Appendix A: Derivations | 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 A: Derivations

Anthony Brassil, Christopher G Gibbs and Callum Ryan

April 2025

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A.1 Unconstrained optimal policy

The unconstrained optimal policy problem is to maximise (9) subject to (10) to (13). The first-order conditions for this problem are

(A1)

π_{t} + μ_{t}^{π} - λ μ_{t - 1}^{π} - \frac{λ}{β σ} \sum_{j = 0}^{t} μ_{t - 1 - j}^{x} - ρ g μ_{t}^{ω π} = 0

(A2)

α x_{t} - κ μ_{t}^{π} + μ_{t}^{x} - λ (\frac{1}{β} - 1) \sum_{j = 0}^{t} μ_{t - 1 - j}^{x} - ρ g μ_{t}^{ω x} = 0

(A3)

- (1 - λ) β^{2} ρ 𝔼_{t} μ_{t + 1}^{π} - (1 - λ) \frac{β ρ}{1 - β ρ} \frac{1}{σ} 𝔼_{t} μ_{t + 1}^{x} + μ_{t}^{ω π} - β ρ (1 - g) 𝔼_{t} μ_{t + 1}^{ω π} = 0

(A4)

- (1 - λ) \frac{(1 - β) β ρ}{1 - β ρ} 𝔼_{t} μ_{t + 1}^{x} + μ_{t}^{ω x} - β ρ (1 - g) 𝔼_{t} μ_{t + 1}^{ω x} = 0

(A5)

\frac{1}{σ} \sum_{j = 0}^{t} μ_{t - j}^{x} = 0

From (A5), we have $μ_{t}^{x}$ = 0. This is Result 1. When policy is unconstrained, the IS curve is not a binding constraint. The central bank can generate whatever output gap it wants.

Substituting $μ_{t}^{x}$ = 0 into (A4) gives $μ_{t}^{ω x}$ = 0. Learners' output gap expectations enter only the IS curve. Therefore, when policy is unconstrained, there is no reason for the central bank to try to influence learners' output gap expectations.^[21]

Combining (A1) and (A2) to eliminate $μ_{t}^{π}$ gives Equation (14) in the main text. Similarly, using (A1) to eliminate $μ_{t}^{π}$ in (A3) gives Equation (15). The remainder of the derivation for unconstrained optimal target criterion (17) is in the main text.

A.1.1 Derivation of Equation (19)

First, we rewrite $μ_{t}^{ω π}$ in terms of expected output gaps by substituting (14) into the second right-hand side term in (15) and iterating forward to get

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

Then, inverting the optimal target criterion (17) gives

(A6)

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

The equilibrium system of (10), (12) and (A6) has the following minimum state variable solution

μ_{t}^{ω π} = (1 - λ) (a_{x} x_{t} - a_{ω} ω_{t}^{π} - a_{u} u_{t}) = (1 - λ) (a_{x} x_{t} - a_{ω} ρ g π_{t} - a_{ω} ρ (1 - g) ω_{t - 1}^{π} - a_{u} u_{t})

for some constants $a_{x}, a_{ω}$ and a_u, where the second equality uses the law of motion of $ω_{t}^{π}$ .

A.2 Constrained optimal policy

Suppose we add to (10) to (13) some constraint on the policy rate. This could be an information constraint (21), or a ZLB constraint (32). For this optimal policy problem, the first four first-order conditions, (A1) to (A4), remain the same. But instead of (A5), we have (22), reproduced here:

(22)

\frac{1}{σ} \sum_{j = 0}^{t} μ_{t - j}^{x} - μ_{t}^{i} = 0

where $μ_{t}^{i}$ is the Lagrange multiplier on the policy rate constraint.^[22]

Using (A1) to eliminate $μ_{t}^{π}$ from (A2) to (A4) and then substituting in $σ (1 - L) μ_{t}^{i} = μ_{t}^{x}$ from (22) gives (25), (34) and (35), which we reproduce here:

(25)

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

(34)

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

(35)

