has virtually no effect on potential GDP at the end of the simulations.
- If multipliers vary with the output gap (Alternative 2):
  - PV of GDP through 2027 is 0.3 percent higher than baseline.
  - Potential output in 2027 is permanently 0.2 percent higher than baseline.
- If multipliers vary with GDP growth (Alternative 3):
  - Both the PV of GDP and potential GDP in 2027 are lower under delayed consolidation than in the baseline.
  - Mechanism: initial stimulus increases subsequent consolidation magnitude and duration; lower growth during consolidation raises the fiscal multiplier (because multiplier varies negatively with growth), creating a downward spiral of lower growth and higher multipliers that more than offsets initial benefits.

### Intuition on multiplier variability and "second-derivative effects"
- When multipliers vary with growth (Alternative 3):
  - Stimulus has diminishing marginal benefits (multiplier shrinks as stimulus boosts growth).
  - Consolidation has increasing marginal costs (multiplier increases as consolidation reduces growth).
  - Rapid reaction of growth to policy changes amplifies adverse dynamics.
- When multipliers vary with the output gap (Alternative 2):
  - Output gap is more sluggish than growth, so "second-derivative effects" are weaker; benefits of stimulus when the output gap is negative can dominate.
- Empirical uncertainties:
  - Whether multipliers depend on output gap or growth, and their exact functional forms, remain open questions. Auerbach and Gorodnichenko (2012) provide some evidence favoring growth-dependence but results are inconclusive.

### Sensitivity to output-gap persistence and hysteresis
- Varying persistence of the output gap (ρ):
  - If multipliers are constant (Alternative 1): delaying consolidation negligible regardless of ρ.
  - If multipliers vary over the cycle (Alternatives 2 and 3): higher persistence amplifies both gains (Alternative 2) and losses (Alternative 3).
- Varying hysteresis strength (h):
  - Stronger hysteresis magnifies gains and losses from asymmetric multipliers; little effect when multipliers are constant.
- Note on hysteresis modeling:
  - Hysteresis modeled as linear in the output gap. Time-varying or asymmetric hysteresis (large adverse effects when output gap negative, no positive effects when positive) could magnify gains from delaying consolidation into periods with positive output gaps, but achieving debt sustainability while waiting may be challenging under large negative initial gaps and structural deficits.

### Declining marginal utility and robustness
- PV of utility computed using CRRA utility with coefficient of two and consumption as a constant share of output.
- Result: PV of utility differs from PV of GDP by less than a tenth of one percent in any scenario — no material change to conclusions.

### Liquidity traps and discontinuous multipliers
- Stylized experiment: multiplier normally 0.2, rising to 1.5 when output gap < -2 percent (deep slump). Other parameters as baseline.
  - "Temporary stimulus" scenario: a 1 percent of GDP temporary fiscal stimulus can push economy out of slump, shrinking multiplier back to 0.2; resulting PV of GDP boost is about 2 percent of GDP.
  - "Slower consolidation" scenario: smaller stimulus insufficient to exit slump; multiplier remains high during consolidation, prolonging wide output gap/high multiplier state and lowering PV of GDP relative to baseline.
- Caveats:
  - Debate exists whether zero lower bound is binding; unconventional monetary policies could keep monetary policy effective, invalidating the discontinuous-multiplier assumption.
  - Liquidity trap is more strictly linked to the full-employment interest rate; a richer model capturing interest-rate dynamics could yield different outcomes. Stimulus that raises full-employment rates might later require reversal that re-enters the liquidity trap.

### Summary of simulation results
- Delaying consolidation yields no substantive gains if multipliers are constant, even with hysteresis.
- If multipliers vary over the business cycle and are higher during stimulus than during consolidation, delaying consolidation can produce meaningful gains.
- The sign and magnitude of gains depend crucially on:
  - Whether multipliers vary with the output gap or with growth.
  - The persistence of output gaps (ρ).
  - The strength and functional form of hysteresis (h).
  - Other macroeconomic and fiscal conditions.

