Discussion Papers In Economics
And Business

Income Taxation, Interest-Rate Control and Macroeconomic
Stability with Balanced-Budget

Seiya Fujisaki and Kazuo Mino

Discussion Paper 08-20

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANIncome Taxation, Interest-Rate Control and Macroeconomic
Stability with Balanced-Budget

Seiya Fujisaki and Kazuo Mino

Discussion Paper 08-20

April 2008

この研究は「大学院経済学研究科・経済学部記念事業」
基金より援助を受けた、記して感謝する。

Graduate School of Economics andOsaka School of International Public Policy (OSIPP)
Osaka University, Toyonaka, Osaka 560-0043, JAPANIncome Taxation, Interest-Rate Control and Macroeconomic
Stability with Balanced-Budget
Seiya Fujisaki∗and Kazuo Mino†
Graduate School of Economics, Osaka University
April 2008

Abstract
This paper studies stabilization eﬀects of fiscal and monetary policy rules in thecontext of a standard real business cycle model with money. We assume that the fiscal
authority adjusts the rate of income tax subject to the balanced-budget constraint,
while the monetary authority controls the nominal interest rate by observing inflation. Inspecting macroeconomic stability of the steady state equilibrium of the model
economy, we demonstrate that whether or not policy rules eliminate the possibilityof sunspot-driven fluctuations critically depends upon the appropriate combination ofprogressiveness of taxation and activeness of interest-rate control.
JEL classification code E52, E62, E63
Keywords: balanced budget, interest rate control, determinacy of equilibrium∗

Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka, 563-0043,

JAPAN (email: ege010fs@mail2.osaka-u.ac.jp)†
Corresponding Author : Graduate School of Economics, Osaka University, 1-7, Machikaneyama, Toyonaka, Osaka, 563-0043, JAPAN (email: mino@econ.osaka-u.ac.jp)

1

Introduction

Income taxation under balanced-budget rule and interest-rate control have been consideredmost eﬀective tools for establishing macroeconomic stability. If it is appropriately selected,
each policy rule may stabilize the economy by mitigating income fluctuations. It is, however, rather unclear whether or not those fiscal and monetary policy rules strengthen theirstabilizing eﬀects each other, if the fiscal authority and the central bank adopt specificactions simultaneously. Although stabilization eﬀects of policy rules have been discussed
extensively, the main stream literature has investigated the stabilization roles of income
taxation and interest control rules separately. Therefore, these studies fail to answer the
relevant question mentioned above.
The purpose of this paper is to explore the interactions between income taxation andinterest rate control rules under the balanced-budget discipline in a prototype model of realbusiness cycle theory. Unlike most of the foregoing studies, this paper treats both fiscal
and monetary policy rules in a single model. More specifically, we introduce money into
the baseline real business cycle model with flexible price via a cash-in-advance constraint.
We assume that the fiscal authority adjusts income tax endogenously in each moment
subject to the balanced-budget rule. In the main part of the paper, we follow the taxation
scheme assumed by Guo and Lansing (1998) in which the rate of income tax depends
on the individual income relative to the average income in the economy at large. The
key distinction in this policy rule is whether taxation on individual income relative tothe average income is progressive or regressive. Given such a fiscal action, the monetary
authority adopts an interest-control rule under which the nominal interest rate responds tothe current rate of inflation relative to the target level of inflation. As usual, the eﬀect ofinterest rate control on macroeconomic stability depends on the sensitivity of interest rate
to a change in inflation. Introducing those fiscal and monetary actions into the baselinemodel, we examine the dynamic behavior of the model economy.
Our study presents three main findings. First, if progressive income taxation is combined with active interest rate control rule (i.e. nominal interest rate responds to inflation
more than one for one), then the economy exhibits equilibrium determinacy, so that wewill not observe expectations-driven fluctuations. Second, if the interest rate control is

1passive, equilibrium indeterminacy could emerge even under progressive income taxation.
Third, if income taxation is regressive, then the interest rate control rule may play a pivotal role for establishing macroeconomic stability. In this case, indeterminacy may emerge
under both active and passive interest rate control rules. However, if the interest rate isrelatively insensitive to inflation, then equilibrium indeterminacy can be eliminated evenin the presence of strong regressiveness of income taxation. Those findings claim that in
the general equilibrium settings with money and capital, it is critically relevant to findappropriate combinations of fiscal and monetary policy rules. Even though the balanced
budget rule and progressive income taxation may contribute to establishing aggregate stability, it is still important to select a suitable monetary policy rule to avoid depressing
stabilization power of fiscal actions.
In the existing literature, Guo and Lansing (1998) show that progressive tax may eliminate the possibility of equilibrium indeterminacy even in the presence of strong degreeof external increasing returns. Schmitt-Grohé and Uribe (1997) and Guo and Harrison(2004) examine the interrelationship between balanced-budget rule and determinacy ofequilibrium. While Schmitt-Grohé and Uribe (1997) emphasize that the balanced-budget
with a fixed government spending and endogenous taxation may generate sunspot-driven
fluctuations, Guo and Harrison (2004) claim that such an unstable behavior can be eliminated if the balanced budget is maintained by adjusting government expenditure under
fixed rates of income tax. These studies utilize the baseline real business cycle models
without money. As for monetary policy rules, there has been a large body of literaturethat investigates stabilization eﬀect of interest-rate control rule à la Taylor (1993). Although many authors (e.g. Benhabib et al. 2001) point out that the interest control
rule may easily produce expectations driven fluctuations in the economies without capital, more recent studies show that the role of interest rate rules for aggregate stability
is less relevant in an economy with capital formation: see, for example, Carlstrom andFuerst (2005) and Meng and Yip (2004). These studies, however, ignore the role of fiscal
policy. The present paper integrates these two lines of research on the stabilization rolesof taxation and interest control.
It is to be noted that several authors have examined interactions between fiscal andmonetary policy rules in the context of new Keynesian models with sticky price adjustment.
2

Following Leeper’s (1991) modelling, Kurozumi (2005), Linnenmann (2006) and Lubik(2005) consider the eﬀects of interest rate rule when the fiscal authority adjusts the rate ofincome tax to maintain a target level of the government debt. Those studies, therefore, do
not assume the balanced-budget rule in its strict sense. Edge and Rudd (2007) explore how
the presence of distortionary taxation on interest income aﬀects the sensitivity of interest
rate control to inflation and income necessary for avoiding equilibrium indeterminacy.
Although Edge and Rudd (2007) utilize a sticky price model with fixed rates of income
tax, the primary concern of their study is close to ours.
The next section constructs the base model. Section 3 presents our main discussion on
the stability property of the model economy in the presence of interaction between fiscal
and monetary policy rules. Section 4 briefly considers alternative formulations of policy
rules. Section 5 concludes.

