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Progressive Taxation and Robust Monetary Policy∗

Kazuki HiragaaKohei Hasuib

August 1, 2022

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.

Keywords: Monetary policy, Progressive taxation, Robust policy

JEL Classiﬁcation: E50; E52

∗This work is accepted for publication in The B.E. Journal of Macroeconomics. We are deeply grateful to two

anonymous referees, Masataka Eguchi, and Hiroshi Fujiki for their valuable suggestions and comments. Hiraga

acknowledges ﬁnancial support from JSPS KAKENHI Grant Number 19K13727. Hasui acknowledges ﬁnancial

support from JSPS KAKENHI Grant Number 17K13768.

aAssociate Professor, Graduate School of Economics, Nagoya City University, Nagoya, Japan.

E-mail: khiraga581470@gmail.com

bAssociate Professor, Faculty of Economics, Aichi University, Nagoya, Japan.

E-mail: khasui@vega.aichi-u.ac.jp

1

1 Introduction

One of the policy objectives of monetary and ﬁscal authorities is to stabilize the economy in

the face of varying circumstances. Speciﬁcally, ﬁscal authority has an automatic stabilizing

function through a progressive income tax system, while monetary authority adjusts policy

rates to stabilize the economy. Actually, Auerbach and Feenberg (2000) show that a federal tax

system absorbs 8% of the shock to output by empirical estimation, and argue that progressive

taxation is eﬀective in stabilizing output. Dolls et al. (2012) found that automatic stabilizers

absorbed more than 30 percent of the proportional income shock in the US and EU during

the 2008 ﬁnancial crisis. In addition to empirical analyses, some recent theoretical analyses

have shown that progressive labor income taxation has profound implications for monetary

policy in the New Keynesian framework. For instance, Collard and Dellas (2005) show that

progressive taxation weakens the central bank’s perfect price stabilization in optimal monetary

policy. Mattesini and Rossi (2012) reveal that progressive taxation induces a trade-oﬀ between

inﬂation and the output gap in optimal monetary policy.

This paper takes a diﬀerent approach to analyze the relationship between progressive tax-

ation and monetary policy: namely, we focus on the robust monetary policy when model un-

certainty exists. By introducing Hansen and Sargent’s (2008) robust control problem into the

model of Mattesini and Rossi (2012), this paper complements their results in terms of robust

monetary policy. The model uncertainty has a signiﬁcant eﬀect on macroeconomic ﬂuctuation,

because it makes macroeconomic variables respond cautiously or aggressively. Leitemo and

Söderström (2008a,b) show that macroeconomic variables respond aggressively or cautiously

depending on the economic structure and types of the shock under robust policy. Giordani

and Söderlind (2004) and Gerke and Hammermann (2016) showed that some macroeconomic

variables respond aggressively when Hansen and Sargent’s (2008) model uncertainty exists.

As mentioned above, while it has been shown that the automatic stabilizer is eﬀective in

stabilizing the macroeconomic ﬂuctuations, these ﬂuctuations may be caused and ampliﬁed by

the model uncertainty. Therefore, it is important research question whether the progressive

taxation is eﬀective in stabilizing macroeconomic ﬂuctuations which are caused by the model

uncertainty. Moreover, this research question is relevant from a policy perspective, because

progressive taxation is relevant to stabilization problem in monetary policy, as pointed out by

Collard and Dellas (2005) and Mattesini and Rossi (2012). Based on this background, this

paper investigates whether progressive taxation on labor income mitigates or ampliﬁes the ag-

gressive response of macroeconomic variables under a robust monetary policy. By addressing

this question, we reveal whether the progressivity of labor income taxation is eﬀective in sta-

2

bilizing the economy in the presence of model uncertainty.1The main ﬁndings are as follows.

First, the higher the progressivity, the smaller the eﬀect of model uncertainty on the inﬂation

rate. The model uncertainty increases the ﬂuctuation of the inﬂation rate, but the progressive

taxation diminishes this increase. The intuition for this result lies in the slope of the Phillips

curve. The slope of the Phillips curve becomes steeper as progressivity increases. As Mattesini

and Rossi (2012) point out, this is because the higher rate of labor income taxation raises the

marginal cost in the ﬁrm sector. This higher value of slope increases the impact of output

gap on the inﬂation rate with the model uncertainty. Because the central bank knows this

mechanism, monetary policy does not have to aggressively adjust the output gap to absorb the

cost-push shock. Therefore, the higher progressive taxation mitigates the increased responses

of the inﬂation rate, output gap, and interest rate under robust monetary policy.

Second, we show that a suﬃciently high progressivity brings the economy into a determinate

equilibrium even if the model uncertainty is strong. Mattesini and Rossi (2012) point out the po-

tential importance of progressive taxation and monetary policy in terms of model determinacy.2

They show that a higher progressivity brings the economy into equilibrium-indeterminacy un-

der the Taylor rule: When the central bank increases the policy rate in response to an increase

in the inﬂation rate, a higher progressivity of taxation on labor income mitigates the decrease

in labor supply due to a slight increase in net wage income, and hence the output gap does

not decrease suﬃciently. This mitigated decrease in output gap violates the convergence of

inﬂation, and consequently self-fulﬁlling inﬂation occurs.

On the other hand, we show that progressive taxation brings the economy into determinate

equilibrium under robust monetary policy. In a standard New Keynesian model, a strong model

uncertainty brings the economy into equilibrium-indeterminacy (Hasui, 2021). Unlike in the case

of the Taylor rule, robust monetary policy under discretion adjusts the policy rate while taking

into account both robustness and progressivity. We show that the determinacy condition is

proportional to our ﬁrst result that the aggressive response of the inﬂation rate is mitigated

by higher progressivity. This mechanism prevents the economy from falling into a self-fulﬁlling

inﬂation rate. Based on these ﬁndings, we show that the progressive taxation on labor income

plays an eﬀective role in stabilizing the economy in terms of variance and determinacy of the

rational expectation equilibrium (REE) when model uncertainty is considered.

Our paper is related to several strands in the literature. First, our paper is relevant to

the literature on the robust control problem. In the robust control approach, the concept of

model uncertainty is that the policy maker designs the policy supposing the “worst-case” out-

1We refer to the degree of progressive taxation as “progressivity” in the present paper.

2Guo (1999) and Guo and Lansing (1998) showed the nontrivial implication of progressive taxation for

equilibrium determinacy.

3

come. Under Hansen and Sargent’s (2008) robust control problem, the true economic structure

deviates from the so-called reference macroeconomic structure due to the speciﬁcation error

terms. The monetary authority and economic agent cannot know this true economic structure.

Moreover, they cannot formulate the probability of these speciﬁcation error terms. When the

central bank designs a monetary policy supposing the worst-case outcome, it is called “robust

policy.” Accordingly, robust policy can deviate from policy in the REE. In this paper, we ana-

lyze the robust policy following Leitemo and Söderström (2008a,b), who derive the closed form

solution of robust monetary policy under discretion in a standard New Keynesian framework.

On the other hand, Hansen and Sargent’s (2008) solution is shown using a state space method

and structural form method (Giordani and Söderlind, 2004; Dennis, 2008; Dennis et al., 2009;

Dennis, 2010, 2014).3Our paper is not only diﬀerent from these analyses in introducing the

progressive taxation, but also in analyzing the determinate condition under robust monetary

policy. Although Hansen and Sargent’s (2008) robust control problem aﬀects equilibrium de-

terminacy, little attention has been paid to this phenomenon. We show that the robust control

problem of Hansen and Sargent (2008) aﬀects the equilibrium determinacy and a stronger model

uncertainty brings the model economy into equilibrium-indeterminacy.4

Second, our paper is relevant to the literature of monetary policy analyses with progressive

taxation: Mattesini and Rossi (2012) derive the central bank’s welfare loss function around the

distorted steady state following the approach of Benigno and Woodford (2005).5They show

that the progressivity of taxation enters into the loss function and slope of the Phillips curve,

and this worsens trade-oﬀ between the inﬂation rate and output gap. Their linear-quadratic

framework enables us to analyze the robust monetary policy with progressive taxation quite

tractably. On the other hand, Collard and Dellas (2005) introduce progressive labor income

taxation into the New Keynesian model and evaluate the welfare around the eﬃcient steady

state. They show that the introduction of progressive taxation has profound implications for

monetary policy because it weakens the perfect price stability of monetary policy. Engler and

Strehl (2016) also analyze the eﬀect of progressive taxation on social welfare by introducing

the progressive taxation on labor income into the New Keynesian model which incorporates

3Barlevy (2011) surveyed the analyses of robust policy, and Hansen and Sargent (2012) provide three notions

of a robust planner in a continuous time model.

