GCOE Discussion Paper Series
Global COE Program

Human Behavior and Socioeconomic Dynamics
Discussion Paper No.72

Habit Formation, Interest-Rate Control andEquilibrium Determinacy

Seiya Fujisaki

August 2009GCOE Secretariat
Graduate School of Economics

OSAKA UNIVERSITY

1-7 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan

Habit Formation, Interest-Rate Control andEquilibrium Determinacy ∗Seiya Fujisaki†
August, 2009

Abstract
We examine macroeconomic stability of a monetary economy with habit
formation in consumption. We assume that monetary authority controls therate of nominal interest in response to inflation and output gap. We showthat in the presence of habit persistence not only active but also passive monetary policy can generate equilibrium determinacy under empirically plausible
values of the elasticity of intertemporal substitution in felicity.
Keywords and Phrases: equilibrium determinacy, habit formation,
Taylor rule, endogenous labor.
JEL Classification Numbers: E21, E52, O42∗†

I am grateful to Kazuo Mino for his helpful comments on earlier versions of this paper.
Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka, 563-

0043, JAPAN (email: ege010fs@mail2.econ.osaka-u.ac.jp)

1

Introduction

This paper introduces habit persistence in consumption into a cash-in-advance (CIA)model with interest-rate control. The purpose of this paper is to investigate whether
the presence of habit formation may alter the stabilization effects of monetary policy.
The presence of habit persistence in consumption has been confirmed in a numberof empirical studies and the theoretical effects of habit formation have been discussed
extensively. 1 Several authors have already examined monetary dynamic models
with habit formation. For example, Auray et. al. (2002 and 2005) introduce money
via the CIA constraint into macrodynamic models in which real money holdings
bind consumption. They show that a high degree of habit persistence may yield
multiple equilibria. In contrast, Auray et. al. (2004) reveal that in a moneyin-the-utility-function model, the presence of habit does not affect macroeconomic
stability.
The foregoing studies on monetary dynamics with habit formation usually assume the traditional monetary policy in the money and growth literature, that is,
the monetary authority fixes the expansion rate of nominal money supply. It is now
well understood that the exogenous money supply rule does not precisely describe
the central banks’ behaviors in many countries. Rather, as emphasized by Taylor(1993), they control the nominal interest rate by observing inflation and output gap.
Considering this fact, a large number of authors have examined stabilization effectof interest-rate control rules: see, for example, Benhabib et. al. (2001), Meng (2002)
and others. However, these authors ignore habit persistence in consumption. Our
central concern is to reconsider the stabilizing effect of interest-rate control in thepresence of habit formation.
It is to be noted that Graham (2008) constructs a savers-and-spenders modelof monetary economy with habit formation and interest-rate control. He uses a
1

For example, see Abel (1990), Carroll et. al. (1997), Carroll (2000), Constantinides (1990),

Fuhrer (2000), Smith (2002), and Weder (2000).

1

stochastic New Keynesian model that includes imperfect competition, sticky priceadjustment as well as capital accumulation, and his research concern focuses on
numerical experiments rather than analytical implications. In contrast, we use a
simple model of competitive, production economy in order to consider equilibrium
determinacy analytically. Our model is based on Meng (2002) who studies stabilization effect of interest-rate control in a CIA economy with variable labor supply.
We generalize Meng’s (2002) analysis in two aspects.
First, as was emphasized, we introduce habit persistence in consumption into thebase model. When the implicit cost of habit accumulation which represents a negative effect on the utility is taken into consideration under household’s optimization,
passive monetary policy which lowers the real rate of interest with a higher inflation
may easily create determinate equilibrium under plausible values of the elasticity ofintertemporal substitution. This is in contrast to the finding that in models without
habit formation determinacy under passive interest-rate control can hardly emerge.
Second, we assume that the central bank adjusts the nominal interest rate in
response not only to the rate of inflation but also to output gap, while Meng (2002)
assumes that the nominal interest rate depends on inflation alone. 2 Given our policy
rule, a rise in the rate of inflation may lower a real interest rate, even when monetary
policy is active in which the real rate of interest rises with a higher inflation. For
the effect on output via habit formation combining with this type of interest-rate
control, it becomes more difficult to hold determinacy under active policy and highelasticity of intertemporal substitution.
The rest of this paper is organized as follows. Section 2 sets up the base model.
Section 3 examines equilibrium determinacy/indeterminacy conditions for the basemodel. This section also considers the case with outward-looking habit persistence
2

Whether it is significant for macroeconomic stability depends on a structure of the economy.

For example, Meng and Yip (2004) show that it does not affect equilibrium determinacy in the
Ramsey model, while it has an influence on stability in an AK growth economy as shown in Fujisaki
and Mino (2007).

2

where habit formation in consumption is external to each household. Section 4
presents numerical examples for alternative forms of utility functions. Section 5
concludes.

