## _wp07206

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---

### Introduction and motivation
- Opening quotation: "Central banks are often accused of being obsessed with ináation. This is untrue. If they are obsessed with anything, it is with Öscal policy." (Mervyn King, 1995).
- Conventional wisdom challenged:
  - Well-designed monetary policy satisfying the Taylor principle stabilises cost-push shocks; fiscal policy is often seen as limited to ensuring fiscal solvency.
  - Leeper (1991) and related literature show that if fiscal policy fails to ensure debt sustainability, optimal monetary policy may become passive (violating the Taylor principle) — the fiscal theory of the price level.
- Central claim of paper:
  - Under discretionary policy, even with optimally set monetary and fiscal policy, "stabilisation bias" implies conventional wisdom can be inappropriate.

### Stabilisation bias: conceptual findings
- Two forms of time-inconsistency-related inefficiency:
  - Level bias (Barro and Gordon 1983): attempts to attain an excess level for target variables.
  - Stabilisation (dynamic) bias (Currie and Levine 1987, 1993): inability to commit to time-inconsistent policy paths that manipulate private expectations.
- Inflation stabilisation bias:
  - Under commitment, promising future tight policy aids current inflation reduction; under discretion, inability to commit leads to suboptimally slow disinflation and stronger initial interest rate hikes ("ináation stabilisation bias").
- Debt stabilisation bias:
  - Under commitment, government debt may follow a random walk (Benigno and Woodford 2003; Schmitt-Grohe and Uribe 2004; Leith and Wren-Lewis 2007).
  - Under discretion, repeated incentive to cut debt in the first period forces public debt to be returned to its initial value after a shock — a costly "debt stabilisation bias".
  - Mechanism: when debt is above pre-shock level, each period there is an incentive to cut debt because prior inflation expectations are already set.

### Key contributions and novel analysis
- Generalisations of prior work:
  - Adds inflation persistence via rule-of-thumb price setters (Steinsson (2003) approach).
  - Presents analysis of optimal policy under discretion when, under commitment, control of both inflation and debt would be time inconsistent.
- Main results hinge on:
  - Steady-state ratio of debt to output (B).
  - Degree of inflation persistence (!).
- Summary:
  - Low steady-state debt: adjustment burden shared by fiscal and monetary policy; monetary policy fulfills the Taylor Principle; fiscal policy more active under discretion than under commitment (at all levels of inflation persistence).
  - High steady-state debt:
    - Low inflation persistence: optimal discretionary monetary policy may cut interest rates in the first period in response to a cost-push shock (violating the Taylor Principle). Debt stabilisation bias dominates inflation stabilisation bias.
    - High inflation persistence: initial interest-rate cuts become infeasible if inflation cannot be controlled thereafter; beyond a threshold in persistence, interest rates must rise in response to a cost-push shock.
  - Fiscal policy cuts spending very strongly in a high debt economy to assist a debt-constrained monetary authority.
- Threshold property:
  - The threshold of inflation persistence beyond which falling first-period interest rates cease to be optimal is higher for larger steady-state values of debt.

### Model structure and primitives (compact)
- Core setup: microfounded closed-economy New Keynesian model with rule-of-thumb price setters and fiscal policy (G, constant tax rate , nominal debt B).
- Representative consumer (linearised):
  - Euler equation: ^C_t = E_t ^C_{t+1} - (i_t - E_t _{t+1})  (equation (4)).
  - Asset accumulation (linearised): ^A_{t+1} = ^i_t + 1 [^A_t - _t + (1-)/A ( ^Y_t -  ^C_t )]  (equation (5)).
- Price setting and hybrid Phillips curve:
  - Backward-looking pricing rule: P^B_t = P^_{t-1} _{t-1} [Y_{t-1}/Y^n_{t-1}]^  (equation (6)).
  - Hybrid Phillips curve (linearised): ^_t = _f E_t ^_{t+1} + _b ^_{t-1} + _c ^C_t + _{y0} ^Y_t + _{y1} ^Y_{t-1} + ^_t  (equation (7)).
  - Coefficients explicitly: _f = (
 + !(1-
+
)), _b = !/(
 + !(1-
+
)), _c = (1-
) (1-
) (1-!) / ((
 + !(1-
+
)) ( +")) , _{y1} = (1-
)!/(
 + !(1-
+
)) , _{y0} = (1-
) (1-
) (1-!) / ((
 + !(1-
+
)) ( +")) - (1-
) 
 !/(
 + !(1-
+
)) ,  = (1-
) ( + ) / (
  ( + ")), = v_y / v_{yy} Y.
- Aggregate demand and fiscal dynamics:
  - Y_t = C_t + G_t  (equation (8)); steady-state G = (1-) Y.
  - Linearised real debt law of motion: ^B_{t+1} = ^i_t + 1 [  ^B_t - _t + (1-)/B ^G_t - /B ^Y_t ]  (equation (10)).
- Compact linearised system:
  - c_t = E_t c_{t+1} - (i_t - E_t _{t+1})  (equation (11)).
  - _t = _f E_t _{t+1} + _b _{t-1} + _c c_t + _{y0} y_t + _{y1} y_{t-1} + _t  (equation (12)).
  - y_t = (1-) g_t +  c_t  (equation (13)).
  - b_{t+1} = i_t + 1 [  b_t - _t + (1-) B g_t - /B y_t ]  (equation (14)).
- Social welfare / period loss function (second-order approximation):
  - Aggregate loss: W_s = _c c_s^2 + _g g_s^2 + _y y_s^2 + _s^2 + _2 (4_s)^2 + _3 y_{s-1}^2 + _4 y_{s-1} 4_s + O(3)  (equation (15)).
  - Coefficients: _c =   (1-
) (1-
) / ("( + )), _g = (1-)  (1-
) (1-
) / ("( + )), _y = 1/(  (1-
) (1-
) / ("( + )), _2 = ! (1-!), _3 = ! (1-
)^2 ^2 / (2 (1-!) ), _4 = -2 ! (1-
)  / (1-!).

### Social welfare function: structure and interpretation
- Three component interpretation:
  - Demand terms (c, g and y): smoothing private and public consumption and disutility from hours worked.
  - Inflation level term (): cost of ináation via price dispersion with sticky prices.
  - Smoothing terms (4 and y_{-1}): arise with ináation persistence because current ináation depends on past output and ináation for rule-of-thumb price setters.
- Compensating consumption interpretation:
  - Loss is expressed as the required steady-state fall in consumption that would balance the welfare gain from eliminating variability in consumption, government spending and leisure.
  - Appendix 3 derives this compensating consumption measure.