4In addition to Hansen and Sargent’s (2008) robust control approach, other papers have analyzed model

uncertainty with multiplicative uncertainty (Brainard, 1967), min-max approaches for robust policy (Giannoni,

2002; Levin and Williams, 2003; Onatski and Williams, 2003; Tillmann, 2009), parametric uncertainty (Giannoni,

2007), Bayesian parametric uncertainty (Batini et al., 2006; Cogley et al., 2008), and various alternative policy

rules (Orphanides and Wieland, 2013).

5Therefore, we derive the robust monetary policy under discretion with Mattesini and Rossi’s (2012) linear-

quadratic framework around the distorted steady state.

4

Ricardian and Non-Ricardian households. They show that the progressive taxation improves the

social welfare when only the Ricardian household is considered, but the aggregate social welfare

declines when both Ricardian and Non-Ricardian households are considered. Heer and Maußner

(2006) investigate the distributional eﬀects of productivity shock in the heterogeneous agent

overlapping generation (OLG) model with sticky prices. In their model, the ﬁrm’s income is

taxed progressively, and this aﬀects the consumption growth of the retired generation. Vanhala

(2006) introduces the progressive labor income taxation into the labor search-matching model,

and ﬁnds that the higher progressive income taxation generates a trade-oﬀ between income

inequality and unemployment, and that these values at the steady state critically depend on

the initial value of progressivity. The present paper is diﬀerent from these analyses in that we

introduce model uncertainty. Recently, Bilbiie et al. (2020) examined the optimal monetary

and ﬁscal policy mix analytically in a tractable heterogeneous agent New Keynesian (THANK)

model. They showed that the tax distribution, whether progressive or regressive, induces a

trade-oﬀ for monetary policy between stabilizing the real activity and inﬂation.

In terms of equilibrium determinacy, several papers show that progressive income taxation

has profound implications for economic stability. Guo (1999) and Guo and Lansing (1998)

show that higher progressivity is required to obtain an equilibrium in the economy that allows

an increasing return to scale.6Christiano and Harrison (1999) illustrate how the automatic

income tax stabilizer is eﬀective in obtaining the eﬃcient allocation and stabilizing the output

with unique equilibria. Dromel and Pintus (2008) analyze the model determinacy in a model

incorporating constant return to scale. They show that sunspot equilibria occur when the

degree of progressivity is low, although the progressive income tax decreases the frequency of

indeterminate equilibria.

In addition to these theoretical analyses, some papers have empirically analyzed the auto-

matic stabilizer that includes progressive taxation: in one of the leading studies in the literature

of automatic stabilizers, McKay and Reis (2016) analyze the eﬀects of an automatic stabilizer

on the US business cycle both empirically and theoretically. They show that tax-and-transfer

programs are eﬀective in stabilizing demand volatility in a New Keynesian model incorporating

an automatic stabilizer. Heathcote et al. (2020a) model the heterogeneous agent incomplete

market framework and show that the optimal progressivity did not change entirely between

1980 and 2016 in the US.7Fatás and Mihov (2012) analyze the cyclical behavior of ﬁscal pol-

6Recently, Chen and Guo (2013a,b, 2014) analyze the relationship between progressive taxation and equi-

librium (in)determinacy in a model with productive government spending and utility-generating government

purchases.

7Heathcote et al. (2017, 2020b) have analyzed progressive taxation in a heterogeneous agent framework.

Heathcote et al. (2017) considered the optimal degree of progressivity in the US calibrated model incorporating

various trade-oﬀs. Heathcote et al. (2020b) analyzed optimal progressive taxation in the overlapping gener-

5

icy empirically, and show that an automatic budget-balance is more eﬀective in stabilizing the

output ﬂuctuation than discretionary ﬁscal policy.8Dolls et al. (2012) show that the automatic

stabilizers absorbed more than 30% of a proportional income shock in the US and EU during

the 2008 ﬁnancial crisis. Auerbach and Feenberg (2000) show that the federal tax system ab-

sorbs 8% of a shock to output by empirical estimation, and argue that progressive taxation is

eﬀective in stabilizing the output.

The remainder of the present paper is organized as follows. In Section 2, we describe the

New Keynesian model incorporating the progressive taxation and Hansen and Sargent’s (2008)

robust control problem. Section 3 shows analytically and quantitatively how the progressivity of

labor income tax inﬂuences the eﬀects of the model uncertainty in terms of policy function and

determinacy. In Section 4, we show quantitative results under other parameter conﬁgurations

in terms of robustness. Section 5 concludes the paper.

2 The model

Progressive taxation on labor income

The model is the New Keynesian model incorporating progressive taxation on labor income,

proposed by Mattesini and Rossi (2012). In their model, the labor income taxation is introduced

into the budget constraint as follows:9

PtCt+R−1

tBt= (1 −τt)WtNt+Bt−1+Dt−PtTℓ

t,(1)

where Rt,Bt,Wt,Dt, and Tℓ

tdenote gross nominal risk-free interest rate, risk free government

bond, nominal wage, proﬁt income, and lump-sum tax, respectively. τtdenotes the average tax

rate on labor income. Mattesini and Rossi (2012) speciﬁes the form of τtfollowing Guo (1999)

and Guo and Lansing (1998):

τt= 1 −ηYn

Yn,t ϕn

,(2)

where η∈(0,1] and ϕn∈[0,1), and Yn=W N/P and Yn,t =WtNt/Ptdenote the base level

of income and actual level of income, respectively. We assume 0≤ϕn<1and 0< η ≤1,

because it is diﬃcult to ensure 0≤τt<1if the values of ϕnand ηare too large. ϕndenotes the

ation model incorporating the variation of progressivity for age. They revealed that the optimal schedule of

progressivity should be U-shaped for degree of age.

8Fatás and Mihov (2001) argue that the size of government can be a proxy for an automatic stabilizer and

then show empirically that relationship between government size and output volatility are negative in OECD

countries and the US.

9We provide the brief description of Mattesini and Rossi’s (2012) model in Appendix A.3.

6

progressivity of labor taxation, and it is a key parameter in subsequent analyses of the present

paper. For instance, when the actual level of income is equal to the base level of income, the tax

rate is 1−η. However, when the actual level of income is greater than the base level, the tax

rate is greater than 1−η. To ensure this mechanism, we show how an increase in the actual level

of income taxation alters the tax income. The tax on labor income is given by τtWt

PtNt=τtYn,t.

Partial diﬀerentiation of τtYn,t with respect to Yn,t is given as follows:

τm

t=∂τtYn,t

∂Yn,t

= 1 −η(1 −ϕn)Yn

Yn,t ϕn

=τt+ηϕnYn

Yn,t ϕn

,(3)

where we call τm

tthe marginal tax rate of labor income. As Mattesini and Rossi (2012) shows,

Equation (3) is quite intuitive why ϕnindicates the progressivity. If ϕn= 0,τm

t=τt. This

shows that the average tax rate and marginal tax rate is equivalent when ϕn= 0. On the other

hand, the marginal tax rate is greater than the average tax rate when ϕn>0. Therefore, ϕn

denotes the degree of progressivity of taxation, and a higher ϕnmeans a stronger progressivity.

In this environment, the ﬁrst order condition of a household’s labor supply is given as follows:

Cσ

tNϕ

t= (1 −τm

t)Wt

Pt

.(4)

Equation (4) shows that the household’s labor supply decision depends on the marginal tax

rate. Considering τm

tis a function of NtWt/Pt, the log-linearized equation of (4) is derived as

follows:

σct+ϕnt= (1 −ϕn)wt−ϕnnt.(5)

The eﬀects of taxation are indicated by the terms ϕnntand ϕnwt. Combining the ﬁrm’s labor

demand, the progressivity ϕnenters into the marginal cost. Therefore, the progressivity of labor

income taxation enters into the slope of the aggregate supply equation.