2

The Model

The representative household’s optimization problem isZ ∞maxu(c − θh, l)e−ρt dt, 0 < θ < 1,

ρ > 0,(1)0

subject toȧ = (R − π)a − Rm + wz − c − τ,αc ≤ m,ḣ = β(c − h),α > 0,
β > 0,

(2)
(3)
(4)

where a ≡ b + m real financial assets, b bonds, m real money holdings, c is consumption, ρ the time discount rate, l leisure, n ≡ 1 − l labor, R the nominal interest
rate, π the rate of inflation, w the real wage, τ lump taxes, h the stock of consumption habits, θ habit persistence, and β denotes the speed of habit adjustment.
The instantaneous felicity function, u(c − θh, l), is monotonically increasing andstrictly concave in c − θh and l. Additionally, we assume that both leisure and(habit-adjusted) consumption are normal goods, that is,ν1 ≡ u12 u1 − u2 u11 > 0,

ν2 ≡ u12 u2 − u1 u22 > 0, and − (u12 )2 + u11 u22 > 0.

Equation (2) is the household’s flow budget constraint. Condition (3) is the cashin-advance constraint in which cash has to be held in advance of purchasing goods.
The stock ofZ habits equals weighted average of past consumption in such a way thatht = βe−βtte−βτ cτ dτ . Therefore, the dynamic behavior of h is represented as−∞(4). The production function isy = n = 1 − l,3(5)and thus the real wage is normalized to hold w = 1.
We focus on the situation where the cash-in-advance constraint (3) is binding
so that αc = m holds. Additionally, habit formation is assumed to be internal in
the sense that the household takes into account the accumulation of habit stock (4)when it solves the optimization problem. The Hamiltonian function is
H = u(c − θh, l) + q[(R − π)a + (1 − l) − (1 + αR)c − τ ] + λβ(c − h),(6)where q and λ respectively denote the shadow value of assets, a, and habit stock, h.
Since an increase in h lowers utility, the implicit value of h evaluated by utility, λ,has a negative value. The first-ordered conditions are as follows:u1 (c − θh, l) = q(1 + αR) − λβ,(7)u2 (c − θh, l) = q,(8)q̇ = [ρ + π − R]q,(9)λ̇ = (ρ + β)λ + θu1 (c − θh, l).

(10)These conditions are standard except for the effect of habit persistence. Especially, (7) states that the marginal benefit of habit-adjusted consumption equals its
marginal cost that equals the marginal (dis)utility of having an additional unit ofreal financial wealth plus that of an additional unit of habit stock. Additionally, thetransversality conditions are lim e−ρt qt at = 0 and lim e−ρt λt ht = 0.
t→∞

t→∞

As Taylor (1993) originally suggests, we assume the monetary policy rule suchthat the central bank sets the rate of nominal interest according not only to the
inflation rate but also to the output gap:
R(π, y) = φπ (π − π ∗ ) + φy (y − y ∗ ) + R∗ ,φπ > 0,φy ≥ 0.

(11)

In (11), π ∗ and R∗ respectively denote the target rates of inflation and nominalinterest set by the monetary authority, which satisfy π ∗ > −ρ and π ∗ + ρ = R∗ > 0.
We discuss determination of the steady-state value of output y ∗ in Section 3.2. The
4

interest-rate control rule (11) is said to be active if φπ > 1. In this case, the monetary
authority rises a real interest rate with a higher rate of inflation under a given levelof output. Conversely, the rule is passive when φπ < 1.
The government’s budget constraint with zero government purchases in realterms is given by the following: 3ȧ = (R − π)a − Rm − τ.

(12)Combining (12) with (2) gives the goods-market equilibrium conditionc = 1 − l.

3

Equilibrium Dynamics

3.1

Dynamic System(13)

From (7) and (8), we obtainu2 (c − θh, l)1=,u1 (c − θh, l)
1 + αR − βx(14)which means that the marginal rate of substitution between leisure l and the habitadjusted consumption c − θh, equals the real wage in terms of the effective priceincluding the opportunity cost of money holdings with internal habit, 1+ αR −βx >λ1, where x = < 0. We can also interpret that the left-hand side in (14) representsqa labor supply and the right-hand side is labor-demand.
Combining (14) with (13), we derive the demand function of leisureµ¶1
l = l h,,1 + αR − βx3Originally, the budget is expressed asḂ = RB − Ṁ − P τ,

where capital letters are nominal terms and P denotes the price level.

5

(15)

whereθν1(u1 )2∈ (−1, 0) and l2 = −< 0.
ν1 + ν2
ν1 + ν2∂c∂c
Conversely,∈ (0, 1) and> 0 because of goods-market equi∂h∂[1/(1 + αR − βx)]librium (13). These properties are due to the assumption of normal goods. When1
habit increases under a given effective real wage
, the gross consump1 + αR − βxu2constant.
tion should be higher in order to keep the marginal rate of substitution
u1
Since habit persistence satisfies θ ∈ (0, 1), leisure decreases with habit stock lessl1 = −

than one for one. When the effective real wage rises, the labor demand increases sothat the household cuts leisure.
Using (11) and (15), we find that the equilibrium rate of inflation is expressed
by
µ
µ
¶
¶
1
R − R ∗ φy∗−
1 − l h,−y ,π = π(R, x, h) = π +φπφπ
1 + αR − βx∗(16)which satisfiesµ¶
1
αφy l2
βφy l2φy
πR =1−> 0, πx =≤ 0, πh =l1 ≤ 0.
22
φπ(1 + αR − βx)φπ (1 + αR − βx)φπ(17)The equilibrium income becomes higher with a rise in habit stock h and a fall in thecost of accumulation of habit −x, and thus the equilibrium inflation rate falls whenφy > 0. Even though φπ > 1, πR could be larger than one because an increase in anominal interest rate lowers both the effective real wage and production. This meansthat an active interest-rate control may not generate a rise in the real interest ratewith a higher inflation rate. When φy = 0, these effects via the monetary-policy’s
1and πh = πx = 0.
response to output gap disappears so that πR =φπFrom (8) and (15), we obtain the following differential equations:q̇θu12u12 − u22 ˙
=−
ḣ −
l,
q
u2
u2
l˙ = l1 ḣ −l2(αṘ − β ẋ).
(1 + αR − βx)2
6

