Delegating Optimal Monetary Policy Inertia in a Small-Open Economy

Daisuke Ida; Mitsuhiro Okano

doi:10.1515/bejm-2020-0181

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Delegating Optimal Monetary Policy Inertia in a Small-Open Economy

Daisuke Ida und Mitsuhiro Okano

Veröffentlicht/Copyright: 10. Dezember 2020

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Aus der Zeitschrift The B.E. Journal of Macroeconomics Band 21 Heft 2

Abstract

This paper explores the delegation of several targeting regimes in a small open new Keynesian (NK) model and examines how central banks overcome stabilization bias in a small open NK model. Results indicate that both speed limit and real exchange rate targeting can carry the isomorphic properties of optimal monetary policy over to the closed economy. In addition, neither nominal income growth targeting nor CPI inflation targeting replicates a commitment policy. These findings provide new implications for optimal monetary policy in an open economy.

Keywords: optimal monetary policy; targeting regime; commitment; discretion

JEL Classification: E52; E58; F41

Corresponding author: Daisuke Ida, Faculty of Economics, Momoyama Gakuin University, 1-1, Manabino, Izumi, Osaka 594-1198, Japan, E-mail: ida-dai@andrew.ac.jp

We would like to thank Arpad Abraham and the anonymous referees for helpful comments and suggestions. We would like to thank Ryo Kato, Takeki Sunakawa, and Kenya Takaku for helpful comments and suggestions. This paper was supported by JSPS KAKENHI Grant Numbers 16H03618 and 17K13766. All remaining errors are our own.

Funding source: JSPS KAKENHI

Award Identifier / Grant number: 16H03618

Award Identifier / Grant number: 17K13766

Appendix A: Proofs of the Propositions

A.1 Proof of Proposition 1

Before defining the Bellman equation, we obtain the following objective function:

(A.1) λ x t 2 + λ S ( x t − x t − 1 ) 2 + α ( π t c ) 2 + 2 c π x t − 1 π t = α π t 2 + ( λ + λ S + α ν 2 σ ν 2 ) x t 2 + ( λ S + α ν 2 σ ν 2 ) x t − 1 2 + 2 α ν σ ν π t x t − 2 ( λ S + α ν 2 σ ν 2 ) x t x t − 1 + 2 ( c π x t − 1 − α ν σ ν ) π t x t − 1

Then we define the Bellman equation as follows:

V ( x t − 1 ; u t ) = min 1 2 { α π t 2 + ( λ + λ S + α ν 2 σ ν 2 ) x t 2 + ( λ S + α ν 2 σ ν 2 ) x t − 1 2 + 2 α ν σ ν π t x t − 2 ( λ S + α ν 2 σ ν 2 ) x t x t − 1 + 2 ( c π x t − 1 − α ν σ ν ) π t x t − 1 + β E t V ( x t ; u t + 1 ) }

The first-order condition with respect to x t is

(A.2) α ( γ + ν σ ν ) π t + ( λ + λ S + α ν 2 σ ν 2 + α ν σ ν γ ) x t − ( λ S + α ν 2 σ ν 2 + α ν σ ν γ − c π γ ) x t − 1 + β E t ∂ V ( x t ; u t + 1 ) ∂ x t = 0 .

In addition, from the envelope theorem, we obtain

(A.3) ∂ V ( x t − 1 ; u t ) ∂ x t − 1 = − ( λ S + α ν 2 σ ν 2 ) Δ x t + ( c π − α ν σ ν ) π t .

Substituting (A.3) into (A.2) and considering the optimal targeting rule, we obtain the following second-order difference equation:

(A.4) − β [ ( λ S + α ν 2 σ ν 2 ) + ( c π − α ν σ ν ) λ ˜ κ ν ] E t x t + 1 + [ λ + λ S + α ν 2 σ ν 2 + α ν σ ν γ − α λ ˜ κ ν ( γ + ν σ ν ) + β { ( λ S + α ν 2 σ ν 2 ) + ( c π − α ν σ ν ) λ ˜ κ ν } ] x t + [ α λ ˜ κ ν ( γ + ν σ ν ) − ( λ S + α ν 2 σ ν 2 + ν σ ν γ ) + c π γ ] x t − 1 = 0 .

