## _wp08164 ## Source details **Canonical URL:** [_wp08164](https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08164.pdf) ## Other formats - [Markdown version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08164.pdf.md) - [Structured JSON version](/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08164.pdf.json) --- ### I. Introduction — purpose and key contributions - Framework: independent central bank conducts monetary policy; government sets fiscal policy. Monetary policy performs bulk of stabilization; fiscal policy ensures sustainability of public debt. - Literature context: - Dixit and Lambertini (2003: DL): one-period games; Nash can produce a "fight" between policymakers; Stackelberg fiscal leadership generally produces less extreme outcomes; static game limits analysis of public debt and inter-temporal trade-offs. - Beetsma and Bovenberg (1997b: BB): two-period dynamic game; with public debt under discretion less debt accumulated in first period than under commitment; myopic government accumulates more first-period debt to induce later inflation. - Kirsanova, Stehn and Vines (2005: KSV): infinite-horizon backward-looking model; Nash with myopic government can produce a "fight"; model lacks forward-looking expectations and microfounded welfare metric. - Contribution of current paper: - Uses an infinite-horizon New Keynesian model with government debt, microfounded, forward-looking. - Key summarized results: - With lump-sum taxes available and no public debt: Nash leads to too much government spending and excessively high interest rates; fiscal leadership reinstates cooperative discretion outcome. - Without lump-sum taxes and with government debt: Nash still entails too much public spending but leads to lower interest rates than cooperative discretionary policy because debt must be returned to pre-shock level under discretion and monetary policy contributes to required debt adjustment. - Fiscal leadership does not avoid this costly outcome without lump-sum taxes. - Imposing a debt penalty on a myopic government raises welfare substantially; appointing a conservative central bank is less effective. ### II. The model — agents, constraints, and welfare - Time period: quarter. - Consumers: - Representative infinitely lived; maximize E_t sum_{s=t}^∞ β^{s-t} [ u(C_s) + f(G_s) - v(H_s(z)) ]. - Budget constraint (nominal): P_t C_t + E_t R_{t;t+1} A_{t+1} = A_t + (1-τ) (W_t(z) H_t(z) + Π_t(z)) + T, with E_t(R_{t;t+1}) = 1/(1 + i_t). - Euler equation (linearised): ĉ_t = E_t ĉ_{t+1} - σ ( i_t - E_t π_{t+1} ) (equation (4)). - Price setting: - Calvo pricing with probability (1-θ) of re-optimizing. - Phillips curve: π̂_t = β E_t π̂_{t+1} + κ Ŝ_t, with κ = (1-θβ)(1-θ)/θ (1+ε̄)/ε̄. - Real marginal cost: Ŝ_t = 1/φ [ ŷ_t + 1/σ ĉ_t + μ̂_t ] (equation (5)). - Aggregate demand and fiscal policy: - Y_t = C_t + G_t (equation (6)); linearised ŷ_t = χ ĉ_t + (1-χ) ĝ_t (equation (7)). - Government: constant income tax rate τ; issues one-period non-indexed nominal debt B̄. - Nominal debt evolution: B_{t+1} = (1 + i_t) B_t + P_t G_t - τ P_t Y_t - T (equation (8)). - Linearised real debt evolution: b̂_{t+1} = i_t + 1/β b̂_t - π̂_t + (1-χ) B ĝ_t - τ B ŷ_t (equation (9)). - System (gap variables x_t = ^X_t - ^X_t^n; with Â_t = B̂_t): - c_t = E_t c_{t+1} - σ ( i_t - E_t π_{t+1} ) (10) - π_t = β E_t π_{t+1} + κ [ χ/σ c_t + 1/φ y_t ] + μ_t (11) - y_t = (1-χ) g_t + χ c_t (12) - b_{t+1} = i_t + 1/β χ b_t - π_t + (1-χ) B g_t - τ B y_t (13) - Social welfare: - Period loss function W_s = λ_π π_s^2 + λ_c c_s^2 + λ_y y_s^2 + λ_g g_s^2 (15). - Coefficients: λ_π = ((ε̄ + 1)/ε̄) (1-θβ)(1-θ)/θ; λ_c = χ / σ; λ_y = 1/φ; λ_g = (1-χ)/σ. - Policy objectives: - Central bank benevolent, maximizes W^M_s = W_s subject to (10)–(12). - Government myopic due to probability (1-p) of being voted out; effective discount p β. Myopic government objective uses λ^F_π = ((ε̄ + 1)/ε̄) (1-θ p β)(1-θ)/θ, and places lower weight on future and on inflation terms when p < 1. - Calibration and shocks: - Parameters: β = 0.99; σ = 0.5; φ = 2; ε̄ = 7; θ = 0.75; χ = 0.75; B = 0.1. - Cost-push shock: i.i.d. with standard deviation 0.005. - Baseline myopia: p = 0:75. - Robustness checks performed for alternative calibrations. ### III. Solution approach — cooperative and non-cooperative rules - Cooperative (centralised) policy: - Single policymaker chooses i_t and g_t to minimize social loss subject to (10)–(12). - Distinguish commitment (ability to manipulate expectations via promises) and discretion (time-consistent policy). - Optimal discretionary reaction functions respond to cost-push shock and, when lump-sum taxes unavailable, to existing debt stock. - Non-cooperative policy under discretion: - Setup: government and central bank minimize individual loss functions choosing g_t and i_t respectively, subject to economy dynamics. - Two strategic regimes: - Nash: simultaneous move; each treats other's actions parametrically; optimal feedback rules respond to the shock and, without lump-sum taxes, the debt stock. - Fiscal leadership (Stackelberg): government moves first, internalizes central bank reaction; government reaction includes cost-push shock and existing debt; central bank rule responds to government spending. - Numerical solution via extension of Blake and Kirsanova (2006); leader-follower matrices found iteratively using generalized Schur decomposition and first-order conditions (equations (27), (28)). ### IV. Optimal policy when lump-sum taxes are available (no debt accumulation) - Cooperative policy: - Commitment: - Policymaker raises nominal interest rate persistently to reduce inflation; fiscal policy inactive on impact; stabilisation left to monetary policy. - Commitment is time-inconsistent: temptation to