(18)
(19)

Combining these equations with (9), we see that R changes according toṘ =βẋ − E1 {[ρ + π(R, x, h) − R] − E2 ḣ},
α

(20)

where
E1 =

ν1 + ν2
θ[−(u12 )2 + u11 u22 ]> 0.
; E2 =
αu2 (u12 − u22 )ν1 + ν2

Note that sign[E1 ] =sign[u12 − u22 ]. Additionally, dynamic equations of x and h arerespectively given by the following: 4ẋ = [β(1 − θ) − π(R, x, h) + R]x + θ(1 + αR),
·
µ
¸
¶
1
ḣ = β 1 − l h,−h .
1 + αR − βx

(21)
(22)

Consequently, we obtain a complete dynamic system consisting of (20)-(22) withrespect to R, x and h.

3.2

Stability

In the following, asterisks ”∗” denote the steady-state values realized when it holds
that Ṙ = ḣ = ẋ = 0 in (20)-(22). In view of the interest-rate control rule (11), it
also holds that π = π ∗ and y = y ∗ in the steady state. We can show that our model
has a unique steady state. From (20), we deriveR∗ = ρ + π ∗ ,(23)which gives the steady-state rate of nominal interest. Combining this with (21), we
obtain a unique level of x∗ such thatx∗ = −

θ(1 + αR∗ )< 0.
β(1 − θ) + ρ(24)

To determinate the steady-state value of output y ∗ , note that (13), (22) and (24)
yield

µ

β(1 − θ) + ρ
1−l h ,
(1 + αR∗ )(β + ρ)∗

4(21) is derived from¶= h∗ .

ẋλ̇ q̇= − , (7), (9), and (10). Substituting (15) into (4), we obtain (22)x
λ q

as the dynamic of h.

7

Since −1 < l1∗ < 0 for all h∗ > 0, there is a unique level of h∗ satisfying the above
equation and thus y ∗ (= c∗ = h∗ = 1 − l∗ ) is uniquely given as well.
Now, we examine the stability of this economy around the steady state. The
coefficient matrix of the linearized system of the original one (20)-(22) around thesteady state is




Ṙ∗
 R


J =  ẋ∗R

ḣ∗R

Ṙx∗
ẋ∗x
ḣ∗x

Ṙh∗



ẋ∗h  ,

∗
ḣh

where
β ∗
ẋR + E1∗ {(1 − πR ∗ ) + E2∗ ḣ∗R },αββṘx∗ = ẋ∗x − E1∗ (πx ∗ − E2∗ ḣ∗x ), Ṙh∗ = ẋ∗h − E1∗ (πh ∗ − E2∗ ḣ∗h ),
α
α
∗
ṘR
=

ẋ∗R = x∗ ·(1 − πR ∗ ) + θα, ẋ∗x = β(1 − θ) + ρ − x∗ πx ∗ , ẋ∗h = −x∗ πh ∗ ≤ 0,µ ∗ ¶2µ ∗ ¶2∗∗ u2∗
2 ∗ u2ḣR = αβl2 ∗< 0, ḣx = −β l2 ∗
> 0,
u1
u1
·
¸
θν1∗
∗
∗
ḣh = −β(l1 + 1) = −β − ∗+ 1 < 0.
ν1 + ν2∗
There are two jump variables, R and x, and one predetermined variable, h, in the
dynamic system so that the steady state satisfies local determinacy if two eigenvaluesof matrix J have positive real parts. When all eigenvalues have positive real parts,
there is no equilibrium paths. We call this situation ”non-stationary”. Otherwise,
equilibrium indeterminacy holds, that is, there exist multiple equilibrium paths.
Letting µi (i = 1, 2, 3) be the eigenvalues of J, we obtain the following: 5β[β(1 − θ) + ρ]E1∗ (1 + l1 ∗ )(φπ − 1 + φy yπ∗ ),detJ = µ1 µ2 µ3 = −
φπ

(25)

φy
u∗2
βθu∗12−u∗12 − u∗22 φπ u∗12 − u∗22·¸βθ
1 + αR∗φπ − 1∗∗ ∗∗∗+ν + ν2 + u2 (u12 − u11 ), (26)
φπ αu∗1 (u∗12 − u∗22 ) 1β(1 − θ) + ρ

traceJ = µ1 + µ2 + µ3 = ρ +u2u2(1 + αR − βx), u12 − u11 = ν1 − u11 (αR − βx) > 0. On the other hand,u1u1ν2 + u12 u2 (αR − βx)
u12 − u22 =and thus the sign of (u12 − u22 ) is ambiguous when u12 < 0.
u2 (1 + αR − βx)5Since 1 =8where 6yπ∗ ≡αl2∗ [β(1 − θ) + ρ]
< 0.
(1 + l1∗ )(1 + αR∗ )2 (β + ρ)