The optimal delegation parameter is founded by noting that Equation (A.4) evaluated at the conjectured solution ( x t c , π t c ) should be identity. Therefore, to observe the optimal delegation parameter, all the coefficients should be zero. This leads to the following:

(A.5) ( λ S + α ν 2 σ ν 2 ) + ( c π − α ν σ ν ) λ ˜ κ ν = 0

(A.6) λ + λ S + α ν 2 σ ν 2 + α ν σ ν γ − α λ ˜ κ ν ( γ + ν σ ν ) = 0

(A.7) α λ ˜ κ ν ( γ + ν σ ν ) − ( λ S + α ν 2 σ ν 2 + ν σ ν γ ) + c π γ = 0

Solving Equations (A.5)–(A.7), we obtain the optimal delegation coefficients under the speed limit policy.

A.2 Proof of Proposition 2

Before defining the Bellman equation, we arrange the objective function under nominal income growth targeting. Substituting the definition of CPI inflation into the objective function, we obtain the following objective function:

(A.8) λ x t 2 + α π t 2 + ψ ( π t c + x t − x t − 1 ) 2 + 2 c π π t x t − 1 = λ x t 2 + α π t 2 + ψ ( π t + ν σ ν Δ x t + x t − x t − 1 ) 2 + 2 c π π t x t − 1 = ( λ + ψ ( 1 + ν σ ν ) 2 ) x t 2 + ( α + ψ ) π t 2 + ψ ( 1 + ν σ ν ) 2 x t − 1 2 − 2 ψ ( 1 + ν σ ν ) 2 x t x t − 1 + 2 ψ ( 1 + ν σ ν ) π t x t + 2 ( c π − ψ ( 1 + ν σ ν ) ) π t x t − 1

Then, we define the Bellman equation as follows:

V ( ( x t − 1 ; u t ) ) = min 1 2 { ( λ + ψ ( 1 + ν σ ν ) 2 ) x t 2 + ( α + ψ ) π t 2 + ψ ( 1 + ν σ ν ) 2 x t − 1 2 − 2 ψ ( 1 + ν σ ν ) 2 x t x t − 1 + 2 ψ ( 1 + ν σ ν ) π t x t + 2 ( c π − ψ ( 1 + ν σ ν ) ) π t x t − 1 + β E t V ( x t ; u t + 1 ) }

The first-order condition of this optimization is given as follows:

(A.9) ( λ + ψ ( 1 + ν σ ν ) 2 ) x t + ( α + ψ ) γ π t − ψ ( 1 + ν σ ν ) 2 x t − 1 + ψ ( 1 + ν σ ν ) π t + ψ ( 1 + ν σ ν ) γ x t + ( c π − ψ ( 1 + ν σ ν ) ) γ x t − 1 + β E t ∂ V ( x t ; u t + 1 ) ∂ x t = 0 .

It follows from the envelope theorem that

(A.10) ∂ V ( x t − 1 ; u t ) ∂ x t − 1 = − ψ ( 1 + ν σ ν ) 2 Δ x t + [ c π − ψ ( 1 + ν σ ν ) ] π t .