### Calibration (quarterly)
- Parameters:
  -  = 0:99
  -  = 0:5
  - = 2
  - " = 5
  -
 = 0:75 (Calvo parameter; prices on average set once a year).
  - Steady-state share of private consumption in output:  = 0:75.
- Two key parameters varied: proportion of rule-of-thumb price setters (!) and steady-state ratio of debt to output (B).
- Examples of estimated persistence of ináation (b):
  - Gali and Gertler (1999): b = 0:3.
  - Mehra (2004) and Rudebusch (2002): b = 0:7.

### The proportion of rule-of-thumb price setters (!)
- Effects of raising !:
  - Reduces forward-lookingness and increases backward-lookingness in the Phillips curve (reduces f and raises b).
  - Extreme case != 0 yields the standard New Keynesian Phillips curve (f = 1; b =  y1 = 0).
- Decomposition of ináation:
  - t = F_t + B_t + mark-up shock; where F_t = f E_t _{t+1} + c c_t + y0 y_t and B_t = b _{t-1} + y1 y_{t-1}.
- Comparative advantage:
  - Monetary policy relatively more effective at controlling forward-looking inflation; fiscal policy has comparative advantage in controlling backward-looking inflation.
- High ! increases weights 2, 3, 4 in the loss function.
- No solution for polar case != 1 (Phillips curve converges to accelerationist form; social welfare undefined).

### Steady-state debt (B) regimes
- B determines leverage of monetary policy over interest payments and relative effectiveness of policies on debt.
- Two benchmark regimes:
  - "Low" debt economy: B = 0:1 (implies annual debt to GDP ratio of 2:5%).
  - "High" debt economy: B = 0:4 (implies annual debt to GDP ratio of 10%).

### Solving for optimal policy: commitment vs discretion
- Canonical linear-quadratic setup with partitioned states:
  - Predetermined/state variables X1;t = (_{t-1}; _{t-1}; y_{t-1}; b_t)'.
  - Jump variables X2;t = (_t; c_t)'.
  - Instruments U_t = (i_t; g_t)'.
- Commitment:
  - Lagrange multipliers ^ and ^ associated with predetermined and non-predetermined variables.
  - Key property: Lagrange multiplier on debt follows a random walk under commitment ( b_{s+1} =  b_s).
  - Time inconsistency: initial 0 = 0, but _s need not remain zero; magnitude of _s measures time inconsistency.
- Discretion:
  - Time consistency requires _s = 0 for all s.
  - Policymaker chooses a reaction function X2;t = G X1;t K U_t and solves for U_t = F X1;t and X2;t = C X1;t numerically.
  - Discretionary feedback coefficients denoted ^D components (e.g., ^D_ etc.).

### Simulations and comparative statics
- Purpose: impulse responses to a unit cost-push shock under optimal policy.
- Cases simulated:
  - Debt: low (B = 0:1) and high (B = 0:4).
  - Phillips curve specifications: New Keynesian (! = 0), hybrid (! = 0:75), predominantly backward-looking (! = 0:99).
- Reported outcomes (Table 1 summary items preserved):
  - Absolute welfare loss ("Loss") under commitment and discretion.
  - Excess loss over commitment expressed as %W and as %C (percentage of steady-state consumption foregone).
  - Maximum eigenvalue of predetermined-variable system (indicator of speed of debt adjustment).
  - Optimal feedback coefficients under commitment and discretion, including first-period instrument responses to the cost-push shock:  C ,  C ,  D ,  D .

### The low debt economy (B = 0:1) — selected quantitative findings
- Under optimal commitment with a New Keynesian Phillips curve:
  - Taylor Principle is fulfilled in the first period: C >1 (nominal interest rate rise increases real interest rate).
  - Fiscal policy almost inactive: C = -0:11.
  - Debt follows a random walk; CMax = 1.
  - Debt sustainability achieved via permanently lower government spending (C b <0 and C b <0).
- Selected table entries (Low Debt block):
  - WelfareLoss (Low Debt): 0.18 (commitment) and 0.28 (discretion).
  - % W entries: -55.6, -6.9, -0.22.
  - % C entries: -0.09, -0.28, -0.29.
  - Monetary feedback coefficients (shock i examples across columns): 2.99, 6.27, 13.95, 24.63, 17.69, 28.48.
  - Fiscal shock feedback i examples: -0.11, -4.10, -4.48, -5.73, -8.02, -8.40.
  - Monetary inflation feedback i examples: 0.00, 0.00, 3.68, 4.26, 6.10, 6.16.
  - Fiscal debt feedback i b examples: -0.005, -0.107, -0.005, -0.011, -0.005, -0.006.
- Time inconsistency under commitment:
  - Predetermined Lagrange multipliers _s and c_s nonzero during disinflation for all inflation persistence levels.
  - _s negative for s >0: incentive to renege by raising inflation after period 0.
  - c_s negative for s >0 (equivalently  b_s positive): incentive to reduce debt under commitment, generating the random-walk property.
  - Welfare gains from reneging on inflation largest in first period; gains from reneging on consumption/debt follow a random walk.

### Discretionary policy in the low debt economy
- Under a New Keynesian Phillips curve:
  - Discretion produces classic inflation stabilisation bias: less effective inflation control than commitment.
  - Policymaker under discretion raises interest rates strongly in the first period then returns them to zero faster than under commitment — larger initial recession and slower subsequent disinflation.
  - Debt does not follow a random walk under discretion; it returns to initial steady-state level (Dmax = 0:73).
  - Combined biases imply loss equivalent to a 0:09% fall in steady-state consumption.
- Adjustment mechanics:
  - First-period debt adjustment mostly through lower spending in calibration.
  - Fiscal policy far more active under discretion than under commitment to ensure fast debt convergence.

### Inflation persistence effects (low debt)
- Increasing != (more backward-looking):
  - Dynamics under discretion converge toward commitment outcomes.
  - Maximum eigenvalue rises: 0:73 → 0:98 → 0:99 as != goes 0 → 0:75 → 0:99.
  - Welfare costs of discretion rise to 0:28% and 0:29% of steady-state consumption for hybrid and backward-looking Phillips curves respectively.