(27)

From (25), we find that·¸
φπ − 1 + φy yπ∗sign[detJ] = −sign.
u∗12 − u∗22

(28)

The key findings as for local determinacy are summarized in the following propositions and Table 1.
Proposition 1 The necessary condition for determinate (resp. indeterminate) equiφπ − 1 + φy yπ∗librium is thathas a positive (resp. negative) value. 7
u∗12 − u∗22
Proposition 2 Suppose that u∗12 > 0 and φπ > 1. If monetary authority controls
the nominal rate of interest in response to inflation alone (φy = 0), equilibrium islocally determinate. 8 Otherwise, indeterminacy could be generated.
These propositions suggest that the role of habit on equilibrium determinacy isemphasized by the response to output gap in the interest-rate control. When φy = 0,
the effect of habit on equilibrium determinacy depends only on the form of utility
function. Hence, the result in Proposition 2 is similar to one obtained in the model
without habit shown below.

3.3

Intuitive Implication

To obtain intuition behind the Propositions 1 and 2, first remember the main results
obtained in the model without habit formation. If there is no habit persistence, the¶β(1 − θ) + ρ= h∗ . We can derive yπ around the steady(1 + αR∗ )(β + ρ)state in other cases analyzed in Section 3.3 by the same way.
7
Even though detJ < 0, all eigenvalues may be negative when trJ < 0, which implies equilib6µ(27) is derived from 1 − l h∗ ,rium indeterminacy. Non-stationary under detJ > 0 is because of not only habit stock but also
endogenous labor. Since consumption is constant under the endowment economy, habit accumulation depresses if habit stock increases, but this may not hold when consumption is endogenously
determined.
8This is because both detJ > 0 and trJ > 0 necessarily hold so that only one root is stable.

9dynamic equation reduces toṘ = −E1 [ρ + π(R) − R],
where E1 =(29)

ν1 + ν2. Inspecting (29) yields the following:
αu2 (u12 − u22 )Proposition 3 When there is no habit persistence, equilibrium determinacy around
φπ − 1 + φy ȳπ
the steady state 9 holds if and only if> 0, whereū12 − ū22
ȳπ ≡

α¯l2
< 0.
(1 + αR∗ )2(30)

Otherwise, indeterminacy emerges.
The result is also classified in Table 2. Meng (2002) who assumes that φy = 0 showsa special case of this setting.
φπ − 1 + φy yπ> 0 around the steady state in each economy anWe find thatu12 − u22
alyzed in this and previous subsections is a necessary condition for equilibrium
determinacy. To begin with, we investigate the implication of the condition without considering the effects of habit in consumption. If φπ − 1 + φy yπ is positive, ahigher inflation rate rises the real rate of interest and thus consumption decreases,
because it becomes more beneficial to accumulate financial assets. On the other
hand, u12 − u22 represents the effect on marginal utility of leisure when consumption
increases one unit. When consumption decreases with the higher inflation rate, themarginal utility of leisure becomes smaller if u12 > u22 and thus an agent tries to
cut leisure, which contradicts to lower consumption under φπ − 1 + φy yπ > 0. Thisφπ − 1 + φy yπ> 0.
is why determinacy can emerge whenu12 − u22
If we consider the effects of internal habit, this process for equilibrium determinacy may be violated. Habit accumulation is decelerated by a decrease in grossconsumption due to monetary policy such that φπ − 1 + φy yπ > 0. If an agent
9

When the target rate of inflation is π ∗ , the steady state of the variable z in this economy isrepresented as z̄.

10internalizes habit, he recognizes that this lowers the implicit cost of habit accumulation and thus raises consumption. Therefore, when there is habit persistence in
consumption, indeterminacy could emerge even if φπ − 1 + φy yπ > 0 and u12 > u22 .
In general, yπ is negative, because the opportunity cost of holding money increases with the rate of inflation and thus consumption falls, which equals to production and is constrained by real money balances. Since consumption rises with thestock of habit as in Section 3.1, yπ∗ and ȳπ can be different. However, when φy = 0,yπ becomes ineffective in the real interest rate and thus the criterion of monetary
policy is sign[φπ − 1] regardless of the existence of habit. Therefore, the difference
between the case with internal habit and the one without habit comes only fromthe form of the utility function. Consequently, if the nominal interest rate does not
respond to the output gap, stabilization effects of monetary policy rule in the modelwith habits are close to those established in the model without habits.
In order to clarify the magnitude of the effect from habit parameters, θ and β,
on equilibrium determinacy, we have to examine the relation among the values of yπ
and u12 − u22 around the steady state in each economy, but we cannot analyticallyfind this relation. We solve this problem by specifying the utility function in thenext section.