renege once inflation is low. - Discretion: - Inflation controlled less effectively than under commitment (inflation stabilisation bias). - Policymaker raises interest rates strongly in first period inducing large recession, then lowers rates, yielding slower inflation control. - Fiscal policy remains inactive. - Non-cooperative with a myopic fiscal authority (baseline p = 0:75): - Nash: - Government raises spending on impact; fiscal response to shock rises from zero to 0:49 (see Table 1 column (2)). - Central bank monetary response to shock rises from 13:12 to 13:17 (see Table 1 column (2)). - Outcome: higher interest rates and excessive public spending vs cooperative discretion; fight and excess welfare loss small because inflation variability similar across regimes. - As p → 1, Nash converges to cooperative discretion. - Fiscal leadership: - Prevents instrument fight: myopic government internalises monetary response and does not overspend; fiscal feedback on shock remains at zero (see Table 1 column (3)). - Instrument responses and welfare identical to cooperative discretionary policy for all p. - Table 1 (selected entries related to lump-sum taxes; columns labelled D, N, FL): - Welfare Loss (D N FL): 0.4256 0.4258 0.4256 - Optimal monetary feedback shock coefficient (_π): 13.12 13.17 13.12 - Optimal fiscal feedback shock coefficient (_π): 0.00 0.49 0.00 ### V. Optimal policy when lump-sum taxes are not available (debt allowed) - Overview: - Removing lump-sum taxes allows debt accumulation; introduces "debt stabilisation bias." - Key difference: under discretion debt must return to pre-shock level, constraining monetary policy. - Cooperative policy with debt: - Commitment: - Similar monetary strategy: raise interest rates to reduce inflation; fiscal policy inactive on impact, but government spending is permanently lowered to service permanently higher stock of debt. - Debt accumulates under commitment: random-walk result where debt stock remains permanently higher in response to shock because marginal benefits of reducing debt are balanced by costs. - Commitment is time-inconsistent for both inflation and debt. - Discretion: - Two stabilization biases: inflation stabilisation bias and debt stabilisation bias. - Debt stabilisation bias: time-consistent discretionary policymaker requires debt to return to steady-state, so interest rates, inflation, government spending return to steady-state values while debt is actively reduced. - Monetary feedback coefficient falls from 13:12 to 8:88 (see Table 1 column (4)). - Fiscal feedback coefficient falls from 0 to -5:50 (see Table 1 column (4)). - Monetary policy constrained in responding to inflation because raising rates to fight inflation increases debt; fiscal policy must play an active role cutting government spending strongly to reduce debt. - Non-cooperative policy with myopic fiscal authority (no lump-sum taxes): - Nash equilibrium: - Myopic government spends excessively on impact—spending falls less than cooperative discretionary outcome—dampening recession and reducing assistance to central bank in controlling debt and inflation. - Central bank raises interest rate by less than under cooperative discretion; monetary feedback on shock falls from 8:88 to 5:84 (Table 1 column (5)). - Fiscal feedback on shock rises from -5:50 to -4:61 (Table 1 column (5))—i.e., less contractionary fiscal response under Nash. - Result: Nash leads to higher output and consumption variability and higher inflation; welfare loss increases with fiscal myopia. - Mechanism differs from BB: central bank internalizes effect on public debt and therefore responds less forcefully to inflation when government overspends, exacerbating inflation. - Fiscal leadership: - Fiscal leadership fails to avoid bad outcome without lump-sum taxes. - Under both Nash and FL myopic authority spends excessively; central bank raises interest rates by less than cooperative discretion to limit debt accumulation. - Nash and FL yield almost identical outcomes and similar welfare losses (Table 1 columns (5) and (6)). - Reported instrument coefficients for FL and Nash (no lump-sum taxes): monetary and fiscal feedback coefficients on shock equal 4:72 and 5:74 respectively (footnote 32). - Table 1 (selected entries for No Lump-Sum Taxes; columns D N FL): - Welfare Loss (D N FL): 0.4917 0.5056 0.5049 - Optimal monetary feedback shock coefficient (_π): 8.88 5.84 5.74 - Optimal monetary feedback debt coefficient (_b): -0.03 -0.03 - Optimal fiscal feedback shock coefficient (_π): -5.50 -4.61 -4.72 - Optimal fiscal feedback debt coefficient (_b): -0.11 -0.11 ### VI. Institutional remedies and robustness - Fiscal leadership efficacy: - With lump-sum taxes: fiscal leadership reproduces cooperative discretionary outcome. - Without lump-sum taxes: fiscal leadership does not prevent myopic overspending; Nash and FL outcomes similar. - Two institutional interventions evaluated: - A. Debt penalty: - Modeled as myopic government maximizing W_F^s + _B b_s^2 (penalty on deviation of debt from steady-state). - Optimally chosen debt penalty _B = _B is highly effective under both Nash and FL. - Effects: - Prevents myopic overspending and induces inertial fiscal contraction while debt exceeds pre-shock level. - Relieves pressure on monetary policy, enabling stronger first-period interest rate increases, faster disinflation, and improved welfare. - Welfare losses under Nash and FL can fall below cooperative discretionary policy (second-best). - Too strong a debt penalty can be detrimental (excessively tight debt control). - Calibration note: optimal debt penalty approximately equal to one tenth of the weight on consumption (_B ≈ 1/10 _c) in baseline calibration (footnote 37). - Deficit penalties are