2

The Base Model

2.1

Households

There is a continuum of identical, infinitely lived households with a unit mass. The flow
budget constraint for the household isṀ = (1 − τ ) py + pT − pc − pv,
where M nominal stocks of money, p price level, y real income per capita, c consumption,v gross investment for capital, τ rate of factor income tax, and T is the real transfer fromthe government (or lump-sum tax if it has a negative value). Since we have normalized
the number of household to unity, M, y, T. c and v represent their aggregate values aswell. Real income consists of rent from capital and wage revenue:y = rk + wl,
where r is real rate of return to capital, w is real wage rate and l denotes labor supply.
Denoting real money balances m ≡ M/p and the rate if inflation π ≡ ṗ/p, we rewrite thehousehold’s flow budget constraint asṁ = (1 − τ ) (rk + wl) + T − c − v − πm.
3(1)The stock of capital changes according tok̇ = v − δk,(2)where δ ∈ (0, 1) denotes the rate of capital depreciation. In addition, a cash-in-advanceconstraint applies to consumption spending so that pc ≤ M or
c≤m

(3)

in each moment of time. In this paper we assume that investment spending is not subject
to the cash-in-advance constraint.
The instantaneous utility of the representative family depends on consumption andlabor supply. Following the standard specification, we assume that the objective functionof the household is
U=

Z ∞
0

¶
µ
l1+γ
dt, γ > 0, ρ > 0, B > 0.
e−ρt log c − B
1+γ

Given the initial holdings of k0 and m0 , the household maximizes U subject to (1) , (2)and (3) under given sequences of {rt , wt , τt , Tt }∞t=0 .
To derive the optimization conditions for the household, we set up the current-value
Hamiltonian function:
H = log c − B

l1+γ+ λ [(1 − τ ) (rk + wl) + T − c − v − πm]
1+γ+ μ (v − δk) + ζ (m − c) ,where λ and μ respectively denote the costate variables of m and k, and ζ is a Lagrange
multiplier corresponding to the cash-in-advance constraint on consumption spending. In
what follows, we assume that the rate of tax, τ, is depends the level of individual income.
The rate of income tax is thus given byτ = τ (y) = τ (rk + wl) .
Considering such a taxation rule, we find that the necessary conditions for an optimum
involve the following:∂H/∂c = 1/c − (λ + ζ) = 0,
4

(4)

£
¤
∂H/∂l = −Blγ + λ 1 − τ (y) − τ 0 (y) y w = 0,(5)

∂H/∂v = −λ + μ = 0,

(6)

ζ (m − c) = 0, m − c ≥ 0, ζ ≥ 0,(7)

λ̇ = λ (ρ + π) − ζ,

(8)

¤
£
μ̇ = (ρ + δ) μ − λ 1 − τ (y) − τ 0 (y) y) r,(9)

together with the transversality conditions, limt→∞ bt λt e−ρt = 0 and limt→∞ mt μt e−ρt = 0
as well as the initial conditions on m and k. In conditions (4) and (5) , τ (y) + τ 0 (y) yrepresents the marginal tax rate perceived by the household. As in Guo and Lansing(1998), we assume that each household takes the proportional tax rule into account when
deciding their optimal consumption plan.
In this paper we focus on the situation where the cash-in-advance constraint is always
eﬀective, so that c = m holds for all t ≥ 0. First, (6) means that μ = λ so that from (8)and (9) we obtain:
ζ=λThus (4) is written as¤ª©£
1 − τ (y) − τ 0 (y) y r + π .

©
£
¤
ª
1
= λ 1 + 1 − τ (y) − τ 0 (y) y r + π − δ .
c

(10)

(11)

As a result, (5) and (10) yields:clγ B =

[1 − τ (y) − τ 0 (y) y] w.
1 + [1 − τ (y) − τ 0 (y) y] r + π(12)The left-hand side of the above is the marginal rate of substitution between consumption
and the labor and the right-hand side expresses the eﬀective, after-tax rare of real wage
rate. Since we assume that the cash-in-advance constraint always binds, an additional
consumption generates an additional opportunity cost of holding money, which is given by
the after-tax, net rate of return to capital plus the rate of inflation. Thus the right-handside of (12) expresses the real wage rate in terms of the eﬀective price including the costof money holding.

5

2.2

Firms

The production side of the model economy follows the standard formulation. There are
identical, infinitely many firms and the total number of firms is normalized to one. The
production function of an individual firms is given by
y = Akα l1−α ,0 < α < 1,

A > 0.

(13)

In in a competitive economy we will consider, α represents the income share of capital.
tWe focus on the case where α has an empirically plausible value, so that in what follows,
we assume that α is less than 0.5. The commodity market is assumed to be competitive
and thus the rate of return to capital and the real wage equal the marginal products ofcapital and labor, respectively:

2.3yr = αAkf α−1 l1−α = α ,
k

(14)

y
w = (1 − α) Ak α l−α = (1 − α) .
l

(15)Policy Rules

The fiscal and monetary authorities respectively control the rate of income tax, τ, andthe nominal interest rate, R, according to their own policy rules. As assumed by SchmittGrohé and Uribe (1997) and Guo and Lansing (1998), the fiscal authority follows thebalanced budget discipline. To emphasize this assumption, we assume away government
debt. The flow budget constraint for the government is thus given byτ y + ṁ + πm = g + T,
where g denotes the government’s consumption spending. A key assumption of our analysis
is that under the balanced-budget rule the fiscal authority cannot use seigniorage income
to finance the government consumption.1 This means thatg = τy(16)

holds in each moment. As a consequence, the real seigniorage income, Ṁ /p, is transferredback to the households, so that ṁ + πm = T.
1

Hence, fiscal policy is ’passive’ in the sense of Leeper (1991).

