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Progressive Taxation and Robust Monetary Policy

  • Kazuki Hiraga EMAIL logo and Kohei Hasui EMAIL logo

Abstract

Recent monetary policy analyses show the profound implications of progressive taxation for monetary policy. This paper investigates how progressive taxation on labor income changes the effect of model uncertainty by introducing robust control. We obtained the following results: (i) Higher progressive taxation decreases the effect of model uncertainty on the inflation rate, output gap, and interest rate. (ii) A sufficiently higher progressive taxation brings the economy into the determinate equilibrium even if the model uncertainty is strong. According to these results, we conclude that progressive taxation on labor income is effective in mitigating the effects of model uncertainty in terms of variance and equilibrium determinacy.

JEL Classification: E50; E52

Corresponding authors: Kazuki Hiraga, Graduate School of Economics, Nagoya City University, Nagoya, Japan, E-mail: ; and Kohei Hasui, Faculty of Economics, Aichi University, Nagoya, Japan, E-mail:

Award Identifier / Grant number: KAKENHI Grant Number 17K13768

Award Identifier / Grant number: KAKENHI Grant Number 19K13727

Acknowledgement

We are deeply grateful to two anonymous referees, Masataka Eguchi, and Hiroshi Fujiki for their valuable suggestions and comments.

  1. Author contribution: All the authors have accepted responsibility for the entire content of this submitted manuscript and approved submission.

  2. Research funding: Hiraga acknowledges financial support from JSPS KAKENHI Grant Number 19K13727. Hasui acknowledges financial support from JSPS KAKENHI Grant Number 17K13768.

  3. Conflict of interest statement: The authors declare no conflicts of interest regarding this article.

Appendix A

A.1 Proofs for Propositions 1 and 2

In this section, we derive the signs of the coefficients in measure (27) and give the proofs for Propositions 1 and 2 under Assumptions 1 and 2.

A.1.1 Proof for Proposition 1 and Inequality (28)

First of all, we show signs of the partial derivatives of κ(ϕ n ), q π (ϕ n ), q x (ϕ n ), κ(ϕ n )/α(ϕ n ), μ π (ϕ n ), and μ y (ϕ n ) with respect to ϕ n . The signs of ∂μ π /∂ϕ n and ∂μ y /∂ϕ n are give as follows:

(A.1) μ π ( ϕ n ) ϕ n = 1 1 ϕ n + ϕ + ϕ n ( 1 ϕ n ) 2 > 0 ,
(A.2) μ y ( ϕ n ) ϕ n = 1 1 ϕ n + 1 + ϕ + ϕ n ( 1 ϕ n ) 2 > 0 ,

Using (A.2) and (A.1), we obtain the signs of ∂κ(ϕ n )/∂ϕ n and ∂q x (ϕ n )/∂ϕ n :

(A.3) κ ( ϕ n ) ϕ n = λ μ y ϕ n > 0 ,
(A.4) q π ( ϕ n ) ϕ n = ϵ λ 1 Φ ( 1 + ϕ + ϕ n ) 2 < 0 ,
(A.5) q x ( ϕ n ) ϕ n = Φ μ y ϕ n > 0 ,
(A.6) κ ( ϕ n ) α ( ϕ n ) ϕ n = ϵ ( 1 Φ ) ( 1 + ϕ ) 2 [ ( 1 + ϕ ) ( ( 1 ϕ n ) + ϕ n Φ ) + ϕ n Φ ] > 0 .

where Φ = 1 − (ϵη(ϵ − 1))/ϵ < 1 denotes the steady state wedge of the marginal rate of substitution between consumption and leisure and the marginal product of labor.[19]

Now we derive the sign of 2 | c π ( θ , ϕ n ) | θ ϕ n . 2 | c π ( θ , ϕ n ) | θ ϕ n is given as follows:

(A.7) 2 | c π ( θ , ϕ n ) | θ ϕ n = 1 / θ 2 ω ( θ , ϕ n ) 2 q π ( ϕ n ) ϕ n 2 q π ( ϕ n ) / θ 2 ω ( θ , ϕ n ) 3 × κ ( ϕ n ) α ( ϕ n ) κ ( ϕ n ) ϕ n + κ ( ϕ n ) κ ( ϕ n ) ϕ n ϕ n 1 θ q π ( ϕ n ) ϕ n .

