Appendix C: Optimal Policy with Infinite-horizon Phillips Curve | RDP 2025-02: Boundedly Rational Expectations and the Optimality of Flexible Average Inflation Targeting

RDP 2025-02: Boundedly Rational Expectations and the Optimality of Flexible Average Inflation Targeting Appendix C: Optimal Policy with Infinite-horizon Phillips Curve

Anthony Brassil, Christopher G Gibbs and Callum Ryan

April 2025

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C.1 Infinite-horizon Phillips curve

In the main text, we model firms' pricing decisions by taking the standard New Keynesian Phillips curve and substituting our alternative specification for aggregate inflation expectations into the expectation term. This is consistent with the ‘Euler equation’ approach to adaptive learning, and also approximates a range of other behavioural theories. But it is not consistent with the ‘anticipated utility’ approach to adaptive learning. In this appendix, we provide optimal policy results using an ‘anticipated utility’ specification for the Phillips curve:

(C1)

π_{t} = {\hat{𝔼}}_{t} \frac{1}{1 - θ β L^{- 1}} (κ x_{t} + (1 - θ) β π_{t + 1})

Here the parameter $θ$ is the Calvo probability, that is, the probability that a firm cannot change its price in a given period. This Phillips curve follows from the firms' optimal price setting under the standard microfoundations of the New Keynesian model for any arbitrary expectations. Firms' prices depend on their expectations over the entire future sequence of inflation and output gaps, ${π_{T}, x_{T}}_{T = t}^{\infty}$ . If we assume rational expectations, then (C1) collapses to the standard one-step-ahead Phillips curve (2).

Using our specification for agents' beliefs, (3) to (6), we have

(C2)

\begin{array}{l} π_{t} = λ 𝔼_{t} \frac{1}{1 - θ β L^{- 1}} (κ x_{t} + (1 - θ) β π_{t + 1}) + (1 - λ) 𝔼_{t}^{l} \frac{1}{1 - θ β L^{- 1}} (κ x_{t} + (1 - θ) β π_{t + 1}) \\ = λ 𝔼_{t} \frac{1}{1 - θ β L^{- 1}} (κ x_{t} + (1 - θ) β π_{t + 1}) \\ + (1 - λ) (κ x_{t} + \frac{θ β ρ}{1 - θ β ρ} κ ω_{t - 1}^{x} + \frac{(1 - θ) β ρ}{1 - θ β ρ} ω_{t - 1}^{π}) \end{array}

The optimal policy problem is the same as in the main text, but with a different Phillips curve, that is, choose the sequence ${π_{t}, x_{t}, ω_{t}^{π}, ω_{t}^{x}, i_{t}}_{t = 0}^{\infty}$ that maximises the welfare function (9) subject to the Phillips curve (C2), the IS curve (11), the evolution of beliefs (12) and (13), and potentially some information constraint or the ZLB.

The first-order conditions are

\begin{array}{l} π_{t} + μ_{t}^{π} - λ (1 - θ) \frac{1}{1 - θ L} μ_{t - 1}^{π} - \frac{λ}{β σ} \frac{1}{1 - L} μ_{t - 1}^{x} - ρ g μ_{t}^{ω π} = 0 \\ α x_{t} - κ μ_{t}^{π} - λ θ κ \frac{1}{1 - θ L} μ_{t - 1}^{π} + μ_{t}^{x} - \frac{λ (1 - β)}{β} \frac{1}{1 - L} μ_{t - 1}^{x} - ρ g μ_{t}^{ω x} = 0 \\ - (1 - λ) \frac{(1 - θ) β^{2} ρ}{1 - θ β ρ} 𝔼_{t} μ_{t + 1}^{π} - (1 - λ) \frac{β ρ}{σ (1 - β ρ)} 𝔼_{t} μ_{t + 1}^{x} + μ_{t}^{ω π} - β ρ (1 - g) 𝔼_{t} μ_{t + 1}^{ω π} = 0 \\ - (1 - λ) \frac{θ β^{2} ρ}{1 - θ β ρ} κ 𝔼_{t} μ_{t + 1}^{π} - (1 - λ) \frac{(1 - β) β ρ}{1 - β ρ} 𝔼_{t} μ_{t + 1}^{x} + μ_{t}^{ω π} - β ρ (1 - g) 𝔼_{t} μ_{t + 1}^{ω x} = 0 \end{array}

plus an additional condition for $μ_{t}^{x}$ that depends on whether there is an imperfect information or ZLB constraint.