6Given the general principle mentioned above, the monetary authority is assumed tofollow an interest rate control rule such that
R (π) = π + r∗

³ π ´η
π∗r∗ > 0, π ∗ ≥ 0,

,

(17)

where r∗ > 0 is the steady-state level of net rate of return to capital and π ∗ expresses
the target rate of inflation. We assume that the target rate of inflation is positive so thatπ ∗ is a positive constant set by the monetary authority. Under given r∗ and π ∗ , we seethat R0 (π) > 1 (resp. R0 (π) < 1) according to η > 0 (resp. η < 0) . Hence, if η > 0,then the monetary authority adopts an active control rule under which it adjusts thenominal interest rate more than one for one with inflation. Conversely, when when η < 0,
the interest rate control is passive in the sense that the monetary authority changes thenominal interest rate less than one for one with inflation. When η = 0, the monetary
authority controls the nominal interest rate to keep the real interest rate at the rate of r∗ .
Notice that the Fisher equation gives the relation between the nominal and real interestin such a way thatR = r + π.

(18)

³ r ´1(19)

Therefore, (17) and (18) yield:
π=

η

r∗

π∗ ,

which gives the relation between the equilibrium rate of inflation and the real rate of
return to capital. This means that in our setting the nominal interest rate control is toadjust the rate of inflation tax accoroding to a specified rule.2
As for the fiscal rule under balanced budget, we consider two alternative regimes.
One is taxation rule use the formulation by Guo and Lansing (1998). In this regime, thegovernment consumption is adjusted to keep the balanced budget and the rate of income
2

In the presence of distortional income taxation, the Fisher equation may be modified: see, for exam-

ple, Feldstein (1976). For example, the non-arbitrage condition (18) may be replaced with (1 − τ ) R =(1 − τ ) r + π so that R = r + π/ (1 − τ ) . If this is the case, we assume that the central bank adopts aninterest control rule such thatR=

³ π ´η
π
+ r∗
,
1−τ
π∗

which is compatible with the modified Fisher equation in the long-run equilibrium where π = π∗ . As aresult, we obtain () even in the case of modified Fisher condition.

7

is determined by the following taxation rule:µ ∗ ¶φy1−α< φ < 1, 0 < τ0 < 1,, −
τ (y) = 1 − (1 − τ0 )
y
α

(20)

where y∗ denotes the steady-state level of per capita income.3 Given this taxation rule,
the after-tax income is written as[1 − τ (y)] y = (1 − τ0 ) y∗φ y 1−φ .
As a result, if we denote the after-tax real income by I (y) ≡ [1 − τ (y)] y, we obtain
I 0 (y)= 1 − φ.
I (y) /ySince I (0) = 0, the above equation means that if 0 < φ < 1, then I 0 (y) < I (y) /y so thatthe after-tax income is strictly concave in taxable income y, that is, income taxation isprogressive. Conversely, when φ < 0, function I (y) is strictly convex and hence, income
taxation is regressive. When φ = 0, we obtain the liner taxation rule under which the rateof income tax is fixed as τ0 . It is also to be noted that in the steady state where y = y∗ ,the rate of tax is also fixed at τ0 .4
The alternative fiscal rule, which is assumed by Schmitt-Grohé and Uribe (1997), is tokeep the government spending fixed and the rate of income tax is adjusted to balance thebudget. In this case, the rate of income tax is determined by
τ=

g
,
y

where g is fixed at a certain level. Obviously, income taxation in this regime is stronglyregressive, because a higher income reduces the tax rate. While this paper mostly focus
on the first rule, we briefly discuss this second rule in Section 4.
3

As shown in Section 3.2. the restriction φ > − (1 − α) /α ensures that the steady state level ofconsumption has a positive value.
4
Individual tax payment is T (y) = τ (y) y. Given (20) , we have T 0 (y) = 1 − (1 − τ0 ) (1 − φ) y ∗φ y −θ . In

the steady state where y = y ∗ , we see that T 0 (y) = φ (1 − τ0 ) + τ0 . Hence, if
φ>−

τ0
,
1 − τ0then the total tax payment increases with income y. Since income share of capital, α, is less than 0.5 inreality, when φ satisfies φ > −α/ (1 − α) , it in general holds that φ > −τ0 / (1 − τ0 ) .

8

2.4

Capital Accumulation

Combining the flow budget constraints for the household and the government yields thecommodity-market equilibrium condition: y = k̇ + δk + c + g. Under the first fiscal rule thegovernment consumption is endogenously determined, and thereby the market equilibrium
is written ask̇ = (1 − τ ) y − c − δk.

3

(21)

Policy Rules and Macroeconomic Stability

In this section we assume that the fiscal authority uses the taxation rule given by (20). We
first derive the dynamical system that describes the equilibrium dynamics of the model
economy and explore the stability condition around the steady state equilibrium.

3.1

Dynamic System

In order to derive a complete dynamic system that summarizes the model displayed above,
we focus on the behaviors of capita stock, k, and the shadow value of real money balances,λ. First observe that (20) givesµ ∗ ¶φy.
1 − τ (y) − τ (y) y = (1 − τ0 ) (1 − φ)y0Using the above equation, together with (5) and (15) , we may express the equilibrium
level of employment in the following way:∙

(1 − τ0 ) (1 − φ) (1 − α)
l=
B

¸ 1

1+γ

∗

φ

1−φ

1

y 1+γ y 1+γ λ 1+γ .

Inserting the above into the production function (13) and solving it with respect to y, we
obtainy = Âkα(1+γ)∆1−αλ ∆ ≡ y (k, λ) ,where
∆ = α + γ + φ(1 − α),
¸ 1−α
∙
1+γ
(1 − τ0 )(1 − φ)(1 − α) ∆ ∗ φ(1−α) φy ∆ .
Â = A ∆B
9(22)

Equation (22) represents the short-run production function under a given level of y∗ .
Similarly, the real interest rate is expressed asr=α(1−α)(γ+φ)
1−αy∆= αÂk−λ ∆ ,kimplying that the after-tax marginal rate of return to capital is¡¢αγ(1−φ)−(γ+φ)(1−φ)(1−α)∆∆
1 − τ − τ 0 y r = α (1 − τ0 ) (1 − φ) y ∗φ Â1−φ k
λ≡ r̂ (k, λ) .

For determining the equilibrium rate of inflation, in view of (14) , (19) and (22) , we
may express π as a function of k and λ in such a way that
π=π

∗

Ã

αÂ
r∗

!1

η

k

−

(1−α)(γ+φ)η∆

1−αλ η∆ ≡ π (k, λ) .

(23)

Hence, using (23) , we see that the optimal consumption depends on k and λ in thefollowing manner:c=1≡ c (k, λ) .
λ[1 + r̂ (k, λ) + π (k, λ) − δ](24)

Summing up the above manipulation, we find that the dynamic equation of capitalstock is expressed ask̇ = (1 − τ0 )y ∗ φ y(k, λ)1−φ − c (k, λ) − δk,(25)

and the shadow value of capital (real money balances) changes according to
λ̇ = λ [ρ + δ − r̂ (k, λ)] .