Using the envelope theorem and the optimal targeting rule, we obtain the following second-order difference equation:

− β [ ψ ( 1 + ν σ ν ) 2 + λ ˜ κ ν ( c π − ψ ( 1 + ν σ ν ) ) ] E t x t + 1

+ [ λ + ψ ( 1 + ν σ ν ) ( 1 + γ + ν σ ν ) − λ ˜ κ ν ( α γ + ψ ( 1 + γ + ν σ ν ) )

+ β ( ψ ( 1 + ν σ ν ) 2 + λ ˜ κ ν ( c π − ψ ( 1 + ν σ ν ) ) ) ] x t

(A.11) [ γ c π − ψ ( 1 + ν σ ν ) ( 1 + γ + ν σ ν ) + λ ˜ κ ν ( α γ + ψ ( 1 + γ + ν σ ν ) ) ] x t − 1 = 0

Once more, this equation of the optimal delegation parameter requires noting that Equation (A.11) evaluated at the conjectured solution ( x t c , π t c ) should be identity, and all the coefficients should be zero. This leads to the following:

(A.12) ψ ( 1 + ν σ ν ) 2 + λ ˜ κ ν ( c π − ψ ( 1 + ν σ ν ) ) = 0 ,

(A.13) λ + ψ ( 1 + ν σ ν ) ( 1 + γ + ν σ ν ) − λ ˜ κ ν ( α γ + ψ ( 1 + γ + ν σ ν ) ) = 0 ,

(A.14) γ c π − ψ ( 1 + ν σ ν ) ( 1 + γ + ν σ ν ) + λ ˜ κ ν ( α γ + ψ ( 1 + γ + ν σ ν ) ) = 0 .

Solving Equations (A.12)–(A.14) provides the result of Proposition 2.

A.3 Proof of Proposition 3

Before defining the Bellman equation, we obtain the following objective function:

(A.15) λ x x t 2 + π t 2 + λ q Δ q t 2 + 2 c π π t x t − 1 = λ x x t 2 + π t 2 + λ q ( ( 1 − ν ) σ ν Δ x t ) 2 + 2 c π π t x t − 1 = ( λ x + ( 1 − ν ) 2 σ ν 2 λ q ) x t + π t 2 + ( 1 − ν ) 2 σ ν 2 λ q x t − 1 2 − 2 ( 1 − ν ) σ ν λ q x t x t − 1 + 2 c π π t x t − 1

Using this objective function, we can define the Bellman equation as follows:

V ( x t − 1 ; u t ) = min 1 2 { ( λ x + ( 1 − ν ) 2 σ ν 2 λ q ) x t 2 + π t 2 + ( 1 − ν ) 2 σ ν 2 λ q x t − 1 2 − 2 ( 1 − ν ) σ ν λ q x t x t − 1 + 2 c π π t x t − 1 + β E t V ( x t ; u t + 1 ) }

The first-order condition with respect to x t is

(A.16) ( λ x + ( 1 − ν ) 2 σ ν 2 λ q ) x t + γ π t − ( 1 − ν ) σ ν λ q x t − 1 + c π γ x t − 1 + β E t ∂ V ( x t ; u t + 1 ) ∂ x t = 0 .

In addition, from the envelope theorem, we obtain

(A.17) ∂ V ( x t − 1 ; u t ) ∂ x t − 1 = − λ q ( 1 − ν ) 2 σ ν 2 Δ x t + c π π t .

Substituting Equation (A.17) and Equations (9) and (10) into Equation (A.16) and then considering the optimal targeting rule, we obtain the following second-order difference equation:

(A.18) − [ β ( λ ˜ κ ν c π + ( 1 − ν ) 2 σ ν 2 λ q ) ] E t x t + 1 + [ λ x + ( 1 − ν ) 2 σ ν 2 λ q − λ ˜ κ ν γ + β ( λ ˜ κ ν c π + ( 1 − ν ) 2 σ ν 2 λ q ) ] x t + [ c π γ − ( 1 − ν ) 2 σ ν 2 λ q + λ ˜ κ ν γ ] x t − 1 = 0 .

As explained by Bilbiie (2014), in this equation, the optimal delegation parameter is found by noting that Equation (A.18) evaluated at the conjectured solution ( x t c , π t c ) should be identity. Therefore, to identify the optimal delegation parameter, all the coefficients should be zero. This leads to the following:

(A.19) λ ˜ κ ν c π + ( 1 − ν ) 2 σ ν 2 λ q = 0 ,

(A.20) λ x + ( 1 − ν ) 2 σ ν 2 λ q − λ ˜ κ ν γ = 0 ,

(A.21) c π γ − ( 1 − ν ) 2 σ ν 2 λ q + λ ˜ κ ν γ = 0 .

