Eurilton Alves Araújo Júnior

English

Research Papers in Economics

Optimal Monetary Policy and Interest
Income Taxation
Eurilton Araœjo
Ibmec Sªo Paulo
Resumo:
Esse artigo estuda pol￿tica monetÆria ￿tima discricionÆria quando o modelo
Novo-Keynesiano incorpora tributa￿ªo da renda dos juros de t￿tulos pœbli-
cos. No modelo, a elasticidade da in￿ a￿ªo e do hiato do produto em rela￿ªo
aos choques de oferta e demanda sªo fun￿ıes crescentes da al￿quota de trib-
uta￿ªo. Adiconalmente, simula￿ıes numØricas mostram que altos n￿veis de
tributa￿ªo aumentam as volatilidades do hiato e da in￿ a￿ªo, como tambØm
a fun￿ªo de perda do Banco Central.
Palavras-Chave: pol￿tica monetÆria, tributa￿ªo de rendimento de juros e
discricionariedade
Abstract:
This paper studies optimal discretionary monetary policy when the basic
new Keynesian model is extended to incorporate interest income taxation.
The elasticities of in￿ ation and the output gap to supply and demand shocks
are increasing functions of the tax rate. Moreover, numerical simulations
show that high levels of taxation increase in￿ ation volatility, the output gap
volatility and the unconditional expectation of the central bank￿ s loss func-
tion.
Keywords: monetary policy, interest income taxation, discretion
`rea da ANPEC: `rea 3-Macroeconomia, Economia MonetÆria e Finan￿as
JEL: E43, E52, E581 Introduction
Despite the growing literature on the optimal design of monetary policy, the
e⁄ects of taxation on central bank strategies have received relatively little
attention. In some recent papers, ￿scal and monetary policy are designed
jointly in the context of the Ramsey problem; Schmitt-GrohØ and Uribe
(2004a, 2004b) are examples of this approach. In some cases, however, the
central bank has to set monetary policy taking the tax system as given. In
this situation, changes in tax rates have an impact on aggregate dynamics
and, consequently, on how monetary policy should be conducted.
This paper studies how the tax system can in￿ uence the monetary policy
design process. In an extension of the basic new Keynesian model studied
in Rłisland (2003), I derive the optimal monetary policy in the absence of
commitment. Rłisland (2003) analyzes equilibrium determinacy in a model
in which nominal interest income from government bonds and pro￿ts are
taxed at a constant rate. This form of taxation alters the household￿ s budget
constraint and makes the after-tax interest rate a crucial variable for the
understanding of aggregate demand.
In this paper, I show that the elasticities of in￿ ation and the output gap
to supply and demand shocks are increasing functions of the tax rate. In
addition, numerical simulations show that high tax rates reduce economic
welfare and increase the volatilities of in￿ ation and the output gap.
2 The Model
The model presented in Rłisland (2003) changes the basic new Keynesian
framework by assuming that nominal interest income on government bonds
and pro￿ts are taxed at a constant rate ￿, where 0 < ￿ < 1.
Households with a time-separable utility and a discount factor ￿, where
0 < ￿ < 1 , maximize their expected lifetime utility given a sequence of
budget constraints. The period utility is given by:
U(Ct;Nt) =
C
1￿￿
t
1 ￿ ￿
+
￿
1 ￿ ￿
￿
Mt
Pt
￿1￿￿
￿ ￿
N
1+’
t
1 + ’
;
where Ct, Mt
Pt and Nt are consumption, real money balances and employ-
ment, respectively.
At each date, the budget constraint is:
1Bt+1 ￿ Bt + Mt+1 ￿ Mt = WtNt + It(1 ￿ ￿)Bt + (1 ￿ ￿)￿t ￿ PtCt ￿ PtTt
where Bt is the household￿ s nominal government bond holdings, Mt is
nominal money balances, Wt is the nominal wage, It is the nominal interest
rate, ￿t is nominal pro￿ts received from ￿rms, and Tt is a real lump-sum tax.
The government satis￿es its intertemporal budget constraint, not explicitly
considered, by adjusting Tt.
Aggregate demand is derived from the representative household￿ s Euler
equation. After imposing market clearing conditions, the log-linear form of
the Euler equation is:
xt = Et(xt+1) ￿
1
￿
[(1 ￿ ￿)it ￿ Et(￿t+1)] + gt (1)
Firms in a monopolistic competitive environment produce di⁄erentiated
goods with a linear technology using only labor. The Calvo mechanism
describes price decisions, where ￿ is the fraction of ￿rms not adjusting their
price in a given period. In the neighborhood of a zero-in￿ ation steady state,
the new Keynesian Phillips curve characterizes in￿ ation dynamics according
to the following expression:
￿t = ￿Et(￿t+1) + kxt + ut (2)
where k =
(1￿￿￿)(1￿￿)
￿ .
The variables xt; it and ￿t are the output gap, the nominal interest rate