(26)

A pair of diﬀerential equations, (25) and (26) , constitute a complete dynamic system under
the interest rate control and the taxation rule with endogenous government expenditure.
Note that y (k, λ) and r̂ (k, λ) satisfyy (k, λ),k¶1µ
³ r ´11y (k, λ) ηη∗∗ ∗− η
π (k, λ) = π
=π r
.
r∗
k
r̂ (k, λ) = α (1 − τ0 ) (1 − φ)

10

(27)

(28)

3.2

Steady-State Equilibrium

In the steady state where k and λ stay constant over time, it should hold that π = π ∗ ,r = r∗ and y = y ∗ . It is to be noted that in the steady state, we obtain:
1 − τ (y ∗ ) − τ 0 (y ∗ ) y ∗ = (1 − τ0 ) (1 − φ) .
We should also notice that from (22) the production function in the steady state is givenby∆α(1+γ)1−αy ∗ = Â α+γ k∗ α+γ λ∗ α+γ(29)

and the values of k, c and λ satisfy the following conditions:
y∗
ρ+δ
,
=
∗
k
α (1 − τ0 ) (1 − φ)

(30)

c∗
y∗
ρ + δ [1 − α (1 − φ)]
,
=
(1
−
τ
)
0 ∗ −δ =∗kk
α (1 − φ)k∗,
λ∗ k ∗ = ∗c (1 + ρ + π ∗ )

(31)
(32)

In the above, k∗ , c∗ and λ∗ denote their steady-state values. Equation (30) is the modified
Golden-rule condition corresponding to λ̇ = 0, while (31) comes from the long-rum market
equilibrium condition: k̇ = 0. The modified golden-rule condition (30) determines theincome-capital ratio, y∗ /k ∗ , which gives the consumption-capital ratio, c∗ /k∗ by (31) .
Then the steady-state implicit value of capital, k ∗ λ∗ , is given by (32) . The last condition
yieldsλ∗ =α (1 − φ)(1 + ρ + π ∗ ) [ρ + δ(1 − α (1 − φ))] k∗

=

β
,
k∗

where β denotes the coeﬃcient of 1/k ∗ . Using the above relation, together (29) and (30) ,we find that the steady-state level of capital is uniquely determined such that:¸ (1−α)(1+γ) ∙
¸ 1
∙
α+γ
1
1
α (1 − τ0 ) (1 − φ)
(1 − α) (1 − τ0 ) (1 − φ) 1−α 1+γ
∗
1−α
β
.
k =A
ρ+δ
B

(33)

Therefore, the steady-state value of k and λ are uniquely expressed by all the parameters
involved in the model. Once k ∗ and λ∗ are given, the steady state levels of c (= m) and l
are determined uniquely as well.
The steady-state value of capital given by (33) demonstrates that policy parameters,
τ0 , φ, η and π ∗ aﬀect the long-run levels of capital, income, employment and consumption in a complex manner. However, it is rather easy to derive intuitive implications of
11

the eﬀects of a change in policy parameters. First, observe that the degree of activenessof interest-rate control, η, fails to aﬀect the steady state levels of capital, employment
and income. Second, regardless of progressiveness of income tax (i.e. the sign of φ) , thesteady-state capital decreases with τ0 , φ and π ∗ . Third, (30) and (31) show that a changein π ∗ will not aﬀect y ∗ /k ∗ and c∗ /k∗ , so that it alters k ∗, y∗ and c∗ proportionally. Additionally, (31) also shows that a rise in φ increases c∗ /k∗ , while τ0 does not aﬀect c∗ /k∗ .
Finally, by use of (12), (14) , (15) , (30) and (31) , the steady-state rete of employment
satisfies the following relation:l∗γ+1 =

(1 − α) (ρ + δ) (1 − φ),
(1 + ρ + π ∗ ) {ρ + δ [1 − α (1 − φ)]}implying that l∗ decreases with φ and π ∗ , while τ0 does not aﬀect l∗ .
Inspecting the steady state conditions derived above, we may summarize the comparative statics results in the steady state equilibrium as follows:
Proposition 1 The degree of activeness of interest rate control, η, does not aﬀect thesteady state levels of capital, income, consumption and employment. The impacts ofchanges in policy parameters τ0 , φ and π ∗ are shown as the following table.
k∗

l∗

y∗

y ∗ /k∗c∗ /k∗c∗ /y∗

τ0

−

0

−

+

0

−

φ

−

−

−

+

+

+

π∗

0

−

−

0

0

0

Table 1: Policy impacts on the steady state values of key variables.

3.3

Equilibrium Determinacy

In order to examine the equilibrium dynamics near the steady state, let us conduct linear
approximation of (25) and (26) at the steady-state equilibrium. The coeﬃcient matrix ofthe approximated system is given by⎤⎡(1 − τ0 ) (1 − φ) yk (k ∗ , λ∗ ) − ck (k ∗ , λ∗ ) − δ (1 − τ0 ) (1 − φ) yλ (k ∗ , λ∗ ) − cλ (k ∗ , λ∗ )⎦.
J= ⎣−λ∗ r̂λ (k∗ , λ∗ )−λ∗ r̂k (k ∗ , λ∗ )12

Since the shadow value of capital, λ, is an unpredetermined variable, if J has one stableroot, the converging path under perfect foresight is at least locally unique. Thus determinacy of equilibrium is established when the determinacy of J has a negative value. In
contrast, when det J > 0 and the trace of J is negative, there exists a continuum of equilibria around the steady state. As shown in Appendix 1, using the steady-state conditions,
we find that the partial derivatives appeared in J can be expressed by the given parametervalues. The trace and determinant of J are respectively written as:ρ + δ[1 − α(1 − φ)]1
trace J = ρ −∗(1 + π + ρ)∆α(1 − φ)¾½π∗,× [φ(αγ + 1) + γ(1 − α)](ρ + δ) + (1 − α) (γ + φ)η∙¸
ρ + δ [1 − α (1 − φ)] ρ + δ
φ(1 − α)π ∗det J = −(γ + 1)(1 − α(1 − φ)) +.
α (1 − φ)∆η(1 + π ∗ + ρ)

(34)

(35)

where ∆ = α + γ + φ (1 − α) .