ineffective because they lack the inter-temporal incentive mechanism of debt penalties. - B. Conservative central bank (delegation): - Central bank objective modified by weight η on inflation (conservative if η > 1). - Without debt penalty (λ_B = 0): appointing conservative central bank worsens outcomes—greater interest-rate aggressiveness induces myopic government to spend more; a mildly anti-conservative central bank (η_ < 1) may improve welfare by reducing incentive to overspend. - With optimal debt penalty in place (λ_B = λ*_B = _B): central bank should be made mildly conservative (η_ > 1); fiscal policy controls debt more tightly allowing monetary policy to fight inflation; optimal conservatism small and welfare gains limited. - Conclusion: changing government incentives (debt penalty) is more effective than changing central bank incentives. - Robustness: - Stronger price rigidities intensify instrument fight. - Higher elasticity of substitution has ambiguous effects. - Results robust to higher steady-state debt B and different combinations of nominal/indexed debt; with higher B myopic government can force central bank to cut interest rates (violating Taylor principle). - With significant inflation persistence, Nash no longer leads to lower interest rates than cooperative discretion; FL regains some effectiveness. ### VII. Policy implications and conclusions - Main conclusions: - Fiscal leadership can avoid Nash outcomes with lump-sum taxes available. - When lump-sum taxes are unavailable and government debt matters, Nash still implies too much public spending but leads to lower interest rates than cooperative discretion because debt must be returned to pre-shock levels; fiscal leadership does not remedy this. - Institutional remedies: - Imposing a debt penalty on a myopic government under Nash or FL raises welfare substantially and can outperform cooperative discretionary outcomes. - Appointing a conservative central bank is less effective and can be counterproductive without fiscal constraints; with a debt penalty a mildly conservative central bank marginally improves outcomes. - Policy recommendations: - Implement debt-targeting mechanisms (debt penalties) to discipline myopic fiscal authorities when central bank is independent. - Relying solely on conservative central bank appointment is insufficient and may worsen outcomes unless complemented by fiscal constraints. - Avoid deficit-based penalties as they fail to provide necessary inter-temporal incentives. - Suggested extensions: - Allow multi-period debt maturity to assess maturity structure effects on monetary leverage and distortion from myopic government. - Empirically test model predictions on strategic interactions constraining monetary responses to inflation, especially in emerging markets. *Italic: Source: IMF Working Paper _wp08164 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . 36* ### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ### _wp08164 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ### I. Introduction - Framework: independent central bank conducts monetary policy; government sets fiscal policy. Monetary policy performs bulk of stabilization; fiscal policy ensures sustainability of public debt. - Literature context: - Dixit and Lambertini (2003: DL): one-period games; Nash can produce a "fight" between policymakers; Stackelberg fiscal leadership generally produces less extreme outcomes; shortcoming: static game, cannot analyze public debt or inter-temporal trade-offs. - Beetsma and Bovenberg (1997b: BB): two-period dynamic game; inclusion of public debt introduces additional time inconsistency; under discretion less debt accumulated in first period than under commitment; myopic government accumulates more first-period debt to induce later inflation; debt target can raise welfare but not fully replicate commitment unless central bank is conservative. - Kirsanova, Stehn and Vines (2005: KSV): infinite-horizon backward-looking model; Nash with myopic government and benevolent central bank can produce "fight": central bank raises interest rate to reduce inflation, government increases spending to postpone recession, leading to excessively high interest rates, too much spending, and large debt increases; fiscal leadership avoids fight; model lacks forward-looking expectations and a microfounded welfare metric. - Contribution of current paper: - Uses an infinite-horizon New Keynesian model with government debt, microfounded, forward-looking. - Builds on Benigno and Woodford (2003), Schmitt-Grohe and Uribe (2004), Leith and Wren-Lewis (2007), Stehn and Vines (2008), and the algorithm of Blake and Kirsanova (2006). - Key results summarized: - With lump-sum taxes available and no public debt: Nash leads to too much government spending and excessively high interest rates; fiscal leadership reinstates cooperative discretion outcome. - Without lump-sum taxes and with government debt: Nash still entails too much public spending but leads to lower interest rates than cooperative discretionary policy because debt must be returned to pre-shock level under discretion and monetary policy contributes to required debt adjustment. - Fiscal leadership does not avoid this costly outcome. - Imposing a debt penalty on a myopic government raises welfare substantially; appointing a conservative central bank is less effective. - Paper structure: Section II model, Section III solution method, Section IV optimal policy with lump-sum taxes, Section V optimal policy with debt accumulation, Section VI optimal institutions, Section VII conclusion. ### II. The Model - Based on Woodford (2003a) and Kirsanova and Wren-Lewis (2007); extended New Keynesian model including fiscal policy and government debt. - Time