and in￿ ation, respectively. In￿ ation and the nominal interest rate are ex-
pressed in log-deviations from their steady states, which are normalized to
zero.
Demand shocks gt and supply shocks ut are added to the model. These
disturbances follow autoregressive structures:
gt = ￿ggt￿1 + "
g
t (3)
ut = ￿uut￿1 + "
u
t (4)
where 0 < ￿g < 1 and 0 < ￿u < 1 are the autoregressive coe¢ cients.
Both "
g
tand "u
t are white noise, with variances ￿2
g and ￿2
u, respectively.
23 Optimal Monetary Policy under Discretion
The policy problem is to choose time paths for ￿t, xt and it that minimize
the central bank￿ s loss function, which translates the behavior of macroeco-
nomic aggregates into a welfare measure to evaluate di⁄erent policy choices.
Clarida et al. (1999) and Giannoni & Woodford (2003a, 2003b) discuss more
extensively the design of optimal monetary policies in new Keynesian models.
I assume that the policymaker seeks to minimize the objective function
L = E
(
(1 ￿ ￿)
1
2
1 X
t=0
￿
t ￿
￿
2
t + ￿xx
2
t + ￿ii
2
t
￿
)
(5)
subject to the constraints imposed by the structural equations (1) to (4).
Expression (5) can be interpreted as a second-order approximation to
the lifetime utility function of a representative household; Woodford (2003)
discusses this interpretation in detail. The presence of the interest rate vari-
ability is related to costs of transactions and to the fact that the nominal
interest rate has a lower bound at zero. The relative weights placed on the
stabilization of the output gap and the nominal interest rate are ￿x and ￿i,
respectively. These weights are strictly positive.
I assume that a commitment technology is absent. In practice, monetary
authorities do not make any kind of binding commitments concerning the
course of future policy actions. In this context, the central bank chooses its
policy by reoptimizing in every period. This type of policy is known as a
discretionary policy. Since central banks cannot manipulate private agents￿
beliefs, private expectations are taken as given, and the optimal policy is
obtained by solving the following sequence of static optimization problems:
Min
￿t;xt;it
1
2
[￿
2
t + ￿xx
2
t + ￿ii
2
t] + Ft
subject to
xt = ￿
1
￿
(1 ￿ ￿)it + ft (6)
and
￿t = kxt + ht (7)
where ft = Et(xt+1) + 1
￿Et(￿t+1) + gt and ht = ￿Et(￿t+1) + ut
The ￿rst order conditions are:
3￿t + ’1t = 0
￿xxt + ’1t ￿ k’2t = 0
￿iit +
1
￿
(1 ￿ ￿)’1t = 0
where ’1t and ’2t are Lagrange multipliers associated with restrictions
(6) and (7), respectively.
After solving for the Lagrange multipliers, the nominal interest rate is:
it =
(1 ￿ ￿)
￿i￿
(￿xxt + k￿t) (8)
To ￿nd an analytical solution, according to the method of undetermined
coe¢ cients, I posit the following decision rules for in￿ ation and the output
gap:
￿t = a1 gt + a2ut (9)
xt = a3gt + a4ut (10)
Using equations (1) to (4), (8), (9) and (10), I solve for the unknown
coe¢ cients a1, a2, a3 and a4 as a function of the structural parameters. The
results are:
a1 =
k
(1 ￿ ￿￿g)(1 ￿ ￿g) ￿
￿gk
￿ + [k2
￿i + ￿x
￿i (1 ￿ ￿￿g)]z
a2 =
(1 ￿ ￿u) + ￿x
￿i z
(1 ￿ ￿￿u)(1 ￿ ￿u) ￿
￿uk
￿ + [k2
￿i + ￿x
￿i (1 ￿ ￿￿u)]z
a3 =
(1 ￿ ￿￿g)
(1 ￿ ￿￿g)(1 ￿ ￿g) ￿
￿gk
￿ + [k2
￿i + ￿x
￿i (1 ￿ ￿￿g)]z
a4 =
￿u
￿ ￿ k
￿iz
(1 ￿ ￿￿u)(1 ￿ ￿u) ￿
￿uk
￿ + [k2
￿i + ￿x
￿i (1 ￿ ￿￿u)]z
4where z =
￿
1￿￿
￿
￿2.
The following proposition summarizes how changes in the tax rate impact
the above coe¢ cients.
Proposition 1 The elasticities of in￿ation and the output gap to supply and
demand shocks are increasing functions of the tax rate ￿.
Proof. The elasticities are the coe¢ cients a1, a2, a3 and a4. I take the
derivative of each coe¢ cient with respect to ￿. After some algebraic manip-
ulations,
da1
d￿
=
h
2k(1￿￿)
￿2
ih
k2
￿i + ￿x
￿i (1 ￿ ￿￿g)
i
n
(1 ￿ ￿￿g)(1 ￿ ￿g) ￿
￿gk
￿ +
h
k2
￿i + ￿x
￿i (1 ￿ ￿￿g)
i
z
o2
da2
d￿
=
h
2(1￿￿)
￿2
ih
￿x
￿i
￿uk
￿ + (1 ￿ ￿u)k2
￿i
i
n
(1 ￿ ￿￿u)(1 ￿ ￿u) ￿
￿uk
￿ + [k2
￿i + ￿x
￿i (1 ￿ ￿￿u)]z
o2
da3
d￿
=
h
2(1￿￿)(1￿￿￿g)
￿2
ih
k2
￿i + ￿x
￿i (1 ￿ ￿￿g)
i
n
(1 ￿ ￿￿g)(1 ￿ ￿g) ￿
￿gk
￿ +
h
k2
￿i + ￿x
￿i (1 ￿ ￿￿g)
i
z
o2
da4
d￿
=
h
2(1￿￿)(1￿￿￿u)
￿2
ih
￿x
￿i
￿u
￿ + (1 ￿ ￿u) k
￿i
i
n
(1 ￿ ￿￿u)(1 ￿ ￿u) ￿
￿uk
￿ + [k2
￿i + ￿x
￿i (1 ￿ ￿￿u)]z
o2
Since all structural parameters are strictly positive, 0 < ￿ < 1, 0 < ￿ < 1,
0 < ￿g < 1 and 0 < ￿u < 1, the conclusion is that all the derivatives are
strictly positive.
The variances of in￿ ation and the output gap are:
￿
2
￿ = (a1)
2￿
2
g + (a2)
2￿
2
u
￿
2
x = (a3)
2￿
2
g + (a4)
2￿
2
u
The derivatives with respect to ￿ are:
d￿2
￿
d￿
= 2
￿
a1￿
2
g
da1
d￿
+ a2￿
2
u
da2
d￿
￿
5d￿2
x
d￿
= 2
￿
a3￿
2
g
da3
d￿
+ a4￿
2
u
da4
d￿
￿