C.2 Unconstrained case

In the absence of imperfect central bank information or the ZLB, the additional first-order condition is $μ_{t}^{x}$ = 0. Eliminating $μ_{t}^{x}$ and $μ_{t}^{π}$ gives

(C3)

\frac{α}{κ} x_{t} = - \frac{1 - θ (1 - λ) L}{1 - (λ + θ (1 - λ)) L} π_{t} + \frac{1 - θ (1 - λ) L}{1 - (λ + θ (1 - λ)) L} ρ g μ_{t}^{ω^{π}} + \frac{ρ g}{κ} μ_{t}^{ω^{x}}

The multipliers $μ_{t}^{ω^{π}}$ and $μ_{t}^{ω^{x}}$ – that is, the welfare effect of shifts in learners' inflation and output gap expectations – are given by

μ_{t}^{ω^{π}} = β ρ (1 - g) 𝔼_{t} μ_{t + 1}^{ω^{π}} - β^{2} (1 - λ) \frac{(1 - θ)}{1 - β θ ρ} 𝔼_{t} \frac{1 - θ L}{1 - (λ + θ (1 - λ)) L} (π_{t + 1} - ρ g μ_{t + 1}^{ω^{π}})

μ_{t}^{ω^{x}} = κ \frac{θ}{1 - θ} μ_{t}^{ω^{π}}

Solving forward for $μ_{t}^{ω^{π}}$ then gives

\frac{1}{1 - (λ + θ (1 - λ)) L} μ_{t}^{ω^{π}} = - 1 (1 - λ) \frac{(1 - θ) ρ}{1 - β θ ρ} Ω^{'} \frac{1}{1 - {ζ^{'}}_{1} L} 𝔼_{t} \frac{1}{1 - {ζ^{'}}_{2}^{- 1} L^{- 1}} \frac{1 - θ L}{1 - (λ + θ (1 - λ)) L} π_{t + 1}

where $Ω^{'} \equiv \frac{β^{2} {ζ^{'}}_{2}^{- 1}}{β ρ (1 - g) + (1 - λ) g β^{2} \frac{(1 - θ) ρ^{2}}{1 - β θ ρ}} \equiv \frac{β^{2} {ζ^{'}}_{1}}{λ + θ (1 - λ)}$ and $ζ_{1}$ and $ζ_{2}$ are the roots of the characteristic polynomial

\begin{array}{l} (β ρ (1 - g) + (1 - λ) g β^{2} \frac{{(1 - θ)}^{2}}{1 - β θ ρ}) ζ^{2} \\ - (1 + β ρ (1 - g) (λ + θ (1 - λ)) + θ (1 - λ) g β^{2} \frac{(1 - θ) ρ^{2}}{1 - β θ ρ}) ζ \\ + (λ + θ (1 - λ)) = 0 \end{array}

Using this solution to substitute for $μ_{t}^{ω^{π}}$ and $μ_{t}^{ω^{x}}$ in (C3) gives the optimal target criterion:^[25]

(C4)

\begin{array}{l} \frac{α}{κ} x_{t} = - \frac{1 - θ (1 - λ) L}{1 - (λ + θ (1 - λ)) L} π_{t} \\ - (1 - λ) g \frac{ρ^{2}}{1 - β θ ρ} Ω^{'} \frac{1 - θ L}{1 - {ζ^{'}}_{1} L} 𝔼_{t} \frac{1}{1 - {ζ^{'}}_{2}^{- 1} L^{- 1}} \frac{1 - θ L}{1 - (λ + θ (1 - λ)) L} π_{t + 1} \end{array}

This optimal target criterion has the same structure as the analogous one in the main body, (17). It includes a make-up component and a pre-emptive component, and the intuition behind it is the same. The differences are a matter of degree. First, the optimal weights on past outcomes decay more slowly in (C4). This is because the infinite-horizon Phillips curve implies that make-up commitments further into the future have a stronger effect on current-period pricing decisions than is the case with the one-step-ahead Phillips curve.^[26] Second, the optimal pre-emptive response to expected future inflation is stronger. This is because policy-induced changes in learners' beliefs now have a stronger effect on future outcomes, because both their inflation and output gap expectations influence their pricing decisions (with the one-step-ahead Phillips curve, only their inflation expectations were relevant).