period interpretation: a quarter (calibration specified later). A. Consumers - Representative infinitely lived individuals produce differentiated good z, supply effort H(z), consume C, and derive utility from per capita government consumption G. - Individual maximization problem: - max_{ {C_s, h_s} }_{s=t} E_t sum_{s=t}^infty β^{s-t} [ u(C_s) + f(G_s) - v(H_s(z)) ] - Budget constraint (nominal): - P_t C_t + E_t R_{t;t+1} A_{t+1} = A_t + (1-τ) (W_t(z) H_t(z) + Π_t(z)) + T - E_t(R_{t;t+1}) = 1/(1 + i_t) - Aggregation: consumption index C_t = [ ∫_0^1 c_t(z)^{(ε_t-1)/ε_t} dz ]^{ε_t/(ε_t-1)} with stochastic elasticity of substitution ε_t > 1 and mean ε̄. - Transversality and no-Ponzi: lim_{s→∞} E_t [ R_{t;s} A_s ] = 0; net present value bounded. - Euler equation (linearised): - ĉ_t = E_t ĉ_{t+1} - σ ( i_t - E_t π_{t+1} ) (equation (4)) - where σ = -u_C/(u_{CC} C) = -f_G/(f_{GG} G) is intertemporal elasticity of substitution; inflation π_t = P_t/P_{t-1} - 1, steady-state inflation zero. - Aggregate nominal assets accumulation: - A_{t+1} = (1 + i_t) A_t + (1-τ) P_t Y_t - P_t C_t - T (equation (3)) - Notation: steady-state X, log-deviation ^X_t = ln(X_t / X). B. Price Setting - Calvo pricing: firms re-optimize prices with probability (1-θ); otherwise fixed. - Phillips curve: - π̂_t = β E_t π̂_{t+1} + κ Ŝ_t (equation (11) structure) - with κ = (1-θ)(1-β θ)/θ (1+ε̄)/ε̄ (as stated: κ = (1-θβ)(1-θ) θ^{-1} (1+ε̄)/ε̄) - Real marginal cost: - Ŝ_t = 1/φ [ ŷ_t + 1/σ ĉ_t + μ̂_t ] (equation (5)) - where μ̂_t is a cost-push shock. C. Aggregate Demand and Fiscal Policy - National income identity: - Y_t = C_t + G_t (equation (6)) - Linearised: - ŷ_t = χ ĉ_t + (1-χ) ĝ_t (equation (7)) - χ = C/Y steady-state share of private consumption in output. - Government: constant income tax rate τ, issues one-period non-indexed nominal debt B̄. - Nominal debt evolution: - B_{t+1} = (1 + i_t) B_t + P_t G_t - τ P_t Y_t - T (equation (8)) - Linearised real debt evolution: - b̂_{t+1} = i_t + 1/β b̂_t - π̂_t + (1-χ) B ĝ_t - τ B ŷ_t (equation (9)) - real debt B_t = B̄_t / P_{t-1}, steady-state debt-output ratio B. D. The System (gap variables x_t = ^X_t - ^X_t^n) - With Â_t = B̂_t, the system: - c_t = E_t c_{t+1} - σ ( i_t - E_t π_{t+1} ) (equation (10)) - π_t = β E_t π_{t+1} + κ [ χ/σ c_t + 1/φ y_t ] + μ_t (equation (11)) - y_t = (1-χ) g_t + χ c_t (equation (12)) - b_{t+1} = i_t + 1/β χ b_t - π_t + (1-χ) B g_t - τ B y_t (equation (13)) - Model variables: intertemporal Euler, New Keynesian Phillips curve, linearised aggregate demand, government debt evolution. E. Social Welfare - Second-order approximation of aggregate utility yields period loss function W_s and lifetime loss: - (1/2) E_t sum_{s=t}^∞ β^{s-t} W_s = - (1/2) E_t sum_{s=t}^∞ β^{s-t} W_s (equation (14) structure) - Period loss function: - W_s = λ_π π_s^2 + λ_c c_s^2 + λ_y y_s^2 + λ_g g_s^2 (equation (15)) - Coefficients: - λ_π = ((ε̄ + 1)/ε̄) (1-θβ)(1-θ)/θ - λ_c = χ / σ - λ_y = 1/φ - λ_g = (1-χ)/σ - Interpretation: - Quadratic costs in c, g, y reflect smoothing preferences and disutility of labor. - Inflation costs reflect price dispersion under Calvo pricing; λ_π depends on discount rate β. F. Policy Objectives - Assumptions: - Central bank is benevolent and maximizes social welfare W^M_s = W_s subject to economy evolution (10)–(12). - Government maximizes same-period utility but is myopic due to exogenous probability (1-p) ≥ 0 of being voted out; effective discount factor p β. - Myopic government objective (second-order approximation): - E_t (1/2) sum_{s=t}^∞ (p β)^{s-t} W^F_s (equation (16)) - W^F_s = λ^F_π π_s^2 + λ_c c_s^2 + λ_y y_s^2 + λ_g g_s^2 (equation (17)) - λ^F_π = ((ε̄ + 1)/ε̄) (1-θ p β)(1-θ)/θ - Effects of myopia: - Inter-temporal: government places lower weight on future losses. - Intra-temporal: government attaches less weight to inflation terms relative to other terms (for p < 1, λ^F_π < λ_π). G. Calibration - Period: quarter. - Parameter values: - β = 0.99 - σ = 0.5 - φ = 2 - ε̄ = 7 - θ = 0.75 (implies prices set on average once a year) - χ = 0.75 (steady-state share of private consumption in output) - B = 0.1 (steady-state ratio of debt to output) - Shock process: - i.i.d. cost-push shock with standard deviation of 0.005. - Baseline myopia parameter: - p = 0.75 - Robustness checks: sensitivity to calibrations considered in later sections. ### III. Solving for Optimal Policy - Solution method described; details delegated to Appendix C. - Cooperative (centralised) policy: - Single policymaker chooses interest rate and government spending to minimize social loss subject to economy evolution (10)–(12). - Optimal commitment: policymaker can manipulate expectations by committing to any policy plans (including time-inconsistent). Given linear-quadratic problem, optimal policy can be expressed as linear instrument rule responding to predetermined variables. - Optimal discretion: policies must be time consistent. - Reaction functions for interest rate and spending under optimal discretionary policy respond to the cost-push shock and, if lump-sum taxes are unavailable, to existing debt stock. *Italic: Source: IMF Working Paper _wp08164 - References . . . . . . . . . . . . . . . . . . . . . . . . 