### The high debt economy (B = 0:4) — selected findings
- Higher B increases monetary policy leverage over debt via interest payments.
- For a New Keynesian Phillips curve in high debt:
  - Optimal discretionary policy may cut the interest rate in the first period (D <0) in response to a cost-push shock, reducing debt via lower real interest payments and higher inflation.
  - Interaction of biases:
    - Inflation stabilisation bias calls for initial interest rate hike.
    - Debt stabilisation bias (when debt is high) calls for initial interest rate cut; debt bias can dominate under low inflation persistence.
  - Dynamics: interest rates may rise strongly in the second period to control inflation; fiscal policy cuts spending aggressively in the first period.
  - Example calibrated maximum eigenvalue: 0:11 (faster convergence than low-debt case).
  - Debt falls below steady-state on impact then returns from below to initial level.
- Inflation persistence effects in high debt:
  - As != increases, controlling inflation with initial interest-rate cuts becomes harder.
  - For hybrid Phillips curve, interest rates may still fall but require very large spending cuts.
  - For strongly backward-looking Phillips curve, monetary policy cannot cut interest rates in the first period; under discretion the planner raises the interest rate in the first period (C >0 in Table 1).
  - Maximum eigenvalue under discretion remains considerably below unity (e.g., 0:57) even for high !=.
  - Welfare costs larger: for a hybrid Phillips curve excess loss over commitment ≈ 0:79% of steady-state consumption; for predominantly backward-looking ináation cost ≈ 1:02%.

### Comparative results and thresholds (summary)
- Figure 7 description (qualitative results preserved):
  - Under commitment: planner always raises interest rate and cuts spending (C >0; C <0 for all ! and B).
  - Under discretion: fiscal policy more active than under commitment; activism rises with B.
  - For B ≲ 0:2, under discretion planner raises interest rate in response to cost-push shocks and fulfills Taylor principle for all !.
  - For B > 0:2, monetary policy may be forced to cut interest rates to stabilise debt if Phillips curve is predominantly forward-looking; threshold != at which perverse response vanishes depends on B (e.g., for B = 0:25 threshold about != 0:1; for B = 0:3 threshold about != 0:5).
  - For very high B (B > 0:5) violation of Taylor principle in the first period remains optimal at all ! (including != 0:99).
- Figure 8 description:
  - Maximum eigenvalue under discretion (Dmax) moves toward unity for low B as system becomes more backward-looking, but remains well below unity for high B.
  - Excess loss of discretion over commitment (as % of steady-state consumption) is larger for larger B.
  - For empirically plausible hybrid Phillips curve the inability to commit imposes a welfare cost ≈ 0:6% fall in steady-state consumption.
  - As persistence rises further, excess loss may fall for intermediate B, but for very high B inability to commit becomes increasingly costly.

### Policy-relevant conclusions and recommendations
- Under discretion, public debt must be returned to its pre-shock level for time consistency; therefore optimal fiscal policy is more active under discretion than under commitment at all levels of inflation persistence and debt to assist a constrained monetary authority.
- Monetary policy should fulfill the Taylor Principle in an economy with inflation persistence, but exceptions arise:
  - In high-debt, low-inflation-persistence environments, optimal discretionary monetary policy may cut interest rates initially, violating the Taylor Principle.
- Institutional implications:
  - Institutions that promote commitment of both monetary and fiscal policy (e.g., announced targets, long tenure of policymakers) are highly desirable because gains from commitment are larger when public debt is present — especially for high debt.
- Quantitative sensitivity:
  - Thresholds depend critically on modelling features: debt maturity (one-period maturity here), nominal versus real debt denomination, and intertemporal elasticity of substitution.

### Phillips curve derivation (appendix highlights)
- Derivation uses household and firm FOCs and flexible-price equilibrium linearisation.
- Forward-looking optimal price linearised expression involves discounted sums of future inflation and real marginal-cost terms; mark-up shocks ^_s enter the Phillips curve.
- Backward-looking price setters use a rule: P^B_t depends on past prices and past output; combining forward-looking and backward-looking pricing yields the hybrid Phillips curve (equation (7)).
- Key analytic observations:
  - Mark-up shocks influence inflation dynamics.
  - Constant wage-income tax rate ξ does not affect dynamic Phillips curve.

*Source: _wp07206 - References / Appendix excerpts from the IMF working paper content provided.*

### References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   50

### _wp07206 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   50

### Tables and Figures (inventory)
- Table 1. Optimal Monetary and Optimal Fiscal Policy Summary for Commitment (i=C) and Discretion (i=D).
- Figure 1. Optimal Policy with a New Keynesian Phillips curve (!= 0) in a ílowí debt economy (B= 0:1).
- Figure 2. Optimal Policy with a hybrid Phillips curve (!= 0:75) in a ílowí debt economy (B= 0:1).
- Figure 3. Optimal Policy with a predominantly backward looking Phillips curve (!= 0:99) in a ílowídebt economy (B= 0:1).
- Figure 4. Optimal Policy with a New Keynesian Phillips curve (!= 0) in a íhighí debt economy (B= 0:4).
- Figure 5. Optimal Policy with a hybrid Phillips curve (!= 0:75) in a íhighídebt economy (B= 0:4).
- Figure 6. Optimal Policy with a predominantly backward looking Phillips curve (!= 0:99) in a íhighídebt economy (B= 0:4).
- Figure 7. Monetary (i) and Öscal (i) feedback coe¢ cients on the cost-push shock under optimal commitment (i=C) and discretion (i=D).
- Figure 8. The excess loss of discretion over commitment (in%of steady-state consumption foregone) and the maximum eigenvalue (vDmax) under discretion.

### Major themes from Introduction and Motivation
- Opening quotation: "Central banks are often accused of being obsessed with ináation. This is untrue. If they are obsessed with anything, it is with Öscal policy." (Mervyn King, 1995).
- Conventional wisdom: well-designed monetary policy that satisfies the Taylor principle stabilises the economy to cost-push shocks; fiscal policy's role is often limited to ensuring fiscal solvency.
- Leeper (1991) and related literature: when fiscal policy fails to ensure debt sustainability, optimal monetary policy may become passive (violating the Taylor principle) to stabilise debt — the fiscal theory of the price level.
- Paper's central claim: under discretionary policy, even with both monetary and fiscal policy set optimally, "stabilisation bias" implies conventional wisdom can be inappropriate.

### Stabilisation bias (conceptual findings)
- Two forms of time-inconsistency-related inefficiency:
  - Level bias (Barro and Gordon 1983): policymaker attempts to attain an excess level for target variables.
  - Stabilisation (dynamic) bias (Currie and Levine 1987, 1993): inability to commit to time-inconsistent policy paths that would manipulate private sector expectations.
- Inflation stabilisation bias:
  - Under commitment, reducing current inflation can be aided by promising future tight policy — a time-inconsistent promise.
  - Under discretion, inability to make such promises leads to suboptimally slow disinflation and stronger initial interest rate hikes (Woodford (2003b) terminology: "ináation stabilisation bias").
- Debt stabilisation bias:
  - Under commitment, government debt follows a random walk in some models (Benigno and Woodford 2003; Schmitt-Grohe and Uribe 2004; Leith and Wren-Lewis 2007).
  - Under discretion, because of time-inconsistency (incentive to cut debt in the first period repeatedly), public debt must be returned to its initial value after a shock — a costly bias labelled "debt stabilisation bias".
  - The mechanism: when debt is above pre-shock level, there is an incentive each period to cut debt because earlier inflation expectations are already set.