Solving Equations (A.19)–(A.21), we obtain the optimal delegation parameters in Proposition 3.

A.4 Proof of Proposition 4

Before defining the Bellman equation for solving this optimization problem, we calculate the objective function under CPI inflation targeting as follows:

(A.22) λ x 2 + α π t 2 + λ π ( π t c ) 2 + 2 c π π t x t − 1 = λ x 2 + α π t 2 + λ π ( π t + ν σ ν Δ x t ) 2 + 2 c π π t x t − 1 = ( λ + ν 2 σ ν 2 λ π ) x t 2 + ( α + λ π ) π t 2 + ν 2 σ ν 2 λ π x t − 1 2 + 2 ν σ ν λ π π t x t − 2 ν 2 σ ν 2 λ π x t x t − 1 + 2 ( c π − ν σ ν λ π ) π t x t − 1 .

Using this objective function, we can define the Bellman equation as follows:

V ( x t − 1 ; u t ) = min 1 2 { ( λ + ν 2 σ ν 2 λ π ) x t 2 + ( α + λ π ) π t 2 + ν 2 σ ν 2 λ π x t − 1 2 + 2 ν σ ν λ π π t x t − 2 ν 2 σ ν 2 λ π x t x t − 1 + 2 ( c π − ν σ ν λ π ) π t x t − 1 + β E t V ( x t ; u t + 1 ) }

The first-order condition of this optimization is given

(A.23) ( λ + ν 2 σ ν 2 λ π ) x t + ( α + λ π ) γ π t + ν σ ν λ π π t − ν 2 σ ν 2 λ π x t − 1 + ( c π − ν σ ν λ π ) γ x t − 1 + β E t ∂ V ( x t ; u t + 1 ) ∂ x t = 0 .

From the envelope theorem, we obtain the following equation:

(A.24) ∂ V ( x t − 1 ; u t ) ∂ x t − 1 = − ν 2 σ ν 2 λ π Δ x t + ( c π − ν σ ν λ π ) .

Using this envelope theorem and the optimal targeting rule with commitment leads to the following second-order difference equation:

(A.25) − [ β ( ν 2 σ ν 2 λ π + λ ˜ κ ν ( c π − ν σ ν λ π ) ) ] E t x t + 1 + [ λ + ( ν σ ν λ π − λ ˜ κ ν ( α + λ π ( γ + ν σ ν ) ) ) + β ( ν 2 σ ν 2 λ π + λ ˜ κ ν ( c π − ν σ ν λ π ) ) ] x t + [ γ c π − ( ν σ ν λ π − λ ˜ κ ν ( α + λ π ( γ + ν σ ν ) ) ) ] x t − 1 = 0 .

Again, as per the foregoing, the optimal delegation parameter is found by noting that Equation (A.25) evaluated at the conjectured solution ( x t c , π t c ) should be identity. To identify the optimal delegation parameter, all the coefficients should be zero. This leads to the following:

(A.26) ν 2 σ ν 2 λ π + λ ˜ κ ν ( c π − ν σ ν λ π ) = 0 ,

(A.27) λ + ( ν σ ν λ π − λ ˜ κ ν ( α + λ π ( γ + ν σ ν ) ) ) = 0 ,

(A.28) γ c π − ( ν σ ν λ π − λ ˜ κ ν ( α + λ π ( γ + ν σ ν ) ) ) = 0 .

Solving Equations (A.26)–(A.28) leads to the result of Proposition 4.

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Received: 2020-04-29

Accepted: 2020-11-28

Published Online: 2020-12-10

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optimal monetary policy; targeting regime; commitment; discretion