3.4

External Habit Formation

In this subsection, we briefly consider the case where the habit stock represents thesocial average level so that habit formation is outward-looking. In other words, c in(4) is the average consumption in the economy at large so that an agent takes the
motion of h as given when deciding his optimal consumption plans. 10 Then, theHamiltonian function is now given byĤ = u(c − θh, l) + q[(R − π)a + (1 − l) − (1 + αR)c − τ ].

10This assumption is often used in papers related with habit formation.

11

(31)

Using the optimization and equilibrium conditions, we find that the dynamic equations are as follows:Ṙ = −E1 {[ρ + π(R, h) − R] − E2 ḣ},
¶
¸
·
µ
1
−h .
ḣ = β 1 − l h,
1 + αR(32)(33)

E1 and E2 has the same forms as in (20). Since the law of motion of habit is external,
an agent does not consider the cost of habit accumulation −βx so that x disappears
from the reduced dynamic system. Given the target rate of inflation π ∗ , we add”**” to the steady state values of key variables in the case of external habits. We
see that c∗ < c∗∗ and c̄ < c∗∗ , because the following results are satisfied:u2 ((1 − θ)c∗ , 1 − c∗ )β(1 − θ) + ρ
u2 ((1 − θ)c∗∗ , 1 − c∗∗ )
1
=
<
=
,
∗
∗
∗
∗∗
∗∗
u1 ((1 − θ)c , 1 − c )(1 + αR )(β + ρ)
u1 ((1 − θ)c , 1 − c )1 + αR∗u2 (c̄, 1 − c̄)u2 ((1 − θ)c∗∗ , 1 − c∗∗ )
1
=
=
.
∗∗
∗∗
u1 (c̄, 1 − c̄)u1 ((1 − θ)c , 1 − c )1 + αR∗
The coefficient matrix of the linearlized system of the original one (32)-(33)around the steady state is




Jˆ =

Ṙ∗∗
 R

Ṙh∗∗


ḣ∗∗
R

ḣ∗∗
h

,

where
∗∗
ṘR
= E1∗∗ {(1 − πR∗∗ ) + E2∗∗ ḣ∗∗Ṙh∗∗ = −E1∗∗ πh∗∗ + E1∗∗ E2∗∗ ḣ∗∗R },h ,µ ∗∗ ¶2
·
¸
θν1∗∗
∗∗
∗∗ u2
∗∗
∗∗
ḣR = αβl2
< 0, ḣh = −β(l1 + 1) = −β − ∗∗+ 1 < 0.
u∗∗
ν1 + ν2∗∗
1

There is one jump variable, R, and one predetermined variable, h, in the dynamic
system so that the steady state satisfies local determinacy, if one eigenvalues are
positive. When all eigenvalues are positive, non-stationary holds. Otherwise, equilibrium is indeterminate. We find thatβE1∗∗ (1 + l1∗∗ )(φπ − 1 + φy yπ∗∗ )
,
φπ
·
¸
∗∗
θν
1
∗∗
∗∗
∗∗
∗∗
traceJˆ = µ1 + µ2 = E1 {(1 − πR ) + E2 ḣR } − β − ∗∗+1 ,
ν1 + ν2∗∗
detJˆ = µ1 µ2 = −

12

(34)
(35)

whereyπ∗∗ ≡

αl2∗∗< 0.
(1 + l1∗∗ )(1 + αR∗ )2(36)

Table 3 and the following proposition represent the results concerning equilibrium
determinacy:
Proposition 4 In the case of external habit formation, local equilibrium determiφπ − 1 + φy yπ∗∗nacy holds if and only if
> 0.
∗∗
u∗∗
12 − u22Comparing propositions and tables, we again see that the determinacy condition isclose to one for the model without habits.

4

Examples4.1

Non-Separable Utility

We use an example of utility function such that[(c − θh)η l1−η ]1−σ − 1u(c − θh, l) =,1−σ

0 < η < 1,

0 ≤ θ < 1,

σ > 0,(37)

where σ is the inverse of the elasticity of intertemporal substitution. If we assume no
habit formation (θ = 0), then this specification corresponds to that in Meng (2002).
As in (26) and (35), traces are complicated. In this subsection, we see only theφπ − 1 + φy yπvalues ofaround the steady state, which is the necessary conditionu12 − u22
for equilibrium determinacy when it is positive.
Gross consumption function 11 and the corresponding steady-state values in eachcase arec(R, x, h) =θh(1 − η)(1 + αR − βx) + η ∗η:c =,(1 − η)(1 + αR − βx) + η(1 − η)(1 − θ)(1 + αR∗ − βx∗ ) + ηc(R, h) =11

θh(1 − η)(1 + αR) + η ∗∗η:c =,(1 − η)(1 + αR) + η(1 − η)(1 − θ)(1 + αR∗ ) + η

The forms of functions are independent from σ, because we assume the Cobb-Douglas utility

function as (37).

