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Progressive Taxation and Robust Monetary Policy

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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.
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+R1
tBt= (1 τt)WtNt+Bt1+DtPtT
t,(1)
where Rt,Bt,Wt,Dt, and T
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 0ϕn<1and 0< η 1,
because it is difficult to ensure 0τt<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 1η. However, when the actual level of income is greater than the base level, the tax
rate is greater than 1η. 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)wtϕnnt.(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
+T
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
σitEtπt+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 0γc1denote 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=1(ϵ1)η
ϵ,
µπ(ϕn) = ϕ+ϕn
1ϕn
,
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
σhitEtπt+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+jν0,(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
xtt,it
max
νr
tu
t
Et
X
j=0
βj[qπ(ϕn)π2
t+j+qx(ϕn)x2
t+jθ(νr
t+j)2θ(νu
t+j)2],
s.t.
xt=Etxt+1 γc
σhitEtπt+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(ϕn)qπ(ϕn)
For subsequent analyses, we impose the following Assumption:
Assumption 1. We assume 0β < 1,0φ1,ϕ > 0,ϵ > 1,0ϕn<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 0τt<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 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 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
=12
ω(θ, ϕn)2
∂qπ(ϕn)
∂ϕn
2qπ(ϕn)2
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
=12
ω(θ, ϕn)2
qπ(ϕn)κ(ϕn)
α(ϕn)
∂ϕn
κ(ϕn)
α(ϕn)
∂qπ
∂ϕn
2κ(ϕn)(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(ϕn)qπ(ϕn) 0
σ
γc
βκ(ϕn)(ϕn)
1+κ(ϕn)2(ϕn)qπ(ϕn) 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(ϕn)qπ(ϕn) > 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)2
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 β2|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(n)|
|cj(n)|×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). 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 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) 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 become larger
as ϕnincreases. Therefore, stabilization effects of ϕnmay be large when 1 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 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. 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 is too high when the equilibrium is
indeterminate. Figure 4 shows that the equilibrium is indeterminate when 1 is greater than
0.035 (θis 28.57 when 1 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) 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(n)|
|cj(n)|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 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 C1σ/(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
1ϕn
>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) 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)(ϕn),
µπ(ϕn), and µy(ϕn)with respect to ϕn. The signs of µπ/∂ϕnand ∂µy/∂ϕnare give as follows:
∂µπ(ϕn)
∂ϕn
=1
1ϕn
+ϕ+ϕn
(1 ϕn)2>0,(A.1)
∂µy(ϕn)
∂ϕn
=1
1ϕn
+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
1Φ
(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(ϵη(ϵ1)) < 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
=12
ω(θ, ϕn)2
∂qπ(ϕn)
∂ϕn
2qπ(ϕn)2
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, 12
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)(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
=12
ω(θ, ϕn)2
qπ(ϕn)κ(ϕn)
α(ϕn)
∂ϕn
κ(ϕn)
α(ϕn)
∂qπ
∂ϕn
2κ(ϕn)(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)(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
T1
X
t=0 1
2(νw
t)νw
t+ (νw
t)ut,(A.14)
ra=1
T
T1
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(rw0) + freq(ra0)] .(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"C1σ
t
1σN1+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+R1
tBt= (1 τt)WtNt+Bt+DtPtT
t,(A.18)
where Rt,Bt,Wt,Dt,T
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 0τt<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:
Cσ
t=βRtEt"Cσ
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=(1φ)(1φβ)
φ.
Log-linearizing Eqs. (A.22), (A.25), (A.23), and (A.24) yields
σct+ϕnt= (1 ϕn)wtϕnnt,(A.27)
yt=γcct+ (1 γc)gt,(A.28)
yt=at+nt,(A.29)
mct=wtat,(A.30)
Combining these equation yields the following relation:
mct=σ
1ϕn1
γc
yt1γc
γc
gt+ϕ+ϕn
1ϕn
(ytat)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=ytyn
t,
ut=κ(ϕn)(yn
ty
t),
y
t= (1 + ϕ)(1 Φ)at+1 + ϕ
1ϕn
Φat,
yn
t=1 + ϕ
1 + ϕ+ϕn
at.
Finally, the tax rate ˆτtis reduced as follows:
ˆτt=ηϕn
1η1 + 1 + ϕ+ϕn
1ϕnytηϕn
1η
1 + ϕ
1ϕn
1
Ξut.(A.33)
where
Ξ = κ(ϕn)(1 + ϕ)(1 Φ) + Φ
1ϕn
1
1ϕ+ϕn.
23We assume γc= 1 and σ= 1.
31
Fig. A1. Effects of an increase in model uncertainty (1) 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) 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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