The behavior of in￿ ation and the output gap volatilities as a function of ￿
depends upon the signs of the coe¢ cients a1, a2, a3 and a4: If all coe¢ cients
are positive, then the variances of in￿ ation and the output gap increase with
￿. The next proposition shows that some restrictions on the autoregressive
coe¢ cients lead to positive elasticities. These restrictions are:
￿g ￿
1
2
2
4
￿
1 +
1
￿
+
k
￿￿
￿
￿
s￿
1 +
1
￿
+
k
￿￿
￿2
￿
4
￿
3
5 (11)
￿u ￿
1
2
2
4
￿
1 +
1
￿
+
k
￿￿
￿
￿
s￿
1 +
1
￿
+
k
￿￿
￿2
￿
4
￿
3
5 (12)
￿u >
k￿
￿i
z (13)
Proposition 2 If conditions (11), (12) and (13) are satis￿ed, then the vari-
ances of in￿ation and the output gap are increasing functions of ￿.
Proof. A set of su¢ cient conditions for all coe¢ cients to be positive is:
(1 ￿ ￿￿g)(1 ￿ ￿g) ￿
￿gk
￿ ￿ 0, (1 ￿ ￿￿u)(1 ￿ ￿u) ￿
￿uk
￿ ￿ 0 and ￿u > k￿
￿i z. The
￿rst two conditions lead to the following inequality:
F(￿) = ￿
2 ￿
￿
1 + ￿ + k
￿
￿
￿
￿ +
1
￿
￿ 0
Since F(0) = 1
￿ > 0 and F(1) = ￿ k
￿￿ < 0, F(￿) ￿ 0 if and only if
0 < ￿ ￿ ￿, where ￿ is the smallest root of F(￿), which is less than one. After
some algebra, I ￿nd that F(￿) has two real roots since
￿
1 + 1
￿ + k
￿￿
￿2
￿ 4
￿ =
￿
1 ￿ 1
￿
￿2
+ 2
￿
1 + 1
￿
￿
k
￿￿ +
￿
k
￿￿
￿2
> 0, and the smallest root is given by
1
2
"￿
1 + 1
￿ + k
￿￿
￿
￿
r￿
1 + 1
￿ + k
￿￿
￿2
￿ 4
￿
#
.
Proposition 2 derives only su¢ cient conditions such that variances are
increasing functions of ￿. In sum, the implications of changes in ￿ for macro-
economic volatility depend upon speci￿c parameter values.
64 Numerical Results
I simulate the model using a benchmark parameterization to evaluate the
impact of changes in ￿ on macroeconomic volatility and welfare. The pa-
rameters are calibrated following Gianonni and Woodford (2003b). I set
￿ = 0:99, 1
￿ = 0:16, k = 0:024, ￿x = 0:048, ￿i = 0:236 and ￿u = ￿g = 0:35 as
my baseline values. The variances of the shocks are ￿2
g = 0:35 and ￿2
u = 0:17.
I normalize all variances and the central bank￿ s loss associated with the
absence of taxation to 1. Therefore, Figure 1 reports relative values. The
volatilities of in￿ ation and the output gap are increasing functions of ￿.
The unconditional expectation of the central bank￿ s loss function is also an
increasing function of ￿. Thus, increases in interest rate income taxes reduce
social welfare. By contrast, the interest rate volatility decreases as a result
of an increase in ￿. We can see that the impact of changes in ￿ on welfare is
minor under the baseline parameters. Additional simulations, not reported
here, indicate that these results are very robust as long as the shocks are not
extremely persistent.
The magnitude of the impact of changes in ￿ on welfare and volatilities,
however, depends upon the relative importance of the output gap in the mon-
etary policy design process. Figure 2 shows relative volatilities and losses for
￿x = 0:5. As long as the output gap becomes more important for monetary
policy, increases in ￿ cause signi￿cant changes in welfare. High tax rates are
also associated with more volatile output gaps.
5 Conclusion
This paper studies how interest income taxation a⁄ects aggregate dynamics
when the central bank chooses its policy optimally under discretion. I show
analytically that the elasticities of in￿ ation and the output gap to supply
and demand shocks are increasing functions of the tax rate. In addition,
numerical simulations show that high tax rates reduce economic welfare and
increase the volatilities of in￿ ation and the output gap. Therefore, changes in
interest income taxes have important implications for the design of monetary
policy strategies.
7References
Clarida, R., J. Gali and M. Gertler., 1999, The Science of Monetary Policy: A
New Keynesian Perspective, Journal of Economic Literature 37, 1661-1706.
Giannoni, M. and M. Woodford, 2003a, Optimal Interest-Rules I: General
Theory, NBER Working Paper Number 9419
Giannoni, M. and M. Woodford, 2003b, Optimal Interest-Rules II: Applic-
tions, NBER Working Paper Number 9419
Rłisland, ￿., 2003, Capital income taxation, equilibrium determinacy and
the Taylor principle, Economics Letters 81, 147-153.
Schmitt-GrohØ, S. and M. Uribe, 2004a, Optimal Fiscal and Monetary Policy
under Sticky Prices, Journal of Economic Theory 114, 198-230.
Schmitt-GrohØ, S. and M. Uribe, 2004b, Optimal Fiscal and Monetary Policy
under Imperfect Competition, Journal of Macroeconomics 26, 183-209.
Woodford, M., 2003, Interest and Prices: Foundations of a Theory of Mone-
tary Policy. (Princeton University Press, Princeton).
8Figure 1: Relative Volatility-low ￿x
0 0.5 1
1
1.01
1.02
1.03
r
e
l
a
t
i
v
e
 