13c(R) =ηη: c̄ =.
(1 − η)(1 + αR) + η(1 − η)(1 + αR∗ ) + η

Using these results, we obtain the absolute value of yπ :|yπ∗ | =αc∗,
1 + αR∗|yπ∗∗ | =αc∗∗,
1 + αR∗|ȳπ | =α(1 − η)c̄.
(1 − η)(1 + αR∗ ) + η

Additionally, there exists σ̃ such that sign[σ̃ − σ] = sign[u12 − u22 ] around the steadystate. More specifically, we obtain:σ̃ ∗ = 1 +β(1 − θ) + ρ> 1,(1 − η)[αR∗ (β + ρ) + βθ]¯ =1+σ̃ ∗∗ = σ̃1
> 1.
(1 − η)αR∗

We find thatc̄ < c∗ < c∗∗ ,|ȳπ | < |yπ∗ | < |yπ∗∗ |,¯σ̃ ∗ < σ̃ ∗∗ = σ̃,and the results of comparative statics are shown in Table 4 and Figure 1. 12
Circles in Figure 1 emphasize the intersecting point of σ̃ and 1 + φy |yπ | ineach economy, because equilibrium determinacy can hold in the areas where satφπ − (1 + φy |yπ |)> 0 around the steady state. However, we note that thisisfy
σ̃ − σis not a sufficient condition for determinacy when habit is internal. The range
1 ≤ φπ ≤ 1 + φy |yπ | becomes broader with a rise in θ and φy and a decrease in β
and R∗ , which implies higher steady-state consumption and an increase in policy
response to output gap. Then, it becomes difficult to generate determinacy whenσ < σ̃. On the other hand, σ > σ̃ is harder to hold when R∗ is higher. Moreover,an increase in habit parameters θ and β lowers σ̃ when habit is internal, and thus
determinacy under passive policy can emerge more easily.
We substitute a following numerical example into the critical values:(ρ, α, η) = (0.02, 1, 0.7).
Table 5 numerically examines the analytical results in Table 4. Totally, we find thatthe impact of habit persistence in consumption θ on the critical values is stronger
12

Since R∗ = ρ + π ∗ , a rise in R∗ has the same effect as an increase in π ∗ . Thus, the negative¯ is equivalent to Meng (2002).
impact of π ∗ on σ̃

14than that of the adjustment speed of habit β or the target rate of nominal interestR∗ . We show more detailed results in the following.
If φy is enough high and consumption habit exists, the steady-state value of−φy yπ can be more than 0.5. This implies that real rate of interest does not increase
with a higher inflation rate even when φπ = 1.5 which is the value empirically shownin Taylor (1993). Therefore, it becomes harder to emerge determinacy under σ < σ̃
and active monetary policy. As for this effect, whether habit is internal or external
does not seem to be important.
However, unless habit is internal, the value of σ̃ is extremely high. This meansthat monetary authority can easily accomplish macroeconomic stability without
habit by adopting active policy. In contrast, either active or passive interest-rate
control can generate equilibrium determinacy under moderate values of the elasticity
of intertemporal substitution, if habit is internal and habit persistence θ is highenough.

4.2

Separable Utility

In contrast to other cases, sign of trace is important for whether equilibrium pathis uniquely determinate when habit is internal. In order to investigate this issue
clearly, we focus on the additive separable utility (i.e., σ = 1 in (37)). In thissubsection, we restrict our attention to the case of internal habits.
When σ = 1, we can rewirte (26) asµ¶φπ − 1 1 + αR∗
1 − c∗
β+ρ
φy
∗
1+
,
traceJ = ρ − (1 − c ) +φπφπα(1 − θ)c∗ β(1 − θ) + ρ(38)

and the critical value of φπ such that trace is zero is represented byµ¶
1 + αR∗β+ρ
1 − c∗∗φy (1 − c ) +1+α(1 − θ)c∗ β(1 − θ) + ρµ¶φπ (traceJ = 0; σ = 1) =1 + αR∗β+ρ
1 − c∗ρ+1+α(1 − θ)c∗ β(1 − θ) + ρ≡ A1 φy + A2 .

15

Additionally, the critical line of detJ = 0 is
φπ (detJ = 0; σ = 1) =αc∗φy + 1.
1 + αR∗

Since we have already derivedηθ(1 + αR∗ )∗c =and x = −,(1 − η)(1 − θ)(1 + αR∗ − βx∗ ) + ηβ(1 − θ) + ρ∗

we can see that these lines depend on β and θ. Using the lines and the fact that
αc∗
A2 < 1 and A1 <, we draw Figure 2 which shows the relation between
1 + αR∗
macroeconomic stability and monetary policy under σ = 1 in an economy withinternal habit.
We find that monetary policy satisfying φπ − (1 + φy |yπ∗ |) > 0 necessarily makes
equilibrium determinate. Additionally, the slope of φπ (detJ = 0; σ = 1) becomessteeper when β falls and θ rises, which implies that it is harder to hold equilibrium
determinacy. Using numerical example in the previous section and R∗ = 0.05, we
αc∗. When βmake Table 6 which shows the effects of β and θ on A1 , A2 , and1 + αR∗and θ increase, the line φπ (traceJ = 0; σ = 1) moves clockwise because the slope (A1 )is flatter and the intercept (A2 ) rises. That is, the area where either indeterminate or
non-stationary can emerge widens if θ is higher. From Table 6, we can also find thatαc∗
; 0.7. Therefore, equilibrium determinacy
A1 ∈ (0.02, 0.2), A2 ; 0.99 and1 + αR∗can easily hold when (φπ , φy ) = (1.5, 0.5) in which Taylor (1993) empirically shows.