36* ### Appendix 1.). Our simulations will summarise optimal policies under discretion by ### _wp08164 - Appendix 1.). Our simulations will summarise optimal policies under discretion by ### B. Non-Cooperative Policy under Discretion - Setup: - The government and the central bank minimise their individual loss functions by choosing government spending and the interest rate respectively, subject to the evolution of the economy. - Games are analysed under discretion. Solution approach: Blake and Kirsanova (2006) (described in Appendix 2.). - Two strategic regimes solved: - Nash: central bank and government set instruments simultaneously, treating the other's actions parametrically. Optimal feedback rules expressed in responses to the shock and, without lump-sum taxes, the debt stock (see Appendix 2.). - Fiscal leadership (Stackelberg): government moves first and internalises the central bank reaction function. Government reaction to government spending includes the cost-push shock and existing debt, as under Nash and cooperative policy. The central bank interest-rate reaction function includes a response to government spending. Interest-rate feedback rule reported as function of cost-push shock and, without lump-sum taxes, debt. - Key mechanism and outcomes: - Under Nash with a myopic government (baseline degree of fiscal myopia p= 0:75), the government raises spending on impact to dampen the monetary contraction; the central bank raises interest rates slightly more than the cooperative discretionary policy. Monetary and fiscal policy engage in a small "fight" over disinflation. - The fight is small because monetary policy is relatively more effective in controlling real marginal cost than fiscal policy; a slight increase in real interest rates can offset fiscal expansion. - Fiscal leadership (Stackelberg) prevents excessive fiscal expansion: the myopic government internalises the monetary response and does not spend excessively. Fiscal leadership yields instrument responses identical to cooperative policy under discretion for all p, and identical welfare outcomes to cooperative discretionary policy. - Robustness: - Stronger price rigidities (higher γ) flatten the Phillips curve, increasing the relative weight on inflation stabilisation and intensifying the instrument fight: the myopic government spends more and the central bank hikes interest rates by more. - Increasing the elasticity of substitution (higher ) raises consumption responsiveness to the interest rate but lowers the effect of consumption on real marginal cost, increasing relative effectiveness of public spending and intensifying the Nash fight. - Similar qualitative effects arise from raising the elasticity of substitution between goods (") and lowering the elasticity of labour supply ( ). ### IV. Optimal Policy when Lump-Sum Taxes are Available - Context: - Policymaker can choose lump-sum taxes to fulfil the intertemporal budget constraint; debt accumulation unnecessary. - Cooperative benchmark reported (Commitment and Discretion), then non-cooperative games with myopic government. - A. Cooperative Policy 1. Commitment - Under commitment, the policymaker raises the nominal interest rate persistently in response to a rise in inflation. The resulting real rate increase reduces consumption, lowers real marginal cost and inflation. - Optimal fiscal policy is inactive in response to the cost-push shock when lump-sum taxes are available: stabilisation is left to monetary policy because (i) monetary policy is more effective in controlling real marginal cost and (ii) fiscal instrument movements are costly (they induce suboptimal public spending). - Commitment policy is time inconsistent: committing to future tight monetary policy reduces current inflation without a large present recession, but once inflation is low the policymaker is tempted to renege. 2. Discretion - Under discretion, inflation is controlled less effectively than under commitment (inflation stabilisation bias). Unable to promise high future interest rates, the policymaker raises interest rates very strongly in the first period, inducing a large recession, then returns rates more quickly to zero, yielding slower inflation control. - Fiscal policy under discretion remains inactive when lump-sum taxes are available. - B. Non-Cooperative Policy with a Myopic Fiscal Authority 1. Nash - Myopic government under Nash raises government spending on impact (the fiscal response to the shock rises from zero to 0:49, see column (2) in Table 1). - Central bank monetary response to the shock rises from 13:12 to 13:17 (see column (2) in Table 1). - Result: higher interest rates and excessive public spending compared to cooperative equilibrium under discretion; however the magnitude of the fight and excess welfare loss is small because inflation variability (the dominant weight in the loss function) is similar under Nash and cooperative policy. - As p → 1 (no conflicting objectives), Nash outcomes converge to cooperative discretionary outcomes. 2. Fiscal Leadership - Fiscal leadership is effective at avoiding the instrument fight: the myopic government does not spend excessively; fiscal feedback coefficient on the shock remains at zero (see column (3) in Table 1). - Instrument responses and welfare outcomes identical to cooperative policy under discretion for all degrees of fiscal myopia p. - Figures and additional notes: - Figure 1: dynamic responses to a transitory cost-push shock under optimal policy; Figure 2: first-period responses and welfare losses across p. - With a unit cost-push shock, first-period interest rate and public spending movements equal their feedback coefficients onto the cost-push shock; for baseline p= 0:75, feedback coefficients in Table 1 equal first-period policy responses in Figure 2. - The magnitude of the loss rises considerably for auto-correlated cost-push shocks: a process with a first-order autoregressive parameter of 0:8 leads to an increase in the excess loss of Nash by a factor of 15. ### Table 1 (optimal policy simulations for a transitory cost-push shock) - Note: D denotes cooperation under discretion. N and FL denote policy with a myopic government (p= 0:75) under Nash and Fiscal leadership respectively. - Columns: Lump-Sum Taxes (D N FL) and No Lump-Sum Taxes (D N FL) - Welfare