First of all, it is easy to see that if the target rate of inflation, π ∗ , is non negative,
income taxation is progressive (φ > 0) and the interest rate control is active (η > 0) , thendet J has a negative value, so that the steady-state equilibrium is locally determinate.
Similarly, if − α+γ1−α < φ < 0 (so that ∆ > 0) and η < 0, then det J < 0. Thus in this caseindeterminacy of equilibrium will not emerge either. In addition, if φ = 0 and the rate oftax is fixed at τ0 , thendet J = −

ρ + δ(1 − α)(ρ + δ) (γ + 1)(1 − α) < 0,α2

implying that, regardless of the monetary policy rules, the dynamic system exhibits equilibrium determinacy. To sum up, a set of suﬃcient conditions for equilibrium determinacy
are the following:
Proposition 2 (i) Given a positive rate of target inflation, either if income taxation isprogressive and interest rate control is active or if income taxation is regressive to satisfy− α+γ
1−α < φ < 0 and interest rate control is passive, then the steady-state equilibrium islocally determinate. (ii) If income tax is flat (φ = 0) , local determinacy holds regardlessof monetary policy rules.
To focus on the other possibilities of equilibrium (in)determinacy,as clear as possible,
let us assume that the elasticity of labor supply is zero: γ = 0. This case corresponds to13

the real business cycle model with indivisible labor analyzed by Hansen (1985). Given
this assumption, we obtain∙¸π∗ φρ + δ [1 − α(1 − φ)]ρ + δ + (1 − α),trace J = ρ −α(1 − φ)(1 + π ∗ + ρ)
η ∆
∙
¸
1
{ρ + δ [1 − α (1 − φ)]} (ρ + δ)φ(1 − α)π ∗det J = −1 − α (1 − φ) +.
∗α (1 − φ)η(1 + π + ρ) ∆

(36)
(37)

where
∆ ≡ α + φ (1 − α) .
First, assume that income taxation is progressive (φ > 0) . In this case ∆ > 0 and,hence, the necessary and suﬃcient condition for determinacy is
1 − α (1 − φ) +φ(1 − α)π ∗> 0.
η(1 + π ∗ + ρ)

The above conditiuon implies that if π ∗ ≥ 0, equilibrium determinacy is established under
the following conditions:η > 0 or η < −(1 − α)π ∗ φ.
[1 − α (1 − φ)] (1 + ρ + π ∗ )(38)

If η satisfies−(1 − α)π ∗ φ< η < 0,[1 − α (1 − φ)] (1 + ρ + π ∗ )(39)∗then we see that det J > 0. It is easy to see that in this case iwe obtain ρ+δ+(1 − α) πη < 0,and, hence, from (36) the trace of J has a positive value. Therefore, if η satisfies (39) ,the steady state is a source and there is no converging path around it.
Table 2 summarizes the patterns of dynamics under progressive tax.
φ>0
η>0

D

η̂ < η < 0

I or Uη < η̂

D

Table 2: Stability properties under progressive taxation
D :determinacy, I :indeterminacy, U :unstable∗(1−α)π φη̂ = − [1−α(1−φ)](1+ρ+π∗ ) (< 0)14

It is to be pointed out that, as shown by numerical examples presented in Section
3.5, when 0 < φ < 1, condition (39) may not be satisfies for plauseible parameter values.
Therefore, the steady state is mostly unstable for the case of −η̂ < η < 0. To sum up, inthe case of progressive taxation we obtain:
Proposition 3 If income taxation is progressive and the target rate of inflation is nonnegative, the perfect-foresight competitive equilibrium is locally determinate, either if theinterest-rate control is active or it is suﬃciently passive. Equilibrium indeterminacy may
not emerge in this regime.
Next, consider the case of regeressive taxation (φ < 0) . In this case the necessary andsuﬃcient condition for local determinacy is¸∙
1(1 − α)π ∗ φ> 0.
1 − α (1 − φ) +∗η(1 + π + ρ) ∆(40)

This condition is fulfilled, either if−α(1 − α)π ∗ φ< φ < 0 (⇐⇒ ∆ > 0) and η > −≡ η̂ (> 0)1−α[1 − α (1 − φ)] (1 + ρ + π ∗ )

or if
φ<−

α
(1 − α)π ∗ φ(⇐⇒ ∆ < 0) and 0 < η < −≡ η̂ (> 0) .
1−α
[1 − α (1 − φ)] (1 + ρ + π ∗ )In words, if a relatively low degree of regressiveness taxaion, coupled with a high degreeof passive interest-rate control, may produce indeterminacy.
In contrast, the necessary conditions for equilibrium indeterminacy are the following:∙¸(1 − α)π ∗ φ1
1 − α (1 − φ) +
< 0,
(41)
∗
η(1 + π + ρ) ∆
¸
∙
π∗ φ
> 0.
(42)
ρ + δ + (1 − α)η ∆When −α/ (1 − α) < φ < 0 (so ∆ > 0) , then both (41) and (42) are satisfied, if and onlyif−

(1 − α) π ∗ φ(1 − α) π ∗
<η<−
.
ρ+δ
1 + π∗ + ρ

Note that the above condition is necessary but not suﬃcient for establishing equilibrium
indeternacy: if trace J > 0 in (36) , the steady state is a source (unstable). In contrast,15αif − 1−αα < φ < − 1−α (so ∆ < 0) , we find that indeterminacy may emerge more easily.

Table 3 gives a classification of dynamic patterns in the case of regressive taxation.
α
− 1−α
<φ<0

α
− 1−α
α < φ < − 1−α−η̂ < ηD

I or U0 < η < −η̂I or UDη̄ < η < 0D

I or Uη < η̄
D
I
Tanle 3: Stability properties under regressive taxation
D: determinate, I: indeterminate, U : unstable∗∗(1−α)π φη̄ = − (1−α)πη̂ ≡ − [1−α(1−φ)](1+ρ+π∗ ) (> 0),
ρ+δ (< 0) ,The following propoisition summarizes our finding.
Proposition 4 (i) Suppose that income taxation is mildly regressive and the target rateof inflation is positive. Then the steady state is locally determinate, either if interest
rate control is suﬃciently active or if it is passive. (ii) Suppose that income taxation is
suﬃciently regressiveness and the target rate of inflation is positive. Then the steady state
holds equilibrium determinacy only when interest rate control is mildly active.

