Appendix B: Further Results | 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 B: Further Results

Anthony Brassil, Christopher G Gibbs and Callum Ryan

April 2025

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B.1 Optimal policy with non-rational interest rate expectations

B.1.1 When does learning about interest rates constrain policy?

To get a sense of how learning about i_t might alter our results, here we repeat the simple example in Section 2 of Eusepi, Giannoni and Preston (2024) with our hybrid model for expectations. Consider the model

π_{t} = β {\hat{E}}_{t} π_{t + 1} + κ x_{i}

x_{t} = {\hat{E}}_{t} \frac{1}{1 - β L^{- 1}} [(1 - β) x_{t + 1} - \frac{1}{σ} (i_{t} - π_{t + 1} - r_{t}^{n})]

with $r_{t}^{n}$ iid. Suppose the central bank wants to implement $π_{t}$ = 0. Then the interest rate path follows

i_{t} = r_{t}^{n} + \frac{σ β}{κ} {\hat{E}}_{t} π_{t + 1} + {\hat{E}}_{t} \frac{1}{1 - β L^{- 1}} [σ (1 - β) x_{t + 1} - (β i_{t + 1} - π_{t + 1} - β r_{t + 1}^{n})]

Now suppose that nominal interest rate expectations are partially rational and partially learned:

{\hat{E}}_{t} i_{T} = λ 𝔼_{t} i_{T} + (1 - λ) ρ^{T - t} ω_{t - 1}^{i}

ω_{t}^{i} = ρ ω_{t - 1}^{i} + ρ g (i_{t} - ω_{t - 1}^{i})

For simplicity, assume all output gap and inflation expectations are fixed at steady state (the mechanism we are interested in here is the feedback between interest rates and interest rate expectations). Substituting in expectations, the interest rate path needed to implement $π_{t}$ is

i_{t} = r_{t}^{n} - λ β 𝔼_{t} \frac{1}{1 - β L^{- 1}} i_{t + 1} - (1 - λ) \frac{β ρ}{1 - β ρ} ω_{t - 1}^{i}

Substituting in $ω_{t} = \frac{ρ g}{1 - ρ (1 - g) L} i_{t}$ and rearranging gives

(B1)

𝔼_{t} [1 + λ β \frac{L - 1}{1 - β L^{- 1}} + (1 - λ) \frac{β ρ^{2} g}{1 - β ρ} \frac{L}{1 - ρ (1 - g) L}] i_{t} = r_{t}^{n}

A bounded path for i_t and $ω_{t}^{i}$ exists only when^[23]

(B2)

g < \frac{1 + ρ}{ρ} Λ where Λ = \frac{1}{1 + \frac{(1 - λ) \frac{β ρ}{1 - β ρ}}{1 - λ \frac{β}{1 + β}}}

If we substitute in $ρ$ = 1 and $λ$ = 0, then this expression nests the equivalent condition (8) in Eusepi, Giannoni and Preston (2024): $g < 2 (1 - β)$ .

Effect of $λ$ : relative to Eusepi, Giannoni and Preston (2024), introducing rational expectations $(λ > 0)$ affects this constraint in two opposite ways. The reduction in the share of learners means their interest rate expectations have less of an effect (this is the $(1 - λ)$ term in $Λ$ ). But the rational agents recognise that the central bank in the future has to offset the interest rate expectations of learners, which changes their rate expectations, and the central bank now has to offset these rational expectations (this is the $λ$ term in $Λ$ ). But we can rewrite $Λ$ as

Λ (1 - β ρ) (1 + \frac{β ρ λ}{1 + β (1 - (1 + ρ) λ)}) \geq 1 - β ρ

Therefore, the first effect outweighs the second, and so increasing the share of rational agents eases the constraint.

Effect of $ρ$ : if $ρ < 1$ , then the constraint eases because (i) a given interest rate surprise does not affect short-term beliefs as much or for as long, and (ii) a given short-term belief does not affect long-term interest rate expectations as much.

B.1.2 Optimal policy with learning about interest rates

For a simple example, assume prices are fixed, and continue with the assumption that output gap expectations are fixed. The optimal policy problem is

\begin{array}{l} _{{x_{t}, i_{t}, ω_{t}^{i}}}^{\min} \frac{1}{2} 𝔼_{t} \frac{1}{1 - β L^{- 1}} x_{t}^{2} \\ s .t . σ x_{t} = - i_{t} - λ β 𝔼_{t} \frac{1}{1 - β L^{- 1}} i_{t + 1} - (1 - λ) \frac{β ρ}{1 - β ρ} ω_{t - 1}^{i} + r_{t}^{n} \\ ω_{t}^{i} = ρ g i_{t} + ρ (1 - g) ω_{t - 1}^{i} \end{array}

Let $β \to 1$ . Take the first-order conditions and eliminate multipliers to get the targeting rule

(B3)

x_{t} + λ \frac{1}{1 - L} x_{t - 1} + (1 - λ) \frac{ρ^{2} g}{1 - ρ} 𝔼_{t} \frac{1}{1 - ρ (1 - g) L^{- 1}} x_{t + 1} = 0

The intuition in the main text around history dependence and pre-emption depending on the proportion of rational agents and learners extends to the case of interest rate expectations.