Regarding a government budget constraint, Mattesini and Rossi (2012) assume a balanced

budget as follows:

Gt=τt

WtNt

Pt

+Tℓ

t.(6)

The New Keynesian model incorporating progressive taxation

Introducing the mentioned structure of progressive taxation on labor income, Mattesini and

Rossi (2012) derive the following New Keynesian model:

xt=Etxt+1 −γc

σit−Etπt+1 −reff

t,(7)

πt=βEtπt+1 +κ(ϕn)xt+ut,(8)

7

where πt,xt, and itdenote the inﬂation rate, welfare-based output gap, and nominal interest

rate, respectively. Meanwhile reff

tand utdenote the real interest rate in eﬃcient friction-

less equilibrium and the cost-push shock, respectively. We assume that reff

tand utare i.i.d

disturbance terms.10

Equation (7) is an IS curve, which is derived from the household’s intertemporal decision

regarding consumption. This equation shows that the current output gap is determined by

the expected output gap and deviation of real interest rate from eﬃcient natural interest rate.

Equation (8) is the Phillips curve, which is derived from a ﬁrm’s optimal price setting. Equation

(8) shows that the current inﬂation rate is determined by the expected inﬂation rate and output

gap.

Parameters, σ > 0,0< β < 1,κ(ϕn)>0, and 0≤γc≤1denote the inverse of the

intertemporal elasticity of substitution (we also call this “relative risk aversion”) for consumption,

the subjective discount factor, the slope of the Phillips curve, and the steady state fraction of

consumption and output, respectively. κ(ϕn)is given as follows:

κ(ϕn) = λµy(ϕn), λ =(1 −φ)(1 −φβ)

φ, µy(ϕn) = σ+γc(ϕ+ϕn)

γc(1 −ϕn),(9)

where φdenotes the parameter of price stickiness in Calvo-pricing. As mentioned in the previous

subsection, Eq. (9) shows that progressivity ϕnenters into the slope of the Phillips curve.

As Mattesini and Rossi (2012) show, the slope is an increasing function of progressivity, i.e.,

∂κ(ϕn)/∂ϕn>0(cf. Appendix A.1.1).

The progressive taxation also alters the welfare loss function. Mattesini and Rossi (2012)

derive the social welfare loss by approximating the household’s utility second order around the

distorted steady state:

Lt=Et

∞

X

j=0

βjLt+j, Lt=qπ(ϕn)π2

t+qx(ϕn)x2

t.(10)

where

qπ(ϕn) = (1 −Φ) + (1 + µπ(ϕn))Φ

µy(ϕn)ϵ

λ,

qx(ϕn) = γc(1 −Φ)(1 + ϕ)−(1 −σ)

γc

+µy(ϕn)Φ,

Φ=1−(ϵ−1)η

ϵ,

µπ(ϕn) = ϕ+ϕn

1−ϕn

,

10ref f

tand utare a reduced form and they depend on potential output, government spending, and productivity.

However, Mattesini and Rossi (2012) show that reff

tand utcan be treated as completely exogenous.

8

where ϵdenotes the price elasticity of demand for diﬀerentiated goods. The weight on the

inﬂation rate and output gap depend on the progressivity of taxation. As Mattesini and

Rossi (2012) show, the weight on the inﬂation rate is a decreasing function of progressiv-

ity, i.e., ∂qπ(ϕn)/∂ϕn<0, and the weight on the output gap is an increasing function, i.e.,

∂qx(ϕn)/∂ϕn>0(cf. Appendix A.1.1).

2.1 Robust control problem

In this section, we relax the assumption of perfect knowledge by introducing Hansen and Sar-

gent’s (2008) robust control problem. We call the model up to this point “the reference model”,

which is the model under the assumption of perfect knowledge.

In the robust control problem, the true model lies around the reference model, and agents

know the reference model but not the true model. Therefore, a misspeciﬁcation exists between

the true model and reference model. To describe this misspeciﬁcation, the robust control ap-

proach introduces speciﬁcation error terms, denoted νr

tand νu

t. The misspeciﬁed economic

model is given by the following equations:

xt=Etxt+1 −γc

σhit−Etπt+1 −(reff

t+νr

t)i,(11)

πt=βEtπt+1 +κ(ϕn)xt+ (ut+νu

t),(12)

Agents cannot formulate probabilities to νr

tand νu

t. Therefore, the central bank designs the

“robust monetary policy” supposing that the “worst-case” outcome will be realized.

The speciﬁcation error terms are set by an “evil agent” to maximize the welfare loss subject

to

Et

∞

X

j=0

βjνt+jν′

t+j≤ν0,(13)

where νt= [νr

t, νu

t].

The problem of robust monetary policy can be (re-)formulated as the following multiplier

problem (Hansen and Sargent, 2008; Leitemo and Söderström, 2008a,b):

min

xt,πt,it

max

νr

t,νu

t

Et

∞

X

j=0

βj[qπ(ϕn)π2

t+j+qx(ϕn)x2

t+j−θ(νr

t+j)2−θ(νu

t+j)2],

s.t.

xt=Etxt+1 −γc

σhit−Etπt+1 −(reff

t+νr

t)i,

πt=βEtπt+1 +κ(ϕn)xt+ (ut+νu

t)

(14)

where θdenotes the Lagrange multiplier on quadratic terms of speciﬁcation errors. Therefore,

θis the preference or inverse of model uncertainty. In the robust control problem, a lower θ

9

indicates stronger model uncertainty. On the other hand, the model uncertainty does not exists

when θ=∞, i.e., the equilibrium does not deviate from the REE.

3 The eﬀects of the model uncertainty

3.1 Policy function analysis

In this section, we derive the policy function of robust policy under discretion. Therefore, we

treat expected terms as given in optimization problem (14). First, we obtain the following

targeting rule and relation of the misspeciﬁcation term:

xt=−κ(ϕn)

α(ϕn)πt,(15)

νu

t=qπ(ϕn)

θπt,(16)

where α(ϕn) = qx(ϕn)/qπ(ϕn). By substituting Eqs. (15) and (16) into Eqs. (7) and (8), we

derive the policy functions. We guess the policy function as follows:11

πt=cπut,(17)

xt=cxut,(18)

it=ciut+reff

t,(19)

where cπ,cx, and ciare coeﬃcients to be solved. Solving with the undetermined coeﬃcient

method, we obtain the following solution:

cπ(θ, ϕn) = 1

ω(θ, ϕn),(20)

cx(θ, ϕn) = −κ(ϕn)

α(ϕn)

1

ω(θ, ϕn),(21)

ci(θ, ϕn) = σ

γc

κ(ϕn)

α(ϕn)

1

ω(θ, ϕn),(22)

where

ω(θ, ϕn) = 1 + κ(ϕn)2/α(ϕn)−qπ(ϕn)/θ

For subsequent analyses, we impose the following Assumption:

Assumption 1. We assume 0≤β < 1,0≤φ≤1,ϕ > 0,ϵ > 1,0≤ϕn<1, and 0< η ≤1.

11Because of the “divine coincidence” under optimal policy, inﬂation and output gap do not depend on the

eﬃcient interest rate. The interest rate responds one-to-one to the eﬃcient interest rate so that the shock of the

eﬃcient interest rate is absorbed completely.

10

Parameter values in Assumption 1 are plausible in analyses of the New Keynesian framework.

Parameters ϕnand ηare positive and less than 1following Mattesini and Rossi (2012). This is

because if the values of ϕand ηare too large, it is diﬃcult to ensure 0≤τt<1. For the inverse

of the Frisch elasticity, we assume ϕ > 0. This is based on Rotemberg and Woodford’s (1997)

calibration for the US economy (ϕ= 0.47) and Smets and Wouters’s (2007) estimation for the

US economy (ϕ= 1.83). On the other hand, King et al. (1988a,b) showed that ϕshould be

greater than 1in order for the economy to be on the balance growth path. Therefore, we show

numerical results under ϕ= 0.47 and ϕ= 1.83 as well as a benchmark calibration in Section

4.2.