### Key contributions and novel analysis
- Generalises Benigno and Woodford (2003) and Leith and Wren-Lewis (2007) by:
  - Adding inflation persistence via rule-of-thumb price setters (Steinsson (2003) approach).
  - Presenting analysis of optimal policy under discretion when, under commitment, control of both inflation and debt would be time inconsistent.
- Main results on how outcomes under optimal discretion depend on:
  - The steady-state ratio of debt to output (B).
  - The degree of inflation persistence.
- With low steady-state debt:
  - Burden of adjustment shared by fiscal and monetary policy.
  - Monetary policy fulfills the Taylor Principle; fiscal policy is more active under discretion than under commitment (applies at all levels of inflation persistence).
- With high steady-state debt:
  - Two cases:
    - Low inflation persistence: optimal discretionary monetary policy may cut interest rates in the first period in response to a cost-push shock (violating the Taylor Principle), replicating Leith and Wren-Lewis (2007). Debt stabilisation bias dominates inflation stabilisation bias.
    - High inflation persistence: cutting interest rates in the first period becomes infeasible if inflation cannot be controlled thereafter; beyond a threshold of inflation persistence, interest rates must rise in response to a cost-push shock.
  - Fiscal policy cuts spending very strongly in a high debt economy to assist the debt-constrained monetary authority.

### Summary conclusions relevant for policy
- Under discretion, because debt does not follow a random walk, optimal fiscal policy is more active under discretion than under commitment for stabilising inflation and debt — this holds at all levels of inflation persistence and all levels of debt.
- Monetary policy should fulfill the Taylor Principle and fiscal policy should play an active role in stabilising cost-push shocks in an economy with public debt and inflation persistence — though exceptions arise in high-debt/low-inflation-persistence regimes where first-period interest-rate cuts can be optimal under discretion.
- The threshold value of inflation persistence beyond which falling first-period interest rates cease to be optimal is higher for larger steady-state values of debt (i.e., stronger debt-stabilisation constraints raise that threshold).

### Model structure and primitives (compact)
- Core setup: microfounded closed-economy New Keynesian model with:
  - Rule-of-thumb price setters (Steinsson 2003) introducing inflation persistence.
  - Fiscal policy: government spending G, constant income tax rate , nominal debt B evolving over time.
  - Social welfare given by a second-order approximation to aggregate utility.
- Representative consumer problem:
  - Lifetime maximisation: max_{ {C_s,h_s} } E_t sum_{s=t}^∞ _s [u(C_s) + f(G_s) - v(h_s(z))]  (equation (1)).
  - Euler equation (linearised): ^C_t = E_t ^C_{t+1} - (i_t - E_t _{t+1})  (equation (4)).
  - Asset accumulation linearised: ^A_{t+1} = ^i_t + 1 [^A_t - _t + (1-)/A ( ^Y_t -  ^C_t )]  (equation (5)).
- Price setting and hybrid Phillips curve:
  - Mix of Calvo contracting and rule-of-thumb price setters: forward-looking and backward-looking components.
  - Backward-looking pricing rule: P^B_t = P^_{t-1} _{t-1} [Y_{t-1}/Y^n_{t-1}]^  (equation (6)).
  - Hybrid Phillips curve (linearised): ^_t = _f E_t ^_{t+1} + _b ^_{t-1} + _c ^C_t + _{y0} ^Y_t + _{y1} ^Y_{t-1} + ^_t  (equation (7)).
  - Coefficients defined (explicit forms preserved): _f = (
 + !(1-
+
)), _b = !/(
 + !(1-
+
)), _c = (1-
) (1-
) (1-!) / ((
 + !(1-
+
)) ( +")) , _{y1} = (1-
)!/(
 + !(1-
+
)) , _{y0} = (1-
) (1-
) (1-!) / ((
 + !(1-
+
)) ( +")) - (1-
) 
 !/(
 + !(1-
+
)) ,  = (1-
) ( + ) / (
  ( + ")), and labor elasticity parameter  = v_y / v_{yy} Y.
- Aggregate demand and fiscal dynamics:
  - Output identity: Y_t = C_t + G_t  (equation (8)); steady-state G = (1-) Y.
  - Linearised income identity: ^Y_t = (1-) ^C_t +  ^G_t.
  - Nominal debt evolution: B_{t+1} = (1 + i_t) B_t + P_t G_t -  P_t Y_t  (equation (9)).
  - Linearised real debt law of motion: ^B_{t+1} = ^i_t + 1 [  ^B_t - _t + (1-)/B ^G_t - /B ^Y_t ]  (equation (10)).
- Compact system of linearised gap equations:
  - c_t = E_t c_{t+1} - (i_t - E_t _{t+1})  (equation (11)).
  - _t = _f E_t _{t+1} + _b _{t-1} + _c c_t + _{y0} y_t + _{y1} y_{t-1} + _t  (equation (12)).
  - y_t = (1-) g_t +  c_t  (equation (13)).
  - b_{t+1} = i_t + 1 [  b_t - _t + (1-) B g_t - /B y_t ]  (equation (14)).
- Social welfare / period loss function (second-order approximation):
  - Aggregate second-order loss: W_s = _c c_s^2 + _g g_s^2 + _y y_s^2 + _s^2 + _2 (4_s)^2 + _3 y_{s-1}^2 + _4 y_{s-1} 4_s + O(3)  (equation (15)).
  - Coefficients given explicitly in terms of model primitives: _c =   (1-
) (1-
) / ("( + )), _g = (1-)  (1-
) (1-
) / ("( + )), _y = 1/(  (1-
) (1-
) / ("( + )), _2 = ! (1-!), _3 = ! (1-
)^2 ^2 / (2 (1-!) ), _4 = -2 ! (1-
)  / (1-!).

### Equations and numbered references preserved
- The summary preserves equation numbering and referenced results: (1) consumer maximisation; (2)-(5) consumption and asset relations; (6) rule-of-thumb price setting; (7) hybrid Phillips curve; (8)-(10) aggregate demand and fiscal dynamics; (11)-(14) compact system; (15) welfare/loss function.