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

Now substitute (34) and (35) into (25) and collect all the non- $u_{t}^{i}$ terms in ${\tilde{π}}_{t}$ , which is defined in Result 4 in the main text. This gives

\begin{array}{l} {\tilde{π}}_{t} + \frac{σ}{κ} μ_{t}^{i} = \frac{λ}{β} \frac{κ}{σ} \frac{1}{1 - λ L} \frac{σ}{κ} μ_{t - 1}^{i} \\ + (1 - λ) g ρ^{2} Ω \frac{λ}{β} \frac{κ}{σ} \frac{1}{1 - ζ_{1} L} 𝔼_{t} \frac{1}{1 - ζ_{2} L^{- 1}} \frac{1}{1 - λ L} \frac{σ}{κ} μ_{t}^{i} \\ + (1 - λ) g ρ^{2} Ω \frac{1}{β (1 - β ρ)} \frac{κ}{σ} \frac{1}{1 - ζ_{1} L} 𝔼_{t} \frac{1 - L}{1 - ζ_{2} 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 ρ^{2} \frac{(1 - β) β}{1 - β ρ} 𝔼_{t} \frac{1 - L}{1 - β ρ (1 - g) L^{- 1}} \frac{σ}{κ} μ_{t + 1}^{i} \\ = {\frac{λ}{β} \frac{κ}{σ} \frac{1}{1 - λ} + λ (\frac{1}{β} - 1) + 1} \frac{σ}{κ} μ_{t - 1}^{i} \\ + (1 - λ) g ρ^{2} {Ω \frac{λ}{β} \frac{κ}{σ} \frac{1}{1 - ζ_{2} λ} \frac{1}{1 - ζ_{1} L} \frac{1}{1 - λ L} - Ω \frac{1}{β (1 - β ρ)} \frac{κ}{σ} \frac{1}{1 - ζ_{1} L} - \frac{(1 - β) β}{1 - β ρ}} \frac{σ}{κ} μ_{t}^{i} \\ + (1 - λ) g ρ^{2} \frac{(1 - β) β}{1 - β ρ} (1 - β ρ (1 - g)) 𝔼_{t} \frac{1}{1 - β ρ (1 - g) L^{- 1}} \frac{σ}{κ} μ_{t + 1}^{i} \\ + (1 - λ) g ρ^{2} Ω \frac{κ}{σ} {\frac{λ}{β} \frac{1}{1 - ζ_{2} λ} ζ_{2} + \frac{1}{β (1 - β ρ)} (1 - ζ_{2})} 𝔼_{t} \frac{1}{1 - ζ_{2} L^{- 1}} \frac{σ}{κ} μ_{t + 1}^{i} \\ + (1 - λ) g ρ^{2} Ω \frac{κ}{σ} {\frac{λ}{β} \frac{1}{1 - ζ_{2} λ} ζ_{2} + \frac{1}{β (1 - β ρ)} (1 - ζ_{2})} ζ_{1} \frac{1}{1 - ζ_{1} L} 𝔼_{t - 1} \frac{1}{1 - ζ_{2} L^{- 1}} \frac{σ}{κ} μ_{t}^{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_{1} (L^{- 1}) \frac{σ}{κ} μ_{t + 1}^{i} + P_{2} (L) 𝔼_{t - 1} P_{3} (L^{- 1}) \frac{σ}{κ} μ_{t}^{i} \end{array}

where $Θ$ and P(L) are defined in Result 4 and P₁(L^–1), P₂(L) and P₃(L^–1) are defined in Result 5. Finally, define

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}^{1} + 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

(A7)

{\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})

To get the optimal target criterion under imperfect information, Result 4, take the expectation of (A7) with respect to the central bank's information set. Since $Δ_{t}$ is just a multiple of $μ_{t}^{i}$ , (23) implies $Δ_{t | t}$ = 0 and $𝔼_{t} Δ_{t + k}$ = 0 for any $k \geq 1$ .

To get the optimal target criterion with the ZLB, Result 5, just recognise that $Δ_{t}$ is non-zero if and only if the ZLB is binding.

A.2.1 Derivation of Equation (31)

Start the optimal policy conditions (25), (26) and (27). Together with the Phillips curve (10) and the updating rule for learners' inflation expectations (12), these five equations determine the optimal equilibrium paths of ${π_{t}, x_{t}, ω_{t}^{π}, μ_{t}^{ω π}, μ_{t}^{ω x}}$ in terms of the paths of $μ_{t}^{i}$ and the cost-push shock u_t.