Loss - Loss: 0.4256 0.4258 0.4256 0.4917 0.5056 0.5049 - % C: -0.0002 0.0000 -0.0139 0.0132 - Optimal Monetary Feedback Rule (i_t = _π π_t + _b b_t) - Shock (_π): 13.12 13.17 13.12 8.88 5.84 5.74 - Debt (_b): ---- 0.01 -0.03 -0.03 - Optimal Fiscal Feedback Rule (g_t = _π π_t + _b b_t) - Shock (_π): 0.00 0.49 0.00 -5.50 -4.61 -4.72 - Debt (_b): ---- 0.11 -0.11 -0.11 ### V. Optimal Policy when Lump-Sum Taxes are not Available - Overview: - Relax lump-sum taxes assumption and allow debt accumulation. Figures (right-hand panels) and Table 1 columns (4) to (6) present outcomes with government debt. - Introduces the concept of "debt stabilisation bias." - A. Cooperative Policy 1. Commitment - Optimal policy under commitment with government debt is similar to the lump-sum taxes case: monetary policy raises interest rates to reduce inflation and achieves rapid disinflation via commitment. - Fiscal policy is inactive on impact but government spending is permanently lowered to service the permanently higher stock of debt. - Debt accumulates through higher interest payments and a fall in tax receipts (via the constant income tax rate ). Under optimal commitment policy, the debt stock remains permanently higher in response to the shock: a "random walk" result. - Rationale: reducing debt permanently yields benefits (lower permanent interest payments) but entails costs (reducing government spending and/or lowering interest rates is inflationary and costly). Because benefits are discounted and the marginal cost of debt reduction is non-monotonic over time, there exists a margin at which debt remains permanently higher. - Optimal commitment policy is time inconsistent in control of both inflation and debt: a policymaker re-optimising each period would have an incentive to unexpectedly lower debt in every period, making the random-walk outcome time inconsistent. *Italic: Source text from the referenced IMF working paper appendix.* ### 2. Discretion ### 2. Discretion ### Overview - Optimal discretionary policy with government debt exhibits two stabilisation biases: the classic ëináation stabilisation biasí and a ëdebt stabilisation biasí. - Under discretion, ináation is controlled less effectively than under commitment; debt returns to its initial (pre-shock) value rather than following a random walk. ### Debt stabilisation bias: mechanism and implications - The ëdebt stabilisation biasí arises because a time-consistent discretionary policymaker cannot commit not to cut debt in future periods; the only time-consistent solution requires debt to return to its steady-state (pre-shock) level. - Consequence: interest rates, ináation, and government spending return to their steady-state values; debt must also return to its steady-state level to remove incentives for unexpected debt reductions. ### Quantitative adjustments under discretion (comparisons referenced to Table 1 and Figure 1) - Monetary feedback coefficient falls from13:12to8:88, see column (4) in Table 1. - Fiscal feedback coefficient falls from zero to5:50, see column (4) in Table 1. - Given the need to reduce debt back to its initial level, the policymaker raises interest rates substantially less than when lump-sum taxes were available. - Monetary policy is therefore constrained in its response to ináation by the need to control debt tightly. - Fiscal policy becomes optimal to play an active role in reducing government debt and assisting the central bank in controlling ináation by cutting government spending strongly. - Conclusion: debt stabilisation bias constrains monetary policy and necessitates a much more active role for fiscal policy under discretion than under commitment. ### Non-Cooperative Policy with a Myopic Fiscal Authority - The earlier result (with lump-sum taxes) where Nash led to a fight but fiscal leadership prevented it does not continue to hold when lump-sum taxes are unavailable. #### Nash equilibrium behavior - The myopic government under Nash continues to spend excessively in response to the cost-push shock when lump-sum taxes are unavailable; spending falls less than under the cooperative discretionary outcome to dampen the recession induced by monetary tightening. - The myopic government therefore does not sufficiently assist the central bank in controlling debt and ináation. - Unlike when lump-sum taxes were available, the central bank does not respond to excess spending with higher interest rates; instead it raises the interest rate by less than under cooperative discretionary policy. - Figure 2 (referenced) shows excess spending under Nash leads to lower interest rates than under cooperative discretionary policy; this mechanism strengthens with greater fiscal myopia. #### Quantitative changes under Nash (comparisons referenced to Table 1 and Figure 2) - The fiscal feedback coefficient on the shock rises from5:50to4:61, see column (5) of Table 1. - The monetary feedback coefficient on the shock falls from8:88to5:84, see column (5) of Table 1. - Although difficult to see in Figure 1, first-period ináation is higher under Nash (shown in the first row of Figure 2). - The Nash outcome increases social costs: higher output and consumption variability and higher ináation; welfare loss rises with the degree of fiscal myopia (see Figure 2). ### Relation to prior literature and mechanisms - BB (two-period model) show myopic government in Nash raises debt in the first period to induce more ináation later; BB assume the central bank ignores the government budget constraint. - In contrast, in this setup the central bank internalises the effect of the interest rate and ináation on the public debt stock; when the myopic government spends excessively, the central bank responds by raising interest rates by less than