In this subsection, we limit the case under Assumption 2 to derive analytical conditions

for the eﬀects of the model uncertainty and progressivity clearly following Mattesini and Rossi

(2012):

Assumption 2. We assume σ= 1 and γc= 1.

We impose Assumption 2 for several reasons: ﬁrst, the analytic derivation of ∂(κ/α)

∂ϕnis

tractable when we assume σ= 1 and γc= 1, because qxis simpliﬁed as (1 −Φ)(1 + ϕ) + µyΦ.

Second, several studies set the relative risk aversion coeﬃcient at 1to analyze empirically

plausible responses of the nominal interest rate (Keen, 2004; Christiano et al., 2005; Nakajima,

2006). From this perspective, we set σand γcso that γc/σ is equal to 1.

However, many studies have estimated the relative risk aversion and have shown various

estimation results. Using micro-data, Hall (1988) and Barsky et al. (1997) estimated the in-

tertemporal elasticity of substitutions and showed that 1/σ is less likely than 0.2. Rotemberg

and Woodford (1997) calibrated the US economy by setting σat 0.16. On the other hand,

Smets and Wouters (2007) showed that the estimated value of σfor the US economy is 1.38.

Recently, Chen et al. (2017) estimated a σvalue of 2.901 under optimal discretionary monetary

policy with the US data. From the theoretical perspective, King et al. (1988a,b) showed that

σshould be greater than 1in order for the economy to be on the balanced growth path. Based

on these studies, we show numerical results under σ= 1/6.25 = 0.16 and σ= 1.38 as well as a

benchmark calibration in Section 4.1.

Finally, to avoid complication in analyzing the eﬀect of the model uncertainty, we impose

the following assumption (Leitemo and Söderström, 2008a):

Assumption 3. We analyze the eﬀects of small decreases in θstarting from θ=∞.

Assumption 3 means that we analyze the eﬀect of the model uncertainty starting from

the rational expectation. We focus on the situation that the model misspeciﬁcation cannot

be identiﬁed easily by the policy maker. By restricting the analysis to small increases in the

model uncertainty, we are able to avoid the case in which a robust planner looks like a foolish

11

catastrophist. As Giordani and Söderlind (2004) show, it is easy to make a robust policy maker

look like a foolish catastrophist, whose policy function would be implausible when the model

uncertainty is signiﬁcantly large. Assumption 3 is one way to avoid this problem. Another

way to deal with this problem is to use detection error probability, which is explained in a

subsequent Section.

Based on Assumption 3, we evaluate the eﬀect of model uncertainty on the policy function

as follows (Leitemo and Söderström, 2008a):

−∂|cj(θ, ϕn)|

∂θ , j =π, x, i, (23)

Measure (23) shows changes in the coeﬃcient in response to the small decrease in θ. The

absolute values of the coeﬃcients are given as follows:

|cπ(∞, ϕn)|=cπ(∞, ϕn),

|cx(∞, ϕn)|=−cx(∞, ϕn),

|ci(∞, ϕn)|=ci(∞, ϕn).

Applying (23) to cπ, and cx,ci, we obtain the following:

−∂|cπ(θ, ϕn)|

∂θ =1/qπ

ω(θ, ϕn)2

1

θ2>0,(24)

−∂|cx(θ, ϕn)|

∂θ =κ/qx

ω(θ, ϕn)2

1

θ2>0,(25)

−∂|ci(θ, ϕn)|

∂θ =σ

γc

κ/qx

ω(θ, ϕn)2

1

θ2>0,(26)

Inequalities (24)-(26) show that output gap, nominal interest rate, and inﬂation rate respond

to cost-push shocks more aggressively as the model uncertainty becomes stronger. As previous

studies point out, the model uncertainty causes large ﬂuctuation of inﬂation and output gap.

The robust monetary policy adjusts the nominal interest rate more aggressively to decrease this

large ﬂuctuation.

Next, we analyze how the ϕnimpacts the eﬀects of the model uncertainty. To obtain how

ϕnalters the eﬀects of the model uncertainty, we calculate the following cross-derivative (Hasui,

2020, 2021):

∂(−∂|cj(θ, ϕn)|/∂θ)

∂ϕn

=−∂2|cj(θ, ϕn)|

∂θ∂ϕn

, j =π, x, i. (27)

First, we analyze how ϕnalters the eﬀect of the model uncertainty on inﬂation rate. Applying

12

measure (27) to cπ, we obtain the following equation (cf. Appendix A.1.1 for derivation):

−∂2|cπ(θ, ϕn)|

∂θ∂ϕn

=1/θ2

ω(θ, ϕn)2

∂qπ(ϕn)

∂ϕn

−2qπ(ϕn)/θ2

ω(θ, ϕn)3

κ(ϕn)

α(ϕn)

∂κ(ϕn)

∂ϕn

+κ(ϕn)∂κ(ϕn)

∂ϕn

∂ϕn

−1

θ

∂qπ(ϕn)

∂ϕn

<0,

(28)

if ω(θ, ϕn)>0.

Therefore we can make the following Proposition:

Proposition 1. In the worst-case scenario, condition (28) shows that a higher progressive tax

rate decreases the eﬀects of the model uncertainty on inﬂation under Assumptions 1 – 3, and

ω(θ, ϕn)>0.

Proof. See Appendix A.1.1.

The inﬂation rate responds to the cost-push shock aggressively as model uncertainty in-

creases. However, Proposition 1 shows that this aggressiveness is reduced as the progressivity

of labor income taxation increases. Therefore, progressive taxation can be eﬀective for stabiliz-

ing the inﬂation rate even if model uncertainty exists.

Next, we analyze how ϕnalters the eﬀects of the model uncertainty on output gap and

nominal interest rate. Applying measure (27) to cxand ci, we obtain the following equations:

−∂2|cx(θ, ϕn)|

∂θ∂ϕn

=1/θ2

ω(θ, ϕn)2

qπ(ϕn)∂κ(ϕn)

α(ϕn)

∂ϕn

−κ(ϕn)

α(ϕn)

∂qπ

∂ϕn

−2κ(ϕn)/α(ϕn)

ω(θ, ϕn)3

qπ(ϕn)

θ2

κ(ϕn)∂κ(ϕn)

α(ϕn)

∂ϕn

+κ(ϕn)

α(ϕn)

∂κ(ϕn)

∂ϕn

−1

θ

∂qπ(ϕn)

∂ϕn

(29)

−∂2|ci(θ, ϕn)|

∂θ∂ϕn

=−σ

γc

∂2|cx(θ, ϕn)|

∂θ∂ϕn

.(30)

The signs of (29) and (30) depend on ϕn. Therefore, we obtain the following Proposition:

Proposition 2. In the worst-case scenario, Equations (29) and (30) show that the eﬀects of the

model uncertainty on output gap and nominal interest rate depend on progressivity of taxation

on labor income under Assumptions 1 – 3.

Proof. See Appendix A.1.2.

However, as described in Appendix A.1.2, the signs of (29) and (30) are ambiguous. Ac-

cordingly, we analyze how ϕnalters the eﬀects of the model uncertainty on the output gap and

nominal interest rate numerically.

13

Here we interpret Propositions 1 and 2 in terms of generalizability. As Mattesini and

Rossi (2012) showed, the progressive taxation is eﬀective in stabilizing the economy in the

REE. Therefore, Proposition 1 complements their result in terms of robust monetary policy by

showing that the progressive taxation is eﬀective in stabilizing the inﬂation rate in worst-case

outcomes. Proposition 2 also shows that there is a possibility that the progressive taxation is

eﬀective in stabilizing the output gap in worst-case outcomes, although the signs of (29) and

(30) are ambiguous. Therefore, we conclude that the progressive taxation is at least eﬀective in

stabilizing the inﬂation rate in both the REE and worst-case outcomes. As shown in subsequent

numerical results, this eﬀectiveness increases as the model uncertainty increases. However, the

eﬀectiveness is based on the result that the model uncertainty increases the economic ﬂuctuation.

Under Hansen and Sargent’s (2008) robust control problem, economic ﬂuctuations often increase

(Giordani and Söderlind, 2004; Leitemo and Söderström, 2008a,b). On the other hand, the

model uncertainty decreases economic ﬂuctuation under other approaches (Brainard, 1967).