*Source: _wp07206 - References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .   50*

### Appendix 2. provides the details of this derivation. We see that the social welfare function

### _wp07206 - Appendix 2. provides the details of this derivation. We see that the social welfare function

### Social welfare function: structure and interpretation
- The social welfare function consists of three terms in a model with Öscal policy and ináation persistence:
  - Demand terms (c, g and y): arise because the representative consumer has an incentive to smooth both private and public consumption and dislikes áuctuations in hours worked.
  - Inflation level term (): captures the cost of ináation with sticky prices; higher ináation induces greater price dispersion across industries, which is costly.
  - Smoothing terms (4 and y1): arise with ináation persistence because current ináation depends on past output and ináation with rule-of-thumb price setters; past values affect price dispersion similarly to current ináation.
- Compensating consumption interpretation:
  - The loss is expressed in terms of the required steady-state fall in consumption that would balance the welfare gain from eliminating variability of consumption, government spending and leisure.
  - Appendix 3. explains how to derive this compensating consumption measure.

### Calibration (G.)
- Time period: quarter.
- Parameters set as:
  -  = 0:99
  -  = 0:5
  - = 2
  - " = 5
  -
 = 0:75 (Calvo parameter implies prices are on average set once a year).
  - Steady-state share of private consumption in output:  = 0:75.
- Two key parameters varied throughout the paper due to disagreement in the literature:
  - Proportion of rule-of-thumb price setters (!).
  - Steady-state ratio of debt to output (B).
- Estimates of persistence of ináation (b) vary: examples given
  - Gali and Gertler (1999): b = 0:3 (predominantly forward-looking).
  - Mehra (2004) and Rudebusch (2002): b = 0:7 (predominantly backward-looking).

### The proportion of rule-of-thumb price setters (!)
- Raising ! has important model effects:
  - Higher ! reduces degree of forward relative to backward-lookingness in the Phillips curve (reducesf and raisesb).
  - Extreme case ! = 0 produces standard New Keynesian Phillips curve without a backward-looking component (f = 1 andb = y1 = 0).
- Decomposition of ináation:
  - t = forward-looking component (F t =fEtt+1 +c c t +y0 y t), backward-looking component (B t =bt1 +y1 y t1), plus mark-up shock.
  - For given ináation expectations, forward-looking ináation depends on consumption and output gap because Calvo price setters base decisions on real marginal cost.
  - Backward-looking price setters base decisions on past output, not marginal cost.
- Relative effectiveness of monetary vs fiscal policy (holding expectations and consumption constant):
  - F c = B c = c y1 + y0 y1
  - F g = B g = y0 y1
  - Implication: as long as c >0, monetary policy is relatively more effective in controlling forward-looking component of ináation than the backward-looking component compared to fiscal policy. Monetary policy has a "comparative advantage" in controlling forward-looking ináation; fiscal policy has comparative advantage in controlling backward-looking ináation.
- High ! increases weights on terms 
 s and y s1 in the microfounded loss function: weights 2, 3 and 4 dominate for high ! because more firms base pricing on past output and ináation.
- No solution in polar case != 1: Phillips curve converges to accelerationist form but social welfare function is not well defined in this limit; cannot nest simple backward-looking models (e.g., Bean (1998) or Kirsanova et al (2005)).

### The steady-state value of debt (B)
- B determines leverage of monetary policy over interest payments and thus relative effectiveness of monetary vs fiscal policy in affecting debt.
- Two regimes defined:
  - "Low" debt economy: debt to output ratio B = 0:1 (implies annual debt to GDP ratio of 2:5%).
  - "High" debt economy: debt to output ratio B = 0:4 (implies annual debt to GDP ratio of 10%).
- The choice of steady-state debt ratio plays a crucial role in the analysis.

### Solving for optimal policy (III.)
- Policymaker minimises the social loss by choosing interest rate and spending subject to evolution of the economy (equations (11) to (14)).
- Optimal policy differs under commitment and discretion:
  - Under commitment: policymaker can credibly commit to future policies and affect expectations.
  - Under discretion: policymaker re-optimises every period and treats non-predetermined variables parametrically.
- Canonical linear-quadratic setup (following Currie and Levine (1993)):
  - State partition: X1;t (predetermined/state variables) and X2;t (forward-looking/jump variables).
  - Instrument vector Ut dimension k, shock "t+1 white noise.
  - Model in canonical form: X t+1 = A X t + B U t + E " t+1
  - For this model:
    - X1;t = ( t ;  t1 ; y t1 ; b t )0
    - X2;t = ( t ; c t )0
    - U t = (i t ; g t )0
    - " t+1 = (  t+1 ; 0 ; 0 ; 0 )0
  - Period loss L t = G0 t Q G t with G t = C Z t and Z t = (X01;t ; X02;t ; U0 t )0 leading to representation L t = Z0 t  Z t.
  - Weight matrices (including off-diagonals such as 12 and 21) define smoothing weights; Appendix B defines matrices A, B, E and the weight matrices in terms of structural parameters.

### Optimal policy under commitment (B.)
- Constrained loss function H C defined with Lagrange multipliers ^ and ^ associated with predetermined and non-predetermined variables respectively.
- First-order conditions (for s > 0):
  - State variables (X1;s) yield conditions (equations (19)–(22)).
  - Equation (22): @H C @b s =  b s+1  b s = 0, implying the Lagrange multiplier for debt follows a random walk:  b s+1 =  b s.
  - Jump variables (X2;s) yield conditions (24) and (25).
  - Instruments Ut conditions yield (26) and (27).
  - Important relation from (26) and (28):  b s+1 =   c s+1. Along optimal commitment path, non-predetermined Lagrange multiplier on debt is proportional to predetermined Lagrange multiplier of consumption.
- Initial period (s = 0):
  - First-period Lagrange multipliers for jump variables set to zero (0 = 0).
  - First-order conditions for jump variables differ at s = 0: see (29) and (30).
- Solution structure:
  - Certainty-equivalent solution can be written (31) and (32) with matrices  and	 solved using initial conditions X1;0 and terminal conditions for non-predetermined variables; algorithm of Soderlind (1999) can be used.
  - Definition of 11 as feedback coefficients on state variables under optimal commitment: 11 = (   C   C   C y  C b  C   C   C y  C b  ) (example notation preserved).
- Time inconsistency:
  - First-order conditions for non-predetermined variables (29) and (30) highlight time inconsistency: initial imposition 0 = 0 implies that for s > 0,  s need not remain zero; policymaker re-optimising at s > 0 will set  s = 0, reneging on earlier plan.
  - Magnitude of  s measures extent of time inconsistency; two sources: ináation and consumption control.
  - Negative values of   s and  c s indicate social loss under optimal commitment could be reduced by raising ináation and consumption (i.e., raising ináation and lowering debt).
  - Cost-to-go under commitment (L OPC s) can be written as:
    - L OPC s = 1 2 [ tr(   21 X1;s X01;s ) + tr(  22  s 0 s ) ]
  - A gain from reneging at times > 0 exists equal to tr(  22  s 0 s ); Levine (1988) shows diagonal entries of  22 are negative, so incentive to renege exists along trajectory.