v
o
l
a
t
i
l
i
t
y
tax rate
Output Gap
0 0.5 1
1
1.0001
1.0002
1.0003
1.0004
r
e
l
a
t
i
v
e
 
v
o
l
a
t
i
l
i
t
y
tax rate
Inflation
0 0.5 1
0
0.5
1
r
e
l
a
t
i
v
e
 
v
o
l
a
t
i
l
i
t
y
tax rate
Interest Rate
0 0.5 1
1
1.0005
1.001
1.0015
1.002
r
e
l
a
t
i
v
e
 
l
o
s
s
tax rate
Loss Function
9Figure 2: Relative Volatility-High ￿x
0 0.5 1
1
1.05
1.1
1.15
1.2
r
e
l
a
t
i
v
e
 
v
o
l
a
t
i
l
i
t
y
tax rate
Output Gap
0 0.5 1
1
1.0005
1.001
1.0015
1.002
r
e
l
a
t
i
v
e
 
v
o
l
a
t
i
l
i
t
y
tax rate
Inflation
0 0.5 1
0
0.5
1
r
e
l
a
t
i
v
e
 
v
o
l
a
t
i
l
i
t
y
tax rate
Interest Rate
0 0.5 1
1
1.02
1.04
1.06
1.08
r
e
l
a
t
i
v
e
 
l
o
s
s
tax rate
Loss Function
10

Optimal Monetary Policy and Interest Income Taxation

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Optimal Monetary Policy and Interest Income Taxation

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