While this study does not execute the relevant analysis, the eﬀect of progressive taxation may

be small in such cases.

3.2 Determinacy

Mattesini and Rossi (2012) showed the potential importance of determinacy and progressive

taxation by deriving the determinacy condition under the contemporaneous data Taylor rule

and forecast data Taylor rule. In this section, we derive the determinacy condition under robust

monetary policy. The system of the model is expressed with matrices as follows:

Xt=AEtXt+1 +BZt,

A=

β

1+κ(ϕn)2/α(ϕn)−qπ(ϕn)/θ 0

−σ

γc

βκ(ϕn)/α(ϕn)

1+κ(ϕn)2/α(ϕn)−qπ(ϕn)/θ 0

(31)

where Xt= [πt, it]′,Zt= [reff

t, ut]′, and Bis a 2×2matrix of coeﬃcient on shocks. The system

is determinate if all eigenvalues of Aare less than 1. Suppose µ2+m1µ+m2is the characteristic

equation of A, both of the absolute eigenvalues of Aare less than 1 when |m1|<1 + m2and

|m2|<1, where m1=−tr(A)and m2=det(A). Combining |m1|<1 + m2and |m2|<1, the

14

conditions of determinacy are given as follows:12

|m1|=

β

ω(θ, ϕn)

<1.(33)

If ω(θ, ϕn) = 1 + κ(ϕn)2/α(ϕn)−qπ(ϕn)/θ > 0, the absolute value symbol is removed straight-

forwardly. The above determinate condition is reduced into the following inequality:

δ(θ, ϕn)<1,(34)

where δ(θ, ϕn) = β

ω(θ,ϕn). Therefore we obtain the following Proposition:

Proposition 3. The equilibrium is determinate if δ(θ, ϕn)<1under robust policy with the

progressive taxation on labor income, if ω(θ, ϕn)>0.

Next we analyze how a small decrease in θalters the equilibrium determinacy:

−∂δ(θ, ϕn)

∂θ =β

ω(θ, ϕn)2

1

θ2>0.(35)

Equation (35) shows that an increase in model uncertainty brings the economy into indetermi-

nacy. Analogously, we analyze how a small increase in ϕnalters the equilibrium determinacy:

−∂2δ(θ, ϕn)

∂θ∂ϕn

=β/θ2

ω(θ, ϕn)2

∂qπ(ϕn)

∂ϕn

−2βqπ(ϕn)/θ2

ω(θ, ϕn)3

κ(ϕn)

α(ϕn)

∂κ(ϕn)

∂ϕn

+κ(ϕn)∂κ(ϕn)

∂ϕn

∂ϕn

−1

θ

∂qπ(ϕn)

∂ϕn

.

(36)

Equation (36) is identical to −β∂2|cπ(θ,ϕn)|

∂θ∂ ϕn. This means that −∂2δ(θ,ϕn)

∂θ∂ ϕn<0under Assumptions

1 and 2, and ω(θ, ϕn)>0. Therefore we obtain following Proposition:

Proposition 4. A higher progressive tax rate decreases the eﬀects of the model uncertainty that

brings the economy to indeterminacy under Assumptions 1 and 2, and ω(θ, ϕn)>0.

Proof. see Appendix A.1.1.

Proposition 4 shows that the equilibrium would be determinate when the progressivity is

suﬃciently large, even if strong model uncertainty exists.

12Alternatively, Condition (33) can be obtained by substituting Eqs. (15) and (16) into the Phillips curve (12):

πt=β

ω(θ, ϕn)Etπt+1 +1

ω(θ, ϕn)ut.(32)

To prevent Eq. (32) from diverging, |β

ω(θ,ϕn)|must be less than 1. We note that we obtain the policy function

of the inﬂation rate (20) by iterating Eq. (32) forwardly.

15

Our result is opposite to that of Mattesini and Rossi (2012). Mattesini and Rossi (2012)

derive the determinacy condition under the Taylor rule, and show that a higher progressivity

brings the economy into the equilibrium-indeterminacy. They reveal that this is because an

increase in progressivity mitigates the eﬀect of monetary policy on output. Suppose that the

monetary policy raises the policy rate in response to the rise in inﬂation rate. Progressive

taxation on labor income mitigates the decrease in labor supply, because the rise in the real

interest rate increases net wage income slightly. Therefore, a lesser decrease in output gap

violates the convergence of inﬂation; i.e., self-fulﬁlling inﬂation occurs.

On the other hand, unlike under the Taylor rule, robust monetary policy under optimal

discretion adjusts the policy rate considering both robustness and progressivity. The eﬀects of

the model uncertainty and progressivity on the determinate condition, indicated by Eq. (36), are

proportional to the condition of Eq. (28). Therefore, the interpretation is the same as the result

in Section 3.1, in which an increase in progressivity reduced the eﬀect of the model uncertainty

on the inﬂation. This prevents the economy from falling into a self-fulﬁlling inﬂation rate.13

3.3 A numerical example

Policy function

Figure 1 shows the numerical example under the parameter values σ= 1,γc= 1,β= 0.99,

ϕ= 1,η= 0.7(Mattesini and Rossi, 2012), φ= 0.7(Nakamura and Steinsson, 2008), and

ϵ= 9.8(Cogley and Sbordone, 2008). Panels (a)-(c) show |cj(θ, ϕn)|, and Panels (d)-(f) show

|cj(θ,ϕn)|−|cj(∞,ϕn)|

|cj(∞,ϕn)|×100 (for j=π, x, i), which indicates the deviation rate of robust policy from

the REE for degree of strength of the model uncertainty (1/θ). We plot the cases of ϕn= 0.13

(Sweden), 0.18 (United States), and 0.25 (United Kingdom). These values are indicated in

Table 2 of Mattesini and Rossi (2012). In this parameter conﬁguration, the equilibrium is

determinate until 1/θ is greater than 0.044.

Panels (a)-(c) in the ﬁgure show that the coeﬃcients in the policy function of inﬂation rate,

output gap, and nominal interest rate increase as the model uncertainty increases. However, ac-

cording to Panels (d)-(f), these increases grow smaller as the progressivity of taxation increases.

Therefore, a higher progressive taxation on labor income mitigates the large ﬂuctuation caused

by the model uncertainty.

We present an intuition of these results as follows: Under discretionary policy, the central

bank raises the policy rate in response to a positive cost-push shock. With this policy response,

the output gap decreases and the inﬂation rate increases in response to the cost-push shock.

13Though our determinate condition with the model uncertainty and progressive taxation is simple, this result

is based on the assumption of a balanced budget in the government sector. However, recently, Nourry et al.

(2013) show that the balanced budget might be a factor of indeterminate equilibria.

16

0 0.01 0.02 0.03 0.04

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0 0.01 0.02 0.03 0.04

2

3

4

5

6

7

0 0.01 0.02 0.03 0.04

2

3

4

5

6

7

0 0.01 0.02 0.03 0.04

0

50

100

150

200

0 0.01 0.02 0.03 0.04

0

50

100

150

200

0 0.01 0.02 0.03 0.04

0

50

100

150

200

Fig. 1. Eﬀects of an increase in model uncertainty (1/θ) for degrees of ϕn= 0.13,0.18, and 0.25.

Under the robust monetary policy, the central bank raises the policy rate aggressively, because

the central bank supposes larger responses of the output gap and the inﬂation rate. Now we

consider the case when the progressivity of taxation increases. The higher progressive taxation

on labor income makes the slope of the Phillips curve steeper. This is because the higher rate of

labor income taxation raises the marginal cost in the ﬁrm sector. Due to this higher marginal

cost, the inﬂation rate rises. This steeper slope mitigates the aggressive response of policy rate

to the positive cost-push shock. The mechanism of this mitigated aggressiveness is the same as

that of the REE: The central bank decreases the output gap by raising the policy rate. However,

the impact of the output gap on inﬂation is large, because the value of the slope becomes large

when progressivity is high. Therefore, the central bank does not have to decrease the output

gap aggressively when progressivity is high. Consequently, the eﬀect of the model uncertainty

is decreased by higher progressivity. We interpret this to mean that an increase in progressivity

is desirable in terms of robust monetary policy.