3.4

Discussion

To obtain intuitive implication of determinacy/indeterminacy conditions displayed in
Propositions 2, 3 and 4, let us inspect the optimization condition (12) in detail. Usingw = αy/l, this condition is rewritten asclγ B =

(1 − τ0 ) (1 − φ) (1 − α) y∗φ A1−φ k α(1−φ) l(1−α)(1−φ)−1.
1 + r̂ + π(43)Under a given level of consumption, c, the left-hand side of (43) represents the labor supply
curve and the right hand side is considered the labor demand curve. Given c, r̂ and π.
If we assume that γ = 0 for expositional simplicity, the labor supply curve becomes ahorizontal line. Hence, if (1 − φ) (1 − α) − 1 = −∆ < 0, the labor demand curve has anegative slope and thus less steeper than the labor supply curve. In contrast, if ∆ < 0,then the labor demand has a positive slope and steeper than the labor supply curve.
16

Now suppose that the economy initially stays at the steady state equilibrium. Suppose
further that a sunspot-deriven shock makes agents optimistic and households anticipate
that the output and employment will expand. This raises consumption demand and thelabor supply curve shifts upward and, hence, the equilibrium employment decreases, aslong as the labor demand curve does not shift. Remember that the after-tax rate of return,r̂, and the rate of inflation, π, are respectively expressed in the following manner:r̂ = α (1 − τ0 ) (1 − φ) y ∗φ A1−φ k α(1−φ)−1 l(1−α)(1−φ) ,
1

1

α

1−α

π = r∗− η π ∗ y ∗φ A η k η −1 l η .
These expressions show that an expected increase in l raises r̂. It also increases inflation, if
η > 0. As a result, the anticipated increase in employment shifts the labor demand curve
downwards. Then the equilibrium employment decrease further, implying that the initial,
optimistic expectation will not be self-fulfilled, because product will contract rather than
expand. Consequently, such an expectation driven fluctuations cannot be realized so thatequilibrium is determinate.
Notice that if η is negative and its absolute value is small, a rise in employment yields
a suﬃciently large decrease in inflation. This may reduces the after-tax nominal interest
rate, r̂ + π, and therefore, the labor demand curve may shift up. If such a shift large
enough to enhance the equilibrium level of employment, the initial optimistic expectations
can be self-fulfilled. Hence, as Table 2 shows, if the interest rate control is mildly passive,
there may exist multiple converging paths. In contrast, if the interest rate control isstrongly passive so that the absolute value of η is large enough, a decrease in the after-tax
nominal interest rate, r̂ + π, is small. As a result, an upward shift of the labor demand
curve cannot cancel a reduction of employment due to an upward shift of labor supply
curve. Consequently, if η is small enough to fulfill η < −η̂φ, the possibility of equilibrium
indeterminacy is eliminated.
Next, assume that ∆ < 0 so that the labor demand curve has a positive slope. If theafter-tax nominal interest rate, r̂ + π, is constant, then the initial shift of labor supply
curve due to an increase in consumption raises the equilibrium level of employment. If
we consider the eﬀective real wage, labor demand curve may have negative slope even if∆ < 0. In particular, when η has a small positive value, a rise in l yields a large increase17

in π so that the labor demand curve may have a negative slope. In this case the initialexpectation cannot be self fulfilled. At the same time, if η > 0, both r̂ and π are increasedby an expansion of employment, which yields a downward shift of the labor demand curve.
Since the labor demand curve is steeper than the labor supply curve, this shift produces
a further enhancement of the employment level. As a consequence, the initial expectation
can be self-fulfilled and equilibrium indeterminacy emerges. In contrast, if η < 0 and η isclose to zero, a higher employment may lower the after-tax nominal interest rate, implyingthat the labor demand curve shifts upward. If this is the case, the equilibrium level of
employment may not increase, which means that the initial expected change in economic
condition cannot be realized. Thus indeterminacy may not emerge in this situation.

3.5

A Numerical Example

To focus on the roles of key policy parameters, φ and η, more clearly, let us inspect anumerical example. In so doing, we first depict the graphs of the conditions for det J = 0
and trace J = 0 in (φ, η) space. In what follows, we still focus on the case of indivisible
labor (γ = 0).
Remembering that ∆ ≡ α + φ (1 − α) and definingφ̃ ≡ −

α
,
1−α

³
´
we see that sign ∆ = sign φ − φ̃ . Note that½¾
φ(1 − α)π ∗sign det J = sign (1 − α + αφ)η +if ∆η > 0,
(1 + π ∗ + ρ)½¾
φ(1 − α)π ∗sign det J = −sign (1 − α + αφ)η +if ∆η < 0.
(1 + π ∗ + ρ)and that det J = 0 holds when the folloiwing condition is fulfilled:η=−φ(1 − α)π ∗≡ ηd (φ; π ∗ ) .
(1 + π ∗ + ρ)(1 − α + αφ)(44)Eauation (44) shows that, given a positive rate of the target inflation, π ∗ (> 0), η deareases
as φ rises. In addition η decreases (resp. increases) with π ∗ , if φ is positive (resp. negative).
To consider the sign of trace J, defineC(φ; π ∗ ) ≡ρα(1 − φ)(1 + π ∗ + ρ)∆− φ(ρ + δ).
ρ + δ[1 − α(1 − φ)]18

Then it hodls thatsign {trace J} = sign½

C(φ; π ∗ )η − (1 − α)φπ ∗
∆η

¾

.

This condition is rewritten as
C(φ; π ∗ )> 0,∆ηC(φ; π ∗ )sign {trace J} = −sign {η − ηtr (φ; π ∗ )} if< 0,∆η

sign {trace J} = sign {η − ηtr (φ; π ∗ )} if

where the locus of trace J = 0 is given byηtr (φ; π ∗ ) =(1 − α)φπ ∗.
C(φ; π ∗ )We set the convetional magnitude for each parameter:
time discount rate (ρ) = 0.04, income share of capita (α) = 0.4,
capital depreciation rate (δ) = 0.05, 5 .
µ¶1−αGiven those paprameter values, φ ∈ −, 1 = (−1.5, 1). Assuming that the target
αrate of inflation is π ∗ = 0.03, we obtain the following:φ̃ = −0.67,
ηd (φ; 0.03) = −ηtr (φ; 0.03) =18φ.
107(6 + 4φ)

180φ(7 + 2φ).
−12072φ2 − 2876φ + 6848

Using the numerical results displayed above, we depict the graphical results in Figure
1.6 This figure first reveals that in our numerical example equilibrium indeterminacy
may not emerge as long as φ exceeds −0.67 and, hence, progressive taxation ensuresdeterminacy regardless of interest control. Second, in the case of progressive taxation(φ > 0), even though there is a region in which the steady state is totally unstable (so6When depicting graphs in Figure 1, we use the following facts. First, note that C(·) > 0 for φ ∈1−α< φL < φ̃ and 0 < φH < 1. Second, ηtr (·) is an(φL , φH ) , where φL and φH respectively satisfy −α∗increasing function of φ with a positive value of π > 0. Moreover, ηtr (·) may move around the origin
1−α< φ < φL (resp.
in the (φ, η) plane in the clockwise (resp. counterclockwise) as π∗ rises when −αφL < φ < 1).