The first term, 1 / θ 2 ω ( θ , ϕ n ) 2 q π ( ϕ n ) ϕ n is negative because q π ( ϕ n ) ϕ n is negative. The terms in bracket in the second term are all positive, because κ ( ϕ n ) ϕ n > 0 , [ κ ( ϕ n ) / α ( ϕ n ) ] ϕ n > 0 , and q π ( ϕ n ) ϕ n < 0 . Therefore, the second term is also negative if ω(θ, ϕ n ) > 0. By combining these signs, we obtain (28) and Proposition 1:

(A.8) 2 | c π ( θ , ϕ n ) | θ ϕ n < 0 if ω ( θ , ϕ n ) > 0 .

A.1.2 Proof for Proposition 2

Now we derive the sign of 2 | c x ( θ , ϕ n ) | θ ϕ n and 2 | c i ( θ , ϕ n ) | θ ϕ n :

(A.9) 2 | c x ( θ , ϕ n ) | θ ϕ n = 1 / θ 2 ω ( θ , ϕ n ) 2 q π ( ϕ n ) κ ( ϕ n ) α ( ϕ n ) ϕ n κ ( ϕ n ) α ( ϕ n ) q π ϕ n 2 κ ( ϕ n ) / α ( ϕ n ) ω ( θ , ϕ n ) 3 × q π ( ϕ n ) θ 2 κ ( ϕ n ) κ ( ϕ n ) α ( ϕ n ) ϕ n + κ ( ϕ n ) α ( ϕ n ) κ ( ϕ n ) ϕ n 1 θ q π ( ϕ n ) ϕ n
(A.10) 2 | c i ( θ , ϕ n ) | θ ϕ n = σ γ c 2 | c i ( θ , ϕ n ) | θ ϕ n .

First, we ensure the sign of the first term in the brackets:

(A.11) q π ( ϕ n ) κ ( ϕ n ) α ( ϕ n ) ϕ n κ ( ϕ n ) α ( ϕ n ) q π ϕ n = ϵ 2 ( 1 Φ ) λ ( 2 ϕ n + ϕ ϕ n ) ( 1 Φ ) + ϕ 3 + 3 ϕ 2 + 2 ϕ + 2 ϕ ϕ n + ϕ 2 ϕ n ( 1 + ϕ + ϕ n ) 2 [ ( 1 + ϕ ) ( ( 1 ϕ n ) + ϕ n Φ ) + ϕ n Φ ] 2 > 0

Since Φ < 1, the sign of the first term is positive. On the other hand, the sign of the second term is negative, because κ ( ϕ n ) ϕ n > 0 , [ κ ( ϕ n ) / α ( ϕ n ) ] ϕ n > 0 , and q π ( ϕ n ) ϕ n < 0 .

Calculating the summation of the first and second terms is too complicated. Therefore, the sign of 2 | c x ( θ , ϕ n ) | θ ϕ n and 2 | c i ( θ , ϕ n ) | θ ϕ n is ambiguous.

A.2 Detection Error Probability

A.2.1 Approximating Model

Before calculating the detection error probability, we have to derive the approximating model. We can consider the case that the central bank design the policy supposing model misspecification but there are no misspecification. This is called the approximating model in the literature of robust control (Giordani and Söderlind 2004; Leitemo and Söderström 2008a, 2008b). In this paper, we obtain the approximating model by substituting the policy function of nominal interest rate under robust policy i t = c i (θ, ϕ n ) into the non-distorted structural equation (i.e. ν t u = 0 ). The following are the obtained policy function of inflation and output gap under the approximating model, respectively:[20]

(A.12) π t = c π a ( θ , ϕ n ) u t ,
(A.13) x t = c x a ( θ , ϕ n ) u t ,

where

c π a = 1 κ ( ϕ n ) 2 α ( ϕ n ) ω ( θ , ϕ n ) , c x a = κ ( ϕ n ) α ( ϕ n ) ω ( θ , ϕ n ) .