Table C1 compares the performance of the simple target criteria from Table 2 under the infinite-horizon Phillips curve. The equivalent values for the one-step-ahead Phillips curve (i.e. Table D1) are in parentheses. The results are fairly similar. As the previous paragraph would suggest, the main differences are that with the infinite-horizon Phillips curve (i) the optimal weight $γ$ on lagged inflation outcomes in the WAIT is higher, and PLT now outperforms IT; and (ii) a pre-emptive response to expected inflation now offers a small welfare improvement.

Table C1: Performance of Simple Target Criteria with Infinite-horizon Phillips Curve – Unconstrained Case
Rule	Loss (% relative to IT)	$ψ$	$γ$	$γ_{x}$	$ψ_{f}$	$γ_{f}$
5-yr AIT	61.1 (96.5)	16.9 (31.3)
2-yr AIT	16.8 (48.8)	18.5 (13.0)
1-yr AIT	2.8 (7.7)	15.0 (21.5)
IT	0 (0)	10.6 (11.6)
IT + pre-emptive	−0.0 (−4.4)	10.6 (0.03)			0.00 (1.09)	undef (1.50)
PLT	−5.9 (6.6)	6.6 (9.1)	1 (1)
WAIT	−11.0 (−6.6)	6.9 (8.4)	0.70 (0.52)
WAIT + pre-emptive	−12.5 (−6.6)	5.5 (8.2)	0.89 (0.52)		10.92 (0.33)	0.00 (1.02)
WAIT + WAXT	−12.5 (−6.6)	8.0 (8.4)	0.88 (0.53)	0.43 (0.01)
WAIT + WAXT + pre-emptive	−12.5 (−6.6)	7.9 (8.2)	0.88 (0.52)	0.43 (0.00)	0.12 (0.33)	0.03 (1.02)
Optimal	−12.5 (−6.6)
Notes: Welfare losses are normalised relative to the welfare under IT. The parameters for each rule are optimised to minimise welfare. The optimal value of a parameter is undefined (labelled undef) if the parameter has no effect on loss at the optimum. Values in parentheses are the equivalent values with the one-step-ahead Phillips curve (from Table D1).

C.3 Comparison to Eusepi, Giannoni and Preston (2018)

Eusepi et al (2018) study optimal policy in an adaptive learning model (without imperfect information or the ZLB). They show that in this setting, the infinite-horizon Phillips curve under some calibrations implies that optimal policy is history dependent, whereas it is not history dependent under the one-step-ahead Phillips curve.

If we set $λ = 0$ , our model nests theirs. Optimal policy is given by setting $λ = 0$ in (C4):

\frac{α}{κ} x_{t} = - π_{t} - g \frac{β^{2} ρ^{2}}{1 - β θ ρ} 𝔼_{t} \frac{1}{1 - β ρ (1 - g \frac{1 - β ρ}{1 - β θ ρ}) L^{- 1}} π_{t + 1}

Setting $θ = 0$ in this targeting rule gives optimal policy with $λ = 0$ and the one-step-ahead Phillips curve, as studied by Molnár and Santoro (2014). Therefore using the infinite-horizon Phillips curve just increases the coefficient on the average of expected future inflation (while also slightly increasing the weight on nearer-term expectations within that average compared to further out expectations). With $λ = 0$ , optimal policy always involves no (direct) response to any lagged variables.

The difference in the optimal equilibrium stems not from the optimal policy rules, but from the direct effect of beliefs on the Phillips curve. Only inflation beliefs enter the one-step-ahead Phillips curve, so the Phillips curve will always shift up following an inflationary cost-push shock, then return to its original position. In contrast, both inflation and output gap beliefs enter the infinite-horizon Phillips curve. If the loss function places sufficiently low weight on the output gap term, then the central bank's aggressive response will cause the Phillips curve to shift downwards (once the initial shock has dissipated), which means that inflation will overshoot. It is the private sector pricing decisions that are history dependent (via beliefs) and overshoot, not the policy rule.