### Optimal policy under discretion (C.)
- Discretion requires time consistency; policymaker takes non-predetermined variables as given ( s = 0 for all s).
- Public reaction function postulated (guess):
  - X2;t = G X1;t K U t
  - Substitute into Lagrangean and solve; first-order conditions outlined in Appendix 2.
- Certainty-equivalent solution converges to:
  - U t = F X1;t
  - X2;t = C X1;t
  - F and C found numerically; dynamics of X1;t found by substituting into canonical law of motion.
- Definition of discretionary instrument feedback coefficients:
  - F = (   D   D   D y  D b  D   D   D y  D b  )

### Simulating optimal policy (IV.)
- Purpose: analyse behaviour of optimal monetary and fiscal policy via impulse responses to a unit cost-push under optimal policy.
- Cases analysed:
  - Debt regimes: low (B = 0:1) and high (B = 0:4).
  - Three Phillips curve specifications:
    - New Keynesian Phillips curve (! = 0)
    - Hybrid Phillips curve (! = 0:75)
    - Predominantly backward-looking Phillips curve (! = 0:99)
- Reported outcomes (Table 1, summary description):
  - Absolute welfare loss ("Loss") and excess loss over commitment.
  - Excess loss expressed as percentage loss above commitment (%W) and as percentage of steady-state consumption foregone (%C).
  - Maximum eigenvalue of system of predetermined variables (indicative of speed of debt adjustment).
  - Optimal feedback coefficients under commitment and discretion (reaction function coefficients).
  - First-period instrument responses to unit cost-push equal to feedback coefficients onto cost-push shock:
    - Under commitment: first-period interest rate and spending movements are given by  C  and  C  respectively.
    - Under discretion: first-period interest rate and spending movements are given by  D  and  D  respectively.
- Analysis plan:
  - Section A: Optimal policy under commitment (figures 1–3 plot responses labelled C).
  - Section B: Optimal policy under discretion (responses labelled D).
  - Section C: Extension to more general values of debt and ináation persistence.

*Source: _wp07206 - Appendix 2. provides the details of this derivation. We see that the social welfare function*

### 1.  The Low Debt Economy

### 1.  The Low Debt Economy

### Optimal commitment policy in a low debt economy (New Keynesian Phillips curve, B= 0:1)
- Policymaker response:
  - The policymaker raises the nominal interest rate to control inflation; nominal interest rates rise sufficiently strongly to increase the real interest rate (C >1), i.e., the Taylor principle is fulfilled in the first period.
  - Monetary policy is highly effective in achieving disinflation by steering inflation expectations through commitment to and delivery of contractionary policy.
- Fiscal policy response:
  - Optimal fiscal policy is almost inactive to a unit cost-push shock: C is negative but small at -0:11, so the fiscal authority leaves stabilization almost entirely to monetary policy.
  - Fiscal instrument movements are costly because they induce a suboptimal quantity of public goods provision; fiscal policy primarily ensures debt sustainability.
- Debt dynamics and stability:
  - Debt accumulates strongly through persistently higher interest rates and corresponding fall in income tax revenues; following the shock, debt remains permanently higher and follows a random walk.
  - Column (1) in Table 1: maximum eigenvalue for the simulated system equals one (CMax = 1).
  - Debt sustainability is achieved through permanently lower government spending via negative fiscal feedback on debt (C b <0 and C b <0).
  - The policy mix implies that public debt, government spending, consumption, and output converge to a new steady state.

### Key quantitative outcomes reported (low debt, B= 0:1; select entries)
- WelfareLoss (Low Debt): 0.18 (commitment) and 0.28 (discretion) in the table header context.
- % W: -55.6, -6.9, -0.22 (table entries under Low Debt block).
- % C: -0.09, -0.28, -0.29 (table entries under Low Debt block).
- Optimal monetary feedback coefficients (selected):
  - Shock i: 2.99, 6.27, 13.95, 24.63, 17.69, 28.48 (as reported across columns).
  - Inflation i: 0.00, 0.00, 3.68, 4.26, 6.10, 6.16.
  - Output i y: 0.00, 0.00, 0.23, 0.26, 0.37, 0.38.
  - Debt i b: -0.003, -0.012, -0.003, -0.001, -0.002, -0.001.
- Optimal fiscal feedback coefficients (selected):
  - Shock i: -0.11, -4.10, -4.48, -5.73, -8.02, -8.40.
  - Inflation i: 0.00, 0.00, -3.47, -3.62, -6.69, -6.70.
  - Output i y: 0.00, 0.00, -0.21, -0.22, -0.41, -0.41.
  - Debt i b: -0.005, -0.107, -0.005, -0.011, -0.005, -0.006.

### Inflation persistence (hybrid and backward-looking Phillips curves)
- With inflation persistence (hybrid Phillips curve, != 0:75):
  - Monetary policy must raise interest rates significantly more in the first period and induce a larger fall in consumption.
  - Monetary policy feeds back onto past inflation and output in a stabilizing manner (C >0 and C y >0).
  - Fiscal policy becomes active: government spending falls in response to the cost-push shock (C <0) to accelerate disinflation of rule-of-thumb price setters and reduce price dispersion despite fiscal adjustment costs.
  - Debt still follows a random walk; stronger rise in interest rates leads to more rapid accumulation of public debt than without inflation persistence.
- With a predominantly backward-looking Phillips curve (!= 0:99):
  - Interest rates must rise even more and spending must fall more strongly to stabilize inflation, implying more debt accumulation than with a hybrid Phillips curve.

### Time inconsistency under commitment
- Predetermined Lagrange multipliers:
  - Both predetermined Lagrange multipliers ( s and c s) are different from zero during the disinflation process for all levels of inflation persistence, signaling time inconsistency of optimal commitment policy.
- Inflation control time inconsistency:
  -  s is negative for s >0, indicating a social loss reduction is possible by setting a higher inflation rate after period 0 than was optimal at time 0. This creates an incentive to renege on announced tight policy once inflation has fallen.
- Debt control time inconsistency:
  - c s is different from zero for s >0; negative c s is equivalent to positive b s and indicates an incentive to reduce debt under commitment policy.
  - The incentive to cut debt persists over time because the gain from reducing debt is constant while the cost of reducing debt is lower in the first period than in subsequent periods, leading to a random walk in debt under commitment.
- Welfare gains from reneging:
  - The welfare incentive to renege on the announced optimal inflation path is largest in the first period and falls over time as inflation returns to zero.
  - The incentive to renege on the consumption (and hence debt) path follows a random walk.
  - Figure evidence: welfare gains from reneging on inflation are significantly larger than gains from reneging on debt.