A.2.2 Detection Error Probability

Now we describe in detail the calculation of the detection error probability. The overall definition of detection error probability is expressed as follows (Giordani and Söderlind 2004):

p ( θ ) = 1 2 × Prob ( L A > L W | W ) + 1 2 × Prob ( L W > L A | A ) ,

where LA and LW denote the values of the likelihood of the approximating model and worst-case scenario (i.e., robust monetary policy), respectively. The notations A and W denote the approximating model and worst-case scenario, respectively.

Given the data from the model, the probability is calculated as the rate of wrong choices between the worst-case scenario and the approximating model. The detection error probability thus indicates the difficulty of distinguishing between models with and without misspecification.

To obtain the detection error probability, we generate the data of the misspecification term for the worst-case and the approximating model (we express these as ν t w and ν t a , respectively) for sufficiently long periods T.[21]

Then, we calculate the relative likelihood r w and r a as follows:[22]

(A.14) r w = 1 T t = 0 T 1 1 2 ν t w ν t w + ν t w u t ,
(A.15) r a = 1 T t = 0 T 1 1 2 ν t a ν t a ν t a u t .

Finally, we obtain the detection error probability as follows:

(A.16) p ( θ ) = 1 2 f r e q ( r w 0 ) + f r e q ( r a 0 ) .

A.3 Brief Description of Mattesini and Rossi (2012) Model

In this section, we present a brief description of Mattesini and Rossi’s (2012) model.

A.3.1 Households

Households’ preference is given as follows:

(A.17) U t = E 0 j = 0 β j C t 1 σ 1 σ N t 1 + ϕ 1 + ϕ , σ , ϕ > 0 .

where C t and N t denote consumption and supply of labor hours, respectively. The flow budget constraint is given as follows:

(A.18) P t C t + R t 1 B t = ( 1 τ t ) W t N t + B t + D t P t T t ,

where R t , B t , W t , D t , T t denote gross nominal risk-free interest rate, risk free government bond, nominal wage, profit income, and lump-sum tax, respectively. τ t denotes the taxes on labor income. Mattesini and Rossi (2012) specified the form of τ t following Guo (1999) and Guo and Lansing (1998):

(A.19) τ t = 1 η Y n Y n , t ϕ n .

where η ∈ (0, 1] and ϕ n ∈ [0, 1), and Y n = WN/P and Yn,t = W t N t /P t denote base level of income and actual level of income, respectively. The tax rate ensures 0 ≤ τ t < 1 when ϕ n > 0.

The marginal tax rate on labor income is given as follows:

(A.20) τ t m = τ t Y n , t Y n , t = 1 η ( 1 ϕ n ) Y n Y n , t ϕ n = τ t + η ϕ n Y n Y n , t ϕ n .

First order conditions of households are given as follows:

(A.21) C t σ = β R t E t C t + 1 σ Π t + 1 ,
(A.22) C t σ N t ϕ = 1 τ t m W t P t ,

A.3.2 Firms and Government

Firm k’s production function is a constant return to the scale:

(A.23) Y t ( k ) = A t N t ( k ) ,

where k ∈ [0, 1]. The aggregate marginal cost is

(A.24) MC t = 1 A t W t P t .

The resource constraint is

(A.25) Y t = C t + G t .

A.3.3 Log-Linearized Expression

The intermediate goods firms’ optimal price setting yields the following standard Phillips curve with marginal cost expression:

(A.26) π t = β E t π t + 1 + λ m c t .

where λ = ( 1 φ ) ( 1 φ β ) φ .