C.4 General case

If there is a constraint on the policy rate (e.g. due to imperfect central bank information or the ZLB), then $σ (1 - L) μ_{t}^{i} = μ_{t}^{x}$ , where $μ_{t}^{i}$ is the multiplier on this constraint. Substituting this condition in for $μ_{t}^{x}$ and eliminating $μ_{t}^{π}$ from the first-order conditions gives

(C5)

\begin{array}{l} \frac{α}{κ} x_{t} = - \frac{1 - θ (1 - λ) L}{1 - (λ + θ (1 - λ)) L} π_{t} + \frac{1 - θ (1 - λ) L}{1 - (λ + θ (1 - λ)) L} ρ g μ_{t}^{ω^{π}} + \frac{ρ g}{κ} μ_{t}^{ω^{x}} \\ + (\frac{λ}{β} \frac{κ}{σ} \frac{1 - θ (1 - λ) L}{1 - (λ + θ (1 - λ)) L} + λ (\frac{1}{β} - 1) +) \frac{σ}{κ} μ_{t - 1}^{i} \end{array}

The multipliers on learners' beliefs are

\begin{array}{l} μ_{t}^{ω π} = β ρ (1 - g) 𝔼_{t} μ_{t + 1}^{ω π} - (1 - λ) \frac{(1 - θ) β^{2} ρ}{1 - β θ ρ} 𝔼_{t} \frac{1 - θ L}{1 - (λ + θ (1 - λ)) L} (π_{t + 1} - \frac{λ}{β} \frac{κ}{σ} \frac{σ}{κ} μ_{t}^{i} - ρ g μ_{t + 1}^{ω^{π}}) \\ + (1 - λ) \frac{β ρ}{1 - β ρ} 𝔼_{t} (1 - L) μ_{t + 1}^{i} \end{array}

μ_{t}^{ω^{x}} = \frac{θ}{1 - θ} κ μ_{t}^{ω^{π}} + (1 - λ) \frac{β ρ}{1 - β ρ} ((1 - β) - \frac{θ}{1 - θ} \frac{κ}{σ}) 𝔼_{t} \frac{1}{1 - β ρ (1 - g) L^{- 1}} σ (1 - L) μ_{t + 1}^{i}

Solving forward for $μ_{t}^{ω^{π}}$ gives

\begin{array}{l} μ_{t}^{ω^{π}} = - (1 - λ) \frac{(1 - θ) ρ}{1 - β θ ρ} Ω^{'} \frac{1 - (λ + θ (1 - λ)) L}{1 - {ζ^{'}}_{1} L} 𝔼_{t} \frac{1}{1 - {ζ^{'}}_{2}^{- 1} L^{- 1}} \frac{1 - θ L}{1 - (λ + θ (1 - λ)) L} (π_{t + 1} - \frac{λ}{β} \frac{κ}{σ} \frac{σ}{κ} μ_{t}^{i}) \\ + (1 - λ) \frac{ρ Ω^{'}}{β (1 - β ρ)} \frac{κ}{σ} \frac{1 - (λ + θ (1 - λ)) L}{1 - {ζ^{'}}_{1} L} 𝔼_{t} \frac{1 - L}{1 - {ζ^{'}}_{2}^{- 1} L^{- 1}} \frac{σ}{κ} μ_{t + 1}^{i} \end{array}

Using this solution to substitute for $μ_{t}^{ω^{π}}$ and $μ_{t}^{ω^{x}}$ in (C5) and collecting all the non- $μ_{t}^{i}$ terms in ${\tilde{π}}_{t}$ gives

\begin{array}{l} {\tilde{π}}_{t} + \frac{σ}{κ} μ_{t}^{i} = \frac{λ}{β} \frac{κ}{σ} \frac{1 - θ (1 - λ) L}{1 - (λ + θ (1 - λ)) L} \frac{σ}{κ} μ_{t - 1}^{i} \\ + (1 - λ) g \frac{ρ^{2}}{1 - β θ ρ} Ω^{'} \frac{λ}{β} \frac{κ}{σ} \frac{1 - θ L}{1 - ζ^{'}_{1} L} 𝔼_{t} \frac{1}{1 - ζ^{'} {_{2}}^{- 1} L^{- 1}} \frac{1 - θ L}{1 - (λ + θ (1 - λ)) L} \frac{σ}{κ} μ_{t}^{i} \\ + (1 - λ) g \frac{ρ^{2}}{β (1 - β ρ) (1 - θ)} Ω^{'} \frac{κ}{σ} \frac{1 - θ L}{1 - ζ^{'}_{1} L} 𝔼_{t} \frac{1 - L}{1 - ζ^{'} {_{2}}^{- 1} L^{- 1}} \frac{σ}{κ} μ_{t + 1}^{i} \\ + (λ (\frac{1}{β} - 1) + 1) \frac{σ}{κ} μ_{t - 1}^{i} \\ + (1 - λ) g \frac{β ρ^{2}}{1 - β ρ} ((1 - β) - \frac{θ}{1 - θ} \frac{κ}{σ}) 𝔼_{t} \frac{1 - L}{1 - β ρ (1 - g) L^{- 1}} \frac{σ}{κ} μ_{t + 1}^{i} \\ = - (1 - λ) g ρ^{2} Θ^{'} \frac{σ}{κ} μ_{t}^{i} + P^{'} (L) \frac{σ}{κ} μ_{t - 1}^{i} + 𝔼_{t} P^{'}_{2} (L^{- 1}) \frac{σ}{κ} μ_{t + 1}^{i} + P^{'}_{3} (L) 𝔼_{t - 1} P^{'}_{4} (L^{- 1}) \frac{σ}{κ} μ_{t}^{i} \end{array}