We ﬁnd that higher tax progressivity mitigates the eﬀect of cost push shock which is a

negative supply shock. That is, the government decreases the income tax rate more when ϕnis

larger. Labor income tax reduction has a positive supply shock which is opposite to the eﬀect

of the positive cost-push shock, except for the change in relative price of labor, which causes a

substitution eﬀect.

17

0 1 2 3

0

0.5

1

10-3

0123

-3

-2

-1

010-3

0 1 2 3

0

0.005

0.01

0123

-0.04

-0.03

-0.02

-0.01

0

Fig. 2. Impulse responses to 1% cost-push shock

Impulse responses

We also plot the impulse responses to a 1% positive cost-push shock. We set θso that the

detection error probability is 20% when ϕn= 0.18 following Giordani and Söderlind (2004).14

The detection error 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.15

Figure 2 shows results consistent with the policy function analysis. The central bank raises

the nominal interest rate in response to the positive cost-push shock. When model uncertainty

exists, the central bank raises the nominal interest rate aggressively. Therefore, the dashed line

response is larger than the solid line in Panel (c) in the ﬁgure. In response to the positive cost-

push shock and policy response, the inﬂation rate increases and the output gap decreases. As

we found in the policy function analysis, the inﬂation rate and output gap responds aggressively

when model uncertainty exists. Therefore, the dashed lines response is larger than the solid

lines in Panels (a) and (b) in the ﬁgure. However, these aggressive responses are reduced when

the progressivity ϕnincreases. Thus, the dotted lines respond less than the dashed lines in

Panels (a)-(c).

Figure 2 also plots the responses of tax rate (Panel d).16 The tax rate decreases in response

14Therefore, the detection error probability is not 20% when ϕn= 0.25.

15We give a detailed explanation of detection error probability in the Appendix A.2.

16We show the log-linearized equation of tax rate in Eq.(A.33) in the Appendix.

18

to the positive cost-push shock, because real income such as the output decreases. However,

the tax rate decreases more under ϕn= 0.25 than ϕn= 0.18. This is the eﬀect of progressivity

and hence the reduction of the output is reduced as indicated by lesser decreases of the dotted

line than the dashed line (Panel b).

The importance of labor income tax to the business cycle has been shown by several studies

(Braun, 1994; McCallum, 1999). Therefore, institutional changes in the income tax rate should

also be important in business cycles. As Girouard and André (2005) and Holter et al. (2019)

provide, the progressivity of income tax is higher in the UK than US. Combined with their

results, our results show that the income tax system in the UK is more desirable than that in

the US in terms of robust policy.

It should be mentioned, however, that our results do not always hold given historical trends

in income tax rates. Tax reform takes into account not only economic stabilization but also

the tax burden perspective. Income tax rates in the US and UK have depended on political

developments. The most frequently mentioned examples of tax reform in the US and the UK

are from the 1980s. In the US, the top individual income tax rate was reduced from 70% to

50% (the Reagan administration, 1981). In the UK, the individual income tax rate was reduced

from seven levels to two (25% and 40%), which signiﬁcantly ﬂattened the rate (the Thatcher

administration, 1988).

Since then, various tax reforms have been implemented, including lowering tax rates, simpli-

fying of tax levels, and expanding the tax base in both the US and UK. As the most recent tax

reform in the US, the 2017 tax reform simpliﬁed the individual income tax system and reduced

tax rates: The seven-rate structure was maintained, but most of the rates were reduced. The

new rates ranged from 10% to 37%, with the top rate being reduced from 39.6%.

Along with changes in income tax rates, income tax progressivity is believed to have declined.

Piketty and Saez (2007) compared the income tax system of the US with that of the UK in terms

of average and marginal tax rates. They showed that income tax progressivity has declined in

both the US and the UK.

Thus, while the results of our study may be desirable from the perspective of economic stabi-

lization and robust policies, our results may not necessarily be appropriate from the perspective

of tax burden and income inequality.

When stabilization eﬀects are large?

The stabilization eﬀect of ϕnis small in Figure 2. As Mattesini and Rossi (2012) show, the

qualitative eﬀect of ϕndoes not change even if we change the values of deep parameters under

Assumption 1. In addition to this qualitative property, the quantity of the eﬀect of ϕndoes not

change drastically even if we change the values of deep parameters as we show in the Appendix

19

Fig. 3. Impulse responses to cost-push shock. Note: we give a 1% cost-push shock in Panel (A). In

Panel (B), we give a cost-push shock so that the response of inﬂation rate is same as that in Panel (B)

in period 0 when ϕnis 0.18.

of the present paper. However, Fig. 1 shows that the eﬀects of increases in 1/θ become larger

as ϕnincreases. Therefore, stabilization eﬀects of ϕnmay be large when 1/θ is large.

Figure 3 plots impulse responses to a cost-push shock: Fig. 3A shows the benchmark case

(θ= 71), and Fig. 3B shows the case when the degree of the model uncertainty 1/θ is high

(θ= 31). Comparing Panel (A) with Panel (B), the stabilization eﬀect of the progressivity is

larger under the case of θ= 31 than under the case of θ= 71: when ϕn= 0.25, inﬂation and

the output gap are less responsive under the case of θ= 31 than under the case of θ= 71.

However, the detection error probability is 0.0% when θis 31 (on the other hand, under

the rational expectation case, i.e., θ=∞, the detection error probability is 50% ). Considering

that a detection error probability of 20% is suggested in the literature, the model uncertainty

is signiﬁcantly strong. Therefore, we conclude that the stabilization eﬀects of the progressive

taxation are large when the model uncertainty is signiﬁcantly high, but such a situation would

be rare from the perspective of detection error probability.17

20

Fig. 4. Determinate-indeterminate region in (1/θ, ϕn)space.

Determinacy

We also provide a numerical example for determinacy. Figure 4 plots the determinate-indeterminate

region for the degree of ϕnand 1/θ. The ﬁgure shows that the equilibrium is determinate when

the ϕnis suﬃciently large even if strong model uncertainty exists.18

However, the degree of the model uncertainty 1/θ is too high when the equilibrium is

indeterminate. Figure 4 shows that the equilibrium is indeterminate when 1/θ is greater than

0.035 (θis 28.57 when 1/θ is 0.035). Therefore, the equilibrium is determinate under a plausible

value of the model uncertainty, while a higher progressivity mitigates the eﬀects of the model

uncertainty in terms of determinacy.

4 Additional quantitative analysis

In this section, we show quantitative results under other parameter conﬁgurations for the inverse

of the intertemporal elasticity of substitution σand the inverse of the Frisch elasticity ϕin terms

of robustness.

17However, we are able to avoid the case in which a robust planner looks like a foolish catastrophist, because

the deviations of impulse responses between the robust policy and the REE are reasonable.

18Guo (1999) also shows that progressive taxation brings the economy into the determinate equilibrium.

21

Fig. 5. Eﬀects of an increase in model uncertainty (1/θ) for degrees of ϕn= 0.13,0.18, and 0.25 under

σ= 0.16 (Panel A) and σ= 1.38 (Panel B).

22

4.1 The inverse of the intertemporal elasticity of substitution

As we mentioned in Section 3, many studies have estimated the relative risk aversion and have

shown various estimation results. Following Rotemberg and Woodford (1997) and Smets and

Wouters (2007), we show quantitative results under σ= 0.16 (Rotemberg and Woodford, 1997)

and σ= 1.38 (Smets and Wouters, 2007).

Figure 5 shows numerical examples for |cj(θ, ϕn)|and |cj(θ,ϕn)|−|cj(∞,ϕn)|

|cj(∞,ϕn)|under σ= 0.16

(Panels in Fig. 5A) and σ= 1.38 (Panels in Fig. 5B). The ﬁgure shows that the eﬀects of the

model uncertainty decrease as σincreases from 0.16 to 1.38, while progressivity decreases the

eﬀects of mode uncertainty (Panels d-f in Fig. 5A and B). Because σenters into the structural

equations and the welfare loss function, it is not clear which is the driving force behind the

results in Fig. 5. In this context, it is worth mentioning that σin the slope of the NKPC is the

driving force behind the results in Fig. 5. Figure A1 in the Appendix of the present paper plots

the case in which relative risk aversion in the slope (deﬁned as σκ) is set to 1.38, and 0.16 for

all other σexcept σκ. In Fig. A1, the eﬀects of the model uncertainty are small, although all

lines increase as 1/θ increases. The slope κincreases as σincreases as follows:

∂κ

∂σ =λ

γc(1 −ϕn)>0.(37)

The rationale underlying the ﬁnding that a higher slope reduces the eﬀect of model un-

certainty is the same as explained in Section 3.3: The higher relative risk aversion makes the

slope of the Phillips curve steeper. The central bank decreases the output gap by raising the

policy rate in response to a positive cost-push shock. However, the impact of the output gap on

inﬂation is large, because the value of the slope becomes large when σis high. Therefore, the

central bank does not have to decrease the output gap aggressively. Consequently, the eﬀect of

the model uncertainty is decreased when σis high.