5

Concluding Remarks

We analyze the stabilization effects of the interest-rate control rule in the presenceof habit formation in consumption. We assume the monetary policy under which
the nominal rate of interest responds not only to inflation but also to output gap.
Main results are as follows.
First, a necessary condition for determinate equilibrium is the combination ofa higher (resp. lower) elasticity of intertemporal substitution and an interest-rate
16

control such that the real rate of interest rises (resp. falls) with a higher inflation
rate. However, in contrast to the model without habit, this is not sufficient whenan agent takes the implicit cost of habit accumulation into consideration.
Second, we numerically show that the effect generated by the monetary policy’s
response to output gap is larger when habit exists, but whether habit is internalor external may not produce significant differences. Additionally, in the absence ofinternal habits, an extremely small elasticity of intertemporal substitution is necessary to hold that the marginal utility of leisure decreases with a rise in consumption.
We have shown that determinacy under active policy and the high elasticity of intertemporal substitution becomes harder to emerge because of habit in consumption
and of the monetary-policy’s response to output gap: and that not only active but
also passive interest-rate control can make equilibrium uniquely determinate undermoderate values of the elasticity of intertemporal substitution if habit persistence
is high enough and an agent internalizes habit.
We suggest some future themes for the study of stability in an economy with habit
formation and the monetary policy rule of interest-rate control type. As Auray et.
al. (2002, 2004 and 2005) show, the way of introduction of money can be importantfor macroeconomic stability. We have obtained a benchmark for comparing with thecase of the MIUF. Since the timing of money holdings may play a critical role in the
MIUF model, a model of Auray et. al. (2004) with interest-rate control, instead of
the constant growth rate of nominal money supply, would be an interesting setting
to be analyzed. Introducing material capital accumulation may be also an important
issue.

References[1] Abel, A., B., 1990, Asset Prices under Habit Formation and Catching up with
the Joneses, American Economic Review 80(2), 38-42

17

[2] Auray, S., Collard, F., and Féve, P., 2002, Habit Persistence and Beliefs Based
Liquidity Effect, Economics Bulletin 2002;5;2;1-7.
[3] Auray, S., Collard, F., and Féve, P., 2004, Habit Persistence and Money in the
Utility Function, Economics Bulletin 2004;5;10;1-9.
[4] Auray, S., Collard, F., and Féve, P., 2005, Habit persistence, money growth
rule and real indeterminacy, Review of Economic Dynamics 8, 48-67.
[5] Benhabib, J., Schmitt-Grohé, S., and Uribe, M., 2001, Monetary Policy andMultiple Equilibria, American Economic Review 91(1), 167-185.
[6] Carroll, C., D., 2000, Solving consumption models with multiplicative habits,
Economics Letters 68, 67-77.
[7] Carroll, C., D., Overland, J., and Weil, D., N., 1997, Comparison Utility in a
Growth Model, Journal of Economic Growth 2, 339-367[8] Constantinides, G., M., 1990, Habit Formation: A Resolution of the Equity
Premium Puzzle, Journal of Political Economy 98, 519-543.
[9] Fuhrer, C., J., 2000, Habit Formation in Consumption and Its Implications for
Monetary-Policy Models, American Economic Review 90(3), 367-390.
[10] Fujisaki, S., and Mino, K., 2007, Generalized Taylor Rule and Determinacy ofGrowth Equilibrium, Economics Bulletin 2007;5;11;1-7.
[11] Graham, L., 2008. Consumption Habits and Labor Supply, Journal of Macroeconomics 30, 382-395.
[12] Meng, Q., 2002, Monetary Policy and Multiple Equilibria in a Cash-In-Advance
Economy, Economics Letters 74, 165-170.
[13] Meng, Q., and Yip, C., K., 2004, Investment, interest rate rules, and equilibrium
determinacy, Economic Theory 23, 863-878.
18

[14] Smith, W., T., 2002, Consumption and saving with habit formation and durability, Economics Letters 75, 369-375.
[15] Taylor, J. B., 1993, Discretion versus Policy Rules in Practice, CarnegieRochester Conference Series on Public Policy 39, 195-214.
[16] Weder, M., 2000. Can Habit Formation Solve the Consumption Anomaly in the
Two-Sector Business Cycle Model? Journal of Macroeconomics 22(3), 433-444.

19

Table 1: Equilibrium Determinacy in the Economy with Internal Habitu∗12 > u∗22

u∗12 < u∗22

φπ > 1 − φy yπ∗

D, I

I, NS

1 ≤ φπ ≤ 1 − φy yπ∗

I, NS

D, I

φπ < 1I, NS
D
D:determinate, I:indeterminate, NS:non-stationary
Table 2: Equilibrium Determinacy in the Economy without Habitū12 > ū22

ū12 < ū22

φπ > 1 − φy ȳπDI

1 ≤ φπ ≤ 1 − φy ȳπIDφπ < 1I
D
D:determinate, I:indeterminate, NS:non-stationary
Table 3: Equilibrium Determinacy in the Economy with External Habit∗∗u∗∗12 > u22∗∗
u∗∗12 < u22