where ${\tilde{π}}_{t}$ is the residual from the unconstrained optimal target criterion (C4) and $Θ^{'},P^{'} (L), {P^{'}}_{1} (L^{- 1}), {P^{'}}_{2} (L)$ and ${P^{'}}_{3} (L^{- 1})$ are defined as

Θ^{'} = - \frac{Ω^{'}}{1 - β θ ρ} \frac{λ}{β} \frac{κ}{σ} \frac{1 - ζ^{'} {_{2}}^{- 1} θ}{1 - ζ^{'} {_{2}}^{- 1} (λ + θ (1 - λ))} + \frac{Ω^{'}}{β (1 - β ρ) (1 - θ)} \frac{κ}{σ} + \frac{β}{1 - β ρ} ((1 - β) - \frac{θ}{1 - θ} \frac{κ}{σ})

\begin{array}{l} P^{'} (L) = \frac{λ}{β} \frac{κ}{σ} \frac{1 - θ (1 - λ) L}{1 - (λ + θ (1 - λ)) L} + λ (\frac{1}{β} - 1) + 1 \\ + (1 - λ) g ρ^{2} Ω^{'} \frac{κ}{σ} (\frac{λ}{β} \frac{1 - {ζ^{'}}_{2}^{- 1} θ}{1 - {ζ^{'}}_{2}^{- 1} θ (λ + θ (1 - λ))} (\frac{λ (1 - θ)}{1 - {ζ^{'}}_{2}^{- 1} θ} \frac{1 - θ L}{1 - (λ + θ (1 - λ)) L} + ζ_{1} - θ) - \frac{ζ_{1} - θ}{β (1 - β ρ) (1 - θ)}) \frac{1}{1 - ζ_{1} L} \end{array}

\begin{array}{l} {P^{'}}_{1} (L^{- 1}) = (1 - λ) g ρ^{2} Ω^{'} \frac{κ}{σ} (\frac{λ}{β} \frac{{ζ^{'}}_{2}^{- 1} (1 - {ζ^{'}}_{2}^{- 1} θ)}{1 - {ζ^{'}}_{2}^{- 1} (λ + θ (1 - λ))} + \frac{1 - {ζ^{'}}_{2}^{- 1}}{β (1 - β ρ) (1 - θ)}) 𝔼_{t} \frac{1}{1 - {ζ^{'}}_{2}^{- 1} L^{- 1}} \\ + (1 - λ) g ρ^{2} \frac{β}{1 - β ρ} ((1 - β) - \frac{θ}{1 - θ} \frac{κ}{σ}) (1 - β ρ (1 - g)) 𝔼_{t} \frac{1}{1 - β ρ (1 - g) L^{- 1}} \end{array}

{P^{'}}_{2} (L) = (ζ_{1} - θ) \frac{1}{1 - ζ_{1} L}

{P^{'}}_{3} (L^{- 1}) = (1 - λ) g ρ^{2} Ω^{'} \frac{κ}{σ} (\frac{λ}{β} \frac{{ζ^{'}}_{2}^{- 1} (1 - {ζ^{'}}_{2}^{- 1} θ)}{1 - {ζ^{'}}_{2}^{- 1} (λ + θ (1 - λ))} + \frac{1 - {ζ^{'}}_{2}^{- 1}}{β (1 - β ρ) (1 - θ)}) 𝔼_{t} \frac{1}{1 - {ζ^{'}}_{2}^{- 1} L^{- 1}}