### Relation to literature and mechanism notes
- The random-walk result for debt parallels findings in Benigno and Woodford 2003, Schmitt-Grohe and Uribe 2004, and Leith and Wren-Lewis 2007.
- Intuition: contractionary monetary policy in response to cost-push shocks creates debt; reducing that debt is costly (reduces government spending and raises consumption) and also inflationary to the extent it involves lowering interest rates; optimal trade-off leads to permanent higher debt.
- Fiscal policy’s comparative advantage with inflation persistence: fiscal cuts help align rule-of-thumb price setters with Calvo price setters, reducing price dispersion and social loss despite fiscal costs.

*Source: _wp07206 - 1.  The Low Debt Economy*

### 1.  The Low Debt Economy

### 1.  The Low Debt Economy

### Optimal discretionary vs. commitment in a low debt economy
- Under a New Keynesian Phillips curve, inflation is controlled much less effectively under discretion than commitment, producing the classic inflation stabilisation bias described by Currie and Levine (1987, 1993) and Woodford (2003b).
- Policymaker under discretion raises interest rates very strongly in the first period and then returns them to zero more quickly than under commitment, inducing a larger initial recession and slower subsequent disinflation.
- Debt does not follow a random walk under optimal discretionary policy; it returns to its initial (steady-state) value — the "debt stabilisation bias" (Leith and Wren-Lewis (2007)).
- For the low debt economy with a New Keynesian Phillips curve:
  - maximum eigenvalue for the simulated system under discretion: Dmax = 0:73.
  - the combined inflation and debt stabilisation bias induces a loss equivalent to a0:09%fall in steady-state consumption.

### Adjustment mechanics and optimal policy mix (low debt)
- Inflation is partly forward-looking; debt is entirely backward-looking. Thus:
  - Inflation in period 1 can be reduced via expectations of future contractionary policy.
  - Debt in period 1 can only be reduced via lower interest rates and/or lower spending in that period.
- Policymaker chooses to do bulk of debt adjustment in the first period when inflationary costs are smallest.
- The split between lowering interest rates and cutting spending depends critically on the steady-state ratio of debt to output.
- For the low debt calibration:
  - the first-period debt adjustment is done mostly through lower spending.
  - spending under discretionary policy falls much more strongly in response to the cost-push shock than under commitment.
  - strongly negative fiscal and monetary feedback onto the debt stock ensures fast convergence of debt back to its initial level.
- Debt stabilisation bias necessitates a much more active role for fiscal policy under discretion than under commitment.

### Inflation persistence (effects in low debt economy)
- With inflation persistence and rule-of-thumb price setters, monetary policy raises interest rates more strongly under discretion and fiscal policy cuts spending more actively to assist monetary policy in controlling both inflation and debt.
- As inflation persistence (!) increases, dynamics under discretion become more similar to commitment because:
  - the link from expected inflation to current inflation weakens,
  - time-inconsistent promises about future policy become less effective.
- Table 1 results as ! increases from != 0 to != 0:75 and != 0:99:
  - maximum eigenvalue rises from 0:73 to 0:98 and 0:99 respectively.
- Welfare costs of discretionary policy (combined biases) for higher !:
  - rises to 0:28% and 0:29% of steady-state consumption for a hybrid and backward-looking Phillips curve respectively,
  - driven by higher weight on price dispersion in the social welfare function.

### High Debt Economy — key findings
- As steady-state debt to output rises, monetary policy becomes more powerful relative to fiscal policy in controlling the debt stock because leverage over interest payments increases.
- For a New Keynesian Phillips curve in a high debt economy:
  - optimal discretionary policy may cut the interest rate in the first period in response to a cost-push shock (D <0).
  - Cutting interest rates initially reduces debt via lower real interest payments and fuels inflation, which further reduces debt.
- Interaction of biases:
  - Inflation stabilisation bias (on its own) calls for an initial interest rate hike.
  - Debt stabilisation bias (on its own, when debt is high) calls for an initial interest rate cut.
  - When debt is high and inflation persistence is low, debt stabilisation bias dominates, producing the counter-intuitive initial interest rate cut.
- Dynamics after initial cut:
  - interest rates rise strongly in the second period to control inflation,
  - fiscal policy reduces spending aggressively in the first period to assist monetary policy.
- For the high debt calibration:
  - system converges faster than in low debt case: maximum eigenvalue = 0:11 (Table 1) for the example discussed.
  - debt falls below steady-state on impact then returns from below to initial level.

### Inflation persistence (effects in high debt economy)
- As persistence (!) increases:
  - it becomes harder to control inflation even with first-period interest-rate cuts.
  - for a hybrid Phillips curve, interest rates may continue to fall but only with very large spending cuts.
  - for a strongly backward-looking Phillips curve, the effect of future inflation expectations on current inflation is too weak, and monetary policy cannot cut interest rates in the first period.
    - under such strong backward-looking behavior, the social planner under discretion raises the interest rate in the first period (confirmed by C >0 in Table 1).
- System implications:
  - unlike the low debt economy (where dynamics under discretion approached commitment as != 0:99), in the high debt economy debt continues to converge much faster to pre-shock levels even for high ! (maximum eigenvalue remains considerably below unity at 0:57).
  - welfare costs are considerably higher for the high debt economy, even for high values of inflation persistence:
    - for a hybrid Phillips curve the excess loss of discretionary policy over commitment is equivalent to a0:79%fall in steady-state consumption,
    - for a predominantly backward-looking Phillips curve this cost rises to 1:02%.

### Summary across calibrations and policy implications
- Figure 7 (described) shows:
  - under optimal commitment the social planner always raises the interest rate and cuts spending (C >0; C <0 for all ! and B).
  - under discretion, fiscal policy is more active than under commitment; activism rises with steady-state debt ratio.
  - for low steady-state debt (B0:2), the planner under discretion raises the interest rate in response to the cost-push shock and fulfills the Taylor principle for all levels of inflation persistence.
  - for B >0:2, monetary policy may be forced to cut interest rates to stabilise debt — but only if the Phillips curve is predominantly forward-looking; thresholds of != at which the perverse response vanishes depend on B (e.g., for B= 0:25 threshold about != 0:1; for B= 0:3 threshold about != 0:5).
  - for very high debt calibrations (B>0:5) violation of the Taylor principle in the first period remains optimal at all levels of inflation persistence (even for != 0:99).
- Figure 8 (described) shows:
  - maximum eigenvalue under discretion (Dmax) moves toward unity for low steady-state debt as the system becomes more backward looking, but remains well below unity for high debt economies.
  - excess loss of discretion over commitment (in% of steady-state consumption foregone) is large and higher for larger steady-state debt.
  - for an empirically plausible hybrid Phillips curve the inability to commit imposes a welfare cost equivalent to around 0:6% fall in steady-state consumption.
  - as persistence rises further, excess loss may fall for intermediate debt ratios (less forward-looking agents reduce severity of stabilisation bias), but for very high debt economies the inability to commit becomes increasingly costly.