φπ > 1 − φy yπ∗∗

D

I, NS

1 ≤ φπ ≤ 1 − φy yπ∗∗

I, NS

D

φπ < 1I
D
D:determinate, I:indeterminate, NS:non-stationary

Table 4: Comparative Statics
c , |yπ∗ |c∗∗ , |yπ∗∗ |c̄, |ȳπ |

σ̃ ∗

σ̃ ∗∗

¯
σ̃

θ

+

+

0

−

0

0

β

−

0

0

−

0

0

R∗

−

−

−

−

−

−

∗

20

Table 5: Numerical Examples (Section 4.1)
θ

β

R∗

φy

c∗

c∗∗

c̄

∗
−φy yπ∗∗−φy yπ−φy ȳπ

σ̃ ∗

σ̃ ∗∗

¯
σ̃

0.2

0.2

0.03

0.5

0.69850.72900.69380.33910.35870.103113.876112.111112.111

0.2

0.2

0.03

1.5

0.69850.72900.69381.0172

1.07620.309413.876112.111112.111

0.2

0.2

0.07

0.5

0.69040.73160.6856

0.3326

0.34190.1007

11.83048.61948.619

0.2

0.2

0.07

1.5

0.69040.73160.6856

0.96791.02560.3002

11.83048.61948.619

0.2

0.8

0.03

0.5

0.69500.72900.69380.33740.35870.1031

12.918112.111112.111

0.2

0.8

0.03

1.5

0.69500.72900.69381.0122

1.07620.309412.918112.111112.111

0.2

0.8

0.07

0.5

0.68690.73160.68560.32100.34190.1007

11.12048.61948.619

0.2

0.8

0.07

1.5

0.68690.73160.6856

0.96301.02560.3002

11.12048.61948.619

0.5

0.2

0.03

0.5

0.71190.8192

0.69380.3456

0.39770.1031

4.752

112.111112.111

0.5

0.2

0.03

1.5

0.71190.8192

0.69381.03681.19300.30944.752

112.111112.111

0.5

0.2

0.07

0.5

0.70400.81350.68560.32900.38010.10074.466

48.61948.619

0.5

0.2

0.07

1.5

0.70400.81350.6856

0.98701.14040.3022

4.466

48.61948.619

0.5

0.8

0.03

0.5

0.69890.8192

0.69380.33920.39770.1031

4.297

112.111112.111

0.5

0.8

0.03

1.5

0.69890.8192

0.69381.01771.19300.30944.297

112.111112.111

0.5

0.8

0.07

0.5

0.69080.81350.6856

0.32280.38010.10074.061

48.61948.619

0.5

0.8

0.07

1.5

0.69080.81350.6856

0.96841.14040.30944.061

48.61948.619

0.8

0.2

0.03

0.5

0.75550.91890.69380.36670.44610.1031

2.200

112.111112.111

0.8

0.2

0.03

1.5

0.75550.91890.69381.1002

1.33820.30942.200

112.111112.111

0.8

0.2

0.07

0.5

0.74830.91600.6856

0.34970.42800.10072.140

48.61948.619

0.8

0.2

0.07

1.5

0.74830.91600.68561.04911.28410.30222.140

48.61948.619

0.8

0.8

0.03

0.5

0.71320.91890.69380.34620.44610.1031

1.903

112.111112.111

0.8

0.8

0.03

1.5

0.71320.91890.69381.0386

1.33820.30941.903

112.111112.111

0.8

0.8

0.07

0.5

0.70530.91600.68560.3296

0.42800.10071.860

48.61948.619

0.8

0.8

0.07

1.5

0.70530.91600.6856

0.98881.28410.30221.860

48.61948.619

Table 6: Numerical Examples (Section 4.2)
αc∗
θ
β
A1
A2
1 + αR∗
0.2

0.2

0.17206 0.98874

0.69440.2

0.5

0.17182 0.98885

0.69170.2

0.8

0.17175 0.98888

0.69100.5

0.2

0.10987 0.99248

0.70800.5

0.5

0.10711 0.99291

0.69770.5

0.8

0.10631 0.99303

0.69480.8

0.2

0.03343 0.99731

0.75190.8

0.5

0.02822 0.99799

0.71940.8

0.8

0.02674 0.99816

0.709221

φπExternal Habitθ ↑, R∗ ↓, φy ↑1 + φy |yπ∗∗ |θ ↑, β ↓, R∗ ↓, φy ↑1 + φy |yπ∗ |Internal Habit
R∗ ↓, φy ↑1 + φy |ȳπ |Without Habit1θ ↑, β ↑, R∗ ↑
0

1

R∗ ↑
σ̃ ∗

¯ = σ̃ ∗∗σ̃σ̃ ∗Figure 1: An Illustration of Comparative Statics
Circles emphasize the critical intersecting point of σ̃ and 1 + φy |yπ | in each case.
φπ − (1 + φy |yπ |)> 0 around
Determinacy can hold only in the areas where satisfyσ̃ − σthe steady state.

22φπ (detJ = 0; σ = 1)φπφπ (traceJ = 0; σ = 1)
1

φy

0

determinate

indeterminate

indeterminate or non-stationary
Figure 2: Equilibrium Determinacy under Internal Habit (σ = 1)23