Now define $π_{t}^{*}$ and $Δ_{t}$ as in Appendix A.2:

π_{t}^{*} \equiv \frac{1}{1 + (1 - λ) g ρ^{2} Θ^{'}} \frac{σ}{κ} (P^{'} (L) μ_{t - 1}^{i} + 𝔼_{t} {P^{'}}_{1} (L^{- 1}) μ_{t + 1}^{i} + {P^{'}}_{2} (L) 𝔼_{t - 1} {P^{'}}_{3} (L^{- 1}) μ_{t}^{i})

Δ_{t} \equiv π_{t}^{*} - {\tilde{π}}_{t} = (1 + (1 - λ) g ρ^{2} Θ^{'}) \frac{σ}{κ} μ_{t}^{i}

so we have

{\tilde{π}}_{t} + Δ_{t} = π_{t}^{*} \equiv \frac{1}{1 + (1 - λ) g ρ^{2} Θ^{'}} (P^{'} (L) Δ_{t - 1} + 𝔼_{t} {P^{'}}_{1} (L^{- 1}) Δ_{t + 1} + {P^{'}}_{2} (L) 𝔼_{t - 1} {P^{'}}_{3} (L^{- 1}) \frac{σ}{κ} Δ_{t})

The optimal target criterion is then the same as with the one-step-ahead Phillips curve in Result 5, with the newly defined $Θ^{'}, P^{'} (L), {P^{'}}_{1} (L^{- 1}), {P^{'}}_{2} (L)$ and ${P^{'}}_{3} (L^{- 1})$ .

As in the unconstrained case, the main differences between this optimal target criterion and the one with the one-step-ahead Phillips curve are that (i) the weight on lagged misses is larger, because commitments further in the future have a larger effect on current-period inflation via the Phillips curve, and (ii) the optimal pre-emptive response to expected future misses is larger, because the central bank has greater effect on future outcomes now that learners' beliefs about both inflation and the output gap affect inflation.

Since both the make-up and pre-emptive channels of policy are stronger, the interaction between them is also strengthened. But this interaction has offsetting effects, so the change in optimal policy is small.^[27]

Table C2 compares the performance of the simple target criteria from Table 2 under the infinite-horizon Phillips curve with imperfect central bank information. The equivalent values for the one-step-ahead Phillips curve (i.e. Table 3) are in parentheses. The results are very similar.

Table C2: Performance of Simple Target Criteria with Infinite-horizon Phillips Curve – Imperfect Information Case
Rule	Loss (% relative to IT)	$Ψ$	$γ$	$γ_{x}$	$Ψ_{f}$	$γ_{f}$
5-yr AIT	18.2 (24.6)	23.8 (27.4)
IT	0 (0)	20.9 (20.9)
IT + preemptive	−2.3 (−14.8)	27.1 (0.0)			10.04 (1.60)	1.28 (1.75)
2-yr AIT	−5.9 (−4.5)	34.3 (27.9)
1-yr AIT	−12.3 (−15.7)	24.4 (15.6)
PLT	−18.8 (−23.8)	9.2 (20.3)	1 (1)
WAIT	−20.0 (−25.1)	9.6 (17.4)	0.80 (0.75)
WAIT + pre-emptive	−20.0 (−25.1)	9.6 (17.4)	0.80 (0.75)		0.00 (0.00)	undef (undef)
WAIT + WAXT	−29.5 (−35.4)	10.0 (12.3)	0.97 (0.94)	0.69 (0.89)
WAIT + WAXT + pre-emptive	−29.5 (−35.4)	10.0 (12.3)	0.97 (0.94)	0.69 (0.89)	0.00 (0.00)	undef(undef)
Optimal	−30.5 (−36.5)
Notes: Welfare losses are normalised relative to the welfare under IT. The parameters for each rule are optimised to minimise welfare. The optimal value of a parameter is undefined (labelled undef) if the parameter has no effect on loss at the optimum. Values in parentheses are the equivalent values with the one-step-ahead Phillips curve (from Table 3).

C.5 Robustly optimal policy under parameter uncertainty with the infinite-horizon Phillips curve

Tables C3 and C4 replicate the robustly optimal policy exercise in Tables D2 and 4, but with the infinite-horizon Phillips curve. The results are very similar.