Log-linearizing Eqs. (A.22)(A.25) yields

(A.27) σ c t + ϕ n t = ( 1 ϕ n ) w t ϕ n n t ,
(A.28) y t = γ c c t + ( 1 γ c ) g t ,
(A.29) y t = a t + n t ,
(A.30) m c t = w t a t ,

Combining these equation yields the following relation:

(A.31) m c t = σ 1 ϕ n 1 γ c y t 1 γ c γ c g t + ϕ + ϕ n 1 ϕ n ( y t a t ) a t

After some complicated substitution, we obtain the following reduced expression:[23]

(A.32) π t = β E t π t + 1 + κ ( ϕ n ) x t + u t ,

where

x t = y t y t n , u t = κ ( ϕ n ) ( y t n y t * ) , y t * = ( 1 + ϕ ) ( 1 Φ ) a t + 1 + ϕ 1 ϕ n Φ a t , y t n = 1 + ϕ 1 + ϕ + ϕ n a t .

Finally, the tax rate τ ̂ t is reduced as follows:

(A.33) τ ̂ t = η ϕ n 1 η 1 + 1 + ϕ + ϕ n 1 ϕ n y t η ϕ n 1 η 1 + ϕ 1 ϕ n 1 Ξ u t .

where

Ξ = κ ( ϕ n ) ( 1 + ϕ ) ( 1 Φ ) + Φ 1 ϕ n 1 1 ϕ + ϕ n .

A.4 The Effects of Slope in NKPC

In Section 4, we analyzed the effects of the model uncertainty under σ ≠ 1 and ϕ ≠ 1 following Rotemberg and Woodford (1997) and Smets and Wouters (2007). In this section, we show results when we change the values of σ and ϕ in the slope of the Phillips curve κ.

We define the inverse of the intertemporal elasticity of substitution and the inverse of the Frisch elasticity in the slope as σ κ and ϕ κ , respectively:

(A.34) μ y ( ϕ n ) = σ κ + γ c ( ϕ κ + ϕ n ) γ c ( 1 ϕ n ) , κ ( ϕ n ) = λ μ y ( ϕ n ) , λ = ( 1 φ ) ( 1 φ β ) φ .

Figures A1 and A2 plot the policy function and determinate-indeterminate region when we set σ and σ κ at 0.16 and 1.38, respectively. As we mentioned in Section 4, the effects of the model uncertainty become small and all regions are determinate in the plotted area.

Figure A1: 
Effects of an increase in model uncertainty (1/θ) for degrees of ϕ
n
 = 0.13, 0.18, and 0.25 under σ = 0.16 and σ
κ
 = 1.38.
Figure A1:

Effects of an increase in model uncertainty (1/θ) for degrees of ϕ n = 0.13, 0.18, and 0.25 under σ = 0.16 and σ κ = 1.38.

Figure A2: 
Determinate-indeterminate region in the (1/θ, ϕ
n
) space under σ = 0.16 and σ
κ
 = 1.38.
Figure A2:

Determinate-indeterminate region in the (1/θ, ϕ n ) space under σ = 0.16 and σ κ = 1.38.

Figures A3 and A4 plot the policy function and determinate-indeterminate region when we set ϕ and ϕ κ at 0.47 and 1.83, respectively. Analogous to the case of σ κ , the effects of the model uncertainty become small and all regions are determinate in the plotted area.

Figure A3: 
Effects of an increase in model uncertainty (1/θ) for degrees of ϕ
n
 = 0.13, 0.18, and 0.25 under ϕ = 0.47 and ϕ
κ
 = 1.83.
Figure A3:

Effects of an increase in model uncertainty (1/θ) for degrees of ϕ n = 0.13, 0.18, and 0.25 under ϕ = 0.47 and ϕ κ = 1.83.

Figure A4: 
Determinate-indeterminate region in the (1/θ, ϕ
n
) space under ϕ = 0.46 and ϕ
κ
 = 1.83.
Figure A4:

Determinate-indeterminate region in the (1/θ, ϕ n ) space under ϕ = 0.46 and ϕ κ = 1.83.

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Received: 2021-12-02
Revised: 2022-05-21
Accepted: 2022-07-31
Published Online: 2022-08-24

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