The characteristic polynomial for (B3) is very closely related to the characteristic polynomial for (B1). Condition (B2) turns out to be important for implementing this optimal target criterion:^[24]

If (B2) is satisfied, then the target criterion (B3) pins down a unique path for x_t (which converges to x_t = 0). Substituting it into the IS curve gives a non-explosive path for i_t.
If (B2) is not satisfied, then there are infinitely many paths for x_t that satisfy the target criterion. But, substituting into the IS curve, only one of those paths implies a bounded path for i_t and $ω_{t}^{i}$ . Therefore, there still exists a unique bounded solution that is consistent with the optimal target criterion.

If $β < 1$ , then there is a set of parameterisations for which there is no bounded solution consistent with the optimal target criterion. This occurs when $β$ is sufficiently low that the optimal intertemporal trade-off requires x_t = 0, but g is sufficiently high than this cannot be achieved without an explosive interest rate path. It occurs due to a tension between an impatient policymaker, who discounts the future a lot, and significant lags in the transmission of policy, which mean that current policy has significant future effects. If $β$ is close to one, then the set of parameterisations for which this occurs is small.

B.2 Optimal policy under discretion

What would our optimal targeting rules look like under discretion? This is a difficult question to address analytically in our full model, but we can consider a two-period version of our model to get some intuition:

(B4)

π_{0} = β λ 𝔼_{0} π_{1} + κ x_{0} + u_{0}

(B5)

π_{1} = β (1 - λ) g π_{0} + κ x_{1} + u_{1}

B.2.1 Optimal policy under commitment

We start by solving for optimal commitment policy in this two-period model so that we have a benchmark to which we can compare optimal discretionary policy. The first-order conditions under commitment are

\begin{array}{l} π_{0} - μ_{0} + β^{2} (1 - λ) g 𝔼_{0} μ_{1} = 0 \\ π_{1} + λ μ_{0} - μ_{1} = 0 \\ α x_{0} + κ μ_{0} = 0 \\ α x_{1} + κ μ_{1} = 0 \end{array}

Eliminating the multipliers gives the optimal target criteria:

(B6)

\frac{α}{κ} x_{0} = - \frac{π_{0} + β^{2} (1 - λ) g 𝔼_{0} π_{1}}{1 - β^{2} (1 - λ) g λ}

(B7)

\frac{α}{κ} x_{1} = - π_{1} - \frac{λ β^{2} (1 - λ) g}{1 - λ β^{2} (1 - λ) g} 𝔼_{0} π_{1} - \frac{1}{1 - λ β^{2} (1 - λ) g} λ π_{0}

The heterogeneity of expectations and the interaction between rational and learners' expectations is important (i.e. the denominators increase to 1 if $λ$ = 0 or $λ$ = 1). Equation (B6) shows that when $λ \in$ (0,1), greater weight should be placed on current and expected future inflation, because the costs of following through on make-up commitments next period are eased by influencing learners' expectations this period. Equation (B7) shows that for $λ \in$ (0,1), greater weight should also be placed on past inflation and past expectations of inflation, because these make-up commitments ease the costs of past pre-emptive policy.

B.2.2 Optimal policy under discretion

Working backwards, optimal policy under discretion in the second period is

(B8)

\frac{α}{κ} x_{1} = - π_{1}

Then in the first period, the policymaker faces the constraints (B4), (B5) and (B8). The first-order conditions are

\begin{array}{l} π_{0} - μ_{0} + β^{2} (1 - λ) g 𝔼_{0} μ_{1} = 0 \\ π_{1} + λ μ_{0} - μ_{1} - μ_{2} = 0 \\ α x_{0} + κ μ_{0} = 0 \\ α x_{1} + κ μ_{1} - \frac{α}{κ} μ_{2} = 0 \end{array}

Eliminating the multipliers gives the optimal target criterion

(B9)

\frac{α}{κ} x_{0} = - \frac{π_{0} + β^{2} (1 - λ) g 𝔼_{0} π_{1}}{1 - β^{2} (1 - λ) g \frac{λ}{1 + \frac{κ^{2}}{α}}}

Compare (B9) to the optimal commitment policy (B6). Optimal policy under discretion places less weight on both inflation terms. Again heterogeneity is important: this difference is only because of the interaction between rational agents and learners. If $λ$ = 0 or $λ$ = 1, then period 0 optimal policy is identical under commitment and discretion.

The reason why discretion involves less response to inflation is because the future policymaker will not be responding to past inflation. So there is no need to use learners' expectations to ease the cost of doing so. With both learning and rational expectations, there is still some reason to lift learners' inflation expectations, because that will affect future inflation and thereby influence current rational inflation expectations. But the future policymaker will attempt to offset this inflation rather than accomodate it, so the effect is less powerful.

Footnotes

For all parameterisations, the roots of the characteristic polynomial for (B1) are real, and only one is greater than 1. A bounded path for i_t and $ω_{t}^{i}$ exists if and only if the other root is greater than –1, which occurs when condition (B2) is satisfied. [23]

The roots of the characteristic polynomial for (B3) are the reciprocals of the roots of the characteristic polynomial for (B1) (setting $β$ = 1 in (B1)). They are real, and at least one lies inside the unit circle. The other root lies outside the unit circle if and only if condition (B2) is satisfied. [24]