Next we look at determinacy: Figure 6 plots the determinate-indeterminate regions under

σ= 0.16 (Fig. 6A) and σκ= 1.38 (Fig. 6B). The ﬁgure shows that the determinate regions

expand as relative risk aversion increases. The driving force behind the results in Figure 6 is σ

in the slope of the Phillips curve. Analogous to numerical results in policy function, Figure A2

in the Appendix of the present paper plots the case where σκis set to 1.38 and all other σ

is are to 0.16 except σκ. The ﬁgure shows that equilibrium is determinate under the plotted

area. Therefore, relative risk aversion in the slope of the Phillips curve stabilizes the economy

in terms of determinacy. The intuition behind the result is the same as in the explanation for

Fig. 5: Because the value of the slope becomes larger when σis high, the central bank does not

have to decrease the output gap aggressively. This reduces the possibility that the economy falls

into sunspot equilibrium, and then the equilibrium is stabilized in terms of the determinacy.

Why does the slope of the Phillips curve increase as the relative risk aversion increases? We

23

Fig. 6. Determinate-indeterminate region in the (1/θ, ϕn)space under σ= 0.16 (Panel A) and σ= 1.38

(Panel B).

interpret the results in Figs. 5 and 6 as follows. Because the utility function of consumption

is given by C1−σ/(1 −σ), more consumption is required to obtain higher utility for higher σ.

Therefore, households need higher real wages to obtain more consumption. This increases the

ﬁrm’s costs, and then the higher marginal cost appears as a higher value of κin the Phillips

curve. However, this higher value of κmitigates the eﬀects of the model uncertainty under

robust monetary policy.

4.2 Inverse of the Frisch elasticity

Analogous to relative risk aversion, this section shows quantitative results under ϕ= 0.47

(Rotemberg and Woodford, 1997) and ϕ= 1.83 (Smets and Wouters, 2007). Figures 7 and 8

plot the policy function and determinate-indeterminate region under ϕ= 0.47 and ϕ= 1.83,

respectively. Both ﬁgures show similar results to Figs. 5 and 6: Fig. 7 shows that the eﬀects

of the model uncertainty are small when ϕis 1.83, and Fig. 8 shows that determinate regions

expand as ϕincreases from 0.47 to 1.83. The driving force behind these results is ϕin the slope

of the Phillips curve. In the Appendix of the present paper, we show the results when we set

the inverse of the Frisch elasticity in κ, which we deﬁne as ϕκ, at 1.83, and 0.47 for all other

ϕexcept ϕκ(Figs. A3 and A4). Combining the results in Figs. 7 and 8 with those in Figs. A3

and A4, the economy is stabilized more as ϕκincreases in terms of both policy function and

equilibrium determinacy.

The intuition behind these results is also the same as the case of relative risk aversion,

because an increase in ϕmakes the slope of the Phillips curve steeper:

∂κ

∂ϕ =λ

1−ϕn

>0.(38)

Therefore, the eﬀects of the model uncertainty are mitigated via the same rationale as described

24

Fig. 7. Eﬀects of an increase in model uncertainty (1/θ) for degrees of ϕn= 0.13,0.18, and 0.25 under

ϕ= 0.47 (Panel A) and ϕ= 1.83 (Panel B).

25

Fig. 8. Determinate-indeterminate region in (1/θ, ϕn)space under ϕ= 0.47 (Panel A) and ϕ= 1.83

(Panel B).

in Section 4.1. An increase in ϕmeans higher costs for ﬁrms, because households feel the higher

cost of the labor supply. This appears as a higher value of κin the Phillips curve, and then

mitigates the eﬀects of the model uncertainty under robust monetary policy.

5 Conclusion

This paper investigated how progressive taxation on labor income impacts the eﬀect of model

uncertainty by introducing a robust control approach into the model of Mattesini and Rossi

(2012). We revealed that a higher progressivity decreases the eﬀect of the model uncertainty

on the inﬂation rate, output gap, and interest rate. Moreover, suﬃciently higher progressivity

brings the economy into the determinate equilibrium even though the stronger model uncer-

tainty brings the economy to indeterminate equilibria. These results show that progressive

taxation on labor income contributes to stabilizing the economy when model uncertainty exists

by mitigating the eﬀects of the model uncertainty in terms of macroeconomic variance and

equilibrium determinacy.

26

Appendix

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(ϕn),qπ(ϕn),qx(ϕn),κ(ϕn)/α(ϕn),

µπ(ϕn), and µy(ϕn)with respect to ϕn. The signs of ∂µπ/∂ϕnand ∂µy/∂ϕnare give as follows:

∂µπ(ϕn)

∂ϕn

=1

1−ϕn

+ϕ+ϕn

(1 −ϕn)2>0,(A.1)

∂µy(ϕn)

∂ϕn

=1

1−ϕn

+1 + ϕ+ϕn

(1 −ϕn)2>0,(A.2)

Using (A.2) and (A.1), we obtain the signs of ∂κ(ϕn)/∂ϕnand ∂qx(ϕn)/∂ϕn:

∂κ(ϕn)

∂ϕn

=λ∂µy

∂ϕn

>0,(A.3)

∂qπ(ϕn)

∂ϕn

=−ϵ

λ

1−Φ

(1 + ϕ+ϕn)2<0,(A.4)

∂qx(ϕn)

∂ϕn

= Φ ∂µy

∂ϕn

>0,(A.5)

∂κ(ϕn)

α(ϕn)

∂ϕn

=ϵ(1 −Φ)(1 + ϕ)2

[(1 + ϕ)((1 −ϕn) + ϕnΦ) + ϕnΦ] >0.(A.6)

where Φ=1−(ϵ−η(ϵ−1))/ϵ < 1denotes 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|cπ(θ,ϕn)|

∂θ∂ ϕn.−∂2|cπ(θ,ϕn)|

∂θ∂ ϕnis given as follows:

−∂2|cπ(θ, ϕn)|

∂θ∂ϕn

=1/θ2

ω(θ, ϕn)2

∂qπ(ϕn)

∂ϕn

−2qπ(ϕn)/θ2

ω(θ, ϕn)3

κ(ϕn)

α(ϕn)

∂κ(ϕn)

∂ϕn

+κ(ϕn)∂κ(ϕn)

∂ϕn

∂ϕn

−1

θ

∂qπ(ϕn)

∂ϕn

.

(A.7)

The ﬁrst term, 1/θ2

ω(θ,ϕn)2

∂qπ(ϕn)

∂ϕnis negative because ∂qπ(ϕn)

∂ϕnis negative. The terms in bracket

in the second term are all positive , because ∂κ(ϕn)

∂ϕn>0,∂[κ(ϕn)/α(ϕn)]

∂ϕn>0, and ∂qπ(ϕn)

∂ϕn<0.

Therefore, the second term is also negative if ω(θ, ϕn)>0. By combining these signs, we obtain

(28) and Proposition 1:

−∂2|cπ(θ, ϕn)|

∂θ∂ϕn

<0if ω(θ, ϕn)>0.(A.8)

19See Mattesini and Rossi (2012, p.855) for a detailed derivation of Φ.