Policy-relevant conclusions
- Under discretion, public debt must be returned to its pre-shock level for time consistency; this implies:
  - optimal fiscal policy is more active under discretion than commitment at all levels of inflation persistence and debt to assist a constrained monetary authority.
  - in high-debt, low-inflation-persistence environments, optimal discretionary monetary policy may violate the Taylor principle by cutting interest rates initially.
- Monetary policy should fulfill the Taylor principle in an economy with inflation persistence, but the conventional view that monetary policy bears most stabilization burden while fiscal policy merely ensures sustainability is inappropriate under optimal discretionary policy.
- Institutions promoting commitment of both monetary and fiscal policy (e.g., announced targets, long tenure of policymakers) are highly desirable because gains from commitment are larger when public debt is present — especially if debt is high.
- Quantitative thresholds depend critically on model features:
  - debt definition/maturity (one-period maturity here implies large monetary leverage because the entire debt stock is rolled over each period);
  - whether debt is denoted in nominal terms (which would raise the effect of inflation on debt);
  - intertemporal elasticity of substitution (higher elasticity reduces likelihood of Taylor-principle violation by strengthening interest-rate effects on consumption and inflation).

*Italic line: Source — _wp07206 - 1.  The Low Debt Economy*

### 1.  The Phillips Curve

### 1.  The Phillips Curve

### Derivation from household and firm first-order conditions
- Nominal wage from the consumer's first-order conditions:
  - Equation (38): w_t(z) = v_y(y_t(z)) (1ξ) u_C(C_t) P_t
- Production function:
  - y_t(z) = h_t(z)
- Total cost of supplying good z (labour subsidy denoted by _w):
  - 1 _w w_t(z) h_t(z) = 1 _w w_t(z) y_t(z)
- Demand for good z from intra-temporal consumption optimisation:
  - Equation (39): y_t(z) = [p_t(z)/P_t]^{-ε_t} Y_t
- Firm's profit-maximisation problem (expressed in the text) leads to a forward-looking first-order condition for the optimal price p^f_t(z), summarised in the stochastic discounted-sum condition that appears immediately after the maximisation problem (see source for full expression).

### Substitution and definition of mark-up
- Substituting (39) and (38) into the firm's first-order condition yields equation (40), which (in the source) is written as:
  - 0 = E_t ∑_{s=t} (β)^{s−t} p^{−ε_s−1}_{f;s}(z) R_{t;s} Y_s [ (p_s(z)/P_s)^{-ε_s} { p_{f;s}(z) − _s _w / P_s v_y( p_{f;s}(z)/P_s ) ε_s Y_s } (1ξ) u_C(C_s) ]
- The stochastic mark-up of firms is denoted in the source as:
  - _s = "s (1"s)
- The optimal forward-looking price is defined as p_t(z) = p^f_t(z).

### Flexible-price equilibrium and linearisation strategy
- In flexible-price equilibrium the discounted-sum identity simplifies using ∑_{s=t} (β)^{s−t} = 1/(1−β), yielding the equilibrium condition:
  - v_y(y) (1ξ) U_C(C) = _w _ = w/P
- Because that term in square brackets is zero in equilibrium, the linearisation focuses on linearising deviations of that term and evaluating multiplicative factors at steady-state.
- The source linearises the term v_y( p^f;s(z)/P_s ) u_C(C_s) and expresses it using log-deviation notation Z_t = Z̄ [1 + ^Z_t]:
  - The linear expansion given in the source:
    - v_y u_C − " v_{yy} / v_y [ ^p^f;s(z) − ^P_s ] + v_{yy} Y / v_y ^Y_s − u_{CC} C / u_C ^C_s
  - (The term involving ^"s is zero because it is multiplied by ln p^f_t(z)/P.)

### Final linearised forward-looking price expression
- The source derives an expression for ^p^f;t(z) after algebraic manipulation and using ^P_s = ∑_{k=1}^{s−t} π_{t+k}:
  - ^p^f;t(z) = E_t [ ∑_{k=1}^{s−t} (β)^k π_{t+k} ] + (1β) [ 1 + "¯ ] E_t ∑_{s=t} (β)^{s−t} [ 1/φ ^Y_s + 1/ψ ^C_s + ^_s ]
  - The source then simplifies this further to:
    - ^p^f;t(z) = E_t ∑_{s=t} (β)^{s−t} ∑_{k=1}^s (β)^k π_{t+k} + (1β) [ 1 + "¯ ] E_t ∑_{s=t} (β)^{s−t} [ 1/φ ^Y_s + 1/ψ ^C_s + ^_s ]
  - (See source for exact intermediate algebra and definitions of parameters 1/φ and 1/ψ and "¯.)

### Connection to the hybrid Phillips curve and qualitative implications
- Steinsson (2003) is cited in the source as showing how the linearised forward-looking price equation can be used to derive the final specification of the hybrid Phillips curve (equation (7) in the main text).
- Key analytical observations from the derivation in the source:
  - Mark-up shocks ^_s enter the Phillips curve and affect inflation dynamics.
  - The constant wage-income tax rate ξ does not affect the dynamic Phillips curve.

### Backward-looking (rule-of-thumb) price setters
- The source specifies the backward-looking pricing rule (linearised) used by some firms (rule-of-thumb price setters):
  - ^P^B_Hk;t = (1−λ) ln( P^F_Hk;t−1 / P_Hk;t−1 ) + λ ln( P^B_Hk;t−1 / P_Hk;t−1 ) − ln ε_Hk;t + ln ε_Hk;t−1 + ρ ln( Y_k;t−1 / Y^n_k;t−1 )
- The source indicates that, following manipulation similar to Steinsson (2003), Kirsanova and Wren-Lewis (2005) show how to obtain the Phillips curve reported in the main text from the mixture of forward-looking and backward-looking price-setting.

*Italicised source: _wp07206 - 1.  The Phillips Curve (IMF Working Paper PDF content provided).*

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