the cooperative policy under discretion, creating more ináation. - The mechanism of foregoing ináation control to stabilise debt is reminiscent of the ëpassiveí regime (Leeper 1991), but here the constrained monetary response to ináation is the optimal discretionary outcome of strategic interaction between a benevolent central bank and a myopic government, not the consequence of exogenous fiscal insolvency. *Source: _wp08164 - 2. Discretion* ### 2. Fiscal Leadership ### 2. Fiscal Leadership ### Key findings: fiscal leadership with and without lump-sum taxes - Öscal leadership (FL-D) isnote§ective in avoiding the undesirable outcome obtained under Nash when lump-sum taxes are not available. - Under both Nash and Öscal leadership the myopic authority spends excessively, inducing the central bank to raise interest rates by less than the cooperative policy under discretion to curtail debt accumulation. - Without lump-sum taxes, a setup of Öscal leadership doesnotprevent over-spending because the government, taking the central bank response into account, dampens the first-period contraction of output and consumption. - The disciplining mechanism present under Öscal leadership with lump-sum taxes is disabled by the time-consistency requirement to adjust debt back to its initial level. - Nash and Öscal leadership lead to almost the same outcomes and similar welfare losses (see Table 1 and Figure 2). - Öscal and monetary feedback coe¢ cients on the shock equal4:72and5:74respectively (footnote 32). ### Robustness of results - Stronger price rigidities (higher ) intensify the instrument fight (right-hand of Figure 3). - An increase in the elasticity of substitution (higher ) does not have straightforward implications for instrument settings because: - With higher  interest rates need to move less to dis-inflate. - A lower stock of debt is built up and the central bank is less constrained in its response to inflation. - Alternative debt calibrations (bottom rows of Figure 3): - Results continue to hold with higher steady-state debt (larger B): a myopic government under Nash and Öscal leadership constrains monetary policy more severely and can force the central bank to cut interest rates (violating the Taylor principle) in response to the cost-push shock. - Results are largely unchanged for different combinations of nominal and indexed debt (share of nominal debt denoted with ). While interest rates are raised more strongly with a higher share of indexed debt under Nash and Öscal leadership, the effect is very small. - Sensitivity to inflation persistence: - With significant inflation persistence, a Nash game no longer leads to lower interest rates than the cooperative policy under discretion because inflationary consequences become prohibitively costly. - With inflation persistence, Öscal leadership regains some effectiveness in preventing myopic government over-spending. ### Optimal institutions: design options and effects - When lump-sum taxes are available: - Öscal leadership yields the same outcome as the optimal cooperative policy under discretion; ensuring the myopic government moves first (credible budget process) is sufficient. - Without lump-sum taxes: - Both Nash and Öscal leadership produce socially costly outcomes. - Delegation of Öscal policy to a benevolent Öscal policy committee (p = 1) could reproduce cooperative outcomes but is judged practically infeasible. - Two institutional interventions considered: imposing a debt penalty (section A) and appointing a conservative central bank (section B). #### A. A Debt Penalty - Aim: discipline the government by raising the marginal cost of public spending. - Implementation in model: the myopic government maximises W_F^s + _B b_s^2 (penalty on deviation of debt from steady-state). - Interpretation: financial sanctions for missing the debt target (e.g., Excessive Deficit Procedure). - Main effects (Figure 4): - An optimally chosen debt penalty is highly effective under both Nash and Öscal leadership. - Optimally-chosen debt penalty (_B = _B) prevents myopic overspending and raises the fiscal feedback on debt (government spending is slowly smoothed to zero). - Mechanism: debt penalty induces inertial fiscal response—sustained contractionary fiscal policy while debt exceeds pre-shock level—relieving pressure on monetary policy to reduce debt and enabling stronger first-period interest rate increases to fight inflation, yielding faster disinflation and improved welfare. - Welfare losses under Nash and Öscal leadership can fall below those under cooperative policy under discretion (second-best result). - Debt penalties can be detrimental if too strong (lead to overly tight debt control). - A deficit penalty would not be effective because it lacks the inter-temporal incentive mechanism provided by a debt penalty. - Calibration note: for the baseline calibration, the optimal debt penalty is approximately equal to one tenth of the weight on consumption (i.e. _B  1/10 _c) (footnote 37). #### B. A Conservative Central Bank - Context: literature on optimal delegation and conservative central banks; mixed implications when government is myopic and when debt penalties are present. - Setup: central bank maximises social welfare but attaches weight η on inflation (central bank conservative if η > 1). - Findings (Figure 5): - Without a debt penalty (λ_B = 0), appointing a conservative central bank further worsens problems caused by a myopic government under both Nash and Öscal leadership: - A more inflation-averse central bank raises interest rates more aggressively, which induces the myopic government to raise spending even more. - A mildly anti-conservative central bank (η_ < 1) can raise welfare by raising interest rates less strongly and