As in the derivation of Equation (19), we can rewrite $μ_{t}^{ω π}$ in (26) in terms of expected output gaps instead of expected inflation:

(A8)

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

Then combining (27) and (A8) and using the minimum state variable solution to evaluate expectations terms, we can write

\begin{array}{l} μ_{t}^{ω π} + (1 - λ L) \frac{1}{κ} μ_{t}^{ω x} = (1 - λ) ρ β^{2} 𝔼_{t} \frac{1}{1 - ρ β (1 - g) L^{- 1}} \frac{α}{κ} x_{t + 1} \\ - (1 - λ) ρ β^{2} (\frac{λ}{β} (1 - β) + 1 + \frac{1}{β (1 - β ρ)} \frac{κ}{σ} + \frac{1 - β}{β (1 - β ρ)}) \frac{σ}{κ} μ_{t}^{i} \\ + λ (1 - λ) ρ β^{2} \frac{1 - β}{β (1 - β ρ)} \frac{σ}{κ} μ_{t - 1}^{i} \\ = (1 - λ) (a_{x} x_{t} - a_{ω} ρ g π_{t} - a_{ω} ρ (1 - g) ω_{t - 1}^{π} - a_{u} u_{t} - a_{μ, 1} μ_{t}^{i} - a_{μ, 2} μ_{t - 1}^{i}) \end{array}

where a_x, $a_{ω}$ and a_u are the same constants as in (18), and $a_{μ, 1}$ and $a_{μ, 2}$ are new constants. Combining with the updating rule for learners' inflation expectations and using (21) gives the following description of optimal policy

𝔼_{t | t} \frac{1}{1 - \frac{ρ_{0} - ρ_{1} L}{1 - γ L} L} (\frac{α}{κ} x_{t} + ϕ (\frac{Ψ}{1 - γ L} + \frac{1 - Ψ}{1 - ρ (1 - g) L}) π_{t} + ϕ v_{u} \frac{1}{1 - γ L} u_{t}) = 0

Or, simply $𝔼_{t | t} (Q_{x} (L) x_{t} + Q_{π} (L) π_{t} + Q_{u} (L) u_{t}) = 0$ , where $Q_{x} (L), Q_{π} (L)$ and Q_u(L) are power series in L. The new parameters are

ρ_{0} \equiv \frac{λ + \frac{λ}{β} \frac{κ}{σ} + \frac{λ}{β} (1 - β) + 1 - (1 - λ) ρ g a_{μ, 2} \frac{κ}{σ}}{1 + (1 - λ) ρ g a_{μ, 1} \frac{κ}{σ}} - γ

ρ_{1} \equiv \frac{λ (\frac{λ}{β} (1 - β) + 1)}{1 + (1 - λ) ρ g a_{μ, 1} \frac{κ}{σ}}

The first and third power series, Q_x(L) and Q_u(L), are two different weighted averages of two exponentially weighted averages of lags. The second power series, $Q_{π} (L)$ , is a weighted average of these same two exponentially weighted averages of lags, plus a third exponentially weighted average of lags, that is,

Q_{x} (L) \equiv \frac{1}{1 - \frac{ρ_{0} - ρ_{1} L}{1 - γ L} L} \equiv \frac{ϕ_{x}}{1 - ξ_{1} L} + \frac{1 - ϕ_{x}}{1 - ξ_{2} L}

Q_{π} (L) \equiv \frac{1}{1 - \frac{ρ_{0} - ρ_{1} L}{1 - γ L} L} ϕ (\frac{Ψ}{1 - γ L} + \frac{1 - Ψ}{1 - ρ (1 - g) L}) \equiv ϕ (\frac{ϕ_{π 1}}{1 - ξ_{1} L} + \frac{ϕ_{π 2}}{1 - ξ_{2} L} + \frac{1 - ϕ_{π 1} - ϕ_{π 2}}{1 - ρ (1 - g) L})

Q_{u} (L) \equiv \frac{1}{1 - \frac{ρ_{0} - ρ_{1} L}{1 - γ L} L} ϕ v_{u} \frac{1}{1 - γ L} \equiv ϕ v_{u} (\frac{ϕ_{u}}{1 - ξ_{1} L} + \frac{1 - ϕ_{u}}{1 - ξ_{2} L})

where $ξ_{1}$ and $ξ_{2}$ are the roots of $ξ^{2} - (γ + ρ_{0}) ξ + ρ_{1}$ and $ϕ_{π 1}, ϕ_{π 2}, ϕ_{x}$ and $ϕ_{u}$ are constants.

Footnotes

This conclusion would not hold if we had an infinite-horizon Phillips curve, as in Eusepi et al (2018), because output gap expectations would then enter the Phillips curve. [21]

A similar first-order condition would apply if, instead of a constraint on the policy rate, the loss function contained a policy rate stability or smoothing term. Then $u_{t}^{i}$ would be replaced with the derivative of the (intertemporal) loss function with respect to i_t. The structure of the optimal target criterion is therefore identical in all these cases up to the definition of $u_{t}^{i}$ . [22]