27

□

A.1.2 Proof for Proposition 2

Now we derive the sign of −∂2|cx(θ,ϕn)|

∂θ∂ ϕnand −∂2|ci(θ,ϕn)|

∂θ∂ ϕn:

−∂2|cx(θ, ϕn)|

∂θ∂ϕn

=1/θ2

ω(θ, ϕn)2

qπ(ϕn)∂κ(ϕn)

α(ϕn)

∂ϕn

−κ(ϕn)

α(ϕn)

∂qπ

∂ϕn

−2κ(ϕn)/α(ϕn)

ω(θ, ϕn)3

qπ(ϕn)

θ2

κ(ϕn)∂κ(ϕn)

α(ϕn)

∂ϕn

+κ(ϕn)

α(ϕn)

∂κ(ϕn)

∂ϕn

−1

θ

∂qπ(ϕn)

∂ϕn

(A.9)

−∂2|ci(θ, ϕn)|

∂θ∂ϕn

=−σ

γc

∂2|ci(θ, ϕn)|

∂θ∂ϕn

.(A.10)

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

qπ(ϕn)∂κ(ϕn)

α(ϕn)

∂ϕn

−κ(ϕn)

α(ϕn)

∂qπ

∂ϕn

=ϵ2(1 −Φ)

λ

(2ϕn+ϕϕn)(1 −Φ) + ϕ3+ 3ϕ2+ 2ϕ+ 2ϕϕn+ϕ2ϕn

(1 + ϕ+ϕn)2[(1 + ϕ)((1 −ϕn) + ϕnΦ) + ϕnΦ]2>0

(A.11)

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(ϕn)

∂ϕn>0,∂[κ(ϕn)/α(ϕn)]

∂ϕn>0, and ∂qπ(ϕn)

∂ϕn<0.

Calculating the summation of the ﬁrst and second terms is too complicated. Therefore, the

sign of −∂2|cx(θ,ϕn)|

∂θ∂ ϕnand −∂2|ci(θ,ϕn)|

∂θ∂ ϕnis 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 misspeciﬁca-

tion but there are no misspeciﬁcation. This is called the approximating model in the literature

of robust control (Leitemo and Söderström, 2008a,b; Giordani and Söderlind, 2004). In this

paper, we obtain the approximating model by substituting the policy function of nominal in-

terest rate under robust policy it=ci(θ, ϕn)into the non-distorted structural equation (i.e.,

νu

t= 0). The following are the obtained policy function of inﬂation and output gap under the

approximating model, respectively:20

πt=ca

π(θ, ϕn)ut,(A.12)

xt=ca

x(θ, ϕn)ut,(A.13)

20The obtained coeﬃcient ca

xis identical to cxin Eq. (21).

28

where

ca

π= 1 −κ(ϕn)2

α(ϕn)ω(θ, ϕn),

ca

x=−κ(ϕn)

α(ϕn)ω(θ, ϕn).

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(LA> LW|W) + 1

2×Prob(LW> LA|A),

where LAand LWdenote the values of the likelihood of the approximating model and worst-

case scenario (i.e., robust monetary policy), respectively. The notations Aand Wdenote 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 νw

tand νa

t, respectively)

for suﬃciently long periods T.21

Then, we calculate the relative likelihood rwand raas follows:22

rw=1

T

T−1

X

t=0 1

2(νw

t)′νw

t+ (νw

t)′ut,(A.14)

ra=1

T

T−1

X

t=0 1

2(νa

t)′νa

t−(νa

t)′ut.(A.15)

Finally, we obtain the detection error probability as follows:

p(θ) = 1

2[freq(rw≤0) + freq(ra≤0)] .(A.16)

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.

21To calculate the detection error probabilities, we need to add the speciﬁcation error to the approximating

model virtually though no speciﬁcation error exists in the approximating model.

22For detailed explanations of the relative likelihood and derivation of Eqs. (A.14) and (A.15), see Hansen and

Sargent (2008) and Matlab code (ErrDetProb.m) of Giordani and Söderlind (2004).

29

A.3.1 Households

Households’ preference is given as follows:

Ut=E0

∞

X

j=0

βj"C1−σ

t

1−σ−N1+ϕ

t

1 + ϕ#, σ, ϕ > 0.(A.17)

where Ctand Ntdenote consumption and supply of labor hours, respectively. The ﬂow budget

constraint is given as follows:

PtCt+R−1

tBt= (1 −τt)WtNt+Bt+Dt−PtTℓ

t,(A.18)

where Rt,Bt,Wt,Dt,Tℓ

tdenote gross nominal risk-free interest rate, risk free government

bond, nominal wage, proﬁt income, and lump-sum tax, respectively. τtdenotes the taxes on

labor income. Mattesini and Rossi (2012) speciﬁe the form of τtfollowing Guo (1999) and Guo

and Lansing (1998):

τt= 1 −ηYn

Yn,t ϕn

.(A.19)

where η∈(0,1] and ϕn∈[0,1), and Yn=W N/P and Yn,t =WtNt/Ptdenote base level of

income and actual level of income, respectively. The tax rate ensures 0≤τt<1when ϕn>0.

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

τm

t=∂τtYn,t

∂Yn,t

= 1 −η(1 −ϕn)Yn

Yn,t ϕn

=τt+ηϕnYn

Yn,t ϕn

.(A.20)

First order conditions of households are given as follows:

C−σ

t=βRtEt"C−σ

t+1

Πt+1 #,(A.21)

Cσ

tNϕ

t= (1 −τm

t)Wt

Pt

,(A.22)

A.3.2 Firms and Government

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

Yt(k) = AtNt(k),(A.23)

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

MCt=1

At

Wt

Pt

.(A.24)

The resource constraint is

Yt=Ct+Gt.(A.25)

30

A.3.3 Log-linearized expression

The intermediate goods ﬁrms’ optimal price setting yields the following standard Phillips curve

with marginal cost expression:

πt=βEtπt+1 +λmct.(A.26)

where λ=(1−φ)(1−φβ)

φ.

Log-linearizing Eqs. (A.22), (A.25), (A.23), and (A.24) yields

σct+ϕnt= (1 −ϕn)wt−ϕnnt,(A.27)

yt=γcct+ (1 −γc)gt,(A.28)

yt=at+nt,(A.29)

mct=wt−at,(A.30)

Combining these equation yields the following relation:

mct=σ

1−ϕn1

γc

yt−1−γc

γc

gt+ϕ+ϕn

1−ϕn

(yt−at)−at(A.31)

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

πt=βEtπt+1 +κ(ϕn)xt+ut,(A.32)

where

xt=yt−yn

t,

ut=κ(ϕn)(yn

t−y∗

t),

y∗

t= (1 + ϕ)(1 −Φ)at+1 + ϕ

1−ϕn

Φat,

yn

t=1 + ϕ

1 + ϕ+ϕn

at.

Finally, the tax rate ˆτtis reduced as follows:

ˆτt=ηϕn

1−η1 + 1 + ϕ+ϕn

1−ϕnyt−ηϕn

1−η

1 + ϕ

1−ϕn

1

Ξut.(A.33)

where

Ξ = κ(ϕn)(1 + ϕ)(1 −Φ) + Φ

1−ϕn

−1

1−ϕ+ϕn.

23We assume γc= 1 and σ= 1.

31

Fig. A1. Eﬀects of an increase in model uncertainty (1/θ) for degrees of ϕn= 0.13,0.18, and 0.25 under

σ= 0.16 and σκ= 1.38.

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 σand ϕin 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 σκand ϕκ, respectively:

µy(ϕn) = σκ+γc(ϕκ+ϕn)

γc(1 −ϕn), κ(ϕn) = λµy(ϕn), λ =(1 −φ)(1 −φβ)

φ.(A.34)

Figures A1 and A2 plot the policy function and determinate-indeterminate region when we

set σand σκat 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.

Figures A3 and A4 plot the policy function and determinate-indeterminate region when we

set ϕand ϕκat 0.47 and 1.83, respectively. Analogous to the case of σκ, the eﬀects of the model

uncertainty become small and all regions are determinate in the plotted area.

32

Fig. A2. Determinate-indeterminate region in the (1/θ, ϕn)space under σ= 0.16 and σκ= 1.38.

Fig. A3. Eﬀects of an increase in model uncertainty (1/θ) for degrees of ϕn= 0.13,0.18, and 0.25 under

ϕ= 0.47 and ϕκ= 1.83.

33

Fig. A4. Determinate-indeterminate region in the (1/θ, ϕn)space under ϕ= 0.46 and ϕκ= 1.83.

34

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