Table C3: Policy under Parameter Uncertainty with Infinite-horizon Phillips Curve – Unconstrained Case
Rule	Average loss (% relative to IT)	$Ψ$	$γ$	$γ_{x}$	$Ψ_{f}$	$γ_{f}$
5-yr AIT	5.6 × 10¹¹ (535.6)	1.0 (1.2)
2-yr AIT	187.8 (136.1)	1.1 (7.4)
1-yr AIT	86.3 (9.7)	2.6 (19.9)
PLT	0.5 (4.2)	8.0 (10.9)	1 (1)
IT	0 (0)	6.2 (13.7)
IT + pre-emptive	−9.9 (−0.4)	6.4 (3.0)			10.37 (14.9)	0.00 (0.42)
WAIT	−14.6 (−8.1)	7.0 (9.2)	0.53 (0.56)
WAIT + WAXT	−15.4 (−8.1)	6.9 (9.4)	0.73 (0.60)	0.29 (0.08)
WAIT + pre-emptive	−16.5 (−8.9)	5.5 (7.2)	0.74 (0.67)		11.60 (7.4)	0.00 (0.51)
WAIT + WAXT + pre-emptive	−16.6 (−8.9)	6.6 (7.2)	0.73 (0.67)	0.17 (0.00)	6.91 (7.4)	0.00 (0.51)
Optimal	−21.2 (−13.7)
Notes: The parameters reports are set to maximise the average loss across all points in the grid. Average welfare losses are normalised relative to the welfare under IT. Values in parentheses are the equivalent values with the one-step-ahead Phillips curve (from Table D2).

Table C4: Policy under Parameter Uncertainty with Infinite-horizon Phillips Curve – Imperfect Information Case
Rule	Average loss (% relative to IT)	$Ψ$	$γ$	$γ_{x}$	$Ψ_{f}$	$γ_{f}$
5-yr AIT	2.3 ×10¹¹ (118.0)	1.0 (1.4)
2-yr AIT	58.9 (19.8)	0.8 (9.2)
1-yr AIT	27.0 (−16.6)	2.0 (20.7)
IT	0 (0)	4.8 (20.9)
PLT	−16.8 (−22.4)	7.4 (16.8)	1 (1)
IT + pre-emptive	−17.3 (−1.3)	6.7 (14.9)			16.92 (13.0)	1.67 (1.52)
WAIT	−22.7 (−24.0)	6.1 (15.1)	0.66 (0.75)
WAIT + WAXT	−26.9 (−31.0)	4.7 (14.5)	0.83 (0.87)	0.28 (0.69)
WAIT + pre-emptive	−27.0 (−24.0)	12.1 (15.1)	0.65 (0.75)		14.53 (0.0)	0.00 (undef)
WAIT + WAXT + pre-emptive	−33.5 (−31.1)	8.9 (13.3)	0.86 (0.87)	0.51 (0.66)	14.54 (6.8)	0.28 (0.00)
Optimal	−38.2 (−35.7)
Notes: The parameters reports are set to maximise the average loss across all points in the grid. Average welfare losses are normalised relative to the welfare under IT. The optimal value of a parameter is undefined (labelled undef) if the parameter has no effect on loss at the optimum. Values in parentheses are the equivalent values with the one-step-ahead Phillips curve (from Table 4).

Footnotes

An alternative form of this optimal target criterion is

π_{t} = - \frac{α}{κ} x_{t} + (λ \frac{L}{1 - θ L} + (1 - λ) g ρ^{2} \frac{β^{2}}{1 - β θ ρ} 𝔼_{t} \frac{L^{- 1}}{1 - β ρ (1 - g) L^{- 1}}) ((1 - θ) \frac{α}{κ} x_{t} - θ π_{t})

[25]

With the one-step-ahead Phillips curve, commitments about policy more than one period ahead still influence current-period outcomes via the inflation expectations of rational agents. But this effect occurs via a general equilibrium channel and is therefore dampened by the presence of non-rational agents. With the infinite-horizon Phillips curve, these commitments affect the pricing decisions of rational agents directly, via both their inflation and output gap expectations, without this dampening. [26]

The dynamic interaction between rational and learners' expectations in the Phillips curve tends to strengthen the effect of both make-up and pre-emptive policy relative to its cost (this is captured in the first term in $Θ^{'}$ ). But the dynamic interaction between rational interest rate expectations and learners' inflation and output gap expectations lowers the benefits of make-up and pre-emptive policy relative to cost (this is the second and third terms in $Θ^{'}$ ). [27]