reducing incentive for excess spending (consistent with BB). - With an optimal debt penalty in place (λ_B = λ*_B = _B), the central bank should be made mildly conservative (η_ > 1) under both Nash and Öscal leadership: - Fiscal policy is induced to control debt more tightly, allowing interest rates to rise and control inflation more closely. - The optimal degree of conservatism is small because excessive emphasis on inflation increases consumption and output volatility. - Gains from appointing a conservative central bank when a debt penalty exists are small; changing government incentives is more effective than changing central bank incentives. ### Policy implications and recommendations - Constraining governments with debt targets (debt penalties) in countries with independent central banks can yield significant welfare gains and mitigate adverse non-cooperative interactions between fiscal and monetary policy. - Appointing a conservative central bank is less effective by itself and can be counterproductive without accompanying fiscal constraints; when combined with an optimal debt penalty, a mildly conservative central bank marginally improves outcomes. - Deficit-based penalties are unlikely to reproduce the inter-temporal incentives provided by debt penalties and therefore are less effective. ### Conclusion and avenues for future work - Two main contributions: - Öscal leadership can avoid the Nash outcome of excessive spending and high interest rates under optimal discretionary policy when lump-sum taxes are available. - This result breaks down when lump-sum taxes are unavailable: with government debt the Nash equilibrium still entails too much public spending but leads to lower interest rates than the cooperative policy under discretion because debt must be returned to pre-shock levels and monetary policy shoulders much of the adjustment. - Institutional remedies: - Imposing a debt penalty on a myopic government under Nash or Öscal leadership raises welfare significantly. - Appointing a conservative central bank is less effective and can be harmful unless accompanied by fiscal discipline. - Future extensions identified: - Allowing for multi-period debt maturity to assess how maturity structure affects monetary leverage over interest payments and the distortion from a myopic government (reference to Persson et al 2006). - Empirical testing of model predictions: whether strategic interactions between central banks and governments constrain monetary policy responses to inflation, particularly in emerging market economies (references to Baig et al 2006, Mitra 2007). *Source: _wp08164 - 2. Fiscal Leadership (PDF chapter).* ### 2. Optimal Non-Cooperative Policy under Discretion ### 2. Optimal Non-Cooperative Policy under Discretion ### Fiscal Leadership - Setup: Follower (ultimate follower) reaction function is guessed; using the generalised Schur decomposition the current value of the (non-instrument) jump variables as a function of predetermined and instrument variables is: - X2,t = −J X1,t − K_F u_F,t − K_L u_L,t (equation (27)) - Matrices J, K_F and K_L are unknown and determined in equilibrium. - Follower's problem: - Constrained period welfare loss: - H_F^s = β_F^{s−t} [ X_s' Q_F X_s + 2 X_s' U_F u_s + u_s' R_F u_s + λ_{s+1}' (A_11 X1,s + A_12 X2,s + B_1 u_s − X1,s+1) ] - λ_{s+1} is a vector of Lagrange multipliers; β_F is the follower discount rate. - First-order conditions (FOCs): - ∂H_F^s/∂u_F^s + (∂H_F^s/∂X2,s)(∂X2,s/∂u_F^s) = 0 - ∂H_F^s/∂X1,s + (∂H_F^s/∂X2,s)(∂X2,s/∂X1,s) + [ (∂H_F^s/∂u_L^s) + (∂H_F^s/∂X2,s)(∂X2,s/∂u_L^s) ] (∂u_L^s/∂X1,s) + ∂H_F^{s−1}/∂X1,s = 0 - ∂H_F^s/∂λ_{s+1} = 0 - Derivatives from (27): - ∂X2,s/∂X1,s = −J' - ∂X2,s/∂u_F,s = −K_F' - ∂X2,s/∂u_L,s = −K_L' - Follower reaction (linear rule): - u_F,t = −F_F X1,t − L u_L,t (equation (28)) - Matrices F_F and L depend on J, K_F and K_L of the ultimate follower. - Leader's (Stackelberg) problem: - Constrained period welfare loss: - H_L^s = β_L^{s−t} [ X_s' Q_L X_s + 2 X_s' U_L u_s + u_s' R_L u_s + μ_{s+1}' (A_11 X1,s + A_12 X2,s + B_1 u_s − X1,s+1) ] - μ_{s+1} is a vector of Lagrange multipliers; β_L is the leader discount rate. - First-order conditions (FOCs): - ∂H_L^s/∂u_L^s + (∂H_L^s/∂X2,s)(∂X2,s/∂u_L^s) + [ (∂H_L^s/∂u_F^s) + (∂H_L^s/∂X2,s)(∂X2,s/∂u_F^s) ] (∂u_F^s/∂u_L^s) = 0 - ∂H_L^s/∂X1,s + (∂H_L^s/∂X2,s)(∂X2,s/∂X1,s) + [ (∂H_L^s/∂u_F^s) + (∂H_L^s/∂X2,s)(∂X2,s/∂u_F^s) ] (∂u_F^s/∂X1,s) + ∂H_L^{s−1}/∂X1,s = 0 - ∂H_L^s/∂μ_{s+1} = 0 - Leader internalises follower behaviour and economy evolution: - ∂u_F^s/∂u_L^s = −L' - ∂u_F^s/∂X1,s = −F_F' - Leader optimal policy rule (linear): - u_L,t = −F_L X1,t - Matrix F_L depends on J, K_F, K_L of the ultimate follower and F_F, L of the follower. - Solution method: matrices J, K_F, K_L, F_F, L, and F_L found using an iterative numerical procedure. ### Nash - Setup: In Nash equilibrium each policymaker treats the other's reaction function parametrically and does not internalise it. - Constrained period loss for player i (i = F, L): - H_i^s = β_i^{s−t} [ X_s' Q_i X_s + 2 X_s' U_i u_s + u_s' R_i u_s + λ_{i,s+1}' (A_11 X1,s + A_12 X2,s + B_1 u_s − X1,s+1) ] - First-order conditions (identical form for both players): - ∂H_i^s/∂u_i^s + (∂H_i^s/∂X2,s)(∂X2,s/∂u_i^s) = 0 - ∂H_i^s/∂X1,s + (∂H_i^s/∂X2,s)(∂X2,s/∂X1,s) + [ (∂H_i^s/∂u_j^s) + (∂H_i^s/∂X2,s)(∂X2,s/∂u_j^s) ] (∂u_j^s/∂X1,s) + ∂H_i^{s−1}/∂X1,s = 0 - ∂H_i^s/∂λ_{i,s+1} = 0 - Assume reaction functions: - u_i,t = −F_i X1,t - Solution method: optimal feedback coefficients F_i found using an iterative procedure. *Blake and Kirsanova (2006) extension of Currie and Levine (1993) as presented in the source PDF.* --- _Source: https://www.imf.org/-/media/websites/imf/imported-full-text-pdf/external/pubs/ft/wp/2008/_wp08164.pdf_