19

the equilibrium path is nonstationary), such a region in (φ, η) space is considerably small.
Third, the steady state would be totally unstable if η is positive and suﬃciently small forthe case of φ ∈ [−0.67, 0]. Finally when income taxation is regressive enough to satisfy
φ < −0.67, the interest control, i.e. the magnitude of η, critically aﬀects the stability
property of the economy.

4

Alternative Policy Rules

In this section we briefly discuss alternative fiscal and monetary policy rules that would
modify our main findings shown in the previous sections.

4.1

Fixed Government Spending

So far, we have assumed that the government consumption is endogenously determinedto satisfy the balanced-budget rule. The second scheme of fiscal rule is that the fiscal
authority fixes the government expenditure, g, by adjusting the rate of tax, τ, to balanceits budget. Schmitt-Grohe and Uribe (1997) assume such a balanced-budget rule. If thisis the case, the rate of average tax is determined asτ (y) =

g
.
y

(45)

Unlike the first rule, when deciding its optimal plan, the household takes the tax rate τ asgiven, because y in (45) represents the aggregate income rather than an individual income.
In equilibrium, the after tax income is simply given by I (y) = [1 − τ (y)] y = y − g andy
I 0 (y)=> 1,
I (y) /yy−g
implying that income tax is regressive. In this case the after-tax factor prices are given by
y
y−g
=α
,
k
k
y−g
y
.
ŵ = (1 − τ (y)) (1 − α) = (1 − α)llr̂ = (1 − τ (y)) α

(46)
(47)

Since the household considers that τ is exogenously determined, two of the the firstorder conditions for an optimum shown in Section 2.1 are replaced with the following:∂H/∂l = −Blγ + λ [1 − τ (y)] w = 0,
20

(48)

μ̇ = (ρ + δ) μ − λ [1 − τ (y)] r,(49)

where τ (y) = g/y and g (> 0) is given. From (47) and (48) ,the instantaneous equilibrium
level of employment satisfies
Blγ+1+ g = Akα l1−α .
(1 − α) λIf exists, there are at most two values of l satisfying the above. In the following we ignore
the smaller level of l because it produces unconventional results (for example, a higher
government consumption increases employment). The higher equilibrium level of l can be
written asl = l (k, λ; g) ,lk > 0, lλ > 0,lg < 0.

(50)

Using (50) , we find that the equilibrium level of output, the after-tax rate of returnare written as
Akα l (k, λ; g)1−α = y (k, λ; g) , yk > 0, yλ > 0, yg < 0,
¶
µ
¶
µ
y
y (k, λ, g) − gg
α =α= r̂ (k, λ; g) , r̂λ > 0, r̂g < 0.
1−ykkNote that the after-tax rate of return to capital may increase with capital if a higher ksuﬃciently reduces g/k. Consequently, the reduced dynamic system is given byk̇ = y (k, λ; g) − c (k, λ; g) − δk − g,λ̇ = λ [ρ + δ − r̂ (k, λ; g)] ,
where
1
,
λ [1 + r̂ (k, λ; g) + π (k, λ; g)]¶1µ
1
y (k, λ; g) η∗ ∗− η
π (k, λ; g) = π r.
kc (k, λ; g) =The following discussion is essentially the same as that in Sections 3.2 and 3,3. The
key for the analysis is the behavior of the after-tax levels of rate of return and real wage.
In this fiscal policy regime, equation (12) is written asclγ B =

(1 − α) (Akα l−α − g/l),
1 + r̂ + π
21

(51)

whereNotice that³g´,r̂ = α Akα−1 l1−α −k¢1∗− 1 ¡
π = π ∗ r η αAkα−1 l1−α η .
∂ ((1 − τ ) w)
= l2 (g − αy)∂lso that the government consumption is large enough to satisfy g > αy, the labor demand
function represented by the right-hand side of (51) increases with l. Again, assume thatγ = 0. According to the discussion in Section 4.4, if g > αy, then indeterminacy ofequilibrium is easy to be observed. In addition, if η > 0, a higher employment caused bya sunspot driven disturbance increases the after-tax nominal interest rate, r̂ + π. Hence,
the labor demand curve shifts downward, which enhances the possibility of indeterminacy.
If g < αy, then the labor demand curve is negatively sloped. Even in this case, if η isnegative and its absolute value is small, a higher employment reduces the after-tax nominalinterest rate. As a consequence, the labor demand curve shifts up, under which emergenceof multiple equilibrium can remain.

4.2

Factor Specific Taxation

If capital and labor income are taxed separately, we may set the following tax functions:µ ∗ ∗ ¶φw
´ µ r∗ k ∗ ¶φr
³
w l
k
w
, τw (wl) = 1 − (1 − τ0 ),
τk (rk) = 1 − 1 − τ0
rkwl

where 0 < τ0k , τ0r < 1. Notice that in the case of Cobb-Douglas production function, we
obtain
r∗ k∗
y∗
= ,
rk
y

w∗ l∗
y∗
=
wl
y

Again, the equibalnce between labor demand and supply is decribed byclγ B =ŵ,1 + r̂ + π

whereŵ = (1 − τ0w ) (1 − φw ) (1 − α) y ∗φw l∗−φw A1−φw kα(1−φw ) l(1−α)(1−φw )r̂ = α (1 − τ0r ) (1 − φr ) y ∗φ A1−φ kα(1−φ)−1 l(1−α)(1−φr )22Therefore, it is easy to see that indterminacy tends to emerge more easily, if wage income
taxation is regressive (φw < 0) and capital income taxation is progressive (φr > 0) .

4.3

Taylor Rule

Taylor (1993) originally proposes the interet rate control rule under which the nominalinterest rate responeds to real income as well as inflation. In our notation, the original
Taylor rule can be formulated as
R (π) = π + r∗³ π ´η µ y ¶ξ
π∗

y∗

,

ξ < 1.