### Risks and other considerations when evaluating delayed consolidation
- Risks potentially omitted or hard to quantify in the model:
  - Crowding-out: temporarily higher debt-to-GDP may absorb financial resources, reducing private investment; effect likely limited with a negative output gap but may rise as recovery occurs.
  - Higher sovereign risk premia and external borrowing costs: temporary debt increases (roughly 5 percent of GDP in the scenario) could be misread as weak fiscal discipline, raising borrowing costs and harming debt dynamics and the real economy; conversely, large asymmetric multipliers and hysteresis could reduce risk premia by raising the PV of GDP.
  - Political economy: consolidation is often unpopular; delays risk never implementing required consolidation, leading to higher long-run deficits and debt. Political support early in consolidation may argue for acting sooner despite possible economic gains from delay.
  - Loss of option value: postponing consolidation today may reduce fiscal space to respond to future negative shocks; each delay moves the country closer to debt constraints.
  - Risk of negative shocks during delayed consolidation: if shocks hit, countries may be unable to delay further, and higher multipliers during the shock can amplify contractionary effects of consolidation.
- Additional policy-relevant factors beyond PV of GDP:
  - Effects on external balance.
  - Direct effects of public investment on potential GDP, especially if large relative to financing costs.
- Policy implication emphasized:
  - The absolute size of the fiscal multiplier is less decisive than how the multiplier varies across the cycle; the time-variation of multipliers may be more crucial for designing optimal fiscal paths.

*Source: IMF working paper content provided in the supplied PDF excerpt.*

### APPENDIX A

### APPENDIX A

### Summary of the DeLong and Summers (2012) model used to trace interest rate thresholds

- In this model, the time-1 fiscal stimulus ܵ generates a contemporaneous increase in GDP by ߤܵ, where ߤ is the fiscal multipliers.
- The stimulus permanently increases potential GDP from time 2 onwards by ݄ߤܵ, with the parameter ݄ capturing the strength of the hysteresis effect.

### Implications for debt and fiscal balances

- The stimulus increases public debt by ሻߤ߬ሺ1ეܵ, where ߬ is the elasticity of the fiscal balance to GDP.
- Instead of requiring a subsequent phase of consolidation to bring the debt-to-GDP ratio back to the level it would have been at without the stimulus, DeLong and Summers only require the government to pay the growth-adjusted interest rates on the new debt.
- From time 2 onwards the government has to collect additional taxes by ܵ
  ( ߤ߬1ე )
  ሻ݃ეݎሺ, where ݎ and ݃ are respectively the real interest rate on government debt and the GDP real growth rate.

### Net effect on future fiscal revenues and the contractionary effect

- By increasing potential GDP, the fiscal stimulus increases future fiscal revenues by ݄ߤܵ߬, so that the government has to increase taxes only to cover the difference.
- This generates a contractionary effect on GDP equal to ሺܵߦ
  ( ߤ߬1ე )
  ሻ( ݃ეݎ )
  ሻ݄ߤ߬ე, where ߦ is the distortionary impact of taxation, which we set to 0.5, as in DeLong and Summers.

### Present-value derivative and interest rate threshold

- The partial derivative of the PV of GDP with respect to the fiscal stimulus is thus given by:

  ܸ߲ܲ
  ∂S
  ൅ߤൌ
  ݄	ߤ
  ܩეߚ/1
  ߦე
  ሺ
  ߤ߬1ე
  ሻሺ
  ݃ეݎ
  ሻ
  ݄ߤ߬ე
  ܩეߚ/1

- Setting this derivative to zero, we can solve for the interest rate threshold shown in Figure 3:

  ߤ൅݃rൌ
  ሻ߬ߦሺ1ე݄൅ܩეߚ/1
  ߦ
  ሺ
  ߤ߬1ე
  ሻ

### Key parameter values preserved from the model

- Distortionary impact of taxation: ߦ = 0.5
- Model parameters and expressions are reported above exactly as in the source text.

*Source: APPENDIX A*

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_Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2015/_wp1552.pdf_