(52)

In this case, the Fisher equattion, R = r + π, gives
π=π∗³ r ´ 1 µ y ¶− ηξ
η

r∗

y∗

.

Since r = αy/k, the above is rewritten as
1

1−ε

1

ξ

π = α η A η π ∗ r∗− η y∗ η k

α−1−ε
η

l

(1−α)(1−ε)
.
η

1

(53)
1

α

1−α

When ξ = 0, the equilibrium rate of inflation is π = π = A η r∗− η π ∗ y ∗φ k η −1 l η . Therefore, if the intrest rate control is active with respective to real income, i.e. ξ has a positive
value, then a change in labor employment, l, has a smaller impact on the rate of inflation
under (52) than under (17) . Hence, when η < 0, a rise in l yields a smaller decrease in
π in the case of ξ > 0. Therefore, in view of the discussion in Section 3.4, the Taylor
type control rule given by (52) may contribute to reducing the possiblity of equilibrium
indeterminacy.

5Conclusion

We have analyzed the stabilization roles of fiscal and monetary policy rules in a monetary
real business cycle model with flexible price adjustment. In this paper, we have assumed
that the rate of income tax is endogenously adjusted to balance the government budget,
while the monetary authority uses the Taylor type interest rate control scheme. Ourinvestigation reveals that in the context of a simple real business cycle model we use,equilibrium determinacy depends heavily on the taxation rule rather than on monetary23

policy rule. In particular, as suggested by our numerical example, progressive taxation
under balanced budget rule tends to eliminate the possibility of equilibrium indeterminacy
regardless of activeness of interest rate control. On the contrary, if income taxation isregressive, whether interest rate rule is active or passive may be pivotal to hold equilibrium
determinacy. Since the eﬀects of regressive income tax are close to those generated byincreasing returns to scale, our finding suggests that the role of interest rate control would
be more relevant in the non-standard situation like regressive taxation or increasing returnto scale.
It is, however, to be noticed that our main results emphasized above may partly
come from the simplicity of our model. Our conclusion would be modified, if we assume
more general settings. Possible generalization of the model includes non-separable utility
between consumption and labor as well as a more general form of money demand (forexample, distinction between cash goods and credit goods or cash-in-advance constraint
on investment),.and more general form of interest rate rule in which the nominal interest
rate responds to real income as well as to inflation. Re-examining our discussion in thoseextended frameworks deserves further research.

Appendix
The coeﬃcient matrix is expressed as⎡⎤(1 − τ0 ) (1 − φ) yk (k∗ , λ∗ ) − ck (k∗ , λ∗ ) − δ (1 − τ0 ) (1 − φ) yλ (k∗ , λ∗ ) − cλ (k∗ , λ∗ )⎦.
J =⎣−λ∗ r̂λ (k ∗ , λ∗ )−λ∗ r̂k (k∗ , λ∗ )Note that from (22) , (27) and (28) all of the functions y (.) , r̂ (.) and π (.) are of CobbDougla forms. Thus we find that the partial derivatives in J are respectively given by thefollowing:
α (1 + γ) y∗,∆k∗
1 − α y∗yλ (k ∗ , λ∗ ) =,∆ λ∗φ(αγ + 1) + γ(1 − α) r̂∗,r̂k (k ∗ , λ∗ ) = −∆k∗(1 − α) (γ + φ) π ∗πk (k∗ , λ∗ ) = −,η∆k∗yk (k ∗ , λ∗ ) =24

πλ (k ∗ , λ∗ ) =1 − α π∗,η∆ λ∗(1 − φ) (1 − α) r̂∗,∆λ∗r̂k (k∗ , λ∗ ) + πk (k∗ , λ∗ )ck (k ∗ , λ∗ ) = −c∗,
1 + ρ + π∗r̂λ (k ∗ , λ∗ ) =cλ (k ∗ , λ∗ ) = −(c∗ )2 [1 + π ∗ + ρ + λ∗ (r̂λ (k∗ , λ∗ ) + πk (k∗ , λ∗ ))].
As a result, using
y∗
ρ+δ
,
=
∗
k
α (1 − τ0 ) (1 − φ)c∗
ρ + δ [1 − α (1 − φ)]and r̂∗ = ρ + δ,=∗kα (1 − φ)we express the trace and determinant of J in the following way:ρ+δ((1 + γ) − (1 − α)(1 − φ)) − δ∆µ¶φ(αγ + 1) + γ(1 − α) ρ + δ (1 − α) (γ + φ) π ∗c−+
1 + π∗ + ρ
∆
k∗
η∆
k∗
∙
1
ρ + δ[1 − α(1 − φ)]= ρη −∗(1 + π + ρ)∆α(1 − φ)¸
1× {[φ(αγ + 1) + γ(1 − α)](ρ + δ)η + (1 − α) (γ + φ)π ∗ } .
ηtrace J =det J = − [(1 − τ0 ) (1 − φ) yk (k∗ , λ∗ ) − ck (k∗ , λ∗ ) − δ] [λ∗ r̂λ (k ∗ , λ∗ )]+ [(1 − τ0 ) (1 − φ) yλ (k ∗ , λ∗ ) − cλ (k ∗ , λ∗ )] [λ∗ r̂k (k∗ , λ∗ )]∗ ∗∗ ∗ µ (1 − φ) (1 − α) r̂ ∗ ¶∗∗ r̂k (k , λ ) + πk (k , λ )+ (c∗ )2 [1 + π ∗ + ρ + λ∗ (r̂λ (k ∗ , λ∗ )+ λ [−c1 + ρ + π∗
∆
λ∗
¸
∙
∗
φ(αγ + 1) + γ(1 − α) r̂∗∗ ∗∗ (1 − φ) (1 − α) r̂+ πk (k , λ )] −
−
δλ
∆
k∗
∆
λ∗
¶
µ
c∗
φ(1 − α)π ∗(1 − φ)(1 − α)c∗ ρ + δ(ρ + δ) ∗ − ∗+ φ(αγ + 1) + γ(1 − α)
=−
∆
k
k ∆
(1 + π ∗ + ρ)η
∙¸ρ + δ [1 − α (1 − φ)] ρ + δ
φ(1 − α)π ∗=−(γ + 1)(1 − α + αφ) +,α (1 − φ)∆(1 + π ∗ + ρ)η

25

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27

∆=0

1

η

0
-1.5

1φ

-0.67

-1

Figure 1. Equilibrium determinacy
determinate

non-stationary

ηd

ηtr

indeterminate